Les systèmes à sauts : théorie et application
Systems with jumps : theory and applications
OPEN ACCESS
Introduced in the eigthies, the systems with jumps, also called at the beginning hybrid systems, can efficiently modelize a great number of physical systems subject to random jumps in their dynamics . A manoeuvering fighting aircraft, widely studied in the litterature, is a good illustration of this type of systems . Stability, optimal control and stochastic filtering were the major areas of study for these models . In fact this modelization results from a combination of three ingredients : diffusion, jumps and deterministic processes. When restricting ourselves to the first two kinds of processes, we consider models with stochastic diffusions.
So, after having introduced the general mathematical representation for such processes, we consider in this article a typical application in aerospace, the target tracking, to show the significant improvements in performance brought by this approach.
When then conclude, after having recalled the various contributions brought by our group on the stochastic diffusions during the last fifteen years.
Résumé
Introduits dans les années quatre vingt, les systèmes à sauts, plus connus à l'origine sous le nom de systèmes hybrides, offrent un cadre mathématique idéal pour l'étude de systèmes physiques caractérisés par des modifications (sauts), brutales et aléatoires de leur dynamique. Un des exemples les plus couramment rencontrés dans la littérature est celui d'un avion de chasse effectuant diverses manoeuvres . La stabilité, la commande optimale et le filtrage stachastique ont été les trois principaux domaines d'études de ces modèles. Cette modélisation résulte en fait d'un mélange de trois constituants élémentaires que sont les processus de diffusion, les processus à sauts et les processus déterministes . En se restreignant aux deux premiers ingrédients, nous parlerons de modèles à diffusions aléatoires.
Après avoir introduit la modélisation mathématique de tels processus, nous nous intéressons dans cet article à une application type, la poursuite de trajectoire, afin de montrer les améliorations significatives qu'apporte cette approche. Enfin nous conclurons, non sans avoir rappelé les différents travaux accomplis sur les diffusions aléatoires dans notre groupe, au cours de ces quinze dernières années.
Systems with jumps, random diffusion, stochastic filtering, target tracking
Mots clés
Systèmes à sauts, diffusion aléatoire, filtrage stochastique, poursuite de cibles
[1] S . Allam, F. Dufour, and P. Bertrand, Finite fast Fourier transform filter for discrete linear systems with jump parameters . In Proceedings of ECC 99 , 1999 .
[2] F. Bernard, F. Dufour, and P. Bertrand, On the JLQ problem with uncertainty, IEEE Transactions on Automatic Control, 42(6) :869-872, 1997 .
[3] F. Bernard, F . Dufour, and P. Bertrand, Systems with markovian jump parameters : approximations for the nonlinear filtering problem . In Proceedings of ECC 97, Brussels, Belgium, 1997 .
[4] T. Björk, Finite optimal filters for a class of nonlinear diffusions with jumping parameters, Stochastics, 6 :121-138, 1982 .
[5] H .A .P. Blom, A Sophisticated Tracking Algorithm for ATC surveilllance data . In Proceedings of the International Radar Conference, Paris, 1984 .
[6] H .A .P. Blom and Y. Bar-Shalom, The interacting multiple model algorithm for systems with markovian switching coefficients, IEEE Transactions on Automatic Control, 33(8) :780-783, 1988 .
[7] P.E . Caines and H.F. Chen, Optimal adaptive LQG control for systems with finite state process parameters, IEEE Transactions on Automatic Control, 30(2) :185-189, 1985 .
[8] P.E. Caines and J-F. Zhang, Adaptive control for jump parameter systems via non- linear filtering. In Proceedings of the 31st Conference on Decision and Control, p. 699-704, Tucson, Arizona, 1992.
[9] P.E . Caines and J-F. Zhang, On the adaptive control for jump parameter systems via non-linear filtering, SIAM Journal of Control and Optimization , 33(6) :1758-1777, 1995 .
[10] O .L .V. Costa, C .A .B . Raymundo, and F. Dufour, Variational inequalities for the optimal stopping with continuous control of piecewise deterministic markov processes . In Proceedings of ACC 99, 1999 .
[11] M.H.A . Davis, Piecewise-deterministic markov processes : A general class of non-diffusion stochastic models, .1. R. Statist. Soc. B, 46(3) :353-388, 1984 .
[12] F. Dufour, Contribution à l'étude des systèmes linéaires à sauts markoviens , PhD thesis, LSS, Université Paris XI, France, 1994 .
[13] F. Dufour and P. Bertrand, The filtering problem for continuous-time linear systems with markovian switching coefficients, Systems & Control Letters , 23 :453-461, 1994.
[14] F. Dufour and P. Bertrand, Stabilizing control law for hybrid models, IEEE Transactions on Automatic Control, 39(11) :2354-2357, 1994.
[15] F. Dufour and P. Bertrand, An image based filter for discrete-time markovian jump linear systems, Automatica, 32(2) :241-247, 1996.
[16] F. Dufour and O .L .V. Costa, Stability of piecewise-deterministic markov processes, To appear in SIAM Journal of Control and Optimization, June,1999 .
