SYSTEMATIC METHODOLOGY OF FUZZY-LOGIC MODELING AND ...

SYSTEMATIC METHODOLOGY OF FUZZY-LOGIC MODELING AND CONTROL AND APPLICATION TO ROBOTICS

MOHAMMAD REZA EMAMl

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mechanical and Industrial Engineering University of Toronto

O Copyright by Mohammad R. Emami 1997

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TO Zohreh, Ali, Zahra, and my parents

SYSTEMATIC METHODOLOGY OF FUZZY-LOGIC MODELING AND CONTROL AND APPLICATION 70 ROBOTICS Doctor of Philosophy Mohammad Reza Emami Department of Mechanical and Industrial Engineering

University of Toronto

I ABSTRACT (1 This thesis presents a systematic approach to füzzy-logic modeling and control of complex systerns. In the proposed methodology, the füzzy mode1 of the system and control rules are obtained from input-output data with no need of a priori information. The proposed fuzzy modeling methodology has three significant features: (i) a unified parametenzed reasoning formulation; (ii) an improved fuzzy clustering algorithm. and (iii) an efficient strategy of selecting significant system inputs and their membership functions. The proposed fuzzy control stmcture consists of a fuzzy mode1 of the system and robust hzzy rules in order to ensure stability and satisfactory system performance. We develop a generalized formulation of sliding mode control for a class of nonlinear muti-input multi-output systems. This formulation has two distinguish features: (i) it is applicable to "black box" systems with no need to identify interna1 parameters or to assume specific propenies; (ii) it is possible to design the robust controt command for each system state independently while the stabiIity and robustness of the entire system is guaranteed. We apply the generalized formulation to andysis of the stability and robustness of the proposed fuzzy-logic control system. We also derive guidelines for designing the robust fuuy control niles. We apply the methodology to rnodeling and trajectory control of a four degree-of-freedom robot rnanipulator. Results of the proposed fuuy-logic methodology are cornpared with those of a complete analytical simulation and a heunstic f u u y rnodeling technique. A supenor rnodeling performance in t e m s of accuracy and simplicity is obtained. The control performance is also compared with high-gain servo controllers for different trajectories. and a higher performance is achieved.

iii

I wish to thank my CO-supervisorProfessor Andrew A. Goldenberg for providinp me the best environment for study, research, and training. Without his invaluable guidance and advice. this work could never have been completed at this level. 1 am particularly gratehil for his moral support and encourasement in ail aspects of my life. What 1 learnt frorn him is far beyond the technical context. 1 would like to extend my sincere gratitude to my other CO-supervisor. Professor 1.

Burhan Turksen for giving me a deeper understanding of fuzzy set theoiy and fuzzy logic. 1 always enjoyed discussing absuact concepts and theories with him.

During my association with the Robotics and Automation Laboratory. 1 gained a lot of experience from many scientists and experts. iMy thanks are due to d l of them: specially to Professor Nenad M. Kircansky who inuoduced the IRIS facility to me. and was always there when 1 needed help. and to engineers Jacek Wiercienski. Pawei Kuzan. and Rafi Barakat for their technical help in design and implementation. Thanks to al1 my colleagues in R U for providing a peaceful and friendly environment. 1 would dso like to thank The Minisvy of Culture and Higher Education of the Islamic

Republic of Iran for its financiai support during rny snidy. My special thanks are due to Professor Reza Hosseini. the Higher Education Advisor. for his endless cffons to facilitate Our study in Canada. Last but not least. I owe thanks to my family. to whorn this thesis is dedicated. my wife Zohreh for her greatest moral support in ail moments of this research. and my parents for their everlasting patience and encouragement.

ABSTRACT

iii

ACKNOWLEDGMENTS

iv

TABLE OF CONTENTS

v

LlST OF FlGURES

viii

LlST OF TABLES

xii

NOTATION

xiii

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

CHAPTER 1 : INTRQDUCTW

1

1.1 : Motivation.................................................................................. 1.2 : Notion of Fuuy-Logic Modeling and Control................................. 1.2.1 : Fuzzy Sets and Fuzzy Logic .................................................. 1.2.2 : Fuzzy-Logic Modeling.......................................................... 1.2.3 : Fuuy-Logic Control............................................................ 1.3 : Background and Outline of the Thesis ......................................... 1.3.1 : Fuzzy-Logic Modeling.......................................................... 1.3.2 : Fuzzy-Logic Control............................................................. 1.3.3 : Application of FLC to Robotics............................................... 1.4 : Contributions.............................................................................

CHAPTER 2 : W S O N I N G P m C F S S IN

M O D M

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

2.1 : Introduction................................................................................ 2.2 : Fuuy Connectives..................................................................... 2.2.1 : Fuzzy Cornplernent............................................................. 2.2.2 : Fuzzy Set Intersection and Union........................................... 2.2.3 : Extension of Triangular N o m and Conorm Functions.................

2.3 : Implication of Individual Rules..................................................... 2.4 : Aggregation of the Rules........................................................... 2.5 : lnference of the Rule Set............................................................ 2.5.1 : Reasoning Based on Mamdani's Approximation........................ 2.5.2 : Reasoning Based on Fomal Logical Approach......................... 2.5.3 : Unified Pararneterized Fuzzy Reasoning Method.......................

2.6 : Defuztification of the Output....................................................... 2.7 : A Simplified Parameterized Reasoning Formulation...................... 2.8 : Conclusion.................................................................................50

C L U S T W G fl F

CHAPTER 3 : 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

CHAPTER 4 :~

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

mM

W

......... 51

: Introduction................................................................................ : A Brief Background............................................................... : F u n y c-Means Clustering Algorithm ......................................... : Cluster Validity: Specification of the Number of Clusters................ : Selection of Weighting Exponent (m) in Fuuy Clustering.............. : Initial Guess and Local Optimality in FCM Algorithms.................... : Formation of Membership Functions............................................ : Conclusion................................................................................

-

L

O

G

I

C

M

R

l

T

T

H

51

53 54 56

62 68 71 74

........................... M 76

4.1 : Introduction................................................................................ 4.2 : Input Selection in F u u y Modeling................................................ 4.2.1 : Background............................................................................ 4.2.2 : The Proposed Method.............................................................. 4.3 : Assignment of Input mernberships............................................... 4.4 : Fuuy Parameter Identification.................................................... 4.4.1 : FuInference Parameter Optimization.................................. 4.4.2 : Membership Parameter Tuning.............................................. 4.5 : Fuzzy Modeling Algorithm........................................................... 4.6 : Case Study ................................................................................ 4.7 : Conclusion.................................................................................

CHAPTER 5 :SYSTWTIC, D -

AND A u Y S I S _ Q F T a

............**........................*................

94

5.1 : Introduction...............................................................................94 5.2 : Robust Model-Based Fuuy-Logic Control : Design and Analysis ..... 95 5.2.1 : The proposed Fuuy-Logic Control Structure.............................. 95 5.2.2 : Fundamental and Theorems.............................................................. 98 5.2.3 : A Generalized Formulation of Sliding Mode Control.......................... 102 5.2.4 : Design of the Robust F u u y Control Rules ..................................... 110

5.3 : Case Study.................................................................................116 5.4 : Conclusion........................................................................................ 121

CHAPTER 6 : EPPLIWTION TO ROBOTICS; . . . . . . . . . . . . . . . . . .m. . . . . . . . . . . . . . . . . * . * . . . . . . . . .

122

6.1 : Introduction.................................................................................122 6.2 : Experimental Setup...............................................................123 6.3 : Simulation................................................................................ 126

6.3.1 : Desirable Configuration for experiments................................... 6.3.2 : Kinematic and Dynamic Parameter Estimation........................... 6.3.3 : Dynamics Model of the IRIS A m ............................................

6.4 : Fuuy-Logic Modeling and System Identification........................... 6.4.1 : Experiment. Data Acquisition and Analysis ............................... 6.4.1.1 : Setup Preparation.................................................. 6.4.1.2 :Test Plan............................................................. 6.4.1.3 : Data Selection and Processing................................. 6.4.2 : System Identification Procedure.............................................. 6.4.3 : Fuzzy Model Validation ......................................................... 6.5 : Fuuy-Logic Control.................................................................... 6.5.1 : Design and Analysis............................................................. 6.5.2 : Experimenta! Results............................................................ 6.5.3 : Comparison Study of the Results............................................

CHAPTER 7 :C O N ~ S l A Ow~ FUTU-RCY

.......-...........m...... 165

7.1 : Conclusions...................................................................................... 165 7.2 : Future Research............................................................................... 167

vii

L/ST OF FIGURES set t h e o s............................................................ 4

A general vierv ofji4,-

Membership firnction of the lingriistic variable "error"..........................

5

The principal stnictrire of fic~z-logiccontrol.................................... 8 Flow chart offiru7 -stem modeling ........................................................ 11 Parameteraized t - n o m (lefr)and t-conorm f right)for

.

.

p=.0001 (top) p = l (middle) p= 100 (bonorn).........................................

25

The algorithm of calciilating t-conorm.................................................... 28 The nlgorithm of caicrdating t - n o m...................................................... 2

The@-

mode1 of a nonlinenr ?stem ....................................................

