16.Bibliography.Index
.pdfThis version: 22/10/2004
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This version: 22/10/2004
|
Symbol Index |
Symbol |
: Description and page(s) when applicable |
(i) |
: the ith derivative with respect to time, 597 |
|
|
f0(z) |
: derivative of f at z, 619 |
L |
: adjoint of the linear map L, 594 |
S \ A |
: set theoretic complement of A in S, 618 |
f g |
: convolution of f and g, 632 |
z¯ |
: complex conjugate of z, 617 |
fˇ |
: Fourier transform of f, 631 |
k·k |
: norm, 595 |
˙: derivative with respect to time
: empty set
,: equal by definition, i
|·| |
: modulus of a complex number, or absolute value of a real number, 617 |
0n |
: n × n matrix of zeros, 587 |
0m,n |
: m × n matrix of zeros, 587 |
1(t) |
: the unit step function, 607 |
1N,D |
: step response of SISO linear system (N, D) in input/output form, 108 |
A: state dynamics matrix for a SISO linear system, 24
A(ˆı, |ˆ) |
: (n − 1) × (n − 1) matrix obtained by deleting the ith row and the jth |
|
column of A Rn×n, 588 |
adj(A) |
: adjugate of the matrix A, 588 |
b: input vector for a SISO linear system, 24
ct |
: output vector for a SISO linear system, 24 |
C(A, b) |
: controllability matrix for (A, b), 38 |
c.f.r. |
: canonical fractional representative, 612 |
C: complex numbers, 617
C+ |
: complex numbers with positive real part, 610 |
|
C |
+ |
: complex numbers with nonnegative real part, 610 |
C− |
: complex numbers with negative real part, 610 |
|
C |
− |
: complex numbers with nonpositive real part, 610 |
C[ξ] |
: polynomials with complex coe cients in indeterminant ξ, 609 |
|
C(ξ) |
: rational functions with complex coe cients in indeterminant ξ, 611 |
D: feedforward term for a SISO linear system, 24
D(z, r) |
: open disk of radius r centred at z, 618 |
D(z, r) |
: closed disk of radius r centred at z, 618 |
δ(t) |
: the delta-function at 0, 629 |
det A |
: determinant of the matrix A, 587 |
dim(V ) |
: dimension of the vector space V , 586 |
650 |
Symbol Index |
Df |
: the Jacobian matrix for a map f : Rn → Rm, 12 |
e(t) |
: the error for an observer, 430 |
ei |
: ith standard basis vector for Rn, 586 |
F: general field, standing typically for either R or C, 585
F[s]n×n |
: set of n × n matrices with components in F[s], 231 |
F: Fourier transform, 632
F−1 |
: inverse Fourier transform, 632 |
h+ |
: causal impulse response for the SISO linear system (N, D) in in- |
N,D |
|
|
put/output form, 94 |
h+ |
: causal impulse response for the SISO linear system Σ, 49 |
Σ |
|
hN,D− |
: anticausal impulse response for the SISO linear system (N, D) in in- |
|
put/output form, 94 |
hΣ− |
: anticausal impulse response for the SISO linear system Σ, 53 |
hN,D |
: causal impulse response for the SISO linear system (N, D) in in- |
|
put/output form, 94 |
hΣ |
: causal impulse response for the SISO linear system Σ, 49 |
HΣ |
: frequency response for SISO linear system Σ, 120 |
I: a generic interval, 26
id |
: identity map, 592 |
Im |
: imaginary part of a complex number, 617 |
image |
: image of a linear map or matrix, 592 |
In |
: n × n identity matrix, 587 |
iR |
: imaginary axis, 610 |
I: set of intervals, 26
ker |
: |
kernel of a linear map or matrix, 592 |
lim sup |
: |
limit of the least upper bounds, 96 |
L (f) |
: |
two-sided Laplace transform of f, 633 |
L0++ (f) |
: |
the right causal Laplace transform, 634 |
L0+−(f) |
: |
the left causal Laplace transform, 634 |
L0+− (f) |
: |
the right anticausal Laplace transform, 634 |
L0−−(f) |
: |
the left anticausal Laplace transform, 634 |
L −1 |
: |
two-sided inverse Laplace transform, 633 |
n: state dimension of a generic SISO linear system Σ = (A, b, ct, D), or the degree of the denominator polynomial in a generic SISO linear system (N, D) in input/output form, 24
