Professor Michael Grimble, Director of the Industrial Control Centre and Past Chairman of the Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow, UK

• Preface xixAcknowledgements xxi1 Introduction to Optimal and Robust Control 11.1 Introduction 11.1.1 Optimality, Feedback and Robustness 21.1.2 High-integrity and Fault-tolerant Control Systems 31.1.3 Self-healing Control Systems 41.1.4 Fault Monitoring and Detection 51.1.5 Adaptive versus Robust Control 51.1.6 Artificial Intelligence, Neural Networks and Fuzzy Control 51.1.7 Discrete-time Systems 71.2 The H2 and H∞ Spaces and Norms 81.2.1 Graphical Interpretation of the H∞ Norm 91.2.2 Terms Used in H∞ Robust Control Systems Design 91.3 Introduction to H∞ Control Design 91.3.1 Properties of H∞ Robust Control Design 111.3.2 Comparison of H∞ and H2 /LQG Controllers 121.3.3 Relationships between Classical Design and H∞ Robust Control 131.3.4 H2 and H∞ Design and Relationship to PID Control 131.3.5 H∞ Polynomial Systems Synthesis Theory 131.4 State-space Modelling and Synthesis Theory 141.4.1 State-space Solution of Discrete-time H∞ Control Problem 141.4.2 H∞ Control Design Objectives 151.4.3 State-feedback Control Solution 151.4.4 State-feedback Control Problem: Cross-product Costing Case 181.4.5 State-space Solution of Discrete-time H∞ Filtering Problem 191.4.6 Bounded Real Lemma 211.4.7 Output Feedback H∞ Control Problem 241.5 Introduction to H2 or LQG Polynomial Synthesis 291.5.1 System Description 291.5.2 Cost Function and Solution 311.5.3 Minimisation of the Performance Criterion 311.5.4 Solution of the Diophantine Equations and Stability 341.5.5 H2 /LQG Design Examples 351.6 Benchmarking 401.6.1 Restricted Structure Benchmarking 411.6.2 Rules for Benchmark Cost Function Selection 421.7 Condition Monitoring 441.8 Combining H2 , H∞  and ‘ 1 Optimal Control Designs 451.9 Linear Matrix Inequalities 461.10 Concluding Remarks 471.11 Problems 481.12 References 512 Scalar H2 and LQG Optimal Control 572.1 Introduction 572.1.1 Industrial Controller Structures 582.1.2 The 2½-DOF Structure 592.1.3 Restricted Structure Control Laws 602.2 Stochastic System Description 602.2.1 Ideal Response Models 622.2.2 System Equations 622.2.3 Cost Function Weighting Terms 632.3 Dual-criterion Cost-minimisation Problem 642.3.1 Solution of the Dual-criterion Minimisation Problem 662.3.2 Theorem Summarising LQG Controller 712.3.3 Remarks on the Equations and Solution 732.3.4 Design Guidelines 762.3.5 Controller Implementation 772.3.6 LQG Ship-steering Autopilot Application 782.4 LQG Controller with Robust Weighting Function 822.4.1 Youla Parameterisation 822.4.2 Cost Function with Robust Weighting Function 832.4.3 Solution of the Dual-criterion Problem with Robust Weighting 842.4.4 Summary of H2 /LQG Synthesis Problem with Robust Weighting 862.4.5 Comments on the Solution 882.5 Introduction to the Standard System Model 892.5.1 Standard System Model 892.6 The Standard System Model Structure 912.6.1 Polynomial System Models 922.6.2 Reference Model 932.6.3 Cost Function Signals to be Weighted 942.7 Generalised H2 Optimal Control: Standard System Model 952.7.1 Optimal Control Solution of the Standard System Model Problem 962.7.2 Summary of H2 /LQG Controller for Standard System Results 1022.7.3 Remarks on the Solution 1042.8 Concluding Remarks 1052.9 Problems 1052.10 References 1093 H∞ Optimal Control of Scalar Systems 1133.1 Introduction 1133.1.1 Links Between LQG and H∞ Solutions 1143.1.2 Reference and Feedback Controller Designs 1153.2 System Description 1153.3 Lemma Linking H∞ and LQG Control Problems 1153.4 Calculation of the H∞ Optimal Controller 1163.4.1 Simple H∞ Controller Structures and Calculations 1173.4.2 Zero Measurement Noise Case 1173.4.3 Solution for the H∞ Optimal Controller 1183.4.4 Stability Robustness of Mixed-sensitivity H∞ Designs 1213.4.5 One-block H∞ Control Problems 1223.5 The GH∞ Control Problem 1233.5.1 GH∞ Cost Function Definition 1243.5.2 Youla Parameterised Form of the GH∞ Controller 1263.5.3 Calculation of the GH∞ Controller 1283.6 Stability Robustness of GH∞ Designs 1363.6.1 Structure of the Uncertain System 1363.6.2 Rational Uncertainty Structure 1373.6.3 Stability Lemma 1393.6.4 Influence of the Uncertainty Model 1403.6.5 Design Procedure for Uncertain Systems 1403.6.6 H∞ Self-Tuning Controller for Systems with Parametric Uncertainty 1473.7 Standard System and Cost Function Description 1473.8 Calculation of H∞ Controller for the Standard System 1473.8.1 F-iteration Method of Solving the Robust Weighting Equation 1483.8.2 H2 / H∞ Trade-off 1493.9 Probabilistic System Descriptions and H∞ Control 1503.9.1 Uncertain System Model 1513.9.2 Cost Function Definition 1533.9.3 Uncertain System and Polynomial Equation Representation 1553.9.4 Discussion of Probabilistic Uncertainty Modelling and Control 1583.10 Concluding Remarks 1583.11 Problems 1593.12 References 1634 Multivariable H2 /LQG Optimal Control 1674.1 Introduction 1674.1.1 Matrix Fraction Descriptions 1684.2 Multivariable System Description 1684.2.1 Multivariable Sensitivity Matrices and Signal Spectra 1704.2.2 Choice of Noise and Cost Function Weightings 1714.3 LQG Optimal Control Problem and Solution 1714.3.1 Solution of the H2 /LQG Problem 1724.3.2 Solution of the Diophantine Equations 1754.4 Youla Parameterisation and Auxiliary Problem 1824.4.1 Youla Parameterisation for the Auxiliary Problem 1844.4.2 Summary of Multivariable Problem Results with Robust Weighting 1864.5 H 2 /LQG Optimal Control Problem: Measurement Noise Case 1874.5.1 Predictive Optimal Control 1904.5.2 SIMO Predictive Optimal Control Problem 1904.5.3 Probabilistic Description of Uncertainty 1964.6 The GLQG Optimal Control Problem 1964.6.1 Solution of the GLQG Problem 1974.6.2 Modified GLQG Cost Function and Youla Parameterisation 1994.7 Design of Automatic Voltage Regulators 2004.8 Pseudo-state Modelling and Separation Principle 2104.8.1 Introduction to Pseudo-state Methods 2104.8.2 Pseudo-state Discrete-time Plant Model 2114.8.3 Discrete Pseudo-state Feedback Optimal Control 2154.8.4 Solution of the Pseudo-state Feedback Control Problem 2174.8.5 Discrete Pseudo-state Estimation Problem 2224.8.6 Solution of the Discrete-time pseudo-state Estimation Problem 2244.8.7 Output Feedback Control Problem and Separation Principle 2304.8.8 Computational Example 2354.8.9 Pseudo-state Approach Remarks 2404.9 Concluding Remarks 2404.10 Problems 2414.11 References 2455 Multivariable H∞ Optimal Control 2495.1 Introduction 2495.1.1 Suboptimal H∞ Control Problems 2505.2 H∞ Multivariable Controllers 2505.2.1 Derivation of the Weighting Filter W 2515.2.2 Robust Weighting Equation 2525.2.3 Calculation of the H∞ Optimal Controller 2535.2.4 Superoptimality in H∞ Design 2585.2.5 Single-input Multi-output Systems 2595.3 One-block and GH∞ Optimal Control Problems 2595.3.1 One-block Nehari Problems 2595.3.2 Categories of Nehari Problem 2605.3.3 Constraint on the Choice of Weights for Simplified Design 2615.3.4 GH∞ Optimal Control Problem 2625.3.5 Final Remarks on LQG Embedding H∞ Solution 2675.4 Suboptimal H∞ Multivariable Controllers 2685.4.1 System Description and Game Problem 2695.4.2 Linear Fractional Transformation 2715.4.3 Signals and Bounded Power Property 2715.4.4 System and Cost Weighting Function