Hierarchical nonlinear switching control design with applications to propulsion systems

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

255

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Alexander Leonessa, Wassim M. Haddad and VijaySekhar Chellaboina

Hierarchical Nonlinear Switching Control Design with Applications to Propulsion Systems With 28 Figures

~ Springer

Series Advisory Board A. B e n s o u s s a n • M.J. G r i m b l e • P. K o k o t o v i c • A.B. K u r z h a n s k i

H. Kwakernaak • J.L.Massey • M. Morari

Author Alexander Leonessa,AssistantProfessor Department of Ocean Engineering,FloridaAtlanticUniversity,USA Wassim M. Haddad, Professor School of Aerospace Engineering,Georgia Institiuteof Technology, Atlanta, GA 30332-0150, U S A VijaySekhar CheUaboina, Assistant Professor D e p a r t m e n t o f M e c h a n i c a l a n d A e r o s p a c e Engineering, U n i v e r s i t y o f Missouri, Columbia

ISBN 1-85233-335-9 Springer-Verlag London Berlin Heidelberg British Library Cataloguing in Publication Data Leoness~ A. Hierarchical nonlinear switching control design with applications to propulsion systems. - (Lecture notes in control and information sciences; 255) l.Nonlinear systems 2.Nonlinear control theory 3.Switching theory I.TRle II.Hadded, W.M. III.Chellaboina, V. 629.8'36 ISBN 1852333359 Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress Apart from any fair deeling for the purposes of research or private study, 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-Verleg London Limited 2000 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 authors Printed and bound at the Atheneeum Press Ltd., Gateshead, Tyne & Wear 69/3830-543210 Printed on acid-free paper SPIN 10770548

To my fiancee; Keri N. Swaby A.L.

To my family and the memory of my father; Mikhael S. Haddad W. M. H.

To my family and the memory o/my mother; Andhra Jayashree V.C.

Preface

General nonlinear system stabilization is notoriously difficult and remains an open problem. If the operating range of the control system is small and if the system nonlinearities are smooth, then the control system can be locally approximated by a linearized system about a given operating condition and well established linear multivariable control methods can be used to maintain local stability and performance. However, in modern high performance engineering applications such as advanced tactical fighter aircraft, large flexible space structures, and variable-cycle gas turbine engines, to cite but a few examples, the locally approximated linearized system does not cover the operating range of the nonlinear system. In this case, gain scheduled controllers can be designed over several fixed operating points covering the system's operating range and controller gains interpolated over this range. However, due to approximation linearization errors and neglected operating point transitions, the resulting gain scheduled system does not have any guarantees of performance or stability. The main objective of this monograph is to develop a general nonlinear control design methodology for nonlinear uncertain dynamical systems. Specifically, a hierarchical nonlinear switching control framework is developed that provides a rigorous alternative to gain scheduling control for general nonlinear uncertain systems. The main tool for establishing a hierarchical nonlinear switching control framework is a generalized Lyapunov and invariant set framework that address stability of switched feedback systems. The desire for developing an integrated control system-design methodology for advanced propulsion systems has led to significant activity in modeling and control of flow compression systems in recent years. In this monograph we apply the hierarchical switching control framework to axial and centrifugal flow compression systems to address the compressor aerodynamic instabilities of rotating stall and surge. The proposed control framework accounts for the coupling between higher-order modes while explicitly addressing actuator rate saturation constraints and system modeling uncertainty. After the introductory chapter, the presentation is organized in two major parts. The basic hierarchical nonlinear switching control framework is devel-

VIII

Preface

oped in Chapters 2-4 while Chapters 5 and 6 present applications of the proposed hierarchical control framework to axial and centrifugal compression systems, respectively. Specifically, in Chapter 2 we develop generalized Lyapunov and invariant set theorems for nonlinear dynamical systems wherein all regularity assumptions on the Lyapunov function and the system dynamics are removed. In particular, local and global stability theorems are given using lower semicontinuous Lyapunov functions. Furthermore, generalized invariant set theorems are derived wherein system trajectories converge to a union of largest invariant sets contained in intersections over finite intervals of the closure of generalized Lyapunov level surfaces. The proposed results provide transparent generalizations to standard Lyapunov and invariant set theorems. Using the generalized Lyapunov and invariant set theorems developed in Chapter 2, in Chapter 3 a nonlinear control-system design framework predicated on a hierarchical switching controller architecture parameterized over a set of moving system equilibria is developed. Specifically, using equilibriadependent Lyapunov functions, a hierarchical nonlinear control strategy is developed that stabilizes a given nonlinear system by stabilizing a collection of nonlinear controlled subsystems. The switching nonlinear controller architecture is designed based on a generalized lower semicontinuous Lyapunov function obtained by minimizing a potential function over a given switching set induced by the parameterized system equilibria. The proposed framework provides a rigorous alternative to designing gain scheduled feedback controllers and guarantees local and global closed-loop system stability for general nonlinear systems. Furthermore, the hierarchical switching control framework is extended to include inverse optimality notions. Specifically, the hierarchical controller is parameterized with respect to a given system equilibrium manifold wherein an inverse optimal morphing strategy is constructed to coordinate the hierarchical switching. The overall approach is quite different from the quasivariational inequality methods for optimal switching systems developed in the literature in that our results provide hierarchical homotopic feedback controllers guaranteeing closed-loop stability via an underlying Lyapunov function. Finally, in Chapter 4 the proposed control framework is extended to account for system parametric uncertainty wherein the hierarchical switching architecture is parameterized over a set of moving nominal system equilibria. In Chapters 5 and 6 we apply the proposed hierarchical nonlinear control framework to propulsion systems. First, however, we develop models for rotating stall and surge in axial and centrifugal flow compression systems that lend themselves to the application of nonlinear control design. Specifically, a self-contained first principles derivation of the governing fluid dynamic equa-

Preface

IX

tions for axial and centrifugal flow compression systems that is accessible to control-system designers requiring state space models for modern nonlinear control is developed. The hierarchical switching control framework is then applied to control rotating stall and surge in jet engine compression systems. To reflect a more realistic design we account for uncertainty in the pressureflow compressor performance characteristic map as well as impose a rate saturation constraint on the system actuator throttle opening. The appropriate background for this monograph is a first course in state space methods along with a first course on nonlinear systems at the level of Khalil [73]. Chapters 2-4 are suitable for an advanced course in nonlinear switching feedback control design while Chapters 5 and 6 are suitable for students and researchers having an interest in propulsion control. The results reported in this monograph were obtained at the School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, between September 1995 and December 1999. Financial support for this work was in part provided by the National Science Foundation under Grant ECS-9496249, the Air Force Office of Scientific Research under Grant F49620-96-1-0125, and the Army Research Office under Grant DAAH04-96-1-0008.

Boca Raton, Florida, USA, April 2000 Atlanta, Georgia, USA, April 2000 Columbia, Missouri, USA, April 2000

Alexander Leonessa Wassim M. Haddad VijaySekhar Chellaboina

Contents

List of Figures ................................................ .

.

Introduction ..............................................

1

1.1 1.2

1 4

Mathematical Preliminaries .............................. Generalized Stability Theorems .......................... Conclusion ............................................

Nonlinear System Stabilization via Hierarchical Switching Controllers ............................................... 3.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 3.3 3.4 3.5 3.6 3.7

.

N o n l i n e a r C o n t r o l Design: M o t i v a t i o n a n d O v e r v i e w . . . . . . . . B r i e f O u t l i n e of t h e M o n o g r a p h . . . . . . . . . . . . . . . . . . . . . . . . . .

G e n e r a l i z e d L y a p u n o v a n d I n v a r i a n t S e t T h e o r e m s for N o n linear Dynamical Systems ................................ 2.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 2.3 2.4

.

XIII

7 7 8 11 19

21 21

Mathematical Preliminaries .............................. P a r a m e t e r i z e d S y s t e m E q u i l i b r i a a n d D o m a i n s of A t t r a c t i o n . Nonlinear System Stabilization via a Hierarchical Switching Controller Architecture ..................................

23 24

E x t e n s i o n s to N o n l i n e a r D y n a m i c C o m p e n s a t i o n . . . . . . . . . . . Inverse O p t i m a l N o n l i n e a r S w i t c h i n g C o n t r o l . . . . . . . . . . . . . . Conclusion ............................................

35 39 46

N o n l i n e a r R o b u s t S w i t c h i n g C o n t r o l l e r s for N o n l i n e a r U n certain Systems ........................................... 4.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 M a t h e m a t i c a l P r e l i m i n a r i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 P a r a m e t e r i z e d N o m i n a l S y s t e m E q u i l i b r i a , S y s t e m A t t r a c tors, a n d D o m a i n s of A t t r a c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 R o b u s t N o n l i n e a r S y s t e m S t a b i l i z a t i o n v i a a H i e r a r c h i c a l Switching Controller Architecture ........................

26

47 47 48 49 51

XII

Contents

4.5 .

.

7.

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

H i e r a r c h i c a l S w i t c h i n g C o n t r o l for M u l t i - M o d e A x i a l F l o w Compressor Models ....................................... 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Governing Fluid Dynamic Equations for Axial Flow Compression Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Entrance Duct and Inlet Guide Vane Entrance . . . . . . . 5.2.2 Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Exit Duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Governing System Flow Equations . . . . . . . . . . . . . . . . . . 5.2.5 Plenum and Throttle Discharge . . . . . . . . . . . . . . . . . . . . 5.3 Multi-Mode State Space Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Finite Element Multi-Mode State Space Model . . . . . . . . . . . . . . 5.5 Control for Single-Mode versus Multi-Mode Model . . . . . . . . . . 5.6 Stabilization of Multi-Mode Axial Flow Compression System Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Robust Stabilization of Axial Flow Compressors with Uncertain Pressure-Flow Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Uncertain Finite Element Multi-Mode State Space Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Hierarchical Robust Control for Propulsion Systems .. 5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

H i e r a r c h i c a l S w i t c h i n g C o n t r o l for C e n t r i f u g a l F l o w C o m pressor Models ........................................... 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Governing Fluid Dynamic Equations for Centrifugal Compression Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Conservation of Mass in the Plenum . . . . . . . . . . . . . . . . 6.2.2 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Turbocharger Spool Dynamics . . . . . . . . . . . . . . . . . . . . . 6.3 Parameterized System Equilibria and Local Set Point Designs 6.4 Hierarchical Nonlinear Switching Control for Centrifugal Compression Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions ...............................................

57

59 59 62 62 68 69 70 71 72 75 77 80 84 85 87 95

97 97 98 98 100 104 105 107 115 117

Bibliography ..................................................

123

Index .........................................................

131

List of Figures

3.1 3.2

Switching controller architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hierarchical switching control strategy . . . . . . . . . . . . . . . . . . . . . . . .

21 34

4.1

Robust switching controller architecture . . . . . . . . . . . . . . . . . . . . . . .

48

5.1

Schematic of compressor characteristic map for a typical compression system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Compressor system geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Closed-loop state response for one-mode model: Backstepping controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Closed-loop state response for two-mode model: Backstepping controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Closed-loop state response for two-mode model: Switching nonlinear controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Closed-loop state response for two-mode model: Rate saturated versus rate unsaturated control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Control effort and control rate for two-mode model: Rate saturated versus rate unsaturated control . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Level set values kl~ and k2~ as functions of A . . . . . . . . . . . . . . . . . . 5.9 Actual and nominal compressor characteristics . . . . . . . . . . . . . . . . . 5.10 Controlled squared stall amplitudes, flow, and pressure versus time 5.11 Throttle opening versus time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

Centrifugal compressor system geometry . . . . . . . . . . . . . . . . . . . . . . Compressor characteristic maps and efficiency lines for different spool speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase portrait of pressure-flow map . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure rise versus time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass flow versus time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressor spool speed versus time . . . . . . . . . . . . . . . . . . . . . . . . . . Control effort versus time: Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control effort versus time: Throttle opening . . . . . . . . . . . . . . . . . . .

60 62 78 79 84 85 85 92 94 95 96 99 104 110 111 111 112 112 113

XIV

List of Figures

6.9 P h a s e p o r t r a i t of pressure-flow m a p . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 P r e s s u r e rise versus t i m e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 M a s s flow versus t i m e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

114 114 115

6.12 C o m p r e s s o r s p o o l s p e e d versus t i m e . . . . . . . . . . . . . . . . . . . . . . . . . .

115

6.13 C o n t r o l effort versus time: T o r q u e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.14 C o n t r o l effort versus time: T h r o t t l e o p e n i n g . . . . . . . . . . . . . . . . . . .

116 116

1. I n t r o d u c t i o n

1.1 N o n l i n e a r

Control

Design: Motivation

and Overview

Since all physical systems are inherently nonlinear with system nonlinearities arising from numerous sources including, for example, friction (e.g., Coulomb, hysteresis), gyroscopic effects (e.g., rotational motion), kinematic effects (e.g., backlash), input constraints (e.g., saturation, deadband), and geometric constraints, plant nonlinearities must be accounted for in the controlsystem design process. However, since nonlinear systems can exhibit multiple equilibria, limit cycles, bifurcations, jump resonance phenomena, and chaos, general nonlinear system stabilization is notoriously difficult mad remains an open problem. Control system designers have usually resorted to Lyapunov methods [137, 89, 73] in order to obtain stabilizing controllers for nonlinear systems. In particular, for smooth feedback, Lyapunov-based methods were inspired by Jurdjevic and Quinn [70] who give sufficient conditions for smooth stabilization based on the ability of constructing a Lyapunov function for the closed-loop system [134]. Unfortunately, however, there does not exist a unified procedure for finding a Lyapunov function candidate that will stabilize the closed-loop system for general nonlinear systems. This is further exacerbated when addressing robustness in uncertain nonlinear systems. Recent work involving differential geometric methods [25, 66, 26, 117, 28, 27] has made the design of controllers for certain classes of nonlinear systems more methodical. Such frameworks include the concepts of zero dynamics and feedback linearization and require that the system zero dynamics are asymptotically stable assuring the existence of globally defined diffeomorphisms to transform the nonlinear system into a normal form [66]. For this class of systems, feedback linearization techniques usually rely on cancelling out system nonlinearities using feedback and may therefore lead to inefficient designs since feedback linearizing controllers may generate unnecessarily large control effort to cancel beneficial system nonlinearities [47]. Furthermore, a serious drawback of all feedback linearization techniques is the failure to account for system uncertainty since exact cancellation of the nonlinear dynamics via feedback is required and hence an exact knowledge of the dynamics is as-

2

1. Introduction

sumed resulting in non-robust designs. Even though robustness frameworks to parametric uncertainty via feedback linearization techniques involving a two stage design consisting of nominal feedback linearization followed by additional state feedback designed to guarantee robustness have been developed [118, 126, 130, 131], the fact that such approaches do not directly account for system uncertainty can result in severe robustness problems with respect to nonlinear errors internal to the system dynamics. Furthermore, restrictive matching conditions are imposed to the structure of the uncertainty in order to address general feedback linearizable systems [129]. Backstepping control has also recently received a great deal of attention in the nonlinear control literature [72, 76, 77]. The popularity of this control methodology can be explained in a large part due to the fact that it provides a systematic procedure for finding a Lyapunov function for nonlinear closed-loop cascade systems. Furthermore, the controller is obtained in such a way that the nonlinearities of the dynamical system, which may be useful in attaining performance objectives, do not need to be cancelled as in state or output feedback linearization techniques. However, once again this approach is limited to strict-feedback systems [77, 119]. To address optimality issues within nonlinear control-system design, one may consider an optimal control problem in which a performance functional is minimized along the closed-loop system trajectories. This problem leads to the maximum principle [24, 74] which usually provides open-loop optimal control laws, characterized via nonlinear two-point boundary-value problems requiring iterative solution schemes. Alternatively, to obtain feedback optimal controllers, one may formulate the optimal control problem as a dynamic programming problem [21] by considering a value function [127], or return function [24], that minimizes a cost functional among all possible system trajectories. In this case, the value function is given by the solution to the Hamilton-Jacobi-Bellman equation [24, 74]. Computational methods for solving the Hamilton-Jacobi-Bellman partial differential equation have not been developed to a level comparable to solving Riccati equations arising in linear optimal control problems and, in general, it is very difficult to solve. In fact, solutions may not even exist unless one allows a generalized notion of viscosity solutions [19]. In order to avoid the complexity in solving the Hamilton-Jacobi-Bellman equation one may consider an inverse optimal control problem [105, 47, 119] where one does not attempt to minimize a given cost functional, but rather, minimizes a derived cost functional. The basic underlying idea of inverse optimality is the fact that the steady-state solution to the Hamilton-Jacobi-Bellman equation is a Lyapunov function for the nonlinear controlled system [22, 67]. However, in this case the complexity of solving the Hamilton-Jacobi-Bellman equation is shifted to finding a Lyapunov

I.I Nonlinear Control Design: Motivation and Overview

3

function for the closed-loop system limiting this approach to strict-feedback systems. Furthermore, the performance of inverse optimal controllers can be arbitrarily poor when compared to the optimal performance as measured by a designer specified cost functional. If the operating range of the control system is small and if the system nonlinearities are smooth, then the control system can be locally approximated by a linearized system about a given operating condition and linear multivariable control theory can be used to maintain local stability and performance. However, in high performance engineering applications such as advanced tactical fighter aircraft and variable-cycle gas turbine aeroengines, the locally approximated linearized system does not cover the operating range of the system dynamics. In this case, gain scheduled controllers can be designed over several fixed operating points covering the system's operating range and controller gains interpolated over this range [132, 115]. However, due to approximation linearization errors and neglected operating point transitions, the resulting gain scheduled system does not have any guarantees of performance or stability. Even though stability properties of gain scheduled controllers are analyzed in [122, 84] and stability guarantees are provided for plant output scheduling, a design framework for gain scheduling control guaranteeing system stability over an operating range of the nonlinear plant dynamics has not been addressed in the literature. In an attempt to develop a design framework for gain scheduling control, linear parameter-varying system theory has been developed [123, 124, 121, 108]. Since gain scheduling control involves a linear parameter-dependent plant, linear parameter-varying methods for gain scheduling seem natural. However, even though this is indeed the case for linear dynamical systems involving exogenous parameters, this is not the case for nonlinear dynamical systems. This is due to the fact that a nonlinear system cannot be represented as a true linear parameter-varying system since the varying system parameters are endogenous, that is, functions of the system state. Hence, stability and performance guarantees of linear parameter-varying systems do not extend to the nonlinear system. Of course, in the case where the magnitude and rate of the endogenous parameters are constrained such that the linear parameter-varying system hopefully behaves closely to the actual nonlinear system, then stable controllers can be designed using quasi-linear parameter-varying representations [80]. However, in the case of unexpectedly large amplitude uncertain exogenous disturbances and/or system parametric uncertainty, a priori assumptions on magnitude and rate constraints on endogenous parameters are unverifiable. For nonlinear controlled dynamical systems null controllability does not in general imply continuous or smooth stabilizability [10, 113]. Variable struc-

4

1. Introduction

ture control is perhaps the quintessential discontinuous nonlinear stabilization approach for controlling nonlinear systems (see [135, 39] and the numerous references therein). In particular, variable structure control utilizes a high-speed switching control law to drive the system state trajectories onto an invariant sliding manifold. In order to establish the existence of a sliding mode, the controller is constructed such that the system state velocity vector is directed towards the switching surface. In an ideal case, this results in infinitely fast switching. However, in practical implementation, infinitely fast switching leads to high frequency chattering which may excite unmodeled high frequency system dynamics leading to system instabilities. In an attempt to overcome this problem, boundary layer controllers which continuously approximate the discontinuous control action in a neighborhood of the switching surface have been proposed [39]. However, in this case the resulting controller does not guarantee asymptotic stability but rather uniform ultimate boundedness [39]. Finally, variable structure controllers are generally limited to nonlinear affine systems.

