fuzzy pid basic
TRANSCRIPT
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Procee ding of the 2004 American Control Conferen ce
Boston, Massachusetts June
30
-July
2 2004
I WeM14.6
A
Developed Method of Tuning PID ontrollers
with Fuzzy Rules
for
Integrating Processes
Jianming
Zhang, Ning
W a n g
and Shuqing
W a n g
A b s t m c G The proportional integral derivative
(PID)
controllers are widely applied in industrial process, and the
tuning method of PID controller parameters is still a
hot
research area.
A
fuzzy tuning scheme for P ID controller
settings is developed for integrator plus time delay
processes in this paper, in which a fuzzy rule base
reasoning method are utilized on-line to determine a
tuning parameter a based on the error and the first
change of the error of the process. Then this tuning
parameter a is used
to
calculate the PID controller
parameten. Computer simulations are performed for an
example of integrating plants. Comparing to some
reported methods, the performance of the presented
approach
is
shown to be satisfying.
I. INTRODUCTION
N industrial process control area,the proportional-integral-
derivative (PID) controllers are still widely used because
of their simple structure, easy understanding to operators,
robust performance in a wide range o f operating conditions,
and their easy implementation using analogue or digital
hardware. It has been reported that more than 9 5 % of the
controllers in industrial process control applications are of
the PID type controller
[2].
Integrating processes are
frequently encountered in the process industries. Many
chemical processes can be m odeled by a pure integrator plus
time delay model such as kexp(-m)/s
.
Since the model
contains only two parameters (the proportional coefficient k
and the time delay constant
T ,
it
is
very convenient for
estimating the model parameters by relay feedback method
or closed-loop identification
[3],[4].
Recent years, more and more PID tuning methods are
proposed to deal with various integrating processes. Chien
1 . . '
and Fruehauf
[SI
proposed an internal model control (IMC)
method to find the settings for a PI controller, in a process
consisting of an infegrator and a time delay. Tyreus and
Luyben [6] pointed out that the IMC-based PI controller
could lead to a poor'control performance unless care
i s
taken
in selecting the closed-loop time constant, and proposed an
alternative approach based on classical frequency response
methods for PI settings. This tuning rule has been extended to
design PID controllers by L uyben in
[7].
Wang and Cluett [8]
also discussed this control problem and proposed a PID
controller designing method based on specification in terms
of desired control
on
signal trajectory which
is
scaled with
respect to the magnitude of the coefficient in the second term
of the Taylor's series expansion. More recently, Visioli
[9]
proposed a tuning method for integrating systems based on
minimizing
ISE,
ISTE and
ITSE
with a genetic algorithm.
The results are fitted by simple equations related to the
proportional coefficient k and the time delay constant T of
the process. However, the optimization for servo response
results in a PD co tdo ller and on ly the results for regulatory
response can give a PID controller. Chidambaram and Sree
[I] proposed a simple method for the PI, PD and PID
controller settings for such process based
on
matching the
coefficients of corresponding po wers of
s
in the numerator
and that
i n
the denominator of the closed-loop transfer
function. Since there
is
only one adjustable parameter a for
the PID tuner, care must be taken in selecting the tuning
parameter a in order to obtaining a good performance. In
this paper,
a
fuzzy inference method
is
introduced to find a
proper value
a
according to the response oft he closed loop
system. The perfom;ance of the modified method
is
shown
with an example.
The paper is organized as following. The next section
gives a brief introdhction of the method proposed
in [ I ] .
~~
Section
I l l
presents the tuning rules with fuzzy inference for
the tuning parameter
a
omputer simulations and results
are given
in
section IV, and the performance of the proposed
method is compared with that of
[I]
[7]-[9]. Conclusions are
summarized in
Section
v'
Manuscript received Seplember 20,2 00 3.
Jianming Zhang is
the National
Key
Laboratory
of
contra^
Technology, institute of Advanced process control, Zhejiang
University, Hanpzhou,
310027, P.
R. ofChina.
(Phone:
086-571-87951 125;
Fax: 086571-87951445; e m a i l : jmz han ga iipc.zju.edu.cn).
Nine
Wanr
is
with
the National Kev Laboratow
of
Industrial
Control~~~~~~
- -
Technology, Institute of Advanced Process Control, Zhejiang Universiv,
Shuqing Wang is
with
the National Key Laboratory oflndushial Control
Hangzhou, 310027.
(e-mail:
"wan@ iip.zju edu.cn).