Definitions 2725.5 Polynomial System for Suboptimal H∞ Control Problem 2735.5.1 J-spectral Factorisation 2745.5.2 Diophantine Equations for Causal and Noncausal Decomposition 2745.6 Solution of Suboptimal H∞ State Feedback Problem 2755.6.1 Discrete-time Game Problem 2755.6.2 Relationship Between the Game and H∞ Problems 2765.6.3 Standard System Model Equations and Sensitivity 2775.6.4 Completing-the-squares 2775.6.5 Cost Index Terms 2785.6.6 Cost Integrand Simplification 2795.6.7 Contour Integral Simplification 2795.6.8 Optimal Control Law Calculation 2805.6.9 Expression for H0TJH0 2815.6.10 Saddle-point Solution 2825.6.11 Expression for the Minimum Cost 2835.7 Suboptimal H∞ State-feedback Control Problem 2845.7.1 Remarks on the Solution 2855.8 Relationship Between Polynomial and State-space Results 2875.8.1 J-spectral Factorisation Using Riccati Equation 2885.8.2 Relationship between the Polynomial and State-space Equations 2905.9 Solution of Suboptimal Output Feedback Control Problem 2915.9.1 Final Remarks on the Suboptimal H∞ Solution 2915.10 Problems 2925.11 References 2956 Robust Control Systems Design and Implementation 2996.1 Introduction 2996.1.1 The Control Design Problem 3006.1.2 Justification for H∞ Control Design 3026.1.3 Dynamic Cost Function Weightings 3036.1.4 Properties of Sensitivity Functions for Discrete-time Systems 3046.2 Avoiding Impractical H∞ Designs 3066.2.1 Equalising H∞ Solutions and Implications for Multivariable Design 3076.3 Pole-zero Cancellation Properties of LQG and H∞ Designs 3086.3.1 Polynomial Systems Approach 3086.3.2 H2 =LQG Optimal Control Problem 3086.3.3 H∞ Optimal Control Problem 3106.3.4 Cancellation of Minimum-phase Plant Zeros 3116.3.5 Cancellation of Stable Plant Poles 3126.3.6 Sendzimir Steel Rolling Mill Results 3146.4 System Pole and Zero Properties 3146.4.1 Controller Poles and Zeros due to Weightings 3146.4.2 Poles of the Closed-loop System 3156.5 Influence of Weightings on Frequency Responses 3166.5.1 Stability Criterion and Cost Function Weighting Selection 3166.5.2 Influence of the Choice of Weights on the Sensitivity Functions 3176.5.3 Use of Constant Cost Weightings in H∞ Design 3196.5.4 Poor Robustness due to Unrealistic Weightings 3206.6 Loop Shaping Design for Multivariable Systems 3246.6.1 Singular Value Approximations 3246.6.2 Robustness and Loop Shaping 3266.6.3 Stability and Performance Boundaries 3276.6.4 Robust Design for Systems in Standard Model Form 3286.6.5 Structured Singular Values 3306.7 Formalised Design Procedures 3316.7.1 Steps in a H∞ Design Procedure 3316.7.2 Cost Function Weighting Selection for Scalar Systems 3326.8 Mutivariable Robust Control Design Problem 3346.8.1 Problems in Multivariable Control 3356.8.2 Poles and Zeros of Multivariable Systems 3366.8.3 Interaction Measures 3376.9 Multivariable Control of Submarine Depth and Pitch 3376.9.1 Selection of Weights in Multivariable Problems 3376.9.2 Multivariable Submarine Motion Control 3386.9.3 Multivariable Submarine Control Design Results 3406.9.4 Speed of Response and Interaction 3436.9.5 Order of the Weighting Terms 3466.9.6 Two-degree-of-freedom Submarine Control 3466.10 Restricted Structure and Multiple Model Control 3466.10.1 Feedforward and Feedback Polynomial System Plant 3476.10.2 H2 /LQG Restricted Structure Optimal Control Problem 3506.10.3 Numerical Algorithm for Single- and Multi-model Systems 3626.10.4 Hot Strip Finishing Mill Tension Control 3706.10.5 Benefits of Multiple-model Approach 3796.10.6 Restricted Structure Benchmarking 3796.11 Concluding Remarks 3816.12 Problems 3826.13 References 3847 H 2 Filtering, Smoothing and Prediction 3897.1 Introduction 3897.1.1 Standard Signal Processing Model 3907.2 Signal Processing System Description 3907.2.1 Summary of Estimation Problem Assumptions 3917.2.2 Optimal Estimator Transfer-function 3927.2.3 System Equations 3927.2.4 Polynomial Matrix Descriptions 3927.2.5 Spectral Factorisation 3937.3 The Standard H2 Optimal Estimation Problem 3937.3.1 H2 Standard System Model Estimation Problem Solution 3947.3.2 Estimation Error Power Spectrum: Completion of Squares 3947.3.3 Wiener Filtering Solution 3957.3.4 Introduction of the First Diophantine Equation 3967.3.5 Optimal Estimator when Signal Model Stable 3967.3.6 Optimal Estimator when Signal Model can be Unstable 3997.3.7 Optimal Estimator when Signal Model can be Unstable 4047.4 Solution of Filtering, Smoothing and Predication Problems 4087.4.1 State Estimation Problem 4087.4.2 Output Filtering and Prediction 4097.4.3 Deconvolution Estimation 4107.4.4 Robust Weighting Function W 4137.4.5 Extensions of the Estimator Capabilities 4147.5 Strip Thickness Estimation from Roll Force Measurements 4157.5.1 Rolling Mill Model 4167.5.2 Continuous-time Dynamic Mill Model 4167.6 Strip Thickness Estimation Using Force Measurments 4187.7 Strip Thickness Estimation Using X-Ray Gauge Measurements 4217.8 Strip Thickness Estimation Using Gauge Measurements 4227.9 Time-varying and Nonstationary Filtering 4267.9.1 Linear Multichannel Estimation Problem 4287.9.2 Output Estimation Problem 4317.9.3 Relationship to the Kalman Filtering Problem 4357.10 Conclusions 4407.11 Problems 4417.12 References 4428 H∞ Filtering, Smoothing and Prediction 4458.1 Introduction 4458.1.1 The H∞ Filtering Problem 4468.1.2 Smoothing Filters 4478.1.3 Probabilistic Representation of Uncertainty for Filtering Problems 4488.2 Solution of H∞ Optimal Estimation Problem 4488.2.1 Relationship Between H2 and H∞ Minimisation Problems 4488.2.2 Solution Strategy and Weightings 4498.2.3 Derivation of the Weighting Filter W 4508.2.4 Robustness Weighting Diophantine Equation 4518.2.5 H∞ Optimal Estimator for the Generalised System Model 4528.2.6 Properties of the Optimal Estimator 4538.3 H∞ Deconvolution Filtering Problem 4538.3.1 Deconvolution System Description 4548.3.2 Solution of the H∞ Deconvolution Estimation Problem 4558.4 Suboptimal H∞ Multi-Channel Filters 4578.4.1 Discrete-time System and Signal Source Descriptions 4578.4.2 Duality and the Game Problem 4598.4.3 Results for the Suboptimal H∞ Filtering Problem 4608.4.4 Remarks on the Solution 4628.5 Relevance of H∞ Methods to Signal Processing Applications 4638.6 Final Remarks on the Suboptimal H∞ Filtering Problem 4638.7 Problems 4648.8 References 4659 Applications of H2 /LQG Optimal Control 4699.1 Introduction 4699.2 Wind Turbine Power Control Systems 4709.2.1 Definition of Wind Turbine Transfer Functions 4729.2.2 Weighting Function Definitions 4749.2.3 Numerical Results for Wind Turbine Example 4769.2.4 Wind Turbine Feedback Controller Cancellation Properties 4819.2.5 Role of the Ideal-response Models in Design 4839.2.6 Fixed- and Variable-speed Wind Turbines 4849.2.7 Comparison of Wind Turbine Controllers 4849.2.8 Wind Turbine Condition Monitoring 4849.3 Design of an H2 Flight Control System 4859.3.1 System Models 4859.3.2 Design Requirements and Specification 4879.3.3 Flight Control System: Time and Frequency Responses 4909.3.4 Flight Control System Design Including Flexible Modes 4949.3.5 LQG Flight Control Study Design Results 4959.3.6 Classical and LQG Controller Design 4979.4 Thickness Control Systems Design Using Force Feedback 5009.4.1 Optimal Control Solution for the Gauge Control Problem 5029.4.2 Rolling