Advanced Strategies in Control Systems with Input and Output Constraints (Lecture Notes in Control and Information Sciences)

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Lecture Notes in Control and Information Sciences 346 Editors: M. Thoma, M. Morari

Sophie Tarbouriech, Germain Garcia, Adolf H. Glattfelder (Eds.)

Advanced Strategies in Control Systems with Input and Output Constraints

ABC

Series Advisory Board F. Allgöwer, P. Fleming, P. Kokotovic, A.B. Kurzhanski, H. Kwakernaak, A. Rantzer, J.N. Tsitsiklis

Editors Dr. Sophie Tarbouriech

Adolf Hermann Glattfelder ETH Zürich Dept.Informationstechnol./Elekt Inst. Automatik Physikstr. 3 8092 Zürich Switzerland

LAAS-CNRS 7 Avenue du Colonel Roche 31077 Toulouse cedex 4 France

Professor Germain Garcia LAAS-CNRS 7 Avenue du Colonel Roche 31077 Toulouse cedex 4 France

Library of Congress Control Number: 2006930671 ISSN print edition: 0170-8643 ISSN electronic edition: 1610-7411 ISBN-10 3-540-37009-9 Springer Berlin Heidelberg New York ISBN-13 978-3-540-37009-3 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and techbooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper

SPIN: 11616344

89/techbooks

543210

Preface

Physical, safety or technological constraints induce that the control actuators can neither provide unlimited amplitude signals nor unlimited speed of reaction. The control problems of combat aircraft prototypes and satellite launchers offer interesting examples of the difficulties due to these major constraints. Neglecting actuator saturations on both amplitude and dynamics can be source of undesirable or even catastrophic behavior for the closed-loop system (such as loosing closed-loop stability) [3]. Such actuator saturations have also been blamed as one of several unfortunate mishaps leading to the 1986 Chernobyl nuclear power plant disaster [12], [10]. For these reasons, the study of the control problem (its structure, performance and stability analysis) for systems subject to both amplitude and rate actuator saturations as typical input constraints has received the attention of many researchers in the last years (see, for example, [13], [8], [7], [6]). Anti-windup is an empirical approach to cope with nonlinear effects due to input constraints, and override is a related technique for handling output constraints, [6]. The anti-windup approach consists of taking into account the effect of saturations in a second step after a previous design performed disregarding the saturation terms. The idea is then to introduce control modifications in order to recover, as much as possible, the performance induced by a previous design carried out on the basis of the unsaturated system. In particular, anti-windup schemes have been successfully applied to avoid or minimize the windup of the integral action in PID controllers. This technique is largely applied in industry. In this case, most of the related literature focuses on the performance improvement in the sense of avoiding large and oscillatory transient responses (see, among others, [1], [5]).

VI

Preface

More recently, special attention has been paid to the influence of the anti-windup schemes on the stability and the performance of the closed-loop system (see, for example, [2], [9], [11]). Several results on the anti-windup problem are concerned with achieving global stability properties. Since global results cannot be achieved for open-loop exponentially unstable linear systems in the presence of actuator saturation, local results have to be developed. In this context, a key issue concerns the determination of stability domains for the closed-loop system. If the resulting basin of attraction is not sufficiently large, the system can present a divergent behavior depending on its initialization and the action of disturbances. It is worth to notice that the basin of attraction is modified (and therefore can be enlarged) by the anti-windup loop. In [4], or in the ACC03 Workshop “T-1: Modern Anti-windup Synthesis”, some constructive conditions are proposed both to determine suitable anti-windup gains and to quantify the closed-loop region of stability in the case of amplitude saturation actuator. The override technique uses the same basic approach of a two-step design. A linear control loop is designed for the main output first without regard of the output constraints. It normally performs control for small enough deviations form its design operating point. Then one or more additional feedback control loops are designed for the system trajectory to run along or close to those output constraints. The transfer between the loops is automatic (for example by Min-Max-Selectors) and bumpless (by using antiwindup) and constitutes the dominant nonlinear element in the control system, Much less research results have been published on this topic so far, [6]. The book is organized as follows. • Part 1 is devoted to anti-windup strategies and consists of chapters 1 through 6. • Part 2 is devoted to model predictive control (MPC) and consists of chapters 7 through 10. • Part 3 is devoted to stability and stabilization methods for constrained systems and consists of chapters 11 through 15. Note that this partition is somewhat arbitrary as most of the chapters are interconnected, and mainly reflects the editors’ biases and interests. We hope that this volume will help in claiming many of the problems for controls researchers, and to alert graduate students to the many interesting ideas

Preface

VII

Proposal of Benchmarks (Common Application Examples) We decided to provide two benchmark problems, hoping that this will make reading the book more attractive. One of them or both are considered in several chapters. However the given benchmark problems may not be ideally suited to each design method. Then in some chapters, an additional case has been supplemented. Both examples are abstracted from their specific industrial background, but conserve the main features which are relevant in this specific context. The main focus is on plants which are exponentially unstable systems. They are known to be sensitive to constraints. We elected this feature as it seems currently to be the most interesting and also the most challenging one. Hurwitz systems have been covered in many publications in recent years, and that area seems to have matured. In operation, both plants have a strong persistent disturbance z, which moves the steady state value u ¯ of the control variable u close to the respective saturation. As one control engineer put it: ‘operate as close as possible to the top of the hill (to maximize revenue), but do not fall off the cliff on the other side’. The plants are described by models in continuous form. Model variables are scaled and given in ‘per unit’ form. All state variables xi are measured and are available as outputs yi . No provision has been made for fast non-modelled dynamics. However it is recommended to check for possible bandwidth limitations. The same holds for the inevitable measurement noise. The controllers may be implemented in continuous or time discrete form. Integral action or a suitable substitute shall be provided, in order to drive the control error to zero for steady state loads z¯ = 0. A typical electro-hydraulic actuator subsystem is considered. The mechanical end stops of the servomotor piston are at ulo , uhi (stroke constraints), and vlo , vhi represent the flow constraints from the pilot servovalve (slew constraints), and ks is the finite, moderate gain of the P-controller. This model is embedded in the typical cascade structure, where the main controller outputs the position reference uc = rs . In the contribution we expect the contributor to

VIII

Preface 2

1

v_up

u_hi

u_c k_s

1/T_s

u

1 s

v_dn

u_lo k_a_s

-

show that the specifications are met,

-

document the system responses to a given test sequence, and indicate the maximum stabilizable inputs,

-

provide a stability analysis,

-

and optionally to comment on the implementation requirements.

The Inverted Pendulum Consider the inverted pendulum around its upright position for small inclination angles. The pendulum mass m1 shall be concentrated at its center of gravity (cg), and the connecting member of length L to the slider at its bottom is mass-free and rigid. The slider mass m3 may move horizontally without friction. A horizontal force is applied by the actuator subsystem to the slider, in order to control the speed error to zero. Alternatively you may investigate the case of driving the position error to zero. Denote as state variables the horizontal speed of the pendulum cg as x1 , the horizontal displacement between the cg’s of pendulum and slider as x2 and the horizontal speed of the slider as x3 . The load z1 represents a horizontal force on the pendulum cg, which is persistent and with unknown but bounded magnitude. And u represents the force by the actuator subsystem, which is constrained first in magnitude (stroke) only and then in rate (slew) as well. -

The linearized model in appropriately scaled variables is given by

Preface

IX

v1

x1

m1

Fu

Fz

m3 x3

v3

d x1 = −x2 + z1 ; y1 = x1 dt d τ2 x2 = −x1 + x3 ; y2 = x2 dt d τ3 x3 = +x2 + u; y3 = x3 dt τ1

-

with parameter values τ3 = τ2 = τ1 = 1.0 s.

-

The transfer functions have one pole at the origin, one on the negative real axis, and one symmetrical on the positive real axis.

-

The closed loop bandwidth in the linear range is specified at ≥ 2.0 rad/s.

-

The actuator magnitude (stroke) saturations are at ulo = −1.25 and uhi = +1.25 and on the actuator rate (slew) are: (a) |du/dt| ≤ 10.0/s

-

and then (b) |du/dt| ≤ 2.0/s

The test sequence is defined as follows -

always start in closed loop operation at initial conditions x(0) = 0, reference r1 = 0 and load z1 = 0.

-

apply a large reference ‘step’ of size r1 up to max. stabilizable or to r1 = 2.0, whichever is smaller, with slew rate dr1 /dt = +0.5/s, after stabilization apply a small additional reference Δr1 = 0.10 and then back to Δr1 = 0.0. and set back r1 = 0., with slew rate dr1 /dt = −0.5/s

X

Preface

-

apply a large load ‘step’ of size z1 up to z1 = 1.0 or to max. stabilizable, whichever is smaller with slew rate dz1 /dt = +0.25/s after stabilization apply a small additional load ‘step’ Δz1 = 0.10 and then back to Δz1 = 0.0. and set back z1 = 0., with slew rate dz1 /dt = −0.25/s 2

Continuous Stirred Tank Reactor (CSTR) with a Strong Exothermal Reaction Consider a batch reactor with a continuous feed flow of reactant and control of the contents temperature by means of a heating/cooling jacket. The main variable to be controlled is the fluid temperature x2 , to r2 = 1.0, with zero steady state error e2 = r2 − x2 → 0. The main disturbance to the temperature control loop is the thermal power production. It should be as high as possible without the temperature x2 running away, in order to maximize the production rate. The limit to this is set by the maximum cooling heat flow though the wall to the jacket, and thus by the minimum temperature of the heating/cooling fluid (u2 ) entering the jacket. Both magnitude and rate constraints on u2 are specified from the subsystem delivering the heating/cooling fluid. The reactant concentration x1 is to be controlled to its setpoint r1 through the reactant feed flow u1 , with no perceptible magnitude or rate constraints here. Both the temperature controller R2 and the concentration controller R1 are to be designed. The model of the plant shall be given by the nonlinear equations (in ‘per-units’): d x1 = −a1 f τ1 dt

+ u1 ;

d τ2 dt x2 = −a2 f

+ a3 (x3 − x2 ); y2 = x2

y1 = x1

d τ3 dt x3 = −a3 (x3 − x2 ) + a4 (u2 − x3 ); y3 = x3

where f = a0 · x1 · max[0, (x2 − x2s )]

Preface

XI

with x1 for the reactant mass or concentration in the vessel content fluid where the reaction takes place, x2 for the content temperature, x3 for the jacket temperature (lumping both metal walls and heating/cooling fluid temperatures), u2 for the entry temperature of the heating/cooling fluid from the supply subsystem to the jacket, and u1 for the reactant feed flow. Below the ignition temperature x2s , f is zero. Above, f is proportional to the reactant concentration x1 and the fluid contents temperature x2 − x2s . The mass flow disappearing from the mass balance of reactant x1 is set proportional to f (with coefficient a1 ). The thermal power production is also set proportional to f , with coefficient a2 . The exothermal reaction relates to a2 < 0. During production, the contents temperature x2 must always stay above the ignition temperature x2s , which calls for high closed loop performance. The inventory of reactant must also be closely controlled, for safety reasons. The following numerical values are specified: 1 = 10.; r2 − x2s a1 = 1.0; a2 = −1.0; a3 = 1.0; a4 = 1.0;

x2s = 0.90; a0 =

τ1 = 0.20; τ2 = 1.00; τ3 = 0.20; where a small τ1 /τ2 signifies a relatively small inventory of reactant in the reactor, and where a large τ3 /τ2 signifies a relatively large heat capacity of the jacket. The values for τ1 , τ2 , τ3 imply a time scaling of typically 1000 → 1. For the nominal steady state operating conditions at x ¯1 = 1.0; x ¯2 = 1.0; x ¯3 = 0.0; that is u ¯2 = −1.0; u ¯1 = 1.0; the open loop poles are at s1 = +3.4640; s2 = +0.7049; s3 = −10.1734 The closed loop bandwidth for the temperature control in its linear range is specified at ≥ 15.0 rad/s. The actuator stroke is saturated at u2lo = −1.10 and u2hi = +1.10

XII

Preface

and the maximum actuator slew rate is (a) |du2 /dt| = 200.0/s or (b) |du2 /dt| = 5.0/s The proportional gain of the concentration controller R1 is to be: kp1 ≥ 10.0 No slew saturations are specified for both r1 (t) and r2 (t). The test sequence is defined as follows: -

start in closed loop operation with all initial conditions and inputs r1 , r2 at zero,

-

apply a reference step of size r2 = 1.0; and wait for equilibration

-

then add a load step of size r1 up to max. stabilizable or to r1 = 1.00, whichever is smaller.

The control targets are -

to attain the full equilibrium production at r1 = 1.0; r2 = 1.0 as fast as possible,

-

and then control with the closed loop bandwidth specified above, with - an additional reactant inflow step

Δr1 = +0.020

- and after equilibration a step change in a2 of

Δa2 = +0.020

- and after equilibration an additional temperature reference step Δr2 = −0.020 -

and fast shut-down of production by resetting first r1 to zero, and then r2 to zero. 2

Acknowledgements The idea of this edited book was formed through a series of e-mail exchanges and face-to-face discussions when the three of us met at conferences or during visiting stay in each of the lab in each side. First and foremost, we would like to thank all the contributors of the book. Without their encouragement, enthusiasm, and patience, this book would have not been pos-

Preface

XIII

sible. A list of contributors is provided at the end of the book. We also thank also Isabelle Queinnec (LAAS-CNRS) for her help regarding latex problems. We also wish to thank Springer for agreeing to publish this book. We wish to express our gratitude to Dr. Thomas Ditzinger (Engineering Editor), and Ms. Heather King (International Engineering Editorial) for their careful consideration and helpful suggestions regarding the format and organization of the book.

Toulouse, France, June 2006,

Sophie Tarbouriech

Toulouse, France, June 2006,

Germain Garcia

Zürich, Switzerland, June 2006 ,

Adolf H. Glattfelder

References ˚ om and L. Rundqwist. Integrator windup and how to avoid it. In Proc. of the 1. K. J. Astr¨ American Control Conference, Pittsburgh, USA, pp.1693-1698, 1989. 2. C. Barbu, R. Reginatto, A. R. Teel and L. Zaccarian. Anti-windup for exponentially unstable linear systems with inputs limited in magnitude and rate. In Proc. of the American Control Conference, Chicago, USA, 2000. 3. J.M. Berg, K.D. Hammett, C.A. Schwartz and S.S. Banda. An analysis of the destabilizing effect of daisy chained rate-limited actuators. IEEE Trans. Control Syst. Tech., vol.4, no.2, pp.375-380, 1996. 4. Y-Y. Cao, Z. Lin and D.G. Ward. An antiwindup approach to enlarging domain of attraction for linear systems subject to actuator saturation. IEEE Transactions on Automatic Control, vol.47, no.1, pp.140-145, 2002. 5. H. A. Fertik and C. W. Ross. Direct digital control algorithm with anti-windup feature. ISA Transactions, vol.6, pp.317-328, 1967. 6. A.H. Glattfelder and W. Schaufelberger. Control Systems with Input and Output Constraints. Springer-Verlag, London, 2003. 7. J.M. Gomes da Silva Jr., S. Tarbouriech and G. Garcia. Local stabilization of linear systems under amplitude and rate saturating actuators. IEEE Transactions on Automatic Control, vol.48, no.5, pp.842-847, 2003. 8. V. Kapila and K. Grigoriadis. Actuator saturation control. Marcel Dekker, Inc., 2002. 9. N. Kapoor, A. R. Teel and P. Daoutidis. An anti-windup design for linear systems with input saturation. Automatica, vol.34, no.5, pp.559-574,1998.

XIV

Preface

10. M. Kothare. Control of systems subject to constraints. PhD thesis, Division of Engineering and Applied Science, California Institute of Technology, USA, 1997. 11. M. V. Kothare and M. Morari. Multiplier Theory for stability analisys of Anti-windup control systems. Automatica, vol.35, pp.917-928, 1999. 12. G. Stein. Bode lecture: Respect to unstable. In Proc. of the 28th IEEE Conference on Decision and Control, Tampa, USA, 1989. 13. F. Tyan and D. Bernstein. Dynamic output feedback compensation for linear systems with independent amplitude and rate saturation. International Journal of Control, vol.67, no.1, pp.89-116, 1997.

Contents

Anti-windup Augmentation for Plasma Vertical Stabilization in the DIII-D Tokamak Eugenio Schuster, Michael Walker, David Humphreys, Miroslav Krstić . . . . . . .

1

Stable and Unstable Systems with Amplitude and Rate Saturation Peter Hippe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 An Anti-windup Design for Linear Systems with Imprecise Knowledge of the Actuator Input Output Characteristics Haijun Fang, Zongli Lin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Design and Analysis of Override Control for Exponentially Unstable Systems with Input Saturations Adolf Hermann Glattfelder, Walter Schaufelberger . . . . . . . . . . . . . . . . . . . . . . . 91 Anti-windup Compensation using a Decoupling Architecture Matthew C. Turner, Guido Herrmann, Ian Postlethwaite . . . . . . . . . . . . . . . . . . . 121 Anti-Windup Strategy for Systems Subject to Actuator and Sensor Saturations Sophie Tarbouriech, Isabelle Queinnec, Germain Garcia . . . . . . . . . . . . . . . . . . 173

XVI

Contents

Sampled-Data Nonlinear Model Predictive Control for Constrained Continuous Time Systems Rolf Findeisen, Tobias Raff, and Frank Allgöwer . . . . . . . . . . . . . . . . . . . . . . . . . 207 Explicit Model Predictive Control Urban Maeder, Raphael Cagienard, Manfred Morari . . . . . . . . . . . . . . . . . . . . . 237 Constrained Control Using Model Predictive Control J.M. Maciejowski, P.J. Goulart, E.C. Kerrigan . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Risk Adjusted Receding Horizon Control of Constrained Linear Parameter Varying Systems M. Sznaier, C. M. Lagoa, X. Li, A. A. Stoorvogel . . . . . . . . . . . . . . . . . . . . . . . . . 293 Case Studies on the Control of Input-Constrained Linear Plants Via Output Feedback Containing an Internal Deadzone Loop Dan Dai, Tingshu Hu, Andrew R. Teel, Luca Zaccarian . . . . . . . . . . . . . . . . . . . 313 Set Based Control Synthesis for State and Velocity Constrained Systems Franco Blanchini, Stefano Miani, Carlo Savorgnan . . . . . . . . . . . . . . . . . . . . . . 341 Output Feedback for Discrete-Time Systems with Amplitude and Rate Constrained Actuators J.M. Gomes da Silva Jr., D. Limon, T. Alamo, E.F. Camacho . . . . . . . . . . . . . . . 369 Decentralized Stabilization of Linear Time Invariant Systems Subject to Actuator Saturation Anton A. Stoorvogel, , Ali Saberi, Ciprian Deliu, , Peddapullaiah Sannuti . . . . . 397 On the Stabilization of Linear Discrete-Time Delay Systems Subject to Input Saturation Karim Yakoubi, Yacine Chitour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Contents

XVII

List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

Anti-windup Augmentation for Plasma Vertical Stabilization in the DIII-D Tokamak Eugenio Schuster1 , Michael Walker2 , David Humphreys3 , and Miroslav Krstić4 1

Lehigh University [email protected]

2

General Atomics [email protected]

3

General Atomics [email protected]

4

University of California San Diego [email protected]

Summary. In the Advanced Tokamak (AT) operating mode of the DIII-D tokamak, an integrated multivariable controller takes into account highly coupled influences of plasma equilibrium shape, profile, and stability control. Time-scale separation in the system allows a multiloop design: the inner loop closed by the nominal vertical controller designed to control a linear exponentially unstable plant and the outer loop closed by the nominal shape controller designed to control a nonlinear stabilized plant. Due to actuator constraints, the nominal vertical controller fails to stabilize the vertical position of the plasma inside the tokamak when large or fast disturbances are present or when the references coming from the shape controller change suddenly. Anti-windup synthesis is proposed to find a nonlinear modification of the nominal vertical controller that prevents vertical instability and undesirable oscillations but leaves the inner loop unmodified when there is no input saturation.

1 Introduction Demands for more varied shapes of the plasma and requirements for high performance regulation of the plasma boundary and internal profiles are the common denominator of the Advanced Tokamak (AT) operating mode in DIII-D [1]. This operating mode requires multivariable control techniques [2] to take into account the highly coupled influences of equilibrium shape, profile, and stability control. Current efforts are focused on providing improved control for ongoing experimental

2

Eugenio Schuster, Michael Walker, David Humphreys, and Miroslav Krstić

operations, preparing for future operational control needs, and making advances toward integrated control for AT scenarios. Vertical and Shape Control: The initial step toward integrating multiple individual controls is implementation of a multivariable shape and vertical controller for routine operational use which can be extended to integrate other controls. The long term goal is to integrate the shape and vertical control with control of plasma profiles such as pressure, radial E-field, and current profiles using feedback commands to actuators such as gas injectors, pumps, neutral beams (NB), electron cyclotron heating (ECH), and electron cyclotron current drive (ECCD). The problem of vertical and shape control in tokamaks was and is still extensively studied in the fusion community. A recent summary of the existing work in the field can be found in [3]. Several solutions [4, 5, 6, 7, 8, 9, 10] for the design of the nominal controller were proposed for different tokamaks using varied control techniques based on linearized models. Although the saturation of coil currents and voltages (actuators) is a common problem in tokamaks and there were efforts to minimize the control demand for shape and vertical control and to avoid saturation [11, 12], the saturation of the actuators was rarely taken into account in the design of the controllers until recently [13, 14]. Although similar in concept, our work uses a different approach to the problem: anti-windup compensator. The input constraints are not taken into account at the moment of designing the nominal controller. The goal is not the design of the nominal controller but the design of an anti-windup compensator that blends any given nominal controller, which is designed to fulfil some local (saturation is not considered) performance criterion, with a nonlinear feedback designed to guarantee stability in the presence of input saturation but not necessarily tuned for local performance. This nonlinear modification of the nominal controller also aims at keeping the nominal controller well-behaved and avoiding undesirable oscillations. The anti-windup augmentation must in addition leave the nominal closed loop unmodified when no saturation is present. Shape Control Methodology: The shape control methodology at DIII-D is based on “isoflux control”. The isoflux control method, now in routine use on DIII-D, exploits the capability of the real time

Anti-Windup Augmentation in the DIII-D Tokamak

3

Fig. 1. DIII-D controlled plasma parameters in isoflux control (X-point R and Z positions, and flux at control points on control segments).

EFIT plasma shape reconstruction algorithm to calculate magnetic flux at specified locations within the tokamak vacuum vessel. Figure 1 illustrates a lower single null plasma which was controlled using isoflux control. Real time EFIT can calculate very accurately the value of flux in the vicinity of the plasma boundary. Thus, the controlled parameters are the values of flux at prespecified control points along with the X-point R and Z positions, defined in Figure 1. By requiring that the flux at each control point be equal to the same constant value, the control forces the same flux contour to pass through all of these control points. By choosing this constant value equal to the flux at the X-point, this flux contour must be the last closed flux surface or separatrix. The desired separatrix location is specified by selecting one of a large number of control points along each of several control segments. An X-point control grid is used to assist in calculating the X-point location by providing detailed flux and field information at a number of closely spaced points in the vicinity of the X-point. Several problems make practical implementation of shape and vertical position controllers on DIII-D challenging: (P1) Computational speed is insufficient to do both vertical stabilization and shape control with the same controller. (P2) Vertical stability control and shape control share the same actuators, i.e., shaping coils (F-coils)

4


2A, 2B, 6A, 6B, 7A, and 7B (Figure 1). This is a particular problem for the outer coils because they become the only coils which perform shape control for the outer plasma boundary. (P3) The shape control power supply system (choppers) is extremely nonlinear. (P4) Limitations on actuator voltage imply that commands to shaping power supplies (choppers) often saturate, particularly with large or fast disturbances. (P5) Multiple current and voltage constraints imply that the range of accessible plasma equilibria is constrained. Programmed attempts to reach equilibria outside these constraints can lead to problems such as exceeding coil current limits or saturation of actuators leading to loss of control. (P6) Changes in linearized response of the nonlinear plasma during the course of a discharge can degrade control by linear controllers. This chapter is organized as follows. Section 2 introduces the challenges and the strategy used for plasma shape and vertical position control. Section 3 introduces the basics of the anti-windup method. The characteristics of our plant and its controllable region are presented respectively in Section 4 and Section 5. The design of the antiwindup compensator is detailed in Section 6. Section 7 discusses some simulation results and the conclusions are presented in Section 8.

2 Control Strategy Computational Speed Limitation: Time-scale separation of vertical and shape control appears to be critical for DIIID, since multivariable shape controllers can require significant computation [15]. Figure 2 shows the closed loop system comprised of the DIII-D plant and stabilizing controller. This system is stable and the 6 coil currents F2A, F2B, F6A, F6B, F7A, and F7B are approximately controlled to a set of input reference values. As a result, this system can act as an inner control loop for shape control. Figure 3 shows that the inner loop’s inputs are the 6 vertical coil current reference signals rI , the centroid (center of mass of plasma current) vertical position reference signal rZ , and up to 12 shape coil command voltages v.


5

Fig. 2. Inner loop: Vertical Stabilization.

Actuator Sharing: A method implemented for sharing actuators involves constructing a linear controller which simultaneously performs vertical stabilization and shape control (by controlling the currents of the 6 vertical control coils F2A, F2B, F6A, F6B, F7A, and F7B). The former is done on a fast time scale, as seen by comparing Figures 2 and 3. By integrating current control of the vertical control coils into a stabilizing controller, conflicts between shape and vertical control use of these coils is eliminated. “Frequency sharing” is accomplished explicitly with an H∞ loop shaping design by weighting low frequencies to regulate the coil currents and high frequencies to stabilize the plasma. The design technique ensures that the overall system remains robustly stable. Power Supply Nonlinearities: The problem of highly nonlinear outer power supplies (choppers) was addressed previously by constructing closed loop controllers [16] using a nonlinear output inversion. However, this solution, for the outer loop, is not fast enough to be implemented in the inner loop. A possible approach to deal with the inner choppers is shown in Figure 4. To take into account the nonlinear nature of the choppers we incorporate them into an augmented saturation block. The nominal linear vertical controller is synthesized without using a model of the choppers and its output yc is equal to the coil voltages in the absence of saturation. A chopper inverse function, which is part

6


Fig. 3. Outer loop: Shape Control.

of the vertical controller, computes the necessary command voltages Vc within the saturation levels to make u equal to yc . When |yc | is large, the saturation block will obviously result in |u| being smaller than |yc |. Although the saturation levels of the command voltages Vc are still fixed values (±10 V), the saturation levels of the augmented saturation block are functions of time, i.e., of the coil load currents IL and DC charging supply voltage Vps . Constrained Control: In order to make this approach successful the inner controller (vertical controller, Figure 2) must guarantee stability of the plant for all commands coming from the outer controller (shape controller, Figure 3). However, the constraints on the input of the inner plant due to the saturation of the actuators may prevent this goal from being achieved. The saturation of the coil voltages can not only degrade the performance of the inner closed loop system but also impede the vertical stabilization when the synthesis of the nominal inner controller does not account for plant input saturation. Although the saturation of coil currents and voltages is a common problem in tokamaks and there are efforts to minimize the control demand for shape and vertical control and to avoid saturation [12, 13], the saturation of the actuators are generally not taken into account in the design of the nominal controllers in present works.


7

Fig. 4. Plant Architecture.

The inner loop design must take care then of the windup of that loop and ensure vertical stability for any command coming from the outer controller. We understand as windup the phenomenon characterized by degradation of nominal performance and even loss of stability due to magnitude and/or rate limits in the control actuaction devices. The anti-windup synthesis problem is to find a nonlinear modification of the predesigned nominal linear controller that prevents vertical instability and undesirable oscillations but leaves the nominal closed loop unmodified when there is no input saturation. This problem is different from the problem of synthesizing a controller that accounts for input saturation without requiring it to match a given predesigned arbitrary controller locally. Several survey papers [17, 18, 19] describe early ad-hoc anti-windup methods. Recently several other approaches have been proposed [20, 21, 22, 23, 24, 25, 26]. Due to the characteristics of our problem we follow the ideas introduced in the companion papers [27, 28] and also discussed in [29, 30] for exponentially unstable systems. The method is modified to fulfill the performance requirements of our system.

8


3 Anti-Windup Compensator Fundamentals We consider exponentially unstable linear plants with control input u ∈ m and measurements y ∈ p . We write the model of our system in state-space form, (1)

x˙ = Ax + Bu,

separating the stable modes (xs ∈ ns ) from the exponentially unstable modes (xu ∈ nu ),

x˙ s x˙ u

=

As Asu

0 Au

xs xu

+

Bs

Bu

u (2)

y = Cx + Du,

where the dimension of the state vector x is n = ns +nu . The eigenvalues of As have non-positive real part, and the eigenvalues of Au have positive real part. In addition, we consider that a nominal linear controller with state xc ∈ nc , input uc ∈ p , output yc ∈ m and reference r ∈ p , x˙ c = Ac xc + Bc uc + Gr

(3)

yc = Cc xc + Dc uc + Hr

(4)

has been already designed so that the closed loop system with interconnection conditions u = yc ,

(5)

uc = y

is well posed and internally stable. When the controller output is subject to saturation, i.e., the interconnection conditions (5) are changed to u = sat(yc ),

(6)

uc = y,

the synthesis of an anti-windup scheme is necessary. In this case the interconnection conditions are modified to u = sat(yc + v1 ),

(7)

u c = y + v2 ,

where the signals v1 and v2 are the outputs of the anti-windup compensator [29] x˙ aw = Axaw + B[sat(yc + v1 ) − yc ] v1 = (β(xu ) − 1)yc + α (xu − β(xu )(xu − xawu ), β(xu )κ(xaw )) v2 = −Cxaw − D[sat(yc + v1 ) − yc ],

(8) (9) (10)


9

Fig. 5. Anti-windup scheme.

and where the state xaw is also divided into stable (xaws ) and unstable (xawu ) modes. The anti-windup scheme is illustrated in Figure 5. In addition to modifying the nominal controller when input saturation is encountered, the anti-windup compensator modifies the closed loop if the exponentially unstable modes get close to the boundary of some reasonably large subset of the region where these unstable modes are controllable with the given bound on the control (controllable region). The “distance” from this boundary is measured by the function β, defined as [29] 1, xu ∈ χlower β(xu ) = 0, xu ∈ χupper

(11)

and interpolated in between, where χlower ⊂ χupper are subsets of χ, the domain of attraction of the disturbance-free system subject to the saturation of the output controller or what we call controllable region. The freedom to define χlower and χupper is a tool the designer has to deal with the disturbances that although not modeled are present in the system. The smaller χlower and χupper , the bigger the disturbances tolerated by the system without escaping the controllable region χ (at the same time, unnecessarily small choices of χlower and χupper result in the reduction of the region in which desired linear performance is achieved). When the unstable modes get close to the boundary of the controllable region χ, the closed loop is modified by the function α, which takes over control of the plant (β = 0 ⇒ v1 = −yc +α ⇒ u = sat(α)). One choice of the function α : nu × m → m is given by [29]

10


α(ζ, ω) = Ku ζ + ω

(12)

where Ku is such that Au + Bu Ku is Hurwitz. The function κ(xaw ) is designed to improve the performance of the antiwindup scheme when the controller output is not saturating. It is important to note at this point that this scheme requires the measurement or estimation of the exponentially unstable modes xu . Several comments are in order here: • When xu ∈ χupper the unstable mode is close to escape from the controllable region χ. In this case the antiwindup must modify in a nonlinear way the output of the controller in order to guarantee the stability of the system. Noting that xu ∈ χupper implies that β(xu ) = 0, and in turns that v1 = −yc + Ku xu , we can write the dynamics of the exponentially unstable mode as x˙ u = Au xu + Bu sat(yc + v1 ) = Au xu + Bu sat(Ku xu ). The goal is to design the feedback gain Ku to stabilize the unstable modes before they reach the boundary of the controllable region χ. The anti-windup, through the signal v1 , is ensuring that the unstable modes do not escape the controllable region and guaranteeing in this way that the plant is well behaved. • Let us denote P (s) = C(sI − A)−1 B + D. Then, we can write y = P (s)sat(yc + v1 ) v2 = −P (s)[sat(yc + v1 ) − yc ], and consequently note that uc = y + v2 = P (s)yc , which, in view of (5), is the output of the nonsaturated model. Thus, the antiwindup, through the signal v2 , is hiding the saturation from the nominal controller and guaranteeing in this way that the controller is well behaved. • When xu ∈ χlower and there is no saturation of the controller output, i.e., sat(yc + v1 ) = yc + v1 , we do not want the antiwindup either through v1 or through v2 to affect the nominal closed loop system. Noting that xu ∈ χlower implies that β(xu ) = 1, and in turns that v1 = Ku xawu + κ(xaw ), we can write the dynamics of the anti-windup in this case as


x˙ aws x˙ awu

= =

As Asu 0 Au

xaws xawu

As Asu + Bs Ku 0 Au + Bu Ku

+

Bs Bu

xaws xawu

11

v1

+

Bs Bu

κ(xaw )

v1 = Ku xawu + κ(xaw ) v2 = −Cxaw − Dv1 . If κ = 0, since Au + Bu Ku is Hurwitz, we have the freedom through a proper design of Ku , to make xawu , and consequently v1 , converge to zero as fast as desired. However this does not guarantee a fast convergence of v2 because this signal is also a function of xaws which is evolving in open loop. A problem arises when some of the stable eigenvalues are very slow because the system will be affected by the anti-windup for an unnecessarily long time. In this case, a proper design of κ can make v1 and v2 converge to zero as fast as possible.

4 Plant Model Figure 4 illustrates the architecture of our plant. The dynamics of our plant can be written as As Asu xs Bs Es x˙ s = x˙ = Ax + Bu + Ev = + u+ v x˙ u 0 Au xu Bu Eu

(13)

y = Cx + Du + Gv, where there are n ≈ 50 states, the vector u of dimension m = 6 are the voltage commands for power supplies on the vertical coils F 2A, F 2B, F 6A, F 6B, F 7A and F 7B, the vector v of dimension q ≤ 12 are the voltage demands for the shape coils, the vector y of dimension p = 7 consists of the six vertical coil currents IL and the plasma centroid position. Due to the composition of the output vector y it is convenient to write the reference for the nominal controller as r = [rIT

rZ ]T ,

where rI are the current references for the six vertical coils and rZ is the centroid position reference. The main characteristics of our plant can be summarized as: • There is only one unstable eigenvalue, i.e., nu = 1. The ns = n − 1 stable eigenvalues all have strictly negative real parts. However, some of them are very close to zero (slow modes).

12


• Defining the saturation function max

sataamin (b) =

⎧ ⎪ ⎪ amax if amax < b ⎨

b if amin ≤ b ≤ amax ⎪ ⎪ ⎩ amin if b < amin

(14)

the saturation of the channel i of the controller, for i = 1, . . . , m, will be denoted as M max (t)

sat(yci ) = satMimin (t) (yci ) i

where the saturation levels

Mimin (t)

and Mimax (t) are functions of time (i.e., of

coil load current IL (t) and DC supply voltage Vps (t)). • There is no direct measurement of the unstable mode xu . • The control input u is not the only input of the inner plant. In addition to the voltage commands u for the vertical coil from the vertical controller, there are voltage demands v for the shape coils coming from the shape controller.

5 Controllable Region Given the dynamics of the unstable mode (from 13) by the scalar equation x˙ u = Au xu + Bu u + Eu v,

(15)

we can compute the minimum and maximum values of the unstable mode that can be reached without losing control authority to stabilize the system, min

− Eu v − (Bu u) = Au max − Eu v − (Bu u) = , Au

xmax u xmin u

and define the controllable region as

. ≤ xu ≤ xmax χ = xu : xmin u u

(16) (17)

(18)

The maximal and minimal control are given by min

(Bu u)

max

(Bu u)

= =

m i=1 m i=1

Bui gi (−Bui )

(19)

Bui gi (Bui )

(20)


13

6β(xu ) 1

@

@ @

@ @

xmin u

xmin,u u

xmin,l u

xmax,l u

xmax,u u

xmax u

xu

Fig. 6. Beta function.

where Bui is the i-th component of Bu and the function gi is defined as Mimax if a > 0 gi (a) = Mimin if a < 0.

(21)

6 Anti-Windup Design Design of Function β: Once χ is determined, we can define

< xu < xmax,l χlower = xu : xmin,l u u

< xu < xmax,u χupper = xu : xmin,u u u

(22) (23)

= f l xmin , xmax,l = f l xmax , xmin,u = f u xmin , xmax,u = f u xmax , where xmin,l u u u u u u u u and 0 < f l < f u < 1. Once χ, χlower and χupper are defined, the function β in (11) adopts the shape shown in Figure 6. Design of Gain Ku : The feedback gain Ku in (12) is designed such that Au + Bu Ku < 0 sgn(Bui ) = −sgn(Kui )

(24) (25)

| > max(|Mimin |, |Mimax |) |Kui xmax,u u

(26)

| > max(|Mimin |, |Mimax |), |Kui xmin,u u

(27)

14


for i = 1, . . . , m, where Bui and Kui are the i-th components of Bu and Ku respectively. With (26),(27), we guarantee that for xmax,u ≤ xu < xmax (where u u β(xu ) = 0) we have Bu u = Bu sat(Ku xu ) = (Bu u)min

(28)

and consequently that sgn(x˙ u ) = sgn(Au xu + Bu sat(Ku xu ) + Eu v) = sgn(Au xu + (Bu u)min + Eu v) (29)

0.

(30)

Conditions (29) and (30) ensure stabilization of the unstable mode when β(xu ) = 0 through the signal v1 = −yc + Ku xu . Design of the Function κ: We want to make the states xaw of the anti-windup compensator converge to zero as fast as possible when the unstable mode is in the “safe” region, defined by the condition β(xu ) = 1, and no channel of the controller output is saturating. The function κ(xaw ) in (9) can be designed toward this goal. However, at this point we depart from the original method and follow an alternative procedure for simplicity and effectiveness. We make κ(xaw ) = 0 and modify the structure of the anti-windup compensator (8) as follows x˙ aw = Axaw + B[sat(yc + v1 ) − yc ] − [1 − γ(yc + v1 )]δxaw ,

δ > 0 (31)

where the function γ is defined as γ=

m

γi

(32)

i=1

and the function γi , for each channel i = 1, . . . m,, is defined in Figure 7. With min,γ

0 ≤ fγ ≤ 1, we define the threshold values Mi

max,γ

and Mi

as


-

-

1

6γi (a)

?

6

?

15

-

6 -

Mimin,γ

Mimin

Mimax,γ

Mimax

a

Fig. 7. Gamma function for each channel i = 1, . . . , m.

min,γ Mi max,γ

Mi

max

Mi − Mimin Mimin + Mimax − fγ = 2 2 max

min max Mi + Mi − Mimin Mi + fγ = . 2 2

Note that γi becomes one when the signal coming from the controller and antiwindup goes above Mimax or below Mimin . However, the function γi recovers its zero value only when the signal coming from the controller and anti-windup bemax,γ

comes lower than Mi

min,γ

< Mimax or higher than Mi

> Mimin . These his-

teresis loops are introduced to avoid chattering in the scheme. The function γ is an indication of saturation, being zero if none of the input channel is saturating and one otherwise. In this way, when the unstable mode is in the “safe” region and there is no saturation the dynamics of the anti-windup can be written as x˙ aw = Axaw + Bv1 − δxaw As − δIs xaws Asu + Bs Ku = 0 Au + Bu Ku − δIu xawu

(33)

v1 = Ku xawu

(34)

v2 = −Cxaw − Dv1 ,

(35)

where Is and Iu are identity matrices of appropriate dimension. The rate of convergence of xaw to zero can be regulated now by the gain δ. A scheme of the anti-windup design is shown in Figure 8. The Saturation Limits block computes the saturation levels Mimin and Mimax , for i = 1, . . . , m, which are functions of time (i.e., of coil load currents IL (t) and DC supply voltage Vps (t)). These saturation levels are used by the Saturation Indication and Anti-Windup blocks to compute the function γ and the controllable region χ respectively. The Observer block estimates the unstable mode which cannot be measured. This estimate is used

16


Fig. 8. Inner loop anti-windup scheme.

by the Anti-Windup block to compute the function β. The Saturation Indication computes γ to speed up the convergence of slow modes (and v2 ) to zero. Once the controllable region χ and the function β are computed, and the function γ provided, the Anti-Windup block is able to achieve stability through the signal v1 and keep the nominal controller well-behaved through the signal v2 . Rate Limiter: It is necessary at this point to recall that we are controlling the current in the vertical coils by modulating their imposed voltages. It is clear now that there will be a minimum integration time constant which will depend on the inductance of the coils. Given a maximum coil voltage value dictated by the saturation level we will have then a maximum rate of variation for the coil current. Any reference rI (imposed by the shape controller) that exceeds this physical rate limit will only cause performance deterioration due to the windup of the controller.


17

The rate limit on the vertical coil current references aims at preventing the shape controller from asking the system for a response rate that cannot be physically fulfilled. The dynamics of the rate limiter for the vertical coil current references can be written as r˙I = Rsgn(˜ rI − rI )

(36)

where M max

r˜I = satMImin (rI∗ )

(37)

I

is the saturated version of the current references rI∗ imposed by the shape controller. The saturation levels MImin and MImax are the minimum and maximum currents allowed in the coils. As long as rI = r˜I , the output of the rate limiter rI follows the target reference r˜I with a rate limit R = diag(R1 , R2 , . . . , Rm ) according to (36). The components Ri , for i = 1, . . . , m, can be fixed or varied adaptively as it will be shown below. The centroid position reference is also limited by physical constraints due to the finite size of the tokamak represented by MZmin and MZmax , and is written as M max

∗ rZ = satMZmin (rZ ).

(38)

Z

Figure 8 shows that the vertical coil current references and the centroid position reference r∗ = [rI∗

T

∗ T rZ ] imposed by the shape controller go through a double

saturation stage. The first saturation is in magnitude and is due to constraints imposed on the coil currents and centroid position due to physical considerations. The second saturation is in rate and is due to the rate limits imposed on the coil current references as part of the inner anti-windup design (rate limiter). To take advantage of all the actuation force available and increase the response speed of our system, when rI = r˜I we vary Ri , for i = 1, . . . , m, adaptively between maximum and minimum values (Rimax and Rimin ) according to the available control, ⎧ min(Mimax − yci , yci − Mimin ) if Mimin ≤ yci ≤ Mimax ⎪ ⎪ ⎨0 if Ri > Rimax and Mimin ≤ yci ≤ Mimax ˙ if Ri < Rimin and (yci < Mimin or yci > Mimax ) R i = Ki 0 ⎪ max ⎪ yci ) if yci > Mimax and Ri > Rimin ⎩ (Mi −min min min (yci − Mi

)

if yci < Mi

and Ri > Ri

(39) where Ki is a constant gain. The maximum and minimum values Rimin and Rimax define the zone where Ri is allowed to move. These limits are imposed to avoid the windup of Ri . Basically, Ri is increased when there is available control, is decreased when the channel is saturating, and is frozen when one of its limits is reached.

18


Watch Dog: As it is shown in Figure 4 and stated in (13), our plant is also governed by the voltage demands for the shape coils v coming from the shape controller. This represents a potential risk of instability for our plant because the unstable mode can be potentially pushed outside the controllable region by shape control voltages. The goal of the watch-dog is to limit the voltage demands to the shape coils to avoid the loss of controllability of the unstable mode due to the sudden shrinkage of the controllable region. We call this block ”Watch Dog” because it is permanently monitoring and regulating the value of the shape coil voltage demands v to mantain the size of the controllable region of the system above certain level. We present in this section our first approach to the problem. There is room for further development aimed at reducing the conservatism of the algorithm. Figure 8 shows that the shape coil voltage demands v ∗ coming from the shape controller go through a double saturation stage. The first magnitude saturation is due to the watch-dog that is part of the inner anti-windup augmentation. The second magnitude saturation comes from the hardware limitations of the power supply stage. We can write the shape coil voltage demands as M max (t)

v = satMvmin (t) (˜ v) , v

(40)

where N max (t)

v˜ = satNvmin (t) (v ∗ ) v

(41)

is the output of the watch-dog block. The saturation levels Mvmin and Mvmax are determined by the hardware limitations of the power supply stage and are functions of time because they depend on the coil load current and DC supply voltage. The saturation levels Nvmin and Nvmax , imposed by the watch-dog as it will be explained below, are also functions of time because they depend on the value of the unstable mode xu and the controllable region χ. The limiting values of the controllable region given by (16) and (17) are plotted as functions of Eu v in Figure 9. We cannot allow the controllable region to shrink to a null set and we impose a minimum size to it defined as

max . χmin = xu : (xmin ) ≤ x ≤ (x ) max u min u u

(42)

This minimum controllable region defines hard limits on Eu v allowing this signal to lim be between (Eu v)lim min and (Eu v)max . In addition we need to impose variable limits


19

xmax (Eu v) u

aa xmax , xmin u u 6 aa aa aa aa aa aa max (xu )min aa aa lim aa (Eu v)min ? aaa aa 6 a min lim aa Eu v aa (xu )max (Eu v)max a aa aa aa aa aa aa aa xmin (Eu v) a u Fig. 9. Controllable region as a function of Eu v.

on Eu v depending on the position of the unstable mode xu . Toward this goal we define (xmax )allowed = max(fu xu , (xmax )min ) u u max

= − (Bu u) (Eu v)allowed max

− Au (xmax )allowed u

)allowed = min(fu xu , (xmin )max ) (xmin u u min

= − (Bu u) (Eu v)allowed min

− Au (xmin )allowed u

where fu > 1 is a design constant and for each channel i we compute ⎧ (E v)allowed max ⎪ if sgn(Eui ) > 0 ⎨ Nucoils |Euj | max j=1 (Nv )i = (E v)allowed min ⎪ if sgn(Eui ) < 0 ⎩ − Nucoils |Euj | j=1 ⎧ (E v)allowed max ⎪ if sgn(Eui ) < 0 ⎨ − Nucoils |Euj | min j=1 (Nv )i = (E v)allowed min ⎪ if sgn(Eui ) > 0. ⎩ Nucoils j=1

(43) (44) (45) (46)

(47)

(48)

|Euj |

Observer: The stability of the anti-windup scheme is guaranteed when the unstable mode is directly measured [29]. When there is no such direct measurement, a bad estimation of the unstable mode can prevent the anti-windup compensator to stabilize the system. A high gain observer is required in this case to ensure that the estimation is fast enough to prevent any excursion of the unstable mode outside the controllable

20


region. The estimate x û is substituted for xu in the anti-windup dynamics, which, sumarizing (9), (10), (12), (31), is given by x ˆ˙ s =x ˆ˙ = Aˆ x + Bu + Ev + L(y − yˆ) ˙x û

(49)

yˆ = C x ˆ + Du + Gv. The estimate x û is substituted for xu in the anti-windup dynamics, which, sumarizing (9), (10), (12), (31), is given by x˙ aw = Axaw + B[sat(yc + v1 ) − yc ] − [1 − γ(yc + v1 )]δxaw v1 = (β(ˆ xu ) − 1)yc + Ku [ˆ xu − β(ˆ xu )(ˆ xu − xawu )]

(50)

v2 = −Cxaw − D[sat(yc + v1 ) − yc ]. For our simulation studies, a convential Luenberger observer was implemented. However, during the implementation stage other types of observers more suitable for noisy environments will be considered. From the observer we do not need much accuracy, we only need to know if the unstable mode is inside χlower or outside χupper . Certain level of noise in the estimation can be tolerated because it is always possible to compensate the inaccuracy of the observer with a convenient selection (reduction) of the design parameters f l and f u , paying the price on the other hand of reducing conservatively the region χupper where we allow the states to move.

7 Simulation Results Simulations were run to show the effect of the anti-windup. Figure 10 shows the response of the augmented closed loop when rIi , for i = 1, . . . , m, are step functions of magnitude approximately equal to 840 Amps. Compared to the results without anti-windup, the excursion of the controller output yc (Figure 10-a) is reduced and the stability is recovered but the step response for the coil currents (y1 , . . . , y6 ) (Figure 10-c) is far from the desired one. In addition, the centroid vertical position (y7 ) (Figure 10-d) exceeds physical limits (±20cm approximately; 1V = 10cm). For all the simulations in this paper we have made rZ = 0 which means that the centroid vertical position must be regulated around its equilibrium value. However, the nominal vertical controller is not designed for a perfect regulation of this variable because we do not care about the centroid vertical position as long as it stays within

Anti-Windup Augmentation in the DIII-D Tokamak Controller Output (total): y

c

5000

21

Plant input (Chopper output) (total): u

1000

500

0

0

−500

−5000 0.05

0.1

0.15

0.2

0.25

0.3

−1000 0.05

0.35

0.1

0.15

0.2

0.25

Time (sec)

Time (sec)

(a)

(b)

1

1000

6

0.35

Centroid vertical position (total): y7

Vertical F−coil currents (rel) (Amps): y ,...,y 2000

0.3

0

−2

0 −4

−1000

−6

−2000 −3000 0.05

0.1

0.15

0.2 0.25 Time (sec)

0.3

0.35

−8 0.05

0.1

0.15

0.2

(c)

0.3

0.35

(d)

β(xu)

1.2

0.25

Time (sec)

Unstable Mode x −− Controllable Region u

10000 1

0.8

x u min xu max x

5000

u

0.6

0 0.4 0.2 0 0.05

−5000

0.1

0.15

0.2

0.25

Time (sec)

(e)

0.3

0.35

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (sec)

(f)

Fig. 10. System response with anti-windup to step changes of approximately 840 Amps in rIi , for i = 1, . . . , m, at t = 0.1 sec (rZ = 0 and v = 0). Low gain Ku , δ = 0.

22

Eugenio Schuster, Michael Walker, David Humphreys, and Miroslav Krstić v1

6000

v2

400

4000

300

2000 200 0 100 −2000 0

−4000 −6000 0.05

0.1

0.15

0.2

0.25

0.3

0.35

−100 0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (sec)

Time (sec)

(a)

(b)

Vertical F−coil currents (rel) (Amps): y1,...,y6

1500

Centroid vertical position (total): y7

0

1000

−0.5 500

−1

0 −500

−1.5 −1000 −1500 0.05

0.1

0.15

0.2 0.25 Time (sec)

0.3

0.35

−2 0.05

0.1

0.15

(c)

0.3

0.35

(d)

Vc

10

0.2 0.25 Time (sec)

10000

5

Unstable Mode xu −− Controllable Region x u xmin u xmax u

5000

0 0

−5 −5000

−10 0.05

0.1

0.15

0.2

0.25

Time (sec)

(e)

0.3

0.35

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (sec)

(f)

Fig. 11. System response with anti-windup to step changes of approximately 840 Amps in rIi , for i = 1, . . . , m, at t = 0.1 sec (rZ = 0 and v = 0). High gain Ku , δ = 0.

Anti-Windup Augmentation in the DIII-D Tokamak v

Vc

10

400

23

2

300

5

200 0

100 −5

0

−10 0.05

0.1

0.15

0.2 0.25 Time (sec)

0.3

0.35

−100 0.05

0.1

0.15

(a)

1500

0.2 0.25 Time (sec)

0.3

0.35

(b)


Centroid vertical position (total): y

7

0

1000

−0.5 500

−1

0 −500

−1.5 −1000 −1500 0.05

0.1

0.15

0.2

0.25

Time (sec)

(c)

0.3

0.35

−2 0.05

0.1

0.15

0.2

0.25

Time (sec)

0.3

0.35

(d)

Fig. 12. System response with anti-windup to step changes of approximately 840 Amps in rIi , for i = 1, . . . , m, at t = 0.1 sec (rZ = 0 and v = 0). High gain Ku , δ > 0.

the physical limits. The β(xu ) = 1 value (Figure 10-e) shows the intervention of the anti-windup compensator in order to keep the unstable mode inside χlower . It is possible to note that in this case the unstable mode (Figure 10-f) gets really close to the boundary of the controllable region and it requires a great effort from the anti-windup compensator, manifested in the shape of input signal u (Figure 10-b), to return it to the safe region. We can avoid this effect either by reducing the size of χlower , χupper in (22), (23) (by reducing the values of f l and f u ), or by increasing the anti-windup gain Ku in (12).

24

Eugenio Schuster, Michael Walker, David Humphreys, and Miroslav Krstić *

1000

Current Targets (rel) (Amps): r

1000

800

800

600

600

400

400

200

200

0

0.1

0.2

0.3

0

0.4

Current References (rel) (Amps): r

0.1

0.2

0.3

0.4

Time (sec)

Time (sec)

(a)

(b)

Vertical F−coil currents (rel) (Amps): y ,...,y 1

1000 800

6

Vc

10

5

600 0

400 −5

200 0

0.1

0.2

0.3

Time (sec)

(c)

0.4

−10

0.1

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Fig. 13. System response with anti-windup and rate limiter to step changes of approximately 840 Amps in rIi , for i = 1, . . . , m, at t = 0.1 sec (rZ = 0 and v = 0). Fixed rate limit R.

Figure 11 shows the response of the augmented closed loop when rIi , for i = 1, . . . , m, are step functions of magnitude approximately equal to 840 Amps but this time with a higher gain Ku . Comparing Figures 10-f and 11-f, it is possible to note that the excursion of the unstable mode xu is much smaller (β(xu ) = 1 for all time in this case). In addition, the centroid vertical position (y7 ) (Figure 11-d) is kept within it physical limits, and the step response for the coil currents (y1 , . . . , y6 ) (Figure 11-c) is improved. The intervention of the anti-windup compensator through β(xu ) = 1 is inversely proportional to the gain Ku . A high value of Ku helps to keep the unstable mode inside χlower and makes less frequent a value of β(xu )

Anti-Windup Augmentation in the DIII-D Tokamak R

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Fig. 14. System response with anti-windup and rate limiter to step changes of approximately 840 Amps in rIi , for i = 1, . . . , m, at t = 0.1 sec (rZ = 0 and v = 0). Variable rate limit R.

distinct from one. Although better than in Figure 10-c, the step response for the coil currents (y1 , . . . , y6 ) in Figure 11-c is still far from the desired one. It is possible to note that even after the system leaves the saturation zone (the chopper commands Vc (Figure 11-e) come back within their limits (±10V )) the nominal controller exhibits a sluggish behavior. This is explained by the fact that the signal v2 (Figure 11-b) still modifies the nominal closed loop when there is no saturation. From (33), when δ = 0 the stable modes of the anti-windup are evolving in open loop and the time it takes for them and for v2 to converge to zero depends on the magnitude of their eigenvalues. As it was discussed in previous sections, it is possible to note that although the

26

Eugenio Schuster, Michael Walker, David Humphreys, and Miroslav Krstić Vertical F−coil currents (rel) (Amps): y1,...,y6

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Fig. 15. System response with anti-windup to step changes of approximately 280 Amps in rIi , for i = 1, . . . , m, at t = 0.1 sec (rZ = 0 and v = 0). (a) no anti-windup, (b) anti-windup δ = 0, (c) anti-windup δ > 0, (d) anti-windup δ > 0 + rate limiter with variable rate limit R.

signal v1 (Figure 11-a) converges to zero arbitrarily fast once the saturation is absent, the signal v2 converges to zero very slowly due to the presence of very slow modes in our system. Figure 12 shows the importance of the function γ modifying the structure of the antiwindup system. When δ = 0 it is possible to make the signal v2 converge to zero very fast (once the chopper commands Vc (Figure 12-a) come back within their limits (±10V )) as it is shown in Figure 12-b. In this way we allow the system to evolve according to the nominal design. In our system the difference between the δ = 0 and δ = 0 cases is huge because the plant has stable eigenvalues with real part


27

very close to zero. The step response for the coil currents (y1 , . . . , y6 ) (Figure 12-c) is improved greatly and their steady state values are reached much sooner, while the centroid vertical position (y7 ) (Figure 12-d) is kept within it physical limits. Figure 13 shows how smooth the response of the system is when a fixed rate limit is imposed on each channel of the reference rI . The reference r∗ (Figure 13-a) imposed by the shape controller, which in this case turns to be equal to r˜ because of the absence of saturation, is modified by the rate limiter to ensure that the reference signal r (Figure 13-b) reaching the vertical controller has a fixed rate limit. The improvement in the response is manifested by the shape of the coil currents signals (y1 , . . . , y6 ) (Figure 13-c). However the conservatism of this approach can be appreciated when we examine the chopper commands Vc (Figure 13-d). These signals show that there is appreciable available actuation force that is not used. Figure 14 shows how the system becomes much faster thanks to the adaptation of the reference rate limit but keeps the smoothness of the response. A quick comparison between Figure 12-c (or Figure 13-c) and Figure 14-c shows the effectiveness of the approach. It is possible to note how the adaptation of the rate limit R between Rmin and Rmax (Figure 14-a) for each coil according to the available actuation force (Figure 14-d) allows the system to reach the steady state for the coil currents much sooner. The current references in r (Figure 14-b) depend now on the available actuation force in each one of the vertical coils. The rate limit is reduced for those channels more affected by saturation. Although it is not possible to compare the performance of the system without and with anti-windup compensators when the current references are steps of magnitude equal to 840 Amps (because the system without anti-windup compensator is unstable in this case), it is possible to make such comparison when the current references are steps of magnitude equal to 280 Amps as it is shown in Figure 15. When δ = 0 the step response (b) exhibits a deterioration of performance with respect to the step response without anti-windup (a). However, when δ = 0 the step response (c) is smoother and shows a settling time that is approximately the same as shown in the step response without anti-windup (a). When the rate limiter is activated (d), the step response is not only smoother but also the settling time is reduced considerably for some of the coils.

28


8 Conclusions The proposed scheme has been shown in nonlinear simulations to be very effective in guaranteeing stability of the inner loop in the presence of voltage saturation of the vertical coils. The scheme is being implemented and will be tested in experimental conditions. After succeeding in the vertical stabilization of the plasma in experimental conditions, efforts will be concentrated on the design of the outer shape control loop. The necessity of a similar anti-windup scheme for the outer loop is anticipated; not only due to the inherent limitations of its actuators but also due to the fact that the inner loop will modify, through the watch-dog and rate limiter, the control signals of the outer loop in order to preserve stability of the inner plant and improve performance. In this case we will deal with a stable (stabilized by the inner loop design) but nonlinear plant.

References 1. J. L. Luxon, “A design retrospective of the DIII-D tokamak,” Nuclear Fusion, vol. 42, pp. 614-33, 2002. 2. D. A. Humphreys, M. L. Walker, J. A. Leuer and J. R. Ferron, “Initial implementation of a multivariable plasma shape and position controller on the DIII-D tokamak,” Proceedings of the 2000 IEEE International Conference on Control Applications, Anchorage, Alaska, USA, pp. 385-94, September 2000. 3. R. Albanese and G. Ambrosino, “Current, position and shape control of tokamak plasmas: A literature review,” Proceedings of the 2000 IEEE International Conference on Control Applications, Anchorage, Alaska, USA, pp. 412-18, September 2000. 4. Y. Mitrishkin and H. Kimura, “Plasma vertical speed robust control in fusion energy advanced tokamak,” Proceedings of the 40th Conference on Decision and Control, Orlando, Florida, USA, pp. 1292-97, December 2001. 5. M. Ariola, A. Pironti and A. Portone, “Vertical stabilization and plasma shape control in the ITER-FEAT tokamak,” Proceedings of the 2000 IEEE International Conference on Control Applications, Anchorage, Alaska, USA, pp. 401-405, September 2000. 6. M. Ariola, A. Pironti and A. Portone, “A framework for the design of a plasma current and shape controller in next-generation tokamaks,” Fusion Technology, Vol. 36, pp. 263-277, Nov. 1999.


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7. M. Ariola, G. Ambrosino, J. B. Lister, A. Pironti, F. Villone and P. Vyas, “A modern plasma controller tested on the TCV tokamak,” Fusion Technology, Vol. 36, pp. 126-138, Sep. 1999. 8. G. Ambrosino, M. Ariola, Y. Mitrishkin, A. Pironti and A. Portone, “Plasma current and shape control in tokamaks using H∞ and μ-synthesis,” Proceedings of the 36th Conference on Decision and Control, San Diego, California, USA, pp. 3697-3702, December 1997. 9. A. Portone, R. Albanese, Y. Gribov, M. Hughet, D. Humphreys, C. E. Kessel, P. L. Mondino, L. D. Pearlstein and J. C. Wesley, “Dynamic control of plasma position and shape in ITER,” Fusion Technology, Vol. 32, pp. 374-88, Nov. 1997. 10. A. Portone, Y. Gribov, M. Hughet, P. L. Mondino, R. Albanese, D. Ciscato, D. Humphreys, C. E. Kessel, L. D. Pearlstein and D. J. Ward, “Plasma position and shape control for ITER,” Proceedings of the 16th IEEE/NPSS Symposium on Fusion Engineering, Champaign, IL, USA, pp. 345-348, Sep. 1995. 11. G. Ambrosino, M. Ariola, A. Pironti, A. Portone and M. Walker, “A control scheme to deal with coil current saturation in a tokamak,” IEEE Transactions on Control Systems Technology, Vol. 9, no. 6, p. 831, November 2001. 12. R. Albanese, G. Ambrosino, A. Pironti, R. Fressa and A. Portone, “Optimization of the power supply demand for plasma shape control in a tokamak,” IEEE Transactions on Magnetics, Vol. 34, no. 5, p. 3580, September 1998. 13. L. Scibile and B. Kouvaritakis, “A discrete adaptive near-time optimum control for the plasma vertical position in a tokamak,” IEEE Transactions on Control Systems Technology, Vol. 9, no. 1, p. 148, January 2001. 14. J-Y. Favez, Ph. Mullhaupt, B. Srinivasan, J.B. Lister and D. Bonvin, “Improving the region of attraction of ITER in the presence of actuator saturation,” IEEE Conference on Decision and Control, 4616-21, 2003. 15. M.L. Walker, D.A. Humphreys and E. Schuster, “Some nonlinear controls for nonlinear processes in the DIII-D tokamak,” Proceedings of the 42th IEEE Conference on Decision and Control, Maui, Hawaii, USA, pp. 6565-6571, Dec. 2003. 16. M.L. Walker, J.R. Ferron, B. Penaflor, D.A. Humphreys, J.A. Leuer, A.W. Hyatt, C.B. Forest, J.T. Scoville, B.W. Rice, E.A. Lazarus, T.W. Petrie, S.L. Allen, G.L. Jackson, R. Maingi, “Status of DIII-D plasma control,” Proceedings of the 16th IEEE/NPSS SOFE, p. 885, 1995. ˚ om and L. Rundqwist, “Integrator windup and how to avoid it,” Proceedings of 17. K. J. Astr¨ the American Control Conference, Pittsburgh (PA), USA, vol. 2, pp. 1693-98, June 1989. 18. R. Hanus, “Antiwindup and bumpless transfer: a survey,” Proceedings of the 12th IMACS World Congress, Paris, France, vol. 2, pp. 59-65, 1988.

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19. M. Morari, “Some control problems in the process industries,” H. L. Trentelman and J. C. Willems, editors, Essays on Control: Perspectives in the Theory and its Applications, pp. 55-77. Birkhauser, Boston (MA), USA, 1993. 20. E. G. Gilbert and I. V. Kolmanovsky, “Set-point control of nonlinear systems with state and control constraints: A Lyapunov-function, Reference-Governor approach,” Proceedings of the 38th Conference on Decision and Control, Phoenix (AZ), USA, pp. 2507-12, Dec. 1999. 21. A. Bemporad and M. Morari, “Control of systems integrating logic, dynamics, and constraints,” Automatica, vol. 35, pp. 407-427, 1999. 22. L. Scilibe and B. Kouvaritakis, “Stability region for a class of open-loop unstable linear systems: theory and application,” Automatica, vol. 36, pp. 37-44, 2000. 23. J. S. Shamma, “Anti-windup via constrained regulation with observers,” Systems and Control Letters, vol. 40, pp. 1869-83, 2000. 24. A. Zheng, M. V. Kothare and M. Morari, “Anti-windup design for internal model control,” International Journal of Control, vol. 60, no. 5, pp. 1015-1024, 1994. 25. A. Miyamoto and G. Vinnicombe, “Robust control of plants with saturation nonlinearity based on coprime factor representation,” Proceedings of the 36th Conference on Decision and Control, Kobe, Japan, pp. 2838-40, 1996. 26. E. F. Mulder, M. V. Kothare and M. Morari, “Multivariable anti-windup controller synthesis using linear matrix inequalities,” Automatica, vol. 37, no. 9, pp. 1407-16, 2001. 27. A. R. Teel and N. Kapoor, “The L2 anti-windup problem: Its definition and solution,” Proceedings of the 4th ECC, Brussels, Belgium, July 1997. 28. A. R. Teel and N. Kapoor, “Uniting local and global controllers,” Proceedings of the 4th ECC, Brussels, Belgium, July 1997. 29. A. R. Teel, “Anti-Windup for exponentially unstable linear systems,” International Journal of Robust and Nonlinear Control, vol. 9, pp. 701-716, 1999. 30. C. Barbu, R. Reginatto, A.R. Teel and L. Zaccarian, “Anti-windup for exponentially unstable linear systems with rate and magnitude limits,” V. Kapila and K. Grigoriadis (editors), Actuator Saturation Control, Marcel Dekker, 2002.

Stable and Unstable Systems with Amplitude and Rate Saturation Peter Hippe Lehrstuhl für Regelungstechnik, Universität Erlangen-Nürenberg [email protected]

1 Introduction In nearly all control applications, the output signal of the compensator cannot be transferred to the system with unlimited amplitudes and arbitrarily fast. Such limitations in magnitude and rate can lead to performance degradations and they may even cause an unstable behaviour of the closed loop. These undesired effects of actuator saturation are called windup. The problem most intensively investigated is that of amplitude saturation. The prevention of windup effects due to input magnitude restrictions is the subject of a great variety of papers. The references in [3], [8] and [26] give a good overview of the existing literature. Controllers with badly damped or unstable modes tend to “wind up” during input saturation, giving rise to big overshoots and slowly settling transients. Since these effects are related to the dynamics of the controller, they are called controller windup [15]. Most of the early windup literature was devoted to the prevention of this kind of windup. A unified framework incorporating the various existing schemes is presented in [19]. If the observer technique [15], [13] is applied, controller windup is prevented and the dynamics of the controller no longer influence the effects of input saturation. Then all undesired effects of input saturation can be attributed to plant windup [15], caused by the dynamics of the controlled plant which are assigned by constant state feedback. When the characteristic polynomial of the state observer appears in the numerator

32

Peter Hippe

of the reference channel, the conditioning technique yields the same results as the observer technique. In the framework of the conditioning technique, the remaining effects of plant windup are called the “short sightedness of the conditioning technique” [10]. Whereas controller windup can be removed by mere structural methods (appropriately add a model of input saturation to the controller and stabilize it when its output signal becomes constraint), the prevention of plant windup usually calls for additional dynamic elements like the filtered setpoint [25] or the additional dynamic network [15]. Consequently, the prevention of windup is possible in a two-step approach. First remove the influence of the controller dynamics, and if this does not remove all undesired effects of input saturation (i.e., if the danger of plant windup exists), introduce additional dynamic elements. A one-step approach to the joint prevention of controller and plant windup was presented in [31]. This anti-windup scheme contains a model of the plant (additional dynamics), and its major advantage is the automatic elimination of controller windup, regardless of whether the compensator is a classical P, PI or PID controller or an observer-based controller possibly containing signal models for disturbance rejection. The only design parameter of the scheme is the feedback of the model states, assuring the prevention of plant windup. There is a slight drawback insofar, as the additional dynamics are also present if the undesired effects of input saturation can solely be attributed to controller windup. Without additional measures, the above-discussed methods for the prevention of plant windup are only applicable to open loop stable systems. Magnitude constraints always limit the maximum sustainable output amplitudes. In stable systems no control input can drive the outputs beyond these amplitudes, which are finite in the absence of eigenvalues at s = 0. For unstable systems, these amplitudes have to be avoided under all circumstances. If the output of an unstable system were to reach this limit, it could increase exponentially for any controller action. Consequently, stability can only be guaranteed as long as a sufficient safety margin is observed. For “slow” dynamics of the stabilizing control, step-like reference inputs could be applied close to the maximum sustainable amplitudes, i.e., the safety margins could be kept small. For faster dynamics of the closed loop, overshoots can appear, requiring considerably larger margins. This is the effect of plant windup. Virtually all known measures for the prevention of plant windup only become active after a saturation limit has been hit for the first time. Disturbances acting during

Stable and Unstable Systems with Amplitude and Rate Saturation

33

reference transients can obviously destabilize such schemes in the presence of exponentially unstable systems. There are papers on the control of constrained unstable systems. In [17] a reference “governor” is presented which, via a time varying rate operator, shapes the reference input so that the control input does not saturate. The computations for constructing the time varying rate are, however, quite complicated. In [34] it is proposed to approximate the linear constraints on the input (and states) by conservative quadratic constraints. In [27] the set point tracking problem was investigated using an internal model control configuration, and rules of thumb were presented giving bounds on amplitude and frequency for sinusoidal references ensuring stable saturated operation of the nonlinear reference filter. The tracking of constant reference signals and the computation of regions of initial conditions for which stability is guaranteed is considered in [32],[33]. In [32] a nonlinear term is added to the control law whereas in [33], arbitrary LQ control laws are allowed. Disturbance rejection for unstable systems with input saturation has also been investigated by various authors. The rejection of input additive disturbances is discussed for state feedback control in [26] and [21] whereas [20] solves the H∞ -almost disturbance decoupling problem. The windup problem for exponentially unstable systems in the presence of reference and disturbance inputs is considered in [30]. The disturbances, however, are restricted not to act on the exponentially unstable modes. The joint problem of tracking a certain reference level in the presence of (constant) disturbances was investigated in [29]. The objective of this approach is to null the steady state error due to disturbances by using a saturated controller containing nonlinear integrating action. Beyond the above discussed windup effects, there is an additional problem caused by saturating actors in MIMO systems, namely that of directionality. “For MIMO controllers, the saturation may cause a change in the direction of the plant input resulting in disastrous consequences” [7]. There are also various attempts to solve the directionality problem. An anti-windup scheme with stability augmentation and appropriate modifications of the saturation limits to prevent directionality problems was presented in [28]. In [36] and [24] the conditioning technique is enhanced for windup prevention by an artificial nonlinearity, optimized by nonlinear programming such that the realizable reference signal stays to the actual one as closely as possible. A similar approach has also been used in

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Peter Hippe

[23]. An artificial multivariable state dependent saturation element preserving the directionality of the input vector is used in [18]. An optimization of the internal model control setup to minimize the effects of input saturation on the directionality problem is presented in [37]. All these approaches constitute approximate solutions. The only scheme seemingly preserving the decoupled behaviour in spite of input saturation is the directionality preserving control of [4]. Unfortunately it maintains directionality only in very specific applications. Its basic feature, loop gain reduction to prevent input signal saturation, makes the scheme time variant so that stability may be at stake especially in the case of unstable plants. The methods discussed so far have been developed for amplitude saturated actuators. If one has to cope with an additional rate saturation, the prevention of its destabilizing influence calls for novel measures. Also the case of joint amplitude and rate constraints has been investigated intensively (see e.g., the references in [9], [2] or [1], and the discussions of the existing approaches therein). The solutions presented so far use an actuator model incorporating two nonlinear elements (for amplitude and rate saturation). Consequently the problem becomes much more involved as compared to the well-known techniques for amplitude-restricted actuators. In this contribution, a new scheme for the prevention of windup is presented which is applicable both to stable and unstable SISO and MIMO systems. It uses a nonlinear model-based reference shaping filter for feedforward tracking such that feedback signals are not caused by reference inputs. The feedback controller is used for system stabilization and disturbance rejection. Therefore, one can reserve part of the constrained input for disturbance rejection while the remaining amplitude region is available for tracking. This allows a proof of stability for constrained unstable systems subject to reference and disturbance inputs. Also a perfect directionality preservation becomes possible by the presented approach. Also presented is a solution to the problem of joint amplitude and rate saturation, which allows to use the well-known results for amplitude saturated actuators. This is made possible by the introduction of a simple ersatz model for such actuators, consisting of an input saturation and a first-order dynamic element, whose time constant is chosen so that the rate constraint is automatically met. By adding the dynamic element to the plant, the windup prevention for amplitude- and rate-constrained actuators boils down to the well-known problem of amplitude saturation for the augmented plant.


35

Two examples demonstrate the application of the proposed scheme for the prevention of windup.

2 The Problem Considered Given a strictly proper, linear, time invariant MIMO system having state x(t) ∈ n , control input us (t) ∈ p , controlled variables y(t) ∈ p and measurements ym (t) ∈ m , and a completely controllable and observable state space representation x(t) ˙ = Ax(t) + Bus (t) + Bd d(t) y(t) = Cx(t) + Cd d(t)

(1)

ym (t) = Cm x(t) + Cmd d(t) where d(t) ∈ q is a disturbance input. In the sequel, the tracking of constant reference signals is investigated. This is only feasible when the system has no transmission zeros at the origin. Therefore, it is assumed that the (square) transfer function matrix G(s) = C(sI − A)−1 B = N (s)D−1 (s)

(2)

is such that det N (s) does not have zeros at s = 0. There is a saturating nonlinearity us (t) = satu0 (u(t)) at the input of the system and its components are described by ⎧ ⎪ ⎪ u if ui > u0i ⎪ ⎨ 0i satu0i (ui ) = ui if − u0i ≤ ui ≤ u0i ; u0i > 0, i = 1, 2, ..., p ⎪ ⎪ ⎪ ⎩−u if ui < −u0i 0i

(3)

Due to input saturation the maximum sustainable output amplitudes are confined to the regions yimin ≤ yi (∞) ≤ yimax , i = 1, 2, ..., p

(4)

and it is assumed that these limits are finite, which is the case in systems without eigenvalues at s = 0. The case of systems with eigenvalues in the origin of the s-plane will be addressed in Remark 5.

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In order to get a meaningful control problem, the controlled variables must stay within the maximum usable amplitude ranges yil ≤ yi (∞) ≤ yiu , i = 1, 2, ..., p

(5)

characterized by the requirement that arbitrary combinations of the output signals yi (t) can be sustained on the boundaries (5). In single input systems the equalities |yimin | = yil and |yimax | = |yiu | always hold. In MIMO systems, however, one usually has |yimin | > yil and |yimax | > |yiu | and the usable regions not only depend on the saturation limits u0i , but also on each other. Example 1. As a demonstrating example consider the system with transfer matrix ⎡ ⎤ 1 1 ⎢ ⎥ ⎢ s + 4 s + 1⎥ ⎢ ⎥ G(s) = ⎢ ⎥ ⎣ 1 1 ⎦ s+2

s+5

and saturation limits u01 = u02 = 10. The maximum sustainable output amplitudes are y1min = −12.5, y1max = 12.5, y2min = −7 and y2max = 7, because the input us1 (∞) 10 y1 (∞) 12.5 vector gives rise to an output vector . = = 10 7 us2 (∞) y2 (∞) y1l , y1u −7.5, +7.5 One maximum usable output range is, for example, = . In −3, +3 y2l , y2u order to find this range, one has to check for the worst case situation that can still be handled with the given input limits. If both outputs y1∞ = y1 (∞) and y2∞ = y2 (∞) were at the upper or the lower limits, the necessary components of the input vector

t would be |us1∞ | = 10/3 and |us2∞ | = 20/3. A vector y∞ = 7.5, −3 , however, requires an input vector uts∞ = −10, 10 . An output range with symmetric lim −3.75, 3.75 y1l , y1u as the vector its of equal magnitude is characterized by l u = −3.75, 3.75 y 2 , y2 t = 3.75, −3.75 requires an input vector uts∞ = −10, 6.25 . If, as another y∞

example, only positive outputs were desired, the existing in with equal magnitudes l u y 1 , y1 0, 4.5 t put saturation limits allow l u = , because the vector y∞ = 0, 4.5 0, 4.5 y 2 , y2 t requires an input vector us∞ = 10, −2.5 .


