The Control Handbook: Control System Advanced Methods

Contents......Page 8
Preface to the Second Edition......Page 14
Acknowledgments......Page 16
Editorial Board......Page 18
Editor......Page 20
Contributors......Page 22
Section I: Analysis Methods for MIMO Linear Systems......Page 29
1.1 Introduction......Page 30
1.2 Numerical Background......Page 33
Linear Algebraic Equations and Linear Least-Squares Problems......Page 36
Eigenvalue and Generalized Eigenvalue Problems......Page 38
The Singular Value Decomposition and Some Applications......Page 39
Some Typical Techniques......Page 42
Transfer Functions, Poles, and Zeros......Page 43
Controllability and Other \"Abilities\"......Page 44
Frequency Response Calculations......Page 45
Lyapunov, Sylvester, and Riccati Equations......Page 46
Pole Assignment and Observer Design......Page 49
Robust Control......Page 50
General Remarks......Page 51
Mathematical Software in Control......Page 52
References......Page 53
Eigenvectors and Eigenvalues......Page 55
The Matrix Exponential......Page 56
Multivariable Poles......Page 57
Controllability and Observability......Page 58
Other Tests for Controllability and Observability......Page 59
Definition of MIMO Transmission Zeros......Page 60
Calculation of Transmission Zeros......Page 61
Transmission Zeros for Nonsquare Systems......Page 62
2.5 Multivariable Pole-Zero Cancellations......Page 63
References......Page 66
3.1 Introduction......Page 67
3.2 Analysis of Continuous-Time Causal Linear Time-Varying Systems......Page 68
State Model Realizations......Page 69
The State Model......Page 72
Change of State Variables......Page 75
Stability......Page 77
Controllability and Observability......Page 79
Control Canonical Form and Controller Design......Page 82
3.3 Discrete-Time Linear Time-Varying Systems......Page 84
State Model in the General Case......Page 85
Stability......Page 87
Controllability and Observability......Page 88
Change of State Variables and Canonical Forms......Page 89
3.4 Applications and Examples......Page 91
Observer and Controller Design......Page 92
Exponential Systems......Page 94
Stability......Page 95
The Lyapunov Criterion......Page 96
References......Page 97
Further Reading......Page 98
4.1 Introduction......Page 100
The Hankel, Controllability, and Observability Operator......Page 102
Balanced State-Space Realizations......Page 104
Model Reduction......Page 105
Unstable Systems, Closed-Loop Balancing......Page 107
4.3 Balancing for Nonlinear Systems......Page 112
Basics of Nonlinear Balanced Realizations......Page 113
Balanced Realizations Based on Singular-Value Analysis of Hankel Operators......Page 115
Model Order Reduction......Page 119
4.4 Concluding Remarks......Page 120
References......Page 121
5.1 Introduction......Page 124
5.2 Review of Elementary Notions......Page 125
5.3 (A, im B)-Controlled and (A, ker C)-Conditioned Invariant Subspaces and Duality......Page 129
5.4 Algebraic Properties of Controlled and Conditioned Invariants......Page 133
5.5 Maximum-Controlled and Minimum-Conditioned Invariants......Page 134
5.6 Self-Bounded-Controlled and Self-Hidden-Conditioned Invariants and Constrained Reachability and Observability......Page 135
5.7 Internal, External Stabilizability......Page 137
5.8 Disturbance Localization (or Decoupling) Problem......Page 140
5.9 Disturbance Localization with Stability......Page 142
5.10 Disturbance Localization by Dynamic Compensator......Page 143
References......Page 148
6.1 Introduction......Page 150
6.2 Polynomial Matrix Fraction Descriptions......Page 152
6.3 Fractional Degree and MacMillan Degree......Page 160
6.4 Smith–MacMillan Form, ARMA Models, and Stable Coprime Factorization......Page 166
References......Page 170
7.1 Motivations and Preliminaries......Page 171
7.2 Description of the Uncertainty Structures......Page 173
7.3 Uncertainty Structure Preservation with Feedback......Page 174
7.4 Overbounding with Affine Uncertainty: The Issue of Conservatism......Page 175
7.5 Robustness Analysis for Affine Plants......Page 177
Value Set Construction for Affine Plants......Page 178
The DC-Electric Motor Example Revisited......Page 179
7.6 Robustness Analysis for Affine Polynomials......Page 180
Value Set Construction for Affine Polynomials......Page 181
Interval Polynomials: Kharitonov\'s Theorem and Value Set Geometry......Page 182
Algebraic Criteria for Robust Stability......Page 183
7.7 Multiaffine Uncertainty Structures......Page 184
7.8 General Uncertainty Structures and Controller Synthesis......Page 186
References......Page 187
8.1 Modeling MIMO Linear Time-Invariant Systems in Terms of Transfer Function Matrices......Page 189
8.2 Frequency Response for MIMO Plants......Page 190
8.3 Mathematical Detour......Page 191
Introduction to Complex Vectors and Complex Matrices......Page 192
The Singular Value Decomposition......Page 193
8.4 The SVD and MIMO Frequency Response Analysis......Page 197
Singular Value Plots (SV Plots)......Page 198
Computing Directional Information......Page 199
Classical Unity-Feedback Systems......Page 200
A More General Setting......Page 205
References......Page 208
9.1 Introduction......Page 209
Sources of Uncertainty......Page 210
Multiplicative Representation of Unstructured Uncertainty......Page 211
