596 Chapter 8 / PID Controllers and Modified PID Controllerswhere
A(s)=As+z 1 BAs+z 2 BpAs+zmB
B(s)=sNAs+pN±1BAs+pN±2BpAs+pnB
whereNmay be 0, 1, 2andnm.Assume also that Gc1is a PID controller followed
by a filter 1/A(s),or
andGc2is a PID, PI, PD, I, D, or P controller followed by a filter 1/A(s).That is
where some of a 2 ,b 2 , and g 2 may be zero. Then it is possible to write as
(8–3)
wherea,b, andgare constants. Then
Because of the presence ofsin the numerator, the response y(t)to a step disturbance
input approaches zero astapproaches infinity, as shown below. Since
if the disturbance input is a step function of magnituded,or
and assuming the system is stable, then
= 0
=limsS 0
sKA(0)d
sB(0)+bK
y(q)=limsS 0 sc
sKA(s)
sB(s)+Aas+b+gs^2 BK
d
d
s
D(s)=
d
s
Y(s)=
sKA(s)
sB(s)+Aas+b+gs^2 BK
D(s)
=
sKA(s)
sB(s)+Aas+b+gs^2 BK
Y(s)
D(s)
=
Gp
1 +AGc1+Gc2BGp
=
K
A(s)
B(s)
1 +
as+b+gs^2
s
K
B(s)
Gc1+Gc2=
as+b+gs^2
s
1
A(s)
Gc1+Gc2
Gc2(s)=
a 2 s+b 2 +g 2 s^2
s
1
A(s)
Gc1(s)=
a 1 s+b 1 +g 1 s^2
s
1
A(s)
Openmirrors.com