Practical Control System Design

Adrian Medioli is the automation engineer for Whiteley Corp. Pty. Ltd. After graduating, he spent 11 years as a senior automation engineer before completing his PhD. in Electrical Engineering in 2008 at the University of Newcastle, Australia. From 2008-2021 he was employed as a research academic for Complex Dynamic Systems and Control at the University of Newcastle. Graham Goodwin is Emeritus Laureate Professor, University of Newcastle, Australia. He is a Fellow of the Royal Society, a Foreign Member of the Royal Swedish Academy of Sciences and in 2021 he was awarded the American Control Council John Ragazzini Education Award.

• Preface xixAbout the Authors xxiAcknowledgements xxiiiAbout the Companion Website xxivPart I Modelling and Analysis of Linear Systems 11 Introduction to Control System Design 31.1 Introduction 31.2 A Brief History of Control 41.3 Digital Control 51.4 Our Selection 51.5 Thinking Outside the Box 61.6 How the Book Is Organised 61.7 Testing the Reader’s Understanding 61.8 Revision Questions 7Further Reading 72 Control as an Inverse Problem 92.1 Introduction 92.2 The Elements 92.3 Using Eigenvalue Analysis 102.4 The Effect of Process and Disturbance Errors 112.5 Feedback Control 112.6 The Effect of Measurement Noise 122.7 Sensitivity Functions 142.8 Reducing the Impact of Disturbances and Model Error 142.9 Impact of Measurement Noise 142.10 Other Useful Sensitivity Functions 142.11 Stability (A First Look) 152.12 Sum of Sensitivity and Complementary Sensitivity 152.13 Revision Questions 16Further Reading 163 Introduction to Modelling 173.1 Introduction 173.2 Physical Modelling 173.2.1 Radio Telescope Positioning 173.2.2 Band-Pass Filter 193.2.3 Inverted Pendulum 193.2.4 Flow of Liquid out of a Tank 203.3 State-Space Model Representation 213.3.1 Systems Without Zeros 223.3.2 Systems Which Depend on Derivatives of the Input 233.3.3 Example: State-Space Representation 243.4 Linearisation and Approximation 253.4.1 Linearisation of Inverted Pendulum Model 263.5 Revision Questions 27Further Reading 284 Continuous-Time Signals and Systems 294.1 Introduction 294.2 Linear Continuous-Time Models 294.3 Laplace Transforms 304.4 Application of Laplace Transforms to Linear Differential Equations 314.4.1 Example: Angle of Radio Telescope 324.4.2 Example: Modelling the Angular Velocity of Radio Telescope 334.5 A Heuristic Introduction to Laplace Transforms 334.6 Transfer Functions 344.6.1 High-Order Differential Equation Models 344.6.2 Example: Transfer Function for Radio Telescope 354.6.3 Transfer Functions for Continuous-Time State-Space Models 354.6.4 Example: Inverted Pendulum 364.6.5 Poles, Zeros and Other Properties of Transfer Functions 364.6.6 Time Delays 364.6.7 Heuristic Development of Transfer Function of Delay 374.6.8 Example: Heating System 374.7 Stability of Transfer Functions 384.7.1 Example: Poles of the Radio Telescope Model 384.8 Impulse Response of Continuous-Time Linear Systems 384.8.1 Impulse Response 384.8.2 Convolution and Transfer Functions 394.9 Step Response 394.10 Steady-State Response and Integral Action 404.11 Terms Used to Describe Step Responses 404.12 Frequency Response 414.12.1 Nyquist Diagrams 434.12.2 Bode Diagrams 434.12.3 Example: Simple Transfer Function 444.13 Revision Questions 45Further Reading 465 Laboratory 1: Modelling of an Electromechanical Servomechanism 475.1 Introduction 475.2 The Physical Apparatus 475.3 Estimation of Motor Parameters 495.3.1 Motivation for Building a Model 505.3.2 Experiment: Why Build a Model? 