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ELSEVIER Fuzzy Sets and Systems 101 (1999) 353-362

FUZZY sets and systems

Extracting fuzzy rules for system modeling using a hybrid of genetic algorithms and Kalman filter

Liang W a n g * , J o h n Yen

Department of Computer Science, Center for Fuzzy Logic, Robotics, and Intelligent Systems, Texas A&M University, College Station, TX 77843-3112, USA

Received November 1996; received in revised form February 1997

Abstract

This paper proposes a hybrid algorithm for extracting important fuzzy rules from a given rule base to construct a "parsimonious" fuzzy model with a high generalization ability. This algorithm combines the advantages of genetic algorithms' strong search capacity and Kalman filter's fast convergence merit. Each random combination of the rules in the rule base is coded into a binary string and treated as a chromosome in genetic algorithms. The binary string indicates the structure of a fuzzy model. The parameters of the model are then estimated using the Kalman filter. In order to achieve a trade-off between the accuracy and the complexity of a fuzzy model, the Schwarz-Rissanen Criterion is used as an evaluation function in the hybrid algorithm. The practical applicability of the proposed algorithm is examined by computer simulations on a human operator modeling problem and a nonlinear system modeling problem. © 1999 Elsevier Science B.V. All rights reserved.

Keywords: Fuzzy modeling; Genetic algorithms; Rule extraction

1. Introduction

A fuzzy model is a set of if-then rules that maps inputs to outputs. Theoretically, the more the number of fuzzy rules, the better the approximation ability of the model [28]. In practice, however, a good fit on training data does not always assure good performance on future data, especially if the number of adjustable parameters in the model is large compared to the available training data. A fuzzy model containing a large number of rules is liable to encounter the risk of"overfitting", capable of fitting training data completely but incapable of generalizing reasonably on future data.

A trade-offis thus required in constructing a fuzzy model. On the one hand, the number of fuzzy rules must be sufficient to provide the approximation capacity; on the other hand, the number of fuzzy rules must be "parsimonious" to guarantee the generalization ability. In a recent paper by the authors [-27], a statistical information criterion, known as the Schwarz-Rissanen Criterion (SRC) [18, 17] was used to achieve such

* Corresponding author.

0165-0114/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S0165-01 1 4 ( 9 7 ) 0 0 0 9 8 - 5

354 L. Wang, ,£ Yen / Fuzzy Sets and Systems 101 (1999) 353 362

a trade-off. This paper further extends the above research by introducing a hybrid technique based on genetic algorithms (GAs) and Kalman filter in which the SRC is used as an evaluation function.

Numerous work has been done in the application of GAs to the design and implementation of fuzzy systems. Roughly, this work can be divided into three classes: (1) Using GAs one determines the membership functions with a fixed number of fuzzy rules. For example, Karr and Gentry [7] designed a fuzzy controller consisting of 14 rules for an industrial pH control process in which a GA was used to determine the parameters of fuzzy membership functions. Krishnakumar and Satyadas [9] used a similar approach to construct a fuzzy controller for an aircraft control problem. Also, in [15], a GA was used to tune a dimension-known fuzzy relation matrix for controlling a DC series motor. (2) Using GAs one finds fuzzy rules with known membership functions. This is the work done in [21] where a GA was used to generate fuzzy rules (which were represented in a decision table) in order to center a cart of mass on a one-dimensional track. The membership functions were assumed to be triangular and have known parameters. (3) Using GA one finds both membership functions and fuzzy rules. For example, Karr [6] first used a GA to select fuzzy rules from a given (large) rule base. Following this stage, Karr used another GA to tune membership functions. Unlike Karr's work, Lee and Takagi [11], Homaifar and McCormick [3], and Ishibuchi et al. [4] used GAs to determine fuzzy rules and membership functions simultaneously. Our method falls into the third class. However, different from these already established results, our method is a hybrid algorithm in which a GA is combined with a Kalman filter to find fuzzy rules and estimate model parameters.

