Analytic Solution of a System of Linear Distributed Order Differential Equations in the Reimann-Liouville Sense

Abstract

In this paper, the solution to a system of linear distributed order di erential equations in the Riemann-Liouville sense is analytically obtained. The distributed order relaxation equation is a special case of the system investigated in this paper. The solution of the mentioned system is introduced on the basis of a function, which can be considered as the distributed order generalization of the matrix Mittag-Le er functions. It is shown that this generalized function in two special cases of the weight function can be expressed in terms of the multivariate Mittag-Le er functions and the Wright functions.

Tuning the Implementable Structures of Fractional-Order PID Controllers for Control of FOPDT Processes

“…To find the optimal tuning rules for the free parameters of the implementable fractional-order controller in Equation (10), firstly, the normalized form of the controller given in Equation (9) is optimally tuned for the normalized process transfer function in Equation (7)…”

“…This field includes the applications of fractional-order differentiation/integration operators in modeling of real-world processes and proposing effective control laws. Fractional operators, on one hand, provide a framework for more exact modeling of the processes from different areas such as electrical engineering [1,2], mechanical engineering [3][4][5], medicine [6] and relaxation processes [7] with fewer parameters in comparison with integer-order models. On the other hand, considering their unique characteristics are used to design controllers that are more robust to process variations in comparison with traditional integer-order controllers.…”