Dissipative Systems Analysis and Control

Transcription

1 Bernard Brogliato, Rogelio Lozano, Bernhard Maschke and Olav Egeland Dissipative Systems Analysis and Control Theory and Applications 2nd Edition With 94 Figures 4y Sprin er

2 1 Introduction Example 1: System with Mass Spring and Damper Example 2: RLC Circuit Example 3: A Mass with a PD Controller Example 4: Adaptive Control 6 2 Positive Real Systems Dynamical System State-space Representation Definitions Interconnections of Passive Systems Linear Systems Passivity of the PID Controllers Stability of a Passive Feedback Interconnection Mechanical Analogs for PD Controllers Multivariable Linear Systems The Scattering Formulation Impedance Matching Feedback Loop Bounded Real and Positive Real Transfer Functions Examples Mechanical Resonances Systems with Several Resonances Two Motors Driving an Elastic Load Strictly Positive Real (SPR) Systems Frequency Domain Conditions for a Transfer Function to be SPR Necessary Conditions for H{s) to be PR (SPR) Tests for SPRness Interconnection of Positive Real Systems Special Cases of Positive Real Systems Applications 62

3 viii SPR and Adaptive Control Adaptive Output Feedback Design of SPR Systems 65 3 Kaiman-Yakubovich-Popov Lemma The Positive Real Lemma PR Transfer Functions A Digression to Optimal Control Duality Positive Real Lemma for SPR Systems Descriptor Variable Systems Weakly SPR Systems and the KYP Lemma KYP Lemma for Non-minimal Systems Spectral Factors Sign-controllability State Space Decomposition A Relaxed KYP Lemma for SPR Functions with Stabilizable Realization SPR Problem with Observers The Feedback KYP Lemma Time-varying Systems Interconnection of PR Systems Positive Realness and Optimal Control General Considerations Least Squares Optimal Control The Popov Function and the KYP Lemma LMI A Recapitulating Theorem On the Design of Passive LQG Controllers Summary A Digression on Semidefinite Programming Problems The Lur'e Problem (Absolute Stability) Introduction Well-posedness of ODEs Aizerman's and Kalman's Conjectures Multivalued Nonlinearities Dissipative Evolution Variational Inequalities The Circle Criterion Loop Transformations The Popov Criterion Discrete-time Systems The KYP Lemma The Tsypkin Criterion Discretization of PR Systems 175

4 ix 4 Dissipative Systems Normed Spaces C p Norms Relationships Between C\, 2 and L^ Spaces Review of Some Properties of C p Signals Example of Applications of the Properties of p Functions in Adaptive Control Linear Maps Induced Norms Properties of Induced Norms Extended Spaces Gain of an Operator Small Gain Theorem Dissipative Systems Definitions The Signification of ß Storage Functions (Available, Required Supply) Examples Regularity of the Storage Functions Nonlinear KYP Lemma A Particular Case Nonlinear KYP Lemma in the General Case Time-varying Systems Nonlinear-in-the-input Systems Dissipative Systems and Partial Differential Inequalities The linear invariant case The Nonlinear Case y = h(x) The Nonlinear Case y = h(x) + j(x)u Recapitulation Inverse Optimal Control Nonlinear Discrete-time Systems PR tangent System and dissipativity Infinite-dimensional Systems An Extension of the KYP Lemma The Wave Equation The Heat Equation Further Results Stability of Dissipative Systems Passivity Theorems One-channel Results Two-channel Results Lossless and WSPR Blocks Interconnection Large-scale Systems Positive Definiteness of Storage Functions 266

5 x 5.3 WSPR Does not Imply OSP Stabilization by Output Feedback Autonomous Systems Time-varying Nonlinear Systems Evolution Variational Inequalities Equivalence to a Passive System Cascaded Systems Input-to-State Stability (ISS) and Dissipativity Passivity of Linear Delay Systems Systems with State Delay Interconnection of Passive Systems Extension to a System with Distributed State Delay Absolute Stability Nonlinear Hoo Control Introduction Closed-loop Synthesis: Static State Feedback Closed-loop Synthesis: PR Dynamic Feedback Nonlinear H^ More on Finite-power-gain Systems Popov's Hyperstability Dissipative Physical Systems Lagrangian Control Systems Definition and Properties Simple Mechanical Systems Hamiltonian Control Systems Input-output Hamiltonian Systems Port Controlled Hamiltonian Systems Rigid Joint-Rigid Link Manipulators The Available Storage The Required Supply Flexible Joint-Rigid Link Manipulators The Available Storage The Required Supply A Bouncing System Including Actuator Dynamics Armature-controlled DC Motors Field-controlled DC Motors Passive Environment Systems with Holonomic Constraints Compliant Environment Nonsmooth Lagrangian Systems Systems with C Solutions Systems with BV Solutions 365

6 xi 7 Passivity-based Control Brief Historical Survey The Lagrange-Dirichlet Theorem Lyapunov Stability Asymptotic Lyapunov Stability Invertibility of the Lagrange-Dirichlet Theorem The Lagrange-Dirichlet Theorem for Nonsmooth Lagrangian Systems (BV Solutions) The Lagrange-Dirichlet Theorem for Nonsmooth Lagrangian Systems (C Solutions) Conclusion Rigid Joint-Rigid Link Systems: State Feedback PD Control PID Control More about Lyapunov Functions and the Passivity Theorem Extensions of the PD Controller for the Tracking Case Other Types of State Feedback Controllers Rigid Joint-Rigid Link: Position Feedback P + Observer Control The Paden and Panja + Observer Controller The Slotine and Li + Observer Controller Flexible Joint-Rigid Link: State Feedback Passivity-based Controller: The Lozano and Brogliato Scheme Other Globally Tracking Feedback Controllers Flexible Joint-Rigid Link: Output Feedback PD Control Motor Position Feedback Including Actuator Dynamics Armature-controlled DC Motors Field-controlled DC Motors Constrained Mechanical Systems Regulation with a Position PD Controller Holonomic Constraints Nonsmooth Lagrangian Systems Controlled Lagrangians Adaptive Control Lagrangian Systems Rigid Joint-Rigid Link Manipulators Flexible Joint-Rigid Link Manipulators: The Adaptive Lozano and Brogliato Algorithm Flexible Joint-Rigid Link Manipulators: The Backstepping Algorithm 452

7 xii 8.2 Linear Invariant Systems A Scalar Example Systems with Relative Degree r = Systems with Relative Degree r = Systems with Relative Degree r > Experimental Results Flexible Joint Manipulators Introduction Controller Design The Experimental Devices Experimental Results Conclusions Stabilization of the Inverted Pendulum Introduction System's Dynamics Stabilizing Control Law Simulation Results Experimental Results Conclusions 504 A Background Material 507 A.l Lyapunov Stability 507 A.l.l Autonomous Systems 507 A.l.2 Non-autonomous Systems 511 A.2 Differential Geometry Theory 515 A.2.1 Normal Form 517 A.2.2 Feedback Linearization 518 A.2.3 Stabilization of Feedback Linearizable Systems 519 A.2.4 Further Reading 520 A.3 Viscosity Solutions 520 A.4 Algebraic Riccati Equations 523 A.4.1 Reduced Riccati Equation for WSPR Systems 525 A.5 Some Useful Matrix Algebra 531 A.5.1 Results Useful for the KYP Lemma LMI 531 A.5.2 Inverse of Matrices 533 A.5.3 Jordan Chain 534 A.5.4 Auxiliary Lemmas for the KYP Lemma Proof 534 A.6 Well-posedness Results for State Delay Systems 537 References 539 Index 571