Control of Uncertain Systems with Bounded Inputs (Lecture Notes in Control and Information Sciences)

Lecture Notes in Control and Information Sciences Editor: M. Thoma

227

SophieTarbouriechandGermainGarcia(Eds)

Control of Uncertain Systemswith

Bounded Inputs

~ Springer

Series Advisory Board A. Bensoussan • M.J. Grimble • P. Kokotovic • tt. Kwakernaak J.L. Massey • Y.Z. Tsypkin Editors Sophie Tarbouriech Germain Garcia LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 Toulouse Cedex 4, France

ISBN 3-540-76183-7 Springer-Verlag Berlin Heidelberg New York British Library Cataloguing in Publication Data Control of uncertain systems with bounded inputs, (Lecture notes in control and information sciences ; 227) 1.Feedback control systems 2.Uncertainty (Information theory) LTarbouriech, Sophie II.Garcia, Germain ISBN 3540761837 Library of Congress Cataloging-in-Publication Data Control of uncertain systems with bounded inputs / Sophie Tarbouriech and Germain Garcia, eds. p. cm. - - (Lecture notes in control and information sciences ; 227) ISBN3-540-76183-7 (pbk. : alk. paper) 1. Real-time control. 2. Systems analysis. I. Tarbouriech, Sophie. II. Garcia, Germain. ItL Series. TJ217,7.C66 1997 97-14914 629.8'312- -de21 CIP Apart from any fair dealing for the purposes of research or private stud),, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms oflicences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. © Springer-Verlag London Limited 1997 Printed in Great Britain The use of 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 laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Typesetting: Camera ready by editors Printed and bound at the Athenaeum Press Ltd, Gateshead 6913830-543210 Printed on acid-free paper

Preface

In a practical control problem, many constraints have to be handled in order to design controllers which operate in a real environment. The first step in a control problem is to find an appropriate model for the system. It is wellknown that this step, if successfully applied (that is, if the model gives an accurate representation of physical phenomena) leads usually to a satisfactory control law design, satisfactory meaning that the observed behavior of the real controlled system is conform with the desired results. A model can be derived in several ways. The most direct approach consists in applying general physical laws, decomposing the modelling problem into subproblems, solving each of them and by some more or less simple manipulations, deriving a complete model. Obviously, this method is based on some strong a priori knowledges and then some approximations are usually considered. For some systems, the previous approach is difficult to implement because the application of physical laws is practically impossible or simply because only a partial knowledge on the system is available. In this case, the system is considered as a black box and a model is elaborated from experimental data (identification). Some crucial choices have to be done in order to derive a satisfactory model, these choices concerning essentially the input, the model order and model structure. It is also possible to combine the two previous methods. An a priori knowledge on the system is then combined with identification tools. The model structure results from the a priori knowledge while the model parameters are obtained by an identification method. In conclusion, to obtain a model using one of the above approaches, it is often necessary to approximate or neglect some phenomena, or to choose some key parameters. A direct consequence is that the derived model is affected by some uncertainties. To find a control operating in a real environment, uncertainties have to be appropriately described and their effects considered in the control law design (Robust Control). Some potential results on robust control have been widely developed these last decades and although, some intensive works continue to be developed, this domain has attained a certain maturity degree.

VI Concerning the control, practically the control is bounded and saturations can occur, these problems being the consequence of actuators limitations. It is also important to include them in the control law design (Constrained Control). A large amount of works has been done in this way and severM approaches were developed in the literature. It seems to be fundamental to combine the results obtained in these two fields in order to derive some methodologies with practical interest and therefore to design some controllers capable of achieving acceptable performances under uncertainty or disturbance and design constraints. For two or three years, a significative effort is done in this sense. This book entitled "Control of Uncertain Systems with Bounded Inputs" is aimed to give a good sample, not exhaustive of course, of what it was done up to now in the field of robust and constrained control design. The idea is to propose a collection of papers in which some fundamental ideas and concepts are proposed, each paper constituting a chapter of the book. The book is organized as follows. C h a p t e r 1. F e e d b a c k C o n t r o l of C o n s t r a i n e d D i s c r e t e - T i m e Systems by E. De Santis In this chapter, the author considers a linear discrete-time system with exogenous disturbances and with bounded inputs and states. The problem of controlling such a system is addressed. Conditions are proposed for the existence of state feedback controllers that achieve a level of performance with respect to some given criteria. An additional requirement is that the undisturbed system is asymptotically stable. If the problem has solution, the subclass of controllers that achieve the highest level of robustness, with respect to given parametric uncertainties in the system matrices, is determined. C h a p t e r 2. £:2-Disturbance A t t e n u a t i o n for L i n e a r S y s t e m s w i t h B o u n d e d C o n t r o l s : a n ARF_~-Based A p p r o a c h by R. Su£rez, J. Alvarez-Ram~rez, M. Sznaier, C. Ibarra-Valdez Linear continuous-time systems with additive disturbance and bounded controls are considered. A technique for obtaining a bounded continuous feedback control function is proposed in order both to make globally stable the closed-loop system and to satisfy an £:2 to £:2 disturbance attenuation in a neighborhood of the origin. The solution is given in terms of solutions to an MgebrMc parametrized Riccati equation. The proposed control is then a linear-like feedback law with state-dependent gains.

VII C h a p t e r 3. Stability Analysis of Uncertain Systems with Saturation Constraints by A.N. Michel and L. Hou This chapter addressed new sufficient conditions for the global asymptotic stability of uncertain systems described by ordinary differential equations under saturation constraints. Systems operating on the unit hypercube in ~'* (where all states are subject to saturation constraints) and systems with partial state saturation constraints (where only some of the states are subject to constraints) are studied. These types of systems are widely used in several areas of applications, including control systems, signal processing, and artificial neural networks. The usefulness of the proposed results is shown by means of a specific example. C h a p t e r 4. M u l t i - O b j e c t i v e B o u n d e d C o n t r o l o f U n c e r t a i n Nonlinear Systems : an Inverted P e n d u l u m Example by S. Dussy and L. E1 Ghaoui Considering a nonlinear parameter-dependent system, an output feedback controller is sought in order for the closed-loop system to satisfy some specifications as stability, disturbance rejection, command input and output peak bounds. The controller state space matrices are allowed to depend on a set of measured parameters and/or states appearing in the nonlinearities. The specifications have to be robustly satisfied with respect to the remaining (unmeasured) parameters and/states appearing in the nonlinearities. Sufficient conditions are derived to ensure the existence of such a mixed gainscheduled/robust controller. These conditions are LMIs, associated with a set of nonconvex conditions. An efficient heuristic to solve them is proposed. The design method is illustrated with an inverted pendulum example. C h a p t e r 5. S t a b i l i z a t i o n of L i n e a r D i s c r e t e - T i m e S y s t e m s w i t h Saturating Controls and Norm-Bounded Time-Varying Uncertainty by S. Tarbouriech and G. Garcia Discrete-time systems with norm-bounded time-varying uncertainty and bounded control are considered. From the solution of a discrete Riccati equation a control gain and a set of safe initial conditions are derived. The asymptotic stability of the closed-loop system is then locally guaranteed for all admissible uncertainties. The connections between these results and the disturbance rejection problem are investigated. The class of perturbations which can be rejected in presence of saturating controls is characterized. The results are illustrated with the discretized model of the inverted pendulum. Furthermore, the approach by LMIS is discussed.

VIII C h a p t e r 6. N o n l i n e a r C o n t r o l l e r s for C o n s t r a i n e d Stabilization of Uncertain Dynamic Systems by F. Blanchini and S. Miani The problem of determining and implementing a state feedback stabilizing control law for linear continuous-time dynamic systems affected by timevarying memoryless uncertainties in the presence of state and control constraints is addressed. The properties of the polyhedral Lyapunov functions, i.e. Lyapunov functions whose level surfaces aretheir capability of providing an arbitrarily good approximation of the maximal set of attraction, which is the largest set of initial states which can be brought to the origin with a guaranteed convergence speed. First the basic theoretical background needed for the scope is recalled. Some recent results concerning the construction of the mentioned Lyapunov functions and the controller implementation are reported. Finally, the results of the practical implementation on a two-tank laboratory system of a linear variable-structure and a quantized control law proposed in literature is presented. C h a p t e r 7. Hoo O u t p u t F e e d b a c k C o n t r o l w i t h S t a t e C o n s t r a i n t s by A. Trofino, E.B. Castelan, A. Fischman In this chapter, a biconvex programming approach is presented for the design of output feedback controllers for discrete-time systems subject to state constraints and additive disturbances. The method proposed is based on necessary and sufficient conditions for the existence of stabilizing static output feedback controllers. Mixed frequency and time domain specifications for the closed-loop system like Hoo performance requirements and state constraints in the presence of disturbances are investigated. C h a p t e r 8. D y n a m i c O u t p u t F e e d b a c k C o m p e n s a t i o n for S y s t e m s with Input Saturation by F. Tyan and D.S. Bernstein This chapter deals with optimization techniques to synthesize feedback controllers that provide local or global stabilization along with suboptimal performance for systems with input saturation. The approach is based upon LQG-type fixed-structure techniques that yield both full and reduced-order, linear and nonlinear controller. The positive real lemma provides the basis for constructing nonlinear output feedback dynamic compensators. A major aspect of the presented approach is the guaranteed subset of the domain of attraction of the closed-loop system. The results are then illustrated with several numerical examples.

IX C h a p t e r 9. Q u a n t i f i e r E l i m i n a t i o n A p p r o a c h to F r e q u e n c y D o m a i n Design by P. Dorato, W. Yang, C. Abdallah Quantifier-elimination methods are proposed for the design of fixed structure compensators which guarantee robust frequency domain bounds. It is shown for example, that robust stability and robust frequency domain control-effort constraints, can be reduced to system of multivariable polynomial inequalities, with logic quantifiers on the frequency variable and plantparameter variables. Quantifier-elimination software can then be used to eliminate quantifier variables and to obtain quantifier-free formulas which define sets of admissible compensators parameters. C h a p t e r 10. Stabilizing F e e d b a c k Design for Linear S y s t e m s w i t h Rate Limited Actuators by Z. Lin, M. Pachter, S. Banda, Y. Shamash This chapter considers two design techniques recently developed for linear systems with position limited actuators : piecewise-linear LQ control and lowand-high gain feedback. These techniques are combined and applied to the design of a stabilizing feedback controller for linear systems with rate-limited actuators. An open-loop exponentially unstable F-16 class fighter aircraft is used to demonstrate the efficiency of the proposed control design method. The proposed combined design takes advantages of these techniques while avoiding their disadvantages.

We hope that this book would highly contribute to significative developments in the field of robust and constrained control, and would be a reference for future investigations in this field. Toulouse, April 8, 1997.

Sophie Tarbouriech L.A.A.S.-C.N.R.S.

Germain Garcia L.A.A.S.-C.N.R.S. I.N.S.A.T

Table of C o n t e n t s

List o f C o n t r i b u t o r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XIII

C h a p t e r 1. F e e d b a c k C o n t r o l of C o n s t r a i n e d D i s c r e t e - T i m e Systems ......................................................

1

C h a p t e r 2. L 2 - D i s t u r b a n c e A t t e n u a t i o n for L i n e a r S y s t e m s w i t h B o u n d e d C o n t r o l s : an A R E - B a s e d A p p r o a c h . . . . . . . . . . . . . 25 C h a p t e r 3. S t a b i l i t y A n a l y s i s of U n c e r t a i n S y s t e m s w i t h S a t u ration Constraints ........................................ 39 C h a p t e r 4. M u l t i - O b j e c t i v e B o u n d e d C o n t r o l o f U n c e r t a i n Nonl i n e a r S y s t e m s : an I n v e r t e d P e n d u l u m E x a m p l e . . . . . . . . . . 55 C h a p t e r 5. S t a b i l i z a t i o n of L i n e a r D i s c r e t e - T i m e S y s t e m s w i t h S a t u r a t i n g C o n t r o l s a n d N o r m - B o u n d e d T i m e - V a r y i n g Uncertainty .................................................. 75 C h a p t e r 6. N o n l i n e a r C o n t r o l l e r s for C o n s t r a i n e d S t a b i l i z a t i o n of Uncertain Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 C h a p t e r 7. Hoo O u t p u t F e e d b a c k C o n t r o l w i t h S t a t e C o n s t r a i n t s 119 C h a p t e r 8, D y n a m i c O u t p u t F e e d b a c k C o m p e n s a t i o n for Systems with Input Saturation ............................... 129 C h a p t e r 9. Q u a n t i f i e r E l i m i n a t i o n A p p r o a c h to F r e q u e n c y Dom a i n Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 C h a p t e r 10. Stabilizing F e e d b a c k Design for L i n e a r S y s t e m s with Rate Limited Actuators ............................. 173

List of Contributors

Siva B a n d a Flight Dynamics Dr. (WL/FGIC) Wright Laboratory Stony Brook, NY 11794-3600, USA.

Franco Blanehini Dipartimeto di Matematica e Informatica, Universita Degli studi di Udine, Via Zannon 6, 33100 Udine, Italy. E-mail : [email protected] Denis S. B e r n s t e i n and Feng Tyan Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI 48198, USA. E-mail : [email protected] Elena De Santis University of L'Aquila, Department of Electrical Engineering, 67040 Poggio di Roio (L'Aquila), Italy. E-mail : desantisQdsiaql.ing.univaq.it

P e t e r Dorato, Wei Yang and Chaouki Abdallah Department of Electrical and Computer Engineering, University of New Mexico, Albuquerque, NM 87131-1356, USA. E-mail : [email protected]

St~phane Dussy and Laurent E1 Ghaoui Laboratoire de Math~matiques Appliqu~es, Ecole Nationale Sup~rieure Techniques Avanqdes, 32 Boulevard Victor, 75739 Paris, France. E-mail : [email protected] Arfio Fischman Laboratoire d'Automatique de Grenoble (URA CNRS 228), ENSIEG, BP 46, 38402 St.-Matind'H~res, France. Zongll Lin Dept. of Applied Math. &5Stat. SUNY at Stony Brook Stony Brook, NY 11794-3600, USA. E-mail : [email protected] Stefano Miani Dipartimento di Elettronica e Informatica, Universith degli Studi di Padova, Via Gradenigo 6/a, 35131 Padova, Italy. E-mail : [email protected] Anthony N. Michel and Ling Hou Department of Electrical Engineering, University of Notre Dame,

XIV

List of Contributors

Notre Dame, IN 46556, USA. E-mail: ant [email protected] Meir Pachter Dept. of Elect. & Comp. Sci. Air Force Inst. of Tech. Wright-Patterson, AFB, OH 45433, USA. Yacov S h a m a s h College of Engr. & Applied Sci. SUNY at Stony Brook Stony Brook, NY 11794-2200, USA. R o d o l f o Su~rez, Jos~ AlvarezR a m i r e z , C. I b a r r a - V a l d e z Divisi6n de Ciencias B£sica e Ingenierfa, Universidad Aut6noma Metropolitana - Iztapalapa, Apdo Postal 55-534, 09000 Mdxieo D.F., M6xico. E-mail : [email protected] M a r i o Sznaier Department of Electrical Engineering, The Pennsylvania State University, University Park, PA 16802, USA. E-mail: [email protected] Sophie T a r b o u r i e c h a n d Germain Garcia LAAS-CNRS 7 avenue du colonel Roche, 31077 Toulouse cedex, France. E-mail: [email protected] A l e x a n d r e Trofino a n d E u g e n i o Castelan Laborat6rio de Controle e Microinform£tica (LCMI/EEL/UFSC) Universidade Federal de Santa Catarina PO 476, 88040-900,

Florian6polis (S.C.), Brazil. E-mail : [email protected]

Chapter 1. Feedback Control of Constrained D i s c r e t e - T i m e Systems* Elena De Santis Department of Electrical Engineering, University of L'Aquila, Monteluco di Roio, 67040 L'Aquila. FAX+39-862- 434403. E-mail: [email protected]

1.

Introduction

and

problem

statement

Let us consider the system x(t + 1) = Ax(t) + Bu(t) + D6(t)

(1.1)

where the vectors x(t) E R n , u(t) E R m and 6(t) e R ~ represent respectively the state, the input and the disturbance, and let us consider the sets Xo C R n, ~ C_ R n, U C R m and A C R d , w h e r e X 0 i s b o u n d e d , 0 E A , 0 E X 0 , 0 E ~ U , and X0 C ~. Let us define the following problem: PROBLEM (P1): Given a set H ' C R~_, where R~_ denotes the nonnegative orthant of the space R ~, find a control law and two parameters 7 and p E / / ¢ such that:

a) u(t) e p v vt b) vx0 e x0, x(t) e z, v~(t) e 4' vt > 0 c) if 6(t) = 0, t >_ f, the evolution starting from x(t-) tends to the origin with some rate of convergence, Vz(t-) E Xo. The theory of positive invariant sets plays a central role in our approach. Therefore we recall now the main definitions: D e f i n i t i o n 1.1. Given a set V and the constraint u E V, a set X with 0 E X is )t.contractive controllable (shortly)~ . c.c.) ifVx E X, 3u E V : A x + Bu E AX. If 3~=I, the set X is positively invariant controllable (shortly p.i.c.). D e f i n i t i o n 1.2. Given a set V and the constraint u E V, a set X is Apositive invariant controllable (shortly A- p.i.c) if Vx E X, 3u E U : A z + Bu + D A C X . Problem (P1) above can be formulated as a problem of existence of some set. To this aim we can state the following proposition:

° Work supported by MURST-40 and MURST-60

2

Elena De Santis

P r o p o s i t i o n 1.1. Problem (P1) has a solution if the following problem (P2) has a solution: Given the set H = 11' x {A :0 < A < 1}, find a triple (A, 7, p) E 1I, and a bounded set ~(A, 7, P) such that:

0 Xo c

p) c

ii)Vx E ~(A, 7,p) 3u E pU : A x + Bu E A~()L'Lp) and A ~ + B u + Dza C

p)

7

--

Given a triple (A, 7, P) E 11, denote by XaTp the family of all the sets ,U(A, 7, P) with properties i and ii defined in Proposition 1.1. L e m m a 1.1. The family X:~p, if nonempty, has a maximal element

P r o o f : It suffices to notice that if X1 E Xxw0 and X2 E Xxwo, the set X I U X 2 belongs to the same family X),wp. [] A number of subproblems can be formulated, relaxing or particularizing our requirements. We cite now only some main examples: i) if 27 = X0, A = {0} and p = 1 the problem reduces to the problem studied in [12]. ii) if U = R m, Xo = {0}, A = {5: 11511~ < 1}, 27 = {x : llxl[oo _< 7"}, where 7" is the minimum7 such that IIz(.)tloo _< "ztlS(-)lloo, = 0, Problem (P1) reduces to a classical I1 optimal control problem (see e.g. [3]). We recall that, as it was shown in [9], even in the case of full state feedback control, 11 optimal and near optimal linear controllers may be dynamic and of arbitrarily high order. In [14] a constructive algorithm is presented for the computation of near optimal nonlinear controllers which are static. Our approach goes in this direction, but while the above cited algorithm recursively increase the performance, until a value close to the optimal performance is obtained, our technique may be considered "one-shot", in a sense that will be clear later. Obviously, if U is some set in R m we have an input constrained 11 problem. iii) if X0 = {0}, U = R m, 7 = ~, A = A problem (P2) reduces to the problem studied in [10]. iv) if 27 = R ~, A = {0}, p = 1, X0 is a given but arbitrarily large set, the problem becomes a problem of constrained exponential stabilization as studied e.g. in [13]. v) if A = {0}, X = ~, p = 1 and X0 is to be determined, the problem is that of finding the largest set in 27, such that for each initial point in the set, the evolution exponentially converges to the origin, with assigned rate of convergence A, being the input constrained in U.