(N, D) |
: generic SISO linear system in input/output form, 91 |
NΣ |
: matrix whose eigenvalues are the spectrum for the zero dynamics, 46 |
O(A, c) |
: observability matrix for (A, c), 34 |
F(s)n×n |
: set of n × n matrices with components in F(s), 216 |
Re |
: real part of a complex number, 617 |
R: real numbers, 585
R+ |
: |
positive real numbers, 585 |
R[ξ] |
: |
polynomials with real coe cients in indeterminant ξ, 609 |
|
Symbol Index |
651 |
652 |
Symbol Index |
R(ξ) |
: rational functions with real coe cients in indeterminant ξ, 611 |
|
|
|
RN,D |
: ramp response of SISO linear system (N, D) in input/output form, 108 |
|
|
|
Rc(f) |
: region of absolute convergence for f, 633 |
|
|
|
Σ: SISO linear system of the form (A, b, ct, D) (might on occasion be MIMO), 24
σmax(f) |
: maximum abscissa of absolute convergence of f, 633 |
σmin(f) |
: minimum abscissa of absolute convergence of f, 633 |
ΣN,D |
: canonical minimal realisation of input/output system (N, D), 93 |
Sn |
: set of permutations of (1, . . . , n), 588 |
span {S} |
: smallest subspace containing S, 587 |
spec(A) |
: spectrum of A Rn×n, 593 |
spec(P ) |
: set of roots of a polynomial P , 610 |
T (A, b) |
: largest subspace of Rn on which every curve can be followed by Σ = |
|
(A, b, c, D), 42 |
At |
: transpose of the matrix A, 587 |
TN,D |
: transfer function for SISO linear system (N, D) in input/output form, |
|
91 |
tr(A) |
: trace of the matrix A, 587 |
T: the vector space of test functions, 628
TΣ |
: transfer function of a SISO linear system Σ, 79 |
u: input to a SISO linear system, 24
U: set of admissible input, 26
ΩΣ |
: set of imaginary axis poles for TΣ, 119 |
x: state vector for a SISO linear system, 24
y: output of a SISO linear system, 24
yh |
: |
homogeneous solution to a scalar ordinary di erential equation, 597 |
yp |
: |
particular solution to a scalar ordinary di erential equation, 598 |
Z: set of integers, i
Zk |
: sequence of subspaces defined for computing the spectrum of the zero |
|
dynamics, 45 |
ZΣ |
: subspace used in computation of zero dynamics, 45 |
This version: 22/10/2004
Index
abscissa of absolute convergence, 122, 363, 641
maximum, 633 minimum, 633
adjoint, 594 adjugate, 79, 588
admissible, 26, 49, 53, 93, 94, 116, 423, 425, 426, 443, 444, 526, 564
allowable, 147, 148 amplifier, 15
analogue computer, 93 analytic, 619
anticausal, 26, 53, 94, 98, 567, 574 appended system, 230
arc, 620 argument, 617
asymptote, 130, 150 asymptotic observer, 430
bandwidth, 347, 374, 375, 477, 483
and gain crossover frequency, 348, 477, 483
and performance, 321, 325 and rise time, 321, 325–327 for first-order systems, 321 for general systems, 326, 347 for second-order systems, 325
basis, 586 bijective, 592
Blaschke product, 377
block diagram, 3, 76, 92, 93, 112, 116, 244, 279, 352, 353, 420
and signal flow graphs, 212, 271 for PID control, 261, 268
for state-space models, 79, 250, 255, 259, 357, 418
negative feedback, 64, 77, 213 parallel, 63, 76, 213
series, 63, 76, 213
unity gain, 247, 281, 333, 342, 344, 361, 368, 382
Bode integral formulae, 369, 375, 382 and performance, 372
Bode plot, 124, 124, 132, 135, 138, 152, 154–156, 276, 308, 348, 352, 353, 389–392, 554
and minimum phase, 137, 156
and Nyquist plot, 135, 285, 293, 294, 475, 490
and the H∞-norm, 146 by hand, 129
for closed-loop transfer function, 374 for first-order systems, 321
for lead/lag compensation, 470 for PID compensator, 473
for second-order systems, 153, 324 for sensitivity function, 373–375
magnitude, 124, 132, 133, 135, 146, 148, 154, 156, 308, 311, 321, 348, 352, 353, 373–375, 377, 383, 390, 391, 480, 554
phase, 124, 135, 137, 144, 154, 294, 306, 325, 353, 470, 491
Bode’s Gain/Phase Theorem, 138, 143, 157, 279, 361
