CHAPTER 3 FUZZY INFERENCE SYSTEM

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1 CHAPTER 3 FUZZY INFERENCE SYSTEM Fuzzy inference is the process of formulating the mapping from a given input to an output using fuzzy logic. There are three types of fuzzy inference system that can be implemented in fuzzy logic tool box: Mamdani-type, Sugeno-type and The Standard Additive Model (SAM). We applied Mamdani model to solve the temperature control problem. [3.1]. THE MAMDANI MODEL. Mamdani model is one of the most useful models which consist of the following rules that describe a mapping from U 1 x U 2 xu 3. U n to W. is and is and is Then is (3.1) Where ( J=1,2,..n) are the input variables, y is the output variable and and are fuzzy sets for and y respectively. Given inputs of the form:, where. are fuzzy subsets of U 1, U 2,U 3. U n, the contribution of rule R i to a mamdani model s output is a fuzzy set whose membership function is computed by (3.2) Where α i is known as matching degree of rule R i and is the matching degree between x J and R i S condition respect to x J. ( ) (3.3) 73

2 the operator ^ denotes the min operator. The final output of the model is the aggregation of outputs from all rules using the max operator: (3.4) The output results are fuzzy outputs that could be defuzzified into crisp output by using defuzzification methods.[49],[50] [3.2]. THE TSK MODEL. The Takagi-Sugeno-Kang (TSK) model was introduced by T.Takagi and M.Sugeno.This model reduce the number of rules required by Mamdani model, specially for complex and high dimensional problems. To achieve this goal, the TSK model replaces the fuzzy sets in the consequent part (then-part) of the Mamdani rule with a linear equation of the input variables, For example Two input and one output TSK model consist of rules in the form of IF x is and y is THEN z = ax+by+c (3.5) Where a,b,c are numerical constants. In general, rules in a TSK model have the form IF is and y is THEN y = (,,.. ) = (3.6) Where is the linear model and are real valued parameters. The inference performed by the TSK model is an interpolation of all the relevant linear models. The degree of relevance of a linear model is determined by the degree the input data belongs to the fuzzy subspace associated with the linear model. These degree of relevance become the weight in the interpolation process. 74

3 The total output of the model is given by the equation below where αi is the matching degree of rule Ri, which is analogous to the matching degree of the Mamdani model (3.7) The inputs to a TSK model are crisp (non fuzzy) numbers. Therefore, the degree the input x 1 = a 1, x 2 =a 2, x r = a r, matches i th rule is typically computed using the min operator: (3.8) However the product operator can be used : = x (3.9) Let us consider a TSK model consisting of the following three rules : IF x is Small THEN y = L 1 (x), IF x is Medium THEN y= L 2 (x), (3.10) IF x is Large THEN y= L 3 (x), The output of such a model is (3.11) The TSK model provides a powerful tool for modeling complex systems. It can express highly nonlinear function using a small number of rules. The potential applicaton of TSK models, hence is very broad. The great advantage of the TSK model is its representative power; it can describe a highly nonlinear system using a small number of rules. 75

4 Moreover, due to the explicit functional representation form, it is convenient to identify its parameters using some learning algorithms.[51] [3.3].THE STANDARD ADDITIVE MODEL (SAM) The structure of fuzzy rules in SAM is identical to that of the Mamdani model. It was introduced by B.Kosko [52].There are four differences between the inference scheme of these two models: (1) SAM assumes the inputs are crisp, while while Mamdani model handles both crisp and fuzzy inputs. (2) SAM uses the scaling inference method while Mamdani uses the clipping method. (3) SAM uses addition to combine the conclusion of fuzzy rules, while the mamdani model uses max. (4) SAM includes the centroid defuzzification technique, while the Mamdani model does not insist on a specific defuzzification method. Let us consider a standard Additive model consisting of rules of the form of IF x is AND y is THEN z is. (3.12) Given crisp inputs x= x 0, y = y 0, the output of the model is x x (3.13) Where centroid is the function that performs the centroid defuzzification. This is formerly stated in the theorms below Theorem: Suppose a SAM model that describes a mapping from U x W to W contains rules in the form of IF x is and y is THEN z is. Then the model s output from the inputs x=x 0 ; y = y 0 is 76

