1
A Table Lookup Scheme for Fuzzy Logic Based Model Identification Applied to Time Series Prediction Prof. M Farooq Department of Electrical & Computer Engg. Royal Military College of Canada
Khatoon Shahida & Prof. Ibraheem & Prof. Moinuddin Department of Electrical Engg. Faculty of Engg. Tech., J.M.I, New Delhi-25,India
[email protected]
Kingston, Ontario K7K7B4 farooq@rmca
Abstract - Zadeh introduced Fuzzy sets in 1965 to represent and manipulate data and information that possess nonstatistical uncertainty. Since this date, fuzzy logic has been applied to many fields such as industry, medicine, economics and so on. The reason for this rapid growth in the use of fuzzy logic worldwide is that fuzzy logic provides an appropriate mechanism to describe the static and/or dynamic behavior of complex physical systems that are difficult to yield their conventional mathematical models. We can consider a fuzzy set as a fuzzy model of human concept. In this paper, we consider fuzzy modeling as an approach to form a system model using a description language based on fuzzy logic with fuzzy predicates. This paper presents a general approach to modeling and identification of dynamic systems based on fuzzy logic. A table –lookup scheme is presented to generate fuzzy rules from numerical data. This method determines a mapping from input space to output space based on the combined fuzzy rule base using defuzzifying procedure. Application to time series prediction problem is also presented. Key words: Fuzzy Logic, Identification model, Time series, fuzzy inference engine, Table-lookup scheme, COG method of defuzzification
1
Introduction
Fuzzy modeling is the most important issue in fuzzy theory. We can consider a fuzzy set as a fuzzy model of a human
concept[1]. In this paper, we consider fuzzy modeling to be an approach to form a system model using a description language based on fuzzy logic with predicates. We can interpret the fuzzy modeling as a linguistic modeling scheme by which we linguistically behavior using a natural language [2]. The fuzzy modeling is a system description with fuzzy quantities. Fuzzy quantities are expressed in terms of fuzzy numbers or fuzzy sets associated with linguistic labels. Therefore, the relation between input and output variables can be viewed as a set of fuzzy logical rules or fuzzy-set associations. Since functional variables are stored in a distributed rule-based fashion, the value of the function at any point in the input space is derived by aggregating the consequences of fuzzy logical rules. It has been shown that fuzzy systems are capable of approximating any real continuous function to any desired degree of accuracy [25-26]. Fuzzy modeling can be applied to various domains of problems such as signal processing, pattern recognition, and control [3-11]. The main problem of fuzzy control is to design a fuzzy controller where we usually take an expertsystem-like approach [13]. That is, we derive fuzzy control rules from a human operator’s experience and /or engineer’s knowledge, which are mostly based on their qualitative knowledge of an objective system. There are three other approaches to generate the rules. Usually, the design procedure is thus something like the following: first, we build linguistic control rules; second, we adjust the parameters of fuzzy sets by which the linguistic terms in the control rules are quantitatively interpreted.
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2 Apart from fuzzy control, the studies of fuzzy modeling are divided into two groups. The studies of the first group deal with fuzzy modeling of a system itself or fuzzy modeling for simulation. The studies of the second group deal with fuzzy modeling of a plant for control. Just as with the modern control theory, we can design a fuzzy controller based on a fuzzy model of a plant if a fuzzy model can be identified [11,12]. Let us consider the problem of modeling. Modeling is classified from the view point of a description language. However, there are some common problems to be solved in modeling independently of both the description language and data type. Thus we refer to the systems theory. The most complicated problems arise when we take a black box approach to modeling. In the black box approach, we have to build a dynamical model using only input-output data. This stage of modeling is usually referred to identification. From the afro-mentioned, system identification involves finding a model that may be regarded as equivalent to the objective system with respect to input-output data. The identification for the fuzzy model has two aspects as usual: structure identificationa and parameter identification. This problem will be discussed in general in section 3.0.