F t i ; ~ otirprlt of the nonlinear ?stem for input set [ I . X . 3.0. 4.21........ Defil~ifiedoirrput of the nonlinear ?stem for input set [Z.Zj. 3.0, 4.21

Approximate

(

7' J and exact f y' ) dejçzified orrtpprrrjor irtpiplrt set

[1.25. 3.0. 4-21........................................................................................ Yager's ( 7; . ) and proposed sinzplijïedji~nctionf

7' )for e.rnmpie 2.1 ..

Sugeno 's ( 7 ; ) and proposed simpiified firnction f The algorithm ofjr?

7' )for e-mrnple 2.I

c-means clustering .............................................

Trace of fi^^ total scatter rnatrik as n fiinction of rnfor Normal4 data set............................................................................................................ Cluster validity index s.a

fitnction of c for Normal-4 data S..........

Trace off ri;^ total scatter rnatrk as n fimction of rn for the data of example 3.2............................................................................................. Clusfer v a l i d i ~index s, as n firnction of cfor the data of e-ample 3.2

viii

8

LIST OF FIGURES

î7te M C algorithm for assigning the initial cluster centers.................... Index scs for rhe data in Eromple 3.2 rvitli AHC algorithmfor choosing

.

.

VO (m=3) . ................................ .

,

The k - N N f u q classification algorithm................................................. Membership frinctions for the the system output in Erainple 3.2 wit/zo~it classification........................ ..................................................................... Membership functions for the the -rem

outpitt in Erornple 3.2 rvith

classification...................... . . ................................................................. Q~ialitativeillitsrration of the effect of input variable .r, ou the i" nde... "Peak" points should be the sarne for input and orltp~irclrtsters............ n t e algorithin of significnnt input selectioiz............................................ One oicrpur cluster with nvo corresponding inpl 1 :

For O < Uik < 1, we have:

and tr(ST) is strictly monotone decreasing function (evaluated at optimal pairs of (U.V))

of m on the interval (1,m). In conclusion, as rn varies from one

CO

infinity, t r ( S ~ )

monotonically decreases from tr(STh)to zero. Consequently. an appropriate value for m is

.

what holds tr(ST) somewhere in the middle of its domain (tr(STh) O). reminding that tr(STh)is a constant value depending only on the data set. There is another restriction for >n from the fuzzy modeling point of view. As mentioned earlier. as ln becomes Iarger, the membership assignment becomes "fuzzier". such that, for large values of ni, ail membership grades tend to become Uc.

[n

fuzzy

modeling, Our desire is to have fuzzy clusters specific enough to have at least one data point with membership grade equai or close to one. Let nt,, be the maximum value of m which satisfies this condition for al1 partitions solved by FCM algorithm. Now. if the value of in, selected such that tr(ST) lies somewhere in the middle of its domain, is less than mm, we can trust our cluster validity by s, and that

ni

would be Our choice for the

FCM algorithm. Othenvise, mm, should be selecteci for FCM algorithm. Since tr(Sr, is a function of number of clusten c as well as rn, the process of choosing ln and c should be

performed iteratively, starting from choosing an initial m. then deriving s, for several

FUZZY CLUSTERING IN FUZZY MODELING

CHAPTER

nurnbers of clusters. getting the optimum c. and finally checkmg if for this c and m. u(ST) is satisfactorily far from its limits; otherwise, the process should be repeated by a new m.

EXAMPLE 3.1 As a practical example. we use Normal4 data set. which is a sarnple of N=800 points

consisting of 200 points each from the four components of a mixture of ~ 4 M-variate . nomals. with the population mean of p = 3e and covariance rnatrix of Z=L for each component. Pal and Bezdek (Pal, 1995) implement this data set to inspect the reliability of some cluster validity indices. In Figure 3.2, h i u y total scatter matrix is plotted as a function of rn for different numbers of clusters. It is observed that the reliable domain for VI

(values of m which make tr(Sr) be in the rniddle of its range) is [ l . j . 2.51 for c=I-IO.

For c=2, the appropriate choice would be m=2. Figure 3.3 shows the behavior of cluster Li

validity index s,, as a function of c, out of the reliable domain for two values of m. Le.. rn=3.0 and m= 1.2. From Figure 3.3. It is observed that

would be the optimum number

of ciusters for rn=2.

1

2

3

4

5

6

7

WElGHTlNG EXPONENT (m)

Figure 3.2 : Trace offriu.~total scatter rnatrl~as afrtnction of nz for Normal4 data set

NUMBER OF CLUSTERS (c)

Figure 3.3 : CIrrster validic index s,, as a jimctinn of c jor ~Vormal-4dam set

-

ln (Sugeno. 1993). Sugeno and Yasukawa introduce a nonlinear static system with two input variables x, and .Y:

. and a single output

as follows:

50 input-output data are availabie. and the output data are to be clustered by FCM algorithm. For the same data, the trace of the fuzzy total scatter matrix is shown in Figure 3.4 for different number of clusters as rn varies. In order to choose initial Vo of the FCM

aigorithm. a number of random searches should be perfomed to obtain the optimum for each c (this will be discussed in the next section). For this case, the reliable zone is

.

rnc [2.7 3.61. Figure 3.5 presents , s as a fùnction of c for three different values of m. For

m=3, the optimum number of clusters is cIearly c=7.

3.6 INITIAL GUESS AND LOCAL OPTlMALlTY IN FCM ALGORITHMS The third problem in FCM algorithms &ses from the fact that these algorithms may produce only local minima or partial optima points. Therefore. a different initiai guess for rnean vectors vi may conclude different optimum results. It should also be mentioned that "convergent' algorithms do not necessady stop at local minima. i.e.. they may stop at

saddle points or local maximizers. as well (Isrnail, 1991). This fact affects cluster vdidity as well as cluster andysis. In order ro show how serious this problem could be. let us consider the following example.

EXAMPLE 3.3 Returning to Exarnple 3.2. in order to assign the optimum nurnber of clusters. using the Fukuyama-Sugno criterion. Sugeno and Yasukawa obtained curve # 1 shown by the solid line in Figure 3.6. They conclude that the best number of clusters is c=6. By using the same data. and the same criterion. and the same power exponent m=2 (although it was shown in the previous section that for this data set. m=7 is out of the reliable domain). but with different initial values of Vo, we obtain a different optimum number of clusters as shown by curve #î in Figure 3.6. i-e.. c=8. In fact. by trying several random initial guesses and choosing one which minimizes the Fukuyama-Sugeno index. we will have curve #3 in Figure 3.6 which is quite different from curve #1. Two solutions have been suggested for this problem. so far. One is to select the initial prototypes by using some knowledge about the data (Pal. 1995). which is contextdependent, and is not always possible. The second is the repeated use of the FCM algorithm for different randornly chosen initial prototypes. and then selecting the best arnong the generated solutions (Kamel, 1994). T h s method is time-consuming and even

after a long search. it is not guaranteed that the best solution is achieved. because there is no guideline for selecting initial mean vecton in this method.

FUZZY CLUSTERING IN FU=

CIUPTER

L

i

1

1- 1

MODELING

cuve # 1 : one random Vo curve #2 : different randorn Vo

r----:

NUMBEI? OF CLUSTER ( c )

Figure 3.6 : The F~lkrqarna-Srrgenoindex for rhe data of E-rarnple 3.2 rrsith differertr V,] !rn=2)

In order to efficientiy obtain a preference for initial locations of ciuster prototypes. we implement an Agglomerative Hierarchical Clustering (AHC) algorithm t Duda. 1973) as

an introductory procedure to find proper suggestion for the initial locations of cluster prototypes for the FCM aigorithm. The AHC al=orithm places each of the N aata vectors in individual clusters. Then. by defming a dissimilarity matrix (depending on the method). it starts to merge two or more of the trivial clusters. getting the second level of data partition. The process is repeated to form a sequence of nested clustering in w h c h the number of clusters decreases as the sequence progresses until the required number of

clusters c are obtained. The specific method used in this research is Ward's method (Ward. 1963). Having unlabeled object data X={xl. x:.. ...xx J, the basic algorithm for this method is s h o w in

Figure 3.7.

STEP 1 :CHOOSE number of clusters c; the mauix of dissimilarities D=[dij] as the following Euclidean-based distance:

where, Vhi and vh, are mean vectors of hard clusten Xi and X,, respectively.

STEP 2 : LOOP FOR t = N t o c

;

andXi.,=xi. i=1 .2....fl

FIND the pair of distinct clusters which have the minimum dij,say X,., and Xj.1: MERGE X,., and X,., :

DELETE X,.,; iWXT t -

-

-

--

-

-

-

Figure 3.7 :The A HC algorirhrnfor assigning the inirinl cluster cenrers

The result of the above process is c hard clusten for the data. which would be a good start for fuzzy clustering procedure. By this method. we have a preference to choose the initial prototypes without any knowledge about the data n priori: and it is more efficient

than random search arnong different initial gesses. This strategy has an effect on Our cluster validity criterion.,s

By choosing the hard

clusters that corne out of the agglomerative hierarchicai clustering as the initial guess.

SC,

has a monotonic decreasing tendency for large values of c. Therefore. for cluster validity. it is recornrnended that one should plot s,(U,V;X) as a function of c. and then select the

starting point of the decreasing epoch as the maximum c (c,). Then the optimum value of c is obtained by minimizing s, over c-2,3,.. ..cm. Figure 3.8 shows the cluster validity for data in Example 3.3 (m=3. as discussed in Section 3.5). Decreasing tendency after cmK=22 is observed and therefore. the optimum number of clusters between 2 and c,,

is c=8.

