Page 241 - Electrical Engineering Dictionary

is the output vector. in the following manner:

2 m F(k, k) = I nxn for k ∈ Z
u ∈ L ([t 0 , ∞), R )
loc
F(k, j) = F(k, j + 1)A(j)
is an admissible control, A(t) is n×n dimen- = A(k − 1)A(k − 2)...A(j + 1)A(j)
sional matrix, with piecewise-continuous el-
ements, B(t) is n × m dimensional matrix, for k> j.
with piecewise-continuous elements, C(t) is
q × n dimensional matrix with piecewise- dynamical linear stationary continuous-
continuous elements, D(t) is q × m dimen- time finite-dimensional system a sys-
sional matrix with piecewise-continuous ele- tem described by the linear differential state-
ments. The solution of the state equation has equation
the form
0
x (t) = Ax(t) + Bu(t) (1)
x(t, x(t 0 ), u) = F(t, t 0 )x(t 0 )
Z t and the linear algebraic output equation
+ F(t, s)B(s)u(s)ds
y(t) = Cx(t) + Du(t)
t 0
where F(t, s) is n×n dimensional transition where
matrix for a dynamical system. x(t) ∈ R n

dynamical linear nonstationary discrete- is the state vector,
time finite-dimensional system a system
m
described by the linear difference state equa- u(t) ∈ R
tion
is the input vector,
x(k + 1) = A(k)x(k) + B(k)u(k) (1) q
y(t) ∈ R
and the linear algebraic output equation
is the output vector, A, B, C, and D are
constant matrices of appropriate dimensions.
y(k) = C(k)x(k) + D(k)u(k)
The transition matrix of (1) has the form
n F(t, s) = e A(t−s) .
where x(k) ∈ R is the state vector, u(k) ∈
m
q
R is a control vector, y(k) ∈ R is an out-
dynamical linear stationary discrete-time
put vector, and A(k), B(k), C(k), and D(k)
finite-dimensional system a system de-
are matrices of appropriate dimensions with
scribed by the linear difference state equation
variable coefficients. Solution of the differ-
ence state equation (1) has the form x(k + 1) = Ax(k) + Bu(k) (1)
x(k, x(k 0 ), u) = F(k, k 0 )x(k 0 ) and the linear algebraic output equation
j=k−1
X
+ F(k, j + 1)B(j)u(j) y(k) = Cx(k) + Du(k)
j=k 0 n m
where x(k) ∈ R is a state vector, u(k) ∈ R
q
where F(k, j) is n×n dimensional transition is a control vector, y(k) ∈ R is an output
matrix defined for all vector, and A, B, C, and D are constant ma-
trices of appropriate dimensions. The tran-
k ≥ j sition matrix of (1) has the form F(k, j) =
A k−j .

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2000 by CRC Press LLC