1.2 B r i e f O u t l i n e

of the Monograph

The main objective of this monograph is to develop a general nonlinear control design methodology for nonlinear uncertain systems with input saturation constraints. The results are then applied to the control of rotating stall and surge in jet engine compression systems. The main contents of the monograph are as follows. In Chapter 2 we develop generalized Lyapunov and invariant set theorems for nonlinear dynamical systems wherein all regularity assumptions on the Lyapunov function and the system dynamics are removed. In particular, local and global stability theorems are presented using generalized Lyapunov functions that are lower semicontinuous. In the case where the generalized Lyapunov function is taken to be a C 1 function, our results collapse to the standard Lyapunov stability and invariant set theorems. The present formulation provides new invariant set stability theorem generalizations by explicitly characterizing system limit sets in terms of lower semicontinuous Lyapunov functions. The generalized Lyapunov and invariant set theorems developed in Chapter 2 are used in Chapter 3 to develop a nonlinear control design framework predicated on a hierarchical switching controller architecture. Specifically, using equilibria-dependent Lyapunov functions, or instantaneous (with respect to a given parameterized equilibrium manifold) Lyapunov functions, a hierarchical nonlinear control strategy is developed that stabilizes a given nonlinear system using a supervisory switching controller that coordinates lower-level

1.2 Brief Outline of the Monograph

5

stabilizing subcontrollers. The hierarchical switching nonlinear controller architecture is developed based on a generalized lower semicontinuous Lyapunov function obtained by minimizing a potential function, associated with each domain of attraction, over a given switching set induced by the parameterized system equilibria. In Chapter 4 the hierarchical switching nonlinear controller architecture is extended to address the problem of robust stabilization for nonlinear uncertain systems. In Chapter 5 we develop a self-contained first principles derivation of a multi-mode model for rotating stall and surge in axial flow compression systems, that is accessible to control-system designers requiring state space models for modern nonlinear control design. Then, we apply the hierarchical switching nonlinear control framework to mitigate the aerodynamics instabilities of rotating stall and surge in multi-mode axial compressor models while accounting for the effects of system parametric uncertainty and a rate saturation constraint on the system actuator throttle opening. In Chapter 6 we address the problem of nonlinear stabilization for centrifugal compression systems. First, we obtain a three-state lumped parameter model for surge in centrifugal flow compression systems. The low-order centrifugal compression system model presented involves pressure and mass flow compression system dynamics using principles of conservation of mass and momentum. Furthermore, in order to account for the influence of speed transients on the compression surge dynamics, turbocharger spool dynamics are also considered. Next, we develop globally stabilizing control laws for the lumped parameter centrifugal compressor surge model with spool dynamics using the nonlinear switching control framework. Inverse optimal nonlinear switching controllers are also developed. Both switching nonlinear control frameworks are directly applicable to centrifugal compression systems with amplitude and rate saturation constraints. Finally, in Chapter 7 conclusions are discussed.

2. Generalized Lyapunov and Invariant Set Theorems for Nonlinear Dynamical Systems

2.1 Introduction One of the most basic issues in system theory is stability of dynamical systems. The most complete contribution to the stability analysis of nonlinear dynamical systems is due to Lyapunov [95l. Lyapunov's results, along with the Barbashin-Krasovskii-LaSalle invariance principle [18, 75, 82], provide a powerful framework for analyzing the stability of nonlinear dynamical systems. In particular, Lyapunov's direct method can provide local and global stability conclusions of an equilibrium point of a nonlinear dynamical system if a smooth (at least C 1) positive definite function of the nonlinear system states (Lyapunov function) can be constructed for which its time rate of change due to perturbations in a neighborhood of the system's equilibrium is always negative or zero, with strict negative definiteness ensuring asymptotic stability. Alternatively, using the Barbashin-Krasovskii-LaSalle invariance principle [18, 75, 82] the strict negative definiteness condition on the Lyapunov derivative can be relaxed while assuring asymptotic stability. In particular, if a smooth function defined on a compact invariant set with respect to the nonlinear dynamical system can be constructed whose derivative along the system's trajectories is negative semidefinite and no system trajectories can stay indefinitely at points where the function's derivative identically vanishes, then the system's equilibrium is asymptotically stable. Most Lyapunov stability and invariant set theorems presented in the literature require that the Lyapunov function candidate for a nonlinear dynamical system be a C 1 function with a negative-definite derivative (see [59, 71, 73, 83, 137, 141] and the numerous references therein). This is due to the fact that the majority of the dynamical systems considered are systems possessing continuous dynamics and hence Lyapunov theorems provide stability conditions that do not require knowledge of the system trajectories [59, 71, 73, 83, 137, 141]. However, in light of the increasingly complex nature of dynamical systems such as biological systems [90], hybrid systems [140], sampled-data systems [58], discrete-event systems [110], gain scheduled systems [111, 96, 86], and constrained mechanical systems [11], system

8

2. Generalized Lyapunov and Invariant Set Theorems

discontinuities arise naturally. Even though standard Lyapunov theory is applicable for systems with discontinuous system dynamics and continuous motions, it might be simpler to construct discontinuous "Lyapunov" functions to establish system stability. For example, in gain scheduled control it is not uncommon to use several different controllers designed over several fixed operating points covering the system's operating range and to switch between them over this range. Even though for each operating range one can construct a C 1 Lyapunov function, to show closed-loop system stability over the whole system operating envelope for a given switching control strategy, a generalized Lyapunov function involving combinations of the Lyapunov functions for each operating range can be constructed [86, 96, 111]. However, in this case, as shown in Chapter 3, the generalized Lyapunov function is non-smooth and non-continuous [86, 96, 111]. In this chapter we develop generalized Lyapunov and invariant set theorems for nonlinear dynamical systems wherein all regularity assumptions on the Lyapunov function and the system dynamics are removed. In particular, local and global stability theorems are presented using generalized Lyapunov functions that are lower semicontinuous. Furthermore, generalized invariant set theorems are derived wherein system trajectories converge to a union of largest invariant sets contained on the boundary of the intersections over finite intervals of the closure of generalized Lyapunov level surfaces. In the case where the generalized Lyapunov function is taken to be a C 1 function, our results collapse to the standard Lyapunov stability and invariant set theorems. Finally, we note that nondifferentiable Lyapunov functions have been considered in the literature. Specifically, continuous and lower semicontinuous Lyapunov functions have been considered in [23, 8, 140, 6, 7, 9], with [8, 7, 9] focusing on viability theory and differential inclusions. Furthermore, significant extensions of LaSalle's invariance principle for continuous Lyapunov functions are developed in [29, 116]. However, the present formulation provides new invariant set stability theorem generalizations not considered in [8, 29, 116] by explicitly characterizing system limit sets in terms of lower semicontinuous Lyapunov functions.

2.2 Mathematical

Preliminaries

In this section we establish definitions, notation, and several key results used later in the chapter. Let R denote the set of real numbers, let Rn denote the set of n x 1 real column vectors, and let (.)w denote transpose. Furthermore, let OS, S, and 8 denote the boundary, the interior, and the closure of the set S C Rn , respectively. A set S C_ R'* is connected if there does not exist open sets 01 and 02 in 1~n such that S C O1UO2, SNOl ~ 0, SnO2 ~ 0,

2.2 Mathematical Preliminaries

9

and S n O1 N 02 = O. Recall that S is a connected subset of R if, and only if, S is either an interval or a single point. Let [[. [[ denote the Euclidean vector norm, let V'(x) denote the gradient of V at x, and let D + V ( x ) denote the lower Dini derivative of V at x [114]. Finally, let C o denote the set of continuous functions and C r denote the set of functions with r-continuous derivatives. In this chapter we consider the general nonlinear dynamical system = f(x(t)),

z(0) = zo,

t

Zxo,

(2.1)

where z(t) q 79 C R", t E Zxo, is the system state vector, Ixo C_ lit is the maximal interval of existence of a solution x(.) of (2.1), 79 is an open set, 0 E D, and f : 79 ~ R". A function z : Ixo -~ D is said to be a solution to (2.1) on the interval Zxo C_ R with initial condition x(0) = xo, if z(t) satisfies (2.1) for all t E Ixo. Note that we do not assume any regularity condition on the function f('). However, we do assume that for every y E 7) there exists a unique solution z(.) of (2.1) defined on Z~ satisfying x(0) = y. Furthermore, we assume that all the solutions x(t), t E Zxo, to (2.1) are continuous functions of the system initial conditions x0 E 7) which, with the assumption of uniqueness of solutions, implies continuity of solutions x(t), t e Zxo, to (2.1) [60, p. 24]. If f(.) is Lipschitz continuous on 79 then there exists a unique solution to (2.1). In this case, the semi-group property s(t + v, x0) = s(t,s(r, Xo)), t,T E Zzo, and the continuity of s(t,.) on 79, t E Zxo, hold, where s(.,Xo) denotes the solution of the nonlinear dynamical system (2.1) with initial condition x(0) = x0. Alternatively, uniqueness of solutions in time along with the continuity of f(.) ensure that the solutions to (2.1) satisfy the semigroup property and are continuous functions of the initial condition x0 E 79 even when f(-) is not Lipschitz continuous on 79 (see [35, Theorem 4.3, p. 59]). More generally, f(.) need not be continuous. In particular, if f(-) is discontinuous but bounded and x(-) is the unique solution to (2.1) in the sense of Filippov [44], then the semi-group property along with the continuous dependence of solutions on initial conditions hold [44]. Next, we introduce several definitions and two key results that are necessary for the main results of this chapter. Definition 2.1. Let De C_ 79 and let V : De ~ R. For a E ~,, the set V - l ( a ) -~ {x E 79c : V ( x ) = a} is called the a-level set. For a , ~ e ~, a < 8, the set Y - l ( [ a , fl]) a= {x e 79c : a < V ( x ) < 1~} is called the [a,/?]-sublevel set. D e f i n i t i o n 2.2. The trajectory x(t) e 79 C R n, t E Zxo, of (2.1) denotes the solution to (2.1) corresponding to the initial condition x(O) = Xo evaluated

10

2. Generalized Lyapunov and Invariant Set Theorems

at time t. The trajectory z(t), t E I,o, is bounded on ~ o C_ R if there exists 7 > 0 such that IIx(t)ll < "y, t E :r,o. Definition 2.3. A set J~4+ C_ l) C_ R n (resp., Jk4-) is a positively (resp., negatively) invariant set for the nonlinear dynamical system (~.1) if zo 6 .~vt+ (resp., .Vt-) implies that [0, +oo) C_ 27~o (resp., (-oo,0] C_ 77xo) and z(t) E J~+ (resp., . M - ) for all t > 0 (resp., t < 0). A set All C_ ~D C_ R" is an invariant set for the nonlinear dynamical system (~. I) if xo E JVl implies that Y',o = R and z(t) E .Vt for all t E R. Definition 2.4. p E ~ C R n is a positive limit point of the trajectory z(t), t E Z~o, if [0, +oo) C_ Sxo and for all 6 > 0 and finite time T > 0 there exists t > T such that IIx(t) - Pll < e. The set of all positive limit points of z(t), t E ~,o, is the positive limit set 79+ ofz(t), t E I~ o . Note that IIx(t) - pll < e for all e > 0 and t > T > 0 is equivalent to the CO existence of a sequence {t,},=o, with t , ~ c~ as n ~ ~ , such that z( t , ) -~ p as n ~ oo. The following result on positive limit sets is fundamental and forms the basis for all later developments. L e m m a 2.1 responding to positive limit invariant set.

([73]). Suppose the forward solution z(t), t > O, to (2.1) coran initial condition z(O) = Zo exists and is bounded. Then the set 79+0 of z(t), t E Ixo, is a nonempty, compact, connected Furthermore, z(t) ~ 79+0 as t ~ oo.

It is important to note that Lemma 2.1 holds for time-invariant nonlinear dynamical systems (2.1) possessing unique solutions with solutions being continuous functions of the system initial conditions. More generally, Lemma 2.1 holds if s(t + r, zo) = s(t, s(r, Zo)), t, r E Izo, and s(., Xo) is a continuous function of x0 E D. Finally, Lemma 2.1 also holds for all discrete-time, timeinvariant nonlinear dynamical systems that have (unique) solutions which are continuously dependent on the system initial conditions. Of course, in this case, 79+o is not connected. The following definition introduces three types of stability as well as attraction of (2.1) with respect to a compact positively invariant set. Definition 2.5. Let Do C D be a compact positively invariant set for the nonlinear dynamical system (2.1). Do is Lyapunov stable if ]or every open neighborhood 01 C_ l) of 7~o, there exists an open neighborhood O~ C_ 01 of Do such that x(t) E 01, t > O, for all zo E 02. Do is attractive if there exists an open neighborhood 03 C D of Do such that 79+0 C_ Do for all Xo E 03. Do is asymptotically stable if it is Lyapunov stable and attractive. Do is globally asymptotically stable if it is Lyapunov stable and 79+0 C_ Do for all xo E ~". Finally, Do is unstable if it is not Lyapunov stable.

2.3 Generalized Stability Theorems

Ii

Next, we give a set theoretic definition involving the domain, or region, of attraction of the compact positively invariant set 1)c of (2.1). Definition 2.6. Suppose the compact positively invariant set l)o C l) of (2.1) is attractive. Then the domain of attraction 1)^ of 1)o is defined as =

{xo 9

1) :

P-o c_1)o}.

(2.2)

Recall that 1)^ is an open, connected invariant set [23, Proposition 4.15,

p. ssl. Next, we present a key theorem due to Weierstrass involving lower semicontinuous functions on compact sets. For the statement of this result the following definition is needed. Definition 2.7. Let De C l). A function V : Dc -~ R is lower semicontinoo uous on 1)e if for every sequence {xn}n=o C De such that limn-~oo Xn = X, Y(x) < lim infn-.cr V(xn).

Note that a bounded function V : De --+ R is lower semicontinuous at x e :De if, and only if, for each e > 0 there exists 6 > 0 such that IIx - Yll < 6, y 9 De, implies V(x) - V(y) < ~. Furthermore, if :De C D is compact and V : De -~ R is lower semicontinuous at x 9 :De, then for each a 9 R the set {x 9 De : Y(x) < a} is compact. T h e o r e m 2.1 ([114]). Suppose De C D is compact and V : De -+ R is lower semieontinuous. Then there exists x* e De such that V(x*) < V(x), z e1)e.

2.3 Generalized

Stability

Theorems

As discussed in Section 2.1, most Lyapunov stability theorems presented in the literature require that the Lyapunov function candidate for a nonlinear dynamical system be a C 1 function with a negative-definite derivative along the system trajectories. In this section, we present several generalized stability theorems where we relax both of these assumptions while guaranteeing local and global stability of a nonlinear dynamical system. The following result gives sufficient conditions for Lyapunov stability of a compact positively invariant set with respect to a nonlinear dynamical system. T h e o r e m 2.2. Consider the nonlinear dynamical system (2.1), let Doo be a compact positively invariant set with respect to (2.1) such that Do C D, and let x(t), t E Y-xo, denote the solution to (2.1) with Xo E l). Assume that

12

2. Generalized Lyapunov and Invariant Set Theorems

there exists a lower semicontinuous function V : D -~ R such that V(.) is continuous on Do and v ( z ) = o,

x

V(x) > O,

x E 79,

vcx(t)) < v(x(T)),

(2.3)

Vo,

x ~. 790, 0
0 since 790 O 0Oz = 0 and V(x) > 0, x E 79, x r :Do. Next, using the facts that V(x) = 0, x e :Do, and V(.) is continuous on D0, it follows that the set 02 _a {x E O1 : V(x) < a} ~ is not empty. Now, it follows from (2.5) that for all x(0) E O2, v ( x ( t ) ) < v(x(0)) < a,

t > 0,

which, since V(x) > a, x E OOx, implies that x(t) r 001, t >_ O. Hence, for every open neighborhood O1 C_ 79 of 790, there exists an open neighborhood 02 C_ O1 of 79o such that, if x(0) E 02, then x(t) E Oz, t >_O, which proves Lyapunov stability of the compact positively invariant set 79o of (2.1). 9 A lower semicontinuous function V(.), with V(-) being continuous on :Do, satisfying (2.3) and (2.4) is called a generalized Lyapunov function candidate for the nonlinear dynamical system (2.1). If, additionally, V(.) satisfies (2.5), V(.) is called a generalized Lyapunov function for the nonlinear dynamical system (2.1). Note that in the case where the function V(.) is C z on 79 in Theorem 2.2, it follows that V(x(t)) r > 0, is equivalent to ~'(x) ~ W ( x ) f ( x ) _< 0, x E 79. In this case conditions (2.3)(2.5) in Theorem 2.2 specialize to the standard Lyapunov stability conditions [59, 73, 83]. Next, we generalize the Barbashin-Krasovskii-LaSalle invariant set theorems [18, 75, 82, 83, 73] to the case in which the function V(.) is lower semicontinuous. For the remainder of the results of this chapter define the notation 7~7 ~ n V-Z([%c]), c>'y

(2.6)

for arbitrary V : 79 C_ IRn --+ R and 7 E R, and let 2~4.y denote the largest invariant set (with respect to (2.1)) contained in T&r T h e o r e m 2.3. Consider the nonlinear dynamical system ($.1), let x(t), t E Zxo, denote the solution to (2.1), and let 79c C 79 be a compact positively invariant set with respect to (2.1). Assume that there exists a lower

2.3 Generalized Stability Theorems

13

semicontinuous function V : De --r R such that V(x(t)) < V ( x ( r ) ) , for all 7" E [0, t] and xo E De. If Xo E De, then z(t) ~ ./vt a UT~R .s as t ~ c~. P r o o f . Let x(t), t E Zzo, be the solution to (2.1) with xo E De so that [0, +c~) _C Zxo. Since V(.) is lower semicontinuous on the compact set :De, there exists/~ E IR such that V(x) > 8, x E :De. Hence, since V(x(t)), t > O, is nonincreasing, 7xo & limt~r162 V(x(t)), :Co E De, exists. Now, for all p E 79+ there exists an increasing unbounded sequence {tn}n~__o,with to = 0, such that lim,,_.~ x(tn) = p. Next, since V(z(tn)), n > O, is nonincreasing it follows that for all n > O, %0 < V(x(t~)) < V ( x ( t ~ ) ) , n > N , or, equivalently, since De is positively invariant, Z(tn) E V-l([7~o, V(X(tN))]), n > N. Now, since l i m , ~ x ( t n ) = p it follows that p E V-l([f~o,V(x(tn))]), n > O. Furthermore, since l i m n - ~ V(x(t,,)) = 7~Q it follows that for every c > 7xo, there exists n > 0 such that 7xo _< V(x(tn)) < c which implies that for every c > 7~o, P E V-1([7~o,c]). Hence, p E ~7-o which implies that :P+o C_ 7~.r.~ . Now, since De is compact and positively invariant it follows that the forward solution x(t), t > O, to (2.1) is bounded for all xo E De and hence it follows from Lemma 2.1 that 79+ is a nonempty compact connected invariant set which further implies that 79~+is a subset of the largest invariant set contained in ~'r.o, that is, 79+0 C_ " ~ - o " Hence, for all xo E De, 79+0 C_ .s Finally, since x(t) ---r 79+0 as t ~ oo it follows that x(t) ~ .s as t ~ oo. 9 If, in Theorem 2.3, Jk4 contains no other trajectory other than the trivial trajectory x(t) ~ O, then the zero solution x(t) =- 0 to (2.1) is attractive and De is a subset of the domain of attraction. Furthermore, note that if V : :De -r IR is a lower semicontinuous function such that all the conditions of Theorem 2.3 are satisfied, then for every x0 E :De there exists q'~o -< V(xo) such that 79+ _C " ~ - o C .s Finally, since Y -1 ([7, c]) = {x E De : V(x) > 3'} n {x E 7)r : V(x) < c} and {x E Dc : V(x) < c} is a closed set, it follows that 7~.r,~ C {x E :De : V(x) < 7}, where 7~7,~ =a V_a([7,c]) \ V_1([7,c]) ' c > 7, for a fixed 7 E IR. Hence,

where 7~7 =~ he>7 7~,c, is such that V ( x ) < 7, x E 7~.~. Finally, if V(.) is C O then 7~.r,c = O, "r e ~ c > 3', and hence 7~.y = V -1 (7). It is important to note that even though the stability conditions appearing in Theorem 2.3 are system trajectory dependent, in Chapter 3 a hierarchical switching nonlinear control strategy is developed using Theorem 2.3 without requiring knowledge of the system trajectories. Furthermore, note that as in standard Lyapunov and invariant set theorems involving C 1 functions, Theorem 2.3 allows one to characterize the invariant set . ~ without knowledge

14

2. Generalized Lyapunov and Invariant Set Theorems

of the system trajectories x(t), t E Zx o. Similar remarks hold for the rest of the theorems in this section. To illustrate the utility of Theorem 2.3 consider the simple scalar nonlinear dynamical system given by ~(t) = - x ( t ) ( x ( t ) - 1)(xCt) + 2),

x(O) = x0,

t > O,

(2.7)

with generalized Lyapunov function candidate V ( x ) given by (x + 2 ) 2 , x < 0,

V(x)=

( x - l ) 2,x>_0.

