Soft Computing Techniques Module II PDF

MRT307 Soft Computing Techniques Module – II Fuzzy Inference System Fuzzy inference system is a popular computing framework based on the fuzzy set theory, fuzzy if-then rules and fuzzy reasoning. Applications: Automatic control, decision analysis, Expert System, data classification and rob...

MRT307 Soft Computing Techniques Module – II Fuzzy Inference System Fuzzy inference system is a popular computing framework based on the fuzzy set theory, fuzzy if-then rules and fuzzy reasoning. Applications: Automatic control, decision analysis, Expert System, data classification and robotics etc Basic Components in FIS Three Conceptual Components a) Rule base: Consists of a set of fuzzy rules b) Database or Dictionary: Defines the Membership function. c) Reasoning Mechanism: Performs the inference Procedure. Defuzzification A method used to extract a Crisp value that represents a fuzzy set. Diagrammatic Representation of FIS Mamdani Fuzzy Models The Mamdani fuzzy inference system was proposed as the first attempt to control a steam engine and boiler combination by a set of linguistic control rules obtained from experienced human operators. Figure 1 is an illustration of how a two-rule Mamdani fuzzy inference system derives the overall output z when subjected to two crisp inputs x and y.  Fig : Mamdani Fuzzy inference System In Mamdani's application, two fuzzy inference systems were used as two controllers to generate the heat input to the boiler and throttle opening of the engine cylinder, respectively, to regulate the steam pressure in the boiler and the speed of the engine. Since the plant takes only crisp values as inputs, we have to use a defuzzifier to convert a fuzzy set to a crisp value. Defuzzification Defuzzification refers to the way a crisp value is extracted from a fuzzy set as a representative value. In general, there are five methods for defuzzifying a fuzzy set A of a universe of discourse Z (Here the fuzzy set A is usually represented by an aggregated output. Advantages of the Mamdani Method ▪ It is intuitive. ▪ It has widespread acceptance. ▪ It is well suited to human input. ▪ Centroid of Area (COA) Bisector of Area (BOA) Mean of Maximum (MOM) Smallest of Maximum (SOM) Largest of Maximum (LOM) Diagrammatic Representation: Single input and Single Output Mamdani model: It can be expressed as: Two input and Single Output Mamdani fuzzy model Expressed as: Expressed as: Sugeno Fuzzy model The Sugeno Fuzzy model (also known as the TSK fuzzy model) was proposed by Takagi, Sugeno, and Kang in an effort to develop a systematic approach to generating fuzzy rules from a given input-output dataset. A typical fuzzy rule in a Sugeno fuzzy model has the form: where A and B are fuzzy sets in the antecedent, while z=f(x,y) is a crisp function in the consequent. Usually f(x, y) is a polynomial in the input variables x and y, but it can be any function as long as it can appropriately describe the output of the model within the fuzzy region specified by the antecedent of the rule. When f(x, y) is a first-order polynomial, the resulting fuzzy inference system is called a first-order Sugeno fuzzy model, which was originally proposed. When f is a constant, we then have a zero-order Sugeno fuzzy model, which can be viewed either as a special case of the Mamdani Fuzzy inference system The output of a zero-order Sugeno model is a smooth function of its input variables as long as the neighbouring MFs in the antecedent have enough overlap. In other words, the overlap of MFs in the consequent of a Mamdani model does not have a decisive effect on the smoothness; it is the overlap of the antecedent MFs that determines the smoothness of the resulting input-output behaviour. Advantages of the Sugeno Method It is computationally efficient. It works well with linear techniques (e.g., PID control). It works well with optimization and adaptive techniques. It has guaranteed continuity of the output surface. It is well suited to mathematical analysis. Also have 2 types: Single input and Single output Two input and Single output Tsukamoto Fuzzy Model The consequent of each fuzzy if-then rule is represented by a fuzzy set with monotonical MF. As a result, the inferred output of each rule is defined as a crisp value induced by the rules‟ firing strength. The overall output is taken as the weighted average of each rule‟s output. Tsukamoto fuzzy model is illustrated as: Example: Single-input Tsukamoto fuzzy model A single-input Tsukamoto fuzzy model can be expresses as – If X is small then Y is C1 – If X is medium then Y is C2 – If X is large then Y is C3 Where C1, C2, C3 are monotonical membership function. Derivative based optimization Optimization means minimizing a function without reducing its functionality. In this optimization is done based on the derivative information. Mainly Two techniques are used. They are: a) Descent Method b) Newton Method Descent Method Let E be the real valued objective function. There is a n dimensional input space θ= θ* =[θ1,θ2,…,θn]T. Our main is to find a value of θ such that E(θ1,θ2,…,θn) is minimum on θ = θ*. The search of this minimum is performed through a certain direction d starting from an initial value θ= θ0 (iterative scheme). Θnext is found out by the formula: Where θnow is the Current value and d is the direction vector. It can be also defined as: Here ƞ is called as step size which determines the number of steps needed. It is also called as learning rate. The principal differences between various descent algorithms lie in the first procedure for determining successive directions; Once d is determined, η is computed as (line minimization) Descent methods are mainly divided in to two. They are: a) Gradient Based method b) Method of steepest descent Gradient Based Methods Method is defined as: Graphically it is represented as: Feasible descent directions are determined by deflecting the gradients using the equation Equation 6.9 G is the set of deflected gradients. Stopping Criteria used: Method of Steepest Descent When G = ƞI, then Where ƞ is a positive step size value and I is the identity matrix Classical Newton Method