Robust Control: Youla Parameterization Approach

Cover
Title Page
Copyright
Contents
Preface
Acknowledgments
Introduction
About the Companion Website
Part I Control Design Using Youla Parameterization: Single Input Single Output (SISO)
Chapter 1 Review of the Laplace Transform
1.1 The Laplace Transform Concept
1.2 Singularity Functions
1.2.1 Definition of the Impulse Function
1.2.2 The Impulse Function and the Riemann Integral
1.2.3 The General Definition of Singularity Functions
1.2.3.1 “Graphs” of Some Singularity Functions
1.3 The Laplace Transform
1.3.1 Definition of the Laplace Transform
1.3.2 Laplace Transform Properties
1.3.3 Shifting the Laplace Transform
1.3.4 Laplace Transform Derivatives
1.3.5 Transforms of Singularity Functions
1.4 Inverse Laplace Transform
1.4.1 Inverse Laplace Transformation by Heaviside Expansion
1.4.1.1 Distinct Poles
1.4.1.2 Distinct Poles with G(s) Being Proper
1.4.1.3 Repeated Poles
1.5 The Transfer Function and the State Space Representations (State Equations)
1.5.1 The Transfer Function
1.5.2 The State Equations
1.5.3 Transfer Function Properties
1.5.4 Poles and Zeros of a Transfer Function
1.5.5 Physical Realizability
1.6 Problems
Chapter 2 The Response of Linear, Time‐Invariant Dynamic Systems
2.1 The Time Response of Dynamic Systems
2.1.1 Final Value Theorem
2.1.2 Initial Value Theorem
2.1.3 Convolution and the Laplace Transform
2.1.4 Transmission Blocking Response
2.1.5 Stability
2.1.6 Initial Values and Reverse Action
2.1.7 Final Values and Static Gain
2.1.8 Time Response Metrics
2.1.8.1 First‐Order System (Single‐Pole Response)
2.1.8.2 Second‐Order System (Quadratic Factor)
2.1.9 The Effect of Zeros on Transient Response
2.1.10 The Butterworth Pattern
2.2 Frequency Response of Dynamic Systems
2.2.1 Steady‐State Frequency Response of LTI systems
2.2.2 Frequency Response Representation
2.2.3 Frequency Response: The Real Pole
2.2.4 Frequency Response: The Real Zero
2.2.5 Frequency Response: The Quadratic Factor
2.2.6 Frequency Response: Pure Time Delay
2.2.7 Frequency Response: Static Gain
2.2.8 Frequency Response: The Composite Transfer Function
2.2.9 Frequency Response: Asymptote Formulas
2.2.10 Physical Realizability
2.2.11 Non‐minimum Phase, All‐Pass, and Blaschke Factors
2.3 Frequency Response Plotting
2.3.1 Matlab Codes for Plotting System Frequency Response
2.3.1.1 Bode Plot
2.3.1.2 Polar Plot/Nyquist Diagram
2.4 Problems
Chapter 3 Feedback Principals
3.1 The Value of Feedback Control
3.1.1 The Advantages of the Closed Loop
3.2 Closed‐Loop Transfer Functions
3.2.1 The Return Ratio
3.2.2 Closed‐Loop Transfer Functions and the Return Difference
3.2.3 Sensitivity, Complementary Sensitivity, and the Youla Parameter
3.3 Well‐Posedness and Internal Stability
3.3.1 Well‐Posedness
3.3.2 The Internal Stability of Feedback Control
3.3.2.1 The Closed‐Loop Characteristic Equation and Closed‐Loop Poles
3.3.2.2 Closed‐Loop Zeros
3.3.2.3 Pole–Zero Cancellation and The Internal Stability of Feedback Control
3.4 The Youla Parameterization of all Internally Stabilizing Compensators
3.5 Interpolation Conditions
3.6 Steady‐State Error
3.7 Feedback Design, and Frequency Methods: Input Attenuation and Robustness
3.7.1 The Frequency Paradigm
3.7.2 Input Attenuation and Command Following
3.7.3 Bode Measures of Performance Robustness
3.7.4 Graphical Interpretation of Return, Sensitivity, and Complementary Sensitivity
