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Adaptive fuzzy modeling and control of chaotic dynamical systems

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Title:
Adaptive fuzzy modeling and control of chaotic dynamical systems
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Applebaum, Doris Ellen
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Denver, CO
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University of Colorado Denver
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English
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xx, 247 leaves : illustrations ; 28 cm

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Chaotic behavior in systems -- Mathematical models ( lcsh )
Adaptive control systems ( lcsh )
Fuzzy systems ( lcsh )
Adaptive control systems ( fast )
Chaotic behavior in systems -- Mathematical models ( fast )
Fuzzy systems ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Bibliography:
Includes bibliographical references (leaves 240-247).
Thesis:
Applied mathematics
General Note:
Department of Mathematical and Statistical Science
Statement of Responsibility:
by Doris Ellen Applebaum.

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|University of Colorado Denver
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Auraria Library
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All applicable rights reserved by the source institution and holding location.
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44093157 ( OCLC )
ocm44093157
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LD1190.L622 1999d .A66 ( lcc )

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Full Text
ADAPTIVE FUZZY MODELING AND CONTROL
OF
CHAOTIC DYNAMICAL SYSTEMS
by
Doris Ellen. Applebaum
B.A., Hunter College, 1971
M.S., Rutgers University, 1974
M.S., University of Colorado at Boulder, 1986
M.S., University of Colorado at Denver, 1991
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Applied Mathematics
1999


1999 by Doris Ellen Applebaum
All rights reserved.


This thesis for the Doctor of Philosophy
degree by
Doris Ellen Applebaum
has been approved
Shel Dalton
Randall Tag{
&
CJi-6
Jan Bialasiewicz
Kent Goodrich
£4
/HI
Date


Applebaum, Doris Ellen (Ph.D., Applied Mathematics)
Adaptive Fuzzy Modeling and Control of Chaotic Dynamical Systems
Thesis directed by Professor William Briggs
ABSTRACT
This research investigates the modeling and adaptive fuzzy control of chaotic
dynamical systems using fuzzy rules for the description of the underlying plant.
The fuzzy rule description becomes the basis for building an indirect adaptive
fuzzy controller. The bisection and homogeneity algorithm is introduced as a
modification and an extension of a recursive partitioning algorithm that gener-
ates rules directly from the data. Experimental results in the fuzzy modeling
of chaos are presented for the three-dimensional autonomous Lorenz attractor
and for the nonautonomous chaotic periodically perturbed pendulum. A variable
step size Eulers method performs trajectory reconstruction over multiple Stan-
dard Additive Model fuzzy systems. Domain decomposition splits regions prior
to running the algorithm. Domain decomposition enforces a bound on the train-
ing time, the time necessary for fuzzy rule generation, and, for nonautonomous
system modeling, it imposes a temporal ordering on fuzzy rules. Research results
indicate that domain decomposition, Standard Additive Model fuzzy inference,
IV


and a one-step Eulers method produce smooth trajectory approximations on
the order of Runge-Kutta numerical simulations. To answer the question, How
does one know that numerically generated computer maps of chaos are real?, the
fuzzy shadowing property is introduced. Sufficiency conditions, in the form of two
corollaries and a constructive theorem, are proven. These state conditions for an
orbit, generated by am additive fuzzy logic system, to be e-shadowed by a true
orbit of the dynamical system. The final emphasis of this research is the building
of an indirect adaptive fuzzy controller to train the chaotic pendulum to follow
a periodic reference trajectory. Fuzzy functional decomposition is introduced as
a method for decomposing fuzzy rules based upon conditional expectations for
modeling unknown functions of a second-order system of relative degree two. A
gradient projection method is introduced into the algorithm for adapting system
parameters for control. Experimental results demonstrate that the fuzzy adap-
tive, controller has a mean square tracking error that converges asymptotically
to zero with a convergence rate on the order of one thousand times faster than
conventional state feedback linearizing controllers.
v


This abstract accurately represents the content of the candidate s thesis.
I recommend its publication.
vi


DEDICATION
In Praise of
HASHSM
To the memory of
Max and Molly Ellenbogen, beloved grandparents,
Saul Ellenbogen, beloved uncle,
Edwin Leonard Ackerhalt, beloved father,
The Ancestor, guardian ad light em, killed in a synagogue ablaze with Torah,
Whos hand unseen has shaped the way


ACKNOWLEDGEMENT
Many thanks to Bill Briggs, the chairperson of my thesis committee, who
through many years has been my teacher and advisor and who as a great teacher
has allowed me to investigate unknown territory of applied fuzzy set theory. Many
thanks to Shel Dalton, my thesis advisor, who has stood by me over the past three
years. Many thanks to Randy Tagg for introducing me to the crane problem.
Many thanks to John Starrett, for all his insights into the chaotic dynamics of
the periodically perturbed pendulum. Id like to also thank John for introducing
me to the shadowing property of chaotic systems.
Especially, Id like to thank my husband, Leonard Ira Appiebaum, for giving
me all the support that I needed to make it through graduate school.


CONTENTS
Figures................................................................ xv
Chapter
1. Introduction.......................................................... 1
2. Mathematical Foundations of Fuzzy Modeling............................ 7
2.1 Introduction................................................... 7
2.2 System Identification: a General Overview......................... 8
2.3 Fuzzy Models..................................................... 13
2.3.1 Mamdani-type fuzzy models........................................ 15
2.3.2 Takagi-Sugeno (TS) fuzzy models.................................. 17
2.4 Product Space Clustering Methods for
Identification of Fuzzy Rules.................................... 18
2.5 Basic Configuration of Fuzzy Logic Systems ...................... 21
2.5.1 Fuzzy Logic Operations for Fuzzy Inference....................... 23
2.5.2 Defuzzification.................................................. 25
2.6 Fundamentals of Standard Additive Model
(SAM) Fuzzy Logic Systems........................................ 27
2.7 The Standard Additive Model
as a Conditional Expectation..................................... 32
ix


2.8 Fuzzy Interpolation using
SAM Fuzzy Systems ............................................. 37
2.9 Analytic Justification for Combining
Multiple SAM Fuzzy Systems....................................... 38
2.10 Conclusions....................................................... 41
3. Fuzzy Modeling of Complex Systems.................................... 42
3.1 Introduction...................................................... 42
3.2 The Bisection and Homogeneity Algorithm
for Structural Identification ................................... 44
3.3 Modifications to Hiew and Tsangs
Recursive Partitioning Algorithm................................. 47
3.4 Top Level Pseudo-code Description
of the Bisection and Homogeneity Algorithm....................... 50
3.5 Two-Dimensional Example of
Bisection and Homogeneity Algorithm.............................. 52
3.6 Heuristic for Choosing Homogeneity Parameters..................... 56
3.7 Construct Rule Procedure.......................................... 57
3.8 Domain Decomposition over
Multiple SAM Fuzzy Systems....................................... 61
3.9 Trajectory Reconstruction......................................... 62
3.9.1 Step One: Correlation-Product Inference........................... 64
3.9.2 Step Two: Variable Step Size Eulers Method....................... 65
3.10 Avenue for Future Research........................................ 67
3.11 Conclusions....................................................... 68
x


4. Experimental Results: Fuzzy Modeling............................ 70
4.1 Introduction................................................... 70
4.2 Experimental Results for Two-dimensional
Autonomous Dynamical Systems................................... 72
4.2.1 Two-dimensional Undamped Pendulum.............................. 74
4.2.2 Two-dimensional Van der Pol Oscillator......................... 78
4.2.3 Choosing Homogeneity Parameters................................ 80
4.3 Fuzzy Modeling of the
Three-dimensional Lorenz Attractor............................. 83
4.3.1 Domain Decomposition to Reduce
Computational Complexity of Lorenz............................. 88
4.3.2 Choosing Homogeneity Parameters................................ 90
4.4 Structural Identification for
a Chaotic Regime of the Pendulum............................... 94
4.4.1 Fuzzy Trajectory Reconstruction................................ 97
4.4.2 Domain Decomposition for
Temporal Ordering of Fuzzy Rules............................... 98
4.4.3 Quarter Phase Partitioning.....................................104
4.5 Fuzzy Reconstruction of Poincare Maps..........................Ill
4.6 Conclusions....................................................113
i
i
XI


5. The Fuzzy Shadowing Property.....................................115
5.1 Introduction..................................................115
5.2 Numerical Approximations of Chaos.............................118
5.3 The Shadowing Property of Dynamical Systems...................125
5.3.1 Examples of Hyperbolic Attractors ............................130
5.4 Fuzzy Shadowing Property
of Fuzzy Approximations.......................................132
5.5 Sufficiency Condition for Fuzzy Shadowing.....................133
5.6 Computer Proofs for
e-Shadowing in Non-Hyperbolic Systems.........................138
5.7 Conclusions...................................................145
6. Indirect Adaptive Fuzzy Control..................................146
6.1 Introduction..................................................146
6.2 History of Adaptive Control...................................148
6.3 Basic Concepts in Adaptive Control............................149
6.4 Control of Chaos..............................................153
6.5 State Feedback Linearizing Control
of the Parametrically Perturbed Pendulum......................155
6.6 Statement of the Indirect
Adaptive Fuzzy Control Problem................................159
6.7 Fundamentals of Adaptive
Fuzzy Controller Design
for the Parametrically Perturbed Pendulum.....................163
6.8 Functional Decomposition for Adaptive Control.................165
xn


6.9 Overall System Architecture
of Indirect Adaptive Fuzzy Control................................169
6.10 Lyapunov Analysis for
Indirect Adaptive Fuzzy Control...................................171
6.10.1 Preliminaries:
Development of Lyapunov Function..................................173
6.10.2 Construction of a Supervisory Control /zs.........................176
6.10.3 Construction of an Adaptive Law...................................180
6.10.4 Adapt_SystemJParameters().........................................184
6.11 On-line Adaptation Algorithm.......................................187
6.12 Choosing Design Parameters.........................................192
6.13 Conclusions........................................................194
7. Experimented Results................................................. 196
7.1 Introduction:
Control of Chaotic Pendulum ......................................196
7.2 Linearizing State Feedback
Control of Nonlinear Mathieu Equation.............................197
7.2.1 Linearizing State Feedback with
Supervisory Control...............................................202
7.3 Indirect Adaptive Fuzzy Control
From Chaos to Periodicity:
The Parametrically Perturbed Pendulum.............................212
7.3.1 Motivation Cranes Aboard Ships at Sea............................212
7.3.2 Derivation of Nonlinear Mathieu Equation...........................214
7.3.3 Adaptive Fuzzy Control for Tracking sin(i):
Comparison with Linearizing State Feedback........................216
xiii


7.3.4 Adaptive Fuzzy Control for Tracking of sin (cot):
Comparison with Linearizing State Feedback........................227
7.4 Conclusions....................................................... 234
8. Conclusions...........................................................237
Bibliography 240
xiv


FIGURES
Figure
2.1 The partitioning of a continuous domain into three fuzzy sets slow,
medium and fast for the linguistic variable speed........ 14
2.2 Fuzzy cluster partition of the input-output space using fuzzy Mam-
dani rule: If x is A$, then y is B 2.3 Regression surface y = f(x) and product space clustering of four
fuzzy if-then rules............................................ 20
2.4 Configuration of fuzzy logic system with fuzzifier and defuzzifier. . 21
2.5 Antecedent fuzzy sets NL and ZE and consequent PL for rule Rj
of the inverted pendulum....................................... 22
2.6 a. Correlation-minimum inference, b. Correlation-product infer-
ence procedure........................................................ 24
2.7 Computation of the fuzzy centroid.............................. 26
2.8 Architecture of standard additive model (SAM) fuzzy system ... 28
2.9 Regression line c{x)............................................. 36
3.1 Key features of bisection and homogeneity algorithm.............. 44
3.2 Architecture of Generate-Rule(r,d) procedure..................... 45
3.3 The generation of fuzzy rules with two antecedents............... 52
xv


3.4 Architecture of the Construct-Rule(r,d) procedure................. 58
3.5 Pseudo-code description of Construct-Rule(r,d).................... 59
3.6 Architecture of fuzzy trajectory reconstruction................... 63
4.1 Trajectories generated from 1001 iterations of fourth order Runge-
Kutta simulations of undamped pendulum............................ 75
4.2 Fuzzy rules modeling trajectories of the undamped pendulum, (y =
10"7;............................................................. 76
4.3 Fourth order Runge-Kutta simulations of Van der Pol............... 78
4.4 Fuzzy rules modeling trajectories of the Van der Pol oscillator, (y =
10-7)............................................................. 79
4.5 Van der Pol histogram ofdx/dt..................................... 81
4.6 Van der Pol histogram of dy/dt.................................... 82
4.7 Runge-Kutta numerical simulation of x-z plane of Lorenz system
generating 2600 points. (xo,yo,zo) = (5.0,5.0,5.0)................ 84
4.8 Trajectory reconstruction of x-z plane of Lorenz system........... 85
4.9 Runge-Kutta numerical simulation ofx-y plane of Lorenz system. 86
4.10 Fuzzy rule reconstruction ofx-y plane of Lorenz system........... 86
4.11 Runge-Kutta simulation ofy-z plane of Lorenz system.............. 87
4.12 Fuzzy rule reconstruction ofy-z plane of Lorenz system........... 87
4.13 Histogram for Lorenz dx/dt based upon 79,800 sample points, x =
0.9, r = J = 1.11................................................. 90
4.14 Histogram for Lorenz dz/dt based upon 79,800 sample points,
X = 0.9, r = ^ = 1.11............................................ 91
4.15 Histogram for Lorenz dy/dt based upon 79,800 sample points. . 92
xvi