Mill Model 5029.4.3 Continuous-time Mill Models 5029.4.4 Definition of the Polynomial Models for the Standard System 5039.4.5 Cost Function Definition 5049.4.6 BUR Eccentricity Problem Results 5069.4.7 Mismatched Eccentricity Model Conditions 5109.5 Thickness Control Using Gauge Measurement 5109.5.1 Transport Delay in Thickness Measurement 5129.5.2 Feedback System Models in Polynomial Form 5169.5.3 Choice of Cost Function Weightings for Gauge Feedback Control Problem 5169.5.4 Degree of Stability 5179.6 Ship Roll Stabilisation 5189.6.1 Fin Control Unit 5199.6.2 Speed Adaptation 5209.6.3 Models for the Ship Stabilisation System 5219.6.4 Weighting Selection for LQG Roll Stabilisation Design 5219.6.5 Frequency Responses 5229.6.6 Advantages of the Optimal System in Comparison with Classical Methods 5249.6.7 Rudder-roll Stabilisation and Ship Steering 5259.7 Concluding Remarks 5259.8 Problems 5269.9 References 52610 Industrial Applications of H∞ Optimal Control 52910.1 Introduction 52910.1.1 Applications where H∞ Robust Control Design is Applicable 53010.1.2 Safety Critical Control Systems 53010.1.3 Flight Control Systems 53010.2 H∞ Flight Control Systems Design 53210.2.1 Design Requirements and Specification 53410.2.2 Definition of Cost Function Weightings 53410.2.3 Generalised LQG and H∞ Controller Time- and Frequency-responses 53510.2.4 Introducing a Measurement Noise Model 54010.2.5 Comparison of Controllers 54310.3 H∞ Gauge Control System Design Using Force Feedback 54310.3.1 Thickness Control System Frequency- and Time-responses 54610.3.2 Mismatched Eccentricity Model and Robustness 55110.3.3 Thickness Profile Control 55210.4 Submarine Depth and Course-keeping H∞ Design 55410.4.1 Forces and Moments 55410.4.2 Depth Control 55510.4.3 Sea-state and Sea Current Disturbances 55610.4.4 Submarine Motion Dynamics 55810.4.5 Submarine Depth and Pitch Control Design 56110.4.6 Submarine Depth-keeping Controllers 56210.4.7 Submarine Model Responses 56310.4.8 Model Tuning 56810.4.9 Summary of the Output and Input Disturbance Models 57110.4.10 Submarine Depth and Pitch Control 57210.4.11 Summary of the Selected Weighting Terms 57310.4.12 Scalar Design and Responses: Depth Control 57410.4.13 Scalar Design and Responses: Pitch Control 57810.4.14 Improving the Scalar System Time-responses 58010.5 H∞ Control of Remotely Operated Underwater Vehicles 58010.5.1 Design of ROV Controllers 58410.6 H∞ Control of Surface Ships 58510.6.1 H∞ Fin Roll Stabilisation System Design 58510.6.2 H∞ Ship Track-keeping Control 58810.7 Concluding Remarks 59110.8 Problems 59210.9 References 59311 Time-varying and Nonlinear Control 59511.1 Introduction 59511.2 Optimal Control of Time-varying Linear Systems 59611.2.1 Linear Time-varying and Adjoint Operators 59711.2.2 The Quadratic Cost Index 59811.2.3 Solution of the Time-varying Linear Quadratic Control Problem 59911.3 Modelling and Control of Nonlinear Systems 60211.3.1 Nonlinear Systems Modelling 60311.3.2 Hard Nonlinearities 60411.3.3 Typical System Structures 60511.3.4 Feedback Linearisation 60511.4 NLQG Compensation and Control 60711.4.1 Nonlinear Control Example 60811.4.2 Polynomial Versions of Plant Transfer-function Operators 60911.4.3 Use of Time-varying Cost Function Weighting 61011.4.4 The NLQG Algorithm and Properties 61111.5 NLQG Example with Input and Output Nonlinearities 61211.5.1 System and Cost Function Description 61311.5.2 Simulation Results 61311.5.3 Frequency-domain Results 61411.5.4 Improving NLQG Control Using Future Change Information 62011.6 Nonlinear Generalised Minimum Variance Control 62211.6.1 Nonlinear System Description 62311.6.2 Nonlinear and Linear Subsystem Models 62511.6.3 Signals 62711.7 Nonlinear Generalised Minimum Variance Problem 62711.7.1 Solution of the Nonlinear Feedback/Feedforward Control Problem 62911.7.2 Polynomial Models for the Feedback/Feedforward Control Problem 63011.7.3 Diophantine Equations 63011.7.4 Optimisation 63211.7.5 Alternative Control Solution and Stability 63411.7.6 Closed-loop System Stability 63611.7.7 Simplifying the Controller 63611.7.8 Effect of Bias or Steady-state Levels 63711.8 Nonlinear GMV Control Problem 63911.9 Nonlinear Smith Predictor 64411.9.1 Weighting Selection Based on an Existing Controller 64711.10 Concluding Remarks 64811.11 References 669Appendix 1 Notation and Mathematical Preliminaries 653Notation 653Partitions 654Infimum and Supremum 654A1.1 Vectors 654A1.2 Matrices 655A1.2.1 Matrix Inverse Relationships 657A1.2.2 Matrix Singular Value Relationships 658A1.2.3 Matrix Norm Relationships 659A1.3 Polynomial Matrices 661A1.3.1 Polynomial Equations 662A1.4 Transfer-function Matrices 663A1.4.1 Adjoint, All-pass and Inner Functions 664A1.4.2 Transfer-function Matrix for the Standard System Model 665A1.5 Vector and Normed Spaces 665A1.5.1 Hardy Spaces and Norms 667A1.6 References 669

Professor Michael Grimble, Director of the Industrial Control Centre and Past Chairman of the Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow, UK

• Preface xixAcknowledgements xxi1 Introduction to Optimal and Robust Control 11.1 Introduction 11.1.1 Optimality, Feedback and Robustness 21.1.2 High-integrity and Fault-tolerant Control Systems 31.1.3 Self-healing Control Systems 41.1.4 Fault Monitoring and Detection 51.1.5 Adaptive versus Robust Control 51.1.6 Artificial Intelligence, Neural Networks and Fuzzy Control 51.1.7 Discrete-time Systems 71.2 The H2 and H∞ Spaces and Norms 81.2.1 Graphical Interpretation of the H∞ Norm 91.2.2 Terms Used in H∞ Robust Control Systems Design 91.3 Introduction to H∞ Control Design 91.3.1 Properties of H∞ Robust Control Design 111.3.2 Comparison of H∞ and H2 /LQG Controllers 121.3.3 Relationships between Classical Design and H∞ Robust Control 131.3.4 H2 and H∞ Design and Relationship to PID Control 131.3.5 H∞ Polynomial Systems Synthesis Theory 131.4 State-space Modelling and Synthesis Theory 141.4.1 State-space Solution of Discrete-time H∞ Control Problem 141.4.2 H∞ Control Design Objectives 151.4.3 State-feedback Control Solution 151.4.4 State-feedback Control Problem: Cross-product Costing Case 181.4.5 State-space Solution of Discrete-time H∞ Filtering Problem 191.4.6 Bounded Real Lemma 211.4.7 Output Feedback H∞ Control Problem 241.5 Introduction to H2 or LQG Polynomial Synthesis 291.5.1 System Description 291.5.2 Cost Function and Solution 311.5.3 Minimisation of the Performance Criterion 311.5.4 Solution of the Diophantine Equations and Stability 341.5.5 H2 /LQG Design Examples 351.6 Benchmarking 401.6.1 Restricted Structure Benchmarking 411.6.2 Rules for Benchmark Cost Function Selection 421.7 Condition Monitoring 441.8 Combining H2 , H∞  and ‘ 1 Optimal Control Designs 451.9 Linear Matrix Inequalities 461.10 Concluding Remarks 471.11 Problems 481.12 References 512 Scalar H2 and LQG Optimal Control 572.1 Introduction 572.1.1 Industrial Controller Structures 582.1.2 The 2½-DOF Structure 592.1.3 Restricted Structure Control Laws 602.2 Stochastic System Description 602.2.1 Ideal Response Models 622.2.2 System Equations 622.2.3 Cost Function Weighting Terms 632.3 Dual-criterion Cost-minimisation Problem 642.3.1 Solution of the Dual-criterion Minimisation Problem 662.3.2 Theorem Summarising LQG Controller 712.3.3 Remarks on the Equations and