Fuzzy self tuning of PID controllers

Fuzzy Sets and Systems 56 (1993) 37-46
North-Holland

37

Fuzzy self-tuning of PID controllers
Shi-Zhong He 1, Shaohua Tan
Feng-Lan Xu:
Department of Electrical Engineering, National University of
Singapore, 10 Kent Ridge Crescent, Singapore 0511

Pei-Zhuang Wang 3
Institute of System Science, National University of Singapore,
Heng Mui Keng Terrace, Kent Ridge, Singapore 0511
Received August 1992
Revised October 1992

Abstract." This paper presents a novel fuzzy self-tuning PID
control scheme for regulating industrial processes. The
essential idea of the scheme is to parameterize a
Ziegler-Nichols-like tuning formula by a single parameter a,
then to use an on-line fuzzy inference mechanism to self-tune
the parameter. The fuzzy tuning mechanism, with process
output error and error rate as its inputs, adjusts c~ in such a
way that it speeds up the convergence of the process output
to a set-point Yr, and slows down the divergence trend of the
output from Yr. A comparative simulation study on various
processes, including a second-order process, processes with
long dead-time and non-minimum phase processes, shows
that the performance of the new scheme improves
considerably, in terms of set-point and load disturbance

responses, over the PID controllers well-tuned using both the
classical Ziegler-Nichols formula and the more recent
Refined Ziegler-Nichols formula.

Keywords: Fuzzy self-tuning; fuzzy control; adaptive control.

1. Introduction

Despite the advent of many sophisticated
control theories and techniques, the majority of
industrial processes nowadays are still regulated
by PID controllers. This status quo not just
indicates the cautious attitude of the practicing
Correspondence to: Shaohua Tan, Department of Electrical Engineering, National University of Singapore, 10 Kent
Ridge Crescent, Singapore 0511.
1 On leave from Department of Automation, Tsinghua
University, Beijing 100084, China.
2 On leave from Department of Electrical Engineering,
Tsinghua University, Beijing 100084, China.
3 On leave from Department of Mathematics, Beijing
Normal University, Beijing 100088, China.

world towards the new invention, it does reveal
the rich potential of this extremely simple
(almost primitive, perhaps, in the eyes of some
control theorists) control strategy for meeting
various specifications for a vast variety of
industrial processes.
A crucial issue in the PID control is the setting
of the controller parameters, the so-called tuning

problem. The conventional way to do the tuning
is to study the mathematical models of processes,
and try to come up with a simple tuning law that
will establish a set of constant PID parameters
based on the models.
It is not hard to show theoretically that the
PID is adequate for the processes modelled
perfectly by linear first or second order systems.
Tuning laws can easily be established in these
cases. Unfortunately, real industrial processes
can never be modelled perfectly as simply as the
linear first and the second order systems. They
may have such marked characteristics as
high-order, dead-time, nonlinearity, etc., and
may be affected by noise, load disturbance and
other ambient conditions that cause parameter
variation and sudden model structural change.
The existing theories can no longer provide
systematic and robust tuning laws for these
complex situations. Thus, many of the PID
tuning laws actually incorporate empirical
evidences to compensate for the model complexity and variation. As can be expected, these
tuning laws are often ad hoc in nature, and may
only be useful for a certain class of processes or
under certain conditions. A typical example of
the model-based tuning laws is the famous
Ziegler-Nichols tuning formula [15]. Apart from
the fact that it may completely fail to tune the
processes with, for example, relatively large
dead-time, its tuning will have to be supplemented with purely experience-based fine-tuning

to meet the response requirements.
Along the line of empirical investigation and
approximate analysis, the work on improving the
PID tuning has been going on especially in the
past decade. There has been attempt to revise

