Game Theory Approach to H_∞-Optimal Discrete-Time Fixed-Point and Fixed-Lag Smoothing

Optimal H_∞-fixed-point and fixed-las 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_∞-optimal fixed-point smoothing. Differently from the usual case in H_∞-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 H2 solution. Unlike the H₂ case, the recursive application of the H_∞fixed-point smoothing algorithm does not lead to fixed-lag smoothing in the H_∞-norm sense. The Hoo fixed-lag smoothing filter is derived by augmenting the state vector of the system with additional delayed states.