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Robust Control

Youla Parameterization Approach

Assadian, Farhad / Mallon, Kevin R.

Wiley-ASME Press Series

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1. Auflage Februar 2022
464 Seiten, Hardcover
Wiley & Sons Ltd

ISBN: 978-1-119-50036-0
John Wiley & Sons

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Robust Control

Robust Control

Youla Parameterization Approach

Discover efficient methods for designing robust control systems

In Robust Control: Youla Parameterization Approach, accomplished engineers Dr. Farhad Assadian and Kevin R. Mallon deliver an insightful treatment of robust control system design that does not require a theoretical background in controls. The authors connect classical control theory to modern control concepts using the Youla method and offer practical examples from the automotive industry for designing control systems with the Youla method.

The book demonstrates that feedback control can be elegantly designed in the frequency domain using the Youla parameterization approach. It offers deep insights into the many practical applications from utilizing this technique in both Single Input Single Output (SISO) and Multiple Input Multiple Output (MIMO) design. Finally, the book provides an estimation technique using Youla parameterization and controller output observer for the first time.

Robust Control offers readers:
* A thorough introduction to a review of the Laplace Transform, including singularity functions and transfer functions
* Comprehensive explorations of the response of linear, time-invariant, and dynamic systems, as well as feedback principles and feedback design for SISO
* Practical discussions of norms and feedback systems, feedback design by the optimization of closed-loop norms, and estimation design for SISO using the parameterization approach
* In-depth examinations of MIMO control and multivariable transfer function properties

Perfect for industrial researchers and engineers working with control systems, Robust Control: Youla Parameterization Approach is also an indispensable resource for graduate students in mechanical, aerospace, electrical, and chemical engineering.

List of Figures xv

List of Tables xxv

Preface xxvii

Acknowledgments xxxi

Introduction xxxiii

About the Companion Website xli

Part I Control Design Using Youla Parameterization: Single Input Single Output (SISO) 1

