Infinite-Dimensional Lur'e Systems: Input-To-State Stability and Convergence Properties

Abstract

We consider forced Lur'e systems in which the linear dynamic component is an infinite-dimensional well-posed system. Numerous physically motivated delay-and partial-differential equations are known to belong to this class of infinitedimensional systems. We investigate input-to-state stability (ISS) and incremental ISS properties: our results are reminiscent of well-known absolute stability criteria such as the complex Aizerman conjecture and the circle criterion. The incremental ISS results are used to derive certain convergence properties, namely the convergentinput convergent-state (CICS) property and asymptotic periodicity of the state and output under periodic forcing. In particular, we provide sufficient conditions for ISS and incremental ISS. The theory is illustrated with examples.

Remarks on input-to-state stability of collocated systems with saturated feedback

“…This is combined in the notion of input-to-state stability (ISS), which has recently been studied for infinite-dimensional systems e.g. in [7,9,19,20] and particularly for semilinear systems in [5,6,23], see also [18] for a survey. The effect of feedback laws acting (approximately) linearly only locally is known in the literature as saturation and first appeared in [24,25] in the context of stabilization of infinite-dimensional linear systems, see also [10].…”

Infinite-dimensional Lur’e systems with almost periodic forcing

“…With regard to stability properties, our main result is Theorem 3.4, which is reminiscent of the complex Aizerman conjecture [19,20] (familiar from finite-dimensional control theory) and constitutes a refinement of [16,Theorem 4.1]. The main novelty here is that we obtain an incremental ISS estimate which is in terms of the Stepanov norm…”

“…We remark that the concept of almost periodicity in the sense of Stepanov generalizes that of Bohr, which, in the following, will be simply referred to as almost periodicity. Adopting the set-up considered in [16], we study the forced Lur'e system shown in Fig. 1, where Σ is a well-posed 1 linear infinite-dimensional system and f is a static nonlinearity.…”

“…Classical absolute stability theory comes in two flavours: in a state-space setting, unforced (v = 0) finite-dimensional systems are considered and the emphasis is on global asymptotic stability, whilst the input-output approach (initiated by Sandberg and Zames in the 1960s) focusses on L 2 -stability and, to a lesser extent, on L ∞ -stability, see [13,44]. A more recent development is the analysis of state-space systems of Lur'e format in an input-to-state stability (ISS) context, thereby, in a sense, merging the two strands of the earlier theory [3,16,21,22,35]. The ISS concept was introduced (for general nonlinear control systems) in [37] and further developed across a huge range of papers, see, for example, the survey articles [12,38].…”

“…So far, the ISS approach to Lur'e systems is very much restricted to finitedimensional systems with [16] being one of the very few exceptions. In fact, in [16] a number of incremental ISS results are derived (the underlying concept inspired by that introduced in [2]) and are then applied to obtain convergence properties including the converging input-converging output property and the asymptotic periodicity of the state and output trajectories under periodic forcing. In this paper, we provide a refinement of the incremental ISS results in [16] and use them to analyse the asymptotic behaviour of the Lur'e system shown in Fig.…”