NASH STRATEGY PARAMETER DEPENDENT CONTROL FOR POLYTOPIC SYSTEMS M. Jungers * P. L. D. Peres ** E. B. Castelan *** E. R. De Pieri *** H. Abou-Kandil * * SATIE, ENS Cachan, CNRS, UniverSud 61 av du President Wilson, F-94230 Cachan, France [Jungers,Abou-Kandil]@satie.ens-cachan.fr ** DT/FEEC/UNICAMP CP 6101, 13081-970, Campinas SP Brazil peres@dt.fee.unicamp.br *** DAS/CTC/UFSC 88049 900 Florian´ opolis SC Brazil [eugenio,edson]@das.ufsc.br Abstract: This paper deals with designing Linear-Parameter Varying state feed- back controls for systems including structured uncertainties described by a poly- tope, in the multiobjective framework. These controls are obtained by noncon- vex coupled Semi-Definite Programs for linear-quadratic nonzero-sum differential games on infinite time horizon. An example illustrates the proposed generic algorithm. Copyright c  IFAC 2007 Keywords: Game Theory, Nash Strategy, Coupled Algebraic Riccati Type Equations, Polytopic Uncertain Systems, LMI, SDP, Robustness. 1. INTRODUCTION Game Theory (Ba¸sar and Olsder, 1995) is an adapted field to design controllers for exactly known systems governed by several inputs (or players) and where each player aims to minimize his/her own cost function. In particular, when the different players have the same hierarchical position (interchangeable players), Nash strate- gies offer a nice framework to design cautious controls: no player can improve his/her payoff or criterion by deviating unilaterally from his/her Nash strategy once the equilibrium is attained. Unfortunately, the tools which enable to design such controllers use variational calculus and there- fore are not well adapted to uncertain systems. 1 Partially supported by CAPES/COFECUB n ◦ 489/05 Program, ARCUS Program (ARCUS 2005 Project Ile-de- France Brazil), FAPESP and CNPq, Brazil A way to treat uncertainties is to interpret pertur- bation as an exogenous input, that is a fictitious player (Chen et al., 1997; van den Broek et al., 2003). An extended definition of Nash equilibrium is proposed in (Tanaka and Yokoyama, 1991). For a bounded energy disturbance, the cost value deviates from the nominal one by a distance which increases in function of the disturbance energy (Jimenez and Poznyak, 2006). In addition, if the disturbance is periodic and of known period, using a learning lapse of time, it is possible to estimate the effect of the disturbance and a pure Nash equilibrium is obtained. However, the associated equations are not easy to solve and the character- istics of the disturbance should be known. In (Jungers et al., 2006a), a reformulation of Cou- pled Algebraic Riccati type Equations (CARE) associated with Nash equilibrium, in terms of non- convex coupling problem between different Semi-