boundary point, 618 bounded, 618 branch, 212
tree, 10
break frequency, 130, 321, 472
c.f.r., see canonical fractional representative cancellation, 81, 83–87, 93, 120, 136, 189,
241–243, 246, 250, 276, 284, 288, 290, 398, 405, 560
canonical form, 54, 59 controllability, 70 controllable, 407
controller, 55, 56, 59, 70, 93, 253, 358, 398, 406, 526, 527, 541, 542, 545, 552, 573
first Luenberger-Brunovsky, 70, 71 Jordan, 73, 274
observability, 71
observer, 57, 58, 59, 93, 116, 256, 400, 404, 411
second Luenberger-Brunovsky, 56, 58 canonical fractional representative, 612 canonical minimal realisation, 93, 108, 115,
248, 259, 272, 274, 418, 428, 575 capacitor, 10
Cauchy’s Integral Theorem, 621
654
Cauchy-Riemann equations, 619 causal, 26, 48, 94, 95, 153, 635
Cayley-Hamilton Theorem, 36, 37, 40, 41, 56, 61, 593
change of basis matrix, 593
change of coordinates, 54, 57–59, 67, 81, 407, 542, 545, 592
for linear maps, 593 for vectors, 593
characteristic equation, 597
characteristic polynomial, 55, 63–65, 78, 82, 137, 249, 252, 257, 271, 272, 274, 526, 593
for interconnected systems, 237, 240–242 Chipart, 199
closed circuit, 114 closed set, 618
closed-loop, 6, 211, 244, 251, 253, 255, 256, 258, 259, 279, 333, 397, 417, 425
poles, 246, 252, 253, 268, 273, 275, 276, 281, 364, 409, 497, 499
response, 247, 258, 261, 364, 367, 393, 469, 480, 496, 499, 523, 544
cofactor
of a matrix, 588
of a signal flow graph, 219, 222, 223 columnspace, 592
complete, 41, 43, 92, 115, 120, 275, 409, 573, 574, 576
complex conjugate, 617 complex numbers, 585 components
of a linear map, 592 of a vector, 587
connected, 218, 618 continuous, 618, 628
contour, 140, 279, 281, 286, 313, 369, 621 controllability, 38, 38, 41, 42, 44, 45, 48, 55,
57, 59–61, 66–68, 70, 73, 92, 93, 97, 114–116, 156, 164, 168, 252, 260, 273, 358, 398–400, 403, 404, 406, 408, 410, 415, 417, 441, 442, 542, 545, 547, 552, 555, 576
and coprime polynomials, 81, 83–87 for interconnected systems, 243
for MIMO systems, 43 Gramian, 576
matrix, 38, 41 operator, 575
controllability canonical form, 70
Index
controlled output, 329
controller, 1, 6, 7, 146, 150, 244, 246, 249, 250, 273, 304, 307, 315, 333, 338, 342, 362, 374, 382, 384, 386, 389, 391, 392, 394, 397, 416–418, 420, 443, 469, 475, 477, 483, 486, 488, 557, 558, 561, 581
optimal, 525, 543, 544, 550
proper, 260, 354, 425, 426, 444, 557, 560, 562, 570
stabilising, 405, 418, 420, 422, 423, 426, 427, 443, 444, 487, 562, 564
strongly stabilising, 429
controller canonical form, 55, 56, 59, 70, 93, 253, 358, 398, 406, 526, 527, 541, 542, 545, 552, 573
convergence
of test functions, 628
convolution, 27, 49, 95, 102, 103, 107, 112, 152, 165, 177, 632, 641
for Laplace transforms, 636 convolution for Fourier transforms, 632 coprime, 81, 83–87, 91, 93, 114, 123, 136,
250, 276, 334, 403, 535, 538, 610, 611, 612, 612, 616
factorisation, 422, 423–425, 444, 521, 560–562, 581
fractional representative, 421, 423–427, 444, 521, 561, 562, 581
rational functions, 421, 421, 422, 423 coupled masses, 15, 67, 115, 155, 205, 207,
275 coupled tanks, 18
Cramer’s Rule, 223, 588 curve, 620
closed, 620 di erentiable, 620 Jordan, 620 length, 621 simple, 620
cuto frequency lower, 347 upper, 347
dB/decade, 130
DC servo motor, 2, 11, 15, 211, 275, 276 decibel, 124
degree, 609 delta-function, 629 derivative, 619
of a distribution, 629
Index
derivative control, 263, 264, 266, 276, 339, 443
derivative time, 261, 263, 474, 499 design problem
for dynamic output feedback, 260 for input/output systems, 246 for static output feedback, 256 for static state feedback, 251
detectable, 429 determinant
of a matrix, 193, 587
of a signal flow graph, 219, 220, 223, 242, 281, 284
diagonalisable, 73, 594, 600 di erentiable, 619 dimension, 586
