João P. Hespanha. January 16, 2009
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1 LINEAR SYSTEMS THEORY João P. Hespanha January 16, 2009 Disclaimer: This is a draft and probably contains a few typos. Comments and information about typos are welcome. Please contact the author at hespanha@ece.ucsb.edu. c Copyright to João Hespanha. Please do not distribute this document without the author s consent.
2 Index adjoint matrix, 30 aircraft roll dynamics, , , 221 algebraic equivalence, 33, 36 algebraic Riccati equation (ARE), 81, 172, , 199 applied forces vector, 14 ARE, see algebraic Riccati equation BIBO, see Bounded-input, bounded-output stability block diagram, 7 9, 55, 109, bounded-input, bounded-output stability (BIBO), 164, 166 continuous time, discrete time, 78 Bryson s rule, 170, 173 canonical basis, 38 cascade interconnection, 7, 55 causality, 21, 24, 35, 41 Cayley-Hamilton theorem, 45, 94, 97, 101, 109, 117 centrifugal/coriolis/friction matrix, 14 certainty equivalence, 197 characteristic polynomial, 31, 45 46, 116, 143, 152 cheap control, closed-loop poles, LQR cost, closed-loop state estimator, 136 cofactor, 30, 35 comparison lemma, 64 complementary sensitivity function, 183 complete sequence, 216 conservative force, 14 conservative forces vector, 14 constructibility Gramian, 135, 198 continuous time, constructible system continuous time, 123, 124 discrete time, 126 continuous-time system, 5 controllability Gramian continuous time, infinite time, 102 controllability matrix, 93 98, 107, 116, 144 controllability test controllability matrix, 100 eigenvector, 100, 166, 180, 208 Lyapunov, Popov-Belevitch-Hautus (PBH), 101 controllable canonical form, 32, 33, 35, 105, 117, 143 controllable decomposition, controllable-from-the-origin subspace, see reachable subspace controllable state component, 109 controllable subspace, 104, 108, 133 continuous time, controllable system, 111, 117, 139, 166 continuous time, 99 controllable-to-the-origin subspace, see controllable subspace controlled output, 169, 210, 214, 218 control specification, , convexity, 215, 222 convolution continuous time, 24 discrete time, 26 Laplace transform, 27 coprime fraction, 143 polynomials, 143, 149, 152 CVX, cvx_begin, 218 cvx_end, 218 minimize, 221 variable, 218, 221 degree of a transfer function, 143 detectability test eigenvector, 134 Lyapunov, 135 Popov-Belevitch-Hautus (PBH), 135 detectable system, , 180 diagonalizable matrix, 51, 144 discrete-time system, 6 disturbance rejection, 192 domain of the Riccati operator, 175 duality controllability/observability, reachability/constructibility,
3 LQR/LQG Optimal Control 225 dual algebraic Riccati equation, 199, 208 dual ARE, see dual algebraic Riccati equation eigenvalue assignment, 116 output injection, 136 state feedback, 115 eigenvector test controllability, 100 detectability, 134 observability, 128 stabilizability, 112 equilibrium point continuous time, 11 discrete time, 13 equivalence transformation, see similarity transformation exogenous input, 210, 214, 218 feasibility problem, 215, 216 feedback invariant, 79 81, feedback linearization, forced response, 24, 39 fundamental theorem of linear equations, 90, 100, 101 gain margin, 182 generalized coordinates vector, 14 generic property, 54 greatest common divisor (gcd), 150 Hamiltonian matrix, 83, , , 194 homogeneous response, 23, 29, 39 homogeneous system, 37, 43, 61, 66 hovercraft, 15 Hurwitz matrix, see stability matrix H-infinity norm, see root mean square gain image of a matrix, 90 impulse response, 49 continuous time, 23 25, 27, discrete time, 26, 30 infinite-dimensional systems, 22 initial condition, 5 7, 11, 13, 22, 30, 33, 34, 37, 39, 61, 65, 73, 78 input-output model, 5, 6 internal stability, see Lyapunov stability invariant subspace, 100, 104, 133, 180 invariant zero polynomial, 158 inverse system, left inverse, 162 right inverse, 162 inverted pendulum, 15, 18, 19, 69 irreducible realization, see realization Jordan normal