An integer order approximation method based on stability boundary locus for fractional order derivative/integrator operators.

This paper introduces an integer order approximation method for numerical implementation of fractional order derivative/integrator operators in control systems. The proposed method is based on fitting the stability boundary locus (SBL) of fractional order derivative/integrator operators and SBL of integer order transfer functions. SBL defines a boundary in the parametric design plane of controller, which separates stable and unstable regions of a feedback control system and SBL analysis is mainly employed to graphically indicate the choice of controller parameters which result in stable operation of the feedback systems. This study reveals that the SBL curves of fractional order operators can be matched with integer order models in a limited frequency range. SBL fitting method provides straightforward solutions to obtain an integer order model approximation of fractional order operators and systems according to matching points from SBL of fractional order systems in desired frequency ranges. Thus, the proposed method can effectively deal with stability preservation problems of approximate models. Illustrative examples are given to show performance of the proposed method and results are compared with the well-known approximation methods developed for fractional order systems. The integer-order approximate modeling of fractional order PID controllers is also illustrated for control applications.

Disturbance rejection performance analyses of closed loop control systems by reference to disturbance ratio.

This study investigates disturbance rejection capacity of closed loop control systems by means of reference to disturbance ratio (RDR). The RDR analysis calculates the ratio of reference signal energy to disturbance signal energy at the system output and provides a quantitative evaluation of disturbance rejection performance of control systems on the bases of communication channel limitations. Essentially, RDR provides a straightforward analytical method for the comparison and improvement of implicit disturbance rejection capacity of closed loop control systems. Theoretical analyses demonstrate us that RDR of the negative feedback closed loop control systems are determined by energy spectral density of controller transfer function. In this manner, authors derived design criteria for specifications of disturbance rejection performances of PID and fractional order PID (FOPID) controller structures. RDR spectra are calculated for investigation of frequency dependence of disturbance rejection capacity and spectral RDR analyses are carried out for PID and FOPID controllers. For the validation of theoretical results, simulation examples are presented.