TakagiSugeno

Takagi-Sugeno, often written as Takagi-Sugeno-Kang (TSK) or Takagi-Sugeno, refers to a widely used fuzzy inference system (FIS) and modeling framework in fuzzy logic. It was developed by Takagi and Sugeno in 1985 and later extended by Kang, leading to the common designation Takagi-Sugeno-Kang. The approach is particularly valued for its mathematical tractability and suitability for model-based control and system identification.

In a Takagi-Sugeno system, the rule base consists of if-then rules with fuzzy antecedents and consequents that are linear or constant functions of the input variables. A typical first-order rule has a consequent of the form z = a1 x1 + a2 x2 + ... + an xn + b, where the inputs x1, x2, ... are the linguistic variables corresponding to measured signals. The outputs of all rules are aggregated, usually by a weighted average, to produce a single crisp output after defuzzification. The scheme enables nonlinear, multivariate modeling by combining several local linear models, each active in a region of the input space.

Zero-order and first-order variants are common: zero-order uses a constant output for each rule, while first-order uses linear functions of the inputs. Training involves estimating the parameters of the linear consequents and the membership functions of the antecedents, often through least squares or optimization methods. Because the output is a weighted sum of linear models, TS models are well-suited to optimization, adaptive control, and neural-network-based neuro-fuzzy hybrids.

Applications span process and automated control, robotics, automotive systems, signal processing, and time-series modeling, where nonlinear dynamics can be effectively captured by a network of local linear models.