Fuzzy Pid Basic

7/24/2019 Fuzzy Pid Basic

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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).

0-7803-83354/04/ 17.00 02004 AACC

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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

1112


5/6

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]

1113


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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.

Enc. Pror.. vol. IO. 33-41. 1990.

[SI

_ . .

[6] B. D. Tyreus and

W.

L.

Luyben

Tuning

PI

c~ntro l l en far

integratorldead-lime

processes,

I d .

Emg.

Chem.

Res., vol.

31,

2625-2628, 1992.

Luyben, W .

L.

Tuning Proportional-Integral-Derivative ontrollers

for

IntegratodDeadtime

Processes.

Ind. Eng. Chem. Res. 1996, 35,

[7]

3480-3483.

L. Wane. and W . R. Cluen Tuning PID ~ ~ n t r ~ l l e r ~or intee.ratine.

81

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processis,

IEE

Proc. CTA, vol. 14i, 385-392, 1997.

191

A

Vir iol i . Ootimal tunine of PTD controllers

for

inteeml and

.

unstable processes, IEE Proe

Conrrol

nteory Appl , YOI 48,

180-184.2001

[ O ] L

A

Zadeh, F u q Sets,lnformonon

ondConrro1,

voI 8,338-353,

1965.

[ I

]

Z. -Y. Zhaa,

M.

Tomizuka and S. sda, F u q gain

scheduling

of

PID

ontrollers,

IEEE

Trans.

on Syslem,

M m nd Cy6ernericJ.vol.

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