Reduced complexity dynamic programming solution for Kalman filtering of linear discrete time descriptor systems

Abstract

We consider linear discrete time descriptor systems that are described by state and measurement equations that have both stochastic and purely deterministic components. We suggest an estimation algorithm that operates by decomposing the system into stochastic and deterministic parts, and processing each part separately. It solves the deterministic subsystem using the pseudo-inverse according to the Moore-Penrose definition [1], and then minimizes the Kalman filter objective function by exploiting the orthogonal subspace defined by the deterministic subsystem. A simulation example is given for estimating tray composition for a distillation column by linearization over a trajectory of a non-linear differential algebraic model. Compared to the method of R. Nikoukhah et.al [2], the reduction in time produced by our method for this example is 87%. The reason is that our algorithm requires only 1-block matrix inversion that does not involve any singular blocks, whereas the algorithm in [2] requires 3-block matrix inversions containing possibly singular matrix blocks arising from singular covariance matrices.