Now, note that

D+v(.)[-.(. =

- 1)(x + 2)]

- 2 x ( x - l)(x + 2) 2, x < O, < O, - 2 x ( z - l)2(x + 2), z >__O, -

X

E R,

which implies that V(x(t)), t >_ O, is nonincreasing along the system trajectories. Next, note that ~ = V-1(7), 7 E R\{4}, and T~4 = V-1(4)O{0}. Since the only invariant sets contained in 7"&r are the equilibrium points xel = - 2 , xe2 = 0, xes = 1, it follows that )vl~ = O, 3' ~ {0, 1, 4}, r = {-2, 1}, ~4x = {0}, and .~,44 = {0} which implies that .~4 = {-2, 0, 1}. Hence, it follows from Theorem 2.3 that for every x0 E R the solution to (2.7) approaches the invariant set .A~ = {-2, 0, 1} as t --roo which can be easily verified. As shown by the above example, Theorem 2.3 allows for a systematic way of constructing system Lyapunov functions by piecing together a collection of functions. The following corollary to Theorem 2.3 presents sufficient conditions that guarantee local asymptotic stability of a compact positively invariant set with respect to the nonlinear dynamical system (2.1). Corollary 2.1. Consider the nonlinear dynamical system (2.1), let l)c and 79o be compact positively invariant sets with respect to (2.1) such that 790 C o

D o Dc C D, and let x(t), t E :T.xo, denote the solution to (2.1) corresponding to ~o E De. Assume that there exists a lower semicontinuous function V : Dc --+ R such that V(.) is continuous on :Do and V(x) = 0,

x E 790,

V(x) > 0,

x E Dc,

V(z(t)) < V(z(r)),

(2.8) x r :Do,

0 < r < t,

(2.9) (2.10)

and assume that .Ad a__ nT>o .A4.r C_ Do. Then :Do is locally asymptotically stable and De is a subset of the domain of attraction.

2.3 Generalized StabilityTheorems

15

P r o o f . The result is a direct consequence of Theorems 2.2 and 2.3.

9

Next, we specialize Theorem 2.3 to the Barbashin-Krasovskii-LaSalle invariant set theorem wherein V(.) is a C 1 function. C o r o l l a r y 2.2. Consider the nonlinear dynamical system (2.1), let Dc C Z) be a compact positively invariant set with respect to (2.1) and let z(t), t E Zzo, denote the solution to (2.1) corresponding to xo E De. Assume that there exists a C 1 function V : De -4 R such that W ( x ) f ( x ) r. Now, since V(.) is C 1 it follows that 7 ~ = V-1(7), ~, E R. In this case, it follows from Theorem 2.3 that for every Xo E De there exists %0 E R such that 79+ C_ M~.o, where M ~ . o is the largest invariant set contained in 7 ~ . o = V-1(%o) which implies that V ( x ) = 7zo, x E P + . Hence, since M ~ . o is an invariant set it follows that for all x(0) E M ~ . o, x(t) E Jt4~, o, t _> 0, and thus V'(x(0)) =t~ dV(z(t))dt t = O

"-

V'(x(O))f(x(O)) = 0, which implies the A4~. o is contained in A4 which is the largest invariant set contained in T~. Hence, since x(t) ~ 79+0 C Jt4 as t -~ cx~, it follows that x(t) --~ ./t4 as t --~ 0o. 9 Next, we sharpen the results of Theorem 2.3 by providing a refined construction of the invariant set M . In particular, we show that the system trajectories converge to a union of largest invariant sets contained on the boundary of the intersections over finite intervals of the closure of generalized Lyapunov level surfaces. T h e o r e m 2.4. Consider the nonlinear dynamical system (2.1), let l)c and Do be compact positively invariant sets with respect to (2.1) such that Do C Dc C D, and let x(t), t E Zz o, denote the solution to (2.1) corresponding to Xo E Dc. Assume that there exists a lower semicontinuous function V : 7)c --~ R such that

V ( x ) -- 0,

x E Do,

V ( x ) > O,

x E Dc,

V(x(t)) < V ( x ( r ) ) ,

0

(2.11)

x ~. 7)0,

(2.12)

t.

(2.13)

< v
O. Furthermore, if zo E Dc, then x(t) --} )Q ~= U.reg/v[~ as t ~ oo, where ~ ~= {'y > 0 : o

R~ N Do ~ qg}. If, in addition, Do C De and V(.) is continuous on Do then 7)o is locally asymptotically stable and De is a subset of the domain of attraction. P r o o f . Since De is a compact positively invariant set, it follows that for all x0 E De, the forward solution x(t), t > O, to (2.1) is bounded. Hence, it follows from Lemma 2.1 that, for all x0 E De, P+o is a nonempty, compact, connected invariant set. Next, it follows from Theorem 2.3 and the fact that V(.) is positive definite (with respect to :De \Do), that for every Xo E De there exists 7xo >_ 0 such that P+o C_ 2~4~.~ C_ 7s ~ . Now, given x(0) 9 V - l ( f x o ) , 7zo > 0, (2.14) implies that there exists tl > 0 such that V(x(tl)) < %0 and x(tl) r V-a(%o). Hence, V-X(Txo) C T&r.o does not contain any invariant set. Alternatively, if x(0) 9 7 ~ . ~ then V(x(O)) < "rxo and (2.14) implies that x(t) r V -x (%o), t >_ 0. Hence, any invariant set contained in 7~7. o is a subset of 7~.r.o, which implies that Jvt.r. o C 7~. o , 7xo > 0. If "~ > 0 is such that r %0, for all Xo 9 :De, then there does not exist xo 9 De such that P+o C_ 7~q and hence A,t~ = O. Now, ad absurdum, suppose Do N79+o = O. Since V(.) is lower semicontinuous it follows from Theorem 2.1 that there exists k 9 P+o such that a = Y(~) < V(x), x 9 79+o" Now, with x(0) = ~ r Do it follows CO from (2.14) that there exists an increasing unbounded sequence {tn}n=o, with to = 0, such that V(x(tn+l)) < V(x(tn)), n = 0, 1,..., which implies that there exists t > 0 such that V(x(t)) < a which further implies that x(t) r 79+0 contradicting the fact that 79+0 is an invariant set. Hence, there exists q 9 Do such that q 9 79+ C_ Ts ~ which implies that ~7*o N Do r O. Thus, 7zo 9 ~ for all Xo 9 De which further implies that 79x+ C_ JQ. Now, since x(t) -r 79+0 C_r as t --roo it follows that x(t) ~ M as t -~ oo. o

If V(.) is continuous on Do C De, then Lyapunov stability of the compact positively invariant set Do follows from Theorem 2.2. Furthermore, from the continuity of V(.) on :Do and the fact that V(x) = 0 for all x 9 Do, it follows that ~ = {0} and )Q = .Mo. Hence, 79+0 C_ Do for all xo 9 De establishing local asymptotic stability of the compact positively invariant set Do of (2.1) with a subset of the domain of attraction given by De. 9 In all of the above results we explicitly assumed that there exists a compact positively invariant set 7)c C 7) with respect to (2.1). Next, we provide a result that does not require the existence of such :De.

2.3 Generalized Stability Theorems

17

T h e o r e m 2.5. Consider the nonlinear dynamical system (2.1), let 79o be a compact positively invariant set with respect to (2.1) such that Do C 19, and let x(t), t E Zzo, denote the solution to (~.I) corresponding to xo E 7). Assume that there exists a lower semicontinuous ]unction V : 79 ~ IR such that

v ( x ) = o,

x e 790,

v ( ~ ) > o,

x e 79,

v ( , ( t ) ) _< v(,(7-)),

(2.15) x r Do,

0 < 7- < t.

(2.16) (2.17)

Then all forward solutions x(t), t > O, to (2.1) that are bounded approach .M _a Or>o M r as t --r co. If, in addition, for all xo E D, Xo r Do, there oo exists an increasing unbounded sequence {tn}~=o, with to = O, such that

n = 0, 1,...,

V(x(tn+,)) < V(x(tn)),

(2.18)

then, either M-t C 7~r ~ R r \ V-X(')'), or M r = O, 7 > O. Furthermore, all .forward solutions x(t), t > O, to (2.1) that are bounded approach 2(4 ~= Ur~ M r as t ~ c~, where ~ a= {7 > 0 : 7~ n Do ~ O}.

Proof. The proof is a direct consequence of Theorems 2.3 and 2.4 with 9c given by the union of all bounded trajectories of (2.1). 9 Next, we present a generalized global invariant set theorem for guaranteeing global attraction and global asymptotic stability of a compact positively invariant set of a nonlinear dynamical system. T h e o r e m 2.6. Consider the nonlinear dynamical system (2.1) with D = Rn and let x(t), t E Ixo, denote the solution to (2.1) corresponding to Xo E ~n Assume that there exists a compact positively invariant set Do with respect to (2.1) and a lower semieontinuous fanction V : R n -~ R such that V(x) = O,

x E 790,

V ( x ) > 0,

x EtRn,

V(x(t)) < V(x(7-)),

y(x)

- ~ c o a s tlzll ~

x r 7)0,

(2.19) (2.20)

0 < 7" < t,

(2.21)

o0.

(2.22)

Then for all xo E ~n, x(t) -r J~4 ~- Ur_>0A/Ir, as t -~ co. If, in addition, for all Xo E ~n, xo ~- 790, there exists an increasing unbounded sequence {t n}n=0, with to = O, such that

v(~(t~+~)) < v(~(t,)),

~ = o, 1 , . . . ,

(2.23)

then, either .~4r C 7~r ~ 7~ \ V-1(7), or .M r = 0, 7 > O. Furthermore, x(t) -~ .~4 ~= U-regMr as t --r oo, where ~ ~= {3' > 0 : 7~-tDT)o ~ O}.

18

2. Generalized Lyapunov and Invariant Set Theorems

Finally, if V (.) is continuous on Do then the compact positively invariant set I)o of (~.I) is globally asymptotically stable. P r o o f . Note that since V(z) -+ co as [[x[[ -+ co it follows that for every /3 > 0 there exists r > 0 such that V(x) >/3 for all Ilxl[ > r, or, equivalently, v-x([0,/3]) c_ {x: Ilxll < r} which implies that V-1([0,/3]) is bounded for all /3 > 0. Hence, for all xo E R", V-l([0,/3zo]) is bounded, where/3xo ~ V(zo). Furthermore, since V(.) is a positive-definite lower semicontinuous function, it follows that V -x ([0,/3xo]) is closed and, since V(x(t)), t > O, is nonincreasing, V -x ([0,/3xo]) is positively invariant. Hence, for every xo ERn, V-l([0,/3xo]) is a compact positively invariant set. Now, with De = V -x ([0,/3xo]) it follows from Theorem 2.3 that there exists %0 E [0,/3ffio] such that Pz+o C_ .~vl~.~ C 7~o o which implies that z(t) -+ r as t -+ co. If, in addition, for all zo E Rn , zo ~ :Do, there exists an increasing unbounded sequence {tn}n=O, with to = 0, such that (2.23) holds then it follows from Theorem 2.4 that x(t) -+ . ~ as t-+co. Finally, if V(.) is continuous on Do then Lyapunov stability follows as in the proof of Theorem 2.4. Furthermore, in this case, G -- {0} which implies that f14 = A4o. Hence, P + C_ :Do establishing global asymptotic stability of the compact positively invariant set :Do of (2.1). 9 OO

^

If in Theorem 2.4 (resp., Theorem 2.6) the function V(.) is C 1 on De (resp., Rn), :Do =- {0}, and Y'(x)f(x) < O, x E De (resp., x E R"), x # 0, then every increasing unbounded sequence {tn}n~__0, with to = 0, is such that V(x(tn+x)) < V(x(tn)), n = 0,1, .... In this case, Theorems 2.4 and 2.6 specialize to the standard Lyapunov stability theorems for local and global asymptotic stability, respectively. Note that the results in this chapter also hold for nonlinear discrete-time dynamical systems described by time-invariant difference equations whose (unique) solutions are continuous functions of the initial conditions. Specifically, in this case all of the above results and proofs proceed exactly as in the continuous-time case by replacing t E IR with k E Z, where 7~ is the set of integers. It is important to note that even though the stability conditions appearing in Theorems 2.3-2.6 are system trajectory dependent, in Chapter 3 we present a hierarchical switching nonlinear controller guaranteeing nonlinear system stabilization without requiring knowledge of the closed-loop system trajectories. Finally, we note that the concept of lower semicontinuous Lyapunov functions has been considered in the literature. Specifically, lower semicontinuous Lyapunov functions have been considered in [23, 8], with [8] focusing on viability theory and differential inclusions. However, the present formulation provides new invariant set stability theorem generalizations characterizing

2.4 Conclusion

19

system limit sets in terms of lower semicontinuous Lyapunov functions not considered in [23, 8].

2.4 Conclusion Generalized Lyapunov and invariant set stability theorems for nonlinear dynamical systems were developed. In particular, local and global stability theorems were presented using generalized lower semicontinuous Lyapunov functions providing a transparent generalization of standard Lyapunov and invariant set theorems.

3. Nonlinear S y s t e m Stabilization via Hierarchical Switching Controllers

3.1 Introduction In this chapter a nonlinear control design framework predicated on a hierarchical switching controller architecture parameterized over a set of moving system equilibria is developed. Specifically, using equilibria-dependent Lyapunov functions, or instantaneous (with respect to a given parameterized equilibrium manifold) Lyapunov functions, a hierarchical nonlinear control strategy is developed that stabilizes a given nonlinear system using a supervisory switching controller that coordinates lower-level stabilizing subcontrollers (see Figure 3.1) [68, 65, 30, 36]. Each subcontroller can be nonlinear

.i

i

..,..! ......~:,~ .........! .....................~

.......

2-

............................................

'v ........................... ~.....' ....................

~.,..

'.: .:...'...".?..'............~

..... J

Fig. 3.1. Switching controller architecture and thus local set point designs can be nonlinear. Furthermore, for each parameterized equilibrium manifold, the collection of subcontrollers provide guaranteed domains of attraction with nonempty intersections that cover the

22

3. Nonlinear System Stabilization

region of operation of the nonlinear system in the state space. A hierarchical switching nonlinear controller architecture is developed based on a generalized lower semicontinuous Lyapunov function obtained by minimizing a potential function, associated with each domain of attraction, over a given switching set induced by the parameterized system equilibria. The switching set specifies the subcontroUer to be activated at the point of switching, which occurs within the intersections of the domains of attraction. The hierarchical switching nonlinear controller guarantees that the generalized Lyapunov function is nonincreasing along the closed-loop system trajectories with strictly decreasing values at the switching points, establishing asymptotic stability. In the case where one of the parameterized system equilibrium points is globally asymptotically stable, the proposed nonlinear stabilization framework guarantees global attraction to any given system invariant set. If, in addition, a structural topological constraint is enforced on the switching set, then the proposed framework guarantees global asymptotic stability of any given system equilibrium on the parameterized system equilibrium manifold. Furthermore, since the proposed switching nonlinear control strategy is predicted on a generalized Lyapunov function framework with strictly decreasing values at the switching points, the possibility of a sliding mode is precluded. Hence, the proposed nonlinear stabilization framework avoids the undesirable effects of high-speed switching onto an invariant sliding manifold which is one of the main limitations of variable structure controllers. Finally, we note that the present framework provides a rigorous alternative to designing gain scheduled controllers for general nonlinear systems by explicitly capturing plant nonlinearities and quantifying the notion of slow-varying system parameters which place fundamental limitations on achievable performance of gain scheduling controllers. Limited to system analysis, related but different approaches to the proposed hierarchical switching control design framework are given in [92, 93, 111, 96]. Specifically, analysis of switched linear systems in the plane (I1~2) are given in [92, 93]. More recently, asymptotic stability analysis of m-linear systems using Lyapunov-like functions is given in [111]. Stability of a multicontroller switched system is analyzed using Lyapunov functions and sliding surfaces in [96]. A hybrid stabilization strategy for nonlinear systems controlled by linear controllers is discussed in [51] wherein domains of attraction are enlarged by the use of a switching strategy. However, this analysis is limited to linearly controlled systems in the plane. Even though the approach can be extended to higher-order systems, the computational complexity needed to analyze the direction of the closed-loop system flows render the approach impractical. The special issues on hybrid control systems [2, 3] present an excellent analysis expansion on switching systems with controller

3.2 Mathematical Preliminaries

23

design methods limited to specific applications. We note that the concept of equilibria-dependent Lyapunov functions was first introduced by the authors in [85] where a globally stabilizing control design framework for controlling rotating stall and surge in axial flow compressors was developed. In parallel research to [85] a related but different approach was introduced in [101] wherein control Lyapunov functions for nonlinear systems linearized about a finite number of "trim" points to guarantee stability of a range of operating conditions were given. In the case where we specialize the switching set to a finite number of equilibrium points, we recover the results of [101]. Furthermore, we note that the authors in [101] mainly focus on how to choose the switching points so as to construct a domain of attraction that covers an a priori designated region in the state space. Hence, in the special case where the switching set contains a finite number of equilibrium points, the results of [101] can be used within the present framework to construct feasible switching sets. Finally, we emphasize that our approach is constructive in nature rather than existential. In particular, we provide an explicit construction for a hierarchical switching controller for nonlinear system stabilization and in this case our constructive conditions are complementary to existential results on asymptotic controllability via discontinuous feedback [33].

