Small-gain stability theorems for positive Lur'e inclusions

Guiver, Chris; Logemann, Hartmut; R�ffer, Bj�rn

Abstract

Stability results are presented for a class of differential and difference inclusions, so-called positive Lur{\textquoteright}e inclusions which arise, for example, as the feedback interconnection of a linear positive system with a positive set-valued static nonlinearity. We formulate sufficient conditions in terms of weighted one-norms, reminiscent of the small-gain condition, which ensure that the zero equilibrium enjoys various global stability properties, including asymptotic and exponential stability. We also consider input-to-state stability, familiar from nonlinear control theory, in the context of forced positive Lur{\textquoteright}e inclusions. Typical for the study of positive systems, our analysis benefits from comparison arguments and linear Lyapunov functions. The theory is illustrated with examples.

Citation

Guiver, C., Logemann, H., & Rüffer, B. (2019). Small-gain stability theorems for positive Lur'e inclusions. Positivity, 23, 249-289. https://doi.org/10.1007/s11117-018-0605-2

• Journal Article Type: Article
• Acceptance Date: Aug 2, 2018
• Online Publication Date: Aug 23, 2018
• Publication Date: 2019-04
• Deposit Date: Jul 23, 2020
• Publicly Available Date: Jul 24, 2020
• Journal: Positivity
• Print ISSN: 1385-1292
• Electronic ISSN: 1572-9281
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 23
• Pages: 249-289
• DOI: https://doi.org/10.1007/s11117-018-0605-2
• Keywords: Differential inclusion, Exponential stability, Input-to-state stability, Lur’e systems, Population biology, Positive systems
• Public URL: http://researchrepository.napier.ac.uk/Output/2677275