1 Review of the Laplace Transform 3

1.1 The Laplace Transform Concept 3

1.2 Singularity Functions 3

1.2.1 Definition of the Impulse Function 4

1.2.2 The Impulse Function and the Riemann Integral 5

1.2.3 The General Definition of Singularity Functions 5

1.2.3.1 "Graphs" of Some Singularity Functions 5

1.3 The Laplace Transform 7

1.3.1 Definition of the Laplace Transform 7

1.3.2 Laplace Transform Properties 8

1.3.3 Shifting the Laplace Transform 8

1.3.4 Laplace Transform Derivatives 10

1.3.5 Transforms of Singularity Functions 12

1.4 Inverse Laplace Transform 13

1.4.1 Inverse Laplace Transformation by Heaviside Expansion 13

1.4.1.1 Distinct Poles 13

1.4.1.2 Distinct Poles with G(s) Being Proper 13

1.4.1.3 Repeated Poles 14

1.5 The Transfer Function and the State Space Representations (State Equations) 16

1.5.1 The Transfer Function 16

1.5.2 The State Equations 16

1.5.3 Transfer Function Properties 17

1.5.4 Poles and Zeros of a Transfer Function 18

1.5.5 Physical Realizability 19

1.6 Problems 21

2 The Response of Linear, Time-Invariant Dynamic Systems 25

2.1 The Time Response of Dynamic Systems 25

2.1.1 Final Value Theorem 25

2.1.2 Initial Value Theorem 26

2.1.3 Convolution and the Laplace Transform 27

2.1.4 Transmission Blocking Response 29

2.1.5 Stability 31

2.1.6 Initial Values and Reverse Action 35

2.1.7 Final Values and Static Gain 36

2.1.8 Time Response Metrics 38

2.1.8.1 First-Order System (Single-Pole Response) 38

2.1.8.2 Second-Order System (Quadratic Factor) 39

2.1.9 The Effect of Zeros on Transient Response 41

2.1.10 The Butterworth Pattern 42

2.2 Frequency Response of Dynamic Systems 43

2.2.1 Steady-State Frequency Response of LTI systems 43

2.2.2 Frequency Response Representation 45

2.2.3 Frequency Response: The Real Pole 45

2.2.4 Frequency Response: The Real Zero 47

2.2.5 Frequency Response: The Quadratic Factor 49

2.2.6 Frequency Response: Pure Time Delay 50

2.2.7 Frequency Response: Static Gain 53

2.2.8 Frequency Response: The Composite Transfer Function 53

2.2.9 Frequency Response: Asymptote Formulas 54

2.2.10 Physical Realizability 54

2.2.11 Non-minimum Phase, All-Pass, and Blaschke Factors 55

2.3 Frequency Response Plotting 55

2.3.1 Matlab Codes for Plotting System Frequency Response 56

2.3.1.1 Bode Plot 56

2.3.1.2 Polar Plot/Nyquist Diagram 56

2.4 Problems 57

3 Feedback Principals 61

3.1 The Value of Feedback Control 62

3.1.1 The Advantages of the Closed Loop 63

3.2 Closed-Loop Transfer Functions 64

3.2.1 The Return Ratio 65

3.2.2 Closed-Loop Transfer Functions and the Return Difference 65

3.2.3 Sensitivity, Complementary Sensitivity, and the Youla Parameter 66

3.3 Well-Posedness and Internal Stability 70

3.3.1 Well-Posedness 70

3.3.2 The Internal Stability of Feedback Control 71

3.3.2.1 The Closed-Loop Characteristic Equation and Closed-Loop Poles 72

3.3.2.2 Closed-Loop Zeros 72

3.3.2.3 Pole-Zero Cancellation and The Internal Stability of Feedback Control 73

3.4 The Youla Parameterization of all Internally Stabilizing Compensators 76

3.5 Interpolation Conditions 80

3.6 Steady-State Error 83

3.7 Feedback Design, and Frequency Methods: Input Attenuation and Robustness 83

3.7.1 The Frequency Paradigm 84

3.7.2 Input Attenuation and Command Following 84

3.7.3 Bode Measures of Performance Robustness 85

3.7.4 Graphical Interpretation of Return, Sensitivity, and Complementary Sensitivity 88

3.7.5 Weighting Factors and Performance Robustness 89

3.8 The Saturation Constraints 90

3.8.1 Bandwidth and Response Time 90

3.8.2 The Youla Parameter and Saturation 91

3.9 Problems 93

4 Feedback Design For SISO: Shaping and Parameterization 95

4.1 Closed-Loop Stability Under Uncertain Conditions 95

4.1.1 Harmonic Consistency 95

4.1.2 Nyquist Stability Criterion: Heuristic Justification 96

4.1.3 Stability Margins and Stability Robustness 98

4.1.4 Margins, T(j omega) and S(j omega), and H infinity Norms (Relationships Between Classical and Neoclassical Approaches) 99

