                       Internal Lecture Notes

        Feedback Control Systems: Analysis and Design using MATLAB

        Authors:
                    Professor Dingyu XUE
         School of Information Science and Engineering
              Northeastern University, P R China

                   Professor Derek P Atherton
                      School of Engineering
             Sussex University, Brighton BN1 9QT, UK

                     Professor Xinhe XU
                       Graduate School
              Northeastern University, P R China

Contents
Please note that some part of the notes is still under written and the
contents file is subject to change without notice.

Chapter 1    Introduction to Feedback Control                         1
  1.1  Introduction                                                   1
  1.2  Why Feedback                                                   1
  1.3  Historical Background                                          1
  1.4  System Models, Analysis and Design                             2
  1.5  Introduction to Computer Aided Tools --- MATLAB                2
     1.5.1 Introduction to MATLAB                                     2
     1.5.2 Standard MATLAB Statements and Functions                   2
     1.5.3 Graphics Facilities in MATLAB                              3
     1.5.4 On-line Help Facilities in MATLAB                          3
     1.5.5 MATLAB Toolboxes                                           4
  Problems                                                            5
Chapter 2    Mathematical Modelling of Feedback Systems               7
  2.1  Introduction                                                   7
  2.2  A Modelling Example                                            8
  2.3  Laplace Transform and Transfer Function Description            8
  2.4  Other Mathematical Model Representations                      11
     2.4.1 State Space Modelling                                     11
     2.4.2 Pole-zero-gain Description                                14
  2.5  Modelling of Interconnected Block Diagrams                    15
     2.5.1 Series Connections                                        15
     2.5.2 Parallel Connections                                      16
     2.5.3 Feedback Connections                                      17
     2.5.4 More Complicated Connections                              18
  2.6  Conversion among Different Model Formats                      22
     2.6.1 Conversion from State Space Models to Transfer Functions  22
     2.6.2 Conversion to and from the Pole-zero-gain Model           24
     2.6.3 State Space Realisation                                   26
  2.7  Modelling of Even More Complicated Systems                    33
     2.7.1 SIMULINK Modelling                                        33
     2.7.2 Linearization of Nonlinear Models                         34
  Problems                                                           37
Chapter 3    Properties of Linear Control Systems                    39
  3.1  Introduction                                                  39
  3.2  Stability Analysis                                            39
     3.2.1 Direct Method                                             39
     3.2.2 Indirect Methods                                          41
     3.2.3 Internal Stability of Feedback Control Systems            47
  3.3  Controllability and Observability Analysis                    50
     3.3.1 System Controllability                                    50
     3.3.2 System Observability                                      52
     3.3.3 Controllability and Observability Gramians                54
     3.3.4 Stabilizability and Detectability                         54
  3.4  Kalman Decomposition of Linear Systems                        57
     3.4.1 Kalman Decomposition                                      57
     3.4.2 Revisit to Minimal Realization Problems                   59
  3.5  Time Moments and Markov Parameters                            60
     3.5.1 Expansion about $s=0$ --- time moments                    60
     3.5.2 Expansion about $s=\infty $ --- Markov parameters         62
  3.6  Robustness Analysis of Feedback Control Systems               63
     3.6.1 Norm Measures of Signals and Systems                      63
     3.6.2 ${\cal H}_2$-norm Evaluation                              64
     3.6.3 ${\cal H}_\infty $-norm Evaluation                        66
     3.6.4 Sensitivity and Complementary Sensitivity                 67
     3.6.5 Stability Margins                                         68
  Problems                                                           69
Chapter 4   Time Domain Analysis of Linear Systems                   71
  4.1  Introduction                                                  71
  4.2  Analytical Solutions of System Responses                      71
     4.2.1 Partial Fraction Expansion Approach                       72
     4.2.2 State Space Method                                        77
  4.3  Demonstrative Example: a Second-order System                  84
     4.3.1 Analytical Studies                                        84
     4.3.2 Response Comparisons for Different Parameters             85
     4.3.3 Impulse Response Analysis                                 87
  4.4  Step Response Analysis of Linear Systems                      89
     4.4.1 Time Response Specifications                              89
     4.4.2 Step Response Evaluation                                  91
  4.5  Impulse Response Analysis of Linear Systems                  100