37

For the sake of simplicity, the limits of the maximum usable amplitudes are assumed to be symmetric, i.e., yiu = −yil = y0i , i = 1, 2, ..., p. In the case of stable plants, the limits y0i also define the maximum used ranges. In the case of unstable plants, these limits must be avoided because a non-recoverable unstable behaviour could occur if an output yi (t) reached the limit y0i . In view of safety consideration and of unknown disturbance inputs, the outputs must be confined to the maximum used ranges −r0i < yi (t) < r0i , i = 1, 2, ..., p

(6)

where the r0i > 0 satisfy r0i < y0i . Tis also specifies the regions −r0i ≤ ri (t) ≤ r0i > 0, i = 1, 2, ..., p

(7)

of the maximum applied reference signal changes. The r0i depend on the dynamics assigned to the loop. For slow dynamics, the margins Mi = y0i − r0i may be small, whereas for faster dynamics, they have to be increased. This is the effect of plant windup due to input saturation. Now consider the case where the signals usi (t) not only differ from the ui (t), i = 1, 2, ..., p whenever |ui (t)| ≥ u0i > 0 (amplitude saturation), but also whenever |u˙ i (t)| ≥ uvi > 0 (rate saturation), i.e., there is an actuator with combined amplitude and rate saturation. In [1] a model according to u˙ si (t) = uvi sgn {satu0i [ui (t)] − usi (t)} , i = 1, 2, ..., p

(8)

has been proposed, where satu0i (·) is the above-defined saturation function (3) and sgn(·) is the standard sign function. In numerical simulations, this actuator model causes problems whenever the input to the sign function vanishes and whenever |u˙ i (t)| < uvi . A model better suited for simulations of an amplitude- and raterestricted actuator component usi (t) = satuuvi (ui (t)) is shown in Fig. 1. 0i The gains Ri , i = 1, 2, ..., p should be chosen such that the resulting time constants 1/Ri do not have a significant influence on the behaviour of the closed loop. The nonlinear elements in Fig. 1 are again of the saturation type, namely u ¯si (t) = satu0i (ui (t)) = sign(ui (t)) min {u0i , |ui (t)|} , u0i > 0 and wsi (t) = satuvi (wi (t)) = sign(wi (t)) min {uvi , |wi (t)|} , uvi > 0, i = 1, 2, ..., p.

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Peter Hippe

satu0i ui

satuvi

usi

wsi

wi

u ¯si

1

Ri

s

Fig. 1. Simulation model of an amplitude- and rate-restricted actuator component

In the case of mere amplitude saturation, one inserts a model of the restricted actuator (see (3)) at the output of the compensator. Thus, the actual actuator never saturates, so that only the model has to be taken into account. If the actuator contains an additional rate saturation, the ersatz models shown in Fig. 2 can be inserted at each output ui (t), i = 1, 2, ..., p of the compensator. Due to the saturation element u ¯si (t) = satu0i (ui (t)) the amplitude of the output signal usi (t) never exceeds the limit u0i . When choosing the time constant Tvi of the first-order system as Tvi =

2u0i , i = 1, 2, ..., p uvi

(9)

one also assures |u˙ si (t)| ≤ uvi , because the maximum speed |u˙ si (t)| resulting in the worst case situation (namely a sudden change of u ¯si (t) from one limit to the opposite) is uvi . Consequently, when inserting these models at the compensator output, the plant input signal neither exceeds the rate nor the amplitude constraints, so that the actual actuator nonlinearities do not have to be taken into account. When adding the first-order elements of the p actuator models x˙ v (t) = Av xv (t) + Bv u ¯s (t)

(10)

us (t) = xv (t)

satu0i ui

usi

u ¯si 1 Tvi s + 1

Fig. 2. Ersatz model of an actuator component with joint amplitude and rate constraints


39

t with the ¯ts (t) = [¯ us1 (t), · · · , u ¯sp (t)], vectors xv (t) = [xn+1 (t), · · · , xn+p (t)] , u 1 Tv1 , · · ·

ttv =

, T1vp , and Av = diag(−tv ), Bv = diag(tv ) to the system (1), one

obtains the state equations ¯x(t) + B ¯u ¯d d(t) x ¯˙ (t) = A¯ ¯s (t) + B y(t) = C¯ x ¯(t) + Cd d(t)

(11)

¯(t) + C¯md d(t) y¯m (t) = C¯m x of the augmented system with x ¯(t) =

x(t)

xv (t)

, y¯m (t) =

ym (t)

us (t)

(since the vector

us (t) = xv (t) is measurable) and the parameters

A A¯ = 0

C¯ = C

B

Av 0 ,

,

¯= B

C¯m =

0

Bv Cm 0

¯d = B

,

0 Ip

Bd 0

,

C¯md =

Cmd

0

0

0

This augmented plant has an input nonlinearity u ¯s (t) = satu0 (u(t)), so that the problem of joint amplitude and rate constraints can be reformulated as a problem with mere input saturation for the system (11). Remark 1. The following discussions are valid for systems with amplitude saturation only and for systems with joint amplitude and rate saturation. In the case of mere amplitude restraints, the state equations (1), and in the case of additional rate constraints, the state equations (11) constitute the description of the plant. In order to keep the developments simple, the case of mere amplitude saturation is considered in the sequel, as it is now obvious, how the case of additional rate constraints can be handled. Remark 2. The above-suggested procedure is somehow outside the usual design paradigm of windup prevention, namely: “Design the linear control regardless of the input saturations and then add appropriate measures to remove the problems caused by these restrictions.” However, the introduction of the ersatz model for an amplitude- and rate-restrained actuator considerably facilitates the design of antiwindup measures and it has the advantage of being applicable in the presence of stable and unstable systems.

40

Peter Hippe

When using a slightly modified ersatz model, the usual design paradigm is also applicable in the case of additionally rate-resticted actuators (see [14], Section 7.4). This modified ersatz model, however, should only be applied in the case of stable systems.

3 The Underlying Control Scheme Given the system description (1), it is assumed that there is a nominal state feedback (12)

uC (t) = Kx(t)

This nominal linear control, characterized by the feedback interconnection us (t) = −uC (t) is supposed to stabilize the possibly unstable system, and to give a desired disturbance attenuation such that even worst case disturbances do not give rise to control signals ui (t) with amplitudes beyond the saturation limits u0i . Reference tracking is assured by feedforward control as shown in Fig. 3. The corresponding input signal uCr (t) drives a model of the system in the reference shaping filter (see Section 5) and it is assumed, that this model is correct. Therefore, after the initial transients have decayed, reference inputs lead to coinciding states xM (t) = x(t) in the model and the system. Consequently, the feedback signal uCd (t) vanishes unless xM (t) = x(t) is caused by external disturbances.

d

satu0 uCr

u

us

Linear System

uCd r

x

Nonlinear reference

K

shaping filter xM

Fig. 3. Proposed control scheme

y


41

The reference shaping filter is designed such that only the safely applicable reference signal amplitudes are effective and all control signals uCri (t) stay in predefined amplitude ranges. Assume that the worst case disturbances cause feedback signals |uCdi (t)| ≤ βi u0i , i = 1, 2, ..., p, 0 < βi < 1

(13)

In exponentially unstable systems (13) is a necessary assumption. If disturbances generate feedback signals |uCdi (t)| > u0i , the stability of the loop cannot be guaranteed. By confining the signals uCri (t) to the regions |uCri (t)| ≤ (1 − βi )u0i = uβi , i = 1, 2, ..., p

(14)

the saturation us (t) = satu0 (u(t)) at the input of the system never becomes active in spite of jointly acting reference and disturbance signals. Consequently, even in the presence of exponentially unstable systems the stability of the scheme in Fig. 3 is guaranteed. Remark 3. A design of the scheme in Fig. 3 such that the input saturation us (t) = satu0 (u(t)) never becomes active gives rather conservative results. If one knew the exact forms of the reference and the disturbance signals, one could allow for some over-saturation in the sense of [27]. For arbitrary signal forms, however, a stable behaviour of a loop containing an unstable system can only be guaranteed if input saturation never becomes active (consider, e.g., the case of constant disturbances driving one or more input signals ui (t) beyond the saturation limits). Remark 4. Disturbances acting during reference transients could be allowed to cause saturating input signals ui (t) in stable systems. For a nominal state feedback (12) such that K(sI − A)−1 B violates the circle criterion, this would, however, entail the danger of plant windup. For stable systems, plant windup can be prevented by an additional dynamic element (ADE). In the scheme of Fig. 3 this ADE is equivalent to substituting the input u(t) = uCr (t) − uCd (t) to the saturating element by u(t) = uCr (t) − uCd (t) − η(t) with

(15)

42

Peter Hippe

˙ = (A − BKS )ξ(t) + B [u(t) − us (t)] ξ(t) η(t) = (K − KS )ξ(t)

(16)

where KS is a “safe state feedback” such that KS (sI −A)−1 B meets the circle criterion [13]. This ADE is the time domain version of the “additional dynamic network” in [15]. The maximum amplitude of the output signal of a SISO system due to arbitrary input signals of know maximum amplitudes can be obtained by the L1 -norm of the corresponding transfer function (see also [12]). Given a linear, asymptotically stable, time invariant SISO system with transfer function G(s) and a corresponding impulse response h(t). Then, when an arbitrary input signal u(t) with known maximum amplitude |u(t)| ≤ ulim > 0 is applied to the input, the maximum amplitude ylim of the output of this system is bounded by ∞ ylim ≤ ulim |h(τ)| dτ = ulim G(s) 1

(17)

0

where G(s) 1 is the L1 -norm of G(s) (see, e.g., [5]). For a MIMO system with a p × m-transfer function matrix G(s), the notion G(s) 1 will be used in the sense ⎡

⎤ G11 (s) 1 · · · G1m (s) 1 ⎢ ⎥ .. .. ⎥ G(s) 1 = ⎢ . . ⎣ ⎦

(18)

Gp1 (s) 1 · · · Gpm (s) 1 Denote the transfer function matrix of the closed loop between the disturbances d and the controller output uCd (see Fig. 3) by uCd (s) = Gd (s)d(s)

(19)

Then, given the worst case amplitudes dimax , i = 1, 2, ..., q of the disturbances di (t), the signals uCdi (t) caused by these disturbance inputs are limited by uCdimax =

p j=1

Gij (s) 1 djmax i = 1, 2, ..., p

(20)


43

4 Modifications in the Presence of State Observers and Signal Models for Disturbance Rejection The scheme shown in Fig. 3 uses constant state feedback control. Usually such controllers also incorporate state observers and signal models for disturbance rejection. In view of reduced-order observers, the measurement vector ym (t) of the system (1) is subdivided according to

⎤ ⎡ ⎤ ⎡ ⎤ ⎡ 1 1 1 C C ym (t) ⎦ = ⎣ m ⎦ x(t) + ⎣ md ⎦ d(t) ym (t) = ⎣ 2 2 2 (t) Cm Cmd ym

(21)

2 where the vector ym (t) contains the κ measurements directly used to obtain the state 1 estimate x ˆ(t) and the vector ym (t) the remaining (m − k) measurements.

The (reduced-order) state observer of order nO = n−κ with 0 ≤ κ ≤ m is described by z(t) ˙ = F z(t) + Dym (t) + T Bus (t)

(22)

where the observer technique for the prevention of controller windup has been applied, i.e., the observer is driven by the constrained input signal. For vanishing disturbances d(t) ≡ 0 and if the initial transients have decayed, the state estimate ⎡ ⎤⎡ ⎡ ⎤ ⎤ 2 2 2 ym Cm ym (t) (t) ⎦ = Ψ2 Θ ⎣ ⎦ = Ψym (t) + Θz(t) x ˆ(t) = ⎣ ⎦ ⎣ (23) T z(t) z(t) coincides with the state x(t) of the system if the relations T A − F T = DCm

(24)

ΨCm + ΘT = In

(25)

and

are satisfied, where the abbreviation Ψ = [0n,m−κ Ψ2 ] has been used. Substituting x(t) in Fig. 3 by x ˆ(t), only the disturbance rejection of the loop is modified. The reference behaviour remains unchanged since the constrained input us (t) does not cause observation errors. Consequently, for reference inputs r(t) one has x ˆ(t) = x(t) = xM (t), so that uCd (t) vanishes. This allows the above-discussed subdivision of the limited input signal into one part for disturbance rejection and the remaining part for tracking.

44

Peter Hippe

If constantly acting disturbances, modeled in a signal process with characteristic polynomial det(sI − Sd ) = sq + ψq−1 sq−1 + · · · + ψ0

(26)

have to be compensated asymptotically, one can use a slightly modified version of Davison’s approach [6]. In its original version, Davison’s approach assures reference tracking and disturbance rejection for the same type of signals. On the one hand, this is more than required here and on the other hand, it can have an unfavourable influence on the tracking behaviour [11]. Therefore, Davison’s approach is modified such that the signal model is only used for disturbance rejection. To achieve this, the signal model is no longer driven by Bε [y(t) − r(t)], but by Bε y(t) and Bσ [ˆ uC (t) + us (t)], where u ˆC (t) is the output of the observer-based controller and us (t) is the saturated input signal to the plant. By an appropriate choice of Bσ the dynamics of the modified signal model do not influence the tracking behaviour. The modified signal model has the form v(t) ˙ = Sv(t) + Bε y(t) − Bσ [ˆ uC (t) + us (t)]

(27)

with S = diag [Sd , · · · , Sd ] , Bε = diag [bε , · · · , bε ] and ⎤ 1 0 ··· 0 ⎢ . .. ⎥ .. ⎥ ⎢ . ⎢ . . . ⎥ Sd = ⎢ ⎥, ⎥ ⎢ 0 0 · · · 1 ⎦ ⎣ −ψ0 −ψ1 · · · −ψq−1 ⎡

⎡ ⎤ 0 ⎢.⎥ ⎢.⎥ ⎢.⎥ bε = ⎢ ⎥ ⎢0⎥ ⎣ ⎦ 1

Thus the vector v(t) has pq components. The quantity u ˆC (t) is defined by x ˆ(t) u ˆC (t) = Kx Kv v(t)

(28)

(29)

The matrices Kx and Kv will be chosen to obtain a controlled signal model with desired eigenvalues and a reference behaviour that only depends on the nominal state feedback Kx(t). Given the solution X of the Sylvester equation X(A − BK) − SX = Bε C and the matrix

(30)


45

Bσ = −XB

(31)

With this Bσ choose the feedback matrix Kv such that the eigenvalues of (S − Bσ Kv ) have desired locations. When finally computing the feedback matrix Kx according to K x = K − Kv X

(32)

the eigenvalues of the loop are the zeros of the characteristic polynomial det(sI − A + BK)det(sI − F )det(sI − S + Bσ Kv ), and neither the state observer nor the signal models influence the reference behaviour (for a proof of this see, e.g., [13]). In the absence of disturbances and after the initial transients have decayed, reference inputs cause u ˆC (t) = K x ˆ(t) = Kx(t). Therefore, the scheme in Fig. 3 needs to be slightly modified as shown in Fig. 4. d satu0 uCr

y x˙ = Ax+Bus +Bd d

us

u

y = Cx + Cd d

ym

ym = Cm x + Cmd d

r

Nonlinear uCd

reference

x ˆ

Kx

z˙ = F z +Dym +T Bus x ˆ = Θz + Ψym

shaping filter σ xM v Kv

K

v˙ = Sv+Bε y−Bσ σ

u ˆC

Fig. 4. The control structure of Fig. 3 in the case of observer-based compensators containing signal models for disturbance rejection

5 The Model-based Reference Shaping Filter Given is a model x˙ M (t) = AM xM (t) + BM uas (t) yM (t) = CM xM (t)

(33)

46

Peter Hippe

of the system (1) with an external input uas (t), and we assume that this model is correct, i.e., (AM , BM , CM ) = (A, B, C). At the input of the model is a nonlinearity uas (t) = satuβ (ua (t)), where the saturation limits uβi are defined in (14). Further assume for the moment that AM has no eigenvalues at s = 0. The case of systems having eigenvalues at s = 0 will be discussed in Remark 5. Based on this model of the system, the nonlinear reference shaping filter shown in Fig. 5 is considered. First we design a state feedback ua (t) = −Ka xM (t) + La ubs (t)

(34)

for the model (33) so that in the linear case (i.e., uas (t) = ua (t)) the possibly unstable model is stabilized, and the steady state errors yM i (∞) − ubsi (∞), i = 1, 2, ..., p vanish for constant input signals ubsi (t). This is assured by

−1 La = CM (−AM + BM Ka )−1 BM

(35)

Given the limits uβi for the maximum amplitudes of the input signals uasi (t) = ˆ uCri (t) to the model of the system, the zeros of det(sI − AM + BM Ka ) are chosen such that the ranges

r Reference shaping filter L−1 a Lb

uCr satuβ

satr0 ub

ubs

ua La

L−1 a Kb

yM

uas x˙ M = AM xM +BM uas

CM

Ka

xM

Fig. 5. Reference shaping filter


−r0i ≤ ubsi (t) ≤ r0i , i = 1, 2, ..., p

47

(36)

can be made as large as possible without causing signals |uai (t)| > uβi (see (14)). This tuning of (34) has two important implications. On the one hand, actuator saturation does not become active when using uCr (t) = uas (t) as an input to the system (see Figs. 3 and 4), and on the other hand, also the input saturation uas (t) = satuβ (ua (t)) in front of the model in the scheme of Fig. 5 is never active. Thus, the possibly unstable dynamics of the model are always stabilized by the linear control (34). For arbitrary input signals ri (t), the restriction (36) can be assured by a saturation element ubs (t) = satr0 (ub (t)) as shown in Fig. 5. The transfer behaviour between ubs and uas = ua is characterized by ua (s) = Gua (s)ubs (s) with Gua (s) = −Ka (sI − AM + BM Ka )−1 BM La + La

(37)

Given arbitrary input signals within the limits (36), one obtains the upper limits of the signals uai (t) with the aid of the L1 -norm Gua (s) 1 (see also (17), (18) and (20)) as uaimax

p u G a (s) r0j , i = 1, 2, ..., p = ij 1

(38)

j=1

and these have to meet the restrictions uaimax ≤ uβi . Due to the above construction, the inner loop is a linear stable system with input saturation ubs (t) = satr0 (ub (t)). Consequently, one can add an outer loop

with

−1 ub (t) = −L−1 a Kb xM (t) + La Lb r(t)

(39)

−1 −1 Lb = CM [−AM + BM (Ka + Kb )] BM

(40)

assuring vanishing tracking errors yM i (∞) − ri (∞), i = 1, 2, ..., p for constant reference signals. Now consider the transfer function matrix −1 GL (s) = L−1 BM La a Kb (sI − AM + BM Ka )

(41)

characterizing the transfer behaviour ub (s) = −GL (s)ubs (s) in Fig. 5 (recall that saturation uas (t) = satuβ (ua (t)) never becomes active). If GL (s) satisfies the circle

48

Peter Hippe

criterion [35], the cascaded nonlinear loop in Fig. 5 is globally asymptotically stable for arbitrary reference inputs. This, of course, is only true for vanishing initial conditions xM (0). As r(t) is the only input to the reference shaping filter, a start from xM (0) = 0 can be assured. Remark 5. The above-presented design also holds for systems with eigenvalues at the origin of the s-plane, and it then allows for the chosen maximum reference signal amplitudes r0i . If, however, one wants to exploit the potential of such systems to reach unlimited output amplitudes in spite of input saturation, eigenvalues in the inner loop (i.e., roots of det(sI − AM + BM Ka )) should be placed at s = 0. Only in the second cascade (b) these eigenvalues are also shifted to the left. Such reference shaping filters allow one to obtain unlimited amplitudes yM (∞) in a stable manner. Remark 6. In the case of stable systems, the inner stabilizing loop in the reference shaping filter is no necessity. Then the trajectory generator in Fig. 5 could be realized with one loop only (i.e., Ka = 0, La = Ip , and the nonlinearity ubs (t) = satr0 (ub (t)) removed), and the outer cascade such that Kb (sI−AM )−1 BM satisfies the circle criterion. Then, however, directionality problems (see Section 6) could occur, which is not the case, when the above-discussed filter design is also carried out for stable systems.

6 The Problem of Directionality An important design goal in MIMO systems is to assure a decoupled reference behaviour. The change of one set point should not have a disturbing influence on the remaining ones. The decoupling controller generates specific plant input signals ui (t), defining the direction of the input vector u(t). When one or more input signals saturate, the direction of this input vector is changed, giving rise to possibly strong and undesired coupling effects, the so-called directionality problem. Another important issue related to the decoupling control of constrained systems is the fact, that there are plants that cannot be decoupled without violating all the existing stability criteria for systems with input saturation. Consequently, if saturation becomes active in such a loop, an unstable behaviour can occur (see Example 2).


49

The above-presented design procedure allows an exact preservation of directionality. This becomes possible by designing the inner loop of the reference shaping filter to obtain a decoupled transfer behaviour between ubs and yM such that arbitrary signals ubsi (t) within the ranges −r0i ≤ ubsi (t) ≤ r0i do not cause saturated signals uai (t). Seen from the input ub (t) this leads to p linear decoupled subsystems with input saturations ubsi (t) = satr0i (ubi (t)). When adding p SISO loops ubi (t) = −kbi yM i (t) + li ri (t), a saturation ubsi (t) = satr0i (ubi (t)) does not cause directionality problems. Given the feedback factors kbi , the rows of L−1 a Kb in (39) have the forms diag[kb1 , . . . , kbp ]Cm and L−1 a Lb = diag [l1 , · · · , lp ]. The shaping filter exhibits a stable behaviour for arbitrary reference inputs if the transfer matrix (41) satisfies, i.e., the circle criterion. If the system cannot be decoupled by static state feedback, the necessary prefilter (see, e.g., [22]) could be incorporated in the reference shaping filter and when the prefiltered input signal to the model also drives the system, the scheme of Figure 3 would again be applicable.

7 Two Examples Example 2. Considered is a stable system with ⎡

0

1

0

0

0

0

⎡ 0 ⎢ ⎢1 ⎢ ⎢0 ⎢ B=⎢ ⎢0 ⎢ ⎢ ⎣0 0

⎤

⎥ ⎢ ⎢−3 −2 2 0 −1 −1⎥ ⎥ ⎢ ⎢ 0 0 0 1 0 0⎥ ⎥ ⎢ A=⎢ ⎥, ⎢−2 2 −1 −3 −2 −2⎥ ⎥ ⎢ ⎥ ⎢ 0 0 0 0 0 1 ⎦ ⎣ −1 −3 2 2 0 −1

0 0 0 1 0 0

C = Cm

⎤

⎥ 0⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ ⎥ 0⎦ 1

⎤

⎡ 3 ⎢ =⎢ ⎣−2 1

0

1

1

0

10.5

0

1

1 −18

⎥ 0⎥ ⎦

1

0

0

1

18.5

0

and input saturation limits u01 = 6, u02 = 12 and u03 = 9. If one wants to use symmetric output regions with equal bounds y0i = y0 , the maximum sustainable amplitudes are y0 = 1.248, since the stationary output vector y(∞) = t

[−1.248, 1.248, 1.248] requires an input vector u(∞) where the first component

50

Peter Hippe

is at its boundary u01 = 6. In what follows it is assumed that the maximum applied reference signals are limited by r0i = 1, i = 1, 2, 3. The problems caused by input saturation in this system are first discussed in a standard state feedback setting with u(t) = −Kx(t) + Lr(t). The system can be decoupled by static state feedback, but each decoupling K violates the circle criterion. Assigning, e.g., a decoupling control u(t) = −Kx(t) + Lr(t) with ⎡ ⎤ ⎡ 7 0 18 8 9 1 72.5 9.5 ⎢ ⎥ ⎢ ⎢ ⎥ K=⎢ ⎣−16 0 6 5 −128 −20 ⎦ , L = ⎣ 0 7 −15 −5 −5

1

56

−7

14

0

⎤ 0 ⎥ 0⎥ ⎦ 7

the (linear) closed loop ⎞ has a decoupled reference behaviour of the form y(s) = ⎛ diag⎝

7

7 ⎠r(s). Using a step-like reference input sequence ri (t) = , s+7 s+7 s+7 ,

7

rsi 1(t) − 2rsi 1(t − 5), i = 1, 2, 3, (here 1(t) is the unit step function), the transients of the loop already exhibit an unstable behaviour for amplitudes rs1 = 0.5, rs2 = −0.5 and rs3 = −0.5. Also the directionality preserving approach [4] does not stabilize the transients in this case. A stable behaviour of the nonlinear loop can be obtained when the state feedback satisfies the Kalman-Yakubovich lemma [35] At P + P A = −Qt Q B t P + W t Q = KKY W t W = 2Ip With

⎤

⎡ 0.8

0.4

⎢ Q=⎢ ⎣−0.8 −0.8 0.4

0.4

the solution for KKY is ⎡ 1.290 2.978 ⎢ KKY = ⎢ ⎣−2.176 −2.465 2.900 −1.718

0

0

0

0.8

0.4

0

1.6

0

2.8

0

⎥ 0⎥ ⎦ 0.4

0.8359 −1.333 −2.787 −2.284

⎤

3.195

2.480

3.354

⎥ 1.799⎥ ⎦

1.659

1.799

8.665

4.037

and the corresponding reference injection matrix is


⎡ LKY =

29.51705 −66.76825 −84.80765

1 ⎢ ⎢ 35.5475 98.6925 32 ⎣ 28.503 −39.415

51

⎤

⎥ 85.1105 ⎥ ⎦

−39.539

Figure 6 shows the resulting reactions of the closed loop to the above reference sequences (i.e., for amplitudes rs1 = 0.5, rs2 = −0.5 and rs3 = −0.5). It is deteriorated by large and long lasting overshoots due to unfavorable zeros and severe cross-couplings. A perfect preservation of directionality for all reference signals within the amplitude limits r0i , i = 1, 2, 3 can be obtained by the approach presented in Section 5. In a first step the inner loop is parametrized by ⎤ ⎡ −1.35 1.55 2.55 1 4.775 9.5 ⎥ ⎢ ⎥, Ka = ⎢ −3.4 0 −0.3 −1.3 −14.6 −20 ⎦ ⎣ −1.95 −4.85 1.45 1 7.175 7.7

⎤

⎡ 0.55

⎢ La = ⎢ ⎣ 0 −0.55

0 0.7 0

⎛

assuring a decoupled behaviour yM (s) = diag⎝

0

⎥ 0⎥ ⎦ 0.7

⎞ 0.55

0.7

0.7

⎠ubs (s). , , s + 0.55 s + 0.7 s + 0.7

This tuning was done by trial and error, which is facilitated by the decoupling design having only three free parameters, namely the pole locations in the three channels. The above pair (Ka , La ) leads to a transfer matrix Gua (s) (see (37)), whose L1 norm is

Outputs y1 (t), y2 (t) and y3 (t)

4

y2 (t) 2

y1 (t) 0

−2

−4

y3 (t) 0

2

4

6

8

10

Time

Fig. 6. Reference step responses with a control that satisfies the circle criterion

52

Peter Hippe

⎤ ⎡ 1.44 2.26 2.30 ⎥ ⎢ ⎥ Gua 1 = ⎢ 2.72 2.67 3.83 ⎦ ⎣ 3.49 1.81 3.69 Inserting the r0i = 1, i = 1, 2, 3 in (38), one obtains the upper limits of the resulting input signals uai (t) as ua1max = 6, ua2max = 9.22 and ua3max = 8.99, so that for arbitrary signals ubi (t) the input signals uai (t) never exceed the saturation limits uβi = u0i (i.e., no amplitude margin for disturbance rejection has been reserved). Now adding an outer (decoupling) cascade with ⎤ ⎡ 58.35 19.45 19.45 0 204.225 0 ⎥ ⎢ ⎥, Kb = ⎢ −38.6 0 19.3 19.3 −347.4 0 ⎦ ⎣ −39.05 −0.15 −19.45 0 152.825 19.3

⎤

⎡ 20

⎢ Lb = ⎢ ⎣ 0 −20

0

0

20

⎥ 0⎥ ⎦

0

20

! [2723, 2123, 2123] Cm and diag[l1 , l2 , l3 ] = ! 1 L−1 L = diag [2800, 2200, 2200] . The transfer matrix (41) satisfies the cirb a 77

one obtains L−1 a Kb = diag

1 77

cle criterion, guaranteeing a stable behaviour of the nonlinear reference shaping filter for all input signals r(t). For small reference inputs, the linear behaviour of the ⎛ reference shaping filter ⎞ is characterized by the transfer matrix yM (s) = diag⎝

20

,

20

,

20

s + 20 s + 20 s + 20

⎠r(s).

Figure 7 shows the reference behaviour of this filter, which is identical with that of the controlled system in a scheme of Fig. 3, if, e.g., the feedback matrix K = KKY is used in a “nominal” control (12). There are no directionality problems for any reference signal changes within the regions −1 ≤ ri (t) ≤ 1, i = 1, 2, 3 and the transient behaviour is satisfactory.

Example 3. Considered is the benchmark system of Section A1. Case a) Amplitude constraint u0 = 1 and rate constraint uv = 10. To handle the joint amplitude and rate constraints, the augmented system must be used. With the given values of the constraints, the time constant (9) is Tv = 0.2. Consequently, the state equations of the augmented system (11) are (to comply with the benchmark the notation d(t) = ˆ z1 (t) is used)


53

Outputs y1 (t), y2 (t) and y3 (t)

1

y1 (t)

0.5

0

y2 (t)

−0.5 y3 (t)

−1

0

2

4

6

8

10

Time

Fig. 7. Reference behaviour of the directionality preserving filter

⎡

⎢ ⎢−1 x ¯˙ (t) = ⎢ ⎢ 0 ⎣

⎡ ⎤ ⎡ ⎤ 0 1 ⎢ ⎥ ⎥ ⎢ ⎥ ⎢0⎥ ⎢0⎥ 0 1 0⎥ ⎢ ⎥u ⎥x ⎢ ⎥z1 (t) ¯ ¯ (t) + (t) + s ⎥ ⎢ ⎥ ⎢0⎥ 1 0 1⎦ ⎣0⎦ ⎣ ⎦ 5 0 0 −5 0 ¯(t) 0 0 x

0 −1

0

y(t) = 1 0

0

0

⎤

y¯m (t) = x ¯(t) To allow input disturbances z1 (t) = z1s 1(t) with relatively large amplitudes z1s , a nominal state feedback K x ¯(t) = [−31.66, 35.72, 25.66, 3.26] x ¯(t) is chosen, placing the eigenvalues of the closed loop at s1 = −0.3, s2 = −1 and s3/4 = −10. The integral action in the nominal controller is represented by the signal model v(t) ˙ = y(t) − Bσ [uC (t) + u ¯s (t)] (see (27)), i.e., S = 0 and Bε = 1, and the characteristic polynomial of the controlled signal model is chosen as det(sI − S + Bσ Kv ) = s + 0.05. The Sylvester equation (30) is solved by X =

1 3

[−15.73, 12.83, 2.13, 0.1], and

with this solution (31) gives Bσ = −0.5/3 and consequently Kv = −0.3. Using (32), the feedback Kx = [−33.233, 37.003, 25.873, 3.27] results. Since the states of the system are all measurable, the design of the feedback control is complete.

54

Peter Hippe 1

Output y(t)

0.8

0.6

0.4

0.2

0

0

5

10

15 Time

20

25

30

Fig. 8. Reference step responses

When choosing the parameters in the inner loop of the reference shaping filter as Ka = [−1.24992, 1.6896, 1.168, −0.36] and La = −0.08192 (giving det(sI − ¯ a ) = (s + 0.8)4 ), the L1 -norm of (37) is 1.01892. Restricting the input A¯ + BK signal ubs (t) to r0 = 1/1.01892, the inner saturation uas (t) = satu0 (ua (t)) (i.e., we have assumed β = 0 in (14)) never becomes active for arbitrary reference inputs r(t). Thus, the inner loop of the reference shaping filter is a stable, linear system with input nonlinearity ubs (t) = satr0 (ub (t)). In the second cascade, the feedback is chosen to obtain the characteristic polyno¯ b ) = (s + 1.8)4 , which corresponds to Kb = ¯ a + BK mial det(sI − A¯ + BK [−5.1376, 5.856, 3.12, 0.8] and leads to Lb = −2.09952. With this Kb the transfer function (41) of the linear part of the loop satisfies the circle criterion so that the reference shaping filter is asymptotically stable for all reference inputs. With this parameterization of the trajectory generator, the loop of Fig. 4 exhibits the reference behaviour shown in Fig. 8. The input signals uCr (t) to the plant produced by the reference shaping filter never pass the saturation limit u0 = 1 and the rate of duCr (t) always stays well below the upper limit uv = 10. change dt


55

The broken lines in Fig. 9 show the disturbance step responses up to the maximum allowable amplitude z1s = 0.744. For a step amplitude z1s = 0.745 the loop exhibits an unstable behaviour.

Output y(t)

3.5 3

tuning for z1s = 0.744

2.5

tuning for z1s = 0.5

2 1.5 1 0.5 0 −0.5

0

10

20

30

40

50

Time

Fig. 9. Disturbance step responses with control settings allowing maximum amplitudes z1s = 0.744 and z1s = 0.5

A much better disturbance rejection in view of the resulting amplitudes of y(t) is obtained by the nominal state feedback K = [−25, 31.4, 19, 5.6] placing the eigenvalues of the controlled augmented system at s1 = −30 and s2/3/4 = −1, and when det(sI − S + Bσ Kv ) = (s + 1) is chosen. This leads to Bσ = −0.5/3, Kv = −6 and Kx = [−49.8, 50.4, 25.6, 5.8]. This parameterization, however, reduces the amplitudes of the step disturbances which do not destabilize the loop. The solid lines in Figure 9 show the disturbance step responses with this modified control up to the maximum allowable step amplitude z1s = 0.5.

Case b) Amplitude constraint u0 = 1 and rate constraint uv = 2. With the given constraint values, the time constant (9) is now Tv = 1. Consequently, the description (11) of the augmented system has the form

56

Peter Hippe

⎡

⎢ ⎢−1 x ¯˙ (t) = ⎢ ⎢ 0 ⎣

⎡ ⎤ ⎡ ⎤ 0 1 ⎢ ⎥ ⎥ ⎢ ⎥ ⎢0⎥ ⎢0⎥ 0 1 0⎥ ⎢ ⎥u ⎥x ⎢ ⎥z1 (t) ¯ ¯ (t) + (t) + s ⎥ ⎢ ⎥ ⎢0⎥ 1 0 1⎦ ⎣0⎦ ⎣ ⎦ 1 0 0 −1 0 ¯(t) 0 0 x

0 −1

0

y(t) = 1 0

0

0

⎤

y¯m (t) = x ¯(t) To obtain a stable behaviour for step disturbances up to an amplitude z1s = 0.38, the nominal state feedback K x ¯(t) = [−45, 57, 35, 12] x ¯(t) (giving det(sI − A¯ + ¯ BK) = (s + 1)3 (s + 10)) and det(sI − S + Bσ Kv ) = s + 0.05 were chosen. The Sylvester equation (30) is now solved by the row vector X = [−4.4, 3.5, 1.3, 0.1], and with this solution (31) gives Bσ = −0.1 and, consequently, Kv = −0.5 and (32) leads to Kx = [−47.2, 58.75, 35.65, 12.05]. The model-based reference shaping filter is parameterized by the vector Ka = [−4.71350625, 6.2985, 4.535, 1.6] which yields the characteristic polynomial det(sI− ¯ a ) = (s + 0.65)4 and La = −0.17850625. With this feedback the L1 A¯ + BK norm of (37) is 1.0785 so that r0 = 1/1.0785 assures that the inner saturation uas (t) = satu0 (ua (t)) is never active. When finally choosing Kb = [−14.322, 17.496, 10.08, 3.2] (which leads to det(sI− ¯ BK ¯ a + BK ¯ b ) = (s+1.45)4 and Lb = −4.42050625), the transfer function (41) A+ satisfies the circle criterion. The loop of Fig. 4 now exhibits the reference behaviour shown in Fig. 10. The input signals uCr (t) producing these transients never hit the saturation limit u0 = 1 and the rate of change |u˙ Cr (t)| always stays well below the upper limit uv = 2. Figure 11 shows the disturbance step responses up to the maximum allowable step amplitude z1s = 0.38. Again, the disturbance rejection could be improved in view of the resulting amplitudes of the output y(t), but this would again reduce the maximum allowable amplitude z1s of the disturbance signal. According to the benchmark specifications, the potential of the new scheme to handle the joint action of reference and disturbance signals in a stable manner has not been exploited. If disturbance rejection would require α% of the restricted input signals, this could be handled by reducing the value of uβ (see Fig. 4) by α%.


57

1

Output y(t)

0.8

0.6

0.4

0.2

0

0

5

10

15

20

25

30

Time

Fig. 10. Reference step responses of the loop

2

Output y(t)

1.5

1

0.5

0

0

10

20

30

40

50

Time

Fig. 11. Disturbance step responses up the maximum amplitude z1s = 0.38

8 Conclusions A windup prevention scheme was presented which is equally applicable to stable and unstable constrained SISO and MIMO systems. Beyond windup prevention, the new scheme also allows a perfect directionality preservation for stable and unstable MIMO systems. The scheme uses feedback control for system stabilization and disturbance rejection, and a nonlinear model-based reference shaping filter for feedfor-

58

Peter Hippe

ward tracking. For arbitrary reference inputs this filter generates plant input signals such, that only the safely applicable reference signal amplitudes are transferred to the system, and that the plant input signals stay within user-defined limits. The structure of the closed loop assures vanishing feedback signals for reference inputs, so that the input signals to the system result from a superposition of the signals necessary for disturbance rejection and the signals used for reference tracking. For exponentially unstable systems, a stable behaviour of the loop in spite of jointly acting reference and disturbance inputs can be obtained when splitting the constrained input range into a part used for disturbance rejection while only the remaining part is utilized for reference tracking. For stable systems, disturbances could be allowed to drive the input signals into saturation. If the danger of plant windup exists it can easily be removed by an additional dynamic element. Since the plant input signals generated in the reference shaping filter do not cause saturation, a systematic prevention of the possibly existing directionality problems becomes also feasible in the new scheme. Using a special model for an actuator with joint amplitude and rate saturation, the results presented are readily applicable also to systems with such actuators. A MIMO example and an unstable system were used to demonstrate the design procedures.

References 1. Barbu C, Reginatto R, Teel AR, Zaccarian L (2002) Anti-windup for exponentially unstable linear systems with rate and magnitude constraints. In: Kapila V, Grigoriadis KM (eds), Actuator saturation control. Marcel Dekker, N.Y. 2. Bateman A, Lin Z (2002) An Analysis and Design Method for Linear Systems Under Nested Saturations. In Proc. 2002 ACC, Anchorage, AK:1198–1203 3. Bernstein DS, Michel AN (1995) Intern. J. Robust and Nonlinear Control 5:375–380 4. Campo PJ, Morari M (1990) Computers Chem. Engrng. 14:343–358 5. Dahleh MA, Pearson JB (1987) IEEE Trans. on Automatic Control 32:889–895 6. Davison EJ (1976) IEEE Trans. on Automatic Control 21:25–34 7. Doyle JC, Smith RS, Enns DF (1987) Control of Plants with Input Saturation NonLinearities. In Proc. 1987 ACC, Minneapolis:1034–1039 8. Gökçek C, Kabamba PT, Meerkov SM (2001) IEEE Trans. on Automatic Control 46:1529–1542


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9. Gomes da Silva Jr. JM, Tarbouriech S, Garcia G (2003) IEEE Trans. on Automatic Control 48:842–847 10. Hanus R, Kinnaert M, Henrotte JL (1987) Automatica 23:729–739 11. Hippe P (1992) Automatica 28:1003–1009 12. Hippe P (2003) Automatica 39:1967–1973 13. Hippe P (2005) Windup prevention when using Davison’s approach to disturbance rejection. In Proc. 2005 IFAC World Congress Prague. 14. Hippe P (2006) Windup in control - Its effects and their prevention. Springer Berlin Heidelberg New York London 15. Hippe P, Wurmthaler C (1999) Automatica 35:689–695 16. Johnson CD (1971) IEEE Trans. Automatic Control 16:635–644 17. Kapasouris P, Stein G (1990) Design of Feedback Control Systems for Unstable Plants with Saturating Actuators. In Proc. IFAC Symposium on Nonlinear Control System Design. 18. Kendi TA, Doyle FJ (1997) J. Proc. Control 7:329–343 19. Kothare MV, Campo PJ, Morari M, Nett CN (1994) Automatica 30:1869–1883 20. Lin Z (1997) IEEE Trans. on Automatic Control 42:992–995 21. Lin Z, Saberi A, Teel AR (1996) Automatica 32:619–624 22. Moness M, Amin MH (1988) Int. J. Control 47:1925–1936 23. Mhatre S, Brosilow C (1996) Multivariable Model State Feedback. In Proc. IFAC World Congress, San Francisco:139 24. Peng Y, Vrancic D, Hanus R, Weller SSR (1998) Automatica 34:1559–1565 25. Rönnbäck S, Walgama KS, Sternby J (1991) An Extension to the Generalized AntiWindup Compensator. In Proc. 13th IMACS World Congress on Scientific Computation, Dublin, Ireland. 26. Saberi A, Lin Z, Teel AR (1996) IEEE Trans. on Automatic Control 41:368–378 27. Seron MM, Goodwin GC, Graebe SF (1995) IEE Proc. Control Theory and Applications 142:335–344 28. Shim JH, Goodwin GC, Graebe SF (1996) MIMO Design with Saturating Actuators. In Proc. CESA’96 Multiconference, Lille, France:1026–1031 29. Tabouriech S, Pittet C, Burgat C (2000) Int. J. Robust and Nonlinear Control 10:489–512 30. Teel AR (1999) Int. J. Robust and Nonlinear Control 9:701–716 31. Teel AR, Kapoor N (1997) The L2 Anti-Windup Problem: Its Definition and Solution. In Proc. 4th ECC, Brussels, Belgium. 32. Turner MC, Postlethwaite I, Walker DJ (2000) Int. J. Control 73:1160–1172 33. Turner MC, Postlethwaite I (2001) Int. J. Control 74:1425–1434 34. Verriest EI, Pajunen GA (1992) Analysis of Constrained Tracking Problem. In Proc. 1992 American Control Conference:676–680

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35. Vidyasagar M (1993) Nonlinear Systems Analysis. Prentice Hall, Englewood Cliffs, N.J. 36. Walgama KS, Sternby J (1993) IEE Proc. Control Theory and Applications 140:231–241 37. Zheng A, Kothare MV, Morari M (1994) Int. J. Control 60:1015–1024

An Anti-windup Design for Linear Systems with Imprecise Knowledge of the Actuator Input Output Characteristics Haijun Fang1 and Zongli Lin2 1

Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, P. O. Box 400743, Charlottesville, VA 22904-4743, U.S.A. [email protected]

2

Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, P. O. Box 400743, Charlottesville, VA 22904-4743, U.S.A. [email protected]

Summary. This chapter proposes an anti-windup design for linear systems in the absence of precise knowledge of the actuator nonlinearity. The actuator input output characteristic is assumed to reside in a nonlinear sector bounded by two convex/concave piecewise linear curves. For the closed-loop system with a given anti-windup compensation gain matrix, stability properties, both in the absence and in the presence of external disturbances are characterized in terms of matrix inequalities. The estimation of the domain of attraction with a contractive invariant ellipsoid as well as the assessment of the disturbance tolerance and disturbance rejection properties of the closed-loop system are then formulated and solved as LMI optimization problems. The design of the anti-windup compensation gain matrix for a large domain of attraction and/or a large disturbance tolerance/rejection capability is then carried out by viewing the anti-windup compensation gain as an additional free parameter in these optimization problems. Numerical examples as well as an experimental setup are used to illustrate the proposed analysis and design approach and its effectiveness.

This work was supported in part by NSF grant CMS-0324329.

62

Haijun Fang and Zongli Lin

1 Introduction Saturation nonlinearity is ubiquitous in engineering systems. In a control system, physical limitations on its actuator cause the control input to saturate. It has been long recognized that actuator saturation, when occurs, degrades system performance and even causes the loss of stability. Even though there has been a surge of research on control design that directly takes actuator nonlinearity into account (see, e.g., [2, 7, 10, 12] and the references therein), the anti-windup design remains an important approach to dealing with actuator saturation and continues to attract attention from control research community (see [1, 3, 6, 5, 9, 13, 11, 17, 15, 16] for a small sample of the literature). While the earlier work on anti-windup design was more ad hoc in nature, recent literature has witnessed the emergence of systematic approaches to the design of anti-windup compensators. The both performances and stability are quantified in a rigorous way and the synthesis of the anti-windup compensators are formulated as optimization problems, usually with linear or bilinear matrix inequality constraints (see, e.g., [3, 9, 5, 14, 16]). In the recent literature on control systems with actuator saturation, there is a separate line of work started in [8], which aims to accommodate the uncertainties that commonly exist in the actuator input output characteristics. This is motivated by the fact that the behavior of a physical actuator usually cannot be represented by a standard saturation function and its actual input output characteristics are often not precisely known and may even vary as time. More specifically, in [8], the concept of a generalized sector bounded by two piecewise linear convex/concave curves (see Fig. 1) is introduced and the actuator input output characteristics are assumed to reside in such a generalized sector. By exploring the relationship between the piecewise linear functions and a saturation function, necessary and sufficient conditions under which an ellipsoid is contractively invariant for any nonlinearity that resides in the given generalized sector are derived in terms of linear matrix inequalities. Based on these conditions, a constrained optimization problem is formulated and solved, in which the largest contractively invariant ellipsoid is sought for use as an estimate of the domain of attraction. By allowing the feedback gain as an additional free parameter, this optimization problem is also readily adapted for the synthesis of feedback gain. By its analogy

Robust Anti-windup Design

63

5

v 4

v=ψ (u) 1

3

v=ψ(u)

2

1

v=ψ2(u)

u

0

o −1

−2

−3

−4

−5 −8

−6

−4

−2

0

2

4

6

8

Fig. 1. A generalized sector as bounded by a convex and a concave curve.

with the classical absolute stability theory, such an approach to the stability analysis and design is referred to as a generalized absolute stability approach. Since the generalized sector can describe actuator nonlinearity more flexibly than the classical sector and circle criterion, and the set invariant conditions established in [8] are necessary conditions, less conservative results are expected from the generalized absolute stability approach. This generalized absolute stability approach of [8] has also be adopted recently in the analysis and design for disturbance tolerance and disturbance rejection [4]. The objective of this chapter is to develop a generalized absolute stability approach to the design of an anti-windup gain for a given closed-loop system with imprecise knowledge of the actuator input output characteristics. We will address the problems of stability analysis and disturbance tolerance/rejection assessment. For the closedloop system under a given anti-windup compensation gain and in the absence of disturbance, we will establish the conditions under which an ellipsoid is contractively invariant. The determination of the largest such ellipsoid for use as an estimate of the domain of attraction is then formulated and solved as an LMI optimization problem. The same optimization problem is then adapted for the design of the anti-windup by

64


viewing the anti-windup gain as an additional design parameter. As the optimization problem with this additional free parameter is no longer an LMI problem, an iterative LMI algorithm is derived for its solution. In the presence of disturbances, the goal is to design an anti-windup compensation gain to improve the system disturbance tolerance and rejection capabilities, while ensuring a certain size of the stability region. As with the design for stability, we will start with the assessment of the disturbance tolerance and rejection capabilities of the system under a given anti-windup gain by formulation and solution of LMI optimization problems. The design of the anti-windup gain is again carried out by allowing the anti-windup gain matrix in these optimization problems as an additional free parameter and solving them via iterative LMI algorithms. The remainder of this chapter is organized as follows. Section 2 gives the problem statements and some necessary preliminaries. The analysis of the stability and disturbance tolerance/rejection capabilities is carried out in Section 3. Iterative LMI algorithms for solving the optimization problems involved in the design of the antiwindup gain are developed in Section 4. Numerical examples are used to demonstrate the effectiveness of these algorithms. In Section 5, the proposed design approach is applied to an experimental setup. Finally, conclusions are drawn in Section 6.

2 Problem Statement and Preliminaries We consider the following system, ⎧ ⎪ ⎪ x˙ = Ax + Bψ(u, t) + Kd, ⎨ y = C1 x, ⎪ ⎪ ⎩ z = C x,

(1)

2

where x ∈ Rn is the system state, u ∈ R is the control input, y ∈ Rm is the measurement output, z ∈ Rp is the controlled output, d ∈ Rq is the external disturbance, with ||d||∞ ≤ α, for some α ≥ 0. The function ψ(u, t) is a saturation type function that represents the actuator input output characteristics and is not precisely known. The function ψ(u, t) is however assumed to reside in a generalized sector defined by two piecewise linear concave functions ψ1 (u) and ψ2 (u),


⎧ ⎪ ki0 u, if u ∈ [0, bi1 ], ⎪ ⎪ ⎪ ⎪ ⎨ ψi1 (u) = ki1 u + ci1 , if u ∈ (bi1 , bi2 ), ψi (u) = .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎩ ψ (u) = k u + c , if u ∈ (b , ∞), iNi iNi iNi iNi

65

(2)

where for each i ∈ [1, 2], ki0 > ki1 > ki2 > · · · > kiNi , ki(Ni −1) > 0 (see Fig. 2).

v v = k i0 u

ψ i1(u )

ψ i2(u ) ψ i3(u )

ψi(u )

ki0v3 k i0v2 k i0v1

o

bi1 = v1 v 2

v3

bi2

b i3

u

Fig. 2. Concave piecewise linear functions, ψi (u), i ∈ [1, 2].

We will assume that a linear dynamic compensator of the form x˙ c = Ac xc + Bc y, xc (0) = 0 ∈ Rnc , u = Cc xc + Dc y

(3)

has been designed that stabilizes the system (1) in the absence of actuator nonlinearity, i.e., φ(u, t) ≡ u, and meets the performance specification. The anti-windup compensation we consider in this chapter is a correction term Ec (ψ(u, t) − u) in the above pre-designed dynamic output feedback law. The output feedback compensator with the anti-windup compensation thus takes the following form, x˙ c = Ac xc + Bc y + Ec (ψ(u, t) − u), xc (0) = 0, u = Cc xc + Dc y.

(4)

The closed-loop system (1) under the output feedback compensator (4) can be written as,

66


⎧ ˆx + B(ψ(u, ˆ ˆ ⎪ ˆ˙ = Aˆ t) − u) + Kd, ⎪ ⎨x u = Fˆ x ˆ, ⎪ ⎪ ⎩z = C 0 x ˆ,

(5)

2

where x ˆ= and

Aˆ =

A + BDc C1 BCc Bc C1

Ac

ˆ= ,B

x

xc

B

Ec

,

ˆ = , Fˆ = Dc C1 Cc , K

K 0

.

With the control input substituted, the system (5) can be readily rewritten as ⎧ ⎨x ˆ Fˆ )ˆ ˆ Fˆ x, t) + Kd, ˆ ˆ˙ = (Aˆ − B x + Bψ( ⎩ z = C2 0 x ˆ.

(6)

We will propose design of the anti-windup gain Ec , in the presence and in the absence of the external disturbance, respectively. In the absence of the disturbance, we will design an anti-windup compensation gain Ec such that the closed-loop system has a large domain of attraction. In the presence of the external disturbance, we will design an anti-windup compensation gain to improve the tolerance and rejection capabilities of the closed-loop system. We will measure the disturbance tolerance capability by the bound α on disturbances, under which any trajectory of the closed-loop system starting from a given set of initial conditions will remain bounded. An alternative measurement of the disturbance tolerance capability is the size of the maximum ellipsoid that remains invariant under the influence of disturbances bounded in magnitude by a given number α. We will measure the disturbance rejection capability by the bound on the l∞ norm of the controlled output z. To achieve the above design objectives by the approach summarized in Section 1, we need to recall some preliminaries from our earlier work [4, 8]. Given a positive definite matrix P ∈ R(n+nc )×(n+nc ) and a scalar ρ > 0, denote

ˆT P x ˆ≤ρ , ε(P, ρ) := x ˆ ∈ Rn+nc , x which is an ellipsoid in the state space Rn+nc . For a vector H ∈ R1×(n+nc ) , denote L(H) := x ˆ ∈ Rn+nc ,

|H x ˆ| ≤ 1 ,


67

which is a subset of the state space where the signal H x ˆ is bounded by 1. The convex hull of a given set of vectors x1 , x2 , · · · , xl , is denoted as co{x1 , x2 , · · · , xl }. Given a set XR ∈ R(n+nc )×(n+nc ) and a β > 0, the set βXR is defined as x, βXR := {βˆ

∀ˆ x ∈ XR } .

The following theorem, adapted from [8], forms the basis for the design of antiwindup compensation gain in the absence of external disturbances. Theorem 1. Consider the system (6) in the absence of the disturbance d ≡ 0, and let the positive definite matrix P ∈ R(n+nc )×(n+nc ) be given. If there exist matrices ˆ ij ∈ R1×(n+nc ) , j ∈ [1, Ni ], i ∈ [1, 2], such that, for j ∈ [1, Ni ], i ∈ [1, 2], H ˆ Fˆ )T P + P (Aˆ − B ˆ Fˆ ) < 0, ˆ Fˆ + ki0 B ˆ Fˆ + ki0 B (Aˆ − B (7) ˆ Fˆ + B ˆH ˆ ij )T P + P (Aˆ − B ˆ Fˆ + B ˆH ˆ ij ) < 0, (Aˆ − B # "ˆ Hij −kij Fˆ , then the ellipsoid ε(P, 1) is contractively invariant, and ε(P, 1) ⊂ L cij and thus is an estimate of the domain of attraction. The next two theorems are adapted from [4] and are needed in the design of antiwindup compensation gain in the presence of external disturbances. Theorem 2. Consider the system (6), and let the positive definite matrix P ∈ ˆ ij ∈ R1×(n+nc ) , j ∈ [1, Ni ], i ∈ R(n+nc )×(n+nc ) be given. If there exist matrices H [1, 2] and a positive number η, such that, for j ∈ [1, Ni ], i ∈ [1, 2], ˆ Fˆ )T P +P (Aˆ − B ˆ Fˆ ) + 1 P K ˆ Fˆ + ki0 B ˆK ˆ T P + ηαP < 0, ˆ Fˆ + ki0 B (Aˆ − B η (8) ˆ Fˆ + B ˆH ˆ ij ) + 1 P K ˆK ˆ T P + ηαP < 0, ˆ Fˆ + B ˆH ˆ ij )T P +P (Aˆ − B (Aˆ − B η # "ˆ Hij −kij Fˆ , then the ellipsoid ε(P, α) is an invariant set. and ε(P, α) ⊂ L cij Theorem 3. Consider the system (6), and let P ∈ R(n+nc )×(n+nc ) be a given positive definite matrix and ζ > 0 be a given positive scalar. If there exist matrices ˆ ij ∈ R1×(n+nc ) , j ∈ [1, Ni ], i ∈ [1, 2] and a positive number η, such that, for H j ∈ [1, Ni ], i ∈ [1, 2], ⎧ ˆ Fˆ + ki0 B ˆ Fˆ )T P + P (Aˆ − B ˆ Fˆ ) + 1 P K ˆ Fˆ + ki0 B ˆK ˆ T P + ηαP < 0, ⎪ (Aˆ − B ⎪ η ⎪ ⎪ ⎨ (Aˆ − B ˆ Fˆ + B ˆH ˆ ij ) + 1 P K ˆ ˆ T P + ηαP < 0, ˆ Fˆ + B ˆH ˆ ij )T P + P (Aˆ − B η K ζ2 ⎪ C2T ⎪ ⎪ ⎪ C2 0 ≤ P , ⎩ α 0 (9)

68


and ε(P, α) ⊂ L

"

ˆ ij −kij F H cij

# . Then the maximal l∞ norm of the system output z is

less than or equivalent to ζ.

3 System Analysis under a Given Anti-windup Gain In this section, we assume that the anti-windup compensation gain Ec is given. The problems of estimating the domain of attraction in the absence of disturbances and disturbance tolerance/rejection capabilities are formulated and solved as optimization problems with LMI constraints. The estimation of the domain of attraction: Based on Theorem 1, the estimation of the domain of attraction of the system (6) in the absence of the disturbance can be formulated into the following optimization problem, max

ˆ ij ,j∈[1,Ni ],i∈[1,2] P >0,H

β,

(10)

s.t. a) βXR ⊂ ε(P, 1), b) Inequalities (7), i c) ε(P, 1) ⊂ ∩N j=1 L

$

ˆ ij − kij Fˆ H cij

% ,

where XR is a shape reference which controls the shape of the resulting ellipsoid. The shape reference set XR could be either an ellipsoid ε(R, 1), R ∈ R(n+nc )×(n+nc ) , or a polyhedron XR = co {ˆ x1 , x ˆ2 , · · · , x ˆl } ,

(11)

where x î ∈ R(n+nc )×(n+nc ) , i ∈ [1, l]. ˆ ij Q, j ∈ [1, Ni ], i ∈ To solve the optimization problem (10), let Q = P −1 , Yîj = H [1, 2] and γ = β12 . Then, the condition (7) is equivalent to ⎧ ˆ ˆˆ ˆˆ T ˆˆ ˆ ˆˆ ⎪ ⎪ ⎨ Q(A − B F + ki0 B F ) + (A − B F + ki0 B F )Q < 0, ˆ Fˆ )T + (Aˆ − B ˆ Fˆ )Q + (B ˆ Yîj )T + B ˆ Yîj < 0, Q(Aˆ − B ⎪ ⎪ ⎩ j ∈ [1, N ], i ∈ [1, 2]. i

And the condition c) is equivalent to

(12)


⎡ ⎣

1 (Yîj −kij Fˆ Q)T cij

Yîj −kij Fˆ Q cij

Q

69

⎤ ⎦ > 0, j ∈ [1, Ni ], i ∈ [1, 2].

If XR = ε(R, 1), the condition a) is equivalent to γR I > 0. I Q

(13)

(14)

Thus, the optimization problem (10) can be transformed into the following LMI optimization problem, min

Q>0,Yîj ,j∈[1,Ni ],i∈[1,2]

γ,

(15)

s.t. (14), (12), (13). On the other hand, if XR = co {ˆ x1 , x ˆ2 , · · · , x ˆl }, then the condition a) is equivalent to

γ x ˆTi x î Q

> 0, i ∈ [1, l],

(16)

and the optimization problem (10) is equivalent to the following LMI optimization problem, min

Q>0,Yîj ,j∈[1,Ni ],i∈[1,2]

γ,

(17)

s.t. (16), (12), (13). Disturbance tolerance: Let the set of initial conditions be specified by an ellipsoid ε(S, 1), S ∈ R(n+nc )×(n+nc ) . The disturbance tolerance capability of the closed-loop system can be measured by the maximum level of the disturbances, α, under which all trajectories of the closed-loop system starting from inside ε(S, 1) remain bounded. In view of Theorem 2, the determination of such a maximum value of α can be formulated into the following constrained optimization problem, max

ˆ ij ,j∈[1,Ni ],i∈[1,2],η>0 P >0,H

α,

(18)

s.t. a) ε(S, 1) ⊂ ε(P, α), b) Inequalities (8), i c) ε(P, α) ⊂ ∩N j=1 L

$


% .

70


ˆ ij Q, j ∈ [1, Ni ], i ∈ To solve this optimization problem, let Q = αP −1 , Yîj = H [1, 2] and ηˆ = ηα. Then the constraint a) is equivalent to αS αI > 0, αI Q

(19)

and the condition (8) is equivalent to ⎧ ˆ−B ˆ Fˆ + ki0 B ˆ Fˆ )T + (Aˆ − B ˆ Fˆ )Q + ηˆQ αK ˆ ˆ Fˆ + ki0 B ⎪ Q( A ⎪ ⎪ ≤ 0, ⎪ ⎪ ˆT ⎪ α K −ˆ η ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ T T ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Q(A − B F ) + (A − B F )Q + B Yij + (B Yij ) + ηˆQ αK ⎪ ≤ 0, ⎪ ⎪ ˆT ⎪ αK −ˆ η ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ j ∈ [1, N ], i ∈ [1, 2]. i

(20)

Thus, the optimization problem (18) is equivalent to max

Q>0,Yîj ,j∈[1,Ni ],i∈[1,2],ˆ η >0

α,

(21)

s.t. (19), (20), (13). For a fixed value of ηˆ, all constraints of the problem (21) are LMIs. By sweeping ηˆ over the interval (0, ∞), we can obtain the global maximum of α. An alternative measurement of the disturbance tolerance capability is the size of the largest ellipsoid that remains invariant under the influence of disturbances whose magnitudes are bounded by a given positive number α. In view of Theorem 2, the size of the largest such ellipsoid can be determined by solving the following constrained optimization problem, max

ˆ ij ,j∈[1,Ni ],i∈[1,2],η>0 P >0,H

β,

(22)

s.t. a) βXR ⊂ ε(P, α), b) Inequalities (8), i c) ε(P, α) ⊂ ∩N j=1 L

$


% .

ˆ ij Q, the condition b) in (22) is Under the change of variable, Q = P −1 , Yîj = H equivalent to


71

⎧ 1 ˆ ˆT ˆ ˆˆ ˆˆ T ˆˆ ˆ ˆˆ ⎪ ⎪ ⎨ Q(A − B F + ki0 B F ) + (A − B F + ki0 B F )Q + η K K + ηαQ < 0, ˆ Yˆ T +ki0 B ˆ Yîj + 1 K ˆK ˆ T +ηαQ < 0, (23) ˆ Fˆ )+(Aˆ − B ˆ Fˆ )T Q+ki0 B Q(Aˆ − B ij η ⎪ ⎪ ⎩ j ∈ [1, N ], i ∈ [1, 2], i

and the constraint c) is equivalent to ⎡ ⎣

Yîj −kij Fˆ Q 1 α cij (Yîj −kij Fˆ Q)T Q cij

⎤ ⎦ > 0, j ∈ [1, Ni ], i ∈ [1, 2].

(24)

Let XR be an ellipsoid ε(R, 1), R ∈ R(n+nc )×(n+nc ) . By the additional change of variable, γ =

1 , β2

the optimization problem (22) can be transformed into, min

Q>0,Yîj ,j∈[1,Ni ],i∈[1,2],η>0

s.t.

γαR I I

Q

γ,

(25)

> 0,

Inequalities (23), (24). Similarly, if XR is a polyhedron defined in (11), then the problem (22) is equivalent to min

Q>0,Yîj ,j∈[1,Ni ],i∈[1,2],η>0

s.t.

γα x î x ˆTi Q

γ,

(26)

> 0,

Inequalities (23), (24). In both optimization problems (25) and (26), all constrains are LMIs for a fixed value of η. Therefore, the minimum of γ, hence the maximum of β, can be achieved by sweeping the value of η over the interval (0, ∞). Disturbance rejection: We measure the disturbance rejection capability of a closed-loop system by the maximal l∞ norm of its controlled output within a given set of initial conditions, say an ellipsoid ε(S, 1). In view of Theorem 3, this problem can be formulated into the following optimization problem,

72


max

ˆ ij ,j∈[1,Ni ],i∈[1,2],η>0 P >0,H

(27)

ξ,

s.t. a) ε(S, 1) ⊂ ε(P, α), b) Inequalities (9),

$

i c) ε(P, α) ⊂ ∩N j=1 L


% .

ˆ ij Q, ∀j ∈ [1, Ni ], i ∈ [1, 2], the Under the change of variable, Q = P −1 , Yîj = H optimization problem (27) is equivalent to min

Q>0,Yîj ,j∈[1,Ni ],i∈[1,2],η>0

s.t. a)

b)

(28)

S I I αQ

> 0,

ξ,

C2 0 Q

C2 0

≤

ζ2 I, α

c) Inequalities (23), (24). Again, all the constraints in (28) are LMIs for a fixed value of η. The global minimum of ζ can be obtained by sweeping η over the interval (0, ∞).

4 Anti-windup Compensation Gain Design In this section, we will propose algorithms for designing the anti-windup compensation gain to improve system stability and disturbance tolerance/rejection capabilities. On the basis of optimization problems proposed in Section 3, the design of antiwindup compensation gain can be formulated into optimization problems by viewing Ec as an additional variable. However, with Ec as an additional variable, constraints in these optimization problems can no longer be transformed into LMIs when P , Ec ˆ ij , j ∈ [1, Ni ], i ∈ [1, 2], are all variables. We will develop iterative algorithms and H to solve these optimization problems. Design for a large domain of attraction: We will first consider the design of Ec to achieve a large domain of attraction for the closed-loop system. This involves the optimization problem (10), where constraint (7), with Ec as an additional variable, cannot be transformed into an LMI. Following the idea of [3], let


P=

P1 P12

˜= ,B

T

P12 P22

(n+nc )×(n+nc )

ˆ B

0

, P˜2 =

(n+nc )×1

P12 P22

73

. (29) (n+nc )×nc

Then, (7) can be rewritten as # " ⎧ ˆ + P AˆT + (ki0 − 1) P B ˜ T P + P˜2 Ec Fˆ + Fˆ E T P˜ T < 0, ˜ Fˆ + Fˆ T B ⎪ AP c 2 ⎪ ⎪ ⎪ ⎨ ˆ ˜ TP ˜ Fˆ − H ˆ ij ) − (Fˆ − H ˆ ij )T B AP + P AˆT − P B( (30) ⎪ ˜2 Ec (Fˆ − H ˆ ij ) − (Fˆ − H ˆ ij )T EcT P˜ T < 0, ⎪ − P 2 ⎪ ⎪ ⎩ j ∈ [1, Ni ], i ∈ [1, 2], which are LMIs in P1 and Ec . Thus, for a given P˜2 , the optimization problem (10) can be adapted to the following one for the design of Ec : max β,

(31)

P1 >0,Ec

s.t. a) P > 0, b) βXR ⊂ ε(P, 1), c) Inequalities (30), %T $ ˆ ij − kij Fˆ ˆ ij − kij Fˆ H H , d) P ≥ cij cij Let γ =

1 . β2

j ∈ [1, Ni ], i ∈ [1, 2].

If XR is an ellipsoid ε(R, 1), constraint b) in (31) is equivalent to P ≤ γR.

(32)

On the other hand, if XR is a polyhedron as defined in (11), then b) in (31) is equivalent to γ − xTi P xi ≥ 0,

i ∈ [1, l].

(33)

Consequently, the optimization problem (31) is equivalent to min γ,

(34)

P1 >0,Ec

s.t. a) P > 0, b) (32) or (33), c) Inequalities (30), %T $ ˆ ij − kij Fˆ ˆ ij − kij Fˆ H H , d) P ≥ cij cij which, for a given P˜2 , is an LMI problem.

j ∈ [1, Ni ], i ∈ [1, 2],

74


Motivated by the above development, we propose the following iterative algorithm for the design of an Ec that results in a large domain of attraction. Algorithm 1 (Design of Ec for a large domain of attraction): Step 1. Let XR be the shape reference set and set Ec = 0. If XR is an ellipsoid ε(R, 1), solve the optimization problem (15). If XR is a polyhedron as defined in (11), solve the optimization problem (17). Denote the solution as γ0 , Q and Yîj , j ∈ [1, Ni ], i ∈ [1, 2]. Set XR = √1 XR . γ0

Step 2. Set the initial value for Ec , and let k = 1 and γopt = 1. Step 3. Using the value of Ec obtained in the previous step. If XR is an ellipsoid ε(R, 1), solve the problem (15) for γk , Q, Yîj , j ∈ [1, Ni ], i ∈ [1, 2]. If XR is a polyhedron as defined in (11), solve the optimization problem (17) for γ , Q, Yîj , j ∈ [1, Ni ], i ∈ [1, 2]. If γ > 1, go to Step 2. Else, let γ = γ γ , k

XR =

k

√1 XR , P γk

−1

=Q

opt

k opt

ˆ ij = Yîj P, j ∈ [1, Ni ], i ∈ [1, 2]. ,H

Step 4. If 1 − γk < δ, a pre-determined tolerance, go to Step 6. ˆ ij , j ∈ [1, Ni ], i ∈ [1, 2], P12 , P22 obtained in Step 3, Step 5. Using the value of H solve the LMI problem (34). Let k = k + 1, go to Step 3. Step 6. Let βopt = √γ1

opt

. The solution Ec is a feasible solution. End of the algo-

rithm. Remark 1. To prevent the anti-windup compensation gain Ec to reach an undesirably large value, an additional constraint can be incorporated into theoptimization prob

lems involved in the algorithm above. More specifically, Ec = e1 , e2 , · · · , enc the following constraint can be imposed: −φmin ≤ ei ≤ φmax ,

T

,

i ∈ [1, nc ],

where φmin and φmax are the bounds imposed on the anti-windup compensation gain. Example. Consider the system (1), where 0.1 −0.1 5 A= , B= , C1 = 1 0 , 0.2 −3 0

C2 = 1 1 ,

K=

1 1

,


75

and ψ(u, t) is a locally Lipschitz function that reside in a generalized sector bounded by ψ1 (u) = u and ψ2 (u) of the form (2) with N = 5 and (k20 , k21 , k22 , k23 , k24 , k25 ) = (0.7845, 0.3218, 0.1419, 0.0622, 0.0243, 0), (c21 , c22 , c23 , c24 , c25 ) = (0.4636, 0.8234, 1.0625, 1.2517, 1.4464). Such a function ψ(u) = tan−1 (u) is depicted in Fig. 3) in a dashed curve. The two solid curves in the figure are the bounding function ψ1 (u) and ψ2 (u). 2

v

v=u v=tan−1(u)

1.5

v=ψ2(u)

1

0.5

u 0

o −0.5

−1

−1.5

−2 −10

−8

−6

−4

−2

0

2

4

6

8

10

Fig. 3. The function ψ(u) = tan−1 (u) in the generalized sector defined by ψ1 (u) and ψ2 (u) in the example.

Also, assume that a dynamic controller of the form (3) has been designed with −171.2 27.06 −598.2 Ac = , Bc = , Cc = 0.146 0.088 , Dc = 0, −68 −626.8 −4.567 We would like to design an Ec that results in a large domain of attraction. Each element of this Ec is restricted to bebetween −100 and 100. Using Algorithm 1, we 1 choose the initial value of Ec to be , and obtain 1

76


Ec =

P1 = 10−3

,

0.8102

with

−0.6946

0.1681 −0.0054 −0.0054

0.0002

.

Shown in Figs. 4 and 5 are the cross section of the ellipsoid ε(P, 1) at xc = 0 and V (ˆ x(t)) = x ˆT P x ˆ along several trajectories that start from the boundary of the ellipsoid ε(P, 1). In the simulation, ψ(u) = tan−1 (u). The smaller ellipsoid also shown in Fig. 4 is the cross section of the invariant ellipsoid at xc = 0, which is obtained for the closed-loop system in the absence of the anti-windup compensation, as computed in Step 1 of Algorithm 1.

5000

4000

3000

2000

x2

1000

0

−1000

−2000

−3000

−4000

−5000 −200

−150

−100

−50

0

50

100

150

200

x1

Fig. 4. The cross section of the ellipsoid ε(P, 1) at xc = 0.

Design for large disturbance tolerance capability: We now consider the design of an anti-windup compensation gain Ec to achieve large disturbance tolerance capability. We will first consider the case when the disturbance tolerance capability is measured by the largest bound on the disturbance


77

1

0.9

0.8

0.7

V

0.6

0.5

0.4

0.3

0.2

0.1

0

0

5

10

15

20

25

30

t

Fig. 5. V (ˆ x(t)) along several trajectories that start from the boundary of the ellipsoid ε(P, 1).

under which the closed-loop system trajectories starting from a given set of initial conditions ε(S, 1), S ∈ Rn+nc )×(n+nc ) , remain bounded. The design of such an Ec can be carried out by solving the optimization problem (18) with Ec viewed as an additional variable. To develop an algorithm for the solution of this optimization problem, we first have that constraint (8) is equivalent to # " ⎤ ⎧⎡ ˆ +P AˆT +(ki0 −1) P B ˜ T P + P˜2 Ec +E T P˜ T + ηˆP P K ˆ ˜ Fˆ + Fˆ T B ⎪ AP c 2 ⎪ ⎪ ⎣ ⎦ < 0, " #T ⎪ ⎪ ⎪ ˆ ⎪ P K −ˆ η γ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎡ ⎤ ˆ TP ˆ +P AˆT −P B( ˜ Fˆ − H ˆ ij )−(Fˆ − H ˆ ij )T B (35) AP ˆ⎥ ⎪ P K ⎪ ⎢ ⎪ ˆ ij )−(Fˆ − H ˆ ij )T EcT P˜ T + ηˆP ⎪ ⎥ < 0, ⎢ − P˜2 Ec (Fˆ − H ⎪ 2 ⎪ ⎦ ⎣ " # ⎪ T ⎪ ⎪ ˆ ⎪ −ˆ ηγ PK ⎪ ⎪ ⎪ ⎩ j ∈ [1, Ni ], i ∈ [1, 2], ˜ and P˜2 are as defined in (29). Also, the constraint where γ = α1 , ηˆ ="ηα, and P1#, B i ε(P, α) ⊂ ∩N j=1 L

ˆ ij −kij Fˆ H cij

is equivalent to

78


⎡ ⎣

γ ˆ ij −kij Fˆ )T (H cij

ˆ ij −kij Fˆ H cij

⎤ ⎦ > 0,

P

j ∈ [1, Ni ], i ∈ [1, 2].

(36)

We observe that the inequalities in (35) and (36) are all linear in the variables P1 and Ec . This motivates us to develop the following algorithm for the design of an Ec that leads to a large disturbance tolerance capability, as measured by the level of allowable disturbances without causing unbounded trajectories. Algorithm 2 (Design of Ec for a high level of allowable disturbance): Step 1. Set Ec = 0, and solve the optimization problem (21). Denote the solution as α0 . Step 2. Set the initial value for Ec . Let k = 1, αopt = α0 . Step 3. Using the value of Ec obtained in the previous step, solve the optimization problem (21). Denote the solution as αk , Q and Yîj , j ∈ [1, Ni ], i ∈ [1, 2]. Then, ˆ ij = Yîj P, j ∈ [1, Ni ], i ∈ [1, 2]. If αk < αopt , go to Step 2). P = αk Q−1 , H Else if αk /αopt < δ, a pre-determined tolerance, let αopt = αk and go to Step 5. Else, let αopt = αk and k = k + 1. ˆ ij , j ∈ [1, Ni ], i ∈ [1, 2], P12 , P22 obtained in Step 3, Step 4. Using the value of H solve the following optimization problem min

P1 >0,Ec ,ˆ η

γ,

(37)

s.t. a) P > 0, b) P < αopt S, c) Inequalities (35) and (36). Denote the solution as αk =

1 γ,

P1 and Ec . If αk /αopt < δ, a pre-determined

tolerance, let αopt = αk and go to Step 5. Else, let αopt = αk and k = k + 1. Go to Step 3. Step 5. The estimate of the largest disturbance bound is αopt , with a corresponding Ec . End of algorithm. Remark 2. In Steps 2-4 of Algorithm 2, ∀k > 1, it is easy to observe that

αk αopt

> 1.

Thus, the pre-determined tolerance δ should be a number which is slightly larger than 1. The algorithm ends when the difference between αk and αopt is small enough.


79

Remark 3. In Algorithm 2, the optimization problems (21) and (37) are LMI problems for each fixed value of ηˆ. Thus, both optimization problems can be solved by sweeping ηˆ over the interval (0, ∞). Example (continued). Let T S = I. Following Algorithm 2, we choose the initial value of Ec as −5 −20 , and obtain Ec =

−96.0117 −82.9739

,

with the corresponding αmax = 6.135 and 0.0626 −0.0021 . P1 = −0.0021 0.0001 Shown in Figs. 6 and 7 are the cross section of the ellipsoid ε(P, αmax ) at xc = 0 and V (ˆ x(t)) along several trajectories under the influence of a disturbance bounded by α = 6.0 sin(t). All trajectories start from the boundary of the ellipsoid ε(P, αmax ). 500

400

300

200

x2

100

0

−100

−200

−300

−400

−500 −20

−15

−10

−5

0

5

10

15

x1

Fig. 6. The cross section of the ellipsoid ε(P, αmax ) at xc = 0.

