Cone-bounded nonlinearities and mean-square bounds--Estimation lower bound

This paper is one of a series in which performance bounds are derived for state estimation and regulation problems employing mean-square criteria. The systems considered are partially observed finite-dimensional continuous-time stochastic processes driven by additive white Gaussian noise. The particular systems to which the bounds apply are those which, when modeled by Itô differential equations, contain drift (
\dot{dt}
) coefficients which are, to within a uniformly Lipschitz residual, jointly linear in the system state and externally applied control. The bounds derived in the complete series of papers include both upper and lower bounds for state estimation and for a quadratic regulator problem. The upper bounds pertain to the performance of some simple, almost linear designs reminiscent of the designs which are optimal in the linear case. The lower bounds pertain to the optimum performance attainable within a broad class of admissible control laws. The present paper treats the lower bound for filtering. The estimation performance bounds are independent of the control law, giving a particularly simple form to the control performance bounds which are developed in a companion paper. All the bounds converge with vanishing nonlinearity (vanishing Lipschitz constants) to the known optimum performance for the limiting linear system. Consequently, the bounds are asymptotically tight and the simple designs studied are asymptotically optimal with vanishing nonlinearity.