Page 214 - Modern Control Systems

188 Chapter 3 State Variable Models

Since [si - A] -1 = ¢(^), we have
X(s) = <D(s)B£/(s).

Substituting X(s) into Equation (3.77), we obtain
Y(s) = [C$(s)B + B)U(s). (3.78)

Therefore, the transfer function G(s) = Y(s)/U(s) is
G(s) = C®(s)B + D (3.79)

EXAMPLE 3.5 Transfer function of an RLC circuit
Let us determine the transfer function G(s) = Y(s)/U(s) for the RLC circuit of
Figure 3.4 as described by the differential equations (see Equations 3.18 and 3.19):

x = x + C
1 zR 0
L L
)> = [0 R]y
Then we have

1
s C
[si - A] =
- 1 R
_ L
Therefore, we obtain

[H) -i

*(,) = [,I - A]"' = j s )
A
1

where
1
2
A(s) = s + —s

Then the transfer function is

R
-1 ri"
G(s) = [0 R] A(s) CA(s) c
1 s L°_
L A(*) A(s) _