20

80

Haijun Fang and Zongli Lin 1

0.9

0.8

0.7

V

0.6

0.5

0.4

0.3

0.2

0.1

0

0

2

4

6

8

10

12

14

16

18

20

t

Fig. 7. V (ˆ x(t)) along several trajectories that start from the boundary of the ellipsoid ε(P, α).

Next, we consider the situation when the disturbance tolerance capability is measured by the size of the largest ellipsoid that remain invariant under the influence of disturbances whose magnitudes are bounded by a given number α. This problem can be formulated into the optimization (22) with Ec being an additional variable. Unlike (22) with a given Ec , the optimization problem with Ec as an additional variable is no longer transformable into an LMI problem and an effective algorithm is needed for its solution. To this end, we note that, by Schur complement, (8) is equivalent to ⎧⎡ # " ⎤ T T ˜T T ˜T ˆ ˆ ˆ ˜ ˆ ˆ ˜ ⎪ AP +P A +(ki0 −1) P B F + F B P + P2 Ec +Ec P2 +ηαP P K ⎪ ⎪ ⎪ ⎣ ⎦ < 0, " #T ⎪ ⎪ ⎪ ˆ ⎪ PK −η ⎪ ⎪ ⎪ ⎨ (38) ⎡ ⎤ ⎪ ˆ +P AˆT −P B( ˆ ij )−(Fˆ −kij H ˆ ij )T B ˜ TP ˜ Fˆ −kij H ⎪ AP ⎪ ˆ⎥ ⎪ PK ⎢ ⎪ ⎪ T T ˜T ˆ ˆ ˜ ˆ ˆ ⎢ ⎥ < 0, ⎪ − P2 Ec (F −kij Hij )−(F −kij Hij ) Ec P2 +ηαP ⎪ ⎦ ⎪⎣ " #T ⎪ ⎪ ⎩ ˆ −η PK ˜ where " P1 , P˜2 and # B are as defined in (29). Also, the constraint ε(P, α) ⊂ ˆ ˆ Hij −kij F i is equivalent to ∩N j=1 L cij


⎡ ⎣

ˆ ij −kij Fˆ H 1 α cij ˆ ij −kij Fˆ )T (H P cij

81

⎤ ⎦ > 0,

j ∈ [1, Ni ], i ∈ [1, 2].

(39)

It is easy to see that, for a fixed value of η, inequalities (37) and (38) are LMIs in P1 and Ec . The above development motivates us to develop the following algorithm for the design of Ec that leads to a large invariance set in the presence of bounded disturbances. Algorithm 3 (Design of Ec for a large invariant set in the presence of bounded disturbances): Step 1. Let XR be the shape reference set and Ec = 0. If XR is an ellipsoid ε(R, 1), solve the optimization problem (25). If XR is a polyhedron as defined in (11), solve the optimization problem (26). Denote the solution as γ0 . Set XR =

1 √ γ XR .

Step 2. Set the initial value for Ec . Set k = 1 and γopt = 1. Step 3. If XR is an ellipsoid ε(R, 1), solve the optimization problem (25) for γ , Q, Yîj , j ∈ [1, Ni ], i ∈ [1, 2]. If XR is a polyhedron as defined in (11), solve k

the optimization problem (26) for γk , Q, Yîj , j ∈ [1, Ni ], i ∈ [1, 2]. If γk > 1, ˆ ij = Yîj P, j ∈ go to Step 2. Else let γopt = γk γopt , XR = √1γ XR , P = Q−1 , H k

[1, Ni ], i ∈ [1, 2].

Step 4. If 1 − γopt < δ, a pre-determined tolerance, go to Step 6. Else, let k = k + 1. ˆ ij , j ∈ [1, Ni ], i ∈ [1, 2], P12 , and P22 obtained in Step Step 5. Using the value of H 3, solve the following LMI problem, min

P1 >0,Ec ,η

γ,

s.t. a) P > 0, b) (32) or (33), c) Inequalities (38) and (39). Denote the solution as γk , Ec and P1 . Let γopt = γk and XR =

√1 XR . γk

Go to

Step 3. Step 6. Let βopt =

1 . γ2opt

The solution Ec is a feasible solution. End of the algorithm.

82


T Example (continued). Let α = 4, and the initial value of Ec be −1 −10 . Following Algorithm 3, we arrive at −99.7956 0.2078 −0.0078 Ec = , P1 = . −99.0753 −0.0078 0.0005 Shown in Figs. 8 and 9 are the cross section of the maximized ellipsoid ε(P, α) at x(t)) along several trajectories starting from the boundary of the xc = 0, and V (ˆ ellipsoid ε(P, α) under the influence of a disturbance d(t) = 4 sin(t). Also shown in Fig. 8 for comparison is the cross section of the corresponding maximized ellipsoid when Ec = 0, which is much smaller than the ellipsoid ε(P, α) under the obtained anti-windup gain. Design for disturbance rejection: Finally, we consider the design of Ec to increase the disturbance rejection ability of the closed-loop system. We will do this by minimizing the maximum l∞ norm of the system output z within a given set of initial conditions, say, ε(S, 1), and under the influence of disturbances whose magnitudes are bounded by an α > 0. The 150

100

x2

50

0

−50

−100

−150 −8

−6

−4

−2

0

2

4

6

8

x1

Fig. 8. The cross section of the maximized invariant ellipsoid ε(P, α) at xc = 0.


83

1

0.9

0.8

0.7

V

0.6

0.5

0.4

0.3

0.2

0.1

0

0

2

4

6

8

10

12

14

16

18

20

t

Fig. 9. V (ˆ x(t)) along several trajectories starting from the boundary of ε(P, α).

underlining optimization problem will be (28), with Ec as an additional variable. The following is an algorithm for the solution of this optimization problem. Algorithm 4 (Design of Ec for minimizing the l∞ norm of the output): Step 1. Set Ec = 0, and solve the optimization problem (28). Denote the solution as ζ0 . Step 2. Set the initial value for Ec . Let k = 1 and ζopt = ζ0 . Step 3. Using the value of Ec obtained in the previous step, solve the optimization problem (28). Denote the solution as ζk , Q and Yîj , j ∈ [1, Ni ], i ∈ [1, 2]. Let ˆ ij = Yîj P, j ∈ [1, Ni ], i ∈ [1, 2]. If ζk > ζopt , go to Step 2. Else, P = Q−1 , H if

ζk ζopt

> δ, a predetermined tolerance, let ζopt = ζk , and go to Step 5, else, let

ζopt = ζk and k = k + 1. ˆ ij , j ∈ [1, Ni ], i ∈ [1, 2], P12 and P22 obtained in Step Step 4. Using the value of H 3, solve the following optimization problem,

84


max

P1 >0,Ec ,η

γ,

s.t. a) P > 0, b) P < αS, c) Inequalities (38) and (39), C2T 1 d) γ C2 0 ≤ P. α 0 Denote the solution as ζk =

1 √ γ,

P1 and Ec . If

ζk ζopt

> δ, a predetermined

tolerance, let ζopt = ζk , and go to Step 5). Else let ζopt = ζk and k = k + 1, go to Step 3. Step 5. The maximal l∞ norm of the output z is less than ζopt . The solution Ec is a feasible solution. End of the algorithm. Remark 4. It is clear from the algorithm that ζk < ζopt . Hence the tolerance δ should be set as a positive number slightly smaller that 1. Example (continued). Let S = I and Tα = 4. Following Algorithm 4, we choose the initial value of Ec = and obtain the estimated bound of the −50 0 output z as 3.8368, with T the following anti-windup compensation gain Ec = −99.1173 −96.0495 . Shown in Fig. 10 is the response of the system output, under zero initial condition, to a disturbance d = 4sign(sin(t)). The two straight lines in the figure represent the estimated bound on the output z.

5 Application to an Inverted Pendulum on a Cart: In this section, we will consider the problem of stabilizing an inverted pendulum on a cart. This system was constructed at Carnegie Mellon University3 and is shown in Fig. 11. The parameters of the system are given in the following table. A linearized model for this system is given by, 3

http://www.library.cmu.edu/ctms/ctms/examples/pend/invpen.htm

Robust Anti-windup Design 4

3

2

z

1

0

−1

−2

−3

−4

0

2

4

6

8

10

12

14

16

t

Fig. 10. A trajectory of the output z.

m,I

θ

M F

x

Fig. 11. An inverted pendulum on a cart.

18

20

85

86

Haijun Fang and Zongli Lin Table 1. The parameters of the inverted pendulum and the cart. M mass of the cart

0.5kg

m mass of the pendulum

0.2kg

b friction of the cart

0.1 N/m/sec

l

length to pendulum center of mass 0.3 m 0.006 kgm2

I inertia of the pendulum F force applied to the cart x cart position coordinate θ pendulum angle from vertical

⎡ ⎤ ⎡ 0 1 x˙ 2 ⎢ ⎥ ⎢ )b ⎥ ⎢ 0 I(M−(I+ml ⎢x ¨ +m)+M ml2 ⎢ ⎥=⎢ ⎢ θ˙ ⎥ ⎢ 0 ⎣ ⎦ ⎣0 −mlb θ¨ 0 I(M +m)+M ml2 ⎡ 0 ⎢ I+ml2 ⎢ I(M +m)+M ml2 +⎢ ⎢ 0 ⎣

0

0

0

m2 gl2 I(M +m)+M ml2

0

1

−mgl(M +m) I(M +m)+M ml2

0

⎤

⎤⎡ ⎤ x ⎥⎢ ⎥ ⎥ ⎢ x˙ ⎥ ⎥⎢ ⎥ ⎥⎢θ⎥ ⎦⎣ ⎦ θ˙

⎥ ⎥ ⎥ψ(u), ⎥ ⎦

ml I(M +m)+M ml2

⎡ ⎤ x ⎥ ⎢ ⎢ x˙ ⎥ ⎥ y= 1100 ⎢ ⎢θ⎥, ⎣ ⎦ θ˙

(40)

where ψ(·) is defined in the example in Section 4. The unit of x, ˙ x, θ˙ and θ are m/s, m, rad, and rad/s, respectively. Using the data in Table 1, (40) becomes ⎤ ⎤⎡ ⎤ ⎡ ⎡ ⎤ ⎡ 0 0 1.0000 0 0 x x˙ ⎥ ⎥⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ x˙ ⎥ ⎢ 1.8182 ⎥ ⎥ ⎢ 0 −0.1818 2.6727 ⎢x 0 ¨ ⎥ψ(u), ⎥ ⎢ ⎥+⎢ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎢ θ˙ ⎥ ⎢ 0 0 0 1.0000 ⎦ ⎣ θ ⎦ ⎣ 0 ⎥ ⎦ ⎣ ⎦ ⎣ θ¨

0 −0.4545 31.1818 ⎡ ⎤ x ⎥ ⎢ ⎢ x˙ ⎥ ⎥ y= 1100 ⎢ ⎢θ ⎥. ⎣ ⎦ θ˙

0

θ˙

4.5455

(41)


87

By neglecting the actuator nonlinearity, the following controller is designed such that the closed-loop system (41) is stabilized at the origin, ⎤ ⎤ ⎡ ⎡ −5.6316 −4.6316 0 0 5.6316 ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ 2.5073 ⎢ 12.1866 ⎥ 1.7726 −63.2776 −12.7837 ⎥ xc + ⎢ ⎥ x˙ c = ⎢ ⎢ −51.2478 −51.2478 ⎢ 51.2478 ⎥ y 0 1.0000 ⎥ ⎦ ⎦ ⎣ ⎣ −249.5222 −251.3589 −133.6939 −31.9592 286.2569 u = 8.0816 7.7776 −36.2727 −7.0310 xc . However, as shown in Fig. 12, in the presence of the actuator nonlinearity, the controller fails to stabilize the system at T x(0) = 0 0 0.01 0 .

200

0

−200

states of the system

−400

−600

−800

−1000

−1200

−1400

−1600

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t

Fig. 12. State trajectories of the closed-loop system (41) without anti-windup compensation, x(0) = [0 0 0.01 0]T .

By applying Algorithm 1, we obtain the following anti-windup compensation gain, Ec = 0.0031 1.8963 0.1473 6.0010 ,

88

Haijun Fang and Zongli Lin 0.8

0.6

states of the system

0.4

0.2

0

−0.2

−0.4

−0.6

−0.8

0

1

2

3

4

5

6

7

8

9

10

t

Fig. 13. State trajectories of the closed-loop system (41) with anti-windup compensation, x(0) = [0 0 0.06 0]T . 3

2

1

u and ψ(u)

0

−1

−2

−3

−4

−5

−6

0

1

2

3

4

5

6

7

8

9

10

t

Fig. 14. ψ(u)(solid line) and u(dotted line) of the closed-loop system (41) with anti-windup compensation, x(0) = [0 0 0.06 0]T .


89

under which the domain of attraction is significantly enlarged. Shown in Figs. 13 and 14 are the closed-loop trajectories with T x(0) = 0 0 0.06 0 , and the corresponding control input, respectively.

6 Conclusion In this chapter, we have addressed the problem of anti-windup compensator design for linear systems with imprecise knowledge of the actuator input output characteristics. Several problems in the analysis and design of a control system with anti-windup compensation are considered. These problems are formulated into constrained optimization problems, for which numerical algorithms are proposed. Throughout the paper, numerical examples are used to illustrate the effectiveness of the proposed analysis and design procedure. Finally, the proposed anti-windup design was applied to an inverted pendulum on a cart experimental setup to demonstrate that a carefully designed anti-windup compensation gain leads to significant improvement on the closed-loop stability.

References 1. Astrom KJ, Rundqwist L (1989) Integrator windup and how to avoid it. Proceedings of American Control Conference 1693-1698 2. Bernstein DS, Michel AN (1995) A chronological bibliography on saturating actuators. International Journal of Robust and Nonlinear Control 5:375-380 3. Cao Y, Lin Z, Ward DG (2002) An antiwindup approach to enlarging domain of attraction for linear systems subject to actuator saturation. IEEE Transactions on Automatic Control 47:140-145 4. Fang H, Lin Z, Shamash Y (2005) Disturbance tolerance and rejection for linear systems with saturation: disturbance attenuation using output feedback. Proceedings of the 2005 Joint IEEE Conference on Decision and Control and European Control Conference 82948299 5. Gomes JM, Tarbouriech S (2005) Antiwindup design with guaranteed regions of stability: an LMI-based approach. IEEE Transactions on Automatic Control 50:106-111

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6. Grimm G, Hatfield J, Postlethwaite I, Teel AR, Turner MC, Zaccarian L (2003) Antiwindup for stable linear systems with input saturation: an LMI-based synthesis. IEEE Transactions on Automatic Control 30:1509-1525 7. Hu T, Lin Z (2001) Control Systems with Actuator Saturation: Analysis and Design. Birkhäuser, Boston 8. Hu T, Huang B, Lin Z (2004) Absolute stability with a generalized sector condition. IEEE Transactions on Automatic Control 49:535-548 9. Kothare MV, Campo PJ, Morari M, Nett CN (1994) A unified framework for study of anti-windup designs. Automatica 30:1869-1883 10. Lin Z (1998) Low Gain Feedback. Lecture Notes in Control and Information Sciences 240, Springer-Verlag, London 11. Morabito F, Teel AR, Zaccarian L (2004) Nonlinear anti-windup design applied to eulerlagrange systems. IEEE Transactions on Robotics and Automation 20:526-537 12. Tarbouriech S, Garcia G (1997) Control of Uncertain Systems with Bounded Controls. Springer, London 13. Takaba K, Tomida Y (2000) Analysis of anti-windup control system based on LPV system representation. Proceedings of American Control Conference 1760-1765 14. Teel AR, Kapoor N (1997) The L2 anti-windup problem: Its definition and solution. Proceedings of European Control Conference 15. Turner MC, Postlethwaite I (2004) A new prospective on static and low order anti-windup synthesis. International Journal of Control 77:27-44 16. Wu F, Grigoriadis KM, Packard A (2000) Anti-windup controller design using linear parameter-varying control methods. International Journal of Control 73:1104-1114 17. Zaccarian L, Teel AR (2004) Nonlinear scheduled anti-windup design for linear systems. IEEE Transactions on Automatic Control 49:2055-2061

Design and Analysis of Override Control for Exponentially Unstable Systems with Input Saturations Adolf Hermann Glattfelder and Walter Schaufelberger Automatic Control Lab, ETH Zürich, Switzerland, [email protected], [email protected] Summary. The antiwindup and override design techniques have been developed in industrial design practice for controlling plants with input and output constraints. At present, various analysis and synthesis techniques are well established. These techniques have shown to be both effective and efficient for plants of dominant low order, which are either asymptotically stable or at least marginally stable. However, for exponentially unstable plants with input saturations, the analysis for the standard antiwindup design reveals a strongly limited radius of attraction, which is often smaller than what is required from the specifications. In this contribution it is shown how the standard override technique can be used in a systematic way to extend the radius of attraction significantly, such as to meet the specifications. The design is based on nonlinear stability analysis. The generic design approach developed here is applied to both benchmark cases.

1 Introduction In the last two decades the control of asymptotically stable LTI-plants or open integrator chains with input constraints in the form of actuator stroke s has been extensively investigated. Several analytical and computational approaches have been developed (see the introduction to this book).

92

Adolf Hermann Glattfelder and Walter Schaufelberger

Their current state as extended to the control of exponentially unstable plants is given in the contributions in this book. The main characteristic for this class of plants is a severely restricted basin of attraction (stability radius) of the closed loop system, which often does not cover the specification pertaining to the particular application. One such approach is feedback for input constraints and feedback for output constraints. It has been developed in industrial control systems design and is routinely applied in practice. Basically the approach is an intuitive one. It uses simple but frequent cases as a starting point, and then expands the techniques to more complex situations. – This approach has been been extensively applied in [3]. There, suitable generic structures for both the and cases are described, and corresponding implementation variants often used in practice. It also prepares a framework for the analysis of their nonlinear stability properties, which is based on the . This leads to a simple, straightforward and transparent stability-based design. However the nonlinear loop performance is not addressed in a formal way there, but rather using established linear design methods in conjunction with simulations. However the many cases in [3] demonstrate that this produces results, which are quite acceptable from a practical point of view1 . Finally, the focus there clearly is on exponentially stable plants and on chained-openintegrator plants. The case of exponentially unstable plants has been mentioned (see sect. 6.4.2, ‘stability charts’), but not investigated in further detail. This shall be done here. The aim is to explore how far these basic methods carry in this particular area, and more specifically if they are able to extend the basin of attraction sufficiently in order to cover the typical specifications from practical cases. The basic approach is the same as for the non-exponentially-unstable plants. It is a two step design, mixing both analytic and intuitive elements. It starts from closed loop equilibrium at nominal operating conditions. First, small deviations are applied, such that all elements in the loop evolve within their linear range. Then a linear controller is designed which stabilizes the loop and yields the closed loop performance as specified. 1

Note that the relative strength and weakness regarding stability and performance properties are in contrast to the other available methods, and may be seen to complement them

Antiwindup and Override Control Solutions

93

Next, the deviations are increased until the actuator constraints are met, and the performance will deteriorate. The usual countermove is to augment the linear controller structure by a standard algebraic antiwindup feedback (awf), while the linear controller part is not changed (in order to conserve the small signal performance). This works quite well for plants of dominant first order. But for plants of dominant higher order further modifications are needed to attain an acceptable performance. As the intuitive element of the design procedure relies on details observed on the specific case, the two benchmark cases shall be addressed separately. A more general view may evolve from this later. For each benchmark, a linear controller is designed first to meet the small signal specifications. Then a standard awf is added. If the dynamic response does not meet the large signal specifications, then further structural modifications are designed and tested in simulations. In parallel, the nonlinear stability properties are investigated at each step.

2 Case A: The Inverted Pendulum The control problem has been fully stated in the ‘specifications’ (see the corresponding chapter of this book), and shall not be repeated here. The following is an extended version of [4].

2.1 The Standard AWF-Approach The Control Structure The details are shown in Fig. 1.

94


ka 4

r_1(t)

k0/T0

1 s

r_2 r_4

k1 k2

x00

k3

Setpoint sequencer

k4

1 s

1/T4

2

x40 u_lo to u_hi

v_lo to v_hi

x_1

1

u x_2 z

z(t)

x_3 − x_1

Load sequencer

Plant 3

Fig. 1. The control system, consisting of the plant, the nonlinear actuator and the linear augmented state feedback controller with standard algebraic awf

Within the linear range the transfer function of the plant is 1 1 x1 = 2 u sT1 s T2 T3 − (1 + (T3 /T1 )) & & 1+(T3 /T1 ) 3 /T1 ) with poles at s1 = − ; s2 = 0; s3 = + 1+(T T2 T3 T2 T3 Gu =

(1)

Consider now the steady state conditions for the plant. From Fig. 1 for all r → u = 0; for all z → u = −z;

(2)

Thus, z must be restricted to within the working range of u, while r has no such restrictions. In other words the z(t)-sequence can be expected to be more difficult to stabilize than the r(t)-sequence. The next step is to design the linear feedback controller. The method to be used is to Ω1 , as this relates in a straightforward way to the small signal closed loop bandwidth from the specifications.


95

More specifically (s + Ω1 )4 · (s + Ωa ) := 0 with Ωa := α1 · Ω1

(3)

where Ωa is the pole associated to the actuator loop in Fig. 1. Here α1 := 10. And by way of the closed loop characteristic equation for the feedback gains k4s = (4 + α1 ) · (Ω1 T4 ) k3s = (6 + 4α1 ) · (Ω1 T3 )(Ω1 T4 ) + (T4 /T2 ) · (1 + (T3 /T1 )) k2s = (4 + 6α1 ) · (Ω1 T2 )(Ω1 T3 )(Ω1 T4 ) + k4s · (1 + (T3 /T1 )) k1s = (1 + 4α1 ) · (Ω1 T1 )(Ω1 T2 )(Ω1 T3 )(Ω1 T4 ) k0s = (α1 ) · (Ω1 T0 )(Ω1 T1 )(Ω1 T2 )(Ω1 T3 )(Ω1 T4 ) ; with T0 := T2

(4)

Then the augmented state feedback is transformed to the cascade structure Fig. 1, with - actuator loop r4 → x4 , - inclination feedback r2 → x2 , using x3 , - speed control r1 → x1 , with x0 that is k4 = k4s; k3 = k3s/k4s; k2 = k2s/k3s; k1 = k1s/k2s; k0 = k0s/k1s;

(5)

For the resulting dominant fourth order closed loop transfer function r1 → x1 having four poles at −Ω1 , the −3dB-crossover is at ≈ 0.425Ω1 . Thus to comply with the specification: Ω−3dB > 1.0, set Ω1 := 2.5

The Nonlinear Stability Properties They are investigated here with the , [5]. Its basic form allowing a graphical test is for a single input single output linear subsystem and one nonlinear element with one input and one output. However the actuator in the control system Fig. 1 has two nonlinear elements, that is the stroke s uhi , ulo and the slew s vhi , vlo . This may be circumvented by using an approximate model of the actuator, Fig. 2.

96

Adolf Hermann Glattfelder and Walter Schaufelberger 2

1 a_s

w_lin

u_c

w_hi

w

k_s

u 1/T_s

1 s

w_lo a_s

1

0.8

0.6

0.4

0.2

0

0

0.5

1

1.5

Fig. 2. The actuator subsystem, and its step responses r4 → x4

This model now has a single nonlinear element. It is similar to the one used in [1], but avoids the very high gain. – Let as = 1.0 Then whi = uhi ; wlo = ulo d 1 d 1 uup = whi , udn = wlo and dt τs dt τs

(6)

This determines τs and assumes symmetric values. In numerical values whi = 1.25; → τs = 0.125 s for slew rate 10.0 /s; and τs = 0.625 s for slew rate 2.0 /s. The approximate model is conservative with respect to the rate constraint, especially near to the stroke constraints. But it reproduces the position constraint correctly, see Fig. 2. The assumes the nonlinear subsystem to be asymptotically stable. However here, both the plant and the controller in Fig. 1 are not asymptotically stable. Thus the element must be replaced by a parallel arrangement of a unity gain and a dead-span element with unity gain slopes and breakpoints at wlo , whi . Then for the stability test (cf. [3], p.284ff): Δmin =

whi wlinmax − whi

(7)


97

and for the transfer function of the linear subsystem 1 + ka 1s dG F +1= (s + ka ) = 1 D1 1 + R sG

(8)

where dG , D1 denote the characteristic polynomials of the plant and of the linear closed loop. F + 1 is now asymptotically stable due to D1 , that is to the small signal closed loop design. For the benchmark case: F +1=

(s +

1 T4 )

(s2 −

1+(T3 /T1 ) ) T3 T2

s2 s +

(s + Ω1 )4 (s + Ωa )

kas T0

s

(9)

If a compensating awf gain kas is used, i.e. kas := Ωa T0 = α1 Ω1 T2 ; then F +1=

s(s +

1 T4 )

(s2 −

(s + Ω1

1+(T3 /T1 ) ) T3 T2 )4

(10)

and finally for the cascade structure (see Fig. 1): ka1 = kas/(k1k2k3k4s). The unstable root factor produces a strong phase shift of the Nyquist contour F + 1 into the LHP, which indicates a substantial reduction of the radius of attraction, Fig. 4. And the reduction is stronger for the slow actuator case. 1.5 Ts = 0.125 T = 0.625 s

1

0.5

0

−0.5 −0.5

0

0.5

1

1.5

Fig. 3. For the circle test: Nyquist contours of F + 1 for the standard awf approach

98

Adolf Hermann Glattfelder and Walter Schaufelberger 2.5 x 1 x 2 x 3 x

2

4

z r

1

1.5

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

25

30

35

40

45

50

Fig. 4. Response of the system Fig. 1 to the specified r1 (t)-sequence, slow actuator

Transient Responses to the Input Sequences The case with the slow actuator will be documented here, as it is the more critical one. First the setpoint sequence r1 (t) from the specifications is applied first to the control structure Fig. 1. The size of the first step of r1 may be increased until its maximum value from the specifications without diverging. From Fig. 4 the transient is linear everywhere, and thus the full range for r can be applied. If for instance just the slew rate of r(t) would be increased from ±0.5/s to ±1.0/s then the size of r1 must be reduced considerably, Fig. 5, to 0.78 for the fast actuator and to 0.26 for the slow one. Next the load sequence z(t) is applied. Again the size of the first step of z is increased until close to the border of stability, Fig. 6, to 0.75 for the fast actuator and to 0.71 for the slow one. Figure 7 documents the response for the slow original actuator to the z(t)-sequence up to 0.975, but with its slew rate reduced from the specified ±0.25/s to ±0.125/s. By inspection of Figs. 6 and 7, the response to the first large z ramp is the critical one. During the ramp the controller has x2 converge to z, while x1 stabilizes at a nonzero value, which is proportional to the ramp speed selected (this follows directly from


99

2.5 x 1 x 2 x 3 x

2

4

z r

1

1.5

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

25

30

35

40

45

50

Fig. 5. Response to r1 (t)-sequence with faster ramp

1.5 x 1 x 2 x 3 x

1

4

z r

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

25

30

35

40

45

50

Fig. 6. Response of the system Fig. 1 to the specified z(t)-sequence

the plant equations). After the ramp on z ends, the controller moves to reduce x1 to its setpoint at zero again. – For the nominal ramp speed and the associated large offset on all xi , i = 1...4, the margin on the resulting u to the value ulo is no longer sufficient for stabilization. In other words the controller output moves far beyond the value. Then the control loop is cut open for too long, and cannot recover in time

100


1

4

z r

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

25

30

35

40

45

50

Fig. 7. Response to z(t)-sequence with ramps slowed down to half of specified

to stabilize the unstable inclination dynamics. But this is (barely) sufficient for the slow ramp with its smaller offset on x1 , where the controller output saturates for a relatively shorter time span. – Note that this behavior corresponds to the stability properties following from Fig. 3.

2.2 The Override Approach The Basic Idea However the analysis of the responses in Figs. 6 and 7 suggests a possible solution: If the inclination feedback would stay closed ‘for most of’ the transient (‘for all of it’ would be safest) by simply avoiding the actuator s, then the asymptotically stable closed loop dynamics would be conserved. And this can be implemented by placing a on the inclination setpoint r2 (t), which is the output of the lateral speed controller, see Fig. 1. This idea has been investigated in [6]. It generalizes the cascade-limiter design from DC-servodrives where e.g. the speed constraint is implemented by a corresponding on the speed setpoint, which is the position controller output. Note however that the saturation values must correspond as closely as possible to the actual operating span


101

of the actuator but may not exceed it. In other words this approach requires exact a priori knowledge of additional persistent disturbances, which will lead to u = 0, and thereby shift the actuator operating span. Thus the saturation values must be shifted accordingly. This drawback can be avoided with the technique: the input to the actuator loop is taken as the secondary output yc which is to be constrained by override feedback controllers with appropriate setpoints. In essence this replaces the feedforward concept described above by a feedback one, and as such is less sensitive to persistent disturbances. Note that the different approaches are most popular (see the other chapters of this book), while explicit use of overrides is not common. Note also that both approaches are related, [7].

The Override Control Structure Details are shown in Fig. 8. Note that the feedback acts not on r 2 1 but further downstream on r 4 1. This provides a separate set of feedback gains for its independent tuning.

y_c

r_c_hi

r_c_hi x_2

u_c_hi

x_3 awf

R_c_hi

7

r_1(t)

Setpoint sequencer

k0/T0

ka_1

min

1 s

x_1

1

x_2

2

x_3

3

k1 k2

x3

k3

r_4_1

6

ka_C

r_2_c k0_C/T0 r_c_lo

4

r_4

max

r_2_1

x_4

1 s

k3_C k2_C

z(t)

Load sequencer

z

x_3 − x_1

5

x00_lo

Fig. 8. The structure of the override control system

Plant with Actuator Loop

102


The linear constraint feedback is designed as follows (see also [4]). -

To shorten the design process, the actuator dynamics are neglected. From Fig. 8, the same actuator feedback gain k4 will be inserted here, as it has been determined for the main control loop in the previous section.

-

A stabilizing state feedback for the inclination dynamics Go =

s2 T

1 2 T3 − (1 + (T3 /T1 ))

(11)

is designed. This yields for the closed loop response from r 2 c to yc = r4 in Fig. 8, with feedback gains k2s c on x2 and k3s c on x3 = sT2 · x2 : 1 1 + Go · (sT2 k3s c + k2s c) s2 T2 T3 − (1 + (T3 /T1 )) = 2 s T2 T3 + sT2 k3sc + [k2sc − (1 + (T3 /T1 ))]

Gc =

-

(12)

For the overall constraint loop this is the ‘plant’. It has a negative sign on the constant term, therefore the sign on the controller has to be inverted as well for negative feedback in the overall loop. It also has one zero in the right half plane, ' at + [1 + (T3 /T1 )]/[T1 T2 ].

-

A well established design rule states that a pure integral controller is best suited for such inverse-unstable plants. Then the overall closed loop characteristic equation is 0 = 1+

k0sc · Gc sT0

( ) T 3 = s3 T0 T2 T3 + s2 T0 T2 k3sc − k0sc + ... T0 +sT0 [k2sc − (1 + (T3 /T1 ))] + k0sc (1 + (T3 /T1 )) -

(13)

Another such design rule is that the zero in the right half plane should be matched by one closed loop pole no further left than at its mirror position in the left half ' √ plane, that is at − [1 + (T3 /T1 )]/T1 T2 = − 2. Here this pole is set at Ωz = −1.0.

-

The other two closed loop poles may now be assigned, such as by (s + Ωc )2 := 0.


-

103

With βc := Ωz /Ωc k0sc = (βc ) (Ωc T0 )(Ωc T2 )(Ωc T3 ) ; with T0 := T2 T3 k3sc = (2 + βc )(Ωc T0 ) + k0sc T0 k2sc = (1 + 2βc ) (Ωc T2 )(Ωc T3 ) + [1 + (T3 /T1 )]

(14)

Then the augmented state feedback is transformed into the cascade structure Fig. 7, yielding k0C = k0sc /k2sc ; k2C = k2sc /k3sc ; k3C = k3sc ;

-

The awf gain is set as ‘compensating’, that is kaC = α1 Ωc T4 /k2sc

-

Then the setpoints are selected. Set rchi = uhi − Δrc ; rclo = ulo + Δrc ;

(15)

(16)

where Δrc is an additional design parameter. It should be selected such that the s uhi , ulo are not met for the usual disturbances while the constraint loop is in operation. Nor should it be taken too large, as then the available range on u1 is no longer usable. A numerical example shall illustrate this. Let Δrc = −0.06 · ulo = 0.075. This leads to rclo = −1.175. Now compare this to the maximum z = 1.10, taken from the specification, which leads to u = −1.10. Thus the margins for maneuvering in both directions (down to ulo and up to u from z) are the same. -

Finally the controller outputs must be implemented into the main speed loop. As the plant transfer function sign is negative, the direction of u is reversed, and therefore the output of the rchi -controller must be connected to the Max-selector and the rclo -controller to the Min-selector, see [3], p. 376 ff.

The nonlinearity in Fig. 8 has three inputs. But the canonical structure required for the stability test only allows one input. However the Min-Max-selector block is under some weak assumptions equivalent to a one-input dead span nonlinearity, [3], p. 114ff. Then for the linear subsystem

104


F +1 =

≈

(1 + F )1 (1 + F )c (1+(T3 /T1 )) )·s·s T2 T3 (s + Ωa )(s + Ω1 )4 (s + Ωa )(s + Ωc )2 (s + Ωz ) · 3 /T1 )) (s + T14 )(s2 − (1+(T ) · s T2 T3

(s + ·

1 2 T4 )(s

−

·

s+

kas1 T0

s

·

s s+

kasc T0

(17)

In this equation, the ≈ is due to the approximative2 separation of the closed loop polynomial for

-

the loop into (s + Ωa )(s + Ωc )2 . The open loop poles at −(1/T4 ) from the actuator, and the poles from the in' clination dynamics (s2 − [1 + (T3 /T1 )]/[T2 T3 ] belong to the same physical

-

subsystems, and thus the corresponding root factors in (F + 1)1 and (F + 1)c cancel without generating any controllability or observability problems. -

The same holds for the poles at zero from the s in the main controller and the constraint controllers with those in their respective awf loops.

-

And the awf gain kas1 /T0 has been set to the value Ωa in the corresponding closed loop factor, such that this also cancels in the (1 + F )1 part

-

The same holds for the corresponding root factors in the (1 + F )c part.

Thus ((s + Ωc )2 (s + Ωz ) s · (s + Ω1 )4 1 2 (s + Ωc ) (s + Ωz ) s = (s + Ω1 ) (s + Ω1 )2 (s + Ω1 )

F +1 ≈

Setting now Ωc := Ω1

(18)

3

this reduces further to a very simple relation F +1 ≈ 2 3

(s + Ωz ) s (s + Ω1 ) (s + Ω1 )

(19)

Ωa /Ωc is large but not infinite which is reasonable as the as the dynamic properties of both systems 1 and c are not widely different


105

The first part generates a half circle from the origin to +1; j0 as ω runs from 0 to ∞. In other words it evolves in the right half plane. This indicates an ‘unbounded’ radius of attraction. The second part in the equation above modifies this shape by a lead-lag element (note that Ωz < Ω1 ) in the direction of the left half plane. This gets more pronounced for Ω1 Ωz and ultimately will restrict the radius of attraction. – Note that the approximation from above will show up at high frequencies, ω Ω1 , see Fig. 9. 1.5 T = 0.10 4 T4 = 0.50

1

0.5

0

−0.5 −0.5

0

0.5

1

1.5

Fig. 9. Nyquist contours for the override system,if no saturation is met

The shape of the Nyquist contour in Fig. 9 indicates a very large radius of attraction. This allows either large reference steps r1 or, for step loads, resulting steady state offsets u, which are close to the constraint setpoints rchi ; rclo . Transient Responses to the Specified Input Sequence Only the responses with the slow actuator subsystem are shown here. The setpoint response to the nominal sequence r1 (t) is not shown, as it does not differ significantly from Fig. 4. However the response to the faster setpoint ramp from Fig. 5 with the system is shown in Fig. 10. And Fig. 11 shows the response for the nominal load sequence z(t) from the specifications.

106


2

4

z r

1

1.5

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

25

30

35

40

45

50

Fig. 10. Responses of the override control system Fig. 8 to the r(t)-sequence from the specifications, but with increased slew rate 1.5 x 1 x 2 x 3 x

1

4

z r

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

25

30

35

40

45

50

Fig. 11. Responses of the override control system Fig. 8 to the full z(t)-sequence from the specifications

In other words, the solution now covers the full specifications. From Fig. 11, the critical phase is no longer the first large loading, but rather the last large unloading in the z(t)-sequence. The actuator position x4 is moved to its


107

value as soon as the z-ramp starts for a long time interval. After the ramp has come to its end x4 is driven to the opposite , and therefore barely manages to stabilize the response. This is less critical if the ramp slew rate would be −0.20/s instead of −0.25/s, see Fig. 12 (top). If the slew rate is fixed, then the only way out seems to be to increase the actuator working range, from −1.25 to −1.35, Fig. 12 (bottom). 1.5 x 1 x 2 x 3 x

1

4

z r

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

25

30

35

40

45

50

1.5 x 1 x 2 x 3 x

1

4

z r

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

25

30

35

40

45

50

Fig. 12. Responses of the override control system Fig. 8 to the z(t)-sequence, (top) with slower unload ramp −0.25 /s → −0.20 /s, (bottom) with extended working range −1.25 → −1.35

108


3 Case B: The Continuous Stirred Tank Reactor (CSTR) The statement of the control problem has been given in the specifications (see the corresponding chapter of this book), and shall not be repeated here. The bandwidth specification is changed to > 6 rad/s to comply with the usual −3dB notion, as for case A.

3.1 Designing The Linear Control System The plant given in the specifications is a multivariable one, with two inputs (reactant inflow u1 and heat flow u2 into the jacket subsystem) and two outputs (measured reactant concentration y1 and measured temperature y2 of the content). Thus the control structure should be multivariable as well. Here the separation method often used in practice is applied. It consists of designing two independent single input single output loops, thereby neglecting the interaction within the plant, and separating the closed loop bandwidths of both loops as far as feasible. In this case the first loop is given by safety considerations: the measured concentration y1 is fed back to the reactant inflow u1 in order to keep the inventory of reactant under rather tight control: The controller gain of the concentration loop is specified as kp1 > 10. If the contents temperature x2 were set constant (which would imply an infinite bandwidth on the temperature control loop), then for the x1 -closed loop If x1 = 1.0 ∀ t, then 0 = sT1 + (kcc + 1)

(20)

and setting kcc := 15 yields as the closed loop bandwidth Ω1 = 80 rad/s, which is approx. five times faster than the bandwidth specification for the temperature control loop Ω2 := 15 rad/s, and thus complies with the separation rule. Next the temperature control loop is designed. The basic structure is a cascade arrangement with an inner loop (with gain k4) for the servomotor position x4 . Its pole shall be placed at −10 · Ω2 . The plant response is considerably different during runup mode (where x1 = 0) and during production mode (where x1 ≈ 1.0). This is taken into account by designing two separate controllers, and selecting by the current


k_cc

u_1

109

y_1 1 s

1/T1

1

x10

Tstep_r1

a1 x_2_S

max

a0 a2

0 1 s

2

x20

y_1

1/T2

a3

1 s

1/T3

x30

a4

Tstep_u2

k4

x40 1/T4

1

1 s

u_2

Fig. 13. Model of the plant with the concentration control loop (gain kcc ) and the heat input servo loop (gain k4), with no saturations on slew and stroke

concentration setpoint: if r1 < 0.05, then the runup mode controller is selected, else the production mode controller. In production mode the plant has one additional state variable (x1 (t)), which is however already controlled by way of kcc . A temperature controller with full state feedback and would have to interfere with this loop. Therefore an approximate design procedure is pursued. It considers the concentration to be constant x1 = 1.0 ∀ t (which implies an infinite bandwidth of the y1 -loop, in other words kcc → ∞). This will eliminate both the state variable x1 and the control variable u1 , and the design now becomes the standard one for a scalar control variable (u2 ) 4 . Thus three further closed loop poles have to be assigned for state variables x2 , x3 , and the x0 of the temperature master controller. Here the three poles are assigned at −Ω2 with Ω2 = 15 rad/s. The design procedure is the same as for case A, and its details are omitted here. 4

Note that this again implements the separation design idea

110


Then for the original system with finite kcc , the closed loop poles will move away from the assigned locations, specifically for this case with kcc := 15 to −79.86 −8.23 −18.48 + j8.33 −18.48 − j8.33 From transient responses (not shown here)for small setpoint steps on r2 starting from steady state x2 = 1.0 at x1 = 0.0 and at x1 = 1.0, this is considered to be still acceptable. It also complies with the specifications (> 6 rad/s at −3dB).

3.2 The Standard Anti Windup Feedback Approach The Control Structure Details are shown in Fig. 14.

7 concentration setpoint ramper

k_cc Plant with actuator

r_1 (t)

1

x_2

2

x_3

3

x_4

4

u_1

6

kaa

1 s

r_2 (t)

k0a/T0

x00 x20 k2a

u_2

x30

u_lo to u_hi

k4a k3a

5

x_4

Temp. Controller for run_up x_3

x_2

r_2 (t)

u_2 Lin

Switch

u_2 (awf)

temp setpoint ramper

x_1

Fig. 14. The control system with standard awf


111

The Nonlinear Stability Properties The general result of [3], p. 70ff, is applied. For the Nyquist contour F +1=

1 + ka2 (1/sT0 ) d2 · sT0 sT0 + ka2 = · 1 + R2 G2 D2 sT0

(21)

where d2 is the polynomial with the open loop roots, and D2 is the closed loop polynomial including the . The second fraction is from the awf loop. First, the terms sT0 in the numerator and denominator cancel, as they are due to the same integral action. Then consider d2 . Its roots are functions of kcc . For kcc = 15: +8.7037 −10.2695 −79.4341 −5.0 The fastest mode is associated with the concentration loop, the mode with pole at ≈ −10 with the jacket dynamics, and the unstable mode with the content temperature dynamics and the exothermal reaction. The root at −5.0 is from the slow actuator open loop. Further for the closed loop polynomial D2 : D2 :≈ T0 T2 T3 T4 (s + Ω2 )3 · (s + αΩ2 )

(22)

The ≈ is due to the fact that the to three times −Ω2 is performed for kcc → ∞ instead of its finite design value. Further, α := 10 is associated with the actuator positioning feedback, following the usual design rule in practice. Finally design the awf loop as with ‘compensating awf gain’ ka∗2 : ka∗2 := (s + αΩ2 ) → ka∗2 = αΩ2 T0 s+ T0

(23)

Figure 15 shows the Nyquist contour. For the runup phase of the temperature loop it starts in the right hand half plane. And the yields an unbounded radius of attraction. However for the production phase it starts for ω = 0 well in the left hand plane, which indicates a strongly delimited region of attraction, see [3], p.290 ff. This is even more so as the loop operates at its production equilibrium u = −1.0 close to the saturation ulo = −1.10.

112

Adolf Hermann Glattfelder and Walter Schaufelberger 1.5

0.15

0.1

1

0.05

0.5

ω=∞ 0

0 ω=0

−0.05 −0.1

−0.08 −0.06 −0.04 −0.02

0

0.02

ω=0

0.04

0.06

0.08

0.1

−0.5 −0.5

0

0.5

1

1.5

Fig. 15. Nyquist contour for the graphic stability test (circle criterion) of the control system with standard awf with compensating gain ka∗2 and slow actuator (left) for the runup phase (x1 = 0): (right) for the production phase (x1 = 1.0)

Transient Responses to the Benchmark Input Sequence A first intuitive solution to avoid run away would be to limit the concentration runup rate. Here this would have to be set to ≈ +1/6 s−1 . But such a clamp on runup rate of r1 is only needed in the final approach phase. Therefore it may be set larger for concentrations below say 0.6 of nominal. This simple feedforward strategy has been implemented in Fig. 16. It reduces the runup time from 6 s to 3 s. However the simulation also shows that the subsequent small disturbances to the nominal production given in the specifications lead to divergent responses. To avoid this they have to be reduced as follows: for the concentration (Δr1 )

from 0.02 to 0.01

and for the reaction rate (Δa21 ) from 0.02 to 0.01 Thus a better solution is needed.


113

1 0.8 0.6 0.4 0.2 0 x1 x2 x3 x4 u2 r2 r1

−0.2 −0.4 −0.6 −0.8 −1

0

5

10

15

20

1.1

1.05

1

x1 x2 x3 x4 u2 r2 r1

0.95

0.9

0.85

0

5

10

15

20 x 4 u

−0.95

2

−1

−1.05

−1.1 0

5

10

15

20

Fig. 16. Transient response of the control system with standard awf, slow actuator (top) overall view, (center) zoom-in around +1.0, (bottom) zoom-in at −1.0

114


3.3 The Override Approach The Basic Idea The basic mechanism leading to instability is that if the concentration increases too fast then the temperature control is pushed to the actuator , and then the temperature can no longer be controlled and will diverge. To avoid this the concentration must be manipulated such that the temperature controller will no longer push the actuator beyond its saturation, but keep it within its operating range. In other words this means constraining the actuator position x4 to a value sufficiently close to the saturation value. And this can be done -

by standard feedback techniques,

-

designating as constrained output yc the reference to the u2 -actuator loop,

-

setting the constraint controller setpoints to rclo = ulo + Δ; rchi = uhi − Δ;

-

with Δ := 0.01, such that the loss in operating performance is small,

-

and modifying by selection the concentration reference input r1 .

Note that here (in contrast to case A) the ‘disturbance’ is accessible for control. Designing the Override Loop The design procedure is demonstrated on a reduced order system, where the dynamics of the concentration loop and the temperature actuator loop are neglected and the jacket dynamics are disregarded, that is T1 = T4 = T3 := 0, and thus 1 G2r (s) = sT2 − a21

r c _ R + r

c

y c

2

+ _

+ R

2

G +

2

y 2

Fig. 17. The reduced order override loop

(24)


115

Read from Fig. 17 for the closed loop characteristic equation ( ) R2 0 = 1 − Rc − 1 + R2 G2r k1 sTc + k0c k12 sT0 + k02 · 2 = 1+ c sTc s T0 T2 + sT0 [k12 − a21 ] + k02 = s3 Tc T0 T2 + s2 Tc T0 [k12 − a21 + k1c k12 ] ( ) T0 + sTc k02 + k1c k02 + k12 k0c + k0c k02 Tc

(25)

Using now 1 k12 − a21 k1c k12 and 1 k02 k1c k02 +

T0 k1 k0 Tc 2 c

(26)

and setting Tc = T0 = T2 := 1.0 yields in the low frequency range, which is of interest here: 0 ≈ s2 Tc T0 [k1c k12 ] + sTc [k1c k02 + k12 k0c ] + k0c k02

(

) k1c k12 k1c k12 2 = s Tc T0 · + sTc + +1 k0c k02 k0c k02

(27)

Also note k1c 2 = k0c Ωc T k1 2 := 2Ω2 T ; that is 2 = k02 Ω2 T

for Rc : k0c := Ω2c T 2 ; and k1c := 2Ωc T ; that is and for R2 : k02 := Ω22 T 2 ; and k12 and select Ωc := Ω2 finally 0 ≈ 4

s Ω2

2 +4

s Ω2

( +1 =

s (Ω2 /2)

)2 (28) +1

for the characteristic equation of the closed loop while the is active. This simple but approximate design rule shall be applied from here on, but it will have to be validated by the transient responses.

The Override Control Structure Details are shown in Fig. 18. As for case A, a sign inversion is required in the controller to conserve negative feedback, and the constraining action for the low setpoint must be applied to the Min-selection.

116

Adolf Hermann Glattfelder and Walter Schaufelberger 7

Override controller

ka_c

1 s

k0_c/Tc

− r_c_lo

k_cc

x_c0 k1_c

min

x_1

−1

1

u_1 r_1 (t)

Concentration setpoint sequencer

x_2

2

x_3

3

x_4

4

5

r_2(t)

y_c Temperature setpoint generator

u_2

kaa

6

1 s

k0a/T0

Switch runup to production at r_1 = +0.05

x00 x20 k2a

Plant with actuator

k4a

x30

8

k3a

awf in

x_4

x_2

r2(t)

Temperature controller for runup

x_3

u2_runup

u_lo to u_hi

Fig. 18. The control system with override feedback control from the temperature controller output (i.e. actuator position setpoint) to the concentration setpoint r1 (t)

Nonlinear Stability Properties Applying the general result for systems (without active s) from [3], p.142, F +1=

1 + (kam /sT0 ) 1 + Rc Gc 1 = (F + 1)m · · 1 + Rm Gm 1 + (kac /sTc ) (F + 1)c

(29)

with index m for the main loop and index c for the loop. Here the part with index m is a special case, as there is no feedback loop: Rm = 0; kam = 0; → (F + 1)m = 1.0; Dc sTc Dc and thus F + 1 = · = dc sTc sTc + kac dc (sTc + kac )

(30)


117

where for Dc from the previous section: Dc ≈ s3 Tc T0 T2 + · · · + k0c k02

k02 k0c 3 = Tc T0 T2 s + · · · + 2 Tc Tc T0 T2 ! 3 2 = Tc T0 T2 s + · · · + (Ωc ) Tc (Ω2 )2

(31)

and for dc · (sTc + kac ): dc ≈ s2 T0 T2 + · · · + k02

! ka = Tc T0 T2 s2 + · · · + (Ω2 )2 · (s + c ) Tc

(32)

From this intuitively set ka∗c := (Ωc Tc )2 to obtain a low frequency gain of (F + 1) of about 1. The Nyquist contour for F + 1 is plotted in Fig. 19. It starts on the positive real axis (at +2 + j0), and evolves well into the right half plane, except for frequencies above the closed loop bandwidth (ω > 10Ω2 ), where F (jω + 1) crosses over into the left half plane. So stability can no longer shown by the on-axis . However if the less conservative Popov test is used (see right hand plot in Fig. 19), then an inclined straight line may be plotted through the origin, and approx. parallel to the Popov contour there, which does not intersect the contour anywhere. – And this indicates an unbounded radius of attraction. 15 2.5

2

ω to ∞

10 1.5

1

5 ω=0

ω=0

0.5

0

0

ω to ∞

−0.5

−5 −1

−1.5

−10 −2

−15

0

5

10

15

20

25

−2.5 −0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Fig. 19. Graphic stability test for the override control system override, slow actuator kac := (Ωc Tc )2 ; (left) Nyquist contour, (right) Popov contour zoom-in to the origin

118


1 0.8 0.6 0.4 0.2 0 x1 x2 x3 x 4 yc r2 r1

−0.2 −0.4 −0.6 −0.8 −1

0

5

10

15

20 x1 x2 x3 x4 yc r2 r1

1.1

1.05

1

0.95

0.9

0.85

0

5

10

15

20 x 4 y c u

−0.95

2

−1

−1.05

−1.1 0

5

10

15

20

Fig. 20. Transient response, override control system, slow actuator, kac := (Ωc Tc )2 ; (top) overall view, (center) zoom-in around +1.0, (bottom) zoom-in at −1.0


119

Transient Response to the Specified Input Sequence Figure 20 shows that the rise time from hot standby to production has been reduced from 3 s to approx. 1.8 s, that is to about 60%. It also documents convergence for the full specified variations to the nominal operating conditions. The smooth switch-in and transient response of the loop is documented in Fig. 20 (bottom), at 7.5 s, 12.5 s and at 14 to 15 s.

4 Summary Both cases A and B have been successfully solved and the specifications given for the benchmarks have been fulfilled. This has been achieved in both cases by applying the technique in a suitably modified way, whereas the standard awf design well known in practice has been shown to fail. – Note that the design process used here is not a purely algorithmic or automatic one, but essential elements arise from careful inspection of the specific system’s behavior. However, insight alone is not sufficient, as the number of design parameters and choices is such that analytic parts become mandatory in order to produce rationally founded choices and solid parameter values. For this, nonlinear stability theory has again proven to be a powerful tool. But again it cannot produce all necessary design elements by itself. It must be supplemented by carefully designed experiments for transient responses. Thus solutions by applying the and techniques so far always arise through strong interaction of insight, intuition, analysis and experiments.

References 1. Barbu C., R. Reginatto, A.R. Teel, and L. Zaccharian (2002), Anti-Windup for Exponentially Unstable Linear Systems with Rate and Magnitude Input Limits, in Actuator Saturation Control, Marcel Dekker Inc., ISBN 0-8247-0751-6, Editors V. Kapila and K.M. Grigoriadis, p.1 – 31 2. Tarbouriech S., and G. Garcia (2002), Output Feedback Compensators for Linear Systems with Position and Rate Bounded Actuators, in Actuator Saturation Control, Marcel Dekker Inc., ISBN 0-8247-0751-6, Editors V. Kapila and K.M. Grigoriadis, p. 247 – 272

120


3. Glattfelder A.H., and W. Schaufelberger (2003), Control Systems with Input and Output Constraints, Springer Verlag Limited, ISBN 1-85233-387-1, London 4. Glattfelder A.H., and W. Schaufelberger (2005), Antiwindup and Override Control for Exponentially Unstable Systems with Actuator Constraints, Proceedings 16th IFAC World Congress, Prague, Tu-A20-TO, 2005, paper 02398 5. Khalil H.K. (2002), Nonlinear Systems, 3rd ed., Prentice Hall 6. Hippe P. (2003), Windup Prevention for Unstable Systems, Automatica 39, 1967 - 1973 7. Glattfelder A.H., and W. Schaufelberger (2004), A Path from Antiwindup to Override Control, Proc. 6th IFAC Symposium - Nolcos, Stuttgart, 1379 - 1384

Anti-windup Compensation using a Decoupling Architecture Matthew C. Turner, Guido Herrmann, Ian Postlethwaite Control and Instrumentation Research Group, Department of Engineering, University of Leicester, Leicester, LE1 7RH, UK. Email: {mct6,ixp,gh17}@le.ac.uk Summary. This chapter describes the theory and application of a ‘decoupled’ approach to anti-windup compensation. One of the salient features of this approach is that the anti-windup problem can be posed in a manner which decouples the nominal linear closed-loop from the nonlinear stability problem associated with actuator saturation and the anti-windup compensator. This process also reveals a logical and intuitive performance criterion which one can optimise by means of linear matrix inequalities. Details of how the anti-windup problem can be solved using various forms of compensator are given, together with some robustness considerations. Initially, the problem is posed for globally asymptotically stable plants, although this condition is later relaxed to allow the consideration of plants with exponentially unstable modes. Finally, the results are demonstrated on a multivariable aircraft model and the benchmark inverted pendulum.

1 Introduction 1.1 “Anti-Windup” and Its Philosophy The problem of “windup” was probably first discovered by practical control engineers when designing integral and PID controllers for tracking problems with strict

122

Matthew C. Turner, Guido Herrmann, Ian Postlethwaite

limits on the control signal amplitude. In these types of situation, if saturation occurs it can cause the integrator state to attain a large value (or to “windup”) during saturation and then this energy has to be dissipated once saturation has ceased. Such saturation problems were generally observed as excessive overshoot and/or a slow settling time at the plant output. This early interpretation of the word “windup” was easily justified as it corresponded to the accumulation and dissipitation of charge on the capacitor of an analogue integrator. Thus, at first, windup tended to be associated with saturation problems with integrator type controllers, particularly the popular PID controller. To overcome these saturation problems, a number of largely ad hoc methodologies - so called “anti”-windup techniques - were developed and most have had a fair amount of success attributed to them. Among these ad hoc techniques are high gain anti-windup, reset anti-windup and Hanus anti-windup. In high gain anti-windup, the controller is prevented from producing too large a control signal during saturation by means of a large feedback gain ‘steered’ by a deadzone nonlinearity. In reset antiwindup, the integral state is reset to zero following saturation to prevent the windup problem. Finally, the Hanus technique works by partially inverting the controller, in an attempt to force the controller to produce an output no greater than the saturation level. A good summary of these can be found in [4]. The basic anti-windup architecture is shown in Figure 1. Up until, roughly, the time of reference [7], all anti-windup techniques were ad hoc, had no stability or performance guarantees but were simple to implement and hence were useful to the practising engineer. A particularly useful property of these so-called anti-windup techniques is that they were designed to work in conjunction with the existing controllers and only became active once saturation occurred. Research in optimal control in the 60s and 70s and then again in the mid 90s pointed to a different source of saturation problems. In papers such as [5], [30] and [22], it was observed that the fundamental limitation in (linear) control systems subject to input saturation was the plant not the controller (This observation was also made in less formal manner in the anti-windup literature - see [18]). Importantly, it was established that global stability is only possible for systems with poles only in the closed-left-half complex plane. Furthermore, global asymptotic stability with finitegain stability is not possible using linear control laws if the system contains repeated poles on the imaginary axis (see [27] for a good summary of this). Also, if a system

Anti-windup Compensation using a Decoupling Architecture

123

disturbance

reference Controller

anti−windup compensator

Plant

+ _ saturation detection (deadzone)

Fig. 1. Basic anti-windup architecture

contains poles in the open-right-half complex plain, the most which can be achieved is local stability, with local gain properties ([30]). In contrast to the anti-windup techniques though, the solutions offered by these optimal control techniques tended to be complex, unintuitive and, typically, one-step solutions, i.e. they would not work with an existing controller, they would replace it. Hence, although more theoretically sound, these techniques found less favour with the practising engineer than the aforementioned anti-windup techniques. Some of the weaknesses of the “first generation” of anti-windup compensators were lack of stability guarantees, restriction to single-input-single-output systems (typically) and imprecise tuning rules. In addition, this generation of anti-windup compensators often made the inaccurate connection that saturation problems were always associated with integrators in the controller. The “second generation” of anti-windup compensators attempted to address at least the first two of these problems by appealing to some of the control theory developed for the one-step solutions to the antiwindup problem. That is, the basic architecture of the anti-windup problem would be retained, but ideas from optimal and nonlinear control (see [7] for an early reference) would be used to guarantee stability of the resulting (nonlinear) closed-loops and, as these tools were not necessarily insistent on SISO systems, the second restriction was also removed. Some of the first examples of “second generation” anti-windup techniques can be found in [4], [23], [24] and [31], for example. Today, we are probably still in the era of “second generation” anti-windup technology. Many methods have been proposed based on the underlying idea of augmenting

124


a linear controller during saturation to obtain improved performance [37, 25, 2]. Indeed recently, constructive techniques have become available which allow the tuning and optimisation of various aspects of closed-loop system behaviour e.g. L2 gain, performance recovery, basin of attraction [9, 35, 8]. Probably the main weakness in this generation of AW compensators is that most have been demonstrated in simulation or on toy laboratory examples and few have been tried on real engineering systems (One exception is [14] where the results in [35] were tested on a multivariable hard disk servo system). A possible reason for this is that some of this generation of AW compensators have discarded the simplicity and intuition found in many earlier schemes and have instead embraced mathematical rigour. Our intention in this chapter is to show that the two are not mutually exclusive and to describe and demonstrate techniques which are both practically appealing and endowed with stability and performance guarantees.

1.2 The Goal The aim of this chapter is to motivate and introduce a particular form of anti-windup compensator which we believe is flexible, intuitive and suitable for implementation in real engineering systems. The basic architecture which we use is taken from [38] in which a special parameterisation of anti-windup compensators is proposed (this is related to the work of [21] and [2], and [31] as well). The structure of the chapter is as follows. After the notation is introduced, we describe the architecture of the problem we consider, and show how this has an appealing structure. The next section is devoted to global anti-windup, particularly for open-loop stable plants and a number of different types of anti-windup compensator are proposed (full-order, reduced-order and static). Robustness is also considered. The following section introduces local anti-windup compensation, which must be used if the plant in question is unstable. Based on reduced sector bounds (used in [17] and [26]) similar results are developed for local stability/performance guarantees. Two example sections are then given, where the theoretical results are demonstrated in detail. The chapter concludes with some final remarks.


125

1.3 Notation The notation is standard throughout and mainly complies with that listed at the beginning of this book. However, we specifically define: • The L2 norm of a vector-valued function x(t) as x 2 :=

&*

∞ 0

x(t) 2 dt. Fur-

thermore, whenever x 2 < ∞, we say that x ∈ L2 . • The induced L2 norm or the L2 gain of a (nonlinear) operator Y is defined as Y i,2 := sup

x∈L2

Y x 2 x 2

(1)

If this quantity exists, we say that Y is finite-gain L2 stable, or simply L2 stable.

• Given the sets Dx := x(t) ∈ Rnx : supt≥0 x(t) ≤ βx and

Dy := y(t) ∈ Rny : supt≥0 y(t) ≤ βy , and if Y : Dx → Dy is a nonlinear operator and W ⊂ Dx , then we define the local induced L2 norm, or small signal L2 gain as Y i,2,W :=

sup x∈L2 ,x∈W

Y x 2 x 2

(2)

If this quantity exists, we say that Y is small-signal finite-gain L2 stable, or simply L2 stable. Note that sup x∈L2 ,x∈W

Y x 2 Y x 2 ≤ sup x 2 x∈L2 x 2

(3)

implies that Y i,2,W ≤ Y i,2 and therefore we can expect the small-signal L2 gain to be smaller than the actual L2 gain (which may not exist for certain systems). • We denote the space of real rational p × q transfer functions by R p×q ; the subset of these which permit analytic continuity in the closed-right-half complex plane is denoted RH p×q ∞ ; the space of all such real rational transfer functions is RH ∞ . Note that the H∞ norm of a real rational transfer function, G(s) ∈ RH ∞ , can be defined either in the frequency domain as G ∞ := supω σ ¯ (G(jω)) or in the time domain as G ∞ := G i,2 where G is the time-domain linear operator associated with G(s) and σ ¯ (.) denotes maximum singular value.

126


2 Architecture Consider the system with input saturation and anti-windup compensation shown in Figure 2. This is a reasonably generic form of anti-windup compensation and allows the anti-windup compensator Θ(s) to inject signals, which modify the controller’s behaviour, at the controller output and also one of its inputs. Here G(s) = [G1 (s)

G2 (s)] is the finite-dimensional linear time invariant (FDLTI) plant which

we have assigned the following state-space realisation G(s) ∼

x˙ p = Ap xp + Bp um + Bpd d

(4)

y = Cp xp + Dp um + Dpd d

where xp ∈ Rnp is the plant state, um ∈ Rm is the control input to the plant, y ∈ Rq is the plant (measurement) output and d ∈ Rnd is the disturbance. The FDLTI controller K(s) is assumed to be implemented as K(s) = [K1 (s)

K2 (s)]

and is assigned the following state-space realisation x˙ c = Ac xc + Bc ylin + Bcr r K(s) ∼ ulin = Cc xc + Dc ylin + Dcr r

(5)

where xc ∈ Rnc is the controller state, ulin ∈ Rm is the linear controller output r ∈ Rnr is the disturbance on the controller, normally the reference. The overall controller output is given by u = ulin + θ1 ∈ Rm where θ1 is a signal produced by

d

r K(s)

+ -

u

um G(s)

θ1 + + +

θ2

Θ (s)

Fig. 2. Basic anti-windup architecture

y


127

the anti-windup compensator; if the anti-windup compensator is inactive, u = ulin . ylin ∈ Rq is the linear input to the controller and is given by y + θ2 ∈ Rq ; if the anti-windup compensator is inactive ylin = y. The plant input, um and the controller output are related through the nonlinear function sat : Rm → Rm which is defined as

⎡

⎤ sat1 (u1 ) ⎢ ⎥ .. ⎥ sat(u) = ⎢ . ⎣ ⎦ satm (um )

where sati (ui ) = sign(ui ) min(|ui |, u ¯i )

(6)

∀i and Dzi (ui ) = sign(ui ) max(0, |ui | −

u ¯i ) ∀i. Also u ¯i > 0 ∀i ∈ {1, . . . , m}. The anti-windup compensator is driven by the function Dz : Rm → Rm

⎡

⎤ Dz1 (u1 ) ⎢ ⎥ .. ⎥ Dz(u) = ⎢ . ⎣ ⎦ Dzm (um )

(7)

sat(u) = u − Dz(u).

(8)

and

Both the deadzone and saturation functions satisfy ‘sector bounds’. A nonlinear function N (.) is said to belong to the Sector[0, E] for some diagonal matrix E if the following inequality holds for some diagonal matrix W (see [20] for a full description): N (u) W [Eu − N (u)] ≥ 0

∀u ∈ Rm

(9)

Both the saturation and deadzone functions lie in the Sector[0, I], as they are decentralised functions each of which has a graph lying below the unity gradient line. It is often convenient to use the set U := [−¯ u1 , u ¯1 ] × [−¯ u2 , u ¯2 ] × . . . × [−¯ um , u ¯m ] This set has the interpretation that sat(u) = u,

∀u ∈ U, and Dz(u) = 0,

(10) ∀u ∈

U. As it stands, the scheme depicted in Figure 2 does not have any illuminating features. However, consider a particular case of the generic scheme, as shown in Figure 3 (introduced in [38]), where the anti-windup compensator Θ(s) has been parameterised in terms of a transfer function M (s) ∈ RH ∞ and a copy of the plant G2 (s). Using the identity (8), Figure 3 can be re-drawn as the decoupled scheme in Figure 4. Notice that this system exhibits an attractive decoupled structure:

128

Matthew C. Turner, Guido Herrmann, Ian Postlethwaite d r K

u lin + − ud M−I

y lin + +

y

G

+ −

yd

G2 M

~ u

Fig. 3. Conditioning with M (s) M−I

ud

Disturbance Filter

Nonlinear Loop

− ~ u

+

r

d K

G

u lin

yd

G 2M

+ ylin

− y

Nominal Linear Transfer Function

Fig. 4. Equivalent representation of Figure 3

1. Nominal linear system. This represents the linear system which would have resulted if no saturation was present. 2. Nonlinear loop. This contains the nonlinear stability problem and assuming that the linear loop is stable and the linear operator G2 M is also stable, the entire nonlinear stability problem is contained within this part of the system 3. Disturbance filter. This part of the system determines how the system recovers after a saturation event has stopped. This part is responsible for both the speed and manner of recovery after saturation has ceased. In [39], it was shown that most anti-windup schemes can be interpreted as certain choices of M (s) and therefore schemes such as the Hanus conditioning scheme ([12]) and the high gain approach ([4, 19]) can be analysed in terms of Figure 4. The advantages of viewing anti-windup in terms of Figure 4 is that the nominal lin-


129

ear performance is separated from the nonlinear part of the scheme. Moreover, the stability of the scheme is dependent on the stability of the nonlinear loop, assuming stability of the nominal linear closed loop and stability of the plant. This leads to our first assumption. Assumption 1 •

The poles of

I

−K2 (s)

−G2 (s)

I

−1 are in the open left-half

complex plane. • The limit lims→∞ (I − K2 (s)G2 (s))−1 exists.

The first point in this assumption ensures that all nominal closed loop transfer functions are asymptotically stable (in RH ∞ ) and if r = 0 and d = 0, then limt→∞ [xp (t) xc (t)] = 0. The second assumption ensures that the nominal linear system is well-posed, i.e. unique solutions exist to the feedback equations. These assumptions will be satisfied in most circumstances. From Figure 4, it can be seen that the performance of the overall closed loop system, including the anti-windup compensator, is closely related to the mapping Tp : ulin → yd . This mapping represents the deviation from nominal linear behaviour in response to a saturation event and can be used as a measure of the anti-windup compensator’s performance. If some appropriate norm of this mapping is small, then the anti-windup compensator is successful at keeping performance close to linear (which we assume is the desired performance). In fact, as the AW compensator is parameterised by M (s), the choice of M (s) dictates the system’s stability properties under saturation. This chapter ultimately proposes different approaches to choosing M (s) such that performance during saturation is improved. In [35] (see also [32, 13]), the L2 gain of Tp was minimised using a system of linear matrix inequalities and, furthermore, M (s) was chosen such that it corresponded to static or low order anti-windup compensators. The result of [35] demonstrated, using suitable examples, that direct minimisation of Tp was central to good anti-windup performance, and compensators designed according to the ideas in [35] seemed to perform at least as well, and often better, than most other anti-windup compensators. This idea was extended to more general cases in [15].

130


3 Global Anti-Windup Results In this section, we treat the case where we are interested in global stability of the closed-loop system and in the global (possibly sub-optimal) minimisation of the L2 gain of the operator Tp . From Figure 4, note that in order to have the widest possible choices of M (s), it would seem sensible to require that G2 (s) ∈ RH ∞ , thus M (s) could be chosen as any M (s) ∈ RH ∞ such that the nonlinear loop is asymptotically stable and well-posed. In fact, this intuition is in agreement with results cited earlier where it was proven that in order to obtain global results we must ensure that G2 (s) ∈ RH ∞ 1 . Our second assumption is thus. Assumption 2 Re(λmax (Ap )) < 0 We say that an anti-windup compensator has solved the anti-windup compensation problem if it is stable, not active until saturation occurs, and if, after saturation has occured, it allows the system to recover, asymptotically, linear behaviour. Formally, this is defined as follows: Definition 1 The anti-windup compensator Θ(s) is said to solve the anti-windup problem if the closed loop system in Figure 4 is internally stable, well-posed and the following hold: 1. If dist(ulin , U) = 0,

∀t ≥ 0, then yd = 0,

∀t ≥ 0 (assuming zero initial

conditions for Θ(s)). 2. If dist(ulin , U) ∈ L2 , then yd ∈ L2 . The anti-windup compensator Θ(s) is said to solve strongly the anti-windup problem if, in addition, the following condition is satisfied: 3. The operator Tp : ulin → yd is well-defined and finite gain L2 stable. Remark 1: In previous papers ([35], [32]), we have also considered more general Lp norms, p ∈ [0, ∞). The results given in this chapter are easily extendable to these cases, although the optimisation is simple with the L2 norm. 1

Recall that we require a finite L2 gain to hold; this is only possible globally if G2 (s) ∈ RH ∞ , although local L2 gains hold for a wider class of systems


131

Note that in our parameterisation of anti-windup compensators, stability of the system in Figure 4 is equivalent to stability of the system in Figures 2 and 3.

3.1 Full Order Compensators An appealing choice of M is as a part of a right comprime factorisation of G2 (s) = N (s)M (s)−1 where M, N ∈ RH ∞ . This means that the disturbance filter is given as yd (s) = N (s)˜ u(s) and is hence a stable transfer function. The central reason for the appeal of this choice of M (s) is that this means that the anti-windup compensator M (s) − I Θ(s) = (11) N (s) has order equal to that of the plant, providing the coprime factors share the same state-space. A state-space realisation of the operator Tp is then easily calculated (using [42], for example) as 2

M (s) − I N (s)

⎡

⎤⎡

⎤

Ap + Bp F Bp x˙ ⎢ 2⎥⎢ ⎥ x2 ⎥⎢ ⎥ ∼⎢ u F 0 d ⎣ ⎦⎣ ⎦ u ˜ yd Cp + Dp F Dp

(12)

where we have used the state x2 ∈ Rnp to represent the anti-windup compensator’s state and to distinguish it from the plant state. Another endearing property of this choice of AW compensator is that there is no direct-feedthrough from u ˜ to ud which prevents algebraic loops. The following is the main result of the section: Theorem 1. Under Assumptions 1 and 2, there exists a full anti-windup compensator Θ(s), described by equations (11) and (12), which solves strongly the anti-windup problem if there exist matrices Q > 0, U = diag(μ1 , . . . , μm ) > 0,L ∈ R(m×n and a postive real scalar γ such that the following LMI is satisfied 2

We could in fact use a more general coprime factorisation as discussed in [36] and [15], although it is not clear that this gives us much advantage over the simple factorisation described above

132


⎡ ⎢ ⎢ ⎢ ⎢ ⎣

QAp + Ap Q + L Bp + Bp L Bp U − L 0 QCp + L Dp

−2U

I

U Dp

−γI

0

−γI

⎤ ⎥ ⎥ ⎥ 0 which satisifies J=

d x2 (t) P x2 (t) + yd 2 − γ2 ulin 2 < 0 ∀[x2 dt

u ˜

ulin ] = 0

(14)

it follows that 1. If ulin = 0, then there exists a P > 0 such that v(x2 ) = x2 P x2 is a Lyapunov function for the system; hence the system is globally stable. 2. Integrating J from 0 to T and taking the limit as T → ∞ y 22 < γ2 ulin 22 + (x2 (0) P x2 (0) − x2 (∞) P x2 (∞))

(15)

which implies L2 gain less than γ from ulin to yd (and thus Item 3). The remainder of this part of the proof involves simplifying the expression for J. First, note that as the deadzone function is such that Dz ∈ Sector[0, I] we have that ˜2i , u ˜ i ui ≥ u

∀i ∈ {1, . . . , m}

(16)

From this, it follows that for some matrix W = diag(w1 , . . . , wm ) > 0 ˜) ≥ 0 u ˜ W (u − u Thus it follows that a sufficient condition for J < 0 is if

(17)


d J˜ = x2 (t) P x2 (t)+ yd 2 −γ2 ulin 2 +2˜ u W (u−˜ u) < 0 ∀[x2 dt

u ˜

133

ulin ] = 0 (18)

where u = ulin − ud . Hence using the state-space realisation (12), this becomes ⎡ ⎢ ⎢ ⎣

(Ap + Bp F ) P + P (Ap + Bp F ) + (Cp + Dp F ) (Cp + Dp F )

P BP + (Cp + Dp F ) Dp − F W −2W +

Dp Dp

⎤ 0

⎥ W ⎥ ⎦ 0, U = diag(μ1 , . . . , μm ) > 0,L ∈ R(m+q)×m and a postive real scalar γ such that the following LMI is satisfied ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

¯ + AQ ¯ QA

⎤ ¯ − QC ¯1 B0 U + BL 0 QC2 ¯ 1 − D ¯ 2 ⎥ ¯ 1 L I U D02 ⎥ −2U − U D01 − D01 U − L D + L D ⎥ 0, L ∈ R(m+q)×m and a positive real scalar γ such that the LMI (27) is satisfied (with the state-space realisation of equation (34)). Further˜ achieving Tp i,2 < γ is given by more, if inequality (27) is statisfied, a suitable Θ ˜ = LU −1 . Θ Proof: The proof is similar, mutatis mutandis, to that of Theorem 2.

As with static AW compensators, there is no guarantee that a low-order compensator will exist for a given plant-controller combination, although there is probably more likelihood of one existing. There is no exact formula for computing a low-order compensator although we have found that one of the following procedures can work well ˆ 2 (s) such that deg(G ˆ 2 (s)) < 1. Compute a reduced order approximation of the plant, G ˆ 2 (s). deg(G2 (s)). Set F2 (s) as G 2. Choose F1 (s) as a low pass filter and set its bandwidth to ensure high frequency anti-windup action is prevented. ˜2 . ˜ 1 and Θ 3. Use Theorem 3 to choose Θ 4. Iterate, using different filter bandwidths and higher order plant approximations. or ˆ 2 (s) such that 1. Compute a reduced order approximation of the plant G ˆ 2 (s)) < deg(G2 (s)). deg(G ˆ −1 (s)N ˆ (s). ˆ 2 (s) = M 2. Form a right comprime factorisation G ˆ (s) − I; set F2 (s) = N ˆ (s). 3. Set F1 (s) = M ˜ 2. ˜ 1 and Θ 4. Use Theorem 3 to choose Θ 5. Iterate using higher order plant approximations and different coprime factorisations Note that the second procedure could also involve the synthesis of F in the same way as with the full-order case.