form, 51 55, 62, 71, 180 Kalman decomposition theorem, equality, , 184, 194 filter, 201 inequality, Kalman decomposition, 161 kernel of a matrix, 90 L 1 norm, 214, see peak gain L 2 norm, 214 lag compensator, Laplace transform, 25, 29, 46 convolution, 27 derivative, 27 lead compensator, least common denominator (lcd), 152 left-inverse matrix, 141 Lienard equation, 72 linearity, 22, 35, 41 linear matrix inequality (LMI), 63, 103, 113 linear quadratic Gaussian estimation (LQG), 201 linear quadratic regulation (LQR), 79 82, , linear time-invariant (LTI) system, 6 linear time-varying (LTV) system, 6 L norm, 214 LMI, see linear matrix inequality local linearization, around equilibrium point in continuous time, 12, 18, around equilibrium point in discrete time, 13, 71 around trajectory, 14, loop shaping, loop transfer recovery (LTR), , LQG, see linear quadratic Gaussian estimation LQG/LQR controller, 202 Q-augmented, see Q-augmented LQG/LQR controller LQR, see linear quadratic regulation LTI, see linear time-invariant system LTR, see loop transfer recovery LTV, see linear time-varying system Lyapunov equation, 102, 104 continuous time, 63, 72, 103 discrete time, 66 Lyapunov inequality, 114 continuous time, 63, 103 discrete time, 66 Lyapunov stability asymptotic stability, 61, 62, 65, 66, 103, 111, 115, 134, 136 continuous time, discrete time, exponential stability, 61, 63, 65, 66, 68, 103 instability, 61, 65, 66, 69
4 226 João P. Hespanha marginal stability, 61, 65 stability margin, 72 Lyapunov stability theorem continuous time, 62 65, 103 discrete time, 66 Lyapunov test controllability, 102 detectability, 135 observability, 128 stabilizability, 113 Markov parameters, mass matrix, 14 MATLAB append, 9 bode, 186, 208 ctrbf, 110 ctrb, 98 dlyap, 71 eig, 31, 62, 71 expm, 48 feedback, 9 freqresp, 220 ilaplace, 48 impulse, 220 iztrans, 49 jacobian, 48 jordan, 52 kalman, 207, 208 laplace, 48 linmod, 218 lqr, 82, 172, 186 lyap, 71 minreal, 144, 218 norm, 59 nyquist, 190 obsvf, 137 obsv, 130 parallel, 9 pinv, 144 place, 115 poly, 31 reg, 207, 208 series, 9 sigma, 190 ssdata, 218 ss, 6, 34, 35, 186, 208, 217 step, 219 svd, 59, 98 syms, 47 tf, 34, 35, 217 zpk, 34, 35 ztrans, 49 Simulink, 8 symbolic computation, matrix exponential, 43 46, 49, 53 54, 71 matrix norm, 59 60, 71 submultiplicative, 60, 71 matrix power, 47, 49, McMillan degree, 152, 160, 164 measured output, 169 mechanical systems, fully actuated, 16 MEE, see minimum-energy estimation memoryless system, 5 MI, see multiple-input system MIMO, see multiple-input, multiple-output system minimal realization, see realization minimum-energy control, estimation (MEE), minimum-phase system, 164 MISO, see multiple-input, single-output system MO, see multiple-output system monic polynomial, 143 multiple-input, multiple-output system (MIMO), 5 multiple-input, single-output system (MISO), 5 multiple-input system (MI), 5 multiple-output system (MO), 5 negative-definite matrix, 62 negative-feedback interconnection, 8 negative-semidefinite matrix, 62 noise rejection, 193 norm -norm, 59 Frobenius norm, 59 H-infinity, see root mean square gain L 1, 214, see peak gain L 2, 214 L, 214 one norm, 59 two norm, 59 nullity of a matrix, 90 null space of a matrix, see kernel of a matrix Nyquist plot, 182, 191 stability criterion, 191 observability Gramian, 135 continuous time, observability matrix, 128, 131, 144 observability test eigenvector, 128 Lyapunov, 128 Popov-Belevitch-Hautus (PBH), 128 observable canonical form, 33, 35, 36
