# Robust $H_\infty$ Filtering for Nonlinear Discrete-time Stochastic Systems

## TL;DR

This paper develops a new approach for $H_$ filtering in nonlinear discrete-time stochastic systems using a stochastic bounded real lemma and Hamilton-Jacobi inequalities, providing verifiable conditions for filter design.

## Contribution

It introduces a novel stochastic bounded real lemma and a Hamilton-Jacobi inequality for $H_$ filtering of nonlinear stochastic systems, advancing existing theoretical methods.

## Key findings

- Derived a nonlinear stochastic bounded real lemma.
- Established a sufficient condition for $H_$ filtering via HJI.
- Validated results with practical engineering examples.

## Abstract

This paper mainly discusses the $H_{\infty}$ filtering of general nonlinear discrete time-varying stochastic systems. A nonlinear discrete-time stochastic bounded real lemma (SBRL) is firstly obtained by means of the smoothness of the conditional mathematical expectation, and then, based on the given SBRL and a stochastic LaSalle-type theorem, a sufficient condition for the existence of the $H_\infty$ filtering of general nonlinear discrete time-varying stochastic systems is presented via a new introduced Hamilton-Jacobi inequality (HJI), which is easily verified. When the worst-case disturbance $\{v^*_k\}_{k\in {\mathcal N}}$ is considered, the suboptimal $H_2/H_\infty$ filtering is studied. Two examples including a practical engineering example show the effectiveness of our main results.