Editing Fuzzy logic (section)

=== Takagi–Sugeno–Kang (TSK) ===
The TSK system<ref>{{cite journal |last1=Takagi |first1=Tomohiro |last2=Sugeno |first2=Michio |title=Fuzzy identification of systems and its applications to modeling and control |journal=IEEE Transactions on Systems, Man, and Cybernetics |date=January 1985 |volume=SMC-15 |issue=1 |pages=116–132 |doi=10.1109/TSMC.1985.6313399 |s2cid=3333100 }}</ref> is similar to Mamdani, but the defuzzification process is included in the execution of the fuzzy rules. These are also adapted, so that instead the consequent of the rule is represented through a polynomial function (usually constant or linear). An example of a rule with a constant output would be:<syntaxhighlight lang="text">
IF temperature IS very cold = 2
</syntaxhighlight>In this case, the output will be equal to the constant of the consequent (e.g. 2). In most scenarios we would have an entire rule base, with 2 or more rules. If this is the case, the output of the entire rule base will be the average of the consequent of each rule i (Y<sub>i</sub>), weighted according to the membership value of its antecedent (h<sub>i</sub>):

<math>\frac{\sum_i (h_i \cdot Y_i)}{\sum_i h_i}</math>

An example of a rule with a linear output would be instead:<syntaxhighlight lang="text">
IF temperature IS very cold AND humidity IS high = 2 * temperature +  1 * humidity
</syntaxhighlight>In this case, the output of the rule will be the result of function in the consequent. The variables within the function represent the membership values after fuzzification, '''not''' the crisp values. Same as before, in case we have an entire rule base with 2 or more rules, the total output will be the weighted average between the output of each rule.

The main advantage of using TSK over Mamdani is that it is computationally efficient and works well within other algorithms, such as PID control and with optimization algorithms. It can also guarantee the continuity of the output surface. However, Mamdani is more intuitive and easier to work with by people. Hence, TSK is usually used within other complex methods, such as in [[adaptive neuro fuzzy inference system]]s.