0165-0114/93/$06.00 © 1993--Elsevier Science Publishers B.V. All rights reserved


38

Shi-Zhong He et al. / Fuzzy self-tuning of PID controllers

the half-century-old Ziegler-Nichols formula to
enhance its performance and applicability,
resulting in the so-called Refined ZieglerNichols tuning formula [6]. These refinements,
useful as they may be in improving certain
aspects of the responses for certain processes,
may perform worse for certain other processes.
In a sense, they hit upon the delicate boundary
of performance-process tradeoff, and the complexity may never allow a clear cut. Taking, as
an example, the processes with long dead-time, if
they are controlled by the PID plus the
Ziegler-Nichols tuning, the first overshoot in the
set-point response will be excessively high, which
is considered unacceptable for many applications. The Refined Ziegler-Nichols formula can
be employed in this case to reduce the
overshoot. In doing so, however, the response
time will be slowed down, sometimes considerably. Further, in the case of mild nonlinearities, it is hard to tell for a particular
process if the original formula or its refinement

should be used. All these may be contributed to
the single fact of requiring the PID parameters
to be fixed throughout the control. In other
words, with fixed parameters, the controllers of a
simple structure, such as the PID, cannot go
beyond a certain limit in handling the model
complexity and uncertainty. If we still insist on
using the same controller structure, the controller parameter adaptation with time seems to
be the only way to extend beyond the limit.
Carrying on with the thought, a natural step
ahead is to consider self-tuning PID control,
which tunes the PID parameters on-line to adjust
the controller actions for meeting the real-time
need. This is precisely the direction pursued by a
number of researchers [3, 10, 1]. The idea of the
existing PID self-tuning schemes essentially
follows that of the conventional self-tuning
controllers, i.e., the tuning at any time instance is
based on a structurally-fixed parameter-evolving
process model produced by an on-line identification procedure. The momentary tuning itself will
still have to be done by using some design
formula, or just the Ziegler-Nichols formula.
Thus, these schemes can be seen as trying to deal
with the model complexity and uncertainty
problem by localizing (on the time scale) the
conventional tuning methods. Aside from the
often-cited problem of high computational

demand, a major difficiency for the schemes
seems to be that it does not change the

model-based nature of non-adaptive PID controllers. By assuming a model, the consequent
robustness issue still needs to be settled (even at
localized time instances), which proves to be
difficult. Further, because an a priori assumed
model will have to be necessarily simple, it often
cannot accommodate the structural disturbance,
such as load disturbance on processes. If such a
disturbance happens, the identified model will be
highly inaccurate, leading to the momentarily
degraded controller performances [7].
Against this backdrop, we propose to use a
fuzzy inference based self-tuning scheme for PID
controllers. The essence of the scheme is that at
every time instance, the controller evaluates the
trend of the controlled process output to detect
the possible deviation from a prescribed course.
If a deviation is found, an appropriate control
action according to the nature of the deviation
will be generated instantaneously to correct it.
Compared to the existing model based selftuning schemes, our scheme is empirical based,
and acts more like what we do when we, for
example, try to steer a controller manually to
keep the output of a process on a fixed course.
Our own experience tells that a precise
description of the process (in the form of a
dynamical model) is often irrelevant for our
steering actions. What is more important is the
instant observations of the error, and subsequent
rational actions for bringing it back to course.
There are two key ideas in our scheme. First,

the Ziegler-Nichols formula is parameterized by
a single parameter a. This a is arranged so that
its increase (decrease) will lead to the increase
(decrease) in the proportional term and the
decrease (increase) in both the integral and
differential terms in the PID controller. Such an
arrangement is intended to divert the trend of
the process output using the knowledge of the
qualitative relationship between the proportions
of the PID feedbacks and the profiles of the
process output. Secondly, the on-line tuning
formula for a is a discrete dynamical equation
driven by a fuzzy inference procedure. A simple
fuzzy map is formed in such a way that it updates
a in accordance with the current regulation error
and error rate. Specifically, it speeds up the
convergence of the process output to a set-point


