List of Symbols

Each equation in the book ... would halve the sales. —Stephen Hawking

General

At transpose of matrix A

A complex-conjugate transpose of matrix A

A-1 inverse of square nonsingular matrix A

|| A|| Euclidean (Frobenius) norm of a real-valued matrix A:

diag(a;-) diagonal matrix with elements ai along the diagonal eig(A) eigenvalue of a square matrix A

Ai (A) ith eigenvalue of a square matrix A

Anax (A) maximal eigenvalue of a square matrix A

ai (A) ith singular value of a matrix A

°max (A) maximal singular value of a matrix A

In identity matrix, n x n

0nxm zero matrix, n x m

Linear Systems

(AB,CD)

quadruple of the system state-space representation

(A,B,C)

triple of the system state-space representation

( Ad, Bd, Cd )

discrete-time state-space representation

( Alqg , Blqg , Clqg )

LQG controller state-space representation

( Aœ , Bœ , Cœ )

controller state-space representation

( Ao, Bo, Co )

closed-loop state-space representation

G

transfer function

Gd

discrete-time transfer function

Hi

Hankel matrix

H 2

shifted Hankel matrix

hk

kth Markov parameter

U

input measurement matrix

Y

output measurement matrix

x

system state

xe

system estimated state

u

system (control) input

y

system (measured) output

z

performance output

w

disturbance input

Bi

matrix of disturbance inputs

B2

matrix of control inputs

Ci

matrix of performance outputs

C2

matrix of measured outputs

IGII2

continuous-time system H2 norm

IGII „

continuous-time system H„ norm

IGllh

continuous-time system Hankel norm

IGdl I2

discrete-time system H2 norm

Gd L

discrete-time system H„ norm

IGdllh

discrete-time system Hankel norm

c

controllability matrix

0

observability matrix

c

controllability grammian

o

observability grammian

Yi

/th Hankel singular value

Y max

the largest Hankel singular value of a system

r

matrix of Hankel singular values

CARE

controller algebraic Riccati equation

FARE

filter (or estimator) algebraic Riccati equation

HCARE

H„ controller algebraic Riccati equation

HFARE

Hœ filter (or estimator) algebraic Riccati equation

Sc

solution of CARE

Se

solution of FARE

œc

solution of HCARE

œe

solution of HFARE

M,

ith LQG singular value

Mcoi

ith Hœ singular value

M

matrix of the LQG singular values, M = diag(u, )

matrix of the Hœ singular values, Mœ = diag(^œ, )

P

parameter of the Hœ controller

Kc

controller gain

Ke

estimator gain

£

tracking error

t

time sequence

At

sampling time

N

number of states

s

number of inputs

r

number of outputs

Structures

D

damping matrix

K

stiffness matrix

M

mass matrix

Dm

modal damping matrix

K

modal stiffness matrix

Mm

modal mass matrix

q

structural displacement (nodal)

qm

structural displacement (modal)

qab

structural displacement (almost-balanced)

q,

displacement of the ith degree of freedom

qmi

displacement of the ith mode

qabi

displacement of the ith almost-balanced mode

ti

ith structural mode

$abi

almost-balanced ith structural mode

o

modal matrix

® ab

almost-balanced modal matrix

(O,

ith natural frequency

D.

matrix of natural frequencies

^ /th modal damping

Z matrix of modal damping coefficients

Bo nodal input matrix

Coq nodal displacement output matrix

Cov nodal velocity output matrix

Bm modal input matrix

Cmq modal displacement output matrix

Cmv modal velocity output matrix

Cm modal output matrix, Cm - CmqQ_1 + Cmv bm/ input matrix of the /th mode, /th row of Bm cm/ output matrix of the /th mode, /th column of Cm

||Cm 112 modal output gain, ||Cm 112 = ||Cmq^ |[ + ||Cmv 112

||bm/112 input gain of the /th mode

Afy /th half-power frequency, Aq =

<T2iJ- H2 placement index for the /th actuator (sensor)

and the kth mode

(t^j H placement index for the /th actuator (sensor)

and the kth mode

S2 H2 placement matrix

H placement matrix

I (k) membership index of the kh sensor pole shift factor nd number of degrees of freedom n number of modes

N number of states

5 number of inputs r number of outputs

S number of candidate actuator locations

R number of candidate sensor locations

+1 0

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