Page 149 - Modern Control Systems

Section 2.9 The Simulation of Systems Using Control Design Software 123

£ 0W s+ 1 V{s)
/?(.*) G c(s) G(s) = 2 •*> Y(s)
s + 2 500 s

(a)

»numg=[1]; deng=[500 0 0]; sys1=tf(numg,deng);
»numc=[1 1]; denc=[1 2]; sys2=tf(numc,denc);
»sys3=series(sys1 ,sys2);
»sys=feedback(sys3,[1 ])

Transfer function:
s + 1 Y(s) ._ G c(s)G(s)
FIGURE 2.66 5O0s 3 + 1000s 2 + + 1 4 R(s) I + G c(s)G(s)
A
A
s
(a) Block diagram.
(b) Application of
the feedback
function. (b)

^ £«(*) Process
/?(.v) G(s) • Y(s)
FIGURE 2.67
A basic control
system with the Controller
controller in the « « —
feedback loop.

EXAMPLE 2.20 The feedback function with unity feedback
Let the process, G(s), and the controller, G c(s), be as in Figure 2.66(a). To apply the
feedback function, we first use the series function to compute G c(s)G(s), followed
by the feedback function to close the loop. The command sequence is shown in
Figure 2.66(b). The closed-loop transfer function, as shown in Figure 2.66(b), is

G c(s)G(s) s + 1
T(s) 3 2 sys.
1 + G c(s)G(s) 500s + 1000^ + .? + 1

Another basic feedback control configuration is shown in Figure 2.67. In this case,
the controller is located in the feedback path. The closed-loop transfer function is

m = G(s)
1 =F G(s)H(s)'

EXAMPLE 2.21 The feedback function
Let the process, G(s), and the controller, H(s), be as in Figure 2.68(a). To compute
the closed-loop transfer function with the controller in the feedback loop, we use