Solution 732.3.4 Design Guidelines 762.3.5 Controller Implementation 772.3.6 LQG Ship-steering Autopilot Application 782.4 LQG Controller with Robust Weighting Function 822.4.1 Youla Parameterisation 822.4.2 Cost Function with Robust Weighting Function 832.4.3 Solution of the Dual-criterion Problem with Robust Weighting 842.4.4 Summary of H2 /LQG Synthesis Problem with Robust Weighting 862.4.5 Comments on the Solution 882.5 Introduction to the Standard System Model 892.5.1 Standard System Model 892.6 The Standard System Model Structure 912.6.1 Polynomial System Models 922.6.2 Reference Model 932.6.3 Cost Function Signals to be Weighted 942.7 Generalised H2 Optimal Control: Standard System Model 952.7.1 Optimal Control Solution of the Standard System Model Problem 962.7.2 Summary of H2 /LQG Controller for Standard System Results 1022.7.3 Remarks on the Solution 1042.8 Concluding Remarks 1052.9 Problems 1052.10 References 1093 H∞ Optimal Control of Scalar Systems 1133.1 Introduction 1133.1.1 Links Between LQG and H∞ Solutions 1143.1.2 Reference and Feedback Controller Designs 1153.2 System Description 1153.3 Lemma Linking H∞ and LQG Control Problems 1153.4 Calculation of the H∞ Optimal Controller 1163.4.1 Simple H∞ Controller Structures and Calculations 1173.4.2 Zero Measurement Noise Case 1173.4.3 Solution for the H∞ Optimal Controller 1183.4.4 Stability Robustness of Mixed-sensitivity H∞ Designs 1213.4.5 One-block H∞ Control Problems 1223.5 The GH∞ Control Problem 1233.5.1 GH∞ Cost Function Definition 1243.5.2 Youla Parameterised Form of the GH∞ Controller 1263.5.3 Calculation of the GH∞ Controller 1283.6 Stability Robustness of GH∞ Designs 1363.6.1 Structure of the Uncertain System 1363.6.2 Rational Uncertainty Structure 1373.6.3 Stability Lemma 1393.6.4 Influence of the Uncertainty Model 1403.6.5 Design Procedure for Uncertain Systems 1403.6.6 H∞ Self-Tuning Controller for Systems with Parametric Uncertainty 1473.7 Standard System and Cost Function Description 1473.8 Calculation of H∞ Controller for the Standard System 1473.8.1 F-iteration Method of Solving the Robust Weighting Equation 1483.8.2 H2 / H∞ Trade-off 1493.9 Probabilistic System Descriptions and H∞ Control 1503.9.1 Uncertain System Model 1513.9.2 Cost Function Definition 1533.9.3 Uncertain System and Polynomial Equation Representation 1553.9.4 Discussion of Probabilistic Uncertainty Modelling and Control 1583.10 Concluding Remarks 1583.11 Problems 1593.12 References 1634 Multivariable H2 /LQG Optimal Control 1674.1 Introduction 1674.1.1 Matrix Fraction Descriptions 1684.2 Multivariable System Description 1684.2.1 Multivariable Sensitivity Matrices and Signal Spectra 1704.2.2 Choice of Noise and Cost Function Weightings 1714.3 LQG Optimal Control Problem and Solution 1714.3.1 Solution of the H2 /LQG Problem 1724.3.2 Solution of the Diophantine Equations 1754.4 Youla Parameterisation and Auxiliary Problem 1824.4.1 Youla Parameterisation for the Auxiliary Problem 1844.4.2 Summary of Multivariable Problem Results with Robust Weighting 1864.5 H 2 /LQG Optimal Control Problem: Measurement Noise Case 1874.5.1 Predictive Optimal Control 1904.5.2 SIMO Predictive Optimal Control Problem 1904.5.3 Probabilistic Description of Uncertainty 1964.6 The GLQG Optimal Control Problem 1964.6.1 Solution of the GLQG Problem 1974.6.2 Modified GLQG Cost Function and Youla Parameterisation 1994.7 Design of Automatic Voltage Regulators 2004.8 Pseudo-state Modelling and Separation Principle 2104.8.1 Introduction to Pseudo-state Methods 2104.8.2 Pseudo-state Discrete-time Plant Model 2114.8.3 Discrete Pseudo-state Feedback Optimal Control 2154.8.4 Solution of the Pseudo-state Feedback Control Problem 2174.8.5 Discrete Pseudo-state Estimation Problem 2224.8.6 Solution of the Discrete-time pseudo-state Estimation Problem 2244.8.7 Output Feedback Control Problem and Separation Principle 2304.8.8 Computational Example 2354.8.9 Pseudo-state Approach Remarks 2404.9 Concluding Remarks 2404.10 Problems 2414.11 References 2455 Multivariable H∞ Optimal Control 2495.1 Introduction 2495.1.1 Suboptimal H∞ Control Problems 2505.2 H∞ Multivariable Controllers 2505.2.1 Derivation of the Weighting Filter W 2515.2.2 Robust Weighting Equation 2525.2.3 Calculation of the H∞ Optimal Controller 2535.2.4 Superoptimality in H∞ Design 2585.2.5 Single-input Multi-output Systems 2595.3 One-block and GH∞ Optimal Control Problems 2595.3.1 One-block Nehari Problems 2595.3.2 Categories of Nehari Problem 2605.3.3 Constraint on the Choice of Weights for Simplified Design 2615.3.4 GH∞ Optimal Control Problem 2625.3.5 Final Remarks on LQG Embedding H∞ Solution 2675.4 Suboptimal H∞ Multivariable Controllers 2685.4.1 System Description and Game Problem 2695.4.2 Linear Fractional Transformation 2715.4.3 Signals and Bounded Power Property 2715.4.4 System and Cost Weighting Function Definitions 2725.5 Polynomial System for Suboptimal H∞ Control Problem 2735.5.1 J-spectral Factorisation 2745.5.2 Diophantine Equations for Causal and Noncausal Decomposition 2745.6 Solution of Suboptimal H∞ State Feedback Problem 2755.6.1 Discrete-time Game Problem 2755.6.2 Relationship Between the Game and H∞ Problems 2765.6.3 Standard System Model Equations and Sensitivity 2775.6.4 Completing-the-squares 2775.6.5 Cost Index Terms 2785.6.6 Cost Integrand Simplification 2795.6.7 Contour Integral Simplification 2795.6.8 Optimal Control Law Calculation 2805.6.9 Expression for H0TJH0 2815.6.10 Saddle-point Solution 2825.6.11 Expression for the Minimum Cost 2835.7 Suboptimal H∞ State-feedback Control Problem 2845.7.1 Remarks on the Solution 2855.8 Relationship Between Polynomial and State-space Results 2875.8.1 J-spectral Factorisation Using Riccati Equation 2885.8.2 Relationship between the Polynomial and State-space Equations 2905.9 Solution of Suboptimal Output Feedback Control Problem 2915.9.1 Final Remarks on the Suboptimal H∞ Solution 2915.10 Problems 2925.11 References 2956 Robust Control Systems Design and Implementation 2996.1 Introduction 2996.1.1 The Control Design Problem 3006.1.2 Justification for H∞ Control Design 3026.1.3 Dynamic Cost Function Weightings 3036.1.4 Properties of Sensitivity Functions for Discrete-time Systems 3046.2 Avoiding Impractical H∞ Designs 3066.2.1 Equalising H∞ Solutions and Implications for Multivariable Design 3076.3 Pole-zero Cancellation Properties of LQG and H∞ Designs 3086.3.1 Polynomial Systems Approach 3086.3.2 H2 =LQG Optimal Control Problem 3086.3.3 H∞ Optimal Control Problem 3106.3.4 Cancellation of Minimum-phase Plant Zeros 3116.3.5 Cancellation of Stable Plant Poles 3126.3.6 Sendzimir Steel Rolling Mill Results 3146.4 System Pole and Zero Properties 3146.4.1 Controller Poles and Zeros due to Weightings 3146.4.2 Poles of the Closed-loop System 3156.5 Influence of Weightings on Frequency Responses 3166.5.1 Stability Criterion and Cost Function Weighting Selection 3166.5.2 Influence of the Choice of Weights on the Sensitivity Functions 3176.5.3 Use of Constant Cost Weightings in H∞ Design 3196.5.4 Poor Robustness due to Unrealistic Weightings 3206.6 Loop Shaping Design for Multivariable Systems 3246.6.1 Singular Value Approximations 3246.6.2 Robustness and Loop Shaping 3266.6.3 Stability and Performance Boundaries 3276.6.4 Robust Design