Optimal stochastic estimation with exponential cost criteria

The expected value of the exponential of a weighted quadratic sum of the squares of the estimation error is minimized with respect to the state estimate subject to a Gauss-Markov system. The state estimates are assumed to be a function of the measurement history up to the stage time of the state vector. The estimator which optimizes this exponential cost criterion is linear but is not a conditional mean estimator such as the Kalman filter. This shows that the implications of Sherman´s theorem are restricted to functions of the estimates which have access to the same measurement history such as in smoothing problems. In the solution process the expectation operation is replaced by an extremization operation allowing the formulation of a deterministic discrete time game. The saddle point estimator resulting from this game is the same as that obtained from the solution of an associated disturbance attenuation problem. This optimal stochastic estimator which generalizes the Kalman filter can feature the estimation error of certain states over others by the choice of the quadratic weighting matrices in the cost criterion. Correlation between the measurement and process noises is included