Dissipative Systems Analysis and Control: Theory and Applications

Contents......Page 7
1 Introduction......Page 15
1.1 Example 1: System with Mass Spring and Damper......Page 16
1.2 Example 2: RLC Circuit......Page 17
1.3 Example 3: A Mass with a PD Controller......Page 19
1.4 Example 4: Adaptive Control......Page 20
2 Positive Real Systems......Page 22
2.1 Dynamical System State-space Representation......Page 23
2.2 Definitions......Page 24
2.3 Interconnections of Passive Systems......Page 27
2.4 Linear Systems......Page 28
2.6 Stability of a Passive Feedback Interconnection......Page 37
2.7 Mechanical Analogs for PD Controllers......Page 38
2.8 Multivariable Linear Systems......Page 40
2.9 The Scattering Formulation......Page 41
2.10 Impedance Matching......Page 44
2.11 Feedback Loop......Page 47
2.12 Bounded Real and Positive Real Transfer Functions......Page 49
2.13.1 Mechanical Resonances......Page 60
2.13.2 Systems with Several Resonances......Page 63
2.13.3 Two Motors Driving an Elastic Load......Page 64
2.14 Strictly Positive Real (SPR) Systems......Page 66
2.14.1 Frequency Domain Conditions for a Transfer Function to be SPR......Page 67
2.14.2 Necessary Conditions for H(s) to be PR (SPR)......Page 69
2.14.4 Interconnection of Positive Real Systems......Page 70
2.14.5 Special Cases of Positive Real Systems......Page 71
2.15.1 SPR and Adaptive Control......Page 75
2.15.2 Adaptive Output Feedback......Page 77
2.15.3 Design of SPR Systems......Page 78
3 Kalman-Yakubovich-Popov Lemma......Page 82
3.1.1 PR Transfer Functions......Page 83
3.1.2 A Digression to Optimal Control......Page 89
3.1.3 Duality......Page 91
3.1.4 Positive Real Lemma for SPR Systems......Page 92
3.1.5 Descriptor Variable Systems......Page 104
3.2 Weakly SPR Systems and the KYP Lemma......Page 108
3.3 KYP Lemma for Non-minimal Systems......Page 113
3.3.1 Spectral Factors......Page 115
3.3.2 Sign-controllability......Page 117
3.3.3 State Space Decomposition......Page 119
3.3.4 A Relaxed KYP Lemma for SPR Functions with Stabilizable Realization......Page 120
3.5 The Feedback KYP Lemma......Page 126
3.6 Time-varying Systems......Page 128
3.7 Interconnection of PR Systems......Page 129
3.8.1 General Considerations......Page 132
3.8.2 Least Squares Optimal Control......Page 133
3.8.3 The Popov Function and the KYP Lemma LMI......Page 138
3.8.4 A Recapitulating Theorem......Page 142
3.8.5 On the Design of Passive LQG Controllers......Page 143
3.8.6 Summary......Page 146
3.8.7 A Digression on Semidenite Programming Problems......Page 147
3.9.1 Introduction......Page 148
3.9.2 Well-posedness of ODEs......Page 150
3.9.3 Aizerman\'s and Kalman\'s Conjectures......Page 153
3.9.4 Multivalued Nonlinearities......Page 155
3.9.5 Dissipative Evolution Variational Inequalities......Page 165
3.10 The Circle Criterion......Page 173
3.10.1 Loop Transformations......Page 175
3.11 The Popov Criterion......Page 179
3.12.1 The KYP Lemma......Page 183
3.12.2 The Tsypkin Criterion......Page 186
3.12.3 Discretization of PR Systems......Page 188
4 Dissipative Systems......Page 190
4.2 L[sub(p)] Norms......Page 191
4.3 Review of Some Properties of L[sub(p)] Signals......Page 193
4.3.1 Example of Applications of the Properties of L[sub(p)] Functions in Adaptive Control......Page 199
4.3.4 Properties of Induced Norms......Page 201
4.3.6 Gain of an Operator......Page 203
4.3.7 Small Gain Theorem......Page 204
4.4.1 Definitions......Page 206
4.4.2 The Signification of β......Page 210
4.4.3 Storage Functions (Available, Required Supply)......Page 214
4.4.4 Examples......Page 224
4.4.5 Regularity of the Storage Functions......Page 230
4.5.1 A Particular Case......Page 235
4.5.2 Nonlinear KYP Lemma in the General Case......Page 236
4.5.3 Time-varying Systems......Page 242
4.5.4 Nonlinear-in-the-input Systems......Page 243
4.6.1 The linear invariant case......Page 244
4.6.2 The Nonlinear Case y = h(x)......Page 248
4.6.3 The Nonlinear Case y = h(x) + j(x)u......Page 251
4.6.5 Inverse Optimal Control......Page 256
4.7 Nonlinear Discrete-time Systems......Page 260
4.8 PR tangent system and dissipativity......Page 262
4.9.1 An Extension of the KYP Lemma......Page 265
4.9.2 The Wave Equation......Page 266
4.10 Further Results......Page 268
5.1.1 One-channel Results......Page 270
5.1.2 Two-channel Results......Page 272
5.1.3 Lossless and WSPR Blocks Interconnection......Page 276
5.1.4 Large-scale Systems......Page 277
5.2 Positive Definiteness of Storage Functions......Page 279
5.3 WSPR Does not Imply OSP......Page 283
