The converging-input converging-state property for Lur’e systems

Abstract

Using methods from classical absolute stability theory, combined with recent results on input-to-state stability (ISS) of Lur'e systems, we derive necessary and sufficient conditions for a class of Lur'e systems to have the converging-input converging-state (CICS) property. In particular, we provide sufficient conditions for CICS which are reminiscent of the complex Aizerman conjecture and the circle criterion and connections are also made with small gain ISS theorems. The penultimate section of the paper is devoted to non-negative Lur'e systems which arise naturally in, for example, ecological and biochemical applications: the main result in this context is a sufficient criterion for a so-called "quasi CICS" property for Lur'e systems which, when uncontrolled, admit two equilibria. The theory is illustrated with numerous examples. Keywords Absolute stability · Circle criterion · Converging-input converging-state property · Input-to-state stability · Lur'e systems · Non-negative systems Mathematics Subject Classification

Stability and convergence properties of forced infinite-dimensional discrete-time Lur'e systems

“…As is well-known, global asymptotic or exponential stability of an equilibrium of an unforced nonlinear system does, in general, not guarantee any stability or boundedness properties of the system in the presence of forcing. A recent line of enquiry [2,3,4,19,20,42,43,44] has been investigating to what extent classical absolute stability criteria can be modified to ensure ISS and state convergence properties of forced Lur'e systems. Indeed, a key finding has been that existing absolute stability criteria, under slightly stronger assumptions, are sufficient for ISS in many cases.…”

“…The papers [2,19,20,42,43,44] derive, in finite-dimensional settings, sufficient conditions for ISS which are reminiscent of the complexified Aizerman conjecture and the circle criterion. Furthermore, ISS properties underpin the paper [4], which considers the converginginput converging-state (CICS) property for finite-dimensional, continuous-time Lur'e systems.…”

A circle criterion for strong integral input-to-state stability

“…Recently, absolute stability ideas have been merged with inputto-state stability (ISS) theory to obtain ISS criteria which resemble classical absolute stability results, see [2,11,19,20,29,30,31]. The ISS criteria obtained in [29] have been used in [5] to prove converging-input converging-state properties for Lur'e systems.…”

Semi-global incremental input-to-state stability of discrete-time Lur’e systems

“…Recently, in [13,14], sufficient conditions have been determined which guarantee that certain infinitedimensional Lur'e systems are exponentially incrementally ISS. Moreover, in [13,14], exponential incremental ISS is used to infer convergence properties such as the converging-input converging-state (CICS) property (see, for example, [5]) and the asymptotic periodicity of the response to asymptotically periodic inputs.…”