  4.6  Time Response to Arbitrary Inputs                            104
     4.6.1 Linear System Analysis with MATLAB                       104
     4.6.2 Nonlinear System Analysis with MATLAB                    106
     4.6.3 System Response Evaluation with SIMULINK                 108
  4.7  Time Response of Systems with Time Delays                    110
     4.7.1 Pad\'e Approximations to Delay Terms                     110
     4.7.2 Approximate Analysis to Closed-loop Systems              113
  Problems                                                          115
Chapter 5   Complex and Frequency Domain Analysis of Linear Systems 117
  5.1  Introduction                                                 117
  5.2  Root Locus Analysis of Linear Systems                        117
     5.2.1 Root Locus of a Second-order System                      118
     5.2.2 Principles of Root Locus Sketching                       120
     5.2.3 MATLAB Tools for Root Locus Drawing                      121
     5.2.4 The Dominant Poles and Constant $\zeta $ and $\omega _n$ 
           Contours                                                 129
     5.2.5 Root Locus for Delay Systems                             131
  5.3  Frequency Domain Analysis and Nyquist Plots                  132
     5.3.1 Introduction to Frequency Domain Analysis                132
     5.3.2 Nyquist Plots for Second-order Systems                   135
     5.3.3 Nyquist Plots and Stability Theorem                      137
     5.3.4 Nonlinear Transformation to Ordinary Nyquist Plots       141
  5.4  Closed-loop Frequency Responses of Linear Systems            142
     5.4.1 Constant $M$ and $N$ Circles                             142
     5.4.2 Closed-loop Frequency Response Characteristics           144
  5.5  Bode Diagrams Analysis of Linear Systems                     145
     5.5.1 Bode Diagram of Linear Systems                           145
     5.5.2 Bode Diagrams of a Second-order System                   145
     5.5.3 Asymptotic of Bode Diagrams                              146
     5.5.4 Stability Issues from Bode Diagrams                      147
     5.5.5 Bode Diagram Plotting Using MATLAB                       147
  5.6  Nichols Charts                                               149
     5.6.1 Constant $M(\omega )$ Contour and Constant 
           $\alpha (\omega )$ Contour                               149
     5.6.2 System Analysis Using the Nichols Charts                 150
  5.7  More on Frequency Domain Analysis                            152
     5.7.1 Gain and Phase Margins                                   152
     5.7.2 Frequency Response of Non-minimal Phase Systems          157
     5.7.3 Frequency Response of Time-delay Systems                 158
     5.7.4 Frequency Domain Representations of Sensitivity 
           Functions                                                161
  Problems                                                          161
Chapter 6   Classical Design Techniques                             163
  6.1  Introduction                                                 163
  6.2  Cascade Lead-lag Compensator Design                          163
     6.2.1 Introduction to Lead-lag Synthesis                       163
     6.2.2 Lead-lag Synthesis by Phase Margin Assignment            171
  6.3  Linear Quadratic Optimal Control                             176
     6.3.1 Linear Quadratic Optimal Control Stretages               176
     6.3.2 Linear Quadratic Regulator Problems                      178
     6.3.3 Weighting Matrices Selection                             181
     6.3.4 Observers and Observer Design                            185
     6.3.5 State Feedback and Observer-based State Feedback         189
  6.4  Pole Placement Design                                        190
     6.4.1 Pole Placement Algorithms                                191
     6.4.2 Placing Some of the Poles                                193
     6.4.3 Observer Design using the Pole Placement Technique       193
     6.4.4 Pole Placement via Output Feedback                       194
  6.5  Model Following Control                                      196
     6.5.1 Standard Model Formats with Different Criteria           196
     6.5.2 Model Following Controller Design                        199
     6.5.3 Low Order Controller Design                              203
  Problems                                                          203
Chapter 7    PID Controller Design                                  204
  7.1  Introduction                                                 204
     7.1.1 The PID Actions                                          204
     7.1.2 PID Control with Derivative in Feedback Loop             208
     7.1.3 Integral Windups and Anti-windup PID Controllers         208
  7.2  Ziegler-Nichols Tuning Formula                               209
     7.2.1 Empirical Ziegler-Nichols Tuning Formula                 209
     7.2.2 Methods of Identifying First-order Models                213
     7.2.3 A Modified Ziegler-Nichols Formula                       217
  7.3  Other PID Controller Tuning Formula                          220
     7.3.1 Chien-Hrones-Reswick PID Tuning Algorithm                220
     7.3.2 Cohen-Coon Tuning Algorithm                              223
     7.3.3 Refined Ziegler-Nichols Tuning                           224
     7.3.4 Predictive PI Tuning Algorithm                           227
  7.4  Advanced PID Tuning Methods                                  228
     7.4.1 Gain and Phase Margin Assignment Algorithm               228
     7.4.2 Internal Model Controller                                230
