Colored decision process Petri nets: modeling, analysis and stability

Julio Clempner

International Journal of Applied Mathematics and Computer Science (2005)

  • Volume: 15, Issue: 3, page 405-420
  • ISSN: 1641-876X

Abstract

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In this paper we introduce a new modeling paradigm for developing a decision process representation called the Colored Decision Process Petri Net (CDPPN). It extends the Colored Petri Net (CPN) theoretic approach including Markov decision processes. CPNs are used for process representation taking advantage of the formal semantic and the graphical display. A Markov decision process is utilized as a tool for trajectory planning via a utility function. The main point of the CDPPN is its ability to represent the mark-dynamic and trajectory-dynamic properties of a decision process. Within the mark-dynamic properties framework we show that CDPPN theoretic notions of equilibrium and stability are those of the CPN. In the trajectory-dynamic properties framework, we optimize the utility function used for trajectory planning in the CDPPN by a Lyapunov-like function, obtaining as a result new characterizations for final decision points (optimum point) and stability. Moreover, we show that CDPPN mark-dynamic and Lyapunov trajectory-dynamic properties of equilibrium, stability and final decision points converge under certain restrictions. We propose an algorithm for optimum trajectory planning that makes use of the graphical representation (CPN) and the utility function. Moreover, we consider some results and discuss possible directions for further research.

How to cite

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Clempner, Julio. "Colored decision process Petri nets: modeling, analysis and stability." International Journal of Applied Mathematics and Computer Science 15.3 (2005): 405-420. <http://eudml.org/doc/207754>.

@article{Clempner2005,
abstract = {In this paper we introduce a new modeling paradigm for developing a decision process representation called the Colored Decision Process Petri Net (CDPPN). It extends the Colored Petri Net (CPN) theoretic approach including Markov decision processes. CPNs are used for process representation taking advantage of the formal semantic and the graphical display. A Markov decision process is utilized as a tool for trajectory planning via a utility function. The main point of the CDPPN is its ability to represent the mark-dynamic and trajectory-dynamic properties of a decision process. Within the mark-dynamic properties framework we show that CDPPN theoretic notions of equilibrium and stability are those of the CPN. In the trajectory-dynamic properties framework, we optimize the utility function used for trajectory planning in the CDPPN by a Lyapunov-like function, obtaining as a result new characterizations for final decision points (optimum point) and stability. Moreover, we show that CDPPN mark-dynamic and Lyapunov trajectory-dynamic properties of equilibrium, stability and final decision points converge under certain restrictions. We propose an algorithm for optimum trajectory planning that makes use of the graphical representation (CPN) and the utility function. Moreover, we consider some results and discuss possible directions for further research.},
author = {Clempner, Julio},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {game theory; optimization; decision process; colored decision process Petri nets; colored Petri nets; Lyapunov methods; stability},
language = {eng},
number = {3},
pages = {405-420},
title = {Colored decision process Petri nets: modeling, analysis and stability},
url = {http://eudml.org/doc/207754},
volume = {15},
year = {2005},
}

TY - JOUR
AU - Clempner, Julio
TI - Colored decision process Petri nets: modeling, analysis and stability
JO - International Journal of Applied Mathematics and Computer Science
PY - 2005
VL - 15
IS - 3
SP - 405
EP - 420
AB - In this paper we introduce a new modeling paradigm for developing a decision process representation called the Colored Decision Process Petri Net (CDPPN). It extends the Colored Petri Net (CPN) theoretic approach including Markov decision processes. CPNs are used for process representation taking advantage of the formal semantic and the graphical display. A Markov decision process is utilized as a tool for trajectory planning via a utility function. The main point of the CDPPN is its ability to represent the mark-dynamic and trajectory-dynamic properties of a decision process. Within the mark-dynamic properties framework we show that CDPPN theoretic notions of equilibrium and stability are those of the CPN. In the trajectory-dynamic properties framework, we optimize the utility function used for trajectory planning in the CDPPN by a Lyapunov-like function, obtaining as a result new characterizations for final decision points (optimum point) and stability. Moreover, we show that CDPPN mark-dynamic and Lyapunov trajectory-dynamic properties of equilibrium, stability and final decision points converge under certain restrictions. We propose an algorithm for optimum trajectory planning that makes use of the graphical representation (CPN) and the utility function. Moreover, we consider some results and discuss possible directions for further research.
LA - eng
KW - game theory; optimization; decision process; colored decision process Petri nets; colored Petri nets; Lyapunov methods; stability
UR - http://eudml.org/doc/207754
ER -

References

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  12. Lakshmikantham V., Matrosov V.M. and Sivasundaram S. (1991): Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems. - Dordrecht: Kluwer. Zbl0721.34054
  13. Massera J.L. (1949): On Lyapunoff's coonditions of stability- Ann. Math., Vol. 50, No. 3, pp. 705-721. Zbl0038.25003
  14. Murata T., (1989): Petri Nets Properties, analysis and applications. - Proc. IEEE, Vol. 77, No. 4, pp. 541-580. 
  15. Passino K.M., Burguess K.L. and Michel A.N. (1995): Lagrange stability and boundedness of discrete event systems. - J. Discr. Event Syst. Theory Appl., Vol. 5, pp. 383-403. Zbl0849.93051
  16. Puterman M.L. (1994): Markov Decision Processes: Discrete Stochastic Dynamic Programming. - New York: Wiley. Zbl0829.90134

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