FUZZY LOGIC
AND I N T E L L I G E N T SYSTEMS
INTERNATIONAL SERES IN INTELLIGENT TECHNOLOGIES
Prof. Dr. Dr. h.c. Han...
65 downloads
2077 Views
16MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
FUZZY LOGIC
AND I N T E L L I G E N T SYSTEMS
INTERNATIONAL SERES IN INTELLIGENT TECHNOLOGIES
Prof. Dr. Dr. h.c. HansJiirgen Zimmermann, Editor European Laboratory for Intelligent Techniques Engineering Aachen, Germany Other books in the series:
Applied Research in Fuzzy Technology by Anca L. Ralescu Analysis and Evaluation of Fuzzy Systems by Akira Ishikawa and Terry L. Wilson
FUZZY L O G I C AND INTELLIGENT SYSTEMS E D I T E D BY Professor Hua Li Texas Tech University Lubbock, Texas, USA
Professor Madan Gupta
University o f Saskatchewan Saskatoon, Canada
KLUWER ACADEMIC PUBLISHERS Boston/London/Dordrecht
Distributors for North America: Kluwer Academic Publishers 101 Philip Drive Assinippi Park Norwell, Massachusetts 02061 USA Distributors for all other countries: Kluwer Academic Publishers Group Distribution Centre Post Office Box 322 3300 AH Dordrecht, THE NETHERLANDS
Library of Congress CataloginginPublication Data Fuzzy logic and intelligent systems 1 edited by Hua Li, Madan Gupta. p. cm.  (International series in intelligent technologies ;
3) ISBN 0792395751 1. Neural networks (Computer science) 2. Fuzzy systems. 3. Expert systems (Computer science) I. Li, Hua, 195611. Gupta, Madan. 111. Series. QA76.87. F89 1995 629.8'36dc20 9516748 CIP
Copyright
1995 by Kluwer Academic Publishers
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher, Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park, Norwell, Massachusetts 02061.
Printed on acidfree paper. Printed in the United States of America
CONTENTS
PREFACE 1
IMPROVEMENT OF FUZZY CONTROL METHODS M. Mimmoto 1 INTRODUCTION 2 MULTIPLE FUZZY REASONING 3 FUZZY CONTROLS 4 FUZZY CONTROLS UNDER VARIOUS AGGREGATION
METHODS NEW FUZZY REASONING METHODS FUZZY CONTROLS UNDER VARIOUS DEFUZZIFIER METHODS 7 CONCLUSIONS REFERENCES
5 6
2
NEURAL NETWORKS AND FUZZY BASIS FUNCTIONS FOR FUNCTIONAL APPROXIMATION Liang Jin, Madan M. Gupta, Peter N. Nikifonrk 1 INTRODUCTION 2 STONEWEIERSTRASS THEOREM AND IMPLICATIONS
3 4 5
TRIGONOMETRIC FUNCTION NETU'ORKS MFNNs AS UNIVERSAL APPROXIMATORS KOLMOGOROV'S THEOREM AND FEEDFORWARD NETWORKS
xiii
Fuzzy LOGIC AND INTELLIGENT SYSTEMS
vi
APPROXIMATION USING POLYNOMIAL NEURAL NETWORKS 7 GAUSSIAN NETWORKS UNIVERSAL APPROXIMATION USING FUZZY BASIS 8 FUNCTION NETWORKS 9 CONCLUSIONS REFERENCES 6
3
ORDERING FUZZY REAL QUANTITIES Robert Lowen 1 INTRODUCTION 2 STOCHASTIC FUZZY REAL NUMBERS ORDERING O F THE STOCHASTIC FUZZY REAL LINE 3
REFERENCES
4
FUZZY LOGIC CONTROLLERS FOR AIRCRAFT FLIGHT CONTROL Jia Lao and Edward Lan 1 INTRODUCTION 2 FUZZY PID CONTROLLER FUZZY FLIGHT CONTROL SYSTEM FOR THE X29 3
AIRCRAFT 4 VALIDATION THROUGH NONLINEAR SIMULATION 5 CONCLUSIONS REFERENCES
5
A CLASSICAL CONTROLLER: A SPECIAL CASE OF THE FUZZY LOGIC CONTROLLER Thomas Brehm and Kvldip Rattan 1 INTRODUCTION 2 FLC ARCHITECTURE AND TERMINOLOGY 3 FLC OUTPUT EQUATION DERIVATION FLC AS A PIECEWISE LINEAR CONTROLLER 4 CLASSICAL CONTROLLER: A SPECIAL CASE OF FLC 5 6 SUMMARY
Contents
REFERENCES
6
REAL TIME FUZZY LOGIC CONTROLLER FOR BALANCING A BEAMANDBALL SYSTEM Nowell Godfrey, Hua Li, Yuandong Ji , William Marcy INTRODUCTION PROBLEM FORMULATION THE DESIGN OF FUZZY LOGIC CONTROLLER 3 COMPARISONS TO OTHER CONTROL STRATEGY 4 5 SUMMARY 6 ACKNOWLEDGEMENT REFERENCES 1 2
7
DESIGN OF FUZZY CONTROLLERS BASED ON FREQUENCY AND TRANSIENT CHARACTERISTICS Kazuo Tanaka 1 INTRODUCTION 2 FREQUENCY AND TRANSIENT CHARACTEMSTICS
3 4 5
PHASELEAD COMPENSATION DESIGN EXAMPLES APPLICATION TO TANK LEVEL CONTROL 6 CONCLUSION REFERENCES
8
FUZZY INFERENCE INTEGRATING 3D MEASURING SYSTEM WITH ADAPTIVE SENSING STRATEGY Koji Shirnojima, Toshio Fukuda, Fumihito Ami, and Hideo Matsuum 1 INTRODUCTION 2 3D MEASUREMENT SYSTEM
3 4 5
ADAPTIVE SENSING STRATEGY EXPERIMENTS AND RESULTS CONCLUDING REMARKS
vii
154
...
Fuzzy LOGIC AND INTELLIGENT SYSTEMS
vlll
REFERENCES
9
ROBOT HANDEYE COORDINATION BASED ON FUZZY LOGIC Sukir Kumaresan, Hua Li, and XingMin Li 1 INTRODUCTION 2 SURVEY O F EXISTING TECHNIQUES 3 MATHEMATICAL FORMULATION ALGORITHM FOR THE COORDINATION BASED ON 4 FUZZY LOGIC 5 EXPERIMENTS 6 OPERATION UNDER CAMERA DISPLACEMENT 7 COMPARISON 8 CONCLUSION REFERENCES
10
USING FPGA TECHNIQUE FOR DESIGN AND IMPLEMENTATION OF A FUZZY INFERENCE SYSTEM Donald Hung 1 INTRODUCTION 2 THE FUZZY INFERENCE ALGORITHM 3 FPGA DESIGN CONSIDERATIONS 4 DESIGN METHODOLOGY AND IMPLEMENTATION 5 TESTING RESULTS 6 CONCLUSION AND DISCUSSION REFERENCES
11
AN EMPIRICAL ANALYSIS OF ONE TYPE OF DIRECT ADAPTIVE FUZZY CONTROL Hugues Bersini and Vittorio Gorrini 1 INTRODUCTION 2 A GRADIENT METHOD FOR SUGENO'S FUZZY SYSTEM 3 A SIMPLE DIRECT ADAPTIVE FUZZY CONTROL 4 AN ADAPTIVE FUZZY PID
Fuzzy LOGIC AND INTELLIGENT SYSTEMS
x
7 APPLICATIONS T O ROBOTIC CONTROL 8 CONCLUSIONS REFERENCES
14
CAMCORDER OPERATION JUDGEMENT USING A NEURAL COMPUTING APPROACH Kitahiro Kaneda and Paul Wang 1 INTRODUCTION 2 PROBLEM FORMULATION AND METHODOLOGY 3 EXPERIMENTAL RESULTS 4 CONCLUSION
REFERENCES
15
MODEL REDUCTION AND CONTROL OF MULTISTAGE FLASH (MSF) DESALINIZATION PLANTS Srinivas Ramamurthy,Jayanta Pal, Ak Sinha, Darwish MK A1 Gobaisi, Ganti Prasada Rao
INTRODUCTION MSF CONTROL SYSTEMS MSF PLANT MODELLING REDUCED ORDER MODELLING NEURAL NETWORK MODEL MODEL REDUCTION 7 ROBUST CONTROLLER DESIGN 8 CONCLUSIONS REFERENCES 1 2 3 4 5 6
INDEX
CONTRIBUTORS
M. Mizutnoto Division of Information and Computer Sciences Osaka ElectroCom.munication University Neyagawa, Osaica 572, Japan Liang Jin, Madan M. Gupta, Peter N . Nikiforuk Intelligent Systems Research Laboratory College of Engineering University of Sswkatchewan Setskatoon, Saskatchewan Canada S7N OWO Robert Lowen Departement Wiskunde en Informatica Universiteit Antwerpen Froenenborgerlauui 171 2020 Antwerpen, Belgium Jia Luo and Edward Lan Department of Aerospace Engineering University of Kansas Lawrence, KA 66045, USA T h o m a s B r e h m and Kuldip R a t t a n * Department of Computer Science and Engineering *Department of Electrical Engineering Wright State University Dayton, OH 45435, USA
Nowell Godfrey, Hua Li, Yuandong Ji*, and William Marcy Computer Science Department Texas Tech University Lubbock, TX 79409, USA •Department of System Engineering Case Western Reserve University Cleveland, OH 44106, USA Kazuo T^naka Department of Mechanical Systems Engineering Kauiazawa University 24020, Kodatsuno, Kanazawa920, Japan Koji Shimojima, Toshio Fukuda, Fumihito Aral, and Hideo Matsuura Department of MecheinoInformatics and Systems Nagoya University Furocho, Chikusaku, Hagoya 46401, Japan Sukir Kumaresan, H u a Li, and XingMin Li* •Computing Center, The Institute of Textile Engineering HeDong, Tianjiu, China Computer Science Department Texas Tech University Lubbock, TX 79409, USA Donald Hung Department of Electrical Engineering Gannon University Erie, PA 16541, USA
xii
Hugues Bersini and Vittorio Gorrini IRDIA  CP 194/6 Universite Libre de Bruxelles 50, Av. Franklin Roosevelt 1050 Bruxelles, Belgium Mattias N y b e r g and YohHan P a o Department of Electrical Engineering and Applied Physics Case Western Reserve University Cleveland, OH 44106, USA Kitahiro K a n e d a and Paul P. Wang* Video Products Development Center and Fuzzy Logic Research Laboratory CANON INC., Tokyo, Japan * Department of Electricsd Engineering Duke University Durham, NC 277080291, USA Srinivas Ramamurthy, Jayanta Pal, Ak Sinha, darwish Al Gobaistfand Ganti R a o Department of Electrical Engineering India Institute of Technology Kharagpur 721302, India tWater and Electricity Department Government of Abu Dhabi, UAE
