Modern Control Engineering

Section 7–12 / Lag Compensation 507

Thus, the transfer function of the lag compensator is

Since the gain Kwas determined to be 5 and bwas determined to be 10, we have

The open-loop transfer function of the compensated system is

The magnitude and phase-angle curves of Gc(jv)G(jv)are also shown in Figure 7–104.

The phase margin of the compensated system is about 40°, which is the required value. The

gain margin is about 11 dB, which is quite acceptable. The static velocity error constant is 5 sec–1,

as required. The compensated system, therefore, satisfies the requirements on both the steady

state and the relative stability.

Note that the new gain crossover frequency is decreased from approximately 1 to 0.5 radsec.

This means that the bandwidth of the system is reduced.

To further show the effects of lag compensation, the log-magnitude-versus-phase plots of the gain-

adjusted but uncompensated system G 1 (jv)and of the compensated system Gc(jv)G(jv)are shown

in Figure 7–105. The plot of G 1 (jv)clearly shows that the gain-adjusted but uncompensated system is

unstable. The addition of the lag compensator stabilizes the system. The plot of Gc(jv)G(jv)is tan-

gent to the M=3dB locus. Thus, the resonant peak value is 3 dB, or 1.4, and this peak occurs at

v=0.5radsec.

Compensators designed by different methods or by different designers (even using the same ap-

proach) may look sufficiently different. Any of the well-designed systems, however, will give similar

transient and steady-state performance. The best among many alternatives may be chosen from the

economic consideration that the time constants of the lag compensator should not be too large.

Gc(s)G(s)=

5(10s+1)

s(100s+1)(s+1)(0.5s+1)

Kc=

K

b

=

5

10

=0.5

Gc(s)=Kc(10)

10s+ 1

100s+ 1

=Kc

s+

1

10

s+

1

100

8

4

0

• 4
  
  
  • 90 
  
  
  
  

24

16

20

12

• 8


• 12


• 16


• 20


• 240 – 210 – 180 – 150 – 120 
  G 1

G

1

in dB

0.6

0.4

0.8

(^1) 0.1
0.2
2
0.6
0.4
0.8
G 1
GcG
3 dB
1
(^42)
Figure 7–105
Log-magnitude-
versus-phase plots of
G 1 (gain-adjusted but
uncompensated
open-loop transfer
function) and GcG
(compensated open-
loop transfer
function).