505.3.3 Step Response Testing 505.3.4 Experiment: Measuring the Open-Loop Gain and Time Constant 515.3.5 Frequency Response 515.3.6 Experiment: Measuring Frequency Response 525.3.7 Experiment: Alternative Measurement of Frequency Response 525.4 Revision Questions 53Further Reading 53Part II Control System Design Techniques for Linear Single-input Single-output Systems 556 Analysis of Linear Feedback Systems 576.1 Introduction 576.2 Feedback Structures 576.3 Nominal Sensitivity Functions 596.4 Analysing Stability Using the Characteristic Polynomial 606.4.1 Example: Pole-Zero Cancellation 616.5 Stability and Polynomial Analysis 616.5.1 Stability via Evaluation of the Roots 616.6 Root Locus (RL) 616.7 Nominal Stability Using Frequency Response 636.8 Relative Stability: Stability Margins and Sensitivity Peaks 676.9 From Polar Plots to Bode Diagrams 686.10 Robustness 696.10.1 Achieved Sensitivities 696.10.2 Robust Stability 696.11 Revision Questions 71Further Reading 727 Design of Control Laws for Single-Input Single-Output Linear Systems 737.1 Introduction 737.2 Closed-Loop Pole Assignment 737.2.1 Example: Steam Receiver 747.3 Using Root Locus 757.3.1 Example: Double Integrator 757.3.2 Example: Unstable Process 767.4 All Stabilising Control Laws 777.5 Design Using the Youla–Kucera Parameterisation 797.5.1 Example: Simple First-Order Model 807.6 Integral Action 807.7 Anti-Windup 817.8 PID Design 827.8.1 Structure 827.8.2 Using the Youla–Kucera Parameterisation for PID Design 847.9 Empirical Tuning 847.10 Ziegler–Nichols (Z–N) Oscillation Method 847.10.1 Example: Third-Order Plant 857.11 Two Degrees of Freedom Design 867.12 Disturbance Feedforward 867.13 Revision Questions 87Further Reading 888 Laboratory 2: Position Control of Electromechanical Servomechanism 898.1 Introduction 898.2 Proportional Feedback 898.2.1 Experiment: Testing a Proportion only Control Law 918.3 Using Proportional Plus Derivative Feedback 918.3.1 Experiment: Testing a PD Control Law 928.4 Tachometer Feedback 928.5 PID Design 928.5.1 Output Disturbances 928.5.2 Input Disturbance 938.5.3 A Simple Design Procedure 948.5.4 Experiment: Testing a PID Control Law 948.6 Revision Questions 95Further Reading 959 Laboratory 3: Continuous Casting Machine: Linear Considerations 979.1 Introduction 979.2 The Physical Equipment 979.3 Modelling of Continuous Casting Machine 999.4 Proportional Control 1029.5 Response to Set-Point Changes 1039.6 Experiments 1039.6.1 Experiment: Model Parameter Estimation 1039.6.2 Low Gain Feedback 1049.6.3 High Gain Feedback 1049.7 Effect of Measurement Noise 1049.7.1 Experiment: Measuring the Impact of Measurement Noise 1059.8 Pure Integral Control 1059.8.1 Experiment: Testing Pure Integral Control 1069.9 PI Control 1069.9.1 Experiment: Testing PI Control 1079.9.2 Experiment: Testing the Response to Varying Casting Speed 1089.10 Feedforward Control 1089.10.1 Experiment: Testing Feedforward Control 1099.10.2 Experiment: Testing Sensitivity to the Feedforward Gain 1109.11 Revision Questions 110Further Reading 11010 Laboratory 4: Modelling and Control of Fluid Level in Tanks 11310.1 Introduction 11310.2 The Controllers 11310.3 Physical Modelling 11310.3.1 Experiment: Estimating Plant Gain and Time Constant 11710.4 Closed-Loop Level Control for a Single Tank 11710.4.1 Proportional Only Control 11710.4.2 Experiment: Testing Proportional Control 11710.4.3 Integral Only Control 11810.4.4 Experiment: Testing Integral Control 11810.4.5 Proportional Plus Integral Control 11910.4.6 Experiment: Testing PI Control 11910.4.7 Experiment: Alternative PI Controller 11910.5 Closed-Loop Level Control of Interconnected Tanks 11910.6 Revision Questions 