4.16 Histogram for Lorenz dy/dt range*0.9 = 843 bins; homogeneity
parameter x 0.9, r = ^ = 1.11................................. 93
4.17 Trajectory reconstruction with multiple SAM/COG fuzzy systems. 97
4.18 Runge-Kutta simulation of 50 drive cycles of periodically perturbed
pendulum. I. C. {0,0, ) = (1.03724,1.00712,0.0).............. 99
4.19 Cycle 1: For each sample (0,0, 0) in the subdomain, time stamp 0
is between 0 and 2ir............................................101
4.20 Fuzzy modeling of the chaotic pendulum using cyclic domain de-
composition. .....................................................102
4.21 Three-dimensional grid over 15,000 points of Runge-Kutta gener-
ated chaotic pendulum.............................................105
4.22 Quarter phase partitioning along drive cycle of verticle support,
0 < 0 < 2tt.....................................................106
4.23 Trajectory reconstruction of fifty drive cycles using quarter phase
partitioning, SAM/COG fuzzy systems and a one-step Eulers method.
Grid size:(A0, A0, At) = (0.06,0.06,0.2)................... 107
4.24 Histogram of time rates of change along the x axis for first quarter
phase partition of periodically perturbed pendulum................108
4.25 Histogram of time rates of change along the y axis for first quarter
phase partition of periodically perturbed pendulum................109
4.26 Poincare section for 100 cycles of Runge-Kutta simulation. Ini-
tial Condition: (0,0,0) = (1.017238,1.007119,0.0).................Ill
4.27 Fuzzy Poincare section for 100 cycles of periodically perturbed pen-
dulum, generated by SAM/COG systems and quarter phase parti-
tioning. ...................................................113
5.1 Runge-Kutta simulation of fifty drive cycles of periodically per-
turbed pendulum. I.C. (0,0,0) = (1.03724,1.00712,0.0)............ 119
5.2 SAM/COG fuzzy model of chaos.......................................120
xvu


5.3 A Fuzzy Logic System F:X -f X for the discrete map F(xn) xn+J..123
5.4 A <5-pseudo-orbit {xn} of /........................................127
5.5 e-shadowing true orbit {?/} of <$-pseudo-orbit {xn}..................128
5.6 Splitting of the tangent space at point xn of the 6-pseudo orbit. . . 129
5.7 F : X -f X 3 F(xn) = xn+1.............................................138
5.8 Equation: x + sin(x) = 2.4 cos(£).....................................143
5.9 Computer proof of hyperbolidty: S = 10_w..............................144
6.1 A model-reference direct adaptive fuzzy controller....................151
6.2 A basic indirect adaptive fuzzy controller............................152
6.3 Closed-loop feedback linearizing controller. .........................158
6.4 Computation of the centroid of symmetric triangular fuzzy set A3. 167
6.5 System architecture of an indirect adaptive fuzzy controller. .... 170
6.6 Construction of projection operator {V} for on-line adaptation on
constraint surface h(C)..............................................189
6.7 Gradient projection method............................................191
7.1 One hundred cycles of nonlinear Mathieu equation using fourth
order Runge-Kutta....................................................197
7.2 Linearizing state feedback control of nonlinear Mathieu equation:
twenty five hundred iterations of closed-loop system tracking signal
r(t) = sin(t). fx(0) = yc(0) = 0.7774............................. 198
7.3 Linearizing state feedback control: performance of MSE over 2500
iterations. fx(0) = ^c(0) = 0.7774. r(t) = sin(t).................199
7.4 Linearizing state feedback: performance of MSE over 900 itera-
tions. fj.{0) = fJLciff) 0.7774, r(t) = sin(i)....................200
xviii


7.5 Linearizing state feedback control: oscillations of the control p{t)
over 2500 iterations of closed-loop system. p(Q) = pc(0) = 0.7774 201
7.6 Linearizing state feedback control: performance of p(t) over 900
iterations of closed-loop system. p(Q) = pc(0) = 0.7774. r(f) =
sin(t).........................................................202
7.7 Linearizing feedback control: twenty five hundred iterations of closed-
loop system tracking reference signal r(t) = sin(t). p = pc p9.. 203
7.8 Architecture of state feedback linearizable system controller. .... 205
7.9 Schematic representation of linearizing feedback control...........206
7.10 MSE for linearizing control of parametrically perturbed pendulum.
Oscillatory behavior of controller p = pc + p9 over twenty five
hundred iterations.............................................208
7.11 MSE for nine hundred iterations of linearizing control p = pc + p9
of parametrically perturbed pendulum.................................209
7.12 The control p(t) as a periodic function of time t: reference signal
r(t) = sin(i). p = Pce+Ps............................................209
7.13 The first 900 iterations of the linearizing state feedback control p(t):
reference signal r(t) = sin(t). p = pc + p3.......................211
7.14 Derivation of nonlinear Mathieu equation of motion................215
7.15 Fuzzy trajectory reconstruction for one hundred cycles of the chaotic
pendulum. Initial condition (6, v, ) = (1.01724,1.007119, 24t) . 219
7.16 Indirect adaptive fuzzy control: eight hundred fifty one iteraions of
closed-loop system: reference trajectory r(t) = sin(t)...............220
7.17 Indirect adaptive fuzzy control: MSE over eight hundred fifty one
iterations for reference trajectory r(t) = sin(t)....................223
7.18 Comparison of indirect adaptive fuzzy control: MSE over eight hun-
dred fifty one iterations versus supervised linearizing controller (+)
over nine hundred iterations for r(t) = sin(i).................224
xix


7.19 Indirect adaptive fuzzy control: performance of control p(t)(...) for
reference trajectory r(t) = sin(t)................................225
7.20 Indirect adaptive fuzzy control: performance of adaptive fuzzy con-
trol pit) versus performance of supervisory linearizing control p =
pc + Ps...........................................................226
7.21 Indirect adaptive fuzzy control: tracking reference trajectory r(t) =
sin(ut) (+).......................................................227
7.22 Indirect adaptive fuzzy control: MSE over fifteen hundred itera-
tions: Reference trajectory r(t) = sin(ujt).........................228
7.23 Indirect adaptive fuzzy control: performance of control p(t) for ref-
erence trajectory r(t) = sin(art)...................................229
7.24 Linearizing state feedback control: tracking over fifteen hundred
iterations: Reference trajectory r{t) = sin(utf) (+)..............230
7.25 Linearizing feedback control: MSE over fifteen hundred iterations:
Reference trajectory r{t) = sin(u^).................................231
7.26 MSE performance comparison of indirect adaptive fuzzy control
over fifteen hundred iterations versus state feedback linearizing con-
troller pc{t){--) for r(t) = sin(utf).........................232
7.27 Linearizing feedback control: performance of control p(t) = pc(t)
for reference trajectory r(t) = sin(a;£)..........................233
7.28 Linearizing feedback control p(t) = pc(t) vs adaptive fuzzy control
for reference trajectory r(t) = sin(a;£)............................234
xx


1. Introduction
The motivation for the current research is the control of the chaotic pendulum-
like motions of cranes aboard ships at sea. The pendulum with periodic vertical
support is considered as a working model of a more complex spherical system: a
crane undergoing vertical vibrations (the sway of a ship) with a small load at-
tached to its cable. Crane operators routinely use cranes to transfer small loads
between seafaring vessels. Currently, the crane operators control the transfer
manually. No control data exists for modeling the control strategy and human
expertise is not immediately accessible for control rule design. The addition of
a human-machine interface to a pendulum-motion controller would greatly facil-
itate the operation of the crane for novice crane operators.
Previous analytic control strategies have considered only local linearized OGY-
control [78]. Recently, researchers have also demonstrated the control of an exper-
imental chaotic pendulum using a neural network and a semi-continuous control
extension of the OGY method [7]. These methods control chaos by stabilization
of unstable periodic orbits hidden within the chaotic attractor. Previous fuzzy
crane control strategies have relied extensively on acquiring expert knowledge for
initiating the control rule design [92, 93].
The bisection and homogeneity algorithm, introduced in this dissertation, is a
method of identifying the underlying chaotic pendulum dynamics directly from
the data. It is a recursive partitioning algorithm that modifies and extends the
1


recursive partitioning algorithm of Hiew and Tsang [35, 36] for modeling chaos.
It is an unsupervised learning approach that models the underlying dynamics
directly from the data using fuzzy rules. It is applicable to systems for which the
dynamics are not fully understood.
The bisection and homogeneity algorithm has many advantages. It shares
with feedforward neural network models an advantage over polynomial models,
in that the output is bounded and hence, instabilities in the model do not occur.
An advantage over piecewise polynomial estimators and B-spline networks is the
non-uniform structure of the state space that emerges as the algorithm runs.
With the bisection and homogeneity algorithm, a finer fuzzy rule partition occurs
over chaotic regions than over regions with more regular dynamics. Piecewise
polynomial estimators and B-spline networks assume, for system identification,
a uniform grid that is not based upon the underlying statistics of the model.
Two advantages of the bisection and homogeneity algorithm over neural networks
are its bounded training time and its avoidance of false local minima. The
adaptive fuzzy approach of generating rules directly from the data captures the
underlying dynamics of a complex system without the benefit of human expertise
for extracting modeling or control rules. It allows for parametric uncertainty and
a global control strategy that trains the system to follow a periodic and stable,
in the sense of bounded, reference trajectory.
The bisection and homogeneity algorithm is a member of a class of unsupervised
learning schemes that are referred to as product space clustering methods [6].
Product space clustering is suitable when enough knowledge of the underlying
2


structure of the system exists to allow the identification of state space variables.
For state space modeling, a fuzzy rule maps the current input and state of a sys-
tem to the time rate of change of the state or the derivative in the continuous time
case. The bisection and homogeneity algorithm generates fuzzy rules that model
the continuous time dynamics. Product space clustering methods include c-means
clustering [10], subtractive clustering [18], the Gustafson-Kessel algorithm [32],
fuzzy maximum likelihood estimation algorithm [27], possibilistic clustering [50],
and the mountain method [91, 8]. The c-means clustering and subtractive clus-
tering algorithms require prototypical cluster centers. The remaining methods
also require prototypical points. The mountain method is based upon a selection
of a grid of data points. A member of the grid of points is chosen as a potential
cluster center. The bisection and homogeneity algorithm also starts with a grid
of data points. It differs from the mountain method in that the potential pro-
totype of homogeneity is an entire region of the state space. In a sense, the
recursive partitioning algorithm is a top-down approach to clustering while the
mountain method starting from a prototypical grid point is bottom up. Expert
knowledge is often heuristic and top down. The bisection and homogeneity al-
gorithm is applicable for an iterative system identification procedure that mixes
human knowledge with the statistical properties of the data set.
Chapter 2 of the dissertation gives an overview of the mathematical founda-
tions of fuzzy modeling. In particular, the Standard Additive Model (SAM) of
Kosko [48] is presented. This model forms the basis of the fuzzy inference pro-
cedure that is used for trajectory reconstruction. A SAM system is described in
3


terms of producing the optimal nonlinear estimator function in the least mean
square sense [64]. Chapter 2 also describes the SAM fuzzy model as, according
to a theorem of Kosko [48], a convex combination of conditional expectations of
the input. This property and the linearity property of conditional expectations
are used, in this dissertation, to develop a method of functional decomposition of
fuzzy rules for the modeling of the unknown functions in indirect adaptive fuzzy
control design.
Chapter 3 describes the bisection and homogeneity algorithm. The bisection
and homogeneity algorithm performs both structural identification, based upon
the criterion of homogeneity of a region, and parameter identification. Para-
meter identification refers to the size and shape of antecedent and consequent
membership functions of the fuzzy rules.
Chapter 4 presents the experimental results of fuzzy modeling two and three-
dimensional dynamical systems. Domain decomposition, which splits the state
space into smaller regions prior to running the bisection and homogeneity algo-
rithm, is shown to reduce the computational complexity of the algorithm when
it is applied to the three-dimensional chaotic Lorenz attractor and the chaotic
parametrically perturbed pendulum.
Chapter 5 introduces the fuzzy shadowing property. It extends the literature
on fuzzy chaos [22, 21, 23, 44, 82, 26] by introducing into the language of fuzzy
approximation theory, the notion of e-shadowing of pseudo-orbits generated by
additive fuzzy systems. Two corollaries and a constructive theorem demonstrate
sufficiency conditions for fuzzy shadowing to occur. Chapter 5 also discusses
4


a computer proof of Yorke and Sauer [71] for determining the presence of e-
shadowing for 5-pseudo-orbits generated by an ODE solver.
Chapters 6 and 7 describe the indirect adaptive fuzzy control of the parametri-
cally perturbed pendulum. Both chapters describe the functional decomposition
of the fuzzy rules for representing acceleration and velocity-damping components
of the pendulum dynamics for system identification during running the closed-
loop controller. Chapter 6 presents the Lyapunov analysis that produces the
adaptive control law and the supervisory control law that were originally devel-
oped by Wang [89]. This chapter then introduces a gradient projection method
to guarantee that the parameters of the controller remain within a constraint
set. Chapter 7 presents experimental results of constructing an indirect adaptive
fuzzy controller for the chaotic parametrically perturbed pendulum. This chapter
compares the results of running the closed-loop indirect adaptive fuzzy controller
with conventional state-feedback linearizing control laws.
Chapter 8 summarizes the conclusions of this research and suggests avenues for
further research. This dissertation has demonstrated the strategy for and effec-
tiveness of using the bisection and homogeneity algorithm for indirect adaptive
fuzzy control of a chaotic regime of the pendulum. The research has demon-
strated a global control strategy based upon the choice of reference trajectory
and mild assumptions of smoothness of the signal to be tracked. The adaptive
fuzzy control strategy eliminates the trial and error approach to searching for
unstable periodic orbits that extensions of local OGY-control require. Domain
decomposition reduces the computational complexity of running the algorithm
5


and imposes a temporal ordering on fuzzy rales. Functional decomposition yields
optimal tracking results in the mean square error sense. This dissertation has
also extended the literature on fuzzy chaos and fuzzy approximation theory by
introducing the fuzzy shadowing property of additive fuzzy systems.
Avenues of future research include rule base refinement. There is a tradeoff
between the minimum number of fuzzy rales and the approximation accuracy
required in the modeling of system dynamics. In the area of fuzzy modeling of
chaos, the further analytic and experimental identification of hyperbolic invariant
sets may assist in choice of trajectory for fuzzy modeling and control and in the
selection of piecewise polynomial estimator for fuzzy rale base refinement. This
research suggests a further review of domain decomposition and the performance
of fuzzy inference on the boundaries of subdomains. The body of research on
domain decomposition for finite element meshes may be extended to the fuzzy
modeling of complex systems. Other research directions include the study of the
gradient projection method for adaptive fuzzy control over complex constraint
surfaces for on-line parameter estimation.
6