9.3 Conditions for Stability Robustness......Page 217
Stability Robustness for SISO Systems......Page 218
Stability Robustness for MIMO Systems......Page 226
9.4 Impact of Stability Robustness on Closed-Loop Performance......Page 229
9.5 Other Representations for Model Uncertainty......Page 235
Division Uncertainty......Page 236
Representation for Parametric Uncertainty......Page 237
Notes......Page 238
References......Page 239
10.1 Introduction and Motivation......Page 240
Overview......Page 241
Nominal Signal Response......Page 242
Differential Sensitivity......Page 243
Robust Stability......Page 244
Summary of Feedback System Performance Specification......Page 245
Relationship between Closed-Loop Transfer Functions and Design Specifications......Page 246
Relation between Open and Closed-Loop Specifications......Page 247
Bode Gain-Phase Relation and Interpretations......Page 249
The Bode Sensitivity Integral......Page 250
Complementary Sensitivity Integral......Page 253
Introduction......Page 255
Limitations for Nonminimum Phase Systems......Page 256
Limitations for Unstable Systems......Page 260
Summary......Page 261
Right-Half-Plane Poles and Zeros......Page 262
Limitations in Discrete-Time and Sampled-Data Control......Page 264
Limitations in Control with Communication Constraints......Page 266
Cheap Control and Achievable H2 Performance......Page 267
MIMO Systems......Page 268
Time-Varying and Nonlinear Systems......Page 270
Alleviation of Tracking Performance Limitations......Page 271
10.8 Summary and Further Reading......Page 272
References......Page 273
11.1 Introduction......Page 276
11.2 Characterization of Uncertain Signals......Page 277
11.3 Characterization of Uncertain Plant Models......Page 278
Unstructured Plant Uncertainty Models......Page 279
Structured Plant Uncertainty......Page 285
11.5 Further Reading......Page 290
References......Page 291
Section II: Kalman Filter and Observers......Page 292
12.1 Introduction......Page 293
Basics......Page 294
Expectation......Page 295
Example: Discrete-Time Gaussian Stochastic Processes......Page 297
Stationarity, Ergodicity, and White Noise......Page 298
Transforms......Page 300
Vector Discrete-Time Stochastic Processes and LTI Systems......Page 303
Expectations......Page 306
Example: Continuous-Time Gaussian Stochastic Processes......Page 308
Stationarity, Ergodicity, and White Noise......Page 309
SISO Continuous-Time Linear Systems......Page 310
Vector Continuous-Time Stochastic Process and LTI Systems......Page 312
12.5 Notation......Page 315
References......Page 316
13.1 Introduction......Page 317
13.2 Problem Definition......Page 318
The State Estimation Problem......Page 319
The Kalman Filter Dynamics......Page 320
Properties of Model-Based Observers......Page 322
Guaranteed Stability......Page 323
13.5 The Accurate Measurement Kalman Filter......Page 324
The Main Result......Page 325
References......Page 326
Further Reading......Page 327
14.1 Introduction......Page 328
14.2 Optimal Control and Filtering: Motivation......Page 329
14.3 Riccati Differential Equation......Page 330
General Solutions......Page 333
Symmetric Solutions......Page 335
Definite Solutions......Page 338
14.5 Limiting Behavior of Solutions......Page 340
14.6 Optimal Control and Filtering: Application......Page 342
Invariant Subspace Method......Page 346
Concluding Remarks......Page 347
References......Page 348
Continuous-Time Systems......Page 349
15.3 Linear Reduced-Order Observers......Page 353
15.4 Discrete-Time Systems......Page 357
Delayed Data......Page 358
15.5 The Separation Principle......Page 359
15.6 Nonlinear Observers......Page 362
Reduced-Order Observers......Page 365
Extended Separation Principle......Page 367
Extended Kalman Filter......Page 369
References......Page 370
Section III: Design Methods for MIMO LTI Systems......Page 372
16.1 Introduction......Page 373
16.2 Eigenstructure Assignment Using Output Feedback......Page 374
F-18 HARV Linearized Lateral Dynamics Design Example......Page 376
16.3 Eigenstructure Assignment Using Constrained Output Feedback......Page 378
Eigenvalue/Eigenvector Derivative Method for Choosing Gain-Suppression Structure......Page 379
F-18 HARV Linearized Lateral Dynamics Design Example......Page 380
F-18 HARV Linearized Lateral Dynamics Design Example......Page 382
16.5 Robust, Sampled Data Eigenstructure Assignment......Page 384
Pseudocontrol and Robustness Results......Page 385
FPCC Yaw Pointing/Lateral Translation Controller Design Using Robust, Sampled Data, Eigenstructure Assignment......Page 386
References......Page 389
Appendix......Page 391
17.1 Introduction......Page 393
17.2 The Time-Invariant LQR Problem......Page 394
Physical Motivation for the LQR......Page 395
Designing LQR Controllers......Page 397
Robustness Properties......Page 399
Asymptotic Properties of LQR Controllers......Page 400
17.4 LQR Gain Selection Tools......Page 401
Cross-Weighted Costs......Page 402
The Stochastic LQR Problem......Page 403
Sensitivity Weighted LQR......Page 404
LQR with Frequency Weighted Cost Functionals......Page 409
17.5 Mini-Max and H∞ Full-State Feedback Control......Page 413
Synthesizing Mini-Max Controllers......Page 415
References......Page 417
18.2 The Modern Paradigm......Page 418
