Solutions of TS Fuzzy Model Based Time-Delay Dynamic Systems via Chebyshev Series Approach

Abstract

A shifted Chebyshev series approach for solve the Takagi-Sugeno (TS) fuzzy model based time-delay dynamic equations (TSFMTDE) is developed in this article. The new method simplifies the procedure of solving the TSFMTDE into the solution of a system of recursive formulae involving only matrix algebra. An algorithm based on these recursive formulae, and including only straightforward algebraic computation is also proposed in this article. The new approach proposed is non-iterative, non-differential, non-integral, straightforward, and well-adapted to computer implementation. Two numerical examples are provided. The first shows that the proposed method based on the shifted Chebyshev series may yield close-to-exact solutions. The second (a nonlinear mass-spring-damper mechanical time-delay system with a fuzzy parallel-distributed-compensation controller) demonstrates the application of the proposed approach.

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“…A shifted Chebyshev series approach for solving the Takagi-Sugeno (TS) fuzzy model based time-delay dynamic equations (TSFMTDE) was later developed by Hsu and Chou (2007). The new method simplified the procedure of solving the TSFMTDE into the solution of the systems of recursive formulae involving only matrix algebra.…”

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“…Finally, the CTP has been used in tests of misspecification based on the concurrent validity of the CTP (Hong & White, 1995;Bierens, 1982); to solve time-delay dynamic equations (Hsu & Chou, 2007); to solve infinite triangular arrays of points that constitute the roots of the unit (Ivanov, Rivlin, & Saff, 1990); to develop a model of smooth partitioning of blocks from a set (Mansour, 2009); using the barycentric form of the Lagrange interpolation formula and in confluent divided differences (Salzer, 1971); to solve numerically linear and nonlinear differential and integral equations (Sweilam & Khader, 2010); to develop an algorithm to solve the overdetermined system of complex linear equations (Watson, 1988); and to accelerate the convergence of iterative solutions of simultaneous equations originated while solving partial differential equations (Wrigley, 1963).…”