1392
IEEE TRANSACTIONS
ON
SYSTEMS, MAN, AND CYBERNETICS.
VOL.
23,
NO.
5.
SEPTEMBERIOCTOBER
1993
Fuzzy Gain Scheduling
of
PID
Controllers
Zhen-Yu Zhao,
Member,
IEEE,
Masayoshi Tomizuka,
Member,
IEEE,
and Satoru Isaka,
Member,
IEEE
Abstract-This paper describes the development of a fuzzy
gain scheduling scheme of PID controllers for process control.
Fuzzy rules and reasoning are utilized on-line to determine the
controller parameters based on the error signal and its first
difference. Simulation results demonstrate that better control
performance can be achieved in comparison with Ziegler-
Nichols controllers and Kitamori’s PID controllers.
I. INTRODUCTION
HE BEST-KNOWN controllers used in industrial
T
control processes are proportional-integral-derivative
(PID) controllers because of their simple structure and
ro-
bust performance in a wide range of operating conditions.
The design of such a controller requires specification of
three parameters: proportional gain, integral time con-
stant, and derivative time constant.
So
far, great effort has
been devoted to develop methods to reduce the time spent
on optimizing the choice of controller parameters [8],
[15]. The PID controllers in the literature can be divided
into two main categories. In the first category, the con-
troller parameters are fixed during control after they have
been tuned
or
chosen in a certain optimal way.
The
Ziegler-Nichols tuning formula is perhaps the most well-
known tuning method [5],
[
191. Some other methods exist
for
the PID tuning (see e.g., [l],
[6],
[7]). The PID con-
trollers of this category are simple, but cannot always ef-
fectively control systems with changing parameters, and
may need frequent on-line retuning. The controllers of the
second category have a structure similar to PID control-
lers, but their parameters are adapted on-line based on
parameter estimation, which requires certain knowledge
of the process, e.g., the structure of the plant model [2],
[
171. Such controllers are called adaptive PID controllers
in order to differentiate them from those of the first cate-
wry.
The application
of
knowledge-based systems in process
control is growing, especially in the field
of
fuzzy control
[9], [lo],
[
121-[ 141. In fuzzy control, linguistic descrip-
tions of human expertise in controlling
a
process are rep-
resented as fuzzy rules
or
relations. This knowledge base
is used by an inference mechanism, in conjunction with
some knowledge
of
the states of the process (say, of
measured response variables) in order
to
determine con-
Manuscript received April
10,
1992; revised October
3,
1992.
Z.-Y.
Zhao was with the Department
of
Mechanical Engineering, Uni-
versity
of
California at Berkeley, Berkeley, CA 94720, and is currently
with Omron Advanced Systems, Inc., in Santa Clara, CA.
M. Tomizuka is with the Department
of
Mechanical Engineering, Uni-
versity
of
California at Berkeley, Berkeley, CA 94720.
S.
Isaka
is
with Omron Advanced Systems, Inc., Santa Clara,
CA
95054.
IEEE
Log
Number 920968
1.
trol actions. Although they do not have an apparent struc-
ture of PID controllers, fuzzy logic controllers may be
considered nonlinear PID controllers whose parameters
can be determined on-line based on the error signal and
their time derivative
or
difference.
In this paper, a rule-based scheme
for
gain scheduling
of PID controllers is proposed
for
process control. The
new scheme utilizes fuzzy rules and reasoning to deter-
mine the controller parameters, and the PID controller
generates the control signal. It is demonstrated in this pa-
per that human expertise on PID gain scheduling can be
represented in fuzzy rules. Furthermore, better control
performance can be expected in the proposed method than
that of the PID controllers with fixed parameters.
11. PID CONTROLLER
The transfer function of a PID controller has the fol-
lowing form:
(1)
where
K,,
Ki,
and
Kd
are the proportional, integral, and
derivative gains, respectively. Another useful equivalent
form of the PID controller is
G,(s)
=
Kp
+
Ki/s
+
Kds
G,(s)
=
Kp(l
+
l/(T,s)
+
Tds)
(2)
where
Ti
=
K,/Ki
and
Td
=
Kd/Kp.
T,
and
Td
are known
as the integral and derivative time constants, respectively.
The discrete-time equivalent expression for PID control
used in this paper is given as
n
u(k)
=
K,e(k)
+
KIT,
c
e(i)
+
-
Kd
Ae(k).
r=l
TS
Here,
u(k)
is the control signal,
e(k)
is the error between
the reference and the process output,
Ts
is the sampling
period for the controller, and
Ae(k)
e(k)
-
e(k
-
1).
The parameters of the PID controller
Kp,
Ki,
and
Kd
or
K,,
Ti,
and
Td
can be manipulated to produce various re-
sponse curves from a given process. Finding optimum ad-
justments of a controller for a given process is not trivial.
In the following section, an on-line gain scheduling
scheme of the PID controller based on fuzzy rules is in-
troduced.
111.
Fuzzy
GAIN SCHEDULING
Fig. 1 shows the PID control system with a fuzzy gain
scheduler. The approach taken here is to exploit fuzzy
rules and reasoning to generate controller parameters.
It is assumed that
K,,
Kd
are in prescribed ranges
[K,.
min,
Kp,
,,,I
and
[Kd.
minr
K,,
,ax1
respectively. The aP-
0018-9472/93$03,00
0
1993 IEEE