4.1.4.1 Neoclassical Approach 101

4.2 Mathematical Design Constraints 103

4.2.1 Sensitivity/Complementary Sensitivity Point-wise Constraints 103

4.2.2 Sensitivity, Complementary Sensitivity, and Analytic Constraints 104

4.2.2.1 Non-minimum Phase Constraints on Design 104

4.3 The Neoclassical Approach to Internal Stability 104

4.4 Feedback Design And Parameterization: Stable Objects 106

4.4.1 Renormalization of Gains 108

4.4.2 Shaping of the Closed-Loop: Stable SISO 108

4.4.3 Neoclassical Design Principles 109

4.5 Loop Shaping Using Youla Parameterization 110

4.5.1 LHP Zeros of Gp 111

4.5.2 Non-minimum Phase Zeros 112

4.5.3 LHP Poles of Gp 114

4.5.4 Unstable Poles 115

4.6 Design Guidelines 116

4.7 Design Examples 117

4.8 Problems 125

5 Norms of Feedback Systems 129

5.1 The Laplace and Fourier Transform 129

5.1.1 The Inverse Laplace Transform 129

5.1.2 Parseval's Theorem 131

5.1.3 The Fourier Transform 132

5.1.3.1 Properties of the Fourier Transform 133

5.1.3.2 Inverse Fourier Transformation By Heaviside Expansion 133

5.2 Norms of Signals and Systems 134

5.2.1 Signal Norms 134

5.2.1.1 Particular Norms 135

5.2.1.2 Properties of Norms 136

5.2.2 Norms of Dynamic Systems 137

5.2.3 Input-Output Norms 138

5.2.3.1 Transient Inputs (Energy Bounded) 138

5.2.3.2 Persistent Inputs (Energy Unbounded) 139

5.3 Quantifying Uncertainty 140

5.3.1 The Characterization of Uncertainty in Models 140

5.3.2 Weighting Factors and Stability Robustness 141

5.3.3 Robust Stability (Complementary Sensitivity) and Uncertainty 142

5.3.4 Sensitivity and Performance 145

5.3.5 Performance and Stability 146

5.4 Problems 147

6 Feedback Design By the Optimization of Closed-Loop Norms 149

6.1 Introduction 149

6.1.1 Frequency Domain Control Design Approaches 150

6.2 Optimization Design Objectives and Constraints 151

6.2.1 Algebraic Constraints 151

6.2.2 Analytic Constraints 152

6.2.2.1 Nonminimum Phase Effect 152

6.2.2.2 Bode Sensitivity Integral Theorem 153

6.3 The Linear Fractional Transformation 154

6.4 Setup for Loop-Shaping Optimization 156

6.4.1 Setup for Youla Parameter Loop Shaping 158

6.5 H infinity -norm Optimization Problem 160

6.5.1 Solution to a Simple Optimization Problem 161

6.6 H infinity Design 163

6.7 H infinity Solutions Using Matlab Robust Control Toolbox for SISO Systems 164

6.7.1 Defining Frequency Weights 164

6.8 Problems 168

7 Estimation Design for SISO Using Parameterization Approach 173

7.1 Introduction 173

7.2 Youla Controller Output Observer Concept 175

7.3 The SISO Case 177

7.3.1 Output and Feedthrough Matrices 178

7.3.2 SISO Estimator Design 178

7.4 Final Remarks 182

8 Practical Applications 183

8.1 Yaw Stability Control with Active Limited Slip Differential 183

8.1.1 Model and Control Design 183

8.1.2 Youla Control Design Using Hand Computation 187

8.1.3 H infinity Control Design Using Loop-shaping Technique 188

8.2 Vehicle Yaw Rate and Side-Slip Estimation 195

8.2.1 Kalman Filters 195

8.2.2 Vehicle Model - Nonlinear Bicycle Model with Pacejka Tire Model 196

8.2.3 Linearizing the Bicycle Model 197

8.2.4 Uncertainties 197

8.2.5 State Estimation 198

8.2.6 Youla Parameterization Estimator Design 198

8.2.7 Simulation Results 200

8.2.8 Robustness Test 201

8.2.8.1 Vehicle Mass Variation 201

8.2.8.2 Tire-road Coefficient of Friction 203

Part II Control Design Using Youla Parametrization: Multi Input Multi Output (MIMO) 205