Control of Uncertain Systems with Bounded Inputs

3

vi) If A = {0}, and if, for a given A = A, it is required to minimize the parameter p in such a way that conditions i) and ii) in proposition 1.1 are fulfilled, the problem is that of locally stabilize the system, with prescribed rate of convergence, with minimum effort. The problem studied in [1] is a variant of problem (P1). In fact in that paper the problem was the following: for a given set X0 = ~U and a given A, find a set ZA and a control law, such that ~U~ E ,U, Vx E ~ 3 u 6 U : A x ÷ B u + D A C AZ~. In the same paper this problem was solved also in the case of matrices A and B dependent on unknown parameters, bounded in a known set. If we analyze Problem (P2), we can say that, in general, if the problem has solution, this solution is not unique. In fact we have a set • 6 H of feasible values (A, 7, P), i.e. of values such that a set ~(A, 7, P) with properties defined in the formulation of Problem (P2) exists. Therefore, our objective is not only to find a solution of P2, but also to optimize our choice with respect to some given criteria. Let us call P3 the class of these optimization problems. In this respect, the main characteristic of our approach is that, choosing a s e t / / a s large as possible, (which means setting initial requirements as loose as possible), we use a technique which allows to obtain a control law and values of the parameters in the set ~, starting from feasible values ()~0, ~/0, P0), optimizing with respect to some criteria. This in substance means that the performance of the controlled system is not a priori fixed, but the choice of a satisfactory level of the parameters A and/or -/ and/or p is case driven, taking also into account numerical limitation. As for the criteria to choose a solution in the set ~, the idea is that of associating to each parameter a priority: e.g., for a fixed p = fi, the problem is that of obtaining a satisfactory level of contractivity factor A and for that choice one wants to minimize the parameter % In this example we have associated to the parameter p the highest priority (it is the case in which hard constraints are to be considered for the input); we have associated to the parameter ~/the lowest priority (this means that the level of the disturbances in the system is not critical). In this case H = {)~, 7, P : 0 _ A < 1;0 < 7;0 p _< fi}. With this setting obviously it is well possible that the problem has not solution, but, as we'll see later in section 4, this in general does not imply a complete reinitialization of the algorithm. In contrast to our approach, notice that in [10] and in [1], respectively the values (A,~) and ~ are a priori fixed and, if the problem cannot be solved with those choices, new values of A and/or ~ are assumed, reinitializing the algorithm and iterating the procedure until feasible values are found. A similar remark applies also to [14].

4

Elena De Santis

In this paper we make the assumptions that X0, S and U have nonempty interior, the origin belongs to the interior of each of them and X0 and ,U are bounded. Notice that the assumption that U has interios implies no loss of generality. In fact, it this was not the c a s e , i t should be always possible to compute a matrix B and a set ~" such that BU - BU, a n d / ) has nonempty interior. The assumptions on X0 and 2~ are made to assure the stability of the closed loop system. Under these assumptions, a necessary condition under which Problem P2 has solution is that system (1.1) is stabilizable. This condition is equivalent to say that system (1.1), with u(t) E pU, Vt >_ O, for all positive values of parameter p, admits a bounded )~-c.c. convex set, with the origin in its interior, for some A, 0 < ), < 1. This implies that system (1.1), with u(t) E pU, Vt > 0, Vp > 0 admits a bounded A-c.c. polytope, with the origin in its interior, for all values of parameter A, such that A < A _< 1 (see [1], theorem

3.2). If ~U and U are convex polytopes, we are able to explicitly compute a triple (A, 7, P) E • and the set ~U*(A, 7, P). We'll see that ff ,U and U are polyhedra, in general the controls that solve a problem in the class (P3) are static nonlinear state feedback controls. In fact, given a problem, we'll show that u(t) is a solution, if and only if: Vu(t) 0, where V is a constant matrix, depending on the the matrices A and B of the dynamical system and on the coefficient matrices defining the polyhedra ~ and U. The vector f(x(t), X, 7,P) is the bound vector, which, for fixed values of the parameters A, 7 and p depends on the state vector at time t. We make use of a duality result, which will be recalled now for reader's convenience. Let G be a s x n matrix and let v be a vector in R n. Denote by 7~(G) the range of the matrix G and by P the nonnegative orthant of R 8, i.e., for simplicity, P = R~.. With a slight abuse of notations, in the sequel we'll denote the nonnegative orthant of any space with the same symbol P. The convex cone T~(G) ± n P is polyhedral and pointed. A set of vectors formed taking a nonzero vector from each of its extreme rays is called set of generators of the cone. At this point we can state the following T h e o r e m 1.1. [~] The convex polyhedron {x : Gx < v} is nonvoid in and

only if

Qv>_O where (rows of Q} - (generators of the cone : 7~(G)" n P}.

Control of Uncertain Systems with Bounded Inputs

5

Given some linear programming problem, it can be solved as a parameterized feasibility problem as follows: max ( f , x ) = m a x h : Q (

Yh ) > 0

(1.2)

where {rows of Q} =

of the

7~

-J

P

In [5] this conical approach to linear programming is extensively studied and moreover an implementation in MODULA-2 of an efficient algorithm for the computation of the generators of the cone T~(G) J- n P is given. We'll see in the sequel that the computation of these generators plays a central rote in our approach. The paper in organized as follows: in the section 3 we study some properties of a set in the family Xa7p and we describe a technique to compute the set S* (A, 7, P), given the parameters (),, 7, P). In section 4 we give the tools to solve problem (P3). A numerical example is also developed. Finally in section 5 we address the problem of evaluating the degree of robustness of a given solution, with respect of a given description of the uncertainties in the system matrix A.

2. P r e l i m i n a r y

results

In the following theorem we study how some properties of a set X E Xx-yp depend on properties of the set BU and of the set ,4. The set ~, as we have defined in the previous section, is the set of fesible parameters in H. The symbol C(X) denotes the convex hull of some set X T h e o r e m 2.1. Given a triple (A, 7, P) E • and a set X E Xx~p we have: i) q B Y and S are convex, C(X) ~ x ~ ii) if BU and S are convex and O-symmetric, C(X U ( - X ) ) E Xxnp. Proof: i) If X E Xx~p we can write: Vxl E X 3ul E pU : Axl + Bux E )tX and Axl + B u 1 + D ~-L E X, V~I E ,4 7 Vx2 E X 3u2 E pU : Ax2 + Bu2 E )~X and Ax~ + Bu2 + D 6-'~ E X, V$2 E ,4 7

6

Elena De Santis

therefore 3ul, u2 E BU such that :

A ( a z l + flx~) + w e he(X) and

A(a,~ + ~:~) + (o,B,,, + ,3B,,,) + D~,a e e(x) z l , x 2 E X , Va, f l : a + f l = l ,

a_>0, fl_>0

and hence, because X0 C C(X), C(X) C Z and z2 C C(A), it is proved that C(X) e X~p. ii) we have in this case that BC(U U (-U)) = BU and therefore:

Vxl E X 3 u l E pU : Axl + Bul + D ~-2-1E X V¢51 E A 7 AXl if" BUl E AX Vz2 E - X 3 u 2 E - p U : Az~ + Bu~ - D d-E=E XV6~ E z2 7 Az2 + Bu2 E - A X therefore

A(aazl + l?z2) + B(aul + flu2) + D c '(A U ( - A ) ) E C(X U ( - X ) ) 7 A(azl + ~x2) + B(o, ul + 5u2) ~ X zl ~ X, x2 E - X , Va,//: a + ~ = 1, a _> 0,/~ >_>0 Finally, because X0 C C(X U ( - X ) ) , C(X U ( - X ) ) C Z and A C C(A U (--A)), it follows that C(X U ( - X ) ) E X;~p.H A s s u m p t i o n 2.1. The sets X0, 22 and U are convex, have nonempty interior and the origin belongs to the interior of each of them. Moreover X0 and are bounded. We now state a preliminary result, to solve problems P2 and P3. Let be 220 the maximal p.i.c, set in ~, which, because of necessary condition of stabilizability for system (1.t), is a nonempty set with the origin in its interior. Necessarily we have X0 C 220, otherwise problem P2 has not solution. Let us define the following sequences of sets, depending on the parameters (),,v,p) E / / : &(A,7,p)

= 220 k = o

~k(A,'~,p) = Z:o k = 0 Dk (A, 7, P) = {x: 3u E pU : (Ax + Bu E A~k-1 (A, 7, P) and DA A~+B~+ c_ ~k_l(~, v, p))} k > 0

rk(X,v,p) = & ( ~ , 7 , p) n Z0 k > 0

(2,1)

Control of Uncertain Systems with Bounded Inputs

7

Remark 2.1. similar backward recursions have been used in literature, to solve problems in the context of constrained control and/or optimal control: see e.g. [1], [10], [14], but our technique to compute the sets in the recursion differs from the others used in the above papers because the dependence from the parameters )~, ~/and p is made explicit in the description of each set (see in this respect the papers [6] and [7]). This fact is of paramount importance in our approach, as you can see in the following section 4. L e m m a 2.1. If the sets ~ and B U are convex, each of the sets EkC)t,7,p) is convex, V()t, 7, P) E 1-1. If ~ and B U are moreover O-symmetric, the same is true for each of the sets ~k ()t, 7, P), V()~, 7, P) e II. P r o o f : The convexity of the sets ~k (A, 7, P) can be proved by means of the following chain of implications: ~ convex --+ E0 convex --+/21 (~, 7, P) convex -+ E1 convex --+ [22 convex and so on. If we apply statement (ii) of theorem 2.1, we have that E0 is 0-symmetric, which implies that [21 is 0-symmetric and so on. [] It is easy to prove that, if for some k and for some parameters CA,7, P) E ~,

Xo C Z~ (A, 7, P) C ~2~+1 ()~, 7, P) then the set ~ CA,7, P) belongs to the family X~.~p. Moreover if (A,7,p) E ~, the sequence {hY1,(~,%p), k = 0, 1,...} converges to the set 57" (A, 7, P), otherwise, if (~, %p) does not belong to the feasible parameter set ~, we have that, for some k, the set 2Yk(~, 7, P) becomes empty and/or X0 is not a subset of ~kC]k,7,p) . If ~U0 is a polytope and U is a polyhedron, we are able to explicitly describe the sets defined in (2.1), as we can see in the following theorem 2.2. In the sequel we'll show that it is possible to remove the above assumption on ~U0, because, in our framework, the fact that ~U is a polytope implies that G0 is a polytope, too (see remark 3.1). T h e o r e m 2.2. If ~o = {x : Gx < v} and V = {u : F u < w}, the sets ~k (~, 7, P) and f2~ ()t, 7, P), if nonempty, are polyhedra, for all values of ~, 7, P, and have the following expression:

= { x : O(k)


)d > A, 7 ~ > 7, P~ > P there emsts a k such that Sk (A, 7, P) E Xx,~,¢. P r o o f : The set 57* (A, 7, P) will be denoted in this proof by the symbol 57* for simplicity. Let us make the following positions: A' /~p = m a x p : Yx E I~S* 3u E p'U : A x + B u E AtJ57* DA / ~ = m a x p < Up : Vx E I~S* 3u E p'U : A z + Bu + " 7 " C_ 27" It is easy to see that #x > 1, /~p > 1 and /~ > 1. In fact px > 1 by definition of A',/Jp > 1 because if the set 27* is A- contractive controllable, with the constraint u(t) E pU Vt, being a = e: p > 1, we can write: Vx E 27* 3u E pU : a A x + a B u E AaE* which implies that Vx E aS* 3u E p'U : Ax + B u E AaS* Therefore/Jp > a and hence/~p > 1. Finally, because 57* C XxTp, a set 2~ E AS* exists, such that ~u E pU : Ax + Bu E

7 Therefore some A < 1 depending on 2: exists such that ~ + D 'a C A57". Taking some a = ~ > 1, a a necessarily we have P7 > 1. Let us make the position # = min{#~, pp, PT}. If ~* is a convex compact set with the origin in its interior, some k exists such that ,U* C ~Uk(A,7,p ) C_ fiZ*. The set fi,U* is A- contractive controllable with the constraint u(t) E plU, Vt, and hence

W e ~'k(~,7,p) 3u e / U : A z + Bu E #A,Y'* C_ #;~AZ* = A'X'* C_ A'Z'k(A,7,p ) Moreover Vx E #~* 3u E pU : A z + Bu + D ~ E ~*, which implies that

Vz E Zk(A,7,p) Su E pU : A: + B~ + D~ e E* C_Sk(~,'y,p) If X0 C Z:* (A, 7, P), X0 C ~Uk(A, 7, P) and therefore we can conclude that

~k(A,7,P) E Xx,,.y,,¢. [] We give now a simple lemma, which will be useful in the sequel. Let us consider two nonempty convex polyhedra P1 and P2 described respectively by the inequalities Gz _< v and F z s where, denoting by Mi the ith row vector of some matrix M, the ith component of the vector z(A) is: zi(A) =

max ,(G(k + 1))ix xE.~k (,x,"lo,po)

and the j t h component of the vector s is: .5 = max(O(k))

EXo

.

Because using the approach recalled in equation (1.2) we are able to explicitly express each of the maxima in above definition of z(A) as a function of the parameter A, which appears only in the bound vector of the inequality defining the set 27k(A,70,P0), the problem reduces to the minimization the value of the scalar parameter A, subject to a set of nonlinear inequalities, which defines an open segment, bounded below. The same technique applies also to the sequences {Tk} and {ilk}It is clear that the number of linear inequalities which describe the sets ~k(A, 7, P) and ~k (A, 7, P) increases as k increases, and also the numerical effort required to compute the sequences defined in (3.1) increases, step by step. Therefore it is important to have an efficient algorithm which detects and eliminates all redundant inequalities, step by step. This task is hard if the bound vector of the inequalities depends on a parameter, as in our case. Some research effort is in progress on these computational problems. At this point we can state the following T h e o r e m 3.2. For all

7' pl

>

7 p

7 p

E ~, for all

7' p'

E ~, such that

, the set ~* (A', 7', P') is nonempty and it is polyhedral.

P r o o f : The proof is constructive. In fact we show that an algorithm exists which converges to the set S*()¢,7',p' ) in a finite number of steps. The algorithm is the following: A l g o r i t h m 3.1. STEP 1: Initialize ,~0 = 1, 70 = 0% p0 = e% h = 0, k = 0. STEP 2: Compute S* (A0, 70, p0)- As shown in remark 3.1, this last set is polyhedral, and it can be computed in a finite number of steps, say h0. Set h=h+ho.

14

Elena De Santis

STEP 3: Compute Xl,, increasing k until Xk = ),', k = k. The sequence {Xk} converges to a value less or equal to )~, as k goes to ~ . In fact, if ~ = lim~_.~o{Xk} we have that, given some X, ,U*(X,c¢,~) is empty if X < X~ and it is nonempty if X > X~o. Moreover, because (~, 7, P) E 4, the set 57" (,~, 7, P) is nonempty, and it is a subset of S* (~, ~ , c¢). Therefore Xoo < ),. Because the sequence is nonincreasing (lemma 3.1), the above value k exists and it is finite. Set h = h + k , k = 0, ~0 = A'. 57"(~0, 70,P0) = 2Yr,(~o,7o,po). STEP 4: Compute 7k, increasing k until 7k = 7', k = k. The sequence {Tk} converges to a value less or equal to 7 j, and because the sequence is nonincreasing (lemma 3.1), the above value f~ exists and it is finite. Set h = h + kl k = 0, 70 = 7'.

=

p0).

STEP 5: Compute/Sk, increasing k until fik = p~,k = f~. The sequence {ilk} converges to a value less or equal to p~, and because the sequence is nonincreasing (lemma 3.1), the above value k exists and it is finite. Set h = h + k, po = V. 70, po) = 70, po). STEP 6: The required set ,U* ()~', 3/, pt) is equal to 57~ (~0,70, P0), which is nonempty, and, because it has been computed in h steps of the recursion (2.1), it is a polyhedron. STOP. [] Using the same symbols defined in theorem

2.2, we have:

7 E 4 a triple such that the set ~* ()t, 7, P) is P polyhedral. The set of all and nothing but the inputs that solve problem P2 with this choice of parameters is defined by: T h e o r e m 3.3.

