Index

\ (backslash), 119 basin of attraction, 273 basis, 58, 65, 86, 87, 89, 97, 101 ' (single apostrophe), 116 definition of, 58 of a linear functional, 196 : (colon), 117 of eigenvectors, 155, 158, 245 orthogonal, 87 orthonormal, 73, 81 A vectors, 71 Bass-Gura formula, 416 a posteriori estimate, 483 Bellman’s principle of optimality, a priori estimate, 483 459 ACKER, 413, 530 bias point, 30 Ackermann’s formula, 413, 416, BIBO stability, 298 437 of discrete-time systems, 295 additivity, 5 of time-invariant systems, 297 adjoint operator, 116 testing for, 298 definition of, 116 BIBS stability, 298 adjoint system, 265 bijective mapping, 60 affine system, 36 bilinear form, 195, 222 A-invariant subspaces as inner products, 199 definition of, 147 definition of, 197 algebra, 114 block matrices, 515 algebraic multiplicity, 158, 159, bottom-up method, 159 160 bounded operator, 116 analytic function, 30, 210 angle, 70 anticipatory system, 5 C associativity, 50, 83, 516 augmented system, 433, 437, 502 C2D, 258, 531 automobile suspension, 38 CANON, 155, 326, 531 canonical form, 18, 155 CARE, 473, 474, 478, 532 B Cauchy-Schwarz inequality, 69, balanced model, 401 275 balanced realization, 387, 388 causality BALREAL, 388, 530 definition of, 5 Banach space, 71 Cayley-Hamilton theorem, 209, 210, 212, 313, 329 562 Fundamentals of linear State Space Systems

CDF2RDF, 377, 533 geometric interpretation, 331 center, 245, 274 grammian, 340, 343, 388, 390 chains of generalized eigenvectors, index, 422-424 162 matrix, 313 change of basis, 62, 64, 86, 236, of discrete-time systems, 127 247, 425 of time-varying systems, 339, of a linear operator, 108 343, 344 of a quadratic form, 202 output, 364 change of basis testing, 312, 319 of a linear functional, 198 under sampling, 357 change of basis matrix, 65, 109, using Jordan forms, 324 111, 199, 203, 334 controllable canonical form, 326, change of variables, 19, 21 329, 410, 412 characteristic polynomial, 150, controllable canonical realization, 209, 216 373 CHOL, 533 controllable modes, 334 Cholesky decomposition, 523 controllable subspace, 333, 424 circuit conservation, 7, 8 controller, 433 closed-loop controller, 436 CONV, 534 closed-loop observer, 431 convolution integral, 230 closure coordinate direction, 46, 52, 58, 60, multiplication, 48 72 cofactor, 518 coprime polynomials, 372 colored noise, 505 cost function, 460 commutative diagram, 114 covariance, 205 commutativity, 50 covariance matrix, 205, 482 COMPAN, 327, 533 Cramer’s rule, 116 companion form, 327, 426 critical points, 270 compensator, 447 CROSS, 534 complete space, 71 cross product, 47 composite rotations, 102 CTRB, 313, 369, 413, 427, 534 computer graphics, 106 CTRBF, 334, 370, 436, 535 COND, 179, 533 current estimator, 443, 444, 483 condition number, 179, 430 conic sections, 206 constitutive relationships, 8 D control, 405 damped pendulum, 273 control energy, 389 damper, 9, 10 controllability, 311, 359 DARE, 474, 535 after feedback, 420 deconv, 400 comparison to pole placement, DECONV, 536 418 definition of, 312 decoupled system, 190, 237 effect of similarity degeneracy, 66, 67, 87, 139 transformations, 330 DET, 54, 536

