Page 328 - Mechanical Engineers' Handbook (Volume 2)

4 Standard Forms for Linear Models 319

y (t) g [x (t),x (t),..., x (t),u (t),u (t),... , u (t)]
1
n
2
1
2
p
1
1
y (t) g [x (t),x (t),..., x (t),u (t),u (t),..., u (t)]
2
2
1
2
p
n
1
2

y (t) g [x (t),x (t),..., x (t),u (t),u (t),..., u (t)]
q
1
q
2
2
n
p
1
These equations are expressed more compactly as the two vector equations
˙ x(t) ƒ[x(t),u(t)]
y(t) g[x(t),u(t)]
where
˙ x(t) n 1 state vector
u(t) p 1 input or control vector
y(t) q 1 output or response vector
and ƒ and g are vector-valued functions.
For linear systems, the state equations have the form
˙ x (t) a (t)x (t) a (t)x (t) b (t)u (t) b (t)u (t)
1 11 1 1n n 11 1 1p p
˙ x (t) a (t)x (t) a (t)x (t) b (t)u (t) b (t)u (t)
2 21 1 2n n 21 1 2p p

˙ x (t) a (t)x (t) a (t)x (t) b (t)u (t) b (t)u (t)
n n1 1 nn n n1 1 np p
and the output equations have the form
y (t) c (t)x (t) c (t)x (t) d (t)u (t) d (t)u (t)
1 11 1 1n n 11 1 1p p
y (t) c (t)x (t) c (t)x (t) d (t)u (t) d (t)u (t)
2 21 1 2n n 21 1 2p p

y (t) c (t)x (t) c (t)x (t) d (t)u (t) d (t)u (t)
n q1 1 qn n q1 1 qp p
where the coefficients are groups of parameters. The linear model is expressed more com-
pactly as the two linear vector equations
˙ x(t) A(t)x(t) B(t)u(t)
y(t) C(t)x(t) D(t)u(t)
where the vectors x, u, and y are the same as the general case and the matrices are defined
as
A [a ]isthe n nsystemmatrix
ij
B [b ]isthe n pcontrol, input,or
jk
distribution matrix
C [c ]isthe q n output matrix
lj
D [d ]isthe q p output distribution matrix
lk
For a time-invariant linear system, all of these matrices are constant.