Existence and Completeness of Bounded Disturbance Observers: A Set-Membership Viewpoint

TL;DR

This paper presents a set-membership approach to analyze and synthesize bounded disturbance observers for linear discrete-time systems, establishing existence conditions, developing a filter-based observer, and demonstrating worst-case optimality.

Contribution

It introduces a necessary and sufficient condition for the existence of bounded DOs and develops a set-membership filter-based DO with proven completeness and optimality.

Findings

Existence condition for bounded DOs verified

Set-membership filter-based DO developed and proved complete

Proposed DO achieves worst-case optimality

Abstract

This paper investigates the boundedness of the Disturbance Observer (DO) for linear discrete-time systems. In contrast to previous studies that focus on analyzing and/or designing observer gains, our analysis and synthesis approach is based on a set-membership viewpoint. From this viewpoint, a necessary and sufficient existence condition of bounded DOs is first established, which can be easily verified. Furthermore, a set-membership filter-based DO is developed, and its completeness is proved; thus, our proposed DO is bounded if and only if bounded DOs exist. We also prove that the proposed DO has the capability to achieve the worst-case optimality, which can provide a benchmark for the design of DOs. Finally, numerical simulations are performed to corroborate the effectiveness of the theoretical results.