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.
@article{bwmeta1.element.bwnjournal-article-amcv15i3p405bwm,
author = {Clempner, Julio},
title = {Colored decision process Petri nets: modeling, analysis and stability},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {15},
year = {2005},
pages = {405-420},
zbl = {1169.93371},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv15i3p405bwm}
}
Clempner, Julio. Colored decision process Petri nets: modeling, analysis and stability. International Journal of Applied Mathematics and Computer Science, Tome 15 (2005) pp. 405-420. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv15i3p405bwm/
[000] Bellman R.E. (1957): Dynamic Programming. - Princeton, N.J.: Princeton Univ. Press.
[001] Clempner J. (2005): Optimizing the decision process on Petrinets via a Lyapunov-like function. - Int. J. Pure Appl.Math., Vol. 19, No. 4, pp. 477-494. | Zbl 1111.91006
[002] Clempner J., Medel J. and Carsteanu A. (2005): Extending games with local and robust Lyapunov equilibrium and stability condition.- Int. J. Pure Appl. Math., Vol. 19, No. 4, pp. 441-454. | Zbl 1151.91328
[003] Howard R.A. (1960): Dynamic Programming and Markov Processes. - Cambridge: MIT Press. | Zbl 0091.16001
[004] Jensen K. (1981): Coloured Petri Nets and the Invariant Method. - North-Holland Publishing Company. | Zbl 0475.68035
[005] Jensen K. (1986): Coloured Petri Nets. - Tech. Rep., Computer Science Department, Aarhus University, Denmark. | Zbl 0632.68058
[006] Jensen K. (1994): An Introduction to the Theoretical Aspects of Coloured Petri Nets. - Lecture Notes in Computer Science, Vol. 803, Berlin: Springer.
[007] Jensen K. (1997a): Coloured Petri Nets, Vol. 1. - Berlin: Springer. | Zbl 0632.68058
[008] Jensen K. (1997b): Coloured Petri Nets, Vol. 2. - Berlin: Springer. | Zbl 0632.68058
[009] Kalman R.E. and Bertram J.E. (1960): Control system analysis and design via the 'Second Method' of Lyapunov. - J. Basic Eng., Vol. 82, pp. 371-393.
[010] Lakshmikantham V., Leela S. and Martynyuk A.A. (1990): Practical Stability of Nonlinear Systems. - Singapore: World Scientific. | Zbl 0753.34037
[011] Lakshmikantham V., Matrosov V.M. and Sivasundaram S. (1991): Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems. - Dordrecht: Kluwer. | Zbl 0721.34054
[012] Massera J.L. (1949): On Lyapunoff's coonditions of stability- Ann. Math., Vol. 50, No. 3, pp. 705-721. | Zbl 0038.25003
[013] Murata T., (1989): Petri Nets Properties, analysis and applications. - Proc. IEEE, Vol. 77, No. 4, pp. 541-580.
[014] 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. | Zbl 0849.93051
[015] Puterman M.L. (1994): Markov Decision Processes: Discrete Stochastic Dynamic Programming. - New York: Wiley. | Zbl 0829.90134