A mixed passivity/small-gain theorem for Sobolev input-output stability

Guiver, Chris; Logemann, Hartmut; Opmeer, Mark R.

Abstract

A stability theorem for the feedback connection of two (possibly infinite-dimensional) time-invariant linear systems is presented. The theorem is formulated in the frequency domain and is in the spirit of combined passivity/small-gain results. It places a mixture of positive realness and small-gain assumptions on the two transfer functions to ensure a certain notion of input-output stability, called Sobolev stability (which includes the classical L2-stability concept as a special case). The result is more general than the classical passivity and small-gain theorems: strong positive realness of either the plant or controller is not required and the small gain condition only needs to hold on a suitable subset of the open right-half plane. We show that the “mixed” stability theorem is applicable in settings where L2-stability of the feedback connection is not possible, such as output regulation and disturbance rejection of certain periodic signals by so-called repetitive control.

Citation

Guiver, C., Logemann, H., & Opmeer, M. R. (2024). A mixed passivity/small-gain theorem for Sobolev input-output stability. SIAM Journal on Control and Optimization, 62(6), 3042-3075. https://doi.org/10.1137/24M1643128

• Journal Article Type: Article
• Acceptance Date: Aug 1, 2024
• Online Publication Date: Nov 12, 2024
• Publication Date: 2024-12
• Deposit Date: Aug 1, 2024
• Publicly Available Date: Nov 12, 2024
• Print ISSN: 0363-0129
• Electronic ISSN: 1095-7138
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 62
• Issue: 6
• Pages: 3042-3075
• DOI: https://doi.org/10.1137/24M1643128
• Keywords: Feedback control, Output regulation, Passivity theorem, Positive realness, Small-gain theorem, Sobolev stability
• Publisher URL: https://www.siam.org/publications/siam-journals/siam-journal-on-control-and-optimization/