2
Generation of Fuzzy Rules
process to be controlled and his general control engineering sense. There are some disadvantages in this method of design. Firstly, sometimes an operator cannot tell linguistically what actions he takes in such and such situation. Secondly, when a process is complex, it is difficult to design a fuzzy controller even from a control engineer’s point of view. Another point to note is that this method is heuristic in nature, and very difficult to be generalized.
2.2
Based on Modeling the Operator’s control Actions
When an operator’s skill is important, it is very useful to derive fuzzy control rules by modeling the operator’s control actions [4]. For instance, consider when we drive a car. We know car-driving techniques by our hands and legs rather than by our brain [33-34]. If it is possible to model an operator’s control actions in terms of fuzzy implications using the input-output data concerned with control actions, we can use the obtained model as a fuzzy controller. In this method, it is not difficult to find the input variables of the model by asking an operator what kind of information he uses in his control. Also we can find approximately the number of fuzzy subspaces of the input space.
2.3
Based on the Fuzzy Model of a Process
The two previous methods for generating fuzzy rules work well only in the case where an expert plays an The most important topic to discuss about fuzzy systems is the way to design and derive fuzzy rules. A important role in controlling the process. The first method number of studies have been devoted to this problem [3-4, is somehow based on a rough idea about the characteristics 9-11, 25]. There are, in general, four methods of design and of a process. For example, the output increases as the input derivation of fuzzy rules [4, 13,31]. Sometimes, it seems does, the process has a time lag, and so on. The second that a combination of them would be necessary to construct method uses only the necessary variables available to an effective method for the derivation of fuzzy rules. Until control a process, to which an operator can refer. In fact this now there exists no systematic method for the general controller works better than the expert does, since it does design of fuzzy rules. The following paragraphs will not make errors. However, if we cannot rely on an operator and if we want to have better results than any operator gives, explain these methods. have to find another method of design. 2.1 Expert Experience and control we In the linguistic approach, the linguistic description of the Engineering Knowledge dynamic characteristics of a controlled process may be Most fuzzy controllers have been designed so far viewed as a fuzzy model of the process. By fuzzy by referring to human operators’ experience and/or control modeling, we mean to represent the characteristics of a engineer’s knowledge [23]. In fact we can say that fuzzy process by a set of its fuzzy behaviors which are also control was the first practical application of expert systems expressed by using fuzzy implications concerned with The format of fuzzy implications clearly makes them inputs, state variables, and outputs. suitable as a descriptive language to express an expert’s This method is somewhat more complicated, it yields better thinking, which is essentially fuzzy in its nature. In our performance and reliability. However, this method to the daily life most of the information on which our decisions are design of fuzzy rules has not as yet been fully developed. based is linguistic rather than numerical in nature. Seen in Figure 1 shows an idea for a fuzzy controller consists of a this perspective, fuzzy control rules provide a natural set of control rules and the fuzzy process model consists of framework for the characterization of human behavior and set of process behaviors. decisions analysis. Many experts have found that fuzzy control rules provide a convenient way to express their F u zz y domain knowledge [32]. C o ntro In many cases where an operator plays an important role in C o ntro l P ro cess R u le s process control, it is very useful to find his know-how on control by interviews and to express it in terms of fuzzy implications. It is also possible for a control engineer to list a number of protocols based on his knowledge about a Fig. 1 Fuzzy Controller Based on Modeling a Process
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3 a system which affect the output. In this case we select a finite number out of a finite number and so there are some systematic ways to solve this problem. In a conventional black box approach in systems theory, this type of identification is, however, not explicitly discussed, and models are based on pre-assigned input-output variables. The structure identification of type II, which is concerned with input-output relations, is divided into the subtypes II(a) and II(b). In II(a) , we have to determine the number of fuzzy rules in a fuzzy model. By structure identification in ordinary systems theory, what we mean is 2.4 Based on Learning: Many FLCs have been built to emulate human to find the relations between the inputs and the outputs. decision-making behavior, but few are focused on human Given a description language for modeling, it means the learning [25,36], namely, the ability to create fuzzy control determination of the order of a model. Whereas in fuzzy rules and to modify them based on experience. The Self model, the structure identification of this kind is stated in a Organizing Controller (SOC) [37] has a hierarchical different way. A fuzzy model consists of a number of ifstructure which consists of two rule bases. The first one is then rules. The number of rules in a fuzzy model the general rule base of an FLC. The second one is corresponds to the