139

5 Robustness The techniques described in the foregoing sections should allow the synthesis of antiwindup compensators which perform well during saturation; the results, however, do not guarantee good performance in the presence of saturation and plant uncertainty, which is inevitable in practice. Preserving our decoupled and coprime factor based ˜ is the true thinking, we must now consider the situation depicted in Figure 5, where G ˜ plant given by G(s) = [G1 (s) G2 (s) + ΔG (s)], where G(s) = [G1 (s) G2 (s)] is the model of the plant with which we work and ΔG is additive uncertainty to the feedback part4 which is assumed to be stable and linear. Other types of uncertainty ˜ 2 (s) = (I + Δo (s))G2 (s), and such as output-multiplicative uncertainty, where G ˜ 2 (s) = G2 (s)(I + Δi (s)) could be used input multiplicative uncertainty, where G instead. However, it is easy to see that both these uncertainties can be captured by additive uncertainty (ΔG = Δo G2 or ΔG = G2 Δi ), although the converse is not always true (unless G2 is invertible), so we prefer to work with additive uncertainty. We assume that little is known about the arbitrary, linear time invariant uncertainty ΔG (s) other than the fact that it belongs to a certain ball of uncertainty Δ defined as follows

+ , 1 Δ = Δ ∈ RH ∞ : Δ(s) ∞ < γr

(36)

When uncertainty is present in the system, the appealing decoupled structure of the original scheme is lost. Figure 6 shows an equivalent representation of Figure 5 . Note that the mapping from u ˜ to yΔ (i.e. ΔG (s)M (s)) destroys the decoupling and therefore even if the uncertainty linear system is stable, and the nonlinear loop is stable, this does not guarantee stability of the whole closed loop system. It can be observed, that in order to tackle the problem of anti-windup for uncertain systems we have to work under different assumptions to the standard anti-windup problem and solve the problem in a slightly different manner. Thus, Assumption 3 is Assumption 3 The system in Figure 6 is well-posed and 1. G(s) ∈ RH ∞ 4

It is likely that there will also be a perturbation of the disturbance feedforward portion of the plant, G1 , although this will have no bearing on stability, so for simplicity we do not consider it

140

Matthew C. Turner, Guido Herrmann, Ian Postlethwaite d r

~ G

u lin + − ud

K

M−I

y lin + +

y

+ −

yd

G2 M

~ u

Fig. 5. Anti-windup with uncertainty

M−I

Disturbance Filter

ud Nonlinear Loop

− +

~ u

M − +

r

d G K

u lin

G 2M

ΔG ++

+

yΔ ylin

− y d y

Fig. 6. Equivalent representation of Figure 5

2. Δ(s) ∈ Δ 3. The poles of all Δ ∈ Δ.

I −K2 (s) ˜ 2 (s) −G I

−1 are in the open left-half complex plane for

The first point of the assumption is identical to Assumption 2; the second is necessary for us to obtain the small-gain type results we consider here. The third ensures that the linear system itself (i.e. with no saturation) is stable for all relevant Δ(s) essentially this assumes that a K(s) is a suitable robust linear controller. If Assumption 3 is satisfied, four features are evident from Figure 6: 1. If ΔG is small in some sense, then the robustness of the anti-windup scheme is similar to that of the nominal, unconstrained linear system (via a Small Gain argument).


141

2. If the mapping from ulin → M u ˜ is small, again, the robustness of the antiwindup system is similar to that of the nominal linear system (again using a Small gain argument). So in other words the map ulin → M u ˜ contains important robustness information. 3. The robustness of the system with anti-windup compensation can never be better than the robustness of the linear system. Denoting the ‘modified’ uncertainty by ˜ G : ulin → yΔ , this follows by noting that ΔG ∞ = ΔG i,2 ≤ Δ ˜ G i,2 Δ (by using a contradiction argument). So, in a sense, the retention of the linear system’s robustness can be considered as an optimal property (discussed in more detail later). 4. In order to obtain ‘maximal’ robustness for a given Δ, it is desirable that the map from ulin to the input of the uncertainty is as small as possible in some appropriate norm.

Stability Robustness Optimisation As noted above, in order to achieve maximal robustness we would like the map from ulin to the input of the uncertainty to have as small gain as possible. This can be depicted graphically in Figure 7. Thus, we would like to minimise the map Tr i,2 := [I − M F (ulin )] i,2 where F (.) : Rm → Rm denotes the nonlinear map from ulin to u ˜. As mentioned in [35], this optimisation is typically a difficult problem to solve, so instead we seek to ensure a certain L2 gain bound holds for the map Tr .

M−I ud u lin

+

− ~ u

M

uΔ − +

z

Fig. 7. Robustness optimisation for general anti-windup schemes: graphical representation of Tr

142


To make our discussion brief, we only consider full-order anti-windup compensation explicitly; similar (although not as strong) results can be obtained for low-order and static anti-windup compensation. Thus we choose M (s) to be part of a right-coprime factorisation of G2 (s) = N (s)M −1 (s) and we let the anti-windup compensator have the state-space realisation given in equation (12). In this case, it is obvious that the map F has state-space realisation: ⎤ ⎤ ⎡ ⎡ Ap + Bp F Bp x˙ 2 ⎥ x2 ⎥ ⎢ ⎢ M (s) − I ⎥ ⎥=⎢ ∼⎢ u F 0 d ⎣ ⎦ u ⎦ ⎣ M (s) ˜ F uΔ I

(37)

To ensure robustness, that is to ensure that Tr i,2 < γr , it suffices for the following inequality to hold for sufficiently small γr : J=

d x P x2 + z 2 − γ2r ulin 2 < 0, dt 2

∀ [x2 u ˜ ulin ] = 0

(38)

where x2 is the state vector associated with the realisation of [M − I, M ]. As shown in, for example [10, 35] this ensures that the L2 gain from ulin to z is less than γr and that the system in Figure 7 is asymptotically stable. As before, we use the sector bound (17) to get d J˜ = x2 P x2 + z 2 − γ2r ulin 2 + 2˜ u W (ulin − ud − u ˜) dt

(39)

If J˜ < 0, this implies that J < 0. Evaluating J˜ in a similar manner to before (see [33]) yields the following LMI ⎤ ⎡ QAp + Ap Q + L Bp + Bp L Bp U − L 0 L ⎥ ⎢ ⎥ ⎢ −2U I U ⎥ 0, U = W −1 = diag(ν1 , . . . , νm ) > 0, L, μ > 0. Satisfaction of this LMI means that inequality (39) is satisfied and hence that the √ L2 gain from ulin to z is less than γr = μ and a suitable choice of F is given by −μI −I F = LQ−1 . From the term of this LMI we can see that, as anticipated −I earlier, the L2 gain can be no less than unity, which is the robustness of the nominal linear loop without uncertainty.


143

Remark 4: An important example of a robust anti-windup compensator is the much maligned IMC scheme. In [33] (see also [1]) it was proved that the IMC scheme corresponded to an ‘optimally robust’ anti-windup compensator. In the framework considered here, the IMC scheme results from choosing M (s) = I. This implies that Tr (ulin ) = sat(ulin ) (see Figure 7) and thus this choice corresponds to the

recovery of ‘linear’ robustness.

Optimisation for Robustness and Performance The primary goal of anti-windup compensation is to provide performance improvement during saturation, but optimising the LMI (40) alone does not guarantee this. Indeed, there is little point in optimising (40) when an optimal solution can be found by inspection as the IMC anti-windup solution. The real use of (40) and the arguments of the previous subsection is to use them in conjunction with performance optimisation, the goal being to optimise performance and robustness together, although there will often be a trade-off. Earlier, we argued that a logical choice of operator for which we would like to minimise the L2 gain was given by Tp , that is the map from ulin to yd . For good performance and robustness, it therefore follows that we would like to optimise according to min Tp i,2

s.t. Tr i,2 < γr

This is generally difficult so what we shall be content to ensure that W T p p 0, L, γ > 0. Wp and Wr are positive definite weighting matrices which reflect the relative importance of performance and robustness respectively and are chosen by the designer. The derivation of this LMI is carried out in a similar way to that of the previous section in the spririt of that done in [35]. Remark 5: Throughout this section, we have mainly discussed full-order antiwindup compensation for two reasons: (i) A full-order anti-windup compensator always exists, and (ii) the expressions and derivations of formulae for static and low-order anti-windup compensators are more complex, although the same ideas are certainly applicable to these types of compensator.

Remark 6: Another advantage of using LMI (44) to synthesise full-order compensators is that it tends to prevent fast poles appearing in the compensator dynamics. If a robustness weight (Wr ) was not included in the optimisation - or if Wr was only chosen small - the poles of the anti-windup compensator would tend to be rather fast, lying far to the left of the imaginary axis. Obviously this would require a very high sampling frequency for implementation, which is not always possible in practice. However, when simultaneously optimising performance and robustness using (44), the poles are placed in regions more comparable to that of the controller. This feature is reminiscent of solving ‘singular’ H∞ problems with LMI’s, where poles tend to get placed far from the imaginary axis.

Similar results can be obtained for robustness of static and low order compensators although the results are not as strong. As low order and static AW compensators which guarantee L2 gains are not guaranteed to exist, it follows that robust versions of these compensators will not exist either. However, AW compensators which guarantee locally robust performance may well exist, see [33] for more remarks on this.


145

6 Local Anti-windup The previous sections considered the design of global anti-windup compensators, that is AW compensators which ensure global (quadratic) stability of the complete closed-loop system together with certain L2 gain properties. Unfortunately, for many systems, it is impossible to obtain global results. This class of systems includes all anti-windup problems in which the plant is exponentially unstable, as well as static anti-windup problems which are (globally) infeasible. Also, it is often possible to obtain improved local performance by relaxing the restriction of global feasibility. In practice, this is a fairly mild assumption as states and exogenous inputs are normally restricted to some subset of the space in which they are defined. Thus, antiwindup compensators which only guarantee local stability and performance warrant consideration.

6.1 Sharper Sector Bounds Essentially, we use ideas introduced in [17] and [26], where it was observed that, locally, the deadzone nonlinearity inhabits a narrower sector than Sector[0, I]. Consider Figure 8. Dz i(u i)

_β_ i ui

__ ui

Sector(0,α i)

Sector(0,α i) _ ui

_ βi ui

ui

Fig. 8. The deadzone and Sector bounds

For any u, the deadzone nonlinearity remains in the Sector[0, I], that is, the graph of each component remains below the unity gradient line. However, let us suppose that ui never becomes greater than βi u ¯i ,

∀i, where βi > 1,

∀i, then locally we

can see that the graph of Dzi (ui ) remains below the gradient αi :=

βi −1 βi

< 1.

Thus for all u u ¯ it follows that the deadzone actually inherits the ‘sharper’ sector

146


Sector[0, A] where A := diag(α1 , . . . , αm ) and αi :=

βi −1 βi

∈ (0, 1). This can be

seen in Figure 8. Note that A < I. Thus, it follows that if we can ensure that our control signal is such that u β¯ u

∀t ≥ 0 (where β = [β1 β2 . . . βm ] ), then our deadzone can be considered to

be in Sector[0, A] and we can obtain tighter bounds on the small-signal L2 gain and, potentially, prove local stability for systems where global stability is not achievable. Let us define the set V (β) := {u ∈ Rm : u β¯ u} .

(45)

Thus, for small signal L2 gain we need to consider sets W of the following form W = {ulin ∈ Rm : u(t) ∈ V (β) ∀t ≥ 0} .

(46)

Remark 7: W cannot simply be considered as the set W× := ulin ∈ Rm :

ulin β¯ u because u = ulin − ud and hence W and W× will not have the same geometry and may not satisfy any simple subset properties.

Similarly in order to prove local stability we need to consider positively invariant sets, X such that when ulin = 0 for all xnl (0) ∈ X , u β¯ u, where xnl ∈ Rnnl is the state vector of the nonlinear loop. Hence we define X := {xnl (0) ∈ Rnnl : u(t) ∈ V (β) ∀t ≥ 0}

(47)

where xnl (t) is the state of the dynamic system M (s) in the nonlinear loop. This would be x2 in the case of full-order AW, or x ¯ in the case of static or low order AW. It is now appropriate to define the local anti-windup problem Definition 2 The anti-windup compensator Θ(s) is said to solve the A-local anti-windup problem if the closed loop system in Figure 4 satisfies the following properties: 1. It is well-posed. 2. There exists a non-empty compact set Y such that the closed-loop system is internally stable for all xnl (0) ∈ Y ⊂ X 3. If dist(ulin , U) = 0, conditions for Θ(s)).

∀t ≥ 0, then yd = 0,

∀t ≥ 0 (assuming zero initial


147

4. There exists a non-empty set W such that if dist(ulin , U) ∈ L2 ∩ W , then yd ∈ L2 . The anti-windup compensator Θ(s) is said to solve strongly the A-local anti-windup problem if, in addition, the following condition is satisfied: 5. The operator Tp : ulin → yd is well-defined and small signal finite gain L2 stable, that is Tp i,2,W ≤ γ for some γ > 0. The next section demonstrates how these concepts can be used in the synthesis of full-order anti-windup compensators. Remark 8: Trivially, the set Y¯ = {xnl (0) ∈ Rnnl : u(t) ∈ U, ∀ t ≥ 0} ⊂ X is a true subset of {xnl (0) ∈ Rnnl : u(t) ∈ V (β) ∀t ≥ 0} (actually V (1) = U) and the trivial solution for Y is Y = Y¯ in Definition 2. However, it is required that Y is a true superset of Y¯ , i.e. Y ⊂ Y¯ for Definition 2 to be useful.

6.2 Full Order Anti-Windup Synthesis As before, a logical choice for M (s) is to be part of a coprime factorisation of G(s) and thus the state-space realisation of the nonlinear-loop and disturbance filter remains the same as before. However, in contrast to the last section we shall assume that we require local results, that is for systems where the control signal u remains in the set V (β). The main result is the follwing theorem. Theorem 4. Under Assumption 1, there exists a full anti-windup compensator described by equations (11) and (12) which solves strongly the A-local anti-windup problem if there exist matrices Q > 0, U = diag(μ1 , . . . , μm ) > 0, L ∈ R(m+q)×m and a positive real scalar γ such that the following LMI is satisfied ⎤ ⎡ QAp + Ap Q + L Bp + Bp L Bp U − L A 0 QCp + L Dp ⎥ ⎢ ⎥ ⎢ −2U A U D p ⎥ 0 we have ˜) ≥ 0 u ˜ W (Au − u

(50)

This can then be adjoined to equation (49) to get d J˜ = x2 (t) P x2 (t) + yd 2 − γ2 ulin 2 + u ˜ W (Au − u ˜) < 0 ∀[x2 u ˜ ulin ] = 0 dt (51) As in the foregoing proofs, it can be proved that J˜ < 0 if the LMI (48) is satisfied. Noting that J˜ ≥ J, and picking v(x2 ) = x2 P x2 > 0 as a Lyapunov function, local stability and small-signal L2 gain follow if J˜ < 0. This will be discussed next. First, local exponential stability will be shown. With ulin = 0, it follows from the

LMI (48) that there exists a level set L = x2 ∈ Rnp : x2 P x2 ≤ l2 , l > 0, for which

dv dt

≤ −δv for small δ > 0. Hence, local exponential stability follows. This

implies that there is a nontrivial, compact, positively invariant set Y , L ⊆ Y ⊂ X , which ensures local internal stability. Local exponential stability of the anti-windup compensated system is implies, from Theorem 6.5 of [20], that the operator ulin → yd is small-signal finite L2 gain stable. Thus for small enough ulin , it follows that u = ulin − F x2 ∈ V (β) and hence that there exists a non-empty set W such that a small-signal finite L2 gain holds.

Remark 9: Note that if Ap is exponentially unstable, there will never exist a globally stabilising anti-windup compensator; if Ap is critically stable, a compensator achieving a small-signal L2 gain only is guaranteed, although a globally stabilising compensator will exist. Thus, for such cases, it is convenient (and necessary) to consider a locally stabilising AW compensator. Such a compensator is guaranteed to exist for small enough βi > 1.


149

Remark 10: Many authors have, quite logically, taken as a measure of anti-windup compensator performance, the size of the domain of attraction of the system. However, it can be difficult to relate this back to the behaviour of the saturating control signal. Furthermore, such a measure is complicated in the case of dynamic antiwindup compensators and, for state vectors with large dimensions, the analysis of such regions of attraction becomes fraught with further difficulty. Having said this, in principle such an approach is relatively easy to accommodate within our framework. To estimate the region of attraction for our system we follow the approach of [8] and assume that we seek an ellipsoid approximation of the form E = {x2 ∈ Rnp : x2 P x2 ≤ 1}

(52)

Then, in order that our control signal resides in the reduced sector we must have, when ulin = 0, that ¯i |ud,i | = |Fi x2 | ≤ βi u

∀i

(53)

¯2i ⇔ x2 Fi Fi x2 ≤ β2i u

∀i

(54)

Now if x2 ∈ E, the above holds if ¯2i x2 Fi Fi x2 ≤ x2 P x2 β2i u P Fi ⇔ ≥ 0 ∀i β2i u ¯2i I

∀i

Using the congruence transformation diag(P −1 , I), we have the LMI’s: Q Li ≥ 0 ∀i β2i u ¯2i I

(55) (56)

(57)

These can easily be incorporated into the LMI optimisation schemes advocated earlier, perhaps with the aim of minimising the following objective Γ = ηγ + (1 − η)det(Q)

η ∈ [0, 1]

(58)

where η is used to trade of the importance between local performance and region of attraction and, as the volume of an ellipsoid is inversely proportional to det(Q), minimising this quantity helps to maximise the region of attraction.

Remark 11: Even if a globally stabilising AW compensator exists (i.e. for all exponentially stable plants), it may not always be desirable to use this in practice. This is

150


because it may be accompanied by unacceptably large L2 gains which may be indicative of poor performance. In practise though, if we impose a limit on the control signal magnitude and hence consider local results, it is possible to improve significantly the local performance at the expense of abandoning global stability guarantees.

It is possible to guarantee local robustness i.e. to guarantee robust stability within the set W in a similar way to before, if we replace the LMI in Theorem 4 with an LMI similar to that in inequality (44).

6.3 Static and Low-Order Anti-Windup Synthesis The motivation behind static and low-order anti-windup synthesis is similar to that behind full-order AW synthesis: locally stabilising AW compensators may exist when globally stabilising ones do not; and improved local performance can often be obtained at the expense of discarding global stability guarantees. The main results of the section are the following two Theorems. Theorem 5. There exists a static anti-windup compensator Θ = [Θ1 Θ2 ] ∈ R(m+q)×m which solves strongly the A-local anti-windup problem if there exist matrices Q > 0, U = diag(μ1 , . . . , μm ) > 0,L ∈ R(m+q)×m and a postive real scalar γ such that the following LMI is satisfied ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

¯ + AQ ¯ QA

⎤ ¯ − QC ¯1 A B0 U + BL 0 QC2 ¯ 1 A − A D ¯ 2 ⎥ ¯ 1 L A U D02 ⎥ −2U − U D01 A − AD01 U − L D + L D ⎥ 0, U = diag(μ1 , . . . , μm ) > 0, L ∈ R(m+q)×m and a positive real scalar γ such that


151

the LMI (59) is satisfied (with the state-space realisation of equation (34)). Further˜ achieving Tp i,2,W < γ is given more, if inequality (59) is statisfied, a suitable Θ ˜ = LU −1 . by Θ Proofs: These theorems are proved in a similar way to Theorems 2 and 3.

Remark 12: It is important to point out that, for an arbitrary linear closed loop system, there is no guarantee that a static or low order AW compensator yielding global performance exists (see [9]) - even if the plant is asymptotically stable. However, it can easily be shown (see [34]) that for a small enough sector bound, that is for small enough β, there will always exist a locally stabilising AW compensator, with local performance

Remark 13: As before, these performance based AW compensators can be combined with the ideas on robustness from the foregoing section to get compensators which are locally robustly stable.

6.4 Limiting the Control Signal Magnitude The last three theorems essentially provide a compensator which guarantees that the closed loop system will be stable provided the magnitude of the control signal u does not exceed a certain bound and, therefore, stray outside the local sector, Sector[0, A]. As the control signal is a function of the anti-windup compensator output, u = ulin − ud in the full order case, it appears logical to attempt to minimise, in a weighted sense, this signal in the optimisation process. In other words, we would like to minimise the L2 gain of the operator Wp y d ulin → . Wu (ulin − ud ) where Wp ∈ Rp×p is a static weighting matrix used to penalise the performance map and Wu ∈ Rm×m is a static weighting matrix used to penalise the control signal. Thus for the full order case, the following LMI can be used in Theorem 4

152

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣


QAp + Ap Q + L Bp + Bp L Bp U − L A 0 QCp + L Dp

L

−2U

A

U Dp

0

−γI

0

−I

−γWp−1

0

−γWu−1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ B means that A − B is positive definite. A denotes the transpose of A. For any vector u of m one defines each component of satu0 (u) by satu0 (u(i) ) = sign(u(i) )min(u0(i) , |u(i) |), with u0(i) > 0, i = 1, ..., m.

2 Problem Statement The saturated closed-loop system under consideration is described in Fig. 1. The open-loop model is given by: x˙ = Ax + Bu y = saty0 (Cx)

(1)

176

Sophie Tarbouriech, Isabelle Queinnec, and Germain Garcia uc

controller (η)

yc

u

nonlinear actuator

sensor ±y0

system (x)

y

Fig. 1. Saturated closed-loop system

where x ∈ n , u ∈ m and y ∈ p are the state, the input and the measured output vectors, respectively. A, B and C are real constant matrices of appropriate dimensions. Let us consider the controller dynamics in nc : η˙ = Ac η + Bc uc

(2)

yc = Cc η + Dc uc

where η ∈ nc is the state of the controller, yc ∈ m is the output of the controller and uc ∈ p is the input of the controller. Ac , Bc , Cc , Dc are matrices of appropriate dimensions. Such a controller has generally been designed such that the linear closed-loop system resulting from the linear interconnection conditions (3)

u = yc ; uc = Cx is asymptotically stable.

The real interconnections are however obtained through the actuator and sensor elements. The nonlinear actuator scheme is given in Fig. 2. The actuator model is described as follows: v˙ = Tv satv0 (Tyc (yc − satu0 (v))) + Tu (satu0 (v) − v) yc

+

Tyc −

±v0

Tv

+

v˙

u

v ±u0

+

− Tu

Fig. 2. Nonlinear actuator

(4)

+

Anti-Windup Strategy for Systems Subject to Actuator and Sensor Saturations

177

and the real interconnection, for which some windup problems may arise when saturations occur, is: u = satu0 (v) ; uc = y = saty0 (Cx)

(5)

The vectors u0 , v0 and y0 are the positive saturation levels. Hence, from the above description, the complete closed-loop system reads: x˙ = Ax + Bsatu0 (v) v˙ = Tv satv0 (Tyc (yc − satu0 (v))) + Tu (satu0 (v) − v) η˙ = Ac η + Bc saty0 (Cx)

(6)

yc = Cc η + Dc saty0 (Cx) The set of measured or accessible variables of this system is: yc ; u ; y ; v

(7)

i.e. the output of the controller, actuator and sensor, and the internal state of the actuator. By extension, one can consider that the input and output signals at the bounds of the saturation block v0 are known. On the other hand, one cannot consider that the input of the sensor saturation block is accessible. However, a way to use this information is to build a state observer described as: x ˆ˙ = Aˆ x + Bu + L(saty0 (C x ˆ) − saty0 (Cx))

(8)

=x ˆ−x

(9)

with the error

Then, in order to avoid the undesirable effects of the windup, or at least to mitigate them, we want to build static anti-windup loops, which act on the dynamics and possibly on the output of the controller. Thus, consider the addition of terms related to the nonlinearities: φv0 = satv0 (Tyc (yc − satu0 (v))) − Tyc (yc − satu0 (v)) φu0 = satu0 (v) − v φy0 = saty0 (C x ˆ) − C x ˆ = saty0 (Cx + C) − (Cx + C) and denote also: φy0 = saty0 (Cx) − Cx

178

Sophie Tarbouriech, Isabelle Queinnec, and Germain Garcia

Tu yc + − Tyc

±v0 −

Tv

+ + v˙

+ − ±u0

v −

+

u

system

+

±y0

observer

AW block

controller

±y0 −

+

Fig. 3. Saturated closed-loop system with anti-windup strategy

Then, the saturated closed-loop system with anti-windup strategy can be described in Fig. 3, and reads: x˙ = Ax + Bv + Bφu0 v˙ = Tv Tyc (yc − v − φu0 ) + Tv φv0 + Tu φu0 η˙ = Ac η + Bc Cx + Bc φy0 + Ec φv0 + Fc φu0 + Gc φy0

(10)

˙ = A + LC + Lφy0 − Lφy0 with yc = Cc η + Dc Cx + Dc φy0 + Hc φv0 + Jc φu0 + Kc φy0

(11)

Remark 1. It could be possible to consider a simpler observer as follows: x ˆ˙ = Aˆ x + Bu + L(C x ˆ − saty0 (Cx))

(12)

In such a case, from (9) and (12), the part due to in (10) would read: ˙ = (A + LC) − Lφy0

(13)

Remark 2. This system involves nested saturations (φv0 = f (φu0 , φy0 )), but also, if the gain Hc is not set equal to 0, an implicit function (φv0 = f (φv0 )), which increases the complexity of the conditions. We will see in the sequel how to deal with such a complexity.


179

Thus, by defining the extended vector ⎡ ⎤ x ⎢ ⎥ ⎢v⎥ 2n+m+nc ⎥ ξ=⎢ ⎢η⎥ ∈ ⎣ ⎦ and the following matrices ⎤ ⎡ A B 0 0 ⎥ ⎢ ⎢ Tv Tyc Dc C −Tv Tyc Tv Tyc Cc 0 ⎥ ⎥ ∈ (2n+m+nc )×(2n+m+nc ) A=⎢ ⎥ ⎢ B C 0 A 0 ⎦ ⎣ c c 0 0 0 A ⎤ ⎡ ⎤ ⎡ 0 B ⎥ ⎢ ⎥ ⎢ ⎢ Tv ⎥ ⎢ T u − T v T yc ⎥ (2n+m+n )×(m) c ⎥ ⎥ ∈ (2n+m+nc )×(m) Bv0 = ⎢ ; Bu0 = ⎢ ⎥ ⎢ 0 ⎥∈ ⎢ 0 ⎦ ⎣ ⎦ ⎣ 0 0 ⎤ ⎡ ⎡ ⎤ 0 0 (14) ⎥ ⎢ ⎢ ⎥ ⎢ Tv Tyc Dc ⎥ ⎢ ⎥ ⎥ ∈ (2n+m+nc )×(p) ; RL = ⎢ 0 ⎥ ∈ (2n+m+nc )×n By0 = ⎢ ⎥ ⎢ B ⎢0⎥ ⎦ ⎣ ⎣ ⎦ c 1 0 ⎤ ⎡ ⎡ ⎤ 0 0 ⎥ ⎢ ⎢ ⎥ ⎢ T v T yc ⎥ ⎢ ⎥ ⎥ ∈ (2n+m+nc )×n ; R2 = ⎢ 0 ⎥ ∈ (2n+m+nc )×nc R1 = ⎢ ⎢ 0 ⎥ ⎢1⎥ ⎦ ⎣ ⎣ ⎦ 0 0 CL = 0 0 0 C ∈ p×(2n+m+nc ) the closed-loop system reads: ξ˙ = (A + RL LCL )ξ + (Bv0 + R1 Hc + R2 Ec )φv0 + (Bu0 + R1 Jc + R2 Fc )φu0

(15)

+ (By0 − RL L)φy0 + (RL L + R1 Kc + R2 Gc )φy0 The basin of attraction of system (15), denoted Ba , is defined as the set of all ξ ∈ 2n+m+nc such that for ξ(0) = ξ the corresponding trajectory converges asymptotically to the origin. In particular, when the global stability of the system is ensured the basin of attraction corresponds to the whole state space. However, in the general case, the exact characterization of the basin of attraction is not possible.

180


In this case, it is important to obtain estimates of the basin of attraction. For this, consider first the following definition. Definition 1. A set S0 , containing the origin in its interior, is said to be a region of asymptotic stability for the system (15) if for all ξ(0) ∈ S0 the corresponding trajectory converges asymptotically to the origin. From Definition 1, if S0 is a region of stability, then one gets S0 ⊆ Ba . Hence, regions of asymptotic stability can be used to estimate the basin of attraction [15]. Moreover, in some practical applications one can be interested in ensuring the stability for a given set of admissible initial conditions. This set can be viewed as a practical operation region for the system or a region where the states of the system can be steered by the action of temporary disturbances. The problem we want to solve can be summarized as follows. Problem 1. Determine the anti-windup matrices Ec , Fc , Gc , Hc , Jc , Kc and the observer gain L in order to enlarge the basin of attraction of the closed-loop system. Since in general (even for very simple systems), the analytical characterization of the basin of attraction is not possible, the exact solution of Problem 1 is impossible to obtain. The problem can therefore be indirectly addressed by searching for antiwindup gains Ec , Fc , Gc , Hc , Jc , Kc and the observer gain L that lead to a region of asymptotic stability as large as possible. Problem 2. Given a set of admissible initial states X0 , determine the anti-windup gains Ec ∈ nc ×m , Fc ∈ nc ×m , Gc ∈ nc ×p , Hc ∈ m×m , Jc ∈ m×m , Kc ∈ m×p , the observer gain L ∈ n×p and an associated region of asymptotic stability S0 , such that X0 ⊂ S0 . Remark 3. Note that in Problem 2, the convergence of the observer is implicitly guaranteed. Moreover, when the state x of the system is assumed to be accessible, the observation block may be removed and the problem to be solved can then be stated as follows;


181

Problem 3. Given a set of admissible initial states X0 , determine the anti-windup gains Ec ∈ nc ×m , Fc ∈ nc ×m , Gc ∈ nc ×p , Hc ∈ m×m , Jc ∈ m×m , Kc ∈ m×p and an associated region of asymptotic stability S0 , such that X0 ⊂ S0 . By exploiting quadratic Lyapunov functions and ellipsoidal regions of stability, the maximization of the region of stability can be accomplished by using well-known size optimization criteria. On the other hand, when the open-loop system is asymptotically stable, it can be possible to search for anti-windup matrices Ec , Fc , Gc , Hc , Jc , Kc (and observer gain L) in order to guarantee the global asymptotic stability of the closed loop system. In this last case the basin of attraction will be the whole state space.

3 Preliminaries Let us consider the generic nonlinearity ϕ(α) = satα0 (α) − α, ϕ(α) ∈ m and the following set: S(α0 ) = {α ∈ m , ω ∈ m ; −α0 α − ω α0 }

(16)

Lemma 1. [24] If α and ω are elements of S(α0 ) then the nonlinearity ϕ(α) satisfies the following inequality: ϕ(α) S −1 (ϕ(α) + ω) ≤ 0

(17)

for any diagonal positive definite matrix S ∈ m×m . Moreover, in order to treat in a potentially less-conservative framework the possibility of considering structural conditions, the technique developed in the sequel is based upon the use of the Finsler’s Lemma, recalled below [6]. Lemma 2. Consider a vector ζ ∈ n , a symmetric matrix P ∈ n×n and a matrix B ∈ m×n , such that rank(B) < n. The following statements are equivalent: (i) ζ P ζ < 0, ∀ζ such that Bζ = 0, ζ = 0. (ii)(B ⊥ ) PB ⊥ < 0. (iii)∃μ ∈ : P − μB B < 0. (iv)∃F ∈ n×m : P + FB + B F < 0.

182


4 Theoretical Anti-Windup Conditions 4.1 Quadratic Anti-Windup Conditions First denote the following: Ku = 0 1 0 0 ; Ky = C 0 0 0 K = C 0 0 C ; Kv = Tyc Dc C −Tyc Tyc Cc 0 The following proposition is stated based upon the use of Lemma 1. Proposition 1. If there exist a symmetric positive definite matrix W , matrices Yu , Yv , Yy , Y , ZE , ZF , ZG , ZH , ZJ , ZK , Xu , Xv , Xy , X , L, and four diagonal matrices Su , Sv , Sy , S , all of appropriate dimensions, satisfying1 : ⎤ ⎡ W (A + RL LCL ) + (A + RL LCL )W ⎥ ⎢ ⎥ ⎢ −2Su ⎥ ZF R2 + ZJ R1 + Su Bu0 − Yu ⎢ ⎢ ⎥ ⎢ ⎥ −2Sv ⎢ ⎥ ⎢ −Xu ⎥ 1, another transform T2∗ 0 T2 = (36) 0 I can be constructed such that T2∗ A11 T2∗−1 =

As11 A∗12

0 Au11

(37)

with Au11 strictly unstable and the Eigenvalues of As11 inside or on the unit disc. By applying T2 , the decomposition (33) is finally obtained.

Theorem 3 (Stabilisable Set for Partially Constrained Systems). Consider the system (33) subject to (34) and (35). Assume (A, B) stabilisable. Let KNus be a stabilisable set for the subsystem (A22 , B22 ) subject to Hz z2 (k) Kz and Hu u(k) Ku . Then, KNs = ns × KNus is a stabilisable set for the complete system.

250

Urban Maeder, Raphael Cagienard, and Manfred Morari

Proof. The proof is performed by constructing a control sequence which drives the state to the origin. By definition, for every z = [z1 z2 ] ∈ K∞ , there exists an admissible input sequence which drives z2 to the origin. We therefore consider an arbitrary state z(0) = [z1 (0) 0] and show how to drive the state to the origin from z(0). For the unconstrained (both state and input) case, we consider the class of input sequences U = [u(0) , . . . u(N − 1) ] with Horizon N = n, such that z(N ) = 0 (i.e. terminal state is the origin) with z(N ) = AN z(0) + [AN −1 B, . . . , B]U . Such a sequence exists due to controllability of the system and it is given by U = −[AN −1 B, . . . , B]−1 AN z(0). For the constrained case, we note that the sequence U0 = [0, . . . , 0] is strictly feasible and the state evolves strictly inside the feasible set because the feasible sets X and U both contain the origin in their interior. Consider the sequence αU with 1 ≥ α ≥ 0. The sequence is an interpolation between U0 and U . By convexity, there exists an α > 0 such that αU is a feasible sequence, both for state- and input constraints. Applying this sequence, z(N ) = AN z(0) + α[AN −1 B, . . . , B]U . Insertion yields z(N ) = (1 − α)AN z(0) and hence z1 (N ) = (1 − α)AN 11 z1 (0) for the upper subsystem, since z2 (0) = z2 (N ) = 0. By repeatedly choosing an αi for every N 2k steps, the evolution of the state thus is z1 (kN ) = AkN 11 i=1 (1 − αi ), and hence z1 (kN ) → 0 as k → ∞. Remark 3 (Stability Margin). For unstable systems, the border of the maximum control-invariant set is not necessarily stabilisable. The nearer the state is to the border, the more of the available control authority is spent on merely keeping the state inside the set, but no authority remains for regulating the system away from the border. The system, though never leaving the set, might have very poor performance. Therefore, it is sensible to employ a stability margin for unstable systems, to avoid stickiness. If KNs is computed by the procedure (9), the margin can be increased by selecting a lower Ns , since KNs ⊆ KNs +1 .

Application to the Pendulum For the benchmark system (20), the similarity transform z = T x ˜ with

Explicit Model Predictive Control

⎡

⎤

−0.787 −0.599 0.122

0

−0.358 0.843

0

⎢ ⎢ 0.394 ⎢ ⎢ T = ⎢ −0.062 ⎢ ⎢ −0.350 ⎣ 0.316

251

0.086

⎥ −0.080 ⎥ ⎥ ⎥ 0.261 0.229 0 0.936 ⎥ ⎥ 0.495 0.350 0.670 −0.247 ⎥ ⎦ −0.447 −0.316 0.742 0.223

(38)

leads to decomposition (33) with nu = 2, which means the dimension of the set computation is reduced from 5 to 2. As pointed out above, the choice of the number of iterations defines the margin of control authority; the number of iterations Ns = 25 was chosen by trial. In Figure 1, cuts of the invariant set through x(1) = x(3) = u = 0 and x(2) = d = u = 0 are shown. It has 52 facets and is unbounded, containing the set of equilibrium states (18). Most of the facets only appear in the

1.5

1.5

1

1

0.5

0.5

x(3)

d

u/d cut, which is not very interesting otherwise and is thus not reproduced here.

0

0

−0.5

−0.5

−1

−1

−1.5 −1.5

−1

−0.5

0

x(2)

0.5

1

1.5

−1.5 −1.5

−1

−0.5

0

x(1)

0.5

1

1.5

Fig. 1. K25 , cuts with x(1) = x(3) = u = 0 (left) and x(2) = d = u = 0 (right)

2.8 Controller For the pendulum, the N-step MPC scheme (10) is applied Np −1 ∗ JN (˜ x0 ) p

s.t.

=

min

Δu0 ,...,ΔuNp −1

Ñ x x Ñp Q ˜ p Np

+

˜ xi + Δui RΔui ) (˜ xi Q˜

i=0

˜xi + BΔu ˜ i x ˜i+1 = A˜ ˜i ∈ KNs , ∀ i ∈ {1, . . . , Np }. |Δui−1 | < 2Ts , x

(39)

252


The cost function is constructed as in (27) with the weights

Q = diag [1 1 1 0.1 0] ,

R = 0.1,

(40)

yielding ⎤

⎡ 1 0 0 0

⎢ ⎢0 1 ⎢ ⎢ ˜ Q = ⎢0 0 ⎢ ⎢0 0 ⎣ 0 -1

0

⎥ 0 0 -1 ⎥ ⎥ ⎥ 1 0 0 ⎥. ⎥ 0 0.1 0.1 ⎥ ⎦ 0 0.1 1.1

(41)

The weights are chosen by trial to obtain satisfying large-signal (constrained) behaviour, i.e. when the benchmark test sequences are applied. The small-signal bandwidth requirement is separately treated below. ˜ N , the algebraic Riccati equation is For the computation of the terminal weight Q p solved iteratively, because the plant is technically not controllable, which results in some standard solvers failing. ˜ ˜ PN B) ˜ −1 (B ˜ + Q, ˜ ˜ PN A) PN +1 = A˜ PN A˜ − (A˜ PN B)(R +B

(42)

˜ Iteration (42) is applied until PN converges. The value of PN is then with P1 = Q. applied as terminal weight. The optimisation problem (39) is solved for Np = 25, employing multi-parametric quadratic programming. The piece-wise affine control law is obtained ˜ x(k)). Δu(k) = K(˜

(43)

˜ consists of 55 polyhedrons; cuts through x(1) = x(3) = u = 0 The partition of K(·) and x(2) = d = u = 0 are shown in Figure 2.

Bandwidth The controller (30) is tuned for robust large-signal behaviour, it does not meet the bandwidth requirement in the linear range. In order to improve this behaviour, a fast local controller is created which overrides the global controller near the origin.

1.5

1.5

1

1

0.5

0.5

x(3)

d


0

0

−0.5

−0.5

−1

−1

−1.5 −1.5

−1

−0.5

0

x(2)

0.5

1

1.5

253

−1.5 −1.5

−1

−0.5

0

x(1)

0.5

1

1.5

Fig. 2. Controller partition, cuts through x(1) = x(3) = u = 0 (left) and x(2) = d = u = 0 (right)

First, a linear state feedback controller is computed as in (42). The cost function is constructed with a strong emphasis on the pendulum velocity Q = diag [50 1 1 0.1 0] < big),

R = 0.1.

(44)

For the resulting linear controller, ˜ + R) ˜ −1 (B ˜ ˜ PN B ˜ PN A), Kloc = (B

(45)

the maximum admissible set Xloc is computed as in (8). The local controller is then employed when the state is in Xloc , whereas the global controller is active otherwise. Since the system will stay within Xloc once it enters it, there are no switching issues; the system will be stable if the global controller stabilises it since the state eventually enters the linear region. This controller achieves a tracking magnitude of −2.6db at 2rad/s, thus fulfilling the requirement.

2.9 Stability Analysis Since the explicit controller computed above does not guarantee asymptotic stability by design, it has to be checked a posteriori. In [17] and [12], it is argued that a system with saturating input cannot be globally exponentially stabilised if it has poles on or outside the unit circle.

254


For the pendulum, similar arguments apply. Asymptotic stability can be achieved by definition for the maximum stabilisable set K∞ . Since in this particular case, the set is unbounded in the direction of an integrator mode, exponential stability can not be achieved over it. Clearly, the analysis algorithm in Section 1.4 can therefore not be applied directly. The following algorithm is devised to remove this limitation by choosing a subset of the state space where exponential stability is attained. Algorithm 5 (Local Stability Analysis of Unbounded Partitions) -R Assume an invariant partition XN = r Pr 1. Select large μ > 0. n 2. Calculate new partition {Pr }R r=1 such that Pr = Pr ∩ {x ∈ ; |x|∞ ≤ μ}.

Store indices of affected regions I = {i ∈ {1, . . . , R}; Pi = Pi } 3. Find Lyapunov function as in (12) subject to (13) on partition {Pr }R r=1 . 4. If I = {}, then V = ∞. Break. 5. Solve V =

min

x∈Pi , i∈I

P W Qi (x),

s.t. |x|∞ = μ.

(46)

In Step 2, the partition is bounded by a hypercube. The parameter μ should be chosen such that only unbounded regions are affected. In step 5, the smallest level of the Lyapunov function is determined for which there is a x that leaves the hypercube. Solving for V in Step 5 amounts to solving 2n QPs for every region in I . The complexity of the algorithm is negligible compared to the complexity of finding the Lyapunov funtion in Step 3. Theorem 4 (Stability). If a Lyapunov function is found for the PWA system (11) by Algorithm 5, then the system is exponentially stable for x = 0 in the region XN = XN ∩ {x ∈ n ; P W Q(x) ≤ V }. Proof. By intersecting the partition with a hypercube of size μ, its invariance property is lost. Invariance is re-established by Step 4 after the Lyapunov function is computed. Trajectories can only exit the partition through μ-hyperplanes due to the invariance of the original partition {Pr }R r=1 . Since V is the smallest value for which


255

P W Q(x) touches such a μ-hyperplane, and since P W Q(x) < V , ∀x ∈ XN and P W Q(fP W A (x)) − P W Q(x) < 0, x ∈ XN , trajectories starting in XN cannot leave it, hence XN is invariant. Therefore, Theorem 2 applies.

Application to the Inverted Pendulum Stability analysis was performed for the case without disturbance. Not every combination of the number of iterations for computing the invariant set Ns , and the prediction horizon Np actually yield a stabilising controller. Moreover, because of the conservatism discussed in Remark 1, it is not possible to find a Lyapunov function for every stabilising controller. By selecting a larger Ns – and thus a larger set KNs – the system is allowed to move closer to the stability border, where it might get stuck [16]. In such a case, the prediction horizon may be increased appropriately, which increases the likelihood of the controller to be stable – at the cost of greater complexity in terms of number of regions. Table 1 shows the minimum prediction horizon Np necessary to obtain a stabilising controller for various Ns . Lyapunov analysis was restricted to horizons no larger than 10 due to computational complexity. Additionally, stability was checked by simulating trajectories for a series of initial conditions. The starting points where generated by taking some extreme vertices of the controller regions, and by additionally limiting |x0 |∞ < 10. If convergence was encountered, the system was considered ‘trajectory stable’. It is interesting to note that for some choices of parameters, the system is stable in simulation, but a Lyapunov function cannot be found. For example, choosing Np = 1, Lyapunov stability can be proved for up to Ns = 10. On the other hand, Ns = 10 leads to infeasibility when the disturbance benchmark trajectory is applied, since the controller does not react decisively enough. A larger Ns is required. The pair Ns = 25 and Np = 1 was found to yield satisfying behaviour. However, a Lyapunov function could not be obtained for this controller. Table 2 shows the number of regions resulting from varying prediction horizons. For larger prediction horizons (Np > 8), controller computation takes several hours, the Lyapunov stability check days to complete, which seriously limits application of these controllers.

256

Urban Maeder, Raphael Cagienard, and Manfred Morari Table 1. Minimum prediction horizon for stability Number of

Min. Pred.

Iterations for

Horizon Np for

Invariant Set,

Lyapunov

Ns

Stability

5

1

1

10

1

1

15

5

1

25

> 10

1

40

> 10

3

Min. Pred. Horizon Np for Stable Trajectory

Table 2. Prediction horizon vs. number of regions for Ns = 25 Prediction

Number of

Horizon Np

Regions

1

55

2

171

3

471

5

1509

7

3432

9

6977

10

9585

To sum up, the quick growth in complexity for finding the Lyapunov function, and the discrepancy between stable simulation behaviour, but unprovable stability limit the potential of the method in practice. One often is interested in finding the least complex controller satisfying the design specifications.

2.10 Simulation Results In Figures 3 and 4, the benchmark reference and disturbance trajectories are applied. The parameters for computing the controller were Ns = 25 and Np = 1. It comprises 55 regions.


257

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 0

5

10

15

20

25

30

35

40

45

50

Fig. 3. System response to velocity reference test sequence. y(t) (-), r(t) (--), u(t) (-.),

d dt

u(t)

(..). Controller with Ns = 25, Np = 1; 55 regions.

1.5

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

25

30

35

40

45

50

Fig. 4. System response to disturbance test sequence. y(t) (-), d(t) (--), u(t) (-.),

d dt

u(t) (..).

Controller with Ns = 25, Np = 1; 55 regions.

In Figure 5, trajectories are shown for varying Ns , while keeping Np = 1. Increasing Ns results in quicker control action. On the other hand, the controller eventually enters limit cycles, e.g. for Ns = 40.

258


4.5

1

4 0.8

3.5 0.6

y(t)

y(t)

3

2.5

2

0.4

0.2

1.5 0

1 −0.2

0.5

0

0

2

4

6

8

10

12

14

16

18

20

−0.4

0

2

4

6

8

10

12

14

16

Fig. 5. Step responses for varying Ns , and Np = 1 for all trajectories. Left reference, right disturbance. Ns = 10, 29 regions (-), Ns = 25, 55 regions (-.), Ns = 40, 55 regions (--).

2.11 Conclusion The concepts of control-invariant sets and MPC have been applied to the benchmark problem. An extension to the usual techniques for computing control-invariant sets has been shown to be efficient for partially constrained systems. The explicit solution to the N-step low-complexity MPC scheme has been computed and shown to work well for the reference and disturbance trajectories. The main features of this type of controller are its simplicity and efficiency (less than 100 regions to be evaluated at runtime) and its large feasible region.

2.12 Relation to Furuta Pendulum The Furuta pendulum [6] is a particular and popular instance of the inverted pendulum, which was originally developed for educational purposes, but has received quite some interest by researchers. In the following, we will apply the method used for the benchmark problem to the linearised model; due to the similarity to the benchmark pendulum, this is a straightforward task. The results obtained are compared to those in [1]. The system equations of the Furuta pendulum are ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ˙θ 0 θ 010 ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ θ¨ ⎥ = ⎢ α 0 0 ⎥ ⎢ θ˙ ⎥ + ⎢ β ⎥ u ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ δ φ˙ γ 00 φ¨

(47)


259

with α = 31.3, β = −71.2, γ = −0.5888 and δ = 191 and the input constraint |u| ≤ 0.3.

(48)

√ The system’s eigenvalues are λ1 = 0, λ2,3 = ± α. Like the benchmark example, the Furuta pendulum has an integrator mode and a stable/unstable pair of real poles. The same algorithm used for the benchmark is applied to the Furuta pendulum. Computation of the invariant set is reduced to dimension 1, since no slew rate constraints are present. The resulting set has two facets; it is identical to the set obtained by adhoc methods in [1]. The resulting controller partition comprises 5 regions. Closedloop behaviour under reference tracking is shown in Figure 6. It is essentially identical to the plots in [1]. The strength of our method lies in its generality, since it can be applied both to higher dimensional systems and also when additional state constraints are present.

0.6 0.4 0.2 0 −0.2 −0.4 0

5

10

15

20

0

5

10

15

20

0

5

10

15

20

20 10 0 −10 −20

0.3 0.2 0.1 0 −0.1 −0.2 −0.3

˙ (..); middle: φ(t) ˙ Fig. 6. Step responses of Furuta pendulum. Top: θ(t) (-), θ(t) (-), r(t) (..); bottom: u(t)

260


3 Continuous Stirred Tank Reactor (CSTR) with a strong exothermal reaction 3.1 Modelling the CSTR The nonlinear, continuous-time model of the plant is transformed into a description, which consists of two linear, time-invariant, discrete-time models so that our techniques can be applied. When the plant is heated up and cooled down, no reactant is inside the vessel, i.e. the reactant concentration x1 as well as the reactant feed inflow u1 are zero. The concentration dynamics can be omitted and the nonlinearity f vanishes because x1 = 0. Therefore, for x1 = 0, u1 = 0, a model of the plant with only 2 states and 1 input −1 1 0 xh (t) + uh (t), (49) x˙ h (t) = 5 −10 5 is considered, where xh (t) = [x2 (t) x3 (t)] and uh (t) = u2 (t). After heating up the plant to the nominal temperature, reactant is fed into the vessel and the exothermal reaction starts. The nonlinearity cannot be omitted. We linearize the equations around the nominal operating point x ¯p = [1 1 0] , u ¯p = [1 −1] . The content temperature is assumed to stay above the ignition temperature, i.e. x2 > x2s , as long as reactant is contained inside the vessel. For x1 = 0, x2 > x2s the linearized model with 3 states and 2 inputs is given by ⎤ ⎡ ⎡ −5 −50 0 5 ⎥ ⎢ ⎢ ⎢ x˙ p (t) = ⎢ 1 ⎥ ⎦ xp (t) + ⎣0 ⎣1 9 5 −10

0

⎤ 0 ⎥ 0⎥ ⎦ up (t), 05

(50)

¯p , u ¯p , i.e. xp (t) = where xp (t), up (t) denote the deviations from the operating point x [x1 (t) x2 (t) x3 (t)] − x ¯p and up (t) = [u1 (t) u2 (t)] − u ¯p . In the following we will refer to the areas of operation assigned to the two different models of the plant as Modes, i.e. Mode 1:

x1 = 0, u1 = 0 : model (49),

(51a)

Mode 2:

x1 = 0, x2 > x2s : model (50).

(51b)


261

We will consider the more stringent slew rate constraints on the input u2 as given in case b) of the problem formulation, i.e. |du2 (t)/dt| ≤ 5s−1 . The continuous time models are converted to discrete-time models with sampling time Ts = 0.04s and a zero-order hold. The sampling period is chosen 10 times faster than the required closed loop bandwidth for the temperature control, which is specified at 15 rad/s. The model for Mode 1 in the discrete-time state space description is given as xh (k + 1) =

0.9642 0.0323 0.1616 0.6734 /0 1 . Ah

xh (k) +

0.0035 0.1651 . /0 1

uh (k),

(52)

Bh

where xh (k) = [x2 (k) x3 (k)] and uh (k) = u2 (k). The discrete-time model for Mode 2 is

⎤ ⎤ ⎡ ⎡ 0.7792 −2.1688 −0.0372 0.1786 −0.0025 ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ xp (k+1) = ⎣0.0434 1.3901 0.0397 ⎦ xp (k)+⎣0.0042 0.0040 ⎥ ⎦ up (k), (53) 0.0037 0.1983 0.6738 0.0003 0.1651 . . /0 1 /0 1 Ap

Bp

¯p and up (k) = [u1 (k) u2 (k)] − u ¯p . where xp (k) = [x1 (k) x2 (k) x3 (k)] − x

3.2 Prediction Model The models derived in the previous section are augmented to be able to cope with slew rate constraints on inputs, to apply time varying reference tracking and to implement control with integral action.

Mode 1 To account for slew rate constraints on the cooling/heating fluid entry temperature, the input u2 is formulated in the form u2 (k) = u2 (k − 1) + Δu2 (k) and the rate of change Δu2 (k) is considered as the new input to the system. Model (52) is augmented by the state u2 (k − 1) to keep track of the previous input, i.e. u2 (k) = u2 (k − 1) + Δu2 (k).

(54)

262


Time varying reference tracking is implemented by augmenting the prediction model with an additional state for each state to be tracked. For Mode 1 the content temperature x2 has to be steered to the reference value r2 . The additional state r2 is kept constant during the predicted behavior of the plant: (55)

r2 (k + 1) = r2 (k). Thus the augmented state vector x ˜h (k) is x ˜h (k) = [x2 (k) x3 (k) u2 (k − 1) r2 (k)] , and the update equations for the prediction model are given by ⎞ ⎛ ⎛ ⎞ Ah Bh 0 B ⎟ ⎜ ⎜ h⎟ ⎟ ˜h (k) + ⎜ 1 ⎟ u x ˜h (k + 1) = ⎜ ⎝ 0 1 0⎠ x ⎝ ⎠ ˜h (k), .

0

0 1 /0 1 ˜h A

(56)

(57)

0 . /0 1 ˜h B

where u ˜h (k) = Δu2 (k) and 0 is the null matrix of appropriate dimensions.

Mode 2 As for Mode 1, input u2 is formulated in Δu-form to account for slew rate constraints: u2 (k) = u2 (k − 1) + Δu2 (k).

(58a)

There are no constraints on the rate of change of the reactant feed inflow u1 . Additionally to the content temperature x2 , in Mode 2 the reactant concentration x1 is steered to the reference r1 , i.e. the plant (53) is augmented by the two reference states r1 , r2 : r1 (k + 1) = r1 (k),

(59a)

r2 (k + 1) = r2 (k).

(59b)

The content temperature x2 is required to be tracked with zero steady state error, i.e. e2 = r2 − x2 → 0, this is considered by including an integrator state i2 in the problem formulation:


i2 (k + 1) = i2 (k) + (r2 (k) − x2 (k)).

263

(60)

The augmented state vector xp (k) is referred to the nominal operating point x ¯p = [1 1 0 −1 1 1 0]T , i.e. x ˜p (k) = [x1 (k) x2 (k) x3 (k) u2 (k − 1) r1 (k) r2 (k) i2 (k)]T − x ¯p . The update equations for the linearized prediction model around x ¯2 are ⎞ ⎛ ⎛ ⎞ Ap Bp2 0 0 Bp ⎟ ⎜ ⎜ ⎟ ⎜ 0 1 0 0⎟ ⎜Π1 ⎟ ⎟ x ⎜ ⎟u ˜ (k) + x ˜p (k + 1) = ⎜ p ⎜ 0 0 I 0⎟ ⎜ 0 ⎟ ˜p (k) ⎝ ⎝ ⎠ 2 ⎠ Π2 0 Π3 1 0 . /0 1 . /0 1 ˜p A

(61)

(62)

˜p B

where u ˜p (k) = [u1 (k) Δu2 (k)] − u ¯p and u ¯p = [1 0] , Bp2 is the second column of Bp , i.e. Bp2 = Bp ·[0 1] , Π1 = [0 1], Π2 = [0 −1 0] and Π3 = [0 1]. I2 is the identity matrix of dimensions 2 × 2 and 0 is the null matrix of appropriate dimensions.

3.3 Cost function We first formulate quadratic costs l1 (˜ xh , u ˜h ) and l2 (˜ xp , u ˜p ) for Mode 1 and for Mode 2 for one single time step, respectively. Summation over the prediction horizon Np yields the cost functions J1,Np (˜ xh (k)) and J2,Np (˜ xp (k)) to be minimized in the optimization problem. For Mode 1, the state x2 is steered to the reference value r2 , which is realized by penalizing (x2 −r2 ) with the weight Qh1 . The input u1 required at the given reference point is not known in advance and different from zero. Therefore, in the Δu(k)formulation the previous value u2 (k − 1) is not penalized, but only its deviation Δu2 (k − 1). Furthermore, we do not penalize the jacket temperature x3 , which will allow it to take a value yielding a steady state point corresponding to the reference r2 . The quadratic cost l1 (˜ xh , u ˜h ) for one single time step is l1 (˜ xh , u ˜h ) = (x2 − r2 ) Qh1 (x2 − r2 ) + Δu2 Rh1 Δu2 .

(63)

Summing up the cost l1 (˜ xh , u ˜h ) over each single time step i in the prediction horizon N and appending a terminal cost QhNp = Qh1 yields the cost function J1,N (˜ xh (k)) for Mode 1:

264

Urban Maeder, Raphael Cagienard, and Manfred Morari Np −1

J1,Np (˜ xh (k)) :=

! l1 (˜ xh,i , u ˜h,i ) + x ˜h,Np QhNp x ˜h,Np ,

(64)

i=0

where x ˜h,i denotes the predicted state at time k + i obtained by starting from the measured state x ˜h,0 = x ˜h (k) at time k and by applying to the system (57) the input sequence {˜ uh,0 , . . . , u ˜h,i−1 }. In Mode 2, beside the state x2 , the state x1 is steered to the reference value r1 , i.e. (x1 − r1 ) and (x2 − r2 ) are weighted with Qp1 and Qp2 , respectively. Penalizing x3 is omitted. The inputs required at the steady state are not known in advance, therefore the state u2 (k − 1) and the input u1 are not penalized. Note that for the weight Rp1 belonging to u1 a small value has to be chosen instead of zero to fulfill Rp > 0 (see (4e)). The integrator state i2 is weighted with Qp3 . The quadratic cost l2 (˜ xp , u ˜p ) for one single time step is given by ⎡ ⎤ ⎡ ⎤⎡ ⎤ (x1 − r1 ) (x1 − r1 ) Qp1 0 0 ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ xp , u ˜p ) = ⎢ l2 (˜ (x 0 Q (x − r ) 0 − r ) 2 ⎦ ⎣ p2 2 ⎦ ⎣ 2 ⎦⎣ 2 i2 0 0 Qp3 i2 u1 Rp1 0 u1 + . 0 Rp2 Δu2 Δu2

(65a)

(65b)

xp (k)) is and the resulting cost function J2,Np (˜ Np −1

xp (k)) := J2,Np (˜

! l2 (˜ xp,i , u ˜p,i ) + x ˜p,Np QpNp x ˜p,Np .

(66)

i=0

3.4 Constrained Optimal Control Formulation Given the prediction model and the cost function we formulate an optimal control problem, where the cost function is minimized subject to physical and design constraints. Physical constraints are given on the magnitude and the rate of change of input u2 . The prediction model is considered as a design constraint, which defines the state update equations for the predicted behavior of the plant. For Mode 1, we formulate the optimization problem

Explicit Model Predictive Control ∗ J1,N (˜ xh (k)) := p

min

u ˜h,i ,...,˜ uh,Np −1

(67a)

J1,Np (˜ xh (k))

subj. to − 1.1 ≤ x ˜h(3),i ≤ 1.1, − 0.5 ≤ u ˜h,i ≤ 0.5,

265

i = 0, . . . , Np ,

(67b)

i = 0, . . . , Np − 1,

(67c)

˜h u ˜h,i + B ˜h,i , x ˜h,i+1 = A˜h x

x ˜h,0 = x ˜h (k),

(67d)

˜h,i . For Mode 2, the where x ˜h(3),i denotes the third element of the column vector x physical constraint that reactant can only be fed into the plant and not extracted has to be considered additionally in the optimization problem: ∗ J2,N (˜ xp (k)) := p

min

u ˜p,i ,...,˜ up,Np −1

subj. to − 1.1 ≤ x ˜p(4),i ≤ 1.1, 0≤u ˜p(1),i

(68a)

J2,Np (˜ xp (k)) i = 0, . . . , Np ,

i = 0, . . . , Np − 1,

− 0.5 ≤ u ˜p(2),i ≤ 0.5,

i = 0, . . . , Np − 1,

˜p u ˜p,i + B ˜p,i , x ˜p,i+1 = A˜p x

x ˜p,0 = x ˜p (k).

(68b) (68c) (68d) (68e)

Although problems (67) and (68) look similar, problem (68) is more difficult to solve than (67) because problem (68) comprises 7 states and 2 inputs compared to (67) with 4 states and 1 input. Note that in this formulation no terminal set with an appropriate terminal weight is included as in problem (4). Thus feasibility and stability are not guaranteed.

3.5 Explicit Feedback Control Law As described in Section 1.2, problems (67) and (68) can be considered as multiparametric Quadratic Programs and solved for all x ˜h (k), x ˜p (k) within a polyhedral set of values. Because the state space in (67), (68) is only partially constrained, the exploration of the solution is restricted to the space where the plant is operating. For example, the content temperature x2 has to be steered to the reference values r2 = 1 and r2 = 0.98, therefore the solution is derived in a neighborhood of x2 = 1, i.e. for 0.9 ≤ x2 ≤ 1.1. Otherwise, the controller complexity is increased unnecessarily. The parameters to be tuned are the prediction horizon Np , the weights Qh1 , Rh1 for Mode 1 and Qp1 , Qp2 , Qp3 , Rh2 for Mode 2. As mentioned in section 3.3, Rp1 is set to 1e−6 to fulfill Rp > 0, and not considered as tuning parameter. The parameter

266


values are a trade off between controller performance and its complexity in terms of number of regions resulting from the state feedback solution. The closed loop bandwidth for the temperature control in the linear range is designed to be 15 rad/s. The explicit state feedback solution for the optimal control problems (67), (68) are derived for the following sets of parameters Mode 1 : Np = 2, Qh1 = 10, Rh1 = 1,

(69a)

Mode 2 : Np = 2, Qp1 = Qp2 = 7.5, Qp3 = 3, Rh1 = 1.

(69b)

Table 3 summarizes the controllers complexity. Table 3. Number of regions of controllers. controller states inputs number of regions R Mode 1

4

1

14

Mode 2

7

2

82

3.6 Stability Analysis In Section 3.4 conditions resulting in a controller with feasibility and stability guaranties were omitted on purpose in order to obtain a controller of low complexity. Because the explicit solution is at our disposal, these properties can be analyzed a posteriori. We will investigate the state feedback solution around the nominal operating point, i.e. the Mode 2 controller and fixed reference values r1 = 1, r2 = 1. Reference tracking is omitted, and the reduced state space consists of x ˘p (k) = [x1 (k) x2 (k) x3 (k) u2 (k − 1) i2 (k)] − x ¯p and u ˘p (k) = [u1 (k) Δu2 (k)] − u ¯p , where x ¯p = [1 1 0 −1 0] and u ¯p = [1 0] . Our aim is to show feasibility and stability of the origin, which corresponds to the nominal operating point due to the coordinate transformation. The explicit solution is derived in the reduced state space x ˘p (k), u ˘p (k) with the parameters (69b). The resulting controller is identical to the controller in the previous section and r1 = 1, r2 = 1. The state feedback solution is used to close the loop to an autonomous PWA dynamic system of the form

1.1

1.08

1.08

1.06

1.06

1.04

1.04

1.02

1.02 2

1.1

1

x

x

2


1

0.98

0.98

0.96

0.96

0.94

0.94

0.92

0.92

0.9

0

0.1

0.2

0.3

0.4

0.5

0.6 x

0.7

0.8

0.9

1

(a) Cut x3 = 1, u2 = 1, i2 = 0.

1

1.1

267

0.9

0

0.1

0.2

0.3

0.4

0.5

0.6 x

0.7

0.8

0.9

1

1.1

1

(b) Cut x3 = 0, u2 = −1, i2 = 0.

Fig. 7. Polyhedral partition of the invariant set XI intersected with the x3 -, u2 - and i2 -planes: (a) plant heated up at nominal temperature without reactant inside, (b) nominal operating point.

˘r , if x x ˘2 (k + 1) = A˘r x ˘2 (k) + B ˘2 (k) ∈ Pr ,

r = 1, . . . , R,

(70)

˘p f˘r , B ˘r = B ˘p g˘r and f˘r , g˘r define the affine control law on the where A˘r = A˘p + B feasible set of states XN . We first identify the invariant subset XI of XN , since positive invariance of the set XN is not fulfilled by design, and afterwards search for a PWQ Lyapunov function on XI . Then, Theorem 2 states that the origin of the closed loop system (70) is exponentially stable on XI . In [8] an algorithm is reported for computing the invariant set XI , where trajectories which exit the feasible set XN are removed iteratively until no such states can be found. The invariant set found covers the state space from where production starts up to the nominal operating point. In Figure 7, cuts of the invariant set XI trough x3 = 1, u2 = 1, i2 = 0 and x3 = 0, u2 = −1, i2 = 0 are shown, which correspond to the plant heated up at nominal temperature without reactant inside and to the nominal operating point, respectively. For any initial state within XI , all constraints will be met at all future time-instants, but no guarantee is given that the state vector will ever reach the origin. Therefore, a PWQ Lyapunov function has to be identified on XI . Using the MPT Toolbox [15] an appropriate Lyapunov function was found. Consequently, the closed-loop system is feasible and stable on the set XI around the nominal operating point.

268


3.7 Simulation Results The given test sequence is applied to the system. The plant is heated up using Mode 1, operates in Mode 2 when reactant is in the vessel and is cooled down in Mode 1, again. We switch between the controllers depending on the conditions (51). The inputs are obtained by identifying the region which contains the actual state vector xh (k), xp (k) and subsequent evaluation of the corresponding affine function. They are applied to the nonlinear, continuous time plant with a zero-order hold element. The evaluation is repeated with a sampling interval of Ts = 0.04s. Figure 8 depicts the evolution of the states and the inputs by applying the reference sequence. The upper plot of Figure 8 shows the three states x1 (t), x2 (t), x3 (t) of the nonlinear continuous time plant, the lower plot of Figure 8 shows the inputs

1

0.5

0

−0.5

0

2

4

6

8

10

12

14

16

18

10

12

14

16

18

t

1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0

2

4

6

8 t

Fig. 8. Trajectory evolution of the full test sequence. Upper plot: reactant concentration x1 (t) (--), content temperature x2 (t) (-), jacket temperature x3 (t) (-.). Lower plot: reactant feed flow u1 (t) (--), temperature of heating/cooling fluid u2 (t) (-), magnitude constraints on u2 (t) (..).


269

1.1 1.08 1.06 1.04 1.02 1 0.98 0.96 0.94 0.92 0.9

6

7

8

9

10

11

12

13

14

15

t

Fig. 9. Trajectory evolution around nominal operating point. Reactant concentration x1 (t) (-), content temperature x2 (t) (-), jacket temperature x3 (t) (-.), reference values r1 (t), r2 (t) (..).

u1 (t), u2 (t) and in dotted lines the magnitude constraints on u2 (t). Note that the input u1 (t) is neither constrained in magnitude nor in rate of change, except that reactant can only be fed into and not extracted from the plant (u1 (t) ≥ 0). In Figure 9 the trajectories x1 (t), x2 (t), x3 (t) around the nominal operating point are displayed. Additionally, the references r1 (t), r2 (t) are plotted in dotted lines. We see that the content temperature x2 (t) is controlled with zero steady state error, contrary to the reactant concentration x1 (t). 3.8 Conclusion For the CSTR benchmark example a linearized description has been derived, which consists of two LTI models. The two models have been extended to cover time varying reference tracking, slew rate constraints on inputs and control with integral action. For each augmented model a controller has been derived with the emphasis to keep the controller complexity low, but with the drawback that no stability and feasibility guarantees are given by construction. Because the explicit solution was at disposal, an a posteriori analysis verified that the controller is stable and feasible.

References ˚ ˚ om K. J., Safe Manual Control of the Furuta Pendulum. Proceedings of 1. Akesson J., Astr¨ the 2001 IEEE International Conference on Control Applications, Mexico City, Mexico, (2001)

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2. Barbu C., Galeani S., Teel A. R., Zaccarian L., Non-linear anti-windup for manual flight control. International Journal of Control, 78(14):1111–1129 (2005) 3. Bemporad M., Morari M., Dua V., Pistikopoulos E. N., The explicit linear quadratic regulator for constrained systems. Automatica, 38(1):3–20 (2002) 4. Borrelli F., Constrained Optimal Control Of Linear And Hybrid Systems. Lecture Notes in Control and Information Sciences, Springer (2003) 5. Ferrari-Trecate G., Cuzzola F. A., Mignone D. and Morari M., Analysis of discrete-time piecewise affine and hybrid systems. Automatica, 38(12):2139–2146 (2002) 6. Furuta K., Yamakita M., Kobayashi S., Nishimura M., A new Inverted Pendulum Apparatus for Education, IFAC Symposium on Advances in Control Education, Boston, USA, 191–196 (1991) 7. Gilbert E.G. and Tan K.T., Linear systems with state and control constraints: the theory and applications of maximal output admissible sets. IEEE Trans. Automatic Control, 36(9):1008–1020 (1991) 8. Grieder P. , Lüthi M., Parillo P. and Morari M., Stability & Feasibility of Receding Horizon Control. European Control Conference, Cambridge, UK (2003) 9. Grieder P., Morari M, Complexity Reduction of Receding Horizon Control. IEEE Conference on Decision and Control, Maui, Hawaii, 3179–3184 (2003) 10. Hanley J. G., A Comparison of Nonlinear Algorithms to Prevent Pilot-Induced Oscillations caused by Actuator Rate Limiting. Air Force Institute of Technology (2003) 11. Kerrigan E.C., Robust Constraints Satisfaction: Invariant Sets and Predictive Control. PhD thesis, Department of Engineering, The University of Cambridge, Cambridge, England (2000) 12. Ling Z., Saberi A., Semi-global exponential stabilization of linear discrete-time systems subject to input saturation via linear feedbacks System and Control Letters 24, 125–132 (1995) 13. Maciejowski J.M., Predictive Control with Constrains. Prentice Hall, 2002. 14. Mayne D.Q., Rawlings J.B., Rao C.V., Scokaert P.O.M., Constrained model predictive control: Stability and optimality. Automatica, 36(6):789–814 (2000) 15. Kvasnica

M.,

Grieder

M.,

Baotić

M.,

Multi-Parametric

Toolbox

(MPT).

http://control.ee.ethz.ch/ mpt/ (2004) 16. Pachter M., Miller R.B., Manual Flight Control with Saturating Actuators, IEEE Control Systems Magazine, 18(1):10–19 (1998) 17. Sontag E.D., Sussmann E.J., Nonlinear Output Feedback Design for Linear Systems with Saturating Controls. Proc. 29th IEEE Conference Decision and Control, Honolulu, Hawaii, 3414–3416 (1990) 18. Tøndel P., Constrained Optimal Control via Multiparametric Quadratic Programming. PhD thesis, Department of Engineering Cybernetics, NTNU ,Trondheim, Norway (2000)


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19. Tøndel P., Johansen T.A. and Bemporad A., An algorithm for multiparametric quadratic programming and explicit MPC solutions. Automatica, 39(3) (2003) 20. Yakubovich V.A., Nakaura S., Furuta K., Tracking Domains for Unstable Plants with Saturating-Like Actuators. Asian Journal of Control, 1(4):229–244 (2000)

Constrained Control Using Model Predictive Control J.M. Maciejowski1 , P.J. Goulart1 , and E.C. Kerrigan2 1

Cambridge University Engineering Department, Cambridge CB2 1PZ, England {jmm,pjg42}@eng.cam.ac.uk

2

Imperial College London, Dept. of Electrical and Electronic Engineering and Dept. of Aeronautics, London SW7 2AZ, England [email protected]

1 Introduction The most common way of dealing with constraints in control systems is to ignore them, pretend that the system is linear, and fix things up in a more-or-less ad-hoc fashion after performing a linear design. There are some rather systematic ways of ‘fixing things up’, including certain anti-windup techniques, but logically they are an afterthought, a way of dealing with the nuisance of constraints after the central work has been done. Until recently, things could not be done differently, because almost all the control design techniques available to us were linear techniques.3 This situation has been changed by the emergence of Model Predictive Control (MPC) as a successful technique for controlling systems subject to constraints — several books are already available [6, 7, 10, 11, 17, 19, 20, 21]. MPC allows constraints to be included as part of the problem formulation, and thus faced up to from the outset. Furthermore, this is true for output or state constraints (arising from safety or product quality considerations, etc), as well as input constraints (arising from actuator limitations). It can be applied to typical control problems in which there are set-points as well as constraints, but also to problems in which there are only constraints, and no set-points (such as level control in a buffer tank). A very nice aspect of MPC is that there is no particular difficulty in dealing with multivariable systems 3

Although quite a lot of nonlinear methods have been developed, the proportion of problems to which they can be applied is very small.

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J.M. Maciejowski, P.J. Goulart, and E.C. Kerrigan

in which some output variables have set-points, some have constraints, and some have both set-points and constraints. The technological foundation of MPC is repeated online optimisation. It is therefore usually considered to be a technique suitable only for low-bandwidth applications with low sampling and update rates, such as those found in process control. But modern computing power is so great that in fact MPC can be used in applications usually considered to be relatively high-bandwidth, such as aerospace, automotive, and robotics. This will be illustrated in this chapter by a (simulated) application to the control of an inverted pendulum.

2 Outline of MPC 2.1 The Basic Idea Figure 1 shows the basic idea of predictive control, for the simplest case of a singleinput, single-output plant. We assume a discrete-time setting, and that the current time is labelled as time step k. At the current time the plant output is yk , and the figure shows the previous history of the output trajectory. Also shown is a set-point trajectory, which is the trajectory that the output should follow, ideally. The value of the set-point trajectory at any time t is denoted by st . Optionally, there may be a reference trajectory rk+i|k which is distinct from the setpoint trajectory. This starts at the current output yk , and defines an ideal trajectory along which the plant should return to the set-point trajectory, for instance after a disturbance occurs. The reference trajectory therefore defines an important aspect of the closed-loop behaviour of the controlled plant. It is not necessary to insist that the plant should be driven back to the set-point trajectory as fast as possible, although that choice remains open. It is frequently assumed that the reference trajectory approaches the set-point exponentially from the current output value, with the ‘time constant’ of the exponential. The notation rk+i|k indicates that the reference trajectory depends on the conditions at time k, in general. A predictive controller has an internal model which is used to predict the behaviour of the plant, starting at the current time, over a future prediction horizon. This predicted behaviour depends on the assumed input trajectory u ˆk+i|k (i =

Constrained Control Using Model Predictive Control

275

st rt|k N yt yˆt|k f yˆt|k

k

k+N

Time

k

k+N

Time

Input

Fig. 1. MPC: The Basic Idea

0, 1, . . . , N − 1) that is to be applied over the prediction horizon, and the idea is to select that input which promises the best predicted behaviour. We shall assume that the internal model is linear; this makes the calculation of the best input relatively straightforward. The notation u ˆ rather than u here indicates that at time k we only have a prediction of what the input at time k + i may be; the actual input at that time, uk+i , will probably be different from u ˆk+i|k . Note that we assume that we have the output measurement yk available when deciding the value of the input uk . This implies that our internal model must be strictly proper. In the simplest case we can try to choose the input trajectory such as to bring the plant output at the end of the prediction horizon, namely at time k + N , to the required value rk+N . In this case we say, using the terminology of Richalet [20], that we have a single coincidence point at time k + N . There are several input trajectories {ˆ uk|k , u ˆk+1|k , . . . , u ˆk+N −1|k } which achieve this, and we could choose one of them, for example the one which requires the smallest input energy. But it is usually adequate, and in fact preferable, to impose some simple structure on the input trajectory,

276


parametrised by a smaller number of variables. Figure 1 shows the input assumed to vary over the first three steps of the prediction horizon, but to remain constant thereafter: u ˆk+2|k = u ˆk+3|k = · · · , u ˆk+N −1|k , so that there are three ‘parameters’ to choose: u ˆk|k , u ˆk+1|k , u ˆk+2|k . These parameters (or other similar ones) are chosen so as to minimise some cost function. Once a future input trajectory has been chosen, only the first element of that trajectory is applied as the input signal to the plant. That is, we set uk = u ˆk|k , where uk denotes the actual signal applied. Then the whole cycle of output measurement, prediction, and input trajectory determination is repeated, one sampling interval later: a new output measurement yk+1 is obtained; a new reference trajectory rk+i|k+1 (i = 2, 3, . . .) may be defined; predictions are made over the horizon k + 1 + i, with i = 1, 2, . . . , N ; a new input trajectory u ˆk+1+i|k+1 , with i = 0, 1, . . . , N − 1 is chosen; and finally the next input is applied to the plant: uk+1 = u ˆk+1|k+1 . Since the prediction horizon remains of the same length as before, but slides along by one sampling interval at each step, this way of controlling a plant is often called a receding horizon strategy. This idea remains valid if the plant being controlled is multivariable, that is, if there are several inputs and/or several outputs. In such cases the variables u and/or y become vectors, and we shall henceforth assume that this is the case. The only other major ingredient of MPC that we need is constraints. The choice of future trajectory u ˆk+i|k , (i = 0, 1, . . . , N − 1) should be such that the input signals and their rates of change remain within allowed constraints, and such that the outputs — and possibly inferred variables, namely values which are not measured directly — also remain within allowed constraints.

