Nonlinear approximate game-theoretic estimation

Over a finite-time interval, for a class of linear time-varying dynamical systems with a small nonlinearity, an approximate nonlinear optimal estimation scheme is derived based on a deterministic game-theoretic criterion. Using the calculus of variation approach, this game-theoretic criterion is first maximized by the process disturbance and initial state vectors. The resulting optimality condition is expanded with respect to a small parameter ε and the expression of the worst case state and the Lagrange multiplier vectors are determined. Subsequently, the approximate game-theoretic estimator is derived by minimizing each term in the series of cost criterion over the corresponding element of state estimate vector expansion. The estimator Riccati differential equations (RDE) necessary for the first and higher order correction terms are the same as in the zeroth-order case. The first-order and higher-order correction terms are computed on-line based on nonlinear functions evaluated along the minimax trajectory of the zeroth-order state estimate which has to be updated, through a backward integration, as each new measurement becomes available. The infinite-order approximate minimax estimator is shown to be a priori disturbance attenuating. The Nth-order approximate minimax estimator achieves disturbance attenuation with an incremental increase in the bound proportional to C^N+1