Shi-ZhongHe et al. / Fuzzyself-tuningof PID controllers
Yr, and slows down the divergence trend of the
output from Yr. This is, in fact, the fuzzy
adaptation mechanism we have used successfully
in one of our early works on adaptive fuzzy
controller [9].
Many forms of adaptive fuzzy control schemes
exist, see [12,13,9]. Virtually all of these
schemes are genuine fuzzy control schemes in
the sense that the controllers are actually fuzzy,
although the adaptation mechanisms may sometimes employ non-fuzzy tuning laws. The

proposed fuzzy auto-tuning PID is different in
nature from all these schemes in that it is a
non-fuzzy controller tuned with a fuzzy inference
mechanisms. Further, because of its connection
with the Ziegler-Nichols formula, the proposed
control scheme is not completely model-free. A
simple initialization procedure will have to be
used to obtain the ultimate gain and ultimate
period of the process to be controlled in order to
start the fuzzy adaptation. This limited degree of
model dependence reflects the consideration that
if certain information on the process can be
acquired easily and directly, the control scheme
should be able to make use of it. This is in sharp
contrast to a genuine fuzzy controller, which are
completely model-free and all the control rules
are supposed to come directly from the
experiences. Another rationale behind our
scheme is that it is often hard to directly acquire
the knowledge or possess direct human experiences for controlling a complex process.
However, if the controller structure is fixed to be
the PID control, then the experiences are
narrowed down to more specialized experiences
of choosing a few PID parameters. This latter
problem has been under scrutiny for so long a
time that there have been a great amount of
knowledge and experiences accumulated on the
subject. In this context, it seems more meaningful to keep the PID structure and let the
self-tuning part be handled by the fuzzy logic
approach.

The main objective of the present paper is to
propose this new type of fuzzy self-tuning
control scheme, provide the details of the design
procedure, and conduct a simulation analysis to
compare the scheme with two tuning schemes,
namely, the Ziegler-Nichols tuning and the
Refined Ziegler-Nichols tuning. The general
conclusion of the simulation analysis is that the

39

new fuzzy self-tuning PID controller outperforms the PID controllers tuned by the two
fixed tuning laws.
The paper is organized as follows. In Section
2, the new fuzzy self-tuning PID controller is
described in detail, the exposition covers the
basic structure of the controller, fuzzy tuner, as
well as the initialization of the controller. The
simulation analysis is carried out in Section 3
followed by Section 4, which contains further
discussions and conclusions.

2. The controller and the fuzzy adaptation

Basic structure
To begin with, we assume that the process to
be controlled has single input u(t) and single
output y(t), and the control objective is to bring
the process output y(t) to a prescribed set-point
Yr. The scheme can actually be extended to the

tracking problems where y~ is a time-varying
target output. However, this extension will not
be discussed here to keep our exposition concise.
As mentioned in the previous section, the
fuzzy self-tuning PID controller consists of a
standard PID controller and a fuzzy tuning
mechanism used for the on-line adaptation of the
PID parameters (Figure 1). The PID controller,
which generates a control u(t) based on the
closed-loop error e(t)= Yr- y(t), has the following standard form

u(t)= Kc[e(t)+ Tdde~t) +~ fe(t)dt ] ,

(1)

where Ko Td, T~ are, respectively, the proportional gain, the derivative time and the integral
time of the controller, which are to be adjusted
on-line.
One of the key ideas of the control scheme is
to parametrize the three PID parameters by a
single parameter c~ as shown below
Kc-- 1.2c~ku,
1
T~= 0.75 1--~-a t°,

Ta = 0.25T~,

(2)

where ku, tu are, respectively, the ultimate gain

and the ultimate period of the underlying
process, which will be determined shortly.


Shi-Zhong He et al. / Fuzzy self-tuning of PID controllers

40
f-

Fuzzy Self-tuning Mechanism

I
I
I
I

]FuzzyAdaptatio~-----~ DPasrgm~oerrimZelda

I
[-

m

_

_1

-

Yr

Controller

Process

Fig. 1. The basic structure of the fuzzy self-tuning PID controller.

The form of the parameterization is inspired
by the Ziegler-Nichols formula, and in fact
reduces to it when a = ½. Thus we can think of
the controller as compensating the basic control
of the Ziegler-Nichols by biasing all the
parameters on-line in order to adjust the process
output to a prescribed course. In what follows,
both the fuzzy adaptation and the initialization
will be discussed in detail.

