# Statistical Consistency of Set-Membership Estimator for Linear Systems

## TL;DR

This paper proves that the set-membership estimator remains statistically consistent for linear systems even when measurements are nonsequential, addressing a gap in existing proof techniques.

## Contribution

It establishes the statistical consistency of the set-membership estimator for linear systems with nonsequential measurements, a previously unresolved issue.

## Key findings

- Set-membership estimator is statistically consistent with nonsequential data
- Numerical simulations confirm strong consistency
- Addresses limitations of existing proof techniques

## Abstract

Suppose we can choose from a set of linear autonomous systems with bounded process noise, the dynamics of each system are unknown, and we would like to design a stabilizing policy. The underlying question is how to estimate the dynamics of each system given that measurements of each system will be nonsequential. Though seemingly straightforward, existing proof techniques for proving statistical consistency of system identification procedures fail when measurements are nonsequential. Here, we prove that the set-membership estimator is statistically consistent even when measurements are nonsequential. We numerically illustrate its strong consistency.