System Norms......Page 419
18.3 Output Feedback H∞ and H2 Controllers......Page 422
H2 Design......Page 427
H∞ Design......Page 429
18.5 Aircraft Design Example......Page 430
References......Page 436
19.1 Introduction......Page 437
Motivation......Page 438
Examples......Page 439
Formulation......Page 441
Prototypes......Page 442
Stability and Performance Robustness......Page 444
19.3 Computability......Page 445
l1 Performance Objective with Fixed Input Constraints......Page 446
References......Page 449
20.2 Shortcomings of Simple Robustness Analysis......Page 450
20.3 Complex Structured Singular Value......Page 454
Purely Complex µ......Page 455
Mixed µ: Real and Complex Uncertainty......Page 458
Frequency Domain, Robust Stability Tests with µ......Page 459
20.4 Linear Fractional Transformations and µ......Page 460
20.5 Robust Performance Tests Using µ and Main Loop Theorem......Page 461
Characterization of Performance in µ Setting......Page 462
Frequency-Domain Robust Performance Tests......Page 463
Robust Performance Example......Page 464
20.6 Spinning Satellite: Robust Performance Analysis with µ......Page 467
20.7 Control Design via µ Synthesis......Page 468
20.8 F-14 Lateral-Directional Control System Design......Page 470
Nominal Model and Uncertainty Models......Page 472
Controller Design......Page 474
References......Page 477
21.1 Introduction......Page 478
21.2 Systems and Signals......Page 479
21.4 Feedback Systems......Page 480
21.5 Parameterization of Stabilizing Controllers......Page 482
21.6 Parameterization of Closed-Loop Transfer Functions......Page 483
21.7 Optimal Performance......Page 484
21.8 Robust Stabilization......Page 487
21.9 Robust Performance......Page 489
21.10 Finite Impulse Response......Page 491
21.11 Multivariable Systems......Page 492
21.12 Extensions......Page 496
Further Reading......Page 497
Quantitative Feedback Theory......Page 499
What Can QFT Do?......Page 500
MISO System......Page 501
J LTI Plant Models......Page 503
Plant Templates of Pj(s), [ ] P( jωi)......Page 504
Optimal Bounds Bo( jωi) on Lo( jωi)......Page 505
Synthesizing (or Loop-Shaping) Lo(s) and F(s)......Page 506
Prefilter Design......Page 508
Introduction......Page 509
Assumptions......Page 510
Synthesizing Lmo(w)......Page 511
Prefilter Design......Page 513
z-Domain......Page 515
Derivation of m2 MISO System Equivalents......Page 516
Tracking and Cross-Coupling Specifications......Page 518
Determination of Tracking, Cross-Coupling, and Optimal Bounds......Page 519
Synthesizing the Loop Transmission and Prefilter Functions......Page 521
22.5 QFT Application......Page 522
References......Page 524
22.6 Appendix A......Page 526
22.7 Appendix B......Page 527
Plant Model......Page 528
Class of Tracking/Disturbance Signals......Page 529
23.3 Main Results......Page 530
Robust Servomechanism Controller......Page 533
Complementary Controller......Page 534
Observer-Based Stabilizing Controller......Page 535
23.4 Applications and Example Calculations......Page 536
CAD Approach......Page 547
Case Study Problem—Distillation Column......Page 550
Decentralized Robust Controller......Page 554
23.6 Defining Terms......Page 555
References......Page 557
Further Reading......Page 558
24.1 Introduction......Page 559
24.2 LMIs and Convexity......Page 560
24.3 Numerical Solutions of LMIs......Page 562
24.4 Stability Characterizations with LMIs......Page 563
H2-Performance......Page 565
H∞-Performance......Page 567
The Kalman–Yakubovich–Popov (KYP) Lemma......Page 568
Variants......Page 569
24.6 Optimal Performance Synthesis......Page 570
Output-Feedback Synthesis......Page 571
General Synthesis Procedure......Page 573
Observer and Estimator Synthesis......Page 574
Time-Invariant Parametric Uncertainty......Page 575
Time-Dependent Parametric Uncertainty......Page 576
Robust Performance......Page 577
24.8 Robust State-Feedback and Estimator Synthesis......Page 578
24.9 Gain-Scheduling Synthesis......Page 579
Robustness Analysis with IQCs......Page 581
Examples and Extensions......Page 584
Robust Synthesis......Page 585
24.11 Conclusions......Page 586
References......Page 587
25.1 Introduction to Optimal Control Design......Page 589
The Philosophy of Optimal Control Design......Page 590
The General Continuous-Time Optimal Control Problem......Page 591
Continuous-Time Linear Quadratic Regulator......Page 594
Steady-State and Suboptimal Control......Page 598
Frequency-Domain Results and Robustness of the LQR......Page 602
25.3 The Tracker Problem......Page 604
Optimal LQ Tracker......Page 605
A Practical Suboptimal Tracker......Page 607
Nonlinear Minimum-Time Problems......Page 610
Linear Quadratic Minimum-Time Design......Page 611
Constrained-Input Design and Bang-Bang Control......Page 612
Digital Control of Continuous-Time Systems......Page 615
25.6 Optimal LQ Design for Polynomial Systems......Page 617
Dynamic Programming for DT Systems......Page 620
Dynamic Programming for Continuous-Time Systems......Page 622
Further Reading......Page 623
26.1 Introduction......Page 624
26.2 The Decentralized Control Problem......Page 625
26.3 Plant and Feedback Structures......Page 627
26.4 Decentralized Stabilization......Page 628
Decentralized Inputs and Outputs......Page 629
Structural Analysis......Page 630
