MOMENT-LINEAR STOCHASTIC SYSTEMS

Sandip Roy, George C. Verghese, Bernard C. Lesieutre

Abstract

We introduce a class of quasi-linear models for stochastic dynamics, called moment-linear stochastic systems (MLSS). We formulate MLSS and analyze their dynamics, as well as discussing common stochastic models that can be represented as MLSS. Further studies, including development of optimal estimators and controllers, are summarized. We discuss the reformulation of a common stochastic hybrid system——the Markovian jumplinear system (MJLS)—as an MLSS, and show that the MLSS formulation can be used to develop some new analyses for MJLS. Finally, we briefly discuss the use of MLSS in modeling certain stochastic network dynamics. Our studies suggest that MLSS hold promise in providing a framework for modeling interesting stochastic dynamics in a tractable manner.

References

  1. Asavathiratham, C. (2000). The In uence Model: A Tractable Representation for the Dynamics of Networked Markov Chains, Ph.D. Thesis. EECS Department, Massachusetts Institute of Technology.
  2. Asavathiratham, C., Roy, S., Lesieutre, B. C., and Verghese, G. C. (2001). The in uence model. IEEE Control Systems Magazine.
  3. Baiocchi, A., Melazzi, N. B., Listani, M., Roveri, A., and Winkler, R. (1991). Loss performance analysis of an ATM multiplexer loaded with high-speed on off sources. IEEE Journal on Selected Areas of Communications, 9:388-393.
  4. Basu, S., Choudhury, T., Clarkson, B., and Pentland, A. (2001). Learning human interactions with the in uence model. MIT Media Lab Vision and Modeling TR#539.
  5. Bose, A. (2003). Power system stability: new opportunities for control. Stability and Control of Dynamical Systems with Applications.
  6. Catlin, D. E. (1989). Estimation, Control, and the Discrete Kalman Filter. Springer-Verlag, New York.
  7. Ching, W. K. (1997). Markov-modulated Poisson processes for multi-location inventory problems. International Journal of Production Economics, 52:217-233.
  8. Chizeck, N. J. and Ji, Y. (1988). Optimal quadratic control of jump linear systems with Gaussian noise in discrete-time. Proceedings of the 27th IEEE Conference on Decision and Control, pages 1989-1999.
  9. Costa, O. L. V. (1994). Linear minimum mean square error estimation for discrete-time markovian jump linear systems. IEEE Transactions on Automatic Control, 39:1685-1689.
  10. Durrett, R. (1981). An introduction to in nite particle systems. Stochastic Processes and their Applications, 11:109-150.
  11. Fang, Y. and Loparo, K. A. (2002). Stabilization of continuous-time jump linear systems. IEEE Transactions on Automatic Control, 47:1590-1603.
  12. Fang, Y., Loparo, K. A., and Feng, X. (1991). Modeling issues for the control systems with communication delays. Ford Motor Co., SCP Research Report.
  13. Heskes, T. and Zoeter, O. (2003). Generalized belief propagation for approximate inference in hybrid bayesian networks. Proceedings of the Ninth International Workshop on Arti cial Intelligence and Statistics.
  14. Kelly, F. P. (1979). Reversibility and Stochastic Networks. John Wiley and Sons, New York.
  15. Loparo, K. A., Buchner, M. R., and Vasuveda, K. (1991). Leak detection in an experimental heat exchanger process: a multiple model approach. IEEE Transations on Automatic Control, 36.
  16. Mazor, E., Averbuch, A., Bar-Shalom, Y., and Dayan, J. (1998). Interacting multiple model methods in target tracking: a survey. IEEE Transactions on aerospace and electronic systems, 34(1):103-123.
  17. Mendel, J. M. (1975). Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications. Proceedings of the IEEE, 3:278-305.
  18. Meyn, S. and Tweedie, R. (1994). Markov Chains and Stochastic Stability. http://black.csl.uiuc.edu/ meyn/pages/TOC.html.
  19. Nagarajan, R., Kurose, J. F., and Towsley, D. (1991). Approximation techniques for computing packet loss in nite-buffered voice multiplexers. IEEE Journal on Selected Areas of Communications, 9:368-377.
  20. Rabiner, L. R. (1986). A tutorial on hidden markov models and selected applications in speech recognition. Proceedings of the IEEE, 77(2):257-285.
  21. Rothman, D. and Zaleski, S. (1997). Lattice-Gas Cellular Automata: Simple Models of Complex Hydrodynamics. Cambridge University Press, New York.
  22. Roy, S. (2003). Moment-Linear Stochastic Systems and their Applications. EECS Department, Massachusetts Institute of Technology.
  23. Roy, S., Verghese, G. C., and Lesieutre, B. C. (2004). Moment-linear stochastic systems and networks. Submitted to the Hybrid Systems Computation and Control conference.
  24. Segall, A., Davis, M. H., and Kailath, T. (1975). Nonlinear ltering with counting observations. IEEE Transactions on Information Theory, IT-21(2).
  25. Snyder, D. L. (1972). Filtering and detection of doubly stochastic poisson processes. IEEE Transactions on Information Theory, IT-18:91-102.
  26. Swami, A. and Mendel, J. M. (1990). Time and lag recursive computation of cumulants from a state space model. IEEE Transactions on Automatic Control, 35:4-17.
  27. Sworder, D. D., Boyd, J. E., and Elliot, R. J. (2000). Modal estimation in hybrid systems. Journal of Mathematical Analysis and Applications, 245:225-247.
  28. Zehnwirth, B. (1988). A generalization of the Kalman lter for models with state-dependent observation variance. Journal of the American Statistical Association, 83(401):164-167.
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Paper Citation


in Harvard Style

Roy S., Verghese G. and Lesieutre B. (2004). MOMENT-LINEAR STOCHASTIC SYSTEMS . In Proceedings of the First International Conference on Informatics in Control, Automation and Robotics - Volume 3: ICINCO, ISBN 972-8865-12-0, pages 190-197. DOI: 10.5220/0001143401900197


in Bibtex Style

@conference{icinco04,
author={Sandip Roy and George C. Verghese and Bernard C. Lesieutre},
title={MOMENT-LINEAR STOCHASTIC SYSTEMS},
booktitle={Proceedings of the First International Conference on Informatics in Control, Automation and Robotics - Volume 3: ICINCO,},
year={2004},
pages={190-197},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0001143401900197},
isbn={972-8865-12-0},
}


in EndNote Style

TY - CONF
JO - Proceedings of the First International Conference on Informatics in Control, Automation and Robotics - Volume 3: ICINCO,
TI - MOMENT-LINEAR STOCHASTIC SYSTEMS
SN - 972-8865-12-0
AU - Roy S.
AU - Verghese G.
AU - Lesieutre B.
PY - 2004
SP - 190
EP - 197
DO - 10.5220/0001143401900197