598 Chapter 8 / PID Controllers and Modified PID ControllersDetermination of Gc 2. Now that the coefficients of the transfer functionY(s)/R(s)
are all known and Y(s)/R(s)is given by
(8–4)
we have
SinceGc1is a PID controller and is given by
Y(s)/R(s)can be written as
Therefore, we choose
so that
(8–5)
The response of this system to the unit-step reference input can be made to exhibit the
maximum overshoot between the chosen upper and lower limits, such as
2 %<maximum overshoot< 10 %
The response of the system to the ramp reference input or acceleration reference input
can be made to exhibit no steady-state error. The characteristic of the system of Equa-
tion (8–4) is that it generally exhibits a short settling time. If we wish to further shorten
the settling time, then we need to allow a larger maximum overshoot—for example,
2 %<maximum overshoot< 20 %
The controllerGc2can now be determined from Equations (8–3) and (8–5). Since
Gc1+Gc2=
as+b+gs^2
s
1
A(s)
Gc1=
a 1 s+a 0 +a 2 s^2
Ks
1
A(s)
Kg 1 =a 2 , Ka 1 =a 1 , Kb 1 =a 0
Y(s)
R(s)
=
KAa 1 s+b 1 +g 1 s^2 B
sn+^1 +an sn+an- 1 sn-^1 +p+a 2 s^2 +a 1 s+a 0
Gc1=
a 1 s+b 1 +g 1 s^2
s
1
A(s)
=
Gc1 sKA(s)
sn+^1 +an sn+an- 1 sn-^1 +p+a 2 s^2 +a 1 s+a 0
=
Gc1 sKA(s)
sB(s)+Aas+b+gs^2 BK
Y(s)
R(s)
=Gc1
Y(s)
D(s)
Y(s)
R(s)
=
a 2 s^2 +a 1 s+a 0
sn+^1 +an sn+an- 1 sn-^1 +p+a 2 s^2 +a 1 s+a 0
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