CONTRIBUTORS
PREFACE
Building an intelligent system is challenging work, it can involve many different aspects of learning, a d a p t a t i o n , and control under uncertainty. Neural networks and fuzzy logic have been the major tools for scientists and engineers working in this field. Over the past several years, we have witnessed the rapid growth of utilizing fuzzy logic b o t h in t h e theory and applications in many different engineering fields. Unlike most of the traditional techniques, fuzzy logic algorithms provide a tool to deal with uncertainty, random disturbances, and yet with relatively modest computational effort. T h e world is full of uncertainty. Everyday when we drive to work, we may experience different weather conditions and traffic p a t t e r n s . Everytime we park at the same parking space, but we may not b e able to park the car at exactly the same spot cis we did the day before. T h e uncertainty and imprecision seem always come together, at least at many situations, and it seems t h a t the well balanced solution between the uncertain conditions and imprecision, or adaptive decisionmaking serves our daily life very well. It is then n a t u r a l to ask how much preciseness or impreciseness can we tolerate in the real world situation? at what cost? Fortunately, fuzzy logic as a tool to deal with uncertainty, r a n d o m disturbances, a n d illdefined problems (the problems t h a t can not be well described by closed form equations,) allows us to tackle these problems. Fuzzy logic algorithms offer many attractive features and they have been gaining popularity for solving many real engineering application problems, especially, the problems or systems which are highly nonlinear, which involve many p a r a m e t e r s and many of these parameters are changing or drifting in time. As many people have realized t h a t one of the major obstacles for us to build a real intelligent machine is to deal with r a n d o m disturbances, to process huge amount of imprecise d a t a , to interact with a dynamically changing environment, and to cope with uncertainty. W i t h the use of neuralfuzzy techniqties, it is now possible to attack and solve a class of certain problems by using a customer developed algorithm with an intelligent behavior. Working in the cirea of neural networks, fuzzy logic, and their applications, we felt strongly that there is a need to review the stateoftheart development in
xiv
FUZZY LOGIC
AND INTELLIGENT SYSTEMS
this emerging field. During the 1993 IEEE International Conference on FYizzy Logic jointly held with the 1993 IEEE International Conference on Neural Networks in San Francisco (March 28April 3, 1993), Kluwer Academic Publishers invited us to put a book together. Since then, we have been working on this book project with the researchers who have made significant contributions to this emerging field. Now we are pleased to present this volume as a result of this work. This edited book consists of fifteen chapters by authors from US, Canada, Japan, Europe, and India. The subjects presented in this book reflect the most recent developments in neural networks, fuzzy logic and their applications in intelligent systems. In addition, the balcince between theoretical work and applications makes this book not only suitable for researchers, engineers, but also for graduate students as well. Without the hard work of many contributors, this book would have not been possible. In particular, it is our pleasure to acknowledge the inspiring contributions that Professor Lofti A. Zadeh has made in fuzzy logic, which has inspired many researchers in the field of building intelligent systems. Diiring the editorial phase of this book many students from Texas Tech University have shared various responsibilities. In particular, we would like to thank Samuel Huang who helped to coordinate the peer review process, Nowell Godfrey who painstakingly converted some manuscripts to Latex format and helped to plan the layout of tables and figures to arrange them in an orderly fashion. We would also like to thank Xiaohui Meng, Lazslo Moldovan, and Dongming Liang at Texas Tech University who have helped the preparation of typesetting. The help from Mr. Zachary Rolnik, Senior Editor of Kluwer Academic Publishers, is appreciated.
Hua Harry Li Texas Tech University LubbocJc, Texas, USA Madaji M. Gupta University of Saskatchewan Saskatoon, Saskatchewan, CANADA
1 IMPROVEMENT OF FUZZY CONTROL METHODS M. Mizumoto Division of Information and Computer Sciences Osaka ElectroCommunication University Neyagawa, Osaka 572, Japan
ABSTRACT This chapter introduces improvement methods for fuzzy controls. At first, we introduce new aggregation operators for obtaining a fuzzy set of control actions. A number of new fuzzy reasoning methods such as productsumgravity method, minsumgravity method, (min/product)maxgravity method and others are also introduced. Finally, we show several defuzzification procedures for obtaining a representative point of a fuzzy set of control actions.
1
INTRODUCTION
A number of studies on fuzzy logic controllers has been reported since M a m d a n i [3] implemented a fuzzy logic controller on a boiler steam engine. For most of the existing fuzzy logic controllers, the center of gravity m e t h o d is widely used as a defuzzification m e t h o d which decides an actual control action in a fuzzy set of control actions aggregated by using max. operator from fuzzy sets inferred from fuzzy control rules. In this cJiapter, we propose several new fuzzy control methods and compare control results of these methods. At first, new aggregation operators are introduced which obtain a fuzzy set of control actions. If averaging operators of arithmetic mean, dual of geometric mean and dual of harmonic mean are used as the aggregation operators, better control results can be obtained compared to the case of using m a x operator as in the "minmaxgravity m e t h o d " by M a m d a n i [3]. In this connection, by introducing tnorms and 1
CHAPTER 1
tc.onorms we propose new kinds of fuzzy logic reasoning methods which include productsumgravity method, minsumgravity method, (minproduct)maxgravity method, bounded productbounded sumgravity method. Then we introduce several defuzzifier procedures for obtaining a representative point of a fuzzy set of control actions. Finally, we show that two defuzzification methods called height method and area method provide better control results than that of the widely used center of gravity method.
2
MULTIPLE FUZZY R E A S O N I N G
We shall consider the following multiple fuzzy reasoning form with several fuzzy rules combined with "else",
Rulel :
Ai and Bi => Ci
else
Rule2 :
A2 and B^ => C2
eZ.se (1.1)
Rule n : Fact :
An and B„ =^ Cn XQ and 1/0
Cons where Ai,i — I, • ,n are fuzzy sets in X; B, in Y; and d, C in Z and XQ € X,yo e Y. FVizzy rule \Ai and Bi =S> CjJ,i = 1, • • • ,n is defined as
fiA. andB,=>C.(^,y,z) = /tA^ix) A /iB.iv) A/ic. (2)
(12)
where A stands for min. The inference result C,' inferred from the fact [XQ and yoj and fuzzy rule \Ai and Bi => C J is given as
Mc;(^) = /*A,(a;o) A/iB,(yo) A//c. (^)
(1.3)
The consequence C" of (1.1) at aio and yo is given as follows by interpreting "else" union (U).