Let be

( G(k)B ) ( ~#(x(t))f~(k)-G(k)Ax(t) ) G(k)B u(t) k,~a~. If fi~ = /5, set p0 = t3,h = h + fc,k = 0, compute Z~(A0,70,p0) and go to STEP 4, otherwise STOP. The value krna$ is the maximum admissible number of iterations, and it takes account of dimension of the problem and of computational limitations. STEP 4: Compute ~k, increasing k until a satisfactory level Ak = )¢, k = [¢, is obtained. Set h = h + f¢, k = 0, )~o = )~'. E* (Ao, 70, po) = ~ ( ~ o , 70, Po). STEP 5: Compute 7k, increasing k until a satisfactory level % = 7 r, k = k, is obtained. Set h = h +/% 3'0 = 3". S*(/k0,70,Po) = 27f,(,k0,7o, po). STEP 6: Given the polyhedral set 22* (Ao, 70, P0) = ~* (A', 7', at step 5, apply theorem 3.3, and find a control law. STOP

P), computed

With respect to the above algorithm we remark that only the value p is a priori known, because it represents hard constraints on the input variables. Notice moreover that, because the minimum attainable value of ~k at step

16

Elena De Santis

4 depends on ~, and because the minimum attainable value of 7k at step 5 depends on fi and on A~, obtained at step 4, it is clear that the order in which we perform the minimization of the parameters is relevant, with repsect to the emphasis we want give to each parameter. If the above Mgorithm fails, i.e. if it ends at STEP 3, two cases are possible: (1,c~,~) ~ • or the set ,U*(1,oo, t5) cannot be computed in a number of iterations, compatible with our computation capability. In both cases we have to reformulate a new problem, with a smaller set of initial states and/or with a larger value of ft. A reinitialization of the algorithm is not required: in fact, because we have computed the sequence {ilk, k = 0, 1,...kraal), we can assume one of the elements #~ of the sequence as a new feasible value of fi, and the set ~U~(1, oo,/5) can be immediately derived. Similarly, given some < kraal, we can compute a subset of X0, contained in ,U~(1,0% ~). To end this section, we give a simple numerical example, where we explicitly compute the sequence {Ak}: EXAMPLE 1: Let the set X0 be a neighboround of the origin, described by the inequalities:

(I o) 1I (11) (1) 0 -1

0

1 0

1 1

x_<e

-1

O<e_ 1 P~ = ~(r, 1)P~(1)~(v, 1), where ~5(v, 1) denotes the transition matrix of At(r). Observe that uniqueness of the solution to (2.4) implies Pr (1) = - c T c . Therefore, it follows that Pr is a negative definite matrix, as we wanted to prove. The problem in considering differential Riccati equations like (3.10) is that their solutions can have finite escape times. Therefore, more conditions are required to obtain global stability. In particular, the solution (3.10) does not have finite escape times in (1, oo) if B T B - ~ E E T > 0 ([2]) Remark 3.3. Note that u(x) defined by (3.5) never saturates. This is an important property because a design based on this approach avoids adverse control behavior.

E x a m p l e . Consider the following one-dimensional system ([36]) 5: = ax + bu + wl Z2=~

The corresponding Riceati equation (2.4) becomes c~p2 + 2ap + 1 = O,

where we introduce the 3"-dependent variable ol = (~- - b ~) = are, in general, two solutions:

l_b~,./~ There

%

-a

Pl = Pl(7) =

+ v ~ -

~

-a

;

P2 =P2(~/) --

-

x/a s -

(3.11)

we shall consider the three possible cases, depending on whether a = 0, a > 0 ora O,

• .., h~:.

< xi _< 1, i = i , . . . , n ~ , ,J

anjxj

(1.2)

j----1

j=l

and

h,,( ~ aijx~ -- ~ O' n

j=l

[ Ej=I

aijxj,

I x i l = l and otherwise.

(° Y]~j=laiJxJ ) x i > O

(1.3)

We assume that in system (1.1), the matrix A is known to belong to an interval matrix, i.e., A E [An, AM]. (An interval matrix [An, A M] with Am = [a~] E R n×n, A M = [aM]ER n×", and a~ < a,M for all i and j, is defined by

[AnAM ] :z~ {C - [cij]ER n×n : a,.m . < vii < a iM 1 < i,j j -j , _

• _

n}.)

In this paper, we also investigate the stability properties of systems with partial state saturation, described by differential equations of the form

( ~ = Allx+A12y ~1 h(A21x+A22y)

(1.4)

where n>_m, z E R (n-m), y E D m = { y = ( y l , . . . , y m ) T E R rn:-l_ O,

40

Anthony N. Michel and Ling Hou

n-m

m

(n-m

eljxj+ k=ld, y,),

h(A21x+A22y)= [hm(

j=l

~]T

m

k j =cl jxj+ k=l d .Y )l --

--

"~

(ks) and

h y~

clj xj + j=l

dik Y k=l

= {~

]yd:l and

(~j~:__~CijXj+~k~__ld,kyk)y,>O

;-~1 CijXj JCCk=im dikyk,

otherwise,

(1.6)

where A21 = [cij] • R rex(n-m) and A22 = [dij] E R mxm. We assume that in system (1.4), the matrix

A12]E[Am,AM]. [A21 A2~J

A ~ [All

2. Existence of Solutions Before proceding, it is necessary to consider questions of existence, uniqueness, and continuation of solutions of (1.1), rasp., (1.4). In the following we consider (1.1) without loss of generality. We will first show that for any x(t0) • D n, there is a local solution for (1.1). (A) From well-known results (e.g., [2]), we know that for any x(t0) in the interior of D '~, there exists 6~(to)> 0 such that eA(t-t°)x(to) is in the interior of D '~ for all t • [to, t0 + J~(to)). Therefore, the solution of equation (1.1) exists in this case and is unique and continuous on [to, to + 5~(to)). (B) For any x(to) e ODn, let ~r: {1,..-, n} -+ {1,..-, n} be a permutation such that for some l, k and m such that 0 < k < l < m < n, we have

xa(i)(to) --- E ; = I aa(i)jxj(to),

Ix 0 such that the right side of the above equation is negative for all x such that IIz - z(t0)ll < di. If z(t) with t E [t0,t0 + r ) , v > 0, and Ha(t) - a(t0)l[ < di is a solution of equation (1.1), then ~ is of opposite sign of hi(to). Hence, lzi(t)[ is strictly decreasing and it must be true that tai(t)l < 1 for all t E (to,to + 7-). Similarly, if ~d2xi(to) a itI t 0) > 0, then there exists d~ > 0 such that the right side of the above equation is positive for all x such that [Ix - x(t0)ll < d/. If a(t), t E [to, to + r), r > 0, and IIx(t) - a(t0)[[ < di is a solution of equation (1.1), then it must be true that xi(t) = z,(to) and ( ~ = i aijzj(t))x~(t) > 0 dx dX+ 1 . for all t E (to,t0 + I-). If -d~rxi(to) = 0 then consider dt---r-4:rZi(to) = J a i l , " ' , a i k ] ( A l , l ) ) ~ - t ( A l , l X l + A I , H a H ) , A = 2 , ' ' ' , l, and apply a similar a r dx gument as above. If ~-irai(to) = [ a i l , . . ' , aik](Az,i)~-2Al,zal + AZ,HaH) = 0 for all A = 1 , - . . , l , then ~ x i ( t o )

= 0 for all A and we let di = 1.

For i E { r e + l , . . . , n}, there exists di > 0 such that (~-'-'-~'=1 aa(1)jzj)zo(i)(to) > 0 for all z such that I[z - z(t0)ll < di. If z(t) with t E [t0,t0 + r), r > 0, and IIz(t) - z(t0)ll < di is a solution of equation (1.1), then it has to be true t h a t xi(t) = ai(t0) and ( ~ = 1 a~(oizJ(t))a~,(o(t) > 0, for all t E [to, t0 + r). We now determine the solution of (1.1). Without loss of generality, we assume that there exist r and s such that I < r < s < m and such that for iE{I+I,...,r}, there exists Ai > 1 such that ~ dt~ - - -- 0 when A 1 such that ~ dt ^ and

dXixi(to)

~xitto)

~. \

> O. Let

n

M=maz{I ~=~

-~- 0 when )~ < $i a~jzjl, i = 1 , . . . , n, z e D " } and

let d = min{di,i = 1 , . - . , n } . Let ~t = { x i , - - . , x ~ } T, ~ n = { x r + l , - " , n } T, = [aij]l 0. In the argument above, we have shown that in a neighborhood of x(to), h ~ ( ) - ~ = 1 aijxj) has to equal ~ = 1 aijxj n for i E { 1 , . . . , r } and h ~ ( ~ = l a i j z j ) has to be 0 for i E { s + l , - - - , n } . /i n For the same reasons, we have that h ~ ( ~ j = l aijz~) = ~-]~j=l a~jx~ for i E rt t { 1 , . - . , r } and h~ ()-~j= 1 ai~xi) = 0 for i E { s + l , . . - , n } . For i q { r + l , . . - , s } , h~, ( ~ ' = 1 a~jxj) = ~-,~=1 aijxj = 0 along x(t, to, x(to)). Then h

=

{ 0,

n

E j = l aij(xj - xj), Therefore,

o

if x i / E j _ 1 aijx~l = 0 rt

!

if z i ~ E j = l aijx}) = Y~.j=I aijxj.

Control of Uncertain Systems with Bounded Inputs

hx, for all i = 1 , . . . , n ,

aijx h - h~,

aijx

43

< K l l z ( t ) - x'(t)[l

where K = maz{la~jl: 1 < i , j < n} T h e n ,

IIx(t) - ~'(t)ll = llh(Ax(t)) - h(Az'(t))ll < nKIIz(t) - x'(t)ll. Integrating both sides of the above inequality from to to t o+ rain (r, d/M), and applying the Gronwall Inequality (see, e.g., [2], p.43), we conclude that x'(t) = x(t), for all t E [to, to + rain(r, d/M)), i.e., x(t) is the unique solution of equation (1.1). Combining (A) and (B) above, we conclude that there exists a unique

local solution of equation (1.1) for any initial point in D n. By continuation (see, e.g., [2]), we can extend the local solution to U~[ti,ti+l) = [to, oo) for any initial condition. On each [ti, ti+l), the solution is continuously differentiable (at ti, we mean the right derivative), so that h(Az(t)) is continuous in (ti, ti+l) and continuous from the right at ti.

3. S y s t e m s

Operating on a Closed Unit Hypercube

Before presenting the main results, we introduce some necessary notation. A square matrix is said to be positive definite if it is symmetric and if all its eigenvalues are positive. A matrix P = [Pij] E R '~×n is said to be diagonally dominant, if n

P,i >_ ~ j=l,

lPj'I, i = 1,...,n,

(3.1)

j#i

where 1" I denotes absolute value. Let /2 = {(I, J, I', J')

:

IU J = I '

U J'={1,2,...,n}

a n d / f q J = I fq Jl=O~. J For every (I,J,I',J')E[2

IJrJ,_r aij

-~

,

we let Azj J . J , -_ _t r a l Jijl I J ' l

aiM a~j

l ~~ nn n × n

(3.2)

, where

if(i,j) EI'xlUJ'xJ if ( i , j ) e I ' × J U J ' × I

with the convention that I' × I = 0 when I'=~ or when I = t L

(3.3)

44

Anthony N. Michel and Ling Hou

A point xe E R n is called an equilibrium of system (1.1) if h(Ax~)=0. In particular, xe = 0 is an equilibrium of system (1.1). For the various definitions of stability of an equilibrium in the sense of Lyapunov, and for stability results, refer, e.g., to [1]. In the present context, the equilibrium xe = 0 of system (1.1) is said to be globally asymptotically stable if x~ = 0 is stable and the domain of attraction of xe = 0 is all of D n, since system (1.1) is defined only on D n . The following lemma [2] is required in establishing our results. L e m m a 3.1. Assume that for matrix A = Ao there exists V: Dn --+ R satisfying the following assumptions:

[ ov (x~ • ov ]T ( A - l ) V is positive definite on D '~, ~-, ( x ) = IOn,, ,, " , o-~ (x)j exists for

[ all x E D n, and ! ~ggd(x)]I TAx is negative definite on D" (A-2) For all x E D ~ it is true that

Then the equilibrium x~ = 0 of system (1. i) with A = Ao is globally asymptotically stable. Proofi

In view o f (1.2) and (1.3), it follows readily that along the so-

lutions of (1.1), h(Ax(t)) is continuous from the right in the sense that

limt-~t,,t>t~ h ( A x ( t ) ) = h ( A x ( t @ for any tl. Therefore, along the solutions of system (1.1), the right-hand derivative of V(x(t)), given by FOV

T

x t )'

exists for Mat > to = o. Now by assumption (A-2), V(x(t))

aa(i)jxj(t /V=:

J=:,J#~

j=l,j#i

a~(ojxj(t

]~'=1

xo(i)(t) 21

for all m < i < n.

(3.12)

In the last two steps we have made use of (3.1) and (3.7) and the fact that rearrangement of Pjo(i), J = 1,..., n. It now follows that

Po(j)o(O,J = 1,..., n is a

n

x(t)Tp[Ax(t)-

h (Am(t))] =i=m~+t[(po(i)o(0xa(0

+~

vo(~).(,)x~(t

j=l,jT~i

,)e

ao(,)~x~(,)

," ,3.=1

)1

> o

(3.x3)

and therefore, in view of (3.11) and (3.13), we have that

V(x(t)) -0'

Tu

I

6 S(r) and Y such that

T

>-0'

(4.3)

A T p + PA+CTqSCq +C TCa + 2aP ,

PBp +CTSDqp +C T D~e D~'pSDqp - S + DTzpS Dzp

PBw +CTSDqw

*

*

T DqwSDqw -- 72I

AQ+QAT+B~y+yTB. + B p T B T + 7 --2 B ~ B ~T + 2aQ

* ....

*

T Dqp S Dqw

T QC4T + Y T Dqu

Af < O,

(4.4) T QCT~ + Y T m..

T + 7 --2B~Dq~ T +BpTDqp

BpTDT~

DqpTDqT - T --2 T +7 DqwDqw * < O,

DqpTD~p DzpTDzT - I

(4.5) Su Tu = Ivu.

(4.6)

Control of Uncertain Systems with Bounded Inputs

63

4.3 C o m m a n d i n p u t b o u n d T h e o r e m 4.2. A sufficient condition to have llu(t)tt2 < Urea×for everyt > 0 is that the matrices P, Q, Y as defined in § 4.2 and variables )% p > 0 also satisfy x0 6 £p,~, (4.7) 2 2 Ft()~ -}-/2"/ Wmax) "- 1, (4.8)

yT 0

Q I

I P

> O.

(4.9)

P r o o f : We defined in § 4.2 that P and Q are the upper-left n × n blocks of/5 and/5-1. Precisely,/5 and Q are parameterized with arbitrary matrices M and L as follows (see [12] for more details)

P=

Mr

~

,

O=

LT

-~

,

where P and Q are such that C~-I and C~-, are respectivelythe projection of the ellipsoids Cp and C~ on the subspace of the controllerstate. Then, the matrix Y definedin § 4.2 also depends on the the parameterizationmatrices via Y = -CuLT. Let assume that the above conditions (4.7), (4.8) and (4.9) hold. Then, using Schur complement[5], (4.9) is equivalent to P > 0 and

yT

Q

-1

>_ O.

Then, with Y =-CuL T, we readily obtain P > 0 and

~?~max ---=T Cu L-I(Q

:;

_t)L_T

]

> O.

With-~ = L T (Q - P - i ) - i L (whichstems from the matrix equality/5(~ = I), we may infer the followingequivalent conditions P>0and

[01 Cu

#U2max > 0 .

Using Schur complements, we obtain p>o

T---1 andV~ER,~ ~ Q

--~T--

~_gT

~>0. ]ZUmax

The assumption x0 6 Sp,~ implies that ~ 6 £p,~. Therefore, the state of the controller will never escape the ellipsoid obtained by projecting the statespace of the augmented system on the subspaze of the controller space. In other words, that means that ~ 6 E~-,,~. Then, with u = C~Z, we can write

64

St~phane Dussy and Laurent E1 Ghaoui Vx0 ~

gp,~,

uTu < u~n~×.

[] R e m a r k : As seen in [5], the condition x0 E £p,x for every x0 E Co{v1,..., VL} can be written as the following LMIs

vTPvj O, IIQ(t)I]2 =

tlEix(t)]12 0 also satisfies (d.7), (4.8) and #¢~max -- ETQEi >- O, i = 1.... g.

(4.10)

Proof." Let assume that the above conditions hold. Then using Schur complements with the additional condition Q > 0, (4.10) is equivalent to

Ei

2

Pq,max

>0, --

i:1

...N.

Another Schur complement transformation leads to

V x E R n, x T Q - l x >- x T ETE~ 2 x, #q,m~×

i=l,...N.

Now, if x0 E £p,x, then x E £p,~. Furthermore, we know from (4.3) that p >_ Q-t, which implies that x E £Q-1,~. This achieves the proof since then Vx E R n s.t. x0 E gP, X, Q,max ~_ xTETEix = gTgi, i = 1 , . . . N . [] 4.5 Solving for t h e s y n t h e s i s c o n d i t i o n s Every condition above is an LMI, except for the non-convex equations (4.6) and (4.8). When (4.3) holds, enforcing these conditions can be done by imposing TrSuTu + #(A + W~max72)= uu + 1 with the additional constraint [P I

/ ] ,~+72W2max _> 0.

(4.11)

In fact, the problem belongs to the class of "cone-complementarity problems", which are based on LMI constraints of the form

F ( V , W , Z ) >- O ,

[ VI

WI ] >- 0,

(4.12)

Control of Uncertain Systems with Bounded Inputs

65

where V, W, Z are matrix variables (V, W being symmetric and of same size), and F(V, iV, Z) is a matrix-valued, affine function, with F symmetric. The corresponding cone-complementarity problem is minimize TrVW subject to (4.12).

(4.13)

The heuristic proposed in [11], which is based on solving a sequence of LMI problems, can be used to solve the above non-convex problem. This heuristic is guaranteed to converge to a stationary point. A l g o r i t h m 7i: 1. Find V, W, Z that satisfy the LMI constraints (4.12). If the problem is infeasible, stop. Otherwise, set k = 1. 2. Find Vk, Wk that solve the LMI problem minimize Tr(Vk-IW + Wk-IV) subject to (4.12). 3. If the objective Tr(Vk-lWk + Wk-lVk) has reached a stationary point, stop. Otherwise, set k = k + 1 and go to (2). 4.6 R e c o n s t r u c t i o n o f t h e c o n t r o l l e r When the algorithm exits successfully, ie when Su T~ "-" I and #(A+W2m~x72) _~ 1 (note that a stopping criterion is given in [11]), then we can reconstruct an appropriate controller by solving another LMI problem [12, 8] in the controller variables K only, Cu being directly inferred from the variables Y, P, Q. Note that analytic controller formulae are given in [13].

