Quasi-equilibrium in LQ differential games with bounded uncertain disturbances: robust and adaptive strategies with pre-identification via sliding mode technique

Abstract

A finite time multi-persons linear-quadratic differential game with bounded disturbances and uncertainties is considered. When players cannot measure these disturbances, it is demonstrated that the standard feedback Nash strategies bear to a quasi Nash-equilibrium depending on an uncertainty upper bound that confirms the robustness property of such standard strategies. In the case of periodic disturbances, another concept, namely adaptive concept, containing three different versions is suggested. They are sliding modes, second order sliding modes and window integration. All of them realize the identification of unknown periodic disturbances during an “identification period” when all participants apply the standard feedback Nash strategies with the, so-called, “shifting signal” generated only by a known external exciting signal. After that period the complete standard strategies with “pre-identification” including the recalculated shifting signal are activated. A numerical example dealing with a two participants game shows that the cost functional for each player achieves better values when the adaptive approach is applied.