Discrete Event Dynamic Systems

, Volume 22, Issue 4, pp 479–510 | Cite as

A pseudometric in supervisory control of probabilistic discrete event systems

Article

Abstract

The focus of this paper is the pseudometric used as a key concept in our previous work on optimal supervisory control of probabilistic discrete event systems. The pseudometric is employed to measure the behavioural similarity between probabilistic systems, and initially was defined as a greatest fixed point of a monotone function. This paper further characterizes the pseudometric. First, it gives a logical characterization of the pseudometric so that the distance between two systems is measured by a formula that distinguishes between the systems the most. A trace characterization of the pseudometric is then derived from the logical characterization, characterizing the similarity between systems from a language perspective. Further, the solution of the problem of approximation of a given probabilistic generator with another generator of a prespecified structure is suggested such that the new model is as close as possible to the original one in the pseudometric. The significance of the approximation is then discussed, especially with respect to previous work on optimal supervisory control of probabilistic discrete event systems.

Keywords

Supervisory control Probabilistic systems Pseudometric Optimal control 

References

  1. Arnold A (1994) Finite transition systems. Prentice HallGoogle Scholar
  2. Blackwell D (1962) Discrete dynamic programming. Ann Math Stat 33(2):719–726MathSciNetMATHCrossRefGoogle Scholar
  3. Chattopadhyay I, Ray A (2008) Structural transformations of probabilistic finite state machines. Int J Control 81(5):820–835MathSciNetMATHCrossRefGoogle Scholar
  4. Chattopadhyay I, Mallapragada G, Ray A (2009) ν  ⋆ : a robot path planning algorithm based on renormalized measure of probabilistic regular languages. Int J Control 82(5):849–867MathSciNetMATHCrossRefGoogle Scholar
  5. de Alfaro L, Henzinger TA, Majumdar R (2003) Discounting the future in systems theory. In: Baeten JCM, Lenstra JK, Parrow J, Woeginger GJ (eds) Proceedings of international colloquium on automata, languages and programming. Lecture Notes in Computer Science, vol 2719. Springer, pp 1022–1037Google Scholar
  6. Deng Y, Du W (2009) The Kantorovich metric in computer science: a brief survey. Electron Notes Theor Comput Sci 253:73–82CrossRefGoogle Scholar
  7. Deng Y, Chothia T, Palamidessi C, Pang J (2006) Metrics for action-labelled quantitative transition systems. Electron Notes Theor Comput Sci 153(2):79–96CrossRefGoogle Scholar
  8. Desharnais J, Gupta V, Jagadeesan R, Panangaden P (1999) Metrics for labeled Markov systems. In: Baeten JCM, Mauw S (eds) Proceedings of the 10th international conference on concurrency theory. Lecture Notes in Computer Science, vol 1664. Springer, pp 258–273Google Scholar
  9. Desharnais J, Jagadeesan R, Gupta V, Panangaden P (2002) The metric analogue of weak bisimulation for probabilistic processes. In: Proceedings of the 17th annual IEEE symposium on logic in computer science, IEEE Computer Society, Washington, DC, USA, pp 413–422CrossRefGoogle Scholar
  10. Desharnais J, Gupta V, Jagadeesan R, Panangaden P (2004) Metrics for labelled Markov processes. Theor Comp Sci 318(3):323–354MathSciNetMATHCrossRefGoogle Scholar
  11. Ferns N, Panangaden P, Precup D (2004) Metrics for finite Markov decision processes. In: Proceedings of the 20th conference on uncertainty in artificial intelligence (UAI-04). Banff, Canada, AUAI Press, Arlington, Virginia, pp 162–169, 07–11 July 2004Google Scholar
  12. Ferns N, Panangaden P, Precup D (2005) Metrics for Markov Decision Processes with infinite state spaces. In: Proceedings of the 21st conference in uncertainty in artificial intelligence. Edinburgh, Scotland, AUAI Press, Cambridge, MA, USA, pp 201–208, 26–29 July 2005Google Scholar
  13. Ferns N, Castro PS, Precup D, Panangaden P (2006) Methods for computing state similarity in Markov decision processes. In: Proceedings of the 22nd conference on uncertainty in artificial intelligence, AUAI Press, Cambridge, MA, USA, pp 174–181, 13–16 July 2006Google Scholar
  14. Garg V (1992a) An algebraic approach to modeling probabilistic discrete event systems. In: Proceedings of 31st IEEE conference on decision and control, Tucson, AZ, USA, pp 2348–2353Google Scholar
  15. Garg V (1992b) Probabilistic languages for modeling of DEDS. In: Proceedings of 26th conference on information sciences and systems, vol 1. Princeton, NJ, pp 198–203Google Scholar
  16. Giacalone A, Jou C, Smolka S (1990) Algebraic reasoning for probabilistic concurrent systems. In: Broy M, Jones CB (eds) Proceedings of the working conference on programming concepts and methods, North-Holland, Sea of Gallilee, Israel, pp 443–458Google Scholar
