Kalman Filtering and Riccati Equations for Descriptor Systems

In this paper we consider a general formulation of a discrete-time filtering problem for descriptor systems. It is shown that the nature of descriptor systems leads directly to the need to examine singular estimation problems. Using a dual approach to estimation we derive a so-called 3-block form for the optimal filter and a corresponding 3-block Riccati equation for a general class of time-varying descriptor models which need not represent a well-posed system in that the dynamics may be either over- or under-constrained. Specializing to the time-invariant case we examine the asymptotic properties of the 3-block filter, and in particular analyze in detail the resulting 3-block algebraic Riccati equation, generalizing significantly the results in 23, 28, 33. Finally, the noncausal nature of discrete-time descriptor dynamics implies that future dynamics may provide some information about the present state. We present a modified form for the descriptor Kalman filter that takes this information into account.