MATH319 Slides Lancaster University 2018/9 2 Abstract 3 Useful software and books 4 Chapter 1: Linear systems and their description 5 Feedback 6 Negative Feedback 7 Positive feedback 8 Diagrams 9 Block diagrams 10 Linear differential operators 11 Differential equations as feedback systems 12 Damped harmonic oscillator 13 Matrix form of the damped harmonic oscillator 14 Reduction of order 15 Proof of reduction of order 16 Constant coefficient case 17 Chapter 2: Solving linear systems by matrix theory 18 Cofactors 19 Characteristic polynomial 20 All monic polynomials are characteristic polynomials 21 Companion matrix 22 Norm of a vector 23 Cauchy–Schwarz inequality 24 Norm of a matrix 25 Functions of a matrix 26 Characteristic equation 27 Polynomials of a diagonable matrix 28 Diagonalizing the matrix 29 Functions of the matrix 30 Cayley–Hamilton Theorem 31 Matrix exponential exp(A) or expm (A) 32 Proof of properties of exponential 33 Proof of properties of exponential, continued 34 Proof of properties of exponential, concluded 35 Exponential of a diagonable matrix 36 Jordan blocks 37 Eigenvalue terminology 38 Computing eigenvalues 39 Jordan Canonical Form 40 Exponentials of complex matrices 41 Reducing to Jordan blocks 42 Exponential of a Jordan block 43 Conclusion of proof of Corollary 44 SISO (A,B,C,D) 45 MIMO (A,B,C,D) 46 Domestic MIMO 47 MIMO dominoes 48 Solving the basic linear differential equation 49 Motivation 50 Proof by checking 51 Solving MIMO 52 Terminology concerning solutions 53 Rational functions 54 The transfer function of (A,B,C,D) (see also 92) 55 The transfer function of SISO (A,B,C,D) 56 The transfer function of MIMO (A,B,C,D) 57 Realization with a SISO 58 Proof of realization 59 Cofactors 60 Conclusion of realization 61 Conclusion of the proof 62 Example of realising a proper rational function by a SISO 63 Checking the solution 64 Checking the solution, concluded 65 A determinant formula for realization by SISO 66 MIMO as a feedback system, without differentiators 67 Realising MIMO 68 Chapter 3 Laplace transforms 69 Laplace transform table 70 Examples of Laplace transforms 71 Remarks on some functions 72 Examples of Laplace transforms, continued 73 Properties of the Laplace transform 74 Proof of properties 75 Integration 76 Laplace transform of derivative 77 Differentiating Laplace transforms 78 Conclusion 79 Holomorphic Laplace transform 80 Laplace transform of exponentials and powers 81 Laplace convolution 82 Bounds on convolution formula 83 Proof of convolution formula 84 Conclusion of proof: associativity 85 Lerch’s Uniqueness Theorem 86 Example 87 Example, concluded 88 Laplace transform of a differential equation 89 Proof 90 Example of Laplace transform of differential equation 91 Example of Laplace transform of differential equation, concluded 92 Solving MIMO by Laplace transforms 93 Proof 94 Laplace transform of (A,B,C,D) 95 Exponential and Inverses 96 Exponential and Inverse, concluded 97 Partial fractions 98 Partial fractions and Laplace transforms 99 Laplace transforms which are strictly proper rationals 100 Laplace transform calculation 101 Realization of a linear system 102 Proof of realization 103 Transfer function of a Laplace transformed system 104 Frequency Response 105 Example of gain and phase 106 Gain and phase 107 Chapter 4: Stability of MIMO via transfer functions 108 Matrix form of the damped harmonic oscillator 109 Cases of the damped harmonic oscillator 110 Stability cases 111 Growth of exponentials of diagonable matrices 112 Exponentials of diagonable matrices 113 Conclusion of Proof 114 BIBO stability 115 Undamped harmonic oscillator 116 Marginal stability 117 Resonance 118 Bounded exponentials of matrices 119 BIBO stability in terms of eigenvalues of A 120 Proof 121 Conclusion of proof 122 Positive definite matrices 123 Lyapunov’s criterion 124 Proof 125 Sylvester’s equation 126 Cases 127 Sylvester’s criterion 128 An integral solution of Sylvester’s equation 129 Solution of Sylvester’s equation 130 A solution of Lyapunov’s equation Chapter 5 Feedback control 131 Differential rings 132 Maxwell’s problem 133 Necessary condition for stability 134 Stable rational functions 𝒮 135 The stable rational functions form a differential ring 136 Partial fractions 137 Poles of the transfer function of the damped harmonic oscillator 138 Stability for systems and transfer functions 139 T not stable implies not BIBO stable 140 T stable implies BIBO stable 141 Nyquist’s locus 142 Nyquist’s criterion for stability of T 143 Argument principle 144 Feedback Control 145 Feedback controllers 146 Simple feedback loop 147 Stability problem 148 Example of proportional feedback 149 Example of proportional-integral feedback 150 Highest common factor and common zeros 151 Euclidean algorithm 152 Changes of variable in the rational functions 153 Coprime factorisation in the stable rational functions 154 Example 155 Proof 156 Conclusion of proof 157 Coprime factorisation algorithm 158 Controlling rational systems 159 Proof 160 Wellposedness of the SFL 161 Criterion for well posed SFL 162 Internal stability of SFL 163 Youla’s stabilising controllers 164 Proof 165 Appendix: MATLAB commands for matrices 166 Appendix A: MATLAB commands continued 167 Appendix A: MATLAB commands concluded 168 Appendix B: R commands for matrices 169 Appendix B: R commands for matrices (continued) 170 Appendix C: Wolfram alpha matrix operations 171 Glossary of linear system terminology 172 Glossary of linear system terminology