2. A hybrid algorithm for the extraction of fuzzy rules

2.1. Why hybr id i z ing?

The main strength of GAs is their wide applicability [12]. Since they work through function evaluation rather than differentiation or other such means, GAs are capable of searching solutions in a poorly understood and irregular space. But unfortunately, GAs are tediously slow in convergence. On the contrary, most conventional search algorithms (e.g., the gradient descent algorithm) have a limited application domain but a rapid convergent speed. If an optimization problem is well formulated (e.g., the problem is linear in the parameters or the gradient information is available), these conventional optimization techniques can usually provide a fast and reliable solution. Even if a GA has to be used to solve an optimization problem, it is highly advocated that where possible, the positive features of the conventional algorithms should be incorporated into the GA [2].

Indeed, researchers have been exploring the feasibility of hybridizing GAs with some conventional optimization algorithms. For example, Renders and Bersini [16] investigated two ways of hybridizing GAs with the hill-climbing algorithm. The first one involved two interwoven levels of optimization where the GA and the hill-climbing cooperated in the global process. The second one consisted of modifying the GA by the introduction of new genetic operators in such a way that these new operators captured basic mechanisms of the hill-climbing method. Yen et al. [25] proposed a hybrid algorithm which grafted a concurrent version of Nelder and Mead's downhill simplex with the GA by using the simplex as an additional genetic operator. This hybrid algorithm has been successfully applied to a complex metabolic system modeling and control problem and has shown a fast convergence compared to the pure application of GAs [26].

All this work has focused on hybridizing GAs themselves. For fuzzy modeling, we attempt to explore another form of hybridization. Specifically, we intend to hybridize the modeling process. The hybridization is not arbitrary, however; rather, it is motivated by the unique feature of the fuzzy modeling problem. It is known that the output of a r-input fuzzy model can be expressed as

S.'L1 w,(xl, x2 . . . . . x r ) f , ( x l , x2 . . . . , x r )

Y -- Y, lm=l Wi(Xl, X2 . . . . ,Xr) ' (1)

L. Wang, J. Yen / Fuzzy Sets and Systems 101 (1999) 353-362 355

where m is the number of fuzzy rules in the model, xi is the ith input variable, J] is the output of the ith rule, and wi is the truth degree of the ith rule determined by combining the membership functions /la,j (x j), i = 1, 2, . . . , m, j = 1, 2 . . . . . r, of the input variables according to

wi(xl, x2, . . . , xr) = min(kla,l(xl), ].lAi2(X2) . . . . . ]lAir(Xr)). (2)

There are various forms of f/(see, e.g., [10]). The most widely used form is, however, using a constant or a linear regression equation (see, e.g., [1, 5, 20, 23, 24]. This is also the form employed in our research.

The most difficult aspect in identifying the above model is not in the determination of the parameters in I~A,j(Xj) and f~; rather, it is in the determination of the number of fuzzy rules (i.e., m) and the positions of these rules in the rule base. Once m and the positions of the rules in the rule base are available, these parameters can be estimated without any difficulty by using some conventional optimization technique such as the gradient descent algorithm (see, e.g., [5]). In this case there is no strong reason why GAs should be preferred.

In practice, the most commonly used method to determine m is through trial and error. This method is feasible for a small-sized rule base, but it is usually difficult to give an optimal or near-optimal solution for a rule base consisting of a large number of fuzzy rules. Inspired by the strong search ability of GAs, we let a GA to find the important rules from a given (large) rule base (the most difficult aspect in building a fuzzy model) and leave other less difficult things to the conventional optimization methods. Specifically, by pre-determining the parameters of I~laii (X j), the other things are limited to the estimation of the parameters in f~. Note in this case the estimation problem becomes linear in the parameters, so some well-established linear search algorithms will be at our disposal. Here the Kalman filter is preferred because of its recursive and fast convergent properties.

2.2. Algorithm description

The proposed hybrid algorithm uses a GA to search the important fuzzy rules in a given rule base and a Kalman filter to compute the parameters in f~. Each rule in the rule base is allocated a binary value 1 or 0. A value 1 indicates the inclusion of the rule in the model and a value 0, on the other hand, indicates the exclusion of the rule in the model. Thus, a binary string, say [1, 1, 1, 0, 0, 0], means that the first three rules in a 6-rules rule base are used to construct a fuzzy model.

