Assessment and Future Directions of Nonlinear Model Predictive ...

Optimal Online Control of Dynamical SystemsUnder UncertaintyRafail Gabasov 1 , Faina M. Kirillova 2 , and Natalia M. Dmitruk 21 Belarussian State University, 4 Nezavisimosti av., Minsk 220050, Belarus2 Institute of Mathematics, 11 Surganov str., Minsk 220072, Belaruskirill@nsys.minsk.by, dmitruk@im.bas-net.bySummary. A problem of synthesis of optimal measurement feedbacks for dynamicalsystems under uncertainty is under consideration. An online control scheme providinga guaranteed result under the worst-case conditions is described.1 IntroductionAn up-to-date methodology for control of constrained systems is model predictivecontrol (MPC) [1]. MPC feedback strategies are constructed as a result of openloopoptimization of a nominal model for a given state. However, closed-loopperformance can be poor due to uncertainties present in the system such asdisturbances or modeling inaccuracies, and moreover, the exact measurementof the state can be unavailable. For these reasons, in recent research attentionhas been given to output feedback [2] and robust MPC techniques design [3].In the latter two approaches can be distinguished. One method is to optimizea nominal system subject to tightened constraints [4], while a game approachleads to min-max formulations of MPC [5, 6].This paper deals with the optimal synthesis problem for dynamical systemsunder uncertainty of set-membership type which output is available with a limitedaccuracy. In this case the feedback constructed is rather a measurementfeedback [7] than an output feedback [2], and a problem of set-membership estimationarises. According to classical formulation, feedbacks have to be constructedin advance for all possible future positions of the system. Due to enormouscomputational burden such closed-loop strategies are rarely calculatedeven for linear determined problems. In this paper we implement receding horizoncontrol principle and construct an optimal feedback as a result of repeatedonline optimization under the worst-case uncertainty realization. For linear systemstwo types of problems are solved in the course of the control process: a)optimal observation problem and b) optimal control problem for a determinedcontrol system. Numerical methods for problems a) and b) elaborated by theauthors are briefly discussed.R. Findeisen et al. (Eds.): Assessment and Future Directions, LNCIS 358, pp. 327–334, 2007.springerlink.com c○ Springer-Verlag Berlin Heidelberg 2007