Communications and Control Engineering

Transcription

1 Communications and Control Engineering

2 Published titles include: Stability and Stabilization of Infinite Dimensional Systems with Applications Zheng-Hua Luo, Bao-Zhu Guo and Omer Morgul Nonsmooth Mechanics (Second edition) Bernard Brogliato Nonlinear Control Systems II Alberto Isidori L 2 -Gain and Passivity Techniques in Nonlinear Control Arjan van der Schaft Control of Linear Systems with Regulation and Input Constraints Ali Saberi, Anton A. Stoorvogel and Peddapullaiah Sannuti Robust and H Control Ben M. Chen Computer Controlled Systems Efim N. Rosenwasser and Bernhard P. Lampe Control of Complex and Uncertain Systems Stanislav V. Emelyanov and Sergey K. Korovin Robust Control Design Using H Methods Ian R. Petersen, Valery A. Ugrinovski and Andrey V. Savkin Model Reduction for Control System Design Goro Obinata and Brian D.O. Anderson Control Theory for Linear Systems Harry L. Trentelman, Anton Stoorvogel and Malo Hautus Functional Adaptive Control Simon G. Fabri and Visakan Kadirkamanathan Positive 1D and 2D Systems Tadeusz Kaczorek Identification and Control Using Volterra Models Francis J. Doyle III, Ronald K. Pearson and Bobatunde A. Ogunnaike Non-linear Control for Underactuated Mechanical Systems Isabelle Fantoni and Rogelio Lozano Robust Control (Second edition) Jürgen Ackermann Flow Control by Feedback Ole Morten Aamo and Miroslav Krstić Learning and Generalization (Second edition) Mathukumalli Vidyasagar Constrained Control and Estimation Graham C. Goodwin, María M. Seron and José A. De Doná Randomized Algorithms for Analysis and Control of Uncertain Systems Roberto Tempo, Giuseppe Calafiore and Fabrizio Dabbene Switched Linear Systems Zhendong Sun and Shuzhi S. Ge Subspace Methods for System Identification Tohru Katayama Digital Control Systems Ioan D. Landau and Gianluca Zito Multivariable Computer-controlled Systems E.N. Rosenwasser and B.P. Lampe

3 Bernard Brogliato, Rogelio Lozano, Bernhard Maschke and Olav Egeland Dissipative Systems Analysis and Control Theory and Applications 2nd Edition With 94 Figures 123

4 Bernard Brogliato, PhD INRIA Rhône-Alpes BIBOP Project ZIRST Montbonnot St. Ismier Cedex France Bernhard Maschke, PhD Laboratoire d Automatique et de Génie des Procédés Université Claude Bernard Lyon1 UFR Génie Electrique et des Procédés CNRS UMR 5007 CPE Lyon Villeurbanne Cedex France Rogelio Lozano, PhD Heuristique et Diagnostic des Systèmes Complexes UMR-CNRS 6599 Université de Technologie de Compiègne Centre de Recherche de Royalieu BP Compiègne France Olav Egeland, PhD Department of Engineering Cybernetics NTNU Faculty of Information Technology, Mathematics and Electrical Engineering NO-7491 Trondheim Norway Series Editors E.D.Sontag M.Thoma A.Isidori J.H.vanSchuppen British Library Cataloguing in Publication Data Dissipative systems analysis and control : theory and applications. - 2nd ed. - (Communications and control engineering) 1. System analysis 2. Automatic control - Mathematics I. Brogliato, Bernard, ISBN Library of Congress Control Number: Communications and Control Engineering Series ISSN ISBN (ebook) 2nd edition Printed on acid-free paper ISBN nd edition ISBN st edition Springer-Verlag London Limited 2007 First published 2000, Second edition 2007 MATLAB and Simulink are registered trademarks of The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA , U.S.A. Apart from any fair dealing 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 of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. 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 Springer Science+Business Media springer.com

5 Preface Thank you for your interest in the second edition of our book on dissipative systems. The first version of this book has been improved and augmented in several directions (mainly by the first author supported by the second and third authors of the second version). The link between dissipativity and optimal control is now treated in more detail, and many proofs which were not provided in the first edition are now given in their entirety, making the book more self-contained. One difficulty one encounters when facing the literature on dissipative systems is that there are many different definitions of dissipativity and positive real transfer functions (one could say a proliferation), many different versions of the same fundamental mathematical object (like the Kalman-Yakubovich-Popov Lemma), and it is not always an easy task to discover the links between them all. One objective of this book is to present those notions in a single volume and to try, if possible, to present their relationships in a clear way. Novel sections on descriptor (or singular) systems, discrete-time linear and nonlinear systems, some types of nonsmooth systems, viscosity solutions of the KYP Lemma set of equations, time-varying systems, unbounded differential inclusions, evolution variational inequalities, hyperstability, nonlinear H, input-to-state stability, have been added. Conditions under which the Kalman-Yakubovich-Popov Lemma can be stated without assuming the minimality of the realization are provided in a specific section. Some general results (like well-posedness results for various types of evolution problems encountered in the book, definitions, matrix algebra tools, etc.) are presented in the Appendix, and many others are presented in the main text when they are needed for the first time. We thank J. Collado and S. Hadd who made us some remarks, and we remain of course open to any comments that may help us continue to improve our book. Montbonnot, April 2006 Bernard Brogliato

