Guaranteed cost solution for discrete-time uncertain/nonlinear dynamic games

Motivated by an example of fiscal and monetary policy interaction of a national economy, the problem of uncertain/nonlinear two players discrete-time noncooperative games is investigated. Since the models of the systems are uncertain, the notion of Nash equilibrium solution is not suitable, instead, new Nash guaranteeing strategies and Nash guaranteed costs are defined. The system's uncertainties and/or nonlinearities are assumed to be of quadratically bounded type. First, conditions of the Nash guaranteeing strategies are derived for general uncertain nonlinear systems. These results are specified for systems that have linear nominal part and quadratic cost functions. Approximate solutions are obtained by tractable quadratic matrix inequalities. To illustrate the application of the proposed method, two numerical examples are given.