Experiments on using fuzzy clustering for fuzzy control system design.pdf

7/23/2019 Experiments on using fuzzy clustering for fuzzy control system design.pdf

http://slidepdf.com/reader/full/experiments-on-using-fuzzy-clustering-for-fuzzy-control-system-designpdf 1/7

RIMENTS ON USING FUZZY CLUSTERING FOR FUZZY CONTROL

SYSTEM DESIGN

Andreja Mikulcic and Jianhua Chen

Computer Science Department

Louisiana State University

Baton Rouge, LA 70803

{mikulcic, jianhua}@ bit.csc.lsu.edu

We report experiments on using fuzzy clustering

method for fuzzy control system design. We compare, in

the context of the inverted pendulum problem, the exact

control surface generated by numerical integration and

modeled control surface generated by the fuzzy rules of a

fuzzy controller. The fuzzy rules are obtained by apply-

ing fuzzy clustering to a small random set of learning

points from the exact surface. Small difference between

the modeled and exact surface, mostly less than 1%, indi-

cates that fuzzy clustering method is capable of capturing

the essence of the inherently non-linear and unstable sys-

tem using only a small learning set. The fuzzy controller

is also tested through numerical simulation. The results

indicate that the control system reaches the stable state

rapidly without any overshoot, and that it is robust even in

the presence of disturbances.

1.

INTRODUCTION

Fuzzy control system identification and design has

been a subject of intense research in recent years. The

heart of a fuzzy control system consists of a set of fuzzy

If-then'' rules, which models the process state-control

action relationship in the control domain. The design of

appropriate fuzzy rules is a critical bottleneck in develop-

ing powerful fuzzy control systems. Typically, the speci-

fication of a real-world control problem is given by a

finite set of input-output data points p r =

x r ,

i), and the

design task is to extract a good set of fuzzy rules that best

represent the input-output relationship in the data. In this

paper, we address the design issue by demonstrating good

performance of the fuzzy clustering-based method, which

identifies clusters in the input-output data and constructs a

fuzzy rule for each cluster.

Various approaches have been proposed for the

fuzzy system identification task. Takagi and Sugeno [l]

proposed a heuristic method for designing TakagiBugeno

type of fuzzy rules from

I/O

data, using some perfor-

mance index, Subsequently, Sugeno and Tanaka reported

[2]

a

method for on-line identification of a fuzzy system.

Recently, several proposals are made to use fuzzy cluster-

ing for fuzzy system identification. Sugeno and

Yasukawa [3]uses the fuzzy C-means algorithm to cluster

the output data points to determine the fuzzy rules, and

use some heuristics to select the most relevant input va

ables in fuzzy system design. Bezdek [4] combines

ideas in [3] and [ 5 ] to develop a neural network fuz

control system which also utilizes clustering.

Sin a

deFigueiredo [6] use fuzzy c-means algorithm to clus

the input data points in finding the fuzzy rules. Not

that in these works, fuzzy clustering is mainly used to fi

either the antecedent part of the rules, or the number

fuzzy rules, however, it will not directly produce fuz

rules

-

subsequent processing is needed to generate

rules. In contrast, Kundu and Chen propose

[7 ]

to use

fuzzy linear clustering (by Bezdek and Hathaway) [8]

generate Takagi-Sugeno type fuzzy rules directly from I

data. The advantage of the proposal in

[7]

is that

fuzzy clusters on the input space are discovered simul

neously as the algorithm learns the consequent part (o

put pattern) of each fuzzy rule, represented as a lin

function of the input. Some preliminary experiments

function approximation were reported in

[7]

indicating

promise of the fuzzy linear clustering approach to fuz

system design.