6 Contents 1 Introduction Example1:SystemwithMassSpringandDamper Example 2: RLC Circuit Example3:AMasswithaPDController Example4:AdaptiveControl Positive Real Systems Dynamical System State-space Representation Definitions InterconnectionsofPassiveSystems LinearSystems PassivityofthePIDControllers Stability of a Passive Feedback Interconnection MechanicalAnalogsforPDControllers MultivariableLinearSystems TheScatteringFormulation ImpedanceMatching FeedbackLoop Bounded Real and Positive Real Transfer Functions Examples MechanicalResonances SystemswithSeveralResonances TwoMotorsDrivinganElasticLoad StrictlyPositiveReal(SPR)Systems Frequency Domain Conditions for a Transfer Function tobespr Necessary Conditions for H(s)tobePR(SPR) TestsforSPRness InterconnectionofPositiveRealSystems SpecialCasesofPositiveRealSystems Applications... 62

7 viii Contents SPRandAdaptiveControl AdaptiveOutputFeedback DesignofSPRSystems Kalman-Yakubovich-Popov Lemma ThePositiveRealLemma PRTransferFunctions ADigressionto OptimalControl Duality PositiveRealLemma forsprsystems DescriptorVariableSystems WeaklySPRSystemsandtheKYPLemma KYPLemmaforNon-minimalSystems SpectralFactors Sign-controllability StateSpaceDecomposition A Relaxed KYP Lemma for SPR Functions with Stabilizable Realization SPRProblemwithObservers TheFeedbackKYPLemma Time-varyingSystems InterconnectionofPRSystems PositiveRealnessandOptimalControl GeneralConsiderations LeastSquaresOptimalControl The Popov Function and the KYP Lemma LMI ARecapitulatingTheorem On the Design of Passive LQG Controllers Summary A Digression on Semidefinite Programming Problems The Lur e Problem (Absolute Stability) Introduction Well-posednessofODEs Aizerman sandkalman sconjectures MultivaluedNonlinearities Dissipative Evolution Variational Inequalities TheCircleCriterion LoopTransformations ThePopovCriterion Discrete-timeSystems TheKYPLemma TheTsypkinCriterion DiscretizationofPRSystems...175

8 Contents ix 4 Dissipative Systems NormedSpaces L p Norms Relationships Between L 1, L 2 and L Spaces Review of Some Properties of L p Signals Example of Applications of the Properties of L p FunctionsinAdaptiveControl LinearMaps InducedNorms PropertiesofInducedNorms ExtendedSpaces GainofanOperator SmallGainTheorem DissipativeSystems Definitions The Signification of β Storage Functions (Available, Required Supply) Examples RegularityoftheStorageFunctions NonlinearKYPLemma AParticularCase Nonlinear KYP Lemma in the General Case Time-varyingSystems Nonlinear-in-the-input Systems Dissipative Systems and Partial Differential Inequalities Thelinearinvariantcase The Nonlinear Case y = h(x) The Nonlinear Case y = h(x)+j(x)u Recapitulation InverseOptimalControl NonlinearDiscrete-timeSystems PRtangentsystemanddissipativity Infinite-dimensional Systems AnExtensionoftheKYPLemma TheWaveEquation TheHeatEquation FurtherResults Stability of Dissipative Systems PassivityTheorems One-channelResults Two-channelResults LosslessandWSPRBlocksInterconnection Large-scaleSystems Positive Definiteness of Storage Functions

9 x Contents 5.3 WSPRDoesnotImplyOSP Stabilization by Output Feedback AutonomousSystems Time-varyingNonlinearSystems EvolutionVariationalInequalities EquivalencetoaPassiveSystem CascadedSystems Input-to-State Stability (ISS) and Dissipativity PassivityofLinearDelaySystems SystemswithStateDelay InterconnectionofPassiveSystems Extension to a System with Distributed State Delay Absolute Stability Nonlinear H Control Introduction Closed-loop Synthesis: Static State Feedback Closed-loop Synthesis: PR Dynamic Feedback Nonlinear H MoreonFinite-power-gainSystems Popov s Hyperstability Dissipative Physical Systems LagrangianControlSystems DefinitionandProperties SimpleMechanicalSystems HamiltonianControlSystems Input-output Hamiltonian Systems PortControlledHamiltonianSystems RigidJoint RigidLinkManipulators TheAvailableStorage The Required Supply FlexibleJoint RigidLinkManipulators TheAvailableStorage The Required Supply ABouncingSystem IncludingActuatorDynamics Armature-controlledDCMotors Field-controlledDCMotors PassiveEnvironment SystemswithHolonomicConstraints CompliantEnvironment NonsmoothLagrangianSystems Systems with C 0 Solutions SystemswithBVSolutions...365