Table 1. The fuzzy map from E and R to H
H

E

R
-3

-2

-1

0

1

2

3

-3
-3
-2
-2

-3
-2
-2
-1

-2
-2
-I
-1

-2
-1
-1
0

-1
-1
0

1

-1
0
1
1

0
1
1
2

1

1

-1

0

1

1

2

2

2
3

-1
0

1
1

1
2

2
2

2
3

3
3

-3
-2
-1
0

0
1

Fuzzy adaptation
As shown in Figure 1, the fuzzy self-tuning
mechanism will generate an a(t) given the

instant values of e(t) and O(t) at time t. It is
composed of two parts: a fuzzy core and a
conditional updating formula for a. The fuzzy
core starts with fuzzifying e(t) and O(t) into two
fuzzy variables E, respectively, R. To ensure a
speedy fuzzy inference, both the range of
interest for e(t) and that for ~(t) are covered by
seven different fuzzy sets as shown below
E = {NL, NM, NS, ZO, PS, PM, PL}
R = {NL, NM, NS, ZO, PS, PM, PL},

(3)

where, as usual, the meanings of the acronyms
used in (3) are, respectively, PL for positive
large, PM for positive medium, PS for positive
small, Z O for zero, NS for negative small, NM
for negative medium, and N L for negative large.
For ease of the notation, we shall assign the
integers for the fuzzy sets as
PL=3,
NS=-I,

PM=2,
NM=-2,

PS=I,

ZP=0,

NL=-3,

and denote the fuzzy sets by their corresponding
numbers.
The second part of the fuzzy core is the fuzzy
mapping from E and R to H, where H is another
fuzzy variable whose defuzzified version will be
used for the later updating equation of a. The
range of H similarly consists of seven fuzzy sets
H = { - 3 , - 2 , - 1 , 0, 1, 2, 3},

(4)

and it is linked to E and R by a fuzzy map (a
particular example that we shall use is given in
Table 1). The fuzzy inference is the standard
CRI procedure, hence all the fuzzy rules, such as
those shown in Table 1, will be involved in every
single inference.
We also assume that the membership functions for all fuzzy sets have the following
standard form
2

re(x) = e ( ~ ),

(5)

where xi, o- denote, respectively, the centre and
the spread of each fuzzy set, and their choices
are somehow subjective.

41

Shi-Zhong He et al. / Fuzzy self-tuning of PID controllers
The last part of the fuzzy core is the
defuzzification of H into a real variable h(t). For
smooth opeation, we choose to use the centre of
gravity method.
After h(t) is obtained, it is used in the
following recursive equation to update a
c~(t + 1)
= {a(t) + yh(t)(1 - a(t))
( a ( t ) + vh(t)ce(t)

for ol(t) >0.5,
for c~(t) ~<0.5,

(6)

where y is a positive constant used to modify the
convergence rate of the updating formulae. The
range of 7 is wide, and is typically chosen to be
within the interval [0.2,0.6] for most of the
processes. Note that a(0) is not arbitrary and has
to be set at 0.5. It also follows from (6) that a(0)
is a cut-off value whereby the two updating
formulae switch from one to the other. Such an
arrangement along with the fuzzy map guarantees the smooth and bounded (between 0 and
1) variation of ~, which in turn leads to smooth

and bounded adaptation of the PID parameters.
Initialization of the controller
As the tuning formula involves the ultimate
gain ku and the ultimate period tu, they will have
to be determined prior to the use of the
controller. This process is called initialization.
Several methods for initializing the PID
controllers are available. Because of its intuitive
appeal and ease of operation, the relay feedback
method proposed in [3, 5] is used to initialize the
parameters of the fuzzy self-tuning PID
controller.
The method of relay feedback is based on the

fact that dynamical processes typically encountered in process control will exhibit limit cycle
oscillation under relay feedback. The frequency
of the limit cycle is approximately the ultimate
frequency where the process has a phase lag of
180 °. The period of the oscillation tu is easily
obtained by measuring the time between zero
crossings. The amplitude may be determined by
measuring the peak to peak values of the output.
If d is the relay amplitude and a is the process
output amplitude, the ultimate gain is approximately given by
k.