directed path, 218 discrete, 26
disk
closed, 618 open, 617
distribution, 629 simple, 633
disturbance, 1, 3, 6, 214, 248, 336–338, 340, 354, 364, 477, 478, 480, 482, 485, 490
disturbance type, 336–339, 341, 480, 481 disturbed output, 336
domain, 618
double pendulum, 16, 65, 68, 69, 115, 155, 205, 206, 442, 554
dynamic output feedback, 258, 260, 416, 420, 426
eigenspace, 82, 122, 399, 401, 593 eigenvalue, 47, 73, 85, 89, 120–122, 152, 154,
162, 164, 166, 168, 181, 183, 205, 315, 399, 401, 418, 545, 551, 593, 600
algebraic multiplicity, 162, 419, 593, 600, 602
closed-loop, 252, 417, 442, 544, 554, 555 generalised, 415
geometric multiplicity, 162, 593, 600 eigenvector, 42, 82, 162, 593
generalised, 600
electric circuit, 10, 17, 67, 114, 115 for lag compensation, 489
for lead compensation, 489 for notch filter, 489
equations of motion for an interconnected SISO linear system, 231
655
equilibrium point, 8, 12, 12, 16, 17, 31, 65, 66, 180
error, 2–4, 6, 214, 244, 248, 261, 329, 331, 333, 336, 340, 344, 345, 353, 358, 362, 363, 368, 382–384, 430, 431, 494
essentially bounded, 26 even, 588
feedback, 2, 6, 113, 164, 211, 244, 273, 362 dynamic output, 258, 260, 416, 420, 426 for state-space models, 250, 255, 258,
272–274, 357, 398, 405, 417, 441, 442, 527, 542, 546, 554
negative, 15, 64, 77, 213, 214, 219, 230, 241, 243, 244
static output, 255, 255, 256, 275, 357, 408, 416, 447
static state, 250, 251, 251, 273, 274, 315, 357, 398, 402, 406, 416, 440–442, 527, 542, 546, 550
unity gain, 78, 247, 283, 299, 304, 313, 333, 334, 342, 344, 357, 361, 362, 367–369, 382, 418, 443, 476, 500
feedforward, 79, 96 final value problem, 53
Final Value Theorem, 637
first Luenberger-Brunovsky canonical form,
70, 71
Fourier transform, 631 L1, 631
L2, 632 inverse, 632
frequency response, 119, 135, 138, 148, 207, 279, 383, 469
and impulse response, 122
and periodic solutions, 120, 275 and the transfer function, 120, 122 for first-order systems, 321
for input/output systems, 122
for lead/lag compensator, 470, 477 for PID compensator, 473, 483
for second-order systems, 153, 324 for SISO linear systems, 120
graphical representations, 124, 130, 135, 138
rearranging for Bode plots, 129
gain, 4, 15, 275, 344, 447, 474, 476, 495 L2 → L2, 567
Lp1 → Lp2 , 174
in a signal flow graph, 212, 218, 272 gain crossover frequency, 297, 297, 477, 483
656
and bandwidth, 348, 477, 483
gain margin, 299, 300, 420, 469, 478, 487, 550, 555
limitation of, 300 lower, 297, 297 upper, 297, 297
general solution, 599
graph transmittance, 224, 230, 236
Hardy
function, 173 space, 172, 175
Hermite, 195 criterion, 195, 197
reduced, 196, 197 matrix, 195
Hermite matrix, 196 Hermitian, 594
Hermitian inner product, 594 homogeneous solution, 104, 109, 122, 152,
597
Horowitz area formula, 373, 374
Hurwitz, 163, 186, 189, 190, 193, 202, 207, 208, 261, 274, 316, 398, 400, 408, 413, 415–417, 428, 440, 447, 449, 527, 533, 545, 554, 555, 577
criterion, 193, 199, 203, 208, 272, 290, 293, 313, 322, 476
matrix, 193, 415
identity matrix, 587 image, 592 imaginary part, 617
improper, 91, 169, 233, 311, 444, 560, 570,
612
impulse response, 48, 49, 53, 69, 94, 103, 108–110, 116, 165
and convolution, 49, 53, 102, 103, 107, 112, 152
and frequency response, 122
and the transfer function, 95, 96, 122 anticausal, 53, 574
as a solution to an initial value problem, 108
as limit of realisable outputs, 51, 207 causal, 49
for input/output systems, 94 inductor, 10 infinite-dimensional, 586
initial value problem, 27, 99, 102, 105, 108, 116, 152, 527, 599
injective, 592
Index
inner product, 594 innovation, 431
input, 1, 2, 24, 25–28, 33, 35–37, 39, 44, 49, 61, 79, 90, 105, 107, 152, 153, 226, 230, 242, 248, 259, 260, 345, 368, 383, 526, 548, 567
bounded, 165, 167, 174, 180, 205, 206, 258
