- Volume 20 Issue 3
DOI QR Code
AN OPTIMIZATION APPROACH FOR COMPUTING A SPARSE MONO-CYCLIC POSITIVE REPRESENTATION
- KIM, KYUNGSUP (DEPARTMENT OF COMPUTER ENGINEERING, CHUNGNAM NATIONAL UNIVERSITY)
- Received : 2016.08.03
- Accepted : 2016.09.09
- Published : 2016.09.25
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.
Supported by : Chungnam National University
- 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
- 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
- L. Farina and S. Rinaldi. Positive Linear Systems: Theory and Applications. Wiley Interscience, 2000.
- 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
- 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
- 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
- G. Horvath, P. Reinecke, M. Telek, and K. Wolter. Efficient Generation of PH-distributed Random. Lecture Notes in Computer Science, 7314 (2012), 271-285.
- 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.
- 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
- 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
- K. Kim. A constructive positive realization with sparse matrices for a continuous-time positive linear system. Mathematical Problems in Engineering, 2013 (2013), 1-9.
- 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
- 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
- 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
- I. Horvath and M. Telek. A heuristic procedure for compact Markov representation of PH distributions. In ValueTools, 2014.
- 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
- D. J. Hartfiel. A simplified form for nearly reducible and nearly decomposable matrices, 1970.
- 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
- J. M. Ortega. Numerical Analysis: A Second Course. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, 1990.
- 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
- 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