=

4d
--.

na

(7)

The relay auto-tuner principle is shown in Figure
2.
With both ku and tu in place, and with a(0) set
to be 0.5, the initial values for the PID
parameters naturally follow from (2)
Kc = 0.6ku,

T~= 0.5tu,

Tj : 0.125tu,

which is precisely the Ziegler-Nichols formula.
It is thus clear that except for the initialization
where an implicit model assumption on the
initial status of the process is made, our fuzzy
self-tuning PID controller is a model-free control
scheme.
A close examination of both (2) and (8) shows
that our adaptation scheme can be interpreted as
using a single parameter to bias the PID
controller parameters away from their ZieglerNichols settings in order to compensate for any
inadequacy. This idea of adaptation should be
contrasted with the conventional PID self-tuning
schemes whereby at each time instance a

_•

Fuzzy
Self-tuning

t
Y~ ' C )

l-

•

PID

(8)

,Q~.~

• Process

Fig. 2. Block diagram of the relay auto-tuner.

y


42

Shi-Zhong He et al. / Fuzzy self-tuning of PID controllers

pre-specified process model will have to be
identified, and the tuning either modifies ku and

tu, or changes directly Kc, T~, Td based on the
identified process model. To highlight their
difference, we can think of the two types of
adaptations as either elastic or plastic. The
adaptation we have proposed is elastic in the
sense that it only 'deforms' from a standard set
of controller parameters (the Ziegler-Nichols in
our case) to counter for the process complexity
and variation. The 'deformation' may bounce
back and forth causing all the controller
parameters to vary around the standard set of
controller parameters. Whereas the conventional
way of adaptation is plastic in the sense that the
'deformation' of the controller parameters at
every time instance is dictated by the identified
model and thus irreversible unless the
coefficients of the identified model are moving
up and down around a standard set of
coefficients, which is highly unlikely. It is in
general difficult to tell which form of adaptation
is superior. In the present context, however, the
model-free nature of our elastic adaptation
appears to be conceptually simpler and more
efficient than the conventional model-based
plastic adaptation.
With the preceding design details of the new
control scheme, let us provide a rough account
of how it actually works. Note that this fuzzy
self-tuning mechanism is the one that we
proposed in one of our early works [9]. As being

explained in [9], the essential idea of the
adaptation is to provide appropriate c~ for
several real-time scenarios defined on the profile
of y in relation to Yr- Here we are only interested
in four possible scenarios: y approaches to Yr

from above or below, and y diverts up and down
away from Yr. When y approaches Yr from above
or below, the combined effort of both the fuzzy
map in Table 1 and the updating equations (6)
will increase a, (2) will consequently ensure that
such an increase will push Kc higher and Ti, Td
lower, which in turn speeds up the approaching
of y to Yr- Similarly, when y diverges up (or
down) from Yr, c~ will be decreased, thus
decreasing Kc and increasing T~, Td, consequently
causing the divergence to be slowed down.
Because there is only one parameter c~
involved, the rules for generating all the changes
for the four possible scenarios are easy to form.
Table 1 is only one example of appropriate fuzzy
maps that can be used for the purpose. The
construction of the kind of fuzzy map requires
the analysis of response profiles and their
relationships to c~, and is also affected by the
rate of change we desire on a. The fuzzy map in
Table 1 only effects a moderate a change rate.
High rate of change tends to cause oscillatory
behaviour, and thus is not always desirable.
Theoretical verification of the preceding

statements in the context of model complexity
and uncertainty is not yet available, although
such a verification is possible for a given simple
process model. But this latter type of verification
has specialized nature and does not illustrate our
point. For the time being we mostly rely on the
empiricial studies and simulation analysis for
such a verification. Figure 3 is typical of the
simulation results, which shows how c~ changes
given the corresponding profile of y. This plot
confirms that the particular updating formula (6)
and the fuzzy map indeed modifies c~ as
prescribed.

