Infinite-dimensional Lur’e systems with almost periodic forcing

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 infinite-dimensional systems. We present refinements of recent incremental input-to-state stability results (Guiver in SIAM J Control Optim 57:334–365, 2019) and use them to derive convergence results for trajectories generated by Stepanov almost periodic inputs. In particular, we show that the incremental stability conditions guarantee that for every Stepanov almost periodic input there exists a unique pair of state and output signals which are almost periodic and Stepanov almost periodic, respectively. The almost periods of the state and output signals are shown to be closely related to the almost periods of the input, and a natural module containment result is established. All state and output signals generated by the same Stepanov almost periodic input approach the almost periodic state and the Stepanov almost periodic output in a suitable sense, respectively, as time goes to infinity. The sufficient conditions guaranteeing incremental input-to-state stability and the existence of almost periodic state and Stepanov almost periodic output signals are reminiscent of the conditions featuring in well-known absolute stability criteria such as the complex Aizerman conjecture and the circle criterion.

Incremental input-to-state stability for Lur'e systems and asymptotic behaviour in the presence of Stepanov almost periodic forcing

“…Before presenting the final result of this section, we pause to compare Theorem 4.3 to related results in the literature. The most relevant results in this context are [13, Theorem 3.2.9], [15,Theorem 4.3], [16,Theorem 4.5], [32,Theorem 2] and [41,Theorem 1]. The papers [32,41] are restricted to scalar nonlinearities, that is, m = p = 1) and in [15,32,41] the forcing functions are assumed to be almost periodic in the sense of Bohr.…”

“…The papers [32,41] are restricted to scalar nonlinearities, that is, m = p = 1) and in [15,32,41] the forcing functions are assumed to be almost periodic in the sense of Bohr. A Lyapunov approach is used in [13,15,41], whilst the analyses in [16,32] are based on input-output methods. The paper [16] considers a large class of infinite-dimensional continuous-time systems with the underlying linear system being well-posed in the sense of [36,38], whilst [15] considers finite-dimensional discrete-time systems.…”

“…is sufficient for semi-global incremental iISS (with linear incremental iISS-gain), where, in (1.1), K is a stabilizing output feedback matrix for Σ, Σ K denotes the corresponding linear feedback system and g(Σ K ) denotes the L 2 -gain of Σ K (that is, the H ∞ -norm of the transfer function of Σ K ). We emphasize that this hypothesis is considerably less restrictive than those given in [14,15,16,19]; in particular, the quotient of the LHS of (1.1) and y − y 2 , y 1 = y 2 , is not required to be bounded away from 1, and therefore, (1.1) could be described as a "weak small-gain" condition.…”

“…Lyapunov approaches have been used to deduce global asymptotic stability of unforced (that is, v = 0) Lur'e systems (see, for example, [21,23,26,39]), and input-output methods, pioneered by Sandberg and Zames in the 1960s, have been used to infer L 2 and L ∞ stability (see, for example, [12,39]). More recently, forced Lur'e systems have been analysed in the context of input-to-state stability (ISS) theory, with attention focussed on the extent to which results from classical absolute stability theory can be generalised to ensure certain ISS properties [4,14,16,17,19,22,23,33]. Originating in the paper [34], ISS and its variants, including integral input-to-state stability (iISS), are properties of general controlled nonlinear systems and, roughly, ensure a natural boundedness property of the state, in terms of initial conditions and inputs, see also the survey papers [10,35].…”

“…For all practical purposes, semi-global stability notions seem to be at least as interesting as global stability concepts. We mention that, under a classical incremental L 2 -type small-gain condition, a stronger incremental ISS property holds, namely (global) exponential incremental ISS, even in the infinite-dimensional case, see [14,16,19].…”