Decentrally Stabilizable Structures......Page 632
Vector Lyapunov Functions......Page 633
26.5 Optimization......Page 635
26.6 Adaptive Decentralized Co......Page 637
26.7 Discrete and Sampled-Data Systems......Page 639
LBT Decompositions......Page 640
Acyclic IO Reachable Decompositions......Page 641
Nested Epsilon Decompositions......Page 643
Overlapping Decompositions......Page 644
References......Page 645
Further Reading......Page 646
27.1 Introduction......Page 648
Diagonal Decoupling......Page 649
Diagonal Decoupling with Internal Stability......Page 650
Block Decoupling......Page 654
Decoupling Nonsquare Systems......Page 657
Triangular Decoupling......Page 659
Static Decoupling......Page 660
References......Page 661
Further Reading......Page 662
28.1 Introduction......Page 663
The Idea of Moving Horizon Control......Page 664
Multistep Prediction......Page 665
State-Space Formulation......Page 669
Objective Function......Page 670
Constraints......Page 671
Formulation of Control Problem as a Quadratic Program......Page 673
Moving Horizon Algorithm......Page 675
Solving the QP......Page 676
Proper Constraint Formulation......Page 677
Choice of Horizon Length......Page 679
28.4 Features Found in Other MPC Algorithms......Page 680
The Funnel Approach......Page 681
Use of Other Norms......Page 682
Prioritization of CVs and MVs......Page 683
Bi-Level Optimization......Page 684
Better Identification......Page 685
References......Page 686
Section IV: Analysis and Design of Hybrid Systems......Page 688
29.1 Introduction......Page 689
Systems without Disturbances......Page 690
Systems with Disturbances......Page 692
Reachability Problem......Page 699
Level Set Method......Page 700
Polytope Method......Page 702
d/dt......Page 703
CheckMate......Page 704
Ellipsoidal Method......Page 705
29.4 Ellipsoidal Method......Page 706
Steering the System to a Target......Page 710
Switching System......Page 711
Hybrid System......Page 714
29.6 Conclusion......Page 717
References......Page 718
30.1 Introduction......Page 720
30.2 Arbitrary Switching......Page 721
Common Lyapunov Function......Page 722
Necessary and Sufficient Conditions......Page 724
30.3 Restricted Switching......Page 726
Slow Switching......Page 727
Multiple Lyapunov Functions......Page 728
Piecewise Quadratic Lyapunov Functions......Page 731
30.4 Switching Stabilization......Page 733
Quadratic Switching Stabilization......Page 734
Piecewise Quadratic Switching Stabilization......Page 735
References......Page 738
31.1 Introduction......Page 739
Model of a Two-Switched System......Page 741
The Embedded Optimal Control Problem......Page 742
31.3 Sufficient and Necessary Conditions for Solvability of the EOCP......Page 745
31.4 Optimal Control of a Switched System with Three Modes of Operation......Page 747
31.5 Two-Gear Car Example with Application of Necessary Conditions......Page 749
31.6 Unicycle Example with Direct Collocation......Page 752
Unicycle Model......Page 753
Simulation Results and Discussion......Page 754
31.7 Concluding Remarks......Page 755
31.8 Appendix A: Modeling with Autonomous Switches......Page 757
31.9 Appendix B: Numerical Solution Using Direct Collocation......Page 758
References......Page 760
Section V: Adaptive Control......Page 762
32.2 Design Methods......Page 763
Specifications......Page 764
Feature-Based Techniques......Page 765
Tuning Based on Gain and Phase Margins......Page 769
Analytical Methods......Page 770
Loop-Shaping......Page 773
Optimization Methods......Page 774
Use of the Adaptive Techniques......Page 776
Automatic Tuning......Page 777
Gain Scheduling......Page 778
Adaptive Control......Page 779
32.4 Some Commercial Products......Page 780
References......Page 781
33.1 Introduction......Page 783
Examples of Unknown and Time-Varying Systems......Page 784
33.2 Some Simple Methods......Page 785
A Plant with an Unknown Time-Varying Gain......Page 787
33.3 Plant Models......Page 789
Incorporating Disturbances......Page 791
33.4 Recursive Prediction Error (RPE) Estimators......Page 792
Forgetting Factors......Page 795
Forgetting with Multiparameter Models......Page 797
33.5 Predictive Models......Page 798
33.6 Minimum-Variance (MV) Control......Page 801
33.7 Minimum-Variance Self-Tuning......Page 803
33.8 Pole-Placement (PP) Self-Tuning......Page 808
33.9 Long-Range Predictive Control......Page 810
33.10 The Generalized Predictive Control (GPC) Cost Function......Page 814
33.11 Robustness of Self-Tuning Controllers......Page 815
References......Page 817
34.1 Introduction......Page 818
34.2 MRAC Schemes......Page 819
Model Reference Control......Page 820
Direct MRAC......Page 823
Indirect MRAC......Page 826
Robust MRAC......Page 827
Scalar Example: Adaptive Regulation......Page 828
Scalar Example: Adaptive Tracking......Page 830
Example: Direct MRAC without Normalization (n* = 1)......Page 832
Example: Direct MRAC without Normalization (n* = 2)......Page 833
Example: Direct MRAC with Normalization......Page 834
References......Page 835
Further Reading......Page 836
35.1 Introduction......Page 837
Brief History......Page 838
Direct and Indirect Adaptive Control......Page 840
Online Parameter Estimation......Page 841
Model Reference Adaptive Control......Page 843
Adaptive Pole Placement Control......Page 844
