TakagiSugenoKang

Takagi-Sugeno-Kang (TSK) fuzzy model is a type of fuzzy inference system named after Takagi, Sugeno, and Kang. Developed in the 1980s, it provides a framework for modeling nonlinear systems by combining fuzzy rules with mathematical consequents.

In a TSK model, each rule has a fuzzy antecedent: IF x1 is A1 and x2 is A2 ... THEN y = f(x1, x2, ..., xn). The function f is usually a polynomial, most commonly a linear function, or sometimes a constant. The rules are assembled into a rule base; the input is fuzzified via membership functions.

Inference: For a given input, each rule fires with a certain strength (the product of membership grades). The final output is the weighted average of all consequents: y = sum(wi * fi(x)) / sum(wi). This yields a smooth, differentiable output and a globally linear model in each region.

Training and variants: TS models can be identified from data by estimating the premise parameters (membership functions) and consequent parameters, often via hybrid learning that combines least-squares estimation with gradient descent. ANFIS is a well-known implementation. Kang contributed to the practical formulation and applications in control and system identification.

Applications and significance: TS fuzzy models are widely used in nonlinear system modeling, control, process engineering, forecasting, and robotics due to their interpretability, modular structure, and strong approximation capabilities.