Modern Control Engineering

The first step in the design is to adjust the gain Kto meet the steady-state performance spec-

ification or to provide the required static velocity error constant. Since this constant is given as

20 sec–1, we obtain

or

With K=10, the compensated system will satisfy the steady-state requirement.

We shall next plot the Bode diagram of

Figure 7–95 shows the magnitude and phase-angle curves of G 1 (jv). From this plot, the phase

and gain margins of the system are found to be 17° and ±qdB, respectively. (A phase margin of

17° implies that the system is quite oscillatory. Thus, satisfying the specification on the steady state

yields a poor transient-response performance.) The specification calls for a phase margin of at

least 50°. We thus find the additional phase lead necessary to satisfy the relative stability re-

quirement is 33°. To achieve a phase margin of 50° without decreasing the value of K, the lead com-

pensator must contribute the required phase angle.

Noting that the addition of a lead compensator modifies the magnitude curve in the Bode di-

agram, we realize that the gain crossover frequency will be shifted to the right. We must offset the

increased phase lag of G 1 (jv)due to this increase in the gain crossover frequency. Considering

the shift of the gain crossover frequency, we may assume that fm,the maximum phase lead re-

quired, is approximately 38°. (This means that 5° has been added to compensate for the shift in

the gain crossover frequency.)

Since

sinfm=

1 - a

1 +a

G 1 (jv)=

40

jv(jv+2)

=

20

jv(0.5jv+1)

K= 10

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

Ts+ 1

aTs+ 1

G 1 (s)=slimS 0

s4K

s(s+2)

=2K= 20

Section 7–11 / Lead Compensation 497

12 4 8

v in rad/sec

40

20

0

• 20


• 40
  0 °


• 90 °


• 180 °
  10 20 40 60 100

17 °

dB

Figure 7–95
Bode diagram for
G 1 (jv)=10G(jv)

=40/Cjv(jv+2)D