Index 563 detectability, 361, 476 eigensystem realization algorithm, definition of, 339 394, 395 determinant, 54, 517 simplified, 396 DIAG, 536 eigenvalues, 147, 153, 187, 188, diagonal matrix, 156, 173, 523 190 diagonal system, 237 complex, 152 diagonalization, 156, 215 definition of, 148 difference equation, 4, 23, 24 frozen-time, 302 state space, 25 of time-varying systems, 279 differential eigenvectors, 147, 151, 153, 187, of a vector, 524 188, 190 differential equation, 4, 13 complex, 152 discretized, 26 definition of, 148 partial, 248 electric fields, 149 differentiation ellipsoid, 204, 206 of a matrix, 524 energy, 280, 388 of matrix products, 526 equilibria of vector-matrix products, 525 multiple, 273 dimension, 88 equilibrium point, 30, 35, 269 definition of, 57 definition of, 270 discrete-time system, 23, 26, 27 error covariance, 482, 484-490, 493 solutions of, 257 error dynamics, 441, 443, 484, 492 time-varying, 260 error signal, 125, 431 discretization, 23, 258, 261, 357 error system, 436, 441 discretized differential equations, ESTIM, 431, 487, 539 26 estimation, 457 dissipative systems, 280 estimator, 431 distinct eigenvalues, 158, 167, 214 euclidean norm, 115, 177 distributed-parameter systems, 248 exogenous input, 407, 409 distributivity, 50, 83, 516 expected value, 483 disturbance rejection, 501 experimental data analysis, 130 DLQR, 537 EXPM, 213, 539 DLYAP, 290, 537 external stability, 272, 292 DOT, 46, 538 EYE, 94, 540 double integrator, 231, 509 duality, 128, 313, 315, 339, 345 dynamic programming, 460, 506 F feedback, 332 E feedback gain, 406, 407, 443 feedforward, 16 EIG, 148, 538 feedthrough matrix, 15, 22, 28 eigenfunctions, 248 Fibonacci sequence, 400 eigenspace field, 48 definition of, 158 finite Fourier series, 79

564 Fundamentals of linear State Space Systems

flexible structures, 246 controllability, 340, 343, 388 flow variable, 6, 9, 10 observability, 345, 357, 388 free variables, 120 Gram-Schmidt orthonormalization, Frobenius norm, 116, 179 71, 78, 88 full order observer, 433, 438 function on a vector space, 195 H function space, 68, 153 hamiltonian, 123, 203, 469, 470 functional observer, 445, 446 Hamilton-Jacobi-Bellman, 469, functionals, 195 470 functions HANKEL, 541 of a matrix, 208, 212 Hankel matrix, 383-386, 395, 396, fundamental solution matrix, 252, 400, 401, 522 253, 254 infinite, 384 fundamental test for controllability, modified, 396 312, 360 harmonic oscillator, 26 fundamental test for observability, Hautus test, 319-322 312, 360 hermitian FUNM, 212, 540 functional (form), 198 matrix, 169, 223, 517 G operator, 116, 190 hermitian matrix gain, 406 eigenvectors of, 170 gain matrix, 408, 428, 429, 448 Hilbert space, 70 choosing, 409 homogeneity, 5 gaussian elimination, 116 homogeneous equation, 234 gaussian noise, 495 homogeneous part general solution of a solution of simultaneous of a solution of simultaneous equations, 119, 120, 143 equations, 119 homogeneous system, 5, 232, 252 generalized eigenvector, 159-161, Hooke’s law, 10 166, 191, 240 hyperplanes, 147 chain of, 162 hypersurfaces, 76 generalized inverse, 521 Moore-Penrose, 521 geometric multiplicity, 159, 165, I 173, 218, 322 idempotent, 522 geometry identity, 94 of quadratic forms, 203 ill-conditioned, 179 gradient, 96, 525 image, 94 GRAM, 340, 345, 351, 388, 540 impulse-response matrix, 293, 297 Gram determinant, 90 indefinite, 201 Gram matrix, 53 index of eigenvalue, 163, 166, 173 grammian, 53 definition of, 162