3.2 Mathematical

Preliminaries

In this chapter we consider nonlinear controlled dynamical systems of the form ~(t) = F(x(t), u(t)),

x(0) = x0,

t e Zxo,

(3.1)

where x(t) E 7) C_ R n, t E Zx0, is the system state vector, Z~o C_ ~ is the maximal interval of existence of a solution x(.) of (3.1), 7) is an open set, 0 E 7), u(t) E lg C_ Rm, t E Zzo, is the control input, U is the set of all admissible controls such that u(.) is a measurable function with 0 E U, and F : 7) x / g -~ Rn is continuous on 7) x U. D e f i n i t i o n 3.1. The point Yc E 7) is an equilibrium point of (3.1) if there exists fi E L! such that F(~, fi) = 0. Here we assume that given an equilibrium point ~ E D corresponding to E U and a mapping ~ : 7) x A ~ / 4 , A C_ Rq, 0 E A, such that ~(~, 0) = fi, there exist neighborhoods Do C 7) of ~ and Ao C A of 0, and a continuous function r : Ao --+ 7)0 such that 9 = r and, for every A e Ao, xx -- r is an equilibrium point; that is, F(r162 = 0, A E Ao. This is

24

3. Nonlinear System Stabilization

a necessary condition for parametric stability with respect to Ao as defined in [64, 107]. Note that the connected set A C_ Rq corresponds to a parameterization set with the function r parameterizing the system equilibria. In the special case where q = m and ~o(x, A) = A, it follows that the parameterized system equilibria are given by the constant control u(t) ==- ~. A parameterization that provides a local characterization of an equilibrium manifold, including in neighborhoods of bifurcation points, is given in [81]. Alternatively, the well known Implicit Function Theorem provides sufficient conditions for guaranteeing the existence of such a parameterization under the more restrictive condition of continuous differentiability of the mapping

r T h e o r e m 3.1 ([60]). Assume the function F(x,)~) ~ F(x,~o(x,~)), x E D, )~ E A, is C x at each point (x,)~) E 79• A. Suppose ~'(~,0) = O for o~" - O) is full rank. Then there exist ( ~ , O) E 7)• A and the Jacobian matrix -5g(x, open neighborhoods Do C D of 9 and Ao C A of 0 such that F(x, )~) = O, ,~ E Ao, has a unique solution xx E Do. In particular, there exists a unique C 1 mapping r : Ao --~ Do such that xx = r ~ E Ao, and 9 = r Next, we consider nonlinear feedback controlled dynamical systems. A measurable mapping r : D --~ L/satisfying r = fi is called a control law. Furthermore, if u(t) = r where r is a control law and x(t), t e Z~ o, satisfies (3.1), then u(.) is called a feedback control law. Here, we consider nonlinear closed-loop dynamical systems of the form

~c(t) = F(x(t), r

x(0) = x0,

t 9 :/:xo-

(3.2)

A function x : Z~o ~ D is said to be a solution to (3.2) on the interval 77~o C_ I~ with initial condition x(0) = x0, if x(t) satisfies (3.2) for all t 9 Ix o. Note that we do not assume any regularity condition on the function r However, we do assume that for every y 9 D there exists a unique solution x(.) of (3.2) defined on Zu satisfying x(0) = y. Furthermore, we assume that all the solutions x(t), t 9 Z~o, to (3.2) are continuous functions of the system initial conditions x0 9 D which, with the assumption of uniqueness of solutions, implies continuity of solutions x(t), t 9 77xo, to (3.2) [60, p. 24].

3.3 Parameterized System Equilibria and Domains of Attraction The nonlinear control design framework developed in this chapter is predicated on a hierarchical switching nonlinear controller architecture parameterized over a set of system equilibria. It is important to note that both

3.3 Parameterized System Equilibria and Domains of Attraction

25

the dynamical system and the controller for each parameterized equilibrium can be nonlinear and thus local set point designs are in general nonlinear. Hence, the nonlinear controlled system can be viewed as a collection of controlled subsystems with a hierarchical switching controller architecture. In this section we concentrate on nonlinear stabilization of the local set points parameterized in/). Specifically, we consider the nonlinear controlled dynamical system (3.1) with the origin as an equilibrium point corresponding to the control u --- 0, that is, F(0, 0) = 0. Furthermore, we assume that given a mapping ~o : /) x A ~ U, ~(0, 0) = 0, there exists a continuous function r : Ao -~ Do, where /)o C_ T), 0 E l)o, and Ao C_ A, 0 E Ao, such that F(x~,~(x~,)~)) = 0 with x~ = r e :Do for all )~ E Ao. As discussed in Section 3.2, this is a necessary condition for parametric stability with respect to Ao as defined in [64, 107], while Theorem 3.1 provides sufficient conditions for guaranteeing the existence of such a parameterization. Next, we consider a family of stabilizing feedback control laws given by ~-{r

/)~U:r

O, r

AsC_Ao, (3.3)

such that, for Cn(.) E ~, )~ E As, the closed-loop nonlinear feedback system ~c(t) = F ( x ( t ) , r

x(0) = x0,

t E Zzo,

(3.4)

has an asymptotically stable equilibrium point x~ E Do C_/). Hence, in the terminology of [64, 107], (3.4) is parametrically asymptotically stable with respect to As C_ Ao. Here, we assume that for each ~ E As, the linear or nonlinear feedback controllers r (.) are given. In particular, these controllers correspond to local set point designs and can be obtained using any appropriate standard linear or nonlinear stabilization scheme with a domain of attraction for each ~ E As. For example, appropriate nonlinear stabilization techniques such as feedback linearization [66], nonlinear Hoo control [145], constructive nonlinear control [119], optimal nonlinear control [24], and nonlinear regulation via state-dependent Riccati techniques [34], as well as linear-quadratic stabilization schemes based on locally approximated linearizations, can be used to design the controllers Cn(.) for each A E As. Furthermore, for an asymptotically stable equilibrium point xn E 7:)o _C T), )~ E As, the domain of attraction 7)x of xn is given by /)~ -~ (Xo E / ) : if x(O) = xo then lim x(t) -- x~}.

(3.5)

Next, given a stabilizing feedback controller r for each ~ E As, we provide a guaranteed subset of the domain of attraction D~ of x~ using classical Lyapunov stability theory.

26

3. Nonlinear System Stabilization

T h e o r e m 3.2 ([59]). Let A E As. Consider the closed-loop nonlinear system

(3.4) with Cx(') e r and let x~ e Do C_ D be an equilibrium point of (3.4). Furthermore, let X~ C D be a compact neighborhood of x~. Then x~ e Do C_ D is locally asymptotically stable if, and only if, there exists a C 1 ]unction V~ : Xa -+ ]R such that

= o, Vx(x) > O,

y (x)

V

(3.6) x E Xx, (x)F(x,

x ~ xx, < 0,

(3.7) x c

x #

(3.S)

In addition, a subset of the domain of attraction of x~ is given by :D~ -~ V~-' ([0, cx]),

(3.9)

where cA ~=max{~ > 0: Vx-'([0,fl] ) C_ Xx}. It follows from Theorem 3.2 that for all x0 E :D~, limt-.oo V~(x(t)) = 0 or, equivalently, for each ~ > 0 there exists a finite time T > 0 such that Vx(x(t)) < ~, t >_ T. Hence, given the initial condition x0 E :D~, it follows that for every ~ > 0 there exists a finite time T > 0 such that x(t) E V~-I ([0, ~]), t > T. We stress that the aim of Theorem 3.2 is not to make direct comparisons with existing methods for estimating domains of attraction, but rather in helping to provide a streamlined presentation of the main results of Section 3.4 requiring estimates of domains of attraction for local set point designs. Since D~ given in Theorem 3.2 gives an estimate of the domain of attraction using closed Lyapunov sublevel sets, it may be conservative. To reduce conservatism in estimating a subset of the domain of attraction several alternative methods can be used. For example, maximal Lyapunov functions [136], Zubov's method [142, 59], ellipsoidal estimate mappings [37], Carlemann linearizations [94], computer generated Lyapunov functions [102], iterative Lyapunov function constructions [32], trajectory-reversing methods [73], and open Lyapunov sublevel sets [56], can be used to construct less conservative estimates of the domain of attraction.

3.4 N o n l i n e a r S y s t e m S t a b i l i z a t i o n Switching Controller Architecture

via a Hierarchical

In this section we develop a nonlinear stabilization framework predicated on a hierarchical switching controller architecture parameterized over a set of moving system equilibria. Specifically, using equilibria-dependent Lyapunov ]unctions, or instantaneous (with respect to a given parameterized equilibrium manifold) Lyapunov functions, a hierarchical nonlinear control strategy

3.4 Hierarchical Switching Control

27

is developed that stabilizes a given nonlinear system by stabilizing a collection of nonlinear controlled subsystems while providing an explicit expression for a guaranteed domain of attraction. A switching nonlinear controller architecture is developed based on a generalized lower semicontinuous Lyapunov function obtained by minimizing a potential function, associated with each domain of attraction, over a given switching set induced by the parameterized system equilibria. In the case where one of the parameterized equilibrium points is globally asymptotically stable with a given subcontroller and a structural topological constraint is enforced on the switching set, the proposed nonlinear stabilization framework guarantees global asymptotic stability of any given system equilibrium on the parameterized equilibrium manifold. To state the main results of this section several definitions and a key assumption are needed. Recall that the set As C_ Ao, 0 E As, is such that for every ~ E As there exists a feedback control law r E (I) such that the equilibrium point xx E 7)0 C_ 7) of (3.4) is asymptotically stable with an estimate of the domain of attraction given by 7)x. Since xx, A E As, is an asymptotically stable equilibrium point of (3.4), it follows from Theorem 3.2 that there exists a Lyapunov function Vx(.) satisfying (3.6)-(3.8), and hence, without loss of generality, we can take 7)x, )~ E As, given by (3.9). Furthermore, we assume that the set-valued map 9 : As ~ 2z), where 2~) denotes the collection of all subsets of 79, is such that 7)x = @(A), ~ E As, is continuous. Here, continuity of a set-valued map is defined in the sense of [8, p. 56] and has the property that the limit of a sequence of a continuous setvalued map is the value of the map at the limit of the sequence. In particular, since 7)x, A E As, is given by (3.9), the continuity of the set-valued map ~(.) is guaranteed provided that V~(x), x E 7)x, and cx are continuous functions of the parameter A E As. Next, let S C_ As, 0 E S, denote a switching set such that the following key assumption is satisfied. A s s u m p t i o n 3.1. The switching set S C_ As is such that the following properties hold: i) There exists a continuous positive-definite function p : S -~ ~ such that for all )t E S, )t r O, there exists )q E S such that 0

p( x)
_O, of the controlled system approaches x~, then there exists a finite time T > 0 such that the trajectory enters D ~ . Finally, it is important to note that the switching set $ is arbitrary. In particular, we do not assume that 8 is countable or countably infinite. For example, the switching set S can have a hybrid topological structure involving isolated points and closed sets homeomorphic to intervals on the real line. Next, we show that Assumption 3.1 implies that every level set of the potential function p(.) is either empty or consists of only isolated points. Furthermore, in a neighborhood of the origin every level set of p(.) consists of at most one isolated point. For the statement of this result, let Bx(r), A E/40, r > 0, denote the open ball centered at xx with radius r, that is,

6 (r)

{x

IIx - x ll < r}.

P r o p o s i t i o n 3.1. Let S C_ As be such that Assumption 3.1 holds. Then for every a > O, p-1 (a) is either empty or consists of only isolated points. Furthermore, there exists/3 > 0 such that ]or every a < 8, P-l(a) consists o] at most one isolated point. P r o o f . Suppose, ad absurdum, that there exists ~ E p-1 (a), a > 0, such that is not an isolated point in p - l ( a ) . Now, let J~f C p - l ( a ) be a neighborhood of ~ and note that, by continuity of r and the fact that ~ E p - l ( a ) is not an isolated point, for every e > 0, there exist A1, A2 E oJ(f such that ]]x~ -x;~ 2 ]] < e, and p(A1) = P ( ) ~ 2 ) = Ct. However, since xx, E / ) ~ , it follows that there exists r > 0 such that B~ (r) _C T)~. Now, choosing e < r yields x~ 2 E B~l(r) C_ 7)~ and x~ 2 E T)~: which implies that :D~1 n T)~2 ~ O contradicting i/) of Assumption 3.1. Hence, if p - l ( a ) , a > 0, is non-empty, it must consist of only isolated points. Next, suppose, ad absurdum, that for all 5 > 0 there exist two isolated points A1,A2 E A~f6 -~ {A E S : IIAll < 5} such that p(A1) = p(A2). Now, repeating the above arguments leads to a contradiction. Hence, there exists > 0 such that if A1 e A~f$, then J(f$ np-l(p(A1)) = {A1}. Now, since p(-) is continuous and positive definite, it follows that there exists/~ > 0 such that p - l ( a ) C_~'$, a 0, with 0 9 S corresponding to 0EK Now, for every x 9 Dc ~ Uxes :Dx, define the viable switching set Ys(x) {A 9 x 9 Dx},whichcontainsallA 9 S s u c h t h a t x 9 Dx. Note that oo C Vs(X), that is, x 9 :Dx~, such that if we consider a sequence {An),=1 limn-~cr An = ~, it follows from the continuity of the set-valued map ~(.) that x 9 :DX- Thus, ~ 9 Ys(x) which implies that Ys(x) is a non-empty closed set since it contains all of its accumulation points. Next, we introduce the switching function As(x), x 9 :De, such that the following definitions hold

V(x) ~p(As(x)),

As(x) ~ argmin{p(A) : A 9 Ys(x)),

x 9 :Dc. (3.11)

In particular, As(x), x E :De, corresponds to the value at which p(A) is minimized wherein A belongs to the viable switching set. The following proposition shows that "min" in (3.11) is attained and hence V(x) is well defined.

Proposition 3.2. Let S C_ As and let p : ,9 ~ • be a continuous positivedefinite function such that Assumption 3.1 holds. Then, for all x E 7)c, there exists a unique As(z) e Ys(X) such that p(As(x)) = min{p(A) : A e Ys(X)}. P r o o f . Existence follows from the fact that p(-) is lower bounded and ~s(x), x E Dc, is a non-empty closed set. Now, to prove uniqueness suppose, ad absurdum, As(x) is not unique. In this case, there exist A1, A2 E S, A1 ~ A2, such that p(A1) = p(A2) and x e Dxl n DA2 ~ 0 which contradicts ii) in Assumption 3.1. 9 The next result shows that V(.) given by (3.11) is a generalized Lyapunov function candidate, that is, V(.) is lower semicontinuous on Dc. T h e o r e m 3.3. Let 8 C_ As be such that Assumption 3.1 holds. Then the function V(Xo) = p(As(x)), x E Dc, is lower semicontinuous on Dc and continuous on D~s(x ). Co P r o o f . Let the sequence {xn}n=o C :Dc be such that limn-4oo xn = ~ and define ~ ~ liminfn--,oo As(xn). Here we assume without loss of generality that {As(xn))~=0 converges to ~; if this is not the case, it is always possible to construct a subsequence having this property. Since p(-) is continuous (and hence p(limn-~oo As (x,~)) = limn--,oo p(As (xn))), it suffices to show that V(&) _< p(~). Suppose, ad absurdum, that V(&) > p(~). In this case, there exists a positive integer no such that V(~) > V(xn), n >__no. Now, since by

30

3. Nonlinear System Stabilization

definition As(Z) minimizes p(A) for A 9 ~)s(~), it follows that V(~) < p(A), A 9 ~;s(X). Hence, since Y(~) > V(xn) = p(As(xn)), n > no, it follows that AS(Xn) g[ Vs(~) and ~ r DXs(x~), n _> no. Now, define the closed set =~ UntO=noDxs(~) such that {x n}n=no ~ C 7~. Since 7) is closed, it follows that ~ 9 7) which implies that there exist nl _> no such that ~ 9 79xs(z.1 ) which is a contradiction. o To show that V(x) is continuous on DXs(e) it need only be shown that V(~) is upper semicontinuous on 7~xs(e), or, equivalently, V(~) > p(A). Since o limn--,oo xn = ~c and ~ 9 DXs(e), there exists a positive integer n2 such that xn 9 Dxs(e), n _> n2. Hence, As(Z) 9 Ys(xn) and V(Y:) >_ Y(xn), n >_ n2, which implies that V(~) __ p(A). I Next, we show that with the hierarchical nonlinear feedback control strategy u = CXs(z)(x), x 9 Dc, V(.) given by (3.11) is a generalized Lyapunov function for the nonlinear feedback controlled dynamical system (3.2). The controller notation r denotes a switching nonlinear feedback controller where the switching function As(x), x 9 Dc, is such that definition (3.11) of the generalized Lyapunov function V(x), x 9 Dc, holds for a given potential function p(.) and switching set S satisfying Assumption 3.1. Furthermore, note that since r (x)(x) is defined for x 9 De, it follows that the solution x(-) to (3.2) with x0 9 De and u = Cxs(x)(x) is defined for all values of t 9 77xo such that x(t) 9 79c. However, as will be shown, since Dc is a positively invariant set, [0,+c~) C_ 27~o, while if x0 9 Dc is such that x(t), t < 0, is always contained in De, then Iz o = R- Finally, note that since the solution x(t), t 9 27xo, to (3.2) with x0 9 Dc and u = r is continuous, it follows from Theorem 3.3 that V(x(t)), t 9 5~o, is right continuous. Hence, using the continuity of p(.) and the definition of V(x), x 9 De, it follows that As(x(t)), t 9 Ixo, is also right continuous. Now, the continuity of F(., .) and r A 9 As, imply that F(x(t), Cxs(x(t))(x(t))), t 9 Y-~o, is right continuous. T h e o r e m 3.4. Consider the nonlinear controlled dynamical system given by (3.1) with F(0,0) = 0 and assume there exists a continuous function r : Ao --+ Do, 0 E Ao, parameterizing an equilibrium manifold of (3.1), such that xx = r A E Ao. Furthermore, assume that there exists a C O feedback control law r A E As C_ Ao with 0 E As, that locally stabilizes x~ with a domain of attraction estimate Dx and let S C As, 0 E S, be such that Assumption 3.1 holds. If As(x), x 9 De, is such that V(x), x 9 Dc, given by (3.11) holds and x(t), t 9 :T.xo, is the solution to (3.1) with x(O) = xo 9 Dc and feedback control law u = r

x E Dc,

(3.12)

3.4 Hierarchical Switching Control

31

then De is positively invariant and V(x(t)), t 9 Z~o, is nonmcreasin#. Furthermore, /or all tx,t2 9 I~ o, V(x(t)) = V(z(tl)), t 9 [t,,t2], if, and only if, )~s(x(t)) = )~s(x(h)), t 9 [ta,t2]. Finally, /or all t 9 :Z:~o such that As(x(t)) # O, there exists a finite time T > 0 such that V(x(t+T)) < V(x(t)). P r o o f . First, note that x E 07Pc implies x 6 07)xs(~ ). Since Cxs(x) stabilizes XXs(z) with domain of attraction :Dxs(~), it follows that, for all x 6 ado, the flow of F(x,r is directed towards the interior of Dxs(~ ) and consequently towards the interior of De, which proves positive invariance of De. Next, let x(t), t 6 Z~o, satisfy (3.1) with u(t) = Cx,(x(t)), where At A= As(x(t)), and let, for an arbitrary time tl 9 Zxo, the feedback controI law u = r (x) asymptotically stabilize the equilibrium point xx,, of (3.1) with domain of attraction Dx,,. Since x(tl) 9 79x,1 , it follows from Theorem 3.2 that there exists a C 1 Lyapunov function Vx,1 (.) such that Vx,, (x(t)) ~ V~,, (x(t))F(x(t),r t 9 I~ o, and Vx,, (x(tl)) = V~,, (x(tl))Fx,~(x(tl)) < 0. Next, since F(x(t),r t 9 Ix o, is right continuous, it follows that there exists J > 0 such that Vx,~(x(t)) < 0, t 9 [tl,tx + 6], which implies that Vx,,(x(t)) < Vx,~ (x(tl)), t 9 [tl,tl + 5]. Hence, x(t) 9 7)x,a , t 9 [tl,tl + 6], and )% 9 ]3s(x(t)), t 9 [tl,tl + 5], which implies that Y(x(t)) tl, or, equivalently, At = At, 9 S \ {0}, t _> tl. Then the feedback control law Cx, (') = Cx,~ (') stabilizes the equilibrium point x x , . In this case, it follows from Assumption 3.1 that there exists ,~ ~ At~ such that p()q) < V(X(tl)) and xx,~ 6 79x~, which implies that there exists a > 0 such that xx,1 9 Vx~: ([0 , hi) C_ :Dx,. Hence, it follows that x(t) approaches the level set V ~ ( a ) in a finite time T > 0 so that Y(x(tl + T)) < p()q) < Y(X(tl)), which contradicts the original supposition. [] Next, we show that the hierarchical switching nonlinear controller (3.12) guarantees that the generalized Lyapunov function (3.11) is nonincreasing along the closed-loop system trajectories with strictly decreasing values only

32

3. Nonlinear System Stabilization

at the switching times which occur when the closed-loop system trajectory enters a new domain of attraction with an associated lower potential value. C o r o l l a r y 3.1. Consider the nonlinear controlled dynamical system given by (3.1) with F(0,0) = 0 and assume the hypothesis of Theorem 3.4 hold. Then V(x(t)), t > O, is strictly decreasing only at the switching times which occur when the trajectory x(t), t E Zzo, enters a new domain of attraction with an associated lower potential value. 0

P r o o f . First, we consider the case where x(t2) E D~, 2 , with At2 ~ As(x(t2)) and t2 > 0. It follows from the continuity of the closed-loop system trajectories x(-) that there exists tl < t2 such that x(ti) E Dx, 2 , which implies that At2 E ]2s(x(tl)) and V(x(tl)) < Y(x(t2)). Since V(x(t)), t > O, is a nonincreasing function of time, it follows that V(x(t)) = V(x(t2)), t e [tl, t2]. Alternatively, assume that x(t2) E 0Dx, 2 , and suppose, ad absurdum, that there exists tl < t2 such that x(tl) E D~, 2. Then At = Ate, t E [tl,t2], and, since Vx,2 (x), x E D~,2, attains its maximum at x(t2) E 0Dx, 2 , it follows that Vx,~(x(t)) 0, is a decreasing function of time. Hence, x(t) • Dx,2 and V(x(t)) < V(x(t2)), for all t < t2. II Finally, we present the main result of this section. Specifically, we show that the hierarchical switching nonlinear controller given by (3.12) guarantees that the closed-loop system trajectories converge to a union of largest invariant sets contained on the boundary of intersections over finite intervals of the closure of the generalized Lyapunov level surfaces. In addition, if the switching set S is homeomorphic to an interval on the real line and/or consists of only isolated points, then the hierarchical switching nonlinear controller establishes asymptotic stability of the origin. T h e o r e m 3.5. Consider the nonlinear controlled dynamical system given by (3.1) with F(0, O) = 0 and assume there exists a continuous ]unction r : Ao --+ :Do, 0 9 Ao, parameterizin9 an equilibrium manifold of (3.1), such that x~ = r A 9 Ao. Furthermore, assume that there exists a C o feedback control law r A 9 As _C Ao with 0 E As, that locally stabilizes x~ with a domain of attraction estimate Da and let S C As, 0 9 S, be such that Assumption 3.1 holds. In addition, assume As(x), x E Dc, is such that V(x), x 9 De, given by (3.11) holds, and, ]orxo 9 De, x(t), t E Zxo, is the solution to (3.2) with the feedback control law u = r

x 9 De.