2
1.5
I
0.5
0

0

1

3

4

5

Time vs yp and alpha, yp(-), alpha(--)
Fig. 3. The trend of o~ in relation to the process output y.

7

8


Shi-Zhong He et al. / Fuzzy self-tuning of" PID controllers"

self-tuning PID controller, Y2, Y3 are those of the
PID controllers tuned using the Refined
Ziegler-Nichols
formula,
respectively, the
Ziegler-Nichols formula. We shall use the same
notations for the rest of the simulation examples.
Figure 4 clearly shows the remarkable
set-point response performance of the fuzzy
self-tuning controller over the other two
controllers with shorter rise time, shorter settling
time, and less overshoot. Observe that unlike
normal fuzzy controllers where lowering thc
overshoot is often at the expense of slowing
down considerably the rise time, the new control
scheme seems to reconcile these two requirements. The reason is roughly that the self-tuning
nature of the controller allows the selection of
different PID controllers for controlling the
process at different stages of the responses. The
controller is designed so that it picks up an

appropriate PID controller at each stage.
Figure 4 also shows that the improvement in
the set-point response of the new controller is
not at the cost of the load disturbance response,
although there is no obvious improvement in the
load disturbance response with respect to those
by the other two controllers. For conventional
controllers, these two responses tend to undo
each other, leading to a compromised design in
which none of the responses is at its best. The
new control scheme seems to have provided yet
another reconciliation towards this problem. As
we shall see again in the later simulation
examples, the fuzzy self-tuning PID controller
tends to enhance the set-point response while
keeping the load disturbance at an acceptable
level.

3. Simulation analysis
To cover typical kinds of common industrial
processes, three groups of representative processes are chosen for our simulation analysis. For
each of the latter two types of processes,
different coefficients are set for the same process
models just to reflect the severity of the
time-delay or non-minimum phase characteristics. To evaluate the performance of the new
fuzzy self-tuning PID controller, we compare its
set-point as well as the load disturbance
responses with the PID controllers tuned by the
Ziegler-Nichols formula and the Refined
Ziegler-Nichols formula.

A second-order process

The first process is chosen to have the
following simple second-order characteristics

G(s) =

k,,
(T,s + 1)(~s + 1 )

43

(9)

The parameters of the process a r e kp = 1, T l = l
and ~ = 0.5. The choice of the sampling time t~
is based on the process time constants T~, T2,
and here we set ts to be 0.1. The new set-point y,
is 1, and a static load disturbance is also
introduced into the process at t = 20s.
This process is regulated, separately, by the
fuzzy self-tuning PID controller, and by the two
PID controllers tuned using the Ziegler-Nichols
formula, and the Refined Ziegler-Nichols tuning
formula. The respective set-point responses and
load disturbance responses are summerized in
Figure 4, where y~ is the response of the fuzzy

Closed,-Loop Response of p,rocess

1.5r

0.5

0

~

i

I

i

i

i

i

i

0

5

10

15

20

25

30

35

40

yl(-), y2(-.) and y3(--) vs time
Fig. 4. The set-point and load disturbance responses of the second-order process.


44

Shi-Zhong He et al. / Fuzzy self-tuning of PID controllers

, Closed-LoopResponse,ofprocess

1.5

s",
1 ~ V

-~r'~'''~ -

~~,,~,~--,---'

0.5

00

i

i

10

20

i

i

30
40
50
yl(-), y2(-.) and y3(--) vs time

i

i

60

70

Fig. 5. The set-point and load disturbance responses of the process with small dead-time.
Figures 5 and 6 allow us to draw similar

conclusions: The set-point response improves
considerably while the load disturbance responses are either comparable to those obtained
by the other two controllers (in the case of small
dead-time), or only improves marginally (in the
case of large dead-time). It is also interesting to
observe that in both cases, the fuzzy self-tuning
P I D controller acts m o r e like a further refined
Ziegler-Nichols P I D control, capable of curbing
the excessive overshoot.

