Preface xix
Acknowledgements xxi
1 Introduction to Optimal and Robust Control 1
1.1 Introduction 1
1.1.1 Optimality, Feedback and Robustness 2
1.1.2 High-integrity and Fault-tolerant Control Systems 3
1.1.3 Self-healing Control Systems 4
1.1.4 Fault Monitoring and Detection 5
1.1.5 Adaptive versus Robust Control 5
1.1.6 Artificial Intelligence, Neural Networks and Fuzzy Control 5
1.1.7 Discrete-time Systems 7
1.2 The H2 and Hâ Spaces and Norms 8
1.2.1 Graphical Interpretation of the Hâ Norm 9
1.2.2 Terms Used in Hâ Robust Control Systems Design 9
1.3 Introduction to Hâ Control Design 9
1.3.1 Properties of Hâ Robust Control Design 11
1.3.2 Comparison of Hâ and H2 /LQG Controllers 12
1.3.3 Relationships between Classical Design and Hâ Robust Control 13
1.3.4 H2 and Hâ Design and Relationship to PID Control 13
1.3.5 Hâ Polynomial Systems Synthesis Theory 13
1.4 State-space Modelling and Synthesis Theory 14
1.4.1 State-space Solution of Discrete-time Hâ Control Problem 14
1.4.2 Hâ Control Design Objectives 15
1.4.3 State-feedback Control Solution 15
1.4.4 State-feedback Control Problem: Cross-product Costing Case 18
1.4.5 State-space Solution of Discrete-time Hâ Filtering Problem 19
1.4.6 Bounded Real Lemma 21
1.4.7 Output Feedback Hâ Control Problem 24
1.5 Introduction to H2 or LQG Polynomial Synthesis 29
1.5.1 System Description 29
1.5.2 Cost Function and Solution 31
1.5.3 Minimisation of the Performance Criterion 31
1.5.4 Solution of the Diophantine Equations and Stability 34
1.5.5 H2 /LQG Design Examples 35
1.6 Benchmarking 40
1.6.1 Restricted Structure Benchmarking 41
1.6.2 Rules for Benchmark Cost Function Selection 42
1.7 Condition Monitoring 44
1.8 Combining H2 , Hâ and â 1 Optimal Control Designs 45
1.9 Linear Matrix Inequalities 46
1.10 Concluding Remarks 47
1.11 Problems 48
1.12 References 51
2 Scalar H2 and LQG Optimal Control 57
2.1 Introduction 57
2.1.1 Industrial Controller Structures 58
2.1.2 The 2½-DOF Structure 59
2.1.3 Restricted Structure Control Laws 60
2.2 Stochastic System Description 60
2.2.1 Ideal Response Models 62
2.2.2 System Equations 62
2.2.3 Cost Function Weighting Terms 63
2.3 Dual-criterion Cost-minimisation Problem 64
2.3.1 Solution of the Dual-criterion Minimisation Problem 66
2.3.2 Theorem Summarising LQG Controller 71
2.3.3 Remarks on the Equations and Solution 73
2.3.4 Design Guidelines 76
2.3.5 Controller Implementation 77
2.3.6 LQG Ship-steering Autopilot Application 78
2.4 LQG Controller with Robust Weighting Function 82
2.4.1 Youla Parameterisation 82
2.4.2 Cost Function with Robust Weighting Function 83
2.4.3 Solution of the Dual-criterion Problem with Robust Weighting 84
2.4.4 Summary of H2 /LQG Synthesis Problem with Robust Weighting 86
2.4.5 Comments on the Solution 88
2.5 Introduction to the Standard System Model 89
2.5.1 Standard System Model 89
2.6 The Standard System Model Structure 91
2.6.1 Polynomial System Models 92
2.6.2 Reference Model 93
2.6.3 Cost Function Signals to be Weighted 94
2.7 Generalised H2 Optimal Control: Standard System Model 95
2.7.1 Optimal Control Solution of the Standard System Model Problem 96
2.7.2 Summary of H2 /LQG Controller for Standard System Results 102
2.7.3 Remarks on the Solution 104
2.8 Concluding Remarks 105
2.9 Problems 105
2.10 References 109