for Systems in Standard Model Form 3286.6.5 Structured Singular Values 3306.7 Formalised Design Procedures 3316.7.1 Steps in a H∞ Design Procedure 3316.7.2 Cost Function Weighting Selection for Scalar Systems 3326.8 Mutivariable Robust Control Design Problem 3346.8.1 Problems in Multivariable Control 3356.8.2 Poles and Zeros of Multivariable Systems 3366.8.3 Interaction Measures 3376.9 Multivariable Control of Submarine Depth and Pitch 3376.9.1 Selection of Weights in Multivariable Problems 3376.9.2 Multivariable Submarine Motion Control 3386.9.3 Multivariable Submarine Control Design Results 3406.9.4 Speed of Response and Interaction 3436.9.5 Order of the Weighting Terms 3466.9.6 Two-degree-of-freedom Submarine Control 3466.10 Restricted Structure and Multiple Model Control 3466.10.1 Feedforward and Feedback Polynomial System Plant 3476.10.2 H2 /LQG Restricted Structure Optimal Control Problem 3506.10.3 Numerical Algorithm for Single- and Multi-model Systems 3626.10.4 Hot Strip Finishing Mill Tension Control 3706.10.5 Benefits of Multiple-model Approach 3796.10.6 Restricted Structure Benchmarking 3796.11 Concluding Remarks 3816.12 Problems 3826.13 References 3847 H 2 Filtering, Smoothing and Prediction 3897.1 Introduction 3897.1.1 Standard Signal Processing Model 3907.2 Signal Processing System Description 3907.2.1 Summary of Estimation Problem Assumptions 3917.2.2 Optimal Estimator Transfer-function 3927.2.3 System Equations 3927.2.4 Polynomial Matrix Descriptions 3927.2.5 Spectral Factorisation 3937.3 The Standard H2 Optimal Estimation Problem 3937.3.1 H2 Standard System Model Estimation Problem Solution 3947.3.2 Estimation Error Power Spectrum: Completion of Squares 3947.3.3 Wiener Filtering Solution 3957.3.4 Introduction of the First Diophantine Equation 3967.3.5 Optimal Estimator when Signal Model Stable 3967.3.6 Optimal Estimator when Signal Model can be Unstable 3997.3.7 Optimal Estimator when Signal Model can be Unstable 4047.4 Solution of Filtering, Smoothing and Predication Problems 4087.4.1 State Estimation Problem 4087.4.2 Output Filtering and Prediction 4097.4.3 Deconvolution Estimation 4107.4.4 Robust Weighting Function W 4137.4.5 Extensions of the Estimator Capabilities 4147.5 Strip Thickness Estimation from Roll Force Measurements 4157.5.1 Rolling Mill Model 4167.5.2 Continuous-time Dynamic Mill Model 4167.6 Strip Thickness Estimation Using Force Measurments 4187.7 Strip Thickness Estimation Using X-Ray Gauge Measurements 4217.8 Strip Thickness Estimation Using Gauge Measurements 4227.9 Time-varying and Nonstationary Filtering 4267.9.1 Linear Multichannel Estimation Problem 4287.9.2 Output Estimation Problem 4317.9.3 Relationship to the Kalman Filtering Problem 4357.10 Conclusions 4407.11 Problems 4417.12 References 4428 H∞ Filtering, Smoothing and Prediction 4458.1 Introduction 4458.1.1 The H∞ Filtering Problem 4468.1.2 Smoothing Filters 4478.1.3 Probabilistic Representation of Uncertainty for Filtering Problems 4488.2 Solution of H∞ Optimal Estimation Problem 4488.2.1 Relationship Between H2 and H∞ Minimisation Problems 4488.2.2 Solution Strategy and Weightings 4498.2.3 Derivation of the Weighting Filter W 4508.2.4 Robustness Weighting Diophantine Equation 4518.2.5 H∞ Optimal Estimator for the Generalised System Model 4528.2.6 Properties of the Optimal Estimator 4538.3 H∞ Deconvolution Filtering Problem 4538.3.1 Deconvolution System Description 4548.3.2 Solution of the H∞ Deconvolution Estimation Problem 4558.4 Suboptimal H∞ Multi-Channel Filters 4578.4.1 Discrete-time System and Signal Source Descriptions 4578.4.2 Duality and the Game Problem 4598.4.3 Results for the Suboptimal H∞ Filtering Problem 4608.4.4 Remarks on the Solution 4628.5 Relevance of H∞ Methods to Signal Processing Applications 4638.6 Final Remarks on the Suboptimal H∞ Filtering Problem 4638.7 Problems 4648.8 References 4659 Applications of H2 /LQG Optimal Control 4699.1 Introduction 4699.2 Wind Turbine Power Control Systems 4709.2.1 Definition of Wind Turbine Transfer Functions 4729.2.2 Weighting Function Definitions 4749.2.3 Numerical Results for Wind Turbine Example 4769.2.4 Wind Turbine Feedback Controller Cancellation Properties 4819.2.5 Role of the Ideal-response Models in Design 4839.2.6 Fixed- and Variable-speed Wind Turbines 4849.2.7 Comparison of Wind Turbine Controllers 4849.2.8 Wind Turbine Condition Monitoring 4849.3 Design of an H2 Flight Control System 4859.3.1 System Models 4859.3.2 Design Requirements and Specification 4879.3.3 Flight Control System: Time and Frequency Responses 4909.3.4 Flight Control System Design Including Flexible Modes 4949.3.5 LQG Flight Control Study Design Results 4959.3.6 Classical and LQG Controller Design 4979.4 Thickness Control Systems Design Using Force Feedback 5009.4.1 Optimal Control Solution for the Gauge Control Problem 5029.4.2 Rolling Mill Model 5029.4.3 Continuous-time Mill Models 5029.4.4 Definition of the Polynomial Models for the Standard System 5039.4.5 Cost Function Definition 5049.4.6 BUR Eccentricity Problem Results 5069.4.7 Mismatched Eccentricity Model Conditions 5109.5 Thickness Control Using Gauge Measurement 5109.5.1 Transport Delay in Thickness Measurement 5129.5.2 Feedback System Models in Polynomial Form 5169.5.3 Choice of Cost Function Weightings for Gauge Feedback Control Problem 5169.5.4 Degree of Stability 5179.6 Ship Roll Stabilisation 5189.6.1 Fin Control Unit 5199.6.2 Speed Adaptation 5209.6.3 Models for the Ship Stabilisation System 5219.6.4 Weighting Selection for LQG Roll Stabilisation Design 5219.6.5 Frequency Responses 5229.6.6 Advantages of the Optimal System in Comparison with Classical Methods 5249.6.7 Rudder-roll Stabilisation and Ship Steering 5259.7 Concluding Remarks 5259.8 Problems 5269.9 References 52610 Industrial Applications of H∞ Optimal Control 52910.1 Introduction 52910.1.1 Applications where H∞ Robust Control Design is Applicable 53010.1.2 Safety Critical Control Systems 53010.1.3 Flight Control Systems 53010.2 H∞ Flight Control Systems Design 53210.2.1 Design Requirements and Specification 53410.2.2 Definition of Cost Function Weightings 53410.2.3 Generalised LQG and H∞ Controller Time- and Frequency-responses 53510.2.4 Introducing a Measurement Noise Model 54010.2.5 Comparison of Controllers 54310.3 H∞ Gauge Control System Design Using Force Feedback 54310.3.1 Thickness Control System Frequency- and Time-responses 54610.3.2 Mismatched Eccentricity Model and Robustness 55110.3.3 Thickness Profile Control 55210.4 Submarine Depth and Course-keeping H∞ Design 55410.4.1 Forces and Moments 55410.4.2 Depth Control 55510.4.3 Sea-state and Sea Current Disturbances 55610.4.4 Submarine Motion Dynamics 55810.4.5 Submarine Depth and Pitch Control Design 56110.4.6 Submarine Depth-keeping Controllers 56210.4.7 Submarine Model Responses 56310.4.8 Model Tuning 56810.4.9 Summary of the Output and Input Disturbance Models 57110.4.10 Submarine Depth and Pitch Control 57210.4.11 Summary of the Selected Weighting Terms 57310.4.12 Scalar Design and Responses: Depth Control 57410.4.13 Scalar Design and Responses: Pitch Control 57810.4.14 Improving the Scalar System Time-responses 58010.5 H∞ Control of Remotely Operated Underwater Vehicles 58010.5.1 Design of ROV Controllers 58410.6 H∞ Control of