CRAPTER

FUZZY CLUSTERING IN FUZZY MODELING

Figure 3.8 : Index ,s for rhe data of Erampk 3.2 rr-irhAHC algorirhm for choosing V Ilrn=31

3.7 FORMATION OF MEMBERSHIP FUNCTIONS After assigning the appropriate fuzzy clusters for the output sample data. the next necessary step in fuzzy rnodeling is to f o m the membership functions for the entire output space. One method is to directly estimate the mernbership grades uiJ derived from the fuzzy clustering by suitable trapezoids as it is implemented in (Sugeno. 1993). and to use the approximate trapezoidal functions as a classification for the entire output space.

If the number of sample data used for clustering is adequately large. and if the behavior of the system output is srnooth enough. we may be able to extend the partition obtained from the sample data to the entire space. However. in general. there should be a classification step between the two steps of formation of rnembenhip functions for the output space, Le., clustering step and approximation by trapezoidal functions. We should emphasize the difference between ''cllistering" and "clnssificrrrion". In clusterin_oprocess.

we make suitable partitioning for data set X c K

'.whereas in classification procedure.

FUZZY CLUSTERING IN FUZZY MODELING

CHAPTER

every data point in the entire space R h is labeled. Therefore. the problem of rnembership function formation for the entire output space lies in the category of classification probierns. Since classifier design is usually perfonned by labeled data. clustering the

sample output data is a good tooi to design the appropnate classifiers for the entire output space. From the probabilistic point of view. in order $0classifj the output space. without any assumption of knowing the probabilities and the state conditional densities of al1 classes. the k-Nearest Neighbor (k-NN) algorithm is a sub-optimal procedure. i.e.. its use will usually lead to a probability of error in classification decision making close to minimum possible error rate (Duda. 1973). Keller et al. Keller. 1985) introduce a fuzzy generaiization of k-NN classifying as the algorithm shown in Figure 3.9.

STEP 1 : DENSE

X={xi.x2.....xx}c 2 * of labeled data set wirh membership grade U=[u,,] . For any x of unknown classification.

STEP 2 : CHOOSE number of neiehbors k : 1 < k 5 N : n o m for the disrance Ilx - x,il j= 1.2.....3.

STEP 3 : FND k-nearest neighbors to x among X= { XI .x,.. ...xx } .

STEP 3 : LOOP FOR i=l t o c CALCULATE the rnembership grade to x in class i as :

NEXT i Figure 3.9 : The k-NNftr~? classrficarion algo rithm

CHAPTER

CLUSTERING IN FUZZY MODELING

FU=

73

After partitioning the entire space. in order to obtain simple membership functions, we c m approximate the classified data by trapezoidal functions in a way that. for each fuzzy cluster, convex points are picked up and a trapezoid is fitted to them (Nakanishi. 1993).

EXAMPLE 3.4 We apply the above strategy to data of the nonlinear static system in Example 3.2.

Figures 3.10 and 3.1 1 are output membership functions when the fuzzy clusters are directly approximated by trapezoids. and when the classification process has been performed in between. respectively. Al1 other parameters are identical for both cases (nz=3, c=8). The difference in the results is obvious.

I

OUTPUT ( y )

Figure 3.10 :Mernbersliip frutctiotrs for the systern ortrprtt in Ecarnple 3.2 \r.itltortt classification

- -

-

- -

--

Figure 3.11 :membership fiincrions for the ?stem oltrpur in Erample 3.2 witlz class~catiori

3.8 CONCLUSION in this chapter, we discussed the rule generation phase of fuzzy modelinp. By using fuzzy c-means clustering algorithm, we assign the output clusters that are fuzzy sets of the consequent parts of hizzy mode1 niles. In other words, we speciQ the 'THEN" part of the fuzzy IF-THEN mies. In the following chapter, we focus on how to identify the input

fuzzy sets, Le.. the "IF' parts of fuzzy niles. Three major problems in fuzzy c-means clustenng algorithm were considered in this chapter. Le.. selecting number of clusters. weighting exponent. and initial location of fuzzy cluster centers. An intuitive approach to choosing the optimum number of clusters is to make the

fuzzy clusters (i) compact and (ii) far from each other. at the same tirne. We extended the idea of scatter matrices for hard clustering to fuzzy clustering. The result was a vdidity index which cm be considered as a modification of the Fukuyama-Supno cluster validity index. Like ozher indices. the validity of this index depends on suitable selection of weighting exponent

m

for each data set. Limit analysis and generalization of scatter

cntena pave us an opportunity to introduce another index as fuzzy total scatter matrix to get an understanding of the variation of m in its domain: and to choose it to be far from its extremes in order to make sure that the cluster validity index correctly shows the optimum number of fuzzy clusters. In this regard. we developed a relation beween optimum number of clusters and power exponent for each data set to be partitioned. Exarnples show that the reliable zone for m depends on the data. Hence, dthough the domain of rn may be a part of the interval [1.5 , 3.51, each data set should be separately examined by the proposed strategy to select the suitable values of m and c. Initial-value problem is another bottle-neck in the FCM algorithm which arises from the local optimality of the algorithm. In order to estimate the initial locations of

~UZZY

cluster centers, using the altemating algorithm is essential. Choosing random points in the data as the initial prototypes is by no means reliabie; different results iri both cluster validity and cluster analysis are obtained by choosing different random points, as

demonstrated in Exarnple 3.3. Implementation of a hierarchical hard clustenng method would be a good idea to estimate the initial Vo. This strategy is more efficient and reliable

than a random search amocg different selections of initial prototypes. We used an agglomerative hierarchical algorithm for this purpose. This strategy has an effect on cluster vdidity index s, such that. for large values of m. the index has a monotonically decreasing tendency. The decreasing epoch usually starts at the number of clusters very

far from the optimum c. Finally. a relevant issue in fuzzy rnodeling through fuzzy clustenng was addressed. which is mostly ignored in the literature. In order to f o m the membership functions for the system output. we proposed a classification process to extend the fuzzy partition to the entire output space. Exarnple 3.4 illustrated the effect of this process on the final output membership functions for a typical nonlinear system.

THE FUZZY-LOGIC MODELING ALGORITHM

4.1 INTRODUCTION Retuming to the flow chart of fuzzy modeling represented in Chapter 1 (Figure 1.4). and after investigating the reasoning mechanism in Chapter 2 and mie generation step in

Chapter 3, in this chapter. Our focus is on the remaining steps of fuzzy modeling procedure, Le., input selection and membership assignment, identification of the optimum inference parameters, and membership function adjustment. Moreover, we summarize the proposed systematic fuzzy modeling methodology as an algorithm.

In section 4.2, we propose a novel approach to assigning the significant input variables arnong a finite set of candidates. This approach is not combinatonal, and rherefore requires no iteration. Based on this approach, in section 4.3, we introduce a new concept in the context of f u u y clustenng, ' 7 i c i 3 line clustering" which helps us to assign the convex input mernbership functions. Section 4.4 discusses the pararneter identification phase in two subsections: inference parameter identification (section 4.4.1), and rnembership parameter tuning (section 4.4.2). The final fuzzy modeling algorithm is presented in section 4.5. Two examples are illustrated in section 4.6, followed by conclusions in section 4.7.

CHAPTER

THE FUZZY-LOGE MODELING ALGORITHM

4.2 INPUT SELECTION IN FUZZY MODELING 4.2.1 Background The phase of input selection in system identification is to find the most dominant input variables which affect the output arnong a finite number of input candidates. Theoretically, this problem belongs to a more general field of data analysis. i.e.. dimension reduction. In the anaiysis of multivariate data, it is common practice to look

for the dimension reduction via linear combinations of the initial variables. Classical techniques, such as principal components (Duda. 1973,. discriminant analysis (Friedman. 1967), and canonical correlation (Jain, 1988). are exarnples of this approach. From a practical point of view, another type of dimension reduction is selecting a subset of the variables. The main advantage of this approach is that there is an actual reduction in the nurnber of measured variables. In this way. we can avoid the interpretationai difficulties which could aise in looking at linear combinations of very different kinds of variables. Although it is a common practice to check the weights of variables in a linear combination and to discard those that have "negligible" weights. this is not dways easy to do nor are negligible weights always guaranteed.

