User manual MATLAB FUZZY LOGIC TOOLBOX 2

[. . . ] Fuzzy Logic ToolboxTM 2 User's Guide How to Contact The MathWorks Web Newsgroup www. mathworks. com/contact_TS. html Technical Support www. mathworks. com comp. soft-sys. matlab suggest@mathworks. com bugs@mathworks. com doc@mathworks. com service@mathworks. com info@mathworks. com Product enhancement suggestions Bug reports Documentation error reports Order status, license renewals, passcodes Sales, pricing, and general information 508-647-7000 (Phone) 508-647-7001 (Fax) The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 For contact information about worldwide offices, see the MathWorks Web site. Fuzzy Logic ToolboxTM User's Guide © COPYRIGHT 1995­2010 The MathWorks, Inc. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. [. . . ] Note When entering the name of the . fis file in the blocks, you must enclose it in single quotes. Example: Cart and Pole Simulation The cart and pole simulation is an example of a FIS model auto-generated by the Fuzzy Logic Controller block. 2-96 Working in Simulink® Environment Type slcp at the MATLAB prompt to open the simulation. This model appears. Right-click on the Fuzzy Logic Controller block, and select Look under mask from the right-click menu. The following subsystem opens. 2-97 2 Tutorial Follow the same procedure to look under the mask of the FIS Wizard subsystem to see the implementation of your FIS. This following figure shows part of the implementation (the entire model is too large to show in this document). 2-98 Working in Simulink® Environment As the figure shows, the Fuzzy Logic Controller block uses built-in Simulink blocks to implement your FIS. Although the models can grow complex, this representation is better suited than the S-function sffis for efficient code generation. 2-99 2 Tutorial Sugeno-Type Fuzzy Inference In this section. . . "What is Sugeno-Type Fuzzy Inference?" on page 2-100 "An Example: Two Lines" on page 2-104 "Comparison of Sugeno and Mamdani Methods" on page 2-106 What is Sugeno-Type Fuzzy Inference? The fuzzy inference process discussed so far is Mamdani's fuzzy inference method, the most common methodology. This section discusses the so-called Sugeno, or Takagi-Sugeno-Kang, method of fuzzy inference. Introduced in 1985 [16], it is similar to the Mamdani method in many respects. The first two parts of the fuzzy inference process, fuzzifying the inputs and applying the fuzzy operator, are exactly the same. The main difference between Mamdani and Sugeno is that the Sugeno output membership functions are either linear or constant. A typical rule in a Sugeno fuzzy model has the form If Input 1 = x and Input 2 = y, then Output is z = ax + by + c For a zero-order Sugeno model, the output level z is a constant (a=b =0). The output level zi of each rule is weighted by the firing strength wi of the rule. For example, for an AND rule with Input 1 = x and Input 2 = y, the firing strength is wi = AndMethod( F1 ( x), F2 ( y)) where F1, 2 (. ) are the membership functions for Inputs 1 and 2. 2-100 Sugeno-Type Fuzzy Inference The final output of the system is the weighted average of all rule outputs, computed as wi zi Final Output = i=1 N N wi i=1 where N is the number of rules. 2-101 2 Tutorial A Sugeno rule operates as shown in the following diagram. 1. Apply implication method (prod). 1. If poor rancid z 1 (cheap) z1 service is poor or food is rancid then tip = cheap 2. good If service is good rule 2 has no dependency on input 2 z 2 (average) z2 then tip = average 3. If excellent delicious z 3 (generous) z3 service is excellent or food is delicious then tip = generous service = 3 food = 8 input 1 input 2 output tip = 16. 3% The preceding figure shows the fuzzy tipping model developed in previous sections of this manual adapted for use as a Sugeno system. Fortunately, it is frequently the case that singleton output functions are completely sufficient for the needs of a given problem. As an example, the system tippersg. fis is 2-102 Sugeno-Type Fuzzy Inference the Sugeno-type representation of the now-familiar tipping model. If you load the system and plot its output surface, you will see that it is almost the same as the Mamdani system you have previously seen. a = readfis('tippersg'); gensurf(a) 20 tip 15 10 10 8 6 4 2 food 0 0 2 service 6 4 8 10 The easiest way to visualize first-order Sugeno systems is to think of each rule as defining the location of a moving singleton. That is, the singleton output spikes can move around in a linear fashion in the output space, depending on what the input is. This also tends to make the system notation very compact and efficient. Higher-order Sugeno fuzzy models are possible, but they introduce significant complexity with little obvious merit. Sugeno fuzzy models whose output membership functions are greater than first order are not supported by Fuzzy Logic Toolbox software. Because of the linear dependence of each rule on the input variables, the Sugeno method is ideal for acting as an interpolating supervisor of multiple linear controllers that are to be applied, respectively, to different operating conditions of a dynamic nonlinear system. [. . . ] aggregation The combination of the consequents of each rule in a Mamdani fuzzy inference system in preparation for defuzzification. antecedent The initial (or "if") part of a fuzzy rule. consequent The final (or "then") part of a fuzzy rule. defuzzification The process of transforming a fuzzy output of a fuzzy inference system into a crisp output. [. . . ]