(3.13)

If xo 9 De, then x(t) -4 ](4 ~- U~e~ J ~ as t --~ 00, where ~ ~- {~ >_ 0 : Rq, nDo r 0}. If, in addition, ,-qo ~ {A 9 S : DxnDo r O} is homeomorphic

3.4 Hierarchical Switching Control

33

to [0, a], a > O, with 0 E So corresponding to 0 E ~, or So consists of only isolated points, then the zero solution x(t) =- 0 to (3.2) is locally asymptotically stable with an estimate of domain of attraction given by De. Finally, if D = R '~ and there exists ~ E S such that the feedback control law r globally stabilizes x i , then the above results are global.

P r o o f . The result follows from Theorems 2.4, 3.3, and 3.4. Specifically, Theorem 3.4 implies that if x(~ E Do for an arbitrary t > 0, then V(x(t)) = V(x(t~) = O, t > t. Hence, ~s(x(t)) = O, t > i, and the feedback control law u = Co(x) asymptotically stabilizes the origin with an estimate of the domain of attraction given by D0. In this case, 79o is a compact positively invariant set of (3.2) with the feedback control law (3.13). Next, it follows from Theorems 3.3 and 3.4 that V(-) is a generalized Lyapunov function defined on De. Now, it follows from Theorem 2.4 that, for all xo E De, x(t) -~ J(4 as t --r ~ . Next, if So is homeomorphic to [0,a], a > 0, with 0 E So corresponding to 0 E R, so that Assumption 3.1 is satisfied, it follows from the continuity of the set-valued map ~(.) restricted to So that V(-) is continuous on 790. Now, it follows from Theorem 2.4 and the fact that the origin is the largest invariant set contained in 7"~ that A~ = s~4o = {0}. Hence, x(t) ~ 0 as t ~ ~ establishing local asymptotic stability with an estimate of domain of attraction given by De. Alternatively, if So consists of only isolated points with finite pairwise distance, it follows that ~ consists of the isolated values of p(.) evaluated on the elements of So. Hence, since 7~7 -~ 7~7 \ V-1(7), 3' E G, is bounded, V(-) can only assume a finite number of distinct values on 7~.r, 7 E ~, including the zero value. Now, it follows from Theorems 2.4 and 3.4, respectively, that 3A~ C 7~7 and, for all x0 E De \ 790, there exists an increasing unbounded sequence {t n},~=o, with to = 0, such that V(x(tn+l)) < V(x(t)), n = 0, 1,.... Thus, no forward trajectories can be entirely contained in 7~7, 3' E G \ {0}. Hence, A~7 = O, 3' E G \ {0}, and 3/[ = 3,4o = {0} establishing local asymptotic stability with an estimate of domain of attraction given by 79r Finally, let D = It~n and assume Sg & {~ E S : 79x = IRn } is not empty. In particular, if ~ E Sg, then the feedback control law r ('! globally stabilizes x i. Now, Assumption 3.1 implies that if A1, ~2 E Sg, ~1 ~ ~2, thenp(~l) ~ p(~2). Furthermore, since 79i~ = Di2 = Rn, we obtain that ~1, ~2 E Vs (x) for all x E IR~. Next, assume without loss of generality that P(~I) < P(~2), and note that since )~s(x) minimizes p(.) over Vs(x), we obtain that ,~,s(x) ~ ~2, x E IRn . It follows that V(x) < min{p(~) : ~ E Sg} for all x E ~'~ and only the (unique) value ,~ E Sg that minimizes p(.) over Sg is assumed by the switching function ,~s('), so that all the other elements of Sg can be discarded from S. Hence, without loss of generality, assume that there exists a unique ~ E S such that r globally stabilizes x i. Now, define ,~ ~ {,k ~ S : p(A) < p(~)} ^

34

3. Nonlinear System Stabilization

and ~r =" U~e ~ ~D~ which is a compact positively invariant set. Hence, if

x0 E ~c, it follows from the first part of the theorem that .M is a local attractor and, if So is homeomorphic to an interval on lt~ or consists of only isolated points, the origin is asymptotically stable, with (in both cases) an estimate of the domain of attraction given by ~c. Now, global attraction to 32t as well as global asymptotic stability of the origin is immediate by noting that if x0 ~ ~c, then the forward trajectory of (3.2) approaches ~r in a finite time. If, in fact, x ~ 7}~, then ,~s(x) = ~ which implies that for all x ~ ~r the 9 feedback control law (3.13) stabilizes x~ and, by Assumption 3.1, x~ e ~ . In this case, it follows that for all xo ~ Dc there exists a finite time T > 0 such that x(T) E 7}c. Hence, global attraction as well as global asymptotic stability of the origin is established for the respective cases. 9

-~

__

Systemtrajectory

Fig. 3.2. Hierarchical switching control strategy The switching set 8 is quite general in the sense that it can have a hybrid topological structure involving isolated points and closed sets homeomorphic to intervals on the real line. In the special case where the switching set S consists of only isolated points, the hierarchical switching control strategy given by (3.13) is piecewise continuous (see Figure 3.2). Alternatively, in the special case where the switching set ,S is homeomorphic to an interval on R, the hierarchical switching control strategy given by (3.13) is not necessarily continuous. The continuous control case will be discussed in Section 3.5. In the case where the switching set S is homeomorphic to an interval on R and

3.5 Extensions to Nonlinear Dynamic Compensation

35

a stabilizing controller Co(') for the origin cannot be obtained, that is, Co = 0, 0 9 Dc still holds. Hence, Theorem 3.5 guarantees attraction of the origin if O

0 9 ODe. Alternatively, if 0 9 De, then the origin is asymptotically stable. Finally, it is important to note that since the hierarchical switching nonlinear controller u = Cxs (z)(x), x 9 Dc, is constructed such that the switching function As(x), x 9 De, assures that V ( x ) , x 9 De, defined by (3.11) is a generalized Lyapunov function with strictly decreasing values at the switching points, the possibility of a sliding mode is precluded with the proposed control scheme. In particular, Theorem 3.4 guarantees that the closed-loop state trajectories cross the boundary of adjacent regions of attraction in the state space in a inward direction. Thus, the closed-loop state trajectories enter the lower potential-vaiued domain of attraction before subsequent switching can occur. Hence, the proposed nonlinear stabilization framework avoids the undesirable effects of high-speed switching onto an invariant sliding manifold.

3.5 Extensions to Nonlinear Dynamic Compensation In this section we provide an online procedure for computing the switching function As(x), x 9 Dc, such that (3.11) holds, by constructing an initial value problem for As (x) having a fixed-order dynamic compensator structure. Specifically, we consider a switching set S diffeomorphic to an interval on the real line R and assume that the switching function As(x), x 9 De, is C 1. To present this result, consider the nonlinear controlled dynamical system given by (3.1) with a nonlinear feedback dynamic controller of the form =

= r162

xc(O) = Xr

t 9 I=o,

(3.14) (3.15)

where Xc(t) E C C ll(nc , t E E=o, is the controller state vector, C is an open set, Fr : 7:) x C -~ If(TM , and r : 7) • C -~/4. Note that we do not assume any regularity condition on the mappings Fr .) and r "). However, we do assume that the nonlinear feedback controlled dynamical system given by (3.1), (3.14), and (3.15) is such that the solutions of the closed-loop system on I=0 are unique and continuously dependent on the closed-loop system initial conditions. To construct dynamic controllers of the form (3.14), (3.15) we assume that the switching set S is such that there exists a closed interval [0, a], a > 0, and a diffeomorphism a : [0, a] -~ As, such that a(s) E S, s E [0, a], and a(0) = 0. Fhrthermore, we assume that Vx(.) and cx are C 1 functions of A E S. Next, recall that Theorem 3.5 guarantees that the feedback control law u(x) = Cxs(x)(x), x 9 Dc, where As(x), x 9 Dc, is given by (3.11),

36

3. Nonlinear System Stabilization

locally asymptotically stabilizes the origin of the nonlinear feedback controlled dynamical system (3.2), and De is a subset of the domain of attraction. Furthermore, note that if x0 9 De \ :Do it follows by continuity of the closed-loop trajectories x(.) that there exists a finite time T=o > 0 such that x(Txo) 9 07)0 and As(x(T=o)) = 0. Now, define At ~ As(x(t)) and note that since 8 is connected and the set-valued map g2(-) is continuous, it follows that 7 ~ = V-I(~f), 7 _> 0, which implies that V(x(t)), t >_ O, can be constant on the interval [h, t2] C_ [0, T=o] only if x(t) 9 ODx,, t E [tl, t2], which contradicts the fact that I;'x, (x(t)) < 0. Hence, it follows that V(x(t)), t 9 [0, Txo] , is a strictly decreasing function and Theorem 3.4 further implies that x(t) 9 ODx, = v ~ l ( c x , ) , t 9 [0,T~o]. Thus, (=(t))

=

(3.16)

t 9 [o, T,o],

relates the state trajectories x(.) to the switching function As(x(.)). For the statement of the main result of this section the following definitions and proposition are needed. Define vx(x) a_ O(c~ - Vx(x)) OA ,

a da(s) I w~ -

ds

,

(x,A) 9 Dc x 8, (3.17)

[s=~_~(~)

and Qx(x) =a ~ Vw; ( ( ' x ) which, as shown in Proposition 3.3 below, is well defined for all x 9 De \ Do and A -- As (x). In the case where x 9 Do, define = O.

3.3. Assume a : [0, a] -~ 8 is a diffeomorphism, As : De ~ 8 is a C 1 function, and V~(.) and cA are C 1 functions of A 9 8. Then vx(x) and w~, (x, A) 9 De x 8, defined in (3.17), are such that V~s(z)(x)W~s(= ) ~ 0 for all x E Dc \ Do. Proposition

Proof. The result follows by differentiating both sides of (3.16) and noting that for all x0 9 De, VAo(Xo) < 0, Ao = As(xo). Specifically, differentiating both sides of (3.16) yields =

t 9 [O,T,o].

(3.18)

Next, let s(t), t 9 [0,Tzo], be such that At = a(s(t)), t 9 [0,Tzo], and note that (3.18) evaluated at t = 0 yields =

=

< 0.

(3.19)

Now, noting that vx o (Xo)W~o and h(0) are scalars, it follows that vx o (xo)w~ o 0, z0 9 De \ Do. 9

3.5 Extensions to Nonlinear Dynamic Compensation

37

Next, we present the main result of this section which provides an online procedure for computing the switching function As (x(t)), t E Zzo. Specifically, differentiating both sides of (3.16) with respect to time yields a Davidenkotype differential equation that defines an initial value problem for the switching function and hence the function As(x(t)), t E Z~o, is characterized via a homotopy map. 3.6. Assume a : [0, a] --r S is a diffeomorphism, As : 7)r S is a C 1 function, and Vx(.) and cx are C 1 functions of A E S. Then, the solutions x(t) and A(t), t E Z~o, of the closed-loop nonlinear feedback controlled dynamical system Theorem

x(t) = F ( x ( t ) , Cx(t)(x(t))),

x(0) = Xo,

~(t) = Qx(t)(x(t))F(x(t), Cx(t)(x(t))),

Ao = As(xo),

t E 27~o, (3.20) (3.21)

are such that A(t) = As(x(t)), t E Z~ o, or, equivalently, Vx(t)(x(t)) = cx(t), t E Z~o.

Proof. The result follows by differentiating both sides of (3.16) with respect to time and noting that At = wx,~(t). 9 Note that the switching function dynamics characterized by (3.21) defines a fixed-order dynamic compensator of the form given by (3.14), (3.15). Specifically, defining the compensator state as Xc(t) _a A(t) so that nc = q, the dynamic compensator structure is given by Joe(t) = Q~r162 u(t) = Czc(t)(x(t)).

xc(O) = As(xo),

t e Z~o, (3.22)

(3.23)

Now, it follows from Theorems 3.5 and 3.6 that for all Xo E 7)e the nonlinear feedback controlled dynamical system given by (3.1), (3.22), and (3.23), drives Xc(t) to 0 E S in a finite time Tzo > 0. Note that this result does not violate the assumption of uniqueness of solutions of (3.22) since x(T~o) E ado at the finite time Txo > 0 does not correspond to a system equilibrium point. Next, since A(t) = O, t > T~o, x(t) reaches 0 E 7) asymptotically which guarantees asymptotic stability of the origin with an estimate of the domain of attraction given by 7)c. Similar arguments hold for global asymptotic stability in the case where 7) - 7)c = ]Rn. The compensator dynamics (3.21) characterize the fastest admissible rate of change of the switching function for which the feedback control (3.23) maintains stability of the closed-loop system. As discussed in the Introduction, this quantifies the notion of slow-varying system parameters which has been one of the major shortcomings of gain scheduling practice. It is not

38

3. Nonlinear System Stabilization

surprising to note that this rate is an explicit function of the gradient of the equilibria-dependent Lyapunov functions and the gradient of the domains of attraction estimates with respect to the parameterized equilibrium manifold. Furthermore, the rate of change of the switching function also depends on the gradient of the diffeomorphism evaluated along the switching set S; such a dependence can be used to enforce desirable structural properties of the switching set. In particular, in Section 3.6, a hierarchical inverse optimal control framework for computing a diffeomorphism online will be presented. Finally, as noted in the Introduction, the proposed framework can be used to address practical actuator limitations such as control rate saturation constraints. Specifically, constraining the rate at which the dynamics of the switching function A(t) can evolve on the equilibrium manifold, it is possible to address input rate saturation constraints. In particular, since the parameterization of the equilibrium manifold introduced in Section 3.2 is obtained by setting u = ~(r A), A 9 Ao, it follows that, differentiating both sides of this parameterization with respect to time,

i~(t) where

S(A(t)) 9

~xq

=

S(A(t)))~(t),

t 9 E=o,

(3.24)

is given by

S(A) ~- [c9~(x,A) -de(A) - + L ~-x dA

1

Now, choosing ~o(., .) so that q = m, that is, the control and parameter spaces have the same dimensions, and constructing the switching set S so that S(A), A E S, is not singular, it follows from (3.24) that a constraint on the rate at which the dynamics of the parameter A(t) can evolve on the equilibrium manifold can be enforced effectively placing a rate constraint on the control

u(t). Finally, to elucidate the hierarchical switching nonlinear controller presented in this section and Section 3.4, we present an algorithm that outlines the key steps in constructing the hierarchical switching feedback controller.

Algorithm

3.1. To construct the hierarchical switching feedback control dpXs(x(t))(x(t)) , t > O, perform the following steps:

Step 1. Construct the equilibrium manifold of (3.1) using u = ~(x, A), where ~(., .) is an arbitrary function of A E Ao. Use F(x, qo(x, A)) = 0 to explicitly define the mapping r such that xx = r A E Ao, is an equilibrium point of (3.1} corresponding to the parameter value A. We note that the above parameterization can be constructed using the approaches given in [81, 64, 107].

3.6 Inverse Optimal Nonlinear Switching Control

39

Step 2. Construct the set As C_ Ao such that, for each equilibrium point xa, )~ E As, there exists a stabilizing controller Ca(') and an associated domain of attraction Da corresponding to the level set ca and Lyapunov ]unction Va(.). Here, the controllers Ca('), A E As, can be obtained using any appropriate standard linear or nonlinear stabilization scheme. Step 3. Construct the switching set S C_As and a potential ]unction p : S R + such that Assumption 3.1 is satisfied. In particular: 3a. If )~ E S is an isolated point of S with corresponding equilib~um point xa, then there exists )~1ES such that P()~I) _ O, search for solutions to v a ( x ( t ) ) = cA, ~ 9 S .

4a. If no solution exists, )~s(x(t)) is unchanged. 4b. If one solution ~1 exists and p(~l) < p(,~) then switch )~s(x(t)) to )~1. 4c. If more than one solution exists, repeat Step 4b with )~1 replaced by the solution that minimizes p(.). Note that multiple solutions can be avoided by modifying the ca 's. Step 5. Construct the hierarchical switching feedback controller tas(,(t))(x(t)), where ,~s(x(t)), x 9 De, constructed in Step 4 is such that (3.11) holds. Note that the existence of a switching set S and a potential function p(.) such that Step 3 is satisfied, can be guaranteed by modifying the first part Step 4 as follows:

Step 4'. Given the state space point x(t) at t = tk ~- k A T , where A T > 0 and k = O, 1,..., search for the solutions of V~(x(tk)) < c~, )~ 9 As. In this case, the switching set S C_ As need not be explicitly defined and is computed online. Furthermore, the case where AT ~ 0 recovers the continuous framework described in this section.

