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Build Mamdani Systems (Code)

Tipping Problem from the Command Line

The tipping problem is one of many Fuzzy Logic Toolbox™ examples of fuzzy inference systems. The FIS is always cast as a MATLAB® structure. To load this system, type:

a = readfis('tipper.fis')

This command returns the following result:

a = 
            name: 'tipper'
            type: 'mamdani'
       andMethod: 'min'
        orMethod: 'max'
    defuzzMethod: 'centroid'
       impMethod: 'min'
       aggMethod: 'max'
           input: [1x2 struct]
          output: [1x1 struct]
            rule: [1x3 struct]

The labels on the left of this listing represent the various components of the MATLAB structure associated with tipper.fis. To access the various components of this structure, type the component name after a at the MATLAB prompt. For example:

a.type

returns the following result:

ans =
mamdani

The function

getfis(a)

returns almost the same structure information that typing a, alone does:

Name      = tipper
      Type      = mamdani
      NumInputs = 2
      InLabels  = 
            service
            food
      NumOutputs = 1
      OutLabels = 
            tip
      NumRules = 3
      AndMethod = min
      OrMethod = max
      ImpMethod = min
      AggMethod = max
      DefuzzMethod = centroid

Some of these fields are not part of the structure a. Thus, you cannot get information by typing a.Inlabels, but you can get it by typing

getfis(a,'Inlabels')

Similarly, you can obtain structure information using getfis in this manner.

getfis(a,'input',1)
getfis(a,'output',1)
getfis(a,'input',1,'mf',1)

The structure.field syntax also generates this information. For more information on the syntax for MATLAB structures and cell arrays, see Create a Structure Array and Create a Cell Array in the MATLAB documentation.

For example, type

a.input

or

a.input(1).mf(1)

The function getfis is loosely modeled on the Handle Graphics® function get. The function setfis acts as the reciprocal to getfis. It allows you to change any property of a FIS. For example, if you wanted to change the name of this system, you could type

a = setfis(a,'name','gratuity');

However, because a is already a MATLAB structure, you can set this information more simply by typing

a.name = 'gratuity';

Now the FIS structure a has been changed to reflect the new name. If you want a little more insight into this FIS structure, try

showfis(a)

This syntax returns a printout listing all the information about a. This function is intended more for debugging than anything else, but it shows all the information recorded in the FIS structure

Because the variable, a, designates the fuzzy tipping system, you can display any of the GUIs for the tipping system directly from the command line. Any of the following functions will display the tipping system with the associated GUI:

  • fuzzyLogicDesigner(a) displays the Fuzzy Logic Designer.

  • mfedit(a) displays the Membership Function Editor.

  • ruleedit(a) displays the Rule Editor.

  • ruleview(a) displays the Rule Viewer.

  • surfview(a) displays the Surface Viewer.

If, in addition, a is a Sugeno-type FIS, then neuroFuzzyDesigner(a) displays the Neuro–Fuzzy Designer.

When you open any of these GUIs, you can access any of the other GUIs using the pull-down menu rather than the command line.

System Display Functions

There are three functions designed to give you a high-level view of your fuzzy inference system from the command line: plotfis, plotmf, and gensurf. The first of these displays the whole system as a block diagram much as it would appear on the Fuzzy Logic Designer.

plotfis(a)

After closing any open MATLAB figures or GUI windows, the function plotmf plots all the membership functions associated with a given variable as follows.

plotmf(a,'input',1)

returns the following plots

plotmf(a,'output',1)

These plots appear in the Membership Function Editor GUI, or in an open MATLAB figure, if plotmf is called while either of these is open.

Finally, the function gensurf plots any one or two inputs versus any one output of a given system. The result is either a two-dimensional curve, or a three-dimensional surface. When there are three or more inputs, gensurf must be generated with all but two inputs fixed, as is described in gensurf.

gensurf(a)

Building a System from Scratch

You can build a fuzzy inference system using Fuzzy Logic Toolbox commands as an alternative to the GUI tools. For example, to build the tipping system entirely from the command line, you use the commands newfis, addvar, addmf, and addrule.

Probably the most difficult part of this process is learning the shorthand that the fuzzy inference systems use for building rules. Use the command line function, addrule to do this.