3 Hâ Optimal Control of Scalar Systems 113
3.1 Introduction 113
3.1.1 Links Between LQG and Hâ Solutions 114
3.1.2 Reference and Feedback Controller Designs 115
3.2 System Description 115
3.3 Lemma Linking Hâ and LQG Control Problems 115
3.4 Calculation of the Hâ Optimal Controller 116
3.4.1 Simple Hâ Controller Structures and Calculations 117
3.4.2 Zero Measurement Noise Case 117
3.4.3 Solution for the Hâ Optimal Controller 118
3.4.4 Stability Robustness of Mixed-sensitivity Hâ Designs 121
3.4.5 One-block Hâ Control Problems 122
3.5 The GHâ Control Problem 123
3.5.1 GHâ Cost Function Definition 124
3.5.2 Youla Parameterised Form of the GHâ Controller 126
3.5.3 Calculation of the GHâ Controller 128
3.6 Stability Robustness of GHâ Designs 136
3.6.1 Structure of the Uncertain System 136
3.6.2 Rational Uncertainty Structure 137
3.6.3 Stability Lemma 139
3.6.4 Influence of the Uncertainty Model 140
3.6.5 Design Procedure for Uncertain Systems 140
3.6.6 Hâ Self-Tuning Controller for Systems with Parametric Uncertainty 147
3.7 Standard System and Cost Function Description 147
3.8 Calculation of Hâ Controller for the Standard System 147
3.8.1 F-iteration Method of Solving the Robust Weighting Equation 148
3.8.2 H2 / Hâ Trade-off 149
3.9 Probabilistic System Descriptions and Hâ Control 150
3.9.1 Uncertain System Model 151
3.9.2 Cost Function Definition 153
3.9.3 Uncertain System and Polynomial Equation Representation 155
3.9.4 Discussion of Probabilistic Uncertainty Modelling and Control 158
3.10 Concluding Remarks 158
3.11 Problems 159
3.12 References 163
4 Multivariable H2 /LQG Optimal Control 167
4.1 Introduction 167
4.1.1 Matrix Fraction Descriptions 168
4.2 Multivariable System Description 168
4.2.1 Multivariable Sensitivity Matrices and Signal Spectra 170
4.2.2 Choice of Noise and Cost Function Weightings 171
4.3 LQG Optimal Control Problem and Solution 171
4.3.1 Solution of the H2 /LQG Problem 172
4.3.2 Solution of the Diophantine Equations 175
4.4 Youla Parameterisation and Auxiliary Problem 182
4.4.1 Youla Parameterisation for the Auxiliary Problem 184
4.4.2 Summary of Multivariable Problem Results with Robust Weighting 186
4.5 H 2 /LQG Optimal Control Problem: Measurement Noise Case 187
4.5.1 Predictive Optimal Control 190
4.5.2 SIMO Predictive Optimal Control Problem 190
4.5.3 Probabilistic Description of Uncertainty 196
4.6 The GLQG Optimal Control Problem 196
4.6.1 Solution of the GLQG Problem 197
4.6.2 Modified GLQG Cost Function and Youla Parameterisation 199
4.7 Design of Automatic Voltage Regulators 200
4.8 Pseudo-state Modelling and Separation Principle 210
4.8.1 Introduction to Pseudo-state Methods 210
4.8.2 Pseudo-state Discrete-time Plant Model 211
4.8.3 Discrete Pseudo-state Feedback Optimal Control 215
4.8.4 Solution of the Pseudo-state Feedback Control Problem 217
4.8.5 Discrete Pseudo-state Estimation Problem 222
4.8.6 Solution of the Discrete-time pseudo-state Estimation Problem 224
4.8.7 Output Feedback Control Problem and Separation Principle 230
4.8.8 Computational Example 235
4.8.9 Pseudo-state Approach Remarks 240
4.9 Concluding Remarks 240
4.10 Problems 241
4.11 References 245
5 Multivariable Hâ Optimal Control 249
5.1 Introduction 249
5.1.1 Suboptimal Hâ Control Problems 250
5.2 Hâ Multivariable Controllers 250