3.6 Inverse Optimal Nonlinear Switching Control In this section we develop optimality notions for the hierarchical switching controller architecture developed in Section 3.4. Specifically, in the case where the switching set induced by the parameterized system equilibria is diffeomorphic to a closed interval on the real line, we construct an inverse

40

3. Nonlinear System Stabilization

optimal hierarchical switching strategy that is characterized by a Davidenkotype differential equation. The inverse optimal hierarchical control framework developed herein is quite different from the quasivariational inequality methods for optimal switching systems developed in the literature (e.g., [41, 19]). Specifically, quasivariational methods do not guarantee asymptotic stability via Lyapunov functions and do not necessarily yield feedback controllers. In contrast, our results provide hierarchical homotopic feedback controllers guaranteeing closed-loop stability via an underlying Lyapunov function. To provide an optimal online procedure for computing the switching function As(x), x E ~)r such that (3.11) holds, we extend the results of Section 3.5 by constructing an initial value problem for the switching function As(x) having a fixed-order dynamic compensator structure that minimizes a der/red cost functional. Specifically, to address optimality notions within our nonlinear switching control framework we consider the following minimization problem. O p t i m a l Switching Control P r o b l e m . Consider the nonlinear controlled dynamical system given by (3.1) with u(x) = r x 9 De, where As(x), x 9 7)r is given by (3.11), and let S be diffeomorphic to a closed interval on the real line. Then, determine As(x), x 9 :De, by solving

p(It) = minp(A),

A 9 ])s(X(t)),

t 9 [0,Txo],

(3.25)

subject to (3.16). It follows from (3.11) and Assumption 3.1 that At is the unique solution of the Optimal Switching Control Problem. Furthermore, since {A : Vx(x) = c~,A 9 S , x 9 T)r C_ ])s(X) C_ 8, it follows that Vs(X(t)) in (3.25) can be equivalently replaced by the switching set S. Since the Optimal Switching Control Problem requires that the switching set $ be given, we define an extended minimization problem wherein the switching set is computed online. E x t e n d e d O p t i m a l Switching Control P r o b l e m . Consider the nonlinear controlled dynamical system given by (3.1) with u(x) = Cxs(z)(x), x 9 7)c, where As(x), x 9 :De, is given by (3.11). Then, determine As(x), x 9 :Dc, where S __aUte[0,T,o] At, by solving p(At) = minp(A),

A 9 As,

t 9 [0,Tzo],

(3.26)

subject to (3.16). Note that since At, t 9 [0, Txo], is a C 1 function, the switching set S is diffeomorphic to an interval on the real line. Next, we present the main result of this section which provides an inverse optimal online procedure for computing the switching function As(x(t)), t 9 [0, Tzo]. For the statement of this result define

v:~(x) A dcx dA

OVA(x) 0A '

(3.27)

3.6 Inverse Optimal Nonlinear Switching Control

MA(x, q) =

d2p(A) dA2

d2cA 02VA(x) q-~-~- + q ~-~ ,

41 (3.28)

for x E :De, A E S, and q E K Note that since the Hessian of the potential function p(.) explicitly appears in the definition of MA(., .), we assume, without loss of generality, that p(.) is such that MA(x, q) is nonsingular for all x E :Dc, AES, and q E ~ Furthermore, define

RA(x, q) A

V; (x) + q vA ( x ) M ; ' (x, q) 02 VA (x______)) OAOx vA ( x ) M ; l (x, q)vW (x)

QA(x,q) g M ; ' ( x , q )

(

02VA(x)) vW(x)RA(x,q) -- q OAO-------x'- '

(3.29) (3.30)

where vA(x) and QA(x, q) are such that vA(x)QA(x, q) = V](x). T h e o r e m 3.7. For a fixed x e :De, assume that VA(x) : S --~ ~+, cA : S -~ ]~+, andp : S -~ I~+ are C 2 functions of A 9 S. Then the solutions x(t), A(t), and q(t), t 9 [0, Txo], of the nonlinear feedback controlled dynamical system

~(t) = f ( x ( t ) , CA(t)(x(t))), A(t) = QA(t)(x(t), q(t))F(x(t), CA(t)(x(t))),

x(O) = xo, A(0) = Ao,

(3.31)

q(t) = RA(t)(x(t), q(t))f(x(t), r

q(O) = qo,

(3.33)

(x(t))),

(3.32)

where A(0) = A0 and q(O) = qo satisfy the Extended Optimal Switching Control Problem at t = O, are such that A(t), t 9 [0, Tx0], solves the Extended Optimal Switching Control Problem. P r o o f . To solve the Extended Optimal Switching Control Problem, form the Lagrangian L~(A,q(x)) ~ p(A) - q(x)[cA - VA(x)], A 9 As, x 9 :Dc, where q : :De -~ R is a Lagrange multiplier. If At, t 9 [0,Txo], solves the Extended Optimal Switching Control Problem it follows that

0 = Os

[dp(A)] w 0A

= i--'d~J

A-A, -

q(x(t))v~, (x(t)),

(3.34)

~=A,

0-

Os Oq

q=q(z(0) = cA, - VAt (x(t)),

t 9 [0, Txo].

(3.35)

Differentiating both sides of (3.34) and (3.35) with respect to time and denoting q(x(t)) and At by q(t) and A(t), respectively, it follows that

0 = MA(O (x(t), q(t))A(t) - vW(t)(X(t))~t(t) +q(t)

azVx (x)l~____~l~iF(x(t),r OAOx

(3.36)

42

3. Nonlinear System Stabilization o = v co

OVxco(x) I" F(x(t),r Ox ~=~(t)

),

t e [0,T~o].

(3.37)

Now, since Mx(o(x(t),q(t)) is invertible, (3.32), (3.33) are a direct consequence of (3.36), (3.37), respectively. Finally, the initial conditions A(0) = Ao and q(0) = qo are computed by solving the algebraic system of equations given by (3.34) and (3.35) for t = 0. 9 The update parameters A(t) and q(t), t E [0, T~o], in (3.31)-(3.33) should be interpreted as A(x(t)) and q(x(t)), t G [0,T~o], since they are implicit functions of time parameterized via the system trajectory x(t), t E [0, T~o]. This minor abuse of notation considerably simplifies the presentation. Note that the switching function and Lagrange multiplier dynamics characterized by (3.32) and (3.33), respectively, define a fixed-order compensator of the form given by (3.14) and (3.15). Specifically, defining the compensator state xc(t) = [XWl(t)Xc2(t)]T, where Xr ) -~ ~(t) and xc2(t) ~ q(t) so that nc = q + 1, the dynamic compensator structure is given by ~r

= Q~c,CO(X(t),Xc2(t))F(x(t),r

:i;c2(t) = RzclCt)(x(t),xc2(t))F(x(t),r

xcl(0) = ~o, (3.38) xr

= qo, (3.39) (3.40)

u(t) = r

The next result gives an explicit expression for the time derivative of the Lyapunov function V(x), x E De. P r o p o s i t i o n 3.4. Assume the switching set S is diffeomorphic to an interval on the real line and, for a fixed x E De, assume Vx (x) : S ~ R + , cx : S --+ II+ , and p : S ~ R+ are C 2 functions of )~ E S. Let q : De --4 R be the Lagrange multiplier for the Extended Optimal Switching Control Problem. Then V(x) = q(x) [V~ (x)F(x, Cx (x))]x=xsC,).

(3.41)

Proof. Since V(x) = p()~s(x)) and, by (3.34), ~ x x x=Xs(x) = q(z)vxscz)(z)' it follows that ~'(x) - dp(A)dA

As(x) = q(x)vxs(x)~s(x).

(3.42)

X=,~s(z)

Now, using )~.s(x) = Qxs(~:) (x, q(x))F(x, Cxs (z)(x)) and noting that vx (x) 9Qx(x, q) = V~,(x), (3.41) is immediate. 9 The following proposition gives an implicit characterization for the Lagrange multiplier q(x) 9 ~ x 9 :De, such that its dynamics satisfy (3.33).

3.6 Inverse Optimal Nonlinear Switching Control

43

Proposition 3.5. The Lagrange multiplier q : Dc ~ R for the Extended Optimal Switching Control Problem is given by

[ dA J~=~s(~) q(x) =

C(x)vTs(~) (X)

'

X E/)c,

(3.43)

where c : De --~ R l• is an arbitrary row vector such that c(x)vT(x) # O, x E De, A E S. For c(x) = va(x)Ms x E De, (3.43) implicitly defines the solution q(x) of (3.33). Proof. Forming c(x)(3.34) and solving for q(x) yields (3.43). Next, differentiating (3.43) with respect to time gives , , 02v ( ) [

q(x)= c(x)Mxs(x)(x'q(x))Jts(x) etx) O--~x X=~s(z) c(x)vTs(z)(x ) +q(x) C(x)vTs(x)(X ) F(x,r (3.44) Now, substituting Vxs(~)(x)M;sl(~)(x,q(x)) for c(x) into (3.44) and using (3.37), yields (3.33). 9 It is important to note that (3.43), with c(x) = Vxs(z)(x)M;sl(z)(x, q(x)), x E 7)c, implicitly characterizes the Lagrange multiplier q(x), x E De, since q(x) appears in M;ls(x)(x, q(x)). The next result provides an explicit characterization for the Lagrange multiplier q(x), x E I)c. Proposition 3.6. Consider the polynomial in q(x), x E De, of degree p given by

p[

0=E

k=l

ck(x,q(x))

(

) M~-~(x,q(x))vT (x)] , q(x)v~(x) - -dP(A) dA :~=~s(x) (3.45)

where ck (x, q(x) ), x E 7)c, k = O, 1,... , p, are the coefficients o] the characteristic polynomial associated with Mxs (x)(x, q(x)), x E 7)c. Then, the Lagrange multiplier q : 7)r -+ R [or the Extended Optimal Switching Control Problem is the root of (3.45) such that q(xo) = qo. Proof. With c(x) = V~s(z)(x)M~ls(x)(x,q(x)) it follows from (3.43) that O---[(q(x)v~(x)

dp(A)) dA M~- t (x'q(x))vW(x )]

(3.46)

44

3. Nonlinear System Stabilization

Next, using the Cayley-Hamilton theorem [31] it follows that

M;sl(z) (x, q(x)) = -

1 P Co(x,q(x)) Z Ck(X'q(x))M~i}z)(x'q(x))'

k----1

x 9 De, (3.47)

where co(x,q(x)) = det(M~s(z)(x,q(x)) ) ~ O. Substituting (3.47)into (3.46) yields (3.45). Now, noting that M~(x, q(x)) is affine in q(x), x 9 Dc, it follows that ck(x,q(x)), x 9 :Dc, k = O,l,...,p, is a polynomial in q(x), x 9 De, of degree p - k and hence (3.45) is a polynomial in q(x), x 9 De, of degree p. 9 In the case where p = 2 it follows that

Cl(X,q(x)) = -trM~s(z)(x,q(x)) ,

c2(x,q(x)) = 1,

x 9 Dc,

(3.48)

where "tr" is the trace operator. Now, defining A d2P(A) r Ax = tr - - ~ - ~ 2 Bx (x) = tr \ d,k2

d2p()t) dA2 , 0A2

(3.49)

] 12 - \ d,~2

0A2

,

x 9 De, (3.50)

where/2 is the 2 x 2 identity matrix, it follows from (3.45) that q(z), z 9 De, satisfies

0 = q(x)vx (x)(Ax - q(x)Bx (x))v T (x) - dP(A----~) ( A X d A - q(x)Ba(x))vT(x) a=~s(x)"

(3.51)

Next, we show that the controller 4)Xs(z)(x), x 9 Dc, minimizes a derived nonlinear-nonquadratic cost functional that explicitly depends on the Lagrange multiplier q(x), the nonlinear closed-loop system dynamics, and the gradient of the equilibria-dependent Lyapunov functions evaluated at the switching function. For the statement of this result define the set of asymptotically stabilizing controllers by

S(Xo) a {u(.) : u(.) is admissible and Do is an asymptotically stable positively invariant set of (3.1)},

(3.52)

and consider the performance functional

J(xo, u(.)) = where L : Dc x U -~ IR.

fo ~ L(x(t), u(t)) dt,

(3.53)

3.6 Inverse Optimal Nonlinear Switching Control

45

T h e o r e m 3.8 ([22]). Consider the nonlinear controlled dynamical system given by (3.1) with F(0,0) = 0 and performance functional (3.53). Assume S is diffeomorphic to an interval on the real line, let As(x), x E 79e, be a C 1 function satisfying (3.11), and assume that there exists a C 1 function V : De --+ ~ such that V(x) = O,

x E l)o,

(3.54)

v(x) > o,

x e ve \ 790

(3.55)

~'(x) ~ V'(x)F(x, Cxs(x)(x)) < 0,

x E :De \ 790.

(3.56)

Then the positively invariant set 79o of the closed-loop system x(t) = F(x(t), Cxs (~(t))(x(t))),

x(O) = xo,

t e :s

(3.57)

is locally asymptotically stable with an estimate of the domain of attraction given by De, and the performance functional (3.53), with L(x, Cxs(x)(x)) = -~'(x), is minimized in the sense that J(xo,r

=

min J(xo,u(.)), u(.)eS(zo)

xo E De.

(3.58)

Finally, J(xo,r

= V(xo),

xo E De.

(3.59)

Proof. The proof of closed-loop stability is a direct consequence of Lyapunov's theorem as applied to the closed-loop system (3.57). Optimality follows from Theorem 4.1 of [22] with Hamiltonian H(x,u) = L(x,u) + V'(x)F(x,u), x E 79c, u e 11. 9 It follows from Proposition 3.4 and Theorem 3.8 that the dynamic compensator (3.38)-(3.40) guarantees inverse optimality with respect to the performance functional (3.53) with L(x,u) given by L(x,r = -q(x)Vxs(x)(x)F(x, Cxs (x)(x)), which by (3.41) and (3.56), is positive. Finally, we note that Algorithm 3.1 can be used to construct inverse optimal hierarchical controllers as presented in this section. However, in this case Step 3 in the algorithm is unnecessary if we substitute Step 4 with: Step 4". Given the state space point x(t), the parameter value A(t), and the Lagrange multiplier q(t) at t >_ O, update A(t) and q(t) using (3.32) and (3.33). In this case, the switching set S C_As need not be explicitly defined and is computed online.

46

3. Nonlinear System Stabilization

3.7 Conclusion A nonlinear control-system design framework predicated on a hierarchical switching controller architecture parameterized over a set of system equilibria was developed. Specifically, a hierarchical switching nonlinear control strategy is constructed to stabilize a given nonlinear system by stabilizing a collection of nonlinear controlled subsystems. The switching nonlinear controller architecture is designed based on a generalized Lyapunov function obtained by minimizing a potential function over a given switching set induced by the parameterized system equilibria. An online procedure for computing the switching scheme was proposed by constructing an initial value problem having a fixed-order dynamic compensator structure. Furthermore, an inverse optimal control strategy was obtained by constructing a hierarchical controller parameterized with respect to a given system equilibrium manifold wherein an inverse optimal morphing strategy is developed to coordinate the hierarchical switching. Finally, we note that the results presented in this chapter also hold for nonlinear discrete-time dynamical systems described by time-invariant difference equations with (unique) solutions being continuous functions of the initial conditions. Specifically, in this case all of the results proceed exactly as in the continuous-time case by replacing t E R with k E Z, where Z denotes the set of nonnegative integers. Of course, in this case, the topology of the switching set 5 is such that it only consists of countable or countably infinite isolated points.

4. Nonlinear Robust Switching Controllers for Nonlinear Uncertain Systems

4.1 I n t r o d u c t i o n In Chapter 3, a nonlinear control design framework predicated on a hierarchical switching controller architecture parameterized over a set of moving system equilibria was developed. In this chapter we extend these results to address the problem of robust stabilization for nonlinear uncertain systems. Specifically, using equilibria-dependent Lyapunov functions, or instantaneous (with respect to a given nominal parameterized equilibrium manifold) Lyapunov functions, a hierarchical robust nonlinear control strategy is developed that stabilizes a compact positively invariant set of a nonlinear uncertain system using a supervisory robust switching controller that coordinates lowerlevel stabilizing subcontrollers (see Figure 4.1). Each robust subcontroller can be nonlinear and thus local set point designs can be nonlinear. Furthermore, for each nominally parameterized equilibrium manifold, the collection of the robust subcontrollers provide guaranteed domains of attraction with nonempty intersections that cover the region of operation over the prescribed range of system uncertainty of the nonlinear uncertain system in the state space. A hierarchical robust switching nonlinear controller architecture is developed based on a generalized lower semicontinuous Lyapunov function obtained by minimizing a potential function, associated with each domain of attraction, over a given switching set induced by the parameterized nominal system equilibria. The hierarchical robust switching nonlinear controller guarantees that the generalized Lyapunov function is nonincreasing along the closed-loop system trajectories for all parametric system uncertainty with strictly decreasing values at the switching points, establishing robust asymptotic stability of a compact positively invariant set. Furthermore, as in Chapter 3, since the proposed robust switching nonlinear control strategy is predicated on a generalized Lyapunov framework with strictly decreasing values at the switching points, the possibility of a sliding mode is precluded. Hence, the proposed nonlinear robust stabilization framework avoids the undesirable effects of high-speed switching onto an invariant sliding manifold which is one of the main limitations of variable structure controllers. Finally,

48

4. Nonlinear Robust Switching Controllers

___~ UnceminlyL

..............................................................................

I

*~,

...........

I

......... !. ........ :.......I

'r ' ~ >I

~+

"i"-'+hi~ I "~i co.m.+ I

""

I I *~,

I I *~,

I I *~

t

t

t

T

............. .:. ...............

I

...............................................................-..............................-..............................-...........'....'....'...'....'+..'....................... .

Fig. 4.1. Robust switching controller architecture since the theory for the robust switching controller framework very closely parallels the theory for the switching control framework developed in Chapter 3, many of the results are similar. Hence, the comments in this chapter are brief and the proofs are omitted.

4.2 Mathematical

Preliminaries

In this section we establish definitions and notation used later in the chapter. Specifically, in this chapter we consider nonlinear controlled uncertain dynamical systems of the form #c(t) = F ( x ( t ) , u ( t ) ) ,

x(0) = Xo,

F(., .) E ~-,

t E 2:~o,

(4.1)

where x(t) E 7) C R n, t E 2:zo, is the system state vector, 2:xo C_ IR is the maximal interval of existence of a solution x(.) of (4.1), 7:) is an open set, 0 E 23, u(t) E 11 C_ R m , t E Zxo, is the control input,/4 is the set of all admissible controls such that u(.) is a measurable function with 0 E U, and ~- C { F : T}x U -~ IR'+ : F(., .) E C O} denotes the class of uncertain nonlinear dynamics. Furthermore, we introduce the nominal controlled dynamical system ~(t) = F n ( x ( t ) , u ( t ) ) ,

x(O) = xo,

t E Ixo,

(4.2)

4.3 Parameterized Nominal System Equilibria

49

where Fn(., .) E yr represents the nominal system dynamics. Here, we consider nonlinear closed-loop uncertain dynamical systems of the form

5c(t) = F(x(t), r

x(0) = x0,

F(., .) E yr,

t E 2:~o. (4.3)

The following definition introduces three types of robust stability as well as attraction of (4.3) with respect to a compact positively invariant set. Definition 4.1. Let :Do C :D be a compact positively invariant set for the nonlinear feedback controlled uncertain dynamical system (4.3). Do is robustly Lyapunov stable if for every open neighborhood 01 C_ :D of :Do, there exists an open neighborhood 02 C_ 01 of :Do such that x(t) e 01, t >> O, for all Xo E 02 and F(., .) E yr. :Do is robustly attractive if there exists an open neighborhood 03 C :D of :Do such that 7)+ C_ :Do for all Xo E 03 and F(., .) E yr. :Do is robustly asymptotically stable if it is robustly Lyapunov stable and robustly attractive. Do is robust globally asymptotically stable if it is robustly Lyapunov stable and 7)+0 C :Do for all xo E R n and F(., .) E yr. Finally, :Do is unstable if it is not robustly Lyapunov stable.

4.3 Parameterized Nominal System Equilibria, System Attractors, and Domains of Attraction The nonlinear robust control design framework developed in this chapter is predicated on a hierarchical robust switching nonlinear controller architecture parameterized over a set of nominal system equilibria. It is important to note that both the nominal dynamical system and the robust controller for each parameterized nominal equilibrium can be nonlinear and thus local set point designs are in general nonlinear. Hence, the nonlinear controlled uncertain dynamical system can be viewed as a collection of controlled uncertain subsystems with a hierarchical robust switching controller architecture. In this section we concentrate on robust nonlinear stabilization of compact positively invariant sets, parameterized in D, of the nonlinear closed-loop uncertain subsystems. Specifically, we consider the nonlinear controlled uncertain dynamical system (4.1) with the origin being an equilibrium point of the nominal system corresponding to the control u = 0, that is, Fn (0, 0) = 0. Furthermore, we assume that given a mapping ~o : D x A ~ b/, ~0(0, 0) = 0, there exists a continuous function r : Ao -~ Do, where :Do C_ :D, 0 E :Do, and Ao C A, 0 E Ao, such that Fn(x~,~o(x~,A)) = 0 with x~ = r E :Do for all A E Ao. As discussed in Chapter 3, this is a necessary condition for nominal parametric stability with respect to Ao as defined in [64, 107] while Theorem 3.1 provides sufficient conditions for guaranteeing the existence of such a parameterization for the nominal system.

