Game theory approach to H/sub ∞/-optimal discrete-time fixed-point and fixed-lag smoothing

Abstract

Optimal H/sub /spl infin//-fixed-point and fixed-lag discrete-time smoothing estimators are developed by applying a game theory approach. A deterministic discrete-time game is defined where the estimator plays against nature. Nature determines the system initial condition, the driving input, and the measurement noise, whereas the estimator tries to find an estimate that brings a prescribed cost function that is based on the error of the estimation at a fixed time instant, to a saddle-point equilibrium. The latter estimate yields the H/sub /spl infin//-optimal fixed-point smoothing. Differently from the usual case in H/sub /spl infin//-optimal estimation and control, the critical value of the scalar design parameter of the smoothing game is obtained in closed form, explicitly in the terms of the corresponding H/sub 2/ solution. Unlike the H/sub 2/ case, the recursive application of the H/sub /spl infin// fixed-point smoothing algorithm does not lead to fixed-lag smoothing in the H/sub /spl infin//-norm sense. The H/sub /spl infin// fixed-lag smoothing filter is derived by augmenting the state vector of the system with additional delayed states. >