3.7.5 Weighting Factors and Performance Robustness
3.8 The Saturation Constraints
3.8.1 Bandwidth and Response Time
3.8.2 The Youla Parameter and Saturation
3.9 Problems
Chapter 4 Feedback Design For SISO: Shaping and Parameterization
4.1 Closed‐Loop Stability Under Uncertain Conditions
4.1.1 Harmonic Consistency
4.1.2 Nyquist Stability Criterion: Heuristic Justification
4.1.3 Stability Margins and Stability Robustness
4.1.4 Margins, T(jω) and S(jω), and H∞ Norms (Relationships Between Classical and Neoclassical Approaches)
4.1.4.1 Neoclassical Approach
4.2 Mathematical Design Constraints
4.2.1 Sensitivity/Complementary Sensitivity Point‐wise Constraints
4.2.2 Sensitivity, Complementary Sensitivity, and Analytic Constraints
4.2.2.1 Non‐minimum Phase Constraints on Design
4.3 The Neoclassical Approach to Internal Stability
4.4 Feedback Design And Parameterization: Stable Objects
4.4.1 Renormalization of Gains
4.4.2 Shaping of the Closed‐Loop: Stable SISO
4.4.3 Neoclassical Design Principles
4.5 Loop Shaping Using Youla Parameterization
4.5.1 LHP Zeros of Gp
4.5.2 Non‐minimum Phase Zeros
4.5.3 LHP Poles of Gp
4.5.4 Unstable Poles
4.6 Design Guidelines
4.7 Design Examples
4.8 Problems
Chapter 5 Norms of Feedback Systems
5.1 The Laplace and Fourier Transform
5.1.1 The Inverse Laplace Transform
5.1.2 Parseval's Theorem
5.1.3 The Fourier Transform
5.1.3.1 Properties of the Fourier Transform
5.1.3.2 Inverse Fourier Transformation By Heaviside Expansion
5.2 Norms of Signals and Systems
5.2.1 Signal Norms
5.2.1.1 Particular Norms
5.2.1.2 Properties of Norms
5.2.2 Norms of Dynamic Systems
5.2.3 Input–Output Norms
5.2.3.1 Transient Inputs (Energy Bounded)
5.2.3.2 Persistent Inputs (Energy Unbounded)
5.3 Quantifying Uncertainty
5.3.1 The Characterization of Uncertainty in Models
5.3.2 Weighting Factors and Stability Robustness
5.3.3 Robust Stability (Complementary Sensitivity) and Uncertainty
5.3.4 Sensitivity and Performance
5.3.5 Performance and Stability
5.4 Problems
Chapter 6 Feedback Design By the Optimization of Closed‐Loop Norms
6.1 Introduction
6.1.1 Frequency Domain Control Design Approaches
6.2 Optimization Design Objectives and Constraints
6.2.1 Algebraic Constraints
6.2.2 Analytic Constraints
6.2.2.1 Nonminimum Phase Effect
6.2.2.2 Bode Sensitivity Integral Theorem
6.3 The Linear Fractional Transformation
6.4 Setup for Loop‐Shaping Optimization
6.4.1 Setup for Youla Parameter Loop Shaping
6.5 H∞‐norm Optimization Problem
6.5.1 Solution to a Simple Optimization Problem
6.6 H∞ Design
6.7 H∞ Solutions Using Matlab Robust Control Toolbox for SISO Systems
6.7.1 Defining Frequency Weights
6.8 Problems
Chapter 7 Estimation Design for SISO Using Parameterization Approach
7.1 Introduction
7.2 Youla Controller Output Observer Concept
7.3 The SISO Case
7.3.1 Output and Feedthrough Matrices
7.3.2 SISO Estimator Design
7.4 Final Remarks
Chapter 8 Practical Applications
8.1 Yaw Stability Control with Active Limited Slip Differential
8.1.1 Model and Control Design
8.1.2 Youla Control Design Using Hand Computation
8.1.3 H∞ Control Design Using Loop‐shaping Technique
8.2 Vehicle Yaw Rate and Side‐Slip Estimation
8.2.1 Kalman Filters
8.2.2 Vehicle Model – Nonlinear Bicycle Model with Pacejka Tire Model
8.2.3 Linearizing the Bicycle Model
8.2.4 Uncertainties
8.2.5 State Estimation