5.4.1 Autonomous Systems......Page 285
5.4.2 Time-varying Nonlinear Systems......Page 286
5.4.3 Evolution Variational Inequalities......Page 287
5.5 Equivalence to a Passive System......Page 289
5.6 Cascaded Systems......Page 294
5.7 Input-to-State Stability (ISS) and Dissipativity......Page 295
5.8.1 Systems with State Delay......Page 301
5.8.2 Interconnection of Passive Systems......Page 303
5.8.3 Extension to a System with Distributed State Delay......Page 304
5.8.4 Absolute Stability......Page 307
5.9.1 Introduction......Page 308
5.9.2 Closed-loop Synthesis: Static State Feedback......Page 313
5.9.3 Closed-loop Synthesis: PR Dynamic Feedback......Page 315
5.9.4 Nonlinear H[sub(∞)]......Page 318
5.9.5 More on Finite-power-gain Systems......Page 320
5.10 Popov\'s Hyperstability......Page 323
6.1 Lagrangian Control Systems......Page 327
6.1.1 Definition and Properties......Page 328
6.1.2 Simple Mechanical Systems......Page 336
6.2.1 Input-output Hamiltonian Systems......Page 338
6.2.2 Port Controlled Hamiltonian Systems......Page 343
6.3 Rigid Joint–Rigid Link Manipulators......Page 352
6.3.1 The Available Storage......Page 353
6.3.2 The Required Supply......Page 354
6.4 Flexible Joint–Rigid Link Manipulators......Page 355
6.4.2 The Required Supply......Page 358
6.5 A Bouncing System......Page 359
6.6.1 Armature-controlled DC Motors......Page 361
6.6.2 Field-controlled DC Motors......Page 366
6.7.1 Systems with Holonomic Constraints......Page 370
6.7.2 Compliant Environment......Page 373
6.8.1 Systems with C[sup(0)] Solutions......Page 375
6.8.2 Systems with BV Solutions......Page 377
7.1 Brief Historical Survey......Page 384
7.2.1 Lyapunov Stability......Page 386
7.2.2 Asymptotic Lyapunov Stability......Page 387
7.2.3 Invertibility of the Lagrange-Dirichlet Theorem......Page 389
7.2.4 The Lagrange-Dirichlet Theorem for Nonsmooth Lagrangian Systems (BV Solutions)......Page 390
7.2.5 The Lagrange-Dirichlet Theorem for Nonsmooth Lagrangian Systems (C[sup(0)] Solutions)......Page 395
7.2.6 Conclusion......Page 396
7.3.1 PD Control......Page 397
7.3.2 PID Control......Page 402
7.3.3 More about Lyapunov Functions and the Passivity Theorem......Page 404
7.3.4 Extensions of the PD Controller for the Tracking Case......Page 409
7.3.5 Other Types of State Feedback Controllers......Page 416
7.4.1 P + Observer Control......Page 419
7.4.2 The Paden and Panja + Observer Controller......Page 421
7.4.3 The Slotine and Li + Observer Controller......Page 423
7.5.1 Passivity-based Controller: The Lozano and Brogliato Scheme......Page 425
7.5.2 Other Globally Tracking Feedback Controllers......Page 429
7.6.1 PD Control......Page 433
7.6.2 Motor Position Feedback......Page 435
7.7.1 Armature-controlled DC Motors......Page 437
7.8 Constrained Mechanical Systems......Page 439
7.8.1 Regulation with a Position PD Controller......Page 440
7.8.2 Holonomic Constraints......Page 441
7.8.3 Nonsmooth Lagrangian Systems......Page 442
7.9 Controlled Lagrangians......Page 443
8 Adaptive Control......Page 446
8.1.1 Rigid Joint–Rigid Link Manipulators......Page 447
8.1.2 Flexible Joint–Rigid Link Manipulators: The Adaptive Lozano and Brogliato Algorithm......Page 453
8.1.3 Flexible Joint–Rigid Link Manipulators: The Backstepping Algorithm......Page 463
8.2.1 A Scalar Example......Page 467
8.2.2 Systems with Relative Degree r = 1......Page 468
8.2.3 Systems with Relative Degree r = 2......Page 471
8.2.4 Systems with Relative Degree r ≥ 3......Page 472
9.1.1 Introduction......Page 477
9.1.2 Controller Design......Page 478
9.1.3 The Experimental Devices......Page 479
9.1.4 Experimental Results......Page 483
9.1.5 Conclusions.......Page 493
9.2.1 Introduction......Page 506
9.2.2 System\'s Dynamics......Page 507
9.2.3 Stabilizing Control Law......Page 510
9.2.5 Experimental Results......Page 513
9.3 Conclusions......Page 514
A.1.1 Autonomous systems......Page 517
A.1.2 Non-autonomous Systems......Page 521
A.2 Differential Geometry Theory......Page 525
A.2.1 Normal Form......Page 527
A.2.2 Feedback Linearization......Page 528
A.2.3 Stabilization of Feedback Linearizable Systems......Page 529
A.3 Viscosity Solutions......Page 530
A.4 Algebraic Riccati Equations......Page 533
A. 4.1 Reduced Riccati Equation for WSPR Systems......Page 535
A.5.1 Results Useful for the KYP Lemma LMI......Page 541
A.5.2 Inverse of Matrices......Page 543
A.5.4 Auxiliary Lemmas for the KYP Lemma Proof......Page 544
A.6 Well-posedness Results for State Delay Systems......Page 547
References......Page 549
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