Linear minimum-variance estimation and control in systems with state-dependent noise

A recursive, minimum-variance linear filter and controller is derived for systems in which white state-dependent noise appears in the system dynamics and measurements. The filter without control is a generalization of the Kalman filter and possesses many of its desirable properties. First, the discrete form of the filter is derived. By taking a formal limit, a continuous filter with convergence in distribution to an Ito representation is obtained. The concept of a perfect controller is given, showing the formal duality of the filter and controller with the stochastic controller derived by Wonham. Finally, some of the properties of the filter-controller system are illustrated through the use of a scalar example. It is shown that a filter-controller designed by neglecting the state-dependent noise can destabilize a dynamically stable system.