Modern Control Engineering

618 Chapter 8 / PID Controllers and Modified PID Controllers

Design a control system such that the response to any step disturbance will be damped out

quickly (in 2 to 3 sec in terms of the 2%settling time). Choose the configuration of the closed-loop

poles such that there is a pair of dominant closed-loop poles. Then obtain the response to the

unit-step disturbance input. Also, obtain the response to the unit-step reference input.

Solution.The PID controller has the transfer function

For the disturbance input in the absence of the reference input, the closed-loop transfer function

becomes

(8–14)

The specification requires that the response to the unit-step disturbance be such that the settling

time be 2 to 3 sec and the system have a reasonable damping. We may interpret the specification

as and vn=4radsec for the dominant closed-loop poles. We may choose the third pole

ats=–10so that the effect of this real pole on the response is small. Then the desired charac-

teristic equation can be written as

(s+10)As^2 +2*0.5*4s+4^2 B=(s+10)As^2 +4s+16B=s^3 +14s^2 +56s+160

The characteristic equation for the system given by Equation (8–14) is

s^3 +(3.6+Kab)s^2 +(9+Ka+Kb)s+K=0

Hence, we require

3.6+Kab=14

9+Ka+Kb=56

K=160

which yields

ab=0.065, a+b=0.29375

The PID controller now becomes

With this PID controller, the response to the disturbance is given by

=

s

(s+10)As^2 +4s+ 16 B

D(s)

Cd(s)=

s

s^3 +14s^2 +56s+ 160

D(s)

=

10.4As^2 +4.5192s+15.385B

s

=

160 A0.065s^2 +0.29375s+ 1 B

s

Gc(s)=

KCabs^2 +(a+b)s+ 1 D

s

z=0.5

=

s

s^3 +(3.6+Kab)s^2 +(9+Ka+Kb)s+K

Cd(s)

D(s)

=

s

sAs^2 +3.6s+ 9 B+K(as+1)(bs+1)

Gc(s)=

K(as+1)(bs+1)

s

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