Each variable, input, or output, has an index number, and each membership function has an index number. The rules are built from statements such as the following:

If input1 is MF1 or input2 is MF3, then output1 is MF2 (weight = 0.5)

This rule is turned into a structure according to the following logic. If there are m inputs to a system and n outputs, then the first m vector entries of the rule structure correspond to inputs 1 through m.

  • The entry in column 1 is the index number for the membership function associated with input 1.

  • The entry in column 2 is the index number for the membership function associated with input 2, and so on.

  • The next n columns work the same way for the outputs.

  • Column m + n + 1 is the weight associated with that rule (typically 1) and column m + n + 2 specifies the connective used (where AND = 1 and OR = 2).

The structure associated with the preceding rule is

1 3 2 0.5 2

This sample code shows one way you can build the entire tipping system from the command line, using the MATLAB structure syntax.

a=newfis('tipper');
a.input(1).name='service';
a.input(1).range=[0 10];
a.input(1).mf(1).name='poor';
a.input(1).mf(1).type='gaussmf';
a.input(1).mf(1).params=[1.5 0];
a.input(1).mf(2).name='good';
a.input(1).mf(2).type='gaussmf';
a.input(1).mf(2).params=[1.5 5];
a.input(1).mf(3).name='excellent';
a.input(1).mf(3).type='gaussmf';
a.input(1).mf(3).params=[1.5 10];
a.input(2).name='food';
a.input(2).range=[0 10];
a.input(2).mf(1).name='rancid';
a.input(2).mf(1).type='trapmf';
a.input(2).mf(1).params=[-2 0 1 3];
a.input(2).mf(2).name='delicious';
a.input(2).mf(2).type='trapmf';
a.input(2).mf(2).params=[7 9 10 12];
a.output(1).name='tip';
a.output(1).range=[0 30];
a.output(1).mf(1).name='cheap'
a.output(1).mf(1).type='trimf';
a.output(1).mf(1).params=[0 5 10];
a.output(1).mf(2).name='average';
a.output(1).mf(2).type='trimf';
a.output(1).mf(2).params=[10 15 20];
a.output(1).mf(3).name='generous';
a.output(1).mf(3).type='trimf';
a.output(1).mf(3).params=[20 25 30];
a.rule(1).antecedent=[1 1];
a.rule(1).consequent=[1];
a.rule(1).weight=1;
a.rule(1).connection=2;
a.rule(2).antecedent=[2 0];
a.rule(2).consequent=[2];
a.rule(2).weight=1;
a.rule(2).connection=1;
a.rule(3).antecedent=[3 2];
a.rule(3).consequent=[3];
a.rule(3).weight=1;
a.rule(3).connection=2

    Tip   You can also build the FIS structure using MATLAB workspace variables. For example, to specify the range of the input, type:

    r1= [0 10]
    a.input(1).range = r1;

Alternatively, you can build the entire tipping system from the command line using Fuzzy Logic Toolbox commands. These commands are in the mktipper.m file.

a=newfis('tipper');
a=addvar(a,'input','service',[0 10]); 
a=addmf(a,'input',1,'poor','gaussmf',[1.5 0]);
a=addmf(a,'input',1,'good','gaussmf',[1.5 5]);
a=addmf(a,'input',1,'excellent','gaussmf',[1.5 10]);
a=addvar(a,'input','food',[0 10]);
a=addmf(a,'input',2,'rancid','trapmf',[-2 0 1 3]);
a=addmf(a,'input',2,'delicious','trapmf',[7 9 10 12]);
a=addvar(a,'output','tip',[0 30]);
a=addmf(a,'output',1,'cheap','trimf',[0 5 10]);
a=addmf(a,'output',1,'average','trimf',[10 15 20]);
a=addmf(a,'output',1,'generous','trimf',[20 25 30]);
ruleList=[ ...
1 1 1 1 2
2 0 2 1 1
3 2 3 1 2 ];
a=addrule(a,ruleList);

Specifying Custom Membership and Inference Functions

You can create custom membership and inference functions as described in Specifying Custom Membership Functions, and Specifying Custom Inference Functions, and specify them for building fuzzy inference systems at the command line.