120Further Reading 12111 Laboratory 5: Wind Power (Mechanical Components) 12311.1 Introduction 12311.2 Yaw Control 12311.2.1 Experiment: Estimating the Yaw Time Constant 12711.2.2 Design of Yaw Controller 12711.2.3 Experiment: Testing the Yaw Controller 12811.3 Rotational Velocity Control 12911.3.1 Experiment: Testing the Rotational Velocity Control Law 13311.4 Pitch Control 13311.5 Experiment: Testing the Pitch Controller 13411.6 Revision Questions 135Further Reading 135Part III More Complex Linear Single-Input Single-Output Systems 13712 Time Delay Systems 13912.1 Introduction 13912.2 Transfer Function Analysis 13912.3 Classical PID Design Revisited 14012.4 Padé Approximation 14012.5 Using the Youla–Kucera Parameterisation 14012.6 Smith Predictor 14112.7 Modern Interpretation of Smith Predictor 14212.8 Sensitivity Trade-Offs 14212.9 Theoretical Analysis of Effect of Delay Errors on Smith Predictor 14312.10 Revision Questions 144Further Reading 14513 Laboratory 6: Rolling Mill (Transport Delay) 14713.1 Introduction 14713.2 The Physical System 14713.3 Modelling 14913.3.1 Description of the Process 14913.3.2 Sensors and Actuators 14913.3.3 Disturbances 14913.3.4 Aims of the Control System 14913.4 Building a Model 15013.4.1 The Mill Frame 15013.4.2 Strip Deformation 15013.4.3 Composite Model 15113.4.4 Open-Loop Steady-State Performance 15213.5 Basic Control System Design 15213.6 Linear Control Ignoring the Time Delay 15313.6.1 Experiment: Testing a PI Controller 15413.7 Linear Control Based on Rational Approximation to the Time Delay 15513.7.1 Experiment: Testing PID Design 15613.8 Control System Design Based on Smith Predictor 15613.8.1 Experiment: Testing Smith Predictor 15713.9 Use of a Soft Sensor 15813.9.1 The BISRA Gauge 15813.9.2 Experiment: Testing the BISRA Gauge 15913.10 Robustness of BISRA Gauge 15913.10.1 Experiment: Testing Sensitivity to Mill Modulus 15913.10.2 Experiment: Alternative Solution to Achieve Steady-State Tracking 15913.11 Revision Questions 159Further Reading 16014 Control System Design for Open-Loop Unstable Systems 16114.1 Introduction 16114.2 Some Simple Examples of Open-Loop Unstable Systems 16114.3 All Stabilising Control Laws for Systems Having Undesirable Open-Loop Poles 16314.4 Revision Questions 164Further Reading 16515 Laboratory 7: Control of a Rocket 16715.1 Introduction 16715.2 Dynamics of a Rocket in 2D Flight 16715.2.1 Coordinate Systems 16715.2.2 Forces 16915.2.3 Translational Dynamics 17015.2.4 Rotational Dynamics 17015.2.5 Composite Model 17115.3 Equilibrium 17115.4 Linearised Model 17115.5 Open-Loop Flight 17215.6 Controller Design for the Rocket 17215.6.1 Simplified Design of PID 17215.6.2 Frequency Domain Design 17315.7 Experiment: Testing the Control Law 17415.7.1 Testing the Design Mode in Section 15.6.1 17415.7.2 Testing the Design Made in Section 15.6.2 17515.8 Revision Questions 175Further Reading 17516 Bode Sensitivity Trade-Offs 17716.1 Introduction 17716.2 System Properties 17716.3 Bode Integral Constraints 17816.3.1 Open-Loop Stable Systems 17816.4 Examples of Bode Sensitivity Trade-Offs 17816.4.1 Open-Loop Unstable Systems 18016.5 Bode Complementary Sensitivity Integrals 18016.5.1 Minimum Phase Plants 18016.5.2 Non-minimum Phase Plants 18016.6 Bode Sensitivity for Time-Delay Systems 18016.7 Revision Questions 181Further Reading 181Part IV Sampled Data Control Systems 18317 Principles of Sampled-Data Control System Design 18517.1 Introduction 18517.2 A/D Conversion 18517.3 Sampled Output Noise 18517.4 D/A Conversion 18617.5 Sampled-Data Models 18717.6 Shift Operator Models 18717.7 Divided Difference