Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2015, Article ID 989542, 14 pages
http://dx.doi.org/10.1155/2015/989542
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.

Linked References

  1. T. T. Hartley and C. F. Lorenzo, “Order-distributions and the Laplace-domain logarithmic operator,” Advances in Difference Equations, vol. 2011, article 59, 19 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. W. Chen, “A speculative study of 2/3-order fractional Laplacian modeling of turbulence; some though and conjectures,” Chaos, vol. 16, Article ID 023126, 7 pages, 2006. View at Publisher · View at Google Scholar
  3. K. Weron and M. Kotulski, “On the Cole-Cole relaxation function and related Mittag-Leffler distribution,” Physica A, vol. 232, no. 1-2, pp. 180–188, 1996. View at Publisher · View at Google Scholar · View at Scopus
  4. D. Baleanu, Z. B. Guvenc, and T. J. A. Machado, Eds., New Trends in Nanotechnology and Fractional Calculus Applications, Springer, Dordrecht, The Netherlands, 2010.
  5. R. L. Magin, B. S. Akpa, T. Neuberger, and A. G. Webb, “Fractional order analysis of Sephadex gel structures: NMR measurements reflecting anomalous diffusion,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 12, pp. 4581–4587, 2011. View at Publisher · View at Google Scholar · View at Scopus
  6. M. K. Bouafoura and N. B. Braiek, “PIλDμ controller design for integer and fractional plants using piecewise orthogonal functions,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 5, pp. 1267–1278, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. M. Caputo, Elasticida e Dissipazione, Zanichelli, Bologna, Italy, 1969.
  8. M. Caputo, “Mean-fractional-order-derivative differential equation and filters,” Annali dell' Universita di Ferrara, vol. 41, no. 1, pp. 73–84, 1995. View at Google Scholar · View at MathSciNet
  9. R. L. Bagley and P. J. Torvik, “On the existence of the order domain and the solution of distributed order equation (part I, II),” International Journal of Applied Mechanics, vol. 2, no. 1, pp. 865–987, 2000. View at Google Scholar
  10. T. M. Atanackovic, S. Pilipovic, and D. Zorica, “Time distributed-order diffusion-wave equation. I. Volterra-type equation,” Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences, vol. 465, no. 2106, pp. 1869–1891, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. T. M. Atanackovic, S. Pilipovic, and D. Zorica, “Time distributed-order diffusion-wave equation. II. Applications of Laplace and Fourier transformations,” Proceedings of the Royal Society A, vol. 465, no. 2106, pp. 1893–1917, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. M. Caputo, “Distributed order differential equations modeling dielectric induction and diffusion,” Fractional Calculus and Applied Analysis, vol. 4, no. 4, pp. 421–442, 2001. View at Google Scholar
  13. F. Mainardi and G. Pagnini, “The role of the Fox-Wright functions in fractional sub-diffusion of distributed order,” Journal of Computational and Applied Mathematics, vol. 207, no. 2, pp. 245–257, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. C. F. Lorenzo and T. T. Hartley, “Variable order and distributed order fractional operators,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 57–98, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. H. Sheng, Y. Q. Li, and Y. Chen, “Application of numerical inverse Laplace transform algorithms in fractional calculus,” Journal of the Franklin Institute, vol. 348, no. 2, pp. 315–330, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. K. Diethelm and N. J. Ford, “Numerical analysis for distributed-order differential equations,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 96–104, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  17. Z. Jiao, Y. Q. Chen, and I. Podlubny, Distributed Order Dynamic System Stability, Simulation and Perspective, Springer, Berlin, Germany, 2012, http://www.mathworks.com/matlabcentral/fileexchange/36570.
  18. C. H. Wang, “On the generalization of block pulse operational matrices for fractional and operational calculus,” Journal of the Franklin Institute, vol. 315, no. 2, pp. 91–102, 1983. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. A. H. Bhrawy, D. Baleanu, L. M. Assas, and J. A. T. Machado, “On a generalized laguerre operational matrix of fractional integration,” Mathematical Problems in Engineering, vol. 2013, Article ID 569286, 7 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  20. S. V. Lapin and N. D. Egupov, Theory of Matrix Operator and Its Application in Automatic Control, BMSTU Press, Moscow, Russia, 1997, (Russian).