The algorithm begins with a population of randomly selected fuzzy rules. Each member in the population is coded into a binary string and treated as a chromosome in the GA. A binary string consisting of all l's, which corresponds to the complete rules in the rule base, is always included in the initial population (the first generation). By means of reproduction, crossover, and mutation, the GA produces the individuals (fuzzy models) for the next generation. During the process, the Kalman filter works on the new individuals found by the GA to estimate their parameters. The fitness of each individual is evaluated using SRC, which is represented by

SRC(m) = log(8 "2) + - - log (n) m

(3)

^ 2 where m, as before, denotes the number of fuzzy rules in the model, n is the number of training data, and at is the estimated variance of model residuals. This criterion tries to achieve a trade-off between the accuracy and the complexity of a model. A fuzzy model with a large number of rules may have a low ~ 2 in the first term, but it will usually accompany a large m in the second term. Only the model that has the minimum SRC value is selected as the "best" model.

The process of using Kalman filter to estimate the parameters of individuals, measuring their fitness using SRC, and applying genetic operators to generate a new population of individuals is repeated over many

356 L. Wang, J. Yen / Fuzzy Sets and Systems lOt (1999) 353-362

generations. If all goes well, each generation will tend to contain more of the features that were found useful in the previous generation, and an improvement in overall performance can be realized over the previous generation.

The GA operators in the hybrid algorithm include a roulette wheel selection strategy, a uniform crossover, and a mutation. A crossover rate of 0.5 and a mutation rate of 0.01 are chosen. An elitist strategy [8] is also used whereby the best individual at a given generation is promoted directly to the next generation.

3. Computer simulations

Two examples are provided in this section to illustrate the practical applicability of the proposed hybrid algorithm. The first one is a human operator modeling example that is taken from 1-19]. The second one is a nonlinear system modeling example that is borrowed from [24]. The population size for both examples is set to 10.

The fuzzy rule used here is assumed to have a crisp output, i.e., the f~ in Eq. (1) is a constant value I-5]. The membership functions of the input variables are assumed to be the Gaussian form

I l ( x j - c i j ~ 2 ~ ' (4, /ta,j(xj)=exp - 2 \ aij / _1

where ci~ and air are real-valued parameters. For computational simplicity, cij and aij are predetermined using a k-means clustering algorithm and a nearest-neighbor heuristic, respectively (see, e.g., 1,22]), as suggested in [13].

Example 1. This example is used to show how to build a model of a human operator's control action in a chemical plant. The plant is for producing a polymer by the polymerization of some monomers. Since the start-up of the plant is very complicated, a human operator has to manually operate the system.

The structure of the human operation is shown in 1,19]. Five input candidates, ul: monomer concentration, u2: change of monomer concentration, u3: monomer flow rate, u4 and Us: local temperatures inside the plant, are available, to which a human may refer. The output is the set point for the monomer flow rate.

In this plant the human operator determines the set point for the monomer flow rate and the actual value of the monomer flow rate to be put into the plant is controlled by a PID controller. There are 70 data points available for each of the above six variables. Fig. i shows the actual set point for the monomer flow rate. Like in [19], we select u~, u2 and u3 as the input variables of the fuzzy model. The initial number of fuzzy rules is arbitrarily set to 40. Beginning with a population of randomly selected fuzzy rules, after 100 generations the hybrid algorithm generates the following individual which has a minimum SRC value:

1 , 1 1 1 1 0 0 1 0 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0 0 1 ] .

The number of t's in the binary string indicates the number of fuzzy rules used in the model and their positions indicate which rules in the rule base should be chosen. Here 25 rules have been selected to construct a fuzzy model. The mean squared error (MSE) of the model is 2287 and its SRC value is 9.2526. Fig. 2 gives the SRC values and the number of fuzzy rules by the best individual in each generation. Fig. 3 shows the error between the real set point and the output of the fuzzy model. As a comparison, Fig. 3 also shows the error between the real set point and the output of the fuzzy model constructed using the complete 40 rules which has a MSE of 1797 and a SRC value of 9.9214. It can be seen that the model with the complete fuzzy rules has fitted the real set point slightly better than the model consisting of 25 fuzzy rules. However, as pointed out in Section 1, although a fuzzy model with a large number of fuzzy rules may fit training data very well, it usually has a poor generalization ability. This point will be seen in the following example.

