Estimation for Boundary-Value Descriptor Systems

In this paper we consider models for noncausal processes consisting of discrete-time descriptor dynamics and boundary conditions on the values of the process at the two ends of the interval on which the process is defined. We discuss the general solution and well-posedness of systems of this type and then apply the method of complementary processes to obtain a specification of the optimal smoother in terms of a boundary-value descriptor Hamiltonian system. We then study the implementation of the optimal smoother. Motivated by the Hamiltonian diagonalization results for non-descriptor systems, we show how the descriptor Hamiltonian dynamics can be transformed to two lower-order systems by the use of transformation matrices involving the solution of two generalized Riccati equations. We present several examples illustrating our results and the nature of the smoothing solution and also present equations for covariance analysis of boundary-value descriptor processes including the smoothing error. In addition we discuss several open problems and connections with other related results.