Page 92 - Modern Control Systems

Chapter 2 Mathematical Models of Systems

Therefore, solving Equation (2.40) for I(s) and substituting in Equation (2.41), we have
(1/CsMs)
VM =
R + l/Cs •
Then the transfer function is obtained as the ratio V 2(s)/Vi (s), which is
_ Vi(s) _ 1 1 1/r
G(s) (2 42)
~VLW~ RCS + l " 77TT " 7TT/? '
where T = i?C, the time constant of the network. The single pole of G(s) is
s = — 1/T. Equation (2.42) could be immediately obtained if one observes that the
circuit is a voltage divider, where

-=-^- = ^-^- (2 43^
v,(s) z,{s) + z ( y K • }
2 s
and Z x(s) = R,Z 2 = l/Cs.
A multiloop electrical circuit or an analogous multiple-mass mechanical sys-
tem results in a set of simultaneous equations in the Laplace variable. It is usually
more convenient to solve the simultaneous equations by using matrices and deter-
minants [1, 3,15]. An introduction to matrices and determinants can be found on
the MCS website.
Let us consider the long-term behavior of a system and determine the response
to certain inputs that remain after the transients fade away. Consider the dynamic
system represented by the differential equation
n l
n l
n 2
n
d y d ~ y d ~ r d ~ r ^ AA^
2
2 44
- P^
-? + ^ 1 ^ + *" + ™ = ^ 1 ^ + P "~ ^ + + < ' )
where y(t) is the response, and r(t) is the input or forcing function. If the initial con-
ditions are all zero, then the transfer function is the coefficient of R(s) in
n 2
P(s) D , , p^s"- 1 + p n- 2s - + ••• + p 0
yn i
= ~ :
Y(s) = G(s)R(s) = ——R(s) ) = , ^ 1 ; , , _ ™R(s). (2.45)
2%R( 5
q(s) s» + q^s"- + ••• + q 0
The output response consists of a natural response (determined by the initial
conditions) plus a forced response determined by the input. We now have
q(s) q(s)
where q(s) = 0 is the characteristic equation. If the input has the rational form
n(s)
d(s)
then
.
y (s) = '^ E<fl^l y l{s) + y M + Y3is)< (2 46)
+
q(s) q(s) d(s)