5. N u m e r i c a l

results

Our design results are based on two sets of LMIs: the first set focuses on the s-stability of the system for given initial conditions. It corresponds to the case v -- 0 in § 4.1. The second one guarantees a maximum/:2-gain for the system, as defined in §4.2, with (~ -- 0. (It may be too conservative to try and enforce both a decay-rate and an/:2-gain condition via the same set of LMIs.) The bounds (respectively in rad.s -1 and in tad) on the states that appear in the nonlinearities are Zmax -= [0max ~max] T : [0.5 1]T, and the system parameters are M -- 1, l -- 0.6, g = 10, rn - 0.2(1 4- 0.4(frn) with ]5,n] _< 1. The units used for the system parameters and for the plots are kilogram for weights, meter for lengths, second for time, radian for angles and Newton for forces. The design parameters are c~ = 0.05, X0 -- [00 00 (0 ~0]T = [0 0.2 0 0.2] T. We seek to minimize the bound Um~x on the command input and to achieve good disturbance rejection. The numerical results were obtained using the public domain toolbox MRCT [9, 7] built on top of the software l m ± t o o l [10] and the Semidefinite Programming package S P [19]. The output-feedback

66

St~phane Dussy and Laurent E1 Ghaoui

2:t l

1.5

t

k k

a

.,

0

•

30 ~me , $e~ds

0.25

o.2

0.15

~

.

t" ~

0.1

i o.o5 ', ~

0 •.41.05

4s ~

-O.t

-0.15

i 1

, = 2

3

i 4

i * 5 6 Time, seconds

i 7

i 8

10

Fig. 5.1. Closed-loop system responses (cart position and angle position) with w 0 and nonzero initial conditions. Extremal responses (OF) in plain line, nominal response (OF) in dotted line, nominal response (SF) in dashed line.

Control of Uncertain Systems with Bounded Inputs

67

0.2

>~-0.2 ,l

-0,6 -0.8

,

~

6

Time, seconds

~

8

;

1'0

|' ,~.

2

L

-3~

Time.seconds

8

9

Fig. 5.2. Closed-loop system responses (angle velocity and command input) with w -----0 and nonzero initial conditions. Extremal responses (OF) in plain line, nominal response (OF) in dotted line, nominal response (SF) in dashed line.

68

Stdphane Dussy and Laurent El Ghaoui

4

~ .........

3

2

E

~o

-2

%

i

;o

2o

i

3o

T~ne, a e c o n d s

20

go

60

0.25

0,2 0,15 ~. .

0.1 0,05

o;

,If~Ill

-o.,~[, V v -0,2

0

5

;

v v V VVV 10

15

....

V 20 25 Time, seconds

30

35

40

Fig. 5.3. Closed-loop system responses (cart position and angle position) with disturbance w(t) and zero initial conditions. Extremal responses (OF) in plain line, nominal response (OF) in dotted line, nominal response (SF) in dashed line.

Control of Uncertain Systems with Bounded Inputs

69

4r

~ O t l l l l ~ l

M

0

5

10

Itl

_----

15

~

25

~

~

Time, e . ~ o ~ 1

tiJ:H/ 0.8 0.6 0.4 0.2

~.6 ~'4f/ ~.8

T~,

seconds

Fig. 5.4. Closed-loop system responses (command input and perturbation) with disturbance w(t) and zero initial conditions. Extremal responses (OF) in plain line, nominal response (OF) in dotted line, nominal response (SF) in dashed line.

7"0

Stdphane Dussy and Laurent E1 Ghaoui

controller is given by the LFR (3.3), where the gain matrices are described in Appendix A.2. We have shown the closed-loop system responses with w -= 0 and a nonzero initial condition 00 = 0.2tad, z0 = 0.2m in Fig. 5.1-??, and with zero initial conditions and w(t) = 0.1cos(~rt), 0 < t < 20 in Fig. ??. We plotted in plain line the envelopes of the responses for different values of the weight m varying from - 4 0 % to +40% of the nominal value. We also plotted the response of the nominal system (m = 0.2kg) with the outputfeedback law (dotted line), and with a state-feedback control law (dashed line) u = [17.89 4.65 0.06 0.40]x computed as described in [8]. Both robust stabilization and disturbance rejection are achieved for a 40% variation of the weight around its nominal value. We can note that the bounds on 6(t) and on ~(~), respectively 0max = 0.5tad and ~m~x = lrad/s, are enforced. Also note that the state-feedback controller requires less energy (urea SF × = 3.SNewton) OFx = 6.2Newton). than the output-feedback one (urea

6. Concluding remarks In this paper, an LMI-based methodology was described to ensure various specifications of stability, disturbance rejection, input and output bounds, for a large class of uncertain, perturbed, nonlinear systems. The proposed control law is allowed to depend on parameters or states that are measured in real time, while it remains robust in respect to the unmeasured ones. This methodology has been successfully applied to the control of an uncertain, perturbed inverted pendulum. It is very systematic and handles multiobjective control. One important aspect of the method is that it allows computing trade-off curves for multiobjective design. Recently, accurate robustness analysis tools based on multiplier theory have been devised, see e.g. Megretsky and Rantzers [14] and Balakrishnan [2]. A complete methodology for nonlinear design should include these tools for controller validation.

References 1. P. Apkarian, P. Gahinet, and G. Becker. Self-scheduled "//~0 control of linear parameter-varying systems: a design example. Automatica, 31(9):1251-1261, September 1995. 2. V. Balakrishnan. Linear Matrix Inequalities in robustness analysis with multipliers. Syst. ~ Contr. Letters, 25(4):265-272, July 1995. 3. G. Becker and A. Packard. Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback. Syst. (J Contr. Letters, 23(3):205-215, September 1994.

Control of Uncertain Systems with Bounded Inputs

71

4. B. Bodenheimer and P. Bendotti. Optimal linear parameter-varying control design for a pressurized water reactor. In Proc. IEEE Conf. on Decision Contr., pages 182-187, New Orleans, LA, December 1995. 5. S. Boyd, L. E1 Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequality in systems and control theory. SIAM, 1994. 6. J.C. Doyle, A. Packard, and K. Zhou. Review of E}rTs, LMIs and p. In Proc. IEEE Conf. on Decision ~ Contr., pages 1227-1232, Brighton, England, December 1991. 7. S. Dussy and L. El Ghaoui. Multiobjective Robust Control Toolbox (MRCT): user's guide, 1996. Available via http ://www. enst a. f r/~gropco/st af f/dussy/gocpage, html. 8. S. Dussy and L. El Ghaoul. Robust gain-scheduled control of a class of nonlinear parameter-dependent systems: application to an uncertain inverted pendulum. In Proc. Conf. on Contr. ~ Applications, pages 516-521, Dearborn, MI,

September 1996. 9. S. Dussy and L. E1 Ghaoui. Multiobjective Robust Control Toolbox for LMIbased control. In Proc. IFAC Symposium on Computer Aided Control Systems Design, Gent, Belgium, April 1997. 10. L. E1 Ghaoui, R. Nikoukhah, and F. Delebecque. L M I T O O L : A front-endfor LMI optimization, user's guide, February 1995. Available via anonymous ftp to ftp. ensta, fr, under/pub/elghaoui/imitool. II. L. El Ghaoui, F. Oustry, and M. Ait Rami. A n LM|-based linearization algorithm for static output-feedback and related problems. IEEE Trans. Ant.

Contr., May 1997. 12. L. E1 Ghaoui and G. Scorletti. Control of rational systems using LinearFractional Representations and Linear Matrix Inequalities. Automatica, 32(9):1273-1284, September 1996. 13. T. Iwasaki and R.E. Skelton. All controllers for the general 7ioo control problems: LMI existence conditions and state space formulas. Automatica, 30(8):1307-1317, August 1994. 14. A. Megretski and A. Rantzer. System analysis via integral quadratic constraints. In Proc. IEEE Conf. on Decision ~ Contr., pages 3062-3067, Orlando, FL, December 1994. 15. Y. Nesterov and A. Nemirovsky. Interior point polynomial methods in convex programming: theory and applications. SIAM, 1993. 16. A. Packard. Gain scheduling via Linear-Fractional Transformations. Syst. gJ Contr. Letters, 22(2):79--92, February 1994. 17. A. Packard, K. Zhou, P. Pandey, and G. Becker. A collection of robust control problems leading to LMI's. In Proc. IEEE Conf. on Decision ~J Contr., pages 1245-1250, Brighton, England, December 1991. 18. G. Scorletti and L. El Ghaoui. Improved Linear Matrix Inequalities conditions for gain-scheduling. In Proc. IEEE Conf. on Decision ~ Contr., pages 36263631, New Orleans, LA, December 1995. 19. L. Vandenberghe and S. Boyd. SP, Software for semidefinite programming, user's guide, December 1994. Available via anonymous ftp to is1. s t a n f o r d , edu under/pub/boyd/semidef _prog.

72

St6phane Dussy and Laurent E1 Ghaoui

A. Linear

Fractional

Representation

A.1 L F R of t h e o p e n - l o o p s y s t e m The LFR of the inverted pendulum is described matrices. 0 1 0 0 00 00 0001 0 0 0 0

A=

0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 O0 O0 O0 O0 0 0 0

=

3 Bp--- 4 M

0 0 0 0 0 Vqp

~-

o 0 0 0 ~0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 O0 O0 O0 O0 1

0 00 0 0 0 0 00 -100 0 0 0 1 0 0 00 00 0 oo 0 0 0 0 0 0 0 0 1 0

Bu

0 0 o

=

by (3.1) with the following

, B,~= ~

, 0 0

0 0 0

-3 4M

1

0 0

T

0

-3 4Ml

1

,

Dqu

-go

=

~ Vqw

=

0 0 0 0

0

1

0 0 -1 0 0

0 0 0 0 ~-~ 0 0 00 0 01 0 0 o 0o 0 0 0 0 0 0 0 1 ~- 0

00 0 0 00 00 0 0 0

0 0 0 0 0 0

0 0

0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

--~

0

M

0

0

0 0

0 0 g 0 3~ 0 0 0 0 0 0 0 0 -g Ml

o

0

0

0 0

0 -~

0 0 0 1 0 1 IM 0 0 0 0 -4 31M 0

,

0

0

0 1

0 0

,ool 0 0 0 0 0 3 4M 0 0 0 0 -1 M 0

0 0 4Ag 0 0 0 0 0 0 0 0 -~

Control of Uncertain Systems with Bounded Inputs

73

A.2 LFR of the controller The L F R of the controller is described by (3.3) with the following gain matrices.

A=

-8.8

-3.7

-1.2

-54.2 39.3 -1115

-19.7 19.0 -443

-7.6 -24.1 -181

Bu =

n

Bp = 0.01

-395 1159 8868 -732

0.I -0.4 -6.8 6.3

44 280 -157 5780

0.5 -2.5 -22.4 20.4

Cq = 0.01

D-'-qp = 10 -3

-30 12.0 -27 -4.8 2,2 -0.0 -0.I

-12.7 -13.0 -168.3 -1062

'

-3.2 12.5 114 -89 -0.6 4.7 -3.9 -8,1 -5.7 0.1 -0.1

-4.8 -18 -15 -3.4 -7.2 -0.0 -0.i

, C . = 10 - 3

-22.9 94.8 -53.7 -82.5 -315 13.4 -1.4

D-qu = 0.1

24.0 -53.5 -267 111.3 -0.3 1.2 0.4 -8.2 -3.7 0.0 -0.0

22.5 -30 -10.5 -2.3 -14 -0.0 -0.0

-1.7 3.3 22.7 -10.4

0.6 -2.7 -0.4 13.2 6.2 0.1 0.I 15.9 -22 1,1 3.1 8.5 0.0 0.I

0.9 -8.4 9.2 -0.8 7.5 0.4 0,3 -8.1 13.7 -1.3 -0.5 2.6 -0.0 -0.i

28.9 -9.6

0.1 -1.4 1.5 1.6 1.4 -0.1 0.0 -1.7 3.9 21.4 -10.6

--0.0 0.I -4.0 2.4

,

-0.1 0,6 -0.5 -0.3 -1.1 -0.0 -0.I

-0.0 0.6 0.9 0.2 0.2 0.0 -0.0

'

Chapter 5. Stabilization of Linear DiscreteTime Systems with Saturating Controls and Norm-Bounded Time-Varying Uncertainty Sophie Tarbouriech and Germain Garcia L.A.A.S.- C.N.R.S., 7 avenue du colonel Roche, 31077 Toulouse cedex 4, France. E-mail : tarbour~laas.fr , garcia~laas.fr

1. I n t r o d u c t i o n In practical control problems, numerous constraints have to be handled in order to design controllers which operate in a real environment. Concerning the system modelling step, usually the model is uncertain, the uncertainty resulting from simplifications or simply from parameter inaccuracies [6]... Concerning the control, in most of the pratical situations, the control is bounded and saturations can occur [4]. Usually, these two specificities are considered separately and a large amount of literature deals with them. For robust control, see [6], [3], [7] and references therein. For the saturating controls see for example [4], [5], [15] and references therein. From a practical point of view, it would be interesting to elaborate some specific methods which take into account these two aspects and then which allow to design efficient controllers. The aim of this chapter is to address this problem. Some preliminary results are proposed for the problem of robust stabilization of linear discretetime systems subject to norm bounded uncertainty and bounded controls. The concepts of robust local, semi-global or still global stabilization are considered. To solve these, the quadratic stabilizability concept is used. In the local context, the idea is to derive both a stabilizing state feedback matrix K and a set of local stability as large as possible. In the semi-global context, that is, when the set of admissible initial states is an a priori given bounded set, some preliminary results are furnished. Finally, in the global context, that is, when the set of admissible initial states is the whole state space some remarks to show the possible restriction of the results are expressed. The chapter is organized as follows. In the following section, the problem is stated, the principal notations and concepts introduced. Section 3 deals with the robust local stabilization with saturations. An algorithm based on simple calculations is proposed. In subsection 3.4, the connection between the results of subsection 3.2 and the disturbance rejection problem is investigated. The class of perturbations which can be rejected with a saturated

76

Sophie Tarbouriech and Germain Garcia

control is characterized. Finally, these results are applied to a discretized model of an inverted pendulum. Sections 4 and 5 address the robust semiglobal stabilization and robust global problem, respectively. Finally, section 6 presents some concluding remarks and perspectives.

2. P r o b l e m

formulation

Consider the following discrete-time system : Xk+l = (A + AA)zk + (B + A S ) u k

(2.1)

where Xk E ~'~, Uk E ~m, and matrices A, B are of appropriate dimensions. We suppose that system (2.1) satisfies the following assumptions : ( P 1 ) . The control vector uk belongs to a compact set /2 C ~m defined

by $2 = {u E ~ m ; _ w < u < w}

(2.2)

where w is a positive vector of ~m. By convention, the inequalities are componentwise. ( P 2 ) . Pair (A, B) is assumed to be stabilizable. ( P 3 ) . Matrices AA and AB are defined as [ AA

AB ]=OF(k)[

E1

E2 ]

(2.3)

where D, E1 and E2 are constant matrices of appropriate dimensions defining the structure of uncertainty. Matrix F(k) E .7: C ~l×r is the parameter uncertainty, where 2~ is defined by -=- {F(k) G ~t×~;F(k)tF(k) w ~ Iiixk if --wi < Kixk < wi , Vi = 1,...,m - w i if Kixk 0, Vj. j=l

(3.7)

80

Sophie Tarbouriech azM Germain Garcia

The matrices AJ(c~o) and AAJ(ao) are defined as follows :

AJ(ao)

=

AAJ(o~O)

-"

A + BF(flj)K AA + A B F ( f l i ) K

(3.8) (3.9)

for all j = 1, ...,2 m. Matrix F(~j) is a positive diagonal matrix of ~m×m composed of the scalars flji which can take the value a~ or 1, for i = 1, ..., rn [5]. It is worth to notice that the vector a0 defined in (3.4) allows to generate the polyhedral set S(K, ao) C ~n defined as : s(K,

so)

=

e

Wi < - Kixk

Wi

"

< - -~'a'o>O,Vi=l,...,m}

(3.10)

8(K, so) contains 7)0 and is the maximal set in which for any xk E £(K, so), the scalars ai(xk) defined in (3.2) satisfy (3.5). Hence, model (3.6) is only valid for states xk belonging to 8 ( K , a0). Clearly for x k E $(K, ao), matrix A(a(xk)) + AA(a(xk)) in (3.1) belongs to a convex polyhedron of matrices whose vertices are AJ(ao)+ AAJ(ao). The trajectories of system (2.5) or (3.1) are represented by those of system (3.6) so long as for some x0 E S(K, a0) one gets x(k, xo) E S(K, ao), Vk, that is; if matrices A(a(x(k, xo))) + AA(o~(x(k;xo))) in (3.1) belong to the convex hull co{A 1(ao) + AA 1(ao),..., Al(ao) + A A t (ao) }, with g = 2m. This cz)rresponds to the fact that S(K, a0) must be a positively invariant set with respect to the trajectories of (3.6), which is not generally the case. Hence, the importance of the existence of a positively invariant set 7)0 C S(K, a0) appears in order for system (3.6) to represent system (2.5) or (3.1). 3.2 R o b u s t local s t a b i l i z a t i o n A necessary and sufficient condition for the quadratic stabilizability of system (2.1) by linear state feedback when the control vector is not bounded is given in the following theorem [8]. T h e o r e m 3.1. System (2.1), without control constraints, is quadratically

stabilizable by the static state feedback : K = -(R~ + B t ( p -1 - e D D t ) - l B ) - l B t ( p -1 - e D D t ) - l A _e_t(Re + B t ( p _ 1 _ eDDt)_I B)_I E~E 1

(3.11)

if and only if there exist c > 0 and a positive definite matrix P E ~,,×n satisfying the following discrete Riccati equation fit(P-1 + BR71Bt - e n n t ) - x ft - P +e-~ S~ (I~ - e-I EuR-[1E~)E1 + Q = 0

(3.12)

e-lit - D t P D > 0

(3.13)

with

Control of Uncertain Systems with Bounded Inputs

81

where Q E ~,~xn and R E ~rnxm are positive definite symmetric matrices and matrices .4 and R, are defined as : R, = 7t =

R+e-IE~E; A - e-IBR'~IE~E1

(3.14) (3.15)

Hence, by considering the case of system (2.1) with control constraints, to avoid the saturations on the controls, Theorem 3.1 can be used in order to determine a positively invariant and contractive set Do included in S(K, w). It suffices, for example, to determine the largest ellipsoid $(P,/z) defined by S(P,#) = {x E ~ n ; x t P x < / a ; # > 0}

(3.16)

where the positive scalar # is determined such that

8(19,p) C_S(K, w)

(3.17)

However, we are mainly interested by the possibility for the controls to saturate. Therefore we are interested in determining a positively invariant and contractive set in the sense of Definition 2.2.