Index 565 individual observability indices, Jordan form, 215 451 infinite-dimensional spaces, 248 infinite-horizon control, 473 K INITIAL, 233, 541 KALMAN, 495, 542 initial condition response, 232, 241, Kalman controllability 243 decomposition, 436 initial conditions, 27 Kalman decomposition, 334, 336, injective, 94 370, 399 inner product, 46, 67, 70, 72, 169, Kalman filter, 458, 487, 500, 503, 199 511, 512 definition of, 67 comparison to linear quadratic space, 68, 90 regulator (LQR), 496 input, 4 continuous-time, 491, 495 input matrix, 15 discrete-time, 483 inset, 273 example, 497 integral table, 527 properties of, 495 integrating factor, 253 simplifications, 488 integration steady-state behavior, 496 of a matrix, 524 Kalman gain, 487, 492, 512 integrator diagram, 373 kinematic singularity, 184 integrator realization, 372, 375 kinematics, 181 integrators, 17, 24 Kirchoff’s current law (KCL), 7 internal stability, 271 Kirchoff’s voltage law (KVL), 8 internally balanced, 390, 392 KRON, 543 INV, 45, 94, 542 Kronecker product, 508, 516 invariant space, 238 invariant subspaces, 233 inverse, 518 L operator, 112, 114 inverted pendulum, 33, 451 LaGrange multiplier, 123, 203 invertible, 94 LaPlace transform, 28, 265, 299, irreducible realization 368 definition of, 370 LaSalle’s theorem, 285 isomorphism, 60 leading principal minors, 202 least-squared error, 143, 181 least-squared error solution, 130, J 131 left eigenvector, 319 jacobian, 183-185, 205, 525 LeGendre polynomials, 89 JORDAN, 173 linear algebra Jordan block, 172, 189, 216 definition of, 83 Jordan canonical form, 159, 172, linear combination, 52 175, 188, 241, 261, 302 linear dependence, 52, 57 Jordan canonical realizations, 375 definition of, 52

566 Fundamentals of linear State Space Systems

linear functional, 222 Lyapunov’s direct method, 280, definition of, 195 286 linear independence, 52, 66, 86 Lyapunov’s first method, 302 definition of, 52 Lyapunov’s linearization method, of eigenvectors, 167 302 of functions, 53, 80, 90 linear operator, 93 definition of, 94 M matrix representation of, 97, 99, manipulability ellipses, 185 109 Markov parameters, 378-383, 386, linear quadratic gaussian (LQG), 393-396, 400 500, 501 mass conservation, 7 example, 501 matrix exponential, 213, 218, 223, linear quadratic regulator (LQR), 231, 232, 252 457, 472, 507 matrix inverse, 138, 179 comparison to Kalman filter, 496 matrix inversion lemma, 134, 489, continuous-time, 465 520 discrete-time, 460, 463 matrix polynomial, 209 example, 477 matrix representation linear region, 30 of a linear operator, 99, 101, linear system 138, 154 definition of, 3 of the projection operator, 104 linear transformation, 93 matrix Riccati equation, 463 linearity McClaurin series, 400 definition of, 5 measurement noise, 482, 492 linearization, 29, 30 measurement update, 444, 487 of differential equations, 32, 33 measurements, 131 linear-quadratic-gaussian (LQG), memory, 4 458 metric, 70 LQE, 543 metric space, 70 LQGREG, 487 MIMO systems, 15 LQR, 472, 544 controllability, 324 LSIM, 231, 260, 544 observability, 324 LU decomposition, 524 minimal polynomial, 220 LYAP, 286, 390, 545 definition of, 219 Lyapunov equation, 286, 363, 390, minimal realization, 368, 371, 383, 401, 475 384, 395, 399 discrete-time, 290 definition of, 370 Lyapunov function, 282-286, 303, minimum energy, 509 304 minimum-norm solutions, 123 of time-invariant systems, 285 modal controllability, 331 of time-varying systems, 282 modal decomposition, 245, 247, Lyapunov stability, 269 248 discrete-time, 261

Index 567 modal matrix, 156, 157, 167, 172, NULL, 94, 119, 120, 151, 158, 270, 174, 218, 247 546 modal observability, 331 null space, 94, 96, 139, 151 model, 393 definition of, 95 model reduction, 387 of a projection, 95 modes, 245-247 nullity, 118 discrete-time, 261 definition of, 66 modeshapes, 248 moment of inertia, 171 motor, 38 O multi-input pole placement observability, 311, 359 example, 426 after feedback, 420 multilinear functionals, 197 definition of, 312 multiple and nonexistent solutions geometric interpretation, 331 of simultaneous equations, 118 grammian, 345, 346, 357, 388- multivariable canonical forms, 421 391 index, 446, 451 N matrix, 313 of discrete-time systems, 128, negative definite, 201, 283 353 negative semidefinite, 201, 283 of time-varying systems, 339, Newton’s law, 7, 11 347 nilpotent, 255, 522 under sampling, 357 noise, 482, 491, 501 using Jordan forms, 321, 324 noise floor, 503 observable canonical form, 326, nonexistent solutions 432 of algebraic equations, 122 observable canonical realization, nonlinear equations, 29 374 nonlinear system, 6, 270, 273, 284, observable modes, 334 303 observation energy, 389 nonminimal realization, 399 observer, 405, 430 nonsingular matrix, 66 discrete-time, 442 nonstrictly proper transfer function, full versus reduced, 441 28, 375 gain, 435, 443, 492 norm, 68, 177 open-loop, 431 definition of, 68 poles, 433 euclidean, 69, 74, 115 reduced order, 438, 439 Frobenius, 116 observer and controller lk, 69 example, 437 matrix, 200 OBSV, 313, 547 of an operator, 115, 277 OBSVF, 335, 547 NORM, 47, 68, 115, 545 ODE23, 547 normed linear spaces, 69 Ohm’s law, 12 notch filter, 503 one-to-one, 94

