methods for automated design for singleton fuzzy logic controller

8/11/2019 Methods for Automated Design for Singleton Fuzzy Logic Controller

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Proceedings

of

the 1998 IEEE

International Conference on Control Applications

Trieste, Italy 1-4 September 1998

TA04

esign of a Singleton Fuzzy

Logic

~ o ~ ~ r o l ~ ~ r

Stjepan Bogdan, Zdenko K o v a W

Universiq

o Zagreb

Faculty o

Electrical

Engineering and Computing

Unska

3, HR-IO000 Zagreb;CROATIA

URL:

http://www.

asip.

er .

hr/flrrcg

Abstract

-

Three easy-to-implement fuzzy logic contro ller

(FLC) design methods are proposed; FLC emulation of PI

controller, model reference based d esign a nd de signing of

an'FLC by using a state trajectories. The methods have

been tested in case

of

an unknown control process and

experimental results are given.

The FLC parameters are usually defined based on a

heuristic knowledge.

How

to translate this knowledge to

a set of membershp functions and a set of fuzzy rules may

be a very &cult task. That is why the problem as well as

the various solutions

of

FLC design have been addressed

in numerous papers.

In [11 the gradient descent method has been proposed

for tuning of

n

Takagi-Sugeno set offuzzy rules whde in

[2] the same method has been applied

to

the fuzzy rule

base with output singletons.

An FLC tuning method based on the Hooke&Jeeves

pattern search algorithm has been explained in

[ 3 ] .

Implementation

of

proposed algorithm shows that method

is

able to tune an

FLC

(with

9

and 25 rules) to catch

behavior of a PD con troller.

One o f the best-known neural net model, Kohonen's

self-organizing map, has been introduced as a fuzzy

version in 141. Kohonen's learning laws are used for

tuning the centers

of

the fuzzy sets and to initialize the

f i zzy

rules. A good accuracy and convergence

of

algorithm have been proved by computer simulation.

Aiin of this paper is to propose three tuning methods

that intent to be the first step in the two-inputs-one-output

FLC design. For the predefined number and shape of the

membership functions tuning methods form a control

surface by changing output singletons.

2 FLC

emulation

of PI

algorithm

The form of

a

discrete

PI

algorithm

is

determined by:

~ ( k ) = ~ ( k -) + A u ~ ) (1)

where

and K is a controller gain, T, is a sampling interval and

TI is an integral time constant.

In

order to emulate discrete PI algorithm FLC must

have e(k) a nd A e(k) as inputs and A u(k) as an output. By

knowing m in and max values of these variables one can

determine a universe of discourse for the FLC inputs and

output. Due

to

the

linear

shape of discrete

PI

algorithm the

best emulation is obtained if triangular membership

functions are chosen for input va riables an d i f only two

of

them overlap.

In

that case only non linearity contained in

FL is the center of gravity defuzzification method.

To make FLC simpler for implementation we define

singleton sets over the output universe of discourse of the

FLC.

Let

TE,

and

TDE,

be the i-th and j-th fuzzy sets of e(k)

and Ae k) respectively, and let A, be the q-th fuzzy output

singleton. Tha n the q-th

FLC

rule has

a

form

0

F

e E

TE,

ND Ae E TDEJ

THEN

ziFC=A

The number of rules is define d by the nu mber of input

variables and the number of fuzzy subsets ober their

mv erses of dmo urse For two input tanables with n and

m fuzzy subsets number of rules is nxm

Since we assume that only two neighboring input

membershp

funmons

overlap. four out of nxm FLC rules

contribute to the fina l value of the contro ller output If we

choose e, (k) and Ae,

(k)

such that

pe,(q)=l

and pA:(A5)=l

(which means that e, k) and Ae, (k) correspond with

centroids of I-th and j-th input membership functions)

than the FLC output

is

de tem ne d by only one rule and its

value is equal to

By using equation (2)

and

3 ) we obtain

4)

Equation

4)

directly defines the value

of

q-th FLC

output singleton. The FLC control surface (i.e. all output

singletons) can be found by inserting values of e,(k) and

Ae, (k) for i= l . 2 , ... n and j=1,2. _ . . ~n in equation (4) .

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IEEE

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3 Reference model based FLC design

The following method is based on th e assump tion th at

the unknown control process is

SISO

and stable.

Furthermore it is assumed that the process can be

represented with the second order approxim ation of the

form

Y, 4 = q J k -

1)

+a,g -2) +b,,u k- 1) 5 )

where

yA

is an output

of

the process and

U

is an input of

the process. Former is true for very large class

of

linear as

well

as

nonlinear systems. The process approximation

parameters

a,,,

aA2 nd bA,can be calculated from input-

output data by the one of standard process identification

methods (least square for example).

The goal of the FLC design is to find a controller

which would be able to keep a difference (i.e.

a

tracking

error) between the reference model and the process as

sinall as possible. Since the controller synthesis is tied on

the approximation of the unknown process it is re asonable

to expect that the value of a tracking error will decrease

with accuracy of approximation.