2.2 Computing the Inputs If we assume that the plant model is linear, and that the cost function is quadratic, then the problem which has to be solved is a standard finite-horizon linear quadratic (or ‘LQ’) problem, for which a complete theory exists [5]. If linear inequality constraints on inputs, states or outputs are added, finding the optimal input trajectory becomes a quadratic programming (or ‘QP’) problem. If the cost function is changed from being quadratic to being ‘1-norm’ or ‘∞-norm’, namely one which penalises


277

absolute values of errors, or the peak error, respectively, then finding the optimal solution becomes a linear programming (or ‘LP’) problem. It is standard to assume that the plant is linear and time-invariant, and that after discretisation of time at a single sampling rate it can be modelled as a strictly-proper state-space system: xk+1 = Axk + Buk + Ewk yk = Cxk + vk

(1) (2)

where x is a state vector, u and y are vectors of control inputs and measured outputs, respectively, and w and v are vectors of unknown state disturbances and measurement errors, respectively. A, B, C, E are constant matrices of appropriate dimensions. Other linear model forms are often assumed for MPC, but these can all be obtained from a state-space model, so no loss of generality is involved in assuming this form. (In particular, a conversion of MPC problems from the ‘GPC’ variant of MPC to a state-space form is described in [17].) Plants with time delays can also be modelled by this form, providing that all delays are multiples of the sampling interval. Given a future input trajectory u ˆk+i|k (i = 0, 1, 2, . . .) predicted state and/or output trajectories x ˆk+i|k , yˆk+i|k can be calculated using the model (1), (2) together with any assumptions that are made about the disturbances wk+i and measurement errors vk+i . The future input trajectory is chosen as one that minimises a cost function of the form J=

Hp

(ˆ yk+i|k , rk+i , u ˆk+i−1|k ) + F (ˆ xk+Hp +1|k )

(3)

i=1

in which the function (·, ·, ·) is a stage cost on the output, reference, and input signals during a prediction horizon of length Hp , and the function F (·) is a terminal cost on the state. This minimisation is performed subject to satisfying constraints on the inputs and states, typically in the form of linear inequalities: ⎡ ⎤ x ˆ ⎢ k+i|k ⎥ M⎢ ˆk+i|k ⎥ ⎣ u ⎦≤μ

(4)

Δˆ uk+i|k where Δˆ uk+i|k = u ˆk+i|k − u ˆk+i−1|k , M is a matrix and μ is a vector. In ‘robust MPC’ the minimisation of the cost is sometimes minimisation of the worst case

278


cost, for a range of allowable disturbances and/or plant uncertainties. Thus a straight minimisation problem is replaced by a min-max problem, which usually requires much more computational effort to solve. Common choices of stage cost are: Quadratic: Absolute values: Peak values:

¯) 22 (y, r, u) = Q1/2 (y − r) 22 + R1/2 (u − u

(5)

¯) 1 (y, r, u) = Q(y − r) 1 + R(u − u

(6)

¯) ∞ (y, r, u) = Q(y − r) ∞ + R(u − u

(7)

¯ is an equilibrium value where Q = QT ≥ 0 and R = RT > 0 typically, and u of u that is compatible with the set-point r. (The precise conditions on Q and R may vary.) The first of these leads to a quadratic programming problem to be solved, while the second and third lead to a linear programming problem. (In the ‘robust’ case, if min-max problems are to be solved, several QP or LP problems may need to be solved at each step.) Note that the computation of the optimal control input is not much more complicated if the model, the constraints, and the cost function vary with time, although the analysis of closed-loop properties becomes much more difficult in such cases. (The only additional computational complexity arises because the data for the optimisation problem has to be re-assembled at each step. This is usually a minimal overhead in comparison with the solution of the optimisation problem.)

2.3 Closed-Loop Stability MPC has a reputation for usually giving closed-loop stability. Certainly if one chooses a problem specification at random, one is much more likely to obtain a stable closed loop with an MPC controller than one would find if one chose the parameters of a ‘classical’ (even simple PI) controller at random. But it is possible to construct examples in which the MPC solution gives an unstable control loop, so the stability property should not be taken for granted. The reputation of MPC in this regard was established in industrial applications, in particular in the petrochemical industry. Most such applications involve a benign stable plant, and the control requirements are relatively undemanding, allowing a considerable degree of ‘de-tuning’, which tends to give closed-loop stability when used with stable plants. As MPC becomes


279

used for a wider range of applications, with less stable plants, more aggressive controller requirements, and more complex patterns of active constraints, closed-loop stability is likely to be an increasing problem. Fortunately, there is a well-developed theory regarding stability of MPC, which leads to a number of ways of formulating the MPC problem which guarantee closed-loop stability. The two basic ideas are: • Impose a terminal constraint at the end of the prediction horizon, or • Use an infinite prediction horizon. Suppose that the set-point is constant, and that we change coordinates such that the set-point corresponds to an equilibrium at x = 0. Imposing a terminal constraint of the form x ˆk+Hp |k = 0 always gives closed-loop stability, providing that the resulting optimization problem is always feasible, and that the global optimal solution is found and applied at each step [13]. But this is an unnecessarily strong constraint, and it can be relaxed to a constraint of the form x ˆk+Hp |k ∈ XF , providing that the terminal set XF has suitable properties. For example, if a conventional control exists which stabilises 0 and for which XF is contained in the domain of attraction of 0, and all the constraints are satisfied everywhere on XF , then that is enough to guarantee closedloop stability, if one postulates a dual-mode control, which applies the MPC solution for x ∈ XF and the conventional (typically linear) control for x ∈ XF . Allowing the terminal state x ˆk+Hp |k to belong to a set rather than a point makes it more likely that the MPC problem will have a feasible solution at each step. Using an infinite horizon is an alternative way of ensuring closed-loop stability. For if the closed-loop were unstable then the cost function would be infinite, and hence the controller would not be optimal. This argument assumes that the plant is stabilisable, and that the MPC problem is feasible, and that the global optimum is found at each step. An alternative argument can be made invoking Bellman’s principle of optimality, and using the value function (that is, the optimal value of J — see (3)) as a Lyapunov function [17, Chapter 6]. But the difficulty with using an infinite horizon is that there are infinitely many decision variables {ˆ uk+i|k : i = 0, 1, 2, . . .} to be chosen. This difficulty can be overcome by reparametrising the MPC problem in such a way as to reduce the number of decision variables to a finite number. The usual way of doing this is to assume that there is a horizon length N , such that the constraints can be assumed to be inactive for i ≥ N . Then one only

280


chooses {ˆ uk+i|k : i = 0, 1, 2, . . . , N − 1}, and assumes some fixed feedback law uk+i = μ(xk+i ) for i ≥ N . In particular, if the stage cost in (3) is taken to be quadratic ˆk+i|k ) = Q1/2 x ˆk+i|k 22 + R1/2 u ˆk+i|k 22 (ˆ xk+i|k , rk+i , u

(8)

(assuming rk+i = 0 for all i) and the horizon is taken to be infinite, then the optimal value of the cost is J 0 = xTk P xk

(9)

where P is the symmetric positive semi-definite solution of the algebraic Riccati equation P = AT P A − AT P B(B T P B + R)−1 B T P A + Q

(10)

(assuming the linear plant model (1)–(2)). Consequently, the following cost function has been suggested [9]: J=

N "

Q

1/2

x ˆk+i|k 22

+ R

1/2

u ˆk+i|k 22

#

+x ˆTk+N +1|k P x ˆk+N +1|k

(11)

i=1

A thorough survey of techniques for ensuring stability of MPC controllers is given in [18].

2.4 Robust Feasibility and Stability Solutions to MPC problems are relatively straightforward if model uncertainty is ignored. But they become significantly more complicated if uncertainty is taken into account, and an effort is made to ensure that constraints are satisfied despite the uncertainty. This is of course a requirement in any practical application of MPC. In general model uncertainty can be very ‘unstructured’ — the real system is likely to be nonlinear, time-varying, of unknown state dimension, etc. But useful results can be obtained by assuming more restricted forms of uncertainty, such as uncertain parameter values in a model of the form (1) – (2). In this chapter we shall assume an even more restricted form of uncertainty, namely that everything in the model is known in advance, except for the values of the state disturbance sequence {wk }. We shall also assume that the full state vector can be measured, with negligible measurement noise. The state disturbance values will be assumed to belong to a set wk ∈ W , with


281

W being a known convex polytope. Note that a consequence of these assumptions is that it is possible to infer the value of wk−1 from a measurement of xk . The strategy for ensuring robustness is to establish that a feasible trajectory {uk+i , xk+i+1 : i = 0, 1, . . . , N } can be found, from the current state xk , to a terminal set xk+N +1 ∈ XF , whatever disturbance trajectory is encountered, and that the terminal set has an appropriate robust invariance property. Of course if xk is such that this is impossible, then the constraints in the problem specification cannot be satisfied — but the aim is to ensure that this can never arise. An essential idea is that this robust feasibility problem typically has no solution if one tries to determine in advance what the solution trajectory should be, because there is usually no single trajectory which is appropriate for all possible disturbance realizations. But it may have a solution providing that one allows the trajectory from each time onwards to depend on the particular disturbances that have actually occurred up to that time — in other words, if one decides on a feedback policy rather than a specific trajectory [18]. But optimization over feedback policies is in general very difficult if constraints have to be taken into account. Most proposals for achieving this using finite-dimensional optimization are computationally intractable, because the size of the optimization problem grows exponentially with the size of the problem data (ie with the horizon length, the number of constraints, etc) [22, 17]. One possibility is to pre-compute (off-line) the solution to a suitable robust constrained finite-horizon problem [2, 6, 14]. This solution turns out to be a time-varying piecewise-affine control law; once computed, it can be stored in a table and the appropriate “piece” deployed, depending on the current state. Unfortunately this ‘explicit’ approach is currently feasible only for small problems, because the computation of the solution can grow exponentially with the problem data. A popular approach to overcoming the computational complexity of optimization over feedback policies is to somehow fix a stabilising linear state-feedback law, and then to use on-line optimization to find perturbations to the resulting control sequence in order to satisfy constraints [1, 8, 15]. This approach attempts to reduce the effects of uncertainty by means of the state feedback, so that optimization of a single trajectory becomes feasible despite the huge variety of possible disturbance sequences that might be encountered. But the choice of the linear state-feedback law is essentially ad hoc, and it is not clear how best to choose it. Attempts to improve on this by simultaneously optimising the state-feedback law and the perturbations to

282


it required for feasibility usually result in a non-convex problem, because state and input predictions depend nonlinearly on the state-feedback law. We have recently proposed a way of overcoming this non-convexity [12]. The key to this is a re-parametrization of the control policy as an affine function of past disturbances: ui =

i−1

Mij wj + vi ,

for i = 0, 1, . . . , N − 1.

(12)

j=0

where {Mij } is a set of matrix gains which are to be chosen, and {vi } is a set of perturbations which are also to be chosen. A similar parametrization was also proposed in [23, 16, 4]. It is shown in [12, 3] that this parametrization of the control policy is equivalent to the more conventional state-feedback parametrization: ui =

i

Lij xj + gi

for i = 0, 1, . . . , N − 1.

(13)

j=0

But the crucial difference is that, given xk (and hence w0 , . . . , wk−1 ), the set of admissible Mij and vi in (12) (for i = k, k+1, . . . , k+N −1 and j = 0, 1, . . . , k−1) is convex, whereas the set of admissible Lij and gi in (13) is not.

3 Application to Inverted Pendulum We apply MPC to the inverted pendulum benchmark problem. In this case we found that we could easily meet the performance requirements and satisfy the constraints, without needing any of the machinery from section 2.4. We ran MPC in discrete time, using a time-step of 0.15 sec, and simulated its performance on a continuoustime model of the pendulum. Figure 2 shows a Simulink block diagram of the inverted pendulum, with both an MPC controller and a basic conventional controller available as options. The output of the MPC controller at each update instant is a change in the required force, as required by the rate-commanded actuator. A cost function of the form (11) was used, but modified to: J=

N "

# xk+i|k − x ¯) 22 + R1/2 Δˆ uk+i−1|k 22 + Q1/2 (ˆ

i=1

¯)T P (ˆ xk+N +1|k − x ¯) (14) (ˆ xk+N +1|k − x

rref

rref

y

MPC Controller

Rate Commanded Actuator u_dot

y

DT linear controller

u_dot

pendulum rref

u

u


z y

u_dot

283

time MPC controller. A conventional linear controller is also shown as an alternative option.

Fig. 2. A Simulink model of the continuous-time invered pendulum controlled by a discrete-

zload

284


where x ¯ is an equilibrium value of the state vector that corresponds to the current set-point (see equations (18)), and Δˆ uk+i|k = u ˆk+i|k − u ˆk+i−1|k is the predicted change in the control input signal. We used a horizon N = 35 steps (ie 5.25 sec), with weighting matrices ⎡

100 0

0

0

⎤

⎥ ⎢ ⎢ 0 500 0 0 ⎥ ⎥ Q=⎢ ⎢ 0 0 10 0 ⎥ ⎦ ⎣ 0 0 0 0.01

and

R = 0.01

(15)

where the first 3 states are as defined in the benchmark definition, namely the pendulum speed, the displacement between the pendulum and the slider, and the slider speed. The fourth state variable was the last input applied to the system: x(4)k = uk−1 , which is needed in order to keep track of the predicted values u ˆk+i|k during the prediction horizon — these are needed to ensure that actuator limits are not violated. The corresponding terminal weight P that appears in (11) (and is the solution of (10)) was ⎡

2211.6 −1018.3 −179.12 −15.432

⎢ ⎢ −1018.3 1928.5 P =⎢ ⎢ −179.12 271.34 ⎣ −15.432 23.043

271.34 77.455 7.0224

⎤

⎥ 23.043 ⎥ ⎥ 7.0224 ⎥ ⎦ 1.0311

(16)

Of course the model has to be modified appropriately, to be consistent with the newly-defined state vector, and the use of Δu instead of u as the input. But this is straightforward and standard [7, 17]. Figure 3 shows the internal structure of the MPC controller. At each step a simple disturbance estimator is run, assuming that the state disturbance is given by the difference between the latest measured state and the previously expected state: dˆk = xk − x ˆk|k−1

(17)

(Recall that it is assumed that all the states are measured.) This is then used to estimate the equilibrium value of the force u ¯ required from the actuator, assuming that the disturbance will remain constant at this estimated value: d¯ = dˆk . x ¯ and u ¯ are found as the solutions to the simultaneous equations: ¯ x ¯ = A¯ x + Bu ¯ + d,

Cx ¯ = rk

(18)

xbar

sfunsetpt

NOT MPC controller

Stop Simulation STOP


set point calculator disturbance estimation

ubar du

dbar

dbar

1 u_dot dbar

sfunmpc y

2 y

rref

1 rref

equilibrium set-point calculator.

285

Fig. 3. Internal structure of the MPC controller, showing the disturbance estimator and the

x

286


where rk is the current set-point for the pendulum speed. The resulting value u ¯ then becomes the fourth element of the augmented x ¯. Note that this results in offsetfree control in the presence of unknown but constant disturbance forces, even if the steady-state gains in the model are not accurate. To demonstrate that this design meets the requirements, we simulated a single run, over 120 sec of simulated time. The specified reference trajectory for the pendulum speed was applied over the first 60 seconds, and the specified disturbance force was applied over the last 60 sec, as shown in Figure 4. 2.5

2

1.5

1

0.5

0

0

20

40

60 Time (sec)

80

100

120

Fig. 4. Reference trajectory for pendulum speed (solid line) and disturbance force (dashed line) during simulations.

With the higher limit on the actuator rate, namely |u| ˙ ≤ 10, the results of the simulation are shown in Figure 5. The top and middle graphs show the speeds of the pendulum (with set-point) and the slider, respectively, and it can be seen that the pendulum speed follows the set-point closely, with no steady-state offset. The slider’s speed trajectory is similar to that of the pendulum, except for brief accelerations/decelerations in order to produce accelerations/decelerations of the pendulum. The second graph shows the distance between the slider and the pendulum. This is held to zero (except


287

x1 4 2 0 −2 x

2

1.5 1 0.5 0 −0.5 x

3

4 2 0 −2 −4

u 2 1 0 −1 −2

du/dt 10 5 0 −5 −10

0

20

40

60 Time

80

100

120

Fig. 5. Results of the simulation with |u| ˙ ≤ 10. From the top, the graphs show (1) x1 , the pendulum speed (with set-point), (2) x2 , the distance between the pendulum and the slider, (3) x3 , the slider speed, (4) u, the force applied by the actuator, (5) u, ˙ the rate of change of the force.

288


during transients) when a constant speed set-point is being followed, since it is assumed that there is no friction. But when the constant external disturbance is present, then the slider and the pendulum are a constant distance apart, in order to produce a counteracting force on the pendulum. The fourth graph shows the force applied by the actuator on the slider. It will be seen that this is limited at ±1.25, as required. The bottom graph shows that the rate limits are also respected. It is assumed that the controller does not anticipate set-point changes. If it did have advance awareness of set-point changes then the delay in tracking the controller speed could be eliminated. (Note that this feature is not unique to MPC, despite common belief [5].) If the slew rate of the actuator force is reduced to |u| ˙ ≤ 2 the results are shown in Figure 6. It can be seen that the results are essentially the same as in Figure 5, so that the reduction of the slew rate limit is not of great significance in this problem. Note, however, that the control signals generated by the MPC controller in the two cases are not the same; Figure 5 shows that u˙ = 10 at t ≈ 108 sec, so that the full range of allowed rates is used when the higher limit is available. The quadratic programming problem involved in implementing the MPC solution was solved using an interior point method, implementing the specific ‘banded matrix’ scheme described in [17, section 3.3.1]. The time taken to run the simulation in Simulink was about 15 sec, on a Pentium 4 with 3GHz clock speed and 1 Gbyte of RAM, running Linux. That is, the simulation runs more than 10 times faster than real time. Note that our running speed was limited by network communications, by writing the simulation results to a ‘scope’ while running the simulation, and by simulating the plant as well as the controller. Furthermore the simulation was run within Matlab version 7, making much use of the object-oriented mechanisms, thus incurring considerable run-time overhead. Thus an MPC controller could easily be made to run much faster than this for this problem. A significant feature of MPC is that it can often be made to work quite easily. This was the case for the inverted pendulum benchmark. Once the form of the cost function and the structure of the MPC controller had been chosen, the only tuning that had to be done was the choice of the horizon N and of the weights Q and R (see (14)). Finding a suitable combination of these was straightforward and did not take much time.


289

x1 4 2 0 −2 x

2

1.5 1 0.5 0 −0.5 x

3

4 2 0 −2

u 2 1 0 −1 −2

du/dt 2 1 0 −1 −2

0

20

40

60 Time

80

100

120

Fig. 6. Results of the simulation with |u| ˙ ≤ 2. From the top, the graphs show (1) x1 , the pendulum speed (with set-point), (2) x2 , the distance between the pendulum and the slider, (3) x3 , the slider speed, (4) u, the force applied by the actuator, (5) u, ˙ the rate of change of the force.

290


4 Conclusions Model Predictive Control (MPC) is particularly well-suited to the control of constrained systems. It allows constraints to be included in the problem formulation from the start. The basic form of MPC is easy to understand and to implement, and is no longer confined to application to slow processes. Formulations which ensure stability, at least from some region of attraction about the desired equilibrium, are standard. Formulations which ensure feasibility, and hence stability, from some prescribed set of initial conditions are also available, but typically require much more computation to implement them. However, as we have shown on the inverted pendulum benchmark problem, such advanced formulations are frequently not needed for many problems.

References 1. A. Bemporad. Reducing conservativeness in predictive control of constrained systems with disturbances. In Proc. 37th IEEE Conference on Decision and Control, pages 1384– 1391, Tampa, FL, December 1998. 2. A. Bemporad, F. Borrelli, and M. Morari. Min-max control of constrained uncertain discrete-time linear systems. IEEE Transactions on Automatic Control, 48(9):1600–1606, September 2003. 3. A. Ben-Tal, S. Boyd, and A. Nemirovski. Extending scope of robust optimization: Comprehensive robust counterparts of uncertain problems. Mathematical Programming, 107(1–2):63–89, June 2006. 4. A. Ben-Tal, A. Goryashko, E. Guslitzer, and A Nemirovski. Adjustable robust solutions of uncertain linear programs. Mathematical Programming, 99(2):351–376, March 2004. 5. R.R. Bitmead, M. Gevers, and V. Wertz. Applied Optimal Control: The Thinking Man’s GPC. Prentice-Hall, Englewood Cliffs, NJ, 1990. 6. F. Borrelli. Constrained Optimal Control of Linear and Hybrid Systems. Lecture Notes in Control and Information Sciences. Springer, 2004. 7. E.F. Camacho and C.Bordons. Model Predictive Control. Advanced Textbooks in Control and Signal Processing. Springer, London, 1999. 8. L. Chisci, J.A. Rossiter, and G. Zappa. Systems with persistent state disturbances: predictive control with restricted constraints. Automatica, 37:1019–028, 2001. 9. D. Chmielewski and V. Manousiouthakis. On constrained infinite-time linear quadratic optimal control. Systems and Control Letters, 29:121–129, 1996.

Constrained Control Using Model Predictive Control 10. R. Dittmar and B.-M. Pfeiffer.

Modellbasierte prädiktive Regelung.

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Oldenbourg,

München, 2004. 11. G.C. Goodwin, M.M. Seron, and J.A. De Dona. Constrained Control and Estimation: an Optimisation Approach. Springer, London, 2005. 12. P.J. Goulart, E.C. Kerrigan, and J.M. Maciejowski. Optimization over state feedback policies for robust control with constraints. Automatica, 42(4):523–533, April 2006. 13. S.S. Keerthi and E.G. Gilbert. Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: stability and moving-horizon approximations. Journal of Optimization Theory and Applications, 57:265–293, 1988. 14. E.C. Kerrigan and J.M. Maciejowski. Feedback min-max model predictive control using a single linear program; robust stability and the explicit solution. International Journal of Robust and Nonlinear Control, 14:395–413, 2004. 15. Y.I. Lee and B. Kouvaritakis. Constrained receding horizon predictive control for systems with disturbances. European Journal of Control, pages 1027–1032, August 1999. 16. J. Löfberg. Minimax Approaches to Robust Model Predictive Control. PhD thesis, Linköping University, April 2003. 17. J.M. Maciejowski. Predictive Control with Constraints. Prentice-Hall, Harlow UK, 2002. (Japanese translation by S. Adachi and M. Kanno published by Tokyo Denki University Press, 2005.). 18. D.Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scokaert. Constrained model predictive control: stability and optimality. Automatica, 36:789–814, 2000. 19. D.M. Prett and C.E. Garcia. Fundamental Process Control. Butterworths, Boston, 1988. 20. J. Richalet, G. Lavielle, and J. Mallet. La Commande Prédictive. Eyrolles, Paris, 2004. 21. J.A. Rossiter. Model-based Predictive Control: A Practical Approach. CRC Press, Boca Raton, 2003. 22. P.O.M. Scokaert and D.Q. Mayne. Min-max feedback model predictive control for constrained linear systems. IEEE Transactions on Automatic Control, 43(8):1136–1142, 1998. 23. D.H. van Hessem and O.H. Bosgra. A conic reformulation of model predictive control including bounded and stochastic disturbances under state and input constraints. In Proc. 41st IEEE Conference on Decision and Control. December 2002.

Risk Adjusted Receding Horizon Control of Constrained Linear Parameter Varying Systems M. Sznaier1 , C. M. Lagoa2 , X. Li3 , and A. A. Stoorvogel4 1

EE Dept., The Pennsylvania State University. University Park, PA 16802, USA, [email protected]

2


3


4

Department of Information Technology and Systems Delf University of Technology P.O. Box 5031, 2600 GA Delft, The Netherlands. email: [email protected]

Summary. In the past few years, control of Linear Parameter Varying Systems (LPV) has been the object of considerable attention, as a way of formalizing the intuitively appealing idea of gain scheduling control for nonlinear systems. However, currently available LPV techniques are both computationally demanding and (potentially) very conservative. In this chapter we propose to address these difficulties by combining Receding Horizon and Control Lyapunov Functions techniques in a risk–adjusted framework. The resulting controllers are guaranteed to stabilize the plant and have computational complexity that increases polynomially, rather than exponentially, with the prediction horizon.

Key words: Model Predictive Control, LPV Systems, Risk Adjusted Control.

294

M. Sznaier, C. M. Lagoa, X. Li, and A. A. Stoorvogel

1 Introduction A widely used practice to handle nonlinear dynamics is to linearize the plant around several operating points and then use gain–scheduled linear controllers. However, while intuitively appealing, this idea has several pitfalls [15]. Motivated by these shortcomings, during the past few years considerably attention has been devoted to the problem of synthesizing controllers for Linear Parameter Varying Systems, where the state–space matrices of the plant depend on time–varying parameters whose values are not known a priori, but can be measured by the controller. This research has resulted in controller synthesis methods guaranteeing worst case performance bounds (for instance in an H2 or H∞ sense, see e.g. [1] and references therein). While successful in many situations, these techniques are potentially very conservative in others, since they are based on sufficient conditions. In addition, these methods are computationally very demanding, requiring the solution of a set of functional matrix inequalities. Obtaining computationally tractable problems requires using both finite expansion approximations as well as a gridding of the parameter space, leading to further conservatism. The present chapter seeks to reduce both the computational complexity and conservatism entailed in currently available LPV synthesis methods by combining risk– adjusted (see eg. [5, 6, 13]) and Receding Horizon (see [12] and references therein) ideas. Our main result shows that, by searching over a set of closed–loop strategies the problem can be reduced to finding a solution to a finite set of Linear Matrix Inequalities (LMIs). Finding the exact solution to this problem has computational complexity that grows exponentially with the horizon length. To circumvent this difficulty, we propose to use a stochastic approximation algorithm that is guaranteed to converge to the solution with probability one, and whose computational complexity grows only polynomially with the horizon. The chapter draws inspiration, in addition to [16] and [9], from [14], [7] and [3]. The main difference with [14] and [7] is the use of Receding Horizon techniques and parameter dependent Lyapunov functions. Compared with [3], we consider the case of case of LPV dynamics and we obtain a controller that minimizes the worst case performance, rather than its expected value, over all trajectories compatible with the current parameter value. Finally, the use of closed–loop strategies, based on the solution of a set of LMIs, results in substantial reduction of computational complexity.

Risk Adjusted Receding Horizon Control

295

In this sense the work presented here is related to the earlier work in [18] advocating the use of Riccati based receding horizon controllers for H∞ control of Linear Time Varying systems, and to the work in [11], proposing an LMI–based optimization of closed–loop control strategies.

2 Preliminaries 2.1 The LPV Quadratic Regulator Problem In this chapter we consider the following class of LPV systems: x(t + 1) = A [ρ(t)] x(t) + B2 [ρ(t)] u(t) z(t) = C1 [ρ(t)] x(t) + D12 [ρ(t)] u(t)

(1)

where x ∈ Rnx , u ∈ Su ⊆ Rnu , and z ∈ Rmz represent the state, control, and regulated variables respectively, Su is a convex set containing the origin in its interior, ρ denotes a vector of time–varying parameters that can be measured in real time, and where all matrices involved are continuous functions of ρ. Further, we will assume that the set of parameter trajectories is of the form: FΘ = {ρ : ρ(t + 1) ∈ Θ [ρ(t)] , t = 0, 1, . . .}

(2)

where P ⊂ Rnρ is a compact set and Θ : P → P is a given set valued map . Our goal is, given an initial condition xo , and an initial value of the parameter ρo , to find an admissible parameter dependent state–feedback control law u[x(t), ρ(t)] ∈ Su that minimizes the index: J(xo , ρo , u) =

sup

∞

z T (k)z(k), x(0) = xo

(3)

ρ∈FΘ ,ρ(0)=ρo k=0 T D12 = In the sequel, for simplicity, we make the following standard assumptions: D12

I, C1T D12 = 0. In addition, the explicit dependence of matrices on ρ will be omitted, when it is clear from the context. Definition 1 A function Ψ : Rnx × P → R+ that satisfies the following condition: ,, + + 0 such that: ⎡ ⎤ −Y (ρ) + B2 (ρ)B2T (ρ) Y (ρ)AT (ρ) − B2 (ρ)B2T (ρ) Y (ρ)C1T (ρ)

. ⎢ M (ρ) = ⎣A(ρ)Y (ρ) − B2 (ρ)B2T (ρ)

−Y (θ)

0

0

−I

C1 (ρ)Y (ρ)

⎥ ⎦≤0 (5)

for all ρ ∈ P and θ ∈ Θ(ρ), then V (x, ρ) = xT Y −1 (ρ)x is a parameter dependent CLF for system (1), with associated control action given by u(x, ρ) = −B2T (ρ)Y −1 (ρ)x

(6)

Moreover, the corresponding trajectory satisfies: sup

∞

z T (t)z(t) ≤ xTo Y −1 (ρo )xo

(7)

ρ∈FΘ ,ρ(0)=ρo t=0

Proof. Given any admissible parameter trajectory ρ(.) ∈ FΘ , let T T −1 T −1 T T v = x(t) Y [ρ(t)] x(t + 1) Y [ρ(t + 1)] x(t) C1 [ρ(t)]

(8)

Pre/postmultiplying M [ρ(t)] by v T and v yields: xT (t + 1)Y −1 [ρ(t + 1)]x(t + 1) − xT (t)Y −1 [ρ(t)]x(t) ≤ −z T (t)z(t)

(9)

Summing this inequality along the trajectories of the closed–loop system yields ∞ T T −1 [ρ(0)]xo . Since Y > 0 is a continuous function of 0 z (t)z(t) ≤ xo Y ρ(t) ∈ P compact, it follows that there exist some constant c such that Y −1 (ρ) < ∞ cI. Thus 0 z T (k)z(k) is bounded, which implies that z T (t)z(t) → 0 as t → ∞. Finally, it is easy to show that uniform observability of {A, C}, combined with continuity of all the matrices involved and the fact that ρ ∈ P compact, implies uniform observability of the closed loop system {A − B2 B2T Y −1 , C1 − D12 B2T Y −1 }. Thus z(t) → 0 implies that x(t) → 0.


297

2.2 Motivation: A Conceptual Algorithm for Receding Horizon Control of Constrained LPV Systems In this section, motivated by the work in [16] we introduce a conceptual receding horizon control law for constrained LPV systems. This law provides the motivation for the risk–adjusted approach pursued latter in the chapter. Let Ψ : Rnx × P → R+ be a CLF for system (1) such that it satisfies the additional condition: ,, + + T max min max Ψ[A(ρ)x + B2 (ρ)u, θ] + z z − Ψ(x, ρ) ≤0 ρ∈P

u∈Su

θ∈Θ(ρ)

(10)

and, given a horizon N , define (recursively) the following function J(x, ρ, n, N ): J(x, ρ, N, N ) = Ψ(x, ρ) , + T J(x, ρ, i, N ) = min z (i)z(i)+ max J[A(ρ)x + B(ρ)u, θ, i + 1, N ] , i < N u∈Su

θ∈Θ(ρ)

(11) Finally, let x(t), ρ(t) denote the present state and parameter values, and consider the following Receding Horizon control law: uRH [x(t), ρ(t)] = argmin J[x(t), ρ(t), t, t + N ]

(12)

u∈Su

Theorem 1 Assume that the pair {A(.), C(.)} is uniformly observable for all admissible parameter trajectories ρ(t) ∈ FΘ . Then (i) the control law uRH renders the origin an asymptotically stable equilibrium point of (1), and (ii), as N → ∞, its performance approaches optimality monotonically. Proof: Begin by noting that (10) implies that J(x, ρ, t + N, t + N + 1) , + T = min z (t + N )z(t + N ) + max J(Ax + Bu, θ, t + N + 1, t + N + 1) u∈Su θ∈Θ(ρ) , + = min z T (t + N )z(t + N ) + max Ψ(Ax + Bu, θ) u∈Su

θ∈Θ(ρ)

≤ Ψ(x, ρ) = J(x, ρ, t + N, t + N ) It follows that

298


J(x, ρ, t + N − 1, t + N + 1) , + T = min z (t + N − 1)z(t + N − 1) + max J(Ax + Bu, θ, t + N, t + N + 1) u∈Su θ∈Θ(ρ) , + T ≤ min z (t + N − 1)z(t + N − 1) + max J(Ax + Bu, θ, t + N, t + N ) u∈Su

θ∈Θ(ρ)

= J(x, ρ, t + N − 1, t + N ) A simple induction argument shows now that J(x, ρ, t, t + N + 1) ≤ J(x, ρ, t, t + N )

(13)

For a given initial condition and parameter trajectory, let V (t) = J[xcl (t), ρ(t), t, t + N ] where xcl (t) denotes the closed–loop trajectory corresponding to the control law uRH . From the definition of J it follows that V (t) = z T (t)z(t) + max J[xcl (t + 1), θ, t + 1, t + N ] θ∈Θ[ρ(t)]

≥ z T (t)z(t) + J[xcl (t + 1), ρ(t + 1), t + 1, t + N ]

(14)

≥ z T (t)z(t) + V (t + 1) where the last inequality follows from (13). As before, asymptotic stability of the origin follows from a combination of standard Lyapunov and observability arguments. Property (ii) follows from (13) combined with Bellman’s optimality principle [4]. Remark 1 In the unconstrained case (e.g. Su = Rnu ) a CLF is readily available from Lemma 1 since it can be easily shown that Ψ = xT Y −1 (x, ρ)x, where Y satisfies the AMIs (5), satisfies the inequality (10). In addition, by summing ( 14) along the trajectories and using (13) it follows that this CLF choice yields the following worst–case performance bound: sup ρ∈FΘ

∞

z T (k)z(k) ≤ V (0) ≤ xTo Y −1 (ρo )xo

k=0

3 Risk Adjusted Receding Horizon As shown in Theorem 1, the receding horizon control law uRH is guaranteed to stabilize the system while optimizing a (monotonically improving with N ) approximation


299

to the performance index. However, solving the min–max optimization problem (11) is far from trivial, even when making several approximations such as assuming memoryless, arbitrarily fast time–varying parameters5 . To avoid this difficulty, motivated by the work in [18], in the sequel we will search over closed–loop strategies, rather than control actions. For simplicity, we consider first the unconstrained control case and defer the treatment of constraints until section 3.3

3.1 The Unconstrained Control Case In the sequel, we will denote by XN : Θ × {n, n + 1, . . . , n + N } → P the set of all bounded matrix functions that map N –length admissible parameter trajectories to P , the class of all symmetric positive definite matrices. With this definition, consider now the following receding horizon type control law: Algorithm 6 0.- Data: A horizon N , a CLF Y (ρ) that satisfies (5). 1.- Let ρ(n) denote the measured value of the parameter at time n and solve the following LMI optimization problem in X ∈ XN : min γ

subject to:

X ∈XN

−γ xT (n)

x(n) −X(n)

⎡

≤0

A11

A12

A13

AT13

0

−I

⎤

. ⎢ ⎥ M (ρ) = ⎣AT12 −X(n + i + 1) 0 ⎦ ≤ 0

(15)

A11 = −X(n + i) + B2 [ρ(n + i)]B2T [ρ(n + i)] A12 = X(n + i)AT [ρ(n + i)] − B2 [ρ(n + i)]B2T [ρ(n + i)]) A13 = X(n + i)C1T [ρ(n + i)] for all ρ(n + i + 1) ∈ Θ[ρ(n + i)], i = 0, 1, . . . , N − 1

with boundary condition X(n + N ) = Y [ρ(n + N )], and where, with a slight notational abuse, we denote X[ρ(t), t] simply as X(t). 5

In this case the problem becomes essentially equivalent to synthesizing a controller for a system subject to parametric uncertainty, and it is well known that these problems are generically NP hard.

300


2.- At time n use as control action u[ρ(n), x(n), N ] = −B2T [ρ(n)]X −1 [ρ(n), n]x(n)

(16)

3.- Set n = n + 1 and go to step 1. Theorem 2 Assume that the pair {A(ρ), C(ρ)} is uniformly observable for all admissible parameter trajectories. Then, the control law (16) renders the origin an asymptotically stable point of the closed loop system for all ρ(.) ∈ FΘ . Moreover, performance improves monotonically with the horizon length N . Proof To establish stability, let x(n), ρ(n) and X(n + i), i = 1, . . . , N , denote the present state and parameter values and the corresponding solution to (15). Since {X(n + 1), X(n + 2), . . . , Y [ρ(n + N )], Y [ρ(n + N + 1)} is a feasible solution for (15) starting from any ρ(n + 1) ∈ θ[ρ(n)], we have that xT (n + 1)X −1 [ρ(n + 1)]x(n + 1) ≤ xT (n)X −1 [ρ(n)]x(n) − z T (n)z(n)

(17)

From an observability argument it can be easily shown that this, coupled with the fact that X > 0, implies that xT (n)X −1 (n)x(n) → 0 as n → ∞. Combined with the fact that X −1 (n) is bounded away from zero (since X is bounded), this implies that x(n) → 0. Finally, the second property follows from the fact that if {X(n), X(n + 1), . . . , Y [ρ(n + N )]} is a feasible solution for (15) with horizon N , then {X(n), X(n + 1), . . . , Y [ρ(n + N )], Y [ρ(n + N + 1)]} is feasible for N + 1. Thus γ(N + 1) ≤ γ(N ). Remark 2 Note that, for any N , the sequence: {Y [ρ(n)], Y [ρ(n + 1)], . . . , Y [ρ(n + N )]}

(18)

is a feasible solution for (15). Thus, if there exist a sequence of matrices X(ρ, i) that yields a lower value of the bound xT (n)X −1 [ρ(n), n]x(n) the optimization (15) will find it. Hence the proposed controller will outperforms the standard LPV control


301

law (6). However, its computational complexity is comparable (or worse), since it requires finding feasible solutions to both (5) and (15). However, since the solution to (5) is used only as a boundary condition for (15) at t = n+N , even if a conservative solution is used here6 , performance degradation is considerably less severe than the one incurred by using this conservative solution to compute the control action, for instance via (6). Further, as we show in the sequel, the computational complexity of solving (15) can be substantially reduced using a risk–adjusted approach.

3.2 Stochastic Approximation In principle, infinite–dimensional optimization problems of the form (15) can be (approximately) converted to a finite dimensional optimization by using a finite exm pansion X(ρ, t) = i=1 Xi (t)hi (ρ), where hi (.) are known continuous functions ([1]). However, the computational complexity of the resulting problem grows exponentially with the horizon length N . In this chapter, we propose to avoid this difficulty by pursuing a stochastic approximation approach, whose complexity grows polynomially with N . To this effect, assume that ρˆ = [ρ(n + 1)ρ(n + 2) · · · ρ(n + N )] has a non–zero probability density for all ρ ∈ Fθ . Note that, to compute u(n), one only needs X[ρ(n)]. This observation allows for reformulating the optimization problem (15) to eliminate the need to explicitly compute X[ρ(n+1)]...X[ρ(n+N )] as follows: Given a fixed instant n, ρ(n) and ρˆ ∈ Fθ , define

⎡

−γ xT (n)

⎤

⎢ ⎢ x(n) −X(n) ⎢ . ⎢ M [ρ(n)] M (n) = ⎢ ⎢ .. ⎢ . ⎣

⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦ M [ρ(n + N − 1)]

In terms of M , the constraints in the optimization problem (15) can be expressed as: . λ (γ, X(n), ρ(n), ρˆ) = 6

min

X(n+1),...X(n+N )

5 6 λmax M (n) < 0.

Such a solution can be found using risk–adjusted methods following the approach used in [7].

302


Thus, in this context, one does not need to compute the explicit value of X(i), i = n + 1, . . . , n + N , as long as the minimum above can be computed. Note in passing, that this reformulation preserves convexity, since the function λ(·) is a convex function of its arguments. To complete the approximation of the original optimization problem by a stochastic one, given ζ > 0, define g˜(x) =

eζx − 1 ζ

and collect all the optimization variables (e.g., γ and the entries of X(n)) in a vector xn . Define the following functions . f0 (xn , ρ(n), ρˆ) = γ; . f1 (xn , ρ(n), ρˆ) = g˜ [λ (γ, X(n), ρ(n), ρˆ)] . Consider now the following convex problem min Eρˆ [f0 (xn , ρ(n), ρˆ)] = min γ s.t. Eρˆ [f1 (xn , ρ(n), ρˆ)] ≤ 0;

(19)

where Eρˆ[·] denotes the expected value with respect to the random variable ρˆ. It can be easily shown that the solution to this problem tends to the solution of the problem (15) as ζ → ∞. We are now ready to provide the main result of this section: an algorithm for solving problem (19) in polynomial time. For technical reasons, in the sequel we will assume that the solution to this problem is known to belong to a given compact convex set X (where the matrix X(n) has bounded entries and is positive definite). Let π(·) denote the projection onto X ; i.e., ˜ 2 . π(x) = arg min x − x ˜ ∈X x

and consider the following algorithm: Algorithm 7 1. Generate a feasible solution Y (ρ) to (5), using for instance the procedure proposed in [7]. 2. Initialization: Determine x0n , yi0 ; i = 0, 1 and z10 . Let k = 0. 3. Generate a sample ρˆk = [ρk (n + 1), . . . , ρk (n + N − 1)]


303

4. Let If z1k ≤ −γ0 /k γ then 6 5 k k = π x − b y xk+1 k n n 0 . 6 5 k k = π x − b y Otherwise, xk+1 k n n 1 .. $

5. Let yik+1 = yik + ak

% ∂fi (xn , ρ(n), ρˆk ) − yik ∂xn xn =xk n

i = 0, 1 and

! z1k+1 = z1k + ak f1 (xn , ρ(n), ρˆk ) − z1k ;

6. If z1l < 0 and |γk−1 − γk | < for l = k − Ngood + 1, . . . , k + 1 stop. Otherwise, let k = k + 1 and go to step 3. Theorem 3 Let ak =

α0 kα ;

bk =

β0 , kβ

where α0 , α, β0 and β are positive constants.

Furthermore, assume that γ0 and γ are also positive. Then, if ∞

ak =

k=0

∞

bk = ∞;

k=0

∞

bk =0 k→∞ ak

a2k < ∞; lim

k=0

and 2β − α − 2γ > 1 the sequence xkn converges with probability one to the solution of the problem (19). Proof: Direct application of Theorem 1 in [8].

3.3 Adding Control Constraints In this section we briefly indicate how to modify the algorithm presented above to handle constraints in the control action. A difficulty here is that, while it is relatively easy to incorporate these constraints when searching for a control action in the minimization of (3), this is not the case when dealing instead with closed–loop control strategies of the form (16). Following [17], we propose to accomplished this indirectly, by suitably scaling some of the terms in (15). Specifically, assume that the control constraint set is of the form Su {u : u ∞ ≤ 1} and consider the following control law:

304


Algorithm 8 0.- Data: A horizon N , an initial condition x(0) = xo 0.1- Find a CLF Y (ρ, τ, x) that satisfies the following (functional) LMI, parametric in ρ, x and the scalar τ: ⎡

A12

AT12

−Y (θ, τ, x)

0

AT13

0

−τ · I

B2T (ρ)

0

0

⎢ ⎢ ⎢ ⎣

max ρ∈P

⎤

−Y (ρ, τ, x)

θ ∈ Θ(ρ)

A13 B2 (ρ)

⎥ ⎥ ⎥≤0 0 ⎦ 0

(20)

−τ · I

A12 = Y (ρ, τ, x)AT (ρ) − B2 (ρ)B2T (ρ) A13 = Y (ρ, τ, x)C1T (ρ) 1 xT 1 bTi ≥ 0, ≥ 0, i = 1, nu x Y (ρ, τ, x) bi Y (ρ, τ, x)

(21)

where bi denotes the ith column of B2 (ρ). 0.2- Let τmin (ρ, xo ) = inf {τ : (20)–(21) are feasible} with x = xo and define τo = maxρ∈P τmin (ρ, xo ). 1.- Let ρ(n) denote the measured value of the parameter at time n and solve the following LMI optimization problem in X ∈ XN and the scalar γ: min γ, subject to:

X ∈XN

−γ xT (n) x(n) −X(n)

1

≤0

(22)

τ(n + i) ≤ τo (xo )

T

x (n)

≥ 0,

x(n) X(n)

bTj (n

1

+ i)

≥ 0;

bj (n + i) X(n + i)

(23)

j = 1, . . . , nu , i = 0, 1, . . . , N − 1 ⎡

−X(n + i)

A12

A13

B2 (n + i)

AT12

−X(n + i + 1)

0

0

AT13

0

−τ(n + i) · I

0

B2T (n + i)

0

0

−τ(n + i) · I

⎢ ⎢ ⎢ ⎣

⎤ ⎥ ⎥ ⎥≤0 ⎦

(24)


305

for all ρ(n + i + 1) ∈ Θ[ρ(n + i)], with boundary condition X(n + N ) = Y [ρ(n + N ), τ(n + N ), x(n + N )], where A12 = X(n + i)AT (n + i) − B2 (n + i)B2T (n + i) A13 = X(n + i)C1T (n + i) and i = 1, 2, . . . , N and where Y (.) satisfies the AMI (20). 2.- At time n use as control action u(n, N ) = −B2T [ρ(n)]X −1 [ρ(n), n]x(n)

(25)

3.- Set n = n + 1 and go to step 1. Theorem 4 Assume that the origin is an exponentially stable equilibrium point of A[ρ(.)] for all ρ ∈ FΘ and the pair {A(ρ), C(ρ)} is uniformly observable for all admissible parameter trajectories. Then, the control law (25) (i) is admissible, in the sense that it satisfies the control constraints, and (ii) it renders the origin a globally exponentially stable point of the closed loop system for all admissible parameter trajectories. Proof. See the Appendix. Remark 3 If the open loop system is not exponentially stable but the pairs (A, B2 ) and (A, C1 ) are uniformly controllable and observable respectively, the algorithm above will locally stabilize the system in some neighborhood of the origin, which 7 can be estimated by computing the set S = ρ∈P S(ρ), where S(ρ) = x : lim xT X −1 (ρ, τ)x < ∞ τ→∞

Remark 4 As before, the computational complexity of the optimization problem above grows exponentially with the horizon, even when approximating X by a finite expansion. However, the stochastic approach used in Algorithm 7 can also be used here, with minimal modifications, to obtain polynomial growth approximations.

306


4 Illustrative Example Consider the following discrete time LPV system: 1 0.1 A(ρ(t)) = , B2 (ρ(t)) = 0 0.0787 0 1 − 0.1ρ(k) C1 (ρ(t)) =

1 , D12 (ρ(t)) = √ 1 1 2 −1 0 1 0

with admissible parameter set FΘ = {ρ(t) : ρ(t + 1) ∈ [−1, 1], t = 0, 1, . . .} .

(26)

It can be verified that the matrix function: Y (ρk ) = Y0 + Y1 ρk + Y2 ρk 2 0.0296 −0.0196 Y0 = −0.0196 0.0290 0.0003 −0.0016 Y1 = −0.0016 0.0044 −0.0691 0.0302 Y2 = 10−3 0.0302 0.2542 satisfies(5). Thus V (x, ρ) = xT Y −1 (ρ)x is a parameter dependent CLF for the plant. Figure 1 shows a typical parameter trajectory and the corresponding cost evolution.

In this case the initial condition was x0 = [0.1 0.1] , and the following values were used for the risk adjusted RH controller: N = 10, ζ = 30, α = 0.6, α0 = 1, β = 1, β0 = 10−3 , γ = 0.15 and γ0 = 10−6 . At each time instant, an approximate solution to (15) was obtained using 100 iterations of the stochastic approximation. On a Intel Pentium III 1.6GHz, the total CPU time to perform this iteration is 81.88 seconds. As reference, solving the parametric LMI (5) takes 474.44 seconds, although this needs to be done only once, off–line. The overall cost of the trajectory is J = 1.0886. For comparison, the standard LPV controller (6) results in a cost J = 1.2805. Thus, in this case the risk adjusted controller yields roughly a 20% performance improvement. Similar results were obtained for other initial conditions and parameter trajectories.


307

Fig. 1. Top: Parameter Trajectory. Bottom: Cost Evolution

5 Conclusions Practical tools for synthesizing controllers for LPV systems have emerged relatively recently and are still far from complete. Among others, issues not completely solved yet include non–conservative handling of performance specifications and overall computational complexity.

308


In this chapter we take some steps towards removing these limitations by combining Receding Horizon and Control Lyapunov Function ideas in a risk–adjusted framework. Motivated by some earlier results on regulation of LTV and LPV systems ([18, 16]) the main idea of the chapter is to recast the infinite horizon regulation problem into a (approximately) equivalent finite horizon form, by searching over a suitable set of closed–loop strategies. As shown here, this leads to a globally stabilizing control law that is guaranteed to outperform techniques currently used to deal with LPV systems. However, in principle this is achieved at the expense of computational complexity, since this law requires the on–line solution of a set of functional LMIs. We propose to address this difficulty by using a risk–adjusted approach, where in exchange for a slight probability of constraint violation, one obtains a substantial reduction in computational complexity. Moreover, this approach scales polynomially, rather than exponentially, with system size ([10, 19]). These results were illustrated with a simple example where a risk-adjusted receding horizon controller was used to control a second order LPV plant. As shown there the proposed risk-adjusted receding horizon controller improves performance vis-a-vis a conventional LPV controller, while substantially reducing the computational effort required by a comparable Receding Horizon controller. Research is currently under way seeking to extend these results to the output feedback case and to address the issue of model uncertainty, both parametric and dynamic.

Acknowledgements This research was supported in part by NSF under grants ANI-0125653, ECS-9984260, ECS-0115946, ECS-0221562, and IIS-0312558.

References 1. G. BALAS et al. (1997). Lecture Notes, Workshop on Theory and Application of Linear Parameter Varying Techniques, 1997 ACC. 2. B. R. Barmish and C. M. Lagoa (1997). “The Uniform Distribution: A Rigorous Justification for its Use in Robustness Analysis”, Mathematics of Control, Signals and Systems,10, 203–222.


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3. I. Batina, A. A. Stoorvogel and S. Weiland (2001). “Stochastic Disturbance Rejection in Model Predictive Control by Randomized Algorithms,” Proc. 2001 ACC, Arlington, Va, pp. 732–737. 4. R. E. Bellman and S. E. Dreyfus, (1962). Applied Dynamic Programming, Princeton University Press. 5. G. Calafiore, F. Dabbene and R. Tempo (2000). “Randomized Algorithms for Probabilistic Robustness with Real and Complex Structured Uncertainty,” IEEE Trans. on Autom. Contr., 45,12, 2218–2235. 6. X. Chen and K. Zhou (1997). “ A Probabilistic Approach to Robust Control”, Proc. 36 IEEE CDC, pp. 4894–4895. 7. Y. Fujisaki, F. Dabbene and R. Tempo (2001) “Probabilistic Robust Design of LPV Control Systems,” Proc. 2001 IEEE CDC, Orlando, FL, pp. 2019–2024. 8. A. M. Gupal (1974). “A Certain Stochastic Programming Problem with Constraints of a Probabilistic Nature,” Kibernetika (Kiev), n. 6, pp. 94–100. 9. A. Jadbabaie, J. Yu and J. Hauser (1999). “Receding horizon control of the Caltech ducted fan: a control Lyapunov function approach,” Proc. 1999 IEEE CCA, pp. 51–56,. 10. P. Khargonekar and A. Tikku (1996). “Randomized Algorithms for Robust Control Analysis and Synthesis Have Polynomial Complexity”, Proceedings of the 35th IEEE CDC, pp. 3470–3475. 11. M. Kothare, V. Balakrishnan and M. Morari (1996). “Robust Constrained Model Predictive Control Using Linear Matrix Inequalities,” Automatica, 32, 10, pp. 1361–1379. 12. D. Q. Mayne, J. B. Rawlings, C.V. Rao and P. O. M. Scokaert (2000). “Constrained Model Predictive Control: Stability and Optimality,” Automatica, 36, 6, pp. 789–814. 13. L. Ray and R. Stengel (1993). “A Monte Carlo Approach to the Analysis of Control System Robustness”, Automatica,29,1, 229–236. 14. B. T. Polyak and R. Tempo (2000). “Probabilistic Robust Design with Linear Quadratic Regulators,” Proc. 39 IEEE CDC, pp. 1037–1042. 15. J. S. Shamma and M. Athans (1992). “Gain Scheduling: Potential Hazards and Possible Remedies,” IEEE. Contr. Sys. Mag., 12, 1, pp. 101–17. 16. M. Sznaier (1999). “Receding Horizon: An easy way to improve performance in LPV systems,” Proc. 1999 ACC,” pp. 2257–2261. 17. M. Sznaier and R. Suarez (2000). “Control of Constrained Systems using Receding Horizon Control Lyapunov Functions,” Workshop on Systems with Time–Domain Constraints, Eindhoven University of Technology, Eindoven, The Netherlands. 18. G. Tadmor (1992). “Receding horizon revisited: and easy way to robustly stabilize an LTV system,” Systems and Control Letters, 18, pp. 285–294. 19. R. Tempo, E. W. Bai, and F. Dabbene, F. (1996). “Probabilistic Robustness Analysis: Explicit Bounds for the Minimum Number of Samples,” Proc. 35th IEEE CDC, pp. 3424– 3428.

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A Computing Subgradient of λmax (M (x) ) In this section we describe how to compute a subgradient of λmax (M (x) ). To com˜, pute such a subgradient first note that, given x λmax (M (˜ x) ) = max y T M (˜ x)y = y ∗ T M (˜ x)y ∗

y 2 =1

where y ∗ is an eigenvector of euclidean norm one associated with maximum eigenvalue of M (˜ x). Given that the maximum above is achieved by y ∗ then ∗T ∗ ∂λmax (M (x) ) ∂y M (x)y = . ∂x ∂x x=˜ x x=˜ x

This can be easily computed since, in the case addressed in this chapter, M (x) is an affine function of x.

B Proof of Theorem 4 In order to prove this theorem we need the following preliminary result: Lemma 2 Assume that the origin is an exponentially stable equilibrium point of the open loop system A [ρ(t)] for all admissible parameter trajectories and the pair {A, C} is uniformly observable. Let Y [ρ(t), τ] denote a feasible solution to (20) corresponding to a given τ. Then, given τ1 > τ2 , (20) admits a solution Y [ρ(t), τ1 ] > Y [ρ(t), τ2 ]. Proof: Let δ = τ2 − τ1 . From the hypotesis it follows that for all admissible parameter trajectories ρ ∈ Fθ there exists some Δ [ρ(t)] > 0 that satisfies the inequality: ⎤ ⎡ −Δ [ρ(t)] Δ [ρ(t)] AT [ρ(t)] Δ [ρ(t)] C1T [ρ(t)] 0 ⎥ ⎢ ⎥ ⎢ A [ρ(t)] Δ [ρ(t)] −Δ[ρ(t + 1)] 0 0 ⎥ Y [ρ(t), τ2 ]. Adding (27) and (20) we have that Y [ρ(t), τ1 ] satisfies (20) for the value τ = τ1 .


311

Proof of Theorem 4 Controllability of the pair{A(.), B(.)} implies that for some τ > 0 large enough, (20) has a solution Y > 0. From Lemma 2 it follows then that as τ → ∞, the LMI (20) always admits a solution Y → ∞, which in turn implies that, for any initial condition xo , there always exist some finite τ such that (21) is feasible. Since P is compact, τo in step [0.2] is well defined. Consider now a fixed value of τ ≤ τo and let v = x(t)T Y −1 [ρ(t)] x(t + 1)T Y −1 [ρ(t + 1)]

x(t)T C1T [ρ(t)] B2T [ρ(t)]Y −1 [ρ(t)]x(t) τ τ

(28) Pre/postmultiplying (20) by v and v T yields, after some algebra: T

x (t + 1)Y

−1

[ρ(t + 1), τ]x(t + 1) − x (t)Y T

−1

z(t) 2 [ρ(t)] x(t) ≤ − τ

(29)

From the definition of τo it follows that Y (ρ, τo ) satisfies (20) for all admissible parameter trajectories ρ ∈ FΘ . This fact, together with (29) implies that the LMIs(22)– (24) are feasible for all parameter trajectories, since X = Y is a feasible solution. A compactness argument shows that along the trajectories of the closed loop system c1 I > X(t) > c2 I for some constants c1 , c2. Finally, pre/postmultiplying (24) by x(t)T C1T (t) B2T (t)X −1 (t)x(t) T −1 T −1 v = x(t) X (t) x(t + 1) X (t + 1) τ τ and v T yields: xT (t + 1)X −1 (t + 1)x(t + 1) − xT (t)X −1 (t)x(t) ≤ −

z(t) 2 τ(t)

(30)

From this point on, the proof of asymptotic stability proceeds along the lines of the proof of Theorem 4 using (30) instead of (17). To show that the control law (25) satisfies the constraints |ui (.)| ≤ 1, note that |ui (n)| = | − bTi (n)X −1 (n)x(n)| ≤ bTi (n)X − 2 (n) 2 X − 2 (n)x(n) 2 ≤ 1 1

1

where the last inequality follows from the LMIs (23), coupled with the fact that, along the closed loop trajectories xT (n + i)X −1 x(n + i) ≤ xT (n)X −1 (n)x(n) ≤ 1 .

312


Note in passing that from the proof above it follows that ∞ zT z 1 T ≤ xTo X −1 (xo , ρo , τo )xo z (t)z(t) ≤ τo t=0 τ(t) t

This gives an intuitive explanation of τ: it is a parameter used to scale the performance index so that the control constraints are satisfied.

Case Studies on the Control of Input-Constrained Linear Plants Via Output Feedback Containing an Internal Deadzone Loop Dan Dai1 , Tingshu Hu2 , Andrew R. Teel1 , and Luca Zaccarian3 1

Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, USA diane|[email protected]

2

Department of Electrical and Computer Engineering, University of Massachusetts Lowell, MA 01854, USA [email protected]

3

Dipartimento di Informatica, Sistemi e Produzione, University of Rome, Tor Vergata, 00133 Rome, Italy [email protected]

Summary. In this chapter we address several case studies using LMI optimization methods for designing output feedback control laws to achieve regional performance and stability of linear control systems with input saturation. Algorithms are developed for minimizing the upper bound on the regional L2 gain for exogenous inputs with L2 norm bounded by a given value, and for minimizing this upper bound with a guaranteed reachable set or domain of attraction. Based on the structure of the optimization problems, using the projection lemma, the output feedback controller synthesis is cast as a convex optimization over linear matrix inequalities. The problems are studied in a general setting where the only requirement on the linear plant is detectability and stabilizability.

keywords: Output feedback control, input saturation, L2 gain, reachable set, domain of attraction, LMIs

1 Introduction The behavior of linear, time-invariant (LTI) systems subject to actuator saturation has been extensively studied for several decades. More recently, some systematic

314

Dan Dai, Tingshu Hu, Andrew R. Teel, and Luca Zaccarian

design procedures based on rigorous theoretical analysis have been proposed through various frameworks. Most of the research efforts geared toward constructive linear or nonlinear control for saturated plants can be divided into two main strands. In the first one, called anti-windup design, a pre-designed controller is given, so that its closed-loop with the plant without input saturation is well behaved (at least asymptotically stable but possibly inducing desirable unconstrained closed-loop performance). Given the pre-designed controller, anti-windup design addresses the controller augmentation problem aimed at maintaining the pre-designed controller behavior before saturation and introducing suitable modifications after saturation so that global (or regional) asymptotic stability is guaranteed (local asymptotic stability already holds by the properties of the saturation nonlinearity). Anti-windup research has been largely discussed and many constructive design algorithms have been formally proved to induce suitable stability and performance properties. Many of these constructive approaches (see, e.g., [4, 5, 8, 9, 10, 11, 12, 20, 24, 31]) rely on convex optimization techniques and provide Linear Matrix Inequalities (LMIs) [2] for the anti-windup compensator design. The second research strand, can be called “direct design”, to resemble the fact that saturation is directly accounted for in the controller design and that no specification or constraint is imposed on the behavior of the closed-loop for small signals. Direct designs for saturated systems range from the well-known Model Predictive Control (MPC) techniques [21], especially suitable for discrete-time systems) to sophisticated nonlinear control laws which are able to guarantee global asymptotic stability for all linear saturated and globally stabilizable plants (see, e.g., the scheduled Riccati approach in [22] and the nested saturations of [26, 28]). Several LMI-based methods for direct controller design for linear plants with input saturation have also been proposed (see, e.g., [6, 19, 23, 25]). It is not our scope to mention here all the extensive literature on direct design for saturated systems, but it is worth mentioning that several constructive methods are available that differ in simplicity, effectiveness and formality. Compared to anti-windup control, direct design is a simpler task to accomplish, because there’s no constraint on the closed-loop behavior for small signals. Therefore, the designer has full freedom in the selection of the controller dynamics. Antiwindup design, on the other hand, allows to guarantee that a certain prescribed unconstrained performance is met by the closed-loop as long as the saturation limits are

Output Feedback for Input-Constrained Linear Plants

315

not exceeded (this performance often consists in some linear performance measure when a linear plant+controller pair is under consideration) and that this performance is gradually deteriorated as signals grow larger and larger outside the saturation limits. In this chapter, we will discuss and illustrate on several examples a synthesis method for the construction of output feedback controllers with an internal deadzone loop. This type of structure corresponds to the typical framework used since the 1980’s for the design of control systems for saturated plants. See for example the work in [29, 3, 30, 13, 27, 6] and other references in [1]. In our approach we will use the same tools used in our recent papers [17] for static anti-windup design, and we will recast the underlying optimization problem for the selection of all the controller matrices (whereas in [17] only the static anti-windup gain was selected and the underlying linear controller matrices were fixed). This approach parallels the approach proposed in [23] where classical sector conditions were used and extra assumptions on the direct input-output link of the plant were enforced. A similar assumption was also made in the recent paper [7] which uses similar tools to ours to address both magnitude and rate saturation problems in a compensation scheme with lesser degrees of freedom than ours. Here we use the regional analysis tool adopted in [17], and we extend the general output feedback synthesis to characterize the regional L2 gain and reachable set for a class of norm bounded disturbance inputs, as well as the estimate of domain of attraction. The overall synthesis is cast as an optimization over LMIs, and under a detectability and stabilizability condition on the plant, the proposed design procedure will always lead to regionally stabilizing controllers if the plant is exponentially unstable, to semi-global results if the plant is non-exponentially unstable, and to global results if the plant is already exponentially stable. An interesting advantage of the approach proposed here is that due to the type of transformation that we use, it is possible to derive system theoretic interpretation of the feasibility conditions for the controller design (such as stabilizability and detectability of the pant). This result is novel and was not previously observed in [23]. This chapter is organized as follows: In Section 2 we formulate three problems that will be addressed in the chapter; in Section 3 we state the LMI-based main conditions for output feedback controller synthesis and the procedure for the controller construction; in Section 4 we give the feasible solutions for the problems we presented

316


in Section 2; in Section 5 we illustrate the proposed constructions on several case studies. Notation For compact presentation of matrices, given a square matrix X we denote HeX := X + X T . For P = P T > 0, we denote E(P ) := {x : xT P x ≤ 1}.

2 Problem Statement Consider a linear saturated plant, ⎧ ⎪ ⎪ ⎨ x˙ p = Ap xp + Bp,u sat(yc ) + Bp,w w P y = Cp,y xp + Dp,yu sat(yc ) + Dp,yw w ⎪ ⎪ ⎩z = C x + D sat(y ) + D w p,z p

p,zu

c

(1)

p,zw

where xp ∈ np is the plant state, yc ∈ nu is the control input subject to saturation, w ∈ nw is the exogenous input (possibly containing disturbance, reference and measurement noise), y ∈ ny is the measurement output and z ∈ nz is the performance output. The goal of this chapter is the synthesis of a plant-order linear output feedback controller with internal deadzone loops: x˙ c = Ac xc + Bc y + E1 dz(yc ) C yc = Cc xc + Dc y + E2 dz(yc ),

(2)

where xc ∈ nc (with nc = np ) is the controller state, yc ∈ nu is the controller output and dz(·) : nu → nu is the deadzone function defined as dz(yc ) := yc − sat(yc ) for all yc ∈ nu with sat(·) : nu → nu being the symmetric saturation function having saturation levels u ¯1 , . . . , u ¯nu with its i-th component satu¯i (·) depending on the i-th input component yci as follows: ⎧ ⎪ ¯i , if yci ≥ u ¯i , ⎪ ⎨u satu¯i (yci ) = yci , if − u ¯i ≤ yci ≤ u ¯i , ⎪ ⎪ ⎩ −¯ u , if y ≤ −¯ u. i

ci

(3)

i

The resulting nonlinear closed-loop (1), (2), is depicted in Figure 1. The same output feedback controller structure was considered in [23], where convex synthesis methods for global (rather than regional, as we consider here) stability and

Output Feedback for Input-Constrained Linear Plants w C

yc

317

z P

y

Enhanced Controller

Fig. 1. The linear output feedback control system with deadzone loops.

performance were developed. In [23], it was assumed for simplicity that Dp,yu = 0 (we will remove this assumption here). It is well known that linear saturated plants are characterized by weak stabilizability conditions. In particular, since by linearity the controller authority becomes almost zero for arbitrarily large signals, then global asymptotic stability can only be guaranteed for plants that are not exponentially unstable, while global exponential stability can only be guaranteed if the plant is already exponentially stable. Due to this fact, global results are never achievable (not even with a nonlinear controller) when wanting to exponentially stabilize plants that are not already exponentially stable. On the other hand, local and regional results are always achievable and semiglobal ones are achievable with non-exponentially unstable plants. The following three regional properties are then relevant for the controller design addressed here: Property 1. Given a set Sp ⊂ np , the plant (1) is Sp -regionally exponentially stabilized by controller (2) if the origin of the closed-loop system (1), (2) is exponentially stable with domain of attraction including Sp × Sc (where Sc ⊂ nc is a suitable set including the origin). Property 2. Given a set Rp ⊂ np and a number s > 0, controller (2) guarantees (s, Rp )-reachability for the plant (1) if the response (xp (t), xc (t)), t ≥ 0 of the closed-loop system (1), (2) starting from the equilibrium point (xp (0), xc (0)) = (0, 0) and with w 2 < s, satisfies xp (t) ∈ Rp for all t ≥ 0. Property 3. Given two numbers s, γ > 0, controller (2) guarantees (s, γ)-regional finite L2 gain for the plant (1) if the performance output response z(t), t ≥ 0 of the closed-loop system (1), (2) starting from the equilibrium point (xp (0), xc (0)) = (0, 0) and with w 2 < s, satisfies z 2 < γ w 2 .

318


Based on the three properties introduced above, in this paper we are interested in providing LMI-based design tools for the synthesis of an output feedback controller of the form (2) guaranteeing suitable stability, reachability and L2 gain properties on the corresponding closed-loop. In particular, we will address the following problems: Problem 1. Consider the linear plant (1), a bound s on w 2 , a desired reachability region Rp and a bound γ on the desired regional L2 gain. Design a linear output feedback controller (2) guaranteeing (s, Rp ) reachability, (s, γ)-regional finite L2 gain and which maximizes the exponential stability region Sp of the closed-loop (1), (2). Problem 2. Consider the linear plant (1), a bound s on w 2 , a desired stability region Sp and a bound γ on the desired regional L2 gain. Design a linear output feedback controller (2) guaranteeing Sp regional exponential stability, (s, γ)-regional finite L2 gain and which minimizes the (s, Rp ) reachability region of the closed-loop (1), (2). Problem 3. Consider the linear plant (1), a bound s on w 2 , a desired stability region Sp and a desired reachability region Rp . Design a linear output feedback controller (2) guaranteeing Sp regional exponential stability, (s, Rp ) reachability and which minimizes the (s, γ)-regional finite L2 gain of the closed-loop (1), (2).

3 LMI-Based Design In this section, a set of main feasibility conditions for solving Problems 1 to 3 will be presented in addition to giving a constructive procedure to design a state space representation of the linear output feedback controller (2).

3.1 Main Feasibility Theorem The results that we will derive are based on the sector description of the deadzone originally developed in [15, 14]. The main idea of the description is as follows. For a scalar saturation function satu¯ (·), if |v| ≤ u ¯ then satu¯ (u) is between u and v for all u ∈ . Applying this description to deal with the saturating actuator in Figure 1, we will have satu¯i (yci ) between yci and Hi x as long as |Hi x| ≤ u ¯i . Here x is the


319

combined state in Figure 1 and Hi can be any row vector of appropriate dimensions. It turns out that the choice of Hi can be incorporated into LMI optimization problems. It should be noted that [6] and [9] exploit the same idea to deal with saturations and deadzones, however in our approach yc and x are related to each other in a more general way, rather than yc = F x, as in [6, 9]. The following theorem will be used in the following sections to provide solutions to the problem statements given in Section 2. Theorem 1. Consider the linear plant (1). If the following LMI conditions in the T unknowns Q11 = QT11 > 0, P11 = P11 > 0, γ2 > 0, Yp ∈ nu ×np , K1 ∈ np ×npy ,

K2 ∈ npy ×npy , K3 ∈ npz ×npy are feasible: ⎡ ⎤ A Q + Bpu Yp 0 Bpw ⎢ p 11 ⎥ γ2 I ⎥ 0. I P11 2 2 u ¯i /s Ypi ≥ 0, YpiT Q11

(4a)

K2 Dp,yw − I/2

⎤ 0 0

Dp,zw + K3 Dp,yw −γ2 I/2

⎥ ⎥ 0 diagonal. in the unknowns Ω ¯ U in Step 4. Computation of the controller matrices. From the matrices U and Ω ¯ as Step 3, compute the matrix Ω ¯ := Ω

A¯c C¯c

¯c B ¯c D

¯1 E ¯2 E

¯ U diag(I, I, U −1 ). := Ω Finally, the controller parameters in Ω can be determined applying the following transformation: Ω :=

Ac Bc E1

Cc Dc E2 ⎡ ⎤ I 0 0 ⎢ ⎥ ¯ ⎢ −XD ¯ p,yu C¯c X ¯ XD ¯ p,yu (I − E ¯2 ) ⎥ = Ω ⎣ ⎦ 0

¯ c )−1 . ¯ := (I + Dp,yu D where X

0

I


323

4 Feasibility of the Output Feedback Synthesis Problems In this section, the feasibility conditions established in Theorem 1 will be used to provide conditions for the solvability of Problems 1 to 3.

4.1 Feasibility of and Solution to Problem 1 We first use the result of Theorem 1 to give a solution to Problem 1. To this aim, we use the guaranteed L2 performance and reachability region of items 1 and 3 of Theorem 1 and maximize the size of the guaranteed stability region by maximizing the size of the ellipsoid E((s2 Q11 )−1 ) which, according to item 2 of Theorem 1 is an estimate of the domain of attraction. We state the corollary below for a generic measure αR (·) of the size of the ellipsoid E((s2 Q11 )−1 ). This is typically done with respect to some shape reference of the desired stability region Sp . Corollary 1. Given s, Rp and γ, a solution to Problem 1 is given (whenever feasible) by applying Procedure 1 to the optimal solution of the following maximization problem: sup Q11 ,P11 ,K1 ,K2 ,K3 ,Yp

,γ2

αR (E((s2 Q11 )−1 )), subject to

(4a), (4b), (4c), (4d),

(10a)

2 E(Q−1 11 /s ) ⊂ Rp

(10b)

The formulation in Corollary 1 can be easily particularized to the problem of maxi2 2 −1 mizing the volume of E(Q−1 )) = det(s2 Q11 ). 11 /s ) by selecting αR (E((s Q11 )

Alternative easier selections of αR can correspond to maximizing the size of a region which has a predefined shape. For example, when focusing on ellipsoids, one can seek for stability regions of the type Sp = E(Sp−1 ) = {xp : xTp (αSp )−1 xp ≤ 1},

(11)

where α is a positive scalar such that larger values of α correspond to larger sets Sp . Then the optimization problem (10) can be cast as

324


α, subject to

sup Q11 ,P11 ,K1 ,K2 ,K3 ,Yp ,γ

(4a), (4b), (4c), (4d), (10b)

(12a)

αSp ≤ s2 Q11 .

(12b)

Similarly, if one takes a polyhedral reference region: Sp = αco{x1 , x2 , ..., xnp },

(13)

then constraint (12b) can be replaced by 2 xpi T Q−1 11 xpi ≤ αs ,

or

αs2 xTpi xpi Q11

(14)

≥ 0,

i = 1, . . . , np .

(15)

We finally note that the constraint (10b) on the guaranteed reachability region can be expressed by way of different convex (possibly LMI) conditions depending on the shape of the set Rp . Guidelines in this direction are given in the following section. Remark 2. Based on Corollary 1, reduced LMI conditions can be written to only maximize the estimate of the domain of attraction without any constraint on the other performance measures: sup

αR (E((s2 Q11 )−1 )), subject to

Q11 ,P11 ,K1 ,Yp

He[Ap Q11 + Bpu Yp ] < 0

(16a)

He[P11 Ap + K1 Cpy ] < 0

(16b)

(4c), (4d)

(16c)

From (16) it is straightforward to conclude that if the plant is exponentially stable, then global exponential stability and finite L2 gain can be achieved by the proposed output feedback controller (Yp = 0 and K1 = 0 are sufficient). If the plant is not exponentially unstable, semiglobal results are obtainable. Regional results can always be obtained in the general case and the size of the maximal feasible domain of attraction depends on the particular problem.


325

4.2 Feasibility of and Solution to Problem 2 We now use the result of Theorem 1 to give a solution to Problem 2 following similar steps as the ones in the previous section. When focusing on reachable sets, smaller estimates are desirable, so that there’s a guaranteed bound on the size of the state when the system is disturbed by external inputs. Since by Theorem 1 the reachability region estimate coincides with the estimate of the domain of attraction, the goal addressed in Problem 2 is in contrast with the goal addressed in the previous section. The corresponding equivalent to Corollary 1 is the following (where αR (·) is a measure of the size of the ellipsoid E((s2 Q11 )−1 )): Corollary 2. Given s, Sp and γ, a solution to optimal stability region Problem 2 is given (whenever feasible) by applying Procedure 1 to the optimal solution to the following minimization problem: min

Q11 ,P11 ,K1 ,K2 ,K3 ,Yp ,γ2

αR (E((s2 Q11 )−1 ), subject to

(4a), (4b), (4c), (4d),

(17a)

Sp ⊂ E((s2 Q11 )−1 )

(17b)

Similar to the previous section, the volume of the reachability set can be minimized by selecting αR (E((s2 Q11 )−1 )) = det(s2 Q11 ) in (17). Alternative easier selections of αR correspond to focusing on ellipsoids when choosing Rp = E(αRp−1 ) = {xp : xTp (αRp )−1 xp ≤ 1},

(18)

where Rp = RpT > 0, then the optimization problem (17) becomes min

Q11 ,P11 ,K1 ,K2 ,K3 ,Yp ,γ2

α, subject to

(4a), (4b), (4c), (4d), (17b)

(19a)

s2 Q11 ≤ αRp

(19b)

Similarly, Rp can be selected as the following unbounded set: Rp (α) = {xp : |Cxp | ≤ α},

326


where C ∈ 1×np is a given row vector. Then E((s2 Q11 )−1 ) ⊂ Rp (α) if and only if CQ11 C T ≤ α2 /s2 . If both (19a) and CQ11 C T ≤ α2 /s2 are enforced to hold in the LMI optimization, then it follows that |Cxp (t)| ≤ α for all t if w 2 ≤ s. Therefore, if our objective is to minimize the size of a particular output Cxp , we may formulate the following optimization problem: min

Q11 ,P11 ,K1 ,K2 ,K3 ,Yp ,α

α, subject to

(4a), (4b), (4c), (4d), (17b)

(20a)

CQ11 C T < α2 /s2 ,

(20b)

where the guaranteed L2 gain is be incorporated in (4a) and (4b) and the guaranteed stability set is incorporated in (17b). Remark 3. If there is no interest for a guaranteed L2 gain, then the problem to minimize the desirable reachable set Rp can be simplified to the feasibility of the LMIs in (4) and (17b), by removing the second block row and the second block column of the matrices in (4a) and the third block ones in (4b).