2. Mathematical Foundations of Fuzzy Modeling
2.1 Introduction
Prominent researchers, Sugeno and Yasukawa, in the field of qualitative model-
ing, consider fuzzy modeling as the most important issue in fuzzy theory [61]. The
research concerned with fuzzy modeling has a history of more than 20 years [96].
There are many interpretations of fuzzy modeling. In the narrow sense, as Sugeno
suggests [80], fuzzy modeling is an approach to modeling that uses a system de-
scription based on fuzzy logic with fuzzy predicates. The system descriptive lan-
guage is composed of fuzzy quantities. Fuzzy quantities are expressed in terms
of fuzzy numbers or fuzzy sets that may be associated with linguistic labels. In
this narrow interpretation of fuzzy modeling, a fuzzy set usually does not have a
tight relation with a linguistic label.
In a broader sense, a fuzzy model is a qualitative modeling scheme by which one
qualitatively describes system behavior using a natural language. This research
addresses system identification and fuzzy modeling for control in the narrow sense.
System identification refers to the problem of finding a suitable model structure
for identifying structural elements such as state space dimension and state vari-
ables. Section 2.2 describes the types of model structures that one may choose
from depending upon how much prior knowledge one has of system dynamics.
Parameter estimation refers to fitting a model within a given structure. Chapters
7


3 and 4 describe a recursive partitioning algorithm, the bisection and homogene-
ity algorithm, which in the narrow sense, performs both system identification and
parameter estimation. It is anticipated that in future research, combining and
simplifying the fuzzy rule base will allow a natural language description of chaotic
pendulum dynamics.
This chapter describes the basic concepts of fuzzy modeling for system iden-
tification and parameter estimation of dynamic processes. In particular, it de-
scribes the architecture and fuzzy operations associated with the Standard Addi-
tive Model (SAM) fuzzy logic system. The SAM fuzzy system forms the basis for
system identification and parameter estimation for control of the parametrically
excited pendulum. This chapter states several key theorems that 1) simplify the
computational complexity of designing and implementing the SAM fuzzy logic
system, 2) explain the SAMs fuzzy approximation accuracy in terms of optimal
least mean square error solutions, and 3) justify the combining of independent
SAM fuzzy models for reconstruction of state space descriptions.
2.2 System Identification: a General Overview
Model structures of dynamical systems fall into three broad classes based upon
the amount and type of a priori knowledge known about system dynamics. The
three classes are:
White-box models. These models are based on differential or difference
equations, derived from physical laws without any use of measurements of
the system. In this case, there is no uncertainty with regard to system
8


structure and parameter identification.
Grey-box models. For these models, some physical insight is available, but
several parameters remain to be determined from observed data.
Black-box models. These models are derived entirely from data using no
physical insight. The model structure belongs to families of models that
are known to be flexible and to show good results in practice [72].
Neural networks, wavelets, and fuzzy logic systems are generally viewed as
black-box models. In practical applications, fuzzy logic systems combine the
black box approach with physical or verbal modeling in such a way that certain
prior knowledge from the system is taken into account. Once heuristic information
is added in the form of fuzzy rules, the fuzzy models fit easily into the category
of grey-box modeling approaches.
The first step in modeling is to choose a model class. One makes a choice
between black-box models and the physically parameterized approaches of white
and grey-box modeling. One also chooses between linear and nonlinear structures.
The next consideration is the size of the model class: the number of variables and
combination of variables to be used in the model. For example, in modeling
the parametrically excited pendulum, the variables used for measuring system
dynamics axe angular displacement, angular velocity, and phase. The order of the
dynamical system is also determined. With the above information, the designer
has the general specifications of a model class from which the search for a model
will be carried out. The next step is to consider how to parameterize the model
9


class so that estimation algorithms can find reasonable parameter values. The
bisection and homogeneity algorithm to be discussed in chapter 3 is one such
estimation algorithm.
For the fuzzy modeling of the parametrically excited pendulum and the other
dynamical systems presented in Chapter 4, this research relies heavily on a black-
box modeling approach. The system identification problem for black-box struc-
tures has been described by Sjoberg et.al. in [72]. The key features of black-box
modeling are summarized as follows: Let fjf denote {/z(l) a set of
observed inputs of a dynamical system up to and including time t. Let yt denote
Q/(l) y(2).. .y(£)}, a set of observed ouputs of a dynamical system up to and
including time t. We are looking for a relationship between past observations
{/zt-1,yt_1} and future outputs y(t):
y(t) = yt_1) + (2-1)
where u(t) represents a system disturbance which is assumed to be additive. The
function g is the system model approximation of the output of the dynamical sys-
tem at time t. The goal is that the additive term v(t) be so small that yt_1)
is a good prediction of y(t) from past data. In practical applications, one may find
it relevant to minimize the variance in u(t) to achieve desired performance [28].
To define a model for system identification, we parameterize a family of function
models with a vector 6:
9(fS l,yt l,0)-
10


The quality of 6 can be assessed by means of the fit between the model and
the data:
N
Hlly(t) -90** l.e 1,0)ll2,
t= 1
(2-2)
where the data set is {fiN, yN}-
Chapters 3 and 4 describe fuzzy modeling for reconstruction of fuzzy trajecto-
ries directly from the data. In this context, a mean square error is not applied
to the evaluation of trajectories. The function that is being approximated by the
fuzzy model is considered unknown. Rules are generated directly from the data,
which classifies the learning of fuzzy rules as an unsupervised learning method.
In supervised learning, the function is assumed known and a mean square error
measurement can be applied. Hiew and Tsang, in their later work [36], assume
that the function / is known and they demonstrate the application of a mean
square error measurement in determining the partitioning of fuzzy rule patches
for fuzzy rules. In Chapter 6, which deals with fuzzy modeling for control, a mean
square error has been applied to the evaluation of the fuzzy logic controller. In
this case, the mean square error is applied to the desired reference trajectory.
For solving the modeling problem for system identification, Sjoberg et.al. [72]
state that it may be more useful to express g in equation (2.2) as a concatenation
of two mappings: one that takes the increasing number of past observations
and maps them into a vector 4>(t) and one that takes this vector to the
output space:
9(fS 1,yt \0) =
(2.3)
11


where
4>(t) = The vector (t) is referred to as the regression vector and its components axe
referred to as regressors. Sometimes the regression vector depends on all the
model parameters and
4>{t) = yl~l, 9). (2.5)
The modeling problem for nonlinear systems breaks down into two subproblems:
1. The choice of the dynamic regression vector <£(/zt-l,yt-1,0).
2. The choice of static nonlinear mapping g((j>(t),9).
In black-box modeling, the parameterized function family g((t),6) is often
expressed as a basis function expansion:
9(, 0) = ), (2.6)
where a* 6 7L.
The gk's are referred to as basis functions. Sjoberg et.al. point out that the ex-
pansion (2.6) with different basis functions, together with all the possible choices
of regression vectors 0, establishes a unified framework for investigating most
known nonlinear black-box model structures [72]. The key question becomes how
to choose the basis functions gk- Fuzzy models belong to the model structures
of (2.6). For a fuzzy model, the regression vector 4> represents the scalar input
variables of the model. The basis functions gk axe constructed from the fuzzy
set membership functions and fuzzy inference operations. Fuzzy models will be
discussed in the next section.
12


2.3 Fuzzy Models
A fuzzy model is a type of static mapping that is represented as a set of if-then
rules. There axe three basic types of fuzzy models that use if-then rules for system
identification:
1. Mamdani-type rules defining a linguistic (symbolic) fuzzy model.
2. Takagi-Sugeno-type fuzzy rules defining a Takagi-Sugeno fuzzy model.
3. Singleton-type fuzzy rules defining a singleton fuzzy model.
All three types of fuzzy models employ fuzzy sets for the partitioning of the
domains of the input and output variables into several smaller overlapping regions.
When qualitative information is available, linguistic or symbolic values denote the
fuzzy sets. Linguistic variables are natural language descriptors that denote the
input and output domain variables.
13


Figure 2.1 demonstrates the partitioning of a continuous domain by fuzzy sets.
A'(speed)
Figure 2.1. The partitioning of a continuous domain into three fuzzy sets
slow, medium and fast for the linguistic variable speed.
This figure illustrates two types of fuzzy sets, namely the trapezoid and the
triangle. The Gaussian function represents another type of fuzzy set used in
applications. Fuzzy sets quantize or partition a domain of values. The domain is
referred to as the universe of discourse. The natural language descriptor for the
universe of discourse in Figure 2.1 is the linguistic variable speed.
Fuzzy sets are also referred to as membership functions fip : X f [0 1]. The
fuzzy sets in Figure 2.1 are labeled with linguistic terms slow, medium and
fast. The height h(A) of a fuzzy set A is the largest membership function value
obtained by any element in that set. A fuzzy set is normal when h(A) = 1 (as in
Figure 2.1). A fuzzy set is called subnormal when h(A) < 1. In the body of this
14


research, all fuzzy sets are normal.
Once the fuzzy sets of the fuzzy model form a partition of the input and
output domains, the next step in fuzzy modeling is to establish the relationship
between the linguistic or symbolic values of the input and output variables. These
relationships sure expressed in the form of fuzzy if-then rules. Each fuzzy rule
maps, using fuzzy logic operations, a fuzzy region from the antecedent space (if-
part) of a fuzzy rule to the fuzzy region from the consequent space (then-part) of
this rule.
2.3.1 Mamdani-type fuzzy models
A Mamdani-type fuzzy model [55] that is single-input-single-output (SISO)
consists of fuzzy rules of the following type:
Ri: If a: is Ai, then y is B\
R2* If x is A2, then y is Bo.
The variable a; is an input variable over the universe of discourse X and y is
an output variable defined over the domain Y. Both x and y take their crisp
(pointwise) values on their respective domains. Ai and A2 are antecedent fuzzy
sets that quantize the domain of discourse. Analogously, Bi and B2 are known
as consequent fuzzy sets that quantize the output domain Y.
Figure 2.2 illustrates how each rule defines a fuzzy region in the input-output
space.
15


The set of fuzzy rules partitions the input-output space into several overlapping
fuzzy regions [34].
Y
A, \zAj Aj
Figure 2.2. Fuzzy cluster partition of the input-output space using fuzzy Mam-
dani rule: If x is A3, then y is B?.
The shaded rectangles describe the degrees of membership in the fuzzy region.
The darker rectangles indicate a stronger degree of membership of a crisp input
x in the fuzzy set A3. For example, if a: is at the midpoint of the triangular fuzzy
set A3, the degree of membership of x in A3 is 1. The output triangular fuzzy
set B2 is fired by the degree of membership of x in fuzzy set A3. For an input
x near the comer points of A3, the degree of membership is close to zero. Hence
the shading of the fuzzy region is light in the vicinity of these comer points. The
degree of shading in the fuzzy region reflects the degree to which the output fuzzy
set is scaled or clipped depending upon the inference strategy used.
16


The following two Mamdani-type rules exemplify a fuzzy model for a first-order
continuous dynamical system.
Rx: If x is Ax and p is B2, then x is Cx
R2: If x is Ao and n is Bx, then x is C2
Chapter 4 describes the use of Mamdani-type fuzzy rules for the construction of
fuzzy rule bases for the modeling of the first-order and chaotic parametrically
excited pendulum system.
For the time-discrete case and for a first-order model one may have Mamdani-
type rules as described by Hellendom and Driankov [34]:
Rx: If y(k) is Ax and fi{k) is £2, then y{k + 1) is A3
R2: If y(k) is A* and y.(k) is Ax, then y(k) is Ax.
In the above two fuzzy rules, fi{k) represents the input regressor and y(k), the
output regressor. The rules map the regression vector at time k to the output at
the next time step, k 1.
2.3.2 Takagi-Sugeno (TS) fuzzy models
The TS-fuzzy model in the context of black-box identification is chosen as a
linear regression model. It is less general than a Mamdani-type fuzzy model with
respect to modeling arbitrary nonlinear static mappings. The following rules are
an example of the affine TS-fuzzy model:
Rx: If x is Ax, then y = axx -f- bx
R2: If x is A2, then y = a2x -I- b2,
where a,-, 6,- 6 1Z.
17