Improvement
of Fuzzy Control
Methods
C' = C l U C ^ U    U C ;
fic'iz)
= Mc; (2) V tic',{z) V • • • yfic'Jz)
(1.4)
(15)
where V stands for max. T h e m e t h o d of obtaining a singleton ZQ which is a representative point for the resulting fuzzy set C' of (1.4) is called a deffnzification method. For example, the point which has the largest membership grade of C" can b e taken as the desired singleton. T h e m e t h o d of taking the center of gravity of C as the desired singleton ZQ is widely used in the hizzy controls and is called the centerofgravity method, which is given as
fzfj.c'{z)dz j
^ic'[z}dx.
T h e above fuzzy rea.soning m e t h o d obtained by using (1.3)(1.6) is known as M a m d a n i ' s m e t h o d [3] and referred as ''minmaxgravity method" (a.s illustrated Figure 1)
3
FUZZY CONTROLS
We shall consider a plant model G ( s ) = e ~ ^ ^ / ( l + 2 0 s ) with first order delay, which is used in the discussion in the next section. Fuzzy control rules for the plant model are shown in Table 1 [7, 8] and interpreted as:
CHAPTER 1
1
^*y.
0 /
,
a
\ Xo
1 A2;^ 0
/ , .
Figure 1
\
Minmaxgravity method using equation (1.3), (1.5) £ind (16)
Aq NB
NM
PS
PB
PM
PS
NB NM NS e
zo PS PM PB
Table 1
Ri :
Ae ZO PB PM PS ZO NS NM NB
PS
PM
PB
NS
NM
NB
Fuzzy control rules: e, Ae =^ Aq
e is NB and Ae is ZO ••
R2 : e is NM and Ae is ZO
Aq is PB Aq is PM (1.7)
Ri3 :
e is ZO and Ae is PB =>
Aq is NB
Improvement
of Fuzzy Control
Methods
where e is error and Ae is change in error and Aq is change in action. NB, NM, • • •, P B are fuzzy sets as shown in Figure 2 (width W = 6 ) . W h e n e = e^ and Ae = Aeo are given to a fuzzy controller as premises of (1.7), each control rule /?;(* = 1, •••, 13) infers a fuzzy set C" for A q by using
HC'iM) where Ai,Bi
= //A,(eo) A/XB,(Aeo) A/ic,(A Ci\ is assumed to be denoted as Zi.
6.5
Height Method [1]
T h e height m e t h o d obtains ZQ as a weighted average of the representative points Zi of Ci by the heights hi of C,'. Namely,
hi
^" =
Zi+h2
h ^h
Z2\
\hn'
Zn
J.—n;
.
•
.
(^••^°)
fll + '12 + • • • + " n
This m e t h o d may b e considered as a special case of Sugeno's fuzzy reasoning m e t h o d [8] in which the conclusion part of a fuzzy control rule is a function or a real number rather t h a n a fuzzy set.
6.6
Maximal Height Method
By maximal height method, a representative point Zj of Cj which corresponds to the maximal height hj among hi{i = 1 , . . . , n ) is adopted as ZQ,
xg = Zj (hj the maximal height).
6.7
(131)
Area Method [4]
T h e ^0 is obtained as the weighted average of the representative points ^, by the areas Si of C,',
Si Zi+S2Z2\
+
SnZn
Improvement of Fuzzy Control Methods
6.8
11
Maximal Area Method
A representative point Zj of Cj which corresponds to the maximal area Sj among Si(i = I,... ,n) is selected as zo, that is,
ZQ = Zj {Sj the maximal area).
(1.33)
Figure 7 indicates control results under vaxious defuzzification methods when fuzzy sets in Figure 2 are of width W=6 and 8. It is found from computer simulation that the defuzzification methods of Height Method and Area Method obtain better control results than the center of gravity Method which is widely used in the fuzzy control. Defuzzifier methods such as Average of Maxima Method, Midpoint of Maxima Method, Maximal Height Method and Maximal Area Method whicJi use maximal veilues show bad control results. These defuzzifier methods tend to continue selecting the scime A 0, there is a MFNN with arbitrary hidden layers and the sigmoidal functions which approximates g uniformly on S tuith error < c.
Proof. See Appendix 4. Corollary 2 gives the results of the approximation capabilities of general MFNNs with sigmoidal functions. In fact, the neural activation fimctions in MFNNs may be relaxed to ajiy continuous, bounded and nonconstant functions (Hornik, [12]).
31
Functional Approximation
«1
Figure 4
5
Twolayered network with the McP hidden units.
KOLMOGOROV'S T H E O R E M AND FEEDFORWARD N E T W O R K S
The applications of the Kolmogorov's superposition theorem which concerns the representation of continuous functions defined on an ndimensional cube by sums and superpositions to feedforward neural networks were first studied by HechtNielsen [10], [11]. The study gives an existence of an exact implementation of every continuous function in a structure of threelayered
32
CHAPTER 2
networks. As one of the pioneers in the field of neural networks, HechtNielsen gave some interpretation of the approximation principle of the Kolmogorov's theorem in terms of feedforward neuraJ networks before some more practical achievements of universal approximation capabilities of the feedforward networks were develop independently by Cybenko [5], Funahashi [7], and Hornik, Stinchcombe and White [14]. Most recently, Sprecher [32] presented some new results which may be viewed as a stronger version of the HechtNielsen's results. However, Girosi and Poggio [27] pointed out that the onevariable functions constructed by Kolmogorov [21], and the later improvements by Lorentz [25] and Sprecher [32], are far from being any of the type of functions used in feedforward neurcil networks. Let / = [0,1] denote the closed unit interval and / " = [0,1]" (n > 2) be the Cartesian product of / . The superposition theorem of Kolmogorov (1957) establishes that for each integer n > 2 there are n x (2n + 1) continuous monotonically increasing functions hpg and (2n + 1) continuous functions g^ which can be used to represent exactly every realvalued continuous function f : I^ = [0,1]" —> 9?. The original statement of Kolmogorov can be given as follows.
Theorem 7 (Kolmogorov's Superposition Theorem) There exist a set of increasing continuous functions hp^ : / = [0,1] —^ !ft .so that each continuous function f on I" can be written in the form
2n+l
n
/(x)= X ] g ^ ( ^ h p , ( x p ) ) q=l
(2.7)
p=l
where p, are properly chosen continuous functions of one variable.
Kolmogorov's theorem shows that any continuous function of several variables Ccin be represented exactly by means of a superposition of continuous ftmctions of a single variable and the operation of addition. Moreover, the functions hpq are universal for the given dimension n; they axe independent of a given function / . Only the functions g, are specific for the given function / . Using the leinguage of feedforw£ird neural networks, we may explain Kolmogorov's theorem a.s follows: Any continuous function defined on an ndimensional cube can be implemented exactly by a twolayered feedforward
Functional
33
Approximation
n(2n + l) units
2n + 1 units
Figure 5 The block diagram representing the Kolmogorov's superposition theorem using a twolayered network structure.
network as shown in Figure 5, which has n(2n + 1) units with the increasing continuous functions hpg : I ^ R in the first hidden layer and (2n + 1) units with the continuous functions gg in the second hidden layer.
34
CHAPTER 2
The main improvements to the original Kolmogorov's theorem concentrate on the possibility of replacing the function Qq by a single function g (Lorentz, [25]) and of transforming /ip, into7p/7,i (Sprecher, [32]). Let H be the space with the uniform norm consisting of all the nondecreasing continuous fiuictions on the closed interval I = [0,1] and H^ = H x ... x H he the fcth power of space IT. Kahane modified Kolmogorov's theorem using the following restilts.
Theorem 8 Let Ip (p = 1,... ,n) be. a collection of rationally independent constants. Then for quasi every collection {/ii,... ,/i2n+i} G ^^n+i ^j^^ function f € C{I") can be represented on / " in the form
where g is a continuous
2n+l
n
q=l
p=l
function.
