On the relation between phase-type distributions and positive systems. (English) Zbl 1350.93030

Kim, Kyungsup

Abstr. Appl. Anal. 2015, Article ID 731261, 7 p. (2015).

Summary: The relation between phase-type representation and positive system realization in both the discrete and continuous time is discussed. Using the Perron-Frobenius theorem of nonnegative matrix theory, a transformation from positive realization to phase-type realization is derived under the excitability condition. In order to explain the connection, some useful properties and characteristics such as irreducibility, excitability, transparency, and order reduction for positive realization and phase-type representation are discussed. In addition, the connection between the phase-type renewal process and the feedback positive system is discussed in the stabilization concept.

MSC:

93B15	Realizations from input-output data
15B48	Positive matrices and their generalizations; cones of matrices
93D15	Stabilization of systems by feedback

Keywords:

phase-type representation; positive system realization; discrete-time systems; continuous-time systems; Perron-Frobenius theorem; nonnegative matrix theory

References:

[1]	Benvenuti, L.; Farina, L.; Anderson, B. D. O., Filtering through combination of positive filters, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 46, 12, 1431-1440, (1999)
[2]	Farina, L.; Rinaldi, S., Positive Linear Systems: Theory and Applications, (2000), Wiley · Zbl 0988.93002
[3]	Kaczorek, T.; Sajewski, L., The Realization Problem for Positive and Fractional Systems, 1, (2014), Springer · Zbl 1297.93002
[4]	Kim, K., A construction method for positive realizations with an order bound, Systems & Control Letters, 61, 7, 759-765, (2012) · Zbl 1250.93042
[5]	Kim, K., A constructive positive realization with sparse matrices for a continuous-time positive linear system, Mathematical Problems in Engineering, 2013, (2013) · Zbl 1299.93047
[6]	Cumani, A., On the canonical representation of homogeneous markov processes modelling failure-time distributions, Microelectronics Reliability, 22, 3, 583-602, (1982)
[7]	Latouche, G.; Ramaswami, V., Introduction to Matrix Analytic Methods in Stochastic Modeling, 43, (1999), SIAM · Zbl 0922.60001
[8]	Neuts, M. F., Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach, (1994), New York, NY, USA: Dover Publications, New York, NY, USA
[9]	Commault, C.; Mocanu, S., Phase-type distributions and representations: some results and open problems for system theory, International Journal of Control, 76, 6, 566-580, (2003) · Zbl 1042.60044
[10]	Berman, A.; Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences, (1994), Society for Industrial and Applied Mathematics · Zbl 0815.15016
[11]	Carlson, D., A note on M-matrix equations, Journal of the Society for Industrial & Applied Mathematics, 11, 4, 1027-1033, (1963) · Zbl 0196.30202
[12]	Hershkowitz, D.; Schneider, H., Solutions of Z-matrix equations, Linear Algebra and Its Applications, 106, 25-38, (1988) · Zbl 0652.15016
[13]	Schneider, H., The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties: a survey, Linear Algebra and Its Applications, 84, 161-189, (1986) · Zbl 0613.15017
[14]	Muratori, S.; Rinaldi, S., Excitability, stability, and sign of equilibria in positive linear systems, Systems & Control Letters, 16, 1, 59-63, (1991) · Zbl 0733.93029
[15]	De Leenheer, P.; Aeyels, D., Stabilization of positive linear systems, Systems & Control Letters, 44, 4, 259-271, (2001) · Zbl 0986.93059
[16]	James, G.; Rumchev, V., Stability of positive linear discrete-time systems, Bulletin of the Polish Academy of Sciences Technical Sciences, 53, 1, 1-8, (2005) · Zbl 1194.93127
[17]	Ohta, Y.; Maeda, H.; Kodama, S., Reachability, observability, and realizability of continuous-time positive systems, SIAM Journal on Control and Optimization, 22, 2, 171-180, (1984) · Zbl 0539.93005
[18]	Valcher, M. E., Reachability properties of continuous-time positive systems, IEEE Transactions on Automatic Control, 54, 7, 1586-1590, (2009) · Zbl 1367.93065
[19]	Meyer, C., Matrix Analysis and Applied Linear Algebra, (2000), Society for Industrial and Applied Mathematics
[20]	Victory, J., On nonnegative solutions of matrix equations, Journal on Algebraic and Discrete Methods, 6, 3, 406-412, (1985) · Zbl 0586.15003
[21]	Bobbio, A.; Horvath, A.; Scarpa, M.; Telek, M., Acyclic discrete phase type distributions: properties and a parameter estimation algorithm, Performance Evaluation, 54, 1, 1-32, (2003)