Efficacy of Continued Fraction Expansion technique in the approximation of fractional order systems

Abstract

At a macroscopic level derivatives and integrals are the usual mathematical tools to model real time processes and to perform the basic control and signal processing actions. However, the analysis, design, synthesis and implementation of fractional order differentiator and integrator is a difficult task because of its irrational behaviour. Therefore, for mathematical evaluation of any fractional order system, conversion to its approximate integer order equivalent is essential. In this paper approximated integer order models of fractional differentiator and integrator are developed using the continued fraction expansion technique. A continued fraction is an expression obtained through an iterative process. For any iteration to terminate, a finite numerical value is assigned, which in this paper is equal to the number of frequency points within the desired frequency band. It includes both the lower and upper limit values. A set of coefficients are obtained by finding the gains of the fractional term at respective frequencies and thereby applying the recursive formula. The coefficients thus obtained are substituted in the expression of continued fraction which results in a polynomial function of finite order. The developed models can be directly applied for analysis and realization of fractional order systems. The models are developed for fractional terms 0.1 to 0.9 in steps of 0.1, and also for 0.25 and 0.75. A detailed discussion on the sensitivity analysis is presented, which includes the influence of variable parameters on the accuracy and length of the order. Simulations have been performed in MATLAB. A comparison with both, the ideal values and also with existing methods is performed and tabulated to validate the correctness of the developed models both in terms of accuracy and integer order of the model. It shows that the Matsuda method yield very good results both in terms of magnitude and phase. And, is most suitable for linear phase circuits. Also, the proposed models can be directly used for the realization of customized fractional order Proportional Integral (PI), Proportional-Derivative (PD) and PID controllers. To establish the correctness of CFE based technique for hardware realization, the integer order approximated model of one-tenth and seven-tenth differentiator is decomposed to obtain the circuit parameters resistor (R) and capacitor (C). Then its implementation in OrCAD Capture CIS is performed. It can be seen that the results of realization closely match the actual response.