In order to give a geometric interpretation of Theorem 8, consider the mapping of / " into a (2n + l)dimensional space defined by
Vg =lihg(xi)
+...+lnhg{xn),
q=l,...,2n
+l
(2.9)
This is a continuous onetoone mapping. Otherwise, two points of / " would exist, which are not distinguishable by the family of functions yq{xi,..., x,,), q = 1 , . . . , 2n + i. All functions whidi are representable by Eq. (2.9) would coincide at these two points, and Eq. (2.9) would be impossible for some functions / € (^[J"]. indeed, since / " is compact, its image under mapping is
= I y = (j/i,,2/2n+i) yg = Y^lphg(xp),xei''\
(2.10)
which is also compact, and the mapping (2.9) is a homomorphism between / " and T. It implies that there exists a onetoone relationship between all the
Functional Approximation
35
continuous functions / ( . T I , . . . , x„) on / " and all the continuous fvinctions F{yi,..., y2n+i) on T. Therefore, Theorem 8 can be rewritten as follows : There exists homeomorphic embeddin (2.9). from / " into the (2n + l)dimensional Euclidean space 9?^"+^; that is {/,:/">• R, 7 = 1,... ,2n + 1, so that each continuous function F on the image space T of / " has the form
2n+l
F{yi,...,y2n+i)=Y.9{yp)
(2.11)
p=i
More recently, an improved version of Kolmogorov's theorem due to Sprecher [32] was represented by HechtNielsen [11] as a result concerning the existence of feedforward neural networks. The theorem follows:
Theorem 9 (Kolmogorov's Mapping Neural Network Existence Theorem, HechtNielsen, [11]) Given any continuous function f : I " —> W^ with n > 2, f (x) = y. Then f can he implemented exactly by the following network
2n+l
/.(x) =
53 ^'(^'^)
(2.12)
Zk = Yl\''h{xj
+ kf.) + k
(2.13)
where the real constant X and the continuous real monotonically increasing function h are independent of f although they do depend on n. The real and continuous functions gi are dependent on the function fi and e. The constant e is a rational number 0 < e < {z)
= Wo + 5 ^ t i i i , i i , + 5 ^ w » , i ^ a ; i , x i , + , . . +
J^
Wi,...iva:'iz)ui,...,^
(2.20)
where (f)'{z) = d(f)/dz. Like the backpropagation (BP) algorithm for the MFNN (Rumelhart et al., [29]), an momentum version of the above is easily obtained. Alternatively, since all weights appear linearly in Eq. (2.18) of the higherorder networks, one may use the method for solving linear algebraic equation to carry out the above learning task if the number of the patterns is finite. Doing so, one has to introduce the following two augmented vectors
W = [wo Wi
Xo X i
...
...
Wn W i i
X„ X i
XiXa
...
W„n
...
Xn
...
...
W1...1
X'l
Xi
...
^X2
W„..,„]'^
...
X„]^
Functional Approximation
41
where XQ = 1, so that the network equations (2.17)(2.18) may be rewritten into the following compact vector form
y = (^ ( w'^u(x) j
(2.21)
For the given p pattern pairs {x(k),d(k)} (1 < fc < p), define the following vectors and matrix uT(l) uT(2)
rUd(i)) d =
U =
^^(^(2))
UT'(P)
where u(k) = u(x(k)), 1 < A; < p. Then, the learning problem becomes seeking a solution of the following linear algebraic equation
Uw = d
(2.22)
If the number of the weights is equal to the number of the data and the matrix U is nonsingular, Eq. (2.22) has an unique solution
The more interesting case is that the dimension of the weight vector w is less than the number p of the data, thus, the condition for existence of the exact solution of the above linear is given by
rank
U : d = ranfc[U]
(2.23)
In case the above condition is not satisfied the pseudoinverse solution is usually an option and gives a best fit result. The higherorder networks are
42
CHAPTER
2
capable of dealing with the functional approximation and pattern classification problems. Also, Xu, Oja, Suen [37], Taylor and Coombes [35] demonstrated that they may be effectively applied for problem of using a model of curve, surface, and hypersurface to fit a given data set. This problem is called the nonlinear surface fitting and is often encountered in many engineering applications. Some learning algorithms for solving such problem may be found in their papers. Moreover, if one assumes 0(x) = x in the HONU, the weights exhibit linecirly in the networks and the learning algorithms for the HONs may be characterized as a linear LS procedure, the wellknown local minimum problems existing in many nonlinear neural learning schemes may be avoided.
6.2
SigmaPi Networks
Note that the HON contains simply all linear and nonlinear correlation terms of the input components until the order n. A slight generalized structure of the HON is a polynomial network which includes weighted sums of products of selected input components with an appropriate power. Mathematically, the inputoutput transfer function of this network structure is given by
ui
^l^"''"i)
(2.24)
=
(2.25)
n < "
where Wi, wij £ di, N is the order of the network, and Ui is the output of the jth hidden unit. As has been pointed out by Rumelhart et al. [29], this type of the feedforwcird networks is called the sigmapi networks. It is easy to show that the above network satisfies the StoneWeierstrass theorem if (j){x) = a; is a linear function. Prom the network structure point, a sigmapi network shown in Figure 8 may be considered as a twolayered network with a hidden layer and an output layer, where the units in the hidden layer create the products of selected input components computed with a power operation while, like the conventional weighted combiners, the output unit just makes a weighted summation of all outputs of the hidden units.
43
Functional Approximation
Figure S
Block diagram of the sigmapi network.
Moreover, a modified version of the sigmapi networks was proposed by Hornik [14] and Cotter [4] as follows
= ^ ( H "'•"« I
(2.26)
44
CHAPTER 2
n
= n(p(^;))"'"'
(2.27)
J=l
where i/j;, Wij € fi, and p(ij) is a polynomial of Xj. It is easy to verify that the above network satisfies the StoneWeierstrciss theorem, thus, it may be a choice as an approximator for problems of functional approximation. The sigmapi network (2.24)(2.25) is a special case of the above network while p(xj) is assumed to be a linear function of Xj. In fact, the weights Wij in both the networks (2.24)(2.25) and (2.26)(2.27) may be restricted to integer or nonnegative integer values.
6.3
Ridge Polynomial Networks
To maintain the fast learning and powerful mapping capabilities, and to avoid the combinatorial increase in the number of weights of the higherorder network, some modified polynomial network structures were introdirced recently. One of these is the pisigma network (PSN) (Shin and Ghosh, [30]) which is a regular higherorder striicture and involves a mucii smaller number of weights compared to the higherorder network (HON). The mapping equation of a pisigma network can be represented as
y = 'f>[ nE(^'>^^+^')
(2.28)
The total number of weights for a iVth order pisigma network with n inputs is only (n  1)A^. Compared with the sigmapi structure the number of weights involved in this network is significantly reduced. Unfortunately, when 4>(x) = X the pisigma network does not match the conditions provided by the StoneWeierstrass theorem because the linear .sub.space condition is not satisfied. However, some studies have shown that it is a good network model for smooth functions (Shin and Ghosh, [31]). To modify the structure of the above pisigma networks stich that they satisfy the StoneWeierstrass theorem. Shin and Ghosh [31] suggested to consider the
Functional Approximation
45
socalled ridge, polynomial network (RPN). For the vectors Wg = [ w y i , . . . , Wyn]"^ a n d X = [Xi X2 . . . X „ ] ^ , let < X, Wy > =
n ^ WykXk k=l
represent a inner product between the two vectors. A onevariable continuous function / with form / ( < x,wy >) is called ridge function. A ridge polynomial is a ridge fvmction that can be represented as N
M
for some aij £ 3? and wy 6 SR". The operation equation of a ridge polynomial network (RPN) is given as
N
}
EI
^^
E f l ( <x,wy>t^ji)
(2.29)
when 4>{x) = x the denseness of the above network is easily verified and is described in the following theorem.
T h e o r e m 10 Let fl he the set of all the functions that can he represented by the ridge polynomial network on a compact set S D R":
N
j
n
n j v = { / ( x ) = X ^ n ( 5 ^ w y k X k + »ji):wjik,#jiG!R,xeS j=l1=1 k=l
fi= (J fiw Ar=i
Then Q, is uniformly dense in CIR^]
46
CHAPTER 2
The total number of weights involved in this structure is N(N + l)(n + l ) / 2 . A rompaxison of the number of weights of the above three types of polynomial network structures show that when the networks have the same higherorder terms the weights of RPN is significantly less than that of a HON. In particular, this is a very attractive improvement offered by the RPNs.
7
GAUSSIAN N E T W O R K S
A typical Gaussian network is a twolayered network with an input layer, and a hidden layer of Gaussian units and an output layer of conventional stimmation units as shown in Figure 9. Let x = [xi,X2, ...,xi]'^ and y = [yi,y2i •••)ym]'^ be the input and output of the network, respectively, and u = [ui, U2,..., Un]"^ be the n outputs of the n hidden Gaussian neurons. A Gaussian radial basis function (/>i with a weighted norm is defined by
^(xCiKj=e''(''' [0,1], and is labeled by a linguistic variables F s u d i as '"smair, "medhim", "large", or "very large". T h e most commonlyused fuzzifier is the singleton fiizzifier, which is defined as follows: X e S —> fuzzy set A^ C S with /^^^ (x) = 1 and HA^ ( X ' ) = 0 for X ' € S and x ' / X. T h e fuzzy rule base consists of a set of linguistic rules in the form of "IF a set of conditions are satisfied, T H E N a set of consequences are inferred." Moreover, we consider, in this section, the fuzzy rule base having the M rules with the following forms: Rj (j = 1, 2 , . . . . M): I F i i is A^^ and i z is A{ and . . . and x„ is yl>, T H E N y is B^.