5 (3.14) Where A i is the are under the i th rules.conclusion C i and g i is the centroid of C i (center of gravity) : (3.15) If z is a continuous variable. [3.4]. DESIGNING ANTECEDENT MEMBERSHIP FUNCTION. The membership functions of an input variable s fuzzy sets should usually be designed in such a way that the following two conditions are satisfied: (1) Each membership function overlaps only with the closest neighboring membership functions; (2) for any possible input data, its membership value in all relevant fuzzy sets should sum to 1 or (nearly so) Let us use Ai to denote fuzzy sets of an input variable x. the two guidelines above may be expressed formally as the two equations below: 1: A i A j =ф (3.16) 2: µ Ai (x) 1 Two examples of membership functions that do not follow these design principles are shown in figs 3.1 and 3.2. Fig 3.1 obviously violets the second principle because the membership value 10 in three fuzzy sets do not sum to 1. i.e.,[50] (3.17) 77

6 Figure 3.1 that violets the second principle Figure 3.2 violets both design principles beccause A1 A3 ф Two example of membership function that follows these two guide lines are shown in figure 3.3 and 3.4. The former uses five symmetric membership functions, whereas the latter uses five asymmetric membership functions. 78

7 Fig 3.3 A Symmetric Membership Function Design Following the guidelines Fig 3.4 An Asymmetric membership function design following the guidelines [3.4]. BINARY FUZZY RELATIONS Composition of binary fuzzy relations can be performed conveniently in terms of membership matrices of the relations. Let [ ], [ ] 79

8 and [ ] be membership matrices of binary relations such that R = P0Q. We can then write, using this matrix notation, (i) (ii) Fig 3.5 Example of two convenient representation of a fuzzy binary relation: (i) sagital diagram (ii) membership matrix [ ] [ ] [ ] Where (3.18) 80

9 observe that the same elements of P and Q are used in the calculation of R as would be used in the regular multiplication of matrices, but the product and sum operations are here replaced with the min and max operations, respectively.[53] The following matrix equation illustrates the use of Eq to perform the standard composition of two binary fuzzy relations represented by their membership functions: For example,.8(= ) = max[min(.3,.9), min(.5,.3), min(.8,1)] = max[min(, ),min(, ), min (, )],.4(= ) = max[min(.4,.5), min(.6,.2), min(.5,0)] = max[min(,, min(, ), min (, )] A similar operation on two binary relations, which differs from the composition in that it yield triples instead of pairs, is known as the relation join. For fuzzy relations P(X,Y) and Q(Y,Z), the relational join, P*Q, corresponding to the standard max-min composition is a ternary relation R(X,Y,Z) defined by R(x, y, z) = [P*Q](x,y,z) = min[p(x,y), Q(y,z)] (3.19) For each xєx, yєy, and zєz. 81

10 The fact that the relational join produces a ternary relation from two binary relations is a major differences from the composition, which results in another binary relation. In fact, the max-min composition is obtained by aggregating appropriate elements of the corresponding (i) (ii) 82

11 (iii) Figure 3.6 composition and join of binary relation. Join by the max operator. Formally, [PoQ](x,z) = max[p*q](x,y,z) (3.20) For each xєx, and zєz. [3.5].TEMPERATURE CONTROL PROBLEM. We want to control the room temperature by air flow mixing. The amount of hot air flow and cold air flow is controlled by adjusting the voltage to the pump in the mixing stage. The lowest and highest voltage settings are denoted by V1 and V2. If the voltage is set at V1, maximal cold air flow will be allowed. If the voltage is set at V2, maximal hot air flow will be generated. A voltage between V1 and V2 mixes the hot flow and cold flow proportionally. Rules for the problem are R1: If K is Low Then V=V 1 R2: If K is High Then V=V 2 (3.21) 83