for a signal flow graph, 213 integrable, 536, 536, 538
L1, 631
L2, 631
L∞, 170, 631 Lp, 170
integral control, 264, 267, 288, 340, 478 interconnected SISO linear system, 229, 230,
235, 236, 242
interconnection, 212, 212, 229, 230, 233, 235, 236, 242, 248, 259, 271, 272, 279, 281
negative feedback, 213 parallel, 213
series, 213 interval, 25
invariant subspace, 36, 41, 45, 59, 87, 183, 592, 596
invertible, 588, 592
Jacobian, 12
kernel, 592
Kharitonov’s test, 201, 209 Kirchho ’s laws, 10, 11
Laplace transform, 76, 78, 79, 92, 95, 98, 99, 101–103, 106, 107, 110, 123, 152, 153, 260, 320, 573, 577, 578, 632, 633, 641
and di erential equations, 98, 99, 101, 102, 108
and the matrix exponential, 106, 641 anticausal
left, 634 right, 634
causal, 78, 92, 95, 98, 99, 101–103, 106, 107
left, 78, 92, 95, 98, 99, 101, 634 right, 98, 99, 101–103, 106, 107, 634
inverse, 99, 100, 102, 108, 110, 179, 567, 573, 574, 577, 633, 641
two-sided, 633 Laurent series, 620
least common multiple, 609
Index
length, 218 Liapunov, 180, 545
function, 180, 181, 182, 195, 203 proper, 181
triple, 182, 186, 207, 576 Li´enard, 199
Li´enard-Chipart criterion, 200, 209 linear
map, 591 transformation, 591
linearisation, 8, 12, 12, 16, 17, 31, 31, 65, 66 linearly independent, 586, 600, 602, 604 link, 10
loop, 218, 219
loop gain, 218, 281, 293, 320, 333, 336, 342, 343, 362, 382, 384, 444, 475, 478, 480, 481, 484, 491, 551, 555
bandwidth constraints, 374, 375
poles of, 273, 362, 364, 365, 367, 369, 378 proper, 233, 248, 283–285, 313, 334, 369,
371
stable, 299, 300, 369
strictly proper, 249, 284, 285, 354 unstable, 362, 364, 365, 367–369, 382 zeros of, 273, 362, 364, 365, 367, 369, 378
Mason’s Rule, 223, 230 mass-spring-damper system, 7, 24, 28, 49,
78, 79, 124, 153, 260, 264, 307, 310, 338, 343, 478, 483
matrix exponential, 27, 36, 52, 95, 98, 106, 107, 162, 183, 600, 641
Maximum Modulus Principle, 377, 382, 568, 623
measurable, 26
meromorphic, 172, 280, 566, 619
method of undetermined coe cients, 104, 598
MIMO, 242
MIMO systems, 24, 37, 43, 54, 75, 182, 256, 389, 530, 541, 548, 557
minimum phase, 47, 90, 91, 93, 129, 135, 137, 140, 143, 156, 326, 336, 362, 364, 365, 367–369, 377, 381, 383, 393, 488, 490, 491, 531
minor, 588
principal, 182, 588 modulus, 617
monic, 609
multiplicity, 179, 280, 313, 531, 533, 534, 566, 597, 610
657
algebraic, 162, 237, 419, 593, 600, 602 geometric, 162, 237, 593, 600
negative-definite, 181, 207, 440 negative-semidefinite, 181, 207, 440 neppers, 140
node, 212
noise, 236, 248, 478
nominal performance, 385, 385 nonlinear, 12, 16, 17, 31, 65, 66, 180
nonminimum phase, 47, 90, 91, 93, 129, 135, 137, 140, 143, 156, 326, 336, 362, 364, 365, 367–369, 377, 380, 381, 383, 393, 488, 490, 491, 531
nonsingular, 588 nontouching, 218 norm, 170, 595
2 → 2, 179 H∞, 345
H2, 383, 531
H∞, 146, 172, 175, 179, 302, 376, 381–384, 389, 394, 531, 566
Hp, 172, 173 L1, 345
L2, 345
L∞, 345
L1, 171
L2, 171, 180, 207, 383, 384
L∞, 170, 171, 207, 383, 494 Lp, 170
Lpow, 171, 174, 207 pow → pow, 179 p1 → p2, 174, 175
and stability, 174, 175, 179 for signal in time-domain, 170 for transfer function, 172 Hankel, 578, 578
sup, 170
Nyquist contour, 283, 283, 284, 285, 302, 313, 314, 385, 476, 490
and the Bode plot, 135, 285, 293, 294, 475, 490
and the polar plot, 293 determining the number of
encirclements, 285
for state-space models, 552, 555 imaginary axis poles, 313, 314 plotting of, 285
Nyquist criterion, 279, 281, 284, 299, 304, 313, 314, 416, 420, 444, 476, 488, 490, 551
658
for state-space models, 315, 551
observability, 33, 34, 36, 41, 44, 45, 48, 57, 59–61, 66–68, 70, 73, 93, 97, 114, 115, 136, 137, 155, 156, 167, 168, 182, 186, 207, 246, 260, 273, 398, 400, 401, 403, 404, 408, 417, 429, 430, 432, 442, 554
