Robust Control

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.

• Preface xvAcknowledgments xixIntroduction xxiAbout the Companion Website xxixPart I Control Design Using Youla Parameterization: Single Input Single Output (SISO) 11 Review of the Laplace Transform 31.1 The Laplace Transform Concept 31.2 Singularity Functions 31.2.1 Definition of the Impulse Function 41.2.2 The Impulse Function and the Riemann Integral 51.2.3 The General Definition of Singularity Functions 51.2.3.1 “Graphs” of Some Singularity Functions 51.3 The Laplace Transform 71.3.1 Definition of the Laplace Transform 71.3.2 Laplace Transform Properties 81.3.3 Shifting the Laplace Transform 81.3.4 Laplace Transform Derivatives 101.3.5 Transforms of Singularity Functions 121.4 Inverse Laplace Transform 131.4.1 Inverse Laplace Transformation by Heaviside Expansion 131.4.1.1 Distinct Poles 131.4.1.2 Distinct Poles with G(s) Being Proper 131.4.1.3 Repeated Poles 141.5 The Transfer Function and the State Space Representations (State Equations) 161.5.1 The Transfer Function 161.5.2 The State Equations 161.5.3 Transfer Function Properties 171.5.4 Poles and Zeros of a Transfer Function 181.5.5 Physical Realizability 191.6 Problems 212 The Response of Linear, Time-Invariant Dynamic Systems 252.1 The Time Response of Dynamic Systems 252.1.1 Final Value Theorem 252.1.2 Initial Value Theorem 262.1.3 Convolution and the Laplace Transform 272.1.4 Transmission Blocking Response 292.1.5 Stability 312.1.6 Initial Values and Reverse Action 352.1.7 Final Values and Static Gain 362.1.8 Time Response Metrics 382.1.8.1 First-Order System (Single-Pole Response) 382.1.8.2 Second-Order System (Quadratic Factor) 392.1.9 The Effect of Zeros on Transient Response 412.1.10 The Butterworth Pattern 422.2 Frequency Response of Dynamic Systems 432.2.1 Steady-State Frequency Response of LTI systems 432.2.2 Frequency Response Representation 452.2.3 Frequency Response: The Real Pole 452.2.4 Frequency Response: The Real Zero 472.2.5 Frequency Response: The Quadratic Factor 492.2.6 Frequency Response: Pure Time Delay 502.2.7 Frequency Response: Static Gain 532.2.8 Frequency Response: The Composite Transfer Function 532.2.9 Frequency Response: Asymptote Formulas 542.2.10 Physical Realizability 542.2.11 Non-minimum Phase, All-Pass, and Blaschke Factors 552.3 Frequency Response Plotting 552.3.1 Matlab Codes for Plotting System Frequency Response 562.3.1.1 Bode Plot 562.3.1.2 Polar Plot/Nyquist Diagram 562.4 Problems 573 Feedback Principals 613.1 The Value of Feedback Control 623.1.1 The Advantages of the Closed Loop 633.2 Closed-Loop Transfer Functions 643.2.1 The Return Ratio 653.2.2 Closed-Loop Transfer Functions and the Return Difference 653.2.3 Sensitivity, Complementary Sensitivity, and the Youla Parameter 663.3 Well-Posedness and Internal Stability 703.3.1 Well-Posedness 703.3.2 The Internal Stability of Feedback Control 713.3.2.1 The Closed-Loop Characteristic Equation and Closed-Loop Poles 723.3.2.2 Closed-Loop Zeros 723.3.2.3 Pole–Zero Cancellation and The Internal Stability of Feedback Control 733.4 The Youla Parameterization of all Internally Stabilizing Compensators 763.5 Interpolation Conditions 803.6 Steady-State Error 833.7 Feedback Design, and Frequency Methods: Input Attenuation and Robustness 833.7.1 The Frequency Paradigm 843.7.2 Input Attenuation and Command Following 843.7.3 Bode Measures of Performance Robustness 853.7.4 Graphical Interpretation of Return, Sensitivity, and Complementary Sensitivity 883.7.5 Weighting Factors and Performance Robustness 893.8 The