Game theory approach to finite-time horizon optimal estimation

In this game, a measurement record is given and the first player looks for the best estimate of a prespecified combination of the system states in the presence of a hostile process noise signal and system initial condition that are applied by his adversary. It turns out that the game possesses a saddle-point solution which leads to an optimal smoothed estimate that is identical to the corresponding L₂-optimal estimate. A similar game in which the estimate is restricted to be causal is formulated and solved. This game provides, for the first time, a saddle-point equilibrium interpretation to finite-time H_∞-optimal filtered estimation. The two games are very closely related. It is shown that in the first game the first player´s strategy, which is the optimal smoothed estimate, is a linear-fractional transformation of the H_∞-optimal filter which applies a nonzero free contracting Q parameter. It, therefore, achieves a unity H _∞-norm bound for the operator that relates the exogeneous signals to the estimation error