Numerical solutions of time-varying TS-fuzzy-model-based time-delay dynamic equations via orthogonal functions

In this paper, the approach of orthogonal functions is presented to solve the time-varying Takagi–Sugeno (TS) fuzzy-model-based time-delay dynamic equations (time-varying TSFMTDE). The new method simplifies the procedure of solving the time-varying TSFMTDE into the successive solution of a system of recursive formulae only involving matrix algebra. Based on the presented recursive formulae, an algorithm only including straightforward algebraic computation is also proposed in this paper. The new proposed approach is non-iterative, non-differential, non-integral, straightforward, and well-adapted to computer implementation. Hence, the computational complexity is considerably reduced. The first illustrated example shows that the proposed method, based on the orthogonal functions, can obtain satisfactory results. The second illustrated example, for the pendulum time-delay system with the vibration in the vertical direction on the pivot point having a fuzzy parallel-distributed-compensation controller, is given to demonstrate the application of the proposed approach.