Saturation Constraints 903.8.1 Bandwidth and Response Time 903.8.2 The Youla Parameter and Saturation 913.9 Problems 934 Feedback Design For SISO: Shaping and Parameterization 954.1 Closed-Loop Stability Under Uncertain Conditions 954.1.1 Harmonic Consistency 954.1.2 Nyquist Stability Criterion: Heuristic Justification 964.1.3 Stability Margins and Stability Robustness 984.1.4 Margins, T(j𝜔) and S(j𝜔), and H∞ Norms (Relationships Between Classical and NeoclassicalApproaches) 994.1.4.1 Neoclassical Approach 1014.2 Mathematical Design Constraints 1034.2.1 Sensitivity/Complementary Sensitivity Point-wise Constraints 1034.2.2 Sensitivity, Complementary Sensitivity, and Analytic Constraints 1044.2.2.1 Non-minimum Phase Constraints on Design 1044.3 The Neoclassical Approach to Internal Stability 1044.4 Feedback Design And Parameterization: Stable Objects 1064.4.1 Renormalization of Gains 1084.4.2 Shaping of the Closed-Loop: Stable SISO 1084.4.3 Neoclassical Design Principles 1094.5 Loop Shaping Using Youla Parameterization 1104.5.1 LHP Zeros of Gp 1114.5.2 Non-minimum Phase Zeros 1124.5.3 LHP Poles of Gp 1144.5.4 Unstable Poles 1154.6 Design Guidelines 1164.7 Design Examples 1174.8 Problems 1255 Norms of Feedback Systems 1295.1 The Laplace and Fourier Transform 1295.1.1 The Inverse Laplace Transform 1295.1.2 Parseval’s Theorem 1315.1.3 The Fourier Transform 1325.1.3.1 Properties of the Fourier Transform 1335.1.3.2 Inverse Fourier Transformation By Heaviside Expansion 1335.2 Norms of Signals and Systems 1345.2.1 Signal Norms 1345.2.1.1 Particular Norms 1355.2.1.2 Properties of Norms 1365.2.2 Norms of Dynamic Systems 1375.2.3 Input–Output Norms 1385.2.3.1 Transient Inputs (Energy Bounded) 1385.2.3.2 Persistent Inputs (Energy Unbounded) 1395.3 Quantifying Uncertainty 1405.3.1 The Characterization of Uncertainty in Models 1405.3.2 Weighting Factors and Stability Robustness 1415.3.3 Robust Stability (Complementary Sensitivity) and Uncertainty 1425.3.4 Sensitivity and Performance 1455.3.5 Performance and Stability 1465.4 Problems 1476 Feedback Design By the Optimization of Closed-Loop Norms 1496.1 Introduction 1496.1.1 Frequency Domain Control Design Approaches 1506.2 Optimization Design Objectives and Constraints 1516.2.1 Algebraic Constraints 1516.2.2 Analytic Constraints 1526.2.2.1 Nonminimum Phase Effect 1526.2.2.2 Bode Sensitivity Integral Theorem 1536.3 The Linear Fractional Transformation 1546.4 Setup for Loop-Shaping Optimization 1566.4.1 Setup for Youla Parameter Loop Shaping 1586.5 H∞-norm Optimization Problem 1606.5.1 Solution to a Simple Optimization Problem 1616.6 H∞ Design 1636.7 H∞ Solutions Using Matlab Robust Control Toolbox for SISO Systems 1646.7.1 Defining Frequency Weights 1646.8 Problems 1687 Estimation Design for SISO Using Parameterization Approach 1737.1 Introduction 1737.2 Youla Controller Output Observer Concept 1757.3 The SISO Case 1777.3.1 Output and Feedthrough Matrices 1787.3.2 SISO Estimator Design 1787.4 Final Remarks 1828 Practical Applications 1838.1 Yaw Stability Control with Active Limited Slip Differential 1838.1.1 Model and Control Design 1838.1.2 Youla Control Design Using Hand Computation 1878.1.3 H∞ Control Design Using Loop-shaping Technique 1888.2 Vehicle Yaw Rate and Side-Slip Estimation 1958.2.1 Kalman Filters 1958.2.2 Vehicle Model – Nonlinear Bicycle Model with Pacejka Tire Model 1968.2.3 Linearizing the Bicycle Model 1978.2.4 Uncertainties 1978.2.5 State Estimation 1988.2.6 Youla Parameterization Estimator Design 1988.2.7 Simulation Results 2008.2.8 Robustness Test 2018.2.8.1 Vehicle Mass