Modern Control Engineering

Example Problems and Solutions 367

A–6–13. Consider the model for a space-vehicle control system shown in Figure 6–76. Design a lead

compensatorGc(s)such that the damping ratio zand the undamped natural frequency vnof the

dominant closed-loop poles are 0.5 and 2 radsec, respectively.

Solution.

First Attempt: Assume the lead compensator Gc(s)to be

From the given specifications,z=0.5 and vn=2radsec, the dominant closed-loop poles must

be located at

We first calculate the angle deficiency at this closed-loop pole.

This angle deficiency must be compensated by the lead compensator. There are many ways to

determine the locations of the pole and zero of the lead network. Let us choose the zero of the

compensator at s=–1.Then, referring to Figure 6–77, we have the following equation:

1.73205

x- 1

=tan (90°-70.8934°)=0.34641

=-70.8934°

Angle deficiency=- 120 °- 120 °-10.8934°+ 180 °

s=- 1 ;j 13

Gc(s)=Kc±

s+

1

T

s+

1

aT

≤ (0 6 a 6 1)

R(s) C(s)

Lead

compensator

Gc(s)

Space

vehicle

Sensor

1

s^2

1

0.1s+ 1

+–

Figure 6–76
Space-vehicle control
system.

jv

• 10 s

19.1066°

70.8934°

j1.73205

x

Figure 6–77
Determination of the
pole of the lead
network.