Section 2–5 / State-Space Representation of Scalar Differential Equation Systems 35
Note that
(Refer to Appendix C for the inverse of the 2 2 matrix.)
Thus, we have
which is the transfer function of the system. The same transfer function can be obtained from
Equation (2–16).
Transfer Matrix. Next, consider a multiple-input, multiple-output system. Assume
that there are rinputs and moutputs Define
The transfer matrix G(s)relates the output Y(s)to the input U(s),or
whereG(s)is given by
[The derivation for this equation is the same as that for Equation (2–29).] Since the
input vector uisrdimensional and the output vector yismdimensional, the transfer ma-
trixG(s) is an m*rmatrix.
2–5 STATE-SPACE REPRESENTATION OF SCALAR
DIFFERENTIAL EQUATION SYSTEMS
A dynamic system consisting of a finite number of lumped elements may be described
by ordinary differential equations in which time is the independent variable. By use of
vector-matrix notation, an nth-order differential equation may be expressed by a first-
order vector-matrix differential equation. If nelements of the vector are a set of state
variables, then the vector-matrix differential equation is a stateequation. In this section
we shall present methods for obtaining state-space representations of continuous-time
systems.
G(s)=C(s I-A)-^1 B+D
Y(s)=G(s )U(s )
y= F
y 1
y 2
ym
V, u=F
u 1
u 2
ur
V
u 1 ,u 2 ,p,ur , y 1 ,y 2 ,p,ym.
=
1
ms^2 +bs+k
G(s)=[1 0]
1
s^2 +
b
m
s+
k
m
D
s+
b
m
k
m
1
s
T
C
0
1
m
S
C
s
k
m
- 1
s+
b
m
S
1
1
s^2 +
b
m
s+
k
m
D
s+
b
m
k
m
1
s
T