Modeling and Control of Complex Physical Systems (eBook)

Title Page 2
Foreword 6
Preface 8
Contents 18
Port-Based Modeling of Dynamic Systems 24
Introduction 24
Modeling of dynamic systems 24
History of physical systems modeling of engineering systems 25
Tools needed for the integrated design of controlled physical systems 28
Object-oriented modeling 30
Design phases of engineering systems 30
Multiple views in the design and modeling process 32
Modeling philosophy 33
‘Every model is wrong’ 33
‘A model depends on its problem context’ 34
Physical components versus conceptual elements 34
Ports in dynamical systems models 36
Bilateral bonds versus unilateral signals 36
Dynamic conjugation versus power conjugation 38
Bond graph notation 39
Computational causality 40
System versus environment: system boundary 41
Elementary behaviors and basic concepts 42
Positive orientation and the half-arrow 42
Constitutive relations of elements 43
Storage 44
Irreversible transformation 48
Reversible transformation 49
Supply & demand (sources &
Distribution 51
Summary of elements 52
Modulation and bond activation 52
Causal port properties 53
Fixed causality 53
Preferred causality 54
Arbitrary causality 55
Causal constraints 55
Causal paths 55
Causal analysis: feedback on modeling decisions 56
Sequential Causality Assignment Procedure (SCAP) 56
Example of causal analysis 60
Hierarchical modeling 61
Word bond graphs 61
Multi-bonds 61
Multiport generalizations 62
Example of the use of the port concept 68
Problem context 68
Functional description of the valve 69
Analysis 69
Conclusion 74
Future Trends 74
Port-Hamiltonian Systems 76
From junction structures to Dirac structures 76
From 0- and 1-junctions to Dirac structures 77
Dirac structures 79
Examples of Dirac structures 81
Port-Hamiltonian systems 84
Geometric definition of a port-Hamiltonian system 84
Modulated Dirac structures and port-Hamiltonian systems on manifolds 89
Input-state-output port-Hamiltonian systems 92
Input-state-output port-Hamiltonian systems with direct feed-through 94
Port-Hamiltonian systems with variable topology 95
Relationships with classical Hamiltonian and Euler-Lagrange equations 97
From Euler-Lagrange equations to port-Hamiltonian systems 98
Port-Hamiltonian systems and Legendre transformations 100
Representations of Dirac structures and port-Hamiltonian systems 107
Representations of Dirac structures 107
Representations of port-Hamiltonian systems 111
Elimination of Lagrangian multipliers and constraints 114
Port-Hamiltonian systems in canonical coordinates – Casimirs and algebraic constraints 118
Well-posedness of port-Hamiltonian systems 120
Interconnection of port-Hamiltonian systems 122
Composition of Dirac structures 122
Regularity of interconnections 128
Interconnection of port-Hamiltonian systems 130
Analysis of port-Hamiltonian systems 131
Passivity 131
Casimirs of port-Hamiltonian systems 134
Algebraic constraints of port-Hamiltonian systems 138
Integrability of modulated Dirac structures 139
Scattering representation of Dirac structures and port-Hamiltonian systems 143
What is “scattering”? 143
Scattering representation of ports and Dirac structures 145
Inner product scattering representations 148
Interconnection in scattering representation 151
Port-Based Modeling in Different Domains 154
Modeling of electrical systems 154
Electronic power converter circuits 155
Electromechanical energy conversion in the port-Hamiltonian framework 158
Elementary electromagnet 160
Coupling of the boost converter and the electromagnet 163
Variable structure systems 164
Modeling of mechanical systems 168
Short introduction and motivations 168
Configuration and twist of a rigid body 169
Rigid body dynamics 173
Rigid mechanisms: interconnections of rigid bodies 177
Flexible mechanisms 180
Modeling of simple elastic systems 181
Introduction 181
Simple elasticity 182
The Hamiltonian and Lagrangian picture 187
The linearized scenario 189
Reduction 189
The Euler-Bernoulli beam 192
Summary 194
Port-based modelling and irreversible thermodynamics 195
Basic concepts 195
Distributed parameter systems 197
Lumped parameter systems 206
Constitutive equations 208
Port-based modelling examples 217
Infinite-Dimensional Port-Hamiltonian Systems 233
Modelling origins of boundary port-Hamiltonian systems 233
Conservation law and irreversible thermodynamics 234
Reversible physical systems of two coupled conservation laws 236
Dirac structures underlying dissipative physical systems 246
Stokes-Dirac structures and distributed port Hamiltonian systems 250
Reminder on differential forms 250
Conservation laws and balance equations expressed using k-forms 255
Systems of two conservation laws 257
Stokes-Dirac structures 260
Port Hamiltonian formulation of systems of two conservation laws with boundary energy flow 266
Extension to distributed port variables and scattering boundary variables 270
Extension of port-Hamiltonian systems on Stokes-Dirac structures 276
Timoshenko beam 277
Nonlinear flexible link 281
Ideal isentropic fluid 287
Conclusion 292
Control of Finite-Dimensional Port-Hamiltonian Systems 294
Introduction 294
Energy–balancing control 296
Dissipation obstacle 298
Control by port–interconnection 298
Energy control 301
Stabilization by Casimir generation 302
Port control 304
Achievable Dirac structures with dissipation 306
Achievable Dirac structures 306
Achievable Resistive structures 308
Achievable Dirac structures with dissipation 309
Achievable Casimirs and constraints 311
The role of energy dissipation 313
Casimirs and the dissipation obstacle 314
Casimirs for any resistive relation 315
Casimirs for a given resistive relation 316
Application to control 316
Casimirs and stabilization in finite dimensions 317
Specific Casimirs 317
Casimirs in extended state–space 319
Interconnection and damping assignment passivity based-control (IDA–PBC) 322
Solving the matching equation 325
Energy–balancing of IDA–PBC 328
Power–shaping stabilization 329
From port-Hamiltonian systems to the Brayton–Mosere quations 330
Geometry of Brayton–Moser’s equation 333
Stabilization by power–shaping 335
Stabilization by Casimir generation 338
Remarks 339
Analysis and Control of Infinite-Dimensional Systems 340
Introduction 340
Stability for infinite dimensional systems 342
Arnold’s first stability theorem approach 342
La Salle’s theorem approach 343
Control by damping injection 345
Basic results 345
Control of the Timoshenko beam by damping injection 346
Control by interconnection and energy shaping 350
General considerations 350
Interconnections of Dirac structures for mixed port-Hamiltonian systems 352
Achievable Dirac structures for mixed port-Hamiltonian systems 356
Control by Casimir generation 358
Control by interconnection of a class of mixed port-Hamiltonian systems 361
Control by energy shaping of the Timoshenko beam 376
Model of the plant 376
Casimir functionals for the closed-loop system 378
Control by energy shaping of the Timoshenko beam 380
Appendix A 390
Appendix B 402
Appendix C 419
Author’s Biographies 424
References 432