P r o p o s i t i o n 3.1. Assume that there exist e > 0 and a positive definite matrix P E ~ n x , satisfying the discrete Riccati equation (3.12) and condition (3.13). System (2.1) is locally quadratically stabilizable by the static state feedback matrix K defined in (3.11) in the domain S(P,7) if for aU j = 1, ...,2 m, one gets : - Q + Kt[(Im - r(Zj))(R, + BtWB)(tm - r(/3j)) -r(/3~)Rr'(/35)]K < 0

(3.18)

where W = (p-1 _ eDDt)-l. P r o o f : Compute the decrease of the quadratic Lyapunov function V(xk) = xtkPxk along the trajectories of system (2.5). It follows : 2m

k) = 4

2m

(x )(At (40) + j=l

j=l

(AJ(ao) + AAJ((~0)) - P]xk Then from the convexity of the function V(xk) it follows : 2m

L(Xk, k) < ZAJ(xk)x~[(AJ(ao) + AAJ(ao))tp(AJ(ao) j=l

+AAJ

(ao))

- P]xk

82

Sophie Tarbouriech and Germain Garcia

By setting Lj (xk, k) = xtk[(Aj (ao) + AAJ (c~o))t P(AJ (ao) + AAJ (ao)) - P]xk if Lj(xk, k) < 0 for j = 1,...,2 m, it follows L(xt,,k) < 0. By setting W = (p-1 _ cDDt)-I it follows :

nj(xk, k) < xtk[AJ(ao)tWAJ(ao) - P

+ -1(E1 + E C(ZyC)*(E1 + E

r( yC)],k

Then from (3.11), (3.12), (3.13) and (3.8), (3.9) one obtains:

nj(xk,k) < xtk[-Q + Kt[(F(flj)(e-IE~E2 + BtWB)F(flj) +(Re + BtWB) - F(flj)(Re + BtWB) - ( R , + BtWB).r'(#~)lIilxk Hence if condition (3.18) holds, it follows that Lj(xk, j = 1, ..., 2 m and therefore one gets L(xk, k) < 0 for all xk domain is positively invariant and contractive with respect in the sense of Definition 2.2. Then, since in S(P, 7) system system (2.5), the local asymptotic stability is guaranteed in system (2.5). []

k) < 0 for all e S(P, 7). This to system (3.6) (3.6) represents this domain for

Note that by considering xk E S(P, 7), from (3.4), we can define the resulting a~ as hi0 = min( x/.~v/KiP_iK~, wi 1), Vi = 1 .... , m

(3.19)

and therefore the vectors flj, j = 1, ...,2 m. In fact to apply Proposition 3.1 we have to determine the suitable positive scalar 7 and thus vector a0. The procedure can be stated as follows. A l g o r i t h m 3.1. - Step 1 : Choose any positive definite symmetric matrices Q E ~nxn and R E ~m×rn and a certain starting value Co for e. Step 2 : Determine whether the modified Riccati equation (3.12) has a positive definite symmetric solution satisfying (3.13). If such a solution exists, compute the state feedback matrix K from (3.11) and go to Step 4. Otherwise, go to Step 3. Step 3 : Take e = q < e0. If e is less than some computational accuracy ct : stop. Otherwise, go to Step 2. - Step 4 : Compute, Vi = 1, ..., m, -

-

max xtpx .= rli ar subject to Kia 0.

(5.7)

Control of Uncertain Systems with Bounded Inputs

95

5.1 N u m e r i c a l e x a m p l e Consider the illustrative example borrowed from [5]. System (2.1) is described by the following data : A =

[10 0] [10 ] 0 0

0.9 0.06

-0.06 0.9

; B=

2 3

0 -1

The control takes values in the set $2 defined in (2.2) by w =

[1] 2

"

Consider matrices AA and A B defined in (2.3) described by :

D =

;E,

By choosing Q =

P = K =

0 0 0

: [ o

0 1 0

10

0.02

-o.1

] ; .~

= [ 1

-1

]

0] 0 , the application of Proposition 5.1 gives : 1

0]

0 9.2432 4.2667 ; e = 0.0031 0 4.2667 41.4505 -0.0015 -0.0626 -0.1471 ] -0.0007 0.0211 0.0163

]

6. Conclusion In this chapter, the stabilization of a linear discrete-time norm bounded uncertain system was addressed through a saturated control. The notions of robust local, semi-global and global stabilization were considered. However, we have focused our study on the robust local stabilization problem. The concept of quadratic local stabilizability was defined and used to derive an algorithm allowing both the control law design, based on the existence of a positive definite symmetric matrix solution of a discrete Riccati equation, and a set of safe initial conditions for which asymptotic stability is guaranteed. The size of this set depends on the Riccati equation solution in a complex way and the problem of selecting a matrix which maximizes its size is not obvious and remains open. In this sense, the approach via the satisfaction of some L.M.I. conditions may be considered as an interesting way of research. Some connections with the disturbance rejection problem are also pointed out. The case of state feedbak was considered, but in practice, is often unrealistic. Output feedback stabilization is then more adequate. The results presented in this chapter are only preliminary, some problems have

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t o b e i n v e s t i g a t e d in m o r e d e t a i l s . A m o n g t h e m , t h e o u t p u t f e e d b a c k s t a b i l i z a t i o n a n d t h e role of t h e R i c c a t i e q u a t i o n s o l u t i o n on t h e size o f t h e set of safe i n i t i a l c o n d i t i o n s will be a d d r e s s e d in a f o r t h c o m m i n g issue.

References 1. J. Alvarez-Ramirez, R. Suarez, J. alvarez : Semi-global stabilization of multiinput linear systems with saturated linear state feedback, Systems & Control Letters, 23, pp.247-254, 1994. 2. J.P. Aubin and A. Cellina : Differential inclusions, Springe-Verlag, 1984. 3. B.R. Barmish : Necessary and sufficient conditions for quadratic stabilizability of an uncertain system, J. Optim. Theory Appt., vol.46, no.4, pp. , 1985. 4. D.S. Bernstein and A.N. Michel : A chronological bibliography on saturating actuators, Int. J. of Robust and Nonlinear Control, vol.5, pp.375-380, 1995. 5. C. Burgat and S. Tarbouriech : Non-Linear Systems, vol.2, Chapter 4, Annexes C, D , E, Chapman & Hall, London (U.K), 1996. 6. P. Dorato : Robust Control, IEEE Press Book, 1987. 7. G. Garcia, J. Bernussou, D. Arzelier : Robust stabilization of discrete-time linear systems with norm-bounded time varying uncertainty, Systems and Control Letters, vol.22, pp.327-339, 1994. 8. G. Garcia, J. Bernussou, D. Arzelier : Disk pole location control for uncertain system with 7t2 guaranteed cost, LAAS Report no.94216, submitted for review. 9. Z. Lin and A. Saberi : Semi-global exponential stabilization of linear discrete. time systems subject to input saturation via linear feedbacks, Systems and Control Letters, 24, pp.125-132, 1995. 10. C.C.H. Ma : Unstability of linear unstable systems with inputs limits, J. of Dynamic Syst., Measurement and Control, 113, pp.742-744, 1991. 11. A.P. Molchanov and E.S. Pyatniskii : Criteria of asymptotic stability of differential and difference inclusions encountered in control theory, Systems and Control Letters, 13, pp.59-64, 1989. 12. R. Suarez, J. Alvarez-Ramirez, J. Alvarez : Linear systems with single saturated input : stability analysis, Proc. of 30th IEEE-CDC, Brighton, England, pp.223228, December 1991. 13. S. Tarbouriech and G. Garcia : Global stabilization for linear discrete-time systems with saturating controls and norm-bounded time-varying uncertainty, LAAS Report no.95112, submitted for review. 14. G.F. Wredenhagen and P.R. B~langer : Piecewise-linear LQ control for systems with input constraints, Automatica, vol.30, no.3, pp.403-416, 1994. 15. Y. Yang : Global stabilization of linear systems with bounded feedback, Ph.D. Dissertation, New Brunswick Rutgers, the State University of New Jersey, 1993. 16. K. ~%shida and H. Kawabe : A design of saturating control with a guaranteed cost and its application to the crane control system, IEEE Trans. Autom. Control, vol.37, no.l, pp.121-127, 1992.

Chapter 6. Nonlinear Controllers for the Constrained Stabilization of Uncertain D y n a m i c Systems Franco Blanehini 1 and Stefano Miani 2 i Dipartimento di Matematica e Informatica, Universit& degli Studi di Udine, via delle Scienze 208, 33100 Udine - ITALY 2 Dipartimento di Elettronica e Informatica, Universit£ degli Studi di Padova, via Gradenigo 6/a, 35131 Padova- ITALY

1. I n t r o d u c t i o n In the practical implementation of state feedback controllers there are normally several aspects which the designer has to keep in consideration and which impose restrictions on the allowable closed loop behavior. For instance a certain robustness of the closed loop system is desirable if not necessary to guarantee a stable functioning under different operating conditions which might be for example caused by effectively different set points, component obsolescence, neglected nonlinearities or high frequencies modes. Another issue which has surely to be taken into account is most often the presence of constraints on the control values and on the state variables. The former usually derives from saturation effects of the actuators whereas the latter normally comes from the necessity of keeping the states in a region in which the linearized model represents a good approximation of the real plant or might even be imposed by safety considerations. The constrained control stabilization is by itself a challenging matter and in this contest the designer can either analyze the effects of saturating a stabilizing control law or he can include the constraints in the controller requirements. If stability is the only matter of concern then the first approach is indeed the easiest. The counterpart of this immediateness is unfortunately given by the extremely restricted set of initial states which can be asymptotically driven to the origin [20, 17], say the attraction set. Moreover if state constraints and uncertainties have to be considered then the first approach shows up its deficiencies so that the second approach appears definitely as the most preferable one. In this second class there are several techniques which can be followed to purse the desired performance specification while satisfying the imposed constraints and among these one of the approaches which can be used to overcome these limitations is that based on invariant regions [16, 15, 21, 25, 4, 22, 23, 1]. The key idea which lies behind this approach is that of determining a set of initial conditions starting from which the state evolution can be brought to the origin while assuring that no control and state constraint violation occur.

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This is quite a standard approach and practically amounts to determine a candidate Lyapunov function for the constrained system which can be made decreasing along the system trajectories by a proper choice of the feedback control. Of course there is a certain freedom in the choice of such Lyapunov functions. From the existing literature it turns out that the class of quadratic functions has been the most investigated one mainly due to the elegant and powerful results existing in this area. Although this class is well established and capable of furnishing simple linear control laws, it is not perfectly suited for constrained control synthesis problem due to its conservativity. It is in fact possible to furnish examples of uncertain systems for which quadratic stabilizability cannot be provided although the system is stabilizable [6]. Furthermore it is known that the largest domain of attraction to the origin for a constrained control problem can be arbitrarily closely approximated (or even exactly determined) by means of polyhedrons while ellipsoidal sets may only provide rough approximations. For these reasons in the last years several authors [12, 25, 23, 4, 22, 15] have put their attention on the class of polyhedral functions (say functions whose level surfaces are polyhedrons in ]132) and the associated polyhedral invariant sets. These functions have their force in their capability of well representing linear constralnts on state and control variables while being representable by a finite number of parameters. Moreover it has been recently shown [6] that this class is universal for the robust constrained stabilization problem if memoryless structured uncertainties are considered in the sense that in this case there exists a state feedback stabilizing control law if and only if there exists a polyhedral function which can be made decreasing along the system trajectories by a proper choice of the control law. In this chapter we focus on the problem of determining a state feedback stabilizing control law for constrained dynamic systems (both in the continuous and discrete-time case) affected by structured memoryless uncertainties. In section 2. we will report some preliminary definitions and in the following section we will introduce the class of systems under consideration and we will state the problem. We will reformulate the problem in terms of Lyapunov functions in section 4. where we will also report some known results concerning the constrained stabilization of dynamic systems by means of polyhedral Lyapunov functions. We will present the necessary and sufficient conditions for the existence of a Lyapunov function which will be used in the sections 5. 6. to give a solution to the constrained stabilization problem. Then, based on these results, in section 7. we will focus our attention on the determination of a stabilizing feedback control law for the continuous and discrete-time case. In section 8. we will present an application of the proposed techniques to a two dimensional laboratory system and finally in section 9. we will report some final considerations and the directions for further research in this area. Schematically, the outline of the present paper will be the following: - Definitions

Control of Uncertain Systems with Bounded Inputs

99

- Problem statement Brief summary of theoretical results on constrained control via polyhedral invariant sets. - Construction of a polyhedral Lyapunov function: the discrete-time case. - Construction of a polyhedral Lyapunov function: the continuous-time case Derivation of state feedback stabilizing control laws. - Determination of a polyhedral Lyapunov function and implementation of the cited control laws on a two-tank laboratory system. Analysis of the results and final considerations. -

-

-

2. D e f i n i t i o n s We denote with cony(S) the convex hull of a set S C IR% We will call C-set a closed and convex s e t containing the origin as an interior point. Given a C-set P C fitn we denote with AP = {y = Ax, x E P } and with OP its border. Given r points in fitn we denote with cony(y1, .., Yr) their convex combination and given two C-sets P1 and P2 we denote by cony(P1, P2) = cony(P1 [.J P2) their convex hull. We will be mostly dealing with polyhedral C-sets in view of their advantage of being representable by a finite set of linear inequalities. A polyhedral C-set can indeed be represented in terms of its delimiting planes as

P = {x E fit'* : F i x < 1 , i = 1 , . . . , s } , where each F i represents an n-dimensional row vector as well as by its dual representation

P = cony(v1,.., vk) = cony(V), in terms of its vertex set V = {vl v 2 . . . v r } , (or, with obvious meaning of the notation, its vertex matrix V = [vl v ~ . . . vr]) which will be denoted by vert{P}. For these sets it is possible to introduce a compact notation using component-wise vector inequalities with which the set expression becomes P = {x : F x = ~

0 A!P(z) ~(x)+e(y)

for e v e r y x ¢ 0 , for every A >_ 0, for e v e r y z , y e f i t n .

(2.1)

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Franco Blanchini and Stefano Miani

Every C-set P naturally induces a Gauge function, the Minkowski functional g'p (x), whose expression is given by ~p(x)=min{A : xEAP}. Accordingly to the above every C-set P can be seen as the unit ball of a proper Gauge function P = {x: g,p(x) < 1} Moreover a symmetric C-set P induces a 0-symmetric Gauge function

~p(x) (i.e. such that ~(x) = k~(-x)) which is a norm and every norm induces a symmetric C-set. For a polyhedral set P = {x : Fx 0. The uncertain state matrices belong to the polytopes of matrices p

P

A(w) = ~ w,A,, B(w) = E w,B,, {=I

with

(3.3)

i=l

P

wEW={w:

Ew'=l' i=l

w,_>0},

(3.4)

Control of Uncertain Systems with Bounded Inputs

101

where the given vertex matrices Ai, Bi have appropriate dimensions. In the continuous-time case we will furthermore assume the uncertain function w(t) to be piecewise continuous. In most practical cases the state is also constrained to belong to a Cset X, for example for safety considerations or for linear model validation, and this implies severe restrictions on the choice of the initial conditions and admissible control laws. To the light of this the problem we will focus on is the following: P r o b l e m 3.1. Given the system (3.1) (respectively (3.2)) with the state constrained to belong to the C-set X for every t > 0, determine a set X0 C X and a stabilizing feedback control law u(x(t)) = ~(x(t)) such that for every initial condition x0 E X0 the closed loop evolution is such that: u(t) E U and x(t) E X, lim x(t) = O.

Vt >_ O

t-+q-co

A first solution to this problem can be obviously found by selecting a stabilizing linear static state feedback control law u = K x (for example proceeding along the lines of [2]) and by picking as set of initial states the ellipsoidal region X0 = {x : x T p x ~ d}, where P is chosen in a way such that its derivative is decreasing along the closed loop system trajectories, and d > 0 is the maximal value such that X0 C (Xu N X), being X v = {x : K x E g ) . Unfortunately an inappropriate choice of the gain K might result in a very small set of attraction whereas we are normally interested in determining, given the constraint sets X and U, an as large as possible, according to some criterion, set of initial states which can be asymptotically driven to the origin. Moreover it is possible to furnish examples of dynamic uncertain linear systems which do not admit any ellipsoidal invariant set while they do admit polyhedral invariant sets. In the next section we will recall some of the results concerning set-induced Lyapunov functions for the robust stabilization of uncertain linear dynamic systems in the presence of control and state constraints. We will see how a complete solution to Problem 3.1 can be given by determining a proper polyhedral region of attraction which will result in being a Lyapunov function for the system under consideration.

4. S e t i n d u c e d

polyhedral

Lyapunov

functions

The problem definition, in its actual form, does not shed much light on the choice of the feedback control law. Indeed an alternative and constructive way of proceeding (in the sense that it will provide us with the requested control law) is that of reformulating the problem under consideration by means of Lyapunov functions. To this aim we introduce the following definitions.

