New perspectives on output feedback stabilization at an unobservable target

TL;DR

This paper introduces innovative strategies for stabilizing systems at unobservable targets using perturbations, embeddings, and infinite-dimensional observers, addressing the challenge of decreasing observability near the target.

Contribution

It proposes new methods involving perturbations and infinite-dimensional embeddings to achieve output feedback stabilization at unobservable points.

Findings

Perturbations can improve system observability properties.

Embedding systems into bilinear or infinite-dimensional systems aids stabilization.

The approach extends to systems with nonlinear observations and uses representation theory.

Abstract

We address the problem of dynamic output feedback stabilization at an unobservable target point. The challenge lies in according the antagonistic nature of the objective and the properties of the system: the system tends to be less observable as it approaches the target. We illustrate two main ideas: well chosen perturbations of a state feedback law can yield new observability properties of the closed-loop system, and embedding systems into bilinear systems admitting observers with dissipative error systems allows to mitigate the observability issues. We apply them on a case of systems with linear dynamics and nonlinear observation map and make use of an ad hoc finite-dimensional embedding. More generally, we introduce a new strategy based on infinite-dimensional unitary embeddings. To do so, we extend the usual definition of dynamic output feedback stabilization in order to allow infinite-dimensional observers fed by the output. We show how this technique, based on representation theory, may be applied to achieve output feedback stabilization at an unobservable target.