In this paper, we enhance the work [7] by report

experiments on using fuzzy clustering to learn cont

rules for the inverted pendulum problem. The inver

pendulum problem has been considered a typical rep

sentative of inherently unstable nonlinear control system

thus it is an ideal choice for testing the modeling capab

ity of the fuzzy clustering algorithm. Our simulat

involves several steps. The first step consists of gene

tion of the inverted pendulum problem's exact cont

surface, which is itself of important interest. The seco

step applies fuzzy clustering to learn fuzzy rules from

control surface. The third step utilizes the fuzzy rules

generate modeled control surface and simulate t

inverted pendulum control, with or without perturbatio

The experimental results show that fuzzy clustering i

powerful tool for fuzzy system modeling.

The contributions of the current paper are tw

folds. First, it further provides support for the fuzzy cl

tering approach to fuzzy system design. The success

our experiments suggests that one may further consider

adopt fuzzy clustering for fuzzy modeling in other s

tings (e.g. on-line learning) and thus opens up promisi

directions for future exploration. Second, the approa

0-7803-3645-3/96 $5.0001996 IEEE 2168

http://bit.csc.lsu.edu/





developed in constructing the exact control surface for the

inverted pendulum problem is of independent interest. It

provides a simple, efficient method for generating a

good enough approximation of the inverted pendulum

control surface, whereas it is computationally expensive

to systematically generate the optimal control surface for

such unstable, non-linear systems. This is in contrast to

the cell state space approach [9] where the computation of

the optimal control table takes a global search.

2.

BACKGROUND

2.1.

Fuzzy Control Systems

A fuzzy control system typically consists of a fuzzy

controller and a plant as shown in the following Figure 1.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fuzzy FuzzyRuleBase

C6ntroller

x € x

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure

1.

Fuzzy control system architecture

In this paper, we focus on Takagi-Sugeno type fuzzy rules

of the form

r k : f

X

iS A k ,

then

y i S g k ( X ) , (1)

where x

E

X c Rd for some d 2 1 is the input variable

and

y

E

E

R is the output variable,

A k

is a fuzzy set

over X , g k ( X ) is a linear function of the input variable x.

The purpose of a fuzzy control system is to predict

the (crisp) output value y (the control action) from a spe-

cific (crisp) input value xo (the state of the plant),

so

that

applying

y

will drive the plant to some desirable state

(set-point). This involves computing a (crisp) Y k = g k ( X 0 )

with membership value p A k ( x O )y each rule r k rand then

combining the results from all the rules to generate the

predicted y value by the following equation:

c g k ( X O ) p A k ( X O )

2 )

=l

Y = c

k =l p A k ( X O )

2.2.

Fuzzy

Linear Clustering and Its Application in

Fuzzy Control

In the following we first briefly review the fuzzy

linear clustering method by Hathaway and Bezdek [8],

which Kundu and Chen propose

[7]

to use as a means for

fuzzy control rule design. The main thrust of fuzzy linear

clustering is to simu1tane:ously search for the fuzzy clus-

ters and their linear reipresentative hyperplanes. The

method easily generalizes to the case where the represen-

tative surface is a non-linear function of the input vari-

ables. This can also be viewed as a generalization of the

C-means algorithm [1011(which is also known as fuzzy-

ISODATA algorithm) for fuzzy clustering.

In the fuzzy C

-

means algorithm

[101

we are given

a set of sample data poinltspi= ( x i ,

y i ) :

1

I

n and the

desired number of clusters C (2 2). It produces C fuzzy

clusters A k r

1

k 5 C , b,y giving the membership values

p k i = & ( p i ) for each point

p i

and cluster A k . It chooses

the

p k j

so that the followiing objective function (where m

>

1

is a fixed constant)

(3)

C n

k=l i=l

Jm = %' k i )ml lp i - kll

is minimized under the: constraints that

x p k i

=

1

(summed overall k for each i and each

p k i

2

0.

Here,

Upi-

k l l

denotes the Euclidean distance between the

points p i and

v k

and

v k

i:s visualized as the center of the

cluster A k .

In fuzzy clustering, the membership values pk i and

the coefficients in the (:non)linear functions g k ( x ) are

selected such that the objective function

4),

which is sim-

ilar to eqn. (3), is minimizled:

C n

k=l

i=l

m = xC(pki) n[Yi

g k ( X i ) l 2 , m

> 1.

