This paper presents an overview of some recent results concerning the emerging theory of minimax LQG control for uncertain systems with a relative entropy constraint uncertainty description. This is an important new robust control system design methodology providing minimax optimal performance in terms of a quadratic cost functional. The paper first considers some standard uncertainty descriptions to motivate the relative entropy constraint uncertainty description. The minimax LQG problem under consideration is further motivated by analysing the basic properties of relative entropy. The paper then presents a solution to a worst case control system performance problem which can be generalized to the minimax LQG problem. The solution to this minimax LQG control problem is found to be closely connected to the problem of risk-sensitive optimal control.
@article{bwmeta1.element.bwnjournal-article-amcv16i3p309bwm,
author = {Petersen, Ian},
title = {Minimax LQG control},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {16},
year = {2006},
pages = {309-323},
zbl = {1136.93465},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv16i3p309bwm}
}
Petersen, Ian. Minimax LQG control. International Journal of Applied Mathematics and Computer Science, Tome 16 (2006) pp. 309-323. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv16i3p309bwm/
[000] Boel R.K., James M.R. and Petersen I.R. (2002): Robustness and risk sensitive filtering. - IEEE Trans. Automat. Contr., Vol. 47, No. 3, pp. 451-461.
[001] Collings I.B., James M.R. and Moore J.B. (1996): An information-state approach to risk-sensitive tracking problems. - J. Math. Syst. Estim. Contr., Vol. 6, No. 3, pp. 1-24. | Zbl 1076.93544
[002] Dai Pra P., Meneghini L. and Runggaldier W.J. (1996): Connections between stochastic control and dynamic games. - Math. Contr. Syst.Signals, Vol. 9, No. 2, pp. 303-326. | Zbl 0874.93096
[003] Doyle J., Packard A. and Zhou K. (1991): Review of LFT's, LMI's and μ. - Proc. 30-th IEEE Conf. s Decision on Control, Brighton, England, pp. 1227-1232.
[004] Dupuis P. and Ellis R. (1997): A Weak Convergence Approach to the Theory of Large Deviations. - New York: Wiley. | Zbl 0904.60001
[005] Dupuis P., James M.R. and Petersen I.R. (2000): Robust properties of risk-sensitive control. - Math. Contr. Signals, Syst., Vol. 13, No. 4, pp. 318-332. | Zbl 0971.93081
[006] Horowitz I. (1963): Synthesis of Feedback Systems. - New York: Academic Press. | Zbl 0121.07704
[007] Jacobson D.H. (1973): Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games. - IEEE Trans. Automat. Contr., Vol. 18, No. 2, pp. 124-131. | Zbl 0274.93067
[008] Luenberger D.G. (1969): Optimization by Vector Space Methods. - NewYork: Wiley. | Zbl 0176.12701
[009] Moheimani S.O.R., Savkin A.V. and Petersen I.R. (1997): Minimax optimal control of discrete-time uncertain systems with structured uncertainty. - Dynam. Contr., Vol. 7, No. 1, pp. 5-24. | Zbl 0870.93021
[010] Petersen I.R. and James M.R. (1996): Performance analysis and controller synthesis for nonlinear systems with stochastic uncertainty constraints. - Automatica, Vol. 32, No. 7, pp. 959-972. | Zbl 0861.93033
[011] Petersen I.R., James M.R. and Dupuis P. (2000a): Minimax optimal control of stochastic uncertain systems with relative entropy constraints. - IEEE Trans. Automat. Contr., Vol. 45, No. 3, pp. 398-412. | Zbl 0978.93083
[012] Petersen I.R., Ugrinovski V. and Savkin A.V. (2000b): e Robust Control Design using H^∞ Methods. - London: Springer.
