Linear Feedback Control Analysis and Design with MATLAB
Dingy Xue, u YangQuan Chen, Derek P. Atherton

copyright page

Contents
1 Introduction to Feedback Control 1.1 Introduction. 1.2 Historical Background. 1.3 Structure of the Book. 1.4 A Survival Introduction to MATLAB. 1.4.1 A Brief Overview of MATLAB. 1.4.2 Standard MATLAB Statements and 1.4.3 Graphics Facilities in MATLAB. 1.4.4 On-line Help Facilities in MATLAB 1.4.5 MATLAB Toolboxes. Problems. Functions. 36 37
2 Mathematical Models of Feedback Control Systems 2.1 A Physical Modeling Example. 2.2 The Laplace Transformation. 2.3 Transfer Function Models. 2.3.1 Transfer Functions of Control Systems. 2.3.2 MATLAB Representations of Transfer Functions. 2.3.3 Transfer Function Matrices for Multivariable Systems. 2.3.4 Transfer Functions of Discrete-Time Systems. 2.4 Other Mathematical Model Representations. 2.4.1 State Space Modeling. 2.4.2 Zero-pole-gain Description. 2.5 Modeling of Interconnected Block Diagrams. 2.5.1 Series Connection. 2.5.2 Parallel Connection. 2.5.3 Feedback Connection. 2.5.4 More Complicated Connections. 2.6 Conversion Between Dierent Model Objects. 2.6.1 Conversion to Transfer Functions. 2.6.2 Conversion to Zero-pole-gain Models. 2.6.3 State Space Realization. 2.6.4 Conversion between Continuous and Discrete-time Models 2.7 An Introduction to System Identications. 2.7.1 Identication of Discrete-time Systems. i
ii 2.7.2 2.7.3 2.7.4 Problems Order Selections. Generation of Identication Signals. Identication of Multivariable Systems.

Contents. 45 46

3 Analysis of Linear Control Systems 51 3.1 Properties of Linear Control Systems. 52 3.1.1 Stability Analysis. 52 3.1.2 Controllability and Observability Analysis. 55 3.1.3 Kalman Decomposition of Linear Systems. 59 3.1.4 Time Moments and Markov Parameters. 62 3.1.5 Norm Measures of Signals and Systems. 64 3.2 Time Domain Analysis of Linear Systems. 66 3.2.1 Analytical Solutions to Linear Time Responses. 66 3.2.2 Analytical Solutions to Discrete-Time Systems. 69 3.3 Numerical Simulation of Linear Systems. 70 3.3.1 Step Responses of Linear Systems. 70 3.3.2 Impulse Responses of Linear Systems. 75 3.3.3 Time Responses to Arbitrary Inputs. 75 3.4 Root Locus of Linear Systems. 77 3.5 Frequency Domain Analysis of Linear Systems. 82 3.5.1 Frequency Domain Graphs with MATLAB. 82 3.5.2 Stability Analysis Using Frequency Domain Methods. 86 3.5.3 Gain and Phase Margins of the Systems. 87 3.5.4 Variations of Ordinary Nyquist Plots. 88 3.6 Introduction to Model Reduction Techniques. 91 3.6.1 Pad Approximations and Routh Approximations. 91 e 3.6.2 Pad Approximations to Delay Terms. 94 e 3.6.3 Sub-optimal Reduction Techniques for Systems with Delays. 96 3.6.4 State Space Model Reduction. 99 Problems. Simulation Analysis of Nonlinear Systems 4.1 An Introduction to Simulink. 4.1.1 Commonly Used Simulink Blocks. 4.1.2 Simulink Modeling. 4.1.3 Simulation Algorithms and Control Parameters 4.2 Modeling of Nonlinear Systems by Examples. 4.3 Nonlinear Elements Modelling. 4.3.1 Modeling of Piecewise Linear Nonlinearities. 4.3.2 Limit Cycles of Nonlinear Systems. 4.4 Linearization of Nonlinear Models. Problems. 130
5 Model Based Controller Design 135 5.1 Cascade Lead-lag Compensator Design. 136
Contents 5.1.1 Introduction to Lead-lag Synthesis. 5.1.2 Lead-lag Synthesis by Phase Margin Assignment. 5.2 Linear Quadratic Optimal Control. 5.2.1 Linear Quadratic Optimal Control Strategies. 5.2.2 Linear Quadratic Regulator Problems. 5.2.3 Linear Quadratic Control for Discrete-time Systems. 5.2.4 Weighting Matrices Selections. 5.2.5 Observers and Observer Design. 5.2.6 State Feedback and Observer-Based Controllers. 5.3 Pole Placement Design. 5.3.1 Bass-Guras algorithm. 5.3.2 Ackermanns algorithm. 5.3.3 Numerically robust pole placement algorithm. 5.3.4 Observer Design using Pole Placement Technique. 5.3.5 Observer-based Controller Design Using Pole Placement 5.4 Decoupling Control of Multivariable Systems. 5.4.1 Decoupling Control with State Feedback. 5.4.2 Pole Placement of Decoupling Systems with State Feedback. 5.5 SISOTool An Interactive Controller Design Tool. Problems. 6 PID Controller Design 6.1 Introduction. 6.1.1 The PID Actions. 6.1.2 PID Control with Derivative in Feedback Loop. 6.2 Ziegler-Nichols Tuning Formula. 6.2.1 Empirical Ziegler-Nichols Tuning Formula. 6.2.2 Derivative Action in Feedback Path. 6.2.3 Methods for First-order Plus Delay Model Fitting. 6.2.4 A Modied Ziegler-Nichols Formula. 6.3 Other PID Controller Tuning Formulae. 6.3.1 Chien-Hrones-Reswick PID Tuning Algorithm. 6.3.2 Cohen-Coon Tuning Algorithm. 6.3.3 Rened Ziegler-Nichols Tuning. 6.3.4 The Wang-Juang-Chans Tuning Formula. 6.3.5 Optimum PID Controller Design. 6.4 PID Controller Tuning Algorithms for Other Plants. 6.4.1 PD and PID Parameter Setting for IPD Models. 6.4.2 PD and PID Parameters for FOLIPD Models. 6.4.3 PID Parameters Setting of Unstable FOLPD Models. 6.5 A PID Controller Design Program for FOLPD Models. 6.6 Optimal Controller Design. 6.6.1 Solutions to Optimization Problems with MATLAB. 6.6.2 Optimal Controller Design. 6.6.3 A MATLAB/Simulink Based Optimal Controller Designer and Its Applications.

