A compendium of comparison function results

Wed 11 Apr 2018 13:56:38 AEST

The use of comparison functions has become standard in systems and control theory, particularly for the purposes of studying stability properties. The use of these functions typically allows elegant and succinct statements of stability properties such as asymptotic stability and input-to-state stability and its several variants. Furthermore, over the last 20 years several inequalities involving these comparison functions have been developed that simplify their manipulation in the service of proving more significant results. Many of these inequalities have appeared in the body of proofs or in appendices of various papers. Our goal herein is to collect these inequalities in one place.

Orthonormal basis functions for continuous-time systems and Lp convergence

Wed 11 Apr 2018 12:15:33 AEST

In this paper, model sets for linear-time-invariant continuous-time systems that are spanned by fixed pole orthonormal bases are investigated. These bases generalize the well-known Laguerre and two-parameter Kautz cases. It is shown that the obtained model sets are everywhere dense in the Hardy space H 1(Π) under the same condition as previously derived by the authors for the denseness in the (Π is the open right half plane) Hardy spaces H p(Π), 1

iISS and ISS dissipation inequalities: preservation and interconnection by scaling

Sat 24 Mar 2018 07:10:45 AEDT

In analysis and design of nonlinear dynamical systems, (nonlinear) scaling of Lyapunov functions has been a central idea. This paper proposes a set of tools to make use of such scalings and illustrates their benefits in constructing Lyapunov functions for interconnected nonlinear systems. First, the essence of some scaling techniques used extensively in the literature is reformulated in view of preservation of dissipation inequalities of integral input-to-state stability (iISS) and input-to-state stability (ISS). The iISS small-gain theorem is revisited from this viewpoint. Preservation of ISS dissipation inequalities is shown to not always be necessary, while preserving iISS which is weaker than ISS is convenient. By establishing relationships between the Legendre-Fenchel transform and the reformulated scaling techniques, this paper proposes a way to construct less complicated Lyapunov functions for interconnected systems.