Derivation of the Maximum a Posterori Estimate for Discrete Time Descriptor Systems

TL;DR

This paper derives the MAP state estimator for discrete-time descriptor systems, showing recursive solutions for systems of index 1 and higher, with no model transformations needed under certain conditions.

Contribution

It provides a derivation of the MAP estimator for general descriptor systems using the Kronecker Canonical Transformation, enabling recursive solutions without model transformations.

Findings

MAP estimate for index 1 systems can be computed recursively without transformations.

For higher index systems, recursive MAP estimation is possible if the system is causal and noise-free.

The derivation applies to both causal and non-causal descriptor systems.

Abstract

In this report a derivation of the MAP state estimator objective function for general (possibly non-square) discrete time causal/non-causal descriptor systems is presented. The derivation made use of the Kronecker Canonical Transformation to extract the prior distribution on the descriptor state vector so that Maximum a Posteriori (MAP) point estimation can be used. The analysis indicates that the MAP estimate for index 1 causal descriptor systems does not require any model transformations and can be found recursively. Furthermore, if the descriptor system is of index 2 or higher and the noise free system is causal, then the MAP estimate can also be found recursively without model transformations provided that model causality is accounted for in designing the stochastic model.