Functional Approximation
51
Fuzzy Rule Base
Input X eS
\1 Fuzzifier
crisp Figure 10
fuzzy sets in S >
\1 Fuzzy Inference
fuzzy in^
1
Output
T
Defuzzifier crisp
The schematic representation of a fuzzy system.
where Xi {i =• 1,2,..., n) are the input variables to the fuzzy system, y is the output variable of the fuzzy system, A^ and B^ are linguistic variables characterized by fuzzy membership functions //^j and /iB>, respectively. A simple example is given in Figure 11. Each rule Rj can be viewed as a fuzzy implication
A{X
...xAi—^
B^
which is a fuzzy set in 5 x 3? with
t*A',x...xAi—.BJ(a^i,• • •,a^n,y) = /i^i(a;i) ® . . . (g) fi^j^ ® fisiiv) for X € S and i/ € Jt. The most commonlyused operations for ® are product and min defined as
I^A]i^i)'^^^Aii^2)
=M^iMyij(s2)
52
CHAPTER 2
and
IJ,^,^{xi)®fi^^ix2)
Y
= min[i^i(xi), 11^1(3:2)1
•\ in Ai
IF X IS
B3
THEN y IS III Bi
B2 ^ / V ^
'*""^'^> 0, there exists a f £ Q such that max {x), we may imply
tj}{x),
i>{x),
4'{x)i'{y) =
T A ( 2 l  i ) + VH^2 + f ) ,
if
x,y>
f,
or a ; <  f , y > f
or  f < i < f , 2/>f if ^ > f , J / <  f , or  f < y < f, a; > f */  f < x < f , y >  f if  f < y < f. ^ < f i/  f 0, without loss of generality, assume e < 1. We will now find a finite collection of constants Ui, Wi, and 6i such that
N
sup xeR
2_] Uia{wiX + 6i) — cosig(x)
< f.
Select A^ such that 1/(7V+ 1) < e/2. For « G {1, 2 , . . . , Af} set
Ui =
1 AT+l
Choose M > 0 such that (T{M) < e/2{N + 1) and a(M) > [1  f./2(N + 1)]. Because (T{.) is a sigmoidal function. Furthermore, for i € {1, 2 , . . . , A'^} set
ri = sup{A : cosig{X) =
Jj—j}
and 1 rjv+i = siip{A : cosig{X) = 1 — } 2{N+l)' Since cosig{.) is a continuous squeishing function such r^'s exist. Next, a choice of the constants Wi and 6i will be given. Let
WiVi + 6i = M
(2.44)
and
Wiri+i +0i =
M
(2.45)
63
Functional Approximation
Then, a unique set of Wi and Oi may be determined by the above twoequations as follows
2M
(2.46)
ri  ri+i
0,
=
M{ri + ri+i) ri  ri+i
(2.47)
It is easy to verify that for MJ, Wi and Oi given by Eqs. (1.44)(1.46)
2~]uia(wiX + 6i) — cosig(x]
< f.
1=1
on each of the intervals (—oo,ri], (ri,r2], ... , (r;v,rAr+i], (r;v^.i,+oo).
D
64
CHAPTER 2
Appendix C: The Proof Of Lemma 2
Since 5 is a compact set and A'^ is finite, there is a M > 0 such that for i€{l,2,...,iV}
M < ^
WijXj + 0i< M,
Xe S
;=i
Prom Lemma 6.1, for every e > 0, there is a set of constants U(, wi, and 6i such that N
sup
y
uia{wiX + ^i) — cosig{X) < . / 7 V ^  u . 
1=1
>=i
Hence,
sup xes
E
cosig{^
Ujcr
=1
(=1
WijXj + 9i)
that is.
N
Q
i=l
1=1
Wl(^Wii^j
sup xes
N
+^i) +^(
Let
/(^) =i =EE"^"''^ i ;=i
 E "i'^''**S'(E '^'>^.' "^ ^"'< 1=1
>=1
n
J=l
f.
Functional Approximation
65
Then sup/(a;)5(x) < e. xes a
66
CHAPTER 2
Appendix D: The Proof Of Corollary 2 We need only to prove that the threelayered network with two hidden nenral layers descrihed by
N,
N,
/(^) = !C"!'^
H Vucr(]^ WykXk + ^'j) + li j=l
k=l
can approximate ^ on S with error < e. For every e > 0, using Theorem 6.8, there is a threelayered network N
n
i=l
j=l
such that
/(x)g(x)
r{P)
by setting V/' e M{X),^B
e BY •• r{P)iB)
= P (/"^(B)).
72
CHAPTER 3
2.1
Definition
A stochastic fuzzy real number is a probability measure on J?. T h e set of ail stochjistic fuzzy real numbers therefore is M(3?)Stochastic fuzzy real numbers can also be introduced in a different way. We put Wj(/), (resp. K T  ( / ) ) the set of all leftcontinuous, (resp. rightcontinuous) nondecreasing ftizzy sets /x on 1ft, for which inf /j(x) = 0, SUp/i(l) = 1.
T h e proof of the following theorem can be found in any s t a n d a r d textbook on probability theory, and we therefore refrain from repeating it.
2.2
Theorem
The sets A^(5?), 5R((/), and ^ril) a^s canonically arc descriptions of canonical isomorphisms.
isomorphic.
The
following
1. The map M(!R) —¥ « , ( / )
.P—^F'p
where for all x & 1R,Fp{x) = P{] — cio,a;[), is a bijection. 2. The map M(!R) —^ ^AI)
.P—^F^p
where for all x e Sft, F p ( x ) = P{\ — 00,1]), is a bijection.
•
In the sequel we shall always use that form of the stochcistic fuzzy real line which is best a d a p t e d to our purpo.se. The ordinary real numbers are of course nicely embedded into t h e stochastic fuzzy real line.
2.3
Theorem
The following
are
embeddings.
Ordering Fuzzy Real Quantities
73
1.
Sft —^ MI)
^^Pi
where Fi = li •^]x,oo[' 2. K —> » , ( / ) : I ^
F;
where F^ = l[i,oo[
K —y >f(K) : X —y P^ where for any Borel set B
^/ 1 PAB) " 1 0
if xeB, i/ X3 B.D
Thus we see that we can interpret A^(!R), as an extension of Jt. The points of 3?, or the Dirac measures, are deterministic points, and the points of A1(3?)/!R are points with uncertainty. For instance, the height of man is not a real number. However when using this expression, one thinks and resons with rejil numbers. Mathematically the height of man is represented as a real random variable defined on the set (or on some representative subset) of all men. If one then makes a histogram of this varible one sees that it is distributed approximately normally. Thus the "real quantity" which we call the height of man can represented by a normal probability measure. Now suppose we ask the question: Is the height of man between 1 and 2 meters? Most people will be inclined to answer, yes. However strictly speaking, the question is absurd and the answer is incorrect. The question should have been: What is the probability that a randomly chosen man has a height between 1 and 2 meters? We can use fuzzy sets to model this in a perfectly simple and beautiful way which is much closer to our intuitive feeling. Rather than saying that the height of a man lies in the interval between 1 and 2 meters we shall give a "degree" that it belongs to that interval. We reason as follows. Always thinking of jVl(!R) as an extension of St, we also extend measurable subsets from 5? to M(^). This is where fuzzy sets are really required. Suppose .4 C K is measurable, there is no canonical setextension to M{^), but there is a uniquely defined and canomicaJ fuzzy setextension defined by 6A : Mi?fi) ^
I: P ^
PiA).
74
CHAPTER 3
Now if A is the interval [1,2] and P stands for the normal probability representing the height of man, then =
6A{P)
P[A),
the degree that P belongs to 6A is precisely the probability of having the height of a randomly chosen man lying in the interval A.
3
O R D E R I N G OF T H E S T O C H A S T I C FUZZY REAL LINE
3.1
Definition
Given two stochastic fuzzy real numbers P, Q S A^(S?), we define p{P, Q) = sup P (]  cxD, x{) A Q {]x, oo[). p{P,Q) should be interpreted as the degree that P is strictly smaller than Q.