The problem of variable selection is also refened to as "feantre se le cri on^'. especially in some areas such as pattern recognition and information processing. Three major techniques are suggested for selecting a subset from an initial set of features. Le.. multiple regression (Draper. 1981), discriminant analysis (Seber. 1984). and ciuster analysis (Fowlkes, 1987). A good comparison of these techniques c m be found in (Fowlkes. 1987). Al1 the afore-mentioned methods are expressed in the context of statistical analysis. which, in most cases, benefits from the foxmal analytical background. However. in order io apply these techniques, many conditions such as normal distribution. adquate amount of data, independence, etc. shoulci be satisfied, which is rather crucial in red situations. There are some efforts to use informai techniques such as search method (Almuallim. 199 1), genetic algorithms (Vafaie. 1993), and techniques based on fuzzy sets

CHAPTER

THE FUZZYILOGIC MODELING ALGORITHM

(Pal, 1986) and possibility theory (Di. 1986). in the context of feature selection, in order to relieve the restricted formal conditions of statistical approach. In the context of feature selection. there is no distinction between input and output variables of the investigated system. However, the specific problem of "input selection" can be considered taking this distinction into account. For instance. in the input selection problem, it is quite possible to consider the dependence of the output variabIe(s) to each input variable, separately. in this way. the complexity of the problem at hand would be reduced significantly. Following this more specific approach. in fuzzy rnodeling three basic ideas have been suggested for selecting significant input variables among al1 finite candidates. Sugeno and Yasukawa (Sugeno. 1993) propose a combinatorial approach in which al1 possible combinations of input candidates are considered. For each combination. they build two fuzzy models based on two separated sets of data. and calculate a performance index called i reg da ri^ crilerion" based on a rnethod of analyzing two groups of data in an attempt to cause data independence in model formation (niara, 1980). A cornbination of input variables is chosen which has the minimum value of the performance index. For ro input candidates, the number of fuzzy rnodels to be built and tested for input variable selection is ro(ro+ly2. In another investigation, Takagi and Hayashi (Takagi. 1991) propose a fuzzy reasoning neural network system to identify the significant input variables by eliminating each input candidate and checking a performance index. Those candidates that have less or no improvernent effect on the performance index are considered as non-significant. Again. for ro input candidates. a possible rdro+l)L2 neural nets should be trained in this technique. Building ro(ro+l)R fuzzy or neural neiwork models is quite time-consuming especially for real systems with a large number of potential input variables. Moreover. in our fuzzy modeling methodology, we desire to separate the "input selection" stage from other stages, specifically because the inference rnechanism is not fixed in the proposed methodology . As a matter of fact, unlike Sugeno-Yasukawa approach. we believe that the significance of each input variable in the system is a real property of the system itself and should not depend on a selection of inference method and hence the manner of interpreting the model of the system. The third method of input selection is suggested by

CHAPTER

THE FUZZY-LOGIC MODELING ALGORITHM

79

Lin and Cunningham (Lin, 1994). In their technique, for each input variable, the inputoutput data are plotted and each sample point is hzzified to a Gaussian membership hnction, and then, for each sample point, a f u u y rule is constructed. Next. for potential input values, the defuzzified outputs are derived from the set of mles using Sugeno's heuristic reasoning formulation. As a result of this process, a "jifuzlycurve" is produced in the input-output plane. This prccedure is repeated for other input variables. one at each

time. Significant input variables are supposed to have a wider range for their iuzzy curves. Lin and Cunninghan illustrated the validity of their method by several examples (Lin, 1995).

4.2.2 The Proposed Method In Our proposed fuzzy modeling, for the selection of the significant input variables. we introduce an approach which is compatible with the whole idea of hizzy models. In fuzzy models, unlike anaiyticai models, the input-output relationship is defined for different partitions of input-output space through several IF-THEN rules. Each partition is represented by a membership function. Consider the fuzzy mode1 of a multi-input singleoutput system:

ALSO (3.1)

m........

ALSO IF Ul is B., AND U2 is Bnz AND ... AND Uris B, THEN V is D, In each iule i (i= 1,2,...,n), the input membenhip functions Bi, (i=1.Z,...,r) are aggregated via AND connection which is expressed by a triangular n o m operator: ri(x) = T ' ( B i i ( ~ I ) 9 B i i,... ( ~ ,Bir(xr)) 2) i=1,2,-...n

(4.2 1

where, Tgcan be any kind of t-nom. As expressed in section 2.2.2 of Chapter 2 (property

Fl), for the farnily of t-nom operators, "one" is the neutrd element. Therefore, if for an input variable x,, Bij(xj) is "one" for the whole range of

xj,

then the variable x, does not

have any role in the i" rule. Consequently, if the input variable x, has no effect on the output, its rnembership functions Bij's (i=1,2,...,n) should al1 be b'one" in the entire range

CHAPTER

THE FUnv-LOGlC MûûELlNG ALGORITHM

80

Figure 4.1 : Qualitative illustration of the effect of input variable x, on the ilh ride

of x,. In other words. for each input variable

xj,

the range in which its membership

function Bij is "one" can be a proper index of how effective that input is in the "i rule: if this range is equai to the entire range of

Xj.

then x, has no effect in the i" nile. and if this

happens for the other niles of the f u u y model. then x, has no effect on the system. at dl. and can be removed. This statement is illustrated graphically in figure 4.1. The above discussion can be summarized with the following statement:

COROLLARY 4.1 : In a fuzzy system model. the necessary and sufficient condition for an input variable to be non-significant is that. it has convex membership g a d e equal to "one" al1 over its domain. in al1 E-TIIEN rules. From the above statement. it is desired to define a quantitative index nJas an overall

measure of the "(non)significance" of input variable x, in the fuzzy system as follows:

where. Tij is the range in which membership function Bij(%) is one. and

r, is the entire

range of the variable x,, and n is the number of rules. In the domain of [O.11. small values of rc presents more effective variable and vice versa. While rc, presents the overall effect of x, on the system. each Ti, /

5 c a n provide rneaningful information about the effect of

x, on the ifhmie.

For construction of the input and output membenhip functions. the ideal case would be to partition the (r+l)-dimensional input-output space. where r is the number of

significant input variables. Due to limitation of clustering techniques and potential h g h dimensionality of the problem, this approach is not promising in most of the applications. One practical alternative is to denve the f u u y partition of the output space from the data. in order to obtain the required number of rules for expressing the system behavior, i-e..

CHAPTER

THE FUZZY-LOGlC MODELING A LGORITHM

OUTPUT (

#

5

Figure 4.2 : "Peak" poinrs should be the same for input and orrtput clrtsrers

the output variation. The next step in this approach is to extract the input membership hnctions from the output partirions. After deriving the output space clusters and before discussing the input membership assignment. we are able to implement the proposed

strategy of selecting the significant input variables. The reason is that by output-space ciustering. for each duster (rule), the points whose corresponding output membership grades are equd to one are specified. No matrer how the input membership functions are assigned. these points should have the same input membership grades (equal to one) in the corresponding rules. as illustrated in Figure 4.2. Based on the above reasoning. a simple algorithm is proposed for the input selection as shown in Figure 4.3. One important point should be remarked regarding the above strategy. and the following strategy of input membership assignment. as well. in some applications. there might be more han one input cluster corresponding to one output cluster as illustrated in Figure 4.4. In this case, the input variable having this characteristic is certainly significant, and two (or more) membership functions should be defined for this variable using the suategy explained in the next section. In such cases. the number of mles (n) is more than the nurnber of output clusters ( c ) ,since we have several mles with the same consequent. Without any loss of generality, in the following, we assume that the number of rules is equal ro the number of outpur clusters (c = n). although for the above cases the

same strategy can be applied considenng al1 input clusters corresponding to one output cluster as separate clusters.

THE FUZZY-LOGE MODELlNG ALGORITHM

CHAPTER

STEP 1: GET

the output-space membership matrix

uik,

...

i= 1.2 ..n

.

k=1,2, ...,N. where,n is the number of mies (output clusters) and N is the number of data.

STEP 2: LOOP

FOR j = 1 to ro PUT q = l FOR i = l to n

muD r,,={set of inputs x, with u , k = I . k=l.2 .....Y } FDiD r,={set of al1 x,}

NEXT i NEXT j

STEP 3:COiMPARE the values of rr,. j=12 .....ro:

STEP 4: REMOVE input variables with large value of E, . Figure 4.3 :The algorirlm of significanr inptir selecrion

Figure 4.4 :One orrtptrt clrister with nvo con-esponding input clrtsters

CHAPTER

THE FUZZY-LOGIC MûûELlNG A LGORITHM

83

4.3 ASSIGNMENT OF INPUT MEMBERSHIPS After selecting the significant input variables, suitable membenhip functions should be defined for them. One simple approach is to set the membership grade of each sarnple input equai to its corresponding output membership grade, obtained from the output data clustering process (Sugeno,

1993). Therefore, for each output daturn. dl the

corresponding input variables will have the same membership grade. The problem with this technique is that the membership functions assigned in this way are not convex and further approximation is required to shape the convex membership functions. Moreover. there is no reason for the input membership grades to be the same and equal to the output rnembership grade at each sample point. In Our proposed f u u y modeling methodology. a new technique is suggested based on the proposed input selection strategy. As discussed before (Figure 4.2). assuming that the membership functions are smooth enough. the only points which can be claimed to have

equal input and output membership grades are those which have the unit (or close to unit) membership grade. This was Our basic assurnption in identification of the significant input variables. We use this conclusion to construct the convex input membership functions. In Figure 4.2, suppose that for each output cluster i (i=1.2 .....n). there are several points on the axis of input variable x, (j=l,z .....r). which have output membership

grades equal or close to one. These points lie between vi and v i

( v:,

c vi ). We define

the "distance" of each point x,, (k=l,Z,...,N), located on the axis xj, to the line vkvt with the following func tion:

.v:,) = vq - xjlr

if

x,, < v,,

[ dis(x,, .v;) = x,, - v i

if

x,, > v i

( di&,

NQW. for input data

Xjk

1

1

Q=1,2 ...., r: k=1.2 ,...,N). the membership grades

u;,

corresponding to output cluster i (i= 1.2, ...,n) is formed such that those points which are

-

"closer" to the line vj,vi obtain higher membership grades. Obviously. those points

CHAPTER

THE FUZZYLOGIC MODELING ALGORITHM

84

which are between vl and vf are assigned to have a unit mernbership grade. We cal1 this clustering procedure ''Line F u u y Clustering" algorithm. Although the introduced clustering concept can be generalized to the multi-dimensional space having line or surfacc fuzzy clustering, in this research, we stick to the one-dimensionai case and postpone the generai case to funher research. Anaiogous to the weighted within-group sum of squared errors J, in FCM algonthm, for the line fuzzy clustering algonthm. the following objective function

7,

is defined:

It should be mentioned that in Our application. for each input variable x,. the lines v:,v: (i=1.2. ....n) are known and already specified from output clusters. Therefore. the problem of fuzzy clustering here is to find optimum membership grades u:, such that

5,

becomes minimum. Like FCM algorithm. in line fuzzy clustering algorithm. weighting exponent rn specifies the degree of fuzziness cf the clusters. Solutions to the above optimization problem are directly obtained by distinguishing between two cases when x,, c v:j and x,, > v i

d . and solving --;-Pm) =O

for each case. The final result is

du ,k

obtained as follows:

for i=1,2 ,...,n, j=1,2 ,...,r, and k=1,2 ,...,N ; where, rz is the number of d e s , r is the number of significant input variables, and N is the number of data. The above membership formation procedure is repeated for al1 selected input

variables. Figure 4.5 illustrates the algorithm of input membership formation assignment.