In the analysis

of

control systems it is common to

describe desired behavior of the process 5 ) closed-loop

system with the sec ond order model

y , k ) = a , l ~ m k - l ) + ~ m ~ m ~ - 2 > + b m l ~ , k - l )

6)

Reference model output and input are denoted

ym

and

U,, respectively.

Transfer functions of the process approximation and

the reference model have a form

A lincar pole-zero placement controller is described

with

1

U 2)

=

T(2)

Ur z)

-S Z)YA 2)]

R 4

The orders

of

polynomials R(z), S(z) and T(z) (i.e. the

order of

a

controller) are defined by the request for

causality and stability of

a

closed loop system and

controller (i.e. by the orders of polynomials AA(z), B,(z),

4 L z ) and B )

Solving of Diophant e quation

(9)

where A, z) represents observer dynamics, leads to the

form of polynoinials R(z) and S(z), while solution of the

equation

determines

a

polynomial

T(z).

form

In our case R(z), S(z) and T(z) are of the following

S 2)=slz +so

T 2)=t,z+to

By inserting 11) in

9)

and (10) we have

To find

a

controller let r,=l and r,=O. Then controller

polynomials are

R(z)= z

aAl-um/ aA2-arn2

S 2)=

- +

b A l

Inverse Z-transformation

of 8)

gives

a

recursive

controller equation

In order

to

obtain a form of the con troller suitable for

implementation by using the FLC we have to reorganize

equation 15) in the way to include FLC inputs, error e k)

and change in error Ae(k), into it. A s e(k)=&(k)-y,(k), we

have

aA1 C a A 2 -a ml -a m2 e k )

u(k)=

Assuming that

a

referent input signal

y k)

has

a

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constant value or it changes slowly (i.e. y(k) =y (k- l)) ,

equation

16)

becomes

u(k)=kle(k)

+k2Ae(k)+k3uy(k) 17)

Coefficient k, represents a gain of the feedforw ard

path wlzich is parallel to the FLC . As assumed, the control

process is stable (i.e. does

not

contain integral behavior)

and since the FLC has e and Ae as inputs PD type of

controller ), it

is

necessary to includ e a feed forward p ath as

a part of the controller in order to compensate a static

error. As a consequence, determination of a correct value

for k, is essential. It may be seen from equation

18)

that

in

the case we have a reference model w ith a unity gain k,

is equal CO the inverse value of the process gain.

Coutroller has a form

u(k)

=U ) +k,ur(k)

19)

From 18) and (19) we find that

u,, k)

=k,

(k )+k,Ae(k)

20)

Following the same pattern which has been used for

deterininahon of the FLC that emulates

PI

algorithm we

may choose e, (k) and A eJ(k) such tha t pe,(e,)=l and

FAe,(Ae,)=l n that case the FLC output is determined by

only one rule

uFc7k l e ,

+k2

Aej=A

(21)

As

i n

the previous paragraph, all output singletons

w h c h

form

a control sd a c e can be obtained from (2

1)

for

i=1.2,

...,

n and j=1,2, ..., m.

4. The FLC design based on the state space

trajectories

The method of the FLC design that s hall be described

nest is based on the idea to mimic human operator

decisions while heishe

is

handling

a

process.

By

using triples of the form [u(k),e(k),Ae(k)] or

lAu(k).e(k),Ae(k)J that are acquired from the process

during the

operation

under

different conditions i.e.

referent value changes, disturbance chan ges etc.) one may

form

a

set of state space trajectories that constitute a

control sLuface, deno te it Y, which lies over the e-Ae plane

u=tp[e,Ae] 22)

In case the FLC is used for control we have

r

where

(pJ is a fuzzy

basis function and

A,

represents an

output singleton. The proposed FLC uses the center of

gravity defuzzification method.

To simplify the FLC design we inay predefine the

number of e and Ae membership fun ctions as well as their

shape. In that case the output singletons remain as

parameters that should be used for purpose of tuning the

controller

to

make a model of control surface Y~

Figure 1Determination of the output singletons by using

a state space trajectory.

For the trajectory j (partly shown in Fig. 1) which

contains

5

riples [U( ),e( l),Ae( l)], [u(2),e(2),Ae(2)j,

_ .

,

[u(p)>e(P),Ae(P)l>

to

),e(p+ I),Ae(p+

111,

. .

[u(p+v>,e(p+v>,Ae(p+.l)l,

.

..

lu(i>,e(i>,Ae(Oi

by

using

equation

23)

and assuming that only two neighboring

input membership functions overlap one may w rite down

the following set of equations

Aq.(Pq(P

o+Aq+l* Pq+l P

i +

f o r i=0,1,2

...,

v

The condition for existence

of a

solution of the

problem

24)

is

that every rule has to be fired at least three

times for every trajectory. Often that is not

a

case,

especially if controller has m any ru les. This is the reason

why determination of the FLC output singletons

is

approximative.

