Part A | 10.3
252 Part A Fundamentals
The second case to be examined is when the process exhibits first-order linear dynamics and nonzero
disturbance is present. Assume that G.s/ in (10.100) is
as follows
G.s/ D
K SS
s C 1
:
(10.109)
Then (10.100) with P-control (K.s/ D K p ) yields
Y.s/ D
K SS K p
s C 1 C K SS K p
R.s/
C
K SS
s C 1 C K SS K p
D.s/ :
(10.110)
By assuming that R.s/ D 0, without any loss of generality, the final value theorem of the Laplace transform
(10.6) may be employed in (10.100), to obtain the
closed-loop response when there is a step change in disturbance (d.t/ D u step .t/ , D.s/ D 1=s).
lim
t!1
.y.t// D lim
s!0
.sY.s//
D lim
s!0
Â
s
K SS
s C 1 C K SS K p
D.s/
Ã
+
lim
t!1
.y.t//
D lim
s!0
Â
s
K SS
s C 1 C K SS K p
1
s
Ã
D
K SS
1 C K SS K p
(10.111)
In conclusion in order to minimize the deviation of
y.t ! 1/ from r D 0 a large value for coefficient K p
is needed.
Proportional-Integral Control
For proportional-integral control (PI-control) it holds
that K d D 0. A control law including an integral term
eliminates steady-state error which is the major drawback of P-control. It is for this reason that the PI
controller finds widespread use in industry. A typical
example is speed governors used for regulation of the
rotational speed (rpm, revolutions per second) of engines, turbines, motors, etc.
PI control is defined by the transfer function
u.t/ D K p e.t/ C K i
t
Z
0
e../ d
m
U.s/ D
Â
K p C
K i
s
Ã
E.s/ D
K p s C K i
s
E.s/ :
(10.112)
The effect of PI-control is now investigated in the case
of an LTI–SISO process with a generic transfer function
defined as a ratio of two polynomials p n .s/ and p 0 .s/
with degree m and n, respectively, and m Ä n.
G.s/ D
p n .s/
p 0 .s/
(10.113)
With PI control, (10.100) yields
Y.s/ D
.K i C K p s/p n .s/
sp 0 .s/ C .K i C K p s/p n .s/
R.s/
C
sp n .s/
sp 0 .s/ C .K i C K p s/p n .s/
D.s/ :
(10.114)
As an example, consider the transfer equation in
(10.109). Then, (10.114) yields
Y.s/ D
K SS .K i C K p s/
s.1 C s/ C K SS .K i C K p s/
R.s/
C
K SS s
s.1 C s/ C K SS .K i C K p s/
D.s/ :
(10.115)
Assuming, as in the case of P-control, that R.s/ D 0 and
applying the final value theorem for d.t/ D u step .t/ ,
D.s/ D 1=s, the following is obtained
y.t ! 1/ D lim
s!0
.sY.s//
D
K SS 0
0 .1 C 0/ C K SS .K i C K p 0/
D 0 :
(10.116)
As can be seen, PI control introduces a zero at s D 0 in
the closed-loop transfer function connecting the disturbance to the output signal, resulting to elimination of
the steady-state error.
A word of caution is needed here concerning
closed-loop stability. The closed-loop characteristic
polynomial as can be seen in (10.114) is as follows
p c .s/ D sp 0 .s/ C .K i C K p s/p n .s/ :
(10.117)
For the closed-loop system to be stable all the roots of
the polynomial above must lie in the left-hand complex
plane. However, the degree of this polynomial is (nC1),
that is, higher by 1 than the open-loop, process characteristic polynomial, p 0 .s/. This is the price paid for the
improvement in system’s behavior after introducing PIcontrol.
In the specific case of the transfer function in
(10.109), the closed-loop characteristic polynomials in
252 Part A Fundamentals
The second case to be examined is when the process exhibits first-order linear dynamics and nonzero
disturbance is present. Assume that G.s/ in (10.100) is
as follows
G.s/ D
K SS
s C 1
:
(10.109)
Then (10.100) with P-control (K.s/ D K p ) yields
Y.s/ D
K SS K p
s C 1 C K SS K p
R.s/
C
K SS
s C 1 C K SS K p
D.s/ :
(10.110)
By assuming that R.s/ D 0, without any loss of generality, the final value theorem of the Laplace transform
(10.6) may be employed in (10.100), to obtain the
closed-loop response when there is a step change in disturbance (d.t/ D u step .t/ , D.s/ D 1=s).
lim
t!1
.y.t// D lim
s!0
.sY.s//
D lim
s!0
Â
s
K SS
s C 1 C K SS K p
D.s/
Ã
+
lim
t!1
.y.t//
D lim
s!0
Â
s
K SS
s C 1 C K SS K p
1
s
Ã
D
K SS
1 C K SS K p
(10.111)
In conclusion in order to minimize the deviation of
y.t ! 1/ from r D 0 a large value for coefficient K p
is needed.
Proportional-Integral Control
For proportional-integral control (PI-control) it holds
that K d D 0. A control law including an integral term
eliminates steady-state error which is the major drawback of P-control. It is for this reason that the PI
controller finds widespread use in industry. A typical
example is speed governors used for regulation of the
rotational speed (rpm, revolutions per second) of engines, turbines, motors, etc.
PI control is defined by the transfer function
u.t/ D K p e.t/ C K i
t
Z
0
e../ d
m
U.s/ D
Â
K p C
K i
s
Ã
E.s/ D
K p s C K i
s
E.s/ :
(10.112)
The effect of PI-control is now investigated in the case
of an LTI–SISO process with a generic transfer function
defined as a ratio of two polynomials p n .s/ and p 0 .s/
with degree m and n, respectively, and m Ä n.
G.s/ D
p n .s/
p 0 .s/
(10.113)
With PI control, (10.100) yields
Y.s/ D
.K i C K p s/p n .s/
sp 0 .s/ C .K i C K p s/p n .s/
R.s/
C
sp n .s/
sp 0 .s/ C .K i C K p s/p n .s/
D.s/ :
(10.114)
As an example, consider the transfer equation in
(10.109). Then, (10.114) yields
Y.s/ D
K SS .K i C K p s/
s.1 C s/ C K SS .K i C K p s/
R.s/
C
K SS s
s.1 C s/ C K SS .K i C K p s/
D.s/ :
(10.115)
Assuming, as in the case of P-control, that R.s/ D 0 and
applying the final value theorem for d.t/ D u step .t/ ,
D.s/ D 1=s, the following is obtained
y.t ! 1/ D lim
s!0
.sY.s//
D
K SS 0
0 .1 C 0/ C K SS .K i C K p 0/
D 0 :
(10.116)
As can be seen, PI control introduces a zero at s D 0 in
the closed-loop transfer function connecting the disturbance to the output signal, resulting to elimination of
the steady-state error.
A word of caution is needed here concerning
closed-loop stability. The closed-loop characteristic
polynomial as can be seen in (10.114) is as follows
p c .s/ D sp 0 .s/ C .K i C K p s/p n .s/ :
(10.117)
For the closed-loop system to be stable all the roots of
the polynomial above must lie in the left-hand complex
plane. However, the degree of this polynomial is (nC1),
that is, higher by 1 than the open-loop, process characteristic polynomial, p 0 .s/. This is the price paid for the
improvement in system’s behavior after introducing PIcontrol.
In the specific case of the transfer function in
(10.109), the closed-loop characteristic polynomials in