L. Wang, J. Yen / Fuzzy Sets and Systems 101 (1999) 353-362 357

8O0O

7OO0

I ooo

' 2;0 0o lO

6000

~ 500G E O

O E 4000

3000 O

2000

) H

F J

30 4'0 5o 6'0 k

Fig. 1. Set point for the monomer flow rate in a chemical plant.

70

10 i

9.8

O ~9.6

09.4

9.2 \ _ _

45

N 35

~ 3o O e~

~ 25 Z

20 0

\ - -

0 50 100 50 100 Number of generations Number of generations

Fig. 2. The SRC values and the number of fuzzy rules in the best individual in each generation.

Example 2. Cons ide r the second-o rde r non l inea r difference equa t ion given in [-24]

y(k) = g ( y ( k - 1), y ( k - 2)) + u(k), (5)

where

g ( y ( k - 1), y ( k - 2)) = y ( k - 1)y(k - 2 ) [y (k - 1) - 0.5] 1 + y 2 ( k - 1) + y 2 ( k - 2) (6)

The non l inea r c o m p o n e n t g in this plant , which is usual ly cal led the "unforced sys tem" in con t ro l l i te ra ture (see, e.g., [ 14]), has an equ i l ib r ium state (0, 0) in the state space. This implies tha t while in equ i l ib r ium wi thou t

358 L. Wang, J. Yen / Fuzzy Sets and Systems 101 (1999) 353 362

L U

200

100

0

- 1 O0

-200

200

100

0

-1 O0

-200 0

Fuzzy model with 40 rules

1 ̀O 20 k

Fuzzy model with 25 rules

i i i i

3'0 4'o s'o 6'o 70

1 ro 2̀O 30 4̀O 5̀O 60 70

Fig. 3. Errors between real control actions and model outputs.

1

0.5

0 "T

>' -0.5

-1

- 1 . 5

• °

o t • ". •

; i = ,o ° • • .. -...- . . , ,~.~ . a -

• :..." . ,.-.... , •

° - ° ° ° ~ " ° : .

-1 0 1 y(k-2)

Fig. 4. Trajectory of the unforced system.

an input, the output of the plant is the sequence {0}. Fig. 4 shows the trajectory of the unforced system for different initial conditions in the state space

Our goal is to approximate the unforced system 9 using the fuzzy model. For this purpose, 400 simulated data points are generated from the plant model (5). The first 200 data points are obtained by assuming a random input signal uniformly distributed in the interval [--1.5, 1.5], and the last 200 data points are obtained by using a sinusoid input signal u(k) = sin(2rck/25). The 400 simulated data points are shown in Fig. 5.

L. Wang, J. Yen / Fuzzy Sets and Systems 101 (1999) 353-362 359

1.5

- z ~ 05 i o 0

K l -0 5

-1

-1.5L

-2 0 50 350 400

I l 1 O0 150 200 250 300

k

Fig. 5. Output of the plant model.

Table 1 Comparison of fuzzy models for modeling and testing

Fuzzy model SRC values MSEs (modeling) MSEs (testing)

Fuzzy model with 28 rules -7.2395 3.3299 e - 4 5.9595 e - 4 Fuzzy model with 40 rules -6.9515 3.2884 e - 4 6.9152 e - 4

We use the first 200 data points to build a fuzzy model. The performance of the resulting model is tested using the remaining 200 data points. We select y ( k - 1) and y ( k - 2) as the input variables and arbitrarily set the number of fuzzy rules to 40. Beginning with a population of randomly selected fuzzy rules, after 100 generations the hybrid algorithm generates a fuzzy model consisting of 28 rules which has a minimum SRC value. The positions of these rules in the rule base are indicated by the following binary string

[1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 1 1 0 0 0 0 1 101 1 1 101 1 0 1 0 1 1 101 1].