iii 215 218

iv 6.7 More 6.7.1 6.7.2 6.7.3 Problems Topics on PID Control. Integral Windup and Anti-windup PID Controllers Automatic Tuning of PID Controllers. Control Strategy Selections.

Contents. 299

7 Robust Control Systems Design 7.1 Linear Quadratic Gaussian Control. 7.1.1 Linear Quadratic Gaussian Problem. 7.1.2 LQG Problem Solutions Using MATLAB. 7.1.3 LQG with Loop Transfer Recovery (LTR). 7.2 General Descriptions to the Robust Control Problems. 7.2.1 Small Gain Theorem. 7.2.2 Unstructured Uncertainties. 7.2.3 Robust Control Problems. 7.2.4 Model Representation Under MATLAB. 7.2.5 Dealing with Poles on Imaginary Axis. 7.3 H Controller Design. 7.3.1 Augmentations of Model with Weighting Functions. 7.3.2 Model Augmentation with Weighting Function Under MATLAB. 7.3.3 Weighted Sensitivity Problems A Simple Case. 7.3.4 H Controller Design The General Case. 7.3.5 Optimal H Controller Design. 7.4 Optimal H2 Controller Design. 7.5 The Eects of Weighting Functions in H Control. Problems. 8 Fractional-order Controller An Introduction 8.1 Fractional-order Calculus and its Computations. 8.1.1 Denitions of Fractional-order Calculus. 8.1.2 Properties of Fractional-order Dierentiations. 8.2 Frequency and Time Domain Analysis of Fractional-order Linear Systems. 8.2.1 Fractional-order Transfer Function Modeling. 8.2.2 Inter-connections of Fractional-order Blocks. 8.2.3 Frequency Domain Analysis of Linear Fractional-order Systems. 8.2.4 Time Domain Analysis of Fractional-order Systems. 8.3 Filter Approximation to Fractional-order Dierentiations. 8.3.1 Oustaloups Recursive Filter. 8.3.2 A Rened Oustaloups Filter. 8.3.3 Simulink Based Fractional-order Nonlinear Dierential Equation Solutions. 8.4 Model Reduction Techniques for Fractional-order Systems. 8.5 Controller Design Studies for Fractional-order Systems. Problems.
A CtrlLAB A Feedback Control System Analysis and Design Tool 301 A.1 Introduction. 301 A.1.1 What is CtrlLAB. 301 A.1.2 Installation and Requirements. 302 A.1.3 Execution of CtrlLAB. 302 A.1.4 Copyright and Declaration of CtrlLAB. 303 A.2 Model Entry and Model Conversion. 303 A.2.1 Transfer Function Entry. 303 A.2.2 Entering Other Model Representations. 304 A.2.3 More Complicated Model Entry. 304 A.3 Model Transformation and Reduction. 305 A.3.1 Model Display. 305 A.3.2 State Space Realizations. 307 A.3.3 Model Reduction. 308 A.4 Feedback Control System Analysis. 310 A.4.1 Frequency Domain Analysis. 311 A.4.2 Time Domain Analysis. 313 A.4.3 System Properties Analysis. 315 A.5 Controller Design Examples. 316 A.5.1 Model-Based Controller Designs. 316 A.5.2 Design of PID Controllers. 317 A.5.3 Robust Controller Design. 319 A.6 Graphical Interface Based Tools. 320 A.6.1 A Matrix Processor. 320 A.6.2 A Graphical Curve Processor. 326 Problems. 329 Bibliography Index Index About the Authors 337 338