9 Introduction to Multivariable Feedback Control 207

9.1 Nonoptimal, Optimal, and Robust Control 207

9.1.1 Nonoptimal Control Methods 208

9.1.2 Optimal Control Methods 208

9.1.3 Optimal Robust Control 209

9.2 Review of the SISO Transfer Function 210

9.2.1 Schur Complement 210

9.2.2 Interpretation of Poles and Zeros of a Transfer Function 211

9.2.2.1 Poles 211

9.2.2.2 Zeros 212

9.2.2.3 Transmission Blocking Zeros 213

9.3 Basic Aspects of Transfer Function Matrices 215

9.4 Problems 215

10 Matrix Fractional Description 217

10.1 Transfer Function Matrices 217

10.1.1 Matrix Fraction Description 218

10.2 Polynomial Matrix Properties 219

10.2.1 Minimum-Degree Factorization 220

10.3 Equivalency of Polynomial Matrices 221

10.4 Smith Canonical Form 222

10.5 Smith-McMillan Form 225

10.5.1 Smith-McMillan Form 225

10.5.2 MFD's and Their Relations to Smith-McMillan Form 228

10.5.3 Computing an Irreducible (Coprime) Matrix Fraction Description 229

10.6 MIMO Controllability and Observability 234

10.6.1 State-Space Realization 235

10.6.1.1 SISO System 235

10.6.1.2 MIMO System 236

10.6.2 Controllable Form of State-Space Realization of MIMO System 238

10.6.2.1 Mathematical Details 239

10.7 Straightforward Computational Procedures 243

10.8 Problems 245

11 Eigenvalues and Singular Values 247

11.1 Eigenvalues and Eigenvectors 247

11.2 Matrix Diagonalization 248

11.2.1 Classes of Diagonalizable Matrices 250

11.3 Singular Value Decomposition 253

11.3.1 What is a Singular Value Decomposition? 254

11.3.2 Orthonormal Vectors 255

11.4 Singular Value Decomposition Properties 257

11.5 Comparison of Eigenvalue and Singular Value Decompositions 258

11.5.1 System Gain 259

11.6 Generalized Singular Value Decomposition 262

11.6.1 The Scalar Case 264

11.6.2 Input and Output Spaces 264

11.7 Norms 265

11.7.1 The Spectral Norm 265

11.8 Problems 266

12 MIMO Feedback Principals 267

12.1 Mutlivariable Closed-Loop Transfer Functions 267

12.1.1 Transfer Function Matrix, From r to y 268

12.1.2 Transfer Function Matrix From dy to y As Shown in Figure 12.1 268

12.1.3 Transfer Function Matrix From r to e 269

12.1.4 Transfer Function From r to u 269

12.1.5 Realization Tricks 270

12.2 Well-Posedness of MIMO Systems 270

12.3 State Variable Compositions 271

12.4 Nyquist Criterion for MIMO Systems 273

12.4.1 Characteristic Gains 273

12.4.2 Poles and Zeros 274

12.4.3 Internal Stability 275

12.5 MIMO Performance and Robustness Criteria 276

12.6 Open-Loop Singular Values 278

12.6.1 Crossover Frequency 279

12.6.2 Bandwidth Constraints 280

12.7 Condition Number and its Role in MIMO Control Design 281

12.7.1 Condition Numbers and Decoupling 281

12.7.2 Role of Tu and Su in MIMO Feedback Design 282

12.8 Summary of Requirements 282

12.8.1 Closed-Loop Requirements 282

12.8.2 Open-Loop Requirements 283

12.9 Problems 283

13 Youla Parameterization for Feedback Systems 285

13.1 Neoclassical Control for MIMO Systems 285

13.1.1 Internal Model Control 285

13.2 MIMO Feedback Control Design for Stable Plants 286

13.2.1 Procedure to Find the MIMO Controller, Gc 287

13.2.2 Interpolation Conditions 287

13.3 MIMO Feedback Control Design Examples 287

13.3.1 Summary of Closed-Loop Requirements 290

13.3.2 Summary of Open-Loop Requirements 290

13.4 MIMO Feedback Control Design: Unstable Plants 294

13.4.1 The Proposed Control Design Method 294

13.4.2 Another Approach for MIMO Controller Design 300

13.5 Problems 301

14 Norms of Feedback Systems 303

14.1 Norms 303

14.1.1 Signal Norms, the Discrete Case 303

14.1.2 System Norms 304

14.1.3 The H2-Norm 305

14.1.4 The H infinity -Norm 306

14.2 Linear Fractional Transformations (LFT) 307

14.3 Linear Fractional Transformation Explained 309

14.3.1 LFTs in Control Design 310

14.4 Modeling Uncertainties 312

14.4.1 Uncertainties 312

14.4.2 Descriptions of Unstructured Uncertainty 312

14.5 General Robust Stability Theorem 313