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Franco Blanchini and Stefano Miami

D e f i n i t i o n 4.1. The C-set S C X is a domain of attraction (with speed of convergence/3) for system (3.1) if there exists fl > 0 and a continuous feedback control law u(t) = ~(x(t)) such that for all xo E S the closed loop trajectory x(t) with initial condition x(o) = zo is such that u(t) E U for every t > 0 and ~s (x(t) ) 0 the

Euler Approximating System (EAS) is the following discrete-time system x(k + 1) = [I + rA(w(k))]x(k) + rB(w(k))u(k),

(6.1)

After this simple definition we now summarize the main results concerning the constrained control of a continuous-time dynamic systems. We refer the reader to [6] [7] for the proofs which are omitted here for brevity. The first result we are reporting establishes a close link between the contractive sets of a continuous-time system and those associated to its EAS. F a c t 6.1. If there exists a C-set S which is a domain of attraction for (3.1) with a speed of convergence fl > 0 then for all fl' < fl there exists r > 0 such that the set S is contractive for the EAS with A' = 1 - rf~'. Conversely, if for some 0 < A < 1, there exists a A-contractive C-set P for the EAS (6.1) then P is a domain of attraction for (3.1) with/~ = 1-x T The second main result allows us to determine an 'as close as possible' approximation of the maximal/~-contractive set for the continuous-time system under consideration. Thus the two results (the one just reported and the next) provide a complete solution to the constrained stabilization problem. F a c t 6.2. For every cl, e2 > 0 the set S~ (the largest domain of attraction in X for (3.1) with speed of convergence fl > 0) can always be internally approximated by a polyhedral C-set P such that (1 - el)S~i C P C SZ and such that P is a domain of attraction for (3.1) with speed of convergence fl, with/~ - 42 0 we can get an arbitrarily close approximation of the largest domain of attraction (with speed of convergence/~) for (3.1) by applying the numerical procedure reported in the previous section to the EAS (6.1), for an appropriate choice of the parameter r which finally depends mainly on how close we want this approximation to be. Before going on to the determination of a stabilizing control law for the systems under examination, we would like to let the reader note two things:

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Franco Blanchini and Stefano Miani

- The statement in fact 6.1 can be proved to be false if the exponential approximation (instead of the EAS) is used to compute the maximal 13contractive set. It can in fact be seen by very simple examples [7] that the maximal invariant set for the exponential approximation is not contained in maximal invariant set for the continuous time system. - If we let B = 0 and we assume that the system is stable this result allow us to compute an arbitrarily good approximation of the largest domain of attraction.

7. Linear variable structure and discontinuous control law Once a polyhedral approximation of the domain of attraction for (3.1) or (3.2) with a certain speed of convergence has been found, a feedback control law has to be provided. In this section we will see how it is possible to derive a linear variable structure stabilizing control law proposed in [15] and we will furnish a procedure for its determination. Then we will present a discontinuous control law which is applicable only to continuous-time single input systems. Let then P={x:Fix_ 1 are proper scaling coefficients chosen in such a way the new polytope P = cony(V) is simplicial. In this way, each face o f / 5 is associated to the simplicial cone generated by its n vertices, which is in turn associated to a simplicial sector of the original polytope. A l g o r i t h m 7.1. Stretching algorithm [14] Set k = 0 and/5(k) = p. Label each plane and vertex of/5(k) with consecutive numbers and let Ip(k) = { 1 . . . s ( k ) } be the set of indexes of the planes and let Iv = { 1 . . . r } be the set of the indexes corresponding to the vertices of /5(k) 1. For every i e Ip(k) create the incidence list Adj(i) = { j : F(h)vJ k) = 1} (i.e. of the set of all vertices incident in the i-th plane). 2. If all the adjacency lists contain less then n + 1 elements stop (the polyhedron is simplieial) otherwise 3. Pick the first adjacency list which contains more than n indices and pick a vertex vJ k) from this list 4. Compute the maximum factor # by which the chosen vertex can be stretched while assuring that

vert {conv{v~ k) , .., ,uvJk), .., v(k)} } = vert { p¢k)}. 5. Set ~j = (1 + a~v (k) 2/j 6. Set ver*{bCk+l } = {v{k), .., Oj, ..,

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Franco Blanchini and Stefano Miani

7. Compute the plane representation/5(k+1) = {F(k+l)x < 1} and Ip(kq-1) 8. Set k = k + l a n d g o t o ( 1 ) The above procedure stops in a finite number of steps and furnishes a simplicial supporting p o l y h e d r o n / 5 = {z :/~x < i} with the same number of vertices of the original one. In this case the required mapping is given by h(z) = argmazi Fix. We would like to remark two main aspects. The first is that the new p o l y t o p e / 5 is not contractive in general. It is just an auxiliary polytope whose plane representation allows for the computation of the function h(z). The second comes from the fact that the number of sectors (say the complexity of the compensator) grows up very rapidly as the system dimension increases. We present now an heuristic but efficient technique which can be normally used to speed up the computation of the supporting polyhedron is that of stretching each of the vertices of the original polyhedron P so that the stretched vertices lay on the surface of a casually generated hyper-ellipsoid containing P itself. This can be easily done in the following way A l g o r i t h m 7.2. Heuristic stretching procedure 1. Generate randomly the elements of an n × n real matrix Q. 2. Set S = QQT + pI where I is the n × n identity matrix and p > 0 is a parameter which assures that S > 0 (say ~(x) = x~Sz > 0 for every 3. Pick a constant k such that ~(vi) < k for every vi E vert{P}. 4. For every vi E vert{P} set vi = v i ¢ ~ ) In most of the cases the authors have seen that this results in being a simplicial polyhedron. It is clear that this procedure should be used as preliminary "polishing" for Procedure 7.1. To avoid the burdens deriving from the high number of simplicial sectors involved in the on-line computation of the required linear gain the authors have recently proposed a discontinuous control law applicable to single input continuous-time systems and which relies solely on the contractive region P and which is now reported. We let the reader know that passing from continuous to discontinuous control we must pay attention to some issues. The first is that we must assure the existence of the solution. The second is that although we are considering a control which is discontinuous we assume the existence of a continuous control as in definition 4.1. Suppose a fl-contractive region P = {z : F z ___ i} for a single input continuous-time system has been found and that the control constraint C-set can be written as U = [umin, U,~ax]. Define the mapping ](x) = m i n i iet(~)

Control of Uncertain Systems with Bounded Inputs

111

which associates (arbitrarily) to every x E P a single index corresponding to a sector of P and for every x consider the following min-max problem: u = rain max FI(x)(A(w)x + B(w)u) uEU w E W

and let u[(x) be the control value for which the minimum is reached. Being the above linear in all its terms it is quite clear that this value is either u,~i,, or uma~ (or the value 0 if there is more than one minimizer). In this way the control law u(x) = ui(x) remains defined on the whole state space (this is actually the main reason for the introduction of the mapping f(x)) and can be proved to be stabilizing as reported in the next result. T h e o r e m 7.1. Suppose a polyhedral set P is a fl-contractive domain for a single input continuous time-system as in (3.1). Then the discontinuous control law u(z) = ur(~)

is such that

¢p(.) < e-~'e~(.(0)) for every initial state x(O) E P. The proof of the above theorem needs to be supported by the notion of equivalent control [24] and the reader is referred to [7] for details. As a final comment we would like to point out that the cited control law is suitable to handle the case of quantized control devices (see [7]).

8. Application of the control to the two tank s y s t e m The system we considered is a laboratory two-tank plant whose structure is that reported in the scheme in figure 8.1. It is formed by the electric pump EP whose job is that of supplying water to the two parallel pipes P1 and P2 whose flow can be either 0 or Uma~ and is regulated by two on-off electro-valves EV1 and EV2 which are commanded by the signals coming from the digital board BRD1 (not reported in figure 8.1). The two parallel pipes bring water to the first tank T1 which is connected, through P12, to an identical tank T2 positioned at a lower level. From T2 the water flows out to the recirculation basin BA. The two identical variable inductance devices VID1 and VID2, together with a demodulating circuit in BRD1, allow the computer to acquire the water levels of the two tanks. These levels are the state variables of the system.

112

Franco Blanchini and Stefano Miani

~~VID2 [ I P1 EV1

Fig. 8.1. Plant schematic representation If we denote by hi and h2 the water levels of the two tanks and we choose as linearization point the steady state value [hi0 h2o] T corresponding to the constant input u0 = Um~, and we set *l(t) = hi(t) - hl0(t) and z~(t) = h2(t) - h20(t), we get the linearized time-invariant system

z2

=

2

~

2

~

~

z2

+

0

where the parameters entering the above equations are a = .08409, /3 = .04711, hi0 = .5274, h20 = .4014 and u0 = .02985. To keep into account the effects due to the non linear part of the system we considered the uncertain system described by

]

[1]0

with = .118 :t: .05 = .038 :t: .01. The state and control constraint sets we considered are respectively given by

X = {[xlx2]T: ]xll < .1, }x2l _< .1} and U = {-Ur,,~,:,U,,~a,:}. Starting from X we computed the maximal .2-contractive region by using the corresponding EAS with r = 1 and £ = .8. The region representation is given by P = { x : IFx IO

(A - BKC) ] W

>0

122

Alexandre Trofmo, Eug~nio B. Castelan, and Argo Fischman

Proofi Necessity: From (3.2), we have the following inequalities satisfied: P > 0

P - ( A - B K C ) T p ( A - BKC) > 0

(3.4)

Using the Schur complement we can rewrite (3.4) as:

[

P (A - BKC)

(A-BKC) T ] p-1

> 0

(3.5)

Defining p - 1 ~ W, then (3.5) implies that the conditions (3.3.i)-(3.3.ii) are satisfied and T r ( P W - In) = 0. Since the objective function is non-negative from (3.3.i), the pair (P, W) is an optimal solution of the problem (3.3).

Sufficiency:. Suppose that the optimal solution of the problem (3.3), denoted (P*, W*, K*), satisfies ( P ' W * = I,0. Thus, Tr(P*W* - I , ) = 0 and from (3.3.i) we have W* = P * - ' . This implies that the condition (3.3.ii) is equivalent to (3.5) which is equivalent to (3.4) and (3.2). Boz

Remark 3.1. The optimization problem 79(A, B, C) has convex constraints but it is not globally convex because the optimization function is not convex. However, 7:'(A, B, C) is biconvex in P and W, i.e. it is convex in P for fixed W and in W for fixed P. See [13] for further discussion on these type of problems. In the sequel we show how theorem 3.1 can be used for the determination of H ~ controllers for the system (3.1). Let us consider in (3.1) the auxiliary input vector p(k) E Nmp and the auxiliary performance output vector z(k) ¢ ~q~ such that:

{

x(k + 1) = (A - BKC)x(k) + Bpp(k) z(k) = Z~(k) + Fp(k)

(3.6)

with Bp, E and F being given matrices of appropriate dimensions. Suppose we are interested in the determination of a controller K such that the norm of the transfer function from p(k) to z(k) in (3.6) satisfies: [I E[zI - (A - BKC)]-IBp + F [}~ < 7

(3.7)

Without loss of generality, we assume in the sequel that qz - rap. Since any rank condition is required on matrices E, F and Bp, this assumption can always be met by completing matrices E and F (or B v and F) with some null rows (or columns). A necessary and sufficient condition for (3.7) to be satisfied for some K, is the existence of P > 0 and K such that [8]:

Control of Uncertain Systems with Bounded Inputs

E

F

0

7-~Iq.

E

F

-

0

123

~/Iqz

0 and K satisfying (3.8) (or declare that (3.8) and equivalently (3.7) have no solution). If the objective is the minimization of the H ~ norm of the transfer function in (3.7) then we must apply iteractively theorem 3.1 with smaller values for 7 until its minimum value is achieved. Notice that this problem is affine in 7.

4. M i x e d

H~/State

Constrained

results

As previously quoted, to satisfy requirement (r2) of the problem statement, the controller must be such that the polyhedral set of state constraints, 7~[G, g], is a A-positively invariant set of the closed-loop system (2.7). An internal characterization of the A-positive invariance property, in terms of the extremal points of the sets T~[G,g] and A, is given in [2]. For our purposes, a more convenient characterization is obtained from an external description of the two polyhedral sets and is considered below. P r o p o s i t i o n 4.1. A necessary and sufficient condition to get the A-positive invarianee of Tt[G,g] in closed-loop is the existence of an output feedback matrix K E ~m×q and a non-negative matrix M E ~rxr (M _ 0) such that: M G = G(A - B K C ) Mg + (GBp)+~ + (GBp)-~t -< g where:

(GBp)+=max{(GBp)ij,O } (GBp) 5 = max{-(GBp)ij,O}

(4.1) (4.2)

V i - 1. . . . ,r andVj - 1 , . . . , m

Proof: N e c e s s i t y : In closed-loop, the A-positive invariance property of T~[G, g] can be described as follows:

p(k) j

[o . 1j [ 0

p(k)

Using the extension of Farkas' Lemma presented in [9], (4.3) is satisfied if and only if there exists a matrix 34 = [M K1 K2] ~ 0, with ,M E Nr×(r+2,~p), such that:

124

Alexandre Trofino, Eug~nio B. Castelan, and Argo Fischman

[M I'[z K2]

0

Ira,

0

-/,~,

=

G [A- BKC

B,]

(4.4)

The first equality (4.1) follows directly from (4.4) which also gives GBv = (K1 - K2). By using the definitions of (GBp)+ and (GBp)- and the fact that If1 >- 0 and K2 _ 0, we have: (K1 - K~) = (GB;) + - ( G A p ) - and [K1 - K21 = (GBv)+ + ( G B p ) - -< K1 + K2. Together, these last two relations give (GBp) + -< IQ and (GBp)- -< If2, which allow to obtain (4.2) from (4.5). Sufficiency: Consider that (4.1) and (4.2) are satisfied and let z(k) and p(k) be such that z(k) E 7~[G,g] and p(k) E A From (4.1), Gx(k + 1) = G(A - B K C ) + GBpp(k) = MGx(k) + GBpp(k). But, by assumption, (l _ p(k) ~ ~, , and hence:

[

]

[ ]

Thus, from (4.2), we get: Gz(k + 1) = Mg + (GBp)+(~ + (GBp)-(t ~_ g. D Notice that relations (4.1) and (4.2) reduces to the well known positive invariance relations in the case of unforced systems, that is Bp = 0 (see, for instance, [1], [9]). They can also be specialized to the case of symmetric and non symmetric polyhedral sets as well as to consider the case of linear uncertain systems [10]. Notice also that no rank condition on G is required in the proof of the proposition 1 and, hence, relations (4.1) and (4.2) are also valid in the case rank(G) < n. However, additional stabilizing conditions related to the kernel of G have to be considered in this case of unbounded polyhedral sets [5] [7]. The external characterization of the A-positive invarianee property given above is, in general, computationally more attractive than the one give in [2] in terms of the extremal points of the considered polyhedral sets. Furthermore, (4.1) and (4.2) are linear and convex and, as pointed out in [10], [14], [17], they can be transformed into a linear programming problem to achieve a controller satisfying the state constraints as defined in requirement (r2). In order to achieve both requirements (rl) and (r2), we take advantage of the linearity and convexity property of the A-positive invariance relations and we propose to consider (4.t) and (4.2) as additional constraints to the H a problem shown in (3.8) (see also [16]). In this way, let us define A

=

+

Control of Uncertain Systems with Bounded Inputs and consider matrices G t E ~nxr and N E ~(n-r)xr such that:

N

125 G=

[ I~ ]. In order to reduce the number of constraints and variables of the optimization problem related to the A-invariance relations, we obtain from (4.1):

U = G(A - BIl.

(2..4)

Alternatively, cr(u) can be written as

~(u) =/3(u)u,

(2..5)

where the function/3 : R m --+ (0, 1] is defined by ¢?(u)

=

1,

--

uTv/~U ,

1

uTRu < 1,

(2..6)

uTRu > 1,

(2..7)

Figure 8.1 shows the ellipsoidal function for the case m = 2. The closed-loop system (2.. 1), (2..2) can be represented by the block diagram shown in Figure 8.2. Note that in the SISO case m = 1, the function u - ~r(u) is a deadzone nonlinearity. The following result provides the foundation for our synthesis approach. T h e o r e m 2.1.. Let t)1 E N'~,R2 E P'~,R0 E pro, /30 E [0,1], and assume that (A, C) is observable. Furthermore, suppose there exists /5 E P'~ satisfying

~Tp + PA +/)1 + dTn2d + ½1t)T/5 -- (1 -- Z 0 ) R 0 d ] T R o I [ / } T P -- (1 -- Z0)R0d] = 0.

(2..s)

Then the closed-loop system (2..1) and (2..2) is asymptotically stable with Lyapunov function V(~) = £.T/5~, and the set ~_ ,5 {~'0 E R fi : V(~'0) _ 0. Again, using the assumption that (A, C) is observable, we conclude that the closed-loop system (2..1), (2..2) is asymptotically stable. Next, consider the case V(~(0)) = 0. It follows from (2..10) that u(0) = 0, that is, uT(O)Ru(O) = O. Furthermore, for t > 0, 11C~:(t)) > 0 implies that /9(u(t)) < 1, that is, ur(t)Ru(t) > 1. For t sufficiently close to 0, however, this condition violates the continuity of u(t): It thus follows that there exists To > 0 sufficiently close to 0 such that vc~(t)) _< 0 for all t e (0,T0]. Using similar arguments as in the case rV(.~(0)) < 0, it can be shown that ~>(g:(t)) ¢ 0 for all t ~ (0,T0]. Therefore, V(g:(t)) < 0 for all t e (0,T0]. In particular, V(~(T0)) < 0. Hence we can proceed as in the previous case where vc~(o)) < 0 with the time 0 replaced by To. It thus follows that vcg, ct)) -+ 0 as t --+ oo and the closed-loop system (2..1), (2..2) is asymptotically stable.

