An optimized MATLAB tool for efficient evaluation of fractional-order differential equations of time-varying orders

A usage and benchmarks for a new proposed MATLAB tool for efficient evaluation of fractional-order backward differences, sums, differintegrals, and fractional-order differential equations (FODE) are presented in this paper. The tool also supports solving the generalized differential equations formulated with the time-varying fractional orders (VFODE), commonly employed, e.g., in the algorithm of a variable fractional-order PID controller. Optimized C language implementation overpasses other existing popular solutions written mostly in MATLAB scripting language in terms of the computation time. Comparative performance with the FOTF toolbox has been conducted in this paper analyzing the Grünwald-Letnikov-related numerical methods for three example tasks, essential in the modeling and simulation of fractional-order control systems. These are: obtaining values of binomial coefficients (oblivion function); calculating a fractional-order derivative (FOD); solving a fractional-order differential equation (closed-form). A significant increase in performance starting from 70 % up to 90 % was obtained. Software and binaries described in the paper are available at the referred repository.