12 Fig 3.7 Temperature control system Rule viewer for the problem is as follows Fig 3.8 MATLAB generated rule viewer for The algorithm of fuzzy rule-based inference consists of four basic steps 1.Fuzzy Matching: Calculate the degree to which the input data match the condition of the fuzzy rules. 2.Inference: Calculate the rule s conclusion based on its matching degree. 84

13 3.Combination: Combine the conclusion inferred by all fuzzy rules in to a final conclusion. 4.Defuzzification: It is a process to convert a fuzzy conclusion into a crisp one. Fuzzy Matching: Let us consider fuzzy matching for the flow mixing control rules. The degree to which the input target temperature satisfies the condition of rule R1 target temperature is Low is the same as the degree to which the input target temperature K belongs to the fuzzy set Low. For the convenience we denote the degree of matching between input data and rule R as Matching Degree(I,R).Thus We can conclude Matching Degree( K,R1)= µlow(k) Matching Degree(K,R2)= µhigh(k) Let the input target temperature is 37 0 C i.e. K=37 0 C Fig 3.9 Showing the matching degree 85

14 When a rule has multiple conditions combined using AND (conjuction), we simply use a fuzzy conjunction operator to combine the matching degree of each condition. In general, the degree to which a rule in the form of Matches the input data Is computed by the following formula: (3.22) Inference: After the fuzzy matching step, a fuzzy inference step is invoked for each of the relevant rules to produce a conclusion based on their matching degree. There are two methods to produce the conclusion: (1) Clipping method and (2) scaling method. Both methods generate an inferred conclusion by suppressing the membership function of the consequent. The extent to which they suppress the membership function depends on the degree to which the rule is matched. Fig 3.10 Showing the working of clipping method 86

15 The clipping method cuts off the top of the membership function whose value is higher then the matching degree. In the case of a crisp consequent, the two methods degenerate into an identical one. We recall the rules for the problem R1: If K is Low Then V=V 1 R2: If K is High Then V=V 2 To apply fuzzy inference to these rules with crisp consequents, we need to first convert them into an equivalent fuzzy set representation. For instance, the crisp value V 1 is equivalent to a membership function that assigns value 1 to V 1, and 0 to all other values as shown in figure. Similarly, we can construct the membership function of the crisp value V 2 as shown in figure Fig 3.11 Membership function for crisp values of v1 and v2 It is then straightforward to show that the conclusion V=V 3 inferred by rule R 1 and the conclusion V=V 4 inferred by R 2 have the following membership function for clipping method: (3.23) 87

16 Figure illustrates this for R 3 and a target temperature that is 0.7 degree in the fuzzy set Low. Hence the conclusion inferred by R 3 in this case is Fig 3.12 Fuzzy inference Combining fuzzy Conclusions: A fuzzy rule-based system consists of a set of fuzzy rules with partially overlapping conditions, a particular input to the system often triggers multiple fuzzy rules. Therefore, a third step is needed to combine the inference results of these rules. This is done by superimposing all fuzzy conclusions about a variable. Combining Fuzzy conclusions through superimposition is based on applying the max fuzzy disjunction operator to multiple possibilitybdistributions of the output variable 88

17 Fig 3.13 Output voltage with respect to input temperature Fig 3.14 Showing the control of temperature by voltage supply 89

18 Defuzzification: This is the process to convert fuzzy input into a crisp one. We apply the Center of Area (COA) or Centroid method to get crisp output. This method calculates the weighted average of a fuzzy set (John Yen,Reza Langari et al. 2007).The result of applying COA defuzzification to a fuzzy conclusion Y is A can be expressed by the formula (3.24) If y is discrete and by the formula (3.25) If y is continous Fig 3.15 Fig showing the defuzzified value Putting All Four Steps Together: The result of combining all four steps together is shown in figure. 90

19 Fig 3.16 Showing the overall working of fuzzy system (3.26) The defuzzified output for the flow mixing controller can be expressed as a function of the input target temperature using the above formula. 91