Using the Fractional Differential Equation for the Control of Objects with Delay

Abstract

This paper proposes a new structure and substantiates the choice of a technique for controller design using fractional integration and fractional differentiation. The method of fractional integration is based on extrapolation using a series of integrating and differentiating links, the time constant of which changes symmetrically from one step to another. The new controller structure does not contain the three traditional links (proportional, integrating, and differentiating) but contains six links, three of which are incomplete integration, incomplete differentiation, and redundant (for example, one and a half) integration. The proposed technique consists of modeling a controller with the indicated six paths and using the numerical optimization method to calculate the coefficients of each such path. Alternative controllers are known PID controllers and PIλDµ controllers. Both of these species contain only three tracts. The proposed structure has never been used before, and publications with examples of calculation of such structures were not found in the literature. The effectiveness of the proposed method is confirmed by modeling and examples.