Time-delay processes

The second simulation concerns time-delay
processes of the following general form

G(S)

-

(10)

kpe-°ds

(Ts ~- 1) 2.

The two p a r a m e t e r s kp and T are all set to 1, and
the dead-time 0d, however, will be set to two
different values to examine the ability of the new
controller in handling small or large dead-time.
The sampling rate for the simulation remains to

be 0.1.
0d is first chosen to be 2, corresponding to a
relatively small dead-time. Yr is 1, and a static
load disturbance is introduced at t = 35. The
simulation results are shown in Figure 5. Note
that the notations for the three response curves
Yl, y2 and Y3 are the same as those used in the
first simulation example.
0d is then set to be 6, corresponding to a
relatively large dead-time. This time the static
load disturbance is introduced at t = 55 to allow
the process to settle down. The simulation
results are summarized in Figure 6.

15

1

N o n - m i n i m u m p h a s e processes

The last simulation has to do with nonminimum phase processes. The general form of
the process is given as
kp(1 - ps)
G(s) - -~ + ~ .

(11)

For the first process, the p a r a m e t e r s are chosen
as follows; k p = l , T = I and p = 1 . 4 . Clearly,
this process has a n o n - m i n i m u m phase zero at

s = l i p = 3. The sampling rate is chosen to be

Close-Loop ae~pons~oe process,
; :,/'~.,

,,-~

_

~'."'--~..~_

0.5

00

10

i

i

20

30

i

i

i

i

40
50
60
70
yl(-), y2(-.) and y3(--) vs time

i

i

80

90

100

Fig. 6. The set-point and load disturbance responses of the process with large dead-time.


45

Shi-Zhong He et al. / Fuzzy self-tuning of PID controllers

Close-Loop Response of process

1.5

1

'/

,

,

.........

-°--

0 " 50

-0.5
0

5

10

15
20
25
yl(-), y2(-.) and y3(--) vs time

30

35

--

40

Fig. 7. The set-point and load disturbance responses of the first non-minimum phase process.
0.1, and the load disturbance is introduced
t = 20. Figure 7 shows all the response curves.
For the second n o n - m i n i m u m phase process, p
is set to 2.5, both T and kp remain the same as in
the previous process. The n o n - m i n i m u m phase
zero in this case is at s = 3, much closer to the
origin of the s-plane. All the simulation results
for this process are shown in Figure 8.
It follows from Figures 7 and 8 that while both
the Ziergler-Nichols tuning and the Refined
Ziegler-Nichols tuning cannot provide adequate
control to the n o n - m i n i m u m phase processes,
especially for the second one, the new scheme
works r e m a r k a b l y well in both cases. Observe
that the load disturbance responses also seem to
improve considerably for these processes, thus
contradicting the genral patterns observed in the
preceding simulations. This, perhaps, should not
be understood in terms of the i m p r o v e m e n t
m a d e by the fuzzy self-tuning controller, in fact,
its load disturbance is always acceptable (but not
superior) for various kinds of processes. It
should be understood in terms of the failure of
the other P I D tuning schemes in providing an

1.5

acceptable level of control for certain difficult
processes. The advantage of the new scheme
becomes evident for controlling these processes.

4. Discussions

and conclusions

We have presented a detailed account of a
fuzzy self-tuning P I D controller, its basic
principle, design steps and simulation analysis.
The performance of the new controller is
c o m p a r e d favourably to those of the two existing
conventional P I D tuning formulae.
In a sense, the present work represents an
extension of our early work where exactly the
same adaptation mechanism is used to tune a set
of fuzzy control rules [9]. In both works, our aim
is to show that this so-called elastic adaptation
can be used to great advantage in process
control.
One interesting feature of the present work is
the combination of the conventional P I D
controller with the fuzzy inference method.
Combinations a m o n g other controllers and

CloseA,-LoopResponse of process

__ ,

0.50

05

" ' 0

.