5.2.1 Derivation of the Weighting Filter W 251
5.2.2 Robust Weighting Equation 252
5.2.3 Calculation of the Hâ Optimal Controller 253
5.2.4 Superoptimality in Hâ Design 258
5.2.5 Single-input Multi-output Systems 259
5.3 One-block and GHâ Optimal Control Problems 259
5.3.1 One-block Nehari Problems 259
5.3.2 Categories of Nehari Problem 260
5.3.3 Constraint on the Choice of Weights for Simplified Design 261
5.3.4 GHâ Optimal Control Problem 262
5.3.5 Final Remarks on LQG Embedding Hâ Solution 267
5.4 Suboptimal Hâ Multivariable Controllers 268
5.4.1 System Description and Game Problem 269
5.4.2 Linear Fractional Transformation 271
5.4.3 Signals and Bounded Power Property 271
5.4.4 System and Cost Weighting Function Definitions 272
5.5 Polynomial System for Suboptimal Hâ Control Problem 273
5.5.1 J-spectral Factorisation 274
5.5.2 Diophantine Equations for Causal and Noncausal Decomposition 274
5.6 Solution of Suboptimal Hâ State Feedback Problem 275
5.6.1 Discrete-time Game Problem 275
5.6.2 Relationship Between the Game and Hâ Problems 276
5.6.3 Standard System Model Equations and Sensitivity 277
5.6.4 Completing-the-squares 277
5.6.5 Cost Index Terms 278
5.6.6 Cost Integrand Simplification 279
5.6.7 Contour Integral Simplification 279
5.6.8 Optimal Control Law Calculation 280
5.6.9 Expression for H0TJH0 281
5.6.10 Saddle-point Solution 282
5.6.11 Expression for the Minimum Cost 283
5.7 Suboptimal Hâ State-feedback Control Problem 284
5.7.1 Remarks on the Solution 285
5.8 Relationship Between Polynomial and State-space Results 287
5.8.1 J-spectral Factorisation Using Riccati Equation 288
5.8.2 Relationship between the Polynomial and State-space Equations 290
5.9 Solution of Suboptimal Output Feedback Control Problem 291
5.9.1 Final Remarks on the Suboptimal Hâ Solution 291
5.10 Problems 292
5.11 References 295
6 Robust Control Systems Design and Implementation 299
6.1 Introduction 299
6.1.1 The Control Design Problem 300
6.1.2 Justification for Hâ Control Design 302
6.1.3 Dynamic Cost Function Weightings 303
6.1.4 Properties of Sensitivity Functions for Discrete-time Systems 304
6.2 Avoiding Impractical Hâ Designs 306
6.2.1 Equalising Hâ Solutions and Implications for Multivariable Design 307
6.3 Pole-zero Cancellation Properties of LQG and Hâ Designs 308
6.3.1 Polynomial Systems Approach 308
6.3.2 H2 =LQG Optimal Control Problem 308
6.3.3 Hâ Optimal Control Problem 310
6.3.4 Cancellation of Minimum-phase Plant Zeros 311
6.3.5 Cancellation of Stable Plant Poles 312
6.3.6 Sendzimir Steel Rolling Mill Results 314
6.4 System Pole and Zero Properties 314
6.4.1 Controller Poles and Zeros due to Weightings 314
6.4.2 Poles of the Closed-loop System 315
6.5 Influence of Weightings on Frequency Responses 316
6.5.1 Stability Criterion and Cost Function Weighting Selection 316
6.5.2 Influence of the Choice of Weights on the Sensitivity Functions 317
6.5.3 Use of Constant Cost Weightings in Hâ Design 319
6.5.4 Poor Robustness due to Unrealistic Weightings 320
6.6 Loop Shaping Design for Multivariable Systems 324
6.6.1 Singular Value Approximations 324
6.6.2 Robustness and Loop Shaping 326
6.6.3 Stability and Performance Boundaries 327
6.6.4 Robust Design for Systems in Standard Model Form 328
6.6.5 Structured Singular Values 330