     7.4.3 Pole Placement Tuning                                    232
  7.5  Optimum PID Controller Design                                234
     7.5.1 Set-point PID Tuning                                     234
     7.5.2 Disturbance Rejection PID Tuning                         235
     7.5.3 Gain and Phase Margins Assignment for PID Design         238
     7.5.4 Improved Gain-phase Approach                             238
     7.5.5 Optimum Loop Shaping                                     239
  7.6  Automatic Tuning of PID Controllers                          239
     7.6.1 Determining $\omega _c$ and $K_c$ with Describing 
        Function Method                                             240
     7.6.2 Determining $\omega _c$ and $K_c$ with Tsypkin's Method  241
  Problems                                                          241
Chapter 8    Robust Control Systems Design                          243
  8.1  Introduction                                                 243
  8.2  Linear Quadratic Gaussian (LQG)                              243
     8.2.1 Linear Quadratic Gaussian Problem                        243
     8.2.2 LQG Problem Solutions using MATLAB                       244
     8.2.3 LQG with Loop Transfer Recovery (LTR)                    248
  8.3  General Descriptions to the Robust Control Problems          256
     8.3.1 Small Gain Theorem                                       256
     8.3.2 Unstructured Uncertainties                               256
     8.3.3 Robust Control Structures                                257
     8.3.4 Model Representation under MATLAB                        257
     8.3.5 Dealing with Poles on Imaginary Axis                     259
  8.4  ${\cal H}_\infty $ Controller Design                         261
     8.4.1 Augmentations of Model with Weighting Functions          261
     8.4.2 Model Augmentation with Weighting under MATLAB           263
     8.4.3 Weighted Sensitivity Problems --- A Simple Case          265
     8.4.4 ${\cal H}_\infty $ Controller Design --- A General Case  271
     8.4.5 Optimal ${\cal H}_\infty $ Controller Design             280
  8.5  ${\cal H}_2$ Controller Design                               283
  8.6  More on ${\cal H}_\infty $ Controller Design                 288
     8.6.1 Special ${\cal H}_\infty $ Control Structures            288
     8.6.2 The Effects of Weighting Functions in 
       ${\cal H}_\infty $ Control                                   288
     8.6.3 Weighting Function Selections in Sensitivity Problems    295
  Problems                                                          297
References and Bibliography                                         298
Appendix A   More on Laplace Transformations                        300
  A.1  Basic Transformation Table                                   300
  A.2  Deriving Transformations for Other Functions                 300
  A.3  Laplace Transformation with the Aid of Computers             302
     A.3.1 Laplace Transformation using Mathematica                 302
     A.3.2 Laplace Transformation using Symbolic Toolbox            306
  Problems                                                          308
Chapter B   CtrlLAB --- A Feedback System Analysis and Design Tool  309
  B.1  Introduction                                                 309
     B.1.1 What is CtrlLAB                                          309
     B.1.2 Installation and Requirements                            309
     B.1.3 Execution of CtrlLAB                                     310
     B.1.4 Copyright and Declaration of CtrlLAB                     310
  B.2  Model Entering and Conversion                                311
     B.2.1 Transfer Function Entering                               311
     B.2.2 Other Model Representations Entering                     312
     B.2.3 More Complicated System Entering                         313
  B.3  Model Transformation and Reduction                           314
     B.3.1 Model Display                                            314
     B.3.2 State Space Realization                                  316
     B.3.3 Model Reduction                                          316
  B.4  System Analysis                                              317
     B.4.1 Frequency Domain Analysis                                318
     B.4.2 Time Domain Analysis                                     321
     B.4.3 System Properties Analysis                               323
  B.5  System Design Through Examples                               323
     B.5.1 Classical System Design Facilities                       323
     B.5.2 Design of PID Controllers                                324
     B.5.3 Robust Controller Design                                 326
  B.6  Graphical Interface Based Tools                              328
     B.6.1 A Matrix Processor                                       328
     B.6.2 A Graphical Curve Processor                              332
  Problems                                                          336
Appendix C   More on the Control Systems Toolbox                    333
  C.1  Contents of the Control Systems Toolbox                      333
  C.2  Introduction to Control Systems Toolbox Version 4            335
     C.2.1 Modelling and Model Processing of Linear Systems         335
     C.2.2 System Analysis Facilities                               342
     C.2.3 System Design Facilities                                 344
     C.2.4 LTIView --- a GUI for Linear Time-Invariant Systems      344
Appendix D   MATLAB Functions Developed by the Authors              347
