Journal of the Korean Society for Industrial and Applied Mathematics

  • 제20권3호
  • /
  • Pages.225-242
  • /
  • 2016
  • /
  • 1226-9433(pISSN)
  • /
  • 1229-0645(eISSN)
한국산업응용수학회 (Korean Society for Industrial and Applied Mathematics)
권호 표지
DOI QR코드

DOI QR Code

AN OPTIMIZATION APPROACH FOR COMPUTING A SPARSE MONO-CYCLIC POSITIVE REPRESENTATION

  • KIM, KYUNGSUP (DEPARTMENT OF COMPUTER ENGINEERING, CHUNGNAM NATIONAL UNIVERSITY)
  • 투고 : 2016.08.03
  • 심사 : 2016.09.09
  • 발행 : 2016.09.25
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

The phase-type representation is strongly connected with the positive realization in positive system. We attempt to transform phase-type representation into sparse mono-cyclic positive representation with as low order as possible. Because equivalent positive representations of a given phase-type distribution are non-unique, it is important to find a simple sparse positive representation with lower order that leads to more effective use in applications. A Hypo-Feedback-Coxian Block (HFCB) representation is a good candidate for a simple sparse representation. Our objective is to find an HFCB representation with possibly lower order, including all the eigenvalues of the original generator. We introduce an efficient nonlinear optimization method for computing an HFCB representation from a given phase-type representation. We discuss numerical problems encountered when finding efficiently a stable solution of the nonlinear constrained optimization problem. Numerical simulations are performed to show the effectiveness of the proposed algorithm.

키워드

참고문헌

  1. C. Altafini. Minimal eventually positive realizations of externally positive systems. Automatica, 68 (2016), 140-147. https://doi.org/10.1016/j.automatica.2016.01.072
  2. L. Benvenuti, L. Farina, and B. D. O. Anderson. Filtering through combination of positive filters. Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on, 46(12) (1999), 1431-1440. https://doi.org/10.1109/81.809545
  3. L. Farina and S. Rinaldi. Positive Linear Systems: Theory and Applications. Wiley Interscience, 2000.
  4. B. Nagy, M. Matolcsi, and M. Szilvasi. Order Bound for the Realization of a Combination of Positive Filters. IEEE Transactions on Automatic Control, 52(4) (2007), 724-729. https://doi.org/10.1109/TAC.2007.894540
  5. C. Commault and S. Mocanu. Phase-type distributions and representations: Some results and open problems for system theory. International Journal of Control, 76(6) (2003), 566-580. https://doi.org/10.1080/0020717031000114986
  6. C. A. O'Cinneide. On non-uniqueness of representations of phase-type distributions. Communications in Statistics. Stochastic Models, 5(2) (1989), 247-259. https://doi.org/10.1080/15326348908807108
  7. G. Horvath, P. Reinecke, M. Telek, and K. Wolter. Efficient Generation of PH-distributed Random. Lecture Notes in Computer Science, 7314 (2012), 271-285.
  8. G. Horvath, P. Reinecke, M. Telek, and K. Wolter. Heuristic representation optimization for efficient generation of PH-distributed random variates. Annals of Operations Research, (2014), 1-23.
  9. S. Mocanu and C. Commault. Sparse representations of phase-type distributions. Communications in Statistics. Stochastic Models, 15(4) (1999), 759-778. https://doi.org/10.1080/15326349908807561
  10. K. Kim. A construction method for positive realizations with an order bound. Systems & Control Letters, 61(7) (2012), 759-765. https://doi.org/10.1016/j.sysconle.2012.04.009
  11. K. Kim. A constructive positive realization with sparse matrices for a continuous-time positive linear system. Mathematical Problems in Engineering, 2013 (2013), 1-9.
  12. D. S. Huang. A constructive approach for finding arbitrary roots of polynomials by neural networks. IEEE Transactions on Neural Networks, 15(2) (2004), 477-491. https://doi.org/10.1109/TNN.2004.824424
  13. C. A. O'Cinneide. Phase-Type Distributions and Invariant Polytopes. Advances in Applied Probability, 23(3) (1991), 515-535. https://doi.org/10.1017/S0001867800023715
  14. Q.-M. He, H. Zhang, and J. Xue. Algorithms for Coxianization of Phase-Type Generators. INFORMS J. on Computing, 23(1) (2011), 153-164. https://doi.org/10.1287/ijoc.1100.0383
  15. I. Horvath and M. Telek. A heuristic procedure for compact Markov representation of PH distributions. In ValueTools, 2014.
  16. C. A. O'Cinneide. Phase-type distributions: open problems and a few properties. Stochastic Models, 15(4) (1999), 731-757. https://doi.org/10.1080/15326349908807560
  17. D. J. Hartfiel. A simplified form for nearly reducible and nearly decomposable matrices, 1970.
  18. J. L.Walsh. On the location of the roots of certain types of polynomials. Transactions of American Mathematic Society, 24 (1922), 163-180. https://doi.org/10.1090/S0002-9947-1922-1501220-0
  19. J. M. Ortega. Numerical Analysis: A Second Course. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, 1990.
  20. R. a. Waltz, J. L. Morales, J. Nocedal, and D. Orban. An interior algorithm for nonlinear optimization that combines line search and trust region steps. Mathematical Programming, 107(3) (2006), 391-408. https://doi.org/10.1007/s10107-004-0560-5
  21. Q.-M. He and H. Zhang. A Note on Unicyclic Representations of Phase Type Distributions. Stochastic Models, 21(2-3) (2005), 465-483. https://doi.org/10.1081/STM-200057131