10 Contents xi 7 Passivity-based Control BriefHistoricalSurvey TheLagrange-DirichletTheorem Lyapunov Stability Asymptotic Lyapunov Stability Invertibility of the Lagrange-Dirichlet Theorem The Lagrange-Dirichlet Theorem for Nonsmooth LagrangianSystems(BVSolutions) The Lagrange-Dirichlet Theorem for Nonsmooth Lagrangian Systems (C 0 Solutions) Conclusion RigidJoint RigidLinkSystems:StateFeedback PDControl PIDControl More about Lyapunov Functions and the Passivity Theorem Extensions of the PD Controller for the Tracking Case OtherTypesofStateFeedbackControllers RigidJoint RigidLink:PositionFeedback P+ObserverControl The Paden and Panja + Observer Controller The Slotine and Li + Observer Controller FlexibleJoint RigidLink:StateFeedback Passivity-based Controller: The Lozano and Brogliato Scheme Other Globally Tracking Feedback Controllers FlexibleJoint RigidLink:OutputFeedback PDControl MotorPositionFeedback IncludingActuatorDynamics Armature-controlledDCMotors Field-controlledDCMotors ConstrainedMechanicalSystems Regulation with a Position PD Controller HolonomicConstraints NonsmoothLagrangianSystems ControlledLagrangians Adaptive Control LagrangianSystems Rigid Joint Rigid Link Manipulators Flexible Joint Rigid Link Manipulators: The Adaptive LozanoandBrogliatoAlgorithm Flexible Joint Rigid Link Manipulators: The BacksteppingAlgorithm...452

11 xii Contents 8.2 LinearInvariantSystems AScalarExample Systems with Relative Degree r = Systems with Relative Degree r = Systems with Relative Degree r Experimental Results FlexibleJointManipulators Introduction ControllerDesign TheExperimentalDevices ExperimentalResults Conclusions Stabilization of the Inverted Pendulum Introduction System sdynamics Stabilizing Control Law SimulationResults ExperimentalResults Conclusions A Background Material A.1 Lyapunov Stability A.1.1 Autonomoussystems A.1.2 Non-autonomousSystems A.2 DifferentialGeometryTheory A.2.1 NormalForm A.2.2 FeedbackLinearization A.2.3 Stabilization of Feedback Linearizable Systems A.2.4 FurtherReading A.3 ViscositySolutions A.4 AlgebraicRiccatiEquations A.4.1 Reduced Riccati Equation for WSPR Systems A.5 SomeUsefulMatrixAlgebra A.5.1 ResultsUsefulfortheKYPLemmaLMI A.5.2 InverseofMatrices A.5.3 JordanChain A.5.4 Auxiliary Lemmas for the KYP Lemma Proof A.6 Well-posednessResultsforStateDelaySystems References Index...571

12 Notation IR: the set of real numbers; C the set of complex numbers; IN: thesetof nonnegative integers. IR n (C n ): the set of n dimensional vectors with real (complex) entries. A T : transpose of the matrix A IR n m or C n m. Ā: conjugate of the matrix A C n m. A : conjugate transpose matrix of the matrix A C n m. A>0( 0): positive definite (semi positive definite) matrix. λ(a): an eigenvalue of A IR n m. σ(a) the set of eigenvalues of A IR n m (i.e. the spectrum of A). λ max (A), λ min (A): the largest and smallest eigenvalue of the matrix A, respectively. σ max (A) (σ min (A)): largest (smallest) singular value of A. ρ(a): the spectral radius of A, i.e. max{ λ : λ σ(a)}. tr(a): the trace of the matrix A. A : the Moore-Penrose inverse of the matrix A. ODE: Ordinary Differential Equation; PDE: Partial Differential equation. BV, LBV, RCLBV: Bounded Variation, Local BV, Right Continuous LBV. AC: Absolutely Continuous. I n the n n identity matrix, O n the n n zero matrix. f x (x) IRm n : the jacobian of the function f : IR n IR m at x. f(x) IR n m : the euclidean gradient of the function f : IR n IR m at x ( f(x) = f T x (x)).

13 xiv Notation A function is said to be smooth if it is infinitely differentiable; C 0 denotes the set of continuous functions; C r denotes the set of r-times differentiable functions f( ) withf (r) ( ) continuous. f(t + ): right-limit of the function f( ) att; (f(t ): left-limit). : Euclidean norm in IR n ( x = x T x for all x IR n ). f p : L p -norm of a Lebesgue integrable function f( ). H(jω): the phase of H(jω) C. LTI: Linear Time Invariant (system). Ker(A): kernel of A IR n m ;Im(A): image of A IR n m. dom(f): domain of a function f. K: closure of a domain K IR n (K = K if and only if K is closed). Int(K): interior of a domain K IR n (Int(K) isalwaysopen),i.e. the set of interior points of K (points x of K such that there is a neighborhood of x inside K). Re[ ] denotes the real part and Im[ ] denotes the imaginary part. a.e.: almost everywhere (usually in the Lebesgue measure sense).