Variation 2018.2.8.2 Tire–road Coefficient of Friction 203Part II Control Design Using Youla Parametrization: Multi Input Multi Output (MIMO) 2059 Introduction to Multivariable Feedback Control 2079.1 Nonoptimal, Optimal, and Robust Control 2079.1.1 Nonoptimal Control Methods 2089.1.2 Optimal Control Methods 2089.1.3 Optimal Robust Control 2099.2 Review of the SISO Transfer Function 2109.2.1 Schur Complement 2109.2.2 Interpretation of Poles and Zeros of a Transfer Function 2119.2.2.1 Poles 2119.2.2.2 Zeros 2129.2.2.3 Transmission Blocking Zeros 2139.3 Basic Aspects of Transfer Function Matrices 2159.4 Problems 21510 Matrix Fractional Description 21710.1 Transfer Function Matrices 21710.1.1 Matrix Fraction Description 21810.2 Polynomial Matrix Properties 21910.2.1 Minimum-Degree Factorization 22010.3 Equivalency of Polynomial Matrices 22110.4 Smith Canonical Form 22210.5 Smith–McMillan Form 22510.5.1 Smith–McMillan Form 22510.5.2 MFD’s and Their Relations to Smith–McMillan Form 22810.5.3 Computing an Irreducible (Coprime) Matrix Fraction Description 22910.6 MIMO Controllability and Observability 23410.6.1 State-Space Realization 23510.6.1.1 SISO System 23510.6.1.2 MIMO System 23610.6.2 Controllable Form of State-Space Realization of MIMO System 23810.6.2.1 Mathematical Details 23910.7 Straightforward Computational Procedures 24310.8 Problems 24511 Eigenvalues and Singular Values 24711.1 Eigenvalues and Eigenvectors 24711.2 Matrix Diagonalization 24811.2.1 Classes of Diagonalizable Matrices 25011.3 Singular Value Decomposition 25311.3.1 What is a Singular Value Decomposition? 25411.3.2 Orthonormal Vectors 25511.4 Singular Value Decomposition Properties 25711.5 Comparison of Eigenvalue and Singular Value Decompositions 25811.5.1 System Gain 25911.6 Generalized Singular Value Decomposition 26211.6.1 The Scalar Case 26411.6.2 Input and Output Spaces 26411.7 Norms 26511.7.1 The Spectral Norm 26511.8 Problems 26612 MIMO Feedback Principals 26712.1 Mutlivariable Closed-Loop Transfer Functions 26712.1.1 Transfer Function Matrix, From r to y 26812.1.2 Transfer Function Matrix From dy to y As Shown in Figure 12.1 26812.1.3 Transfer Function Matrix From r to e 26912.1.4 Transfer Function From r to u 26912.1.5 Realization Tricks 27012.2 Well-Posedness of MIMO Systems 27012.3 State Variable Compositions 27112.4 Nyquist Criterion for MIMO Systems 27312.4.1 Characteristic Gains 27312.4.2 Poles and Zeros 27412.4.3 Internal Stability 27512.5 MIMO Performance and Robustness Criteria 27612.6 Open-Loop Singular Values 27812.6.1 Crossover Frequency 27912.6.2 Bandwidth Constraints 28012.7 Condition Number and its Role in MIMO Control Design 28112.7.1 Condition Numbers and Decoupling 28112.7.2 Role of Tu and S u in MIMO Feedback Design 28212.8 Summary of Requirements 28212.8.1 Closed-Loop Requirements 28212.8.2 Open-Loop Requirements 28312.9 Problems 28313 Youla Parameterization for Feedback Systems 28513.1 Neoclassical Control for MIMO Systems 28513.1.1 Internal Model Control 28513.2 MIMO Feedback Control Design for Stable Plants 28613.2.1 Procedure to Find the MIMO Controller, G c 28713.2.2 Interpolation Conditions 28713.3 MIMO Feedback Control Design Examples 28713.3.1 Summary of Closed-Loop Requirements 29013.3.2 Summary of Open-Loop Requirements 29013.4 MIMO Feedback Control Design: Unstable Plants 29413.4.1 The Proposed Control Design Method 29413.4.2 Another Approach for MIMO Controller Design 30013.5 Problems 30114 Norms of Feedback Systems 30314.1 Norms 30314.1.1 Signal Norms, the Discrete Case 30314.1.2 System Norms 30414.1.3 The ℋ 2-Norm 