Modern Control Engineering

Section 8–4 / Design of PID Controllers with Computational Optimization Approach 587

To plot the unit-step response curve of the system with any set shown in the sorted table, we spec-

ify the Kandavalues by entering an appropriate sortsolutioncommand.

Note that for a specification that the maximum overshoot be between 10%and 5%,there

would be three sets of solutions:

K=2.0000, a=0.9000, m=1.0614

K=2.2000, a=0.9000, m=1.0772

K=2.4000, a=0.9000, m=1.0923

Unit-step response curves for these three cases are shown in Figure 8–22. Notice that the sys-

tem with a larger gain Khas a smaller rise time and larger maximum overshoot. Which one of these

three systems is best depends on the system’s objective.

EXAMPLE 8–3 Consider the system shown in Figure 8–23. We want to find all combinations of Kandavalues

such that the closed-loop system has a maximum overshoot of less than 15%, but more than 10%,

in the unit-step response. In addition, the settling time should be less than 3 sec. In this problem,

assume that the search region is

3 K 5 and 0.1a 3

Determine the best choice of the parameters Kanda.

Amplitude

Time (sec)

Unit-Step Response Curves

1.4

1.2

1

0.8

0.6

0.4

0.2

0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

K = 2.4, a = 0.9

K = 2.2, a = 0.9

K = 2, a = 0.9

Figure 8–22
Unit-step response
curves of system with
K=2, a=0.9;
K=2.2, a=0.9;
andK=2.4,
a=0.9.

R(s) C(s)

PID

controller

4

s^3 + 6s^2 + 8s + 4

+

• Plant

(s+ a)^2
K s
Figure 8–23
PID-controlled
system with a
simplified PID
controller.