Modern Control Engineering

Section 8–3 / Design of PID Controllers with Frequency-Response Approach 577

1

K s(s+ 1)(s+5)

(s+ 0.65)^2

s

jv

j 8

j 6

j 4

j 2

• j 6


• j 8


• j 4


• j 2


• 10 – 8 – 6 – 4 – 2 0 2 s

K= 60

K= 30.322

K= 30.322

K= 13.846

K= 13.846

K= 13.846

K= 60

z= 0.358

z= 0.67

+

Figure 8–12
Root-locus diagram
of system when PID
controller has double
zero at s=–0.65.
K=13.846
corresponds to
given by Equation (8–1)
andK=30.322
corresponds to
given by Equation (8–2).

Gc(s)

Gc(s)

the values of Kandaare

K=29, a=0.25

with the maximum overshoot equal to 9.52%and settling time equal to 1.78 sec. Another possible

solution obtained there is that

K=27, a=0.2

with the 5.5%maximum overshoot and 2.89 sec of settling time. See ProblemA–8–12for details.

Approach 8–4 Design of PID Controllers with Computational Optimization

APPROACH

In this section we present a design of a PID controller based on the frequency-response

approach.

Consider the system shown in Figure 8–13. Using a frequency-response approach, de-

sign a PID controller such that the static velocity error constant is 4 sec−^1 , phase margin

is 50° or more, and gain margin is 10 dB or more. Obtain the unit-step and unit-ramp

response curves of the PID controlled system with MATLAB.

Let us choose the PID controller to be

Gc(s)=

K(as+ 1 )(bs+ 1 )

s