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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:
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.
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.
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(j𝜔) and S(j𝜔), and H∞ 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∞-norm Optimization Problem 160
6.5.1 Solution to a Simple Optimization Problem 161
6.6 H∞ Design 163
6.7 H∞ 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∞ 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
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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∞-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∞ Control Design 325
15.2.1 State Feedback (Full Information) ℋ∞ Control Design 327
15.2.2 ℋ∞ Filtering 329
15.3 ℋ∞- 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∞ 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∞ 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
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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
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