Notation and Symbols
Basic Notation
Notation
Meaning
Delimiters for sets, and for statement grouping in
f
g
algorithms in chapter 14.
(
)
Delimiters for expressions.
f : X Y
A function from the set X into the set Y .
!
The empty set.
Conjunction of predicates \and".
^
A norm see page 69. A particular norm is indicated
k
k
with a mnemonic subscript.
@ (x)
The subdi erential of the functional at the point x
see page 293.
=
Equals by de nition.
Equals to rst order.
'
Approximately equal to (used in vague discussions).
<
The inequality holds to rst order.
X
A rst order change in X.
arg min
A minimizer of the argument. See page 58.
The complex numbers.
C
n
The vector space of n-component complex vectors.
C
m n
The vector space of m n complex matrices.
C
X
The expected value of the random variable X.
E
(z)
The imaginary part of a complex number z.
=
383
384
NOTATION AND SYMBOLS
inf
The in mum of a function or set. The reader
unfamiliar with the notation inf can substitute min
without ill e ect.
j
A square root of 1.
;
limsup
The asymptotic supremum of a function see page 72.
(Z)
The probability of the event Z.
Prob
(z)
The real part of a complex number z.
<
The real numbers.
R
+
The nonnegative real numbers.
Rn
The vector space of n-component real vectors.
R
m n
The vector space of m n real matrices.
R
i(M)
The ith singular value of a matrix M: the square root
of the ith largest eigenvalue of M M.
max(M)
The maximum singular value of a matrix M: the
square root of the largest eigenvalue of M M.
sup
The supremum of a function or set. The reader
unfamiliar with the notation sup can substitute max
without ill e ect.
M
The trace of a matrix M: the sum of its entries on the
T
r
diagonal.
M 0
The n n complex matrix M is positive semide nite,
i.e., z Mz 0 for all z
n.
2
C
M > 0
The n n complex matrix M is positive de nite, i.e.,
z Mz > 0 for all nonzero z
n.
2
C
0
The n-component real-valued vector has
nonnegative entries, i.e.,
n+.
2
R
MT
The transpose of a matrix or transfer matrix M.
M
The complex conjugate transpose of a matrix or
transfer matrix M.
M1=2
A symmetric square root of a matrix M = M
0,
i.e., M1=2M1=2 = M.
NOTATION AND SYMBOLS
385
Global Symbols
Symbol
Meaning
Page
A function from the space of transfer matrices to real
53
numbers, i.e., a functional on . A particular
H
function is indicated with a mnemonic subscript.
'
The restriction of a functional to a
252
nite-dimensional domain, i.e., a function from n to
R
. A particular function is indicated with a
R
mnemonic subscript.
A design speci cation: a predicate or boolean function
47
D
on . A particular design speci cation is indicated
H
with a mnemonic subscript.
H
The closed-loop transfer matrix from w to z.
33
Hab
The closed-loop transfer matrix from the signal b to
32
the signal a.
The set of all nz nw transfer matrices. A particular
48
H
subset of (i.e., a design speci cation) is indicated
H
with a mnemonic subscript.
K
The transfer matrix of the controller.
32
L
The classical loop gain, L = P0K.
36
nw
The number of exogenous inputs, i.e., the size of w.
26
nu
The number of actuator inputs, i.e., the size of u.
26
nz
The number of regulated variables, i.e., the size of z.
26
ny
The number of sensed outputs, i.e., the size of y.
26
P
The transfer matrix of the plant.
31
P0
The transfer matrix of a classical plant, which is
34
usually one part of the plant model P.
S
The classical sensitivity transfer function or matrix.
36 41
T
The classical I/O transfer function or matrix.
36 41
w
Exogenous input signal vector.
25
u
Actuator input signal vector.
25
z
Regulated output signal vector.
26
y
Sensed output signal vector.
26
386
NOTATION AND SYMBOLS
Other Symbols
Symbol
Meaning
Page
A feedback perturbation.
221
A set of feedback perturbations.
221
x
The Euclidean norm of a vector x
n or x n,
70
k
k
2
R
2
C
i.e., px x.
u 1
The 1 norm of the signal u.
81
k
k
L
u 2
The 2 norm of the signal u.
80
k
k
L
u
The peak magnitude norm of the signal u.
70
k
k
1
u aa
The average-absolute value norm of the signal u.
74
k
k
u rms
The RMS norm of the signal u.
72
k
k
u ss
The steady-state peak magnitude norm of the signal u.
72
k
k
1
H 2
The 2 norm of the transfer function H.
96 110
k
k
H
H
The
norm of the transfer function H.
112 112
k
k
H
1
1
H a
The a-shifted
norm of the transfer function H.
100
k
k
H
1
1
H hankel
The Hankel norm of the transfer function H.
103
k
k
H pk step
The peak of the step response of the transfer function
95
k
k
H.
H pk gn
The peak gain of the transfer function H.
97 111
k
k
H rms w
The RMS response of the transfer function H when
96
k
k
driven by the stochastic signal w.
H rms gn
The RMS gain of the transfer function H, equal to its 99 112
k
k
norm.
H
1
H wc
A worst case norm of the transfer function H.
97
k
k
The region of achievable speci cations in performance
139
A
space.
AP Bw Bu
The matrices in a state-space representation of the
43
Cz Cy Dzw
plant.
Dzu Dyw Dyu
(u)
The crest factor of a signal u.
91
CF
( )
The dead-zone function.
230
Dz
I (H)
The -entropy of the transfer function H.
113
nproc
A process noise, often actuator-referred.
35
nsensor
A sensor noise.
35
NOTATION AND SYMBOLS
387
u(a)
The amplitude distribution function of the signal u.
76
F
h(t)
The impulse response of the transfer function H at
96
time t.
H(a) H(b)
The closed-loop transfer matrices from w to z achieved
42
H(c) H(d)
by the four controllers K(a) K(b) K(c) K(d) in our
standard example system.
K(a) K(b)
The four controllers in our standard example system.
42
K(c) K(d)
A perturbed plant set.
211
P
Pstd
0
The transfer function of our standard example
41
classical plant.
p q
Auxiliary inputs and outputs used in the perturbation
221
feedback form.
s
Used for both complex frequency, s = + j!, and the
step response of a transfer function or matrix
(although not usually in the same equation).
( )
The saturation function.
220
Sat
sgn( )
The sign function.
97
T1 T2 T3
Stable transfer matrices used in the free parameter
156
representation of achievable closed-loop transfer
matrices.
?
A submatrix or entry of H not relevant to the current
172
discussion.
The minimum value of the function .
311
x
A minimizing argument of the function , i.e.,
311
= (x ).
(f)
The total variation of the function f.
98
Tv
388
NOTATION AND SYMBOLS
List of Acronyms
Acronym
Meaning
Page
1-DOF
One Degree-Of-Freedom
37
2-DOF
Two Degree-Of-Freedom
39
ARE
Algebraic Riccati Equation
122
FIR
Finite Impulse Response
380
I/O
Input/Output
38
LQG
Linear Quadratic Gaussian
278
LQR
Linear Quadratic Regulator
275
LTI
Linear Time-Invariant
29
MAMS
Multiple-Actuator, Multiple-Sensor
40
MIMO
Multiple-Input, Multiple-Output
110
PID
Proportional plus Integral plus Derivative
5
QP
Quadratic Program
345
RMS
Root-Mean-Square
72
SASS
Single-Actuator, Single-Sensor
34
SISO
Single-Input, Single-Output
95
389
390
LIST OF ACRONYMS
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