When the parameters a* hi the TS-fuzzy model are zero, the resulting model
is called a fuzzy singleton model, since a constant 6t- is a singleton fuzzy set.
Prom the above TS-fuzzy rules, one obtains the singleton rules:
Ri: If x is Ai then y = bi G.TZ
R2: If x is A2 then y = 62 72-
In Chapter 7, singleton fuzzy rules are used to construct an indirect adaptive
fuzzy controller for the parametrically excited pendulum.
2.4 Product Space Clustering Methods for
Identification of Fuzzy Rules
Several methods may be used to construct fuzzy rules. Among these methods
are clustering, neural networks, genetic algorithms, and artificial intelligence. A
product space clustering method is a system identification technique for building
fuzzy models directly from data. Chapter 3 describes a recursive partitioning
algorithm, the bisection and homogeneity algorithm, which is a product
space clustering technique used to construct fuzzy rules.
Product space clustering decomposes a nonlinear modeling problem into several
simpler, linear sub-problems. It is a local modeling approach. Global methods for
modeling complex nonlinear systems describe the system under study in terms
of nonlinear functional relationships between the systems variables. Examples
of global methods are nonlinear state space models or input-output black-box
models such as the NARX (Nonlinear Auto-Regressive with eXogeneous input)
structure used in connection with some neural or wavelet networks [6]. Local
modeling methods partition the process domains into fuzzy regions. For each
18


region in the input space, a rule is defined that specifies the output of the model.
The rules can be seen as local submodels of the system. The exact nature of the
submodels, as well as the way they are combined, depends on the particular type
of rules and the fuzzy inference mechanism involved.
Babuska and Verbruggen describe a product space clustering method as a
method that transforms the identification problem into a static nonlinear re-
gression function / : Ttp v ft, where / is said to be the regression function of Y
on X [6, 42] if
E(Y\X) = f(X). (2.7)
For y /(x), the model input x £ 7Zp is called the regression vector ox regressor
and the output y is called the regressand or the response variable. The product
space of the regressor and the regressand, Z = (X x Y) C 7tn is called the
regression space, where n = p-r 1. In this space, the equation y = /(x) defines the
regression surface. By means of the antecedent fuzzy sets, the regression space is
partitioned into smaller regions, over which the regression surface can be locally
approximated. The purpose of identification is to find the parameters of the
regression surface, their locations, and their number over each of the partitioned
regions. Chapter 3 presents a recursive partitioning scheme, the bisection and
homogeneity algorithm, which performs these three tasks.
19


Figure 2.3 shows the product space clustering achieved by four fuzzy rules
for a single-input single-output system (SISO) and a two-dimensional regression
surface y = /(x) for p = 1.
Y
X
Dxrxixixi
a; aTa, a
Figure 2.3. Regression surface y = /(x) and product space clustering of four
fuzzy if-then rules.
The four fuzz rules that partition the product space are:
Ri: If a; is Az, then y is Bz
Rz'- If x is A3, then y is Bz
Rz: If x is At, then y is Bz
Rt: If x is A5, then y is B5
The antecedent fuzzy sets Az, A3, and A4 axe symmetric triangular member-
ship functions. Symmetric triangular membership functions also represent the
consequent fuzzy sets Bz, Bz, and B4. The shaded rectangles suggest the degrees
20


of rule firings which depend on the scalar input x E X. The next section describes
the construction of a fuzzy logic system to approximate the regression surface.
2.5 Basic Configuration of Fuzzy Logic Systems
The output of the system identification procedure is a fuzzy rule base that
describes the dynamic process in terms of if-then fuzzy rules that map fuzzy sets
in the input domain to fuzzy sets in the output domain. The next step is to
construct a fuzzy logic system F : X -+ Y that maps an input signal x to an
output signal y = F(x). Figure 2.4 shows the basic components of a fuzzy logic
system architecture. The input vector x is a vector of crisp scalar values, one
for each dimension of the input-state space X. The output y E 72 is the output
of the defuzzification process.
Figure 2.4. Configuration of fuzzy logic system vnth fuzzifier and defuzzifier.
21


Consider a two-input, one-output fuzzy logic system and the following rule Rf
Rji If xi is A) and xo is Aj, then y is Bj.
The fuzzifier takes the measurements of all variables that represent the dynamic
process. It converts these measurements into appropriate fuzzy sets to express
measurement uncertainties. In practice, the fuzzifier maps x = (xi,xz) to its
fuzzy singleton coordinates. The crisp values Xi and x2 are referred to as fuzzy
singletons. The inference engine uses the membership function aj- : X -* [0 1]
associated with the antecedent fuzzy set Aj of the jth rule to match the fth coor-
dinate of input vector x to Aj. The inference engine matches all rules in parallel.
For each rule Rj in the fuzzy rule base, the fuzzy inference engine fires the if-part
of rule Rj, by the combined degrees to which the fth coordinate of input vector x
belongs to the if-part fuzzy set Aj, for i = 1,..., n, and x = (xi, X2,..., xn)T. Fig-
tire 2.5 shows, for fuzzy modeling of the inverted pendulum [46], the antecedent
fuzzy sets and the consequent fuzzy set for rule Rj.
Rule Rji If x, is A,1 and xz is A^K then y is B,
Figure 2.5. Antecedent fuzzy sets NL and ZE and consequent PL for rule Rj
of the inverted pendulum.
22


The fuzzy sets A} and A~- are labeled with fuzzy-set values NL (Negative Large)
and ZE (Zero). The consequent fuzzy set, Bj, for rule Rj has the linguistic label
PL (Positive Large). These linguistic labels for the antecedent and consequent
fuzzy sets are also referred to as linguistic terms. The linguistic or fuzzy variables
are angle, angular velocity, and motor current. Suppose the input vector x =
(80, 5). For rule Rj, the inference procedure maps x\ with the membership
function for NL to a*?L{xi) = 0.4. This says that x\ belongs to the fuzzy set NL
to degree 0.4. The inference procedure maps with the membership function
for ZE to afE(xz) = 0.7.
2.5.1 Fuzzy Logic Operations for Fuzzy Inference
Once the antecedent membership functions are computed for an input vector
x, the fuzzy logic inference engine applies a fuzzy logic operator, an if-part com-
biner, to combine the values of antecedent membership functions that are joined
by the conjunctive and in the fuzzy rule. This research has used a product
if-part combiner in some examples such as the chaotic parametrically excited
pendulum and a minimum if-part combiner in the fuzzy modeling of other dy-
namic systems. Experiments involving these fuzzy logic models are presented in
Chapter 4. For the fuzzy rule example of Figure 2.5, a product if-part combiner
would yield:
2
a.j(x) = JJ a*(xf) = afL{x{) afE(x2) = 0.4 0.7 = 0.21.
t=i
23


The minimum if-part combiner would yield a pairwise minimum:
aj(x) = a^L{xx) A afE(x2) = min(0.4,0.7) = 0.4.
The next step in the inference process involves the application of a fuzzy impli-
cation operator. The implication operator associates the combined if-part a} (x)
with the rules consequent fuzzy set to fire the fuzzy rule.
Figure 2.6 shows two fuzzy implication operators for rule Rj of Figure 2.5.
a. NL ZE PL
Figure 2.6. a. Correlation-minimum, inference, b. Correlation-product infer-
ence procedure.
Figure 2.6 a. illustrates correlation-minimum inference. The correlation-
minimum inference procedure activates the motor current fuzzy set PL to the
degree aj(x) = 0.4 by taking the pairwise minimum of 0.4 and the PL fuzzy set
24


f*PL,
OAApcPL(y),
for all motor current values y. Correlation-minimum inference clips the conse-
quent fuzzy set PL and produces the fired fuzzy set C[.
Another inference procedure, correlation-product inference is shown in Figure
2.6 b. In this particular example, using the product if-part combiner, the com-
bined if-part value is aj(x) = 0.7. Correlation-product inference activates the
motor current fuzzy set PL to the degree 0.7 by taking the pairwise product of
0.7 and the PL fuzzy set:
0.7fJ.pL{y)
for all motor current values y. Correlation-product inference scales the con-
sequent fuzzy set PL and produces the fired fuzzy set C[ which preserves the
shape of the triangular membership function //p£.
The fined step in fuzzy inference is the combining of fired consequent fuzzy
sets over all rules in the fuzzy rule base. This research uses the sum aggregation
operator. A pointwise sum is taken over all consequent fuzzy sets. Given the fired
fuzzy sets Bj,j = 1 the aggregation sum operator computes the resultant
consequent fuzzy set
B = B'S
j=l
where n is the number of fuzzy rules in the fuzzy rule base.
2.5.2 Defuzzification
Defuzzification is the final step in mapping an input vector x to output of
25


the fuzzy logic controller F(x). The specific type of defuzzification that is used
throughout this dissertation is centroid defuzzification. The SAM Theorem to be
discussed in the following sections of this chapter allows the defuzzified output
to be computed as:

Sj=i aj(x)Vjcj
(2.8)
T,?=laj(x)Vj '
where each then-part consequent fuzzy set Bj has area or volume Vj and centroid
Cj. Figure 2.7 shows the computation of the fuzzy centroid for the fuzzy set B =
B[+B2. Applying equation (2.8), one computes the volumes of the symmetric
triangular fuzzy sets B[ and B'2. The height h of each symmetric triangle may be
interpreted as the resultant if-part combiners aj.(x) and a2(x) of two fuzzy rules.
Evaluating the sums in the denominator and in the numerator, one obtains
denom = 0.28 4.2 + 0.8 12 = 10.776.
numer = 0.28 4.2 75 + 0.8 12 50 = 568.2.
Figure 2.7. Computation of the fuzzy centroid.
The result is the fuzzy centroid 52.7283.
26


The remaining sections of this chapter will review the properties of the Stan-
dard Additive Model (SAM) fuzzy logic system that is the primary fuzzy logic
approach used here to model the chaotic parametrically excited pendulum.
2.6 Fundamentals of Standard Additive Model
(SAM) Fuzzy Logic Systems
This section discusses the basic components of the Standard Additive Model
(SAM) fuzzy logic system. Kosko, in his work on fuzzy engineering, developed
the language and the basic theorems that describe the SAM fuzzy model [48].
An important feature of the SAM fuzzy model, with respect to this research,
is its ability to preserve, through product combiners, the shape of membership
functions. This helps in the modeling of the sensitive fluctuations that occur
in chaotic dynamic systems. A second important feature of a SAM model, is
its ability to produce a functional approximation that yields an optimal mean-
squared modeling error.
This section first discusses the architecture of the SAM fuzzy model. Following
this basic description of SAM components, the section presents theorems that
describe the computational and probabilistic aspects of the SAM model.
Figure 2.8 describes the architecture of a SAM fuzzy logic system. The input
into the SAM model is a crisp scalar or vector x. In practice, the fuzzy set
representing x is a fuzzy singleton set. No explicit fuzzification of x occurs. The
fuzzy rule base consists of rales of the form
If Aj, then Bj,
where the Aj represent antecedent fuzzy sets and Bj represent consequent fuzzy
27


sets for rule j. Once a fuzzy rule fires, the consequent Bj, modified by the rule
firing, becomes the fuzzy set Bj. The Wj are weights, which, in practice, are set
equal to 1.
Figure 2.8. Architecture of standard additive model (SAM) fuzzy system
Correlation-product inference, the aggregation sum operator and centroid de-
fuzzification form the backbone of the SAM fuzzy systems, the Lorenz attrac-
tor, and the chaotic parametrically perturbed pendulum that this study ad-
dresses. In chapter 4, a study of the two-dimensional undamped pendulum sys-
tem, uses correlation-minimum inference to implement the SAM fuzzy model.
Both correlation-product inference and correlation-minimum inference are fuzzy
operations on the consequent fuzzy sets Bj of each rule. The inference operations
on Bj produce the fired consequent fuzzy sets Bj. The aggregation sum oper-
ator performs a pointwise sum over all fired consequent fuzzy sets and produces
28


the fuzzy set
m
i=l
where m is the number of fuzzy rules in the rule base. In the major system
of this study, that of the three-dimensional parametrically perturbed pendulum,
the centroid defuzzifier reduces to the center of gravity (COG) defuzzification
operator.
To explain the basic fuzzy logic operations, consider the fuzzy rule j:
If Xi is Aj and x2 is Aj... and xn is A, then y is Bj.
The min if-part combiner and the product if-part combiner are fuzzy operators
that combine the antecedents of fuzzy rules. Given a crisp input vector x =
(xi,x2, ..,xn), let %(x) be the product of the n membership function values
associated with the n antecedents of a fuzzy rule j. The product if-part combiner
computes ay(x) = n?=i a}(xt-), where ay(x) is the product if-part combiner of rule
j and a*-(xi) is the value of the ith antecedent membership function evaluated at
the ith component of the input vector x. The min if-part combiner computes the
pairwise minimum ay(x) = ay(xx) Afly(x2)A...Adj(xn). Hiew and Tsang, in their
work on the recursive partitioning algorithm [35, 36], used a min if-part combiner
to combine antecedent fuzzy sets. Kosko emphasizes [48] that a key advantage
to using the product if-part combiner over the min if-part combiner is that the
product combiner does not ignore information as does the min combiner. The
product combiner measures the joint strength of the antecedent rule firing. It is
to be noted that, in practice, engineers, in the field of fuzzy pattern recognition,
29


find a benefit in using the min if-part combiner. In certain applications, there
is an advantage in normalizing the fuzzy antecedent values by taking the min
value as the firing strength of the rule. They claim that this smoothes the firing
strength in the case of missing input data. This problem does not occur in the
body of this research.
Correlation-product inference describes the manner in which the resultant value
ay(x), of the product if-part combiner or min if-part combiner, is combined with
the fuzzy output membership function Bj of rule j. The then-part combiner
yields the resultant pointwise product Bj = a,j(x.)Bj. This study used correlation-
minimum inference, in the modeling of the two-dimensional undamped pendulum.
Hiew and Tsang also used correlation-minimum inference in their early work
on recursive partitioning [35]. Correlation-minimum inference clips the output
membership functions Bj by computing the then-part rule firing as a pointwise
minimum: a.j(x.)ABj. A key advantage of using correlation-product inference over
correlation-minimum inference is that correlation-product inference preserves the
shape of the output membership functions through scaling.
SAM fuzzy systems use the aggregation operator sum for combining the re-
sultant output membership functions {Bj}j=l, for a rule base of size n. The
pointwise addition of consequent fuzzy sets preserves a unimodal distribution
and the information content that is contained in the overlap of these sets. The
aggregation operator max, on the other hand, tends to create a uniform distrib-
ution of the output membership functions (a rectangular pulse) with a potential
loss of information [49]. The sum aggregation operator also permits the centroids
30


of the output membership functions to be computed independently which may
lower the computational complexity that arises in computing the defuzzified value
of the output [46]. The SAM Theorem of Kosko addresses the computation of
the output centroid in more detail [48]. Three-dimensional chaotic systems are
sensitive to small changes in initial conditions. Consequently, in designing a fuzzy
approximation scheme for a chaotic system, it is important to preserve the un-
derlying shape of output membership functions and the probability distribution
of inputs. Therefore, for system identification of chaotic trajectories, the sum
operator was used in this study for combining fuzzy rules and for computing the
resulting consequent fuzzy set B = Bj.
The SAM Theorem addresses the computational representation of the standard
additive fuzzy models that approximate the two and three-dimensional dynamic
systems in this study. In the statement of the theorem, the membership functions
bj : BP > [0,1] are associated with the then-part fuzzy sets Bj. Each Cj is the
centroid, also referred to as the center of mass or the average value of y =
(?/i,..., yp) in the universe of discourse of the then-part fuzzy set Bj. Each Uj is
a fuzzy rule weight, which in practice is set equal to I. The Pj(x),j = 1,... ,m
are the discrete probability weights and the V} are the finite positive volumes
of the output membership functions bj. What now follows is a statement of the
theorem taken from [48].
Theorem 2.1 (SAM Theorem) Suppose the fuzzy system F : BP > BP is
a standard additive model: F(x) = Centroid(S) = Centroid(^yl1 Wja.j(x) Bj).
31


