2 - 17
final change in y(t) relative to a unit change in the input x(t). Thus an easy derivation of the steady state gain is to take a unit step input in x(t), or X(s) = 1/s, and find the final value in y(t):
1 b y(∞) = lim [s G(s) X(s)] = lim [s G(s) ] = o s→0 s→0 ao s
(2-32)
The steady state gain is the ratio of the two constant coefficients. Take note that the steady state gain value is based on the transfer function only. From Eqs. (2-31) and (2-32), one easy way to "spot" the steady state gain is to look at a transfer function in the time constant form. Note: (1) When we talk about the poles of G(s) in Eq. (2-29), the discussion is regardless of the input x(t). Of course, the actual response y(t) also depends on x(t) or X(s). (2) Recall from the examples of partial fraction expansion that the polynomial Q(s) in the numerator, or the zeros, affects only the coefficients of the solution y(t), but not the time dependent functions. That is why for qualitative discussions, we focus only on the poles. (3) For the time domain function to be made up only of exponential terms that decay in time, all the poles of a transfer function must have negative real parts. (This point is related to the concept of stability, which we will address formally in Chapter 7.)
2.7
Summary of pole characteristics We now put one and one together. The key is that we can "read" the poles—telling what the form of the time-domain function is. We should have a pretty good idea from our exercises in partial fractions. Here, we provide the results one more time in general notation. Suppose we have taken a characteristic polynomial, found its roots and completed the partial fraction expansion, this is what we expect in the time-domain for each of the terms:
A. Real distinct poles ci Terms of the form , where the pole pi is a real number, have the time-domain function s – pi
ci e
pi t
. Most often, we have a negative real pole such that pi = –ai and the time-domain
function is c i e
– ai t
.
B. Real poles, repeated m times Terms of the form
c i,1 (s – p i)
+
c i,2 (s – p i)
2
+ ... +
c i,m (s – p i) m
with the root pi repeated m times have the time-domain function
c i,1 + c i,2 t +
c i,3 2 c i,m t + ... + tm – 1 e pi t . 2! (m – 1)!
When the pole pi is negative, the decay in time of the entire response will be slower (with respect to only one single pole) because of the terms involving time in the bracket. This is the reason why we say that the response of models with repeated roots (e.g., tanks-in-series later in Section 3.4) tends to be slower or "sluggish."