ࡱ> \^[ac5@  ubjbj22 vXX@4h d~ D"(: ( !!!e'g'g'g'g'g'g'$\)R+'Q#!!##' '%%%#N  e'%#e'% %&& oܕ!$:&1'4'0"(&V,1%V,&V,&!L"6%S","*!!!'' %system modeling using a mamdani rule base Bryan Davis University of Florida 1. Introduction The goal of this project is to model a nonlinear system using a Mamdani rule base. The system we are modeling should be familiar to most people in the US: the automobile. Specifically we are modeling the relationship between the gas pedal displacement, the speed of the vehicle, and the resulting acceleration or deceleration of the vehicle; the same as in [1]. In [1], we evaluated all possible rules based on the correctness of a rule. Correctness was a measure of how well the rule described the actual system behavior over the domain where its antecedent was true. In this project we look how well a Mamdani rule base can model the system, using rules that have a high correctness. The Mamdani rule base is a crisp model of a system, i.e. it takes crisp inputs and produces crisp outputs. It does this with the use of user-defined fuzzy rules on user-defined fuzzy variables. The idea behind using a Mamdani rule base to model crisp system behavior is that the rules for many systems can be easily described by humans in terms of fuzzy variables. Thus we can effectively model a complex non-linear system, with common-sense rules on fuzzy variables. Designing a Mamdani rule base requires three steps: 1) determine appropriate fuzzy sets over the input domain and output range; 2) determine a set of rules between the fuzzy inputs and the fuzzy outputs that model system behavior; 3) create a framework that maps crisp inputs to crisp outputs, given (1) and (2). Steps (1) and (2) are generally dependent on the given application. Step (3) is application independent (for the most part) and is described in section 2, and steps (1) and (2) are application dependent and are discussed in section 3. 2. mamdani rule base The operation of the Mamdani rule base can be broken down into four parts: 1) mapping each of the crisp inputs into a fuzzy variable (fuzzification); 2) determining the output of each rule given its fuzzy antecedents; 3) determining the aggregate output(s) of all of the fuzzy rules; 4) mapping the fuzzy output(s) to crisp output(s) (defuzzification). 2.1. Fuzzification Since the Mamdani rule base models a crisp system, it has crisp inputs and outputs. The rules, however, are given in terms of fuzzy variables. The membership of each fuzzy input variable is evaluated for the given crisp input, and the resulting value is used in evaluating the rules. 2.2. Evaluating the Rules Using the membership values of determined during fuzzification, the rules are evaluated according to the compositional rule of inference. The result is an output fuzzy set that is some clipped version on the user-specified output fuzzy set. The height of this clipped set depends on the minimum height of the antecedents. An alternative to using the minimum height of the antecedents is to use the product of the minimum heights. Thus if two of the inputs are half true and all rest are completely true, then the output is only quarter true, instead of half true. Other possibilities are any of the intersection operators on fuzzy sets. The rule base we implemented lets the user determine which type to use: the minimum, product, bounded difference, drastic intersection, or the Yager intersection. 2.3. Aggregating the Rules After the pervious step, we have a fuzzy output defined for each of the rules in the rule base. We then need to combine these fuzzy outputs into a single fuzzy output. Mamdani defines that the output of the rule base should be the maximum of the outputs of each rule. An alternative is to use any of the union operators defined on fuzzy sets. Our rule base lets the user determine which type to use: the maximum, algebraic sum, bounded sum, drastic union, or the Yager union. Another thing to consider is that some rules might be more important than other rules in determining the system behavior. To account for this, our rule base allows the user to define a weight to each of the rules. The maximum weight is one, and the minimum weight is zero. The fuzzy output of each rule is then multiplied by its weight. 2.4. Defuzzification After the pervious step, we have a fuzzy output defined for the rule base. We need to convert this output into a crisp output. To do this, the centroid (first moment) of the fuzzy output is used. 3. FUZZY SETS and RULEs 3.1. Definition of the Fuzzy Sets Used Figures 2, 3 & 4 show the fuzzy sets defined on the measurable values gas, speed, and acceleration, respectively, for three different fuzzy sets. The three different fuzzy sets have similar mean values, but the shapes triangular, trapezoidal, and crisp, respectively.  