Surface Ships 58510.6.1 H∞ Fin Roll Stabilisation System Design 58510.6.2 H∞ Ship Track-keeping Control 58810.7 Concluding Remarks 59110.8 Problems 59210.9 References 59311 Time-varying and Nonlinear Control 59511.1 Introduction 59511.2 Optimal Control of Time-varying Linear Systems 59611.2.1 Linear Time-varying and Adjoint Operators 59711.2.2 The Quadratic Cost Index 59811.2.3 Solution of the Time-varying Linear Quadratic Control Problem 59911.3 Modelling and Control of Nonlinear Systems 60211.3.1 Nonlinear Systems Modelling 60311.3.2 Hard Nonlinearities 60411.3.3 Typical System Structures 60511.3.4 Feedback Linearisation 60511.4 NLQG Compensation and Control 60711.4.1 Nonlinear Control Example 60811.4.2 Polynomial Versions of Plant Transfer-function Operators 60911.4.3 Use of Time-varying Cost Function Weighting 61011.4.4 The NLQG Algorithm and Properties 61111.5 NLQG Example with Input and Output Nonlinearities 61211.5.1 System and Cost Function Description 61311.5.2 Simulation Results 61311.5.3 Frequency-domain Results 61411.5.4 Improving NLQG Control Using Future Change Information 62011.6 Nonlinear Generalised Minimum Variance Control 62211.6.1 Nonlinear System Description 62311.6.2 Nonlinear and Linear Subsystem Models 62511.6.3 Signals 62711.7 Nonlinear Generalised Minimum Variance Problem 62711.7.1 Solution of the Nonlinear Feedback/Feedforward Control Problem 62911.7.2 Polynomial Models for the Feedback/Feedforward Control Problem 63011.7.3 Diophantine Equations 63011.7.4 Optimisation 63211.7.5 Alternative Control Solution and Stability 63411.7.6 Closed-loop System Stability 63611.7.7 Simplifying the Controller 63611.7.8 Effect of Bias or Steady-state Levels 63711.8 Nonlinear GMV Control Problem 63911.9 Nonlinear Smith Predictor 64411.9.1 Weighting Selection Based on an Existing Controller 64711.10 Concluding Remarks 64811.11 References 669Appendix 1 Notation and Mathematical Preliminaries 653Notation 653Partitions 654Infimum and Supremum 654A1.1 Vectors 654A1.2 Matrices 655A1.2.1 Matrix Inverse Relationships 657A1.2.2 Matrix Singular Value Relationships 658A1.2.3 Matrix Norm Relationships 659A1.3 Polynomial Matrices 661A1.3.1 Polynomial Equations 662A1.4 Transfer-function Matrices 663A1.4.1 Adjoint, All-pass and Inner Functions 664A1.4.2 Transfer-function Matrix for the Standard System Model 665A1.5 Vector and Normed Spaces 665A1.5.1 Hardy Spaces and Norms 667A1.6 References 669

Professor Michael Grimble, Director of the Industrial Control Centre and Past Chairman of the Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow, UK

• Preface xixAcknowledgements xxi1 Introduction to Optimal and Robust Control 11.1 Introduction 11.1.1 Optimality, Feedback and Robustness 21.1.2 High-integrity and Fault-tolerant Control Systems 31.1.3 Self-healing Control Systems 41.1.4 Fault Monitoring and Detection 51.1.5 Adaptive versus Robust Control 51.1.6 Artificial Intelligence, Neural Networks and Fuzzy Control 51.1.7 Discrete-time Systems 71.2 The H2 and H∞ Spaces and Norms 81.2.1 Graphical Interpretation of the H∞ Norm 91.2.2 Terms Used in H∞ Robust Control Systems Design 91.3 Introduction to H∞ Control Design 91.3.1 Properties of H∞ Robust Control Design 111.3.2 Comparison of H∞ and H2 /LQG Controllers 121.3.3 Relationships between Classical Design and H∞ Robust Control 131.3.4 H2 and H∞ Design and Relationship to PID Control 131.3.5 H∞ Polynomial Systems Synthesis Theory 131.4 State-space Modelling and Synthesis Theory 141.4.1 State-space Solution of Discrete-time H∞ Control Problem 141.4.2 H∞ Control Design Objectives 151.4.3 State-feedback Control Solution 151.4.4 State-feedback Control Problem: Cross-product Costing Case 181.4.5 State-space Solution of Discrete-time H∞ Filtering Problem 191.4.6 Bounded Real Lemma 211.4.7 Output Feedback H∞ Control Problem 241.5 Introduction to H2 or LQG Polynomial Synthesis 291.5.1 System Description 291.5.2 Cost Function and Solution 311.5.3 Minimisation of the Performance Criterion 311.5.4 Solution of the Diophantine Equations and Stability 341.5.5 H2 /LQG Design Examples 351.6 Benchmarking 401.6.1 Restricted Structure Benchmarking 411.6.2 Rules for Benchmark Cost Function Selection 421.7 Condition Monitoring 441.8 Combining H2 , H∞  and ‘ 1 Optimal Control Designs 451.9 Linear Matrix Inequalities 461.10 Concluding Remarks 471.11 Problems 481.12 References 512 Scalar H2 and LQG Optimal Control 572.1 Introduction 572.1.1 Industrial Controller Structures 582.1.2 The 2½-DOF Structure 592.1.3 Restricted Structure Control Laws 602.2 Stochastic System Description 602.2.1 Ideal Response Models 622.2.2 System Equations 622.2.3 Cost Function Weighting Terms 632.3 Dual-criterion Cost-minimisation Problem 642.3.1 Solution of the Dual-criterion Minimisation Problem 662.3.2 Theorem Summarising LQG Controller 712.3.3 Remarks on the Equations and Solution 732.3.4 Design Guidelines 762.3.5 Controller Implementation 772.3.6 LQG Ship-steering Autopilot Application 782.4 LQG Controller with Robust Weighting Function 822.4.1 Youla Parameterisation 822.4.2 Cost Function with Robust Weighting Function 832.4.3 Solution of the Dual-criterion Problem with Robust Weighting 842.4.4 Summary of H2 /LQG Synthesis Problem with Robust Weighting 862.4.5 Comments on the Solution 882.5 Introduction to the Standard System Model 892.5.1 Standard System Model 892.6 The Standard System Model Structure 912.6.1 Polynomial System Models 922.6.2 Reference Model 932.6.3 Cost Function Signals to be Weighted 942.7 Generalised H2 Optimal Control: Standard System Model 952.7.1 Optimal Control Solution of the Standard System Model Problem 962.7.2 Summary of H2 /LQG Controller for Standard System Results 1022.7.3 Remarks on the Solution 1042.8 Concluding Remarks 1052.9 Problems 1052.10 References 1093 H∞ Optimal Control of Scalar Systems 1133.1 Introduction 1133.1.1 Links Between LQG and H∞ Solutions 1143.1.2 Reference and Feedback Controller Designs 1153.2 System Description 1153.3 Lemma Linking H∞ and LQG Control Problems 1153.4 Calculation of the H∞ Optimal Controller 1163.4.1 Simple H∞ Controller Structures and Calculations 1173.4.2 Zero Measurement Noise Case 1173.4.3 Solution for the H∞ Optimal Controller 1183.4.4 Stability Robustness of Mixed-sensitivity H∞ Designs 1213.4.5 One-block H∞ Control Problems 1223.5 The GH∞ Control Problem 1233.5.1 GH∞ Cost Function Definition 1243.5.2 Youla Parameterised Form of the GH∞ Controller 1263.5.3 Calculation of the GH∞ Controller 1283.6 Stability Robustness of GH∞ Designs 1363.6.1 Structure of the Uncertain System 1363.6.2 Rational Uncertainty Structure 1373.6.3 Stability Lemma 1393.6.4 Influence of the Uncertainty Model 1403.6.5 Design Procedure for Uncertain Systems 1403.6.6 H∞ Self-Tuning Controller for Systems with Parametric Uncertainty 1473.7 Standard System and Cost Function Description 1473.8 Calculation of H∞ Controller for the Standard System 1473.8.1 F-iteration Method of Solving the Robust Weighting Equation 1483.8.2 H2 / H∞ Trade-off 1493.9 Probabilistic System Descriptions and H∞ Control 1503.9.1 Uncertain System Model 1513.9.2 Cost Function Definition 1533.9.3 Uncertain System and Polynomial Equation Representation 1553.9.4 Discussion of Probabilistic Uncertainty Modelling and Control 1583.10 Concluding