This paper suggests a synergy of fuzzy logic and nature-inspired optimization in terms of the nature-inspired optimal tuning of the input membership functions of a class of Takagi-Sugeno-Kang (TSK) fuzzy models dedicated to Anti-lock Braking Systems (ABSs). A set of TSK fuzzy models is proposed by a novel fuzzy modeling approach for ABSs. The fuzzy modeling approach starts with the derivation of a set of local state-space models of the nonlinear ABS process by the linearization of the first-principle process model at ten operating points. The TSK fuzzy model structure and the initial TSK fuzzy models are obtained by the modal equivalence principle in terms of placing the local state-space models in the rule consequents of the TSK fuzzy models. An operating point selection algorithm to guide modeling is proposed, formulated on the basis of ranking the operating points according to their importance factors, and inserted in the third step of the fuzzy modeling approach. The optimization problems are defined such that to minimize the objective functions expressed as the average of squared modeling errors over the time horizon, and the variables of these functions are a part of the parameters of the input membership functions. Two representative nature-inspired algorithms, namely a Simulated Annealing (SA) algorithm and a Particle Swarm Optimization (PSO) algorithm, are implemented to solve the optimization problems and to obtain optimal TSK fuzzy models. The validation and the comparison of SA and PSO and of the new TSK fuzzy models are carried out for an ABS laboratory equipment. The real-time experimental results highlight that the optimized TSK fuzzy models are simple and consistent with both training data and validation data and that these models outperform the initial TSK fuzzy models. 
Nature-inspired optimal tuning of input membership functions of Takagi-Sugeno-Kang fuzzy models for Anti-lock Braking Systems