A Linear-Quadratic Game Approach to Estimation and Smoothing

An estimator and smoother for a linear time varying system, over a finite time interval, are developed from a linear quadratic (LQ) game approach. The exogenous inputs composed of the measurement and process noise, and the initial state, are assumed to be finite energy signals whose statistics are unknown. The measure of performance is in the form of a disturbance attenuation function and the optimal estimator (smoother) bounds the attenuation function from above. The disturbance attenuation function is converted to a performance measure for a zero-sum LQ game and the exogenous inputs and the estimator are viewed as players in the game; the exogenous inputs attempt to worsen the estimate while the estimator tries to provide the most accurate estimate. The optimal estimator (smoother), restricted to a class of functions dependent on the measurement alone, is found to be unbiased and linear in structure. With a few mild assumptions, the results are extended to a linear time-invariant system on an infinite horizon, and the optimal estimator obtained is shown to satisfy an upper bound on the H ∞ norm.