Abstract
When fractional calculus operators and models are implemented rationally, there exist some problems such as low approximation accuracy of rational approximation function, inability to specify arbitrary approximation frequency band, or poor robustness. Based on the error criterion of the least squares method, a quadratic programming method based on the frequency-domain error is proposed. In this method, the frequency-domain error minimization between the fractional operator s ± r and its rational approximation function is transformed into a quadratic programming problem. The results show that the construction method of the optimal rational approximation function of fractional calculus operator is effective, and the optimal rational approximation function constructed can effectively approximate the fractional calculus operator and model for the specified approximation frequency band.
Highlights
Fractional calculus is a mathematical problem to study integral and differential of arbitrary order
E direct approximation method [10] is to use a function of finite order to approximate fractional operators in z-domain, and different generating functions are usually applied to different discrete operators, for example, power series expansion (PSE) of Euler operator and continuous fractional expansion (CFE) of Tustin operator
French scholar Professor Oustaloup and his colleagues proposed filter of fractional order operator [12]. is kind of filter allows users to choose the frequency band and order of interest and use the integer order transfer function model to approximate the fractional order calculus operator. For those irrational systems which cannot be described by the standard form of fractional transfer functions, the fractional calculus operators can be discretized indirectly by frequency response fitting or Charef filter [13]. e fractional order system can be Mathematical Problems in Engineering approximated by the differential evolution method, and many scholars participated in the research
Summary
Fractional calculus is a mathematical problem to study integral and differential of arbitrary order. Erefore, the fractional order system or operator model is usually approximated by the way of direct approximation in z-domain or by transforming the fractional system into rational transfer function. It should be noted that when the CFE method is used to discretize the closed-loop continuous fractional transfer function, the model may be unstable Some operators, such as Al-Alaoui operator [11], are obtained by trapezoidal interpolation and rectangular integration rules. Is kind of filter allows users to choose the frequency band and order of interest and use the integer order transfer function model to approximate the fractional order calculus operator. Compared with the previous methods, the method proposed in this paper has obvious improvement in approximation accuracy, precision, and algorithm encapsulation and can arbitrarily select the approximation frequency band, which makes the algorithm highly flexible and has great advantages
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