3.2
Proposition
The fuzzy relation p : M{^) X M{^) —> / is an extension of the strict order relation on SR, in the sense that if x,y £'Si then xx
Ordering Fuzzy Real Quantities
3.3
75
Examples
1. If P = P^ for some x £ ^, then p{P.,Q)
= and =
p{P,P,)
Q(b,oo[), P(]oo,i[).
This nicely illustrates the intuitive correctness of the fuzzy order relation which we have defined. The degree that the "deterministic real number'' X is strictly smaller than the "stochastic fuzzy real number" Q equals the Qprobability of finding points strictly larger than x. Analogously, the degree that the "deterministic real number" x is strictly larger than the "stochastic fuzzy real number" P equals to the Pprobability of finding points strictly smaller than x. 2. let us suppose that P is a normal probability measure with mean and standard deviation respectively, mi and si. If we put
^«*;fe/"
exp 2 dt
the distribution function of the normal probability measure with mean and standard deviation respectively 0 and 1, then since
P(loc,x[)
= F ( ^ ) , and
Q(]x,oo[)
=
F ( ^ ^ ) ,
it follows that p(P, Q) = sup P(]  oo, x[) A Q{]x, c»[). i6»
is attained in the point x € fi where
p/x^ZjnA
^1
pf^m2\
Since for any x G R, F{x) = 1  F{—x), it follows that X — mi
X — 7Tl2
76
CHAPTER 3
miS2
+ mjSi
S\ + .S2
Consequently p ( P , Q) = F V •«! + «2 /
= 7= / \/27r y_3o
exp Vd(
Again this example nicely illustrates the meaning and naturality of the fuzzy order relation p. T h u s we have e.g. t h a t lim
^ ( P , Q) =
lim
/
exp
I
lim
p(P,Q)=
lim
>
/
—= /
m3T7i,>oo ^ 2 7 r
' dt = 1,
exp"T"d« = 0,
Voo
and t h a t I
p(P,Q)
=
f° _tl 1 / exp 2 eft = 2 if m i = m2. v27r y_oc
3. lets us suppose t h a t P is a uniform probability measure on an interval / i with length 1 and with a mean m j , and t h a t Q is a uniform probability measure on an interval I2 with length 1 and with a mean 77x2. Then one can easily verify t h a t
{
1
if m2 > nil + 1
i ( m 2 — m i  1) 0
if nil  I < 171,2 < mi + 1, if 7712 < TTli — 1.
This implies t h a t p{P,Q) = 1 as long as the entire interval / i lies on the left of the entire interval 12 T h e value of p ( P , Q ) then gradually decreases as / i moves over I2 towards the right, tmtil it reaches 0 when I2 lies at the left of / i . 4. As a last example let us suppose that P and Q have triangular density fimctions symmetric round the points p € !R and q £ 'Si respectively x  p+ 1 X e\p— l,p] x+p+l i e [ p , p + l] 0 elsewhere.
f
Ordering Fuzzy Real Quantities
77
and
(
xq+l x + q+1 0
xe[ql,q] x€{q,q+l] elsewhere.
Then we find that ' 0 ifl + g  p + i ^ l
PiP,Q)={
f(l+?p^) 1
qp<  2 2
lXn> Roll Conlrolttr
•
^
.
—
V(n>V.
k
riKiyriD Conlrolkr
fen)
6 , (n)^
^t^ ~
ton— Utcral
5,(ii)
^t^
Dtrcrtional Dynunfeaof X29A Advanced Tcchnolosy DcmoiMtrator
p,(*;+i)  Vi,k a n d C/(j+i),it  f/(;+i),(ib+i)) times t h e error .sets peak vahies. T h e constant term uses the peak values for b o t h input fuzzy sets a n d t h e o u t p u t values. Therefore, since b o t h effective gain terms a n d t h e constant t e r m are m a d e up of error a n d ciiange in error peak values, changes in any one of the peak values will affect all terms. T h e same is also true with t h e o u t p u t values. Changes in any one of the four o u t p u t values affects b o t h gain terms a n d t h e constant term.
+
T h e terms in (5.14) through (5.16) only apply to the range of operation between t h e peak vahies Ej and Ej^\ for error and between AEk a n d £Jj+i for change in error. T h e width between t h e peak values for t h e next fuzzy sets may not be t h e same as t h e previous sets. Also, the difference in o u t p u t values may not be the same for the next range of operation. Therefore, t h e effective gain values and t h e constjint t e r m m a y b e different. As an example, the terms for t h e next range operation could be: K. pcff
{Ej+2  Ej+i){AEk+2 ,
K.deff
 Afifc+i)
^Ek+2iUj+2,k+l — Uj+l,k+l) iEj+2  Ej+i){AEk+2 ^Ek+i]
(5.17)
Ej+2iUj+l,k+2  Uj+l,k + l) {Ej+2  Ej+i)(AEk+2  A£fc+i) Ej+l{Uj+2,k+l  Uj+2,k+2)  Ej+i){AEk+2  AEk+i]
+ (E^+2
(5.18)
138
CHAPTER 5
Const'
=
^i*^^^^''+^^^+^^^^^^^>;2y:^^;^^^''*"''^*\ where Co, Cu and C j are fixed values determined by the cJiange in error sets. As the error input changes for the fixed value of cliange in error, the Kp^ff will depend on the error and o u t p u t fuzzy sets. T h e same analysis can b e applied to determine the effective piecewise linear K^ef f for a fixed value of error.
5
CLASSICAL CONTROLLER: A SPECIAL CASE OF FLC
Development of a m a t h e m a t i c a l equation for a F L C based on triangular membership hmctions demonstrates t h a t the F L C approximates a piecewise linear classical controller. However, the P F L C or the P D F L C can a t t a i n a constant gain linear control action if the gain terms can b e m a d e equivalent over all ranges of operation. This special case of the P F L C or P D F L C is equivalent to the classical controller counterpart whose gain is constant over its entire range of operation. This hypothesis is verified by comparing
A Special Case of the Fuzzy Logic Controller
139
controller output equations for the FLC and the classical controller and using graphical analysis teciiniques.
5.1
Classical P Controller: A Special Case of PFLC
The classical P controller output is the input error multiplied by a gain. As shown previously, the output of a PFLC is the error multiplied by an effective gain plus a constant term. The values of the gain and the constant term are dependent on the current operating range. To make the PFLC equivalent to the classical P controller, the gain needs to be constant and the constant term should be zero over all the ranges. Analysis of the PFLC output equation gives the conditions for an equivalent classical P controller. A graphical study of the inputoutput relationship verifies the output equation analysis.
PFLC Output Equation
Analysis
The PFLC equation is expressed as Ej+i  Bj
Ej+i  Ej
where e is the input error, U is the peak value of the output fuzzy set, E is the peak value of a fuzzy set for error and u is the output value. The effective gain is the difference between adjacent fuzzy set peak values for error. To make the effective gain constant over all intervals of operation, the numerator and denominator must be constant. Therefore, the difference between adjacent fuzzy set peak values must be constant over all intervals ( i.e. [Ej+i — Ej) must equal {Ej+2 — Ej+\) etc.). For the numerator, the difference between output values for adjacent rules must also be equivalent (i.e. {Uj+\ — Uj) must equal {Uj+2 — Uj+i) etc. ). The constant term has the same denominator as the effective gain, but the numerator is dependent on the error fuzzy sets ajid the output values. For the constant term to be zero, the numerator, Ej+iUj — EjUj+i, must be zero. This requires that the ratio between adjacent error sets and adjacent output sets must be equal, i.e. & l
= ^ .
,5.20,
140
CHAPTER 5
^
^
Controller
Error Output Figure 5
Classical P controller.
The constraint for constajit effective gain requires the error sets to have equal differences between peak values for adjacent members and the difference between output values for adjacent rules must also be equal. Therefore, if the effective gain is constant over all riinges of operation,the ratio in equation (20) will be constant and the constant term will be zero.
PFLC Graphical Analysis The equivalence of the classical P controller and the PFLC can be verified by showing that they have the same output. Graphical analysis of the control action can be applied to the classical controller as well a.s the FLC. Therefore, the classical P controller as a special case of the PFLC can be visually verified by comparing the output graphs of the two controllers. Consider the classical P controller shown in Figure 5 and expressed as: z = Kp • e
where Kp is the proportional gain. Since Kp is constant, the controller output is a line on a graph. Figure 6 is an example of P controller output where Kp is one and the output is imbounded. However, an actual physical implementation of the P controller has a maximum output. The controller in Figure 7 uses a gain limiter to restrict the maximum output. Since the input error is assumed to be normalized, the actual output is n\m{Kp,Kpe). Kp is the gain of the controller as well as the maximum output. Figure 8 shows the controller output with gain of 1 and a maximum output of ± L The control surface for the classical P controller is identical to the PFLC when all peak values of the input and output fuzzy sets are evenly spaced. As shown in Figure 8, the output for PFLC with 7 fuzzy sets for error is linear
A Special Case of the Fuzzy Logic
: •
t
141
Controller
t
I
I
i
j
i
>
j I
i
i
i
I
I
1
:
i
i
i
/ I
:
!