To include a custom membership function, specify the name of the custom membership function, as shown in the following example:

a=addmf(a,'input',1,'customMF1','custmf1',[0 1 2 4 6 8 9 10]);

To include a custom inference function, specify the name of the custom inference function, as shown in the following example:

a.defuzzMethod='customdefuzz';

FIS Evaluation

To evaluate the output of a fuzzy system for a given input, use the function evalfis. For example, the following script evaluates tipper at the input, [1 2].

a = readfis('tipper');
evalfis([1 2], a)
ans = 
    5.5586

This function can also be used for multiple collections of inputs, because different input vectors are represented in different parts of the input structure.

evalfis([3 5; 2 7], a)
ans =
    12.2184
     7.7885

The FIS Structure

The FIS structure is the MATLAB object that contains all the fuzzy inference system information. This structure is stored inside each GUI tool. Access functions such as getfis and setfis make it easy to examine this structure.

All the information for a given fuzzy inference system is contained in the FIS structure, including variable names, membership function definitions, and so on. This structure can itself be thought of as a hierarchy of structures, as shown in the following diagram.

You can generate a listing of information on the FIS using the showfis command, as shown in the following code sample:

showfis(a)
1.  Name             tipper
2.  Type             mamdani
3.  Inputs/Outputs   [ 2 1 ]
4.  NumInputMFs      [ 3 2 ]
5.  NumOutputMFs     3
6.  NumRules         3
7.  AndMethod        min
8.  OrMethod         max
9.  ImpMethod        min
10. AggMethod        max
11. DefuzzMethod     centroid
12. InLabels         service
13.                  food   
14. OutLabels        tip
15. InRange          [ 0 10 ]
16.                  [ 0 10 ]
17. OutRange         [ 0 30 ]
18. InMFLabels       poor     
19.                  good     
20.                  excellent
21.                  rancid   
22.                  delicious
23. OutMFLabels      cheap   
24.                  average 
25.                  generous
26. InMFTypes        gaussmf
27.                  gaussmf
28.                  gaussmf
29.                  trapmf 
30.                  trapmf 
31. OutMFTypes       trimf
32.                  trimf
33.                  trimf
34. InMFParams       [ 1.5 0 0 0 ] 
35.                  [ 1.5 5 0 0 ] 
36.                  [ 1.5 10 0 0 ]
37.                  [ 0 0 1 3 ]   
38.                  [ 7 9 10 10 ] 
39. OutMFParams      [ 0 5 10 0 ]  
40.                  [ 10 15 20 0 ]
41.                  [ 20 25 30 0 ]
42. Rule Antecedent  [ 1 1 ]
43.                  [ 2 0 ]
44.                  [ 3 2 ]
42. Rule Consequent  1
43.                  2
44.                  3
42. Rule Weight      1
43.                  1
44.                  1
42. Rule Connection  2
43.                  1
44.                  2

The list of command-line functions associated with FIS construction includes getfis, setfis, showfis, addvar, addmf, addrule, rmvar, and rmmf.

Saving FIS Files

A specialized text file format is used for saving fuzzy inference systems. The functions readfis and writefis are used for reading and writing these files.

If you prefer, you can modify the FIS by editing its .fis text file rather than using any of the GUIs. You should be aware, however, that changing one entry may oblige you to change another. For example, if you delete a membership function using this method, you also need to make certain that any rules requiring this membership function are also deleted.

The rules appear in indexed format in a .fis text file. The following sample shows the file tipper.fis.

[System]
Name='tipper'
Type='mamdani'
NumInputs=2
NumOutputs=1
NumRules=3
AndMethod='min'
OrMethod='max'
ImpMethod='min'
AggMethod='max'
DefuzzMethod='centroid'

[Input1]
Name='service'
Range=[0 10]
NumMFs=3
MF1='poor':'gaussmf',[1.5 0]
MF2='good':'gaussmf',[1.5 5]
MF3='excellent':'gaussmf',[1.5 10]

[Input2]
Name='food'
Range=[0 10]
NumMFs=2
MF1='rancid':'trapmf',[0 0 1 3]
MF2='delicious':'trapmf',[7 9 10 10]

[Output1]
Name='tip'
Range=[0 30]
NumMFs=3
MF1='cheap':'trimf',[0 5 10]
MF2='average':'trimf',[10 15 20]
MF3='generous':'trimf',[20 25 30]

[Rules]
1 1, 1 (1) : 2
2 0, 2 (1) : 1
3 2, 3 (1) : 2
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