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

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.