4)

The constraints

pkj

= 1

(summed over all

k

and

p k i

2

0

apply here as well. The essential difference here is the

replacement of the centeir v k for the cluster

A k

by the

hyperplane (hypersurface in the non-liner case) y = g k ( X ) .

The distance

l lpi

- k l l is now replaced by the distance of

p i from the hyperplane y = gk(X),WhiCh is llyi - k ( x i ) l l .

In

[ 7 ] ,

Kundu and Chen proposed to use fuzzy linear clus-

tering for fuzzy rules design. In this work, we present our

experimentations and control simulations for the inverted

pendulum control to verify the viability of the generalized

fuzzy linear clustering approach.

2.3.

The Inverted Pendulum Problem

Very often the quality of a control algorithm is

tested and demonstrated an the inverted pendulum prob-

lem [9] due to its inherent unstability and dynamic char-

acteristics.

As shown in Figure

2,

the inverted pendulum prob-

lem is the problem

of

learning how to balance an upright

pole. Its solution consists of finding the horizontal force

to be applied to the cart in order to balance the pole. The

cart is moving on the track with no friction. Also, the

2169



pole

is

tied up to the cart by a frictionless hinge. Both the

cart and the pole have only one degree of freedom, i.e.

each of them can move in vertical plane only. The state

equations for three-dimensional inverted pendulum sys-

tem are given in Figure 2.

x =

x2

where xl= e denotes the angle of the pole from upright

position in degrees, x

=

6 enotes angular velocity of the

pole, and

x3

denotes velocity of the cart

d s ) .

Other

parameters are: (1) g

=

9.81, acceleration due to gravity;

(2) mc

= 0.9

kg, mass of the cart;

(3)

m p

=

0.1 kg, mass

of the pole; (4)

=

0.5, half pole length; (5) F (in New-

tons), the only control action applied horizontally to the

cart.

3.

EX

ENT

Our experiment is mainly concerned with the angu-

lar position and angular velocity of the inverted pendu-

lum's pole, although cart velocity and displacement are

considered when deciding about the length of time inter-

val in which force F is applied to the cart. The experi-

ment consists of three steps: exact control surface and

learning set generation, fuzzy clustering, and simulation

and modeled surface generation. Since the latter two

steps use output produced by the algorithm for the learn-

ing set generation, we will give it extra attention.

3.1. ~ ~ g o r ~ t ~ ~or learning set generation

Although a significant number of experiments con-

sidering inverted pendulum has been conducted, most

of

the papers do not offer algorithms for generating the con-

trol surface of the system. One of the rare papers that has

partially outlined its algorithm

[9]

for generating control

table is using cell state space method which is very

exhaustive and computationally expensive. In contrast,

this paper proposes a simple algorithm that captures phys-

ical behavior of inverted pendulum system and generates

a good enough' control surface of it. A subset of the

control surface will be used as the learning set by the

fuzzy clustering algorithm.

The goal of proposed algorithm

is

to find the right

(correct) deceleration force under the following condi-

tions: (1) the same force will be applied for the n integra-

tion steps and then set to 0; (2) the pendulum must be in

the shortest amount of time turned around and brought

through equilibrium point with velocity within

5%

of the

initial one; (3) if pendulum

is

initially moving towards the

equilibrium position, no force is applied. The problem

with these conditions is that they contain two unknown

parameters, magnitude of F and n. However, we have

found that the second condition depends heavily on the

number of intervals force is applied: smaller the n, shorter

the time and, smaller the final chart velocity. Based on

this observation, we have chosen to set n = 2, correspond-

ing to the application of the force for 20ms.

Once the boundary conditions are known,

Bi

and

bi

are given, and bf=0.05*8,,where subscripts

i

and denote

initial and final values, respectively, the appropriate value

of F can be determined by numerical integration of sys-

tem of equations, Figure 2, using the Shooting Method

[ll]. The whole process starts with the initial value of the

force which is then incremented in fixed steps until it

becomes either correct or too big, i.e., pendulum passes

through the equilibrium point with the speed above 5% of

the initial one.