Technology, Instirule of Advanced
procUs
cantroi hejiang university,
11. PREVIOUS TUNING METH OD FOR
PID
CONTROLLERS
Chidambaram and Sree U1 proposed a simple method for
Hangzhhou, 310027. (e-mail: rqwang@ iipc.zju.edu.cn).
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the PI, PD and PID controller settings based
on
matching the
coefficients of corresponding powers of
s
in the numerator
and that in the denominator of the closed-loop transfer
function for a servo problem. Their method
is
described
briefly as follows
[I].
For
an
integrator plus time delay process represented by
transfer function kexp(-rs)/s
,
considering the following
PID type controller,
G, =
k,
l + - + T d s .
Where,
k
is
the proportional gain of PID controller,
r
and
rd
are
known
as
the integral and derivative time constants,
respectively. The closed-loop transfer function can be given
[ f s
by
~-q ) -
(K lq + K, +K,q2)exp(-q)
Y , q ) q2
+ ( K l q + K,
+
K3q2)exp(-q)
(1)
where,
s is
the Laplace operator,
(2)
K T
K , = k , k r , K 2 = - , K 3 = K I A , q = s r .
Ti
I.
r
Approximating the time delay term in the denom inator by
a
first-order pade approximation,
1- 0.5q
1+ 0.5q
exp(-q)
~i ~.
(3)
The resulting equation can be w ritten
as
(4)
y q )
= (K ,q +K 2 +K,q2)(1+0.5q)exp(-q)
Y , q )
q 2 1 +
0.5q + ( K l q +
K 2 +
K 3 q 2 ) 1
O S q )
Removing exp(-q)
in
the numerator and introducing
a
parameter a as he coefficients of corresponding powers o f q
of the num erator to that
of
the denominator, one can get the
following set of equations:
( I - a ) K , + O S ( l + a ) K , = 0 ,
( 5 4
O S 1
+ a ) K ,
+
1
- a ) K ,
= a ,
(5b)
( l + a ) K ,
= a .
(SC)
By solving the above equations, the following results can
be obtained for the PID co ntroller parameters,
4a
k ,
=
l + a ) 2 k r
0.5r l+a
T. =
a-1)
where,
a
is the tuning parameter and shou ld be greater than
1. Similar to
IMC
methods, care should be taken in selecting
this tuning parameter. That gives difficult
in
industrial
process applications. In order to avoid this problem,
a
modified method
is
proposed here by a fuzzy tuning method
for finding the appropriate parameter
a
on-line according to
the step response o ft he process.
usec
Fig
I .
Process step response
From the equation
6 ) ,
ne can see that once the parameter
a
increases, the proportional gain k , arises which results
in a
big PID control action; On the contrary, the similar
results can be made too. Therefore, according to the step
response (shown
as
in Fig.l), one can address the following
conclusions for tuning the parameter
a
At the beginning
(point
a,) ,
here
is
a
big system error, therefore,
a
big control
action
is
desired in order to achieve a fast rise time. To
produce
a
big control signal, the PID controller should have
a
large proportional gain, a large integral gain and a small
derivative gain. Thus, according to the relations between the
tuning the parameter
a
and the proportional gain
k, ,
integral gain
Ti
and derivative gain
T d ,
one should give
a
big value a
;
Around the point 6 n Fig. 1, a small control
action is expected to avoid
a
large overshoot, and
a
small
value
a is
requested; Along with the decreasing of system
error gradually, the control action should
go
to a stead state,
therefore, the tuning parameter a should not be regulated
again.
Form the above analysis, the tuning parameter
a
should
be reduced h m big value to a sm all value gradually during
the step response. Since
fuzzy
method has the ability of
reasoning from the system error and the change of error, it
can be utilized on-line to regulate the param eter a properly.
The detail of the approach is to be presented
in
the next
section.