Then F(x) is a convex sum of the m then-part set centroids:
where
F(x) =
ET=iwJaj(x)vJcj
X^7=l WjCLjix)Vj
m
Y,Pj(x)Cj,
i=i
= WjdjWVj
> 'ZjLi wja^Vj
Vj = f bj(yu...,yp)dy1...dyp>0,
JRP
_ Srp V bj(yi,yP)dy1...dyp
Sr? bj(yu , yP)dy\...dyp
(2.9)
(2.10)
(2.11)
(2.12)
Proof: See [48].
When all the volumes Vj are constant, equation (2.9) reduces to the center of
gravity (COG) rule,
PI= E"-i %(*) which produces a SAM/COG fuzzy system.
(2.13)
2.7 The Standard Additive Model
as a Conditional Expectation
Koskos proof that F(x) = E\Y\X = x\ follows directly from the definition of
Centroid(B) and the definition of conditional expectation of a continuous random
variable [48]. It is reproduced here for clarification of these basic definitions:
Theorem 2.2 F(x) = E\Y\X re].
Proof:
F{x) Centroid(B(x))
32


SZtKx^dy
d- r yPB(y\x)dy
J CO
^ E[Y\X = x]
In the above proof, pg(y\x) defines a proper conditional probability density
function, for which, b(x, y) is the joint probability density function and the mar-
ginal density function of x, over all y in the range of F is b(x,y)dy.
For SAM fuzzy systems with correlation-product inference, the fired fuzzy
sets are B'- = aj(x)Bj. The joint density function &'(x, y) is the membership
function associated with the fired fuzzy set Bj. The components of Pj(x) in
Theorem 2.1, have been combined to express Pj as a convex combination of the
volumes of the fired fuzzy sets Bj. The following theorem of Kosko, taken
from [48], expresses a SAM fuzzy system as the convex sum of local conditional
means or centroids.
Theorem 2.3 Suppose F : BF R is an additive fuzzy system such that
F(x) = Centroid(B) and B(x) = WjBj(x). Then
Fix) = f^Pj{x)EB,[Y\X = x], (2.14)
i=i J
where X and Y are the random variables that represent the respective domain
and range spaces of F. The coefficients pj(x) are convex weighted volume ratios
of the fired then-part fuzzy sets By.
Pj > 0 andy^p7 (x) = 1 (convexity property)
j=i
33


Vj(x) = J bj(x,y)dy.
Proof: See [48].
An important consequence of Theorem 2.3 follows from the fact that the SAM
fuzzy logic system F(x), for a given input x, can be expressed as a convex sum of
conditional expectations, which are the centroids of the then-part fuzzy sets Bj.
This property will be useful in the design of an adaptive fuzzy logic controller.
Chapter 6 introduces the indirect adaptive fuzzy rule design. The linearity prop-
erty of conditional expectations will allow the splitting of the consequents of the
fuzzy rules in the formation of two distinct fuzzy rule bases for control.
Another consequence of Theorem 2.3 is that a centroid additive fuzzy system,
F : FF > R, computes an optimal nonlinear mean square estimator. Lemma 2.1
provides the key to the above result. Lemma 2.1 and its proof are derived from
Papoulis discussion concerning optimal mean square estimation and conditional
expected values [64].
Lemma 2.1 The optimal nonlinear mean square estimation of a random variable
Y by a function c(x) over all values x of a random variable X, is
/oo
yf(y\x) dy.
-oo
Proof: To estimate Y by a function c(x), we must find c(x) such that the mean
square value (MS) of the estimation error that satisfies
e = E([Y c(x)]2) = f f [y c(x)]2f(x, y)dxdy (2.15)
J oo J OO
34


is minimum. The joint density distribution f(x,y) = f(y\x)f(x), hence
e = [ f(x) [ [y c(x)]2f(y\x)dydx. (2-16)
J CO JOO
Sincef(x) is a probability density function, the integral f(x)dx = 1 > 0. We
can, therefore, find the minimum e if the inner integral is minimum for every x.
For a fixed x, let c(x) = c. The MS e is minimum for a fixed x if
J oo
is minimum. The inner integrand, I(y,c), is continuous in both variables, y and
c and ¥ is continuous. Then using Leibnitzs rule,
o=£=IZk[v~^mx)dy
-oo
roo
It follows that
= f 2(2/ c)f(y\x)dy
J OO
/OO
{y c)f(y\x)dy.
OO
/oo roo
yf{y\x)dy = c f(y\x)dy
-oo J oo
(2.17)
(2.18)
(2.19)
(2.20)
and
f-ooyf(y\x)dy f-ovf(y\x)dy */ rvl , u
= 'g. 7fiiiw =---------1------= mx =CZ21)
/-oo /(yk)*/
Since a; is arbitrary, this proves that the conditional mean,
/oo
yf(y\x)dy,
-OO
is a critical point of the mean square error e Vrc. To show that for a fixed x, c is
a local minimum of e, consider the second partial derivative d2^,c):
f jr(~2(y ~ c))f(y\x)dy = 2 f f(y\x)dy = 2*1 = 2>0. (2.22)
J oo C7C J-oo
35


The above argument holds for all x. Therefore, the function,
c(x) = E[Y\X = x]= f yf(y\x)dy,
J oo
is the optimal nonlinear estimator of Y in the mean square sense.
The function c(x) = E\Y\X = x\ = y f(y\x)dy is known in the literature
of probability theory as a regression line [64]. Figure 2.9 shows an example of
a regression line c(x). For a given x, the integral yf{y\x)dy is the centroid,
also known as the center of mass, in the vertical strip (x, x + dx).
Figure 2.9. Regression line c(x)
The following corollary to Theorem 2.2 states that an additive centroid fuzzy
system F : RJ1 R computes the optimal nonlinear mean square estimator F(x)
of a random variable Y.
36


Corollary 2.1 Suppose F : BP )- R is an additive fuzzy system such, that F(x) =
Centroid(B) and B{x) = YJjLi WjBj(x). Then F(x) is the optimal nonlinear mean
square error estimator of a random variable Y.
Proof: From Theorem 2.2, it follows that F(x) = E[Y\X = x], where the random
variable Y represents the range of values over which the fuzzy set B is defined.
Therefore, it is an immediate consequence of Lemma 2.1 that F(x) is the optimal
nonlinear mean square error estimator of Y.
2.8 Fuzzy Interpolation using
SAM Fuzzy Systems
Theorem 2.3 states that F(x) = jyjLiPj(.x)-E*#'\Â¥I^ = x\- By definition,
Cj Eb>[Y\X = x] is the centroid of the fired then-part fuzzy set Bj of
rule j. The weights, pj, are convex. That is, Pj{x) > 0 and YJj=\Pj(x) = 1-
Therefore, for a given input x, the weights pi(x), ...,pm(x) define a probability
density function, p(x), over the centroids Cj. Let C = (ci,..., Cm) be the vector of
centroids over the m frizzy rules of the fuzzy rule base. It follows, from Theorem
2.3, that for each input x, the SAM fuzzy system F{x) can be expressed as the
expected value of the m output centroids Cj with respect to the convex weights
pi(x),...,pm(x):
171 r JT
F(x) = -£Pj (rr)c,. V £p(l) [C\. (2.23)
i= i
The SAM fuzzy logic system, for a given input x, interpolates F(x) between the
m output centroids of the fuzzy rules. Kosko shows [48] that the output F(x) is
bounded in the scalar case by the least and greatest centroid value:
Ci < F(x) < Cm.
37


Kosko also shows that for the vector case F : BP ¥ BP, F(x) lies in a p-
dimensional centroid hyperbox. Each component, Fk(x) of F(x), is a convex
combination of the m vector components c£,..., cjj*, for the m fuzzy rules. The
property that
F(x) £ [Cic/t> X--------X \^cfti ^-ighti
holds for SAM fuzzy systems with correlation-product inference, Bj = a.j{x)Bj,
since the fired then-part set Bj has the same centroid as the unfired set Bj. In
the minimum-inference case, Bj min(aJ(x), B_,), the centroids of the fired and
unfired then-part fuzzy sets may not be the same and the interpolation property
may not hold. Kosko uses this interpolation-between-centroids property of F(x),
in his proof of the fuzzy approximation (FAT) theorem. This theorem states
that a SAM fuzzy system uniformly approximates a continuous function on a
compact space. Chapter 5 will formally state the FAT theorem in the context of
e-shadowing of (5-pseudo-orbits of chaotic systems.
This interpolation property between centroids may not hold in systems that
use aggregation operators other than sum. This gives further reason for using a
SAM centroidal system with correlation-product inference over other fuzzy logic
schemes.
2.9 Analytic Justification for Combining
Multiple SAM Fuzzy Systems
Prior to running the bisection and homogeneity algorithm for system iden-
tification and rule base construction, one may decompose the state space into
independent and disjoint computational units. This decomposition is referred to
38


as domain decomposition. Chapter 3 addresses the properties of domain decom-
position and Chapter 4 shows the results of using it for the fuzzy modeling of
chaotic systems. Domain decomposition is effective in reducing the computational
complexity of running a recursive partitioning algorithm. Domain decomposition
plays a key role in the fuzzy modeling of chaotic systems using SAM fuzzy infer-
ence procedures and rules. Its success, in a SAM environment, is due to the fact
that one can combine multiple SAM fuzzy systems to form a single SAM fuzzy
system. Such a decomposition may also strengthen the structural integrity of the
fuzzy rule base and interpolation procedure. The following Corollary to Theorem
2.1 states that two constant volume SAM fuzzy systems can be combined to form
one constant volume SAM fuzzy system.
Corollary 2.2 Let F(x) and G(x) be constant volume SAM fuzzy systems.
Then H = (F + G)(x) is a constant volume SAM fuzzy system.
Proof: Let
... Si-i *j(x)c,-
F(x) SJW*)
and
G(x) =

where vector x; cy is the centroid for the jth output membership function; and p is the
number of rules in the fuzzy rule base for F(x). Analogously, 6fc(x) is the product
if-part combiner for the Ah rule and the current input vector x; d* is the centroid
for the £th output membership function; and q is the number of rules in the fuzzy
39


rale base for C?(x). Let
and
Pj(x) =
Qj(x)
Qfe(x) =
bk(x)
Then F(x) = X)y=i P,-(x)cy and G(x) = Qk(x)dk- It follows that
p-r?=r
(F + G)(x) = F(x) + 1=1
where -R/(x)ej = F/(x)q for Z = 1,2, ...,p and Ri(pz)ei = Qi(x)di for Z = p + l,p 4-
2, ...,p+g. Each Ri(x) is the normalized product if-part combiner of the Zth rule.
Each e; is the centroid of the output membership function of the Zth rule.
The above argument holds for the generalized SAM fuzzy models F(x) and
G(x). The combined fuzzy logic system is a generalized SAM model H = (F +
G)(x). For a generalized SAM, P,(x) = and Qk(x) = ^MVV t where
Vy and Vk are the volumes of their respective then-part fuzzy sets.
Corollary 2.2 justifies the combining of multiple SAM fuzzy systems for trajec-
tory reconstruction over multiple disjoint subdomains. Several methods for com-
bining SAM models have been suggested by Kosko [48]. To construct {f?((x)}, Z =
1,..., r as a discrete probability distribution, such that X) Ff = 1, and H(x) as
a weighted average of SAM systems, let H(x) = |(F(x) + G(x)). The above
corollary applies inductively to any finite number, n, of SAM systems. For a
weighted average of n SAM fuzzy systems, H(x) = ^ XJLi F;(x).
40


2.10 Conclusions
This chapter has described the mathematical foundations of fuzzy modeling for
system identification, parameter estimation, and indirect adaptive fuzzy control.
Theorems of this chapter form the basis of fuzzy modeling and domain decompo-
sition in the remaining chapters of this dissertation. Corollary 2.2 was a major
result of this chapter. It offered a formal justification for the process of domain
decomposition of state space descriptions by showing the additivity property of
SAM fuzzy models. This chapter also described the role that fuzzy modeling
plays in a unified approach to system identification.
An avenue of future research would be to investigate the connectivity on the
boundaries of subdomains. In the example of the Lorenz attractor in Chapter 4,
the state space was split into eight one subdomains. Three of the subdomains
required a small overlap to assure rule firing. For the chaotic pendulum example
of Chapter 4, the drive cycle was split into disjoint subdomains. The goal of a
future investigation would be to refine the size of the rule base, especially on the
boundaries of the subdomains. On the boundary, the combined rule firings may
violate a strict probability distribution of normalized if-part combiners of fuzzy
rules. The problem of refining nodes on the boundaries of subdomains has been
explored by Zipser and Rumelhart in the context of competitive learning winner
take all networks [97]. Further investigation may lead to the formulation of a new
algorithm for reducing both the size of the rule base and the computational time
for running the bisection and homogeneity algorithm across subdomains.
41


3. Fuzzy Modeling of Complex Systems
3.1 Introduction
This chapter describes the bisection and homogeneity algorithm for structural
and parameter identification. It reviews the modifications that the bisection and
homogeneity method has made to the recursive partitioning algorithm of Hiew
and Tsang [35]. Recursive partitioning methods are derived from a mixture of
statistic and heuristic arguments. Such methods are known to give impressive re-
sults in simulations [42, 13]. Unlike recurrent neural networks that also partition
the input space for modeling nonlinear dynamics [56, 70], the domain decomposi-
tion approach, which is used in conjunction with the bisection and homogeneity
algorithm, curtails a computational explosion and creates a bound on the training
time.
Section 3.2 presents the architecture of the bisection and homogeneity algo-
rithm. Section 3.3 describes the modifications that the algorithm has made to
the recursive partitioning approach of Hiew and Tsang. Section 3.4 gives a top
level pseudo-code description of the algorithm. Section 3.5 traces the algorithm
step by step through a two-dimensional example. Section 3.6 describes the heuris-
tic method of choosing homogeneity parameters. Section 3.7 describes the details
of the algorithms method of constructing fuzzy rules. Section 3.8 supplies a gen-
eral overview of domain decomposition and lists its salient features. Section 3.9
42


details the steps of trajectory reconstruction. The final two sections of the chap-
ter discuss avenues for future research and summarize the major contribution of
the bisection and homogeneity algorithm to the research on fuzzy modeling.