8.2.6 Youla Parameterization Estimator Design
8.2.7 Simulation Results
8.2.8 Robustness Test
8.2.8.1 Vehicle Mass Variation
8.2.8.2 Tire–road Coefficient of Friction
Part II Control Design Using Youla Parametrization: Multi Input Multi Output (MIMO)
Chapter 9 Introduction to Multivariable Feedback Control
9.1 Nonoptimal, Optimal, and Robust Control
9.1.1 Nonoptimal Control Methods
9.1.2 Optimal Control Methods
9.1.3 Optimal Robust Control
9.2 Review of the SISO Transfer Function
9.2.1 Schur Complement
9.2.2 Interpretation of Poles and Zeros of a Transfer Function
9.2.2.1 Poles
9.2.2.2 Zeros
9.2.2.3 Transmission Blocking Zeros
9.3 Basic Aspects of Transfer Function Matrices
9.4 Problems
Chapter 10 Matrix Fractional Description
10.1 Transfer Function Matrices
10.1.1 Matrix Fraction Description
10.2 Polynomial Matrix Properties
10.2.1 Minimum‐Degree Factorization
10.3 Equivalency of Polynomial Matrices
10.4 Smith Canonical Form
10.5 Smith–McMillan Form
10.5.1 Smith–McMillan Form
10.5.2 MFD's and Their Relations to Smith–McMillan Form
10.5.3 Computing an Irreducible (Coprime) Matrix Fraction Description
10.6 MIMO Controllability and Observability
10.6.1 State‐Space Realization
10.6.1.1 SISO System
10.6.1.2 MIMO System
10.6.2 Controllable Form of State‐Space Realization of MIMO System
10.6.2.1 Mathematical Details
10.7 Straightforward Computational Procedures
10.8 Problems
Chapter 11 Eigenvalues and Singular Values
11.1 Eigenvalues and Eigenvectors
11.2 Matrix Diagonalization
11.2.1 Classes of Diagonalizable Matrices
11.3 Singular Value Decomposition
11.3.1 What is a Singular Value Decomposition?
11.3.2 Orthonormal Vectors
11.4 Singular Value Decomposition Properties
11.5 Comparison of Eigenvalue and Singular Value Decompositions
11.5.1 System Gain
11.6 Generalized Singular Value Decomposition
11.6.1 The Scalar Case
11.6.2 Input and Output Spaces
11.7 Norms
11.7.1 The Spectral Norm
11.8 Problems
Chapter 12 MIMO Feedback Principals
12.1 Mutlivariable Closed‐Loop Transfer Functions
12.1.1 Transfer Function Matrix, From r_ to y_
12.1.2 Transfer Function Matrix From d_y to y_ As Shown in Figure
12.1.3 Transfer Function Matrix From r_ to e_
12.1.4 Transfer Function From r_ to u_
12.1.5 Realization Tricks
12.2 Well‐Posedness of MIMO Systems
12.3 State Variable Compositions
12.4 Nyquist Criterion for MIMO Systems
12.4.1 Characteristic Gains
12.4.2 Poles and Zeros
12.4.3 Internal Stability
12.5 MIMO Performance and Robustness Criteria
12.6 Open‐Loop Singular Values
12.6.1 Crossover Frequency
12.6.2 Bandwidth Constraints
12.7 Condition Number and its Role in MIMO Control Design
12.7.1 Condition Numbers and Decoupling
12.7.2 Role of Tu and Su in MIMO Feedback Design
12.8 Summary of Requirements
12.8.1 Closed‐Loop Requirements
12.8.2 Open‐Loop Requirements
12.9 Problems
Chapter 13 Youla Parameterization for Feedback Systems
13.1 Neoclassical Control for MIMO Systems
13.1.1 Internal Model Control
13.2 MIMO Feedback Control Design for Stable Plants
13.2.1 Procedure to Find the MIMO Controller, Gc
13.2.2 Interpolation Conditions
13.3 MIMO Feedback Control Design Examples
13.3.1 Summary of Closed‐Loop Requirements
13.3.2 Summary of Open‐Loop Requirements
13.4 MIMO Feedback Control Design: Unstable Plants
13.4.1 The Proposed Control Design Method
13.4.2 Another Approach for MIMO Controller Design
13.5 Problems
Chapter 14 Norms of Feedback Systems
