Bounding the l_2 sensitivity for positive linear observers

TL;DR

This paper derives bounds on the $l_2$ sensitivity of positive linear observers to facilitate differentially private state estimation with minimal added noise.

Contribution

It introduces a method to bound the $l_2$ sensitivity of Luenberger observers for positive systems, aiding privacy-preserving control design.

Findings

Derived a bound for the $l_2$ sensitivity of positive Luenberger observers.

Proposed an approach to minimize the sensitivity bound.

Provided several bounds relevant to differentially private observer design.

Abstract

We consider the design of differentially private Luenberger observers for positive linear systems. In particular, we derive a bound for the $l_2$ sensitivity of Luenberger observers, which is used to quantify the noise required to achieve relaxed differential privacy via the Gaussian mechanism. An approach to minimise this bound for positive observers is described and several bounds relevant to this problem are derived.