Table 1 lists the SRC values and the MSEs of the model in both the training stage and the testing stage. As a comparison, Table 1 also gives the SRC values and the MSEs of the fuzzy model constructed using the complete 40 rules. It can be seen that, compared to the fuzzy model with 28 rules, the fuzzy model with 40 rules has a small MSE in the training stage but a large MSE in the testing stage. This result indicates that the model with 40 rules has been overfitted.

Fig. 6 shows the SRC values and the number of fuzzy rules in the best individual in each generation. Fig. 7 shows the trajectories of the two fuzzy models in the training stage. Comparing Fig. 7 and Fig. 4, it can be

360 L. Wang, J. Yen / Fuzzy Sets and Systems 101 (1999) 353-362

-6.9

-7

~> -7.1 o n,," t.t)

-7.2 /

-7.3

V "~

45

40 > "

~ 35

~_ 30 . ,Q

~ 25 z

20 0

i , - -

L

0 50 1 O0 50 1 O0 Number of generations Number of generations

Fig. 6. The SRC values and the number of fuzzy rules in the best individual in each generation.

0.5

0 i

>'-0.5

Fuzzy model with 40 rules 1 1

• . ~ • . •

.-~,..." -

d ..,~'..~- • . . . : . . . ' , " , . ' . . : .

• , . " : . , . .*,

0.5

0

>'-0.5

-1

Fuzzy model with 28 rules

-1.5

. 1 . . ,

• ,'-'{-: : ' . . • . . - ' . . , ~ : - . - . . .

.• . ~..•.~.,..'...'.,. -. . - . .

-1 0 -1 0 y(k-2) y(k-2)

Fig. 7. Trajectories of two fuzzy models with different number of rules.

seen that the trajectories of the two identified fuzzy models and the real unforced system 9 are almost the same. Fig. 8 shows the errors between the real plant output and the two identified model outputs in the testing stage•

4. C o n c l u s i o n s

GAs are powerful in finding solutions in a complicated search space, but they are computationally expensive• The parameter estimation problem in fuzzy modeling is a standard optimization problem and there is no strong reason that GAs should be used in this aspect. The most difficult aspect in building a fuzzy model, where GAs are thus expected to play their role, is in the identification and finding of the important fuzzy rules that are used to construct the model.

In this paper we investigated how to extract the important fuzzy rules in a given rule base using a hybrid of GA and Kalman filter• This proposed hybrid algorithm combined the advantages of GA's strong-search capacity and Kalman filter's fast-convergence merit• To achieve a trade-off between the accuracy and the complexity of a fuzzy model, the Schwarz-Rissanen criterion was used as the evaluation function of the hybrid algorithm• Computer simulations showed the feasibility of the algorithm•

The proposed algorithm is not necessarily limited to search fuzzy rules with a constant output, it can directly be extended to find fuzzy rules whose outputs consist of a linear regression equation•

L. Wang, J. Yen / Fuzzy Sets and Systems 101 (1999) 353-362 361

0.1 Fuzzy model with 40 rules

i i I I I I [

0.05

~ o W

-0.05

-0.1 0

0.1

I

20 410 610 810 1 O0 120 k

.Fuzzy model with 28 rules

140 160 180 200

005

S 0 U J

-0.05

2 ' 0 4' 6' 8' ' ' ' "010 0 0 0 100 120 140 160 180 200 k

Fig. 8. Output errors between the real system and the identified models in the testing stage.

In the present research, the member sh ip funct ions of the input var iables have been de te rmined using a cluster ing technique. Al ternat ively , they can also be de te rmined toge ther with the consequent parameters . No te that in this case the K a l m a n filter can no longer be used since the p rob l e m has become nonl inear in the parameters . Ins tead, some non l inea r search a lgo r i thm such as the grad ien t decent a lgo r i thm should be

pursued.

Acknowledgements

This research is suppo r t ed by N a t i o n a l Science F o u n d a t i o n Young Inves t iga tor Award IRI-9257293.

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