Rational approximation of fractional order system in delta domain — A unified approach

Abstract

Fractional order operators (differentiators/integrators) or systems (FOS) are mathematically equivalent to infinite dimensional system. For practical uses of these systems an approximate integer order system is required. These call for use of so called approximation methods popularly known as model order reduction technique. From the last three decades FOS are being separately studied for its analog and digital implementation leading to two separate areas in system and control. To alleviate this problem delta operator is used which unify both analog and digital system together at high sampling frequency or lower sampling time limit. Time moment (TM) is a classical technique extensively being used in model order reduction literature for approximation of a higher order model to a reduced order in the complex domain such that the reduced model exhibits the dominant characteristic of the higher order model. Pade approximation is then used in developing reduced order model. A variant of TM, called generalized delta time moment (GDTM) is applied in approximation of a FOS to its reduced order counterpart. The computational algorithm in developing reduced order model using GDTM relies on choice of appropriate real frequency point in the complex delta domain by trial and error. To overcome this problem genetic algorithm (GA) is used to obtain optimal frequency points. At high frequency limit, this reduced model converges to its analog counterpart leading to unification of both analog and digital model reduction theory. Example from the literature is taken to exhibit the efficacy of the proposed methods using MATLAB.