CHAPTER j%

THE FUZZY-LOGIC MODELlNG AL GOWTHM

85

LOOP :

FOR j = l to r FOR i=l to n

FIND

Tlj=

( set Of inputs x, which correspond to output membership

grade u,= L j

PUT vh =

NEXT i CALCULATE u

in(^,^)

*

AND

V:

= ~ax(r,,)

from equation 4.7 . for i= 1.2.....n and k= 12 .....N

NEXT j Figure 4.5 : The algorirhm of input membership function assignrnenr

4.4 FUZZY PARAMETER IDENTIFICATION The stage of parameter identification consists of two steps:

4.4.1 F u u y Inference Parameter Optimization At this step. we speciw the "best" inference mechanism for the system represented by its input-output data. This is a distinguishing feature of Our fuzzy rnodeling methodology. The inference rnechanism (parameters p, q. a. and

P) is optirnized according to the data

such that it provides the most adequate mechanism for modeling the system under investigation, which is not necessarily one of the known extreme inference mechanisms.

In order to find the optimum value of the inference panmeters. we mode1 the problem as the following boundary non-linear optimization problern: Derive the set of parameters [p. q, a, P] such that:

CHAPTER

THE FUIIY'LOGIC MODELING A LGORITHM

86

becomes minimum, subject to the following boundaries:

actual output and where. N is the number of data. y' is the ith

9'

is the

ilh

rnodei output. It

should be mentioned that at this step. the input and output membership functions that are obtained from the structure identification are used. in order to solve the above optimization problem. Function "consri' of MATLAB has been used which is based on Sequential Quaciratic Programming (SQP) method (Grace. 1995).

4.4.2 Membership Functions Parameter Tuning We have already found the input-output membership functions in the structure identification stage. However. it is better to tune the parameters as we do in the ordina? system identification methods. In our methodology. wpezoidal f u v y sets are used as approximations to convex h u y sets. Referring to Figure 2.7:

In order to adjust the trapezoid hinction parameters (y,, yb. y,, and yd) we apply SugenoYasukawa's nining algorithm (Sugeno. 1993) with the modification that a variable adjustment value is used at each ~ n i n gstep. This modification makes the tuning procedure somewhat more efficient. Besides, unlike Sugeno-Yasukawa's method. we adjust the parameters of both input and output rnembership functions. The algorithm s h o w in Figure 4.6 summarizes the tuning process.

THE FUZZY-LOGIC MODELING ALGORITHM

CHAPTER

87

STEP 1: CHOOSE the initial value of adjustment for input (qo)and output (&) membership hnctions (5-

.

10% of the range of the universe of discourse would be a good start) number of

iterations (irer) , number of adjustment changes (div).

STEP 2: E R A T E FOR 1 = I to iter

STEP 2.1: for i=1.2 ....,n (nurnber of rules). and

j=1.2

.....r

(number of input

variabIes), and k=1,2,3,4 :

SET b: as the krhinput mernbership parameter of the jrh h z z y set in the if"rule. SET q = q o

STEP 2.2: LOOP FOR Il = 1 to div SET q = 7/11 . CALCULATE b +q and bl! -q. IF k2.3.4. AND b:-q c

bn-' . THEN 6 ; = b:-':

IF Lz1.2.3, AND b:+q > b:-'. THEN CHOOSE

ELSE

6: = bi-': ELSE

6:

= bi-' - q .

b: = 6:''

+q.

the parameter which shows the least PI (equation 4.8) arnong (6n.bl.bi)

and REPLACE bl! with it.

iF the new PI is less than the old PI. THEN break LOOP Il NEXT Il

STEP 2.3: for i= 1.2 ,.... n (number of d e s ) , and k= 1.LX4 : SET di as the krhoutput mernbership parameter in the i l h rule.

SET

6=j,

STEP 2.4: REPEAT STEP 2.2 for output membership parameters d i . NEXT 1 -

Figure 4.6 : The algorirhm for rrrning inptrr-outprit membership paramerers

CRAPTER

THE FUZZYLOGIC MODELINO ALGORITHM

88

4.5 FUZZY MODELING ALGORITHM Surnrning up the discussions of this chapter and previous ones, we propose the l Û z q modeling algorithm shown in Figure 4.7.

--

-.

-

-

Find suitable weight exponent for clustering output data &

1 Rule Generation; Output Fuzzy Clustering

1 ..

I

Find the optimum number of output clusters

Perform the agglomerative hierarchical hard clustering for the initial prototypes

1

Perform huy clustering for output data

1

-

--

--

-

I o r m the membership hnctions f&the entire output space

1

Input Selection ; Input mernbership Assignment

m

Fuzzy Inference Parameter ~ ~ t i m i z a t i o n l Membership parameter Tuning

r

L

1

1

CanceI ineffective input candidates -

-

-

1

-

m fuzzy line clustenng for input mernbership functions Obtain the optimum value for f ù u y inference parameters

4 Adjust the parameters of input and output fuzzy membership hnctions

Figure 4.7 :The f u ~ ? ?stem rnodeling algorithm

4

1

CHAPTER

THE FUZZY--LOGE MODELING ALGORITHM

89

4.6 CASE STUDY

Example 4.1 in this example, we complete the hizzy modeling process of the nonlinear system

discussed in Exarnple 3.2 of Chapter 3. After deriving the output f u v y clusters and performing the classification for the entire output space in Example 3.1. significant input variables should be identified and their membership functions are to be assigned. The fuzzy system identification is based on 50 input-output data. Two dummy input variables .rj and .rd have been added to check the input selection strategy. Unlike Sugeno-

Yasukawas's approach of dividing die data into rwo sets and using a tirne-demanding combinatorial strategy. we apply the straightfonuard strategy described in section 4.2. The values of n (equation 4.3) for the four input candidates are:

n, =0.60x104

:

rr, =0.69x10J

;

rr3 =5.72x104

;

rc, =4.60x 10'

(4.11)

Clearly. the fint two input variables (x, and -y2) have the minimum rcbswhich are less than those for the other two variables in one order of magnitude. Input membership functions are also assigned through f u v y line clustering as explained in section 4.3. After

approximating the input and output fuzzy clusters by suitable rrapezoidal functions. the first-step rough hzzy rnodel of the sy stem is derived and shown in Figure 4.8. Note that up to here. no inference mechanism is required to denve the input-output clusters. The next step is to select and tune the set of parameters of the fuzzy model. which consists of the inference parameters (p. q. a. and

p), and the input and output membership Function

parameters. At the first step of pararneter identification. the optimum inference parameters are identified as explained in section 4.4.1. The optimum values are :

By optimizing the inference panmeters, the fuzzy model performance index (equation 4.8) is PI=0.171. The second step of parameter identification is to adjust the input and output membership pariameters. This is performed by the tuning aigorithm presented in section 4.2.2. After 5 iterations the error is reduced to PI=.0106 and after 5 more

CHAPTER

THE FUZIY-LOGE MODELING ALGORITHM

90

iterations the performance index becornes PI=0.0040. Figure 4.9 shows the final f u z q model of the system. The performance index of Sugeno-Yasukawa's h u y model of the

same type (position type) starts from PI=0.318, and after 20 iterations. it reaches to

PI=0.079 (Sugeno, 1993). Our methodology shows almost 20 times improvement in this example. Even compared to the Sugeno-Yasukawa position-gradient fuzzy model with

PI=0.010 for this example, the proposed position type fuzzy model shows a better performance.

We apply the proposed algorithm to a famous example of the system identification given by Box and Jenkins (Box, 1970). The process is a gas furnace with single input u(t) (gas flow rate) and single output y(t) (CO2 concentration). For this dynamic system. 10

input candidates y(t- 1)...., y(t-4). u(t- 1 ), .... u(t-6) are considered. We use the same 296 data as in (Sugeno. 1993). Figures 4.10 and 4.11 show the indices trtST) and s,

as

functions of m and c, respectively. m=2.5 and c=6 have been identified for the system. The signifiant input variables are detennined as y(t-1), u(t-2), and u(t-3). The optimum inference parameters for the gas fumace system are derived as:

After 10 iterations. the performance index reduces to a value of PI=0.1584 which is less than both the identified linear model (PI=O.193). and the Sugeno-Yasukawa positiongradient fuzzy model (PI=0.190). In (Sugeno, 1993), no position-type fuzzy model is presented for the process. Figure 4.12 shows the identified f u u y model and Figure 4.13 presents the fuzzy model behavior comparing to the actual process.

