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. [. . . ] 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. For example, the performance 2-103 2 Tutorial of an aircraft may change dramatically with altitude and Mach number. Linear controllers, though easy to compute and well suited to any given flight condition, must be updated regularly and smoothly to keep up with the changing state of the flight vehicle. A Sugeno fuzzy inference system is extremely well suited to the task of smoothly interpolating the linear gains that would be applied across the input space; it is a natural and efficient gain scheduler. Similarly, a Sugeno system is suited for modeling nonlinear systems by interpolating between multiple linear models. An Example: Two Lines To see a specific example of a system with linear output membership functions, consider the one input one output system stored in sugeno1. fis. fismat = readfis('sugeno1'); getfis(fismat, 'output', 1) This syntax returns: Name = output NumMFs = 2 MFLabels = line1 line2 Range = [0 1] The output variable has two membership functions. getfis(fismat, 'output', 1, 'mf', 1) This syntax returns: Name = line1 Type = linear Params = -1 -1 getfis(fismat, 'output', 1, 'mf', 2) 2-104 Sugeno-Type Fuzzy Inference This syntax returns: Name = line2 Type = linear Params = 1 -1 Further, these membership functions are linear functions of the input variable. The membership function line1 is defined by the equation output = (-1) × input + (-1) and the membership function line2 is defined by the equation output = (1) × input + (-1) The input membership functions and rules define which of these output functions are expressed and when: showrule(fismat) ans = 1. If (input is high) then (output is line2) (1) The function plotmf shows us that the membership function low generally refers to input values less than zero, while high refers to values greater than zero. [. . . ] Filev, "Generation of Fuzzy Rules by Mountain Clustering, " Journal of Intelligent & Fuzzy Systems, Vol. [21] Zadeh, L. A. , "Fuzzy sets, " Information and Control, Vol. 338-353, 1965. B-3 B Bibliography [22] Zadeh, L. A. , "Outline of a new approach to the analysis of complex systems and decision processes, " IEEE Transactions on Systems, Man, and Cybernetics, Vol. [23] Zadeh, L. A. , "The concept of a linguistic variable and its application to approximate reasoning, Parts 1, 2, and 3, " Information Sciences, 1975, 8:199-249, 8:301-357, 9:43-80. [. . . ]