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
1 Introduction 1 1.1 Example 1: System with Mass Spring and Damper 2 1.2 Example 2: RLC Circuit 3 1.3 Example 3: A Mass with a PD Controller 5 1.4 Example 4: Adaptive Control 6 2 Positive Real Systems 9 2.1 Dynamical System State-space Representation 10 2.2 Definitions 11 2.3 Interconnections of Passive Systems 14 2.4 Linear Systems 15 2.5 Passivity of the PID Controllers 24 2.6 Stability of a Passive Feedback Interconnection 24 2.7 Mechanical Analogs for PD Controllers 25 2.8 Multivariable Linear Systems 27 2.9 The Scattering Formulation 28 2.10 Impedance Matching 31 2.11 Feedback Loop 34 2.12 Bounded Real and Positive Real Transfer Functions 36 2.13 Examples 47 2.13.1 Mechanical Resonances 47 2.13.2 Systems with Several Resonances 50 2.13.3 Two Motors Driving an Elastic Load 51 2.14 Strictly Positive Real (SPR) Systems 53 2.14.1 Frequency Domain Conditions for a Transfer Function to be SPR 54 2.14.2 Necessary Conditions for H{s) to be PR (SPR) 56 2.14.3 Tests for SPRness 57 2.14.4 Interconnection of Positive Real Systems 57 2.14.5 Special Cases of Positive Real Systems 58 2.15 Applications 62
viii 2.15.1 SPR and Adaptive Control 62 2.15.2 Adaptive Output Feedback 64 2.15.3 Design of SPR Systems 65 3 Kaiman-Yakubovich-Popov Lemma 69 3.1 The Positive Real Lemma 70 3.1.1 PR Transfer Functions 70 3.1.2 A Digression to Optimal Control 76 3.1.3 Duality 78 3.1.4 Positive Real Lemma for SPR Systems 79 3.1.5 Descriptor Variable Systems 91 3.2 Weakly SPR Systems and the KYP Lemma 95 3.3 KYP Lemma for Non-minimal Systems 100 3.3.1 Spectral Factors 102 3.3.2 Sign-controllability 104 3.3.3 State Space Decomposition 106 3.3.4 A Relaxed KYP Lemma for SPR Functions with Stabilizable Realization 107 3.4 SPR Problem with Observers 113 3.5 The Feedback KYP Lemma 113 3.6 Time-varying Systems 115 3.7 Interconnection of PR Systems 116 3.8 Positive Realness and Optimal Control 119 3.8.1 General Considerations 119 3.8.2 Least Squares Optimal Control 120 3.8.3 The Popov Function and the KYP Lemma LMI 125 3.8.4 A Recapitulating Theorem 129 3.8.5 On the Design of Passive LQG Controllers 130 3.8.6 Summary 133 3.8.7 A Digression on Semidefinite Programming Problems.. 134 3.9 The Lur'e Problem (Absolute Stability) 135 3.9.1 Introduction 135 3.9.2 Well-posedness of ODEs 137 3.9.3 Aizerman's and Kalman's Conjectures 140 3.9.4 Multivalued Nonlinearities 142 3.9.5 Dissipative Evolution Variational Inequalities 152 3.10 The Circle Criterion 160 3.10.1 Loop Transformations 162 3.11 The Popov Criterion 166 3.12 Discrete-time Systems 170 3.12.1 The KYP Lemma 170 3.12.2 The Tsypkin Criterion 173 3.12.3 Discretization of PR Systems 175
ix 4 Dissipative Systems 177 4.1 Normed Spaces 178 4.2 C p Norms 178 4.2.1 Relationships Between C\, 2 and L^ Spaces 180 4.3 Review of Some Properties of C p Signals 180 4.3.1 Example of Applications of the Properties of p Functions in Adaptive Control 186 4.3.2 Linear Maps 188 4.3.3 Induced Norms 188 4.3.4 Properties of Induced Norms 188 4.3.5 Extended Spaces 190 4.3.6 Gain of an Operator 190 4.3.7 Small Gain Theorem 191 4.4 Dissipative Systems 193 4.4.1 Definitions 193 4.4.2 The Signification of ß 197 4.4.3 Storage Functions (Available, Required Supply) 201 4.4.4 Examples 211 4.4.5 Regularity of the Storage Functions 217 4.5 Nonlinear KYP Lemma 222 4.5.1 A Particular Case 222 4.5.2 Nonlinear KYP Lemma in the General Case 223 4.5.3 Time-varying Systems 229 4.5.4 Nonlinear-in-the-input Systems 230 4.6 Dissipative Systems and Partial Differential Inequalities 231 4.6.1 The linear invariant case 231 4.6.2 The Nonlinear Case y = h(x) 235 4.6.3 The Nonlinear Case y = h(x) + j(x)u 238 4.6.4 Recapitulation 243 4.6.5 Inverse Optimal Control 243 4.7 Nonlinear Discrete-time Systems 247 4.8 PR tangent System and dissipativity 249 4.9 Infinite-dimensional Systems 252 4.9.1 An Extension of the KYP Lemma 252 4.9.2 The Wave Equation 253 4.9.3 The Heat Equation 255 4.10 Further Results 255 5 Stability of Dissipative Systems 257 5.1 Passivity Theorems 257 5.1.1 One-channel Results 257 5.1.2 Two-channel Results 259 5.1.3 Lossless and WSPR Blocks Interconnection 263 5.1.4 Large-scale Systems 264 5.2 Positive Definiteness of Storage Functions 266