order in a conventional mode. The identification II(b) implies determining how constructed by “meta-rules” which exhibit human-like learning ability to create and modify the general rule base the input space should be partitioned. There are two parts of an if-then rule: the premise part and the consequent part. based on the desired overall performance of the system. So the rules have two structures: the premise structure and the consequent structure. Type II(b) deals with premise 3 Identification in Fuzzy Modeling structure. The premise space of the input variables of fuzzy The identification is divided into two kinds: model is partitioned into several fuzzy subspaces (fuzzy structure identification and parameter identification. sets); where the number of rules corresponds to the number Structure identification can be divided into two types: type I of subspaces. This problem is combinatorial, therefore we and type II, where each type is also divided into two need a heuristic method to find an optional partition together subtypes, a and b. Figure 2 illustrates the classification of with some criterion, i.e. output error. identification. 3.2 Parameter Identification: In ordinary system identification, parameters are Identification the coefficients in a functional system model. In a fuzzy model, the parameters are those in the membership functions of the fuzzy sets. There is not a big difference Parameter Identification Structure Identification between the two except in the number of the parameters, there being many more in a fuzzy model. The structure identification and the parameter identification cannot be Structure Identification Structure Identification separately performed in principle. This fact makes the identification very complicated. However, in some b: Input variables A: Input Candidates a: Number of rules b: Input Space approaches, the parameter identification can be separately performed after the structure identification [11]. Figure 2 Classification of Identification 4 Fuzzy Model Identification Using There are two ideas for designing a fuzzy controller based on fuzzy model. The first one is a heuristic method [35] in which we set fuzzy rules to be compatible with system behavior by considering the control objective. The second idea is the method of identification to determine the structure and the parameters of fuzzy rules model so that the system with a controller satisfies the control objective [3,9,11]. We will investigate this method through simulation in section 3.4.
3.1
Numerical Data
Structure Identification
Generally speaking, the structure identification of a system has to solve two problems: one is to find input variables (type I), the other one is to find input-output relations (type II). Type I consists of I(a) and I(b). In I(a), we find possible input candidates for the inputs to a system. There are of course an infinite number of possible candidates, which should be restricted to a certain number. This type of identification can not be solved in general. Because there is no systematic way to find the exact causes of an unknown phenomenon. We have to take a heuristic method based on experience an/or common sense knowledge. In the structure identification of type I(b), given the possible in put candidates, we find a set of input variables to
In this section, we will deal with fuzzy model identification based on a black box approach. Because even if we can find the local mechanisms of a system, it is often the case that we cannot build the whole system model by arrogating the local mechanisms. In the black box approach, the dynamical model is built using the input-output data of the plant. Existing identification schemes determine a model for a system based on the input-output pairs resulting from existing the system with an input signal and measuring the corresponding outputs. For many industrial system, there is another important information source that is human knowledge, who are familiar with systems and can provide linguistic descriptions about the behavior of the system in terms of vague and fuzzy words. Although these linguistic
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4 descriptions are not precise, they provide important information about the system. For many process control problems, a human operator can determine a set of successful control rules based only on the linguistic description about the process. Existing identification schemes ignore this important source of information and cannot incorporate the linguistic descriptions directly into the identifiers. Even if there are some identification scheme based on combination of linguistic description and numerical information, they still combine these two sources of information in ad hoc way for a specific problem; simulations are then performed to show that this approach works well for the specific problem. This kind of approach has two weak points: 1) it is quite problem dependent, i.e., a method may work well for one problem but is not suited for another problem; and, 2) there is no common framework for modeling, which makes theoretical analyses for these approaches very difficult. The aim of this section is to study and investigate identifiers of nonlinear dynamic systems which combine both linguistic descriptions and input-output pairs in a inform fashion (a fuzzy rule base) into their designs. The method which we will use for identification is called “A Table-Lookup Scheme” [25]. This method performs a one-pass operation on numerical input-output pairs and linguistic fuzzy if-then rules. The key idea of this method is to generate fuzzy rules from input-output pairs, collect the generated rules and linguistic rules into a common fuzzy rule base, and construct a final fuzzy logic system based on the combined fuzzy rule base.
