Modern Control Engineering

542 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method

Consider the system shown in Figure 7–136. Suppose that the element G 1 (s)is unstable. The

complete system may be made stable by choosing a suitable linear element G 2 (s). We apply a si-

nusoidal signal at the input. At steady state, all signals in the loop will be sinusoidal. We measure

the signals e(t), the input to the unstable element, and x(t), the output of the unstable element.

By changing the frequency [and possibly the amplitude for the convenience of measuring e(t)

andx(t)] of the input sinusoid and repeating this process, it is possible to obtain the frequency re-

sponse of the unstable linear element.

A–7–20. Show that the lead network and lag network inserted in cascade in an open loop act as

proportional-plus-derivative control (in the region of small v) and proportional-plus-integral

control (in the region of large v), respectively.

Solution.In the region of small v, the polar plot of the lead network is approximately the same

as that of the proportional-plus-derivative controller. This is shown in Figure 7–137(a).

Similarly, in the region of large v, the polar plot of the lag network approximates the

proportional-plus-integral controller, as shown in Figure 7–137(b).

A–7–21. Consider a lag–lead compensator Gc(s)defined by

Show that at frequency v 1 ,where

the phase angle of Gc(jv)becomes zero. (This compensator acts as a lag compensator for

0<v<v 1 and acts as a lead compensator for v 1 <v<q.) (Refer to Figure 7–109.)

v 1 =

1

1 T 1 T 2

Gc(s)=Kc

as+

1

T 1

bas+

1

T 2

b

as+

b

T 1

bas+

1

bT 2

b

G 1 (s) G 2 (s)

re x c

+–

Figure 7–136

Control system.

Im Im

0 Re

PD controller

Lead network

a

v = 0 v =` v =`v =^0

(a) (b)

PI controller

1

0 Re

1

Lag network

1

b

Figure 7–137

(a) Polar plots of a

lead network and a

proportional-plus-

derivative controller;

(b) polar plots of a

lag network and a

proportional-plus-

integral controller.

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