[013] Savkin A.V. and Petersen I.R. (1995): Minimax optimal control of uncertain systems with structured uncertainty. - Int. J. RobustNonlin. Contr., Vol. 5, No. 2, pp. 119-137. | Zbl 0829.49003
[014] Savkin A.V. and Petersen I.R. (1996): Uncertainty averaging approach to output feedback optimal guaranteed cost control of uncertain systems. - J. Optim. Th. Applic., Vol. 88, No. 2, pp. 321-337. | Zbl 0853.93065
[015] Savkin A.V. and Petersen I.R. (1997): Output feedback guaranteed cost control of uncertain systems on an infinite time interval. - Int. J. Robust Nonlin. Contr., Vol. 7, No. 1, pp. 43-58. | Zbl 0866.93042
[016] Ugrinovskii V.A. and Petersen I.R. (1997): Infinite-horizon minimax optimal control of stochastic partially observed uncertain systems. - Proc. Control 97 Conference, Sydney, Australia, pp. 616-621.
[017] Ugrinovskii V.A. and Petersen I.R. (1999a): Finite horizon minimax optimal control of stochastic partially observed time varying uncertain systems. - Math. Contr., Signals Syst., Vol. 12, No. 1, pp. 1-23. | Zbl 0929.93038
[018] Ugrinovskii V.A. and Petersen I.R. (1999b): Guaranteed cost LQG filtering for stochastic discrete time uncertain systems via risk-sensitive control. - Proc. IEEE Conf. Decision and Control, Phoenix, AZ, pp. 564-569.
[019] Ugrinovskii V.A. and Petersen I.R. (2001a): Minimax LQG control of stochastic partially observed uncertain systems. - SIAM J. Contr. Optim., Vol. 40, No. 4, pp. 1189-1226. | Zbl 1016.93072
[020] Ugrinovskii V.A. and Petersen I.R. (2001b): Robust stability and performance of stochastic uncertain systems on an infinite time interval. - Syst. Contr. Lett., Vol. 44, No. 4, pp. 291-308. | Zbl 0986.93052
[021] Ugrinovskii V.A. and Petersen I.R. (2002a): Robust output feedback stabilization via risk-sensitive control. - Automatica, Vol. 38,No. 6, pp. 945-955. | Zbl 1030.93049
[022] Ugrinovskii V. and Petersen I.R. (2002b): Robust filtering of stochastic uncertain systems on an infinite time horizon. - Int. J. Contr., Vol. 75, No. 8, pp. 614-626. | Zbl 1018.93027
[023] Whittle P. (1981): Risk-sensitive linearquadraticGaussian control. - Adv. Appl. Prob., Vol. 13, No. 4, pp. 764-777. | Zbl 0489.93067
[024] Xie L., Ugrinovski V.A. and Petersen I.R. (2004a): Finite horizon robust state estimation for uncertain finite-alphabet hidden markov models with conditional relative entropy constraints. - Proc. 43-rd IEEE Conf. Decision and Control, Atlantis, Paradise Island,Bahamas, (on CD-ROM).
[025] Xie L., Ugrinovski V.A. and Petersen I.R. (2004b): Probability distances between finite-alphabet hidden markov models. - Proc. 2-nd IFAC Symp. System, Structure and Control, Oaxaca, Mexico, (on CD-ROM).
[026] Xie L., Ugrinovskii V.A. and Petersen I.R. (2005a): Probabilistic distances between finite-state finite-alphabet hidden Markov models. - IEEE Trans. Automat. Contr., Vol. 50, No. 4, pp. 505-511.
[027] Xie L., Ugrinovskii V.A. and Petersen I.R. (2005b): A duality relationship for regular conditional relative entropy. - Proc. 16-th IFAC World Congress, Prague, Czech Republic, (on CD-ROM).
[028] Yoon M. and Ugrinovskii V. (2003): Robust tracking problem for continuous time stochastic uncertain systems. - Proc. 42-nd IEEE Conf. Decision and Control, Hawaii, pp. 282-287.
[029] Yoon M.G., Ugrinovskii V.A. and Petersen I.R. (2004): Robust finite horizon minimax filtering for stochastic discrete time uncertain systems. - Syst. Contr. Lett., Vol. 52, No. 2, pp. 99-112. | Zbl 1157.93531
[030] Yoon M.G., Ugrinovskii V.A. and Petersen I.R. (2005): On the worst disturbance of minimax optimal control. - Automatica, Vol. 41, No. 5, pp. 847-855. | Zbl 1093.93030