and coprime polynomials, 81, 83, 85–87, 93
for interconnected systems, 243 for MIMO systems, 37, 182 Gramian, 576
matrix, 34, 36 operator, 575
observability canonical form, 71 observer, 429
asymptotic, 430 gain vector, 431
observer canonical form, 57–59, 93, 116, 256, 400, 404, 411
observer gain vector, 431 odd, 588
open circuit, 114 open set, 618
open-loop, 1, 3, 4, 6, 15, 211, 244, 246, 247, 253, 254, 257, 258, 275–277, 476
optimal state feedback, 546, 554 stability margins, 550
optimal state feedback vector, 527 order
of a distribution, 629 order of a pole, 619 orientation, 620 originates, 212
output, 1, 24, 33, 36, 37, 79, 90, 122, 152, 180, 226, 230, 242, 244, 248, 255, 258–260, 276, 336, 526, 527, 543, 553
bounded, 165, 258 computation of, 98, 105–107 for a signal flow graph, 213 periodic, 120, 122, 152
unbounded, 165, 167, 205, 206, 258 output compatible, 256, 256
output feedback constant, 256, 400, 409, 414, 415
overshoot, 318, 328, 336, 361, 481, 499 and peaks in Bode plot, 322, 327, 328 and poles at 0, 364
and rise time, 365, 367
and unstable poles, 365, 367
Index
and zeros, 326
for second-order systems, 322
Parseval’s Theorem, 632
partial fraction expansion, 97, 100, 111, 531, 571, 573, 575, 613, 616
particular solution, 122, 329, 598 path, 218
PD control, 443 pendulum, 8, 12, 32
pendulum on cart, 16, 65, 68, 69, 115, 155, 205, 206, 393, 441, 442, 554
performance, 317, 361
and bandwidth, 321, 325
and nonminimum phase zeros, 362, 364, 365, 367, 369, 373, 375, 378
and the sensitivity function, 342, 344, 345, 368, 369, 373–375, 377, 378, 381–384
and the transfer function, 320, 321, 361, 364, 369, 374, 377, 378, 382
and unstable poles, 362, 364, 365, 367, 369, 373, 378
for first-order systems, 320 for second-order systems, 321 for state-space models, 357
frequency-domain criteria, 320, 368, 369, 373–375, 378, 382, 478, 483
time-domain criteria, 318, 335, 336, 361, 362, 364, 365, 367
with additional poles, 327 with zeros, 326
periodic solution, 120, 122, 152, 183, 275 phase crossover frequency, 295, 297
phase margin, 297, 299, 300, 315, 420, 477, 478, 483, 484, 487, 490, 550, 555
limitation of, 300 lower, 297, 297 upper, 297, 297
PI control, 290, 494, 496, 523
PID control, 91, 261, 264, 274, 277, 307, 310, 328, 334, 341, 344, 473, 483, 490, 493 Ziegler-Nicols tuning, 345, 493, 496, 523
piecewise continuous, 26, 49, 94
plant, 1, 2, 7, 146, 149, 157, 244, 246, 249, 250, 258–261, 273, 303, 315, 333, 335, 342, 361, 362, 374, 385, 416, 418, 420, 425, 427, 429, 469, 478, 494, 523
minimum phase, 373, 487
nominal, 146–149, 303, 304, 307, 309,
Index
386, 389, 394, 558, 560, 562 nonminimum phase, 272, 273, 354, 362,
364, 368, 377, 378, 380, 381, 490, 491 stable, 443, 487, 493
unstable, 272, 273, 336, 362, 364, 368, 378, 380, 381, 488, 490, 491
Poisson Integral Formula, 622 Poisson integral formulae, 377, 393 polar plot, 135, 135, 156
and Nyquist plot, 285, 293
pole, 116, 120, 129, 140, 149, 168, 253, 273–276, 279–281, 327, 343, 368, 377, 398, 405, 409, 412, 414, 443,
612, 616, 619
for first-order systems, 320 for second-order systems, 323
imaginary axis, 120, 147, 148, 173, 283–285, 304, 305, 313, 314, 339, 362, 369, 495, 531, 560, 571
placement, 257, 406, 407
stable, 122, 123, 157, 189, 246, 531, 535, 559, 566, 569, 571
unstable, 147, 148, 169, 173, 242, 284, 303, 336, 340, 362, 364, 365, 367–369, 373, 374, 378, 381, 382, 384, 385, 387, 393, 487, 491, 560, 571
poles
closed-loop, 328 Popov-Belevitch-Hautus, 67, 398, 400, 439 positive-definite, 39, 181, 182, 186, 207, 545,
547, 567, 570, 594 positive-semidefinite, 39, 181, 182, 186, 207,
413, 439, 577 power, 180
average, 171 signal, 171, 177 spectrum, 384
Principle of the Argument, 279, 280, 284, 313, 314, 455, 488