3.2 The Bisection and Homogeneity Algorithm
for Structural Identification
Figure 3.1 gives a summary of the key features of the bisection and homo-
geneity algorithm. The first step in modeling a complex system is structural
identification. The following structural identification method creates a piecewise
linear decomposition of the state space. It is a modification and an extension of
the recursive partitioning algorithm of Hiew and Tsang [35].
The Bisection and Homogeneity
Algorithm for Structural and
Parameter Identification
A product-space clustering method.
Recursive partitioning based upon
work of Hiew and Tsang (1994).
Generates a fuzzy rule base for each
dimension of the state space.
Generates one fuzzy rule for each
homogeneous direction of a region.
Constructs a continuous model of the
state space.
Models chaotic regions more finely
than regions with more regular
dynamics._________________________
Figure 3.1. Key features of bisection and homogeneity algorithm.
The algorithm presupposes the ability to decompose a complex system into
44


piecewise linear subsystems [25]. Processing begins with a grid of equally spaced
sample points in a bounded domain. Each sample point is an ordered n-tuple
of system states, where n is the number of dimensions of the state space. Input
into the algorithm also consists of a list of n-tuples of time rates of change for
each sample point. For a variety of applications, the n-tuples of rates of change
translate to measurements of velocity and acceleration [37]. The output of the
bisection and homogeneity algorithm is a collection of fuzzy rule bases, one for
each dimension of the state space. Figure 3.2 describes the components of the
bisection and homogeneity algorithm.
Figure 3.2. Architecture of Generate-Rule(r,d) procedure.
For a specified dimension d, the Generate Rule(r,d) procedure considers all
samples in the bounded region r and compares the rate of change (along dimen-
45


sion d) of each sample with the rates of change of all other samples in that region.
If the rates of change of any two samples are significantly different, the region r is
not homogeneous along its dimension d. In this case, the Split Region(r,d) sub-
routine splits the region along the nonhomogeneous dimension. In its next pass,
Generate Rule(r,d) determines homogeneity in the two newly created regions
along the next modulo (d 4-1) dimension to be considered. Once the algorithm
finds a region that is homogeneous along a given dimension, it calls Construct
Rule(r,d) which creates one rule for that region. Generate Rule(r,d) resets
the dimension d of the region for which it will determine the regions homogeneity
on the regions next pass through the algorithm. The algorithm stops bisecting a
nonhomogeneous region, along a given dimension, once there is only one sample
point remaining in the region or the bound on the width along that dimension is
less than some predefined tolerance, 7. During each pass, Generate Rule(r,d)
determines the homogeneity of regions, splits nonhomogeneous regions and con-
structs rules for the homogeneous ones. The algorithm terminates when no more
nonhomogeneous regions are found.
The algorithm makes an assumption that experimental data is available at any
point of the state space. In the case of the pendulum with movable periodic
support, one can make this assumption since the problem can be modelled. Ex-
perimental pendulum data can closely be matched to a numerical simulation of
a nonlinear Mathieu equation [78, 84].
46


3.3 Modifications to Hiew and Tsangs
Recursive Partitioning Algorithm
There are seven main differences between the bisection and homogeneity algo-
rithm, developed in this dissertation, and the recursive partitioning algorithm of
Hiew and Tsang.
The first difference is that the bisection and homogeneity algorithm modifies
Hiew and Tsangs work by recognizing a bound on the size of intervals and by
allowing bisection to take place along a dimension other than the nonhomogeneous
direction. When the length of the interval to be split is less than a pre-defined
tolerance an alternate direction is chosen for splitting.
The second difference pertains to the modeling of autonomus systems. For
autonomous systems, the bisection and homogeneity algorithm refines the original
grid of samples by computing new samples for centers of empty regions. Hiew
and Tsangs method of recursive partitioning does not describe or commit itself
to a procedure for dealing with empty regions.
The next difference refers to the fact that the bisection and homogeneity al-
gorithm allows homogeneity parameters to be set independently for each state
space dimension. It also describes a heuristic, based upon histograms of rates
of change, for determining homogeneity parameters x and r. Hiew and Tsangs
recursive partitioning algorithm employed a fixed homogeneity parameter x for
determining homogeneity over all dimensions of a region. Their research did not
describe a procedure for choosing the homogeneity parameter.
The fourth difference pertains to rule base explosion. The current research
47


presents a method for combating rule base explosion by using multiple SAM sys-
tems to reconstruct trajectories on smaller segments of the state space. This
method is called domain decomposition. Chapter 2 gave the mathematical justi-
fication for combining multiple SAM fuzzy models.
Another difference which creates an advantage to the modified bisection and
homogeneity algorithm is that domain decomposition of the state space imposes
a temporal ordering of the fuzzy rules. A temporal ordering of rules is necessary
for the modeling of nonautonomous systems. Hiew and Tsangs work did not
address algorithmically the problems of rule base explosion and the modeling of
nonautonomous systems.
The fuzzy modeling approach of this research determines an explicit method for
trajectory reconstruction using fuzzy rules. This modeling approach is believed
to be the first to use a variable step size Eulers method as an integrator for
trajectory reconstruction. The conditional expectations of time rates of change
for each component of the vector field are used to approximate the first deriva-
tives in Eulers method. Hiew and Tsangs work does not address the method
of reconstruction of trajectories once the rates of change are computed by the
centroid defuzzification procedures of the fuzzy logic systems.
The final difference is that this research extends the recursive partitioning
structural identification method of fuzzy modeling to a closed loop system for
indirect adaptive fuzzy control. The closed loop system creates a feedback fuzzy
logic system which has the effect of limiting rule base explosion by modifying
parameters within each fuzzy region of the state space. This topic will be explored
48


fuzzy in Chapter 6 and Chapter 7 of this dissertation.
The advantages of all these modifications are that computational rule base ex-
plosion is contained, temporal ordering can be imposed on the fuzzy modeling
of periodic dynamical systems, and parallel processing of fuzzy regions is also
achievable. Modelers will find that given an explicit method for trajectory recon-
struction, the rapid deployment of the bisection and homogeneity algorithm for
the fuzzy modeling of nonlinear dynamical systems is possible. Section 3.4 gives
a top level pseudo-code description of the algorithm
49


3.4 Top Level Pseudo-code Description
of the Bisection and Homogeneity Algorithm
The following is a top level pseudo-code description of the bisection and homo-
geneity algorithm:
Input: n-tuples of sampled points in the n-dimensional state space, n-tuples of
rates of change for the corresponding sampled points, regions r, defined using the
coordinates of end points of intervals along each dimension of the state space, the
dimension d, for testing homogeneity, associated with each region r.
Output: n fuzzy rule bases, one for each dimension of the state space.
While more nonhomogeneous regions
While more samples s
For each region r
Match(sample s,region r)
If match(s,r)
Determine-Homogeneity(r,d)
End
End
End
While more regions r
If region(r,d) is homogeneous
Construct-rule(r,d)
Set next appropriate dimension d for region r
Else
Split-Region(r,d)
End
End
End
Once the algorithm identifies a region r as being homogeneous along a partic-
ular dimension d, the dimension d is reset to a dimension of region r that has not
yet been tested for homogeneity. In the general case, the algorithm splits a non-
homogeneous region along the new (d = d+1) modulo (n) dimension. An interval
50


length tolerance parameter, 7, is set prior to running the algorithm. When the
length of a interval to be split is less than 7, the algorithm considers an alternate
dimension for splitting. The Split-Region(r,d) procedure has been implemented
slightly differently for the autonomous and nonautonomous dynamic systems that
this research has studied. For the autonomous examples which include the Van
der Pol oscillator and the Lorenz attractor, splitting takes place at the mid-point
of the interval to be split. This is a true bisection of the interval. As a result,
empty regions may be created. The Split-Region(r,d) procedure computes one
sample for the center of the empty region. For the nonautonomous example of
the parametrically perturbed pendulum, splitting takes place along the average
d coordinate value of samples in that region. No new samples are generated and
the input grid of sampled points remains fixed throughout the running of the
algorithm.
51


3.5 Two-Dimensional Example of
Bisection and Homogeneity Algorithm
This section gives insight into how the bisection and homogeneity algorithm
works. It illustrates the steps of the algorithm for a two-dimensional state space
example. Figure 3.3 illustrates structural and parameter identification, using
the bisection and homogeneity algorithm, for the construction of the antecedents
of fuzzy rules that model a two-dimensional state space. The figure shows a
structural decomposition of the state space into four core (disjoint, non fuzzy)
regions.
Figure 3.3. The generation of fuzzy rules with two antecedents.
For Figure 3.3, the algorithm generates these four regions and their respective
fuzzy overlapping regions,and fuzzy rules, depending on the homogeneity of the
input data, as follows:
1. On first pass through the algorithm, the entire bounded domain represents
52


region r and the dimension d = 1 represents the dimension along the y
axis. The region r is equivalent to the pair of intervals, [min x, maxx] and
[min y, max y].
2. Sampled points are matched with region r. Suppose a sample point is
found whose time rate of change along the ^/-coordinate is significantly
different than the time rate of change along the ^-coordinate of another
sample point in region r. The region is not homogeneous with respect to
the d = 1 dimension.
3. The Split-region(r,d) procedure splits region r along the nonhomogeneous
y axis. The dimension d is set to (d +1) modulo 2 = 0, which represents the
state space along the x axis.
4. The algorithm on the second pass takes the two split regions as input. It
determines the homogeneity of each region along the x axis.
5. Suppose a region rl is found to be homogeneous along the x axis. The Con-
struct Rule(rl,d) procedure constructs a symmetric triangular member-
ship function, A)., as an antecedent for the dimension along the x axis and
a symmetric triangular membership function, A^, as an antecedent for the
dimension along the y axis. The base of the triangle along the x(y) dimen-
sion extends beyond the interval [minx,maxx] ([miny,maxy]) according
to a pre-defined overlap rate. The consequent fuzzy membership function,
is constructed for the rule, based upon the minimum, maximum and
average time rates of change, of samples in region rl. This procedure
53


results in rule Rlx =< A^, Ay, Cl > This rule will enter the fuzzy rule base
representing homogeneous rates of change along the x axis. The dimension
d is set to {d + l)modulo 2 = 1. Region rl will be tested for homogeneity
along the y axis on the next pass through the algorithm.
6. Suppose region r2 is not homogeneous along the x axis. Split-region(r2,d)
splits r2 along the x axis into regions r21 and r22. The dimension d is set
to 1 for r21 and r22 (to determine homogeneity for r21 and r22 along the
y axis on the next pass through the algorithm).
7. On the next pass through the algorithm, suppose rl is found to be homo-
geneous along the y axis. Rule Rly =< Bl, B^, > will be added to the
fuzzy rule base representing homogeneous rates of change, along the y
axis of region rl. The algorithm will determine that both dimensions of
region rl have been tested for homogeneity and found to be homogeneous.
As a result, processing will stop for region rl. During this pass, suppose
it is found that r21 is homogeneous along the y axis. The algorithm will
generate a rule R2y =< B%, By, C% > covering the homogeneous region r21
along the y axis. The dimension d will be set to 0 for the region-dimension
pair < r21, d >. During this pass, suppose the algorithm determines that
region r22 is not homogeneous along the y axis. The algorithm will split
the region into regions r221 and r222. The dimension d will be set to 0 for
the next iteration through the algorithm for these two new regions.
8. On the next pass, suppose regions r21, rr221, and r222 are found to be
54


homogenous along the x axis. The algorithm will construct three rules
to be added to the fuzzy rule base for homogeneous rates of change
These rules are: R2X =< A%, C% >, RZX =< A%.,A^,CX > and R4X <
A*, A^, Cx >. These rules cover regions r21, r221 and r222, respectively.
9. On the final pass, suppose the algorithm has found regions r221 and r222
to be homogeneous along the y axis. The algorithm will construct rules
RZy =< B, By, Cy > and R4y =< B*, B*, C* > for regions r221 and r222,
respectively. Processing terminates for regions r221 and r222. No more
incompletely processed regions exist and the algorithm terminates.
The output of the algorithm yields two fuzzy rule base systems, each with
four fuzzy rules. Note that it is not always the case that both fuzzy rule bases
have the same number of rules. The examples of the Van der Pol oscillator, the
Lorenz attractor and the chaotic pendulum will demonstrate the different rule
base sizes in Chapter 4. For instance, if a region R is homogeneous along the
x axis and not homogeneous along the y axis, and the x axis has been tested
first for homogeneity, the algorithm will construct one rule for region R for the
x homogeneous fuzzy rule base and split region R into two subregions. Suppose
both subregions are homogeneous along the y axis. Then the algorithm will
construct two rules for the y homogeneous fuzzy rule base. In this scenario, the x
rule base has one fuzzy rule, covering region R, and the y rule base has two. This
scenario points out one of the basic features of the bisection and homogeneity
algorithm. The algorithm creates fuzzy rules only to the granularization level
55