14.5.1 SVD Properties Applied 314

14.5.2 Robust Performance 315

14.6 Problems 316

15 Optimal Control in MIMO Systems 319

15.1 Output Feedback Control 319

15.1.1 LQG Control 320

15.1.2 Kalman Filter 322

15.1.3 H2 Control 323

15.1.3.1 Kalman Filter 324

15.1.3.2 State Feedback 325

15.2 H infinity Control Design 325

15.2.1 State Feedback (Full Information) H infinity Control Design 327

15.2.2 H infinity Filtering 329

15.3 H infinity - Robust Optimal Control 330

15.4 Problems 332

16 Estimation Design for MIMO Using Parameterization Approach 335

16.1 YCOO Concept for MIMO 335

16.2 MIMO Estimator Design 337

16.3 State Estimation 338

16.3.1 First Decoupled System (Gsm1) 338

16.3.2 Second Decoupled System (Gsm2) 338

16.3.3 Coupled System 339

16.4 Applications 339

16.4.1 States Estimation: Four States 340

16.4.2 Input Estimation: Skyhook Based Control 341

16.4.3 Input Estimation: Road Roughness 342

16.5 Final Remarks 344

17 Practical Applications 345

17.1 Active Suspension 345

17.1.1 Model and Control Design 345

17.1.2 MIMO Youla Control Design 348

17.1.3 H infinity Control Design Technique 350

17.1.4 Uncertain Actuator Model 351

17.1.5 Design Setup 351

17.1.6 Simulation Results 354

17.1.7 Robustness Test: Actuator Model Variations 356

17.2 Advanced Engine Speed Control for Hybrid Vehicles 356

17.2.1 Diesel Hybrid Electric Vehicle Model 357

17.2.2 MISO Youla Control Design 359

17.2.3 First Youla Method 359

17.2.4 Second Youla Method 360

17.2.5 H infinity Control Design 360

17.2.6 Simulation Results 362

17.2.7 Robustness Test 363

17.3 Robust Control for the Powered Descent of a Multibody Lunar Landing System 364

17.3.1 Multibody Dynamics Model 365

17.3.2 Trajectory Optimization 366

17.3.3 MIMO Youla Control Design 367

17.3.4 Youla Method for Under-Actuated Systems 371

17.4 Vehicle Yaw Rate and Sideslip Estimation 374

17.4.1 Background 375

17.4.2 Vehicle Modeling 376

17.4.2.1 Nonlinear Bicycle Model With Pacejka Tire Model 376

17.4.2.2 Kinematic Relationship 376

17.4.2.3 Multi-Input Model 377

17.4.2.4 Linearizing the Bicycle Model for SISO and MIMO Cases 378

17.4.3 State Estimation 378

17.4.3.1 Youla Parameterization Control Design 378

17.4.4 Simulation and Estimation Result 379

17.4.5 Robustness Test 382

17.4.5.1 Vehicle mass variation 382

17.4.5.2 Tire-road coefficient of friction 382

17.4.6 Sensor Bias 382

17.4.7 Final Remarks 386

A Cauchy Integral 387

A.1 Contour Definitions 387

A.2 Contour Integrals 388

A.3 Complex Analysis Definitions 389

A.4 Cauchy-Riemann Conditions 390

A.5 Cauchy Integral Theorem 392

A.5.1 Terminology 394

A.6 Maximum Modulus Theorem 394

A.7 Poisson Integral Formula 396

A.8 Cauchy's Argument Principle 398

A.9 Nyquist Stability Criterion 400

B Singular Value Properties 403

B.1 Spectral Norm Proof 403

B.2 Proof of Bounded Eigenvalues 404

B.3 Proof of Matrix Inequality 404

B.3.1 Upper Bound 405

B.3.2 Lower Bound 405

B.3.3 Combined Inequality 406

B.4 Triangle Inequality 406

B.4.1 Upper Bound 406

B.4.2 Lower Bound 406

B.4.3 Combined Inequality 406

C Bandwidth 407

C.1 Introduction 407

C.2 Information as a Precise Measure of Bandwidth 408

C.2.1 Neoclassical Feedback Control 408

C.2.2 Defining a Measure to Characterize the Usefulness of Feedback 408

C.2.3 Computation of New Bandwidth 409

C.3 Examples 410

C.4 Summary 414

D Example Matlab Code 417

D.1 Example 1 417

D.2 Example 2 419

D.3 Example 3 420

D.4 Example 4 422

References 425

Index 427
Farhad Assadian, PhD, is Professor of Dynamic Systems and Control in the Department of Mechanical and Aerospace Engineering at the University of California, Davis. He teaches courses on dynamics, modelling and simulation, and control theory.

Kevin R. Mallon is a PhD student in the Department of Mechanical and Aerospace Engineering at the University of California, Davis. He previously worked as a robotics engineer at Intelligrated Systems.