I
i X ^
'.
i
I
i
X
i
k
i i
i
1
0
.5
0 Error
Figure 6 Classical P controller output.
Controller Error OuQ)Ut
Limiter Figure 7 Classical P controller with a gain limiter.
142
CHAPTER 5
i
.9
0 Error
J
Figure 8 Output of a gain limited classical P controller and a PFLC with evenly spaced fuzzy sets.
with a maximum value of ± 1 . The slope of the line for each interval is one and the constant term is zero. This plot is identical to the output plot of the cljissical P controller shown in Figure 8. Thus,by comparing the output plots, it is verified that the PFLC with equally spaced rules is equivalent to the classical P controller.
PFLC Numerical Example In this section, the equivalence between the classical P controller and the PFLC is empirically verified with a simulation. Figure 9 shows the block diagrain of a unity feedback control system with the plant transfer function as Gp{s) = ^T^^ij^and the transfer function of the controller that gives a critically damped step response as Gc{s) = 3.24. An FLC was also designed with seven equally spaced error, cjiange in error and output sets. The controller gain were the same as the classical P controller. Figure 10 shows that the response of the system controlled by the PFLC and the gain limited classical P controller are identical.
A Special Case of the Fuzzy Logic Controller
Plint
Controller Input
1 G (s)  3.24 c
Figure 9
5.2
143
°p('>%(s.3.6)
Output
Block diagram of unity feedback control system with P controller.
Classical P D Controller: A Special Case of P D F L C
The PDFLC as a special case of the classical controller can be verified using similar analysis as the PFLC. First, the output equation is examined to determine if all gains can be made the same and if the constant and nonliner term can be set zero. Graphical analysis is used to show that the inputoutput relationships are equivalent.
PDFLC Output Equation Analysis The gain terms for the error and change in error of equation (5.9) are dependent on the distance between input fuzzy set peak values, output fuzzy set peak values and the rule base. The common denominator of all terms in equation (9) in the product of the difference between adjacent peak values of the membership functions. Therefore, the first criteria for obtaining constant gain terms is that all paek values of fuzzy sets must be equally spaced so that the denominator remains constant. The numerators for the error and change in error gain terms consist of the difference between the adjacent output Vcdues multiplied by the set's peak value. The active output values are a 2x2 sub matrix of the rule matrix. If the elements in the sub matrix differ by equal amounts along the rows and along the columns, then e and Ae terms are nonzero but e*Ae and constant term will be zero. If all sub matrices have the same difference between the elements, then the coefficients of e and Ae will be constant for all the applicable rules. This region of the rule matrix is denoted by the plain type font in Table IV. The region denoted by the bold type font in Table IV is where the coefficients
144
CHAPTER 5
\
I
\ i j
j
i
I
"
/\
[
/ i
:
\
ij i
/ 11
!
!
'
j i
i>n.c
/
i
i /
1
i
1 •
•
/j
'^..l
i
i.J i JL..i..I
.J
1
1.S
1
i i
i1
/
;
i
1
0
p
/ /
t

^ /
j _1..J._L .1 J l . i
2
2.5
1 l._
I
I
...I 1 1 1 l..X.l.J>
3.3
4
1 1 1 1 . I > 1
4.9
TlnM
Figure 10 PFLC.
Closed loop system response of the classical P controller and the
A Special Case of the Fuzzy Logic Controller
145
Error
Change in Error
NB NM NS
zo PS PM PB
Table 4
NB NB NB NB NB NM NS ZO
NM NB NB NB NM NS ZO PS
NS NB NB NM NS ZO PS PM
ZO NB NM NS ZO PS PM PB
PS NM NS ZO PS PM PB PB
PM NS ZO PS PM PB PB PB
PB ZO PS PM PB PB PB PB
Rule matrix showing linear, nonlinear and constant regions
of e, Ae, and e*Ae are zero and the constant term is one because the differences between the elements along the rows and along the columns of the 2x2 submatrices are zero. T h e region denoted by the shaded area is a small nonlinear transition region where all four terms of equation (9) have some value in between the two previous regions,. Therefore, except for the small nonlinear region, the P D F L C behaves as a controller with a gain limiter.
PDFLC and Classical Control Surface Analysis Two controllers are equivalent if they have the same control surface. Surface analysis of the control action can be applied to b e the classical controllers as well as the F L C . Therefore, the classical P D controller as a special ca.se of the P D F L C can be visually verified by comparing the two control siirfaces . Consider the classical P D controller shown in Figure 11 and expressed a.s: z = Kd • Ae + Kp • e, where Kj is the derivative gain and Kp is the proportional gain. Since the two control gains are constant, the control surface is a plane as shown in Figure 12. This control surface extends indefinitely in all directions of the plane. T h e physical limitation of a controller model is the m a x i m u m amplifier o u t p u t it can provide. T h e classical controller model in Figure 11 can be modified with a limiter to set a maximum control vahies as shown in Figure 13. T h e error input is assumed to b e normalized and t h e change in error input is multiplied by the gain ^ . T h e o u t p u t is multiplied by Kp. This
146
CHAPTER 5
Error ^
Controller Output
Figure 11
Block diagram of cl^lS8ical PD controller.
1.1. Figure 12
Output of classical PD controller.
A Special Case of the Fuzzy Logic Controller
147
Emr
F i g u r e 13
Model of a physical implementation of a PD controller.
F i g u r e 14
Output of gain limited classical PD controller.
model can be expressed as mixi{Kp, Kp{^ Ae + e). Figure 14 is an example of the controller output for the PD output control surface with both control gains set to one.{Kd = Kp = 1). The limiter values are 1 and 1 which are the same as for the fuzzy controller. As expected, the resultant surface is a plane with cutoff at 1 and 1. The control surface for the classical controller is nearly identical to the FLC when all peak values of the input and output fuzzy sets are evenly spaced. As an example, Figure 15 shows the control surface of a fuzzy controller with seven sets that have equally spaced peak values for error, change in error and outptit. The universe of discourse for the sets is between 1 and 1. Therefore, peak values for the seven membership functions of each control variable and the output values are 1, —§,— 5, 0, j ,  , and 1, respectively. Note that the
148
CHAPTER 5
1.1. F i g u r e 15 fuzzy sets.
Control surface of PPLC with 7 error, change in error and output
control surface is mostly planar with a small nonlinear region at each limit. In the planar region, the control surface is equivalent in gradient and orientation to the classical PD control surface. If the small nonlinear region is ignored, the control action of both controllers are equivalent. As the number of input and output sets and corresponding rules increase, the nonlinear region becomes less significant [1, 2]. All gains for the simulated classical PD controller and PDFLC were set to one. However, if equivalent gains were applied to eacJi controller, the control surface shape and orientation cJianges accordingly. For the PDFLC, modifying the input variable's gain is the same as expanding its universe of discourse. The widths of all membership functions still remain equivalent and all gains will effectively remain constant.
Numerical Example In this section, the equivalence between the classical PD controller and the PDFLC is empirically verified with a simulation. Figure 16 shows the block diagram of a unity feedback control system with the plant transfer function given by Gp{s) = i^^^ g^ and the transfer function of the controller that gives critically damped step response cis Gc{s) — 16.4s + 100. An FLC Wcis also designed with seven equally spaced error, change in error and output sets.
A Special Case of the Fuzzy Logic Controller
CoDtrDlkr
149
Pimt
Input
^
J F i g u r e 16
G(s) 16.4t+100 c
G W rXr?, p^' »(«+3.6)
Output
Block diagram of a feedback control s y s t e m .
The controller gains were the same as the classical PD controller. Figure 17 shows that the response of the system with a PDFLC and the gain limited classical PD controller are identical but the imlimited output classical PD controller has a little faster response. The outputs of each controller as shown in Figure 18 indicates that the imlimited PD controller is able to produce a large control action and gives a faster response. Now consider a system with the plant transfer function as Gp(.s) = gigl.25) and the transfer fimction of the controller that gives a critically damped step response as Gc(s) = 3.94s + 100. Figure 19 shows that the closed loop system response for all three controller are identical. The change in error does not reach as large a value as the previous example so the maximmn controller output is never reached. Figure 20 shows that the outputs of the three controller are similar.