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Feng Tyan and Dennis S. Bernstein

Remark 2.1.. From the proof of Theorem 2.1, it is easy to see that fll indeed is a lower bound of 13(u(t)) for all t >_ 0. Note that if fl~ = 0, then ~ = R ~. R e m a r k 2.2.. Theorem 2.1 can be viewed as an application of the positive real lemma [18] to a deadzone nonlinearity. To see this, define

LT ~ [_(/}wp_ RoC(1 -~o))T(2R0) -½ f v T ~= [(2Ro)½ 0] v,

(/~1 + CTR2C)½] V,

where v T v = I. It is easy to check that the equations

0 = ATP + PA + LTL, 0 /5/~ _ ( I - flo)0WR0 + LTI~r, 0 ----2Ro -- v v ' T w , are satisfied and are equivalent to the Riccati equation (2..8). It thus follows that G(s) is positive real, where 0(@-,

(1-~0)R0C

Ro

"

Remark 2.3.. The small gain theorem can be viewed as a special case of Theorem 2.1. This can be verified by using a simple loopshifting technique. First, note that the closed-loop (2..1), (2..2) can be written as

~(t)

=

u(~) =

( . 4 - ½/}d)~(t) +/}(~r(u(t)) - ½u(t)), ~(0) = x0, (2..12)

~(t),

(2..13)

and it is easy to check that the nonlinearity a(u(t)) - ½u(t) is bounded by 1 the sector [ - ~1 I , ~/]. Next, by choosing 130 = 0, R0 = 2I, R1 = 0, R2 = 0, equation (2..8) can be reduced to the Riccati equation 0 = (-~ - ½/~)T/5 + P ( A - ½BC) + ~ T ~ + ¼/7,/}/}Tp,

(2..14)

which implies that

P r o p o s i t i o n 2.1. Suppose that the assumptions of Theorem 2.1 are satisfied and fl0 _< ½Amin( R 2 R o 1). Then the closed-loop system (2.. 1), (2..2) is globally asymptotically stable. Furthermore, all the eigenvalues of .~ - / ) 6 ' lie in the closed left half plane. Proof. If~0 < ½Amin(R2Rol), then fll = 0. It thus follows directly from Theorem 2.1 that the closed-loop system (2..1}, (2._.2.) is globally asymptotically stable. To show that every eigenvalue of A - B C has nonpositive real part, rewrite (2..8) as 0

:

(A - ~ ) w p

..~ p ( ~ _ B C ) .]_ [ ~ w p -4- (1 + fi0)RoC]T(2R0)-I[}~TP

+(1 + ~0)R0~ + ~, + ~T(R~ - 2Z0Ro)~, (2..16)

Control of Uncertain Systems with Bounded Inputs

135

and note that fl0 0. Then equation (2..8) can be rewritten as

2FP + Pii + ~, + OTR~O +['y1-1(1 - f l o ) ( [ ~ T P - - " / R 2 0 ) T R ~ l ( B T p -- "/R20)

(4..3)

-~--0.

As before, we minimize the cost functional (3..5) subject to (4..3). The following result as well as Proposition 4.1 and later results are obtained by minimizing J(Ac, B~, Co) with respect to Ac, Be, Co, with Ec chosen to have a specific form. These necessary conditions then provide sufficient conditions for closed-loop stability by applying Theorem 2.1. In the previous section equation (3:.20) for Q was obtained by differentiating the Lagrangian with respect to P. However, due to the presence of the nonlinear term involving Ec, the minimization yields inconsistent expressions for A¢ as obtained from equations (2..8) and (3..20) for t5 and Q. To circumvent this problem we take a suboptimal approach involving a specific choice of E~ and an alternative equation for Q which yields consistent expressions for Ac. For convenience define Z ~ B R ~ I B w and ~ ~ 2 -t- ~/(1 -/3o). P r o p o s i t i o n 4.1. Let nc O, ~o E [0, 1], suppose there exist n × n nonnegative-definite matrices P, Q,/5, (~ satisfying (3..13) and o

=

A T p + P A + R1 - ½15-~(1+/3o) 2 - - / - ~ ( 1 - ~ o ) ] P Z P +½5-'(1 + ~o)2rTpZPT±,

(4..4) 0 ---- [A - O r + ¼~-1(1 -/3o)2ZP]T/5 + P[A - Q ~ + ¼"y-'(1 -fl0)22P] + ~ - 1 ( 1 - ~o)~/5,U/5 + ½5-1(I+~o)2(PZP - T~PZPr±), (4..5) 0 = A Q + QA T + Vt - Q-ZQ + r±Q-~QrT, (4..6) 0 --- [A - ½5-1(1 -b f~0)2,UPl(~+ Q[A- ½5-1(1 +/30)2,UP] w +Q-~Q - r±Q-~QT T, (4..7) and let Ac, Be, Co, Ec be given by Ac

=

F A G T + ¼(1+ f10)(3- ~ o ) F B C c - B¢CG T

+ ~1

--1

(1- ~o)2r~pG T,

B~

=

FQCTV~ -~,

Cc Ec

= =

-(I-1(1 +j30)R~IBTpG T, ½(l+/?0)rB.

(4..8) (4..9) (4..10) (4..11)

140

Feng Tyan and Dennis S. Bernstein

Furtherm°re'supp°sethat('4'C) is°bservable'Then/5=

[ P-G~5 +/5 G/SGT-S/GT]

satisfies (2..8). Furthermore, the equilibrium solution i:(t) = 0 of the closedloop system (2..1), (2..2) is asymptotically stable, and 7:) defined by (2..9) is a subset of the domain of attraction of the closed-loop system.

Proof. Let E~ = ½(1 + ~o)CB and require that ~) satisfy

[

[

where ~ =a 1-~, - ~2~0+~o)~. The remaining steps are similar to the proof of Proposition 5.1. Next we consider the full order case nc = n. P r o p o s i t i o n 4.2. Let n~ = n, 7 > O, and ~0 E [0, 1], suppose there exist n x n nonnegative-definite matrices P, Q,/5 satisfying

0 = A T p + P A + R 1 - ½[a-l(l+~o)2-~/-l(1-5o)]P27P, 0 _=. [A _ Q r + ~1/ -1(1 - ~o)2~Tp]T/5+/5[A--Q-2+¼"I-l(1 + g-/~-1(1 - /~0)a/5,U/5 + ½5-1(1 +flo)2pzp, = AQ + QA w + V1 - Q-2Q, 0

(4..13) /~0)2,gP] (4..14) (4..15)

and let Ac, Be, C~, Ec be given by Ac

=

A + ¼(1 + f10)(3+¼7-1

flo)BCc- BcC

(1 - ~0)2~UP,

Uc = QcTv~ q, C~ = Ec

=

-a-1(1 +

~o)R~IBTp,

o)B.

(4_16)

(4..17) (4..18)

(4..19)

Furthermore, suppose that (A, C ) i s observable. T h e n / 5 = [ P_/5+/5 -/5/5 ] satisfies (2..8). Furthermore, the equilibrium solution ~(t) -- 0 of the closedloop system (2..1), (2..2) is asymptotically stable, and :D defined by (2..9) is a subset of the domain of attraction of the closed-loop system.

Proof. The result is a special case of Proposition 4.1 with nc = n and F = GTmr=I.

Remark ~.I. Note that in Proposition 4.2, equations (4..13)-(4..15) are coupled in one direction, so that no iteration is required to solve them.

Control of Uncertain Systems with Bounded Inputs

141

In the full-order case with /?0 = 1, the dynamic compensator given by Proposition 4.2 is an observer based controller with the realization &¢(t) : Axe(t) + B(r(u(t)) + Bc(y(t) - Cx~(t) ),

~(t) c~(t), =

where B~ and C¢ are the standard LQG estimator and controller gains, respectively. Furthermore, since ~0 = i, it can be seen from the proof of Theorem 2A that ~(u(t)) = 1 for all t > 0. Hence the guaranteed domain of attraction of the closed-loop system yields only unsaturated control signals, that is, c~(u(t)) = u(t) for all t > 0.

5. N o n l i n e a r

Controller

Synthesis

II

In this section we consider again the nonlinear controller given by (4..2), and develop an alternative approach for obtaining controller Specifically, in place of (4.. 12) we require that Q satisfy the alternative tion (5..8). P r o p o s i t i o n 5.1. Let nc < n,'/ > 0,fl0 E [0, 1], suppose there exist nonnegative-definite matrices P, Q,/5, (~ satisfying (3..13) and 0 0

= =

A T p + P A + R, - ½15-'(1+~0) 2 - ~ , - ' ( 1 - ~ 0 ) ] P r P +½5-'(1 + ~o)2rTp~P'r±,

=

n x n

(5..1)

[A - Q ~ + ¼7-'(1 - flo)21:P]TP + P[A - Q-Z + ¼"r-~(t - Zo)2ZP] + ~.-l(t

0

(4..1), gains. equa-

_ ~oyPZP

+ ~5 ~ -'

(1+ 13o)~(PSP- r~P~Pr~.),

[A + ¼ ( 7 - ' ( 1 - ~0)2 _ ~ - ' ( 1 - ]30)(1 + ] 3 0 ) ~ ) ~ ( P + ,

(5..2)

P)]O

~'

+Q[A + ~('~ (1 - ]30)2 - 5-'(1 - ]3o)(1 + ]3o)2)Z(P + p)]T 0

=

+v, - Q~Q + r ~ Q ~ O ~ , [A + ¼(.-1 (1 - ]30)2 - 5-' (3 - ]3o)(1 + ]30)2)~P]Q +(~[A + ¼(7-'(1 - ]30)2 - 5-1(3 -/~0)(1 + ]30)2).UP]T

(5..3)

+Q"~Q - r ~ OrQ-r T,

(5..4)

and let Ac, Be, Co, Ec be given by (4..8)-(4..11). Furthermore, suppose that (.~,~)isobservable. T h e n / 5 =

[ P- +G/P3

--/3GT G P G w ] satisfies (2..8). Further-

more, the equilibrium solution ~(t) _= 0 of the closed-loop system (2..1), (2..2) is asymptotically stable, and 7) defined by (2..9) is a subset of the domain of attraction of the closed-loop system. Proof. Defining the Lagrangian

142

Feng Tyan and Dennis S. Bernstein

yields 0 -

OE OAe - 2(pTQ12 + P2Q~),

(5..5)

0

T T T, O~. _ 2P2BcV2 + 2(Pi2Q1, P2QI2)C OBc

(5..6)

-

0

0E =

aCe

[1 + ½7(1 - flo)]R2CeQ2 + [1 - ½(1 -/3o)]Bw(p1Ql2 =

+P12Q2).

(5..7)

Next, let Ec = ½(1 + flo)FB and require that Q satisfy 0=fz +[A - ½(1 -/3o)/~oC + k/~o R~-t/~TP]0 + 0 [ A - ½(1 - rio)BoC + k[~oR21BoT/5] T,

(5..8)

where k =a ¼7-1(1-/3o) 2 - ¼5-1(1-/3o)(1+/3o) 2 and /~o = remaining steps are similar to the proof of Proposition 3.1.

L

0

J

. The

Note that since Q satisfies the alternative equation (5..8) in place of (4..12), the synthesis equations (4..13)-(4..15) and (4..4)-(4..7) are different from (5..9)-(5..11) and (5..1)-(5..4). However, the expressions for Ae, B¢, Ce remain unchanged. P r o p o s i t i o n 5.2. Let nc = n,7 > 0,fl0 E [0, 1], suppose there exist n × n nonnegative-definite matrices P, Q,/5 satisfying 0 =

ATP+PA+R1

i -1 (1+/3o) 2 - 7 - 1 ( 1 - / 3 o ) ] P S P , - ~[$

0 =

[ A - Q S + ¼7-1 (1 -/3o)22P]T/3 +/3[A - Q r + ¼7-' (1 -/3o)~SP] +~7-1(1 -/30)3/52/3 + ½5-1(1 + flo)2PCP,

0

=

(5..9) (5..10)

[ A + ~'("I ~ - I (1 - rio)' - 5-I(i - flo)(1 +/3o)2)Z'(P + P)]Q +Q[A + Z(7 ~ -1 (1 -/3o) 2 - ~-1(1 -/3o)(1 + flo)2)S(P +/3)IT

+V~

-

QrQ,

(5..11)

and let Ac, Bc, Ce, Ee be given by (4..16)-(4..19). Furthermore, suppose that (~,~)isobservable. T h e n / 3 = [ p +_/5 /3

_13 / 3 ] satisfies (2..8). Furthermore,

the equilibrium solution ~(t~ _~ 0 of the closed-loop system (2..1), (2..2) is asymptotically stable, and 79 defined by (2..9) is a subset of the domain of attraction of the closed-loop system. Proof. The proof is similar to the proof of Proposition 5.1 below with nc = n and F = G w = T = I.

Control of Uncertain Systems with Bounded Inputs 6. N u m e r i c a l

143

Algorithm

Here we adopt the numerical algorithm given by [19] to solve the matrix equations given by Propositions 3.2, 3.1, 4.2, 4.1, 5.2, and 5.1. The basic algorithm is demonstrated by means of Proposition 3.1. A l g o r i t h m 6.1. Step 1. Choose iteration number ima, and tolerance ~. Initialize i = 0 and Step 2. Compute P, Q,/5 O given by (3..9)- (3..12). Step 2.1. Solve the Riccati equation (3..9) to obtain P. Step 2.2. Solve equations (3..10) and (3..11) simultaneously to obtain P and Q. Starting with an initial choice of Q, solve (3..10) to obtain /5. Substitute P and/5 into (3..11) and solve for Q. Repeat the above process until/5 and Q converge. Step 2.3. Solve the Lyapunov equation (3..12) to obtain Q. Step 3. Use contragradient diagonalization to update r. First, compute S E R~ ×~' such that STPS

S-1QS -T = D,

:

where D -~ diag(dl,...,dn) and dl > " " > dn > O. Then construct r as r = S ~ S -1,

where ~ ~ diag(~-l,..., Tn) is defined by /'j

= =

1, j n c .

Step 4. If i =/max, go to Step 6. Otherwise, go to Step 5. Step 5. If (tr (~) - nc)/n¢ < ~, go to Step 6. Otherwise, increment i = i + 1 and go to Step 2. Step 6. Compute r=[I~o

O]S-~,

a = [ I ~ o O]SL

and calculate Ac, Be, Cc using (3..14) - (3..16).

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Feng Tyan and Dennis S. Bernstein

7. N u m e r i c a l

Examples

In this section, we reconsider the examples given in [15] to demonstrate the linear and nonlinear full-order and reduced-order compensators given by Propositions 3.2, 4.2, 5.2 and 3.1.

Example 7.1. To illustrate Proposition 3.2, consider the asymptotically stable system =

[_o.o31 o] [o] o 0

y=[1

-0.03 0

0

1 -0.03

x+

o 1

~(.(t)),

0]x,

with the saturation nonlinearity ~(u) given by ~(.)

=

-,

=

sgn(u)4,

1-1 < 4, lul > 4.

50 Choosing R1 = 13, R2 = 100, 1/1 = / 3 , 1/2 = 1, and 13o = 0.3, Ro = 1_-=-~o R2 = 7142.9, Algorithm 6.1 yields the linear controller (3..3), (3..4) with gains (3..25) - (3..27) given by

[-2.87151.00000] Ac =

-3.6223 -2.3579

-0.0300 -0.0841

[2.8415]

1.0000 -0.4114

, Bc =

3.6223 2.3482

,

co = [ -0.0215 -0.1866 -0.8462 ]. By applying Remark 3.1, the set :D is given by 13 = {xo :

xT(p + P)xo
= 0 /\

q2"4 + 20 q2"3 + 8 q l ' 2 q2"2 - 20 ql q2"2 + I0 q2"2 - 80 q1"2 q2 - 200 ql q2 - 100 q2 + 16 q l ' 4 + 80 q l ' 3 + 140 q1"2 + I 0 0 q l + 2 5 < = 0 ] ]

Control of Uncertain Systems with Bounded Inputs

171

and

V(w)[Fl(w,ql, q2) >

0 A F2(w, ql, q2) > 0 A F4(w,ql, q2) > 0]

where Q E P C A D produces the quantifier formula ~V2(ql, q2), [ [ q l - 1 >= 0 / \ q1"2 - 5 0 / \ q2~2 + I0 q2 - 4 q l ' 2 + 10 q l - 5 = o / \

q2 > o / \

q2"4 + 20 q2~3 + 8 q l ' 2 q2"2 + 20 q l q2-2 + 10 q2"2 80 q1"2 q2 + 200 q l q2 - 100 q2 + 16 q l ' 4 - 80 q1"3 + 140 q l ' 2 - 100 q l + 25 = 0 / \ q1"2 - 5 < = 0 ] \ / ql - 1 > = 0 ] ]

from which one can reduce, by computing roots of single-variable polynomials, the following range: 1 < ql < 2.2361. Then we discretize ql within this range and plug into the quantifier-formula kV(ql, q2) a particular discretized value of ql to find what the acceptable range of q2 is for this particular discretized value of ql. The solution for this problem is represented by the table 4.1, for five discrete values of ql. The notation (a, b] in the table produces that q2 is in the range, a < q2 < b. Table 4.1. Design parameter regions for example 4.2 Design parameter ql

Design parameter q2

1.2 1.5 1.7 2.0 2.2

( ( ( ( (

0, 0, 0, 0, 0,

3.6996 3.9588 4.0654 0.1623 0.2721

] ] ] ] ]

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Peter Dorato et al.

5. C o n c l u s i o n The design, in the frequency domain, of robust feedback systems with bounded control effort can be reduced to quantifier elimination problem. However due to computational complexities, only problems of modest size can be solved. Nevertheless it may be possible to solve some practical problems where no analytic design procedures exist.