center of the region and has membership value unity; the other two vertices lie at the centers of the two neighboring respectively, and have membership values equal to zero. Of course, other divisions and other shapes of membership functions are possible.
Step 2: Generate Fuzzy Rules from Given Data Pairs First, determine the degrees of given x1(i) , x2(i)and y in different regions. For example, x1(1) ,in Figure 1.3 has degree 0.8 in B1, degree 0.2 in b2, and zero degrees in all other regions. Similarly, x2(2) in Figure 3 has degree 1 in CE, and zero degrees in all other regions. Second, assign a given x1(i) , x2(i) or y(i)to the region with maximum degree. For example, x1(1) in Figure 1.3 is considered to be B1, and x2(2) in Figure 1.3 is considered to be CE. [x1(1) ( 0.8 in B1, max), x2(1) (0.7 ( x1(1) , x2 (1) ; y(1) ) ⇒ (1) in S1, max); y (0.9 in CE, max)] ⇒ Rule 1 : IF x1 is B1 and x2 is S1 , THEN y is CE; [x1(2) ( 0.6 in B1, max), x2(2) (1 in ( x1(2) , x2 (2) ; y(2) ) ⇒ (2) S1, max); y (0.7 in CE, max)] ⇒ Rule 2 : IF x1 is B1 and x2 is CE , THEN y is B1; The rules generated in this way are “and” rules, that is, rules in which the conditions of the IF part must be met simultaneously in order for the result of the THEN part to occur. (i)
µ (x1) S2 1
4.1 Identification based on a Table-Lookup scheme: Suppose we are given a set of desired input-output data pairs: (x1(1), x2(1) ; y(1)), (x1(2), x2(2) ; y(2)), (x1(3), x2(3) ; y(3)) (1.1)
B1
-1
0
1
B2
X (k) -2
2
4
µ (x1) S2 1
B2
S1
CE
B1
-1
0
1
2
X(k) 4
S1 CE
B1
B2
B3
0
Step 1: Divide the Input and Output Spaces into Fizzy Regions Assume that the domain intervals of x1, x2 and y are [-x1 ,+ x1 ], [- x2, + x2 ] and [- y, +y], respectively, where domain interval of a variable means that most probably this variable will lie in this interval. Divide each domain interval into 2N+1 regions (N can be equal or unequal), denoted by SN (Small N), …, S1( Small 1), CE (Center), B1 (Big 1), …, BN (Big N), and assign each region a fuzzy membership function. Figure 1.3 shown an example where the domain interval of x1 is divided into five regions (N=2), the domain region of x2 is divided into seven regions (N =3), and the domain interval into five regions (N=2). The shape of each membership function is triangular; one vertex lies at the
CE
0 -3.5
where x1 and x2 are inputs, and y is the output. This simple two-input one-output case is chosen in order to clarify the basic ideas of this approach extensions to general multiinput –multi-output cases are straightforward. The task is to generate a set of fuzzy if-then rules from the desired inputoutput pairs of (1.1), and use these fuzzy if-then rules to determine a fuzzy logic system f : ( x1, x2) → y. This approach consists of the following five steps :-
S1
-3.5
-2
µ(x1)
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S3
S2
1 X(k) 0 -3.5 -2.6
-1.7 -0.8
0
1.3 2.2
3.1
4
5
Step 5: Determine a Mapping Based on the combined Fuzzy Rule Base Figure 5
The defuzzification strategy which is used in this method determines the output y for given inputs (x1, x2) as follows: First, for given inputs (x1, x2), we combine the antecedents of the ith fuzzy rule using product operations to determine the degree, of the output corresponding to (x1, x2); that is, µο (yi) = µi (x1) µi(x2)
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Table-Lookup illustration of fuzzy rule base
2
Figure 4
5 4 3 2 1 0 -1 -2 -3 -4
893
B2
812
B1