30514.1.4 The ℋ ∞-Norm 30614.2 Linear Fractional Transformations (LFT) 30714.3 Linear Fractional Transformation Explained 30914.3.1 LFTs in Control Design 31014.4 Modeling Uncertainties 31214.4.1 Uncertainties 31214.4.2 Descriptions of Unstructured Uncertainty 31214.5 General Robust Stability Theorem 31314.5.1 SVD Properties Applied 31414.5.2 Robust Performance 31514.6 Problems 31615 Optimal Control in MIMO Systems 31915.1 Output Feedback Control 31915.1.1 LQG Control 32015.1.2 Kalman Filter 32215.1.3 ℋ 2 Control 32315.1.3.1 Kalman Filter Dynamic Model 32415.1.3.2 State Feedback 32515.2 ℋ ∞ Control Design 32515.2.1 State Feedback (Full Information) ℋ ∞ Control Design 32715.2.2 ℋ ∞ Filtering 32915.3 ℋ ∞- Robust Optimal Control 33015.4 Problems 33216 Estimation Design for MIMO Using Parameterization Approach 33516.1 YCOO Concept for MIMO 33516.2 MIMO Estimator Design 33716.3 State Estimation 33816.3.1 First Decoupled System ( Gsm 1 ) 33816.3.2 Second Decoupled System ( Gsm 2 ) 33816.3.3 Coupled System 33916.4 Applications 33916.4.1 States Estimation: Four States 34016.4.2 Input Estimation: Skyhook Based Control 34116.4.3 Input Estimation: Road Roughness 34216.5 Final Remarks 34417 Practical Applications 34517.1 Active Suspension 34517.1.1 Model and Control Design 34517.1.2 MIMO Youla Control Design 34817.1.3 H ∞ Control Design Technique 35017.1.4 Uncertain Actuator Model 35117.1.5 Design Setup 35117.1.6 Simulation Results 35417.1.7 Robustness Test: Actuator Model Variations 35617.2 Advanced Engine Speed Control for Hybrid Vehicles 35617.2.1 Diesel Hybrid Electric Vehicle Model 35717.2.2 MISO Youla Control Design 35917.2.3 First Youla Method 35917.2.4 Second Youla Method 36017.2.5 H ∞ Control Design 36017.2.6 Simulation Results 36217.2.7 Robustness Test 36317.3 Robust Control for the Powered Descent of a Multibody Lunar Landing System 36417.3.1 Multibody Dynamics Model 36517.3.2 Trajectory Optimization 36617.3.3 MIMO Youla Control Design 36717.3.4 Youla Method for Under-Actuated Systems 37117.4 Vehicle Yaw Rate and Sideslip Estimation 37417.4.1 Background 37517.4.2 Vehicle Modeling 37617.4.2.1 Nonlinear Bicycle Model With Pacejka Tire Model 37617.4.2.2 Kinematic Relationship 37617.4.2.3 Multi-Input Model 37717.4.2.4 Linearizing the Bicycle Model for SISO and MIMO Cases 37817.4.3 State Estimation 37817.4.3.1 Youla Parameterization Control Design 37817.4.4 Simulation and Estimation Result 37917.4.5 Robustness Test 38217.4.5.1 Vehicle mass variation 38217.4.5.2 Tire–road coefficient of friction 38217.4.6 Sensor Bias 38217.4.7 Final Remarks 386A Cauchy Integral 387A.1 Contour Definitions 387A.2 Contour Integrals 388A.3 Complex Analysis Definitions 389A.4 Cauchy–Riemann Conditions 390A.5 Cauchy Integral Theorem 392A.5.1 Terminology 394A.6 Maximum Modulus Theorem 394A.7 Poisson Integral Formula 396A.8 Cauchy’s Argument Principle 398A.9 Nyquist Stability Criterion 400B Singular Value Properties 403B.1 Spectral Norm Proof 403B.2 Proof of Bounded Eigenvalues 404B.3 Proof of Matrix Inequality 404B.3.1 Upper Bound 405B.3.2 Lower Bound 405B.3.3 Combined Inequality 406B.4 Triangle Inequality 406B.4.1 Upper Bound 406B.4.2 Lower Bound 406B.4.3 Combined Inequality 406C Bandwidth 407C.1 Introduction 407C.2 Information as a Precise Measure of Bandwidth 408C.2.1 Neoclassical Feedback Control 408C.2.2 Defining a Measure to Characterize the Usefulness of Feedback 408C.2.3 Computation of New Bandwidth 409C.3 Examples 410C.4 Summary 414D Example Matlab Code 417D.1 Example 1 417D.2 Example 2 419D.3 Example 3 420D.4 Example 4 422References 425Index 427