THE FUZZY-LOGIC MUOELING ALGORrrHM

CHAPTER

; = O

91

p,, ia 3+

0.00 1

na

se-

Figure 4.8 : InitiaZjir--,~ mode1 of rhe nonlinear qstem afier srnrctrtre ident~jïcation

Figure 4.9 : Final firi,?; mode1 of the nortlinear esrem after parameter idenrijication

cHAPTER

THE FUZZY-LOGE MODELING ALGORITHM

-750 Wsighung b o n e n r Im)

Figure 4.10 :Identrfication of rn for gas fumace process

2

3

4

5 5 7 Nurnber of Ciuclers (c)

9

3

1 ;O

Figure 4.1 1 :Specificarion of c for gas firrnace process

- Process

Figure 4.12 : Finalfit-

modei of gas frtrnace process afrer parameter identification

Figure 4.13 :Cornparison offii- rnodei and wai outprtt of gas frrmace process

THE FUZTY-LOGIC MODELING AL GORITHM

93

4.7 CONCLUSION We proposed a systematic approach to h i u y modeling and system identification. The

methodology considers the inference mechanism. as an identifiable object of f u u y systems. as well as their structure and parameters. For the reasoning process, a unified parameterized formulation was developed by which the suitable inference mechanism is adjusted for the system based on the input-output data. Therefore. no selection of inference mechanism is required a priori. and no restriction on any steps of reasoning is necessitated, For structure identification of the fuzry system suitable indices were introduced for identifying the number of rules and level of fuzziness of the f u u y mode[. based on fuzzy c-means clustering technique. Moreover, we used hizzy classification techniques to extend the clusters obtained from the sample data to the entire space. Significant input variables were identified by a new strategy. immediately as a result of output data clustering. Considering only the output space to identiQ the structure of the fuuy system gives us the advantage of simplicity and applicabiliv. By introducing the "hizzy line clustering" problem. we sugpested an appropriate methodology for specifying the input space partition from the output space partition. For parameter identification. an efficient algonthm for tuning mernbership parameters of input and output hzzy sets was introduced. The whole effort was to achieve a systematic and objective technique of fuzzy modeling, which is reliable for a wide range of applications. based on its theoreticai background. The validity of the methodoIogy was examined through two examples. and a comparison study was made with the Sugeno-Yasukawa Fuuy modeling method. The results were quite superior.

SYSTEMATIC DESIGN AND ANALYSIS OF THE FUZZY-LOGIC CONTROL

5.1 INTRODUCTION In this chapter, we introduce an appropriate structure for hzzy-logic control of .MM0 systerns based on their hizzy-logic model. Furthemore. in order to Suarantee the stability and robustness of the system performance, we develop basic guidelines for the derivation

of f u u y control rules using fundarnentals of sliding mode control theory. In chapter 1. a systematic methodology was proposed for developing the Fuzzy-logic mode1 of general nonlinear systems. As a continuation. in this chapter. we use the hizzy-logic model of the nonlinear system for control tasks. This chapter is organized as follows. In section 5.1. we first introduce an architecture for fuzzy-logic convol of complex MM0 systems. Then. we develop a generaiized formulation of model-based sliding mode control for a class of nonlinear MIMO systems in order to prove the effectiveness of the proposed control strategy and to derive guidelines for designing f u u y control mles that ensure stability and robustness. Next, the generalized formulation is applied to obtain the robust fuzzy-logic control mles as a specifîc case of nonlinear controllen. A discussion on how to derive f u v y rules to obtain the desired stability. robustness. and satisfactory performance is included. An exarnple is illustrated in section 5.3, and concluding remarks are presented in section 5.4.

CHAPTER

e

SYSTEMATIC DESIGN & ANALYSIS OF FLC

95

5.2 ROBUST MODEL-BASED FUZZY-LOGIC CONTROL : DESIGN & ANALYSE The key idea of the proposed approach to the design and analysis of the Fuzzy-Logic Control (FLC) system is to consider ETC (with crisp input and output) as a multidimensional nonlinear transfer element with upper and lower limits. The FLC nonlinear characteristics are due to irs computational structure. i.e.. fuuification. inference. and defuzzification. This requires the development of a suitable formulation of sliding mode control for a class of nonlinear MIMO systems that can be applied to fuzzy logic approach. This formulation is crucial to a systematic methodology of FLC design and analysis. In this section. we introduce the structure of the proposed fbzzy-logic control system. The required mathematicai fundamentals and theorems are presented. A generalized formulation of sliding mode control for nonlinear iWiMO systems is developed. and finally, the design of the robust fuzzy IF-THEN rules and membership functions for the FLC is presented.

5.2.1 The Proposed Fuzzy-Logic Control Structure Figure 5.1 illustrates the proposed structure of the FLC for nonlinear LMMOsecond order dynamic systems. The controller consists of two main parts. In the first part, a set of fuzzy IF-THEN rules expresses the dynamic behavior of the system. This "Xmoidedge

base" can be regarded as the fuuy-logic inverse dynamics model which contains the

dynamic interaction between systern States as well as other complicared phenomena in the systern. Unlike analyticai rnodels, the fuzzy-logic model is simple and hence computationaily efficient, and at the same tirne, as we will illustrate for robotic applications in Chapter 6, the fuzzy-logic model can represent complex phenomena of the system behavior more precisely than anaiytical models. Moreover, since the model is directly obtained from input-output data, there is no need to identiQ intemal system parameters for constmcting the model.

CHAPTER

SYSTEMATIC DESIGN & ANA LYSlS OF FLC

96

The second part of the FLC consists of decoupled robust fuzzy IF-THEN rules for each state independently, in order to guarantee system stability. and in order to ensure achievement of the desired performance. As shown in Figure 5.1, two pre-processing units are also required to provide suitable input to FLC which will be specified in the next sub-section. The proposed control structure is intuitive. Based on Our knowledge. which may be incomplete or inaccurate. we try to control the system towards the desired performance based on our knowledge about the system: at the same time. by using some extra rules ( f u u y robustifiers in Fi,we 5.1). we ensure that the system remains stable and does not deviate frorn the desired behavior. In fact, for a simple system. these extra rules might be sufficient by themselves to conîrol the system without any further knowledge as we see in the traditional fuzzy-logic controllers. However. as the complexity (such as Iarge number of input variables. interaction between States. and wide range of disturbance) increases. more information is required which. in Our approach. is formed as the fuzzy-logic knowledge base of the system. in essence. the proposed FLC is a robust model-based control structure in which fuzzy IF-THEN rules are implemented in place of the andyticai formulation to guarantee the desired system stability and performance.

Figure 5.1 : The stntcrrrre of the proposedficzp-fogic control sysrem

cFUPTER

SYSTEMATIC DESIGN B ANALYSE OF FLC

97

Based on the proposed structure, the systematic methodology of design and andysis introduced in this thesis requires the following steps: 1) Development of a fuzzy-logic model. The main knowledge of the system

characteristics is encapsulated in huzy IF-THEN rules. The development of an objective algorithm to extract this knowledge from the system behavior (input-output

data) is the heart of the FLC.This task was acomplished in previous chaptea.

2) Design of the robust f u a y IF-THEN niles for each system state. 3) Proof of stability and completeness of the structure. In the proposed structure. for each system state, robust fuzzy control IF-THEN rules are designed independently. This 'bdecoilpling"charactenstic provides a simple aproach to the design of the robust

f u u y niles. It should be proved that this is sufficient in order to parantee the stability and robust performance of the entire system. The following sub-sections discuss steps two and three.

5.2.2 Fundamental Definitions and Theorems In this section. some basic concepts and theorems of matrix theory are briefly reviewed, and three results which are used in the development of the generalized formulation of sliding mode control, are proved.

DEFINITION 5.1 (Hom, 1985): An nxn real syrnrnetric matrix BER"" positive definite, if for al1 nonzero real vectors

XE

Rn, the real scalar

is called

X ~ B Xis

strictly

positive:

.

Vx é Rn , x # O : IF X*BXr O THEN B is positive definite .

If the above strict inequality is weakened to x TBX defin ite

> O , then B is said to be positive semi-

s

DEFINITION 5.2 (Hom, 1985) : A scalar function f(x) : R -+ R is positive definite if for al1 x E R : a) f(0) = O ,

b) f(x) > O ,

c ) f(x) is continuous,

d)

af

y

is continuous.

If condition (b) becomes f(x) 2 O , then f(x) is positive semi-definite.

DEFINITION 5.3 (Hom, 1985) : Let A9B~RR"" be real symmetrîc matrices. We say A>B if and only if the matrix (A-8) is positive definite; sirnilariy, A 2 B means that (A-

B) is positive semi-definite.

SYSTEMATlC DESIGN & ANAL YSlS OF FLC

CHAPTER

99

DEFINmON 5.4 (Hom, 1985) : A time-varying rnatrix B(t)€Rnm is uniformly positive definite if there exists a positive scaiar a > O such that: Vt 2 O : B(t) > cd, where I is the nxn unity matrix. COROLLARY 5.1 (Hom, 1985) :

A real symmetric matrix BE RWn is positive

definite (semi-definite) if and only if its eigenvalues are positive (non-negative).