50

4. Nonlinear Robust Switching Controllers

Next, we consider a family of stabilizing feedback control laws for the nominal system given by

a (r

:D~/~:r

such that, for r

O,r

AsC_Ao, (4.4)

E q~, A E As, the nonlinear closed-loop nominal system

~(t) = f , ( x ( t ) , r

x(0) = x0,

t E/:~o,

(4.5)

has an asymptotically stable equilibrium point x~ 6 Do C_ D with a corresponding Lyapunov function V~(.). Hence, in the terminology of [64, 107], (4.5) is (nominally) parametrically asymptotically stable with respect to As C_ Ao. Here, we assume that for each A E As, the linear or nonlinear feedback controllers r are given. In particular, these controllers correspond to local set point designs and can be obtained using any appropriate standard linear or nonlinear stabilization scheme with a domain of attraction for each A E As. It is important to note that even though x~, A 6 As, is an equilibrium point of the nominal system (4.2), in general, x~, A E As, is n o t an equilibrium point for the uncertain system (4.1). Hence, V~(.) is not a standard Lyapunov function for the nonlinear closed-loop uncertain system

~(t)=F(x(t),r

x(O)=xo,

F(.,.)62",

te/:~o. (4.6)

However, under an additional assumption on the structure of the system uncertainty, it can be shown that u = r is a robust control law that robustly asymptotically stabilizes a compact positively invariant set Aft, containing the nominal equilibrium point x~, A E As, with domain of attraction :D~. In this case, V~(.) serves as a Lyapunov function of the uncertain system guaranteeing stability with respect to a compact positively invariant set. In particular, defining AF(x, u) a= F(x,u) - F n ( x , u) and assuming that V~(x)AF(x,r < -V~(x)Fn(x,r for all x 6 /)~ such that IIx - xx]l > r, r > 0, it follows that Cx(.) is a robustly stabilizing feedback controller of a compact positively invariant set Afx of (4.6). Next, given a stabilizing feedback robust controller Cx(.) for each A E As, we provide a guaranteed subset of the domain of attraction T)~ of a compact positively invariant set A/'x for the nonlinear closed-loop uncertain system using Lyapunov stability theory. T h e o r e m 4.1 ([59]). Let A E As. Consider the nonlinear uncertain closedloop system (4.6) with Cx(') E 9 and let Af~ be a compact positively invariant set of (4.6). Furthermore, let Xx C 7) be a compact neighborhood olaf,. Then Afx is a robustly asymptotically stable set o] (4.6) ]or all F(., .) E :7:, if, and only if, there exists a C o ]unction V~ : X~ --~ R, with V~ C 1 on X~ \Aft, such that

4.4 Robust Nonlinear System Stabilization

y (x) = o,

x 9 j%,

Vx(x) > O, x 9 X:~ \ Aft, Vx(x) ~- V~(x)F(x,r < O,

51

(4.7) (4.8)

x 9 Xx \Afx,

F(.,.) 9 Y:.

(4.9)

In addition, a subset of the domain of attraction of A/'x is given by (4.10)

where cx ~=max{/~ > 0: V;l([0,/~]) C_ X~}. It follows from Theorem 4.1 that for all Xo 9 D~ and each open set O such that Afx C O c :D~, there exists a finite time T > 0 such that x(t) 9 0 for all t _> T and F(., .) 9 ~'. Alternatively, Theorem 4.1 can be restated by requiring Vx(.) to be a C 1 function on Xx such that Conditions (4.8) and (4.9) hold and Vx(x) > O, x 9 Afx. In this case the compact positively invariant set A/'x is defined byAf~ _a V~l([0,b~]), where bx ~ inf{/~ > 0: l;'x(x) < 0, x 9

v; Note that Conditions (4.7)-(4.9) imply that V~(x) is a Lyapunov function guaranteeing robust stability of a compact positively invaxiant set A/'~ of the closed-loop uncertain system (4.6). However, Condition (4.9) is unverifiable since it is dependent on the uncertain system dynamics F(., .) 9 ~'. This condition is implied by the conditions

V~(x)F(x,r

0,

hA(Xf, Xp)[Xp --Csc(~)] > 0,

i=l,...,nf,

(5.114)

(Xf, Xp) ~ (XfA, XpA).

(5.115)

Note that (5.114) holds for all xfz, i = 1 , . . . , n f , when A > 1. If 0 < A < 1 then (5.114) holds for

xe,-A>-dA, dA----a3(A+I)-x/3(A+3)(1-A) 2

,

i=

1,...,,~,(5.116)

82

5. Hierarchical Switching Control for Axial Flow Compressor Models

while a particular choice of hA(., ") satisfying (5.115) is given by

hA(xf, Xp)

(5.117)

Zp -

In this case IYA(xf,Xp) < 0, (xf, xp) 9 ~,~+1 \ (xf~, xpA), and hence the equilibrium point (xfA, xp~) of (5.106), (5.107) with u = ~b~(xf,xp) is locally asymptotically stable for A 9 (0, 1] and globally asymptotically stable for A > 1. An estimate of the domain of attraction for (5.106), (5.107) with u = CA(xf,Xp) is given by {(xf, xp) :

( R'~ xR,

V~(zf, Xp) ~ cA},

0 < A < 1,

(5.118)

A>I,

where cA = 2 - ~ and p __a (max~{pi~l})_l. The contour level surfaces VA(xt, xp) = cA are defined such that the intersection of the boundary of DA with the plane Xp = ~A is a closed surface contained in the region {xr : - d x < x f ~ - A < dA, i = 1,...,nr} so that l/x(xf, Xp) < 0 for all (xf, xp) E ~Dx \ (xtA, ~A)" Note that ex > 0 for A > 0. To carry out Step 3 of Algorithm 3.1, we consider two topologies for the switching set 8; namely an isolated point topology and a hybrid topology. For S consisting of countably finite isolated points let ,q = {A0,..., Aq)o be such that 0 < Aq < ... < A1 < 1, A0 > 1, and (2"fAi+l,ff~pA~+l) 9 ~)Ai, i 9 {0,...,q - 1}, and let p(A) =-A, A 9 S. To guarantee that PC') satisfies Assumption 3.1 construct A~, k = 0, 1,..., q, online by considering the smallest solution to the equation VAh(x(tk)) = cab, tk A_ k A T , where AT > 0 and k = 0, 1,... ,q, and define ,q __a {Ak}q=0. Now, with the feedback switching control law u = OAs(ZvZp)(Xf,Xp), where As(xf, Xp) is obtained as described in Step 4 of Algorithm 3.1, it follows from Theorem 3.5 that the equilibrium point (xfAq,XpAq) is globally asymptotically stable. In particular, choosing Aq > 0 arbitrarily small, global asymptotic stability of an equilibrium point on the compressor characteristic pressure-flow map arbitrarily close to the maximum pressure point is established. Furthermore, note that As(x(t)), t > 0, is piecewise constant and hence the feedback switching control law u = CAs(ZvZp)(xf,Xp) is piecewise continuous. For S consisting of a hybrid topology let S = [0, 1] U {A}, where A > 1 is such that (xfX,Xpx) 9 :Ds for at least one A 9 [0, 1], and let p(A) = A, A 9 S. Since PC') does not have a local minimum in S (other than the origin) and every A 9 [0, 1] is an accumulation point for S, we are guaranteed, by Step 3b of Algorithm 3.1, that Assumption 3.1 is satisfied. Now global attraction of the origin of the nonlinear dynamical system (5.106), (5.107) is guaranteed by Theorem 3.5 with the feedback control law u = CAs(xf,xp)(Xf,Xp), where As (xf, Xp) is obtained as described in Step 4 of Algorithm 3.1. In particular, if

5.6 Stabilization of Multi-Mode Axial Flow Compression System Models

83

(xf(0), xp(0)) E D ___aOxe[0,1l Dx then As(x(t)), t > O, is a continuous function. Alternatively, if (xf(0), xp(0)) ~( 0 then As(x(t)) = A, t E [0, t'), where i > 0 is such that (xf(t'),Xp(t')) E cOO. In this case, A.~(z(t)), t _> 0, is continuous modulo one discontinuity at t = t. Note that since co = 0 and the origin is on the boundary of/5, the origin is a global attractor but not Lyapunov stable. Next, if (xt(0), Xp(0)) E 0, the on-line fixed-order dynamic compensation procedure given in Section 3.5 can be employed to compute As(z(t)), t >_ {), using the update law (3.21). Specifically, in this case s = A, wx = 1, and i , v-7(FiV~(x). Now, using (3.17), (3.21), (5.111), and (5.116), we obtain 1 =

, - A +

-

+

(5.119)

+

where IYx(xf,Xp) is given by (5.113) and A(0) = As(xf(0),xp(0)). Note that the compensator dynamics given by (5.119) characterize the admissible rate of the compensator state A(t) such that the switching nonlinear controller guarantees that (xf(t), Xp(t)) E O~DA(t), t k 0. It is important to note that the proposed switching nonlinear controller framework can be incorporated to address practical actuator limitations such as control amplitude and rate saturation constraints. Specifically, since the dynamic compensator state A(t) is proportional to the throttle opening (actuator) and since the dynamics given by (5.119) indirectly characterize the fastest admissible rate at which the control throttle can open while maintaining stability of the controlled system, it follows that by constraining the rate at which the dynamics of A(t) can evolve on the equilibrium branch effectively places a rate constraint on the throttle opening. This corresponds to the case where the switching rate of the nonlinear controller is decreased so that the trajectory (xf(t), Zp(t)), t _> 0, is allowed to enter D:~(t). Additionally, amplitude saturation constraints and state constraints can also be enforced by simply choosing )tmax < 1 such that ~)max ~-- I,)0 0, i = 1,..., nf. More generally, there exists A0 _> ~ / ~ - 1 and Aglob,l > ~ / ~ - 1 such that T)Ao collapses to the equilibrium point and :D~o~,~ coincides with the whole state space. Note that A0 and A~obal are dependent on the particular choice of the coefficients a~A, bxA,c ~ , a~A, b~, and c~n. Next, with u(xf, xp) = u~(xf, Xp), we provide an estimate of the domain of attraction for (5.137), (5.138). In particular, define

~

~ ~ {(xr,~p) :

( l~

y ~ ( ~ , ~ ) _< k~A},

~0 < ,~ < ~g~ob~,

A > As~ob,q,

x R,

A/A ~ {(xf, Xp) :

vA(xf, Xp) < k~},

(5.151) (5.152)

~ > ~0,

where - ~

~,

/~ =

m

,

(5.153)

and k2,x ~

1

1

2

max (xf - xfA)TP(xf -- xfA) + ~/3 [Xp -- Xp,x]2, (5.154) (zf,xp)e~)~ ~nf

subject to

~(~f,-~)~A(~f~)+h~(~, ~)[~-r176

2r~ i=l

nf

= ~ ~ m~(~f,). (5.155) i=l

The Lyapunov level surfaces V:~(xf, xp) = kin and V~(xf, Xp) = k2A are constructed such that the intersection of the boundary of :D~ with the plane Xp -- xpA is a closed surface contained in the region {zf : -dA < z~i - A < dA, i = 1,..., nf} and A/~ contains the region where (5.148) is not satisfied, so that lYA(xf,xp) < 0 for all (xf, xp) e ~)A \A/A. Note that for A0 _< A < Aglob,l, kl~ > 0 and k2A > 0. Furthermore, since VA(xf,Xp) is continuous and radially unbounded A/x and Z)~ are compact sets for A E [A0,Aglobal], and hence positively invariant. Thus, if the state space trajectories of (5.137), (5.138) enter :DA, then .MA serves as an attractor. Now, to ensure that A/A C :/:)~ we require that klA > k2x. A typical plot for the level set values klx and k2~ as functions of A is shown in Figure 5.8. Note that there exists Amin such that klA,.~. = k2~,i, and hence Z}A~,. = A/As,.. Hence, requiring A > Amin assures the necessary condition that A/~ C Dx. The coefficients of the two parabolas Plx(') and p2~(') must be such that (5.149) is satisfied along with the above stated necessary conditions. This

92

5. Hierarchical Switching Control for Axial Flow Compressor Models

1o0

0.2

o:3

0:4

0:5

0:6

0:7

0:s

019

{

,:, ,2

Fig. 5.8. Level set values klx and k~ as functions of A leaves some degree of freedom in the choice of the coefficients alA, blA, C1A, a2A, b2A, and c2~, which can be used to maximize the domain of attraction Dx and minimize the attractor Aft. This leads to the following optimization problem for each A: max

(A2_

al~,,bl)~,ClA,a2)~,b2,~,c2~

ci~

(5.156)

alA ] '

subject to al~ + a2~ = 1,

(5.157)

bl~ + b2~ = A + 3,

(5.158)

ClA "[- C2A ---- A(A -~- 3)

1 2,

(5.159)

(q~ - A)2(a2~q~ + b2~q~ + c2~) = 2nf~ 2,

(5.160)

2al~A + bl~ = 0,

(5.161)

al~ < 0,

(5.162)

b~, - 4a1~c1~ > O, b2~ - 4a2~c2~ < 0,

where q~ ~=

2a2~'A-3b2~-V(2a2~A-3b2~')2-16a2~(2c2~-b2~A)and 8a2~

(5.163)

m(xfi ) is chosen

to be a constant value ~ E R, i = 1 , . . . , nf. Note that, under the assumption that Ply(') achieves a maximum at A, the objective function given by (5.156) corresponds to maximizing a~. Furthermore, conditions (5.157)-(5.159) are obtained by equating the coefficients of equal powers in (5.149). Condition (5.160) guarantees that (xf~ - A)2p2~(xfi), i = 1 , . . . , nf, is a convex function

5.7 Robust Stabilization of Axial Flow Compressors

93

for all xfi so that N'~ is minimized, while conditions (5.161)-(5.163) guarantee that Plx(') achieves a maximum at A and plx(A) > 0. Finally, (5.163) guarantees that P2x(') > 0. To carry out Step 3 of Algorithm 4.1, we consider two topologies for the switching set $; namely an isolated point topology and a hybrid topology. For S consisting of countably finite isolated points let S = ~A0,..., Aq~ be such that Amin < Aq < . . . < A1 ~ Aglobal, A0 ~> Aglobai, and N'x~+l C Dx,, i e {0,...,q - 1}, and let p(A) = A, A e S. To guarantee that p(.) satisfies Assumption 4.1 construct Ak, k = 0, 1,... ,q, online by considering the smallest solution to the equation Vxk (x(tk)) = cxh, tk A_ kAT, where AT > 0 and k = 0, 1,... ,q, and define $ _a_{Ak}[=o. Now, with the robust feedback switching control law u = ~b~s(zf,Zp)(Xf,Xp), where As(xf, xp) is obtained as described in Step 4 of Algorithm 4.1, it follows from Theorem 4.4 that the compact positively invariant set A/A, is globally asymptotically stable for all ~r E ~ . Furthermore, note that As(x(t)), t >_ O, is piecewise constant and hence the robust feedback switching control law u = r (~,,,)(xf, xp) is piecewise continuous. For $ consisting of a hybrid topology let S = [~min, Aglobal]t.) {~}, where > Agloba! is such that A/"x 9 :D~, for at least one A 9 [Ami,,Aglobal], and let p(A) = A, A 9 3. Since p(.) does not have a local minimum in 3 (other than the origin) and every A 9 [Amin,Agiobal] is an accumulation point for S, we are guaranteed, by Step 3b of Algorithm 4.1, that Assumption 4.1 is satisfied. Now global robust asymptotic stability of A/'Xm~. for all ~r 9 A is guaranteed by Theorem 4.4 with the feedback control law u = CAs(xf,zp)(xf,xp), where As(xf, Xp) is obtained as described in Step 4 of Algorithm 4.1. In particular, if (xf(0),Xp(0)) 9 D =~ UXe[~,,.,Xg~ob.~]Dx then As(x(t)), t >__0, is a continuous function. Alternatively, if (xe(0),xv(0)) r then As(x(t)) = ~, t 9 [0, t~, where t > 0 is such that (xf(~, Xp(~) 9 0/5. In this case, As(x(t)), t _> 0, is continuous modulo one discontinuity at t = t. Note that since N'~m~. - :D~,~., Afxm~. is a global attractor but not Lyapunov stable. As in the nominal case, the proposed robust switching nonlinear controller framework can be incorporated to address practical actuator limitations such as control amplitude and rate saturation constraints. Specifically, since At ~ As(xf(t), Xp(t)) is proportional to the throttle opening (actuator) and since the dynamics of At indirectly characterize the fastest admissible rate at which the control throttle can open while maintaining stability of the controlled system, it follows that by constraining how fast At can change on the nominal equilibrium branch effectively places a rate constraint on the throttle opening. This corresponds to the case where the switching rate of the nonlinear controller is decreased so that the trajectory (xf(t),xp(t)), t >__O, is allowed to enter :Dx,. Additionally, amplitude saturation constraints and

94

5. HierarchicaI Switching Control for Axial FIow Compressor Models

state constraints can also be enforced by simply choosing ~max 2 is the pressure rise at the exit of the compressor. Next, assuming isentropic process dynamics with a constant specific heat cp, it follows that

([63]) 7o-

'

(6.10)

where T01 is the compressor inlet temperature, T2 is the fluid temperature at compressor rotor exit, and 9, is the specific heat ratio. Now, using Ahideal = Cp(T2-T01), where Ahideal is the ideal change in fluid specific enthalpy which, for a conservative system, is equal to the work done by the compressor rotor, it follows from (6.10) that t)2 = Po

(Ahideal) 1 + - -

...2_ "-1

(6.11)

%Tol

Now, using the fact that the change in angular momentum of fluid is equal to the compressor torque ~'c it follows that ([40, 99]) Tc = mc(r2%2 -- r i c o , ) ,

(6.12)

where rl ~ 88x/D21 + D~I, r2 is the radius of the rotor tip, Dr1 is the inducer tip diameter, Dhl is the inducer hub diameter, and ce, and co~ are the absolute tangential velocity of fluid at the rotor inlet and rotor outlet, respectively. Next, we assume that there is no pre-whirl at the rotor inlet so that %1 = 0 and define the slip factor which is the ratio between the tangential velocity of the fluid at the rotor outlet and the rotor tip velocity by ([40])

6.2 Governing Fluid Dynamic Equations for Centrifugal Compressors

c'2

101

(6.13)

r2o3

where w is the angular velocity of the compressor spool. Here we assume that a is constant. Substituting (6.13) into (6.12), we obtain Vc(mc,W) = ar~mcw,

(6.14)

so that the work done to the fluid by the compressor, which is equal to the change of enthalpy of the fluid, is given by (6.15)

Ah(w) = TcW = ar~w2. mc

Now, we consider incidence losses at the inducer and the diffuser which, respectively, can be expressed as ([48]) ( Ahii(mc,w) = 1 rlw = 89

(

c~ f~sbmc'~ 2 pA ] ' cot Ot2bmc

)'

,

where p is the gas density in the compressor stage, o~2b is the inducer inlet angle, and ~lb is the rotor blade angle. In addition, we consider friction losses at the inducer and the diffuser, denoted as Ahit(mc) and Ahdf(mc), respectively, assumed to be quadratic functions of the mass flow rate at the plenum entrance given by ([431) Ahif(mc) = kifm 2,

Ahdf(me) = kdfm2c,

(6.16)

where kif and kdf are the friction coefficients. Combining (6.15)-(6.16) we obtain the energy delivered to the fluid by the compressor as Ah(mc, w) = Ahideal (W) + Ah,oss (me, w),

(6.17)

where Ahxoss(mc,w) ~ Ahii(mc, w) + Ahdi(mc, W) -t- Ahif(mc) -b Ahdf(mc). (6.18)

In order to capture compressor efficiency, define the isentropic efficiency as ([481) Ahideal(W) ~}c(me, w) ~ Ahloss (mc, w) + Ahideal(W)" Substituting (6.11) and (6.19) into (6.9) yields

(6.19)

102

6. Hierarchical Switching Control for Centrifugal Flow Compressor Models

=

1 + 7/c(mc,w) c - - ~ o1

P0 - Pp 9

(6.20)

Next, defining the nondimensional angular velocity of compressor spool by & _a ~fI and using (6.6), it follows that = b(r162 ~) - r

(6.21)

where b =a "2L2 and r162162 ~) is the compressor characteristic pressure-flow/angular velocity map given by r162

-~ (1 + r}c(r

u

- 1,

(6.22)

where a& 2

~c(r

=

O'~2 -4- l(f15~ -- f2•) 2 -[- l(o'f~ -- f3r 2 + f4(~2 + fS(~2'

(6.23)

and d ___a a 2

CpTo1'

-

-

11

a rl = --,

a A --

r2

a2ptanO~2b,

~--~-]

,

/7o

a2ptan ~Ib

,

f5 ~ kfd ~ a2 ]

(6.24)

9 (6.25)

It is important to note that the compressor characteristic map given by (6.22) holds for the case where the flow through the compressor is positive. In the case of deep surge involving negative mass flow, it is assumed that the pressure rise in the compressor is proportional to the square of the mass flow so that [61, 97]

r162 ~) = ~r + r162

r < o,

(6.26)

where # is a constant and r162

~ r162

~=o = (I + a~cod~.02)--~-~ ~ - 1,

(6.27)

where

~co = 7r162~)

r

=

2a o2 + 2o +/12.