Figure 2  Figure 3  Figure 4 3.2. Determining the Rules Given that there are four fuzzy variables for each input, we can have a total of 16 different fuzzy rules that do not conflict. If humans were to determine the rules for this system they would likely pick an output for each possible input combination, giving a total of 16 outputs. We could have more than 16 possible rules, however, with some of the input combinations being used more than once. For example, we might have the two rules: gas is LIGHT and speed is MOD ! acceleration is SLOW and gas is LIGHT and speed is MOD ! acceleration is NONE to reflect the fact that for most cases, both answers are somewhat true. One thing to avoid, however, is not having a rule that covers a particular input combination. If none of the rules in the rule base fire (produce a non-zero output) at all, then the output fuzzy set will have a membership of zero for all values. The centroid for this set is undefined. So, in summary, we need at least one rule for each input combination, and we can have more than one. The next problem is deciding which rules to pick. My previous paper describes a method for doing this based on the correctness of a given rule. This correctness value is based on defining fuzzy implication (if p then q) in terms of the complement and union (not p or q). Although the Mamdani rule base does not use this definition, it can still be useful in determining how correct a given rule is in describing a system. Since each rule has a correctness, we can use its value to weight the rules. A transformation must be done on the correctness term, since it ranges from -" to ", and the weight must range from 0 to 1. First, we only consider rules with positive correctness, because rules with negative correctness will, give false information about the output (their opposite is more correct). Thus we need a mapping (0, ") to (0, 1). A scaled version of the hyperbolic tangent accomplishes this, as in EMBED Equation.3 , where w is the weight, k is the tanh scale factor, and c is the correctness. Adjusting this scale factor k determines how much the correctness affects the weight. If k is very large, then almost all of the weights will be in the flat section of the tanh curve and all of the weights will get mapped to 1 for rules with positive correctness (weighting has no affect). If the scale factor is very small, then the weights will be in the linear part of the tanh curve, and they will be scaled linearly with the correctness. 4. ANALYSIS and results For comparison, figure 4 shows a contour plot of the actual system. The system inputs fraction of gas, and speed are on the horizontal and vertical axes, respectively. The system output acceleration, is represented by color, with the color mapping displayed on the right of the graph. The units for acceleration (not displayed on the graph) are in g-forces. The other figures are presented in the same format to make them easily comparable.  Figure 4: Plot of Actual System Output As mentioned in section 3, we have many different options to choose in using our rule base to model the given system. Our goal in this section is not just to find the optimal parameters for the Mamdani rule base to model this system, but to explore the effect the parameters have on the model behavior. Since there are far too many possible combinations to illustrate them all, we will look at all possibilities from each, class and choose the best from that class to analyze the other variables. 4.1. Effect of Different Fuzzy Sets In figures 5, 6 & 7, we show the model output with triangular, trapezoidal and crisp fuzzy sets.  Figure 5: Crisp Fuzzy Sets RMS error = 0.137  Figure 6: Trapezoidal Fuzzy Sets RMS error = 0.130  Figure 7: Triangular Fuzzy Sets RMS error = 0.126 From these three figures, the triangular fuzzy sets look to be the best suited to this problem. The crisp fuzzy sets have sharp edges (as expected), and do not model the gradual changes very well. The trapezoidal fuzzy sets are somewhat better, but the triangular fuzzy sets are the smoothest, and we would expect them to model the real system best. For these graphs, all rules with a positive correctness were used, with the weight factor of one we fixed the weight factor to 1, and used the standard intersection and union to evaluate and aggregate our rule outputs in the Mamdani rule base. The root-mean-squared error for the three models is 0.137, 0.130 and 0.126, respectively. 4.2. Effect of Different Rule Base Operations In figures 8, 9 & 10, we show the model output with different rule base operations. There are many possibilities for the rule base operations, so we just chose three. Triangular fuzzy sets are used, and the other variables are fixed to same values as above. Figure 7 (already shown) uses the standard intersection and union for the rule base evaluation and aggregation. Figures 8, 9 & 10, use algebraic product and sum, bounded difference and sum, and drastic intersection and union, respectively. On figures 9 and 10, note that there are white sections on the contour plot. These sections correspond to inputs were the output of the fuzzy set is undefined. This happens when none of the rules in the fuzzy set produce a response. Even though this rule set covers all possibilities of the input space, the output of the intersection operator in the Mamdani rule base can still be zero in places where the input fuzzy set membership is not unity. For the triangular fuzzy sets, this is the case over the entire range of the input variable, except at point singularities. Since the formula for the bounded difference is 0 unless the sum of the inputs exceeds 1, this can easily happen in regions over the triangular space (see figure 9). Figure 10, using the drastic intersection produces no output for any of the rules using the triangular space, so the system output is undefined over the entire space (except a the singularities where the input membership is one). Instead, a picture of drastic rules is shown using the trapezoidal fuzzy sets. Notice that the output is undefined wherever the inputs have non-unity membership for all of the fuzzy sets. Thus it is generally a bad idea to use a fuzzy intersection that produces a zero output for two non-zero inputs (bounded difference and drastic intersection are two that do, and the minimum and product are two that do not). Since bounded difference and the drastic intersection produced undefined results for some inputs, no error is given for them. The algebraic operators result in a root-mean-squared error of 0.120, 0.006 less than the minimum and maximum operators.  