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Linear Systems and Control: a First Course (Course Notes for AAE 564)

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Linear Systems and Control: a First Course (Course Notes for AAE 564) Linear Systems and Control: A First Course (Course notes for AAE 564) Martin Corless School of Aeronautics & Astronautics Purdue University West Lafayette, Indiana [email protected] August 25, 2008 Contents 1 Introduction 1 1.1 Ingredients..................................... 4 1.2 Somenotation................................... 4 1.3 MATLAB ..................................... 4 2 State space representation of dynamical systems 5 2.1 Linearexamples.................................. 5 2.1.1 Afirstexample .............................. 5 2.1.2 Theunattachedmass........................... 6 2.1.3 Spring-mass-damper ........................... 6 2.1.4 Asimplestructure ............................ 7 2.2 Nonlinearexamples............................... 8 2.2.1 Afirstnonlinearsystem ......................... 8 2.2.2 Planarpendulum ............................. 9 2.2.3 Attitudedynamicsofarigidbody. .. 9 2.2.4 Bodyincentralforcemotion. 10 2.2.5 Ballisticsindrag ............................. 11 2.2.6 Doublependulumoncart . .. .. 12 2.2.7 Two-linkroboticmanipulator . .. 13 2.3 Discrete-timeexamples . ... 14 2.3.1 Thediscreteunattachedmass . 14 2.4 Generalrepresentation . ... 15 2.4.1 Continuous-time ............................. 15 2.4.2 Discrete-time ............................... 18 2.4.3 Exercises.................................. 20 2.5 Vectors....................................... 22 2.5.1 Vector spaces and IRn .......................... 22 2.5.2 IR2 andpictures.............................. 24 2.5.3 Derivatives ................................ 25 2.5.4 MATLAB ................................. 25 2.6 Vector representation of dynamical systems . ......... 26 2.7 Solutions and equilibrium states . ...... 28 2.7.1 Continuous-time ............................. 28 2.7.2 Discrete-time ............................... 31 2.7.3 Exercises.................................. 32 iii 2.8 Numericalsimulation . .. .. .. 33 2.8.1 MATLAB ................................. 33 2.8.2 Simulink.................................. 34 2.8.3 IMSL.................................... 35 2.9 Exercises...................................... 38 3 Linear time-invariant (LTI) systems 41 3.1 Matrices...................................... 41 3.2 Linear time-invariant systems . ..... 48 3.2.1 Continuous-time ............................. 48 3.2.2 Discrete-time ............................... 49 3.3 Linearization about an equilibrium solution . ......... 50 3.3.1 Derivativeasamatrix .......................... 50 3.3.2 Linearization of continuous-time systems . ....... 51 3.3.3 Implicitlinearization . .. 54 3.3.4 Linearization of discrete-time systems . ....... 55 4 Some linear algebra 59 4.1 Linearequations ................................. 59 4.2 Subspaces ..................................... 66 4.3 Basis........................................ 67 4.3.1 Span(notSpam) ............................. 67 4.3.2 Linearindependence ........................... 68 4.3.3 Basis.................................... 73 4.4 Rangeandnullspaceofamatrix . .. 74 4.4.1 Nullspace................................. 74 4.4.2 Range ................................... 76 4.4.3 Solving linear equations revisited . ..... 79 4.5 Coordinatetransformations . .... 82 4.5.1 Coordinate transformations and linear maps . ...... 82 4.5.2 Thestructureofalinearmap . 84 5 Behavior of (LTI) systems : I 87 5.1 Initialstuff .................................... 87 5.1.1 CT..................................... 87 5.1.2 DT..................................... 88 5.2 Scalarsystems................................... 89 5.2.1 CT..................................... 89 5.2.2 First order complex and second order real. ..... 90 5.2.3 DT..................................... 91 5.3 Eigenvalues and eigenvectors . ..... 92 5.3.1 Real A ................................... 95 5.3.2 MATLAB ................................. 96 5.3.3 Companionmatrices ........................... 97 5.4 Behavior of continuous-time systems . ...... 98 5.4.1 System significance of eigenvectors and eigenvalues . ......... 98 5.4.2 Solutions for nondefective matrices . ..... 104 5.4.3 Defective matrices and generalized eigenvectors . ......... 105 5.5 Behavior of discrete-time systems . ...... 109 5.5.1 System significance of eigenvectors and eigenvalues . ......... 109 5.5.2 All solutions for nondefective matrices . ...... 110 5.5.3 Some solutions for defective matrices . ..... 110 5.6 Similaritytransformations . ..... 112 5.7 HotelCalifornia................................. 117 5.7.1 Invariantsubspaces . 117 5.7.2 Moreoninvariantsubspaces . 118 5.8 Diagonalizablesystems . 119 5.8.1 Diagonalizablematrices . 119 5.8.2 Diagonalizable continuous-time systems . ....... 