needed and no further. As a result, chaotic regions are more finely partitioned
and are represented by more fuzzy rules than regions that exhibit more regular
dynamics. Generally, for dynamical systems that are represented by position,
velocity, and phase coordinates, sufficient knowledge is available, to suggest to
the modeler which dimension of the problem may be exhibiting the chaotic or
strongly nonlinear dynamics. The modeler would set up the round robin search
of homogeneity coordinates, such that the position coordinate, which reflects, say,
the more regular dynamics (i.e. x = y) of a grid of position-velocity-phase sample
points, would be tested first. The chaotic parametrically perturbed pendulum is
one such example of this behavior. The experiments reported in Chapter 4 are
for the fuzzy modeling of a chaotic regime of the pendulum that exhibits chaotic
behavior in terms of the time rates of change of the y coordinate of angular
velocity. For the fuzzy modeling of the chaotic pendulum, the bisection and
homogeneity algorithm tests the coordinates in a round robin manner starting
with the x coordinate, of angular position, thus achieving the finer partitioning
along the y coordinate axis.
3.6 Heuristic for Choosing Homogeneity Parameters
The rates of change, rl and r2 (|rl| < |r2|, |r2| 0), of two samples axe con-
sidered significantly different if the signs of the rates of change differ or
|rl|/|r2| < x and |rl r2| > r,
56


where x and T are pre-set parameters of the algorithm.
The heuristic method for determining the homogeneity parameters eliminates
the necessity of examining the raw data directly. One inspects the histograms of
rates of change for each dimension of the state space. Since the sample space is
finite, as the partitions become finer, the histograms represent limiting probability
distributions of rates of change. In Chapter 4, histograms are presented for the
examples of the autonomous Van der Pol oscillator and the Lorenz attractor and
for the nonautonomous chaotic pendulum.
Section 3.7 provides a more detailed analysis of the Construct Rule procedure
of the bisection and homogeneity algorithm.
3.7 Construct Rule Procedure
Fuzzy rule construction consists of parameterization of antecedent and conse-
quent membership functions. Each antecedent represents one dimension of the
state space and is described as a symmetric triangular membership function. The
three parameters that determine each membership function are the maximum and
minimum coordinate values of the interval boundary of the homogeneous region
and the mid-point of the range of boundary values. The algorithm uses the
predefined overlap rate to extend the length of the base of the symmetric mem-
bership function, thus creating a fuzzy covering of the region. The overlap rate
is automatically adjusted in cases where the length of the base of a triangular
membership function is less than a certain predefined constant. The extension of
the overlap rate was necessary in order to assure proper rule firing. Rule construc-
57


tion begins once the Generate-Rule(r,d) procedure identifies a homogeneous
region. Figure 3.4 describes the Construct-Rule(r,d) procedure.
RULE (Aj,), Apj > A pj, D)
Figure 3.4. Architecture of the Construct-Rule(r,d) procedure.
58


Figure 3.5 gives a pseudocode description of the Construct Rule procedure:
Construct-rule(r,d):
For i = 1 :n {
Construct A Aifcj^.c^cj+h)
>
Construct A D(dj-o,di,di+o)
Ai symmetric triangular (A) fuzzy
antecedent
D symmetric triangular ( A) fuzzy
consequent
h overlap rate along intervals of
region r
dj average rate of change of samples
in region r
o, a parameter that sets base length of
triangular membership function to 1.
n, the number of state space
dimensions
Ci, the midpoint of interval of homo-
geneous region along dimension i
Figure 3.5. Pseudo-code description of Construct-Rule(r,d).
Inputs into the rule construction process are the homogeneous dimension d
and the range of coordinate values [mini, maxi] along each dimension, i, of the
homogeneous region. The overlap rate, h, is a global input to the system. The
output of the rule construction process is a fuzzy rule in one of the following forms:
If x is A and y is B, then c is C
If x is A and y is B and z is C, then d is D
59


where x, y, and z are coordinates of the state vector and A, B, C and D are fuzzy
sets. The rule with two antecedents is for two-dimensional dynamical system
modeling and the rule with three antecedents is for three-dimensional dynamical
systems.
The consequent of the fuzzy rule is also modeled as a symmetric triangular
membership function. The three parameters that determine the consequent mem-
bership function are the average rate of change and the maximum and minimum
rates of change (along the homogeneous dimension d) within the region. As in
the case of the antecedent membership functions, the algorithm automatically
extends the length of the base of the triangle when the length is below a certain
threshold in order to obtain a dense rule cover.
The generation of fuzzy rules by the bisection and homogeneity algorithm is
directly effected by the grid size, by the size of the data set used in addition to the
grid, by the dimension of the state space, and by the definition of homogeneity
that may be used for a particular application. To reduce the computational
complexity of the algorithm, one may impose a computational decomposition of
the state space that is independent of the bisection and homogeneity algorithms
system and parameter identification process. Section 3.8 addresses the domain
decomposition of the state space to achieve computational efficiency.
60


3.8 Domain Decomposition over
Multiple SAM Fuzzy Systems
Domain decomposition refers to splitting the state space into disjoint subdo-
mains. The algorithm runs independently on each component of the decomposi-
tion. For autonomous systems, domain decomposition reduces the computational
complexity of the algorithm. For nonautonomous systems, domain decomposition
also imposes a temporal ordering on fuzzy rules. A goal in modeling dynamical
systems with periodic forcing or excitations is to preserve the portrait of state
space descriptions (the direction field) over multiple drive cycles. Domain decom-
positions, along the phase axis, accomplish this goal. Section 2.9 presented the
analytic justification of the domain decomposition approach and Corollary 2.2
proved a linearity property for the additive combining of SAM fuzzy systems.
Chapter 4 will describe in detail domain decomposition of the Lorenz attra'.'r .or
into eighty-eight three-dimensional subdomains. This decomposition reduces the
computational complexity of running the algorithm as each subdomain can be
processed independently of other subdomains. Chapter 4 will also describe do-
main decomposition of the periodically perturbed pendulum into quarter phases
of the drive cycle. This decomposition constructs a temporal ordering of fuzzy
rules and allows finer grids of sample points to be used to improve approximation
accuracy and at the same time reduce the computational complexity of running
the algorithm.
The major attributes of domain decomposition can be summarized as follows:
In the first place, by splitting the state space into disjoint subdomains, domain
61


decomposition reduces the computational complexity of rule base construction.
Secondly, the bisection and homogeneity algorithm runs independently on each
component of the decomposition. This allows for the parallel processing and
modularization of the algorithm over the entire state space. Modularization al-
lows one to examine and model the dynamical behavior in local regions of inter-
est. Thirdly, the decomposition allows for the modeling of both autonomous and
nonautonomous dynamical systems. Fourthly, domain decomposition, along the
phase, = a>t, axis, creates a temporal ordering of the fuzzy rules that model
nonautonomous nonlinear dynamics. As shall be seen in Chapter 4, in the case
of the parametrically perturbed pendulum, the decomposition identifies the cycle
number or phase partition. Finally, for the fuzzy modeling of periodic dynami-
cal systems, domain decomposition allows for finer grids, without increasing the
computational complexity. Finer grids preserve the portrait of the state space
over multiple drive cycles.
The final step in the fuzzy modeling of a dynamical system is trajectory re-
construction. Section 3.9 describes the method of combining independent SAM
fuzzy systems (one for each dimension of the state space) to produce the numerical
trajectory.
3.9 Trajectory Reconstruction
Trajectory reconstruction is a two-step procedure. The first step processes
the initial state vector through the inference engine of each fuzzy logic system.
The defuzzified output of the inference procedure is a vector of average time
62


rates of change, one for each state of the system. This vector of rates of change
supplies the coefficients for a variable step size Eulers integrator. Eulers method
computes the next iterate of the state vector for input into the SAM fuzzy systems.
Figure 3.6 describes the architecture of fuzzy trajectory reconstruction for a
three-dimensional dynamical system, using three SAM/GOG fuzzy models.
Figure 3.6. Architecture of fuzzy trajectory reconstruction.
The input vector, x = (x, y, z). Each state variable generates its own SAM
fuzzy system. Recall from Chapter 2, section 2.5, the fuzzy model performs
63


centroidal defuzzification and satisfies:
_ aj(.x)Yjcj /-o i \
(*)Vj 1 '
where each then-part consequent fuzzy set Bj has area or volume Vj and centroid
Cj and where p is the number of fuzzy rules in the SAM fuzzy system. The
SAM fuzzy system consists of centroid defuzzification, product if-part combiner,
correlation-product inference, and the aggregate sum operator as described in
Chapter 2. The output vector,
E(dx)/dt) == (E(dx/dt\x)E(dy/dt\x). E(dz/dt\x)), (3.2)
the expected time rate of change along each dimension of the state space. The
following subsections will describe these steps in detail.
3.9.1 Step One: Correlation-Product Inference
Each fuzzy logic system is an additive fuzzy system that uses correlation-
product encoding [46]. This refers to the fired consequent fuzzy set, B', of a fuzzy
rule as the pointwise product of the value of the if-part combiner, a(x), of the
fuzzy rule and the consequent fuzzy set, B. Correlation-product encoding pre-
serves the shape of the output membership functions through scaling. Fuzzy rule
bases may use the standard intersection, min, or the product if-part combiner, to
combine antecedents of fuzzy rules that are joined linguistically with conjunctive
and. The inference engine uses the aggregation operator, sum, for the follow-
ing reason: Three-dimensional chaotic systems are sensitive to small changes in
initial conditions. The pointwise addition of consequent fuzzy sets preserves a
64


unimodal distribution and the information content that is contained in the over-
lap of these sets. The aggregation operator, max, on the other hand, tends to
create a uniform distribution of the output membership functions (a rectangular
pulse) with a potential loss of information, according to Kosko [49]. Hiew and
Tsang in their research, used a min if-part combiner and correlation-minimum
encoding in their early w'ork [35]. Correlation-minimum encoding clips the out-
put membership function with the value of the antecedent if-part combiner of a
fuzzy rule by taking a pointwise minimum. In their later work [36], they used
correlation-product encoding for firing the consequent fuzzy rules. In modeling
chaos, one wishes to maintain the shape of the output membership functions
and the direction of the input vector of antecedent membership functions [48] to
preserve the information content [21] of the input distribution and the output
signal. Koskos SAM Theorem [48], Theorem 2.1 of Chapter 2, shows that this
encoding method permits the centroids of the output membership functions to
be computed independently, which may lower the computational complexity that
arises in computing the defuzzified value of the output [46].
Eulers method takes as input the vector (3.2) of conditional expectations and
uses it to provide the final step of trajectory reconstruction. Section 3.9.2 de-
scribes this method.
3.9.2 Step Two: Variable Step Size Eulers Method
Domain decomposition and combination of fuzzy rule bases for functional ap-
proximation have also been studied by J.Yen et.al [94]. Their approach explored
the combining of TSK models and the use of a neural network as an integrator.
65


Their objective was to combine fuzzy systems in such a way that the output error
was minimized in the least-squared sense. The proposed fuzzy modeling method,
of this dissertation, maintains an unsupervised learning approach in order to ex-
plore the fuzzy modeling of functions that are not completely understood or well-
defined. Each fuzzy SAM system undergoes defuzzification independently. The
outputs of the defuzzification processes are fed into a one-step Euler integrator
which determines the value of the state variables for the next iteration.
The output of an iteration through each of the fuzzy logic systems is a vector
of average rates of change with respect to time. Each component of the vector is
the defuzzified output from its respective states rule base. Using the center of
area defuzzification method, these average rates of change represent the expected
values of the slopes of the trajectory at a given time step. A variable time step
Eulers method was implemented in this research to perform the reconstruction.
Eulers method is a stable method of trajectory reconstruction [95].
The variable time step used in Eulers method is
_ incr
denom1
where incr is a predefined constant and denom, for the three-dimensional Lorenz
attractor and the chaotic pendulum, is the maximum of the absolute values of the
outputs (slopes) of the defuzzification process. For the two-dimensional examples,
incr was set to 0.001 and for the three-dimensional dynamical systems, it was
set to 0.01. For the two-dimensional examples, denom was chosen as the absolute
value of the conditional expectation of time rate of change of x. The three-
dimensional case preassigns an hmin for a minimum time step of 0.001 and an
66


hmax for a maximum time step of 0.01. An iteration of the numerical method
first evaluates the fuzzy rule bases for the current state vector (xniyn,Zn). The
variable outx, outy and outz are the defuzzified outputs of the respective x, y and
z rule bases. Next, the method computes the step size h:
h zncr
denom
h = ~r~------if h < hmin
denom
h = hmax if h > hmax
Finally, it computes the iterates of x,y and z:
xn+i = h* outx + xn
Vn+i = h* outy + yn
Zn+1 = h* outz + Zn
where outx, outy, and outz are the expected values of the rates of change (the
slopes) of their respective dimensions.
The process of trajectory reconstruction is now complete. Chapter 4 will give
specific examples of applying the above method to two and three-dimensional
dynamical systems.
3.10 Avenue for Future Research
For evaluation of the fuzzy model, once could use the Generalized Cell Mapping
method (GCM) of Hsu [39]. This method of evaluation has been employed by
researchers in the evaluation of fuzzy models for control [75, 17]. The resulting
structural decomposition of the bisection and homogeneity algorithm could be
67


used as a non-uniform grid of cells for the GCM. The GCM could help furnish
meta-rules for rule size reduction and fine turning a feedback fuzzy logic system
based upon the bisection and homogeneity algorithm.
3.11 Conclusions
A non-neural fuzzy rule base approach allows for the incremental addition of
human expertise in the structural design and subsequent refinement of a com-
plex system. It is also an adaptive approach; rule generation depends directly
on the nature of the data. The strength of this structured decomposition is that
subsystems with strong nonlineaxities are modeled more finely than subsystems
with more regular dynamics. Constructing fuzzy rules to model acceleration and
damping-velocity' in this fashion, one could design, using existing work, an indirect
adaptive fuzzy control system for the parametrically driven pendulum [89, 76,16].
This study is believed to be the first formal evaluation of trajectory reconstruc-
tion using fuzzy inference and a one-step Eulers method. The correlation-product
inference procedure and the generation of fuzzy rules directly from the data re-
late the current research to the study of Adaptive Fuzzy Associative Memories
(AFAMs) [46]. The results of this investigation demonstrate that a numerical
method can link independent FAM fuzzy rule bases together to create smooth
trajectory approximations as was predicted by Corollary 2.2 of Chapter 2.
Domain decomposition has been applied, reportedly for the first time, to a re-
cursive partitioning method, through the bisection and homogeneity algorithm,
for the management and containment of computational complexity in fuzzy rule
68


base construction. Domain decomposition can be applied effectively to the tem-
poral ordering of state space descriptions.
Chapter 4 will present the domain decomposition and fuzzy modeling of the
three-dimensional chaotic systems of the Lorenz attractor and the chaotic para-
metrically perturbed pendulum. The fuzzy modeling of two-dimensional dynam-
ical systems will also be presented.
69