4.3 Feasibility of and Solution to Problem 3 Similar to what has been done in the previous two sections with reference to Problems 1 and 2, we use here Theorem 1 to give a solution to Problem 3. Corollary 3. Given s, Rp and Sp , a solution to Problem 3 is given (whenever feasible) by applying Procedure 1 to the optimal solution to the following minimization problem: min

Q11 ,P11 ,K1 ,K2 ,K3 ,Yp

,γ2

γ2 , subject to

(4a), (4b), (4c), (4d),

(21a)

Sp ⊂ E((s2 Q11 )−1 ) ⊂ Rp

(21b)

If we only focus on ellipsoidal reachability and stability sets so that for two given matrices Sp = SpT > 0 and Rp = RpT > 0 Sp := E(Sp−1 ) and Rp := E(Rp−1 ), then the optimization problem (21) can be cast as the following convex formulation:


min

Q11 ,P11 ,K1 ,K2 ,K3 ,Yp ,γ2

327

γ2 , subject to

(4a), (4b), (4c), (4d),

(22a)

Sp ≤ s2 Q11 ≤ Rp

(22b)

Alternative shapes for the guaranteed reachability set Rp and for the guaranteed stability region Sp can be selected by following the indications given in the previous two sections.

5 Examples In this section we illustrate the proposed controller constructions on several case studies. For the different control problems, several examples are organized as follows: in Section 5.1 we address the disturbance rejection problem; in Section 5.2 some cases of set point regulation are studied; in Section 5.3 both the disturbance rejection and the reference tracking problems will be addressed in an exponentially unstable example. 5.1 Disturbance Rejection Example 1. We consider the system used in [10], which has one control input, one disturbance input, four states and one measurement output. The plant state is xp = T ˙ p p˙ θ θ , where p is the horizontal displacement of the cart and θ is the angle of the pendulum. The plant parameters are given by: ⎡ ⎤ 0 1 0 0 0 0 ⎢ ⎥ ⎢ −330.46 −12.15 −2.44 0 2.71762 0 ⎥ ⎥ ⎡ ⎤ ⎢ ⎢ ⎥ 0 0 0 1 0 0 ⎥ Ap Bp,u Bp,w ⎢ ⎥ ⎥ ⎢ ⎢ ⎢ Cp,y Dp,yu Dp,yw ⎥ = ⎢ −812.61 −29.87 −30.10 0 6.68268 15.61 ⎥ ⎥ ⎣ ⎦ ⎢ ⎢ ⎥ ⎢ Cp,z Dp,zu Dp,zw 1 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 1 0 0 0 ⎣ ⎦ 0 0 1 0 0 0 For each s > 0, the achievable L2 gain by a plant order output feedback can be determined with the algorithm based on Theorem 1. If choosing different values of s over (0, ∞), the achievable performance can be obtained as a function of s.

328


The solid line in Figure 2 reports the achievable L2 gain by a suitable plant order output feedback controller, as a function of s. For comparison purposes, we report in the same figure (dashed line) the L2 gain achievable by a dynamic anti-windup compensation when using a specific unconstrained controller (see [16] for details).

200 180 160 140

γ

120 100 80 60 40 20 0 −3 10

−2

10

−1

10

0

10 ||w||

1

10

2

10

3

10

2

Fig. 2. Achievable nonlinear L2 gains. Proposed output feedback (thin solid); Dynamic antiwindup (dashed)

In [10], when the cart-spring-pendulum system is subject to the larger pendulum tap, which is modeled as a constant force of 7.94N with duration 0.01s, the closed-loop response with an LQG controller exhibits undesirable oscillations if the control input is constrained in the range of the D/A converter: [-5, +5] Volts. In [10] a dynamic anti-windup compensator is used to preserve the local LQG behavior and improve the response after saturation. We compare that result to our direct design. In particular, we use Procedure 1 to construct an output feedback controller by fixing s = 0.14. The corresponding optimal L2 gain is γ = 2.26 and the controller matrices are


⎡

Ac Bc E1

329

Cc Dc E2 −215.1

5.7

⎢ ⎢ −1781.3 −1154.1 ⎢ ⎢ = ⎢ −93.8 −53.9 ⎢ ⎢ −4914.0 −2441.5 ⎣ −74.4474 −38.2744

−6.0

137.7

3558.8 −3149.6 −0.6334

⎤

⎥ 360.2 −4122.6 −3202.5 −48.7601 ⎥ ⎥ ⎥ −159.9 21.2 −1047.7 3788.7 −4.4397 ⎥ . ⎥ −416.8 735.1 1491.6 1290.1 −131.6461 ⎥ ⎦ −5.9264 11.7716 120.6582 3.7499 −0.9998 −62.5

θ (radians)

0.5

0

−0.5

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4 5 6 time (seconds)

7

8

9

10

p (meters)

0.1

0

−0.1

u (volts)

5

0

−5

Fig. 3. Simulated response to the large pendulum tap. Constrained response with LQG controller (dashed); Response with the linear output feedback controller (thin solid); Response with the LQG controller and dynamic anti-windup [10] (dash-dotted).

The thin solid curve in Figure 3 represents the response of the closed-loop system with our output feedback controller to the same disturbance that generates the undesirable response arising from the saturated LQG controller (taken from [10]), which is represented by the dashed curve in the same figure. Moreover, for comparison purposes, the dash-dotted curve represents the response when using the dynamic compensator proposed in [10] on top of the LQG controller. Note that the proposed controller guarantees more desirable large signal responses as compared to the anti-windup closed-loop of [10], indeed our controller is not constrained to satisfy the small signal specification as in the anti-windup approach of [10]. On the other hand, it must be recognized that synthesizing the controller by

330


direct design reduces the small signal performance (before saturation) of the overall closed-loop, which can be imposed as an arbitrary linear performance when using the anti-windup tools as in [10] (or the more advanced techniques recently proposed in [17]).

5.2 Reference Tracking In this section, we will include two examples to show how effectively the proposed output feedback controller deals with the reference tracking problem. Before the examples, we introduce a plant augmentation that constrains the arising control system to guarantee zero tracking error. The modified problem corresponds to augmenting Ap with extra integrators that are inserted at the output y according to the block diagram reported in Figure 4. Plant Pnew Reference input

w=r

−1 y−w znew

Control input

u

P

1/s

ynew

Original Palnt

Fig. 4. The augmented plant for set point regulation.

According to the block diagram of Figure 4, not only should the output of the integrator(s) be sent to the controller, but also its input. Namely, the signal ynew is formed by the integrators output and an extra set of signals corresponding to the tracking error y − r. Suppose that the original plant equations are: x˙ p = Axp + Bu u P y = Cxp + Du u

(23)

then, the augmented plant equations, which correspond to the form in (1), are given by the following parameters:


⎡

Ap

Bp,u

Bp,w

Cp,y/z Dp,y/zu Dp,y/zw

A 0 Bu 0

331

⎤

⎥ ⎢ ⎢ C 0 Du −I ⎥ ⎥ ⎢ =⎢ ⎥ ⎣0 I 0 0 ⎦ C 0 Du −I

(24)

As another design criterion, suppose that we wish to recover as much of the linear performance as possible for one output at the expense of the performance in the other output. For this problem, we add a new diagonal matrix δ to the LMI conditions in Theorem 1 (so that the 2 by 2 entry involving γ in (4a) and the corresponding entry in (4b) will be both changed from −γ2 I/2 to −δ−1 γ2 I/2). This extra matrix allows to guarantee the regional finite L2 gain in the form of z T δz ≤ γ2 wT w, thus providing extra degrees of freedom in the performance optimization. If the exogenous input w contains not only references to be tracked but also disturbances, the augmented plant will be different from the form (24). We will discuss this case in the next section. Example 2. Consider the damped mass-spring system used in [16], [31]. The motion equations for this system are given by 0 1 0 x˙ p = xp + u −k/m −f /m 1/m

y = 1 0 xp

T

where xp :=

represents position and speed of the body connected to the q q˙ spring, m is the mass of the body, k is the elastic constant of the spring, f is the damping coefficient, u represents a force exerted on the mass. According to the parameters chosen in [16], [31]: m = 0.12kg, k = 0.8kg/s2 , f = 0.0008kg/s, the plant

matrices are given by: ⎡

Ap

Bp,u

Bp,w


0

1

0

0

⎢ ⎢ −6.667 −0.007 0 8.333 ⎢ ⎢ =⎢ 1 0 0 0 ⎢ ⎢ 0 0 1 0 ⎣ 1 0 0 0

0

⎤

⎥ 0 ⎥ ⎥ ⎥ −1 ⎥ ⎥ 0 ⎥ ⎦ −1

332


For our simulation, the system is starting from the rest position and with the reference switching between ±1 meters every 5s and going back to zero permanently after 15 s. We apply Procedure 1 with the output weight δ−1 = diag(0.42 , 1), and set s = 0.25, such that the optimization problem gives an upper bound of 1.5841 on the weighted L2 gain and the resulting controller matrices are: ⎡ −5.5080 0.1603 −10.7442 1.5543 ⎢ ⎢ −27.1931 −11.3053 −45.9518 −8.5028 Ac Bc E1 =⎢ ⎢ −1.3964 −0.2274 −10.2038 56.6145 Cc Dc E2 ⎣ −0.9526 −0.4371 −1.7830 −0.3329

−5.6132 −1.5255

⎤

⎥ −29.2707 −7.3341 ⎥ ⎥ 6.8613 −0.0527 ⎥ ⎦ −1.1344 0.7160

The corresponding closed-loop response is represented by the thin curve in Figure 5. The mass position is tracking the reference well.

Control input u

Performance output z

Mass position y

The response of damped mass−spring system 2

0

−2

0

5

10

15

20

25

0

5

10

15

20

25

0

5

10

15

20

25

5

0

−5 1

0

−1

Time (s)

Fig. 5. Response to the system in Example 2. Response with linear output feedback controller(thin solid); Reference input (dotted).

Example 3. Consider a fourth-order linear model of the longitudinal dynamics of the F8 aircraft introduced in [18], [10]. The state-space equations are given by


⎡

−0.8 −0.0006 −12

0

⎤

⎡

−19

−3

333

⎤

⎥ ⎥ ⎢ ⎢ ⎢ 0 −0.014 −16.64 −32.2 ⎥ ⎢ 0.66 −0.5 ⎥ ⎥ ⎥u ⎢ ⎢ xp + ⎢ x˙ p = ⎢ ⎥ ⎥ 0 ⎦ ⎣ 1 −0.0001 −1.5 ⎣ −0.16 −0.5 ⎦ 1 0 0 0 0 0 00 0 1 y= xp 0 0 −1 1 The two inputs of the plant are the elevator angle and flaperon angle, each one limited between ±25 degrees in the simulation. The two outputs of the plant are the pitch angle and flight path angle, which are supposed to track the reference in T put w = 10 10 . Similar to the previous example, the original model of the F8 aircraft is first augmented by extra integrators to guarantee zero steady-state error. Therefore, the new output consists in the pitch and flight path angles and the corresponding rates. In the LMI computation of Procedure 1, we note that the output weight helps to balance the performance of the two sets of outputs and select δ−1 = diag(0.1, 0.1, 0.92 , 1.152 ) and s = 1. The upper bound on the weighted L2 gain is 1.3056, and the controller matrices are given by: ⎡

−47.7 −0.9 −86.3 −587.6 −1585.3 221.2

⎤

⎥ ⎢ ⎢ −27.0 −0.3 23.3 −395.6 −875.3 −65.3 ⎥ ⎥ ⎢ ⎢ 6.5 −0.0 −579.4 556.6 −56.7 1532.4 ⎥ ⎥ ⎢ Ac = ⎢ ⎥ ⎥ ⎢ −0.3 0.7 −39.3 6.5 −77.2 104.7 ⎥ ⎢ ⎥ ⎢ −0.6 0.2 2.8 −12.6 −28.1 −7.5 ⎦ ⎣ 0.2 −0.1 −16.6 14.9 −4.3 40.2 ⎤ ⎡ 34.8 −19.7 −226.5 −14.3 ⎥ ⎢ ⎢ −48.3 35.5 −149.0 −2.3 ⎥ ⎥ ⎢ ⎥ ⎢ Bc = ⎢ 16.4 −44.9 43.4 41.2 ⎥ ⎥ ⎢ ⎢ −256.2 −15.5 −77.8 −1.2 ⎥ ⎦ ⎣ 1692.4 171.15 7.2 448.4 0.0217 −0.0004 −0.2902 0.5522 0.6161 0.7687 Cc = −0.0277 0.0039 2.2378 −2.3055 −0.1292 −5.9286

334


Dc = E1 = E2 =

−0.0033 −0.0012 0.1159

0.0331

−0.0159 0.0046 −0.1995 −0.2073 0.0659 0.0488 −0.0981 −0.0029 0.0017 −0.0027

T

0.0595 −0.0025 0.0234

0.2079 0.0163 0.8089 0.9999 0.0004 0.0004 0.9969

The time histories in Figure 6 correspond to the closed-loop response to the reference r = [10 10]T . From the figure it appears that the controller makes wide use of the input authority (the second input is kept into saturation until the flight angle reaches the desired value in a bang-bang type of control action) guaranteeing a fast

12

12

10

10

8 6 4 2 0

0

1

2

3

4

6 4 2 0

1

2

3

4

5

0

1

2 3 time (s)

4

5

30

0

−10

−20

−30

8

0

5

flaperon angle (deg)

elevator angle (deg)

flight angle (deg)

pitch angle (deg)

and desirable output response.

0

1

2 3 time (s)

4

5

20 10 0 −10

Fig. 6. Response of the F8 aircraft with reference tracking.

5.3 Reference Tracking and Disturbance Rejection In the previous examples, we have shown the desirable performance of the proposed output feedback construction. In particular, we have shown its ability to effectively solve both the disturbance rejection and the reference tracking problems. In this section we study an exponentially unstable example and consider the disturbance


335

Plant Pnew r

−1

w

y−r

d

Reference and disturbance input

znew P

u

ynew

1/s

Control input

Original Palnt

Fig. 7. The plant augmented for reference tracking and disturbance rejection.

rejection and reference tracking problems together. Before the example, the block diagram represented in Figure 4 is modified to include both references and disturbances. The counterpart of the original plant model (23) and of the augmented plant parameters (24), correspond to: P

x˙ p = Axp + Bu u + Bd d

with

⎡

Ap

(25)

y = Cxp + Du u + Dd d

Bp,u

Bp,w


A 0 Bu 0 Bd

⎤

⎥ ⎢ ⎢ C 0 Du −I Dd ⎥ ⎥ =⎢ ⎢ ⎥ 0 I 0 0 0 ⎣ ⎦ C 0 Du −I Dd

(26)

r . Using equations (25) and (26) as an intermediate step, can use with w = d Procedure 1 for the following example. Example 4. Consider the inverted pendulum around its upright position for small inclination angles. The pendulum mass m1 shall be concentrated at its center of gravity (cg), and the connecting link of length L to the slider at its bottom is mass-free and rigid. The slider mass m3 may move horizontally without friction. A horizontal force is applied by the actuator to the slider, in order to control the speed error to zero. The linearized model in appropriately scaled variables is given by:

336


τ1 x˙ 1 = −x2 + d; τ2 x˙ 2 = −x1 + x3 ;

(27)

τ3 x˙ 3 = x2 + u; y = x3

with parameter values τ1 = τ2 = τ3 = 1.0s. In (27), the state variables correspond to the horizontal speed of the pendulum cg (represented by x1 ), the horizontal displacement between the cg’s of the pendulum and of the slider (represented by x2 ) and the horizontal speed of the slider (represented by x3 ). The load d represents a horizontal force exerted on the pendulum cg, which is persistent and with unknown but bounded magnitude. The input u represents the force exerted by the actuator, which is constrained in magnitude (stroke) between ±1. To deal with the persistent disturbance while tracking the reference, we first augment the plant model matrices as follows: ⎤ ⎡ ⎡ ⎤ 0 1 0 ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ 0 0⎥ r ⎢ −1 0 1 0 ⎥ ⎢0⎥ ⎥ ⎢ ⎥ ⎢ ⎥ x˙ p = ⎢ ⎢ 0 1 0 0 ⎥ xp + ⎢ 1 ⎥ u + ⎢ 0 0 ⎥ d ⎦ ⎣ ⎦ ⎣ ⎣ ⎦ −1 0 0 0 10 0 ⎡

0 −1 0 0

y=

⎤

0001 0010

xp +

0 0 −1 0

r d

The plant has one pole at the origin and one on the positive real axis: it is exponentially unstable. The test sequence is designed as follows: 1) Start in closed-loop operation from the initial conditions x(0) = 0, reference r = 0 and disturbance d = 0. 2) Apply a step reference of r = 0.08 to the system for 30 s and then reset to zero. 3) At time t = 60s, apply a load step disturbance d = 0.2 to the system. In Procedure 1 we choose s = 0.012 (small region), and solve the LMIs to get the minimized L2 gain γ = 37.78. The controller matrices are given next and the simulation results are shown in Figure 8.


⎡

Ac Bc E1

337

Cc Dc E2 −90.3862 67.9126

1.6317

⎤

⎥ 2.4002 ⎥ ⎥ ⎥ −123.9217 35.4146 44.7308 −13.8450 −116.1458 −5.0009 ⎥ ⎥ −11.3325 7.9340 −13.8678 280.5706 −0.6618 −0.6745 ⎥ ⎦ −47.4669 14.9467 16.8633 0.8502 −46.4682 −1.0000 25.8891 −24.7031 −15.6606 58.9859

Control input u

Performance output z

Horizontal speed y

⎢ ⎢ −61.4551 ⎢ ⎢ = ⎢ 193.4138 ⎢ ⎢ 18.6906 ⎣ 74.7057

−5.3992 −20.5277 −22.1765 35.6479 60.7153

1 0 −1

0

20

40

60

80

100

0

20

40

60

80

100

0

20

40

60

80

100

2 1 0 −1 1 0 −1

time (s)

Fig. 8. Response of the inverted pendulum.

The thin solid curve in Figure 8 represents the response of the closed-loop system with input saturation. The dash-dotted curve represents the reference input and the load disturbance. The upper plot shows that the speed of the slider succeeds in tracking the reference and rejecting the disturbance while exhibiting desirable performance.

6 Conclusions In this chapter we illustrated on several example studies a novel synthesis method for the construction of a linear output feedback controller with an internal deadzone loop. By using regional analysis tools developed in [16], a systematic method for the

338


synthesis of the proposed controller is cast as an optimization problem over LMIs and a variety of regional stability and performance goals can be achieved. For all the proposed examples, the simulations show desirable closed-loop performance.

References 1. D.S. Bernstein and A.N. Michel. A chronological bibliography on saturating actuators. Internat. J. Robust Nonlinear Control, 5(5):375–380, 1995. 2. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. Society for Industrial an Applied Mathematics, 1994. 3. C. Burgat and S. Tarbouriech. Intelligent anti-windup for systems with input magnitude saturation. Int. J. Robust and Nonlinear Control, 8(12):1085–1100, October 1998. 4. Y.Y. Cao, Z. Lin, and D.G. Ward. An antiwindup approach to enlarging domain of attraction for linear systems subject to actuator saturation. IEEE Trans. Aut. Cont., 47(1):140– 145, 2002. 5. S. Crawshaw and G. Vinnicombe. Anti-windup synthesis for guaranteed L2 performance. In Proceedings of the Conference on Decision and Control, pages 1063–1068, Sidney, Australia, December 2000. 6. J.M. Gomes da Silva and S. Tarbouriech. Local stabilization of discrete-time linear systems with saturating controls: An LMI-based approach. IEEE Trans. Aut. Cont., 46(1):119–125, January 2001. 7. J.M. Gomes da Silva Jr, D. Limon, and T. Alamo. Dynamic output feedback for discretetime systems under amplitude and rate actuator contraints. In Joint CDC-ECC, pages 5588–5593, Seville, Spain, December 2005. 8. J.M. Gomes da Silva Jr and S. Tarbouriech. Local stabilization of discrete-time linear systems with saturating actuators: an LMI-based approach. In American Control Conference, pages 92–96, Philadelphia (PA), USA, June 1998. 9. J.M. Gomes da Silva Jr and S. Tarbouriech. Anti-windup design with guaranteed regions of stability: an LMI-based approach. IEEE Trans. Aut. Cont., 50(1):106–111, 2005. 10. G. Grimm, J. Hatfield, I. Postlethwaite, A.R. Teel, M.C. Turner, and L. Zaccarian. Antiwindup for stable linear systems with input saturation: an LMI-based synthesis. IEEE Trans. Aut. Cont. (A), 48(9):1509–1525, September 2003. 11. G. Grimm, A.R. Teel, and L. Zaccarian. Linear LMI-based external anti-windup augmentation for stable linear systems. Automatica (B), 40(11):1987–1996, 2004. 12. G. Grimm, A.R. Teel, and L. Zaccarian. Robust linear anti-windup synthesis for recovery of unconstrained performance. Int. J. Robust and Nonlinear Control (A), 14(13-15):1133– 1168, 2004.


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13. W.M. Haddad and V. Kapila. Anti-windup and guaranteed relative stability margin controllers for systems with input nonlinearities. In Proceedings of the 13th IFAC World Congress, volume H, pages 143–148, San Francisco (CA), USA, June 1996. 14. T. Hu, Z. Lin, and B.M. Chen. Analysis and design for linear discrete-time systems subject to actuator saturation. Systems and Control Letters, 45(2):97–112, 2002. 15. T. Hu, Z. Lin, and B.M. Chen. An analysis and design method for linear systems subject to actuator saturation and disturbance. Automatica, 38(2):351–359, 2002. 16. T. Hu, A.R. Teel, and L. Zaccarian. Nonlinear L2 gain and regional analysis for linear systems with anti-windup compensation. In American Control Conference, pages 3391– 3396, Portland (OR), USA, June 2005. 17. T. Hu, A.R. Teel, and L. Zaccarian. Regional anti-windup compensation for linear systems with input saturation. In American Control Conference, pages 3397–3402, Portland (OR), USA, June 2005. 18. P. Kapasouris, M. Athans, and G. Stein. Design of feedback control systems for stable plants with saturating actuators. In Proceedings of the Conference on Decision and Control, pages 469–479, Austin (TX), USA, December 1988. 19. T.A. Kendi, M.L. Brockman, M. Corless, and F.J. Doyle. Controller synthesis for inputconstrained nonlinear systems subject to unmeasured disturbances. In ACC, pages 3088– 3092, Albuquerque (NM), USA, May 1997. 20. V.M. Marcopoli and S.M. Phillips. Analysis and synthesis tools for a class of actuatorlimited multivariable control systems: A linear matrix inequality approach. Int. J. Robust and Nonlinear Control, 6:1045–1063, 1996. 21. D.Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scokaert. Constrained model predictive control: Stability and optimality. Automatica, 36(6):789–814, 2000. 22. A. Megretski. L2 BIBO output feedback stabilization with saturated control. In 13th Triennial IFAC World Congress, pages 435–440, San Francisco, USA, 1996. 23. E.F. Mulder and M.V. Kothare. Static anti-windup controller synthesis using simultaneous convex design. In ACC, pages 651–656, Anchorage (AK), USA, June 2002. 24. E.F. Mulder, M.V. Kothare, and M. Morari. Multivariable anti-windup controller synthesis using linear matrix inequalities. Automatica, 37(9):1407–1416, September 2001. 25. C.W. Scherer, P. Gahinet, and M. Chilali. Multi-objective output-feedback control via lmi optimization. IEEE Trans. Aut. Cont., 42(7):896–911, July 1997. 26. H.J. Sussmann, E.D. Sontag, and Y. Yang. A general result on the stabilization of linear systems using bounded controls. IEEE Trans. Aut. Cont., 39(12):2411–2424, 1994. 27. K. Takaba. Anti-windup control system design based on Youla parametrization. In Proceedings of the American Control Conference, volume 3, pages 2012–2017, San Diego (CA), USA, June 1999. 28. A.R. Teel. Global stabilization and restricted tracking for multiple integrators with bounded controls. Systems and Control Letters, 18:165–171, 1992.

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29. F. Tyan and D.S. Bernstein. Anti-windup compensator synthesis for systems with saturating actuators. Int. J. Robust and Nonlinear Control, 5(5):521–537, 1995. 30. F. Wu, K.M. Grigoriadis, and A. Packard. Anti-windup controller design using linear parameter-varying control methods. Int. Journal of Control, 73(12):1104–14, August 2000. 31. L. Zaccarian and A.R. Teel. A common framework for anti-windup, bumpless transfer and reliable designs. Automatica (B), 38(10):1735–1744, 2002.

Set Based Control Synthesis for State and Velocity Constrained Systems Franco Blanchini1 , Stefano Miani2 , and Carlo Savorgnan2 1

University of Udine, Dept. of Mathematics and Computer Science, Via delle Scienze 208, 33100, Udine, Italy. email [email protected]

2

University of Udine, Dept. of Electrical, Mechanical and Managerial Eng., Via delle Scienze 208, 33100, Udine, Italy. email [email protected], [email protected]

Summary. In the present contribution, the control synthesis for state and input constrained systems for linear systems via set-valued techniques will be presented. The results will be developed for both uncertain and gain scheduled constrained control problems, where the difference between the two is the knowledge of the uncertain parameter entering the system at each time instant in the gain scheduled case. It will be shown how slew-rate constrained control problems can be recast into standard constrained control problems by means of an extended system. To present such technique, an overview of some of the existing results with some numerical algorithms to derive the stability conditions and the control will be given.

1 Introduction Controlling systems in the presence of constraints is one of the areas which has received great attention in the recent literature [17, 18, 15, 21, 11, 13]. The main ingredients of such control problems are rather standard: the system dynamics, the input limitations, both on the amplitude and the slew–rate, the output or, in general, the state variables constraints and, possibly, the disturbances acting on the system. The final goal of constrained control problems is optimizing some performance measure and, possibly, the requirement of an as simple as possible control law.

342

Franco Blanchini, Stefano Miani, and Carlo Savorgnan

Even for linear systems, constrained control problems are rather difficult to solve. Several techniques have been proposed to face constrained synthesis. Such techniques evolved from the adaptation of analysis results to synthesis problems (e.g. the circle criterion), when the designer was basically verifying the extent of the effects of the nonlinearity in the closed loop system for a linear controller, to a more mature stage when first basic problems on the determination of globally stabilizing control laws for constrained systems were considered [12, 22], and later more complex control laws (e.g. receding horizon and nonlinear controllers) were seeked, keeping in mind from the beginning the presence of constraints and performance requirements [10, 1]. It is worth stressing that most of the recent results have been made available on one side by recent new ideas, on the other side by the availability of reliable numerical algorithms which have made tractable problems which were far from being solvable in the 90’s. In this chapter we will present some results which allow the designer to cope with constrained problems for a specific class of systems. Precisely, the main focus of this chapter will be time-varying systems with structured uncertainty affecting the system matrices, and for these it will be shown how to solve input and slew-rate limitations design problems. Although here, for the sake of simplicity, only the case of symmetric constraints will be considered, it is worth saying that all the results given in the sequel can be extended to the nonsymmetric case.

2 Definitions and Existing Results As already mentioned, the aim of the present work is to show how assessed techniques in the area of constrained control problems can be applied to deal with slewrate constrained problems as well. To this aim, in the present section some of the basic results in the area of constrained control (excluding the presence of rate constraints) are reported. In this section constrained control problems for n dimensional linear time-varying systems of the form x+ (t) = A(w(t))x(t) + B(w(t))u(t) y(t) = C(w(t))x(t) + D(w(t))u(t)

(1)

Set Based Control Synthesis for State and Velocity Constrained Systems

343

are considered. The term x+ denotes the time derivative x(t) ˙ in the continuous-time case or the one step forward delay x(t + 1) in the discrete-time case. In either cases, the system matrices belong to the sets whose expression are A= C=

s

i=1 Ai wi (t), s i=1 Ci wi (t),

B= D=

s i=1

s

Bi wi (t)

i=1 Di wi (t)

(2)

where the matrices Ai ∈ IRn×n , Bi ∈ IRn×m , Ci ∈ IRp×n and Di ∈ IRp×m are given and w(t) is a piecewise constant function which attains its values in W = {w : wi ≥ 0,

s

wi = 1}

(3)

i=1

For the above systems, several control problems can be defined, depending on the availability of information of the state variables, the presence of state and/or input limitations (both on the amplitude and the slew-rate) and the knowledge of the parameter w(t) at each time-instant. In the present context, it will be assumed that the state information is available, and thus all the design problem faced here will eventually lead to state-feedback strategies. In other words, in this work the variable y(t) will be essentially used to characterize the performance measure. It is assumed that the following constraints are given ⎧ ⎪ ⎪ ⎨ u(t) ∈ U x(t) ∈ X ⎪ ⎪ ⎩ y(t) ∈ Y

(4)

where U, X and Y are polyhedral compact sets containing the origin as an interior point. Note that, in principle, the constraint y(t) ∈ Y includes all the mentioned constraints provided that matrices C(·) and D(·) are suitably chosen. In particular we can face state speed constraints x˙ = Ax + Bu ∈ S. Note also that constraints of the form y = C(w)x(t) + D(w)u ∈ Y are equivalent to a finite number or constraints, precisely Ci x + Di u ∈ Y , ∀i We consider the following problems: • when the value of w(t) is not available to the controller, we will talk about robust synthesis;

344


• when such information is available for control purposes, we will talk about gain scheduled synthesis; • when the value of w(t) is known and, moreover, w(t) attains its value only on one of the vertices of the set W defined in (3), say w(t) ∈ Ws = {w : wi = 1, wj = 0 if j = i f or some i}, we will talk about switched gain scheduled synthesis. Remark 1. We let the reader note that, even in the continuous–time case, in view of the fact that the signal w(t) is piecewise-constant, the gain scheduled and switched gain scheduled synthesis problems are rather different. For example, the continuous– time system x+ (t) = x(t) + (w1 (t) − w2 (t))u(t)

(5)

is clearly switched gain scheduling stabilizable (even in the presence of constraints) by the control law u(t) = −2(w1 (t) − w2 (t))x(t), but there is obviously no gain scheduling stabilizing control law due to to the lack of reachability when w1 (t) = w2 (t) = 1/2. To recap, the aim of the present work is that of determining state-feedback gain scheduled or robust controllers for either continuous or discrete-time constrained control problems of the form (1), with performance measure on the variable y(t) (also defined in (1)). Before getting into these issues, some definitions which will be useful in the sequel are reported. We refer the reader to one of the many books on convex analysis [16] for a more detailed exposition on polyhedron, polytopes and their representations. Definition 1. An r × r matrix H belongs to the set H if and only if there exists τ > 0 and a matrix P , whose 1-norm is smaller than 1, such that H = τ−1 (P − I). Definition 2. P ∈ IRn is a polyhedral set or polyhedron if it can be written as follows:

P = x ∈ IRn : fiT x ≤ 1, i = 1, . . . , m

(6)

where fi are vectors in IRn and f T denotes the transpose vector of f . A more compact notation can be obtained by stacking the vectors fiT in the matrix F as to obtain

P = x ∈ IRn : F x ≤ 1

(7)


345

where 1 represents a column vector of appropriate dimensions and whose entries are all equal to 1. A bounded polyhedron P ∈ IRn is a polytopic set or polytope and it can be also represented in terms of its r vertices xj . Indeed, denoting by X ∈ IRn×r the full row rank matrix X = [x1 , . . . , xr ], it is possible to write P as the convex hull of its

vertices: P = Co(X) =

x ∈ IRn : x = Xα,

r

8 αi , αi ≥ 0 .

(8)

i=1

The above is also known as vertex representation. By applying the above definitions it is immediate to see that the sets A and B where A(w) and B(w) attain their values are polytopic matrix sets. The vertices of A and B are respectively Ai and Bi . The reason why polytopic sets have been introduced is that they play a key role in the next fundamental result, which is the core result on which the control of slew–rate constrained systems will be built on in the next sections. Theorem 1. Given the linear time–varying system (1) and the constraints ⎧ ⎪ ⎪ ⎨ x(t) ∈ X u(t) ∈ U ⎪ ⎪ ⎩ y(t) ∈ Y

(9)

where X , U, Y are symmetric polyhedra, the following statements hold. • Continuous–time, robust synthesis [5]. There exists a state feedback controller u(x) = φ(x) such that the closed loop evolution complies with constraints (9) and such that the closed loop state evolution asymptotically converges to 0 if and only if there exist a matrix X ∈ IRn×r , a matrix U ∈ IRp×r and s matrices Hi ∈ H such that, for any i, Ai X + Bi U = XHi

(10)

and each of the state vectors xj corresponding to the columns of the matrix X and each of the input vectors uj corresponding to the columns of the matrix U are such that the following state and input conditions are satisfied

346


⎧ j ⎪ ⎪ ⎨x ∈ X

∀ j = 1, . . . , r

uj ∈ U ∀ j = 1, . . . , r ⎪ ⎪ ⎩ C xj + D uj ∈ Y ∀ j = 1, . . . , r ∀ i = 1, . . . , s i i

(11)

• Continuous–time, switched gain scheduled synthesis [9]. There exists a state feedback controller u(x) = φ(x, i) such that the closed loop evolution complies with constraints (9) and such that the closed loop state evolution asymptotically converges to 0 if and only if there exist a matrix X ∈ IRn×r , s matrices Ui ∈ IRp×r and s matrices Hi ∈ H such that, for any i = 1, . . . s, Ai X + Bi Ui = XHi

(12)

and each of the state vectors xj corresponding to the columns of the matrix X and each of the input vectors uji corresponding to the columns of the matrix Ui are such that the following state and input conditions are satisfied ⎧ j ⎪ ∀ j = 1, . . . , r ⎪ ⎨x ∈ X ⎪ ⎪ ⎩

uji ∈ U ∀ j = 1, . . . , r ∀ i = 1, . . . , s Ci xj +

Di uji

∈Y

(13) ∀ j = 1, . . . , r ∀ i = 1, . . . , s

• Continuous–time gain scheduled synthesis [6]. The existence a state feedback controller u(x) = φ(x, w) such that the closed loop system complies with constraints (9) and such that the closed loop state evolution asymptotically converges to 0 is equivalent to the existence of a control u(x) = φ(x) which achieves the same goal. In simple words, gain-scheduling and robust stabilization are equivalent and they are both equivalent to condition (12) with a single matrix U . Note that according to Remark 1 this equivalence does not hold between robust and switched gain scheduling stabilization unless B is a certain matrix (see also [9] for a detailed analysis of the relations between the different cases). • Discrete–time, robust synthesis [5]. There exists a state feedback controller u(x) = φ(x) complying with constraints (9) and such that the closed loop state evolution asymptotically converges to 0 if and only if there exist a matrix X ∈ IRn×r , a matrix U ∈ IRp×r and s non-negative matrices Pi such that, for any i, Ai X + Bi U = XPi , Pi 1 < 1

(14)

and each of the state vectors xj corresponding to the columns of the matrix X and each of the input vectors uj corresponding to the columns of the matrix U satisfy conditions the conditions (11)


347

• Discrete–time, switched gain scheduled synthesis [9]. There exists a state feedback controller u(x) = φ(x, i) complying with constraints (9) and such that the closed loop state evolution asymptotically converges to 0 if and only if there exist a matrix X ∈ IRn×r , s matrices Ui ∈ IRp×r and s non-negative matrices Pi such that, for any i, Ai X + Bi Ui = XPi , Pi 1 < 1

(15)

and each of the state vectors xj corresponding the columns of the matrix X and each of the input vectors uji corresponding the columns of the matrix Ui satisfy conditions (13). • Discrete–time, gain scheduled synthesis [9]. Differently form the continuous– time case there is no equivalence between the gain scheduled and robust stabilizability. However, when B(w) = B, an if and only if condition for the existence of a state feedback gain scheduled control law complying with the imposed requirements is given by Ai X + BUi = XPi , Pi 1 < 1

(16)

(note that it is slightly different from equation (15)) together with the conditions on the state, input and output limitations. To see the role played by polytopic sets in Theorem 1, we will first briefly concentrate on the discrete-time conditions for robust constrained stabilization, since such case is the easiest to be interpreted. Indeed, by defining the symmetric polytope P = Co(−X, X) (so that the columns of X and −X represent the vertices of P ) and by considering each equation given by the columns of the matrix equation (14), one gets Ai xj + Bi uj = XPij

(17)

where Pij denotes the j-th column of matrix Pi . Now, such condition corresponds to say that for any system vertex i, whenever the state variable corresponds to one vertex of P (x(t) = xj ), there exists a value of the input that does not depend on i (u(t) = uj ) and such that any value that the next step state xj+ can attain (and i which clearly depends on the system vertex i) is strictly included in P . The latter assertion can be immediately verified by rewriting the equation corresponding to the jth column of (14) as

348

Franco Blanchini, Stefano Miani, and Carlo Savorgnan j+

x

j

j

= Ai x + Bi u =

Xpji

=

r

j pk,j i x

(18)

k=1

Since pji is a column of a matrix whose 1-norm is smaller than one, then r k,j j+ ∈ Co(−X, X). Though the above condition k=1 |Pi | < 1 and therefore x holds only on each of the vertices of the set P , it can be shown that, in view of the linearity of the system and the convexity of P , the inclusion x+ ∈ Co(−X, X) is guaranteed for every value of w(t) and for every point of P , say it is possible to derive a state feedback stabilizing control law u = φ(x). Moreover, since the state and input values xj and uj satisfy (11), it can be shown that the derived control law complies with the state, input and output constraints. In the continuous–time case the main condition to be checked, apart from the conditions (11) which are needed for the constraints, is (10). From the definition of Hi , such condition can be written as Ai X + Bi U = Xτ−1 (Pi − I)

(19)

for a proper value of τ, then (I + τAi )X + τBi U = XPi

(20)

where Pi is such that Pi 1 < 1. Thus, equation (20) corresponds to the robust condition for the discrete-time system x+ (t) =

s i=1

wi (t)(I + τAi )x(t) +

s

wi (t)τBi u(t)

(21)

i=1

also known as Euler Approximating System (EAS) of (1) (with time constant τ). Therefore, an interpretation of Theorem 1 is the following: a constrained continuoustime system is robustly stabilizable if and only if there exists τ > 0 such that the corresponding discrete-time EAS is robustly stabilizable system. Similar considerations can be made for the gain scheduling and switched gain scheduling conditions. In this case the matrix U is substituted by Ui . This fact simply implies that the value of u that stabilizes the system depends on the value of the index i.


349

2.1 Algorithms for the Calculation of Contractive Sets So far polytopic sets useful in ascertaining the existence of constrained control laws have been presented as the focal point. The usefulness of the results in Theorem 1 would be anyhow limited if no constructive algorithm for the determination of such sets were available. Fortunately, numerical algorithms for the calculation of such matrices are available [20]. Such numerical algorithms rely on the basic ideas by Bertsekas and Rhodes [3] and essentially consist in the backward calculation of reachability sets. Before getting into this issue, the definition of contractive set in the robust and gain scheduling case as well as that of pre–image set are introduced. Definition 3. P is a robust (resp. gain scheduling) λ-contractive set , with 0 < λ < 1, with respect to the discrete system (1) if for every x ∈ P there exist a value of the system input u (resp. u(w)) such that A(w)x + B(w)u ∈ λP (resp. A(w)x + B(w)u(w) ∈ λP ) for any w ∈ W , where λP = {x : x/λ ∈ P }. To keep things simple and avoid introducing subdifferentials and other mathematichal tools (see [5] for a formal definition), we will say that a set P is β < 0 contractive with respect to the continuous–time system (1) if it is λ = 1 + τβ contractive for the corresponding EAS for a certain τ > 0. The above definition clearly extends to the case of switched gain scheduled systems when w is not allowed to live in the entire W , but just on its vertices. It is rather assessed that the existence of a λ (resp. β in the continuous–time case) contractive set P is equivalent to the fact that, for every x(0) ∈ P , x(t) ≤ γ1 λt x(0) , (resp. x(t) ≤ γ2 eβt x(0) )

(22)

To relate the just introduced definition with what previously presented in Theorem 1, it is worth observing that the sets P introduced in the theorem and described by means of the matrices X (say P = Co(−X, X)), are contractive, but with an unspecified value of λ < 1 (resp. β). The just introduced parameter λ gives a measure of the contractivity and therefore of the convergence speed of the system. A concept on which the algorithm to be introduced is based is the following: Definition 4. Given a polyhedral set P = {x ∈ IRn : F x ≤ 1} and the discrete-time system (1), the one step pre-image of P (referred to as P (1) ) is the set of states which

350


can be brought to P in one step by a proper control action. Such set can be computed as [4, 8]: • in the robust case P (1) = {x ∈ IRn : ∃u such that F (Ai x + Bi u) ≤ 1, ∀i};

(23)

• in the switched gain scheduling case P (1) = {x ∈ IRn : ∀i, ∃ui such that F (Ai x + Bi ui ) ≤ 1}.

(24)

The relation between preimage sets and λ-contractive ones is clarified by the following result. Theorem 2. The set P is λ-contractive with respect to the discrete-time system (1) if P ⊆ preimage(λP ). In [4] it has been shown that a λ-contractive set can be computed starting from any arbitrary P (0) and recursively calculating the pre-image set for the dynamic system x(t + 1) =

B(w(t)) A(w(t)) x(t) + u(t) λ λ

as P (∞) =

9

P (k)

(25)

(26)

k=0,...,∞

It turns out that P (∞) is the largest λ-contractive set in P (0) , say condition (22) cannot be satisfied for any x(0) ∈ P . With this background, we are now ready to present the algorithm to calculate λcontractive sets for robust synthesis problems. We consider the discrete-time system (1) with constraints on input, state and output variables as in Theorem 1. In the algorithm a tolerance is introduced to cope with the possibility of an infinite number of iterations and the intersection (26) is carried out at each set to avoid storing all the pre-image P (k) . Algorithm 9 Set 0 < λ < 1, 0 < < (1 − λ), k = 0. Assign the initial polytope P (0) = X = {x : F (0) x ≤ 1}.


351

1. Calculate the expanded set E (k) ∈ IRn+m E (k) = {(x, u) ∈ IR(n+m) : F (k) [Ai x + Bi u] ≤ λ1, ∀i};

(27)

2. Compute the projected set P r(k) ∈ IRn P r(k) = {x ∈ IRn : ∃u : (x, u) ∈ E (k) , (x, u) ∈ Y , u ∈ U};

(28)

3. Calculate the intersection P (k+1) ∈ IRn P (k+1) = P (k)

9

P r(k)

(29)

(P (k+1) = {x : F (k+1) x ≤ 1}); 4. If P (k+1) is (λ + )-contractive stop, else k = k + 1 and go to step 1. Without getting into the details of the numerical methods needed to properly implement the algorithm, we just focus on the most interesting issues in the algorithm, whose implementation is freely available [20]. • By the assignment P (0) = X , the set calculated when the stopping criterion is met is the largest λ-contractive set compatible with the constraints. • Step 2 is crucial for the algorithm. The projection of E (k) bypasses the problem of calculating simultaneously the contractive set and a proper value of the input that guarantees the contractivity. Indeed, at each step no information about the controller is considered. • The constraints u ∈ U and y ∈ Y are taken into account in (28). Some simple changes are needed to accomplish the computation of a λ-contractive set when the switched gain scheduling synthesis problem is considered. As in the previous algorithm the discrete-time system (1) is considered and the constraints are as in Theorem 1. Algorithm 10 Set 0 < λ < 1, 0 < < (1 − λ), k = 0. Assign the initial polytope P (0) = X = {x : F (0) x ≤ 1}. (k)

1. Calculate the expanded sets Ei (k)

Ei

∈ IRn+m

= {(x, u) ∈ IR(n+m) : F (k) [Ai x + Bi u] ≤ λ1};

(30)

352

Franco Blanchini, Stefano Miani, and Carlo Savorgnan (k)

2. Calculate the projected sets P ri (k)

P ri

∈ IRn (k)

= {x ∈ IRn : ∃u : (x, u) ∈ Ei , (x, u) ∈ Y , u ∈ U};

(31)

3. Calculate the intersection P (k+1) ∈ IRn P (k+1) = P (k)

9

(k)

P ri

(32)

i=1,...,s

(P (k+1) = {x : F (k+1) x ≤ 1}); 4. If P (k+1) is (λ + )-contractive stop, else k = k + 1 and go to step 1. The proposed algorithms (9 and 10) can be extended to consider systems with an unknown input disturbance d(t) ∈ D ∈ IRq , say to consider systems of the form x+ (t) = A(w(t))x(t) + B(w(t))u(t) + Ed(t) y(t) = C(w(t))x(t) + D(w(t))u(t)

(33)

Indeed, since we are in a worst-case setting, it is only needed to change the expansion steps resulting in, for the robust case E (k) = {(x, u) ∈ IR(n+m) : F (k) [Ai x + Bi u] ≤ λ(1 − max F (k) Ed), ∀i} (34) d∈D

and, for the switched gain scheduling case (k)

Ei

= {(x, u) ∈ IR(n+m) : F (k) [Ai x + Bi u] ≤ λ(1 − max F (k) Ed)}. d∈D

(35)

Remark 2. The presented algorithms deal only with discrete-time systems. When a continuous-time system is considered, it is still possible to use the algorithms 9 and 10 by considering the EAS with a proper value of τ > 0. So far, the algorithms presented only give a method to calculate the matrices X in Theorem 1. The calculation of the matrices U , Ui , Hi and Pi can be carried out by using linear programming techniques. In the next section it will be shown how such matrices come in hand to derive stabilizing feedback control laws.

2.2 State Feedback Variable Structure Linear Controllers Once a contractive set is computed it is important to find a control law. Such law is a pure state feedback variable structure linear control of the form u = Φ(x) in the


353

uncertain case and a state feedback variable structure gain scheduling linear control law of the form u = Φ(x, w) in the case of gain scheduled problems. To see how this law is derived, we need to recall that the coupled matrices X and U (or Ui ) in Theorem 1 define the vertices and the corresponding control values to be applied in order to have the inclusion x+ ∈ P . When the state x of the system is not one of the vertices, but belongs anyway to the set P , then such value can be expressed as a (non unique) convex combination of the vertices of P , say x = Xα. The control u(x) = U α can be shown [14, 5] to guarantee the satisfaction of the imposed constraints. A clever way to determine, given x, the vector of multipliers α in a unique way passes through the partition of a polytope in simplicial sectors, as per the next definition: Definition 5. Given a polytopic set P in IRn defined by the convex hull of the vertices xi , we can denote by Xh the full rank matrix obtained by stacking a set of (linearly independent) n vertices: Xh = [xh1 xh2 . . . xhn ] The set

(36)

⎧ ⎫ n n ⎨ ⎬ xhj αj , αj ≤ 1, αj ≥ 0 Sh = x = ⎩ ⎭ j=1

(37)

j=1

is called a simplicial sector of P . It is immediate to see that if a state x belongs to a simplicial sector Sh , then the value α can be computed as

α= x

h1

h2

x

··· x

hn

−1

x

(38)

Now, it can be shown that any polytopic set P can be partitioned in simplicial sectors. More precisely, a particular selection of the vertices is the one given by choosing {h1 h2 . . . hn } such that all the vertices lie on the same face of P , Sh has nonempty 7 interior, Sh Sk = ∅ if h = k and h Sh = P . Once such partition has been determined, it is possible to derive the following variable structure linear controller u = Φ(x) = Kh x

(39)

Kh = Uh Xh−1

(40)

with

354


where the index h corresponds to the simplicial sector Sh to which the state x belongs and the matrix Uh is given by stacking the columns uhj of U . It can be shown [5] that the controller given by equation (39) is robustly stabilizing for the system considered and locally Lipschitz. Moreover, if the matrix X and U are such that the conditions (11) are satisfied, then the proposed control law satisfies the state limitations as well as the input and output constraints. A procedure similar to the one just presented can be used also for the derivation of stabilizing controllers for the other problems considered in Theorem 1. For example, to cope with discrete-time switched system, it is possible to use the equation Ai X + Bi Ui = XPi

(41)

u = Φ(x) = Khi x

(42)

Khi = Uhi Xh−1

(43)

which provides the control

with where h is such that x ∈ Sh and Uhi is the matrix obtained by stacking the columns h

ui j of Ui . Similarly, for discrete–time gain scheduled systems with B certain, it is possible to use the following control law u(x, w) =

s

wi Khi x

(44)

i=1

3 Amplitude and Slew–Rate Constrained Control Problems So far the basic results concerning constrained control systems have been introduced. In this section, it will be shown how the presented results can be applied to the solution of stabilization problems when slew-rate constraints are also present. To this aim, and extended system will be defined and it will be shown how the determination of a constrained control law, as per the results of the previous section, allows to solve the slew-rate constrained control problem. In the present section it will be assumed that the constraints on the state, input and output are given by (9) and that the slewrate constraints are also represented by a polyhedral set u(t) ˙ ∈V

(45)


355

3.1 The Rate Bounding Operator A typical way to face the bounded variation problem is that of inserting in the input channel a suitable operator. In the case of a bounded amplitude system, the typical operator is the saturation one. It is usually assumed that a “desired” input ω is achieved by some feedback. This “desired” input ω then enters an operator which provides the “actual control”. For instance in the scalar case the (symmetric) saturation operator is ⎧ ⎪ ¯ if u ≥ u ¯ ⎪ ⎨ u Satu¯ (u) = u if −¯ u≤u≤u ¯ ⎪ ⎪ ⎩ −¯ u if u ≤ −¯ u

(46)

The saturation operator has an obvious multi–input version. Conversely if the problem is rate–bounded, a more sophisticated model is necessary. An operator model to this purpose is proposed in [19] and another approach has been proposed by [2]. The corresponding operator is now a dynamic one. If w is the desired signal and u the actual signal, assuming |u| ˙ ≤ v¯, the operator depicted in Figure 1 can be expressed as u(t) ˙ = satv¯ [ω − u]

r

ω

(47)

u

v v

F(s)

y

Fig. 1. Rate bounding operator

Given these operators, the problem normally consists on the determination of a controller which produces the “desired control value” ω in such a way that there are no bad consequences once this is saturated. Here, a different approach is used. We produce a control law which satisfies both amplitude and rate constraints. In other words, we do not make use of any saturation

356


operator, but we will derive state feedback control laws which respect the given constraints.

3.2 The Continuous-Time Case To cope with slew–rate limitations and formulate the problem in the settings of state constrained systems previously introduced, we use the following extended system: x(t) ˙ A(w(t)) B(w(t)) x(t) 0 = + v(t) (48) u(t) ˙ 0 0 u(t) I This extended system is simply obtained from the original one by adding an integrator on the input of the original system. Therefore, the control u(t) becomes a new state variable and a new input vector v(t) is defined. By doing so, the original constraint u ∈ U has become a state constraint for the new system and the slew-rate constraint for the original system, u(t) ˙ ∈ V , has now become an input constraint, say v(t) ∈ V . It is clear that the stabilization of the original constrained system (1) with bounded input and slew–rate constraints requires the stabilization of the system without slew– rate constraints, which, to the light of the results of the previous section, is equivalent to the existence of a polytopic contractive set of the form P = {x = Xα, α 1 ≤ 1}

(49)

that doesn’t take into account the slew-rate constraints. In view of the equivalence between the robust and gain scheduling stabilization (and the switched gain scheduling control in the case of uncertain B is certain), we assume constrained robust stabilization and, precisely, we assume the existence of a contractive P , namely the existence of a matrix U ∈ IRp×r , whose columns are in U, and s matrices Hi ∈ H such that the following equation Ai X + Bi U = XHi

(50)

is satisfied as well as the conditions (11). Now, we extend equation (50) to check what happens by considering the slew–rate constraints. To this aim, assuming for brevity that the matrix


M=

X

357

(51)

U

is full row rank3 , it suffices to take s matrices Vi ∈ IRp×r such that, for any i, X Ai Bi X 0 Hi (52) + Vi = U 0 0 U I It is straightforward that the above is simply obtained by choosing (53)

Vi = U Hi

Now, equation (52) implies that the extended system admits a linear variable structure gain scheduled control law of the form (54)

v(t) = φ(x, u, w)

which depends on the parameter w and satisfies the constraints on the state, input and output variables. Because u(t) is nothing but the integral of v(t), this means that by applying u(t) ˙ = v(t), v(t) = φ(x, u, w)

(55)

it is possible to achieve the same closed loop we would achieve by applying the static controller (54) to the augmented system. The above is summarized in the next proposition: Proposition 1. Assume the gain scheduled synthesis problem for system (1), has a solution with a single matrix U (i.e. the robust problem) and that the matrix M is full row rank. Assume furthermore that (52) holds, that the columns of X and U satisfy the state, input and output constraints (9) and that each column vi of V is such that vi ∈ V . Then, for every x(0) ∈ Co(−X, X) there exists u(0) ∈ U such that the closed loop evolution, when the gain scheduled linear variable structure control law (44) is applied: v = Φ(x, u, w) =

s i=1

wi Khi

x u

is such that v(t) ∈ V , x(t) ∈ X , u(t) ∈ U and y(t) ∈ Y , for any w(t) ∈ W and t ≥ 0. 3

If this is not the case, then it is possible to augment such equation, similarly to what has been done in a different setting in [9]

358


The technique proposed has the following shortcomings: • The control is not robust anymore since the knowledge of w is required. Indeed, also if we have a single matrix U satisfying equation (10), the extended system admits an extended contractive set Q represented by the matrix M which satisfies the appropriate equation with matrices Vi depending on i. • When calculating the set P , slew-rate constraints are neglected and therefore there is no guarantee that u˙ ∈ V since this is checked a-posteriori. Indeed, depending on the problem, the columns of V might not belong to V . • For a given x(0), a proper initialization value for u(0) has to be found. The first issue can be solved, in the robust case, by resorting to the results in [6] (and reported in the previous section in Theorem 1) concerning the equivalence between robust and gain scheduling synthesis. Such equivalence relies on the approximation of the extended polytopic region P by a smoothed version P˜ which allows to use a gradient based control law (see also [7]) which is based on the state only (which, in the present case, is composed by the actual state, x, and by the input u) and not on the parameter w. Note that the extended input matrix is [0 I]T which is independent of w and therefore the gradient based controller proposed in [7] can be applied. A more appropriate approach, which takes into account also the second issue, is to start directly by considering the extended system. Precisely we apply the iterative procedure to compute matrices satisfying the equations Ai Bi X X 0 Hi + Vi = U 0 0 U I We assume Vi = V if we are interested in the robust case. The above conditions are equivalent to the solution of the gain scheduled stabilization problem for the extended system and, in view of the equivalence, also to the robust stabilization problem of the extended system, say it is possible to derive a control law v (though not continuous). For a more in depth analysis of the relations between the extended and original system the reader is referred to [9]. Concerning the last point, we postpone the discussion to Section 3.4. Summarizing the presented technique, we have the following synthesis procedure. 1. expand the original system to obtain system (52);


359

2. calculate the largest polytopic contractive set Q (for a given contractivity β) for the system obtained in the previous step by means of the procedures presented Section 2.1; 3. use the controller (56)

u(t) ˙ = v(t) v(t) = φ(x, u)

(or φ(x, u, w)), where φ is a variable structure linear controller for the expanded system. The so derived control law is such that for every initial (extended) state [xT uT ]T the state evolution satisfies the given amplitude and slew–rate constraints and converges to zero as in (22). For a given β, since the so computed contractive is the largest β contractive set, there are no other initial states outside Q starting from which the same convergence is assured while respecting the constraints. 3.3 The Discrete-Time Case In the discrete-time case, the extension which allows to derive a stabilizing controller is

x(t + 1) u(t + 1)

=

A(w(t)) B(w(t)) 0

I

x(t) u(t)

+

0 I

v(t)

(57)

Once again, slew-rate constraints have been transformed into the new input constraints and the original input constraints have been transformed into state constraints and thus it is possible to proceed exactly as in the continuous-time case. Unfortunately, introducing a one step delay in the actual system input is not a payless operation. In fact, the actual input value u(t) can be set only by proper choice of the value of the (fake) input variable at the previous time instant, v(t − 1). The next theorem holds. Theorem 3. Assume the extended system (57) can be stabilized by means of a constrained control law v(t) = φ(x(t), u(t), w(t)) (or in particular v(t) = φ(x(t), u(t))). Then, the original system can be stabilized by a control law complying with the constraints (45). The set of all initial control–states for which the constrained convergence is assured is 8 x X Q = = β, β 1 ≤ 1 . u U

(58)

360


3.4 The Initialization Problem We now wish to go back to the problem of the initial control value. It is indeed quite clear that if control rate are present, the possibility of initializing the control value is a key point in the determination of the set of states that can be brought to the origin without constraints violation (either state, input, output or rate). More precisely, in applying the previous results, we have that convergence is guaranteed under bounds for all states of the extended systems namely for all pair (xT , uT )T ∈ Q . Therefore the convergence to the origin with constraints satisfaction does not depend only on the initial state but also on the initial control value. Therefore we can distinguish two cases: • the control value can be initialized; • the control value is initially assigned. The first cases, which seems more unrealistic, can be indeed justified if one assumes that the “rate bound” is not a hard constrains but rather a soft performance specification which can be instantaneously broken. In this way a “high” rate initial control variation is admitted (a step) to consequently assure a bounded variation. In the second case the control value is given u(0) = u ¯. The two different cases reflect in the set of states which can be brought to the origin without constraint violation, according to the next result: Theorem 4. Assume that for a slew–rate constrained synthesis problem the largest λ (resp. β in the continuous–time case) contractive set Q for the extended system has been found. Then the set X {0} of initial values x(0) starting from which the state evolution converges to the origin at least as in (22) without constraints violation is given by • the projection of Q on IRn : X {0} = {x : [xT uT ]T ∈ Q for some u} • the intersection of Q with the plane u = u ¯. 9 X {0} = X7 = Q {[xT uT ]T : u = u ¯}

(59)


361

Proof. The first item is rather obvious to the light of what has been said in the previous section, and we focus on the second one only. If u(0) = u ¯ is fixed, then since the computed set is the largest λ contractive (resp. β) set, constraint satisfaction and convergence to the origin of the state evolution for the extended system as in (22) is satisfied only for the states [xT uT ]T ∈ Q . Since the u component is fixed, this in turn requires x ∈ X7 .

4 Examples The next very simple example shows how the amplitude–constrained domain of attraction shrinks under slew–rate contraints. Example 1. Consider the continuous–time system (60)

x(t) ˙ = x(t) + u(t) with |u(t)| ≤ 1. A domain of attraction is the set P = {x : |x| ≤ a}

(61)

where a < 1. Let us now investigate on what happens if we impose |u(t)| ˙ ≤ v¯. We need to add the equation (62)

u(t) ˙ = v(t) obtaining the system

x(t) ˙ u(t) ˙

=

11 00

x(t) u(t)

+

0 1

v(t)

(63)

where |v(t)| ≤ v¯. In Figure 2 we can see the domains of attraction of system (63) for v¯ = 2, 1, 0.5, 0.1. Such sets are computed by choosing τ = 0.1 and setting λ = 0.9999 and = 0.00001. The lines resemble continuous curves since the vertices of the polytopes are really close (when v¯ = 0.1 there are more than 350 vertices). If there were no constraints on v(t) (¯ v = ∞) we would have aspected a squared domain of attraction {(x, u) : |x| ≤ 1, |u| ≤ 1}. By choosing a finite value of v¯ and decreasing it we can observe how the domain of attraction shrinks.

362


1 0.8 0.6 0.4

u

0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

−0.8

−0.6

−0.4

−0.2

0 x

0.2

0.4

0.6

0.8

1

Fig. 2. Domains of attraction for system (63) with the given constraints.

In the next example, the behavior of the bounded slew–rate control law derived in the previous sections is applied to a cart-pendulum system. Example 2. Consider the the classical example of the inverted pendulum depicted in Figure 3. The problem that we analyze is stabilizing the position of the pendulum to the top vertical position. The pendulum has a concentrated mass m = 1 Kg and a connecting member length L = g (where g represents the acceleration of gravity). A disturbance z acts on the center of gravity of the pendulum. The cart is characterized by a mass M = 1 Kg. A force u is applied on the cart to control the system. The specification considered are the following • actuator magnitude saturation |u(t)| ≤ 1.25; • actuator rate saturation u(t) ˙ ≤ 10. By choosing the state variables to be the pendulum horizontal speed x1 , the horizontal displacement between the cart and the pendulum x2 and the horizontal speed of the cart x3 , we can linearize the system equations at the origin obtaining the equations


363

Fig. 3. Pendulum state variables.

x˙ 1 (t) = −x2 (t) + z(t)

(64)

x˙ 2 (t) = −x1 (t) + x3 (t)

(65)

x˙ 3 (t) = x2 (t) + u(t)

(66)

To consider the actuator rate saturation we augment the system obtaining ⎡

x˙ 1 (t)

⎤

⎡

⎡ ⎤ ⎡ ⎤ 1 0 ⎢ ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ 0 1 0⎥ ⎥ ⎢ x2 (t) ⎥ + ⎢ 0 ⎥ v(t) + ⎢ 0 ⎥ z(t) ⎢ ⎢0⎥ ⎥ ⎢ ⎥ 1 0 1⎥ ⎣ ⎦ ⎦ ⎣ x3 (t) ⎦ ⎣ 0 ⎦ 0 000 1 u(t)

0 −1 0 0

⎥ ⎢ ⎢ ⎢ x˙ 2 (t) ⎥ ⎢ −1 ⎥ ⎢ ⎢ ⎢ x˙ (t) ⎥ = ⎢ 0 ⎣ 3 ⎦ ⎣ 0 u(t) ˙

⎤⎡

x1 (t)

⎤

(67)

where u(t) becomes a state variable and v(t) is the new input. By using the techniques previously shown, we first investigate on which is a proper value of the bandwith that can be achieved with a variable structure control. For this purpose we first neglect the disturbance input and we impose a bandwidth of B = 2 rad/s. It can be shown that a worst case bound on the bandwidth is given by B=

1−λ τ

(this bound is really conservative). We choose τ = 0.1 and λ = 0.8. The

resulting contractive set is really complex (it is defined by more than 2000) so we lower the requirement on B.

364


For B = 1 rad/s, by choosing τ = 0.1 and λ = 0.9, the resulting contractive set is simpler (it counts 144 inequalities and 618 vertices). To test the control obtained for the calculated set, it can be interesting simulate the system starting from an initial state corresponding to the steady condition where x2 attains the maximal value such that the pendulum reaches the vertical position. This value can be therefore calculated by choosing the greatest value of x2 such that the point (0, 0, x2 , 0) is contained in the contractive set. The value obtained is x2 = 0.168. In Figure 4, the intersection of the contractive set with the hyperplanes x3 = 0 and u = 0 is shown. 1 0.8 0.6 0.4

x2

0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

−0.8

−0.6

−0.4

−0.2

0 x1

0.2

0.4

0.6

0.8

1

Fig. 4. Intersection between the hyperplanes x3 = 0, u = 0 and the domain of attraction (B = 1 rad/s).

Figure 5 shows the simulation with the initial condition (x1 , x2 , x3 , u)0

=

(0, 0.168, 0, 0). It can be observed how the value of v(t) (the solid line in the bottom graph) reaches the minimum allowed. By reintroducing the disturbance we can now investigate on which is the maximal magnitude of z(t) (referred to as zmax ) that is allowed to achieve the stability of the system without breaking the constraints. Using the bisection method, the maximal value is found to be in the open interval (0.09, 0.1). The calculated domain of


365

0.2 0.15

x2

0.1 0.05 0 −0.05

0

1

2

3

4

5 t

6

7

8

9

10

0

1

2

3

4

5 t

6

7

8

9

10

5

u,v

0

−5

−10

Fig. 5. Simulation of system (67) obtained for (x1 , x2 , x3 , u)0 = (0, 0.168, 0, 0) and z(t) = 0. In the top graph x2 (t) is represented by a solid line. In the bottom graph u(t) and v(t) correspond respectively to the dashed and solid line.

attraction (τ = 0.1 and λ = 0.8) is a polytope defined by 102 inequalities and 428 vertices. Figure 6 shows the evolution of x2 (t), z(t), u(t) and v(t) starting form a null initial condition when a persistent disturb z(t ≥ 1) = 0.09 is applied. We can observe that the constraints are far from being broken. This follows from the fact that only certain signals z(t) lead the control input to act close to the constraints. The control we obtained by applying the technique explained in the previous sections guarantees indeed that for every signal |z(t)| ≤ zmax the system is stable and satisfies the constraints. Allowing z(t) to be any signal involves a drawback: when the designer is aware that the disturbance has certain properties, these information can’t be taken into account and the results obtained become quite conservative. To remark this fact, the simulation illustrated in Figure 7 shows that also when |z(t)| ≤ zmax is not verified the control calculated may work well. For example, when 0 f or t < 1 z(t) = 0.5 f or t ≥ 1 the constraints are not broken.

(68)

366

Franco Blanchini, Stefano Miani, and Carlo Savorgnan 0.1 0.08

x2, z

0.06 0.04 0.02 0 −0.02

0

1

2

3

4

5 t

6

7

8

9

10

0

1

2

3

4

5 t

6

7

8

9

10

1

u, v

0.5

0

−0.5

Fig. 6. Simulation of system (67) applying a persistent disturb z(t ≥ 1) = 0.09. In the top graph x2 (t) and z(t) are represented respectively by a dashed and a solid line. In the bottom graph u(t) and v(t) correspond to the dashed and solid line. 0.6

x2, z

0.4

0.2

0

−0.2

0

1

2

3

4

5 t

6

7

8

9

10

0

1

2

3

4

5 t

6

7

8

9

10

6

u, v

4 2 0 −2 −4

Fig. 7. Simulation of system (67) applying a persistent disturb z(t ≥ 1) = 0.5. In the top graph x2 (t) and z(t) are represented respectively by a dashed and a solid line. In the bottom graph u(t) and v(t) correspond to the dashed and solid line.


367

References 1. A. Bemporad, M. Morari, V. Dua, and E. Pistikopoulos. The explicit linear quadratic regulator for constrained systems. Automatica J. IFAC, 38(1):3–20, 2002. 2. J. M. Berg, Hammet K. D., Schwartz C. A., and S. Banda. An analisys of the destabilizing effect of daisy chained rate-limited actuators. Trans. Control Systems Technology, 4(2), 1996. 3. D. P. Bertsekas and I. B. Rhodes. On the minmax reachability of target set and target tubes. Automatica J. IFAC, 7:233–247, 1971. 4. F. Blanchini. Non-quadratic lyapunov function for robust control. Automatica J. IFAC, 31(3):451–461, 1995. 5. F. Blanchini. Set invariance in control – a survey. Automatica J. IFAC, 35(11):1747–1767, 1999. 6. F. Blanchini. The gain scheduling and the robust state feedback stabilization problems. IEEE Trans. Automat Control, 45(11):2061–2070, 2000. 7. F. Blanchini and S. Miani. Constrained stabilization via smooth lyapunov functions. Systems Control Lett., 35:155–163, 1998. 8. F. Blanchini and S. Miani. Stabilization of LPV systems: state feedback, state estimation and duality. SIAM J. Control Optim., 42(1):76–97, 2003. 9. F. Blanchini, S. Miani, and C. Savorgnan. Stability results for continuous and discrete time linear parameter varying systems. In Proc. of the 2005 IFAC conference, page CDROM, Prague, 2005. 10. Y. Cao, Z. Lin, and Y. Shamash. Set invariance analysis and gain-scheduling control for LPV systems subject to actuator saturation. Systems Control Lett., 46(2):137–151, 2002. 11. R. Freeman and L. Praly. Integrator backstepping for bounded controls and control rates. IEEE Trans. Automat Control, 43(2):256–262, 1998. 12. E. G. Gilbert and K. K. Tan. Linear systems with state and control constraints: the theory and the applications of the maximal output admissible sets. IEEE Trans. Automat Control, 36(9):1008–1020, 1991. 13. J. M. Gomes da Silva Jr., S. Tarbouriech, and Garcia G. Local stabilization of linear systems under amplitude and rate saturating actuators. IEEE Trans. Automat Control, 48(5):842–847, 2003. 14. P. Gutman and M. Cwikel. Admissible sets and feedback control for discrete-time linear dynamical systems with bounded controls and states. IEEE Trans. Automat Control, 31(4):373–376, 1986. 15. D. Henrion, S. Tarbouriech, and V. Kucera. Control of linear systems subject to input constraints: a polynomial approach. Automatica, 37(4):597–604, 2001.

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16. J-B. Hiriart-Urruty and C. Lemarcéchal. Convex analysis and minimization Algorithms I and II. Springer-Verlag, 1993. 17. T. Hu and Z. Lin. Control Systems with Actuator Saturation: Analysis and Design. Birkhauser, Boston, 2001. 18. F. Mesquine, F. Tadeo, and A. Benzaouia. Regulator problem for linear systems with constraints on control and its increment or rate. Automatica, 40(8):1387–1395, 2004. 19. A. Saberi, A. Stoorvogel, and P. Sannuti. Communication and control engineering series. In Control of linear systems with regulation and input constraints. Springer-Verlag London, Ltd., London, 2000. 20. C. Savorgnan.

Maxis-g software package.

http://www.diegm.uniud.it/savorgnan or

http://www.diegm.uniud.it/smiani/Research.html, 2005. 21. J. Solis-Daun, R. Suarez, and J. Alvarez-Ramirez. Global stabilization of nonlinear systems with inputs subject to magnitude and rate bounds: a parametric optimization approach. SIAM J. Control Optim., 39(3):682–706, 2000. 22. H. J. Sussmann, E. D. Sontag, and Y. Yang. A general result on the stabilization of linear systems using bounded controls. IEEE Trans. Automat Control, 39:2411–2425, 1994.

Output Feedback for Discrete-Time Systems with Amplitude and Rate Constrained Actuators J.M. Gomes da Silva Jr.1 , D. Limon2 , T. Alamo2 , and E.F. Camacho2 1

Department of Electrical Engineering, UFRGS, Av. Osvaldo Aranha 103, 90035-190 Porto Alegre-RS, Brazil [email protected]

2

Dept. Ingenier´ıa de Sistemas y Automática, Universidad de Sevilla, Camino de los Descubrimentos s/n, 41092 Sevilla, Spain limon,alamo,[email protected]

Summary. The aim of this chapter is the presentation of a technique for the design of stabilizing dynamic output feedback controllers for discrete-time linear systems with rate and amplitude saturating actuators. Two synthesis objectives are considered: the maximization of the domain of attraction of the closed-loop system under some time-domain performance constraints; and the guarantee that the trajectories of the system are bounded in the presence of L2 -bounded disturbances, while ensuring an overbound for the L2 gain from the disturbance to the regulated output. The nonlinear effects introduced by the saturations in the closed-loop system are taken into account by using a generalized sector condition, which allows to propose theoretical conditions to solve the synthesis problems directly in the form of linear matrix inequalities (LMIs). From these conditions, convex optimization problems are proposed in order to compute the controller aiming at the maximization of the basin attraction of the closed-loop system, as well as aiming at ensuring a level of L2 disturbance tolerance and rejection. Numerical examples are provided to illustrate the application of the proposed methodology. Keywords: constrained control, control saturation, output feedback, stabilization, discretetime systems.

1 Introduction The physical impossibility of applying unlimited control signals makes the actuator saturation an ubiquitous problem in control systems. In particular, it is well known

370

J.M. Gomes da Silva Jr., D. Limon, T. Alamo, and E.F. Camacho

that the input saturation is source of performance degeneration, limit cycles, multiple equilibrium points, and even instability. Hence, it was great the interest in studying these negative effects and also in proposing control design procedures, in global, semiglobal and local contexts of stability, taking directly into account the control bounds: see for instance [20], [10], [11], and references therein. It should be pointed out that most of these works consider only input amplitude saturation and state feedback control strategies. Although the proposition of state feedback methods allow to have a good insight into the problem, the practical applicability of these methods is limited. On the other hand the works proposing output feedback strategies consider, in general, observer-based control laws ensuring global or semi-global stabilization. However, when the open-loop system is not null controllable or additional performance and robustness requirements have to be satisfied, local (or regional) stabilization approaches are needed and an implicit additional objective is the enlargement of the basin of attraction of the closed-loop system. In this context, the amount of works proposing output feedback control strategies is even smaller. Works formally addressing the stabilization in the presence of both amplitude and rate saturation started to appear late 90s. Global and semi-global stabilization results using both state feedback and observer-based control laws were proposed in [14], [18] and [19]. Concerning a local stabilizing context, we can cite the results presented in [8], [1], [21], where the synthesis of state feedback control laws are proposed. On the other hand, the synthesis of dynamic output feedback controllers ensuring local stability is considered in [22] and [15]. In [22], a method for designing dynamic output controllers using of the Positive Real Lemma is proposed. The main objective pursued in that paper is the minimization of an LQG criterion. A region of stability is associated to the closed-loop system. However, it should be pointed out that the size and the shape of this region are not taken into account in the design procedure, which can lead to very conservative domains of stability. Furthermore, the controller is computed from the solution of strong coupled Riccati equations which, in general, are not simple to solve. A time-varying dynamic controller is proposed in [15]. Considering amplitude bounded disturbances, the main focus of that paper is the disturbance rejection problem. The problem of characterization of estimates of the region of attraction is not explicitly discussed. The stabilizing conditions are obtained from a polytopic differential inclusion representation of the saturation terms. Hence, considering the parameter that defines the region of validity of the polytopic

Output Feedback with Amplitude and Rate Constrained Actuators

371

representation, the derived conditions are BMIs, which implies the use of iterative LMI relaxation schemes for computing the controller. On the other hand, it should be pointed out that all the references above are concerned only with continuoustime systems and the rate limitation is considered in the modelling of the actuator, i.e. a position-feedback-type model [22] is considered. In this case, the rate saturation is modelled, in fact, as a saturation of the actuator state. Hence, the plant plus the actuator appears as a nonlinear system which renders the formal analysis in the sampled-data control case quite involved. An alternative approach consists therefore in designing a discrete-time nonlinear controller (i.e. considering saturations in the controller) in order to prevent that control signal (to be sent to the actuator) violates the rate and amplitude bounds. The aim of this chapter is the proposition of a technique for the design of stabilizing dynamic output feedback controllers for discrete-time linear systems with rate and amplitude constrained actuators. Two implicit design objectives are particularly considered. The first one concerns the synthesis of the controller aiming at the maximization of the region of attraction of the closed-loop system, while guaranteeing a certain degree of time-domain performance for the system operation in a neighborhood of the origin (equilibrium point). The second one concerns the disturbance tolerance and rejection problems. Since the control signals are bounded, the proposed conditions allows to ensure that the system trajectories are bounded for given L2 -bounded disturbances acting on the system. Furthermore, the synthesis can be carried out in order to minimize the L2 gain from the disturbance to the regulated output. The results apply to both stable and unstable open-loop systems. In order to deal with the rate limitation, we propose the synthesis of a nonlinear dynamic controller which is composed by a classical linear dynamic controller in cascade with input saturating integrators and two static anti-windup loops. It should be pointed out that, differently from the anti-windup approaches (see for instance [12], [9],[7], and references therein), where the controller is supposed to be given, here the idea consists in computing simultaneously the controller and the anti-windup gains. The anti-windup gains appear therefore as extra degrees of freedom in the synthesis problem. The theoretical conditions for solving the synthesis problem are based on a generalized sector condition proposed in [7]. This condition encompasses the classical sector condition, used for instance in [13], and allows (differently from the classical

372


one) the formulation of local stability conditions directly in LMI form. Using then the classical variables transformations as proposed in [17] and [3], it is possible to formulate conditions that allow to compute a dynamic controller that stabilizes the closed-loop system. Convex optimization problems are therefore proposed in order to compute the controller aiming at the maximization of the basin attraction of the closed-loop system, as well as aiming at ensuring a level of L2 disturbance tolerance and rejection. Numerical examples are provided to illustrate the application of the proposed method.