6
SUMMARY
This study uses mathematical expressions for fuzzification and defuzzification to derive an inputoutput equation for a specific PFLC and PDFLC. these equations demonstrate that the PDFLC and PDFLC have similar form as their classical counterparts. However, unlike their classical counterparts, the FLC is piecewise linear. A detail analysis shows that the classical controller is a special case of the FLC. First, the mathematical expression of the PFLC and PDFLC is studied. The PFLC equation consists of an effective gain and a constant term. By
150
CHAPTER 5
F i g u r e 17 Closed loop response for plant controlled by classical PD controller and PDFLC.
A Special Case of the Fuzzy Logic Controller
Figure 18
Claissical PD controller and PDFLC output for example.
151
152
CHAPTER 5
''•J>^—
;
i'
• :
/ i / 1 / / / / / f
/ / 1 / / J \ •_ j  /
.1 l
i
'
I
i 1
i j ' ! ! ! t
1 0
1
uzzy
' 1
i
i
DUfnJtod
•_ j
'
'
l
i
i
i
!
1
i 1
L
i _ 1 . .U . . J .  L
i
U...
1
1
1
1
1 _
Tim*
Figure 19 Closed loop response for plant controlled by classical PD controller and PDFLC.
i
A Special Case of the Fuzzy Logic Controller
153
120
• •
s \ M
1

PO
PO Llm and flj (with the restriction fis > 0), determine the behavior of the filter, which can be used to fine time the compensator on line. FVom equation (6.9)(6.12), we have
K =
aVTO
y/H,^ 1
P =
1 1
Vfh(fh+2Vfhfh)
(6.14)
and the steady state control signal is
w(o = [vTo, v^v^]i(0, and the filter is
(6.15)
Real Time Control of a BeamandBall System
dx{t) dt
/Si+ZVft/Sa
171
/3i+2v'/93/33
!/(^)(6.16)
x(t) + av/IO,/^
av/5ayi0
with x{0)=E[x{Q)]. Note the above two equations are also a dynamic system in state space representation, with y[t) as its input and u{t.) as its output. This observation will simplify the simulation of the control system as described later. Having discussed the case for r = 0, we now generalize the control system to include r 9^ 0, i.e., a nonzero set point. For this, consider the coordinate transformation
'
Xi — r
_
Zl
[ ^2 J
[
X2 J
=
a; —
r
(6.17)
[ UJ
The LQG problem becomes
minE[zf{T)Sx{T)+ »('SI i v
11 t I nn
coniijiik scK NCI
B f l L L  f t N l l  O f f>n S v C H n I N T L K i n C C CO^'VH^CHI fififlCH ig{^)
u;U)CG
194
CHAPTER 7
As mentioned above, it is known t h a t IJJCG and 6m are related to speed of response and damping characteristics, respectively. T h e main purpose of phaselead compensation is to design a control system such t h a t d a m p i n g characteristics are satisfied without losing speed of response, t h a t is, to increase phase margin without changing UJCG By using the transformation m a t r i x of Eq. (7.9), we can increase phase margin without changing OJCGRijii [6] has reported t h a t a phaselead compensation in fuzzy control systems can b e realized if we use the following transformation matrix, cos(—^c) &m{6c)
— sin(—^c) cos(^
 ^ + A())
0+A()
Figure 4
~+A4)
7t+A(})
Fuzzy sets.
The parameters of the fuzzy compensator are a, b, LJCG and ^c in Eq. (7.11) and A $ in Figure 4. The controller design is to determine these parameters. Since Rulel and Rule 2 are related to speed of response and damping characteristics respectively, we should determine a and 6 such that speed of response is satisfied, that is, Tp < Top and uica and 6^ such that the damping characteristics are satisfied, where £p < eop If ^op < Tp or eop < Sp after determining a, 6, wcG and 6c, the value of A $ can be used as a fine adjustment parameter. If the value of A $ is decreased, speed of response is improved. Conversely, if the value of A $ is increased, damping characteristics is improved. Therefore, we should decrease it if Top < Tp. We should increase it if eop < £p. Figure 5 shows fuzzy partitions for each value of the premise parameter A $ . Each region sucJi that membership value of each rule is more
196
CHAPTER 7
than 0.5 is shown. As shown in Figure 5, the regions by changing the premise parameter A $ can be adjusted. The above points will be considered in design procedures. The design procedures will be given in Section 3.2 and 3.3 for linear, or nonlinear systems.
3.2
Design Procedure 1 (The Case Of Linear Plants)
Assume the plant G(.i) is linear. The procedure consists of five steps . [Step 1] Select a desired transient response of control system with a overshoot £op and a time to peak Top. From the value of Sop and Top, calculate the values of ^Oi Wom ^Om and ijJocG by Eqs. (7.1) ~ (7.4). From Theorem 1, judge whether a fuzzy phaselead compensator should be used or not. If a fuzzy compensator should bo used, then go to [Step 2]. [Step 2] Assume that Of. = 0. Then, since n = a' and b = b', the fuzzy controller is reduced to a linear controller Gc(s) = 3±M "phe purpose of this step is to determine a and b in Rule 1. The speed of response is not satisfied if we use a and b calculated from Eqs. (7.5) and (7.6), because the value of a real root is close to those of real parts of complex roots. This means that speed of response is affected by the real root. So, calculate a and 6 by substituting the following C^o' instead of C^g into Eq. (7.5) and (7.6). Co'
(31 h)iJOn
The above equation can be derived by solving
^'"^'"  2 + il + h) for Co' • [Step 3] If the linear controller satisfies damping characteristics, that is, Ep < Eop, then end, else calcalate the phase margin 0^ and the gain crossover frequency uicG of Gc{s)G{s). Next, derive a' and b' from the transformation matrix T{OC,UICG) of Eq. (7.9), where
Frequency and Transient Characteristics
197
[ s t e p 4] Assume t h a t A $ = 0. Investigate whether Tp < Tap and Cp < Cop or not. If b o t h of t h e m are satisfied, then end, else go to [Step 5). [ S t e p 5] A $ if Tp and £p > adjusting [Step 2].
3.3
Adjust the premise p a r a m e t e r A Top and £p < eop. Conversely, increases it if Tp < Top £op If the desired transient response can not be realized by the premise parameter A $ , then h = h + 1 and go back to
Design Procedure 2 (The Case Of NonLinear Plants)
Design Procedure 1 can not b e applied to unknown or nonlinear systems since frequency characteristics can not be used in this case. However, without loss of generality, Design Procedure 1 can be extended to a design procedure for unknown nonlinear controlled objects, because there are clear relations between frequency characteristics and transient characteristics .
[ S t e p 1] Selet a desired transient response of control system with a overshoot £op a n d a time peak Top[ S t e p 2] Assume t h a t 9c = 0. Then, since a = a* and b = b*, the fuzzy controller is reduced to a linear controller Gc{s) = s±M_ Determine a and b in Rule 1 such t h a t Tp < Top. [ S t e p 3] If the linear controller satisfies damping characteristics, t h a t is, £p < eop, then end, else go to [Step 4]. [Step 4] Assume t h a t WQCG = 1 and A $ = 0. Increase gradually the value of 6c of Rule 2 until £p < £op. This means t h a t a* and h* of Rule 2 are adjusted. Investigate whether Tp < Top and (p < ^op or not. If b o t h of t h e m are satisfied, then end, else go to [Step 5]. [ S t e p 5] Adjust the premise p a r a m e t e r A $ . T h e value of A $ decreases if Tp > Top and Cp < eop Conversely, it increases if Tp < Top and £p > £op. If the desired transient response can not b e realized by adjusting the premise p a r a m e t e r A $ , select other value of a and 6 such t h a t the value of Tp is smaller, where 6c = 0, t h a t is, a* = a and h* = b, and go back to [Step 3].
198
CHAPTER 7
4
DESIGN E X A M P L E S
We illustrate some examples of the above design methods.
4.1
Example 1
Let us consider the following pleint j/ =  1 0 y  1 6 y + 2 . 4 5 M .
(7.13)
We design a fuzzy compensator by Design Procedure 1. Fujii [6] reported that a desired transient response is realized by a fuzzy controller. However, we show that the desired transient response is easily realized by a simple linear controller. In other words, it is sufficient to use a linear PI controller in this case. In [6],eop =6. [%) and Top = 1.2[sec.]. FVom Eqs. (7.1)~(7.4), we obtain C = 0.68, 6om = WoCG
=
Won = 3.51, 63.34[deg], 2.35 [rod/sec].
Then, CoWon = 2.39