Page 277 - Modern Control Systems

Section 4.6 Steady-State Error 251
for an open-loop and a closed-loop system. The steady-state error is the error after
the transient response has decayed, leaving only the continuous response.
The error of the open-loop system shown in Figure 4.2 is

Eo(s) = R(s) - Y(s) = (1 - G(s))R(s), (4.46)
when T d(s) = 0. Figure 4.3 shows the closed-loop system. When T (i(s) = 0 and
N(s) = 0, and we let H(s) = 1, the tracking error is given by (Equation 4.3)

EM R 447
= 1 + G1 S)G(S) ^ < >

To calculate the steady-state error, we use the final-value theorem
lim e(t) = lim sE(s). (4.48)

Therefore, using a unit step input as a comparable input, we obtain for the open-
loop system

e„(oo) = lim s(l - G(s))(-J = lim (1 - G(s)) = 1 - G(0). (4.49)

For the closed-loop system we have

(0O) = (4 50)
*<' = SS \TTG!MGM){I) i + ^(0)0(0)- '
The value of G(s) when s = 0 is often called the DC gain and is normally greater
than one. Therefore, the open-loop system will usually have a steady-state error of
significant magnitude. By contrast, the closed-loop system with a reasonably large
DC loop gain L(0) = G c(0)G(0) will have a small steady-state error. In Chapter 5,
we discuss steady-state error in much greater detail.
Upon examination of Equation (4.49), we note that the open-loop control sys-
tem can possess a zero steady-state error by simply adjusting and calibrating the
system's DC gain, G(0), so that G(0) = 1. Therefore, we may logically ask, What is
the advantage of the closed-loop system in this case? To answer this question, we
return to the concept of the sensitivity of the system to parameter changes. In the
open-loop system, we may calibrate the system so that G(0) = 1, but during the
operation of the system, it is inevitable that the parameters of G(s) will change
due to environmental changes and that the DC gain of the system will no longer
be equal to 1. Because it is an open-loop system, the steady-state error will not
equal zero until the system is maintained and recalibrated. By contrast, the closed-
loop feedback system continually monitors the steady-state error and provides an
actuating signal to reduce the steady-state error. Because systems are susceptible
to parameter drift, environmental effects, and calibration errors, negative feed-
back provides benefits. An example of an ingenious feedback control system is
shown in Figure 4.16.
The advantage of the closed-loop system is that it reduces the steady-state error
resulting from parameter changes and calibration errors. This may be illustrated by