14.1 Norms
14.1.1 Signal Norms, the Discrete Case
14.1.2 System Norms
14.1.3 The H2‐Norm
14.1.4 The H∞‐Norm
14.2 Linear Fractional Transformations (LFT)
14.3 Linear Fractional Transformation Explained
14.3.1 LFTs in Control Design
14.4 Modeling Uncertainties
14.4.1 Uncertainties
14.4.2 Descriptions of Unstructured Uncertainty
14.5 General Robust Stability Theorem
14.5.1 SVD Properties Applied
14.5.2 Robust Performance
14.6 Problems
Chapter 15 Optimal Control in MIMO Systems
15.1 Output Feedback Control
15.1.1 LQG Control
15.1.2 Kalman Filter
15.1.3 H2 Control
15.1.3.1 Kalman Filter Dynamic Model
15.1.3.2 State Feedback
15.2 H∞ Control Design
15.2.1 State Feedback (Full Information) H∞ Control Design
15.2.2 H∞ Filtering
15.3 H∞‐ Robust Optimal Control
15.4 Problems
Chapter 16 Estimation Design for MIMO Using Parameterization Approach
16.1 YCOO Concept for MIMO
16.2 MIMO Estimator Design
16.3 State Estimation
16.3.1 First Decoupled System (Gsm1)
16.3.2 Second Decoupled System (Gsm2)
16.3.3 Coupled System
16.4 Applications
16.4.1 States Estimation: Four States
16.4.2 Input Estimation: Skyhook Based Control
16.4.3 Input Estimation: Road Roughness
16.5 Final Remarks
Chapter 17 Practical Applications
17.1 Active Suspension
17.1.1 Model and Control Design
17.1.2 MIMO Youla Control Design
17.1.3 H∞ Control Design Technique
17.1.4 Uncertain Actuator Model
17.1.5 Design Setup
17.1.6 Simulation Results
17.1.7 Robustness Test: Actuator Model Variations
17.2 Advanced Engine Speed Control for Hybrid Vehicles
17.2.1 Diesel Hybrid Electric Vehicle Model
17.2.2 MISO Youla Control Design
17.2.3 First Youla Method
17.2.4 Second Youla Method
17.2.5 H∞ Control Design
17.2.6 Simulation Results
17.2.7 Robustness Test
17.3 Robust Control for the Powered Descent of a Multibody Lunar Landing System
17.3.1 Multibody Dynamics Model
17.3.2 Trajectory Optimization
17.3.3 MIMO Youla Control Design
17.3.4 Youla Method for Under‐Actuated Systems
17.4 Vehicle Yaw Rate and Sideslip Estimation
17.4.1 Background
17.4.2 Vehicle Modeling
17.4.2.1 Nonlinear Bicycle Model With Pacejka Tire Model
17.4.2.2 Kinematic Relationship
17.4.2.3 Multi‐Input Model
17.4.2.4 Linearizing the Bicycle Model for SISO and MIMO Cases
17.4.3 State Estimation
17.4.3.1 Youla Parameterization Control Design
17.4.4 Simulation and Estimation Result
17.4.5 Robustness Test
17.4.5.1 Vehicle mass variation
17.4.5.2 Tire–road coefficient of friction
17.4.6 Sensor Bias
17.4.7 Final Remarks
A Cauchy Integral
A.1 Contour Definitions
A.2 Contour Integrals
A.3 Complex Analysis Definitions
A.4 Cauchy–Riemann Conditions
A.5 Cauchy Integral Theorem
A.5.1 Terminology
A.6 Maximum Modulus Theorem
A.7 Poisson Integral Formula
A.8 Cauchy's Argument Principle
A.9 Nyquist Stability Criterion
B Singular Value Properties
B.1 Spectral Norm Proof
B.2 Proof of Bounded Eigenvalues
B.3 Proof of Matrix Inequality
B.3.1 Upper Bound
B.3.2 Lower Bound
B.3.3 Combined Inequality
B.4 Triangle Inequality
B.4.1 Upper Bound
B.4.2 Lower Bound
B.4.3 Combined Inequality
C Bandwidth
C.1 Introduction
C.2 Information as a Precise Measure of Bandwidth
C.2.1 Neoclassical Feedback Control
C.2.2 Defining a Measure to Characterize the Usefulness of Feedback
C.2.3 Computation of New Bandwidth
C.3 Examples
C.4 Summary
D Example Matlab Code
D.1 Example 1
D.2 Example 2
D.3 Example 3
D.4 Example 4
References
Index
EULA