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Among triples

of

trajectory j there exist v of them

that fire a q-th rule. Find such a triple that

P

V

p,(P a> max

cp, i>

25)

i = p

In that case we may assume that the contribution

of

other rules

to

the controller output may be neglected

Z ((P++Aq'pq((P

a>

(26)

Accordingly, the value of the output singleton

determined by the j-th trajectory is approximately

In case that the rule has been fired by more than one

trajectory the final value of the corresp ondin g singleto n is

equal

to

the mean v alue of singletons calculated by u sing

27)

where

N

is the number of trajectories that fired the q -th

FLC rule

Prob lem arise if the value of the fuzzy basis function

in (27) is sinal1 or it does not df fe r m uch f rom other

fuzzy

basis functions that contribute to the controller output.

Usage of bell shaped membership functions with high

gradienl solves the former problem, while skipping

trajectories

wth

small values of fuz zy basis functio ns inay

help in handling of equation (27).

5. Experimental results

The proposed FLC design methods have been tested

through laboratory experiments. Personal com puter

x386

operarating on

16

MHz with 12-bit

AD

and D/A

converters has been used for im plem entation of alg orithm s

that have performed calculation of the FLC parameters.

To test robustness of proposed methods a white noise has

been added in the system.

The FL,C had seven fuzzy subsets for both inputs w ith

triangular m embership functions in the first and the

second csperirnent (PI cmulation and reference model

based app roach) while in the third experiment (trajectory

based design) membership functions were bell-shaped.

The center of gravity defuzzyfcation method has been

used in all experiments.

Laboratory setup is shown in Fig. 2.

Figure 2

Laboratory setup for testing of automated FLC

design methods.

The process that has been simula ted by analog p rocess

simulator was assumed to be unknown.

In the first experiment the PI controller parameters

have been determined based on the open loop response of

the unknow n controlled process;

K,=3.7

and T,=3 s.

2 4 5 0 0 0 brts

Figure

3

Process responses in case

of

FLC em ulation the

of PI controller

Figure 3shows time responses of the p rocess in case of

FLC emulation of the PI controller. It may be seen that the

difference between the proce ss contro lled by FLC and the

process handled by PI contro ller is tolerable. Final tuning

of

the FLC can be maintained by using on-line learning

methods

16,

7,

XI

The process responses obtained in the second

experiment are shown in figure 4.

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6

Conclusion

~ FLC

~~~

reference

model

~~ ~

open

loop

response

Figure 4 Process responses in the case of FLC design

based on the reference model.

Several experiments have been performed and figure

4shows the worst case. Even in the situation when the

level of measurement noise is high and the reduced-order

process approximation is inaccurate the FLC captures

dynamics of the process very well, but the overshoot is

noticeably higher.

The results o f the third experiment are given in figure

5 .

Figure 5

Process responses in the case of FLC design

based

on

the system trajectory.

The closed loop systein follows desired behavior very

closely. The FLC parameters, derived by using a state

trajectory method, form a con trol surface which ties the

process respo nse close to the d esired behavior.

Paper describes three methods

of FLC

design. One

of

them represents fuzzy emulation of PI algorithm, the

second one is based on the reference model and the third

one is founded on state trajectory.

Even though th e third method shows results superior

to the first two method s, generally speaking a conclusion

that a state trajectory based FLC design beats two other

methods would be wrong. Which method should be used

and which one would give better results depend o n proce ss

characteristics, our knowledge about p rocess, accuracy

of

measurement etc.

A very simple implementation and the experimental

results confirm that a time consuming an d pa id ul ad hoc

determination of FLC parameters can be omitted through

automated FLC design approach.

1 Literature

1.

F. Guely, P. Sian y; A centred formulation

of

Takagi-

Sugeno rules for im proved learning efficiency,Fuzzy

sets andsystems, 62, 1994, 277-285.

2. H. Ishibuchi,

K.

Noyaki, H. Tanaka, Y. Hosaka, M.

Matsuda; Empirical study on learning in fuzzy

systems by rice taste analys is,

Fuzzy sets and systems,

3. H.B. Giirocak,

A.

de San Lazaro;

A

fine tuning

method for fuzzy rule bases, Fuzzy sets and systems,

4. P. Vuoriinaa;Fuzzy self-organizing map, Fuzzy sets

and systems, 66, 1994,223-23

1.

5 .

KovaEiC Z; Bogda n S; CrnoSija

P.

Design and Stability

of Self-organizing Fuzzy Control of High-order

Systems, Proceedings of the 10th IEE E International

Symposium on Intelligent Contr ol, pp. 389-394,

Monterey (USA), 1995.

6. KovaEid

;

ogdan S. Servo Application of a Model

Reference-Based Adaptive Fuzzy Con trol, Proc. of

the

I I th IEEE

nternational Syviposium

on

Intelligent

Control, pp. 49-54, Dearborn, 1996.

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M. Maeda,

S.

Murakami; A self-tuning fuzzy

controller, Fuzzy sets and systems. 51, 1992; 29-40.

8.

S.

Jee,

Y .

Koren; self-organizing fuzzy logic

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methods for automated design for singleton fuzzy logic controller

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