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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
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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
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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)
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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 '
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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.
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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)
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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
. . . . . . . . . . .
. . . . . .
:.
. .
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. . . . . . .
. . . . . .
0 2
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Time (s)
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-
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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