Mixed H ₂/H _∞ filtering by the theory of nash games

Abstract

The aim of this paper is to study an H ₂/H _∞ terminal state estimation problem using the classical theory of Nash equilibria. The H ₂/H _∞ nature of the problem comes from the fact that we seek an estimator which satisfies two Nash inequalities. The first reflects an H _∞ filtering requirement in the sense alluded to in [4], while the second inequality demands that the estimator be optimal in the sense of minimising the variance of the terminal state estimation error. The problem solution exploits a duality with the H ₂/H _∞ control problem studied in [2, 3]. By exploiting duality in this way, one may quickly extablish that an estimator exists which staisfies the two Nash inequalities if and only if a certain pair of cross coupled Riccati equations has a solution on some optimisation interval. We conclude the paper by showing that the Kalman filtering, H _∞ filtering and H ₂/H _∞ filtering problems may all be captured within a unifying Nash game theoretic framework.