5 LQR/LQG Optimal Control 227 observable decomposition, observable state component, 132 observable system, 134, 136, 139, 166 continuous time, 122, 124 discrete time, 126 open-loop state estimator, 135 orthogonal complement, 90 orthogonal matrix, 98, 190, 194 output feedback, 121, 135 output injection, 136 overactuated system, 204 overshoot, 192, 214, 219, 221 parallel interconnection, 7, 55, 123 peak gain, 215, 221 Peano-Baker series, 37, 43 phase margin, 183, 192 poles, 152, 157, 160 SISO, 143, 149 pole placement, see eigenvalue assignment pole polynomial, see characteristic polynomial Popov-Belevitch-Hautus (PBH) test controllability test, 101 detectability test, 135 observability test, 128 stabilizability, 113 positive-definite matrix, 62 positive-semidefinite matrix, 62 prefilter, 213 proofs by contradiction, 64, 75 by contraposition, 75 contraposition, 101 direct, 31 equivalence, 31, 63, 101 set equality, 91 proper rational function, 31 proportional-derivative (PD) control, 16 proportional (P) control, 194 pseudoinverse matrix, 141, 204 Q-augmented LQG/LQR controller, , 218 Q design, Q parameterization, Q system, 209, 215 range of a matrix, see image of a matrix rank of a matrix, 90, 141 reachability Gramian continuous time, infinite time, 102 reachable subspace continuous time, reachable system continuous time, 99 realization, 30 minimal, 83, , , 166 order of, 139, SISO, 32, 35 real polynomial matrix, 150 determinantal divisors, 150 invariant factors, 151 minors, 150 rank, 150 unimodular, 151 real rational matrix, 151 reference tracking, 192 Riemann integral, 27 right-inverse matrix, 141 Ritz approximation, RMS gain, see root mean square gain robot arm, 15 roll-off rate, 185 root locus, 189 root mean square gain, 215, 221 Rosenbrock s system matrix, 158, 204 satellite, 105 Schur stable matrix, 47, 53, 66, 128 semigroup property, see state transition matrix semisimple matrix, see diagonalizable matrix sensitivity function, 183 separation principle, 137, 202, 208 separation theorem, see separation principle set-point control, 187, , 208 settling time, 214, 219 SI, see single-input system similarity transformation, 33, , , SIMO, see single-input, multiple-output system Simulink, see MATLAB single-input, multiple-output system (SIMO), 5 single-input, single-output system (SISO), 5 single-input system (SI), 5 single-output system (SO), 5 SISO, see single-input, single-output system Smith factorization, 151 form, , 158, 160 Smith-McMillan factorization, 152 form, , 157, 160, 163 SO, see single-output system square completion, 81, 114, 115, 172 stability matrix, 46, 54, 61, 63, 71, 102, 103, 128, 172 stabilizability test eigenvector, 112
6 228 João P. Hespanha Lyapunov, Popov-Belevitch-Hautus (PBH), 113 stabilizable system, stabilization output feedback, state feedback, , stabilizing solution to the ARE, 176 stable subspace, 176 standard form for uncontrollable systems, 108, 111 unobservable systems, 132, 134 state estimation, state estimation error, 136 state feedback, 121, 135 state reconstruction Gramian-based in continuous time, Gramian-based in discrete time, 126 state transition matrix, 41 continuous time, discrete time, 40 inverse, 39, 40 semigroup property, 38, 40 Stein equation, see Lyapunov equation, discrete time step response, 214, 219 strict feedback form, 19 strictly proper rational function, 31 strict feedback form, unicycle, 18 uniformly-bounded signal, 74 unobservable state component, 132 unobservable subspace, 131 continuous time, variation of constants formula continuous time, 39, 44, 87, 122, 134 discrete time, 40, 47 Z-transform, 26, 35 zero-state equivalence, 30, 33, 36, 140 zero-state response, 24 zeros invariant, , 165, 204 SISO, 143, 149 transmission, 149, , , , 189, 190, 203, 204 zero polynomial, 152 time invariance, 22, 24, 35 time reversibility controllability/reachability, 93 observability/constructibility, 127 time scaling controllability and reachability, 93 observability/constructibility, 127 time shifting, 22 tracking error, 192 transfer function continuous time, 25 26, discrete time, 26, 30 transmission-blocking property invariant zeros, 159 transmission zeros, transmission blocking property, 204 transpose of a vector/matrix, 7 unconstructible subspace continuous time, uncontrollable state component, 109 uncontrollable system continuous time, 99 under-actuated system, 203 undershoot, 214, 219
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