Figure 8: Algebraic Rule Base RMS error = 0.120  Figure 9: Bounded Rule Base  Figure 10: Drastic Rule Base 4.3. Effect of the Hyperbolic Tangent Scale Factor In figures 11, 12 & 13, we show the model output with different values of the tanh scale factor. Since the best of these uses the triangular fuzzy sets and algebraic intersection and union, those parameters are also used for the following plots. We generate plots values of 10 and 0.1, to show a couple of values for the scaling factor. Remember that the scaling factor is used to modify the relative weights of the rules in terms of their correctness. A high scaling factor tends to give all rules the same weight, while a low scaling factor gives rules a weight proportional to their correctness. During testing, it turned out that the RMS error decreased as the scaling factor went down, indicating that the weighting formula is not very optimum. But we could not find any formula for the weights that worked better than the tanh. 4.4. Changing the Rule Set All previous plots were generated using the rule generation technique described in [1]. The typical method for choosing the rules, however, is manually by trial-and-error. Of course, this method is not well-defined and requires knowledge of the problem domain, and lots of experimentation to get the correct rule set. If rule weighting is taken into account, the problem of choosing the rules becomes even more complicated. To make a rough comparison between the two methods, I used a heuristic that is rather intuitive for the choice of sets: choose one rule for each fuzzy input combination that gives an output that most closely matches the output observed by the real system. Using rules generated by this method, and using no weighting, I get the results shown in figure 13. The root-mean-squared error for this system is 0.144.  Figure 11: Tanh Scaling Factor of 100. RMS error = 0.150  Figure 12: Tanh Scaling Factor of 0.1 RMS error = 0.117  Figure 13: Using One Rule per Input Combination RMS error = 0.144 5. CONCLUSION A Mamdani rule base can be chosen to accurately model a real system where the relation between the inputs and outputs are known. For any given input combination, at least one rule should give a non-zero output. If the set of rules spans all input combinations and the choice for the intersection operator used is one that gives a non-zero output when all of the inputs are non-zero, guarantees this condition is met. For this model, using the product to compute the fuzzy intersection of sets, and using the algebraic sum to compute the fuzzy union of sets resulted in the most accurate best model (in the mean-squared error sense). Also, using triangular fuzzy sets resulted in a smoother output (no flat spots), and better results than the trapezoidal fuzzy sets. The method choosing a rule base based on the results in [1] generate fairly accurate results for our Mamdani model, even though they use different methods of computing implication (Mamdani uses p and q, and [1] uses not p or q). 6. References [1] B. Davis, Evaluation of Implication on Fuzzy Sets in Describing a Given System, University of Florida  HYPERLINK "http://plaza.ufl.edu/badavis/CIS6930_Project1.doc" http://plaza.ufl.edu/badavis/CIS6930_Project1.doc Appendix: description of implemation The code to implement this project was written in MATLAB v5.3. The project was broken down into a few different modules. The module fuzzy_trap uses trapezoid to define a trapezoid shaped fuzzy set on the given variable. The modules fuzzy_comp, fuzzy_union, fuzzy_int, fuzzy_imp perform complement, union, intersection, and implication operations on fuzzy sets defined on one variable. Thus the resulting fuzzy set is also defined on that variable. Rule_graph graphs a set of fuzzy variables defined on the same one-dimensional variable. The modules fuzzy_union2, fuzzy_int2, fuzzy_imp2, perform union, intersection, and implication on fuzzy sets defined on different variables. The resulting fuzzy set is then defined on the Cartesian product of the two variables. Car_accel calculates the acceleration of the car given the gas and speed. Car_vars_tri, cars_vars_trap, and car_vars_crisp defines all of the trivially created variables that are used over again by all of the following modules, for triangular, trapezoidal, and crisp fuzzy sets on the input. Car_implication graphs the contour plots of the membership values of a given compound implication over the car system input space. Car_rules2, evaluates all possible implications over the car input space, and creates a set of rules with positive correctness for the Mamdani rule base, with each rule given a weight calculated from the rules correctness, and the given scaling factor. Mamdani_init initializes a Mamdani rule base given sets of fuzzy sets defined on input and output variables, and a set of rules. 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