Remarks 1583.11 Problems 1593.12 References 1634 Multivariable H2 /LQG Optimal Control 1674.1 Introduction 1674.1.1 Matrix Fraction Descriptions 1684.2 Multivariable System Description 1684.2.1 Multivariable Sensitivity Matrices and Signal Spectra 1704.2.2 Choice of Noise and Cost Function Weightings 1714.3 LQG Optimal Control Problem and Solution 1714.3.1 Solution of the H2 /LQG Problem 1724.3.2 Solution of the Diophantine Equations 1754.4 Youla Parameterisation and Auxiliary Problem 1824.4.1 Youla Parameterisation for the Auxiliary Problem 1844.4.2 Summary of Multivariable Problem Results with Robust Weighting 1864.5 H 2 /LQG Optimal Control Problem: Measurement Noise Case 1874.5.1 Predictive Optimal Control 1904.5.2 SIMO Predictive Optimal Control Problem 1904.5.3 Probabilistic Description of Uncertainty 1964.6 The GLQG Optimal Control Problem 1964.6.1 Solution of the GLQG Problem 1974.6.2 Modified GLQG Cost Function and Youla Parameterisation 1994.7 Design of Automatic Voltage Regulators 2004.8 Pseudo-state Modelling and Separation Principle 2104.8.1 Introduction to Pseudo-state Methods 2104.8.2 Pseudo-state Discrete-time Plant Model 2114.8.3 Discrete Pseudo-state Feedback Optimal Control 2154.8.4 Solution of the Pseudo-state Feedback Control Problem 2174.8.5 Discrete Pseudo-state Estimation Problem 2224.8.6 Solution of the Discrete-time pseudo-state Estimation Problem 2244.8.7 Output Feedback Control Problem and Separation Principle 2304.8.8 Computational Example 2354.8.9 Pseudo-state Approach Remarks 2404.9 Concluding Remarks 2404.10 Problems 2414.11 References 2455 Multivariable H∞ Optimal Control 2495.1 Introduction 2495.1.1 Suboptimal H∞ Control Problems 2505.2 H∞ Multivariable Controllers 2505.2.1 Derivation of the Weighting Filter W 2515.2.2 Robust Weighting Equation 2525.2.3 Calculation of the H∞ Optimal Controller 2535.2.4 Superoptimality in H∞ Design 2585.2.5 Single-input Multi-output Systems 2595.3 One-block and GH∞ Optimal Control Problems 2595.3.1 One-block Nehari Problems 2595.3.2 Categories of Nehari Problem 2605.3.3 Constraint on the Choice of Weights for Simplified Design 2615.3.4 GH∞ Optimal Control Problem 2625.3.5 Final Remarks on LQG Embedding H∞ Solution 2675.4 Suboptimal H∞ Multivariable Controllers 2685.4.1 System Description and Game Problem 2695.4.2 Linear Fractional Transformation 2715.4.3 Signals and Bounded Power Property 2715.4.4 System and Cost Weighting Function Definitions 2725.5 Polynomial System for Suboptimal H∞ Control Problem 2735.5.1 J-spectral Factorisation 2745.5.2 Diophantine Equations for Causal and Noncausal Decomposition 2745.6 Solution of Suboptimal H∞ State Feedback Problem 2755.6.1 Discrete-time Game Problem 2755.6.2 Relationship Between the Game and H∞ Problems 2765.6.3 Standard System Model Equations and Sensitivity 2775.6.4 Completing-the-squares 2775.6.5 Cost Index Terms 2785.6.6 Cost Integrand Simplification 2795.6.7 Contour Integral Simplification 2795.6.8 Optimal Control Law Calculation 2805.6.9 Expression for H0TJH0 2815.6.10 Saddle-point Solution 2825.6.11 Expression for the Minimum Cost 2835.7 Suboptimal H∞ State-feedback Control Problem 2845.7.1 Remarks on the Solution 2855.8 Relationship Between Polynomial and State-space Results 2875.8.1 J-spectral Factorisation Using Riccati Equation 2885.8.2 Relationship between the Polynomial and State-space Equations 2905.9 Solution of Suboptimal Output Feedback Control Problem 2915.9.1 Final Remarks on the Suboptimal H∞ Solution 2915.10 Problems 2925.11 References 2956 Robust Control Systems Design and Implementation 2996.1 Introduction 2996.1.1 The Control Design Problem 3006.1.2 Justification for H∞ Control Design 3026.1.3 Dynamic Cost Function Weightings 3036.1.4 Properties of Sensitivity Functions for Discrete-time Systems 3046.2 Avoiding Impractical H∞ Designs 3066.2.1 Equalising H∞ Solutions and Implications for Multivariable Design 3076.3 Pole-zero Cancellation Properties of LQG and H∞ Designs 3086.3.1 Polynomial Systems Approach 3086.3.2 H2 =LQG Optimal Control Problem 3086.3.3 H∞ Optimal Control Problem 3106.3.4 Cancellation of Minimum-phase Plant Zeros 3116.3.5 Cancellation of Stable Plant Poles 3126.3.6 Sendzimir Steel Rolling Mill Results 3146.4 System Pole and Zero Properties 3146.4.1 Controller Poles and Zeros due to Weightings 3146.4.2 Poles of the Closed-loop System 3156.5 Influence of Weightings on Frequency Responses 3166.5.1 Stability Criterion and Cost Function Weighting Selection 3166.5.2 Influence of the Choice of Weights on the Sensitivity Functions 3176.5.3 Use of Constant Cost Weightings in H∞ Design 3196.5.4 Poor Robustness due to Unrealistic Weightings 3206.6 Loop Shaping Design for Multivariable Systems 3246.6.1 Singular Value Approximations 3246.6.2 Robustness and Loop Shaping 3266.6.3 Stability and Performance Boundaries 3276.6.4 Robust Design for Systems in Standard Model Form 3286.6.5 Structured Singular Values 3306.7 Formalised Design Procedures 3316.7.1 Steps in a H∞ Design Procedure 3316.7.2 Cost Function Weighting Selection for Scalar Systems 3326.8 Mutivariable Robust Control Design Problem 3346.8.1 Problems in Multivariable Control 3356.8.2 Poles and Zeros of Multivariable Systems 3366.8.3 Interaction Measures 3376.9 Multivariable Control of Submarine Depth and Pitch 3376.9.1 Selection of Weights in Multivariable Problems 3376.9.2 Multivariable Submarine Motion Control 3386.9.3 Multivariable Submarine Control Design Results 3406.9.4 Speed of Response and Interaction 3436.9.5 Order of the Weighting Terms 3466.9.6 Two-degree-of-freedom Submarine Control 3466.10 Restricted Structure and Multiple Model Control 3466.10.1 Feedforward and Feedback Polynomial System Plant 3476.10.2 H2 /LQG Restricted Structure Optimal Control Problem 3506.10.3 Numerical Algorithm for Single- and Multi-model Systems 3626.10.4 Hot Strip Finishing Mill Tension Control 3706.10.5 Benefits of Multiple-model Approach 3796.10.6 Restricted Structure Benchmarking 3796.11 Concluding Remarks 3816.12 Problems 3826.13 References 3847 H 2 Filtering, Smoothing and Prediction 3897.1 Introduction 3897.1.1 Standard Signal Processing Model 3907.2 Signal Processing System Description 3907.2.1 Summary of Estimation Problem Assumptions 3917.2.2 Optimal Estimator Transfer-function 3927.2.3 System Equations 3927.2.4 Polynomial Matrix Descriptions 3927.2.5 Spectral Factorisation 3937.3 The Standard H2 Optimal Estimation Problem 3937.3.1 H2 Standard System Model Estimation Problem Solution 3947.3.2 Estimation Error Power Spectrum: Completion of Squares 3947.3.3 Wiener Filtering Solution 3957.3.4 Introduction of the First Diophantine Equation 3967.3.5 Optimal Estimator when Signal Model Stable 3967.3.6 Optimal Estimator when Signal Model can be Unstable 3997.3.7 Optimal Estimator when Signal Model can be Unstable 4047.4 Solution of Filtering, Smoothing and Predication Problems 4087.4.1 State Estimation Problem 4087.4.2 Output Filtering and Prediction 4097.4.3 Deconvolution Estimation 4107.4.4 Robust Weighting Function W 4137.4.5 Extensions of the Estimator Capabilities 4147.5 Strip Thickness Estimation from Roll Force Measurements 4157.5.1 Rolling Mill Model 4167.5.2 Continuous-time Dynamic Mill Model 4167.6 Strip Thickness Estimation Using Force Measurments 4187.7 Strip