6.7 Formalised Design Procedures 331
6.7.1 Steps in a Hâ Design Procedure 331
6.7.2 Cost Function Weighting Selection for Scalar Systems 332
6.8 Mutivariable Robust Control Design Problem 334
6.8.1 Problems in Multivariable Control 335
6.8.2 Poles and Zeros of Multivariable Systems 336
6.8.3 Interaction Measures 337
6.9 Multivariable Control of Submarine Depth and Pitch 337
6.9.1 Selection of Weights in Multivariable Problems 337
6.9.2 Multivariable Submarine Motion Control 338
6.9.3 Multivariable Submarine Control Design Results 340
6.9.4 Speed of Response and Interaction 343
6.9.5 Order of the Weighting Terms 346
6.9.6 Two-degree-of-freedom Submarine Control 346
6.10 Restricted Structure and Multiple Model Control 346
6.10.1 Feedforward and Feedback Polynomial System Plant 347
6.10.2 H2 /LQG Restricted Structure Optimal Control Problem 350
6.10.3 Numerical Algorithm for Single- and Multi-model Systems 362
6.10.4 Hot Strip Finishing Mill Tension Control 370
6.10.5 Benefits of Multiple-model Approach 379
6.10.6 Restricted Structure Benchmarking 379
6.11 Concluding Remarks 381
6.12 Problems 382
6.13 References 384
7 H 2 Filtering, Smoothing and Prediction 389
7.1 Introduction 389
7.1.1 Standard Signal Processing Model 390
7.2 Signal Processing System Description 390
7.2.1 Summary of Estimation Problem Assumptions 391
7.2.2 Optimal Estimator Transfer-function 392
7.2.3 System Equations 392
7.2.4 Polynomial Matrix Descriptions 392
7.2.5 Spectral Factorisation 393
7.3 The Standard H2 Optimal Estimation Problem 393
7.3.1 H2 Standard System Model Estimation Problem Solution 394
7.3.2 Estimation Error Power Spectrum: Completion of Squares 394
7.3.3 Wiener Filtering Solution 395
7.3.4 Introduction of the First Diophantine Equation 396
7.3.5 Optimal Estimator when Signal Model Stable 396
7.3.6 Optimal Estimator when Signal Model can be Unstable 399
7.3.7 Optimal Estimator when Signal Model can be Unstable 404
7.4 Solution of Filtering, Smoothing and Predication Problems 408
7.4.1 State Estimation Problem 408
7.4.2 Output Filtering and Prediction 409
7.4.3 Deconvolution Estimation 410
7.4.4 Robust Weighting Function W 413
7.4.5 Extensions of the Estimator Capabilities 414
7.5 Strip Thickness Estimation from Roll Force Measurements 415
7.5.1 Rolling Mill Model 416
7.5.2 Continuous-time Dynamic Mill Model 416
7.6 Strip Thickness Estimation Using Force Measurments 418
7.7 Strip Thickness Estimation Using X-Ray Gauge Measurements 421
7.8 Strip Thickness Estimation Using Gauge Measurements 422
7.9 Time-varying and Nonstationary Filtering 426
7.9.1 Linear Multichannel Estimation Problem 428
7.9.2 Output Estimation Problem 431
7.9.3 Relationship to the Kalman Filtering Problem 435
7.10 Conclusions 440
7.11 Problems 441
7.12 References 442
8 Hâ Filtering, Smoothing and Prediction 445
8.1 Introduction 445
8.1.1 The Hâ Filtering Problem 446
8.1.2 Smoothing Filters 447
8.1.3 Probabilistic Representation of Uncertainty for Filtering Problems 448
8.2 Solution of Hâ Optimal Estimation Problem 448
8.2.1 Relationship Between H2 and Hâ Minimisation Problems 448
8.2.2 Solution Strategy and Weightings 449
8.2.3 Derivation of the Weighting Filter W 450
8.2.4 Robustness Weighting Diophantine Equation 451
8.2.5 Hâ Optimal Estimator for the Generalised System Model 452
