Chap 4: Fuzzy Inference Systems 2019/5/14 Fuzzy Set Theory Chap 4: Fuzzy Inference Systems ... In this talk, we are going to apply two neural network controller design techniques to fuzzy controllers, and construct the so-called on-line adaptive neuro-fuzzy controllers for nonlinear control systems. We are going to use MATLAB, SIMULINK and Handle Graphics to demonstrate the concept. So you can also get a preview of some of the features of the Fuzzy Logic Toolbox, or FLT, version 2.

Fuzzy If-Then Rules Mamdani style 2019/5/14 Fuzzy If-Then Rules Mamdani style If pressure is high then volume is small high small Sugeno style If speed is medium then resistance = 5*speed By using fuzzy sets, we can formulate fuzzy if-then rules that are commonly used in our daily expressions. Basically, we have two types of fuzzy rules. For Mamdani style, for instance, if pressure is high then volume is small, where igh?and mall?are described by fuzzy sets. For Sugeno style, if the speed of a moving object is medium then the resistance due to atmosphere is 5 times the speed. The basic difference between these two rules is in their THEN part, where Madman style has a fuzzy but Surgeon style has a linear equation. Madman style fuzzy rules were first proposed in the literature; they are more appealing to human intuition. Surgeon style fuzzy rules are proposed later, but they are more suited for mathematical design and analysis. In this talk, wel concentrate on Surgeon style fuzzy if-then rules. medium resistance = 5*speed

Mamdani Fuzzy System page 74 2019/5/14 Mamdani Fuzzy System page 74 Graphics representation: A1 B1 C1 w1 Z X Y A2 B2 C2 w2 Z X Y T-norm C’ Z x is 4.5 X y is 56.8 Y z is zCOA

Fuzzy Inference System (FIS) 2019/5/14 Fuzzy Inference System (FIS) If speed is low then resistance = 2 If speed is medium then resistance = 4*speed If speed is high then resistance = 8*speed MFs low medium high .8 .3 A single fuzzy rule is not very interesting. But if we have a collection of fuzzy rules, we can use them to describe a system behavior. This leads to a fuzzy inference system. For instance, we can describe the resistance experienced by a moving object by the following three rules: .... Then given a crisp speed value, how do we find the resistance value from these three rules? It quite simple and can be done in three steps. In the first step, we find the membership grades for ow? edium? and igh? For instance, if speed is 2, the membership grades for ow? edium?and igh?are .3, .8, and .1, respectively. These numbers also represent how the given input condition peed = 2?satisfies the IF part of the rules. Sometimes these numbers are called the firing strengths of the rules. In the second step, we find the output of each rule, given speed is 2. In the third step, we apply a weighted average method to find the overall resistance, where the weighting factors are equal to the firing strengths of the rules. The whole process to derive the output from a given input condition is called fuzzy reasoning. For a two-input FIS, the process of fuzzy reasoning is better represented by the following diagram. .1 2 Speed Rule 1: w1 = .3; r1 = 2 Rule 2: w2 = .8; r2 = 4*2 Rule 3: w3 = .1; r3 = 8*2 Resistance = S(wi*ri) / Swi = 7.12

TSK Fuzzy System page 81 Rule base Fuzzy reasoning 2019/5/14 TSK Fuzzy System page 81 Rule base If X is A1 and Y is B1 then Z = p1*x + q1*y + r1 If X is A2 and Y is B2 then Z = p2*x + q2*y + r2 Fuzzy reasoning A1 B1 z1 = p1*x+q1*y+r1 w1 In this talk, we are going to use first-order Sugeno fuzzy inference system exclusively, where the output equation of each rule is a linear equation. For example, if we have two fuzzy rules ... We can express the process of fuzzy reasoning by this diagram. First we find the membership grades of the IF parts of the rules; the heights of the dashed line represent these values. Since the pre-conditions in the IF part are connected by AND, so we use multiplication to find the firing strength of each rule. For instance, firing strength w1 for rule 1 is the product of the heights of these two dashed lines. Similar for w2. Once we have w1 and w2, the overall output is again derived by weighted average. X Y A2 B2 z2 = p2*x+q2*y+r2 w2 X Y x=3 y=2 z = w1+w2 w1*z1+w2*z2 P

Tsukamoto Fuzzy System page 84 2019/5/14 Tsukamoto Fuzzy System page 84 Graphics representation: A1 B1 C1 w1 Z X Y z1 A2 B2 C2 w2 Z X Y z2 T-norm z = w1+w2 w1*z1+w2*z2 x is 4.5 X y is 56.8 Y

Zhang-Kandel Fuzzy System 2019/5/14 Zhang-Kandel Fuzzy System Graphics representation: A1 B1 C1 w1 Z X Y za zb A2 B2 C2 w2 Z X Y zc zd T-norm z = w1+w2 w1*z1+w2*z2 x is 4.5 X y is 56.8 Y Z1={Za, Zb} Z2={Zc, Zd}