Page 263 - Modern Control Systems

Section 4.2 Error Signal Analysis 237

In this chapter, we examine how the application of feedback can result in the bene-
fits listed above. Using the notion of a tracking error signal, it will be readily appar-
ent that it is possible to utilize feedback with a controller in the loop to improve
system performance.

4.2 ERROR SIGNAL ANALYSIS

The closed-loop feedback control system shown in Figure 4.3 has three inputs—
R(s), T d(s), and N(s)—and one output, Y(s). The signals T d(s) and N(s) are the
disturbance and measurement noise signals, respectively. Define the tracking
error as

E(s) = R(s) - Y(s). (4.1)

For ease of discussion, we will consider a unity feedback system, that is, H(s) = 1, in
Figure 4.3. In Section 5.5 of the following chapter, the influence of a nonunity feed-
back element in the loop is considered.
After some block diagram manipulation, we find that the output is given by

G c(s)G(s) r/ , G(s) G c(s)G(s)
Y(S} = R{s) + N(s) {42)
1 + G c(s)G(s) l + G c(s)G(s)™ ~ 1 + G c(s)G(s) -

Therefore, with E(s) = R(s) — Y(s), we have

1 G(s) GJs)G(s)
£W + m (43)
= I + G&W,)*® ~ i + G'UMS)™ TTmk -

Define the function

L(s) = G c(s)G(s).

The function, L(s), is known as the loop gain and plays a fundamental role in control
system analysis [12]. In terms of L(s) the tracking error is given by

1 G(s) L(s)
E(s) + N(s) (44)
= m«*w" rn^™ rrm -

We can define the function

F(s) = 1 + L(s).