COROLLARY 5.2 (Hom, 1985) :

A non-singular matrix B€Rn"" is positive definite

(serni-definite) if and only if its inverse B'I is positive definite (serni-definite).

COROLLARY 5.3 (Hom, 1985) :

If A,BER""" are real symmetric matrices and

A 2 B , then for any arbitrary vector XE Rn we have: x TAX 2 x

THEOREM 5.1 (Hom. 1985) :

BX

.

If matnx AER"" is non-singular, and has distinct

eigenvalues, then there exists a similarity transformation such that: A = V-'L,v. where A

is a diagonal matnx of the eigenvalues of A. and V is a non-singular matrix of the

eigenvectors.

THEOREM 5.2 (Hom, 1985) :

If A,BE R ~ " are nxn non-singular mavices with

distinct eigenvalues, and A and B cornmute, Le., AB=BA, then they have the same

,v . The matrices

eigenvectors such that: A = V-'Z ,V and B = V-' t

Z,

and E, are

diagonal matrices of eigenvalues of A and B, respectively.

THEOREM 5.3 (Istratescu, 1987).:

For an nxn

.

positive definite matrix BER^"

and every two arbitrary vectors x , y ~ Rn the following inequality called "generaliced Cauchy-Schwarz inequaliry" holds : X

JxrBx. ~ JyTBy

~ 5B

SYSTEMATIC DESlGN & ANAL YSlS OF FLC

CHAPTER

100

Based on the above theorems, we need to prove the following results that are used in the sequel:

COROLLARY 5.4 : if A,BE Rnm are positive definite (serni-definite), and A and B commute. then their matrix product AB is also positive definite (semi-definite).

PROOF : Since A and B commute, by using Theorem 5.2, we have:

where, LA, = Z,Z,.

Therefore, the eigenvalues of AB are the product of those of A and

B. According to Corollary 5.1, positive definity (semi-definity) of A and B irnplies that al1 their eigenvalues are positive (non-negative), and hence, their products are also positive (or at least non-negative) which also irnplies that AB is positive definite (semi-definite).

+

THEOREM 5.4 : Suppose that for a positive definite nxn matrix BE RnXnthere exists a positive reai scalar for a vector y x

E R n , the

b >O

E Rn

such that

FI 2 B

where I is the nxn unity matrix. Suppose that

there is an upper bound (y/lSp . Then. for any arbitrary vector

following inequality holds:

xTBY bpllxll PROOF : Since

(bI - B)

(5.3) is positive semi-definite. for arbitrary vectors x.y E R n we

have:

According to Theorem 5.3, for the bounded vector y and arbitrary vector x we have:

x T ~5yJXTBX ..,/yTByyTBy

(5.7)

and from 5.4, 5.6, and 5.7, we detennine that:

THEOREM 5.5 : Consider ME R ~ as" an nxn positive definiie macrix and KE R " ~ "as an nxn diagonal positive definite matrix. If there exists a positive real number such that

a2 M . then for every arbitrary vecror x

E

Rn. we have:

-

~ . x ~ M - '2I X'KX (~

PROOF : The assumption definite. i.e.

>0

(5.9)

a2M

means that matrix @ - hl) is positive serni-

@ - M)2 O .

According to Corollary 5.2. since M is positive definite. so is M - ~ Also . since K is diagonal. M%

cornmute. and based on corollary 5 . 4 it is also positive definite.

Furthemore, we have:

(ir-

M)M-[K = M-'K@-

M)

(5.10)

@ - .M)V-'K

Therefore. from Corollary 5.4. it is concluded that de finite:

(Z~-M)M-'KZO

.

Based on Corollary 5.3. for an arbitrary vector x E R n we obtain:

is positive serni-

C H A P T E R ~ SYSTEMATIC DESIGN & ANAL YSIS OF FLC

102

5.2.3 A Generalized Formulation of Sliding Mode Control In order to use sliding mode control theory for the synthesis and analysis of fuzzy control. a generaiized formulation, developed in this section. is required. Without loss of generality and for the sake of clarity, we consider second order dynamic systems. The dynarnics mode1 of such systems with n system states and n input variables is represented as follows:

q = f (q. q;t) + B(q, q; t)u(t) = G(q, 4. u: t)

(5.14)

where, q = [ q , , q , ,..., qnITisthe vector of systern states. and q = [9,,9 z,. ..,q

]T

( I = [ ~ ..... , . ~iln]~T

and

are state velocities and accelerations. respectiveiy. Function P E R n

is a noniinear vector function that represents system dynamics and al1 uncertainties and disturbances, and B E Rn""is also a nonlinear matrix function which acts as a nonlinear control gain. Vector u

é

R n is the control input vector. The acceleration vector of the

entire system can be considered as a nonlinear and tirne-varying vector hinction G E R n -

For further development. we need to introduce the inverse dynamics mode1 of the nonlinear system 5.14, based on the following assumptions:

i) by suitable transformations, it is possible to uansfer the system dynamics into the form presented as equation 5.14 such that the system dynamics is linear in rems of the control input u (although nonlinear in the states).

ii) the conuol gain matrix B is non-singular and bounded positive definite over the entire state space, i.e.,

where.

b

is a positive constant and

b(q,q; t)

is a positive definite function.

Based on the above assumptions, the inverse dynarnics mode1 of the system can be presented as: = M(qT q; t)q+ h(q, q: t) = F(q, q, q; t)

(5.16)

where. M = B", and h = - ~ - ' .f Since B is non-singular, frorn 5.15 it follows that:

1

-

1

where. g = = and m = - . b b

In this analysis, our goal is to derive formulation by using the nonlinear fünction F instead of its components M and h. In this direction. considenng the fact that:

the boundary matrix inequalities 5.17 can be rewritten in the following form of scalar inequalities:

(5.19)

vt 20 : vq.4,ij E R" : mJlq~I's qT[~(q,q,q;t)-~(q,~,o;t)] I&,4;t]14112 The control task is to follow a desired q, and q, in the presence of system parameter variation and uncenainties. The tracking error e = q- q, and the rate of error e = q- q, are to be observed. We define a generalized error vector as follows:

where, P and Q are nxn diagonal matrices. The integral of error is included in the generdized error to ensure zero offset error. Based on the theory of sliding mode control (Appendix A), the tracking control problem c m be formulated as keeping the error vector

e on the sliding surface defined as follows:

CHAPTER

SYSTEMATC D E S W & ANALYSE OF FLC

104

An optimum response for each error state is obtained if the system 5.21 is cntically

darnped where: P=2A;

and

Q=A*

Matrix A is an nxn positive definite diagonal matrix. At this point, we should consider the conditions which guarantee that for each state qi. the system trajectory will approach the sliding surface from any non-zero initial error. within a desired period of time. In order to avoid the chattering effect prevalent in sliding

mode control (Appendix A). the condition is relaxed to asymptotic convergence of system States to a small neighborhood of their corresponding switching surfaces. For each state. this neigborhood is defined as:

The above task can be achieved if the control law u is designed such that for each state a Lyapunov-like condition for system stability holds (Slotine. 1990):

or, in sum:

The pararneter QI is the thickness of the boundary layer and q, is a design pararneter that sets the time the system trajectory requires to reach the boundary layer from an outside initial condition. From equations 5.21 and 5.22. the dynamics of the generaiized error vector s is determined as: S = (ë+ 2Aé+ A? e) =

[a-@, - 2Aè- A' e)]

C H A P T E R ~ SYSTEMATlC DESIGN & ANALYSlS OF FLC

105

The acceleration vector q c m be obtained from systern dynamics. Equation 5.14 c m be rewritten as:

Inserting 5.27 into 5.26 results:

equation 5.28. the te*

[ ~ ( q-, Zh é-

A' e ) + h] is the systern inverse dynarnics with

the input acceleration defined as "reference" acceleration as follows:

Then we define the desired control input as:

Because of the system uncertainty and variation. the inverse dynamics mode1 of the system (in Our case a fhzzy-logic model) is an approximation of the reai system. Hence: Û,,

= ~ ( q , q ; t ) i i , +h(q,q;t)= Ê'(q,&q,:t)

(5.31)

Therefore. we consider that the control input u is defined as :

where. the control t e m u, is the compensation part of the controI due ro model uncertainty. and it should be specified such that the sliding condition 5.75 is satisfied. By replacing u (equation 5.32) in equation 5.28 we have:

where. A F is the uncertainty vector of the inverse dynamics rnodel. Consequently. the left hand side of the sliding condition 5.25 becomes:

CHAPTER

SYSTEMATIC DESIGN & ANAL YSlS OF FLC

106

We assume that the uncertainty A F is bounded as:

By using Theorern 5.4 . for the positive definite matrix B and bounded vector A F. we have:

sT BA F I6pllsll or:

and therefore, equation 5.34 changes to the following inequality:

Considering the fact that:

the following inequality c m be inferred from 5.38:

In order to satisf'y the sliding condition 5.25. we choose a continuous u, such that:

or in another fonn:

Assume that for each state i, we choose uci as a function of

satisfies the following properties:

S,

. uci = Gi(st)

S U C ~that

it

a) Gi(si)is continuous;

b) Gi(si)is rnonotonic and decreasing for O c (s,1 < a,; C)

Gi(0) = 0.

