Modern Control Engineering

Section 7–13 / Lag–Lead Compensation 515

arev=0.7radsec and v=7radsec. Thus, the transfer function of the lead portion of the

lag–lead compensator becomes

Combining the transfer functions of the lag and lead portions of the compensator, we obtain the

transfer function of the lag–lead compensator. Since we chose Kc=1,we have

The magnitude and phase-angle curves of the lag–lead compensator just designed are shown in

Figure 7–111. The open-loop transfer function of the compensated system is

(7–29)

The magnitude and phase-angle curves of the system of Equation (7–29) are also shown in Fig-

ure 7–111. The phase margin of the compensated system is 50°, the gain margin is 16 dB, and the

static velocity error constant is 10 sec–1. All the requirements are therefore met, and the design

has been completed.

Figure 7–112 shows the polar plots of G(jv)(gain-adjusted but uncompensated open-loop

transfer function) and Gc(jv)G(jv)(compensated open-loop transfer function). The Gc(jv)G(jv)

locus is tangent to the M=1.2circle at about v=2radsec. Clearly, this indicates that the com-

pensated system has satisfactory relative stability. The bandwidth of the compensated system is

slightly larger than 2 radsec.

=

10(1.43s+1)(6.67s+1)

s(0.143s+1)(66.7s+1)(s+1)(0.5s+1)

Gc(s)G(s)=

(s+0.7)(s+0.15)20

(s+7)(s+0.015)s(s+1)(s+2)

Gc(s)= a

s+0.7

s+ 7

ba

s+0.15

s+0.015

b= a

1.43s+ 1

0.143s+ 1

ba

6.67s+ 1

66.7s+ 1

b

s+0.7

s+ 7

=

1

10

a

1.43s+ 1

0.143s+ 1

b

M= 1.2

v = 0.15

v = 1

Im

Re

G

0.2 GcG

0.4

• 8


• 7


• 6


• 5


• 4


• 3


• 2


• 1

210

1

• 2 2

2

• 8 – 7 – 5 – 4 – 3 – 1

1

2

• 6

Figure 7–112
Polar plots of G(gain
adjusted) and GcG.