568 Fundamentals of linear State Space Systems

OP-AMP, 40 permanent magnet DC motor, 42, operating point, 30 449 operator norm, 115 phase, 373 optimal control, 430, 457, 458, portrait, 232-234, 240, 264, 273 469, 475 trajectories, 233, 234 optimal estimator, 481, 483 variables, 14, 16, 17 optimal policy, 459 phase-plane, 233 optimization criterion, 457 PINV, 124, 549 ORTH, 71, 548 pitch, roll, and yaw, 102 orthogonal complement, 77, 142 pivot element, 119 orthogonal matrix, 523 pivot variables, 120 orthogonal projection, 76, 80, 88, PLACE, 410, 416, 441, 549 105 plant noise, 492 orthogonal vectors, 71 pole placement, 410 orthogonality pole-zero cancellation, 372 definition of, 71 POLYVALM, 209, 416, 550 orthonormal matrix, 523 Popov-Belevitch-Hautus (PBH) orthonormal vectors, 71 tests, 319 orthonormality population dynamics, 25 definition of, 71 positive definite, 200, 201, 206, outer product, 47 223, 283 output, 4 positive semidefinite, 201 controllability, 364 potential variable, 6, 9, 10 equation, 14, 16, 24 predator-prey system, 25 feedback, 406, 445, 449, 453 prediction, 444 matrix, 15 preimage, 94 variable, 20 principal axes, 172 overdetermined system, 125 principal minors, 202 principle of optimality, 458, 462 process noise, 482 P projection operator, 140, 180 Padé approximation, 400 projection theorem, 75, 77, 96 Parseval’s theorem, 74, 82 pseudo-controllability indices, 397 partial-fraction expansion, 375 pseudoinverse, 124, 126, 133, 144, partial realization, 385 181-184, 521 particular part pseudo-observability indices, 397 of a solution of simultaneous equations, 119, 120, 143 PBH (Popov-Belevitch-Hautus) Q eigenvector test, 319 QR factorization, 523, 550 PBH (Popov-Belevitch-Hautus) quadratic forms, 195, 200, 389 rank test, 320 qualitative analysis Peano-Baker integral series, 254 from phase portraits, 250

Index 569

R Riccati equation, 463, 472-475, 478, 479, 495, 496, 503, 508, RANDN, 551 512 random variable, 482 algebraic, 473 range, 94 differential, 472, 494 range space, 94, 96, 98, 139 right eigenvector, 319 of a projection, 95 robot, 42, 204, 510 rank, 66, 87, 120, 139 robotics, 181 definition of, 66 ROOTS, 553 determination of, 179 rotation matrices, 100 RANK, 66, 427, 551 rotation operator, 141, 152, 187 rational polynomials, 54 Routh-Hurwitz method, 299 reachability, 347 row-reduced echelon form, 119 grammian, 351 RREF, 119, 553 matrix, 349, 350 of discrete-time systems, 348 realizability S definition of, 368 realization, 367 saddle point, 238, 239, 274 sampling period, 23, 258 balanced, 387 sampling time, 26 controllable, 373 savings account, 24 definition of, 368 from Hankel matrix, 384, 385 Schur complement, 519 Jordan canonical, 375 selection vectors, 414 self-adjoint matrix, 517 observable canonical, 374 self-adjoint operator, 188 partial, 385 separation principle, 435, 443 reconstructibility, 347 sesquilinear form, 198 definition of, 348, 355 grammian, 364 Shannon sampling period, 258 matrix, 356 similar operators, 109 similar systems, 190 testing, 356 similarity transformation, 109, 156, recursive least-squares, 133, 135, 216, 218, 236 144 simulation diagram, 16, 17, 27 reduced-order observer, 438, 439 comparison to full-order simultaneous equations, 78, 83, observer, 441 116, 180 singular matrix, 66 REG, 437, 463, 472, 503, 551 singular points, 239 regular eigenvector, 159, 161, 166 singular value decomposition regulator, 406 (SVD), 176, 183, 185, 392, 395 repeated eigenvalues, 157, 214 definition of, 175 representation, 62 singular values, 175, 186 of a vector, 59, 66, 97 applications of, 177 RESIDUE, 375, 552 singular vectors, 176, 185, 244 resolvent, 521 singularity, 66 resolvent identity, 379, 417, 521