Instability Phenomena in Adaptive Systems......Page 846
Robust Adaptive Laws......Page 847
35.3 Nonidentifier-Based Adaptive Control......Page 849
Switching and Multiple Models......Page 850
Unfalsified Adaptive Control......Page 852
Adaptive Control with Mixing......Page 855
35.5 Conclusions......Page 856
References......Page 857
36.1 Introduction......Page 859
Outline......Page 860
36.2 Example: A Servo System......Page 861
36.3 Frequency-Domain Design......Page 864
Frequency-Domain Analysis......Page 865
Tuning Based on Nyquist Analysis......Page 866
H∞ Methods......Page 869
36.4 Generalized Time-Domain System Description......Page 870
Asymptotic Stability......Page 871
Robustness......Page 872
Weighting Matrix Design......Page 873
References......Page 877
Section VI: Analysis and Design of Nonlinear Systems......Page 878
37.1 Input–Output Feedback Linearization......Page 879
37.2 The Zero Dynamics......Page 881
37.3 Local Stabilization of Nonlinear Minimum-Phase Systems......Page 884
37.4 Global Stabilization of Nonlinear Minimum-Phase Systems......Page 886
Bibliography......Page 891
38.1 Introduction......Page 892
38.3 Vector Fields......Page 893
38.4 The Lie Bracket......Page 895
38.5 The Lie Algebras......Page 899
38.6 The Lie Saturate......Page 902
38.7 Applications to Controllability......Page 906
38.8 Rotations......Page 908
38.9 Controllability in SOn(R)......Page 909
Further Reading......Page 912
The Perturbation Method......Page 913
Averaging......Page 915
Singular Perturbation......Page 916
Model Order Reduction......Page 919
39.3 Examples......Page 920
39.4 Defining Terms......Page 923
For Further Information......Page 924
Some Simple Examples......Page 926
Linear Differential Equations......Page 927
40.2 Functional Expansions for Nonlinear Control Systems......Page 928
Bilinear Systems......Page 930
Fliess Series......Page 931
Links between Volterra and Fliess Series......Page 932
40.3 Effective Computation of Volterra Kernels......Page 934
Noncommutative Padé-Type Approximants......Page 936
Analysis of Responses of Systems......Page 937
Optimality......Page 938
40.5 Other Approximations: Application to Motion Planning......Page 940
References......Page 941
41.1 Introduction......Page 944
Notation......Page 945
IQC Modeling......Page 946
Feasibility Optimization and Postfeasibility Analysis......Page 949
IQC Modeling......Page 951
Feasibility Optimization......Page 954
Postfeasibility Analysis......Page 956
41.4 Application of IQC Analysis......Page 957
The IQC Analysis Flow......Page 958
IQC Library......Page 959
Example......Page 960
41.5 Historical Remarks and References......Page 962
References......Page 963
42.1 Introduction......Page 964
42.2 Notation and Examples......Page 965
Kinematic Systems......Page 966
Dynamic Mechanical Systems......Page 967
Controllability Definitions......Page 975
Controllability Tests......Page 976
42.4 Feedback Stabilization......Page 983
Kinematic Example: The Heisenberg System......Page 984
Energy Methods for Nonholonomic Mechanical Systems with Symmetries......Page 988
Optimal Control of the Heisenberg System......Page 993
Search-Based Methods......Page 994
Kinematic Reductions......Page 995
Differentially Flat Systems......Page 996
References......Page 998
Section VII: Stability......Page 1000
43.2 Stability of Equilibrium Points......Page 1001
Lyapunov Method......Page 1002
Perturbed Systems......Page 1004
43.3 Examples and Applications......Page 1005
Feedback Stabilization......Page 1008
43.4 Defining Terms......Page 1009
Further Reading......Page 1010
44.2 Systems and Stability......Page 1011
The Classical Small Gain Theorem......Page 1014
The Classical Passivity Theorem......Page 1017
Simple Nonlinear Separation Theorems......Page 1021
General Conic Regions......Page 1025
Defining Terms......Page 1031
References......Page 1032
Further Reading......Page 1033
45.1 Introduction......Page 1034
Notions of Stability......Page 1035
Gains for Linear Systems......Page 1036
Nonlinear Coordinate Changes......Page 1037
45.2 ISS and Feedback Redesign......Page 1038
Feedback Redesign......Page 1039
A Feedback Redesign Theorem for Actuator Disturbances......Page 1041
Nonlinear Superposition Principle......Page 1042
Robust Stability......Page 1043
Dissipation......Page 1044
Using \"Energy\" Estimates Instead of Amplitudes......Page 1045
45.4 Cascade Interconnections......Page 1046
Dissipation Characterization of iISS......Page 1047
Input-to-Output Stability......Page 1049
Detectability and Observability......Page 1050
Norm-Estimators......Page 1051
45.8 Response to Constant and Periodic Inputs......Page 1052
References......Page 1053
Section VIII: Design......Page 1055
46.1 The Problem of Feedback Linearization......Page 1056
46.2 Normal Forms of Single-Input Single-Output Systems......Page 1059
46.3 Conditions for Exact Linearization via Feedback......Page 1063
References......Page 1070
47.1 Limit Sets......Page 1071
47.2 The Steady-State Behavior of a System......Page 1073
47.3 The Steady-State Response......Page 1074
References......Page 1076
48.1 The Problem......Page 1077
48.2 The Case of Linear Systems as a Design Paradigm......Page 1079
48.3 Steady-State Analysis......Page 1082
48.4 Convergence to the Required Steady State......Page 1085