References 1. C. Abdallah, P. Dorato, W. Yang, R. Liska, and S. Steinberg, Applications o] quantifier elimination theory to control system design, 4th IEEE Mediterranean Symposium on Control & Automation, Chania, Crete, Greece, June 10-14, 1996. 2. B.D.O. Anderson, N.K. Bose, and E.I. Jury, Output feedback and related problems-Solution via Decision methods, IEEE Trans. on Automatic Control, AC-20, pp.53-65, 1975. 3. S. Basu, R. Pollack, and M.F. Roy, On the combinatorial and algebraic complexity o] quantifier elimination, Proc. 35th Symposium on Foundations of Computer Science, Santa Fe, NM, pp. 632-641, 1994. 4. G. E. Collins, Quantifier Elimination in the Elementary Theory of Real Closed Fields by Cylindrical Algebraic Decomposition, Lecture Notes in Computer Science, Spring Verlag, Berlin, Vol. 33, pp. 134-183, 1975. 5. G.E. Collins and H. Hong, Partial cylindrical algebraic decomposition ]or quantifier elimination, J. Symbolic Computation, 12, pp. 299-328, 1991. 6. P. Dorato, W. Yang, and C. Abdallah, Robust multi-objective feedback design by quantifier elimination, submitted to J. Symbolic Computation, 1996. 7. G.Fiorio, S. Malan, M.Milanese, and M. Taragna, Robust Performance Design o] Fixed Structure Controller with Uncertain Parameters, Proc. 32nd IEEE Conf. on Decision and Control, San Antonio, TX, pp. 3029-3031. 8. H. Hong, Improvements in CAD-based Quantifier Elimination, Ph.D Thesis, The Ohio State University, 1990. 9. H. Hong, Simple Solution Formula Construction in Cylindrical Algebraic Decomposition based Quantifier Elimination, ISSAC'92, International Symposium on Symbolic and Algebraic Computation, July 27-29, Berkeley, California (Editor P.S. Wang), ACM Press, New York, pp. 177-188, 1992. 10. R. Liska and S. Steinberg, Applying Quantifier Elimination to Stability Analysis o] Difference Schemes, The Computer Journal, vol. 36, No. 5, pp. 497-503, 1993. 11. A. Seidenberg, A New Decision Method ]or Elementary Algebra, Annals of Math., 60, pp. 365-374, 1954. 12. M. Jirstrand, Algebraic methods ]or modeling and design in control, LinkSping Studies in Science and Technology, Thesis no. 540, LinkSping University, 1996. 13. V.L. Syrmos, C.T. Abdallah, P.Dorato, and K. Grigoriadis, Static Output Feedback: A Survey, Scheduled for publication in Automatica, Feb., 1997. 14. A. Tarski, A Decision Method]or Elementary Algebra and Geometry, 2nd Ed., Berkeley, University of California Press, 1951.

Chapter 10. Stabilizing Feedback Design for Linear Systems with Rate Limited Actuators* Zongli Lin 1, Meir Pachter 2 , Siva Banda 3, Yacov Shamash 4 1 Dept. of Applied Math. & Stat. SUNY at Stony Brook Stony Brook, NY 117943600 Dept. of Elect.& Comp. Sci. Air Force Inst. of Tech. Wright-Patterson AFB, OH 45433 3 Flight Dynamics Dir. (WL/FGIC) Wright Laboratory Stony Brook, NY 11794360O 4 College of Engr. & Applied Sci. SUNY at Stony Brook Stony Brook, NY 117942200

1. I n t r o d u c t i o n Every physical actuator is subject to constraints. These constraints include both position and rate saturation. In the past few years there has been much interest concerning stabilization of linear systems with position saturating actuators, resulting in several promising design techniques. In this paper, we will recourse to the low-and-high gain (LHG) design technique ([7]) and the piecewise linear LQ control (PLC) design technique ([8]). Additional related work might be found in these two papers and in [2]. While actuator position saturation has been addressed in the recent literature, few design techniques are currently available to deal with actuator rate saturation. However, actuator rate saturation often presents a more serious challenge to control engineers, especially flight control engineers. It is known ([1]) that actuator rate saturation could induce a considerable phaselag. Such phase-lag associated with rate saturation has a destabilizing effect. For example, investigators have identified rate saturation as a contributing factor to the recent mishaps of YF-22 [3] and Gripen [4] prototypes and the first production Gripen [6]. For further discussion on the destabilizing effect of actuator rate saturation, see [1]. The objective of this paper is to propose a method of designing stabilizing feedback control laws for linear systems taking into account the effect of actuator rate saturation. The proposed design method views the problem of stabilization with rate saturating actuators as a problem of robust stabilization with position saturating actuators in the presence of input additive uncertainties. The state feedback law is then designed for stabilization with position saturating actuators. Our state feedback design combines the two recently developed design techniques, the PLC and the LHG design techniques. * This work was conducted while the first author was participating in the 1996 AFOSR summer faculty research program. He acknowledges the support of AFOSR.

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ZongliLin, Meir Pachter, Siva Banda, Yacov Shamash

It inherits the advantages of the both design techniques, while avoiding their disadvantages. Thus, the exact knowledge of the dynamics of the actuators will not be needed and the actuator disturbances can be rejected. In particular, in the LHG design, a low gain feedback law is first designed in such a way that the actuator does not saturate in position and the the closed-loop system remains linear. The gain is chosen low to enlarge the region in which the closed-loop system remains linear and hence enlarge the domain of attraction of the closed-loop system. Then, utilizing an appropriate Lyapunov function for the closed-loop system under this low gain feedback control law, a linear high gain feedback control law is constructed and added to the low gain feedback control to form the final LHG feedback control law. Such a linear low-and-high gain feedback control law speeds up the transient response for the state in a certain subspace of the state space and is capable of stabilizing the system in the presence of input-additive plant uncertainties and rejecting arbitrarily large bounded input-additive disturbances. The disadvantage of this control law is that the transient response for the state outside that subspace of the state space remains that of the low gain feedback, which is typically sluggish (due to low feedback gain for a large domain of attraction). On the other hand, the aim of the PLC scheme is to increase the feedback gain piecewisely while adhering to the input bound as the trajectories converge toward the origin. Such a design results in fast transient speed for all states. However, it lacks robustness to large uncertainties and the ability of rejecting arbitrarily large bounded disturbances. The remainder of the paper is organized as follows. In Section 2., we precisely formulate our problem. Section 3. provides the design algorithm and proves that the proposed algorithm results in feedback laws that solve the problem formulated in Section 2.. In Section 4., an F-16 class fighter aircraft model is used to demonstrate the effectiveness of the proposed design algorithm. Concluding remarks are in Section 5..

2. P r o b l e m

Formulation

Consider the linear dynamical system

{ ~ = Ax + Bv, x(O) E X C ]R2 b = satA(--Tlv + T2u + d), v(O) C • CIR.rn

(2.1)

where x E IR~ is the plant state, v E ]Rm is the actuator state and input to the plant, u E IRm is the control input to the actuators, for A = (A1, A2,. " , A t , ) , Ai > 0, the function satza : IRm -+ ]pjn is the standard saturation function that represents actuator rate saturation, i.e., satin(v) = [satA 1(Vl), sata~ (v2),''', sat,a.. (Vrn)], sata, (v~) = sign(vi) min{ Ai, Ivi]}, the positive definite diagonal matrices T1 = d i a g ( r n , v l ~ , - " , r l r n ) and T2 = diag(v21, v22," ", r2m) represent the "time constants" of the actuators and

Control of Uncertain Systems with Bounded Inputs

175

are not precisely known, and finally d : JR+ --+ IRm are the disturbance signals appearing at the input of the actuators. We also make the following necessary assumptions on the system. A s s u m p t i o n 2.1. The pair (A, B) is stabilizable; A s s u m p t i o n 2.2. The nominal value of T1 and T2 is known and is given by T* = d i a g ( r ~ , r ~ , - . . , r * ) with vi > 0. There exist known matrices #1 = d i a g ( # l l , # 1 2 , ' " , p l m ) , kt2 = diag(/-t21,~t22,'",#2m), l#ij[ < 1, and Vl = diag(u11, u12,-" ", ulm), u2 = diag(u21, u 2 2 , ' " , u2m), [ulj[ >_ 1, such that, piT* 0, every trajectory starting from X x ~ enters and remains in W0 after some finite time.

3. A Combined PLC/LHG Design Algorithm As stated earlier, the proposed design algorithm is a combination of the piecewise linear LQ control [8] and the low-and-high gain feedback [7]. Naturally we organize this section as follows. Subsections 3.1 and 3.2 respectively recapitulate the PLC and the LHG design techniques. Subsection 3.3 presents the proposed combined P L C / L H G design algorithm. Finally, in Subsection 3.4, the proposed design algorithm is shown to solve Problem 2.1.

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Zongli Lin, Meir Pachter, Siva Banda, Yacov Shamash

3.1 Piecewlse Linear LQ Control Design Consider the linear dynamical system subject to actuator position saturation, k = Ax + B s a t a ( u ) , x(0) e X C IRn, u E ~:tm

(3.1)

where the saturation function sata : IRm -+ ~,n is as defined in Section 2., and the pair (A, B) is assumed to be stabilizable. The PLC design is based on the following LQ algebraic Riccati equation, AlP + PA - PBR-1BIP + I = 0

(3.2)

where R = diag(e) -- diag(el, e 2 , ' " , era), ei > 0, are the design parameters to be Chosen later. Key to the PLC scheme is the notion of invariant sets. A nonempty subset of e in lRn is positively invariant if for a dynamical system and for any initial condition x(0) e ~, x(t) E e for all t _> 0. For the closed-loop system comprising of the system (3.1) and the LQ control u = - R - 1 B I P x , simple Lyapunov analysis shows that the Lyapunov level set e(P,p) = {x : x ' P x < p}, Vp > 0 is an invariant set, provided that saturation does not occur for all a: E e(P, p). To avoid the saturation from occurring, while fully utilizing the available control capacity, for a given p, e = (el, e2,-.., e,n) will be chosen to be the largest such that

where/?i is the ith column of matrix B and ui is the ith element of u. The existence and uniqueness of such an e are established in [8] and an algorithm for computing such an e is also given in [8]. More specifically, it is shown through the existence of a unique fixed point that the following iteration converges from any initial value to the desired value of e, =

(3.3)

where •

=

1

and for each i = 1 to m, ¢,(e) = --~i~/B~P(~)B, The aim of the PLC scheme is to increase the feedback gain piecewisely while adhering to actuator bounds as the trajectories converge towards the origin. This is achieved by constructing nested level sets, e 0 , e l , . . . , e N , in such a way that as the trajectories traverse successively the surface of each

Control of Uncertain Systems with Bounded Inputs

177

6i and the control law is switched to higher and higher gains as each surface is crossed. The procedure in designing a PLC law is as follows. Given the set of initial conditions X C IR'~, choose an initial level set ¢0 as, 60 = inf{6(P, p) : X C 6(P,p)} p

(3.4)

We denote the value of p associated with 6o as p0, and the corresponding values of e, R and P as E0, R0 and P0 respectively. A simple approach to determining 6o and P0 can also be found in [8]. More specifically, it is shown that the size of 6o grows monotonically as the parameter p grows. Hence, e0 and P0 can be determined by a simple iteration procedure. Here we would like to note that, as explained in [8], increasing p indefinitely for exponentially unstable A will not result in an 6o that grows without bound. To determine the inner level sets 6i's, choose successivefully smaller Pi where pi+l < pi for each i = 1 , 2 , . . - , N. A simple choice of such pi's is the geometric sequence of the form P i = p 0 ( A p ) i, i = O , 1 , 2 , . . . , N where the p-reduction' factor Ap E (0, 1). (Consequently, the values of e, R and P associated with each of these pi's are denoted as ~i, Ri and Pi respectively.) For a discussion of the choice of N and Ap, see [8]. As shown in [8], a critical property of such a sequence of level sets 6i is that they are nested in the sense that 6i C 6i+1 for each i = 0 to N - 1.

3.2 Low-and-High Gain Feedback Design Consider the linear dynamical system subject to actuator position saturation, input additive disturbances and uncertainties,

~=Ax+Bsatzx(u+f(x)+d),

x(O) e X c I R n , u E I R

rn

(3.5)

where the saturation function s a t a : lRrn --~ IKra is as defined in Section 2, the locally Lipschitz function f : lKn -+ lRm represents the input additive plant uncertainties and d the input-additive disturbance. The LHG feedback design for this system is given as follows. First, the level set e0 is determined as in the PLC design. Correspondingly, a state feedback law with a possibly low feedback gain is determined as, UL

--"

_ R o l B' Poz

A high gain state feedback is then constructed as,

UH = - k R o l B ' P o z ,

k >0

The final low-and-high gain state feedback is then given by a simple addition of the low and high gain feedbacks UL and uH, viz.,

178

Zongli Lin, Meir Pachter, Siva Banda, Yacov Shamash u = - ( 1 + k)RotB'Poz,

k>0

Here the design parameter k is referred to as the high gain parameter. As demonstrated in [7] the freedom in choosing the value of this high gain parameter can be utilized to achieve robust stabilization in the presence of input additive plant uncertainties f(x) and input-additive disturbance rejection. Moreover, the transient speed for the states not in the range space of B'Po will increase as the value of k increases. To see this, let us consider the following Lyapunov function,

Yo(z) = x'Poz The evaluation of I) along the trajectories of the closed-loop system in the absence of uncertainties and disturbances gives,

= - x ' z - x'PoBRolB'Poz + 2z'PoB[sata(-(k + 1)RolB'Poz) +Rol B' Pox] m

= - x ' x - x'PoBRolB'Pox - 2 ~

vi[sata,((k + 1)vi) - vi]

i=1

where we have denoted the ith element of v = - R o l B t P o x as vi. By the choice of P0, it is clear t hat lvi I < Ai and hence - vi [sat a, ( ( k + 1) v~) - vi] < 0, for each i = 1 to m. For any x not in the range space of B'Po, that is B'Pox 7£ O, then, for any i such that vi # 0, -vi[sata,((k2 + 1)vi) - vii < - v i [ s a t a , ( ( k l + 1)vi) - vi] if k2 > kl. However, for any x in the null space of

B'Po,

+ 1)vd - vd = 0 for all i.

3.3 Combined PLC/LHG Design In this subsection, we present the proposed combined P L C / L H G state feedback design for linear systems subject to actuator rate saturation (2.1). The feedback control law design is carried out in the following three steps. S t e p 1. Choose a pre-feedback u = v + fi

(3.6)

Let ~ = [x, v]'. Then the system (2.1) under the above pre-feedback is given

by, £" -- A~ -t-/~satA(T2u -I- f(.~) -I" d), ~(0) E X x l; C lit"+m where - [o

and,

+T2]

(3.7)

Control of Uncertain Systems with Bounded Inputs

[A

179

Assumption 2.1, i.e., the pair (A, B) is stabilizable, implies that (A, B) is stabilizable. S t e p 2. Apply the PLC design to the system (3.7), and obtain a sequence of nested level sets ~0, ~1,"" ", eu (and correspondingly, the parameters e0, e2,.. ", eg) and a pieeewise linear feedback law,

f fii=-(,2T*)-lffl[iB'Pi~: =

= -(mT*)-IR?vl

'PNe

for~Eei\ei+l,i=O,...,Y-1 for

e

(3.8) S t e p 3. Design the LHG state feedback based on the PLC feedback law (3.8) and obtain the following combined final P L C / L H G feedback law, f ui = v - (k + 1) (#2T*)-i/~/a/~'/5i£" for ~ • ei \¢i+i, i = 0 , . . . , N - 1 UN = V -- (k + 1)(#2T*)-iRNIB'pN~ for ~ • eN (3.9) where k _> 0 is a design parameter to be specified later. it

3.4 P r o o f In what follows, we will show as a theorem that the combined P L C / L H G state feedback law (3.9) solves Problem 2.1. The effectiveness of this feedback law in comparison with both the PLC and the LHG feedback laws will be demonstrated in the next Section. T h e o r e m 3.1. Let Assumption 2.1 hold. Given the admissible data (D, Wo), there exists a k*(D,I/Vo) > 0 such that, for all k > k*, the combined PLC/LHG state feedback law (3.9) solves Problem 2.1. Moreover, if D = O, k* is independent of •o. P r o o f . The proof is carried out in two steps. In the first step, we show that, for each i = 0 to N - 1, there exits a k[(D) > 0, such that for all k _> k[, in the presence of any d satisfying Assumption 2.3, all trajectories starting from ¢i \ ei+t will remain in ~i and enter into the inner level set ~i+l in a finite time. This in turn implies that, for any k > max{k0, k 2 , . - . , kN-t}, all the trajectories of the closed-loop system starting from W C ~0 will enter the inner-most level set eN in a finite time. The second step of the proof is to show that: if D = 0, there exists a k~v > 0 such that, for all k > k~v, the equilibrium $ = 0 of the closed-lop system is locally asymptotically stable with ¢N contained in its basin of attraction, and if D ~ 0, there exists a k*g(D, W0) > 0 such that, for all k > k~, all the trajectories of the closedloop system starting from £N will remain in it and enter and remain in the set W0 in a finite time. Throughout the proof, we will also notice that in the case that D = 0, all the k*s are independent of D.

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Zongli Lin, Meir Pachter, Siva Banda, Yacov Shamash

Once these two steps are completed, the proof of the theorem is then complete by taking k*(D, 1420) = max{k~, k~,..., k~v}. We start by considering the closed-loop system for ~ E ei \ ei+l, i = 0 to N, = A£' + / ~ s a t ( - ( k + 1)T2(#2T')-l/~-l/~'Pig " + f(~) + d) = (A - / ~ / ~ - l / ~ , p , ) ~ + / ~ [ s a t a ( - ( k + 1)T2(v2T*)-IR'f~.B'P,~. + f(~.) + d)

where eN+l = 0. We next pick the Lyapunov function, v; = ~'P,~

(3.I0)

The evaluation of V/ along the trajectories of the closed-loop system in the presence of uncertainties and disturbances gives, +2~'/3i/~[sata(-(k + 1)T2(lt2T*)-lR71[t'Pifc + f(f:) + el) + k71B'Pi~] 171

= _~,~ _ ~:,p,f~:x~,p,~

_ 2~

vi[sata,((k + 1)~ivi + fi + di) - v, 1

i=1

where we have denoted the ith elements of v -- -/~-l/~'/3ix, f(~) and d respectively as vi, f/ and d~, and 5i -- v2i/p2ir* > 1. By the Construction of 6~, it is clear that ]vii _< Ai for all ~ E 6i. Also note that IIf(~)ll < (lit,2 - ~11 + lira - ~II)IlT*IIII~It = ZlI~II

(3.~1)

Hence we have, I[(k +

1)5i - 1]vii >_ I f / + &l ~

-2vi[sata,((k + 1)$ivi + fi + di) - vi]