Abstract

Background: Because the fractional-order calculus operator is an irrational function of the complex variable, it is required to deal with rational approximation.

Objective: It is difficult to directly implement the fractional-order calculus operator in the numerical simulation and practical application.

Methods: So the minimum amplitude frequency approximation error and phase frequency approximation error of rational approximation function are comprehensively considered in the same rational function class. And the rational approximation function construction method and combined optimal rational approximation method of fractional-order calculus operator based on the function approximation theory and logarithmic frequency characteristic asymptote idea are described in detail. Then, an approximation method of fractional-order calculus operator based on the combined optimal rational approximation is proposed in this paper. The approximation effectiveness of fractional-order calculus operator is analyzed by using s-0.6 . And the combined optimal rational approximation method is applied to the design and digital realization of fractional-order controller. In order to simplify the controller design, the controller structure of parallel connection mode is used and the simplified rational approximation expression and transform of fractionalorder integral operator in the different order is obtained.

Results: The analyzed results show that the combined optimal rational approximation method of fractional-order calculus operator is consistent with the ideal c in amplitude frequency domain and has smallest error for the phase frequency characteristic. And the applied results show that the proposed combined optimal rational approximation method can conveniently control and simulate the practical computer control system by analyzing the simplified rational approximation expression and z transform of fractional-order integral operator.

Conclusion: So the combined optimal rational approximation method can provide better engineering applications of fractional-order controller in the practical control system.

Keywords: Fractional-order calculus, rational approximation of operator, combined optimal rational approximation, discretization, fractional-order PIα Dβ controller, parallel connection structure.

Graphical Abstract