48.5 The Design of the Internal Model......Page 1088
48.6 The Case of Higher Relative Degree......Page 1090
References......Page 1092
49.1 Introduction......Page 1094
49.2 Lyapunov Redesign......Page 1095
49.3 Beyond Lyapunov Redesign......Page 1096
49.4 Recursive Lyapunov Design......Page 1098
49.5 Smooth Control Laws......Page 1100
49.6 Design Flexibilities......Page 1101
Further Reading......Page 1107
50.1 Introduction and Background......Page 1108
Control Structure......Page 1110
Sliding Modes......Page 1111
Conditions for the Existence of a Sliding Mode......Page 1112
50.3 Existence and Uniqueness of Solutions to VSC Systems......Page 1113
50.4 Switching-Surface Design......Page 1114
Regular Form of the Plant Dynamics......Page 1115
Equivalent System Dynamics via Regular Form......Page 1116
Analysis of the State Feedback Structure of Reduced-Order Linear Dynamics......Page 1117
Stability to Equilibrium Manifold......Page 1118
Various Control Structures......Page 1119
50.6 Design Examples......Page 1120
50.7 Chattering......Page 1123
50.8 Robustness to Matched Disturbances and Parameter Variations......Page 1124
50.9 Observer Design......Page 1125
Observer Design 2......Page 1126
References......Page 1128
51.1 Introduction......Page 1130
51.2 Operating Conditions of Nonlinear Systems......Page 1131
Bifurcations......Page 1132
Chaos......Page 1139
51.5 Control of Parametrized Families of Systems......Page 1141
Feedforward/Feedback Structure of Control Laws......Page 1142
Stressed Operation and Break-Down of Linear Techniques......Page 1143
51.6 Control of Systems Exhibiting Bifurcation Behavior......Page 1144
Local Direct State Feedback......Page 1145
Local Dynamic State Feedback......Page 1147
51.7 Control of Chaos......Page 1148
Bifurcation Control of Routes to Chaos......Page 1149
Acknowledgment......Page 1150
References......Page 1151
Further Reading......Page 1152
52.1 Introduction......Page 1154
Noncommuting Vector Fields, Anholonomy, and the Effect of Oscillatory Inputs......Page 1155
Problem Statement......Page 1157
Vibrational Stabilization......Page 1158
52.2 The Constructive Controllability of Drift-Free (Class I) Systems......Page 1161
52.3 Systems with Drift—Stability Analysis Using Energy Methods and the Averaged Potential......Page 1165
First and Second Order Stabilizing Effects of Oscillatory Inputs in Systems with Drift......Page 1166
A Stability Theory for Lagrangian and Hamiltonian Control Systems with Oscillatory Inputs......Page 1167
Acknowledgment......Page 1174
References......Page 1175
53.1 Introduction: Backstepping......Page 1177
Introductory Examples......Page 1179
General Recursive Design Procedure......Page 1184
53.3 Modular Design......Page 1186
Controller Design......Page 1187
Passive Identifier......Page 1190
53.4 Output Feedback Designs......Page 1192
Output-Feedback Design with Tuning Functions......Page 1194
Output-Feedback Modular Design......Page 1197
Unknown Virtual Control Coefficients......Page 1198
53.6 For Further Information......Page 1199
References......Page 1200
54.1 Introduction......Page 1202
Fuzzy Control......Page 1204
Expert and Planning Systems......Page 1206
Genetic Algorithms for Control......Page 1207
54.3 Autonomous Control......Page 1208
The Intelligent Autonomous Controller......Page 1209
The Control-Theoretic View of Autonomy......Page 1210
54.4 Concluding Remarks......Page 1211
References......Page 1212
For Further Information......Page 1213
55.1 Introduction......Page 1214
Philosophy of Fuzzy Control......Page 1215
Summary......Page 1217
Linguistic Rules......Page 1218
Fuzzy Sets, Fuzzy Logic, and the Rule-Base......Page 1219
Fuzzification......Page 1220
The Inference Mechanism......Page 1221
Defuzzification......Page 1222
55.3 Theme Problem: Rotational Inverted Pendulum......Page 1223
Mathematical Model......Page 1224
Balancing Control......Page 1226
Controller Synthesis......Page 1228
Performance Evaluation......Page 1232
Overview......Page 1234
Auto-Tuning for Pendulum Balancing Control......Page 1236
55.6 Concluding Remarks......Page 1238
Acknowledgments......Page 1239
Further Reading......Page 1240
56.1 Introduction......Page 1241
56.2 Approximator Properties......Page 1242
The Domain of Approximation......Page 1243
The Context......Page 1244
Adaptation versus Learning......Page 1245
Universal Approximation......Page 1246
Parameter (Non)Linearity......Page 1247
Localization......Page 1248
Orthogonal Polynomial Series......Page 1249
Localized Basis Influence Functions......Page 1250
Adaptive Approximation Problem......Page 1251
Small-Signal Linearization and Gain Scheduling......Page 1253
Feedback Linearization......Page 1254
56.6 Adaptive Approximation-Based Control......Page 1255
56.7 Reinforcement Learning......Page 1257
56.8 Concluding Remarks......Page 1258
References......Page 1259
Section IX: System Identification......Page 1261
Ten Basic Questions about System Identification......Page 1262
Background and Literature......Page 1263
An Archetypical Problem: ARX Models and the Linear Least-Squares Method......Page 1264
Fitting Models to Data......Page 1268
Model Quality......Page 1269
Measures of Model Fit......Page 1270
Model Structure Selection......Page 1271
Algorithmic Aspects......Page 1272