2 Problem Statement Consider the discrete-time linear system x(t + 1) = Ax(t) + Bu(t) + Bw w(t) y(t) = Cx(t)

(1)

z(t) = Cz x(t) + Dz,u u(t) + Dz,w w(t) where x(t) ∈ n , u(t) ∈ m , y(t) ∈ p , z(t) ∈ r , w(t) ∈ q are the state, the input, the measured output, the regulated output and the disturbance vectors, respectively, and t ∈ N . Matrices A, B, Bw , C, Cz Dz,u and Dz,w are real constant matrices of appropriate dimensions. Pairs (A, B) and (C, A) are assumed to be controllable and observable respectively. The input vector u is subject to limitations defined as follows. • Amplitude constraints: |u(i) (t)| ≤ ρa(i) , i = 1, ..., m

(2)

where ρa(i) > 0, i = 1, ..., m denote the control amplitude bounds. • Rate constraints: |Δu(i) (t)| = |u(i) (t) − u(i) (t − 1)| ≤ ρr(i) , i = 1, ..., m

(3)

where ρr(i) > 0, i = 1, ..., m denote the control rate bounds. The disturbance vector w(t) is assumed to be limited in energy, that is w(t) ∈ L2 and for some scalar δ, 0
η(i) = 0 if −η(i) ≤ α(i) ≤ η(i) i = 1, . . . , m ⎪ ⎪ ⎩α + η (i) (i) if α(i) < −η(i)

(7)

Define now the following vectors and matrices x x ˜ x ˜= ; ξ= v xc A=

A B

B

Bw

0

C 0

; Bw = ; B1 = ; C= ; B= 0 0 Im 0 Im 0 Im L = 0 Im ; Cz = Cz Dz,u ; Dz,u = Dz,u ; Dz,w = Dz,w


A=

A + BDc C BCc

B1

Bw

; B1 = ; Bw = ; B= Ac Ec 0 L = L 0 ; K = Dc C Cc ; C = Cz 0

Bc C

B

375

Fc

Dz,u = Dz,u ; Dz,w = Dz,w From the definitions above, the closed-loop system can be re-written as ξ(t + 1) = Aξ(t) − B1 ψρa (Lξ(t)) − Bψρr (K ξ(t)) + Bw w(t) z(t) = Cz ξ(t) − Dz,u ψρa (Lξ(t)) + Dz,w w(t)

(8)

Considering the generic nonlinearity ψη (α) and defining the set S(η) = {α ∈ m , β ∈ m ; |α(i) − β(i) | ≤ η(i) , i = 1, . . . , m}

(9)

the following lemma can be stated [7]. Lemma 1. If α and β are elements of S(η) then the nonlinearity ψη (α) satisfies the inequality: ψη (α) T (ψη (α) − β) ≤ 0

(10)

for any diagonal positive definite matrix T ∈ m×m . Lemma 2. Consider the following system composed by m-integrators: v(t + 1) = Im v(t) + q(t) u(t) = satρa (v(t)) If |q(t)(i) | ≤ ρr(i) , i = 1, ..., m, it follows that |Δu(i) (t + 1)| = |u(i) (t + 1) − u(i) (t)| ≤ ρr(i) Proof: Considering that the Lipschitz constant of the sat(·) function is equal to 1, it follows directly that |Δu(i) (t+1)| = |satρa(i) (v(i) (t)+q(i) (t))−satρa(i) (v(i) (t))| ≤ |q(t)(i) | ≤ ρr(i) 2

376


4 Stabilization In this section we focus on the synthesis of the controller (5) in order to guarantee the asymptotic stability of the closed-loop system (8). With this aim we consider the disturbance free case, i.e., w(t) = 0, ∀t ≥ 0. A result concerning the local (or regional) stabilization is presented. This result provides LMI conditions that can be used in order to compute the controller matrices and an associated region of stability for the closed-loop system. It can therefore be applied to either stable and unstable open-loop systems. Finally, it is shown how the derived LMI conditions can be cast in optimization problems for computing the controller in order to maximize an ellipsoidal estimate of the region of attraction of the closed-loop system, while ensuring a certain timedomain performance in the linear region of the closed-loop system.

4.1 Theoretical Result Theorem 1. If there exist symmetric positive definite matrices X, Y

(n+m)×(n+m)

ces Aˆ ∈

∈

, positive definite diagonal matrices Sr , Sa ∈ , and matrim×(n+m) (n+m)×(p+m) ˆ ∈ ˆ ∈ m×(p+m) , , Cˆ ∈ ,B ,D m×m

(n+m)×(n+m)

Zr1 , Zr2 , Za1 , Za2 ∈ m×(n+m) , Qr , Qa ∈ (n+m)×m such that the following linear matrix inequalities are verified 3 ⎡ X ⎢ ⎢ In+m Y ⎢ ⎢ Zr1 Zr2 ⎢ ⎢ ⎢ Za1 Za2 ⎢ ⎢ ˆ ⎣ AX + BCˆ A + BDC ˆ Aˆ Y A + BC

2Sr

0

2Sa

BSr B1 Sa Qr

X

Qa In+m

⎡ ⎢ ⎢ ⎣

3

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥>0 ⎥ ⎥ ⎥ ⎦ Y

(11)

⎤ X

In+m

Y

⎥ ⎥ ⎦≥0

(12)

ˆ (i) C − Zr2(i) ρ2 Cˆ(i) − Zr1(i) D r(i)

stands for symmetric blocks; • stands for an element that has no influence on the development


⎡ ⎢ ⎢ ⎣

377

⎤ X

In+m

Y

⎥ ⎥ ⎦≥0

(13)

X(n+i) − Za1(i) [0 Im ](i) − Za2(i) ρ2a(i)

for all i = 1, ..., m, then the dynamic controller (5) with Fc = N −1 (Qr Sr−1 − Y B) Ec = N −1 (Qa Sa−1 − Y B1 ) ˆ Dc = D Cc = (Cˆ − Dc CX)(M )−1 ˆ − Y BDc ) Bc = N −1 (B 5 6 Ac = N −1 Aˆ − (Y AX + Y BDc CX + N Bc CX + Y BCc M ) (M )−1

(14)

where matrices M and N verify N M = In+m − Y X, guarantees that the region

(15) E(P ) = {ξ ∈ 2(n+m) ; ξ P ξ ≤ 1} Y N X M with P = and P −1 = , is a region of asymptotic stability for N • M • the closed-loop system (8). Proof: Consider the closed loop system (8) and the candidate Lyapunov function V (ξ(t)) = ξ(t) P ξ(t), P = P > 0. Assuming w(t) = 0, ∀t ≥ 0, the variation of V (ξ(t)) along the trajectories of system (8) is given by ΔV (ξ(t)) = V (ξ(t + 1)) − V (ξ(t)) = −ξ(t) P ξ(t) + ξ(t) A P Aξ(t) − 2ξ(t) A P B1 ψρa (Lξ(t)) −2ξ(t) A P Bψρr (K ξ(t)) + ψρr (K ξ(t)) B P Bψρr (K ξ(t)) +ψρa (Lξ(t)) B1 P B1 ψρa (Lξ(t)) + 2ψρa (Lξ(t)) B1 P Bψρr (K ξ(t)) (16) Consider matrices Ga , Gr ∈ m×(2(n+m)) and define now the following sets:

Ξ(ρa ) = {ξ ∈ 2(n+m) ; |L(i) ξ − Ga(i) ξ| ≤ ρa(i) , i = 1, ..., m}

Ξ(ρr ) = {ξ ∈ 2(n+m) ; |K(i) ξ − Gr(i) ξ| ≤ ρr(i) , i = 1, ..., m} From Lemma 1, provided that ξ(t) ∈ Ξ(ρa ) ∩ Ξ(ρr ), it follows that ΔV (ξ(t)) ≤ ΔV (ξ(t)) − 2ψρr (K ξ(t)) Tr [ψρr (K ξ(t)) − Gr ξ(t)] −2ψρa (Lξ(t)) Ta [ψρa (Lξ(t)) − Ga ξ(t)]

(17)

378


For ease of notation in the sequel we denote ψr (t) = ψρr (K ξ(t)) and ψa (t) = ψρa (Lξ(t)). Therefore, expression (17) can be put in matrix form, as follows: ⎡ ⎤ ⎡ ⎤ −A P ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ B ⎥ P −A B B1 )θ(t) (18) − ΔV (ξ(t)) ≤ −θ(t) (⎣ −Tr Gr 2Tr ⎥ ⎦ ⎣ ⎦ B1

−Ta Ga 0 2Ta

where θ(t) = [ξ(t) ψr (t) ψa (t)]. From Schur’s complement it follows that ΔV (ξ(t)) < 0 if ξ(t) ∈ Ξ(ρa ) ∩ Ξ(ρr ) and

⎡

P

⎢ ⎢ −Tr Gr 2Tr ⎢ ⎢ −T G 0 2T ⎣ a a a −A Define now a matrix Π = lows that X − Y −1

⎤

⎥ ⎥ ⎥>0 ⎥ ⎦

(19)

B1 P −1

B

X In+m

[17],[3]. Note now that, from (11), it folM 0 > 0. Therefore, In+m − Y X is nonsingular. Thus it is always

possible to compute square and nonsingular matrices N and M verifying the equation N M = In+m − Y X. From the nonsigularity of M it is inferred that Π is nonsingular. Pre and post-multiplying (19) respectively by ⎡ ⎤ ⎡ −Π 0 −Π 0 0 0 ⎢ ⎥ ⎢ ⎢ ⎢ 0 Sr 0 0 ⎥ ⎥ and ⎢ 0 Sr ⎢ ⎢ 0 0 ⎢ 0 0 S 0 ⎥ ⎣ ⎦ ⎣ a 0 0 0 ΠP 0 0

0

0

⎤

⎥ 0 ⎥ ⎥ Sa 0 ⎥ ⎦ 0 PΠ 0

with Sa = Ta−1 and Sr = Tr−1 one gets: ⎡

Π P Π

⎤

⎢ ⎥ ⎢ Gr Π ⎥ 2S r ⎢ ⎥>0 ⎢ G Π ⎥ 0 2S ⎣ ⎦ a a Π P AΠ Π P BSr Π P B1 Sa Π P Π From the definition of Π, it follows that X I BS n+m r Π P Π = ; Π P BSr = In+m Y Y BSr + N Fc Sr

(20)


Π P B1 Sa =

B 1 Sa Y B1 Sa + N E c Sa

; Π P AΠ =

Γ11 Γ12

379

Γ21 Γ22

with Γ11 = (A + BDc C)X + BCc M Γ12 = A + BDc C Γ21 = Y AX + Y BDc CX + N Bc CX + Y BCc M + N Ac M Γ22 = Y (A + BDc C) + N Bc C ˆ Cc M + Dc CX = Consider now the following change of variables Dc = D, ˆ Y AX+Y BDc CX+N Bc CX+Y BCc M +N Ac M = A, ˆ Y BDc +N Bc = B, ˆ C, Y BSr + N Fc Sr = Qr , Y B1 Sa + N Ec Sa = Qa , Gr Π = [Zr1 Zr2 ] and Ga Π = [Za1 Za2 ]. Hence, since Π, Sr and Sa are nonsingular, it follows that if (11) is verified, (19) holds with the matrices Ac , Bc , Cc , Dc , Ec and Dc defined as in (14). On the other hand, considering that ˆ K Π = [Dc CX + Cc M Dc C] = [Cˆ DC] ˜ I] ˜ LΠ = [[0 Im ] 0]Π = [X

(21)

˜ and I˜ correspond respectively to the matrices composed by the last m lines where X pre and post-multiplying inequalities (12) and (13) of matrices X and In+m . Hence −1 (Π ) 0 respectively by and its transpose, it is easy to see that (12) and (13) 0 1 ensures respectively that E(P ) ⊂ Ξ(ρr ) and E(P ) ⊂ Ξ(ρa ) [2]. Thus, if relation (11), (12) and (13) are satisfied, one obtains ΔV (ξ(t)) < 0, ∀ξ(t) ∈ E(P ), ξ(t) = 0, which means that E(P ) is a contractive region for system (8), i.e., if ξ(0) ∈ E(P ), then the corresponding trajectory converges asymptotically to the origin.2

4.2 Optimization Problems According to theorem 1, any feasible solution of the set of LMIs (11), (12) and (13) provides a stabilizing, and probably different, dynamic controller. Between all these solutions, it is interesting to chose one that optimizes some particular objective.

380


The objective considered in this section concerns the maximization of an ellipsoidal estimate of the closed-loop system domain of attraction. In particular, it can be of interest the maximization of the projection of E(P ) onto the states of the plant (i.e., x). This set is denoted as Ex (P ) and is given by Ex (P ) = {x ∈ n ; ∃v ∈ m , xc ∈ n+m ; [x v xc ] ∈ E(P )} −1 = {x ∈ n ; x X11 x ≤ 1}

where X11 ∈ n×n is obtained from X =

X11 X21 X22

.

Note that for any initial state x(0) ∈ Ex (P ), initial values of the states of the dynamic controller v(0) and xc (0) can be found such that ξ(0) ∈ E(P ), i.e. such that the asymptotically stability of the closed-loop system is ensured. In order to maximize the size of Ex (P ), we can for instance adopt one of the following criteria: −1 • Minimization of the trace of X11 . This can be indirectly accomplished by con-

sidering an auxiliary matrix variable R and the following optimization problem: min trace(R) s.t. (11), (12), (13),

R In In X11

>0

(22)

• Minor axis maximization. It can be accomplished by maximizing the smallest eigenvalue of X11 (which corresponds to minimize the largest eigenvalue of −1 X11 ) , as follows:

max λ s.t. (11), (12), (13), X11 ≥ λIn

(23)

• Maximization along certain directions. This can be accomplished by considering a shape set Ξ0 , which describes the directions in which the ellipsoid should be maximized and a scaling factor η > 0 [5],[6]. Considering a polyhedral shape set described by the convex hull of its vertices (i.e. directions in which Ex (P )

should be maximized) Ξ0 = Co{vr ∈ n ; r = 1, . . . , nv }, the optimization problem to be solved in this case is the following:


max η

s.t. (11), (12), (13),

1 ηvr ηvr X11

≥ 0 , r = 1, . . . , nv

381

(24)

• Volume maximization. This can be accomplished from the following convex optimization problem: max log(det(X11 ))) s.t. (11), (12), (13)

(25)

Instead of the maximization of Ex (P ), another option consists in considering the maximization of the cut of E(P ) by the hyperplane plane x, denoted by Ec (P ) and defined as

Ec (P ) = {x ∈ n ; [x 0 0] ∈ E(P )} = {x ∈ n ; x Y x ≤ 1}

Similar optimization problems to (22), (23), (24) and (25), can thus be formulate −1 . considering Y instead of X11

It is worth noticing that performance requirements such as contraction rate, pole placement of the closed-loop system or quadratic cost minimization can be added to the problem. In this case, the resulting optimization problem can be also written as an LMI optimization problem. For instance the controller can be designed in order to ensure some degree of timedomain performance in a neighborhood of the origin. In general we consider this neighborhood as the region of linear behavior of the closed-loop system [8], i.e., the region where saturation does not occur: RL = {ξ ∈ 2(n+m) ; |K(i) ξ| ≤ ρa(i) , |L(i) ξ| ≤ ρr(i) , i = 1, . . . , m} In this case the time-domain performance can be achieved if we consider the pole placement of matrix A in a suitable region inside the unit circle. Considering an LMI framework, the results stated in [17] can be used to place the poles in a called LMI region in the complex plane. For example, if we verify the following LMI, ⎡ ⎤ −σX −σIn+m ⎢ ⎥ ⎢ −σIn+m ⎥ −σY ⎢ ⎥0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ Y ⎦

(28)

0 γIr

⎥ ⎥≥0 ⎦

⎡ ⎢ ⎢ ⎣

(29)

⎤ X

In+m

Y

X(n+i) − Za1(i) [0 Im ](i) − Za2(i) μρ2a(i)

⎥ ⎥≥0 ⎦

(30)

384


for all i = 1, ..., m, then the dynamic controller (5) with the matrices as defined in (14), where matrices M and N verify N M = In+m − Y X, is such that 1. When w = 0: a) the closed-loop trajectories remain bounded in the set E1 = {ξ ∈ 2(n+m) ; ξ P ξ ≤ μ−1 } for any ξ(0) ∈ E0 = {ξ ∈ 2(n+m) ; ξ P ξ ≤ β}, with Y N X M P = , P −1 = N • M • and β = μ−1 − δ−1 ≥ 0, and for any disturbance satisfying (4). b) ||z||22 < γ||w||22 + γξ(0) P ξ(0).

2. When w = 0, the set E1 is included in the basin of attraction of the closed-loop system (8) and it is contractive.

Proof: Consider the closed-loop system (8) and the candidate Lyapunov function V (ξ(t)) =

ξ(t) P ξ(t), P = P > 0, P ∈ 2(n+m)×2(n+m) . Define J (t) = ΔV (ξ(t)) − w(t) w(t) + γ−1 z(t) z(t) with ΔV (ξ(t)) = V (ξ(t + 1)) − V (ξ(t)). If J (t) < 0 along the system trajectories and ξ(0) ∈ E0 , it follows that T −1 t=0

J (t) = V (T ) − V (0) −

T −1

w(t) w(t) + γ−1

t=0

T −1

z(t) z(t) < 0, ∀T,

t=0

whence: • V (T ) < V (0) + ||w||22 ≤ β + δ−1 = μ−1 , ∀T , i.e. the trajectories of the system do not leave the ellipsoid E1 , for ξ(0) ∈ E0 and w(t) satisfying (4); • considering T → ∞, ||z||22 < γ||w||22 + γξ(0) P ξ(0); • if w = 0 then ΔV (ξ(t)) < −γ−1 z(t) z(t) < 0.


385

Computing therefore J (t) along the system trajectories (8) one obtains: J (t) = V (ξ(t + 1)) − V (ξ(t)) = −ξ(t) P ξ(t) + ξ(t) A P Aξ(t) − 2ξ(t) A P B1 ψρa (Lξ(t)) −2ξ(t) A P Bψρr (K ξ(t)) + ψρr (K ξ(t)) B P Bψρr (K ξ(t)) +ψρa (Lξ(t)) B1 P B1 ψρa (Lξ(t)) + 2ψρa (Lξ(t)) B1 P Bψρr (K ξ(t)) +2ξ(t) A P Bw w(t) − 2ψρa (Lξ(t)) B1 P Bw w(t) −2ψρr (K ξ(t)) B P Bw w(t) + w(t) (Bw P Bw − Iq )w(t)

+γ−1 (Cz ξ(t) − Dz,u ψρa (Lξ(t)) + Dz,w w(t)) (Cz ξ(t)− Dz,u ψρa (Lξ(t)) + Dz,w w(t)) Provided that ξ(t) ∈ Ξ(ρa ) ∩ Ξ(ρr ), it follows that J (t) ≤ J (t) − 2ψρr (K ξ(t)) Tr [ψρr (K ξ(t)) − Gr ξ(t)] −2ψρa (Lξ(t)) Ta [ψρa (Lξ(t)) − Ga ξ(t)]

(31)

From here, writing (31) in a matrix form and following the same steps as in the proof of Theorem 1, we conclude that if (28) is satisfied, J (t) < 0 holds provided that ξ(t) ∈ Ξ(ρr ) ∩ Ξ(ρa ). On the other hand, (29) and (30) ensures that E1 ⊂ Ξ(ρr ) and E1 ⊂ Ξ(ρa ). Hence, provided (28), (29) and (30) hold, ξ(0) ∈ E0 and ||w||22 ≤ δ−1 , it follows that ξ(t) never leaves E1 and therefore J (t) < 0, which concludes the proof of item 1. Suppose now that w = 0 and ξ(0) = ξ ∈ E1 , therefore it follows that ξ(0) ∈ Ξ(ρr )∩Ξ(ρa ). In this case, the satisfaction of (28) ensures that ΔV (ξ(t)) < 0, ∀t ≥ 0, ξ(t) = 0. Since this reasoning is valid for all ξ ∈ E1 it follows that this set is a region where the asymptotic stability of the closed-loop system (8) is ensured. This completes the proof of item 2. 2

The idea behind the result stated in Theorem 2 is depicted in Figure 2. It follows that once the initial state is in E0 , provided ||w|| ≤ δ−1 , the corresponding trajectories remain confined in the region E1 . This last set can therefore be seen as an estimate of the reachable domain of the systems trajectories. Note that β, μ and δ are related by the equation β = μ−1 − δ−1 . Hence a trade-off between the size of admissible initial conditions E0 , which is related to β, and the tolerance to the disturbance, related to δ−1 appears: if we want to consider the possibility of larger admissible initial conditions, smaller will be the L2 bound on the tolerated disturbances, and vice-versa.

386


E1

o

E0

ξ(0)

ξ(T )

Fig. 2. Domain of admissible initial states (E0 ) and the reachable set of trajectories (E1 )

5.2 Optimization Problems The conditions in Theorem 2 are LMIs. Hence, if a linear criterion is used, convex optimization problems can be formulated in order to compute the dynamic controller (5). In the sequel, we focus on the design based on two criteria of particular interest: the maximization of the admissible disturbance bound (tolerance disturbance maximization) and the minimization of the L2 gain between the disturbance and the regulated output (disturbance rejection). For simplicity, in both cases we consider that ξ(0) = 0, i.e., the system is supposed to be in equilibrium at t = 0.

Tolerance Disturbance Since we suppose that ξ(0) = 0, in this case μ−1 = δ−1 . The idea is therefore to minimize μ (i.e. maximize the bound on the disturbance) for which a controller can be computed in order to ensure that the closed-loop trajectories are bounded. This can be accomplished by the following optimization problem. min μ s.t. (28), (29) and (30)

(32)

In this case, the value of γ obtained does not matter.

Disturbance Rejection Considering ξ(0) = 0, an upper bound to the L2 -gain between w and z is given by √ γ. Hence, assuming that the admissible disturbance level δ−1 = μ−1 is given, the


387

following problem can be considered. min γ

(33)

s.t. (28), (29) and (30)

6 Numerical Examples 6.1 Example 1 Consider the discrete-time linear system given by 0.8 0.5 0 A= , B= , C= 01 −0.4 1.2 1 This system must be controlled with the following saturating limits |u(t)| ≤ 1,

|Δu(t)| ≤ 0.3

We consider also that, in order to ensure a certain time-domain performance when the system is not saturated, the poles of matrix A must be placed inside a disk of radius 0.9 The stabilizing dynamic controller has been computed solving the optimization problem (22). The obtained controller is given by ⎤ ⎤ ⎡ ⎡ 0.6326 −0.0482 0.0166 14.8846 8.1483 ⎥ ⎥ ⎢ ⎢ ⎥ ; Bc = ⎢ 39.1828 24.2906 ⎥ Ac = ⎢ −0.8236 0.0628 −0.0216 ⎦ ⎦ ⎣ ⎣ 1.9616 −0.1494 0.0515 −42.6207 −142.5520

Cc = 0.0079 −0.0006 0.0002 ; Dc = −0.0582 −0.5910 ⎤ ⎤ ⎡ ⎡ 6.5054 4.0208 ⎥ ⎥ ⎢ ⎢ ⎥ ; Fc = ⎢ −16.2020 ⎥ Ec = ⎢ 30.9151 ⎦ ⎦ ⎣ ⎣ −60.5314

−200.5261

The projection of the stability region E(P ) onto the plant states is given by: 0.0472 −0.0223 Ex (P ) = {x ∈ 2 : x x ≤ 1} −0.0223 0.1742

388

J.M. Gomes da Silva Jr., D. Limon, T. Alamo, and E.F. Camacho 2.5

2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5 −5

−4

−3

−2

−1

0

1

2

3

4

5

Fig. 3. Region Ex (P ) and state portrait of the controlled system.

In Figure 3 this contractive ellipsoid as well as the trajectories of the controlled system for several initial states are shown. For a given initial state, the initial controller states i.e v(0) and xc (0), are chosen in such a way that ξ(0) = [x(0) v(0) xc (0) ] is in E(P ). In Figure 4 the evolution of the system output, y(t), the control action, u(t), and increment of the control action, Δu(t), are depicted when the system starts from x(0) = [2.8 − 1.5] , v(0) = 1.2966, xc (0) = [449.4707 − 709.7774 − 156.1419] . Notice that the limit requirements in u(t) and Δu(t) are satisfied thanks to the proposed saturating dynamic output feedback. Indeed, note that both u(t) and Δu(t) are effectively saturated in the first samples. In Table 1, it is illustrated the trade-off between saturation, the size of the stability region and the time-domain performance in terms of the pole placement of matrix A −1 in a disk of radius σ. Considering different values of σ, the values of trace(X11 ) ' and det(X11 ), obtained from the solution of (22), are shown in two situations: the

first one regards the application of the results of Theorem 1, i.e. the saturation and the nonlinear behavior of the closed-loop system are effectively considered; the second one concerns the linear solution, i.e. the stability region is forced to be contained in the region of linear behavior of the closed-loop system (RL ) in order to avoid saturation. Note that, as expected, smaller is σ (i.e. more stringent is the performance requirement), smaller is the obtained stability region. On the other hand, the solutions considering the saturation and the nonlinear behavior lead to larger domains of stability.


389

y(t)

0 −1 −2 0

5

10

15

20

25

30

0

5

10

15

20

25

30

0

5

10

15 samples

20

25

30

1

u(t)

0.5 0 −0.5

Δ u(t)

0.1 0 −0.1 −0.2 −0.3

Fig. 4. Evolution of y(t), u(t) and Δu(t). Table 1. Trade-off: saturation × region of stability × performance saturated solution linear solution ' ' −1 −1 σ trace(X11 ) det(X11 ) trace(X11 ) det(X11 ) 0.99

0.0715

31.8254

0.0798

28.0904

0.95

0.1193

19.8778

0.1622

14.1177

0.9

0.2214

11.3793

0.3178

7.5111

0.85

0.4161

6.2837

0.5748

4.3165

0.8

0.7846

3.3509

1.0328

2.4725

6.2 Example 2 Consider the pendulum continuous time model presented in the preface. It has been discretized with a sampling time Ts = 0.05 s, obtaining the following discrete-time model matrices ⎡

⎤

1.0013 −0.0500 −0.0013 ⎥ ⎢ ⎥, A=⎢ −0.0500 1.0025 0.0500 ⎦ ⎣ −0.0013 0.0500 1.0013 ⎤ ⎤ ⎡ ⎡ −0.0021 5.0021 ⎥ ⎥ ⎢ ⎢ ⎥ , Bw = 10−2 · ⎢ −0.1251 ⎥ , B = 10−2 · ⎢ 0.1251 ⎦ ⎦ ⎣ ⎣ 5.0021

−0.0021

390


C=

100

010

,

Cz = 1 0 0 ,

Dz,u = 0,

Dz,w = 0.

The amplitude and rate control limits are respectively: |u(t)| ≤ 1.25 , |Δ(u(t))| ≤ 2.0 Regarding the problem of maximizing the region of attraction of the closed-loop system, in Table 2 it is shown the results obtained from the solution of optimization problem (22). As in the previous example, it can be noticed a trade-off between the saturation allowance, the size of the region of stability and the time-performance requirement, expressed as a pole placement in disk of radius σ. Table 2. Trade-off: saturation × region of stability × performance

σ

saturated solution linear solution ' ' −1 −1 trace(X11 ) det(X11 ) trace(X11 ) det(X11 )

0.99

10.0454

591.1106

12.5708

479.7308

0.975

23.1943

184.1258

36.1758

156.9039

0.95

70.7120

94.3923

135.0452

49.1979

0.925 229.6418

0.0576

473.9738

0.0202

0.0049

2345.4

0.0018

0.9

1169.8

Considering, for instance σ = 0.975, the solution of (22) gives the following controller matrices: ⎡

0.3933

0.0002 −0.0006 −0.0001

⎤

⎥ ⎢ ⎢ 806.5808 0.3946 −1.2876 −0.2052 ⎥ ⎥ Ac = ⎢ ⎢ −30.0468 −0.0146 0.0481 0.0070 ⎥ ⎦ ⎣ −523.8979 −0.2556 0.8369 0.1304 ⎤ ⎡ 0.5754 0.6402 −0.0291 ⎥ ⎢ ⎢ 1796.5662 −2838.9892 −26.8975 ⎥ ⎥ Bc = ⎢ ⎢ 1181.7759 −777.2075 9.0466 ⎥ ⎦ ⎣ −1538.1978 2106.5383 50.7020 Cc = 4.8971 0.0024 −0.0078 −0.0012 ; Dc = 10.9580 −17.2724 −1.1629


⎡

−0.0291

⎤

⎡

−0.000001

391

⎤

⎥ ⎥ ⎢ ⎢ ⎢ −0.1133 ⎥ ⎢ 164.382729 ⎥ ⎥ ⎥ ⎢ ⎢ ; Fc = ⎢ Ec = ⎢ ⎥ ⎥ ⎣ 14.8337 ⎦ ⎣ 35.515640 ⎦ 65.9817 93.770018 Considering the controller given by the matrices above, the projection of the stability region E(P ) onto the plant states is given by: ⎤ ⎡ 10.1414 −10.1832 −4.2598 ⎥ ⎢ 3 ⎢ Ex (P ) = {x ∈ : x ⎣ −10.1832 10.5713 4.7669 ⎥ ⎦ x ≤ 1} −4.2598 4.7669 2.4816 In Figure 5 the evolution of the states of the pendulum is shown for the initial condition x(0) = [0.0050, 0.2428, 0.1447]T , v(0) = −3.4855 and xc (0) = [0.1769, −572.9046, −157.2404, 471.0735]T . The corresponding evolutions of u(t) and Δ(u(t)) are depicted in Figure 6. 0.1

x1

0 −0.1 −0.2 0

50

100

150

0

50

100

150

0

50

100

150

0.3

x2

0.2 0.1 0 −0.1

0.2

x3

0 −0.2 −0.4 −0.6 samples

Fig. 5. Evolution of state of the pendulum.

Regarding now the disturbance tolerance problem, from (32), it follows that the maximal value of the bound on ||w||22 , for which it is possible to ensure that the trajectories are bounded is given by μ−1 = δ−1 = 10.2736. In this case, the obtained controller matrices are

392


u

0

−0.5

−1

0

50

0

50

100

150

100

150

0.6 0.5

Δu

0.4 0.3 0.2 0.1 0 −0.1 samples

Fig. 6. Evolution of u(t) and Δu(t).

⎡

0.4493

−0.0402 −0.0000 0.0000

⎤

⎥ ⎢ ⎢ −6.0391 0.5410 0.0000 −0.0000 ⎥ ⎥ ⎢ Ac = ⎢ ⎥ ⎣ 3876.4632 −347.1736 −0.0020 0.0102 ⎦ 97.9460 −8.7760 −0.0001 −0.0003 ⎤ ⎡ −0.0253 0.6773 −0.0326 ⎥ ⎢ ⎥ ⎢ −0.1301 8.3458 0.4154 ⎥ Bc = ⎢ ⎢ 5406.6797 −9120.6486 −135.8883 ⎥ ⎦ ⎣ 136.6425 −230.4732 −10.1011 Cc = 5.4145 −0.4849 −0.0000 0.0000 ; Dc = 7.5517 −12.7393 −1.1898 ⎤ ⎤ ⎡ ⎡ −0.0326 −0.0000 ⎥ ⎥ ⎢ ⎢ ⎢ 0.4146 ⎥ ⎢ −0.0045 ⎥ ⎥ ⎥ ⎢ Ec = ⎢ ⎢ 0.0052 ⎥ ; Fc = ⎢ 715.9553 ⎥ ⎦ ⎦ ⎣ ⎣ −14.8473 −20.6986 Table 3 shows the different values of μ−1 , obtained from (32), with an additional constraint of constraint of pole placement in a disk of radius σ. It can be noticed trade off between time-domain performance and the tolerance to disturbances. Finally, in Table 4, it is shown the values of γ (related to the minimization of the L2 gain) obtained from the solution of optimization problem (33), considering different values of δ. As expected, the larger is δ (i.e. the lower is the bound on the disturbances), smaller is the value of γ, i.e. better is the disturbance rejection.


393

Table 3. Trade-off: σ × maximal bound on the disturbance energy σ

μ−1 = δ−1

1.00

10.2737

0.99

5.3997

0.975

2.5087

0.95

0.9449

0.925

0.3356

0.9

0.0741

Table 4. Trade-off: bound on the disturbance energy × disturbance rejection δ−1

γ

10 17094.7 8 211.8103 6

45.4023

4

14.5536

2

4.7053

1

2.2508

0.5

1.2816

0.1

0.4768

0.01 0.1689

7 Conclusions In this chapter a technique for the design of stabilizing dynamic output feedback controllers for discrete-time linear systems with rate and amplitude constrained actuators was proposed. Two problems have been considered: the stabilization problem in the absence of disturbances; and the tolerance and disturbance problem. In order to cope with the rate saturation problem, a controller structure consisting of a classical linear dynamic compensator in cascade with m input-saturating integrator system and two static anti-windup loops was adopted. The obtained closed-loop system presents in this case two amplitude saturation terms. With this particular structure, it is ensured that the control signal delivered to the actuator will always respect the rate constraints.

394


In order to obtain theoretical conditions allowing to compute a local stabilizing controller, the nonlinear effects of the saturation terms are taken into account by considering a generalized sector condition. The main advantages of such approach are twofold: it leads to conditions directly in LMI form; it allows the simultaneous synthesis of a dynamic compensator and the anti-windup loops. This last fact links, someway, the anti-windup approach (which adds corrections in a pre-computed controller, designed disregarding the saturation effects) with the paradigm of synthesizing a dynamic controller directly taking into account the saturation nonlinearities. Based on the derived conditions, convex optimization problems can be formulated in order to compute the controller considering a synthesis criterion. Three problems have been formulated in this sense. The first one regards the synthesis of the controller with the aim of maximizing an associated region for which the asymptotically stability can be ensured. This problem indirectly addresses the more complex problem of computing a controller in order to maximize the region of attraction of the closed-loop system. Additional performance constraints for the behavior of the system inside the linearity region of the system (where no saturation is active), in terms of pole placement in LMI regions, are considered in the optimization problem. It has been noticed, by means of examples, a clear trade-off between the size of the stability region and the achieved performance in the linearity region: more stringent performances lead to smaller regions of stability. The second formulated problem concerns the computation of the controller aiming at the maximization of an upper bound on the L2 norm of the disturbances for which the corresponding trajectories are guaranteed bounded. Finally, given an L2 bound on the admissible disturbances acting on the system, the problem of determining the controller which minimizes an upper bound on the L2 gain between the disturbance input and the regulated output, while ensuring bounded trajectories and internal stability, is formulated. Another trade-off is observed in this case: smaller are the L2 -norm of the admissible disturbances, smaller is the achieved L2 gain, that is, more effective is the disturbance rejection. The best L2 gain tends to be obtained for small enough disturbances, which do not lead the trajectories outside the linearity region. In this case, the obtained gain is the one of the linear (unsaturated) closed-loop system. On the other hand, by allowing the saturation and take into account the nonlinear behavior of the system it is possible to ”tolerate”, larger disturbances, while ensuring the trajectories are bounded.


395

References 1. A. Bateman and Z. Lin. An analysis and design method for linear systems under nested saturation. System & Control Letters, 48(1):41–52, 2002. 2. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied Mathematics, 1994. 3. M. C. de Oliveira, J. C. Geromel, and J. Bernussou. Design of dynamic output feedback decentralized controllers via a separation procedure. Int. Journal of Control, 73(5):371– 381, 2000. 4. H. Fang, Z. Lin, and T. Hu. Analysis of linear systems in the presence of actuator saturation and L 2 -disturbances. Automatica, 40(7):1229–1238, 2004. 5. J. M. Gomes da Silva Jr. and S. Tarbouriech. Stability regions for linear systems with saturating controls. In Proc. of European Control Conference 1999 (ECC’99), Karlsrhue, Germany, 1999. 6. J. M. Gomes da Silva Jr. and S. Tarbouriech. Local stabilization of discrete-time linear systems with saturating controls: an LMI-based approach. IEEE Trans. on Automatic Control, 46:119–125, 2001. 7. J. M. Gomes da Silva Jr. and S. Tarbouriech. Anti-windup design with guaranteed regions of stability: an lmi-based approach. IEEE Trans. on Automatic Control, 50(1):106–111, 2005. 8. J. M. Gomes da Silva Jr., S. Tarbouriech, and G. Garcia. Local stabilization of linear systems under amplitude and rate saturating actuators. IEEE Trans. on Automatic Control, 48(5):842–847, 2003. 9. G. Grimm, J. Hatfield, I. Postlethwaite, A. Teel, M. Turner, and L. Zaccarian. Antiwindup for stable systems with input saturation: an lmi-based synthesis. IEEE Trans. on Automatic Control, 48(9):1500–1525, 2003. 10. T. Hu and Z. Lin.

Control systems with actuator saturation: analisys and design.

Birkhauser, 2001. 11. V. Kapila and K. Grigoriadis (Editors). Actuator Saturation Control. Marcel Dekker, Inc., 2002. 12. N. Kapoor, A. R. Teel, and P. Daoutidis. An anti-windup design for linear systems with input saturation. Automatica, 34(5):559–574, 1998. 13. T. Kiyama and T. Iwasaki. On the use of multi-loop circle for saturating control synthesis. Systems & Control Letters, 41:105–114, 2000. 14. Z. Lin. Semi-global stabilization of linear systems with position and rate-limited actuators. System & Control Letters, 30:1–11, 1997. 15. T. Nguyen and F. Jabbari. Output feedback controllers for disturbance attenuation with actuator amplitude and rate saturation. Automatica, 36:1339–1346, 2000.

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16. C. Paim, S. Tarbouriech, J. M. Gomes da Silva Jr., and E. B. Castelan. Control design for linear systems with saturating actuators and L 2 -bounded disturbances. In Proc. of the 41st IEEE Conference on Decision and Control, Las Vegas - USA, 2002. 17. C. Scherer, P. Gahinet, and M. Chilali. Multiobjective output-feedback control via lmi optimization. IEEE Trans. on Automatic Control, 42(7):896–911, 1997 1997. 18. G. Shi, A. Saberi, and A. A. Stoorvogel. On Lp performance with global internal stability for linear systems with actuators subject to amplitude and rate saturations. In Proceedings of the 19th American Control Conference (ACC’2000), pages 730–734, Chicago, USA, 2000. 19. A. A. Stoorvogel and A. Saberi. Output regulation for linear systems to amplitude plus rate saturating actuators. Int. J. of Robust and Nonlinear Control, 9(10):631–657, 1999. 20. S. Tarbouriech and G. Garcia (Editors). Control of uncertain systems with bounded inputs. Springer Verlag, 1997. 21. S. Tarbouriech, C. Prieur, and J. M. Gomes da Silva Jr. Stability analysis and stabilization of systems presenting nested saturations. In Proc. of the 43rd IEEE Conference on Decision and Control, Paradise Island , Bahamas, 2004. 22. F. Tyan and D. S. Bernstein. Dynamic output feedback compensation for linear systems with independent amplitude and rate saturation. Int. Journal of Control, 67(1):89–116, 1997.

Decentralized Stabilization of Linear Time Invariant Systems Subject to Actuator Saturation Anton A. Stoorvogel1,2 , Ali Saberi3 , Ciprian Deliu1,2 , and Peddapullaiah Sannuti4 1

Department of Mathematics, and Computing Science, Eindhoven Univ. of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands , [email protected],[email protected].

2

Department of Electrical Engineering, Mathematics and Computer Science, Delft Univ. of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands [email protected],[email protected].

3

School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA 99164-2752, U.S.A. [email protected].

4

Department of Electrical and Computer Engineering, Rutgers University, 94 Brett Road, Piscataway, NJ 08854-8058, U.S.A. [email protected].

Summary. We are concerned here with the stabilization of a linear time invariant system subject to actuator saturation via decentralized control while using linear time invariant dynamic controllers. When there exists no actuator saturation, i.e. when we consider just linear time invariant systems, it is known that global stabilization can be done via decentralized control while using linear time invariant dynamic controllers only if the so-called decentralized fixed modes of it are all in the open left half complex plane. On the other hand, it is known that for linear time invariant systems subject to actuator saturation, semi-global stabilization can be done via centralized control while using linear time invariant dynamic controllers if and only if the open-loop poles of the linearized model of the given system are in the closed left half complex plane. This chapter establishes that the necessary conditions for semi-global stabilization of linear time invariant systems subject to actuator saturation via decentralized control while using linear time invariant dynamic controllers, are indeed the above two conditions, namely (a) the decentralized fixed modes of the linearized model of the given system are in the open left half complex plane, and (b) the open-loop poles of the linearized model of the

Support for this work was provided by the Office of Naval Research under grant number N000140310848.

398

Anton A. Stoorvogel, , Ali Saberi, Ciprian Deliu, , and Peddapullaiah Sannuti

given system are in the closed left half complex plane. We conjecture that these two conditions are also sufficient in general. We prove the sufficiency for the case when the linearized model of the given system is open-loop conditionally stable with eigenvalues on the imaginary axis being distinct. Proving the sufficiency is still an open problem for the case when the linearized model of the given system has repeated eigenvalues on the imaginary axis.

1 Introduction Non-classical information and control structure are two essential and distinguishing characteristics of large-scale systems. The research on decentralized control was formally initiated by Wang and Davison in their seminal paper [17] in 1973, and has been the subject of intense study during the 70’s and 80’s. Most recently there has been a renewed interest in decentralized control because of its fundamental role in the problem of coordinating the motion of multiple autonomous agents which by itself has attracted significant attention. Coordinating the motion of autonomous agents has many engineering applications besides having links to problems in biology, social behavior, statistical physics, and computer graphics. The engineering applications include unmanned aerial vehicles (UAVs), autonomous underwater vehicles (AUVs) and automated highway systems (AHS). A fundamental concept in the study of stabilization using decentralized feedback controllers is that of fixed modes. These are the poles of the system which cannot be shifted by just using any type of decentralized feedback controllers. The idea of fixed modes was introduced by Wang and Davison [17] who also show that decentralized stabilization is possible if and only if the fixed modes are stable. More definitive results are obtained by Corfmat and Morse [4] who present necessary and sufficient conditions under which spectrum assignment is possible in terms of the remnant polynomial of complementary subsystems. Since fixed modes constitute such an important concept in decentralized control, their characterization and determination has been the subject of many papers in the literature. The majority of existing research in decentralized control makes a critical assumption that the interconnections between the subsystems of a given system are unknown but have known bounds. In this regard, tools borrowed from robust control theory and Lyapunov theory are used for the purpose of either synthesis or analysis of decentralized controllers [10, 14, 13]. For the case when the interconnections between

Decentralized Control with Input Saturation

399

the subsystems are known, the existing research is very sparse. In fact, in any case, beyond the decentralized stabilization, no results are yet available dealing with the fundamental control issues such as exact or almost disturbance decoupling, control for various performance objectives, etc. From a different perspective, input saturation in any control scheme is a common phenomenon. Every physically conceivable actuator has bounds on its output. Valves can only be operated between fully open and fully closed states, pumps and compressors have a finite throughput capacity and tanks can only hold a certain volume. Ignoring such saturation effects in any control system design can be detrimental to the stability and performance of controlled systems. A classical example for the detrimental effect of neglecting actuator constraints is the Chernobyl unit 4 nuclear power plant disaster in 1986 [16]. During the last decade and the present one, there has been an intense research activity in the area of control of linear plants with saturating actuators. Such intense research activity has been chronicled in special issues of journals and edited books (e.g. for recent literature see [8, 15, 2]). Fundamental fuel behind such a research activity has been to accentuate the industrial and thus the practical engineering relevance of modern control theory. In this regard, the primary focus of the research activity has been to take into account a priori the presence of saturation nonlinearities in any control system analysis and design. A number of control issues have been considered so far including internal, external, or internal plus external stabilization and output regulation among others. Although not all aspects of these issues have been completely resolved, it is fair to say that a good understanding of these issues exists at present. However, issues related to performance, robustness etc., are very poorly understood and still remain as challenging and complex problems for future research. Having been involved deeply in the past with research on linear systems subject to constraints on its input and state variables, we are now ready to open up a new front line of research in decentralized control by bringing into picture the constraints of actuators. The focus of this chapter is to determine the necessary and sufficient conditions for decentralized stabilization of linear systems subject to constraints on actuators. Obviously, this is related to the seminal work of Wang and Davison [17] but goes beyond it by bringing into picture the input constraints on the top of decentralized constraint.

400


2 Problem Formulation and Preliminaries Consider the linear time invariant systems subject to actuator saturation, ⎧ ν ⎪ ⎨ x˙ = Ax + Bi sat ui Σ: i=1 ⎪ ⎩ yi = Ci x, i = 1, . . . , ν,

(1)

where x ∈ Rn is a state, ui ∈ Rmi , i = 1, . . . , ν are control inputs, yi ∈ Rpi , i = 1, . . . , ν are measured outputs, and ‘sat’ denotes the standard saturation element with the property that for any vector u of arbitrary dimension, sat(u) is a vector of the same dimension as u, and moreover for any positive integer j less than or equal to the dimension of u, the j’th component of sat u, denoted by (sat u)j , compared to the j’th component of u, denoted by (u)j , has the property, ⎧ ⎪ ⎪ 1 if 1 < (u)j , ⎪ ⎨ (sat u)j = (u)j if − 1 ≤ (u)j ≤ 1, ⎪ ⎪ ⎪ ⎩−1 if (u) < −1. j

Here we are looking for ν controllers of the form, z˙i = Ki zi + Li yi , zi ∈ Rsi Σi : u i = Mi zi + Ni y i .

(2)

The controller Σi is said to be i-th channel controller. Before we state the problem we study in this chapter, we would like to recall the concept of semi-global stabilization via decentralized control. Definition 1. Consider a system Σ of the form (1). Then, we say that Σ is semiglobally stabilizable via decentralized control if there exists nonnegative integers s1 , . . . , sν such that for any given collection of compact sets W ⊂ Rn and Si ⊂ Rsi , i = 1, . . . ν, there exist a decentralized set of controllers ν controllers Σi , i = 1, . . . ν, of the form (2) such that the origin of the resulting closed-loop system is asymptotically stable and the domain of attraction includes W × S1 × · · · × Sν . The problem we would like to study in this chapter can be stated as follows: Problem 1. Consider a system Σ of the form (1). Develop the necessary and sufficient conditions such that Σ is semi-globally stabilizable via decentralized control.


401

Remark 1. For the case when ν = 1, the above decentralized control problem retorts to centralized semi-global stabilization of linear time invariant systems subject to actuator saturation. Such a problem has been studied in depth by Saberi and his coworkers. By now it is well known that such a centralized semi-global stabilization problem is solvable by a linear time invariant dynamic controller if and only if the linearized model of the given system is stabilizable and detectable and all the openloop poles of linearized model are in the closed left half complex plane.

3 Review of Decentralized Stabilization of Linear Time Invariant Systems Before we proceed to consider the conditions for the solvability of Problem 1, it is prudent to review the necessary and sufficient conditions for the global decentralized stabilization of linearized model of the given system Σ. To do so, we first write the linearized model of the given system Σ of (1) as, ⎧ ν ⎪ ⎨ x˙ = Ax + Bi ui ¯: Σ i=1 ⎪ ⎩ yi = Ci x, i = 1, . . . , ν.

(3)

The classical decentralized global stabilization problem or more general decentral¯ can be stated as follows: ized pole placement problem for the linearized model Σ Find linear time invariant dynamic controllers Σi , i = 1, . . . ν, of the form (2) ¯ and the controllers Σi , such that the poles of the closed-loop system comprising Σ i = 1, . . . ν, has pre-specified poles in the open left half complex plane. It is easy to observe that, if (A, Bi ) and (A, Ci ) are respectively controllable and observable pairs for some i, the above decentralized pole placement problem can be solved trivially. Wang and Davison in [17] considered the general decentralized pole placement prob¯ Before we state their result, we need to recall the lem for the linearized model Σ. important concept of decentralized fixed modes as was introduced by Wang and Davison: ¯ of the form (3). Then, λ is called a decentralized Definition 2. Consider a system Σ ¯ if for all matrices K1 , . . . , Kν we have that λ is an fixed mode of the system Σ

402


eigenvalue of AK := A +

ν

Bi Ki Ci .

i=1

Wang and Davison proved in [17] that there exist dynamic controllers Σi , i = 1, . . . ν, of the form (2) such that the poles of the closed-loop system comprising ¯ and the controllers Σi , i = 1, . . . ν are at pre-specified locations in the open left Σ ¯ are themselves half complex plane provided that the decentralized fixed modes of Σ in the open left half complex plane and the set of pre-specified locations in the open ¯ This obvileft half complex plane includes the set of decentralized fixed modes of Σ. ously implies that the decentralized stabilization of the linear time invariant system ¯ is possible if and only if the decentralized fixed modes of it are all in open left half Σ complex plane. ¯ play a crucial role The above result implies that the decentralized fixed modes of Σ in decentralized stabilization of linear time invariant systems. As such it is important to know how to compute such fixed modes. One of the easiest procedure to do so is as follows: Since Ki = 0, i = 1, . . . ν, are admissible, in this case AK retorts to A, and hence in view of Definition 2, the decentralized fixed modes are naturally a subset of the eigenvalues of A. Thus the first step is to compute the eigenvalues of A. Second, it can be shown that if Ki , i = 1, . . . ν, are randomly chosen, then with probability one the decentralized fixed modes are common eigenvalues of A and AK . Since algorithms are well developed to determine the eigenvalues of a matrix, the computation of decentralized fixed modes is quite straightforward. After the introduction of the concept of decentralized fixed modes, there has been quite some research on interpretations of this concept. The crucial step in understanding the decentralized fixed modes was its connection to complementary systems as introduced by Corfmat and Morse in the paper [4]. The paper [1] by Anderson and Clements used the ideas of Corfmat and Morse to yield the following characterization of decentralized fixed modes: ¯ of (3). We define, Lemma 1. Consider the system Σ ⎛

" # B = B1 · · · Bν ,

⎞ C1 ⎜ . ⎟ . ⎟ C=⎜ ⎝ . ⎠. Cν


403

Then λ is a decentralized fixed mode if and only if at least one of the following three conditions is satisfied: • λ is an uncontrollable eigenvalue of (A, B). • λ is an unobservable eigenvalue of (C, A). • There exists a partition of the integers {1, 2, . . . , ν} into two disjoint sets {i1 , . . . , iα } and {j1 , . . . , jν−α } where 0 < α < ν for which we have ⎛ ⎞ λI − A Bi1 · · · Biα ⎜ ⎟ ⎜ Cj1 ⎟ 0 · · · 0 ⎜ ⎟ < n. rank ⎜ . .. . . .. ⎟ ⎜ .. ⎟ . . . ⎠ ⎝ Cjν−α 0 · · · 0 Basically the decentralized fixed modes are therefore common blocking zeros of all complementary systems which are, moreover, either unobservable or uncontrollable for each complementary system. For a detailed investigation of blocking zeros we refer to the paper [3]. Other attempts to characterize the decentralized fixed modes can be found in for instance [12, 6, 7]. The above discussion focuses on developing the necessary and sufficient condition under which stabilization of a linear time invariant system by a set of decentralized controllers is possible. The next issue that needs to be discussed pertains to how does one construct systematically the set of decentralized controllers that stabilize a given system assuming that it is possible to do so. In this regard, it is important to recognize that implicit in the proof of pole placement result of Wang and Davison [17] is a constructive algorithm. This algorithm requires as a first step the (possibly random) selection of Ki , i = 1, . . . ν, such that all the eigenvalues of AK = A +

ν

Bi Ki Ci

i=1

are distinct from those of A except for the decentralized fixed modes. Then, dynamic feedback is successively employed to arrive at a dynamic controller Σi , i = 1, . . . ν, placing the poles of resulting closed-loop system that are both controllable and observable eigenvalues of the pairs (A, Bi ) and (A, Ci ) respectively. Also, Corfmat and Morse [4] have studied the decentralized feedback control problem from the point of view of determining a more complete characterization of conditions for stabilizability and pole placement as well as constructing a set of stabilizing decentralized

404


¯ controllers. Their basic approach is to determine conditions under which a system Σ of the form (3) can be made controllable and observable from the input and output variables of a given controller by static feedback applied by the other controllers. Then dynamic compensation can be employed at this controller in a standard way to place the poles of the closed-loop system. ¯ controllable and observable It is not hard to see that a necessary condition to make Σ from a single controller is that none of the transfer functions Ci (sI − A)−1 Bj vanish identically for all i = 1, . . . ν, and j = 1, . . . ν. A system satisfying this condition is termed strongly connected. If a system is not strongly connected, the given system can be decomposed into strongly connected subsystems and each subsystem can be made then controllable and observable from one of its controllers. As outlined in an early survey paper by Sandell et al [11], as a practical design method, the Corfmat and Morse method suffers some defects. At first, it can be noted that even if all the modes of a large scale system can be made controllable and observable from a single controller (or a few controllers if the given system is not strongly connected), some of the modes may be very weakly controllable and observable. Thus, impractically large gains may be required to place all the poles from a single controller. Second, it is unclear that the approach uses the designer’s available degrees of freedom in the best way. Essentially, the approach seems to require that all the disturbances in the system propagate to a single output, where they can be observed and compensated for by the control signals at an adjacent input. Finally, concentration of all the complexity of the control structure at a single (or few) controllers may be undesirable. As pointed out once again in [11], the constructive approach of Wang and Davison also suffers similar drawbacks as mentioned above. Although there is no explicit attempt in their approach to make all of the strongly connected subsystems controllable and observable from a single controller, the generic outcome of the first step of their approach will be precisely this situation. After the early first phase of work of Wang and Davison [17] as well as Corfmat and Morse [4], there has been a lot of second phase of work (see [14, 13] and references there in) on how to construct the set of decentralized controllers for a large ¯ of (3) scale system. These researchers view the given large scale system such as Σ


405

as consisting of ν interconnected subsystems, the i-th subsystem being controlled by the i-th controller Σi . Then, the research in decentralized control is dominated by the point of view of considering the interconnections between the subsystems essentially as disturbances, and then using robust control theory to design strongly robust subsystems in such a way that the effect of such disturbances is minimal. Essentially, the framework of viewing the interconnections as disturbances is fundamentally flawed. Such work belongs to the field of centralized robust control theory. In our opinion, the decentralized control is still in its infancy, and is a very complex and open field.

4 Main Results In this section, we will present the necessary and sufficient conditions for semi-global stabilizability of linear time invariant systems with actuator saturation by utilizing a set of decentralized linear time invariant dynamic controllers. We have the following theorem that pertains to necessary conditions, the proof of which is given in Section 5. Theorem 1. Consider the system Σ given by (1). There exists nonnegative integers s1 , . . . , sν such that for any given collection of compact sets W ⊂ Rn and Si ⊂ Rsi , i = 1, . . . ν, there exist ν controllers of the form (2) such that the origin of the resulting closed loop system is asymptotically stable and the domain of attraction includes W × S1 × · · · × Sν only if ¯ given by (3) are in the open left half complex • All decentralized fixed modes of Σ plane, and • All eigenvalues of A are in the closed left half plane. The following theorem says that besides decentralized fixed modes being in the open left half complex plane, a sufficient condition for semi-global stabilizability of (1) when the set of controllers given by (2) are utilized is that all the eigenvalues of A be in the closed left half plane with those eigenvalues on the imaginary axis having algebraic multiplicity equal to one.

406


Theorem 2. Consider the system Σ given by (1). There exists nonnegative integers s1 , . . . , sν such that for any given collection of compact sets W ⊂ Rn and Si ⊂ Rsi , i = 1, . . . ν, there exist ν controllers of the form (2) such that the origin of the resulting closed loop system is asymptotically stable and the domain of attraction includes W × S1 × · · · × Sν if ¯ given by (3) are in the open left half complex • All decentralized fixed modes of Σ plane, and • All eigenvalues of A are in the closed left half plane with those eigenvalues on the imaginary axis having algebraic multiplicity equal to one. The above theorem is proved in Section 6. Our work done up to now convinces us to state the following conjecture that the necessary conditions given in Theorem 1 are also sufficient for semi-global stabilizability of decentralized linear systems with actuator saturation. Conjecture 1. Consider the system Σ given by (1). There exists nonnegative integers s1 , . . . , sν such that for any given collection of compact sets W ⊂ Rn and Si ⊂ Rsi , i = 1, . . . ν, there exist ν controllers of the form (2) such that the origin of the resulting closed loop system is asymptotically stable and the domain of attraction includes W × S1 × · · · × Sν if and only if ¯ given by (3) are in the open left half complex • All decentralized fixed modes of Σ plane, and • All eigenvalues of A are in the closed left half plane.

5 Proof of Theorem 1 We prove Theorem 1 in this section. Assume that decentralized semi-global stabilization of the given system Σ of (1) is possible. Then, the decentralized stabilization ¯ of Σ as given in (3) is possible. By the result of Wang of the linearized model Σ and Davison [17], this implies that it is necessary to have all the decentralized fixed ¯ in the open left half complex plane. However, we have a simple alternate modes of Σ proof of this fact as follows: Since the linearized model needs to be asymptotically


407

stable, there exists ν linear controllers achieving locally an asymptotically stable system. We define the following matrices for these ν controllers of the form (2): ⎛ ⎞ ⎞ ⎛ K1 0 · · · 0 L1 0 · · · 0 ⎜ ⎟ ⎟ ⎜ ⎜ 0 L . . . ... ⎟ ⎜ 0 K . . . ... ⎟ 2 2 ⎜ ⎜ ⎟ ⎟ K=⎜ . ⎟, L = ⎜ . ⎟, ⎜ . .. .. ⎜ . .. .. ⎟ ⎟ . . 0⎠ . . 0⎠ ⎝ . ⎝ . 0 · · · 0 Kν

⎛ M1 0 · · · 0 ⎜ ⎜ 0 M . . . ... 2 ⎜ M =⎜ . ⎜ . .. .. . . 0 ⎝ .

⎞

⎛

⎟ ⎟ ⎟ ⎟, ⎟ ⎠

0 · · · 0 Mν

0 · · · 0 Lν

N1 0 · · · ⎜ ⎜ 0 N ... 2 ⎜ N =⎜ . ⎜ . .. .. . . ⎝ .

⎞ 0 .. ⎟ . ⎟ ⎟ ⎟. ⎟ 0⎠

0 · · · 0 Nν

For any λ with Re λ ≥ 0 there exists a δ such that (λ + δ)I − K is invertible and the closed loop system when replacing K by K − δI is still asymptotically stable. But then the linearized model of the closed loop system cannot have a pole at λ which implies that we must have that 5 6 ! det λI − A − B M (λI − (K − δI))−1 L + N C = 0. Hence the block diagonal matrix S = M (λI − (K − δI))−1 L + N has the property that det (λI − A − BSC) = 0, and thus λ is not a fixed mode of the system. Since this argument is valid for any λ in the closed right half plane this implies that all the fixed modes must be in the open left half plane. This proves the necessity of the first item of Theorem 1. To prove the necessity of the second item of Theorem 1, assume that λ is an eigenvalue of A in the open right half plane with corresponding left eigenvector p, i.e. pA = λp. Then we have d px(t) = λpx(t) + v(t) dt where v(t) :=

ν i=1

pBi sat ui (t).

408


˜ > 0 such that v(t) ≤ M ˜ for all t > 0. But then There clearly exists an M $ % ˜ ˜ λt M M + |px(t)| > e |px(0)| − Re λ Re λ which does not converge to zero since Re λ ≥ 0, provided the initial condition is such that

˜ M . Re λ Note that this is valid for all controllers and therefore we can clearly not achieve |px(0)| >

semi-global stability.

6 Preliminary Lemmas and Proof of Theorem 2 We will use two lemmas. The first lemma given below is a well-known classical result from Lyapunov theory. Lemma 2. Consider a matrix A ∈ Rn×n , and assume that it has all its eigenvalues in the closed left half plane with those eigenvalues on the imaginary axis having a geometric multiplicity equal to the algebraic multiplicity. Then, there exists a matrix P > 0 such that A P + P A ≤ 0.

(4)

Another useful tool is the following continuity result related to (4). Lemma 3. Assume that we have a sequence of matrices Aδ ∈ Rn×n parameterized by δ and a matrix A ∈ Rn×n such that Aδ → A as δ → 0. Assume that A has all its eigenvalues in the closed left half plane, and that there are p distinct eigenvalues of A on the imaginary axis (i.e. there are p eigenvalues of A on the imaginary axis each with algebraic multiplicity equal to 1). Moreover, assume that Aδ also has all its eigenvalues in the closed left half plane. Let P > 0 be such that (4) is satisfied. Then there exists for small δ > 0 a family of matrices Pδ > 0 such that Aδ Pδ + Pδ Aδ ≤ 0 and Pδ → P as δ → 0.


409

Proof. We first observe that there exists a matrix S such that $ % A 0 11 , S −1 AS = 0 A22 and such that all the eigenvalues of A11 are on the imaginary axis while A22 has all its eigenvalues in the open left half plane. Since A11 and A22 have no common eigenvalues and Aδ → A, there exists a parameterized matrix Sδ such that for δ sufficiently small Sδ−1 Aδ Sδ =

$ A11,δ 0

0

%

A22,δ

where Sδ → S, A11,δ → A11 and A22,δ → A22 as δ → 0. This follows from classical results on the sensitivity of invariant subspaces (see for instance [9, 5]). Given is a matrix P > 0 such that A P + P A ≤ 0. Let us define $ % ¯11 P¯12 P P¯ = S P S = . ¯ P¯12 P22 Obviously, we note that $ % % $ A11 0 A11 0 P¯ + P¯ ≤ 0. 0 A22 0 A22

(5)

Next, given an eigenvector x1 such that A11 x1 = λx1 with Re λ = 0, we have $ % % $ % $ %∗ $ A A11 0 x1 0 x1 11 ¯ + P¯ P = 0. 0 A22 0 A22 0 0 Using (5), the above implies that $ % % $ % $ A x1 0 A11 0 11 P¯ + P¯ = 0. 0 A22 0 A22 0 Since all the eigenvalues on the imaginary axis of A11 ∈ Rv×v are distinct we find that the eigenvectors of A11 span Rv and hence $ % % $ % $ A11 0 I A11 0 P¯ + P¯ = 0. 0 0 A22 0 A22 This leads to

410


$ $ % % $ % A A11 0 0 0 0 11 P¯ + P¯ = ≤ 0. 0V 0 A22 0 A22 This immediately implies that A11 P¯12 + P¯12 A22 = 0 and since A11 and A22 have no eigenvalues in common we find that P¯12 = 0. Thus, we have A11 P¯11 + P¯11 A11 = 0 and A22 P¯22 + P¯22 A22 = V ≤ 0. Next, since A22 has all its eigenvalues in the open left half plane, there exists a parameterized matrix P22,δ for δ small enough such that A22,δ P¯22,δ + P¯22,δ A22,δ = V ≤ 0 while P¯22,δ → P22 as δ → 0. Let A11 = W ΛA W −1 with ΛA a diagonal matrix. Because the eigenvectors of A11 are distinct and A11,δ → A11 , for δ small enough the eigenvectors of A11,δ depend continuously on δ and hence there exists a parameterized matrix Wδ such that Wδ → W while A11,δ = Wδ ΛA W −1 with ΛA diagonal. The matrix P¯11 δ

δ

δ

satisfies A∗11 P¯11 + P¯11 A11 = 0 This implies that ΛP = W ∗ P¯11 W satisfies Λ∗A ΛP + ΛP ΛA = 0. The above equation then shows that ΛP is a diagonal matrix. We know that Λ Aδ → Λ A . We know that ΛAδ is a diagonal matrix whose diagonal elements have real part less than or equal to zero while ΛP is a positive-definite diagonal matrix. Using this, it can be verified that we have Λ∗Aδ ΛP + ΛP ΛAδ ≤ 0. We choose P¯11,δ as P¯11,δ = (Wδ∗ )−1 ΛP Wδ−1 . Obviously, our choice of P¯11,δ satisfies A∗11,δ P¯11,δ + P¯11,δ A11,δ ≤ 0.


411

We observe that P¯11,δ → P¯11 as δ → 0. But then $ % ¯11,δ 0 P Pδ = (Sδ−1 ) Sδ−1 ¯ 0 P22,δ satisfies the conditions of the lemma. This completes the proof of Lemma 3. We proceed now with the proof of Theorem 2. Our proof is constructive and involves a sequential design. We present a recursive algorithm which at each step applies a decentralized feedback law which stabilizes at least one eigenvalue on the imaginary axis while preserving the stability of the stable modes of the system in such a way that the magnitude of each decentralized feedback control is guaranteed never to exceed 1/n. Therefore, after at most n steps the combination of these decentralized feedback laws will asymptotically stabilize the system without ever violating the magnitude constraints of each of the inputs. The basic steps of the algorithm are as formalized below: Algorithm: • Step 0 (Initialization): We first initialize our algorithm at step 0. To do so, let 0 A0 := A, B0,i := Bi , C0,i := Ci , ni,0 := 0, Ni,ε := 0, i = 1, . . . , ν and

x0 := x. Moreover, define P0ε := εP , where P > 0 is a matrix such that A P + P A ≤ 0. Since all the eigenvalues of A on the imaginary axis have multiplicity 1, we know from Lemma 2 that such a matrix P exists. • Step k: For the system Σ given by (3), we have to design ν parameterized decentralized feedback control laws, k k p˙ki = Ki,ε pi + Lki,ε yi , k,ε Σi : k ui = Mi,ε pki + Ni,ε yi + vik in case ni,k > 0, and otherwise Σk,ε : i

pki ∈ Rni,k

k yi + vik , ui = Ni,ε

(6)

(7)

for i = 1, . . . , ν. The closed-loop system comprising the above decentralized feedback control laws and the system Σ of (1) can be written as

412


Σk,ε cl

⎧ ν ⎪ ⎨ x˙ k = Aε xk + Bk,i vik k : i=1 ⎪ ⎩ yi = Ck,i xk , i = 1, . . . , ν,

where xk ∈ Rnk with nk = n +

(8)

ν

ni,k is given by ⎛ ⎞ x ⎜ k⎟ ⎜ p1 ⎟ ⎜ ⎟ xk = ⎜ . ⎟ . ⎜ .. ⎟ ⎝ ⎠ pkν i=1

(9)

In view of (9), we can rewrite ui as k xk + vik ui = Fi,ε k for some appropriate matrix Fi,ε .

The above decentralized feedback control laws given by either (6) or (7) are to be designed in such a way that they satisfy the following properties: 1) The matrix Aεk has all its eigenvalues in the closed-left half plane, and those eigenvalues of Aεk which are on the imaginary axis are distinct. 2) The number of eigenvalues of Aεk on the imaginary axis must at least be one less than the number of eigenvalues of Aεk−1 on the imaginary axis (i.e. at each step of our recursive algorithm we design a decentralized feedback law which stabilizes at least one more eigenvalue on the imaginary axis while preserving the stability of the stable modes of the system designed until then). 3) There exists a family of matrices Pkε such that Pkε → 0 as ε → 0 while for vik = 0, i = 1, . . . , ν, the closed-loop system Σk,ε cl of (8) is such that xk (t) Pkε xk (t) is non-increasing in t for all initial conditions, i.e. (Aεk ) Pkε + Pkε Aεk ≤ 0.

(10)

Moreover, there exists an ε∗ such that for all ε ∈ (0, ε∗ ] we have ui (t) ≤

k n

for all states with xk (t) Pkε xk (t) ≤ n − k + 1.

(11)


413

It is easy to verify that all of the above conditions are true for k = 0. • Terminal step: There exists a value for k, say ≤ n, such that the matrix Aε has all its eigenvalues in the open-left half plane. We set vi = 0 for i = 1, . . . , . The decentralized control laws Σ ,ε i , i = 1, . . . , as given by (6) or (7), all together, represent a decentralized semi-global state feedback law for the given system Σ of (1). More precisely, for any given compact sets W ⊂ Rn , and Si ⊂ Rni, for i = 1, . . . ν, there exists an ε∗ such that the origin of the closed-loop system comprising the given system Σ of (1) and the decentralized control laws Σ ,ε i , i = 1, . . . , as given by (6) or (7) is exponentially stable for any 0 < ε < ε∗ , and the compact set W × S1 × · · · × Sν is within the domain of attraction. Moreover, for all the initial conditions within W × S1 × · · · × Sν , the said closed-loop system behaves like a linear dynamic system, that is the saturation is not activated implying that ui < 1 for all i = 1, . . . , ν. The fact that the decentralized control laws Σ ,ε i , i = 1, . . . , as given by (6) or (7) are semi-globally stabilizing follows from the property 3) as given in step k of the above algorithm. To be explicit, we observe that, for an ε sufficiently small, the set Ωε1 := {x ∈ Rn |x P ε x ≤ 1} is inside the domain of attraction of the equilibrium point of the closed-loop system comprising the given system Σ of (1) and the decentralized control laws Σ ,ε i , i = 1, . . . , as given by (6) or (7). This follows from the fact that for all the initial conditions within Ωε1 , it is obvious from (11) that ui ≤ 1 for all i = 1, . . . , ν. This implies that the said closed-loop system behaves like a linear dynamic system, that is the saturation is not activated. Moreover, this linear dynamic system is asymptotically stable since Aε has all its eigenvalues in the open left half plane, and hence the state converges to zero asymptotically. Next, since P ε → 0 as ε → 0, for an ε sufficiently small, we have that the compact set W × S1 × · · · × Sν is inside Ωε1 . This concludes that the decentralized control laws Σ ,ε i , i = 1, . . . , as given by (6) or (7) are semi-globally stabilizing. This completes the description of our recursive algorithm to design the decentralized feedback control laws having the properties as given in Theorem 2.

414


It remains to prove that the above recursive algorithm succeeds in designing the decentralized feedback control laws having the properties as given in Theorem 2. In order to do so, we assume that the design of decentralized feedback control laws as described in step k can be done, and then prove that the corresponding design in step k + 1 can be done. We proceed now to prove this. After step k we have for the system Σ of (1), ν feedback control laws of the form (6) or (7) such that the system (8) obtained after applying these feedbacks has the properties 1), 2) and 3). We consider the closed-loop system Σk,ε cl of (8). Let λ be an eigenvalue on the imaginary axis of Aεk . We know that decentralized feedback laws do not change the fixed modes and therefore, since λ was not a fixed mode of the original system (1), it is not a fixed mode of the system (8) obtained after applying ν ¯ i such that feedback laws either. Hence there exists a K Aεk

+

ν

¯ i Ck,i Bk,i K

i=1

has no eigenvalue at λ. Therefore, Aεk

+δ

ν

¯ i Ck,i Bk,i K

i=1

has no eigenvalue at λ for almost all δ > 0 (the determinant of λI − Aεk − ν ¯ i Ck,i is a polynomial in δ and is nonzero for δ = 1 and therefore δ i=1 Bk,i K the determinant has a finite number of zeros). Let j be the largest integer such that Aε,δ k

=

Aεk

+δ

j

¯ i Ck,i Bk,i K

i=1

has the same number of eigenvalues on the imaginary axis as Aεk for δ > 0 small enough. This implies that Aε,δ k still has all its eigenvalues in the closed left half plane for δ small enough. We know that (10) is satisfied and hence using Lemma 3 we find that there exists a P¯ ε,δ such that k

¯ ε,δ ¯ ε,δ ε,δ (Aε,δ k ) Pk + Pk Ak ≤ 0

while P¯kε,δ → Pkε as δ → 0. Hence for δ small enough xk P¯kε,δ xk ≤ n − k +

1 2

=⇒ xk Pkε xk ≤ n − k + 1

(12)


415

and for δ small enough we have that ¯ i xk ≤ δK

1 2n

for all xk with xk Pkε xk ≤ n − k + 1.

(13)

For each ε choose δ = δε small enough such that the above two properties (12) and ¯ i , P¯ ε = P¯ ε,δε and (13) are satisfied. We define K ε = δε K i

k

A¯εk := Aεk +

j

k

Bk,i Kiε Ck,i .

i=1

By the definition of j, we know that Aεk

+

j+1

Bk,i Kiε Ck,i

i=1

has less eigenvalues on the imaginary axis than A¯εk . Hence (A¯εk , Bk,j+1 , Ck,j+1 ) has a stabilizable and detectable eigenvalue on the imaginary axis. Choose V such that VV =I

and

ker V = ker Ck,j+1 | A¯εk .

We choose the following decentralized feedback law, vik = Kiε xk + vik+1 ,

(14)

i = 1, . . . , j ,

k p˙ = Aεs p + V Bk,j+1 vj+1 + K(Ck,j+1 V p − yj+1 ) k+1 k vj+1 = Fρ p + vj+1

i = j + 2, . . . , ν.

vik = vik+1 ,

(15) (16)

Equations (14), (15), and (16) together represent our decentralized feedback control laws at step k + 1. Here p ∈ Rs and Aε is such that Aε V = V A¯ε while K is chosen s

such that

Aεs

s

k

+ KCk,j+1 V has all its eigenvalues in the open left half plane while Aεs + KCk,j+1 V and A¯εk have no eigenvalues in common. Moreover, for all ρ the matrix A¯ε + Bk,j+1 Fρ V has at least one eigenvalue less on the imaginary axis than k

Aεk

does, and still has all its eigenvalues in the closed left half plane while Fρ → 0

as ρ ↓ 0. Rewriting the resulting system in a new basis consisting of xk and p − V xk results in


416

x ¯˙ k+1 =

$ A¯εk + Bk,j+1 Fρ V

%

Bk,j+1 Fρ Aεs

0

+ KCk,j+1 V

x ¯k+1 +

ν

¯k+1,i v k+1 B i

i=1

(17)

i = 1, . . . , ν,

yi = C¯k+1,i x ¯k+1 ,

$

where ¯k+1,i = B

%

Bk,i

,

−V Bk,i

# " ¯ Ck+1,i = Ck,i 0

for i = j + 1 while ¯k+1,j+1 = B

$ % Bk,j+1 0

$ C¯k+1,j+1 =

, $

and x ¯k+1 =

xk

p − V xk

Ck,j+1 0 V

%

I

% .

Obviously, the above feedback laws (14), (15), and (16) satisfy at step k + 1 the properties 1), and 2) as mentioned in step k. What remains to show is that they also satisfy property 3). Moreover, we need to write the control laws (14), (15), and (16) in the form of (6) or (7) for step k + 1. In what follows we focus on these aspects. For any ε there exists a Rkε > 0 with (Aεs + KCk,j+1 V ) Rkε + Rkε (Aεs + KCk,j+1 V ) < 0 such that Rkε → 0 as ε ↓ 0. Since Fρ → 0 as ρ → 0, for each ε we have for ρ small enough Fρ e
0 such that (A¯εk + Bk,j+1 Fρ V ) P¯ρε + P¯ρε (A¯εk + Bk,j+1 Fρ V ) ≤ 0 with P¯ρε → P¯kε as ρ → 0. Finally because A¯εk and Aεs + KCk,j+1 V have disjoint eigenvalues we note that for ρ small enough we get that A¯ε + Bk,j+1 Fρ V and Aε + KCk,j+1 V have disjoint k

s

eigenvalues since Fρ → 0 as ρ ↓ 0. But then there exists a Wε,ρ such that


417

Bk,j+1 Fρ + (A¯εk + Bk,j+1 Fρ V )Wε,ρ − Wε,ρ (Aεs + KCk,j+1 V ) = 0 while Wε,ρ → 0 as ρ ↓ 0. Note that this implies that $ %$ %$ % ε ¯ I 0 I −W 0 P ε,ρ ε,ρ ρ P¯k+1 = ε −Wε,ρ I 0 Rk 0 I has the property that: ¯ ε,ρ ¯ ε,ρ ¯ε,ρ (A¯ε,ρ k+1 ) Pk+1 + Pk+1 Ak+1 ≤ 0

for A¯ε,ρ k+1

=

$ A¯εk + Bk,j+1 Fρ V

ε,ρ = lim P¯k+1 ρ↓0

%

Bk,j+1 Fρ Aεs + KCk,j+1 V

0

and

(19)

$ % P¯kε 0 0 Rkε

.

We consider x ¯k+1 such that ε,ρ x ¯k+1 ≤ n − k. x ¯k+1 P¯k+1

(20)

Then we can choose ρ small enough such that xk P¯kε xk ≤ n − k +

1 2

and (p − V xk ) Rkε (p − V xk ) ≤ n − k + 12 .

