Deterministic and stochastic analysis of distributed order systems using operational matrix

Abstract: The fractional order system, which is described by the fractional order derivative and integral, has been studied in many engineering areas. Recently, the concept of fractional order has been generalized to the distributed order concept, which is a parallel connection of fractional order integrals and derivatives taken to the infinitesimal limit in delta order. On the other hand, there are very few numerical methods available for the analysis of distributed order systems, particularly under stochastic forcing. This paper first proposes a numerical scheme for analyzing the behavior of a SISO linear system with a single term distributed order differentiator/integrator using an operational matrix in the time domain under both deterministic and random forcing. To assess the stochastic distributed order system, the existing Monte-Carlo, polynomial chaos and frequency methods are first adopted to the stochastic distributed order system for comparison. The numerical examples demonstrate the accuracy and computational efficiency of the proposed method for analyzing stochastic distributed order systems.