[17] F. Dufour and C . Durieu, A multiple model algorithm to fuse data for localisation of a mobile robot. In Proceedings of the 3rd International Workshop on Advanced Motion Control, p . 230-238, Berkeley, USA, 1994 .
[18] F. Dufour and R . Elliott, Adaptive control for linear systems with Markov pertubations, IEEE Transactions on Automatic Control, 43(3) :351-372, 1998 .
[19] F. Dufour and M . Mariton, Tracking a 3d maneuvring target with passive sensors, IEEE Transactions on Aerospace and Electronic Systems, 27(4) :725-739, 1991 .
[20] R . Elliott, F. Dufour, and F. Sworder, Exact hybrid filters in discrete time , IEEE Transactions on Automatic Control, 41(12) :1807-1810, 1996 .
[21] J . Ezzine and A.D . Haddad, On the largest-lyapunov exponent of assignement and almost sure stabilization of hybrid systems . In Proceedings American Control Conference, p. 805-809, USA, 1989 .
[22] X . Feng, K .A. Loparo, Y. Ji, and H.J . Chizeck, Stochastic stability properties of jump linear systems, IEEE Transactions on Automatic Control, 37(1) :38-53, 1992.
[23] B .E. Griffiths and K .A . Loparo, Optimal control of jump-linear gaussian systems, International Journal of Control, 42(4) :791-819, 1985 .
[24] Y. Ji and H .J . Chizeck, Controllability, stabilizability and continuous-time Markovian jump linear quadratic control, IEEE Transactions on Automatic Control, 35 :777-782, 1990 .
[25] D . Laneuville, Processus à sauts markovien : apport des capteurs imageurs au pistage de cibles manteuvrantes, PhD thesis, LSS, Université Paris XI, France, 1998 .
[26] D . Laneuville, F. Dufour, and P. Bertrand, Imaged based maneuvering target tracking. In Proceedings ofACC 98, 1998 .
[27] D . Laneuville, F. Dufour, and P. Bertrand, A new architecture for maneuvering target tracking : the hybrid imaging filter. In Proceedings of ECC 99, 1999.
[28] R .S . Liptser and A.N . Shiryayev, Statistics of Random Processes I Generl Theory, Springer Verlag, New York, 1977 .
[29] R.S . Liptser and A.N . Shiryayev, Statistics of Random Processes II General Theory, Springer Verlag, New York, 1977 .
[30] K .A. Loparo, M .R . Buchner, and K . Vasudeva, Leak detection in an experimental heat exchanger process : a multiple model approach, IEEE Transactions on Automatic Control, 36 :167-177, 1991 .
[31] K.A . Loparo, Z . Roth, and S.J . Eckert, Nonlinear filtering for systems with random structure, IEEE Transactions on Automatic Control, 31(11) :1064 - 1068, 1986 .
[32] M . Mariton, Almost sure and moments stability of jump linear systems , Sytems & Control Letters, 11 :393-397, 1988 .
[33] M . Mariton, Jump Linear Systems in Automatic Control, Marcel Dekker, New York, 1990 .
[34] M. Mariton and P. Bertrand, Comportement asymptotique de la commande pour les systèmes linéaires à sauts markoviens, C.R . Acad. Sciences, Paris , Série I, 301(13) :683-686, 1985 .
[35] M . Mariton and P. Bertrand, Output feedback fora class of linear systems with stochastic jumping parameters, IEEE Transactions on Automatic Control , 30 :898-900, 1985 .
[36] R. Pinsky and M . Scheutzow, Some remarks and examples concerning the transience and recurrence of random diffusions, Ann. Inst. H. Poincaré, 28 :519-530, 1992 .
[37] R . Rishel, A strong separation principle for stochastic control systems driven by a hidden Markov model, SIAM Journal of Control and Optimization , 32(4) :1008-1020, 1994 .
[38] D . Sworder, Bertrand P., and F. Dufour, Dual-path algorithm for the benchmark tracking problem. In Proceedings of the ACC 94, p. 2088-2092 , Baltimore, USA, 1994 .
[39] D .D . Sworder, Feedback control of a class of linear systems with jump parameters, IEEE Transactions on Automatic Control, 14(1) :9-14, 1969 .
[40] D .D . Sworder, Utilization of repair capability in a stochastic dynamic systems, J. Economic Dynamics and Control, 5 :371-385, 1983 .
[41] D .D . Sworder, Hybrid adaptive control, Applied Mathematics and Computation, 45 :173-192, 1991 .
[42] D .D . Sworder and R .O . Rogers, An LQ-solution to a control problem associated with a solar thermal central receiver, IEEE Transactions on Automatic Control, 28(10) :971-978, 1983 .
[43] W.M . Wonham, Random differential equations in control theory. In A .T. Bharucha-Reid, editor, Probabilistic Method in Applied Mathematics, p . 131-212 . Academic Press, New York, 1971 .
[44] C. Yang, Contribution à l'étude des systèmes à sauts markoviens : détection et commande, PhD thesis, LSS, Université Paris XI, France, 1989 .
[45] C . Yang and M. Mariton, New approximation for the partially observed jump linear quadratic problem, International Journal Systems Science, 22(5) :775 -781, 1991 .