111. THE PROPOSED FUZZY NFERENCE METHOD FOR
TUNING THE PARAMETER (I
Fuzzy sets theory
is
first introduced by Z adeh in 1965 [IO]
and has been applied in the industrial process control
successhlly. Recent years, fuzzy logic has been used to tune
the parameters of
a
PID controller
[ I l l ,
and good
improvement in the process response is achieved. During the
design procedure of fuzzy PID controller, the human
expertise in controlling
a
process
is
represented as fuzzy
rules or relations. This knowledge base is used by an fuzzy
inference mechanism, in conjunction with som e knowledge
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Fuzzy Inference
Setpoint Output
Fig. 2 Fuzzy self-tuning PID controllel
of the state of the process
in
order to determ ine fuzzy control
action on-line. The schem atic diagram of a fuzzy self-tuning
PID controller used
in
this paper is shown in Fig. 2. Where,
the inputs of the fuzzy inference mechanism are the error e
and the change
in
error Ae , and the suitable parameter
for tuning the PID se ttings is given hy the output of the fuzzy
inference system. In this paper, four fuzzy se ts
(Z,
S , M, and
B) are considered for both error and its change. Where,
2
represents approximately zero, S , M and B denote small,
medium and big respectively. Two fuzzy sets (S, B) are used
for the output variable
a
The error (e) and the change
in
error Ae) are normalized with respect to their maximum
values. Therefore, their absolute values can be taken during
the fuzzy inference. The triangular-type membership
function
is
adopted
in
this paper for
e,
Ae and specified as
in
Fig. 3.The sigm oid-type mem bership function
is
used for a
and specified
as
in Fig.
4.
Fig. 3 Membership function o fe and Ae
0.4
a
0
0
0.2 0.4 0 6 0 8 ?
Fig.
4.
Mem bership function of the output a
The grade of the $embership functians p and th e Output
variable a has the following relationship:
p s a )=l/(l+exp(-ZO(a-0.5))) for Big (7a)
p s a ) = -l/(l+exp(-ZO(a-0.5)))for Small, (7b)
The fuzzy rule base can he extracted from operators
expertise or the step response of the process.
In
this paper,
the step response of the process is used. According to the
relationship between tuning parameter a and the step
response, the followhg Fuzzy rules can be addressed. At the
beginning (point a n Fig. I), to produce a big c ontrol action,
the tuning parameter, a can he represe nted by a fuzzy set Big.
And the corresponding rule can be described as
or
IF e isB and Aqis Z THEN a isBIG.
Around the point b ,
in
Fig. 1, a small a is requested for
avoiding a large ov&sh oot and the corresponding fuzzy rule
is :
IF e is Z and Ae
:is
B THEN
a
is
SMALL.
By the analysis, the whole fuzzy rule base for the tuning
parameter a can besummarized in Table 1.
Tableil Fuzzy tuning rules for a
In Table
1,
all of the tuning fuzzy rules have the following
Q:
IF e isA i and Ae is
B,
THEN a
is
Ci
Where, A,, B, and C, are the fuzzy sets.
Using the simplified fuzzy reasoning method [I I] the
agreement of
the
ith rule can be c alculated by the product of
the mem ber function values in the antecedent part ofth e rule:
description:
ht = ~4 ( e ) . p B , ( ) 8)
where
pA
s
the
membership function value o fth e fuzzy set
Ai
determined by the current input value e ; and pB,
s
the
membership functionvalue
of
the fuzzy set
B,
determined
by the current input value A e .
For each
h i ,
the corresponding output of the fuzzy rule
can be calculated from Eq. (7a) or Eq. (7b) as
a i =-0.051n(l/h;-1)+0.5 for a i s B i g
(W
or
a;
=-0.051n(l/(l-h;)-1)+0.5 for
a
isSmall (8b)
Then the centroid method of defuzrification
is
used to get
the crisp value of a from the fuzzy variables. the output
value of f u u y inference system
is
calcu lated by
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a =
hiai
/E hi
(9)
l
Here,
ai
is the value of
a
corresponding to the degree
hi
or the ith rule.
According to the analysis results in previous section, the
tuning parameter
a
should he greater than
1,
and should be
reduced from
a
big value to
a
small value slowly during the
transient response process. Therefore, in this paper, the true
value of a
is
determined
as
following
10)
a k)= 1.1+ A k )
A(k) = A k - l ) . a
where
A 0 )
is
a
parameter selected by the user, it gives
After determining the parameter
a ,
he PID controller
the maxim al value of
a .
parameters can be calculated by using the equation (6).
IV. SIMULATION
RESULTS
Consider the integrating plus time delay process used in
[I], which
is
described by the following transfer hnc tion
e-*
G ( s ) = k -
S
where, in nominal case, k=0.0506 , and 7 = 6 . The
proposed method will be evaluated using this process.
To
compare the performance to the other previous work, the
corresponding PID controller settings are given as
followings, respectively,
Visioli 191: k ,
=
4.5, T, = 8.94 and Td
=
3.54;
Luyben method [7]: k, =2.5639 =56.32 and
Td ~ 3 . 5 6 1 nd with a first-order filter time constant as
0.382;
Chidambaram and Sree[l]: k ,
=
4.0664
, =
27 and
T,
=
2.7 with
a
=
1.25
;
Wang and Cluett
[SI:
k
=
2.0123 ,
T
31.203 and
Td
=
1.5674
;
The performance of above
PID
controller and the
presented method for the nominal process model are shown
in Fig. 5and Fig. 8. The servo responses are shown in Fig.