8.2.6 Properties of the Optimal Estimator 453
8.3 Hâ Deconvolution Filtering Problem 453
8.3.1 Deconvolution System Description 454
8.3.2 Solution of the Hâ Deconvolution Estimation Problem 455
8.4 Suboptimal Hâ Multi-Channel Filters 457
8.4.1 Discrete-time System and Signal Source Descriptions 457
8.4.2 Duality and the Game Problem 459
8.4.3 Results for the Suboptimal Hâ Filtering Problem 460
8.4.4 Remarks on the Solution 462
8.5 Relevance of Hâ Methods to Signal Processing Applications 463
8.6 Final Remarks on the Suboptimal Hâ Filtering Problem 463
8.7 Problems 464
8.8 References 465
9 Applications of H2 /LQG Optimal Control 469
9.1 Introduction 469
9.2 Wind Turbine Power Control Systems 470
9.2.1 Definition of Wind Turbine Transfer Functions 472
9.2.2 Weighting Function Definitions 474
9.2.3 Numerical Results for Wind Turbine Example 476
9.2.4 Wind Turbine Feedback Controller Cancellation Properties 481
9.2.5 Role of the Ideal-response Models in Design 483
9.2.6 Fixed- and Variable-speed Wind Turbines 484
9.2.7 Comparison of Wind Turbine Controllers 484
9.2.8 Wind Turbine Condition Monitoring 484
9.3 Design of an H2 Flight Control System 485
9.3.1 System Models 485
9.3.2 Design Requirements and Specification 487
9.3.3 Flight Control System: Time and Frequency Responses 490
9.3.4 Flight Control System Design Including Flexible Modes 494
9.3.5 LQG Flight Control Study Design Results 495
9.3.6 Classical and LQG Controller Design 497
9.4 Thickness Control Systems Design Using Force Feedback 500
9.4.1 Optimal Control Solution for the Gauge Control Problem 502
9.4.2 Rolling Mill Model 502
9.4.3 Continuous-time Mill Models 502
9.4.4 Definition of the Polynomial Models for the Standard System 503
9.4.5 Cost Function Definition 504
9.4.6 BUR Eccentricity Problem Results 506
9.4.7 Mismatched Eccentricity Model Conditions 510
9.5 Thickness Control Using Gauge Measurement 510
9.5.1 Transport Delay in Thickness Measurement 512
9.5.2 Feedback System Models in Polynomial Form 516
9.5.3 Choice of Cost Function Weightings for Gauge Feedback Control Problem 516
9.5.4 Degree of Stability 517
9.6 Ship Roll Stabilisation 518
9.6.1 Fin Control Unit 519
9.6.2 Speed Adaptation 520
9.6.3 Models for the Ship Stabilisation System 521
9.6.4 Weighting Selection for LQG Roll Stabilisation Design 521
9.6.5 Frequency Responses 522
9.6.6 Advantages of the Optimal System in Comparison with Classical Methods 524
9.6.7 Rudder-roll Stabilisation and Ship Steering 525
9.7 Concluding Remarks 525
9.8 Problems 526
9.9 References 526
10 Industrial Applications of Hâ Optimal Control 529
10.1 Introduction 529
10.1.1 Applications where Hâ Robust Control Design is Applicable 530
10.1.2 Safety Critical Control Systems 530
10.1.3 Flight Control Systems 530
10.2 Hâ Flight Control Systems Design 532
10.2.1 Design Requirements and Specification 534
10.2.2 Definition of Cost Function Weightings 534
10.2.3 Generalised LQG and Hâ Controller Time- and Frequency-responses 535
10.2.4 Introducing a Measurement Noise Model 540
10.2.5 Comparison of Controllers 543
10.3 Hâ Gauge Control System Design Using Force Feedback 543
10.3.1 Thickness Control System Frequency- and Time-responses 546
10.3.2 Mismatched Eccentricity Model and Robustness 551
10.3.3 Thickness Profile Control 552