570 Fundamentals of linear State Space Systems

skew-symmetric matrix, 517 properties, 418 Slash, 553 static, 408, 446 space of functions, 190 state matrix, 15 state space, 13 span state-transition matrix, 252, 257, definition of, 60 264, 265 spectrum, 233 discrete-time, 259, 260 squared-error criterion, 130 properties of, 256 SS, 14, 229, 554 state variables, 3, 13, 16 SS2SS, 109, 392, 425, 555 changing, 19, 20 SS2TF, 29, 368, 555 state-transition matrix, 254, 492 SSDATA, 14, 554 computing, 254 stability, 269 stationary point, 270 asymptotic, 272, 276, 283 statistical error analysis, 204 bounded input/bounded output STEP, 234, 556 (BIBO), 292-294, 304 step response, 234, 237 bounded input/bounded state strictly proper, 375 (BIBS), 293 subspace, 70, 75, 87-90, 140 comparing types, 299 definition of, 75 exponential, 272 proper, 75 external, 272 SUBSPACE, 556 global, 272, 274, 283 sufficient excitation, 394 in the large, 272 supremum, 115 in the sense of Lyapunov, 271, SVD, 176, 557 275, 280 symmetric internal, 271 functional (form), 198 local, 272 matrix, 517 of discrete-time systems, 278 operators, 116 of time-invariant systems, 283 system testing for, 274 causal, 5 uniform, 271, 275, 282 definition of, 3 stabilizability, 361, 475 difference, 4 definition of, 339 differential, 4 stable focus, 243, 274 linear, 5, 37 stable node, 234, 236, 240, 242, nonlinear, 37 274 time-invariant, 37 stacking operator, 516 time-varying, 37 standard basis, 156 system identification, 6, 367 state, 13 for discrete-time systems, 393 state equations, 13, 15 system modes, 331 discrete-time, 23 solutions of, 229 state feedback, 405-407, 428 T dynamic, 446 tangent vector, 233, 240, 244 formulae, 413

Index 571

Taylor series, 29, 32, 205, 210, unstable systems, 291 211, 215, 231, 467 TF2SS, 29, 557 time invariance, 5, 230 V time update, 444, 483, 487 Vandermonde matrix, 522 time-varying system, 5, 24 vector, 45 solution of, 252 euclidean, 46 TOEPLITZ, 418, 558 vector-matrix notation, 14, 21, 32, Toeplitz matrix, 418, 522 98, 129, 132, 197 top-down method, 163 vector space, 48, 50 TRACE, 116, 485, 558 Banach, 71 trace operator, 485, 521 euclidean, 70 transfer function, 3, 369 Hilbert, 70 improper, 29 of functions, 68, 79, 88-90, 153, nonstrictly proper, 375 190 proper, 368 of operators, 114 relationship to state space, 27 of polynomials, 60, 85-90, 109, strictly proper, 375 142, 153, 188, 189 transpose, 116, 517 of rational polynomials, 54 triangle inequality, 68 over a field, 49 triangular matrix, 154, 212, 523 Venn diagram, 299 viscous friction, 38, 42, 273 U unbounded operator, 295 W underdetermined system, 123, 184 weighting function, 200 uniform stability, 271 weighting matrices, 460, 480 uniqueness of solutions, 118, 130, well-conditioned, 179 252 wind turbine, 450 of algebraic equations, 122 unit delay operator, 27 unit vector, 69 Z unitary space, 70 unobservable subspace, 333 zero-input response, 232 unstable node, 274 zeros, 420

572 Fundamentals of linear State Space Systems