A Compositional Approach to Certifying the Almost Global Asymptotic Stability of Cascade Systems
Abstract
In this work, we give sufficient conditions for the almost global asymptotic stability of a cascade in which the subsystems are only almost globally asymptotically stable. The result is extended to upper triangular systems of arbitrary size. In particular, if the unforced subsystems are almost globally asymptotically stable and their only chain recurrent points are hyperbolic equilibria, then the boundedness of forward trajectories is sufficient for the almost global asymptotic stability of the full upper triangular system. We show that unboundedness of such cascades is prohibited by growth rate conditions on the interconnection term and a Lyapunov function for the unforced outer subsystem, and the required structure for the chain recurrent set is enjoyed by classes of systems common in geometric control e.g. dissipative mechanical systems. Our results stand in contrast to prior works that require either time scale separation, prohibitively strong disturbance robustness properties, or global asymptotic stability in the subsystems.
Keywords
Cite
@article{arxiv.2303.15535,
title = {A Compositional Approach to Certifying the Almost Global Asymptotic Stability of Cascade Systems},
author = {Jake Welde and Matthew D. Kvalheim and Vijay Kumar},
journal= {arXiv preprint arXiv:2303.15535},
year = {2023}
}
Comments
This version restructures the last theorem, which now employs truly independent criteria on the three subsystems, staying true to the compositional goals of the paper. We have also added a minor technical assumption for convenience (that the Riemannian metric is complete, making precompactness equivalent to boundedness), and improved the definitions, explanations, and background references