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Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 989542, 14 pages
Research Article

Optimal Design of Stochastic Distributed Order Linear SISO Systems Using Hybrid Spectral Method

1School of Chemical Engineering, Yeungnam University, Gyeongsan 712-749, Republic of Korea
2Chemical & Biological Engineering, University of British Columbia, Vancouver, Canada

Received 20 May 2015; Revised 18 August 2015; Accepted 2 September 2015

Academic Editor: Son Nguyen

Copyright © 2015 Pham Luu Trung Duong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The distributed order concept, which is a parallel connection of fractional order integrals and derivatives taken to the infinitesimal limit in delta order, has been the main focus in many engineering areas recently. On the other hand, there are few numerical methods available for analyzing distributed order systems, particularly under stochastic forcing. This paper proposes a novel numerical scheme for analyzing the behavior of a distributed order linear single input single output control system under random forcing. The method is based on the operational matrix technique to handle stochastic distributed order systems. The existing Monte Carlo, polynomial chaos, and frequency methods were first adapted to the stochastic distributed order system for comparison. Numerical examples were used to illustrate the accuracy and computational efficiency of the proposed method for the analysis of stochastic distributed order systems. The stability of the systems under stochastic perturbations can also be inferred easily from the moment of random output obtained using the proposed method. Based on the hybrid spectral framework, the optimal design was elaborated on by minimizing the suitably defined constrained-optimization problem.