Linear System Descriptions in General......Page 1275
Linear and Ready-Made Models......Page 1277
Frequency Domain Data......Page 1280
Fitting Models to Fourier Transformed Data......Page 1281
Fitting to Frequency Response Data......Page 1282
Transient and Frequency Analysis......Page 1283
Estimating the Frequency Response by Spectral Analysis......Page 1284
Subspace Estimation Techniques for State-Space Models......Page 1286
57.6 Physically Parameterized Models......Page 1288
57.7 Nonlinear Black-Box Models......Page 1290
Nonlinear Mappings: Possibilities......Page 1291
Experiment Design......Page 1294
Model Validation and Model Selection......Page 1295
Software for System Identification......Page 1296
The Practical Side of System Identification......Page 1297
References......Page 1299
Section X: Stochastic Control......Page 1301
58.2 Introduction......Page 1302
58.3 Definitions and Construction......Page 1303
58.4 Properties and Classification......Page 1306
58.5 Algebraic View and Stationarity......Page 1308
58.6 Random Variables......Page 1310
58.7 Limit Theorems: Transitions......Page 1312
58.8 Ergodic Theorems......Page 1314
58.9 Extensions and Comments......Page 1315
References......Page 1317
59.1 Introduction......Page 1318
Stochastic Differential Equations......Page 1319
59.2 White Noise and the Wiener Process......Page 1320
59.3 The Itô Integral......Page 1323
Properties of the Itô Integral......Page 1325
A Simple Form of Itô\'s Rule......Page 1327
59.4 Stochastic Differential Equations and Itô’s Rule......Page 1329
Homogeneous Equations......Page 1330
Linear SDEs in the Narrow Sense......Page 1331
The Langevin Equation......Page 1332
59.6 Transition Probabilities for General SDEs......Page 1333
59.7 Defining Terms......Page 1336
Further Reading......Page 1337
60.1 Introduction......Page 1338
60.2 ARMA and ARMAX Models......Page 1339
60.3 Linear Filtering......Page 1342
60.4 Spectral Factorization......Page 1343
60.5 Yule–Walker Equations and Other Algorithms for Covariance Calculations......Page 1345
60.6 Continuous-Time Processes......Page 1348
60.7 Optimal Prediction......Page 1349
60.8 Wiener Filters......Page 1351
References......Page 1355
Example: The Shortest Path Problem......Page 1356
The Dynamic Programming Method......Page 1357
Observations on the Dynamic Programming Method......Page 1359
61.2 Deterministic Systems with a Finite Horizon......Page 1360
Continuous-Time Systems......Page 1361
61.3 Stochastic Systems......Page 1362
The Discounted Cost Problem......Page 1363
The Average Cost Problem......Page 1365
Connections of Average Cost Problem with Discounted Cost Problems and Recurrence Conditions......Page 1366
Total Undiscounted Cost Criterion......Page 1367
Further Reading......Page 1368
Approximate Dynamic Programming in Relation to Modern Control Approaches......Page 1369
Reinforcement Learning......Page 1370
System/Environment and Control/Behavior Policy......Page 1371
Control System Performance and Bellman\'s Optimality Principle......Page 1372
ADP and Adaptive Critics......Page 1375
Value Function and Q-Function......Page 1376
Key Ingredients for Online Implementation of ADP Methods: Value Function Approximation and TD Error......Page 1377
Policy Update......Page 1381
Terminology–Revisited......Page 1382
Policy Iteration......Page 1383
Value Iteration......Page 1385
Policy Iterations and Value Iterations on the Algebraic Riccati Equation of the LQR Problem......Page 1387
62.3 ADP Algorithms for Continuous-Time Systems......Page 1390
Policy Iteration......Page 1391
Value Iteration......Page 1392
Policy Iterations and Value Iterations and the LQR Problem......Page 1393
Power System and Cost Function......Page 1395
Continuous-Time Value Iteration......Page 1396
Discrete-Time Action-Dependent Value Iteration......Page 1397
References......Page 1400
63.1 Introduction......Page 1402
63.2 Lyapunov Function Method......Page 1406
63.3 The Lyapunov Exponent Method and the Stability of Linear Stochastic Systems......Page 1416
References......Page 1435
64. Stochastic Adaptive Control for Continuous-Time Linear Systems......Page 1438
64.1 Adaptive Control for Continuous-Time Scalar Linear Systems with Fractional Brownian Motions......Page 1447
References......Page 1451
65.1 Motivations and Preliminaries......Page 1452
65.2 Probabilistic Design......Page 1455
65.3 Sequential Methods for Design......Page 1456
Update Rules......Page 1457
Probabilistic Properties......Page 1458
Advanced Techniques......Page 1460
65.4 Scenario Approach to Optimization Problems......Page 1461
65.5 Learning Approach to Nonconvex Optimization......Page 1463
Deterministic Analysis......Page 1465
Probabilistic Analysis......Page 1466
65.7 Randomized Algorithms Control Toolbox......Page 1467
Sample Generation Algorithms......Page 1469
Mixed Deterministic and Probabilistic Setting......Page 1470
Linear Parameter-Varying Systems......Page 1471
Systems and Control Applications......Page 1472
Additional References......Page 1473
66.1 Introduction......Page 1475
Boundedness in Probability......Page 1476
Stability Notion 3......Page 1477
Stability Notion 4......Page 1478
66.3 Stabilization of Stochastic Systems: The Basics......Page 1479
Introductory Example......Page 1481
66.5 Output-Feedback Stabilization......Page 1483