If

the force is too big, it is being decre-

mented in smaller steps until it becomes either correct or

too small, Le., pendulum does not reach the equilibrium

point. If the force is too small, increment step is further

reduced and procedure is repeated. Because of the non-

linear and unstable nature of inverted pendulum system,

some of the previous works were restricted to small initial

angles of the pole. In our case maximal possible initial

angle is 0.873 rad c- 50

.

um

Learning §et Generation

A ~ g o ~ ~ t ~ m

Input: Initial angle @(within 0, .873] rad)

Initial angular velocity e

(within [0, .5] rad/s, clockwise)

2170



1.

2.

3.

4.

5.

6 .

7.

8.

9.

10.

11.

Output:

Force F that should be applied to the cart

Initial force F :=0.

Initial cart velocity x3

:=0;

initial force-inc

:=l.

Save force, F'

:=

F;

x1

:=

8

x2

:=

8,

initial time

t

:=O

initial number of steps STEPS :=O.

If STEPS

2

2 then F

:=0

(F is applied only in the

first two time steps)

Integrate one step, using the state equations in Fig.

2. This will produce new xl, x2,

x3

t

:=

t

+At, STEPS :=STEPS + 1.

If STEPS <MAX-STEPS (the maximal number of

steps) and 0

e 0 <

373 rad, then go to step

4.

Restore force: F

:=

F'.

If

x1

> 0 (current angle is positive), then determine

force-inc > 0), F := F + force-inc, go to step 3 .

Here force-inc is determined as follows:

9.1 If force-inc >

0,

then do nothing.

9.2 Else force-inc := (-l)*FACT*force-inc.

If x1 < 0 and

x2

is not within 5% of 8, then deter-

mine force-inc <

0)

and set F := F

+

force-inc, go

to step 3 . Here force-inc is determined as follows:

10.1 If force-inc

<0,

then do nothing.

10.2 Else force-inc := (-l)*FACT*force-inc.

The right value of

F

is found for the 0 and 8.

Out-

put F and terminate the algorithm.

with MAX_STEPS=512, FACT=0.2, and A,=O.Ol second.

Using the IPLSG algorithm repeatedly with 150

randomly generated pairs of e and 8 as input, we obtain

the learning set (in the form of 150 data points each being

a triplet (0, 8, F)). The IPLSG is also used to generate

exact control surface, i.e. a mesh of 100 by 100 points

with the corresponding forces, see Figure

3

(Exact).

3.2 Fuzzy Rules and Fuzzy Controller

In the second step of the experiment, the learning

set is clustered using fuzzy linear clustering algorithm

[7,8], once with linear form, y = a +alxl

+

~2x2, nd

once with non-linear (transcendental) form, =

a

+

alyo4

+

a 2 x 1

+

a3xf

+a 4 4 +

~ 4x 2 a5x;

+

agx2, of

the hypersurface equation

y

= gk(x,y). To use linear

clustering algorithm in the later case, the equation must be

linearized. That is done by replacing each term on the

RHS

of

the equation by a new independent variable

x;

i.e., -> x i x1 ->

xi,

xf ->

x i

etc., and LHS by y .

Values of the new (primed) variables used in clustering

are calculated accordingly.

3

Once fuzzy rules are determined, the controller out-

put can be generated for any value of input by using eqn.

(2); p A k ' sare taken from the nearest neighbor in the learn-

ing set,

so

called, the nearest neighbor (NN) method.

However, in the non-linear case when

gk

is a function of

y, solving eqn. (2) becomes complicated: for determina-

tion of the nearest neighbor, the value of y has to be

known, which in turn, can not be calculated without

knowing the nearest neighbor first. The solution to the

problem is iteration. In the n-th iteration step, the nearest

neighbor is found by using the current value of y. Then,

eqn. (2), now a transcendental equation, is solved and the

new nearest neighbor

is

determined by using the new

value of y. The algorithm ends if either the new nearest

neighbor is the same as the previous one or the procedure

is repeated a predetermined number of times.

