Takagi Sugeno regulator

The Takagi-Sugeno-controller is on the fuzzy logic beruhender controller . Its behavior is described with rules by fuzzy closing. In the conclusion part of the rules, in contrast to relational fuzzy controllers according to Mamdani, there are no fuzzy sets, but functions of the input variables. The function values ​​are weighted with the degree of compliance with the rules. The result is the precise manipulated variable. No defuzzification is therefore necessary.

Basics

Membership functions

In fuzzy logic, linguistic variables are assigned colloquial terms as a range of values. For example, the linguistic variable control deviation can take on the values negative_small , near_zero or positive_small . The mathematical representation of these terms is the membership function (fuzzy set). They describe the value between 0 and 1 with which a sharp variable belongs to this fuzzy set. Fuzzy logic also provides operators for the logical operations AND and OR.

The behavior of a Takagi Sugeno regulator is determined by rules of form

 ${\displaystyle R_{i}}$ : WENN  ${\displaystyle x_{1}=A_{1l}}$ UND ... UND  ${\displaystyle x_{n}=A_{nk}}$ DANN  ${\displaystyle y_{i}=f(x_{1},...,x_{n})}$

described. Are there

• ${\displaystyle x_{j}}$ sharp input variables.
• ${\displaystyle A_{nk}}$ Value of a linguistic variable.
• ${\displaystyle y_{i}}$Functions of the input variables . ${\displaystyle x_{j}}$

Example: A non-linear system is to be controlled with P controllers. is the control deviation. ${\displaystyle x\,}$

 ${\displaystyle R_{1}}$ : WENN  ${\displaystyle x=NAHENULL}$ DANN  ${\displaystyle y_{1}=0.4\cdot x}$
 ${\displaystyle R_{2}}$ : WENN  ${\displaystyle x=NICHTNULL}$ DANN  ${\displaystyle y_{2}=2.5\cdot x}$

Fuzzification

For each linguistic value, a degree of membership of the sharp input variable is determined via its membership function. The fuzzified input variable is a vector of the degrees of membership. If the membership functions corresponding to the linguistic values and the sharp input variables, then the following applies to the fuzzified input variable : ${\displaystyle \mu _{ij}(x)}$ ${\displaystyle A_{ij}}$ ${\displaystyle x_{i}}$ ${\displaystyle x_{F}\,}$

 ${\displaystyle x_{F}(x_{i})=(\mu _{i1}(x_{i}),...,\mu _{in}(x_{i}))}$

In the example above, we get for for ${\displaystyle x=-0.8}$

 ${\displaystyle R_{1}}$ :  ${\displaystyle x_{F}(x)=0.2}$
 ${\displaystyle R_{2}}$ :  ${\displaystyle x_{F}(x)=0.8}$

Inference

The evaluation of the premises is often done with the minimum operator (see T-Norm ). The degree of fulfillment of the rule is accordingly ${\displaystyle \alpha _{i}}$ ${\displaystyle R_{i}}$

 ${\displaystyle \alpha _{i}=min(\mu _{i1}(x_{i}),...,\mu _{in}(x_{i}))}$.

The following applies to our example

 ${\displaystyle \alpha _{1}=0.2}$
 ${\displaystyle \alpha _{2}=0.8}$

Conclusion

There are no linguistic values ​​in the final part of the rule. Therefore no defuzzification is necessary. The precise output variable results as the mean value of the manipulated variables weighted with the degree of fulfillment of the rules

   ${\displaystyle y={\frac {\sum _{i=1}^{n}y_{i}\cdot \alpha _{i}}{\sum _{i=1}^{n}\alpha _{i}}}}$

For the example we get the sharp manipulated variable

 ${\displaystyle y=-1.664}$

Compared to Mamdani

If you describe a Takagi-Sugeno controller of the 0th order, i.e. one in which all functions are constant functions, this controller is a special case of the Mamdani system, in which all output quantities are singleton variables.

literature

• Holger Lutz, Wolfgang Wendt: Pocket book of control engineering with MATLAB and Simulink , 11th edition, Verlag Europa-Lehrmittel, 2019, ISBN = 978-3-8085-5869-0.
• T. Takagi, M. Sugeno: Fuzzy identification of systems and its applications to modeling and control . IEEE transactions on systems, man, and cybernetics, vol. 15, no. 1, pp. 116-132, 1985.