Modern Control Engineering

Example Problems and Solutions 369

The closed-loop transfer function for the compensated system becomes

Figure 6–79 shows the unit-step response curve. Even though the damping ratio of the

dominant closed-loop poles is 0.5, the amount of overshoot is very much higher than expected. A

closer look at the root-locus plot reveals that the presence of the zero at s=–1is increasing the

amount of the maximum overshoot. [In general, if a closed-loop zero or zeros (compensator zero

or zeros) lie to the right of the dominant pair of the complex poles, then the dominant poles are

no longer dominant.] If large maximum overshoot cannot be tolerated, the compensator zero(s)

should be shifted sufficiently to the left.

In the current design, it is desirable to modify the compensator and make the maximum

overshoot smaller. This can be done by modifying the lead compensator, as presented in the

following second attempt.

Second Attempt: To modify the shape of the root loci, we may use two lead networks, each

contributing half the necessary lead angle, which is Let us choose the

location of the zeros at s=–3.(This is an arbitrary choice. Other choices such as s=–2.5and

s=–4may be made.)

Once we choose two zeros at s=–3,the necessary location of the poles can be determined

as shown in Figure 6–80, or

which yields

y= 1 +

1.73205

0.09535

=19.1652

=tan5.4466°=0.09535

1.73205

y- 1

=tan (40.89334°-35.4467°)

70.8934° 2 =35.4467°.

C(s)

R(s)

=

11.2(s+1)(0.1s+1)

(s+6)s^2 (0.1s+1)+11.2(s+1)

t Sec

071 2 3 4 5 6 8 9 10

Output

0.5

1.5

0

1

Unit-Step Response of Compensated System

Figure 6–79
Unit-step response of
the compensated
system.