A Unified Filter for Simultaneous Input and State Estimation of Linear Discrete-time Stochastic Systems

TL;DR

This paper introduces a unified optimal filter for linear discrete-time stochastic systems that estimates states and unknown inputs simultaneously, ensuring stability and unbiasedness without assumptions on direct feedthrough.

Contribution

It develops a novel unified filter that generalizes existing methods like the Kalman filter, with proven stability and optimality for joint state and input estimation.

Findings

The filter guarantees exponential stability and unbiasedness.

It encompasses existing filters as special cases.

Demonstrated effectiveness through illustrative examples.

Abstract

In this paper, we present a unified optimal and exponentially stable filter for linear discrete-time stochastic systems that simultaneously estimates the states and unknown inputs in an unbiased minimum-variance sense, without making any assumptions on the direct feedthrough matrix. We also derive input and state observability/detectability conditions, and analyze their connection to the convergence and stability of the estimator. We discuss two variations of the filter and their optimality and stability properties, and show that filters in the literature, including the Kalman filter, are special cases of the filter derived in this paper. Finally, illustrative examples are given to demonstrate the performance of the unified unbiased minimum-variance filter.