  17. Hennessy M, Milner R (1985) Algebraic laws for nondeterminism and concurrency. J ACM 32(1):137–161MathSciNetMATHCrossRefGoogle Scholar
  18. Hutchinson JE (1981) Fractals and self-similarity. Indiana Univ Math J 30(5):713–747MathSciNetMATHCrossRefGoogle Scholar
  19. Jou CC, Smolka SA (1990) Equivalences, congruences, and complete axiomatizations for probabilistic processes. In: Baeten JCM, Klop JW (eds) Proceedings of international conference on concurrency theory. Lecture notes in computer science, vol 458. Springer, pp 367–383Google Scholar
  20. Kantorovich L (1942) On the transfer of masses (in Russian). Dokl Akad Nauk 37(2):227–229; translated in Manage Sci, 5:(1–4) (1959)Google Scholar
  21. Koeppl H, Setti G, Pelet S, Mangia M, Petrov T, Peter M (2010) Probability metrics to calibrate stochastic chemical kinetics. In: Proceedings of 2010 IEEE international symposium on circuits and systems (ISCAS), pp 541–544Google Scholar
  22. Kozen D (1985) A probabilistic PDL. J Comput Cyst Sci 30(2):162–178MathSciNetMATHCrossRefGoogle Scholar
  23. Kumar R, Garg V (1998) Control of stochastic discrete event systems: existence. In: Proceedings of 1998 international workshop on discrete event systems, Cagliari, Italy, pp 24–29Google Scholar
  24. Larsen KG, Skou A (1991) Bisimulation through probabilistic testing. Inf Comput 94(1):1–28MathSciNetMATHCrossRefGoogle Scholar
  25. Lawford M, Wonham W (1993) Supervisory control of probabilistic discrete event systems. In: Proceedings of the 36th IEEE Midwest symposium on circuits and systems, IEEE, vol 1, pp 327–331Google Scholar
  26. Li Y, Lin F, Lin ZH (1998) Supervisory control of probabilistic discrete event systems with recovery. IEEE Trans Automat Contr 44(10):1971–1975MathSciNetGoogle Scholar
  27. Mallapragada G, Chattopadhyay I, Ray A (2009) Autonomous robot navigation using optimal control of probabilistic regular languages. Int J Control 82(1):13–26MathSciNetMATHCrossRefGoogle Scholar
  28. Pantelic V (2011) Probabilistic supervisory control of probabilistic discrete event systems. PhD thesis, McMaster University, Hamilton, ON, CanadaGoogle Scholar
  29. Pantelic V, Lawford M (2009) Towards optimal supervisory control of probabilistic discrete event systems. In: Proceedings of 2nd IFAC workshop on dependable control of discrete systems (DCDS 2009). Bari, Italy, pp 85–90Google Scholar
  30. Pantelic V, Lawford M (2010) Use of a metric in supervisory control of probabilistic discrete event systems. In: Proceedings of the 10th international workshop on discrete event systems (WODES 2010), pp 227–232, 30 August–1 SeptemberGoogle Scholar
  31. Pantelic V, Lawford M (2012) Optimal supervisory control of probabilistic discrete event systems. IEEE Trans Automat Contr (in press)Google Scholar
  32. Pantelic V, Postma S, Lawford M (2009) Probabilistic supervisory control of probabilistic discrete event systems. IEEE Trans Automat Contr 54(8):2013–2018MathSciNetCrossRefGoogle Scholar
  33. Postma S, Lawford M (2004) Computation of probabilistic supervisory controllers for model matching. In: Veeravalli V, Dullerud G (eds) Proceedings of allerton conference on communications, control, and computing, Monticello, IllinoisGoogle Scholar
  34. Thorsley D, Klavins E (2010) Approximating stochastic biochemical processes with wasserstein pseudometrics. IET Syst Biol 4(3):193–211CrossRefGoogle Scholar
  35. van Breugel F, Worrell J (2001) An algorithm for quantitative verification of probabilistic transition systems. In: Larsen KG, Nielsen M (eds) Proceedings of international conference on concurrency theory. Lecture Notes in Computer Science, vol 2154. Springer, pp 336–350Google Scholar
  36. van Breugel F, Worrell J (2005) A behavioural pseudometric for probabilistic transition systems. Theor Comp Sci 331(1):115–142MATHCrossRefGoogle Scholar
  37. van Breugel F, Worrell J (2006) Approximating and computing behavioural distances in probabilistic transition systems. Theor Comp Sci 360(1-3):373–385MATHCrossRefGoogle Scholar
  38. van Breugel F, Hermida C, Makkai M, Worrell J (2005) An accessible approach to behavioural pseudometrics. In: Caires L, Italiano G, Monteiro L, Palamidessi C, Yung M (eds) Automata, languages and programming. Lecture Notes in Computer Science, vol 3580. Springer Berlin / Heidelberg, pp 1018–1030Google Scholar
  39. van Breugel F, Hermida C, Makkai M, Worrell J (2007) Recursively defined metric spaces without contraction. Theor Comp Sci 380(1–2):143–163MATHCrossRefGoogle Scholar
  40. Wasserstein L (1969) Markov processes over denumerable products of spaces describing large systems of automata. Probl Inf Transm 5(3):47–52Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Computing and Software, Faculty of EngineeringMcMaster UniversityHamiltonCanada

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