(21)

We choose for each ε a ρ = ρε such that (18) is satisfied while (20) implies that (21) is satisfied and finally Fρ V xk
0 or ni,k = 0 respectively. We can then rewrite the system (17) in terms of the state xk+1 (defined by (9)) instead of x ¯k+1 which requires a basis transformation Tk+1 , i.e. x ¯k+1 = Tk+1 xk+1 . We define ε,ρε ε = Tk+1 Tk+1 P¯k+1 Pk+1

and obviously, for i = 1, . . . , ν, we can write the relationship between yi , vik+1 and ui in the form (6) or (7) depending on whether ni,k+1 = 0 or not. We can now rewrite the control laws (14), (15), and (16) in the form k+1 k+1 p˙k+1 = Ki,ε pi + Lk+1 i i,ε yi , Σk+1,ε : i k+1 ui = Mi,ε pk+1 + Ni,ε yi + vik+1 i

pk+1 ∈ Rni,k+1 i

in case ni,k+1 > 0, and otherwise k+1 : Σk+1,ε yi + vik+1 , ui = Ni,ε i

(22)

(23)

for i = 1, . . . , ν. It is then clear that properties 1), 2) and 3) are satisfied in step k +1. This concludes the proof of Theorem 2.

References 1. B.D.O. A NDERSON AND D.J. C LEMENTS, “Algebraic characterization of fixed modes in decentralized control”, Automatica, 17(5), 1981, pp. 703–712. 2. D.S. B ERNSTEIN AND A.N. M ICHEL, Guest Eds., Special Issue on saturating actuators, Int. J. Robust & Nonlinear Control, 5(5), 1995, pp. 375–540. 3. B.M. C HEN , A. S ABERI , AND P. S ANNUTI, “On blocking zeros and strong stabilizability of linear multivariable systems”, Automatica, 28(5), 1992, pp. 1051–1055. 4. J.P. C ORFMAT AND A.S. M ORSE, “Decentralized control of linear multivariable systems”, Automatica, 12(5), 1976, pp. 479–495.


419

5. G.H. G OLUB AND C.F. VAN L OAN, Matrix Computations, John Hopkins Univ. Press, 2nd Ed., 1989. 6. Z. G ONG AND M. A LDEEN, “On the characterization of fixed modes in decentralized control”, IEEE Trans. Aut. Contr., 37(7), 1992, pp. 1046–1050. ¨ 7. N. G UNDES ¸ AND C.A. D ESOER, Algebraic theory of linear feedback systems with full and decentralized compensators, vol. 142 of Lecture Notes in Control and Information Sciences, Springer Verlag, 1990. 8. V. K APILA AND G. G RIGORIADIS, Eds., Actuator saturation control, Marcel Dekker, 2002. 9. T. K ATO, Perturbation theory for linear operators, Springer Verlag, Berlin, Second Ed., 1976. 10. A.N. M ICHEL AND R.K. M ILLER, Qualitative analysis of large scale dynamical systems, Academic Press, 1977. 11. J R . N. R. S ANDELL , P. VARAIYA , M. ATHANS , AND M.G. S AFONOV, “Survey of decentralized control methods for large scale systems”, IEEE Trans. Aut. Contr., 23(2), 1978, pp. 103–128. 12. H. S ERAJI, “On fixed modes in decentralized control systems”, Int. J. Contr., 35(5), 1982, pp. 775–784. 13. D.D. S ILJAK, Large-scale dynamic systems : stability and structure, North Holland, Amsterdam, 1978. 14.

, Decentralized control of complex systems, Academic Press, London, 1991.

15. A. S ABERI AND A.A. S TOORVOGEL, Guest Eds., Special Issue on control problems with constraints, Int. J. Robust & Nonlinear Control, 9(10), 1999, pp. 583–734. 16. G. S TEIN, “Respect the Unstable”, in IEEE Conference of Decision and Control, Tampa, FL, 1989. Bode prize lecture. 17. S.H. WANG AND E.J. DAVISON, “On the stabilization of decentralized control systems”, IEEE Trans. Aut. Contr., 18(5), 1973, pp. 473–478.

On the Stabilization of Linear Discrete-Time Delay Systems Subject to Input Saturation Karim Yakoubi and Yacine Chitour Laboratoire des signaux et systèmes, Univ. Paris-Sud, CNRS, Supélec, 91192 Gif-sur-Yvette cedex, France. [email protected] [email protected] Summary. This chapter deals with two problems on stabilization of linear discrete systems by static feedbacks which are bounded and time-delayed, namely global asymptotic stabilization and finite gain lp −stabilizability, p ∈ [1, ∞]. Furthermore, an explicit construction of the corresponding feedback laws is given. The feedback laws constructed also result in a closedloop system that is globally asymptotically stable. The results presented here are parallel to our earlier results on the continuous-time delay case, and show that, for the first issue, such stabilization is possible if and only if the system is stabilizable and the transition matrix has spectral radius inside or on the unit circle. For the finite-gain lp −stabilization issue, we assume that the system is neutrally stable. We provide upper bounds for the corresponding lp −gain which is delay-independent.

Key words: Saturated feedback, global stabilization, Lyapunov functions, Linear discrete-time delay systems, Finite-gain stability.

1 Introduction In this chapter we address two issues relative to the stabilization of a linear discrete system subject to input saturation and time-delayed, of the type (S) : x(k + 1) = Ax(k) − rBσ(u(k − h)),

(1)

422

Karim Yakoubi and Yacine Chitour

where (i) x(k) ∈ Rn and u(k) ∈ Rm , with n the dimension of the system and m the number of inputs; (ii) the input bound r ∈ (0, 1] only depend on (S) and there is an arbitrary constant delay h ∈ N∗ appearing in the input; (iii) σ : Rm → Rm represents the actuator saturation (definitions are given in section 2). The results presented here are an analog of those established for the continuoustime counterpart, in the papers [9] and [10]. Although the organization of the current work is tightly patterned after that of [9] and [10], and many of the arguments given here are specific of the discrete-time case and cannot be reduced to corresponding arguments in the continuous-time case. In the zero-delay case, the above problems have been widely investigated in the last years: see for example [11] and reference therein for stabilization and [1], [2] and reference therein for finite-gain stabilizability. It was shown in [1] that, for neutrally stable discrete-time linear systems, finite-gain lp −stabilization can be achieved by linear output feedback, for every p ∈ (1, ∞] except p = 1. In [2], the authors complete the results by showing that they also hold true for the case p = 1. A result due to Yang, Sontag and Sussmann ([11]) shows that such stabilization is possible if and only if the system (S) is stabilizable and the transition matrix has spectral radius less or equal to one. It is trivial to see that these conditions are also necessary in the case of non-zero delay and it seems natural to expect them to be also sufficient. The objective of this chapter is to show that the results of [9] and [10], however, do carry over to linear discrete-time delay systems. More precisely, for the problem of global asymptotic stabilization we show that, a linear discrete-time delay system subject to input saturation can be globally asymptotically stabilized via feedback if and only if the system is stabilizable and all its poles are located inside or on the unit circle. The argument is an extension to the non-zero delay case of that of [11]. Concerning the problem of finite gain lp −stabilizability, p ∈ [1, ∞], we should also mention that, for neutrally stable linear discrete time delay systems subject to actuator saturation, finite gain lp −stabilization can be achieved by linear output feedback for all p ∈ [1, ∞]. We determine a suitable “storage function” Vp and establish for it a “dissipation inequality” of the form ΔVp (k) ≤ −λ1p xu (k) p + λ2p u(k) p , for some constants λ1p > 0 and λ2p > 0 possibly depending on the input bound r and the delay h, with λ1p 0 λ2p (definition is given in section 2). For more discussion on passivity, see [8] for instance. In addition, by choosing carefully the factor r and

Stabilization and Finite-Gain Stabilizability

423

the linear feedback inside the saturation, we are able to provide upper bounds for the lp −gains of (S) which are independent of h > 0. Since the delay h is a constant positive integer, the standard trick to get rid of the delay consists of extending the state space to Xk := (x(k), x(k − 1), · · · , x(k − h)), k ≥ h, and to rewrite the dynamics for Xk . As concerns the stabilization question for the X−system, one may be able to derive similar results as ours even though the corresponding linear algebraic problem does not seem easy. However, as regards the lp −stabilization problem for the X−system, the lp −gain will depend on the delay since the latter appears in the dimension of the state space of X. This is the reason why using the X−system is not relevant for our studies. Organization of the chapter. This chapter is organized as follows. In sections 2 and 3, we provide the main notations used in the chapter and state the Krasovskii Stability Theorem. Section 4 present the finite-gain lp −stabilizability problem. Solution to the delay-independent stabilization based on feedbacks of saturation type is described in section 5. Section 6 contains our result of output problem. The chapter closes with section 7, which draws the conclusions of our current work.

2 Notations and Definition of Saturation Functions For x ∈ Rn , x and xT denote respectively the Euclidean norm of x and the transpose of x. Similarly, for any n × m matrix K, K T and K denote respectively the transpose of K and the induced 2−norm of K. Moreover, λmin (K) and λmax (K) denote the minimal and the maximal singular values of the matrix K. If f (.) and g(.) are two real-valued functions, we mean by f (r) 0 g(r), that there are positive constants ξ1 and ξ2 , independent of r small enough, with ξ1 ≤ ξ2 such that ξ1 g(r) ≤ f (r) ≤ ξ2 g(r). For h > 0, let xk (θ) := x(k + θ), −h ≤ θ ≤ 0. Initial conditions for timedelay systems are elements of Ch := C([−h, 0], Rn ), the Banach space of continuous Rn − valued functions defined on [−h, 0], equipped with the supremum norm, xk h = sup−h≤θ≤0 x(k + θ) . Then, for xk ∈ l∞ ([−h, ∞), Rn ), we have xk l∞ = sup−h≤s 0, there exists a δ > 0 such that, for any Ψ ∈ Ch , with ||Ψ||Ch ≤ δ, there exists k0 ≥ 0, such that the solution xΨ of (Σ)h satisfies ||(xΨ )k ||h ≤ ε, for all k ≥ k0 ; (ii) for all Ψ ∈ Ch , the trajectory of (Σ)h with initial condition Ψ converges to zero as k → ∞. The Krasovskii Theorem gives conditions under which x(k) = 0 is asymptotically stable. As in the continuous time-delay case, the Krasovskii stability theorem is given by: Theorem 1. (Krasovskii Stability Theorem): Suppose that the function f : Ch → Rn takes bounded sets of Ch in bounded sets of Rn and suppose that u(s), v(s) and w(s) are scalar, continuous, positive and nondecreasing functions. If there is a continuous function V : Ch → R+ and a positive number ρ such that for all xt ∈ LV (ρ) := {ψ ∈ Ch : V (ψ) ≤ ρ}, the following conditions hold. 1. u( xk (0) ) ≤ V (xk ) ≤ v( xk h ). 2. ΔV (xk ) ≤ −w( xk (0) ). Then, the solution x(k) ≡ 0 of (Σ)h is asymptotically stable. Moreover, the set LV (ρ) is an invariant set inside the domain of attraction. If items 1 and 2 hold for all x ∈ Rn and u(s) → ∞ as s → ∞, then, the solution x(k) ≡ 0 is GAS.

426


4 On Finite-Gain Stabilizability 4.1 Statement of the Results In this section, we establish finite-gain lp −stabilizability in the various p− norms, using linear state feedbacks stabilizers. we start by giving some definitions. For p ∈ [1, ∞] and 0 ≤ h(≤ T ), we use lp to denote lp (−h, ∞) and we let y lp ( y lp ([−h, T ]) respectively) denote the lp −norm : $ y lp =

∞

$

% p1 y(k) p

, ( y lp ([−h,T ]) =

k=−h

T

% p1 y(k) p

respectively),

k=−h

if p < ∞ and y l∞ =

sup

−h≤k 0 there exists an n × m matrix Fh such that, the following delayed system ! (S)rh : x(k+1) = Ax(k)−rBσ FhT x(k − h) + u1 (k − h) +ru2 (k−h), for k ∈ Z+ , is unrestricted finite-gain lp -stable, for every p ∈ [1, ∞]. Moreover, there exist a constant C0 > 0 and, for every 1 ≤ p ≤ ∞, a real mp > 0 such that, for every h > 0 there is an r∗ (h) ∈ (0, 1], for which the trajectories xu1 ,u2 of (S)rh , r ∈ (0, r∗ (h)], starting at ¯ 0 and corresponding to u1 , u2 ∈ lp with u2 l∞ ≤ C0 , verify xu1 ,u2 lp ≤ mp ( u1 lp + u2 lp ). Remark 1. In the absence of u1 and u2 , the equilibrium point ¯ 0 is globally asymptot! ically stable for the delayed system x(k + 1) = Ax(k) − rBσ FhT x(k − h) . Remark 2. It will be clear from our argument that we can in fact obtain the following stronger ISS-like property ([6] and references there): xψ u1 ,u2 lp ≤ θp ( ψ h ) + Mp ( u1 lp + u2 lp ), where ψ ∈ Ch is the initial condition for the trajectory xψ u1 ,u2 corresponding to u1 , u2 and θp is a K -function (i.e. θp : R+ → R+ , is continuous, strictly increasing and satisfies θp (0) = 0.).

4.2 Proof of Theorem 2 From elementary linear algebra, a neutrally stable matrix A is similar to a matrix $ % A1 0 , (3) 0 A2 where A1 is a q × q convergent or discrete-time Hurwitz (i.e. all its eigenvalues have magnitude less than 1) matrix and A2 is an (n − q) × (n − q) orthogonal matrix. So, up to a change of coordinates, we may assume that A is already in the form (3). In these coordinates, we write

428


$ B=

B1 B2

% ,

where B2 is an (n − q) × m matrix and we write vectors as x = (xT1 , xT2 )T , and u2 = (uT21 , uT22 )T . For h > 0, consider the feedback law (0, FhT ). Then system (S)rh , with this choice of FhT , can be written as ⎧ T ⎪ ⎪ ⎨ x1 (k + 1) = A1 x1 (k) − rB1 σ[Fh x2 (k − h) + u1 (k − h)] + ru21 (k − h), ⎪ ⎪ ⎩ x (k + 1) = A x (k) − rB σ F T x (k − h) + u (k − h)! + ru (k − h). 2 2 2 2 1 22 h 2 Since A1 is discrete-time Hurwitz, it will be sufficient to show that there exists an r∗ (h) ∈ (0, 1], such that the x2 − subsystem is finite-gain lp −stable, for all r ∈ (0, r∗ (h)]. The controllability assumption on (A, B) implies that the pair (A2 , B2 ) is also controllable. Therefore, the theorem is a consequence of the following proposition. Proposition 1. Let σ, u1 , u2 be as in Theorem 2. Let (A, B) a controllable pair with A orthogonal (i.e., AT A = Idn ). Then, for every h > 0, there is an r∗ (h) ∈ (0, 1], such that for every r ∈ (0, r∗ (h)], the system ! (S)rh : x(k + 1) = Ax(k) − rBσ B T Ah+1 x(k − h) + u1 (k − h) + ru2 (k − h), for k ∈ Z+ , verifies the conclusion of Theorem 2. To prove this proposition, first we need to establish a few lemmas carried out in [1]. Lemma 1. For any p > l > 0, there exist two scalars M1 , M2 > 0 such that, for any two positive scalars ξ and ζ, ξ p−l ζ l ≤ M1 ξ p + M2 ζ p . 2 Lemma 2. For any l > 0, there exists a constant Cl > 0 such that (a + b)l ≤ Cl (al + bl ), ∀a, b ≥ 0, Cl := 2l−1 . Consequently, if l ≤ 1, we have (a + b)l ≤ al + Cl bl , ∀a, b ≥ 0.

2


429

Lemma 3. Let A be orthogonal, and suppose that the pair (A, B) is controllable. Then, for every r > 0 such that rB T B < 2In , set Ar := A − rBB T A and Pr the unique symmetric positive-definite solution of the Lyapunov equation ATr Pr Ar − Pr = −In . Then, there exists a r∗ > 0 and χ1 , χ2 > 0 independent of r, such that for every r ∈ (0, r∗ ],

χ1 χ In ≤ Pr ≤ 2 In . r r

2

Remark 3. By Lemma 3, there exists r1∗ ∈ (0, r∗ ] such that λmax (Pr ) 0 λmin (Pr ) 0

1 , r

∀r ∈ (0, r1∗ ].

(4)

Lemma 4. Let Ar and Pr be as given in Lemma 3, then for any p ∈ (1, ∞), there exists a r∗ > 0 such that p

p

[z T ATr Pr Ar z] 2 − [z T Pr z] 2 ≤ −r where ζ > 0 is some constant independent of r.

2−p 2

ζ|z|p , r ∈ (0, r∗ ],

2

Lemma 5. Let A and B be as given in Lemma 3. For any l ∈ [1, ∞) and any r ∈ (0, 1], |σ(B T Az + u)|l ≤ 2l−1 |B|l |z|l + 2l−1 |u|l .

2

Lemma 6. Let A and B be as given in Lemma 3. Pick any z ∈ Rn and d˜ ∈ Rm , any ˜ number η ≥ 3, and any nonnegative real number l. Denote z˜(k) = B T Az(k)+ d(k). Then, provided |z| > rη|Bσ(˜ z )|, we have z ) + M r2 |z|l−2 |Bσ(˜ z )|2 , |Az − rBσ(˜ z )|l ≤ |z|l − lr|z|l−2 z T AT Bσ(˜ for some constant M > 0 which is independent of r.

2

We are now ready to prove Proposition 1. Proof of Proposition 1. Let h > 0 and y be the solution of ⎧ ! T ⎪ y(k + 1) = A − rBB A y(k) + ru2 (k − h), for k ∈ Z+ , ⎪ ⎨ ⎪ ⎪ ⎩y = ¯ 0, on [−h, 0], 0

(5)

430


with u2 ∈ lp and u2 l∞ ≤ C0 , where C0 is a positive constant to be determined later. (By an obvious abuse of notation, lp denotes here lp ([−h, ∞), Rn ).) Let r1∗ > 0 be such that rBB T < 2In . Then, Ar := A−rBB T A is asymptotically stable for all r ∈ (0, r1∗ ], (see [3]). So, for every r ∈ (0, r1∗ ], (5) is lp −stable for any 1 ≤ p ≤ ∞. Set γp denotes its lp −gain, then y lp ≤ γp u2 lp .

(6)

We need the following lemma. Lemma 7. Let γp be the lp −gain of (6). So γp 0 O(1). Proof of Lemma 7: We separate the proof for p ∈ [1, ∞) and for p = ∞. Proof for p ∈ [1, ∞) : The equation (5) may also be written as y(k + 1) = Ar y(k) + ru2 (k − h), y ∈ Rn , u2 ∈ Rn , for k ∈ Z+ . For this system, define the function Vy as " # p2 T Vy (y) = y Pr y ,

(7)

(8)

where Pr is as given in Lemma 3. We next evaluate the increments Vy (y(k + 1)) − Vy (y(k)), which we denote as ΔVy , along any given trajectory of (7). Thus, " #p " #p 2 2 ΔVy = Vy (y(k + 1)) − Vy (y(k)) = y T (k + 1)Pr y(k + 1) − y T (k)Pr y(k) =

y

T

(k)ATr

+

ruT2 (k

p p ! 2 2 T − h) Pr (Ar y(k) + ru2 (k − h)) − y (k)Pr y(k)

p 2 = y T (k)ATr Pr Ar y(k) + 2ry T (k)ATr Pr u2 (k − h) + r2 uT2 (k − h)Pr u2 (k − h) p 2 − y T (k)Pr y(k) ≤ y T (k)ATr Pr Ar y(k) + 2ry T (k)ATr Pr u2 (k − h) + r2 uT2 (k − h)Pr u2 (k − h) p − y T (k)Pr y(k) p2 ν 2 −1 , (9)

for some ν > 0 between ν1 := y T (k)ATr Pr Ar y(k) + 2ry T (k)ATr Pr u2 (k − h) + r2 uT2 (k − h)Pr u2 (k − h) and ν2 := y T (k)Pr y(k) . The last inequality follows p

from finite increments theorem applied to the function s 2 :


431

p p p p s12 − s22 = (s1 − s2 ) ν 2 −1 , 2

for ν between s1 and s2 . By Lemmas 3 and 1, there exist r2∗ ∈ (0, r1∗ ] and a constant π1 > 0 independent of r, such that for any r ∈ (0, r2∗ ], we have # p " ΔVy ≤ − 12 y(k) 2 + π1 u2 (k − h) 2 p2 ν 2 −1

≤

# ⎧" p p 1 2 2 p 2 −1 2 −1 ⎪ − y(k) + π u (k − h) max{ν , ν }, if 1 2 ⎪ 1 2 2 2 ⎨ " # ⎪ ⎪ ⎩ − 1 y(k) 2 + π u (k − h) 2 p min{ν p2 −1 , ν p2 −1 }, if 1 2 1 2 2 2

p 2

− 1 > 0,

p 2

− 1 ≤ 0. (10)

Case 1:

0. By using Lemmas 1-3, there exists r3∗ ∈ (0, r2∗ ] such that for any

r∈

is bounded as follows:

p 2 −1 > (0, r3∗ ], ΔVy

ΔVy ≤

p 2

"

−

1 2 2 y(k)

× ≤

p p 2 −1 2 (3χ2 )

"

"

+ π1 u2 (k − h)

2

χ2 2 r y

−

#

+ 2χ2 y u2 + rχ2 u2 2

1 2 2 y(k)

+ π1 u2 (k − h)

2

#"

# p2 −1

2 1 y 2 r

+ r u2

2

# p2 −1

" # p p ≤ p2 2 2 −2 (3χ2 ) 2 −1 − 12 y(k) 2 + π1 u2 (k − h) 2 " # p−2 p p × 21− 2 y p −1 + r 2 −1 u2 p−2 r2

≤ −Cp1 ≤ −Cp4

y p

p r 2 −1

y p

p r 2 −1

+ Cp2 y p −1 u2 (k − h) 2 + Cp3 r 2 −1 u2 (k − h) p p−2

p

r2

+ Cp5 u2 (k−h) , p −1 p

r2

(11)

where Cpi > 0, 1 ≤ i ≤ 4, are constants independent of r. In deriving (11), the first inequality by Lemma 3, the second and last inequalities by Lemma 1 and The third inequality by Lemma 2. p 2 − (0, r4∗ ],

Case 2: r∈

1 ≤ 0. By Lemmas 9 and 4, there exists r4∗ ∈ (0, r3∗ ] such that for any

432


ΔVy = y T (k)ATr Pr Ar y(k) + 2ry T (k)ATr Pr u2 (k − h)

+ "

≤ y

T

ATr Pr Ar y

≤ −r

2−p 2

≤ −r

2−p 2

≤ −r

2−p 2

p

# p2

ζ y + C

r2 uT2 (k

"

− y Pr y " p 2

T

p2 p2 T − h)Pr u2 (k − h) − y (k)Pr y(k)

# p2

p2 T T 2 T + C p2 2ry Ar Pr u2 + r u2 Pr u2

2χ2 y u2 + rχ2 u2

2

# p2

" # p p p p p ζ y p + C p2 (2χ2 ) 2 y 2 u2 2 + C p2 r 2 χ22 u2 p p

p

p

p

p

ζ y p + C p2 (2χ2 ) 2 y 2 u2 2 + (C p2 )2 r 2 χ22 u2 p

≤ −Cp6 y p + Cp7 u2 p , (12) where Cp6 > 0 and Cp7 > 0 are constants independent of r. Combining case 1 with case 2, we have, for any r ∈ (0, r4∗ ], ΔVy ≤ −ε1 (r) y(k) p + ε2 (r) u2 (k − h) p , where ε1 (r) 0 ε2 (r) 0

⎧ ⎪ ⎪ ⎨

1

p r 2 −1

, if

⎪ ⎪ ⎩ O(1), if

p 2

(13)

− 1 > 0, (14)

p 2

− 1 ≤ 0.

For every k ∈ Z+ , summing both sides of (14) from 0 to k, we have V (y(k)) + ε1 (r) y p(lp )[0,k] ≤ ε2 (r) u2 p(lp )[0,k] .

(15)

Using the fact that V ≥ 0, we conclude that y lp ≤ γp u2 lp ,

(16)

" ε (r) # P1 2 γp := 0 O(1). ε1 (r)

(17)

where, according to (14)

Proof for p = ∞ : From (13) we get for p = 2,


ΔVy ≤ −ε1 y(k) 2 + ε2 u2 2l∞ ,

433

(18)

where ε1 > 0 and ε2 > 0 are constants independent of r. Thus, ΔVy is negative outside the ball centered at the origin, of radius

&

ε2 ε1 u2 l∞ .

It follows that χ1 y(k) 2 ≤ y T (k)Pr y(k) := Vy (y(k)) ≤ sup & r ε2 |ξ|≤

u ε1

Vy (ξ) ≤

2 l∞

χ2 ε2 u2 2l∞ . r ε1 (19)

Thus implies that y l∞ ≤ γ∞ u2 l∞ , where γ∞

" χ ε # 12 2 2 := 0 O(1). χ1 ε1

(20) 2

Let x be the solution of (S)rh starting at ¯ 0 ∈ Ch and corresponding to u1 , u2 . Set z = x − y. Then z satisfies ⎧ ! T h+1 T h+1 ⎪ z(k + 1) = Az(k)−rB[σ B A z(k−h)+B A y(k−h)+u (k−h) − ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ − B T Ay(k)], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩z = ¯ 0, on [−h, 0]. 0

(21) ˜2 (k) = B T Ay(k). Choosing Let u ˜1 (k − h) = B T Ah+1 y(k − h) + u1 (k − h) and u the constant C0 > 0 such that γ∞ B C0 ≤ γ, with γ = lim|ξ|→∞ inf σ(ξ) > 0, we get ˜ u2 l∞ ≤ B y l∞ ≤ γ∞ B C0 ≤ γ.

(22)

Now (21) can be written as ⎧ 5 ! 6 T h+1 ⎪ z(k + 1) = Az(k) − rB σ B A z(k − h) + u ˜ (k − h) − u ˜ (k) , ⎪ 1 2 ⎨ (23)

⎪ ⎪ ⎩z = ¯ 0, on [−h, 0]. 0 From (23), we have z(k) = Ah z(k − h) − r −u ˜2 (i)] = Ah z(k − h) − r

! k−i−1 T h+1 A B[σ B A z(i − h) + u ˜ (i − h) − 1 i=k−h

k−1

h−1 i=0

Ak−i−1 B[σ(B T Ah+1 z(k − 2h + i)+

+u ˜1 (k − 2h + i)) − u ˜2 (k − h + i)].

434


Then, ˜ B T Ah+1 z(k − h) + u ˜1 (k − h) = B T Az(k) + d(k), where h−1 ˜ =u d(k) ˜1 (k − h) + r i=0 B T Ah−i B[σ(B T Ah+1 z(k − 2h + i)+ +u ˜1 (k − 2h + i)) − u ˜2 (k − h + i)]. According to (23), we obtain " # ⎧ T ˜ ⎪ z(k + 1) = Az(k) − rB σ B Az(k) + d(k) − u ˜ (k) 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

:= Az(k) − rB [σ (˜ z (k)) − u ˜2 (k)] , for k ∈ Z+ ,

(24)

0, on [−h, 0], z0 = ¯

˜ We separate the proof for p ∈ (1, ∞), p = 1 and where z˜(k) = B T Az(k) + d(k). for p = ∞. We consider the Lyapunov function V2,p used in [1] (or in [4] and [10]) and given by (25)

V2,p (z) := p,r V0,p (z) + V1,p (z), where p,r > 0 will be chosen later and V0,p (z) := z p+1 ,

p

V1,p (z) = (z T Pr z) 2 ,

(26)

with Pr , the positive definite symmetric matrix, satisfying ATr Pr Ar − Pr = −In , where the Hurwitz matrix Ar defined by A − rBB T A. If W : Rn → R, then ΔW denote the increment W (x(k + 1)) − W (x(k)) along any given trajectory of (24). In the sequel Ci , i = 1, 2, . . . denote constants that are independent of r. Proof for p ∈ (1, ∞) : Along the trajectories of (24) and as in [1], we separate each of the cases z > rη ( Bσ(˜ z ) + B u ˜2 ) and z ≤ rη ( Bσ(˜ z ) + B u ˜2 ) , where η is any number ≥ 3 independent of r. Case 1: z ≤ rη ( Bσ(˜ z ) + B u ˜2 ) . By Lemma 5, for any r ∈ (0, r1∗ ],


435

ΔV0,p := V0,p (z(k + 1)) − V0,p (z(k)) = z(k + 1) p+1 − z(k) p+1 = Az(k) − rB (σ(˜ z) − u ˜2 ) p+1 − z(k) p+1 p+1

≤ Az(k) − rB (σ(˜ z) − u ˜2 ) p+1 ≤ ( z + r ( Bσ(˜ z ) + B u ˜2 )) p+1

≤ [r(η + 1) ( Bσ(˜ z ) + B u ˜2 )] p+1

≤ 2p [r(η + 1) Bσ(˜ z ) ]

p+1

+ 2p [r(η + 1) B u ˜2 ]

˜ p + ˜ ≤ rC1 z p + rC2 ( d u2 p ). (27) In deriving (27), we have used the fact that σ, r and u ˜2 are bounded. case 2: z > rη ( Bσ(˜ z ) + B u ˜2 ) . By Lemmas 6, 5 and 1, there exists r2∗ ∈ (0, r1∗ ] such that for any r ∈ (0, r2∗ ], ΔV0,p = Az − rB(σ(˜ z) − u ˜2 ) p+1 − z p+1 ≤ z p+1 − (p + 1)r z p−1 z T AT B (σ(˜ z) − u ˜2 ) + ! z ) 2 + B u ˜2 2 − z p+1 + 2r2 M z p−1 Bσ(˜ (28) ≤ −(p + 1)r z

p−1 T

p−1 ˜T

z˜ σ(˜ z ) + (p + 1)r z

z )+ d σ(˜

z ) 2 + B u ˜2 + 2M r2 z p−1 Bσ(˜ ˜2 2 + (p + 1)r z p B u

!

" # ˜ p + ˜ ≤ −(p + 1)r z p−1 z˜T σ(˜ z ) + rC3 z p + rC4 d u2 p , where M is defined in Lemma 6. In deriving (28), the first inequality by Lemma 6, the second and third inequalities are the consequence of the fact that σ, r, u ˜2 are bounded and Lemmas 1 and 5. Combining case 1 with case 2, we have, for any r ∈ (0, r2∗ ],

436


ΔV0,p ≤

" # ⎧ p−1 T p p p ˜ ⎪ −(p + 1)r z d , z ˜ σ(˜ z ) + rC z + rC + ˜ u ⎪ 5 6 2 ⎪ ⎪ ⎪ ⎨ if z > rη ( Bσ(˜ z ) + Bσ(˜ z ) ) , ⎪ ⎪ ⎪ " # ⎪ ⎪ p ⎩ rC5 z p +rC6 d ˜ p + ˜ z ) + Bσ(˜ z ) ) , u2 , if z ≤ rη ( Bσ(˜ (29)

where C5 = max{C1 , C3 } and C6 = max{C2 , C4 }. Rewrite the system equation (24) as ⎧ ⎪ z + σ(˜ z ) + d˜ − u ˜2 ], for k ∈ Z+ , ⎪ ⎨ z(k + 1) = Ar z(k) − rB[−˜ (30)

⎪ ⎪ ⎩z = ¯ 0, on [−h, 0], 0

and separate each of the cases z > rη ( Bσ(˜ z ) + B u ˜2 ) and z ≤ rη ( Bσ(˜ z ) + B u ˜2 ) . The increment ΔV1,p along the trajectories of (30) verifies case 1: z ≤ rη ( Bσ(˜ z ) + B u ˜2 ) . By using Lemmas 3, 4, 5 and Jensen’s inequality, ΔV1,p along the trajectories of (30) is bounded as follows: !p !p ΔV1,p = z(k + 1)T Pr z(k + 1) 2 − z(k)T Pr z(k) 2 !p p ≤ Pr 2 Az(k) − rB (σ(˜ z (k)) − u ˜2 ) p − z T (k)Pr z(k) 2 + + z T ATr Pr Ar z

! p2 p

p

p

p

≤ −r

2−p 2

ζ z p + 2p−1 χ22 (η + 1)p r 2 ( Bσ(˜ z ) p + B u ˜2 p )

≤ −r

2−p 2

ζ z p + 2p−1 χ22 (η + 1)p r 2 (2p−1 B 2p z p +

˜ p + B u ˜2 p ) + 2p−1 B p d ≤ −r

2−p 2

p 2

p 2

ζ z + r C7 z + r C8 p

p

"

# p p ˜ d + ˜ u2 ,

r ∈ (0, r2∗ ],

where χ2 and ζ > 0 are as defined in Lemmas 3 and 4, respectively. Case 2: z > rη ( Bσ(˜ z ) + B u ˜2 ) . We next evaluate the increments ΔV1,p along the trajectories of (30). We follow the computations leading to equation (44) of [1] and obtain using ν := 2rz T AT Pr BB T Az − 2rz T AT Pr B (σ(˜ z) − u ˜2 ) +


437

r2 (σ(˜ z) − u ˜2 ) B T Pr B (σ(˜ z) − u ˜2 ) , the existence of r3∗ ∈ (0, r2∗ ] such that for T

every r ∈ (0, r3∗ ], ΔV1,p ≤ −r

2−p 2

+ C11 r

ζ z p + C9 r

2−p 2

"

4−p 2

Pr B z p−1 z˜T σ(˜ z ) + C10 r

2−p 2

z p +

#

(31)

˜ p + ˜ u2 p . d

Summarizing, we may combine case 1 with case 2, to obtain for every r ∈ (0, r3∗ ], ⎧ 2−p 4−p 2−p ⎪ −r 2 ζ z p + C12 r 2 Pr B z p−1 z˜T σ(˜ z ) + C13 r 2 z p + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ " # ⎪ 2−p ⎪ ⎪ p p ˜ 2 ⎪ u2 , if z > rη ( Bσ(˜ +C14 r d + ˜ z ) + B u ˜2 ) , ⎪ ⎨ ΔV1,p ≤ " # ⎪ ⎪ 2−p 2−p 2−p ⎪ p p ˜ ⎪ 2 ξ z p + C 2 z p + C 2 r r + ˜ u −r d , 13 14 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ if z ≤ rη ( Bσ(˜ z ) + B u ˜2 ) , (32) where C12 = C9 , C13 = max{C10 , rp−1 C7 } and C14 = max{C11 , rp−1 C8 }. We choose p,r as p,r :=

C12 2−p r 2 Pr B . p+1

(33)

Using eqs. (29) and (32), we deduce that there exists a r4∗ ∈ (0, r3∗ ] such that for every r ∈ (0, r4∗ ] the estimation of ΔV2,p along the trajectories of (24) verifies " # 2−p 2−p ˜ p + ˜ u2 p . ΔV2,p ≤ −r 2 C15 z p + r 2 C16 d (34) Proof for p = 1 : For z ∈ Rn , consider V1,1 (z) = z T Pr z and V0,1 (z) = z 2 . For clarity, let us repeat here the system equation (24) " # ˜ z(k + 1) = Ar z(k) − rB −˜ z (k) + σ(˜ z (k)) + d(k) − u ˜2 (k) (35) = Az(k) − rB (σ(˜ z (k)) − u˜2 (k)) , ˜ Along trajectories of (35), we obtain where z˜ = B T Az + d. ˜ u2 + ΔV0,1 ≤ −2r˜ z T σ(˜ z ) + 2rd˜T σ(˜ z ) + 2r ˜ z ˜ u2 + 2r d ˜ (36) z ) 2 + 2r2 B u ˜2 2 . + 2r2 Bσ(˜

438


Moreover, for every z ∈ Rn , d˜ ∈ Rm and u ˜2 ∈ Rn we have V1,1 (z(k + 1)) = z(k + 1)T Pr z(k + 1) T

z (k)) − u˜2 (k)) B T ]Pr × = [z T (k)AT − r (σ(˜ (37) [Az(k) − rB (σ(˜ z (k)) − u˜2 (k))] ≤

C17 2 r z

# 2 2 ˜ + C18 r d + ˜ u2 , "

and also, T

V0,1 (z(k + 1)) = [z T (k)AT − r (σ(˜ z (k)) − u˜2 (k)) B T ]× [Az(k) − rB (σ(˜ z (k)) − u˜2 (k))]

(38)

" # ˜ 2 + ˜ u2 2 . ≤ C19 z 2 + d Equation (37) clearly implies " # √ C20 ˜ V1,1 (z(k + 1)) ≤ √ z + rC21 d + ˜ u2 . r 1 2

(39)

˜ + ˜ It is convenient to treat separately two cases namely whether z ≤ r( d u2 ) ˜ + ˜ or not. Assume first that z ≤ r( d u2 ). We clearly have from (39) and (36), respectively that # 1 1 √ " 2 2 ˜ + ˜ ΔV1,1 ≤ V1,1 (z(k + 1)) ≤ C22 r d u2 ,

(40)

˜ + ˜ ΔV0,1 ≤ rC23 ( d u2 ).

(41)

and

For the second inequality, we have used the fact that σ and u ˜2 are bounded. Adding (40) and (41), we get

√ ˜ + ˜ ΔV2,1 ≤ C24 r( d u2 ).

(42)

We deduce that

# √ " √ ˜ + ˜ ˜ + ˜ u2 ) + r r( d u2 ) − z ΔV2,1 ≤ C24 r( d √ √ ˜ + ˜ u2 ). ≤ − r z + C25 r( d

(43)


439

˜ + ˜ We now treat the case where z > r( d u2 ). From (39), it follows that 1 C26 2 V1,1 (z(k + 1)) ≤ √ z , r

(44)

1 1 1 C27 C28 2 2 2 √ z ≤ V1,1 (z(k)) ≤ V1,1 (z(k)) + V1,1 (z(k + 1)) ≤ √ z . r r

(45)

which implies that

After elementary calculus, we get ˜ + ˜ ΔV0,1 ≤ −2r˜ z T σ(˜ z ) + C29 r z + C30 r( d u2 ).

(46)

1 2 ) as As in ([2]), we write Δ(V1,1

ΔV1,1 (z(k))

1 2 Δ(V1,1 )(z(k)) =

1 2

(47)

.

1

2 V1,1 (z(k)) + V1,1 (z(k + 1))

We follow the computations leading to equation (II. 23) of ([2]) and obtain, there is an r5∗ ∈ (0, r4∗ ] such that " #2 1 ˜ + ˜ z T σ(˜ z ) + C32 r d u2 , ΔV1,1 ≤ − z 2 + C31 z ˜ 2

∀r ∈ (0, r5∗ ], (48)

˜

u2 According to (47) and using (45) and the fact that r d + ˜ ≤ 1, we get,

˜ z

# √ √ T √ " ˜ u2 . Δ(V1,1 ) ≤ −C33 r z + C34 r˜ z σ(˜ z ) + C35 r d + ˜ 1 2

∀r ∈ (0, r5∗ ], (49)

Now, consider the following function 1

where

2 V2,1 = 1,r V0,1 + V1,1 ,

(50)

C34 1,r := √ . 2 r

(51)

It follows that there exists r∗ ∈ (0, r5∗ ] such that # √ √ " ˜ + ˜ ΔV2,1 ≤ −C36 r z + C37 r d u2 ,

∀r ∈ (0, r∗ ].

(52)

So, in either case, the Lyapunov function V2,p , 1 ≤ p < ∞, verify # " 1 p 2 p p ˜ u2 , ∀r ∈ (0, r∗ ], ΔV2,p ≤ −θp (r) z + θp (r) d + ˜

(53)

where

440


θp1 (r) 0 θp2 (r) 0 r Now let Hμ (k) = μ

h−1

2−p 2

p ∈ [1, ∞).

,

k−1

(54)

z(θ) p ,

j=0 θ=k−2h+j

where p ∈ [1, ∞) and μ > 0 will be chosen later. Its increments along the trajectories of (24) satisfies, ΔHμ (k) = Hμ (k + 1) − Hμ (k) =μ

h−1 k

p θ=k+1−2h+j z(θ) −

j=0

= μh z(k) p − μ

h−1 j=0

k−1

p θ=k−2h+j z(θ)

(55)

z(k − 2h + j) p .

Finally, consider the following function V (z(k)) = V2,p (z(k)) + Hμ (k). Along the trajectories of (24), from (53) and (55), we have: # " 1 p 2 p p ˜ u2 (k) − ΔV (z(k)) ≤ −(θp (r) − μh) z(k) + θp (r) d(k) + ˜ −μ

h−1 j=0

(56) z(k − 2h + j) p .

˜ it follows that: From the definition of d, ˜ d(k) ≤ ˜ u1 (k − h) + r B 2 max{1, K}

h−1 j=0

+ ˜ u2 (k − h + j) ) + r2 max{1, K} B 3

( ˜ u1 (k − 2h + j) + h−1 j=0

z(k − 2h + j) ,

so, p ˜ ≤ C38 ˜ u1 (k − h) p + rp C39 2(h−1)(p−1) d(k)

h−1

+ ˜ u2 (k − h + j) p ) + r2p C40 2(h−1)(p−1) From (56) and (57), we have:

u1 (k j=0 ( ˜

h−1 j=0

− 2h + j) p +

z(k − 2h + j) p , (57)


441

ΔV (z(k)) ≤ −(θp1 (r) − μh) z(k) p + θp2 (r) max{1, C38 }( ˜ u1 (k − h) p + + ˜ u2 (k) p ) + rp θp2 (r)C39 2(h−1)(p−1)

h−1

u1 (k j=0 ( ˜

− 2h + j) p +

! + ˜ u2 (k − h + j) p ) − μ − r2p θp2 (r)C40 2(h−1)(p−1) × h−1 j=0

z(k − 2h + j) p .

Now, if we choose μ∗ (h) and r∗ (h) ∈ (0, r∗ ] such that, ⎧ 1 θp (r) ∗ (h−1)(p−1) 2 ⎪ ⎪ μ (h) = min , C 2 θ (r) , 40 p ⎪ 2h ⎪ ⎨ +" , ⎪ #1 ⎪ θp2 (r) 2p ⎪ 1 ∗ ∗ ⎪ ,r , ⎩ r (h) = min 2C40 h2(h−1)(p−1) θp1 (r)

(58)

we get for μ ≤ μ∗ (h) and r ≤ r∗ (h), ΔV (z(k)) ≤ − 12 θp1 (r) z(k) p + θp2 (r)[max{1, C38 } ( ˜ u1 (k − h) p + ˜ u2 (k) p ) + rp C39 2(h−1)(p−1)

h−1 j=0

( ˜ u1 (k − 2h + j) p + ˜ u2 (k − h + j) p )]

u1 (k − h) p + ˜ u2 (k) p )+ := −12 θp1 (r) z(k) p +θp2 (r)[max{1, C38 } ( ˜ p ˜ + ˜ n(k) p )], + rp C39 2(h−1)(p−1) ( m(k)

p where m(k) ˜ =

h−1 j=0

˜ u1 (k−2h+j) p and ˜ n(k) p =

h−1 j=0

(59) ˜ u2 (k−h+j) p .

A simple computation shows that 1

1

m ˜ lp ≤ h p ˜ u1 lp and ˜ n lp ≤ h p ˜ u2 lp .

(60)

For every k ∈ Z+ , summing both sides of (59) from 0 to k, we have V (z(k))+ 12 θp1 (r)

k j=0

z(j) p ≤ θp2 (r)[max{1, C38 }

k

u1 (j j=0 ( ˜

− h) p +

+ ˜ u2 (j) p ) + rp C39 2(h−1)(p−1) × k j=0

p ( m(j) ˜ + ˜ n(j) p )].

(61)

442


We conclude that, for every 1 ≤ p < ∞ and k ∈ Z+ , # " V (z(k))+ 12 θp1 (r) z p(lp )[0,k] ≤ θp2 (r)[max{1, C38 } ˜ u1 p(lp )[0,k] + ˜ u2 p(lp )[0,k] + p

(h−1)(p−1)

+r C39 2

"

m ˜ p(lp )[0,k]

+

˜ n2 p(lp )[0,k]

#

]. (62)

Using the fact that V ≥ 0 and the inequality (60), we conclude that ! u1 lp + ˜ u2 lp , z lp ≤ θp3 (r) ˜

(63)

where, according to (54)

# θp2 (r) " 3 p (h−1)(p−1) θp (r) := 2 1 max{1, C38 } + hr C39 2 θp (r)

p1 0 O(1).

(64)

˜2 = B T Ay, we have Since z = x − y, u ˜1 = B T Ah+1 y + u1 and u ⎧ ⎪ ˜ u2 lp ≤ B y lp ≤ B γp u2 lp , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ˜ u1 lp ≤ B γp u2 lp + u1 lp , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z ≥ x − y ≥ x − γ u . lp lp lp lp 2 lp p

(65)

Combining this with (63), we get ! x lp ≤ θp3 (r) u1 lp + γp 1 + 2 B θp3 (r) u2 lp !

(66)

≤ mp u1 lp + u2 lp , where

!

mp = max θp3 (r), γp 1 + 2 B θp3 (r) .

(67)

By using (64) and (6) it is clear that mp 0 O(1),

∀p ∈ [1, ∞).

(68)

So, by choosing rh ≤ 1, the finite-gain mp , 1 ≤ p < ∞ is delay-independent. More precisely, there exists a positive constant C41 independent of r and h such that ! x lp ≤ C41 u1 lp + u2 lp . Proof for p = ∞. Letting p = 2, from (34) and (59) we have


443

2

ΔV (z(k)) ≤ −C42 z(k) 2 + C43 ( ˜ u1 l∞ + ˜ u2 l∞ ) , √ 5 6 ˜ l∞ ≤ h ˜ where C43 := C16 max{1, C38 } + hr2 C39 2h−1 , since m u1 l∞ and √ ˜ n l∞ ≤ h ˜ u& 2 l∞ . Hence, ΔV (z) is negative outside the ball ¢ered at the C43 C43 origin of radius u1 l∞ + ˜ u2 l∞ ) := λR where λ = C42 ( ˜ C42 and R = ( ˜ u1 l∞ + ˜ u2 l∞ ) , from which it follows that, for any state z(k) in the trajectory: V (z(k)) ≤ sup V (ξ) ≤ 2,r λ3 R3 + λ2 R2 |ξ|≤λR

First assume that R ≤ 1. Then, we have χ1 z(k) 2 ≤ z T (k)Pr z(k) ≤ V (z(k)) ≤ r

3h + 1 2 2 χ2 + μh λ R . r 2

(69)

3h + 1 2 + μh λ R2 . 2,r λ + λ r 2 (70) 3

2 χ2

This implies that z l∞ ≤ Kr1 R, where $ Kr1 :=

2 r2,r λ3 + χ2 λ2 + rμh 3h+1 2 λ χ1

If R ≥ 1, we have

2,r z(k) ≤ V (z(k)) ≤ 3

% 12 .

3h + 1 2 + μh λ R3 , 2,r λ + λ r 2 3

2 χ2

(71)

(72)

and we get z l∞ ≤ Kr2 R, where $ Kr2 :=

2 r2,r λ3 + χ2 λ2 + rμh 3h+1 2 λ r2,r

% 13 .

(73)

Let Kr∞ = max{Kr1 , Kr2 }. Then, by using (33), it is clear that Kr∞ 0 O(1).

(74)

So, by choosing rh ≤ 1, the finite-gain Kr∞ is delay-independent. More precisely, there exists a positive constant C44 independent of r and h such that z l∞ ≤ C44 ( ˜ u1 l∞ + ˜ u2 l∞ ) . It is implies that z l∞ ≥ x l∞ − y l∞ ≥ x l∞ − γ∞ u2 L∞ . We conclude that

(75)

444


x l∞ ≤ C44 ( ˜ u1 l∞ + ˜ u2 l∞ ) + γ∞ u2 l∞ ≤ C44 ( u1 l∞ + 2γ∞ B u2 l∞ ) + γ∞ u2 l∞

(76)

≤ m∞ ( u1 l∞ + u2 l∞ ) , where m∞ = max {C44 , γ∞ (1 + 2C44 B )} 0 O(1).

2

5 Delay-Independent GAS using a Stabilizing Feedback of Nested Saturation Type 5.1 Statements of the Results We determine two explicit expressions of globally asymptotically stabilizing feedbacks for general linear discrete-time delay systems, both of nested saturation type, according to the results of the stabilization of delay free-system and continuous-time delay linear systems. The above problem was first studied for delay-free continuoustime systems. It was shown in [7] and [9] that, under standard necessary condition (no eigenvalue of the uncontrolled system have positive real part and that the standard stabilizability rank-condition hold), there exists explicit expressions of globally asymptotically stabilizing feedbacks. Then, it is natural to investigate whether this technique can be extended to the case of linear discrete-time systems where there is a delay in the input. We start by giving some definitions, first introduced in [7] and adapted here to the delay case. Definition 5. (cf. [11]) Given an n−dimensional system (Σ)h : x(k + 1) = f (x(k − h)), k ∈ Z, we say that (Σ)h is IICS (integrable-input converging-state) if, whenever {e(k)}∞ 0 ∈ l1 , every"solution k → x(k) of x(k +1) = f (x(k −h))+e(k) converges to zero as k → ∞. Such a concept is needed in order to state Theorem 3 and an

intermediary result (Lemma 9 given below), which is useful for the induction step in # the proof of Theorem 3 . For a system x(k + 1) = f (x(k), u(k − h)), x ∈ Rn , u ∈ Rm , we say that a feedback u(k − h) := u ¯(x(k − h)) is stabilizing if 0 is a globally asymptotically stable equilibrium of the closed-loop system x(k + 1) = f (x(k), u ¯(x(k − h))) . If, in addition, this closed-loop system is IICS, then we will say that u ¯ is IICS-stabilizing.


445

Definition 6.. (cf. [11]) For an n×n matrix A, let N (A) be the number of eigenvalues z of A such that |z| = 1 and m z ≥ 0, counting multiplicity. In the next theorem, we summarize our results. Theorem 3. Let (S)rh be a linear system x(k + 1) = Ax(k) + Bu(k − h) with state space Rn and input space Rm . Assume that (S)rh is stabilizable and A has no unstable eigenvalues. Let N = N (A). Then, for every ε > 0 and h > 0, there exist a sequence σ = (σ1 , . . . , σN ) of S-functions with σ ≤ ε, an m−tuple l = (l1 , . . . , lm ) of nonnegative integers such that |l| = l1 + . . . + lm = N, a number j j r∗ (h) ∈ (0, 1] and for each 1 ≤ j ≤ m, linear functions fh,i , gh,i : Rn −→ R, 1 ≤

i ≤ lj , such that for all r ∈ (0, r∗ (h)], there are IICS-stabilizing feedbacks j j j j (∗) uj (.) = −rσljj fh,l (x(.)) + α σ j lj −1 lj −1 fh,lj −1 (x(.))+ j j j + . . . + α1 σ1 (fh,1 (x(.))) ,

(77)

where αji ≥ 0, for all i ∈ [1, lj − 1], and " " # # j j j j (∗∗) uj (.) = −r βjlj σljj gh,l (x(.)) + β σ (x(.)) + g j lj −1 lj −1 h,lj −1 + ... −

βj1 σ1j

"

(78)

#

j (x(.)) gh,1

,

where βj1 , . . . , βjlj are nonnegative constants such that βj1 + . . . + βjlj ≤ 1. 5.2 Proof of Theorem 3 The proof of Theorem 3 is based on two lemmas, exactly as in the argument of [11], [7] and [9]. More precisely, Lemma 8 below is Lemma 3.1 of [11] and Lemma 9, which is actually the main technical point, in the nonzero delay version of Lemma 4.1 of [11]. In order to facilitate the analysis of the stabilizability properties by bounded feedback of (S)rh , a linear transformation is carried out in [11]. Lemma 8. (cf. [11]) Consider an n-dimensional linear single-input system (S1 )rh : x(k + 1) = Ax(k) + bu(k − h).

446


Suppose that (A, b) is a controllable pair and all eigenvalues of A have magnitude 1. (i) If λ = 1 or λ = −1 is an eigenvalue of A, then there is a linear change of coordinates T x = (y1 , . . . , yn )T = (¯ y T , yn−1 , yn )T of Rn which transforms (S1 )rh into the form ⎧ ⎪ ⎪ ⎨ y¯(k + 1) = A1 y¯(k) + b1 (yn (k) + u(k − h)) , ⎪ ⎪ ⎩ y (k + 1) = λ (y (k) + u(k − h)) , n n

(79)

where the pair (A1 , b1 ) is controllable and yn is a scalar variable. (ii) If A has an eigenvalue of the form α+iβ, with β = 0, then there is a linear change y T , yn−1 , yn )T of Rn that puts (S1 )rh in the of coordinates T x = (y1 , . . . , yn )T = (¯ form

⎧ ⎪ y¯(k + 1) = A1 y¯(k) + b1 (yn (k) + u(k − h)) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

yn−1 (k + 1) = αyn−1 (k) − β (yn (k) + u(k − h)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y (k + 1) = βy (k) + α (y (k) + u(k − h)) , n

n−1

(80)

n

where the pair (A1 , b1 ) is controllable and yn−1 , yn are scalar variables. The following lemma is the key technical point of the proof. Though its conclusion are similar to Lemma 4 in [9], the proof that we provide is quite different and not totally obvious. Lemma 9. Let a, b be two real constant such that a2 + b2 = 1 and b = 0. Let ej = (ej (0), ej (1), ej (2), . . .), j = 1, 2, be two element of l1 . Then, there exist a constant C0 > 0, a real m∞ > 0 and, for every h > 0 an r∗ (h) ∈ (0, 1] and an 2 × 1 matrix Fh such that, for all r ∈ (0, r∗ (h)], if x = (x1 , x2 )T : R≥0 → R2 is any solution of the control system


447

⎧ ! T ⎪ x (k + 1) = ax (k) − bx (k) + rb[σ F x(k − h) + u(k − h) + 1 1 2 ⎪ h ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + v(k − h)] + re1 (k), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ! r (S2 )h : x2 (k + 1) = bx1 (k) + ax2 (k) − ra[σ FhT x(k − h) + u(k − h) + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + v(k − h)] + re2 (k), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T ¯ on [−h, 0], x0 = ((x1 )0 , (x2 )0 ) = 0, (81) with ¯0 the zero function in Ch , and u, v ∈ l∞ ([−h, ∞), R), with v l∞ ≤ C0 verifies: (i) There exists a finite constant m∞ > 0 independent of r, such that lim sup x(k) ≤ m∞ ( u l∞ + v l∞ + e l∞ ).

(82)

k−→∞

where e = (e1 , e2 )T . (ii) In the absence of u, v and e, the equilibrium (x, y) = (0, 0) is globally asymptotically stable. Remark 4. We will in fact actually obtain the following stronger ISS-like property (see [6] and references there): " # lim sup x(k) ≤ θ∞ ( Ψ h ) + m∞ u L∞ + v L∞ + e L∞ , k−→∞

where Ψ is the initial condition for x and θ∞ is a class-K function (i.e. θ∞ : R≥0 → R≥0 is continuous, strictly increasing and satisfies θ∞ (0) = 0.) Proof of lemma 9. The proof is an immediate consequence of Theorem 2 with an explicit for all The feedback law FhT = B T J h+1 , where $ expression % $ parameters. % a −b −b J= and B = . The Hurwitz matrix Jr is given by b a a ⎛ ⎜ Jr = J − rBB T J = ⎜ ⎝

a −(1 − r)b b (1 − r)a

⎞ ⎟ ⎟, ⎠

448


and the unique symmetric positive-definite solution of the Lyapunov equation JrT Pr Jr − Pr = −Id2 , is ⎛ ⎜ Pr = ⎜ ⎝

2b2 +a2 r 2 b2 r(2−r)

1−r − ab 2−r

1−r − ab 2−r

1+(1−r)2 r(2−r)

⎞ ⎟ ⎟. ⎠

Corollary 1. For n = 1, 2, let J be an n×n matrix, equal to either 1 or −1 if n = 1, or of the form

⎛ ⎜ ⎜ ⎝

α −β β α

⎞ ⎟ ⎟, ⎠

in the case n = 2, with α2 +β2 = 1 and β = 0. Let b = 1 if n = 1, and b = (0, 1)T if n = 2. Then, there exist a constant v0 > 0, independent of r and, for every ε > 0 and h > 0, there is an r∗ (h) ∈ (0, 1], and an n × 1 matrix Fh , such that for any functions v : Z≥−h → R, with v l∞ ≤ v0 , and e : Z≥0 → Rn , e ∈ l1 , if ψ : R≥0 → Rn , is any solution of the system " # x(k + 1) = J x(k) − rσ FhT x(k − h) − ξv(k − h) b + rηv(k − h)b + re(k), defined for all k ≥ 0, where ξ + η = 1, ξη = 0, it follows that there exists T ≥ 0 such that, for r ∈ (0, r∗ (h)], lim sup |ψ(k)| ≤ ε,

∀t ≥ T.

k→∞

Proof of Corollary 1: When n = 2, the conclusion follows from lemma 9. Assume n = 1. Let V (x(k)) = x(k)2 . Let x(.) : [−h, +∞) −→ R be a solution of x(k + 1) = λ [x(k) − rσ (fh x(k − h) − ξv(k − h)) + rηv(k − h)] + re(k), with fh = λh , where λ = ±1. Then, if |e(k)| ≤ e0 , it follows that


449

ΔV (x(k)) = x(k + 1)2 − x(k)2 ! 2 ≤ −2rx(k)σ λh x(k − h) − ξv(k − h) + r2 (K + ηv0 + e0 ) + + 2r|x(k)| (e0 + ηv0 ) ≤ −2rx(k)σ(x(k)) + 2r|x(k)||[σ(x(k)) − σ(λh x(k − h)− 2

− ξv(k − h))]| + r2 (K + ηv0 + e0 ) + 2r|x(k)| (e0 + ηv0 ) .

Since |σ (.)| = 1, we have ΔV (x(k)) ≤ −2rx(k)σ(x(k)) + 2r|x(k)| x(k) − λh x(k − h) + ξv(k − h) + 2

+ r2 (K + ηv0 + e0 ) + 2r|x(k)| (e0 + ηv0 ) ≤ −2rx(k)σ(x(k)) + 2r|x(k)| (rh(K + ηv0 + e0 ) + e0 + v0 ) + 2

+ r2 (K + ηv0 + e0 ) , according to h

x(k) = λ x(k−h)−r

k−1

5 ! 6 λk−j σ λh x(j −h)−ξv(j −h) −ηv(j −h)−λe(j) .

j=k−h

Pick ε > 0, choose constants v0 and r∗ (h) such that hr∗ (h) (K + ηv0 + e0 ) + v0 + e0 ≤

ε , 8

2

(K + ηv0 + e0 ) ≤

ε . 4

Then, if |x(t)| ≥ 4ε , it follows that ΔV (x(k)) ≤ −rx(t)σ(x(t)) + r2 Therefore, |x(k)| ≤

ε 2

ε ε ε ε ≤ −r σ( ) + r2 < 0. 4 4 4 4

for sufficiently large k. We conclude that lim sup x(k) < ε.

2

k−→∞

Proof of Theorem 3. Without loss of generality, making a change of coordinates if necessary, we may assume that the system (S)rh has the following partitioned form:

450


(S)rh :

⎧ ⎪ ⎪ ⎨ x1 (k + 1) = A1 x1 (k) + B1 u(k − h),

x1 (k) ∈ Rn1 ,

⎪ ⎪ ⎩ x (k + 1) = A x (k) + B u(k − h), 2 2 2 2

x2 ∈ Rn2 ,

where n1 + n2 = n, all the eigenvalues of A1 have magnitude 1 and all the eigenvalues of A2 have magnitude less than 1. Here, we take % $ % $ B1 A1 0 ,B = . A= 0 A2 B2 The controllability assumption on (A, B) implies that the pair (A1 , B1 ) is also controllable. Since A2 is asymptotically stable, it will be sufficient to show that if we find an IICS-stabilizing feedback of type (*) or type (**) for the system x1 (k + 1) = A1 x1 (k) + B1 u(k − h) then the same feedback will stabilize (S)rh as well because the second equation, x2 (k + 1) = A2 x2 (k) + B2 u(k − h), can be seen as an asymptotically stable linear system forced by a function that converges to zero. Thus, in order to stabilize (S)rh , it is enough to stabilize the subsystem x1 (k + 1) = A1 x1 (k) + B1 u(k − h). Without loss of generality, in our proof of Theorem we will suppose that (S)rh is already in this form, that is, we assume that all the eigenvalues of A have magnitude 1 and that the pair (A, B) is controllable. i) The single-input case: We start with the single-input case and prove the theorem by induction on the dimension of the system. For dimension zero there is nothing to prove. Consider now a single-input n−dimensional system, n ≥ 1, and suppose that Theorem 3 has been established for all single-input systems of dimensions less than or equal to n − 1. Write N = N (A), and pick any ε > 0. If 1 or −1 is an eigenvalue of A, we apply the first part of Lemma 1 and rewrite our system in the form ⎧ ⎪ ⎪ ⎨ y¯(k + 1) = A1 y¯(k) + (yn (k) + u(k − h)) b1 , ⎪ ⎪ ⎩ y (k + 1) = λ (y (k) + u(k − h)) , n n

(83)

where y¯ = (y1 , . . . , yn−1 )T . In the case when neither 1 nor −1 is an eigenvalue of A which has a pair of eigenvalues of the form α + iβ, with β = 0. So, we apply the second part of Lemma 1. Then we take


⎧ ⎪ y¯(k + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

451

= A1 y¯(k) + (yn (k) + u(k − h)) b1 ,

yn−1 (k + 1) = αyn−1 (k) − β (yn (k) + u(k − h)) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y (k + 1) = βy n n−1 (k) + α (yn (k) + u(k − h)) ,

(84)

where y¯ = (y1 , . . . , yn−2 )T . So, in either case, we can rewrite our system in the ⎧ ⎪ ⎪ ⎨ y¯(k + 1) = A1 y¯(k) + (yn (k) + u(k − h)) b1 ,

form

(85)

⎪ ⎪ ⎩ y˜(k + 1) = J (˜ y (k) + u(k − h)b0 ) ,

where J is as in Corollary 1, (J, b0 ) is a controllable pair, with y˜ = yn , b0 = 1 in the first case and y˜ = (yn−1 , yn )T , b0 = (0, 1)T in the second case. To consider the problem of integrable-input converging-state (IICS), we must study solutions of the following system: ⎧ ⎪ e(k), ⎪ ⎨ y¯(k + 1) = A1 y¯(k) + (yn (k) + u(k − h)) b1 + r¯ (86)

⎪ ⎪ ⎩ y˜(k + 1) = J (˜ e(k), y (k) + u(k − h)b0 ) + r˜

where e˜, e¯ are arbitrary elements of l1 , bounded by e˜0 and e¯0 , and have the same dimensions as y¯, y˜, respectively. We will design a feedback of the form # " T ˜ u(k − h) = r σN Fh y˜(k − h) − ξv(k − h) − ηv(k − h) ,

F˜hT = bT0 J h+1 ,

where ξ and η are constants such that ξη = 0, ξ + η = 1, and v is to be chosen later. From Corollary 1, there exists r1∗ (h) ∈ (0, 1] and for σ = σN , we may pick 0 < v0
0 there exists r2∗ (h) ∈ (0, r1∗ (h)], such that lim sup e¯(k) < ε, ∀r ∈ (0, r2∗ (h)]. k→∞

Note that (A1 , b1 ) is controllable and all eigenvalues of A1 have magnitude 1. Applying the inductive hypothesis to the single-input system of dimension less than or equal to n − 1, y¯(k + 1) = A1 y¯(k) − rv(k − h)b1 + e¯(k), we know that there is a feedback v(k − h) = u ¯(¯ y (k − h)) having (∗) or (∗∗) form (cases ξ = 1, η = 0, and ξ = 0, η = 1, respectively), such that all the trajectories of y¯(k + 1) = A1 y¯(k) − rv(k − h)b1 go to zero as k → ∞, for every vector function e¯ ∈ l1 . The proof for the single-input case is completed. 2 ii) The general case: Next, we deal with the general case of m ≥ 1 inputs and prove Theorem 3 by induction on m. First, we know from the proof above that the theorem is true if m = 1. Assume that Theorem 3 has been established for all κ−input systems, for all κ ≤ m − 1, m ≥ 2, and consider an m−input system (S)rh : x(k + 1) = Ax(k) + Bu(k − h) + e(k) where e(k) ∈ l1 is any decaying vector function. Assume without loss of generality that the first column b1 of B is nonzero and consider the Kalman controllability decomposition of the system (S 1 )rh : x(k + 1) = Ax(k) + b1 u(k − h) (see [5], Lemma 3.3.3). We conclude that, after a change of −1 coordinates y = Syx x, (S 1 )rh has the form ⎧ ⎪ ⎪ y1 (k + 1) = A1 y1 (k) + A2 y2 (k) + ¯b1 u(k − h), ⎨

⎪ ⎪ ⎩ y (k + 1) = A y (k), 2 3 2 Where (A1 , ¯b1 ) is a controllable pair. In these coordinates, (S)rh has the form ⎧ ¯ ⎪ ¯ ¯(k − h) + e1 (k), ⎪ ⎨ y1 (k + 1) = A1 y1 (k) + A2 y2 (k) + b1 u1 (k − h) + B1 u ⎪ ⎪ ⎩ y (k + 1) = A y (k) + B ¯2 u ¯(k − h) + e2 (k), 2 3 2

(88)


453

¯1 , B ¯2 are matrices of appropriate dimensions. So it where u ¯ = (u2 , . . . , um )T and B suffices to show the conclusion for (88). Let n1 , n2 denote the dimensions of y1 , y2 respectively. Recall that N = N (A). Let σ = (σ1 , · · · , σN ) be any finite sequence of saturation functions. Then, for the single-input controllable system y1 (k + 1) = A1 y1 (k) + ¯b1 u1 (k − h), there is a feedback u1 (k − h) = v1 (y1 (k − h)), such that (i)

(89)

v1 is N1 = N (A1 )(∗)−form (or N1 = N (A1 )(∗∗)−form); (ii) the

resulting closed-loop system is IICS. Since (88) is controllable, we conclude that the (m−1)−input subsystem y2 (k+1) = ¯2 u A3 y2 (k) + B ¯(k − h) is controllable as well. By the inductive hypothesis, this subsystem can be stabilized by a feedback T

u ¯(k − h) = v¯2 (y2 (k − h)) = [v2 (y2 (k − h)), . . . , vm (y2 (k − h))] ,

(90)

such that (i) there exists an (m − 1)−tuple k¯ = (N2 , . . . , Nm ) of nonnegative in¯ = N − N1 , such that v¯2 is |k|(∗)−form ¯ ¯ tegers and |k| (or |k|(∗∗)−form); (ii) the resulting closed-loop system is IICS. ¯1 [v2 (y2 (k − h)), . . . , vm (y2 (k − h))]T + e1 (k) still converges Since A2 y2 (k) + B to zero, we conclude that the coordinate y1 in equation (88), with u1 , u ¯ given by (89) and (90), converges to zero. Therefore, the feedback given in (89) and (90) globally stabilizes the system x(k + 1) = Ax(k) + Bu(k − h) and the resulting closed-loop system is IICS.

2

6 Output Feedback Design This section deals with the problem of dynamic output feedback stabilization via linear feedbacks of detectable and stabilizable systems subject to time-delayed and saturated input. Our main result is given in the following theorem. Theorem 4. Consider a linear discrete-time delay input system of the form x(k + 1) = Ax(k) + Bu(k − h) with an Rp −valued output y(k) = Ex(k), p ∈ N, such that all the eigenvalues of A are located inside or on the unit circle, (A, B) is

454


stabilizable and (A, E) is detectable. Then, for any observer system x ˆ(k) = Aˆ x(k)+ Bu(k − h) + L(y(k) − E x ˆ(k)), (L is any matrix such that all the eigenvalues of A − LE are strictly inside the unit circle) and for any bounded IICS-stabilizing feedback u(k) := u ¯(ˆ x(k)) is as given in Theorem 3, the composite system x(k+1) = Ax(k) + B u ¯(ˆ x(k − h)), x ˆ(k) = Aˆ x(k) + B u ¯(ˆ x(k − h)) + L(Ex(k) − E x ˆ(k)) has the origin as a globally asymptotically stable equilibrium. Proof: Set e(k) = x ˆ(k) − x(k). Then, the error e satisfies e(k + 1) = (A − LE)e(k) and e(k) → 0 as k → ∞, since the matrix A − LE is Hurwitz. The equation for x becomes

" # x(k + 1) = Ax(k) + B u ¯ x(k − h) + e(k − h) .

(91)

We know that the trajectories of x(k + 1) = Ax(k) + B u ¯(x(k − h)) go to zero, the problem is whether the same is true for (91). To do this for the feedback defined in Theorem 3, we recall that this feedback is globally Lipschitz. Therefore, we can rewrite (91) as x(k + 1) = Ax(k) + B u ¯ (x(k − h)) + e˜(k − h),

(92)

where e˜(k − h) = B [¯ u (x(k − h) + e(k − h)) − u ¯(x(k − h))] is bounded by a constant multiple of e(k − h). Thus e˜(k − h) → 0 as t → ∞. Then Theorem 3 implies that all solutions of (91) converge to zero which completes our proof.

2

7 Conclusions In this chapter we have dealt with the finite-gain lp −stabilizability, p ∈ [1, ∞] and global asymptotic stabilization of linear discrete systems subject to input saturation and time-delayed. We have shown that these systems under the stabilizability, neutrally stability assumption and the transaction matrix has spectral radius inside or on the unit circle are globally asymptotically stable and finite-gain lp −stabilizable. Moreover, the corresponding lp −gain is delay-independent. The explicit construction of such control laws was also given.


455

References 1. X. Bao, Z. Lin and E. Sontag (2000) “Finite gain stabilization of discrete-time linear systems subject to actuator saturation”, Automatica Vol 36 pp. 269–277. 2. Y. Chitour and Z. Lin (December 2003) “Finite gain lp stabilization of discrete-time linear systems subject to actuator saturation: the case p = 1”, Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA. 3. J. Choi (1999) “On the stabilization of linear discrete-time systems subject to input saturation“, Systems and Control Letters, Vol 36, pp. 241-244. 4. W. Liu, Y. Chitour and E.D. Sontag (1996) “On finite gain stabilizability of linear systems subject to input saturation”, Siam J. Control Opt., Vol 34, NO 4, pp. 1190-1219. 5. E.D. Sontag “Mathematical control theory: Deterministic Finite Dimensional systems” Springer-Verlag, New York. 6. E. D. Sontag (1998) “Comments on integral variants of ISS”, Systems and Control Letters 43, 93-100. 7. H.J. Sussmann, E.D. Sontag and Y. Yang (1994) “A general result on the stabilization of linear systems using bounded controls”, IEEE Trans. Automat. Contr. Vol. 39 pp. 24112425. 8. A.J. Van Der Schaft (1996) “L2 -gain and passivity techniques in nonlinear control”, Lecture Notes in Control and Inform. Springer, London. 9. K. Yakoubi and Y. Chitour, “Linear systems subject to input saturation and time delay: Global asymptotic stabilization”, submitted. 10. K. Yakoubi and Y. Chitour, “Linear systems subject to input saturation and time delay: Finite-gain Lp −stabilization”, to appear in Siam J. Control and Optimization.. 11. Y. Yang, E. D. Sontag and H. J. Sussmann (1997) “Global stabilization of Linear Discrete-Time Systems with Bounded Feedback”, Systems and Control Letters, Vol 30, pp. 273-281.

List of Contributors

Teodoro Alamo (Universidad de Sevilla, Spain) - Chapter 13 Frank Allgower (University of Stuttgart, Germany) - Chapter 7 Franco Blanchini (University of Udine, Italy) - Chapter 12 Raphael Cagienard (ETH, Zurich, Switzerland) - Chapter 8 Eduardo Camacho (Universidad de Sevilla, Spain) - Chapter 13 Yacine Chitour (Supelec, Gif-sur-Yvette, France) - Chapter 15 Dan Dai (University of California, USA) - Chapter 11 Ciprian Deliu (Eindhoven Univ. of Technology, The Netherlands) - Chapter 14 Haijun Fang (University of Virginia, USA) - Chapter 3 Rolf Findeisen (University of Stuttgart, Germany) - Chapter 7 Germain Garcia (LAAS-CNRS, Toulouse, France) - Chapter 6 Adolf H. Glattfelder (ETH, Zurich, Switzerland) - Chapter 4 João Manoel Gomes da Silva Jr. (UFRGS, Porto Alegre, Brazil) - Chapter 13 Paul J. Goulart (Cambridge University, UK) - Chapter 9 Peter Hippe (Universitaet Erlangen-Nuernberg, Germany) - Chapter 2 Guido Herrmann (University of Leicester, UK) - Chapter 5 Tingshu Hu (University of Massachusetts, Lowell, USA) - Chapter 11 David Humphreys (General Atomics, USA) - Chapter 1 Eric C. Kerrigan (Imperial College London, UK) - Chapter 9 Miroslav Krstic (University of California, San Diego, USA) - Chapter 1 Constantino M. Lagoa (The Pennsylvania State University, USA) - Chapter 10 Daniel Limon (Universidad de Sevilla, Spain) - Chapter 13 X. Li (The Pennsylvania State University, USA) - Chapter 10

458

List of Contributors

Zongli Lin (University of Virginia, USA) - Chapter 3 Jan M. Maciejowski (Cambridge University, UK) - Chapter 9 Urban Mäder (ETH, Zurich Switzerland) - Chapter 8 Stefano Miani (University of Udine, Italy) - Chapter 12 Manfred Morari (ETH, Zurich Switzerland) - Chapter 8 Ian Postlethwaite (University of Leicester, UK) - Chapter 5 Isabelle Queinnec (LAAS-CNRS, Toulouse, France) - Chapter 6 Tobias Raff (University of Stuttgart, Germany) - Chapter 7 Walter Schaufelberger (ETH, Zurich, Switzerland) - Chapter 4 Ali Saberi (Washington State University, USA) - Chapter 14 Peddapullaiah Sannuti (Rutgers University, USA) - Chapter 14 Carlo Savorgnan (University of Udine, Italy) - Chapter 12 Eugenio Schuster (Lehigh University, USA) - Chapter 1 Anton A. Stoorvogel (Eindhoven Univ. of Technology, and Delft Univ. of Technology, The Netherlands) - Chapters 10, 14 Mario Sznaier (The Pennsylvania State University, USA) - Chapter 10 Sophie Tarbouriech (LAAS-CNRS, Toulouse, France) - Chapter 6 Andrew R. Teel (University of California, USA) - Chapter 11 Matthew C. Turner (University of Leicester, UK) - Chapter 5 Michael Walker (General Atomics, USA) - Chapter 1 Karim Yacoubi (Supelec, Gif-sur-Yvette, France) - Chapter 15 Luca Zaccarian (Universita’ di Roma, Italy) - Chapter 11

Index

amplitude and rate restriction ersatz model

37

37

amplitude and rate saturation 52

antiwindup

52, 335

38

simulation model example

inverted pendulum

Kalman-Yakubovich lemma linear programming L1 -norm

92, 101, 119

augmented system

277, 278

42, 47, 51

Lyapunov function, PWQ

41, 48, 92, 95, 96, 117

coincidence point

275

control invariant set controller windup cost function

Min-Max-selector

331 103

model predictive control

240 31

MPC

273

multi-parametric programming optimal reachability set

316

optimal regional L2 gain

detectability condition

320

directionality problem

48, 52

disturbance model disturbance rejection

239

326

output feedback construction override

321

92, 101, 103–106, 114–116, 119

327, 334

plant windup

31, 37, 41

pole assignment

94, 109, 111

polyhedral regions 332

280

Popov criterion

324

111

prediction horizon

feasibility conditions

323

44

279

F8 aircraft dynamics feasibility

273

276

deadzone loop

dual-mode

242

39 mass-spring system

circle criterion

50

274, 279

319 quadratic programming

integral action

95, 104, 109, 111

internal model

274

receding horizon

276

276, 278

460

Index

reference shaping filter reference tracking

330, 334

reference trajectory regional L2 gain

saturation

state observer

317

regional reachability

317

317

343, 345, 346

terminal constraint terminal set

344, 346, 347

279

279, 281

357 windup prevention

360

spring-cart-pendulum system 249

344, 346, 347

switched gain scheduled

114, 116 initial input value

43

gain scheduled robust

280

slew–rate constraints

320

synthesis

91, 95, 96, 99, 100, 103, 107,

stabilisable set

278

stabilizability condition

274

regional exponential stability Riccati equation

stability

45

327

stable and unstable systems stable systems

41

40

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