16.Bibliography.Index

This 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