Models 18717.8 Euler Approximate Model 18817.9 Euler Approximate Model in Delta Domain 18817.10 Delta Analysis 18917.11 Historical Notes 18917.12 An Example of Shift and Delta Models 18917.13 Sampled-Data Stability 19017.14 Bode Sensitivity Integrals (Sampled Data Case) 19017.14.1 Z-Domain 19217.14.2 Delta Domain 19217.15 Sampling Zeros 19317.16 Revision Questions 193Further Reading 19418 Laboratory 8: Audio Signal Processing and Optimal Noise Shaping Quantisers 19718.1 Introduction 19718.2 The Physical Apparatus 19718.3 Psychoacoustic Issues 19818.3.1 Experiment: Testing Your Hearing Sensitivity 19918.4 Nearest Neighbour Quantisation 20018.4.1 Experiment: Testing the Nearest Neighbour Quantiser 20018.5 Optimal Noise Shaping Quantiser 20118.5.1 Feedback Quantiser 20118.5.2 Experiment: Test the Feedback Quantiser 20218.6 Utilising Your Own Hearing Sensitivity 20218.6.1 Experiment: Test the Feedback Quantiser Using Your Hearing Sensitivity 20418.7 Audio Quantisation from a Bode Sensitivity Integral Perspective 20418.7.1 Experiment: Spectrum of Errors 20518.7.2 Experiment: Testing Bode Sensitivity Integral 20518.8 Audio Quantisation for More Complex Cases 20518.8.1 Experiment: More Complex Case 20618.9 Revision Questions 206Further Reading 207Part V Simple Multivariable Control Problems 20919 Tools Used for Simple Multivariable Control Problems 21119.1 Introduction 21119.2 Cascade Control 21119.2.1 Example of Cascade Control 21219.3 Imposed SISO Architectures 21419.4 Relative Gain Array 21519.5 An Industrial Example 21519.5.1 The Relative Gain Array 21519.5.2 A Simple MV Transformation 21619.6 Revision Questions 216Further Reading 21620 Laboratory 9: Wind Power (Electrical Components) 21720.1 Introduction 21720.2 Generator Choices 21720.3 Physical Parameters for the Laboratory Wind Turbine 21720.4 The Generator and Grid Side Architectures 21920.5 Background Theory 21920.5.1 Alpha, Beta Coordinates 22020.5.2 dq Frame 22020.5.3 The Inverse Transformation 22120.5.4 First-Order Dynamics in dq Frame 22120.6 Generator Side Model 22220.7 Generator Side Control Law 22320.7.1 Regulation of I Sd 22420.7.2 Regulation of I Sq 22420.7.3 Alignment of dq Frame 22420.7.4 Conversion of V Sd , V Sq Back to Time Domain 22520.8 The Link Capacitor Model 22520.8.1 Current into the Capacitor 22520.8.2 Dynamics of the Capacitor 22520.9 Regulation of the Capacitor Voltage 22620.10 Model for the Grid Side Transformer 22620.11 The Grid Side Control Law 22620.11.1 Regulation of I Cq 22720.11.2 Regulation of I cd 22720.12 Complete Electrical System Control Law 22720.13 Testing the Electrical Control Laws 22920.13.1 Generator Side 22920.13.2 Grid Side 22920.14 Experiments on the Complete System 22920.14.1 Experiment: Testing the Impact of Wind Direction 23020.14.2 Experiment: Testing the Impact of Wind Speed 23120.15 Revision Questions 231Further Reading 23321 Laboratory 10: Cross-Directional Control in Paper Machines: PID Control 23521.1 Introduction 23521.2 Web-Forming Process 23521.3 Basis Weight Control in a Paper Machine 23721.4 Process Model 23721.4.1 Experiment: Measuring the Cross-Directional Profile 24121.4.2 Experiment: Measuring the Machine Direction Dynamics 24121.5 Simple SISO Design Ignoring Coupling 24121.5.1 Experiment: Testing Simple PID Controllers 24221.6 Simple SISO Design Accounting for Coupling 24221.6.1 Experiment: Testing a Decoupled PID Structure 24321.7 Summary 24321.8 Revision Questions 244Further Reading 244Part VI Multivariable Control Systems (More General Methods) 24722 State Variable Feedback 24922.1 Introduction 24922.2 Sampled-Data Control 24922.2.1 