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

Abstract

We present sufficient conditions for semi-global incremental input-to-state stability of a class of forced discrete-time Lur'e systems. The results derived are reminiscent of well-known absolute stability criteria such as the small gain theorem and the circle criterion. We derive a natural sufficient condition which guarantees that asymptotically (almost) periodic inputs generate asymptotically (almost) periodic state trajectories. As a corollary, we obtain sufficient conditions for the converging-input converging-state property to hold.

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 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. A Lyapunov approach is used in [13,15,41], whilst the analyses in [16,32] are based on input-output methods.…”

“…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.…”

“…For background information regarding incremental ISS for general nonlinear systems, we refer the reader to [2]. Recently, in the context of discrete-time Lur'e systems, it has been shown in [15] that a certain "nonlinear" incremental small-gain condition guarantees semi-global incremental ISS and this was exploited to show that the response to almost periodic inputs is asymptotically almost periodic. The concept of semi-global incremental ISS used in [15] means that, for arbitrary bounded sets of initial conditions and input functions there exist comparison functions such that an incremental ISS estimate holds.…”

“…In this paper, we consider finite-dimensional forced continuous-time Lur'e differential equations and the stability property of semi-global incremental iISS which is considerably weaker than the notion of semi-global incremental ISS studied in [15]. The main result of Section 3, Theorem 3.3, asserts that the condition…”