66.6 Adaptive Stabilization in Probability......Page 1489
Backstepping Design......Page 1490
Dominating the Uncertain and Unmeasured Zero Dynamics......Page 1495
66.7 Inverse Optimal Stabilization in Probability......Page 1496
66.8 Extensions......Page 1497
References......Page 1498
Section XI: Control of Distributed Parameter Systems......Page 1499
67.1 Introduction......Page 1500
67.2 State-Space Formulation......Page 1502
67.3 Issues in Controller Design......Page 1508
67.4 LQ Regul......Page 1517
67.5 H∞ Control......Page 1526
References......Page 1534
68.1 Dynamic Elastic Beam Models......Page 1537
68.2 The Equations of Motion......Page 1539
Initially Straight and Untwisted Linear Shearable 3-Dimensional Beams......Page 1540
Nonlinear Planar, Shearable Straight Beams......Page 1541
The Euler–Bernoulli Beam Model......Page 1542
Hyperbolic Systems......Page 1543
Quasi-Hyperbolic Systems......Page 1546
The Euler–Bernoulli Beam......Page 1547
68.4 Stabilizability......Page 1548
Hyperbolic Systems......Page 1549
Linear Models......Page 1550
Boundary Conditions......Page 1552
68.6 Controllability of Dynamic Plates......Page 1553
Controllability of Kirchhoff Plates......Page 1554
Controllability of the Reissner–Mindlin System......Page 1559
Controllability of the von Kármán System......Page 1562
Stabilizability of Kirchhoff Plates......Page 1563
Stabilizability of the Reissner–Mindlin System......Page 1565
References......Page 1566
69.1 Introduction......Page 1568
69.2 Background: Physical Derivation......Page 1569
The Maximum Principle and Conservation......Page 1572
Smoothing and Localization......Page 1573
Linearity......Page 1574
Autonomy, Similarity, and Scalings......Page 1576
A Simple Heat Transfer Problem......Page 1578
Exact Control......Page 1579
System Identification......Page 1580
The Duality of Observability/Controllability......Page 1581
The One-Dimensional Case......Page 1583
Higher-Dimensional Geometries......Page 1584
References......Page 1586
70.1 Comparison with the Finite Dimensional Case......Page 1587
70.2 General Formulation in the Distributed-Parameter Case......Page 1589
70.3 Observation of a Heat Conduction Process......Page 1591
70.4 Observability Theory for Elastic Beams......Page 1594
References......Page 1597
71.1 Introduction......Page 1599
71.2 Unstable Heat Equation......Page 1600
71.3 Observers......Page 1602
Reaction–Advection–Diffusion Systems......Page 1604
71.5 Complex-Valued and Coupled PDEs......Page 1605
71.6 First-Order Hyperbolic Systems......Page 1606
Undamped Wave Equation......Page 1607
Wave Equation with Kelvin–Voigt Damping......Page 1608
71.8 Beams......Page 1609
71.9 Adaptive Control of Parabolic PDEs......Page 1610
71.10 Lyapunov Design......Page 1611
71.11 Certainty Equivalence Design with Passive Identifier......Page 1613
71.12 Certainty Equivalence Design with Swapping Identifier......Page 1615
71.13 Extension to Reaction–Advection–Diffusion Systems in Higher Dimensions......Page 1617
71.14 Plants with Spatially Varying Uncertainties......Page 1619
Transformation to Observer Canonical Form......Page 1620
Filters......Page 1621
Controller......Page 1622
References......Page 1623
72.1 Introduction......Page 1625
72.2 Channel with MHD Flow......Page 1626
Hartmann Equilibrium Profile......Page 1627
The Plant in Wave Number Space......Page 1628
Controlled Velocity Wave Number Analysis......Page 1630
Uncontrolled Velocity Wave Number Analysis......Page 1635
Closed-Loop Stability Properties......Page 1637
72.4 Observer Design......Page 1638
Observer Structure......Page 1639
Observer Gain Design and Convergence Analysis......Page 1640
Observed Wave Number Analysis......Page 1641
Observer Convergence Properties......Page 1646
A Nonlinear Estimator with Boundary Sensing......Page 1647
72.5 For Further Information......Page 1648
References......Page 1649
Section XII: Networks and Networked Controls......Page 1650
73.1 Introduction......Page 1651
73.2 Basic Framework......Page 1652
Deterministic Channel Model......Page 1653
Random Channel Model......Page 1654
Notes on Optimality......Page 1656
73.4 Input to State Stability......Page 1657
References......Page 1658
74.1 Introduction......Page 1660
Information Constraint......Page 1661
74.4 Quadratic Invariance......Page 1663
Structural Constraints......Page 1664
Delays......Page 1665
74.6 Perfectly Decentralized Control......Page 1666
References......Page 1667
75.1 Introduction......Page 1669
Estimation......Page 1670
Control......Page 1671
75.3 Optimal Coding......Page 1674
75.4 Extensions and Open Questions......Page 1678
LQ Control......Page 1680
MMSE Estimation......Page 1682
LQG Control......Page 1683
References......Page 1684
76.1 Introduction......Page 1685
A Unifying Passivity Framework for Internet Congestion Control......Page 1687
CDMA Power Control Game......Page 1688
Passivity-Based Design Procedure for Position Coordination......Page 1690
From Point Mass to Rigid-Body Models......Page 1692
Adaptive Redesign......Page 1693
The Secant Criterion for Cyclic Networks......Page 1695
Generalization to Other Network Topologies......Page 1697
Passivity as a Certificate of Robustness to Diffusion......Page 1699
76.5 Conclusions and Future Topics......Page 1700
References......Page 1701