121 5.8.3 DiagonalizableDTsystems. 123 5.9 Real Jordan form for real diagonalizable systems . .......... 125 5.9.1 CTsystems ................................ 126 5.10 Diagonalizable second order systems . ....... 126 5.10.1 StateplaneportraitsforCTsystems . .... 127 5.11Exercises...................................... 127 6 eAt 133 6.1 Statetransitionmatrix . 133 6.1.1 Continuoustime ............................. 133 6.1.2 Discretetime ............................... 135 6.2 Polynomialsofasquarematrix . 136 6.2.1 Polynomialsofamatrix . .. .. 136 6.2.2 Cayley-HamiltonTheorem . 137 6.3 Functionsofasquarematrix. 139 6.3.1 The matrix exponential: eA ....................... 142 6.3.2 Othermatrixfunctions . 142 6.4 The state transition matrix: eAt ......................... 143 6.5 Computation of eAt ................................ 144 6.5.1 MATLAB ................................. 144 6.5.2 Numericalsimulation. 144 6.5.3 Jordanform................................ 144 6.5.4 Laplacestyle ............................... 144 6.6 Sampled-datasystems ............................. 146 6.7 Exercises...................................... 147 7 Behavior of (LTI) systems : II 149 7.1 Introduction.................................... 149 7.2 Jordanblocks ................................... 150 7.3 Generalizedeigenvectors . .... 153 7.3.1 Generalizedeigenvectors . 153 7.3.2 Chain of generalized eigenvectors . .... 154 7.3.3 System significance of generalized eigenvectors . ......... 155 7.4 TheJordanformofasquarematrix. 156 7.4.1 Specialcase ................................ 156 7.4.2 Block-diagonalmatrices . 158 7.4.3 Alessspecialcase............................. 159 7.4.4 Anotherlessspecialcase . 159 7.4.5 Generalcase................................ 161 7.5 LTIsystems.................................... 162 7.6 Examples on numerical computations with repeated roots ........... 163 8 Stability and boundedness 167 8.1 Continuous-timesystems . 167 8.1.1 Boundednessofsolutions. 167 8.1.2 Stability of an equilibrium state . .... 169 8.1.3 Asymptoticstability . 171 8.2 Discrete-timesystems. 173 8.2.1 Stability of an equilibrium state . .... 174 8.2.2 Asymptoticstability . 174 8.3 Anicetable.................................... 175 8.4 Linearizationandstability . ..... 175 8.5 Exercises...................................... 177 9 Inner products 183 9.1 The usual scalar product on IRn and Cn .................... 183 9.2 Orthogonality ................................... 184 9.2.1 Orthonormalbasis ............................ 185 9.3 Hermitianmatrices ............................... 187 9.4 Positive and negative (semi)definite matrices . .......... 188 9.4.1 Quadraticforms.............................. 188 9.4.2 Definitematrices ............................. 189 9.4.3 Semi-definitematrices . 190 9.5 Singularvaluedecomposition . .... 191 10 Lyapunov theory 197 10.1 Asymptotic stability results . ...... 197 10.1.1 MATLAB.................................. 202 10.2 Stabilityresults ............................... 203 10.3 Mechanicalsystems. .. .. 204 10.3.1 A simple stabilization procedure . ..... 206 10.4DTLyapunovtheory ............................... 209 10.4.1 Stabilityresults . 209 11 Systems with inputs 213 11.1Examples ..................................... 213 11.2 Generaldescription . 213 11.3 Linear systems and linearization . ....... 214 11.3.1 LTIsystems................................ 214 11.3.2 Systems described by higher order differential equations. 214 11.3.3 Controlled equilibrium states . .... 214 11.3.4 Linearization ............................... 215 11.4 ResponseofLTIsystems . 217 11.4.1 Convolutionintegral . 217 11.4.2 Impulseresponse ............................. 218 11.4.3 Response to exponentials and sinusoids . ...... 218 11.5Discrete-time .................................. 219 11.5.1 Linear discrete-time systems . .... 219 11.5.2 Discretizationandzeroorderhold . ..... 219 11.5.3 General form of solution for LTI systems . ..... 222 12 Input output systems 225 12.1Examples ..................................... 225 12.2 Generaldescription . 225 12.3 Linear time-invariant systems and linearization . ............ 227 12.3.1 Linear time invariant systems . 227 12.3.2 Linearization ............................... 227 12.4 ResponseofLTIsystems . 228 12.4.1 Convolutionintegral . 228 12.4.2 Impulseresponse ............................. 230 12.4.3 State space realization of an impulse response . ........ 231 12.4.4 Response to exponentials and sinusoids . ...... 232 12.5Discrete-time .................................. 233 12.5.1 Linear discrete-time systems . .... 233 12.5.2 Discretization and zero-order hold . ...... 233 12.5.3 Generalformofsolution . 234 13 Transfer functions 237 13.1 Transfer functions ........................... 237 ←− ←− 13.1.1 Polesandzeros .............................. 244 13.1.2 Discrete-time
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