Using the method described above, the new control

surfaces based on the same mesh as the exact one are gen-

erated, see Figure

3

(Clustered (linear) and Clustered

(non-linear)). Note that these control surfaces are calcu-

lated using fuzzy rules generated by only 150 randomly

chosen points constituting a learning set. The exact and

clustered surfaces are then compared and relative errors,

titled (Exact-C1ustered)Exact (non-linear) , (Exact-

C1ustered)Exact (linear-detail) , and (Exact-

C1ustered)Exact (non-linear-detail) , are shown in Figure

3 .

In the last two graphs, data for the nine smallest values

of angle and angular velocity is skipped so more details

are visible for the rest of the mesh. These results clearly

demonstrate the significant generalization power of the

clustering algorithm used. Note that the surfaces are plot-

ted only for the positive values of angle and angular

velocity. When both variables are negative force changes

sign but not the magnitude, and when they have opposite

sign force is zero due to the condition ( 3 ) given above.

3.3 Simulation Results

In the third step of our experiment, additional tests

were done

to

show that system really stabilizes in a very

short time both with and without application of distur-

bance force. This is achieved through the simulation of

the inverted pendulum system according to the following

algorithm:

Inverted Pendulum Simulation (IPS) Algorithm

1. Initialize: choose angle and angular velocity ran-

domly.

2. Get F (force) from the controller (see section

above)

3. Randomly generate disturbance

F'

and interval in

which F is applied.

4.

F : = F + F '

2171



5.

6.

ence

Evolution: use angle, angular velocity and F for

integration in time

If system escaped or more than maximum number

of steps elapsed, terminate. Otherwise go to step 2.

The results of simulation with and without pres-

of disturbance force are given in the Figure 4.

These results show the robustness of the developed sys-

tem due its capability to sustain additional disturbance

despite its already unstable and dynamic nature.

4. CONCLUSIONS AND FUTU

In this paper we present results on using fuzzy clus-

tering to learn control rules for the inverted pendulum

problem. We design and implement an algorithm for gen-

erating the learning set of the control problem, then we

apply fuzzy clustering to obtain fuzzy control rules, and

simulate the inverted pendulum system using the learned

rules. The experimental results are highly encouraging,

which support the following conclusions:

The algorithm for generating the control surface

(the learning set) of the inverted pendulum problem

is simple and efficient. The control surface gener-

ated has proven to be effective as the simulation

shows that the system using an approximation of

the control surface does stabilize quickly.

The fuzzy rules obtained from fuzzy clustering

have strong generalization and approximation capa-

bilities. Using only 150 data points as the learning

set, fuzzy clustering produces fuzzy rules which can

approximate the entire space of

10,000

points with

high precision.

The fuzzy inverted pendulum control system

obtained by fuzzy clustering is robust and can resist

quite amount of disturbances.

To sum up, the experimental results clearly demon-

strate that fuzzy linear clustering is a powerful means for

fuzzy system modeling.

Although the non-linear generalization of fuzzy lin-

ear clustering can achieve lower errors, see Figure 3, this

method gives rise to new dilemmas: which powers to

choose for LHS and RHS of the hypersurface equation:

whether to use the transcendental form

or

not; compli-

cated algorithm for determining value of y, etc. All of the

above suggests that one should first use fuzzy linear clus-

tering and only when such attempt does not give the

expected results, turn to the non-linear version.

In order to further improve clustering quality in the

non-linear case, we are planning to implement genetic

algorithms to achieve a better description of cluster repre-

sentative hypersurfaces, i.e., better choice of powers in

the hypersurface equations.

T

We are grateful to Professor Sukhamay Kundu for

helpful discussions on topics related to this work and for

letting us to use his program for fuzzy linear clustering.

The second author's work is supported in part by the NSF

grant ##9409370.

REFERENCES

T. Takagi, M. Sugeno, Fuzzy identification of sys-

tems and its applications to modeling and control,

IEEE

Transactionson Systems, Man, and Cybernet-

ics, Vol. SMC-15, No. 1, 1987, pp. 116-132.