Thickness Estimation Using X-Ray Gauge Measurements 4217.8 Strip Thickness Estimation Using Gauge Measurements 4227.9 Time-varying and Nonstationary Filtering 4267.9.1 Linear Multichannel Estimation Problem 4287.9.2 Output Estimation Problem 4317.9.3 Relationship to the Kalman Filtering Problem 4357.10 Conclusions 4407.11 Problems 4417.12 References 4428 H∞ Filtering, Smoothing and Prediction 4458.1 Introduction 4458.1.1 The H∞ Filtering Problem 4468.1.2 Smoothing Filters 4478.1.3 Probabilistic Representation of Uncertainty for Filtering Problems 4488.2 Solution of H∞ Optimal Estimation Problem 4488.2.1 Relationship Between H2 and H∞ Minimisation Problems 4488.2.2 Solution Strategy and Weightings 4498.2.3 Derivation of the Weighting Filter W 4508.2.4 Robustness Weighting Diophantine Equation 4518.2.5 H∞ Optimal Estimator for the Generalised System Model 4528.2.6 Properties of the Optimal Estimator 4538.3 H∞ Deconvolution Filtering Problem 4538.3.1 Deconvolution System Description 4548.3.2 Solution of the H∞ Deconvolution Estimation Problem 4558.4 Suboptimal H∞ Multi-Channel Filters 4578.4.1 Discrete-time System and Signal Source Descriptions 4578.4.2 Duality and the Game Problem 4598.4.3 Results for the Suboptimal H∞ Filtering Problem 4608.4.4 Remarks on the Solution 4628.5 Relevance of H∞ Methods to Signal Processing Applications 4638.6 Final Remarks on the Suboptimal H∞ Filtering Problem 4638.7 Problems 4648.8 References 4659 Applications of H2 /LQG Optimal Control 4699.1 Introduction 4699.2 Wind Turbine Power Control Systems 4709.2.1 Definition of Wind Turbine Transfer Functions 4729.2.2 Weighting Function Definitions 4749.2.3 Numerical Results for Wind Turbine Example 4769.2.4 Wind Turbine Feedback Controller Cancellation Properties 4819.2.5 Role of the Ideal-response Models in Design 4839.2.6 Fixed- and Variable-speed Wind Turbines 4849.2.7 Comparison of Wind Turbine Controllers 4849.2.8 Wind Turbine Condition Monitoring 4849.3 Design of an H2 Flight Control System 4859.3.1 System Models 4859.3.2 Design Requirements and Specification 4879.3.3 Flight Control System: Time and Frequency Responses 4909.3.4 Flight Control System Design Including Flexible Modes 4949.3.5 LQG Flight Control Study Design Results 4959.3.6 Classical and LQG Controller Design 4979.4 Thickness Control Systems Design Using Force Feedback 5009.4.1 Optimal Control Solution for the Gauge Control Problem 5029.4.2 Rolling Mill Model 5029.4.3 Continuous-time Mill Models 5029.4.4 Definition of the Polynomial Models for the Standard System 5039.4.5 Cost Function Definition 5049.4.6 BUR Eccentricity Problem Results 5069.4.7 Mismatched Eccentricity Model Conditions 5109.5 Thickness Control Using Gauge Measurement 5109.5.1 Transport Delay in Thickness Measurement 5129.5.2 Feedback System Models in Polynomial Form 5169.5.3 Choice of Cost Function Weightings for Gauge Feedback Control Problem 5169.5.4 Degree of Stability 5179.6 Ship Roll Stabilisation 5189.6.1 Fin Control Unit 5199.6.2 Speed Adaptation 5209.6.3 Models for the Ship Stabilisation System 5219.6.4 Weighting Selection for LQG Roll Stabilisation Design 5219.6.5 Frequency Responses 5229.6.6 Advantages of the Optimal System in Comparison with Classical Methods 5249.6.7 Rudder-roll Stabilisation and Ship Steering 5259.7 Concluding Remarks 5259.8 Problems 5269.9 References 52610 Industrial Applications of H∞ Optimal Control 52910.1 Introduction 52910.1.1 Applications where H∞ Robust Control Design is Applicable 53010.1.2 Safety Critical Control Systems 53010.1.3 Flight Control Systems 53010.2 H∞ Flight Control Systems Design 53210.2.1 Design Requirements and Specification 53410.2.2 Definition of Cost Function Weightings 53410.2.3 Generalised LQG and H∞ Controller Time- and Frequency-responses 53510.2.4 Introducing a Measurement Noise Model 54010.2.5 Comparison of Controllers 54310.3 H∞ Gauge Control System Design Using Force Feedback 54310.3.1 Thickness Control System Frequency- and Time-responses 54610.3.2 Mismatched Eccentricity Model and Robustness 55110.3.3 Thickness Profile Control 55210.4 Submarine Depth and Course-keeping H∞ Design 55410.4.1 Forces and Moments 55410.4.2 Depth Control 55510.4.3 Sea-state and Sea Current Disturbances 55610.4.4 Submarine Motion Dynamics 55810.4.5 Submarine Depth and Pitch Control Design 56110.4.6 Submarine Depth-keeping Controllers 56210.4.7 Submarine Model Responses 56310.4.8 Model Tuning 56810.4.9 Summary of the Output and Input Disturbance Models 57110.4.10 Submarine Depth and Pitch Control 57210.4.11 Summary of the Selected Weighting Terms 57310.4.12 Scalar Design and Responses: Depth Control 57410.4.13 Scalar Design and Responses: Pitch Control 57810.4.14 Improving the Scalar System Time-responses 58010.5 H∞ Control of Remotely Operated Underwater Vehicles 58010.5.1 Design of ROV Controllers 58410.6 H∞ Control of Surface Ships 58510.6.1 H∞ Fin Roll Stabilisation System Design 58510.6.2 H∞ Ship Track-keeping Control 58810.7 Concluding Remarks 59110.8 Problems 59210.9 References 59311 Time-varying and Nonlinear Control 59511.1 Introduction 59511.2 Optimal Control of Time-varying Linear Systems 59611.2.1 Linear Time-varying and Adjoint Operators 59711.2.2 The Quadratic Cost Index 59811.2.3 Solution of the Time-varying Linear Quadratic Control Problem 59911.3 Modelling and Control of Nonlinear Systems 60211.3.1 Nonlinear Systems Modelling 60311.3.2 Hard Nonlinearities 60411.3.3 Typical System Structures 60511.3.4 Feedback Linearisation 60511.4 NLQG Compensation and Control 60711.4.1 Nonlinear Control Example 60811.4.2 Polynomial Versions of Plant Transfer-function Operators 60911.4.3 Use of Time-varying Cost Function Weighting 61011.4.4 The NLQG Algorithm and Properties 61111.5 NLQG Example with Input and Output Nonlinearities 61211.5.1 System and Cost Function Description 61311.5.2 Simulation Results 61311.5.3 Frequency-domain Results 61411.5.4 Improving NLQG Control Using Future Change Information 62011.6 Nonlinear Generalised Minimum Variance Control 62211.6.1 Nonlinear System Description 62311.6.2 Nonlinear and Linear Subsystem Models 62511.6.3 Signals 62711.7 Nonlinear Generalised Minimum Variance Problem 62711.7.1 Solution of the Nonlinear Feedback/Feedforward Control Problem 62911.7.2 Polynomial Models for the Feedback/Feedforward Control Problem 63011.7.3 Diophantine Equations 63011.7.4 Optimisation 63211.7.5 Alternative Control Solution and Stability 63411.7.6 Closed-loop System Stability 63611.7.7 Simplifying the Controller 63611.7.8 Effect of Bias or Steady-state Levels 63711.8 Nonlinear GMV Control Problem 63911.9 Nonlinear Smith Predictor 64411.9.1 Weighting Selection Based on an Existing Controller 64711.10 Concluding Remarks 64811.11 References 669Appendix 1 Notation and Mathematical Preliminaries 653Notation 653Partitions 654Infimum and Supremum 654A1.1 Vectors 654A1.2 Matrices 655A1.2.1 Matrix Inverse Relationships 657A1.2.2 Matrix Singular Value Relationships 658A1.2.3 Matrix Norm Relationships 659A1.3 Polynomial Matrices 661A1.3.1 Polynomial Equations 662A1.4 Transfer-function Matrices 663A1.4.1 Adjoint, All-pass and Inner Functions 664A1.4.2 Transfer-function Matrix for the Standard System Model 665A1.5 Vector and Normed Spaces 665A1.5.1 Hardy Spaces and Norms 667A1.6 References 669