proper, 91, 92, 94, 99, 102, 108, 109, 112, 140, 143, 147, 148, 165, 168, 173–175, 179, 230, 233, 272, 273, 277, 283–285, 295, 304, 307, 309–311, 313, 334, 347, 369, 371, 382, 385, 387, 389, 420, 422–427, 443, 444, 491, 531, 557–560, 562, 566, 571, 612
proportional control, 263, 264, 266, 275, 276, 285, 339, 443, 476, 490, 495
proportional gain, 261, 523
659
ramp response, 108, 109, 331
as a solution to an initial value problem, 108
Rank-Nullity Theorem, 592
rational function, 76, 80, 112, 146, 147, 173, 212, 421, 422, 531, 535, 560, 571, 573, 579, 611, 616
real numbers, 585 real part, 617
reference trajectory, 1, 3–5, 244, 248, 250, 255, 318, 328, 331, 338, 364, 383
region, 618
region of absolute convergence, 633 relative degree, 91, 132, 373, 375, 382, 421,
422, 484 reset time, 261, 264 residue, 622
Residue Theorem, 622 resistor, 10
rise time, 318, 361 alternate definition, 318
and additional poles, 327
and bandwidth, 321, 325–327, 477 and overshoot, 365, 367
for first-order systems, 320 for second-order systems, 322
robust performance, 381, 385, 386, 386, 389, 394, 580
for multiplicative uncertainty, 387 graphical interpretation, 389 modified problem, 559, 560, 581 problem, 389, 389, 530, 558, 560, 564
robust stability, 303, 304, 316, 382, 385, 386, 389
for additive uncertainty, 304, 309, 310, 387
graphical interpretation, 309
for multiplicative uncertainty, 304, 307 graphical interpretation, 305
root, 189, 610 Routh, 189
array, 190
criterion, 190, 192, 199, 203, 207, 272, 290, 293, 313, 322, 476
saturation, 349
second Luenberger-Brunovsky canonical form, 56, 58
sensitivity function, 248, 273, 309, 310, 342, 343, 345, 368, 384, 385, 394, 488
660
and performance, 342, 344, 345, 368, 369, 373–375, 377, 378, 381–384
and stability margin, 302, 303 complementary, 248, 304 minimisation, 303, 317, 345, 375, 381,
382 sensor, 2, 244
settling time, 318, 482
and undershoot, 365, 367
for second-order systems, 323 signal, 212
signal flow graph, 212, 212, 219, 223, 229, 271
and block diagrams, 212, 271 negative feedback, 213 parallel, 213
series, 213 signal-to-noise ratio, 275 simple distribution, 633 singular, 588
singular value, 594 singularity
essential, 619 isolated, 619 removable, 619
sink, 212
SISO linear system, 24
SISO linear system in input/output form, 91 SISO nonlinear system, 31, 65, 66 skew-Hermitian, 595
skew-symmetric, 595 skew-symmetric part, 595 source, 212
span, 587 spans, 586
spectrum, 115, 155, 156, 162, 163, 166, 181, 417, 571, 574, 593
for zero dynamics, 88 stability, 180, 181
L2 → L2, 179 Lpow → Lpow, 179 Lp1 → Lp2 , 174
asymptotic, 180, 181
BIBO, 165, 165, 166, 168, 169, 173, 179, 180, 205–207, 272
for interconnected systems, 236 closed-loop, 254, 258, 260, 273, 274, 279,
284, 299, 313, 315, 362, 393, 398, 400, 405, 408, 415, 416, 418, 419, 440, 443, 469, 476, 490, 551
IBIBO, 235–237, 242, 248, 271–273, 281,
Index
284, 299, 304, 313, 314, 336, 337, 353, 362, 393, 418, 420, 425, 443, 444, 447, 476
internal, 162, 162, 163, 165, 182, 205–207 for interconnected systems, 236, 238 internal asymptotic, 162, 162, 163, 168, 182, 205–207, 260, 274, 398, 408,
416, 417
for interconnected systems, 236, 238 Liapunov methods, 180–182, 184, 186,
275, 413, 440
relationship between internal and BIBO, 168, 405
spectral, 164, 205, 206 transfer function, 281
stable
controller, 429
loop gain, 299, 300, 369 plant, 443, 487, 493
transfer function, 91, 122, 129, 140, 143–145, 156, 157, 175, 179, 318, 319, 329, 393, 420, 493, 494, 531
zero dynamics, 47 standard basis, 586
standard Hermitian inner product, 594 standard inner product, 594
state, 24, 27, 28, 33, 35–37, 40, 43, 49, 54, 57, 61, 63–65, 67, 81, 84, 98, 105, 153, 167, 205, 206, 250, 253, 259, 418, 526, 527, 542, 545, 548, 554
controller, 259
