A digital PID controller based on Grünwald-Letnikov fractional-, variable-order operator

The paper presents an example of implementation of a digital PID controller with Griinwald-Letnikov fractional-, variable-order summations and differences. The method of finding values of orders and parameters K
_p
, K
_i
, K
_d
of such controller for considered plant model is presented. Two summation performance criteria are used to evaluate the results achieved with the usage of fractional-, variable-order PID controller tuned by proposed algorithm. Mentioned criteria are integral squared time weighted error (SSTE) and integral squared time-squared weighted error (SST2E). Additionally the results are compared with the ones obtained with PID controller tuned using Ziegler-Nichols method, PID controller tuned with Matlab pidtune function and PID controller which coefficients were tuned by minimising mentioned SSTE and SST2E errors with Nelder-Mead method (implemented by Matlab fminsearch function).