Efficacy of NPC Technique in Approximation of Fractional Order Differentiators and Integrators

Fractional order differentiators and integrators play an important role in control and automation of physical processes. This is because fractional order Proportional-Intergral-Derivative controllers provide two additional parameters for fine tuning which in turn improves the overall system performance. However direct realization of fractional order differentiators and integrators is not easy, so it is first converted into an integer order function and then realized using active/passive elements. NPC is a recent approximation technique proposed in 2018 to derive rational approximation of fractional order differ-integrators. In this paper NPC based integer order models of fractional order differ-integrators are presented for all fractional powers ranging from 0.1 to 0.9 in steps of 0.1 and also for 0.25 and 0.75. The performance of the proposed models is compared with integer models developed using other existing techniques and also with ideal response. This has been done by calculating maximum magnitude and phase errors. It has been found that NPC based rational models are very much close to the ideal frequency response. To establish the correctness of NPC technique these proposed models are applied to develop integer approximation of a fractional plant model of order 0.73 and having 0.5 second delay. The step response results when compared with integer approximation model developed using other existing techniques show that NPC based model is stable and has least steady state error. These proposed models can directly be used for realization of customized fractional order Proportional-Integral, Proportional-Derivative and PID controllers. The simulations have been performed in MATLAB.