Pole Assignment 24922.2.2 Linear Quadratic Regulator (LQR) 24922.3 Dynamic Programming 25022.4 Infinite Horizon Linear Quadratic Optimal Problem 25122.5 Delta-Domain Result 25122.6 Continuous-Time Linear Quadratic Regulator 25222.6.1 Pole Assignment 25222.6.2 Continuous-Time Linear Quadratic Regulator 25222.7 Regulation to a Fixed Set-Point 25322.8 Frequency Domain Insights into the Linear Quadratic Regulator 25422.9 Output Feedback 25522.9.1 A State Estimator (or Observer) 25522.9.2 Certainty Equivalence 25522.10 Separation 25622.11 Achieving Integral Action 25622.11.1 The Problem 25622.11.2 The Remedy 25622.11.3 Properties 25722.12 All Stabilising Control Laws Revisited 25822.12.1 Stable Open-Loop Plants 25922.12.2 Adding Stable Uncontrollable Disturbance States 25922.12.3 Adding Non-stabilisable Disturbance States 26022.13 Model Predictive Control 26022.14 Revision Questions 260Further Reading 26123 The Kalman Filter 26323.1 Introduction 26323.2 Periodic Disturbances 26323.2.1 Continuous-Time Model 26323.2.2 Sampled-Data Process Noise 26423.2.3 Sampled-Data Measurement Noise 26523.2.4 The Full Sampled-Data Model 26523.3 The Best Observer Gain 26623.4 Steady-State Optimal Estimator 26723.5 Treating Non-White Noise 26823.6 Dealing with Constant Disturbances 26823.7 Periodic Disturbances 26823.8 Accounting for Delays 26923.9 Multiple Output Measurements 26923.10 Continuous-Time Kalman Filter 27023.11 Linking Continuous Kalman Filter and Discrete Kalman Filter 27023.12 The Linear Quadratic Regulator Revisited 27123.13 Quantifying the Performance 27123.14 Revision Questions 272Further Reading 27424 Laboratory 11: Rolling Mill Revisited (Periodic Disturbances) 27524.1 Introduction 27524.2 Disturbances 27524.3 Effects of Roll Eccentricity 27624.3.1 Experiment: Measuring the Impact of Roll Eccentricity 27724.4 Tight Feedback Control 27724.4.1 Experiment: Testing the Impact of Eccentricity on the BISRA Gauge 27824.4.2 Analysis of the Effect of Control Law Bandwidth 27824.5 Eccentricity Compensation 27824.5.1 A Simple Eccentricity Predictor 27824.6 Optimal Observer Design 27924.6.1 Experiment: Testing the Eccentricity Estimator 28024.7 Eccentricity Compensation Using the Kalman Filtering 28124.7.1 Experiment: Testing the Kalman Filter for Eccentricity Estimation 28124.8 Conclusion 28224.9 Revision Questions 282Further Reading 283Part VII Introduction to the Modelling and Control of Nonlinear Systems 28525 Modelling and Analysis of Simple Nonlinear Systems 28725.1 Introduction 28725.2 Errors Arising from Large Actuator Movement 28725.3 Nonlinear Correction by Gain Change 28825.4 Nonlinear Correction by Cascade Control 28825.5 Saturation 28925.5.1 Achieving Integral Action via Feedback 28925.5.2 Introducing Anti-Windup in Control Laws Implemented via the Youla–Kucera Parameterisation 29025.5.3 Anti-Windup When an Observer is Used 29025.6 Extension to Rate Limitations 29125.7 Minimal Actuator Movement 29125.8 Describing Function Analysis 29125.9 Predicting the Period and Amplitude of Oscillations 29325.10 Revision Questions 293Further Reading 29426 Laboratory 12: Continuous Casting Machine (Nonlinear Considerations) 29726.1 Introduction 29726.2 The Slide Gate Valve 29726.3 Investigation of Effect of Nonlinear Valve Geometry 29826.3.1 Experiment: Testing Impact of the Nonlinear Geometry of the Valve 29926.3.2 Other Nonlinear Phenomena 30026.4 An Explanation for the Observed Oscillations 30026.5 A Redesign to Account for Slip-Stick Friction 30226.5.1 Experiment: Testing the Impact of Slip-Stick Friction 30226.6 Revision Questions 303Further Reading 30327 Laboratory 13: