Modal consensus observers for distributed parameter systems

• Published in: International journal of robust and nonlinear control 2022-01, Vol.32 (1), p.472-497
• Main Author: Demetriou, Michael A.
• Format: Article
• Language: English
• Subjects: Decomposition; Distributed parameter systems; modal consensus; modal decompositions; Observers; Parabolic differential equations; Partial differential equations; Synchronism; synchronization
• ISSN: 1049-8923; 1099-1239
• DOI: 10.1002/rnc.5840

This article proposes a new type of a consensus protocol for the synchronization of distributed observers in systems governed by parabolic partial differential equations. Addressing the goal of sharing useful information among distributed observers, it delves into the details governing the modal decompositions of distributed parameter systems. Assuming that two different groups of sensors are available to provide process information to the two distributed observers, the proposed modal consensus design ensures that only useful information is transmitted to the requisite modal components of each of the observers. Without any consensus protocol, the observers capture different frequency content of the spatial process in differing degrees, as it relates to the concept of modal observability. Their modal components exhibit different learning behavior toward the process state. In the extreme case, it turns out that certain modal components of the distributed observers occasionally behave as naïve observers. To ensure that, both collectively and modal componentwise, the observers agree both with the process state and with each other, a modal component consensus protocol is proposed. Such a consensus protocol is mono‐directional and provides only useful information necessary to the appropriate modal component of the distributed filters that behaves as a naïve modal observer. This protocol, when ed and applied to different state decompositions can be viewed as mono‐directional projections of information transmitted and received by the participating distributed observers. Detailed numerical studies of advection PDE in one and two spatial dimensions are included to elucidate the details of the proposed modal consensus observers.