lack of in input/output systems, 90, 92, 93, 165, 167, 261
periodic, 120, 152 plant, 259 solution, 27
state compatible, 251, 252
state feedback vector, 251, 358, 542
static output feedback, 255, 255, 256, 275, 357, 408, 416, 447
static state feedback, 250, 251, 251, 273, 274, 315, 357, 398, 402, 406, 412, 416, 440–442, 527, 542, 546, 550
steady-state error, 263, 264, 329, 331, 333, 335, 338, 341, 344, 353, 358, 359, 374, 478, 480
steady-state response, 122, 318 step function, 49, 53, 108, 607 step input, 318, 320, 321, 331, 478
step response, 108, 109, 110, 318, 319, 331, 344, 354, 358, 359, 364, 393, 481,
Index
490, 523
as a solution to an initial value problem, 108
first-order, 320
normalised, 318, 335, 336, 362 second-order, 322
with additional poles, 327 with zeros, 326
steppable, 318, 318, 347
strictly minimum phase, 140, 143
strictly proper, 91, 94, 103, 109, 112, 113, 122, 123, 175, 230, 274, 284, 285, 311, 335, 382, 393, 421–423, 425, 443, 531, 565, 571, 612
structure matrix, 216, 271 subgraph, 218
subspace, 586 summer, 77 support, 627 surjective, 592 symmetric, 594 symmetric part, 595
Taylor series, 619 terminates, 212 test function, 627 time delay, 147, 641 touching, 219 trace, 587
transfer function, 75, 90, 92, 105, 112, 114–116, 154, 156, 230, 273, 352, 358, 393, 409, 442, 489, 490, 545
and changes of coordinate, 81 and controllability, 81, 83, 87, 398 and detectability, 403, 405
and frequency response, 120, 122
and impulse response, 95, 96, 112, 165 and observability, 83, 85, 87, 398
and performance, 361, 364, 369, 374, 377, 378, 382
and stabilisability, 403–405 and zero dynamics, 87, 90
closed-loop, 245, 246, 247, 251, 256, 260, 272, 273, 281, 304, 307, 328, 334–336, 343, 348, 361, 364, 367, 368, 373, 374, 377, 378, 385, 394, 406, 409, 414, 443, 488, 495
first-order, 320
for input/output systems, 90, 91
for interconnected systems, 223, 235, 240, 243
661
for negative feedback interconnection, 113
for parallel interconnection, 113 for series interconnection, 113
for SISO linear systems, 78, 79, 80 improper, 233, 284
invertibility, 88 norms for, 172
open-loop, 244, 247, 279 proper, 102, 113, 284 second-order, 153, 321
stable, 91, 122, 129, 140, 143–145, 156, 157, 175, 179, 318, 319, 329, 393, 420, 493, 494, 531
strictly proper, 102, 113 unstable, 393
transient response, 122, 318, 477 transposition, 588
tree, 10
triangle inequality, 595 type, 329, 331
2 or greater, 335, 336, 374 0, 331, 353 1, 331, 335, 345, 353, 374
and poles at 0, 334
for state-space models, 358, 359
for unity gain feedback loops, 333, 334 of a system, 329
of an input, 329
uncertainty, 2, 4, 145, 149, 150, 303, 304, 307, 316, 385, 386, 389, 391, 394, 406, 550
additive, 148, 149, 303, 308, 310, 385, 386, 389, 391, 558, 560, 562, 581 multiplicative, 147, 147, 149, 303–305,
307, 385, 386, 389, 558, 560–562 structured, 146
unstructured, 146, 147 undershoot, 318
and nonminimum phase zeros, 326, 328, 365, 367
and poles at 0, 364
and settling time, 365, 367 immediate, 393
unstable, 43, 129, 162, 182, 205, 206, 254, 284, 417, 476, 551, 557
loop gain, 362, 364, 365, 367–369, 382 plant, 336, 362, 364, 368, 381, 488, 490,
491
transfer function, 393
662
zero dynamics, 90
vector space, 585
waterbed e ect, 375
well-posed, 233, 233, 248, 255, 259, 272, 284, 369, 376, 426, 443
zero, 116, 129, 140, 271, 275, 280, 284, 313, 326, 336, 343, 368, 374, 377, 398, 405, 421, 531, 612
dynamics, 43, 46, 47, 68, 69, 87, 88, 90, 114, 115, 155, 156
Index
imaginary axis, 533, 559, 560, 565 initial condition, 48, 49, 165, 236 input, 35, 122, 205
minimum phase, 364, 535, 542 nonminimum phase, 242, 284, 326, 362,
364, 365, 367–369, 374, 376, 378, 381, 382, 384, 385, 387, 393, 422, 425, 487, 491
stable, 242
Ziegler-Nicols tuning, 345, 493, 496, 523