Time series prediction is very important practical problem. Applications of time series prediction can be found in the areas of economic and business planning, inventory and production control, weather forecasting, signal processing, control, and lots of other fields. Let x (k) ( k = 1,2,3,….) be a time series. The problem of time- series prediction can be formulated as : given x( k - n + 1) , x( k - n + 2) , … , x(k), determine x( k + l ) where n and l are fixed positive integers. That is, determine a mapping from [x( k - n + 1) , x( k - n + 2) , … , x( k ) ] ε Rn to [x( k+ l )] ε R Now we apply the above method of identification to the time series production problem. Assuming that x(1), x(2), …, x(k) are given, we can form k-n input-output pairs: [ x( k - n ) , … , x( k -1) ; x( k) ] [ x( k – n - 1 ) , … , x( k -2) ; x( k- 1 ) ] … [ x( 1 ) , … , x( n) ; x( n+1) ] (1.5) The plant to be identified is described by the second-order difference equation x( k +1) = g [ x( k ), x( k –1 ) ] + u( k ) (1.6) where g [x( k ) , x ( k-1)] = x( k ) x ( k-1 )[ x( k )+2.5] 1+ x2( k )+ x2( k-1 ) (1.7) and u( k ) = Random Number ε [-2, +1] (1.8)
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CE
Time-Series
650
S1
to
569
S2 S3 S2 S1 CE B1 B2 B3
Application Prediction
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Figure 4 illustrates a table-lookup representation of a fuzzy rule base. We fill the boxes of the base with fuzzy rules according to the following strategy: a combined fuzzy rule base is assigned rules from either those generated from numerical data or linguistic rules; if there is more than one rule in one box of the fuzzy rule base, use the rule that has maximum degree. In this way, both numerical and linguistic rule is an “and” rule, it fills only one box of the fuzzy rule base, but if a linguistic rule is an “or” rule (that is, a rule for which the THEN parts follows if any condition of the IF part is satisfied), it fills all the boxes in the rows or columns corresponding to the regions of the IF part.
5.
407
Step 4: Create a Combined fuzzy rule Base
Where y denotes the center value of the output region of rule I, and n is the number of fuzzy rules in the combined fuzzy rule base.
326
Since there are usually lots of data pairs, and each data pair generates one rule, it is highly probable that there will some conflicting rules, i.e., rules that have the same IF part but a different THEN part. One way to resolve this conflict is to assign a degree to each rule generated from data pairs, and accept only the rule from a conflict group that has maximum degree. In this way not only is the conflict problem resolved, but also number of rules is greatly reduced. We use the following product strategy to assign a degree to each rule: for the rule : IF x1 is A and x2 is B, THEN y is C, the degree of this rule, denoted by D(Rule), is defined as D (Rule) = µ A (x1) µ B (x2) µ C (y) As examples, Rule 1 has degree D (Rule 1) = µ B1 (x1) µ S1 (x2) µ CE (y) = 0.8 × 0.7 × 0.9 = 0.504 (see Figure 3) and rule 2 has a degree D (Rule 2) = µ B1 (x1) µ CE (x2) µ B1 (y) = 0.6 × 1 × 0.7 = 0.42
245
Step 3: Assign a Degree to Each Rule
Then we use the center average defuzzification formula to determine the output = Σ µ ο ( y i ) . yi yο Σµο(yi)
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Division of the input-output space into fuzzy regions and the corresponding membership functions
83
Figure 3
Total time series
we use (1.6) to generate the numerical data. In our simulation used to generate the plant numerical data, we start the generating operation from k=1 upto k = 10000, where we assume the initial value as x(0) = 0 and x(1) = 0.1 . Fig 5 shows the outputs of the plant from k=0 to k= 1000.