Cross-Directional Control (Robustness and Impact of Actuator Saturation) 30527.1 Introduction 30527.2 Effect of Actuator Saturation Without Anti-Windup Protection 30527.2.1 Experiment: Impact of Actuator Saturation 30527.2.2 Experiment: Impact of Actuator Saturation with Decoupled PID Design 30627.3 PI Decoupled Design with Simple Anti-Windup Protection 30627.3.1 Experiment: Testing the Simple Anti-Windup Scheme 30727.4 Conditioning Problems 30827.4.1 Experiment: Testing Actuator Profile 31027.5 PI Decoupled Design with Anti-Windup Protection Limited to Low Spatial Frequencies 31027.5.1 Experiment: Limiting Spatial Frequencies Used in the Controller 31027.6 PI Decoupled Design with Adaptive Spatial Frequency Selection 31127.6.1 Experiment: Testing Adaptive Spatial Frequency Selection 31227.7 Conclusions 31227.8 Revision Questions 312Further Reading 312Part VIII Modelling and Control of More Complex Nonlinear Systems 31528 Modelling of a Rocket in Three-Dimensional Flight 31728.1 Introduction 31728.2 Preliminaries 31728.2.1 Coordinate Systems 31728.2.2 Euler Angles in Three Dimensions 31828.2.3 Time Derivative of Rotation Matrices 32028.2.4 Angular Velocities 32128.2.5 Angular Acceleration 32128.2.6 Cross-Products 32328.3 Translational Dynamics 32328.3.1 Forces 32328.3.2 Model for Translational Dynamics 32428.4 Rotational Dynamics 32428.4.1 Torque 32428.4.2 Model for Rotational Dynamics 32528.5 Stable or Unstable Rocket 32528.6 Revision Questions 326Further Reading 32629 Modelling of a Steam-Generating Boiler 32729.1 Introduction 32729.2 Physical Principles 32829.2.1 Internal Energy and Enthalpy 32829.2.2 Ideal Gases 32829.2.3 Steam 32829.3 Physical Principles Used in Boiler Modelling 32929.4 Mass Balances 32929.5 Constant Volume of Drum, Risers and Downcomers 33129.5.1 Consequence of Constant Volume of the Drum 33229.5.2 Consequence of Constant Volume of the Risers 33229.6 Energy Balances 33329.6.1 Consequence of Drum Energy Balance 33429.6.2 Consequences of Energy Balance in the Risers 33529.7 A Model for Boiler Pressure 33529.8 A Model for Drum Water Level 33629.9 Spatial Discretisation and Homogeneous Mixing in the Risers 33729.9.1 Spatial Discretisation 33829.9.2 Homogeneous Mixing in a Section of the Risers 33929.10 Water Flow in the Downcomers 34029.11 Superheaters 34129.12 Steam Receiver 34129.12.1 Mass Balance 34229.12.2 Energy Balance 34229.12.3 Constant Volume of the Steam Receiver 34229.12.4 Summary of the Model for the Steam Receiver 34329.13 Other Model Components 34329.13.1 Mass Flow out of Drum 34329.13.2 Feedwater Mass Flow 34429.13.3 Total Heat 34429.13.4 Disturbances 34429.13.5 A Preliminary Simulation 34429.14 Revision Questions 344Further Reading 34630 Laboratory 14: Control of a Steam Boiler 34730.1 Introduction 34730.2 Extracting an Approximate Linear Model 34730.2.1 Introduction 34730.2.2 Sine Wave Testing in Closed-Loop (Scalar Case) 34830.2.3 Application to the Boiler Model 34930.2.4 The Steam Receiver 35030.3 The Control Architecture 35130.4 Regulating Steam Flow from the Boiler 35130.5 Boiler Pressure Controller 35130.6 Drum Water Level Controller 35230.6.1 Experiment: Implementing Drum Water Level Control Law 35230.7 Steam Receiver Controller 35330.7.1 Experiment: Testing Steam Receiver Control Law 35330.8 Experiments 35330.8.1 Set Up 35330.8.2 Small Load Change 35430.8.3 Faster Outer Loop 35430.8.4 Slower Outer Loop 35430.8.5 Large Decrease in Load 35530.8.6 Constraints 35530.8.7 Large Load Change with ‘Fast’ Outer Loop 35530.8.8 Large Increase in Load 35530.9 Summary 35530.10 Revision Questions 355Further Reading 356Index 357