(6.28)

Now, for a fixed &, taking the gradient of r162 w) with respect to the nondimensional flow r it follows that the flow corresponding to the maximum

6.2 Governing Fluid Dynamic Equations for Centrifugal Compressors

103

pressure point of the compressor characteristic map is directly proportional to the nondimensional angular velocity of the compressor spool and is given by Cmax = kf~, where kf __a

]1f2 + 0"]3

(6.29)

f2+f2+2A+2

h"

Similarly, for a fixed ~ taking the gradient of ~c(r with respect to the nondimensional flow r we obtain that the maximum value for the isentropic efficiency is given by

+

+ 2(14 + h))

~c~,,x = a(2 + o')f~ - 2o'flf2f3 + (2a + f 2 ) f 2 + 2(2a + 6r2 Jr f2)(f4 Jr f5)" (6.30) Note that ~k.,,= is constant for all spool speeds. This indicates that the compressor achieves the same maximum isentropic efficiency at each maximum pressure point for all spool speeds. However, since these points are critically stable, the need for active control is severe to guarantee stable compression system operation for peak compressor performance. Figure 6.2 shows a typical family of compressor characteristic maps for different spool speeds along with the corresponding constant isentropic efficiency lines. The stone wall depicted in Figure 6.2 corresponds to choked flow at a given cross-section of the compression system. Specifically, assuming that area choking occurs at the impeller and the process is isentropic, it follows [40, p. 211] that Tch To1

_

( 1 + - r -w2

2 7 - 1

(6.31)

2cpTol ) ' ....1_

P01

\~1Ol]

'

(6.32)

where Tch is the stagnation temperature, Pch is the stagnation density, and Pol is the flow density at the compressor inlet. Using a mass balance (6.31) and (6.32) yield inch : AePolaol t

+

i)r w21

~o21-~-~~)

j

,

(6.33)

where inch is the choked mass flow, Ae is the area of the impeller eye, and aol is the speed of sound at the inlet. Using ~ = - ~ , the nondimensional form of (6.33) is given by 7+ 1 where r162=a a--a--m Apo ch and ~ _a ~A. A "

J

'

(6.34)

104

6. Hierarchical Switching Control for Centrifugal Flow Compressor Models 0.5

L

Ii

i

i

i

s

i

i

i

0.45

,,

0.4

o

:.:

i _ 0.55---'r----'~-~ I

I

/I

/

I

J /

0.35 0

"~

0.3

0.25 ~

0.2

0.15

_..~.-o3s,-~--~, /

, I

9

/

I

0.05- /

0

/

/

-

i

/

-

~

/

~

. .~ ~ ; - - !

.-

/

/

~

J"

/ /

/

0.05

/

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

4, Flow Fig. 6.2. Compressor characteristic maps and efficiency lines for different spool speeds 6.2.3 T u r b o c h a r g e r S p o o l D y n a m i c s Using conservation of angular momentum in the turbocharger spool it follows that the spool dynamics are given by dw

I 8 - ~ = rd -- r e ( m e , w ) ,

(6.35)

where Is is the spool mass moment of inertia, rd is the driving torque, and r e ( m e , w ) is the compressor torque. Substituting (6.14) into (6.35) we obtain Is ~ tt = rd -- a r 2 m c w ,

(6.36)

which, using (6.6) and & = -~-, can be written in nondimensional form as = ~ ( r - ale]&),

(6.37)

where _ ~ L3r~po C----- -

-

Isa 2 '

~ r-----

rd

Apor2 "

(6.38)

6.3 Parameterized System Equilibria and Local Set Point Designs

105

Note that in (6.37) the fact that the compressor may enter deep surge has been taken into account. In this case reverse flow can occur, during which the centrifugal compressor can be viewed as a throttling device, and hence can be approximated as a turbine [45].

6.3 Parameterized System Equilibria and Local Set Point Designs In this section we develop Lyapunov-based subcontroller designs for local set points parameterized by the flow through the throttle and the nondimensional driving torque. It is important to note that even though a Lyapunovbased framework [87] can be used to stabilize the compression system, the resulting controller may generate unnecessarily large control amplitude and rate signals that can amplitude and rate saturate the control actuators resulting in system performance degradation and even instability (see [1] and the references therein). To proceed with the local set point designs, first note that with control inputs Ul ~ %hV~ and u2 ~ T it follows from (6.7), (6.21), and (6.37), that a state space model for the centrifugal compressor is given by = ~(r - ul), = b(r162

(6.39) - r

w = ~(u2 - ar

(6.40) (6.41)

To carry out Step 1 of Algorithm 3.1, let q = m = 2 and ~(r r A) = !A where a is a scaling factor and ~ ~ [hiA2] T E R 2, so that the system equilibria are parameterized by the constant control u(t) = ~A, 1 where u(t) a_ [Ul(t)u2(t)]w. In this case, (6.39)-(6.41) have an equilibrium point at (r r 5Jn), where

Next, we carry out Step 2 of Algorithm 3.1. Specifically we show that for A1 > 0 and A2 > 0 there exists a control law such that the equilibrium point (r Cx,&x) of (6.39)-(6.41) is globally asymptotically stable. To show this, define the shifted variables x~l = r - r xn2 = r - r and x~3 = w - &x, so that the given equilibrium point is translated to the origin. Furthermore, with the shifted controls fil g Ul - ~ and fi2 _a u2 - h~ it follows that the parameterized translated nonlinear system is given by ~Xl = a ( x x 2 -- ?~1),

(6.43)

x~2 = b(r

(6.44)

(x~2, x~s) - x~l),

xx3 = ~(fi2 - f ( x ~ 2 , x~3)),

(6.45)

106

6. Hierarchical Switching Control for Centrifugal Flow Compressor Models

where r

a r162 + z~2,&~ + x~3) - r162

(6.46)

f(xx2,x),3) a=alex + x~21(~), + x~3) - aCx&~.

(6.47)

Now, setting ~X : XA2 -- ~I,

(6.48)

fi2 = - k 3 z ~ + ](z~2, x~3),

(6.49)

where k3 > O, and substituting (6.48) and (6.49) into (6.43)-(6.45) yields X~l : ariA,

d:x2 : b(r

(6.50)

x~3) - XXl),

(6.51) (6.52)

XA3 : -k3cxA3.

Next, consider the equilibria-dependent Lyapunov function candidate YA (xA1, xx2, xx3) : -~- (zx1 - r

(ZA2, ZX3) -- k2xA2)2 _{__~x22 + T ~ 3~. ,2

(6.53)

where ks > 0, ai > 0, i = 1,2, 3. The corresponding Lyapunov derivative is given by VA(ZA1, XA2,ZA3) : ~1 (ZA1 -- r

-- k2ZA2)(XA1 -- ~CA(ZA2, ZA3)

--k2XA2) + Ot2XA2XA2 -[- Ot3XA3;TA3 = -(z~1 - r x~3) - k2x~2)(-alafi~ +a1r

(x~2, x~3)z~2 + a~r

(z~2, z~3)~3

_{_(21k2xx 2 + O~2~XA2) -- ot2k2~x~ 2 _ o~3k3c~A3,2

where

Oz~,2 _

0r

(~,~)=(~+x~2, ~+z~3)

70"~ 2

7:i

[r162162 + ~2,~

+ ~3) + 1]~ L(z~2,x~3),

~CA,z~xz(.TA2,XA3) ~_A ~)CA(2:A2,'TA3) -- (~C((~,~) Oxx3 O~z

_ Tad with

[r162 + x~2,Dx + xx3) + 1]88 12(xx2, ZA3),

6.4 Hierarchical Nonlinear Switching Control for Centrifugal Compressors

~ 8~r162 8r

/~(~,x~)

8rIc(~b,~)

I (~,~)=(r

~+=~-~)

o'~2(f2(~/1 - ~b/2) -}- .f3(ff~ - ~b.f3) - 2r

8r

(~

+ 89

- r

107

+ 89

- r

-b .fs))

2+ r

+ ~))~'

and f2(X)~2,~)~3 ) ~ 8 [~J2}7c(r 8 [~2~c(~,~)] 8~.d

--

(~b,~)=(~b~+z.x2,

0"~3( 20"~2 "}" 0~/fl ( ~ f l -- r "b 0r~(a~ -- r (U0~2 "~- l ( ~ f l -- r 2 Jr 1 (6r0~ -- r 2 -~" r -~" .f5)) 2

o.,87r162 -~w~ ~-~

.

Now, choosing the nonlinear control law ~ = - k l (z~l - r 9(r

x~3) - k2z~2) + a {22-

]

(x~2, x~3) - x ~ ) - k 3 e ~ , , ~ . (x~,2, xj,3)xj,3 + ~bxj,2 ,

where k~ > 0, it follows that V~,(x~,l, x~2, x;~3) = - a l k 1 ~ ( x ~ , 1

- r

X.~3) -- k2xA2) 2

-a2k2b:r22 -- ~ 3 k 3 ~ 2 3 0, be such that (Ca(s), r s E [0, a], is ~n equilibrium point of (6.43)-(6.45) in the shifted variables (r162 In particular, s = 0 corresponds to the origin in the shifted variables and hence corresponds to the desired maximum pressure point in the original variables. Furthermore, assume that ca(s) is a C 1 function of s E [0, a] and note that since c~(8) > 0 and V~(s)(') is continuous and radially unbounded, 79~(,) is a compact set for s E [0,a], which further implies that/9~(8) is a positively invariant set of (6.39)-(6.41) with feedback control law r162 Cs, 5~s), s E [0, a]. Finally, since 79~(,) is not empty for all s E [0da ], there exists Sx,S2 > 0 such that s2 < Sl and To carry out Step 3 of Algorithm 3.1, we consider two topologies for the switching set S; namely an isolated point topology and a continuous topology. For S consisting of countably finite isolated points, let S = (A0,..., Aq} where Ak a_ a(sk), k ~ 0 , 1 , . . . , q , and 0 = sq < -.. < So _< a, a > 0, such that (r162 e 7:)x,+,, i G { 0 , . . . , q - 1}, and let p(A) = a-l(A) = s, G S. To guarantee that p(.) satisfies Assumption 3.1 construct sk, k = 0, 1 , . . . , q, online by considering the smallest solution sk _> 0 to the equation V~(s~)(x(tk)) = c~(,~), tk A k A T , where AT > 0 and k = 0 , 1 , . . . , q , and define 8 =a {Ak }k=0, q where Ak = a(sk). Now, with the feedback switching control law u = Cxs(r162162 Cs,&s), where As(r Cs,f~s) is obtained as described in Step 4 of Algorithm 3.1, it follows from Theorem 3.5 that the equilibrium point (era, Cm,~m) is globally asymptotically stable. Furthermore, note that As(r Cs(t), &s(t)), t _> 0, is piecewise constant and hence the feedback switching control law u = r162162162 Cs,ws) is piecewise continuous.

6.4 Hierarchical Nonlinear Switching Control for Centrifugal Compressors

109

For 3 consisting of a continuous topology let S =~ c([0,a]) and let p(A) = a -1 (A) = s, A E 3. By requiring that p(.) does not have a local minimum in 8 (other than the origin) and since every A E 8 is an accumulation point for $, we are guaranteed, by Step 3b of Algorithm 3.1, that Assumption 3.1 is satisfied. Now global attraction of (era, era, ~m) is guaranteed by Theorem 3.5 with the feedback control law u = r162162162162 where As (r Cs, 5Js) is obtained as described in Step 4 of Algorithm 3.1. In particular, if (~s(0), Cs(0),hJs(0)) E D ~ Use[0,o]29~(s) then As(r162 t > 0, is a continuous function. Alternatively, if (r162 r D then As(r162 = a(a), t e [0,t~, where t" > 0 is such that (r162 e 029. In this case, As(r162 t >_ O, is continuous modulo one discontinuity at t = t. Next, if (r (0), Cs(0), ws (0)) 9 29, the online fixed-order dynamic compensation procedure given in Section 3.5 can be employed to compute As (r (t), Cs(t),~s(t)), t 9 [0,Tzo], using the update law (3.21). Note that the compensator dynamics given by (3.21) characterize the admissible rate of the compensator state A(t) such that the switching nonlinear controller guarantees that (r162 9 c929~(t), t 9 [0,Tzo]. Once again, it is important to note that the proposed switching nonlinear controller framework can be incorporated to address practical actuator limitations such as control amplitude and rate saturation constraints. Specifically, since the dynamic compensator state A(t) is proportional to the throttle opening and the nondimensional driving torque and since the dynamics given by (3.21) indirectly characterize the fastest admissible rate at which the control variables can change while maintaining stability of the controlled system, it follows that by constraining the rate at which the dynamics of A(t) can evolve on the equilibrium branch effectively places a rate constraint on the throttle opening and the nondimensional driving torque. This corresponds to the case where the switching rate of the nonlinear controller is decreased so that the trajectory (r162 t >_ O, is allowed to enter 29~(t). Additionally, amplitude saturation constraints and state constraints can also be enforced by simply choosing 8max • 0 such that 29ma~ =AU~e[0,sm.~129~(s) is contained in the region where the system is constrained to operate. In this case, the switching nonlinear controller guarantees attraction of (era, era, win) with an estimate of the domain of attraction given by 29max. Next, we apply the hierarchical nonlinear switching control framework developed in Chapter 3 to the control of surge in centrifugal compression systems. Specifically, we use the three-state centrifugal compressor model derived in this chapter with (~, b, ~,d) -- (9.37,310.81, 23.70, 0.38), (.fl,f2, f 3 , f 4 , f h ) -~ (0.44, 1.07, 2.18,0.17,0.12), 7 : 1.4, # = 5, and a = 0.9.

110

6. Hierarchical Switching Control for Centrifugal Flow Compressor Models

We compare the open-loop response when the compression system is taken from an operating speed of 20,000 rpm to 25,000 rpm, corresponding to the initial conditions (r r 5;0) = (0.305, 0.177, 0.493), with the closed-loop responses obtained using the design parameters (al, a2, a3) = (1, 0.1, 1) and (kl, k2, k3) = (1,3, 1) and the scaling factor a = 10. For the standard nonlinear switching control framework, we use the diffeomorphism a ( s ) = (as,0), s E [0,0.5], and c, = 0.01 + 2s. Furthermore, we consider the closed-loop responses obtained with and without a rate saturation constraint on the throttle opening (l~h,[ _< 5). Figure 6.3 shows the r 1 6 2phase portrait of the state trajectories. The pressure rise, mass flow, and spool speed variations for the open-loop and controlled system are shown in Figures 6.4, 6.5, and 6.6, respectively. Figures 6.7 and 6.8 show the control effort versus time. This comparison illustrates that open-loop control drives the compression system into deep surge while the proposed nonlinear switching controller drives the system to the desired maximum pressure-flow equilibrium point (r r 5~) = (0.656, 0.248, 0.690). Note that the switching controller with a rate saturation constraint guarantees stability with minimal degradation in system performance.

0.32 0.3 0.28 0 r~

"1~ O.26 t~

~ o.24 ~" 0.22 0.2 0.18

-0.2

-0.1

0

0.1

0.2

0.3

(~, Flow Fig. 6.3. Phase portrait of pressure-flow map

0.4

0.5

6.4 Hierarchical Nonlinear Switching Control for Centrifugal Compressors

111

0.32

i,

0.3 fl

0.28 /

"C::

s

9

~ 0.26

I

s /

I I

I ii

I

~O.:N

$

I

I

i i I

0.22

0.2

/

ua

Saturated

,

- - - Unsaturated

tl

II

,~

"

Open-loop 0.18

i

i

oY

I

.5

2

2:5

Time Fig.

6.4.

Pressure rise versus time

0.5

0"4t 0.3

~ Ol

"

'

I I

~

+

I

r~ I "~-

. . . . . . . .

'~+" xu ~,1

I I ii I i I ii I t

#

Saturated

Unsaturated

I I iI

nf

t I i I I i I i I i I i I I I iI iI iI

l~ It

It I

I t

I

I

I

I I II

I I II II II il I I " I I I I iI

il

o

'l

d

,

l Ill

~ II

l II

| II

t

|

'

I t

I I I II I I I I i

,l

I

I

I I I I I i I I I I I I I Ii

~

i+

Open-loop 0.2

oY

I

i

;.+

2

2:5

~, Time

Fig. 6.5. Mass flow versus time For the inverse optimal nonlinear switching control framework developed in Section 3.6, we construct a positive-definite potential function p(-) and solve the Extended Optimal Switching Control Problem by implementing the dynamic controller (3.38)-(3.40). This yields a switching function As (') such that the feedback control law u ~ Cxs (~.,~.,~.)(r Cs, ws) globally asymptotically stabilizes the operating condition (r Cs, ws) = (0, 0, 0). Furthermore, since the dynamic compensator state A(t) is related to the throttle opening

112

6. Hierarchical Switching Control for Centrifugal Flow Compressor Models 0.56

0.54

?

it i

I~

0.52 It

I

....

0.5

i i

"~ 0.48

i

.|

It

I I

I |

I ~

i

l',:,l , t ' : ' . , ' J ~ I

L

v

~

# I

I

,j

i

~fj\l /

t0,46 i /

~ 0.44 /

0.42

Saturated ] --- Unsaturated

/ 0.4 0.38

--0.5

1

1.5 ~, Time

2

Open-loop 2.5

F i g . 6.6. Compressor spool speed versus t i m e 0.17 Saturated I Unsaturated

0.16

0.15 0,14

0.11 0.1 0.09 0.08

0i1

t

0.2

t

0.3 ~, Time

t

0.4

t

0,5

0.6

Fig. 6.7. Control effort versus time: Torque and to the nondimensional driving torque, it follows that by constraining the rate at which the dynamics of A(t) can evolve effectively places a rate constraint on the control variables. Amplitude saturation constraints can be enforced by assigning a higher potential value to the parameters corresponding to the control values that are magnitude limited. Hence by appropriately choosing the potential function, the proposed inverse optimal switching non-

6.4 Hierarchical Nonlinear Switching Control for Centrifugal Compressors

~\

-

-

-

113

Unsaturated

~ \

t~o.'ti \ \ ~~

\

~o.21-, / ...--\.. - - \ - S/

o.,ti:

.......

v

0-;

0:1

0:2

0:3 ~, T ~ e

0.'4

o;

0.6

Fig. 6.8, Control effort versus time: Throttle opening

linear controller can address the practical limitations of control amplitude and rate saturation constraints. To show the efficacy of the proposed inverse optimal switching control approach, we choose cx = 0.4 V~(r era,win) and consider the potential functions

and P2(A1, A2) = ( )~2 + ~22,

~ +~

+h[1-(~:"'~)-(~ - - ~ )2]3,

d()h, )~2) > r,

~(~1,~) _