Rabu, 14 Maret 2012

LYAPUNOV-MAX-PLUS-ALGEBRA STABILITY IN PREDATOR-PREY SYSTEMS MODELED BY TIMED PETRI NET WITH THE ENTIRE HOLDING TIMES ARE CONSIDERED

Zumrotus Sya'diyah, M.Si, Dr. Subiono, MS

Abstract

In this paper, we construct a model of predator-prey systems with timed Petri net and analyze the stabilization of the systems. We discuss a timed Petri net model with the entire holding times are considered. Furthermore, we analyze the periodic behavior of the systems. Using the Lyapunov-max-plus-algebra stability theory, we will obtain the sufficient condition for the stabilization problem. The periodic duration of the oscillation in these systems will be determined. The analysis will also use the interval matrix in max plus algebra. In this case, every holding time in timed Petri net is viewed as an interval value. Keywords and Phrases: Predator-prey Systems, Timed Petri Net, Max-Plus.

INTRODUCTION
Generally, the state of systems changes as time changes. The state spaces are expected to be changed at every tick of the clock. These kinds of systems are called time driven systems. There are some of them which evolve in time by the occurrence of events at possible irregular time intervals, i.e. not necessarily coinciding with clock ticks. In this case, the state transition is a result of the other harmonic events. This kind of systems is called event driven systems [3]. Event driven systems with discrete states are called by discrete event systems. Max plus algebra is the useful approach to represent the discrete event systems. This approaching makes us possible to determine and analyze various kinds of systems properties. The model of them will be linear over max plus algebra. In this kind of systems, event is more decisive than time [9]. We can analyze the systems in max plus algebra easier and simpler than the conventional one because of this linearity [10]. Petri net is a mathematical modeling tool which can be applied to represent the state evolution of the discrete event systems. Petri net is called autonomous if every its transition has at least an input place, i.e. does not have a transition which is always be enabled [2]. In the previous research, the predator-prey systems have been modeled with timed Petri net which is consistent with the real predator-prey behavior in real life [6]. We will modify this model with adding some holding times, condition and event. This Model is inspired by the timed Petri net model of queuing systems with one server that discussed in [10] and the timed Petri net of the predator-prey system discussed in [13]. For this discussion, we need the theory of conventional and interval max plus algebra, timed Petri net, and Lyapunov stability in systems modeled by Petri net. These theories will be discussed in the next section.

Acknowledgement. I wish to give my gratitude to Zvi Retchkimann for giving me the information about this project by his papers he sent me. I also really appreciate in every email and suggestion he gave me related to this project.

REFERENCES
[1]. ADZKIYA, DIEKY: Membangun Model Petri Net Lampu Lalu-lintas dan Simulasinya, Thesis of Mathematics Department FMIPA, Institut Teknologi Sepuluh Nopember (ITS), Surabaya, 2008.
[2]. BACELLI. F, G. COHEN, G.J. ET.AL: Synchronization and Linearity: An Algebra for Discrete Event System, Web-edition, 2001.
[3]. NECOARA, ION: Model Predictive Control for Max-Plus-Linear and Piecewise Affine Systems, Technise Universiteit Delft. Netherland, 2006.
[4]. NOVITASARI, RATNA: Analisis Masalah Generator Dari Possible Dan Universal Eigenvector Pada Matriks Interval Dalam Aljabar Max-Plus, Thesis of Mathematics Department FMIPA, Institut Teknologi Sepuluh Nopember, Surabaya, 2008.
[5]. REJEKI, SRI: Analisis Sistem Jaringan Antrean Dengan Elemen-Elemen Matriks Adjasen Berupa Interval Dalam Aljabar Max-Plus, Thesis of Mathematics Department FMIPA, Institut Teknologi Sepuluh Nopember, Surabaya, 2010.
[6]. RETCHKIMAN, ZVI, Mixed Lyapunov-Max-Plus Algebra Approach to the Stability Problem for a two Species Ecosystem Modeled with Timed Petri Nets, International Mathematical Forum, Vol.5 No.28, 1393-1408, 2010.
[7]. RETCHKIMAN, ZVI, The Stability Problem for Discrete Event Dynamical Systems Modeled with timed Petri Nets Using a Lyapunov-Max-Plus Algebra Approach, International Mathematical Forum, Vol.6 No.11, 541-566, 2011.
[8]. SUBIONO AND J VAN DER WOUDE, Power Algorithms for (Max,+)-and Bipartite (Min,Max,+)-systems, Discrete Event Dynamic Systems: Theory and Applications, 10(4):369-389, 2000.
[9]. SUBIONO: The existence of eigenvalues for reducible matrices in Max-Plus Algebra, Mathematics Department FMIPA- I T S, Surabaya, 2008.
[10]. SUBIONO: Aljabar Max Plus dan Aplikasinya:Model Sistem Antrian. Mathematics Department FMIPA- I T S, Surabaya, 2008.
[11].  SUBIONO AND NUR SOFIYANA, Using Max-Plus Algebra In The   Flow Shop Scheduling, IPTEK, The Journal for Technology and Science, Vol 20, No 3, August 2009, p.83-87, , 2009.
[12]. SUBIONO: Aljabar Max Plus dan Terapannya, Mathematics Department FMIPA, Institut Teknologi Sepuluh Nopember (ITS), Surabaya, 2010.
[13]. SUBIONO AND ZUMROTUS.S: Lyapunov-Max-Plus-Algebra Stability In Predator-Prey Systems Modeled With Timed Petri Net, Mathematics Department FMIPA, Institut Teknologi Sepuluh Nopember (ITS), Surabaya, 2011.

Tidak ada komentar:

Posting Komentar