  21. K. Maleknejad and K. Mahdiani, “Solving nonlinear mixed Volterra-Fredholm integral equations with two dimensional block-pulse functions using direct method,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 9, pp. 3512–3519, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. Z. L. Huang, X. L. Jin, C. W. Lim, and Y. Wang, “Statistical analysis for stochastic systems including fractional derivatives,” Nonlinear Dynamics, vol. 59, no. 1-2, pp. 339–349, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  23. P. L. T. Duong and M. Lee, “Statistical analysis of dead-time system using a deterministic equivalent modeling method,” Asia-Pacific Journal of Chemical Engineering, vol. 6, no. 3, pp. 369–378, 2011. View at Publisher · View at Google Scholar · View at Scopus
  24. M. D. Paola, G. Failla, and A. Pirrotta, “Stationary and non-stationary stochastic response of linear fractional viscoelastic systems,” Probabilistic Engineering Mechanics, vol. 28, no. 4, pp. 85–90, 2012. View at Publisher · View at Google Scholar · View at Scopus
  25. L. Dunn and J. K. Shultis, Exploring Monte Carlo Methods, Elsevier, New York, NY, USA, 2011.
  26. D. P. Kroese, T. Taimre, and Z. I. Botev, Handbook of Monte Carlo Method, John Wiley & Son, Hoboken, NJ, USA, 2011.
  27. D. Xiu, Numerical Method for Stochastic Computation: Spectral Approach, Princeton University Press, Princeton, NJ, USA, 2011.
  28. D. Lucor, C.-H. Su, and G. E. Karniadakis, “Generalized polynomial chaos and random oscillators,” International Journal for Numerical Methods in Engineering, vol. 60, no. 3, pp. 571–596, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. K. A. Pupkov, N. D. Egupov, A. M. Makarenkov, and A. I. Trofimov, Theory and Numerical Methods for Studying Stochastic Systems, Fizmatlits, Moscow, Russia, 2003 (Russian).
  30. D. Xiu and G. E. Karniadakis, “Modeling uncertainty in flow simulations via generalized polynomial chaos,” Journal of Computational Physics, vol. 187, no. 1, pp. 137–167, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. D. Xiu and J. Shen, “Efficient stochastic Galerkin methods for random diffusion equations,” Journal of Computational Physics, vol. 228, no. 2, pp. 266–281, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  32. O. P. Le Maître, O. M. Knio, H. N. Najm, and R. G. Ghanem, “A stochastic projection method for fluid flow. I. Basic formulation,” Journal of Computational Physics, vol. 173, no. 2, pp. 481–511, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. P. L. T. Duong and M. Lee, “Uncertainty propagation in stochastic fractional order processes using spectral methods: a hybrid approach,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 11, pp. 4262–4273, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet
  35. I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999. View at MathSciNet
  36. W. Gaustchi, Orthogonal Polynomials: Computation and Approximations, Oxford University Press, Oxford, UK, 2003.
  37. P. L. T. Duong and M. Lee, “Optimal design of fractional order linear system with stochastic/input/parametric uncertainties by hybrid spectral method,” Journal of Process Control, vol. 24, no. 10, pp. 1639–1645, 2014. View at Google Scholar
  38. R. G. Brown and P. Y. C. Hwang, Introduction to Random Signal and Applied Kalman Filtering with MATLAB Exercises, Wiley, New York, NY, USA, 4th edition, 2012.
  39. V. I. Chernhecki, Analysis Accuracy of Nonlinear Control Systems, Masintroenhie, 1968 (Russian).
  40. C. H. Eab and S. C. Lim, “Fractional Langevin equations of distributed order,” Physical Review E, vol. 83, no. 3, Article ID 031136, 10 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  41. I. Podlubny and M. Kacenak, http://www.mathworks.com/matlabcentral/fileexchange/8738-mittag-leffler-function.
  42. A. C. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue, and V. Feliu, Fractional-Order Systems and Controls, Springer, London, UK, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  43. Y. Li, Y. Chen, and H.-S. Ahn, “Fractional-order iterative learning control for fractional-order linear systems,” Asian Journal of Control, vol. 13, no. 1, pp. 54–63, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  44. M. Grigoriu, “Linear systems with fractional Brownian motion and Gaussian noise,” Probabilistic Engineering Mechanics, vol. 22, no. 3, pp. 276–284, 2007. View at Publisher · View at Google Scholar · View at Scopus
  45. T. E. Duncan, “Some application of fractional Brownian motion to linear system,” in System Theory: Modeling, Analysis and Control, T. E. Djaferis, Ed., Kluwer Academic Publishers, 2000. View at Google Scholar
  46. T. E. Duncan and B. Pasik-Duncan, “Control of some linear with a fractional Brownian motion,” in Proceedings of the 48th IEEE Conference on Decision and Control (CDC '09), pp. 8118–8522, Shanghai, China, 2009.