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6 Steps 1-4 of the above method are used to generate a fuzzy rule base based on the input-output pairs (1.5); then this fuzzy rule base is used to forecast x ( k+l ) for l=1,2, …., using the defuzzifying procedure of step, where the inputs to predictor are x(k+ l- n), x(k+l-n+1), …,x(k+l-1). We assume number of inputs equally to four (n=4), and number of output equally to one (l=1); that is four point values in the series were used to predict the value of the next point. We simulated three cases (I) when the number of fuzzy sets of membership function is equal to 5, (II) when the number of fuzzy sets are 7 and (III) when 9.
µ (x) S4 S3
1 0 0
-1
50
100
150
-2 -3
The membership function for any point in cases (I),(II),(III) are shown in figures 6, 7 and 8 respectively for the numerical predictor. The first 900 points of the series were used as training data, and the final 100 points wee used as test data.
µ (x1) S2 1
S1
CE
B1
-1
0
1
B2
0
X (k) -3.5
-2
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Figure 6 The membership function of all input-output variables in case (I)
µ (x1) S3
S2
S1
CE
B1
B2
B3
1 X(k) 0 -3.5 -2.6 -1.7 -0.8 0 1.3 2.2 3.1 4 Figure 7 The membership function of all input-output variables in case (II)
B2 B3 B4
X(k) -3.5 -2.75 -2 -1.25 -0.5 Figure 8
5.1
x(k) 2
CE B1
1
4 3
S2 S1
0
4 1 1.75 2.5 3.25
The membership function of all input-output variables in case (III)
Simulation Results
We simulated this identification procedure for the time series of figure 5. We started with the fuzzy rule base generated by the data x(1) to x(900), then made a prediction of x(901) to x(1000). Figures 9, 10 and 11 show the results for case (I), (II), and (III) respectively. We show that prediction can be greatly improved by dividing the domain interval into finer regions. Comparing Figures 3.9, 3.10 and 3.11 we see clearly that we obtain better and better results as the domain intervals is more finely divided. Of course, the price paid for doing this is larger rule base. For example: 208, 302 and 501 fuzzy rules are generated in case (I),(II) and (III) respectively from the simulations.
6. Conclusion A fuzzy model consists of a number of if-then rules that describe the behavior of a dynamic system. At the beginning we presented the general methods to generate fuzzy model rules. Then, we discussed the identification problem of a fuzzy model, where we explained the two kinds of identification: structure identification and parameter identification. Based on the table-lookup representation of the fuzzy rule base, we presented a general method to generate fuzzy rules from numerical data. In this sense, we consider this method as one of so-called black-box identification approaches. We applied this method to time series prediction problem. Simulation results show that the generated fuzzy model system is capable of approximating non linear continuous function. It also shows that prediction can be greatly improved by dividing the domain interval into finer regions. We performed three simulations: the first one with five fuzzy sets of the membership function, the second one with seven fuzzy sets of the membership function and third one with nine fuzzy sets of the membership function. The most important advantage o f this method is its simplicity, it just performs a simple one-pass operation on the training data. The price paid for this simplicity is that we have to determine the partitions of the domain intervals and the membership functions in ad hoc manner.
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7
Figure 9
Outputs of the plant (solid line) and the identification model (dashed line) with 5 fuzzy sets.
Figure 10
Outputs of the plant (solid line) and the identification model (dashed line) with 7 fuzzy sets.
Figure 11
Outputs of the plant (solid line) and the identification model (dashed line) with 9 fuzzy sets.
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