M. Sugeno, K. Tanaka, Successive identification of

a fuzzy model and its application to prediction of a

complex system, Fuuy sets and systems, Vol.

42

M. Sugeno, T. Yasukawa,

A

fuzzy logic based

approach to qualitative modeling, IEEE Transac-

tions

on

Fuuy Systems, Vol. 1, No. 1, 1993, pp.

J. Bezdek, Hybrid models for fuzzy control, Proc.

of

the First Int. Symp. on Integrating Knowledge

and Neural Heuristics, May 1994, pp. 3-14.

H. Berenji, Y.Y. Chen, C.C. Lee, J.S. Jang,

S.

Murugesan, A hierarchical approach to designing

approximate-reasoning based controllers for

dynamical physical systems, Proc. of 6th Con on

Uncertainty in AI, 1990, pp. 362.

S.K. Sin and R.J.P. deFigueiredo, Fuzzy system

design through fuzzy clustering and optimal prede-

fuzzi ication, Proc.

of

the 2nd

IEEE

International

Conference on Fuzzy Systems, San Francisco, CA,

S .

Kundu, J. Chen, Fuzzy linear invariant clustering

with applications in fuzzy control, Proc.

of

NASALWAFIPS, 1994.

Hathaway, J. Bezdek, Switching regression models

and fuzzy clustering, IEEE Transactions on

Fuzzy

Systems, Vol. 1,No. 3, Aug. 1993, pp. 195-204.

S

Smith, B. Nokleby, D. Comer, A computational

approach

to

fuzzy logic controller design and analy-

sis using cell state space methods, In: F u u y Control

Systems, Kandel, Langholz, (Eds), pp. 398-426.

J.C. Bezdek, A convergence theorem for the fuzzy

ISODATA clustering algorithms, I Transac-

tions

on

Pattern Analysis and Machine Intelligelace

W.

€3.

Press, S . A. Teukolsky, W. T. Vetterling,B. P.

Flannery, Numerical Recipes in

C ,

Cambridge Uni-

versity Press, Cambridge, 1992

1991, pp. 335-344.

7-3 1.

March 1993, pp. 190-193.

2),1980, pp. 1-8.

2172



Exact Clustered linear)

.8

.8

1

ang. velocity angle ang. velocity angle

50

IO0

50

0

Clustered non-linear) Exact-Clustered)/Exact non-linear)

. .

. . I .

.8

angle

-

._

ang. velocity

Exact-Clustered)/Exact linear-detail)

_.

0.1-

0 2

ang. velocity angle

z

w

.8

ang. velocity angle

Exact-Clustered)/Ex;d non-linear-detail)

. .

, .

. I

.

, . ,

.

..

.

.. ..

, . .

. .

.

. .

, : .

.

.a

Figure3 Comparison

of

control surfaces generated heuristicaly and by clustering

in the last two graphs nine smallest values for angle and ang. velocity weire skipped)

2173



Time s)

0.3-

s o 2

F

- .

O f : .

0.2

-0.2

-1.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . :. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 . - \

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . I

:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.6

0 2 4 6

8

Time s)

0

I

. . . .

_ : . . . . . ._ :

. . . . .

.: . . . . . .

It

....

2 6

Time s)

2

4

6 8

Time (s)

-1.5

0 2 4

6

8

Time

(s)

601 I

I I

. . . . . . . . . . .

. . . . . .

:.

. .

_. .:.I.. . . .

. . . . . . .

. . . . . .

0 2

4 6

8

Time (s)

40

-

0

z

a

: 0

E

-20

.+-

-40

0 2 4 6

8

Time (s)

-

~ i ~ u l a t ~ o nesults for easeswith and without disturbance

initial conditions: angle = 20 deg. and angular velocity = .1 rads)

2174

Experiments on using fuzzy clustering for fuzzy control system design.pdf

Documents

Transcript of Experiments on using fuzzy clustering for fuzzy control system design.pdf