x 5.3 WSPR Does not Imply OSP 270 5.4 Stabilization by Output Feedback 272 5.4.1 Autonomous Systems 272 5.4.2 Time-varying Nonlinear Systems 273 5.4.3 Evolution Variational Inequalities 274 5.5 Equivalence to a Passive System 276 5.6 Cascaded Systems 281 5.7 Input-to-State Stability (ISS) and Dissipativity 282 5.8 Passivity of Linear Delay Systems 288 5.8.1 Systems with State Delay 288 5.8.2 Interconnection of Passive Systems 290 5.8.3 Extension to a System with Distributed State Delay... 291 5.8.4 Absolute Stability 294 5.9 Nonlinear Hoo Control 295 5.9.1 Introduction 295 5.9.2 Closed-loop Synthesis: Static State Feedback 300 5.9.3 Closed-loop Synthesis: PR Dynamic Feedback 302 5.9.4 Nonlinear H^ 305 5.9.5 More on Finite-power-gain Systems 307 5.10 Popov's Hyperstability 310 6 Dissipative Physical Systems 315 6.1 Lagrangian Control Systems 315 6.1.1 Definition and Properties 316 6.1.2 Simple Mechanical Systems 324 6.2 Hamiltonian Control Systems 326 6.2.1 Input-output Hamiltonian Systems 326 6.2.2 Port Controlled Hamiltonian Systems 331 6.3 Rigid Joint-Rigid Link Manipulators 340 6.3.1 The Available Storage 341 6.3.2 The Required Supply 342 6.4 Flexible Joint-Rigid Link Manipulators 343 6.4.1 The Available Storage 346 6.4.2 The Required Supply 346 6.5 A Bouncing System 347 6.6 Including Actuator Dynamics 349 6.6.1 Armature-controlled DC Motors 349 6.6.2 Field-controlled DC Motors 354 6.7 Passive Environment 358 6.7.1 Systems with Holonomic Constraints 358 6.7.2 Compliant Environment 361 6.8 Nonsmooth Lagrangian Systems 363 6.8.1 Systems with C Solutions 363 6.8.2 Systems with BV Solutions 365
xi 7 Passivity-based Control 373 7.1 Brief Historical Survey 373 7.2 The Lagrange-Dirichlet Theorem 375 7.2.1 Lyapunov Stability 375 7.2.2 Asymptotic Lyapunov Stability 376 7.2.3 Invertibility of the Lagrange-Dirichlet Theorem 378 7.2.4 The Lagrange-Dirichlet Theorem for Nonsmooth Lagrangian Systems (BV Solutions) 379 7.2.5 The Lagrange-Dirichlet Theorem for Nonsmooth Lagrangian Systems (C Solutions) 384 7.2.6 Conclusion 385 7.3 Rigid Joint-Rigid Link Systems: State Feedback 386 7.3.1 PD Control 386 7.3.2 PID Control 391 7.3.3 More about Lyapunov Functions and the Passivity Theorem 393 7.3.4 Extensions of the PD Controller for the Tracking Case. 398 7.3.5 Other Types of State Feedback Controllers 405 7.4 Rigid Joint-Rigid Link: Position Feedback 408 7.4.1 P + Observer Control 408 7.4.2 The Paden and Panja + Observer Controller 410 7.4.3 The Slotine and Li + Observer Controller 412 7.5 Flexible Joint-Rigid Link: State Feedback 414 7.5.1 Passivity-based Controller: The Lozano and Brogliato Scheme 414 7.5.2 Other Globally Tracking Feedback Controllers 418 7.6 Flexible Joint-Rigid Link: Output Feedback 422 7.6.1 PD Control 422 7.6.2 Motor Position Feedback 424 7.7 Including Actuator Dynamics 426 7.7.1 Armature-controlled DC Motors 426 7.7.2 Field-controlled DC Motors 428 7.8 Constrained Mechanical Systems 428 7.8.1 Regulation with a Position PD Controller 429 7.8.2 Holonomic Constraints 430 7.8.3 Nonsmooth Lagrangian Systems 431 7.9 Controlled Lagrangians 432 8 Adaptive Control 435 8.1 Lagrangian Systems 436 8.1.1 Rigid Joint-Rigid Link Manipulators 436 8.1.2 Flexible Joint-Rigid Link Manipulators: The Adaptive Lozano and Brogliato Algorithm 442 8.1.3 Flexible Joint-Rigid Link Manipulators: The Backstepping Algorithm 452
xii 8.2 Linear Invariant Systems 456 8.2.1 A Scalar Example 456 8.2.2 Systems with Relative Degree r = 1 457 8.2.3 Systems with Relative Degree r = 2 460 8.2.4 Systems with Relative Degree r > 3 461 9 Experimental Results 467 9.1 Flexible Joint Manipulators 467 9.1.1 Introduction 467 9.1.2 Controller Design 468 9.1.3 The Experimental Devices 469 9.1.4 Experimental Results 473 9.1.5 Conclusions 483 9.2 Stabilization of the Inverted Pendulum 496 9.2.1 Introduction 496 9.2.2 System's Dynamics 497 9.2.3 Stabilizing Control Law 500 9.2.4 Simulation Results 503 9.2.5 Experimental Results 503 9.3 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