Page 355 - Modern Control Systems

Section 5.6 The Steady-State Error of Feedback Control Systems 329

Controller Process
-v E u(s) I
/f(.y) K • Yis)
5 + 2
k
Sensor
FIGURE 5.24 2
A system with a 5 + 4
feedback H(s).

E(s) = [1 - T(s)]R(s).

Let us determine an appropriate gain K so that the steady-state error to a step input
is minimized. The steady-state error is

1
= lim s[l — T(s)]—,
s J
e ss s
5^0 5
where
G c(s)G(s) K(s + 4)
T(s) =
1 + G c(s)G(s)H(s) (s + 2)(5 + 4) + 2K'
Then we have
4K
T(0) =
8 + 2K
The steady-state error for a unit step input is
*„ = 1 - 7-(0).
Thus, to achieve a zero steady-state error, we require that

4K
7-(0) = 1,
8 + 2K
or 8 + 2K = 4K. Thus, K = 4 will yield a zero steady-state error. •

The determination of the steady-state error is simpler for unity feedback systems.
However, it is possible to extend the notion of error constants to nonunity feedback sys-
tems by first appropriately rearranging the block diagram to obtain an equivalent unity
feedback system. Remember that the underlying system must be stable, otherwise our
use of the final value theorem will be compromised. Consider the nonunity feedback
system in Figure 5.21 and assume that ^ = 1. The closed-loop transfer function is

Y(s) G c(s)G(s)
= T(s) =
R(s) 1 + H(s)G c(s)G(sY
By manipulating the block diagram appropriately we can obtain the equivalent
unity feedback system with

Y(s) Z(s) G c(s)G(s)
= T(s) where Z(s) =
R(s) 1 + Z(s) 1 + G c(s)G(s)(H(s) - 1)'