We express Gi(si)as follows:

where, gi(si) is a positive function for ail si, and dsgn(si) is a function defined on the entire R as:

From equation 5.44. the vector u, can be represented as:

where, G is an nxn positive definite diagonal rnatrix with gi(si)as its diagonals and Q is an nxn positive definite diagonal matrix with

A,,

as itr entries if r i

O : otherwiie the

diagonal element corresponding to s, = O is zero.

-

By inserting equation 5.46 into inequality 5.42 and multiplying both sides by -m. we obtain:

Since GR is positive definite (or at Ieast positive semi-definite). according

5.5, we have:

-

msTM%RS

n

> sT GRs = ~g,si.dsgn(sl)

There fore. inequality 5.47 will be satisfied if the following inequaiity holds:

to

Theorem

CHAPTER

SYSTEMATK DESIGN & ANAL YSlS OF FLC

108

Figure 5 3 : The specified domain for robrisr conrrol rerm u,,

It is sufficient that condition 5.49 hoids for each terrn of the surnrnation:

From inequahty 5.5 1. the general condition for u, to parantee compensation for system uncertainty is obtained as:

CHAPTER

SYSTEMATlC DESIGN & ANALYSE OF FLC

109

Figure 5.2 shows the domain in which each 4, can compensate for system uncertainties. We cal1 this domain the "Robusmess Region". This region and properties specified in

5.43 help us to assign the robust control tenn uci for each state, independently.

In our methodology, the control tems Gi are produced by suitable fuzzy IF-THEN rules. The procedure is as follows: we consider the nonlinear .MM0 system 5.14 with the assumptions (i), (ii), and (iii), and with the following inverse dynamics:

First. we pnerate the hzzy-logic inverse dynamics mode1 of the system as:

based on a known bounded enor AF. Then the control input is of the form:

where. q , is defined by equation 5.29. The p n e r d conditions of u, are: for each state i. u,i should satisfy properties 5.43. and be located in the domain defined by 5-52 and iilustrated in Figure 5.2. The robustness region depends on design pararneters h, . q , . and

a , .and pararneters

.m .and p which are defined as foliows:

In conclusion, in this section it was shown that it is possible to design the "decorrpled' robust fuzzy control tems u, (i=1.2. ....n) as illustrated in the control structure of Figure 5.1. Furthemore, we developed the seneral conditions for

Uci

to ensure the stability and

robustness of the entire system. These conditions depend on the bounds of the mode1 error and system parameters which, in our formulation. can be achieved from the inverse dynamics mode1 (equations 5.56 and 5.57). Therefore. in the proposed formulation. the system dynamics is considered as a black box without the necessity to speciS its components specifically.

C H A P T E R ~ SYSTEMATIC DESlGN & ANALYSIS OF FLC

110

5.2m4 Design of the Robust Fuuy Control Rules In section 5.2.3, we have developed an approach to control of a class of nonlinear MIMO systems, which consists of implernenting an inverse dynarnics rnodel and designing a robust control term G,for each system state independently. Each function uci(si)should satisfy conditions 5.43 and 5.52. In this section, we design fuzzy IF-THEN rules to satisQ the afore-mentioned conditions. From figure 5.2, the characteristic relationship between

Uci

and si can be qualitatively

expressed as: "uCiLr inversely as large as si within certain lirnits". W e interpret the above characteristic by the following seven IF-THEN rules:

Positive Big (PB), Positi~7eMedium (PM), Positive Small (PS), Airnost zero (AZ), Negative Small (NS), Negative Mediiim ( N M ) , Negative Large (NL),

THEN THEN THEN THEN THEN THEN THEN

u , is u, is u, is u, is

Negarive Large. Negative Mediion. Negative Smnll, Alrnost Zero. u, is Positive Small, u, is Positive Medilim. u, is Positive Large.

It is possible to apply the unified reasoning formulation proposed in Chapter 2 to robust fuzzy rule set 5.58. However, by having a comprehensive fuzzy rnodel. the robust term of the control input c m have simple characteristics. Therefore. For the sake of simplicity. we use the modified Sugeno's reasoning formulation (Sugeno, 1993) for the inference mechanism of the robust fuzzy rules that provides a simpler and faster result.

is denved as:

Accordingly, given the input si, the crisp output

uci

where, Aik(si)is the membership function of

in the antecedent fuzzy set of the kth d e ,

Si

and bikis the centroid of the consequent fuzzy set of the kth rule. The main objective is tu assign suitable membership functions for generating the robust control rules such that conditions 5.43 and 5.52 are satisfied. It shouid be noted

SYSlEMA TIC DESIGN & ANALYSlS OF FLC

CAAPTER

111

that according to the reasoning formulation 5.59. for the consequent fuzzy sets, only their centroids are required. Considering the input membership functions shown in figure 5.3.

and seven consequent fuzzy set centroids b: for "Almosr Zero" and b: .b:. b: "Positive Small, Medium, Large". and

b: .b'. bi

for

for "Negative Small, Medium, Large*'.

the si-uci relation c m be represented as shown in figure 5.4. For the sake of simplicity and

without loss of generality, the input membenhip functions are arranged such that they always overlap at the degree of membership equal to 0.5. Therefore, for each input s,, two

rules are fired at mosr. Furthemore, a symmetric behavior for uci(si) is assumed. Hence,

-a:

?

=-a;

;

ai3 = - a ,3

.

4

g, =-a,

4

; and

--

;

bl

= -bf :

b,4 = -b:

(5.60)

From figure 5.4, some of the membership parameters can be assigned irnmediately. First, Conditions 5.43 require that:

By using inference formulation 5.59 and membership functions shown in Figure 5.3. a piece-wise linear characteristic is produced for the robust fuzzy control function of each system state i (i= 1.2, ...,n), which c m be formulated as follows:

Figure 5.3 :Membership fitnctiorls of the generalized error sifor robrlsr conrrol rides

c H A P T E R ~ SYSTEMATIC DES/GN & ANALYSIS OF FLC

where, K: = bi"

- bl

.

7

112

-ai ,

conirol und PID conrrol of rhe I R E armfor randorn trajectoty

SIMULATlON AND EXPERIMENT

CHAPTER

158

JOINT #3

5

O

iO

15

JOINT #4

1

20

i

-1wl O

25

5

10

15

20

25

FUZZY CONTROL

FUZM CONTROL

0.5.

-0.4 1

O

5

15 PID CONTROL 10

20

25

O

5

1O

15

20

2:

20

2t

PID CONTROL

15,

1

2,

-2 1 O

5

10

15

I

FUZZY CONTROL

>

-70'

-". rn m

O

1

5

15 PID CONiROL

tO

20

25 Pl0 CONTROL

40

I

1

0

t U U T LUN t KUL

O

-

5

-21

IO

15

20

25

O

10 15 PID CONTROL

20

25

5

10

t5

20

25

PO CONTROL

1

2,

I

O

1

5

5

1O

15

TlME (sec)

20

25

-2'

O

TlME (sec)

Figure 6.23 (cntd.) :Cornparison of the proposedfrc-y conrrol and PID contrul of the IRIS armfor random trajectory

I

SlMULATlON AND EXPERIMENT

CHAPTER

159

JOINT #2

JOINT #1 100

Y -1 L O

-

O

5

5

1

10 PD CONTROL

15

20

10

15

20

-1

O

1

T

I

I

I

L

,

5

10 PID CONTROL

I

15

20

FUZM CONTROL

l

L

-20 O

4

5

10 PI0 COMROL

1S

20

O

5

10 PID CONTROL

1s

20

PID CONTROL

P -50lO

I

I

l

5

10

15

20

FUZZY CONTROL

"

-201 O

I

I

5

1O PlD CONTROL

I

I

l

O

5

10

15

TlME (sec)

I

15

20

I

20

TlME (sec)

Figure 6.24 :Cornpurison of rhe proposedJu--? controi and PID control of the IRIS arnr for sinusuidal trajectory

SIMULATION AND EXPERIMENT

CHAPTER

160

JOINT #3

JOINT #4

100.

1

I

5

-1 1 '

O

5

10

15

10

I

i-ULLY CONTROL

1

PU) CONTROL

20

15

J

20

PID CONTROL

FUZLY CONIROL

FUZLY CONTROL

O

5

10

15

O

5

1O

15

Pli3 CONTROL I

.----------.------*----r------------.--**-------

I

1

I

I

5

1O

15

20

-E

0

l

5 IO! ----...... :-..-------L-----*----

-

2t

.:- - - - - - - - - -

PID CONTROL

P O CONTROL 5

c

-151

O

5

I

I

10

15

TlME (sec)

I

20

O

%

I

1

5

10

1s

TIME (sec)

Figure 6.24 (cntd.) : Cornparison of the proposedftcy control and PID control of the IRIS a m for sinusoidaI trajeciory

1

20

CHAPTER

SIMULATION AND EXPERIMENT

Figure 6.25 :Comporison of the proposedficy control and PID control of the IRIS a m for siep Irojectory

161

SiMULATION AND EXPERIMENT

CHAPTER

162

JOINT #4

JOINT #3

5

1

1O

15

FUZZY CONTROL

20

1

!

-1wl O

25

5

10

l r'

15

25

20

F U Z M CONTROL

L

1

5 4

tO

15

20

2!

20

21

20

25

20

25

20

25

PID CONTROC

,.

--

5

t

FUZZY CONTROL

11

O

PID CONTROL

10

15

FUZZY CONTROL

5

II

1O

15

1

PID CONTROL A