Application of the Approximate Fuzzy Reasoning Based on Interpolation in the Vague Environment of the Fuzzy Rulebase in the Fuzzy Logic Controlled Path Tracking Strategy of Differential Steered AGVs

In most of the practical applications the concept of vague environment [1] gives a simple way for fuzzy approximate reasoning. If the fuzzy partitions (used as primary sets of the fuzzy rulebase) can be described by vague environments [1], the primary fuzzy sets of the antecedent and the consequent parts of the fuzzy rules can be characterised by points in their vague environments. So the fuzzy rules themselves can be characterised by points in their vague environment too. It means, that the question of approximate fuzzy reasoning can be reduced to the problem of interpolation of the rule points in the vague environment of the fuzzy rulebase relation [2,3]. In this paper an approximate fuzzy reasoning method based on rational interpolation in the vague environment of the fuzzy rulebase will be introduced, and as an example of a practical application of the method, a path tracking control strategy for differential steered AGVs (Automated Guided Vehicle) [4] implemented on such a fuzzy logic controller will be introduced. 1 The Vague Environment and the Fuzzy Partition The concept of vague environment is based on the similarity or indistinguishability of the elements. Two values in the vague environment are ε-distinguishable if their distance is grater then ε. The distances in vague environment are weighted distances. The weighting factor or function is called scaling function (or factor) [1]. For finding connections between fuzzy sets and a vague environment we can introduce the membership function μ A x ( ) as a level of similarity a to x, as the degree to which x is indistinguishable to a [1]. So the α-cuts of the fuzzy set μ A x ( ) is the set which contains the elements that are (1−α)-indistinguishable from a (see fig.1.): δ α s ( , ) a b ≤ − 1 , { } ( ) μ δ A s x s x dx ( ) min ( , ), min , = − = −         ∫ 1 1 1 1 a b a b where δ s ( , ) a b is the vague distance of the values a, b, and s(x) are the scaling function of the universe X. Fig. 1. It is very easy to realize (see fig.1.), that this case the vague distance of points a and b (δ s ( , ) a b ) is basically the Disconsistency Measure (SD) of the fuzzy sets A and B (where B is a singleton): ( ) S x D x X A B s = − = ∈ ∩ 1 sup ( , ) μ δ a b if [ ] δ s ( , ) , a b ∈ 0 1 where A B ∩ is the min t-norm, ( ) ( ) ( ) [ ] μ μ μ A B A B x x x ∩ = min , ∀ x ∈ X. The main difference between the disconsistency measure and the vague distance is, that the vague distance is a crisp value in range of [0,∞], while the disconsistency measure is limited to [0,1]. That is why it is useful in interpolate reasoning with insufficient evidence too. So if it is possible to describe all the fuzzy partitions of the primary fuzzy sets (the antecedent and consequent universes) of our fuzzy rulebase, and the observation is a singleton, we can calculate the “extended” disconsistency measures of the antecedent primary fuzzy sets of the rulebase and the observation, and the “extended” disconsistency measures of the consequent primary fuzzy sets and the consequence (we are looking for) as vague distances of points in the antecedent and consequent vague universes. For generating a vague environment of a fuzzy partition we have to find an appropriate scaling function, which describes the shapes of all the terms in the fuzzy partition. A fuzzy partition can be characterised by a vague environment if and only if the membership functions of the terms fulfills the following requirement [1]: s x x d dx ( ) '( ) = = μ μ exists iff { } min ( ), ( ) ' ( ) ' ( ) μ μ μ μ i j i j x x x x > 0 ⇒ = ∀ ∈ i j I , , where s(x) is the vague environment we are looking for. Generally the above condition is not fulfilling, so the question is how to describe all fuzzy sets of the fuzzy partition with one “universal” scaling function. For this reason we propose to use the approximate scaling function [2,3]. The approximate scaling function is an approximation of the scaling functions describes the terms of the fuzzy partition separately. Supposing that the fuzzy terms are triangles, each fuzzy term can be characterised by two constant scaling functions, the scaling factor of the left and the right slope of the triangle. So a triangle shaped fuzzy term can be characterised by three values (by a triple), by the values of the left and the right scaling factors and the value of its core point (e.g., fig.2.). For generating the approximate scaling function we suggest to interpolate the neighbouring scaling factors (e.g., fig.2.). The original fuzzy partition The approximate scaling function The partition is characterised by two triple The approximate fuzzy partition Fig. 2. Approximate scaling function generated by non-linear interpolation, and the original fuzzy partition (A,B) as the approximate scaling function describes it (A’,B’) 2 Approximate Reasoning Based on Vague Environment If the vague environment of a fuzzy partition (the scaling function or the approximate scaling function) exists, the member sets of the fuzzy partition can be characterised by points in the vague environment. (In our case the points are characterising the cores of the terms, while the shapes of the membership functions are described by the scaling function.) If all the vague environments of the antecedent and consequent universes of the fuzzy rulebase are exist, all the primary fuzzy sets (linguistic terms) used in the fuzzy rulebase can be characterised by points in their vague environment. So the fuzzy rules (build on the primary fuzzy sets) can be characterised by points in the vague environment of the fuzzy rulebase too. This case the approximate fuzzy reasoning can be handled as a classical interpolation task. Applying the concept of vague environment (the distances of points are weighted distances), any interpolation, extrapolation or regression methods can be adapted very simply for approximate fuzzy reasoning [2,3]. For example we can adapt the rational interpolation [2,3]. This method generates the conclusion as a weighted sum of the vague consequent values, where the weighting factors are inversely proportional to the vague distances of the observation and the corresponding rule antecedents: ( ) ( ) dist y , y dist y , b