4. Experimental Results: Fuzzy Modeling
4.1 Introduction
The bisection and homogeneity algorithm is an unsupervised learning strat-
egy. The algorithm identifies fuzzy clusters of information directly from the data.
Fuzzy clusters are data points that belong to different regions of the state space
to different degrees. The bisection and homogeneity algorithm creates crisp
disjoint homogeneous regions. The algorithm extends the regions according to a
pre-defined overlap rate. The data points within a region have antecedent sym-
metric triangular membership function values between 0 and 1. The production
of fuzzy clusters is the result of using antecedent triangular membership func-
tions and an overlap rate to define the fuzzy covers of the regions using fuzzy
rules. Fuzzy clustering techniques have been studied in the context of unsuper-
vised pattern recognition [50, 24]. The clustering is based upon a homogeneity
criterion that can be defined for the particular application one has in mind. Many
researchers in working with low level object and pattern recognition problems,
define a homogeneity function in terms of the density of data points which is
measured with a distance function [8]. In this study, homogeneity is defined in
terms of compatibility of time rates of change of sample points in a region [2, 35].
The bisection and homogeneity algorithm is a recursive partitioning algorithm.
It is a top down approach taking into consideration the largest bounded region
TO


possible as a homogeneous region. Other techniques, which assume the presence
of large data sets, are bottom up in their approach to identifying clusters of
information. The mountain method [91] of fuzzy clustering and its extensions [8,
18] are bottom up approaches to the unsupervised learning of fuzzy clusters. The
mountain method relies on a grid of data points for estimating cluster centers.
The density of sample data sets around the grid points is measured using a
mountain function. The recursive partitioning approach of Hiew and Tsang [35]
takes a more direct approach to modeling vector field descriptions of chaotic
dynamical behavior.
Other fuzzy clustering approaches model systems directly from the data. Fuzzy
C-Means clustering was introduced by J. Bezdek in 1981 [10]. According to
this method, initially cluster centers are arbitrarily placed. Each data point has
a membership grade for each cluster. Iteratively, the cluster centers and the
membership grades for each data point are updated. The iteration is based upon
minimizing an objective function that represents the distance from any given
data point to a cluster center. The fuzzy C-Means clustering approach and its
derivative, Subtractive clustering [18], require an initial guess on the part of the
modeler as to location and number of cluster centers. In modeling chaos, it
was decided at the outset of this research, that critical information would be
lost by imposing requirements as to location and number of cluster centers on a
chaotic regime. At the conclusion of this research in modeling and control of the
chaotic parametrically perturbed pendulum, the recursive partitioning approach
continues to appear to be the most viable for the fuzzy modeling of chaos.
71


This chapter reports on the results of modeling two and three-dimensional
systems using the bisection and homogeneity algorithm. Section 4.2 describes
the fuzzy modeling of the autonomous two-dimensional systems: the undamped
pendulum and the Van der Pol Oscillator. Section 4.3 describes the fuzzy mod-
eling of the autonomous three-dimensional chaotic Lorenz attractor. Section 4.4
begins the detailed description of the fuzzy modeling of the nonautonomous para-
metrically perturbed pendulum. Section 4.5 describes the reconstruction of the
Poincare maps of the chaotic pendulum using fuzzy rules.
4.2 Experimental Results for Two-dimensional
Autonomous Dynamical Systems
The structural decomposition method was tested on two-dimensional problems
such as the undamped pendulum and the Van der Pol oscillator. The fuzzy
systems generated trajectories and flows on a par with Runge-Kutta numerical
simulations. The two-dimensional examples gave good approximations with a
55% overlap rate and a minimum triangular function base length for antecedents
and consequents of fuzzy rules of 0.1.
Implementation of the fuzzy trajectory reconstruction for the two-dimensional
systems took place as follows: A C++ program ran the bisection and homogene-
ity algorithm, which generated fuzzy rules for each dynamical system based upon
sun input grid of sample points. The algorithm computed an additional sample
at the center of any empty region. Next, the rules were put into a form for input
into the Matlab fuzzy tool box [31]. The fuzzy tool box generated the fuzzy rule
base systems. The fuzzy inference method was equivalent to a SAM centroidal
72


fuzzy logic system with correlation-minimum inference (clipping of output mem-
bership functions) and min if-part combiner for the undamped pendulum and
correlation-product inference (scaling of output membership functions) and min
if-part combiner for the Van der Pol oscillator. The two-dimensional fuzzy logic
systems did not reduce to a SAM/COG model, since the base length b of the
output triangular membership functions was not constant. The bisection and
homogeneity algorithm produced volumes,
V = 0.5 *b*h
(triangular base length), which varied with each consequent fuzzy set. The height,
h, remained one at the centroid for all membership functions.
A variable time step Eulers method, implemented in C, reconstructed the
trajectories of the two-dimensional systems in the following way: The time step
was
A x incr
=
where
dx
outx = E(),
at
the defuzzified output of the SAM fuzzy logic system representing time rates of
change along the x axis. The Euler approximation gives
xn+i = xn + At outx (4.1)
yn+1 = yn + At outy, (4.2)
where outy E(^), the defuzzified output of the SAM fuzzy logic system rep-
73


resenting time rates of change along the y axis. Equations (4.1) and (4.2) satisfy
rii-rl
Vn+l
xn -f sgn(outx) incr
yn -r (x+1 x,
yn -r (x+1 xn) *
outy
outx
4.2.1 Two-dimensional Undamped Pendulum
The simplest problem to start the experiments with is the undamped pendu-
lum. This was the first test case that was considered in developing and refin-
ing the bisection and homogeneity algorithm. Figures 4.1 and 4.2 compare the
Runge-Kutta numerical simulations of three trajectories with the same trajecto-
ries generated by the three SAM centroidal fuzzy systems and Eulers method.
The following system of ODEs generated the grid of sample points that was used
for the trajectory reconstruction based on the bisection and homogeneity algo-
rithm. The fourth order Runge-Kutta numerical method also used the following
system of equations that satisfy
X = y (4.3)
y sin(x). (4.4)
74


Figure 4.1 shows three numerical Runge-Kutta simulations of the undamped
pendulum for trajectories with initial conditions:
(*o,2fo) = (^0),(a?o!to) = (0, |),(^o,yo) = (tt,0).
Figure 4.1. Trajectories generated from 1001 iterations of fourth order Runge-
Kutta simulations of undamped pendulum.
The Runge-Kutta simulation used a fixed step size, h = 0.1, and generated
1001 points of the trajectory.
Figure 4.2 shows the fuzzy trajectories generated from fuzzy rules using correlation-
minimum inference and centroidal defuzzification. The fuzzy logic system remains
a SAM fuzzy system with the sum aggregation operator combining the fired
consequent fuzzy sets. The tolerance parameter 7 is used by the bisection and
homogeneity algorithm to pre-define the minimum interval length for splitting
a nonhomogeneous region. The bisection and homogeneity algorithm generated
75


803 fuzzy rules for the time rates of change along the x axis (the x rule base) and
976 rules for the time rates of change along the y axis (the y rule base). The input
grid consisted of 6561 equally spaced (0.1) sample points. The bisection and ho-
mogeneity algorithm used a pre-set overlap rate of 0.55 for extending the regions
covered by the fuzzy rules. The overlap rate was the same as that previously
used by Hiew and Tsang in their experiments with the Van der Pol oscillator.
Homogeneity parameters, for the undamped pendulum, were set to x == 0-8 and
r = 0.0001 along both the x and y dimensions. The total number of x rule base
samples remained 6561 at the completion of the bisection and homogeneity al-
gorithm. For trajectory reconstruction, Eulers step size was set to where
incr was set to 0.001 and outx was the defuzzified output of the current iteration
over the x rule base, outx represented the expected time rate of change of x for
the current input vector (x,y).
Figure 4.2. Fuzzy rules modeling trajectories of the undamped pendulum, (j
10~7).
76


Starting from the unstable equilibrium point (x,y) (7r, 0), the fuzzy tra-
jectory wanders off on the unstable manifold when the system is close to the
equilibrium point.
The number of y rule base samples increased to 6580 at completion of the
algorithm. The increase in number of samples was due to the splitting of a
nonhomogeneous region along the midpoint of an interval with all the original
regions samples belonging to one of the newly formed subregions. In this case,
the algorithm computed a sample for the center of the empty subregion. The
Split-Region(r,d) procedure assures that the subregions inherit the homogene-
ity properties of the splitting parent region. Hence, if a parent region is homo-
geneous along the x dimension and nonhomogeneous along the y dimension, the
spitting into subregions will create additional y fuzzy rules. The algorithm will
not construct additional x fuzzy rules for a parent region homogeneous along
the x dimension. As a consequence of this, the bisection and homogeneity algo-
rithm constructs a finer partition of the state space in areas where nonlinear and
chaotic dynamics occur and a courser partition in areas where the dynamics are
more regular.
The next two subsections describe the fuzzy modeling of several trajectories of
the Van der Pol oscillator and introduce the heuristic for choosing homogeneity
parameters x and r. A heuristic for choosing these parameters helps reduce the
time and complexity of analyzing large data sets prior to running the algorithm.
77


4.2.2 Two-dimensional Van der Pol Oscillator
The following set of experiments was suggested by the work of Hiew and
Tsang [35, 36]. These experiments axe on the modeling of a flow of the Van der
Pol oscillator [85] for nonlinear and non chaotic trajectories. Hiew and Tsang
also compared the numerical simulations of flows of the Van der Pol oscillator
with fuzzy rule base reconstruction of trajectories.
Figure 4.3 shows several numerical Runge-Kutta simulations of the Van der
Pol oscillator for trajectories with initial conditions:
(xo,Vo) = (0,1), (x0,y0) = (2,2),(x0,y0) = (4,4),
(*o,yo) = (44)> (^o,yo) = (4, -4), (x0,Po) = (0,2). (4.5)
Figure 4.3. Fourth order Runge-Kutta simulations of Van der Pol.
78


The Van der Pol oscillator satisfies
y = 0.5y(l
x = -y (4.6)
x2)+x (4.7)
This system of ODEs generated a grid of 10,201 equally spaced (0.1) points for
the reconstruction of the flow using fuzzy rules. The algorithm generated 4571
rules for the fuzzy rule base along the ar-dimension and 4334 rules for the fuzzy
rule base along the ^-dimension. The algorithm increased the total number of y
rule base samples from 10,201 to 10,534 by processing empty subregions.
Figure 4.4 shows the fuzzy rule reconstruction of six trajectories for the same
set of initial conditions as (4.5).
2
t
> a
-i
2
*4 4 *3 -2 -I 0 1 2 3 l 5
X
Figure 4.4. Fuzzy rules modeling trajectories of the Van der Pol oscilla-
tor. (y = 10-7J.
The pre-set overlap rate of the fuzzy rules was 0.55. Hiew and Tsang previously
used a fixed grid of sample points for their fuzzy rule construction of the Van der
79


Pol oscillator [35] along with an overlap rate of 0.55. In their experiments, the
recursive partitioning algorithm produced 4096 rules for both x and y fuzzy rule
bases. With regards co their experiment, no information was available detailing
grid size, explicit method of trajectory reconstruction, the choice of homogeneity
parameter, or the method of processing empty regions.
Figures 4.3 and 4.4 indicate that the fuzzy trajectory approximation of the
Van der Pol flow reflects the functional approximation accuracy of the fourth
order Runge-Kutta simulation. The next section will describe the heuristic for
choosing the homogeneity parameters for running the bisection and homogeneity
algorithm over data points representing the Van der Pol oscillator.
4.2.3 Choosing Homogeneity Parameters
As defined in Chapter 3, the homogeneity parameter x represents the ratio
in absolute value of time rates of change, jj£|, along a coordinate axis, of two
samples in a region. The homogeneity parameter r measures distance in terms
of the absolute value of the difference between two time rates of change. For
the former, we are interested in choosing a lower bound and for the latter, an
upper bound. Both bounds give a measure of the allowable coarseness of the
approximation.
For the Van der Pol oscillator, with 10,201 sample points, the histogram of
Figure 4.5 indicates that the rates of change of x with respect to time t, for an
appropriate bin size (partition), have a uniform distribution of values in the limit
as the partition size goes to zero. In this case, the bin size for which one detects
that the distribution is uniform is 200 which is twenty times the range of the
80


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