5

.

10

.

.
.
.
20
25
yl(-), y2(-.) and y3(--) vs time
15

.

30

35

40

Fig. 8. Thc set-point and load disturbance responses of the second non-minimum phase process.


46

Shi-Zhong He et al. / Fuzzy self-tuning of PID controllers

mechanisms have led to encouraging results [14].
We believe that the idea of fuzzy control should
go beyond the present limit of using CRI to pick
up the fuzzy control rules. In fact, there is a
great potential in using the fuzzy inference
method as part of a control scheme, which assists
the control formation rather than generates the
control directly. There have been cases where
the fuzzy logic is used along this line. A
particular example we would like to mention is
to use fuzzy logic to obtain an estimate of the
so-called transient period useful for certain
adaptive control scheme [8]. The present paper
offers yet another non-trivial example of using
fuzzy inference method as a self-tuning mechanism of a conventional controller. As the
adaptation is a major part of any adaptive
control scheme, the investigation along this line
is certainly important in its own right, and may
lead to high-performance adaptive control

schemes.

References
[1] K.J. ~,str6m and B. Wittenmark, Adaptive Control
(Addison-Wesley, Reading, MA., 1989).
[2] K.J. ,~str0m and T. Hagglund, Automatic tuning of PID
controllers, Instrument Society of America, USA, 1988.
[3] K.J. ,~str6m and T. H~igglund, An industrial adaptive
PID controller, Proc. 1989 IFAC Symp. Adaptive
System in Control and Signal Processing, 1989, pp.
293-298.

[4] C.C. Hang and K.J. ~str0m, Refinements of the
Ziegler-Nichols tuning formula for PID auto-tuner,
Proc. of ISA 88 Intern. Conf. and Exhibition, Houston,
X., 1988, pp. 1021-1030.
[5] C.C. Hang and K.J. Astr6m, Practical aspects of PID
auto-tuners based on relay feedback, Proc. Intern.
Syrup. Adaptive Control of Chemical Processes,
Denmark, 1988, pp. 149-154.
[6] C.C. Hang, K.J.A.strOm and W.K. Ho, Refinements of
the Ziegler-Nichols tuning formula, lEE Proc. Part D
138 (1991) 111-118.
[7] C.C. Hang, S.Z. He and T.H. Lee, The normal-modeinaction adaptive PID controller, Preprint 1992 IFAC
Int. Symp. Adaptive Systems in Control and Signal
Processing, France, 1992, pp. 133-138.
[8] S.Z. He, C.C. Hang and T.H. Lee, Incorporating fuzzy
logic into the normal-modeinaction adaptive PID
controller, Proc. 1992 Singapore Intern. Conf. Intelli.
Control Instru., Singapore, Vol. 2, 1992, pp. 1357-1362.

[9] S.Z. He, S. Tan, C.C. Hang and P.Z. Wang, Design of
an on-line rule-adaptive fuzzy control system, Proc.
First IEEE Intern. Conf. Fuzzy Syst., San Diego, 1992,
pp. 83-91.
[10] H.S. Hoopes, W.K. Hawk and R.C. Lewis, A self-tuning
controller, ISA Trans. 22 (1983) 49-58.
[11] T.W. Kraus and T.J. Mayron, Self-tuning PID
controllers based on a pattern recognition approach,
Control Engineering, 1984, pp. 106-111.
[12] T.J. Procyk and E.M. Mamdani, A linguistic selforganization process controller, Automatica 15 (1979)
15-30.
[13] M. Sugeno, Industrial Applications of Fuzzy Control
(North-Holland, Amsterdam, 1985).
[14] S. Tan, A combined PID and neural control scheme for
nonlinear dynamical systems, Proc. 1992 Singapore
Intern. Conf. Intelli. Control Instru., Singapore, Vol. 1,
1992, pp. 377-383.
[15] J.G. Ziegler and N.B. Nichols, Optimum setting for
automatic controllers, Trans. A M S E 65 (1943) 433-444.