Modern Control Engineering

Section 7–12 / Lag Compensation 505

EXAMPLE 7–27 Consider the system shown in Figure 7–103. The open-loop transfer function is given by

It is desired to compensate the system so that the static velocity error constant Kvis 5 sec–1, the

phase margin is at least 40°, and the gain margin is at least 10 dB.

We shall use a lag compensator of the form

Define

Define also

The first step in the design is to adjust the gain Kto meet the required static velocity error con-

stant. Thus,

or

With K=5,the compensated system satisfies the steady-state performance requirement.

We shall next plot the Bode diagram of

G 1 (jv)=

5

jv(jv+1)(0.5jv+1)

K= 5

=slimS 0

sK

s(s+1)(0.5s+1)

=K= 5

Kv=slimS 0 sGc(s)G(s)=limsS 0 s

Ts+ 1

bTs+ 1

G 1 (s)=limsS 0 sG 1 (s)

G 1 (s)=KG(s)=

K

s(s+1)(0.5s+1)

Kc b=K

Gc(s)=Kc b

Ts+ 1

bTs+ 1

=Kc

s+

1

T

s+

1

bT

(b 7 1)

G(s)=

1

s(s+1)(0.5s+1)

1

+– s(s+1) (0.5s+ 1)

Figure 7–103
Control system.