A Contraction Analysis of the Convergence of Risk-Sensitive Filters

TL;DR

This paper presents a contraction analysis of risk-sensitive Riccati equations, showing conditions under which the risk-sensitive filter converges and remains positive definite in multivariable systems.

Contribution

It extends scalar case analysis to multivariable systems, providing a priori estimates for contraction and positivity conditions of risk-sensitive filters.

Findings

The N-fold composition of the Riccati map is strictly contractive for N larger than the number of states.

The range of risk-sensitivity parameters for contraction can be estimated beforehand.

Additional conditions ensure the Riccati solution remains positive definite over time.

Abstract

A contraction analysis of risk-sensitive Riccati equations is proposed. When the state-space model is reachable and observable, a block-update implementation of the risk-sensitive filter is used to show that the N-fold composition of the Riccati map is strictly contractive with respect to the Riemannian metric of positive definite matrices, when N is larger than the number of states. The range of values of the risk-sensitivity parameter for which the map remains contractive can be estimated a priori. It is also found that a second condition must be imposed on the risk-sensitivity parameter and on the initial error variance to ensure that the solution of the risk-sensitive Riccati equation remains positive definite at all times. The two conditions obtained can be viewed as extending to the multivariable case an earlier analysis of Whittle for the scalar case.