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Control Systems Control Systems en.wikibooks.org December 26, 2019 On the 28th of April 2012 the contents of the English as well as German Wikibooks and Wikipedia projects were licensed under Creative Commons Attribution-ShareAlike 3.0 Unported license. A URI to this license is given in the list of figures on page 345. If this document is a derived work from the contents of one of these projects and the content was still licensed by the project under this license at the time of derivation this document has to be licensed under the same, a similar or a compatible license, as stated in section 4b of the license. The list of contributors is included in chapter Contributors on page 337. The licenses GPL, LGPL and GFDL are included in chapter Licenses on page 355, since this book and/or parts of it may or may not be licensed under one or more of these licenses, and thus require inclusion of these licenses. The licenses of the figures are given in the list of figures on page 345. 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Contents 1 Preface 3 2 Introduction 5 2.1 This Wikibook ................................. 5 2.2 What are Control Systems? .......................... 5 2.3 Classical and Modern .............................. 6 2.4 Who is This Book For? ............................. 7 2.5 What are the Prerequisites? .......................... 7 2.6 How is this Book Organized? ......................... 8 2.7 Differential Equations Review ......................... 9 2.8 History ...................................... 11 2.9 Branches of Control Engineering ....................... 12 2.10 MATLAB .................................... 13 2.11 About Formatting ................................ 14 3 System Identification 17 3.1 Systems ..................................... 17 3.2 System Properties ................................ 17 3.3 Initial Time ................................... 17 3.4 Additivity .................................... 18 3.5 Homogeneity .................................. 19 3.6 Linearity ..................................... 20 3.7 Memory ..................................... 21 3.8 Causality ..................................... 21 3.9 Time-Invariance ................................. 22 3.10 LTI Systems ................................... 22 3.11 Lumpedness ................................... 22 3.12 Relaxed ..................................... 23 3.13 Stability ..................................... 23 3.14 Inputs and Outputs ............................... 23 4 Digital and Analog 25 4.1 Digital and Analog ............................... 25 4.2 Analog ...................................... 27 4.3 Digital ...................................... 28 4.4 Hybrid Systems ................................. 29 4.5 Continuous and Discrete ............................ 30 4.6 Sampling and Reconstruction ......................... 30 III Contents 5 System Metrics 33 5.1 System Metrics ................................. 33 5.2 Standard Inputs ................................. 33 5.3 Steady State ................................... 35 5.4 Target Value ................................... 36 5.5 Rise Time .................................... 37 5.6 Percent Overshoot ............................... 37 5.7 Steady-State Error ............................... 38 5.8 Settling Time .................................. 38 5.9 System Order .................................. 38 5.10 System Type .................................. 39 5.11 Visually ..................................... 41 6 System Modeling 43 6.1 The Control Process .............................. 43 6.2 External Description .............................. 43 6.3 Internal Description ............................... 44 6.4 Complex Descriptions .............................. 45 6.5 Representations ................................. 45 6.6 Analysis ..................................... 45 6.7 Modeling Examples ............................... 47 6.8 Manufacture ................................... 47 7 Transforms 49 7.1 Transforms .................................... 49 7.2 Laplace Transform ............................... 49 7.3 Fourier Transform ................................ 58 7.4 Complex Plane ................................. 59 7.5 Euler's Formula ................................. 59 7.6 MATLAB .................................... 60 7.7 Further reading ................................. 60 8 Transfer Functions 61 8.1 Transfer Functions ............................... 61 8.2 Impulse Response ................................ 61 8.3 Convolution ................................... 65 8.4 Convolution Theorem .............................. 66 8.5 Using the Transfer Function .......................... 66 8.6 Frequency Response .............................. 68 9 Sampled Data Systems 71 9.1 Ideal Sampler .................................. 71 9.2 Geometric Series ................................ 71 9.3 The Star Transform ............................... 73 9.4 The Z-Transform ................................ 74 9.5 Star ↔ Z ..................................... 76 9.6 Z ↔ Fourier ................................... 80 9.7 Reconstruction ................................. 81 IV Contents 9.8 Further reading ................................. 87 10 System Delays 89 10.1 Delays ...................................... 89 10.2 Time Shifts ................................... 89 10.3 Delays and Stability .............................. 90 10.4 Delay Margin .................................. 90 10.5 Transform-Domain Delays ........................... 91 10.6 Modified Z-Transform ............................. 92 11 Poles and Zeros 93 11.1 Poles and Zeros ................................. 93 11.2 Time-Domain Relationships .......................... 93 11.3 What are Poles and Zeros ........................... 94 11.4 Effects of Poles and Zeros ........................... 95 11.5 Second-Order Systems ............................. 95 11.6 Higher-Order Systems ............................. 96 12 State-Space Equations 97 12.1 Time-Domain Approach ............................ 97 12.2 State-Space ................................... 98 12.3 State Variables ................................. 99 12.4 Multi-Input, Multi-Output ........................... 99 12.5 State-Space Equations ............................. 100 12.6 Obtaining the State-Space Equations . 103 12.7 State-Space Representation . 105 12.8 Discretization .................................. 107 12.9 Note on Notations ............................... 108 12.10 MATLAB Representation . 108 13 Solutions for Linear Systems 111 13.1 State Equation Solutions ............................ 111 13.2 Solving for x(t) With Zero Input . 111 13.3 Solving for x(t) With Non-Zero Input . 112 13.4 State-Transition Matrix ............................ 113 14 Time-Variant System Solutions 119 14.1 General Time Variant Solution . 119 14.2 Time-Variant, Zero Input . 120 14.3 Time-Variant, Non-zero Input . 124 15 Digital State-Space 125 15.1 Digital Systems ................................. 125 15.2 Digital Systems ................................. 125 15.3 Relating Continuous and Discrete Systems . 126 15.4 Converting Difference Equations . 128 15.5 Solving for x[n] ................................. 129 15.6 Time Variant Solutions ............................. 130 15.7 MATLAB Calculations ............................. 131 V Contents 16 Eigenvalues and Eigenvectors 133 16.1 Eigenvalues and Eigenvectors . 133 16.2 Characteristic Equation ............................ 133 16.3 Exponential Matrix Decomposition . 135 16.4 Non-Unique Eigenvalues ............................ 137 16.5 Equivalence Transformations . 139 17 MIMO Systems 141 17.1 Multi-Input, Multi-Output . 141 17.2 State-Space Representation . 141 17.3 Transfer Function Matrix ............................ 142 17.4 Discrete MIMO Systems ............................ 144 18 System Realization 147 18.1 Realization .................................... 147 18.2 Realization Conditions ............................. 147 18.3 Realizing the Transfer Matrix . 147 19 Gain 149 19.1 What is Gain? .................................. 149 19.2 Responses to Gain ............................... 150 19.3 Gain and Stability ............................... 150 20 Block Diagrams 151 20.1 Systems in Series ................................ 151 20.2 Systems in Parallel ............................... 154 20.3 State Space Model ............................... 154 20.4 Adders and Multipliers ............................. 156 20.5 Simplifying Block Diagrams . 156 20.6 External links .................................. 157 21 Feedback Loops 159 21.1 Feedback ..................................... 159 21.2 Basic Feedback Structure ............................ 159 21.3 Negative vs Positive Feedback . 160 21.4 Feedback Loop Transfer Function . 162 21.5 Open Loop vs Closed Loop . 164 21.6 Placement of a Controller
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