Fractional-, variable-order PID controller implementation based on two discrete-time fractional order operators

In the paper we discuss two kinds of fractional-, variable-order PID controllers, which are based on two different Grunwald-Letnikov fractional-order backward difference definitions. Additionally, the algorithms of finding K p , K i , K d parameters and order values of controllers are described. The algorithms use Nelder-Mead method to find parameters by minimazing given error criteria. The results of unit step-response of both implementations are presented in a graphical form and evaluated using overshoot levels, rise time and two error criteria which are integral squared time weighted error (SSTE) and integral squared time-squared weighted error (SST2E).