5
The servo response of the proposed controller is slightly
better than that of Chidambaram and Sree [l], and is better
than that of Visioli [9]. Th e regulatory responses are shown
in Fig. 8. Th e regulatory response
is
the best for the method
of V isioli [9], since the setting was obtained by m inimizing
ISE for regulatory problem. In Table 2, the comparison of
ISE
values is given for the above method and the modified
method. O bviously, for the regulatory problem, the presented
method is better than that of [7] and
[SI.
The robustness
of
the proposed controller
is
studied by
using perturbation in time delay
T .
The controller settings
are those calculated for the process with nominal time delay
r = 6 ) ,
whereas the actual values in the simulations are
7 =
7
and r = 8 , respectively. Fig. 6 and Fig. 7 show the
results of servo responses. The responses are obtained for th e
regulatory problem as shown in Fig. 9 and Fig. IO Visioli
method gives an oscillatory unstable response when
7 = 8 both in servo response of regulatory response. The
comparison results of
ISE
values are also given in Table 2.
For the servo problem,
the
presented method gives robust
responses and
is
superior to the method of [I], and the
Luybens method gives the best robust performance.
However, for regulatory problems, the performance of the
presented method
is
better than that of [7], [7], and
is
equivalent to that of [I].
From the views of control theory, the servo problem
is
contradictory to the regulatory problem, and their
performance need
t
he trade-off at some degrees during the
design
of
a controller for the given p rocess. From this point
of view, the performance of the presented method in this
paper gives
a
more effective compromise than that of others
mentioned in this paper.
Table 2 Comparison of integral square error for the five
methods
I
ISE
for servo ISE for regulatory I
Remark: PID controllers are designed for =
6 ;
*
means unstable response.
2.5
I- . Present
il
visioli 2ml)
-. Chidambaram
and
SreepOO3)
-
.
Luyben 1997)
nl I
Fig.
. Servo
respanre
of the process
for different
methods 7 =
6
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I
3
I- Present
-. Chidambaram and
SreepOm)
i m
150
50
((sec.)
O
Fig. 6. Servo response of the process for different methods r = 7
MO 250
300
O ,(sec $
Fig. 7. Servo response of the
process
for different methads r
=
8 )
0.6
- . Chidambaram and
Sree 2003)
-0.1
200
250
300
loo
,(sec ?
Fig. 8. Regulatory response ofthe process for different methods r = 6 1
0.E
0.4
2
0.2
- . Chidambaram and Sree Z003)
Wang
and Cluetl 1997)
-0.4
0 50 100
150
200 250 300
' .)
Fig. 9 Regulatory response o e p OCCEE for different methods r = 7
1
-1'
0
50
l a , 150
2
250
3
[(sec.)
Fig. IO. Regulatoryresponse
ofIhe
process
for
different methods
r
=
8 )
V.
CONCLUSIONS
Based
on
the simple method proposed by Chidambaram
and Sree in [l], a modified approach
is
presented
in
this
paper. The m ethod applies fuzzy inference with simple mles
to compute the tuning parameter a on-line. That avoids the
difficulty in selekting a suitable parameter a in
Chidambaram and Sree's m ethod
[I] .
The comparison of
performance for an integrating process is performed. The
results validate that the development performance is
achieved.
REFERENCES
[I]
M. Chidambaram and R. P.Sree, A simple method
of
tuning PID
controller^
for integratoddead-time
processe~,
Computers nd
ChemicolEnginerr ig ,
vol. 27,211-215,2003,
C. C. Yu, uto-IuningofPID Controller. Berl in: Springer, 1990.
2]
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[3]
141
J.
H.
Park,
S.W . Sung and I.
Lee,
lmpmved
relay
auto-tuning with
static loaddisturbance, Aulomorico,
vol.
33,111-715, 1997.
R. Srividya and
M
hidambaram, On-line c~ntr~lleruning
for
integratorplus delay processer,Procesr.
onrrolQuoi.,
vol. 9,59-66
1997.
I. LI Chien and P. S. ruehauf, Consider IMC tuning to improve
oerfomance.Chem.
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[SI
_ . .
[6] B. D. Tyreus and
W.
L.
Luyben
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PI
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]
Z. -Y. Zhaa,
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Tomizuka and S. sda, F u q gain
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