10.4 Submarine Depth and Course-keeping Hâ Design 554
10.4.1 Forces and Moments 554
10.4.2 Depth Control 555
10.4.3 Sea-state and Sea Current Disturbances 556
10.4.4 Submarine Motion Dynamics 558
10.4.5 Submarine Depth and Pitch Control Design 561
10.4.6 Submarine Depth-keeping Controllers 562
10.4.7 Submarine Model Responses 563
10.4.8 Model Tuning 568
10.4.9 Summary of the Output and Input Disturbance Models 571
10.4.10 Submarine Depth and Pitch Control 572
10.4.11 Summary of the Selected Weighting Terms 573
10.4.12 Scalar Design and Responses: Depth Control 574
10.4.13 Scalar Design and Responses: Pitch Control 578
10.4.14 Improving the Scalar System Time-responses 580
10.5 Hâ Control of Remotely Operated Underwater Vehicles 580
10.5.1 Design of ROV Controllers 584
10.6 Hâ Control of Surface Ships 585
10.6.1 Hâ Fin Roll Stabilisation System Design 585
10.6.2 Hâ Ship Track-keeping Control 588
10.7 Concluding Remarks 591
10.8 Problems 592
10.9 References 593
11 Time-varying and Nonlinear Control 595
11.1 Introduction 595
11.2 Optimal Control of Time-varying Linear Systems 596
11.2.1 Linear Time-varying and Adjoint Operators 597
11.2.2 The Quadratic Cost Index 598
11.2.3 Solution of the Time-varying Linear Quadratic Control Problem 599
11.3 Modelling and Control of Nonlinear Systems 602
11.3.1 Nonlinear Systems Modelling 603
11.3.2 Hard Nonlinearities 604
11.3.3 Typical System Structures 605
11.3.4 Feedback Linearisation 605
11.4 NLQG Compensation and Control 607
11.4.1 Nonlinear Control Example 608
11.4.2 Polynomial Versions of Plant Transfer-function Operators 609
11.4.3 Use of Time-varying Cost Function Weighting 610
11.4.4 The NLQG Algorithm and Properties 611
11.5 NLQG Example with Input and Output Nonlinearities 612
11.5.1 System and Cost Function Description 613
11.5.2 Simulation Results 613
11.5.3 Frequency-domain Results 614
11.5.4 Improving NLQG Control Using Future Change Information 620
11.6 Nonlinear Generalised Minimum Variance Control 622
11.6.1 Nonlinear System Description 623
11.6.2 Nonlinear and Linear Subsystem Models 625
11.6.3 Signals 627
11.7 Nonlinear Generalised Minimum Variance Problem 627
11.7.1 Solution of the Nonlinear Feedback/Feedforward Control Problem 629
11.7.2 Polynomial Models for the Feedback/Feedforward Control Problem 630
11.7.3 Diophantine Equations 630
11.7.4 Optimisation 632
11.7.5 Alternative Control Solution and Stability 634
11.7.6 Closed-loop System Stability 636
11.7.7 Simplifying the Controller 636
11.7.8 Effect of Bias or Steady-state Levels 637
11.8 Nonlinear GMV Control Problem 639
11.9 Nonlinear Smith Predictor 644
11.9.1 Weighting Selection Based on an Existing Controller 647
11.10 Concluding Remarks 648
11.11 References 669
Appendix 1 Notation and Mathematical Preliminaries 653
Notation 653
Partitions 654
Infimum and Supremum 654
A1.1 Vectors 654
A1.2 Matrices 655
A1.2.1 Matrix Inverse Relationships 657
A1.2.2 Matrix Singular Value Relationships 658
A1.2.3 Matrix Norm Relationships 659
A1.3 Polynomial Matrices 661
A1.3.1 Polynomial Equations 662
A1.4 Transfer-function Matrices 663
A1.4.1 Adjoint, All-pass and Inner Functions 664
A1.4.2 Transfer-function Matrix for the Standard System Model 665
A1.5 Vector and Normed Spaces 665
A1.5.1 Hardy Spaces and Norms 667
A1.6 References 669
