Noninteractive Fuzzy-Rule-Based Systems

In this paper we have introduced a non interactive model for fuzzy rule based systems A critical aspects of this non interactive model is the introduction of a new set of rules with fewer parameters and without considering the interaction between the functionality of inputs The new non interactive model of the fuzzy rule based system represents the output as a linear combination of the non linear function of individual inputs Introduction A fuzzy rule based system FRBS normally consists of the following components A fuzzi cation stage this stage outputs a fuzzy value when the input is a crisp value This is performed by passing the crisp input into a membership function MF The fuzzy or linguistic values are de ned in an appropriate universe of discourse An inference stage the major function of this stage is to combine the output of the fuzzi cation stage together There are a number of methods for combining the variables e g multiplication A defuzzi cation stage the output of the inference stage is converted back into crisp value When the FRBS contains multi inputs for each rule the MFs of individual inputs must be combined together This combination is usually performed using the conjunctive operator and In this paper we propose a non interactive model for FRBSs a method whereby the and function usually multiplication can be removed completely The power of our new approach becomes apparent when it is used in a situation where the FRBS with multi inputs is not very interactive or when there are too many rules in the FRBS The new non interactive model of the FRBS represents the output as a linear combination of the non linear function of individual inputs At the same time the number of consequent parameters will decrease The organization of the rest of the paper is as follows fuzzy if then rules and related inference mechanisms will be brie y described in Section This is followed by a non interactive model for FRBS for a simple two input one output system in Section to set the background for the proposed method An arti cial neural network ANN structure of the FRBS and subsequent explanation for di erent layers of the ANN will be introduced in Section The general non interactive model of FRBS is given in Section Some numerical examples are presented in Section and conclusions will be drawn in Section Fuzzy Rule Based Systems In this paper fuzzy if then rules of the following con guration are employed for the modeling of linguistic information R If x is A k and xj is Aj and xp is Ap then y is B i where R is the label of i rule xj j p is the j th input variable y is the output Aj j p and k Kj is a fuzzy set and B i is a real number n and p are the numbers of rules and individual inputs respectively The superscript represents the fuzzy values in contra distinction to crisp values The number of individual MFs for a speci c input value xj A j A j A Kj j is Kj Note that Kj n In this paper the MF for the fuzzy values Aj is de ned by a Gaussian function as follows Aj exp xj k j k j A j p k Kj where k j and k j are unknown constant parameters These parameters can be adjusted on line using a gradient descent algorithm We further assume that the universe of antecedent is limited to a speci c domain interval i e xj U j U j j p The decision y t at the t instant as a function of inputs xj t j p is given in the following equation y t Pn i B w Pn i w i where B is the consequent parameters and w is the rule ring strength given by w p Y j Aj xj t k Kj Non Interactive Model For Fuzzy Rule Based Systems In order to emphasis and to clarify the basic idea of our new concept of non interactive model for FRBSs a simple two input one output FRBS is employed Extension to multi input one output case is straightforward and will be explained in the pertinent sections Suppose we are given a FRBS with four rules n two individual MFs for the rst input x and two individual MFs for the second input x i e K K The rules are of the following form R If x is A and x is A then y is B R If x is A and x is A then y is B R If x is A and x is A then y is B R If x is A and x is A then y is B Using the expression for calculating the output y t we have y t B w B w B w B w w w w w where w A x A x w A x A x w A x A x and w A x A x We substitute the consequent parameters B i with four variables C C C and C de ned as follows C C B C C B C C B C C B It is to be noted that the new set of variables are de ned in correlation with the ring strength w for each rule For instance the rst ring strength is w A x A x and the consequent variable B is replaced with C C The output Equation can be rewritten in the following form y t C C A x A x C C A x A x A x A x A x A x A x A x A x A x C C A x A x C C A x A x A x A x A x A x A x A x A x A x C A x C A x A x A x A x A x A x A x C A x C A x A x A x A x A x A x A x C A x C A x A x A x C A x C A x A x A x It is clear that the rst part of the above statements is only a function of the rst input x and the second part is only function of the second input x Therefore the output of FRBS y t which is a non linear function of x t and x t i e y t f x t x t can be expressed as a linear combination of non linear functions of individual inputs y t y t y t f x t f x t where y C A x C A x A x A x and y C A x C A x A x A x The FRBS with rules can be rewritten in the non interactive model if and only if there is a solution exact or approximate for the set of Equations The output y given in Equation represents a fuzzy rule based system with rules and one input x The rules are as follows R If x is A then y is C R If x is A then y is C Similarly the output y given in Equation represents a fuzzy rule based system with rules and one input x Rules are given below R If x is A then y is C R If x is A then y is C To extend the concept of non interactive model for FRBS introduced in this section for multi input systems a NN structure for FRBS is introduced This NN model simpli es the formulation of FRBS Layer 1 Layer 2 Layrer 3 m-1 g m g 6 g 5 g 4 g 3 g 2 g g1