Strong iISS: Combination of iISS and ISS with respect to small inputs

Abstract

This paper studies the notion of Strong iISS, which imposes both integral input-to-state stability (iISS) and input-to-state stability (ISS) with respect to small inputs. This combination characterizes the robustness property, exhibited by many practical systems, that the state remains bounded as long as the magnitude of exogenous inputs is reasonably small but may diverge for stronger disturbances. We provide three Lyapunov-type sufficient conditions for Strong iISS. One is based on iISS Lyapunov functions admitting a radially nonvanishing (class K) dissipation rate. However we show that it is not a necessary condition for Strong iISS. Two less conservative conditions are then provided, which are used to demonstrate that asymptotically stable bilinear systems are Strongly iISS. Finally, we discuss cascade and feedback interconnections of Strong iISS systems.

A region-dependent gain condition for asymptotic stability

“…From Remark 2.4 of [24] Eq. (14) is equivalent to(6) 4. Note that, in contrast to[5,11,12], no information is given about the behaviour of the function s → γ • δ(s), in the interval (0, ∞).…”

Robustness under saturated feedback: Strong iISS for a class of nonlinear systems

“…, they can run unbounded in the presence of arbitrary small constant and even vanishing inputs [4]. Generically, we may expect a bounded state property at most for disturbances whose amplitude is below a given threshold.…”

“…In the case when R = +∞, we recover the classical definition of ISS [16], [19]. However, given a finite R, no guarantee on the behavior of the system can be given when the disturbance magnitude overpasses the threshold R. The very solution of the system may fail to exist if d ≥ R. Hence, a good candidate to evaluate the robustness to exogenous disturbances of systems with saturated feedback seems to be the Strong iISS, recently introduced in [4].…”

“…In this note, we provide sufficient conditions under which a more interesting property, namely Strong iISS, can be achieved by saturated feedback. This property, introduced in [4], not only guarantees ISS with respect to small inputs but also iISS. In particular, it ensures a bounded response to any disturbance whose amplitude is below a given threshold, but also the existence of solutions at all times even for disturbances above that threshold as well as the convergence of the state to the origin in response to any vanishing disturbance.…”

Interconnecting a system having a single input-to-state gain with a system having a region-dependent input-to-state gain

“…The use of a unifying approach is well known in the literature, see [1] for the combination of control Lyapunov functions and [4] for a stability concept uniting ISS and the integral variant of ISS (namely, iISS [17]) properties. This paper is organized as follows.…”