Revisiting the IISS Small‐Gain Theorem through Transient Plus ISS Small‐Gain Regulation

Abstract

Recently, the small-gain theorem for input-to-state stable (ISS) systems has been extended to the class of integral input-to-state stable (iISS) systems. Feedback connections of two iISS systems are robustly stable with respect to disturbance if an extended small-gain condition is satisfied. It has been proved that at least one of the two iISS subsystems needs to be ISS for guaranteeing globally asymptotic stability and iISS of the overall system. Making use of this necessary condition for the stability, this paper gives a new interpretation to the iISS small gain theorem as transient plus ISS small-gain regulation. The observation provides useful information for designing and analyzing nonlinear control systems based on the iISS small-gain theorem.

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

“…Consequently, recalling that it is 0-GAS in view of (11) and invoking [4,Theorem 1], it is also iISS. The conclusion follows by noticing that ISS with respect to small inputs trivially results from (11).…”

“…The feedback interconnection of not necessarily ISS systems has already been addressed in the literature, and Lyapunovbased conditions for iISS have been provided in the literature. See for instance [1] for the case of a feedback interconnection without exogenous input and [12], [9], [11] for the case of systems with inputs. See also [13] for an extension to more than two interconnected subsystems.…”

“…≥0 is the nondecreasing continuous function defined as ν(•) :=ην 1 (•) + ν 2 (•) + λ 1 (R x )λ 2 (•).Since ν is not necessarily a class K function, zero-output dissipativity does not follow yet. Nonetheless,(11) guarantees that |u| ≤ R u ⇒ ∂W ∂x f (x, u) ≤ γ(|u|) .Therefore, definingγ(s) :=    γ(s) if s ≤ R u /2 as + b if s ∈ (R u /2; R u ] ν(s) + 2γ(s) if s > R u ,where a and b are conveniently chosen to ensure continuity,γ is a class K function and it holds that ∂W ∂x f (x, u) ≤γ(|u|)…”

Utility of iISS in Composing Lyapunov Functions for Interconnections

“…For example, we can take the monotone systems approach as demonstrated in Angeli and Astolfi [2007] in the absence of disturbances. It is also possible to exploit the ISS smallgain condition for non-ISS subsystems by assuming that the behavior of the subsystems is ISS after a transient period and that a trajectory estimate of the interconnection during that period is available in a desired manner as discussed in Karafyllis and Jiang [2012] and Ito et al [2013a].…”

Adaptive Integrated Guidance and Control Based on Backstepping and Input‐to‐State Stability

“…This paper proposes an adaptive IGC design approach for homing missiles against maneuvering targets based on a backstepping scheme and input‐to‐state stability (ISS) . In order to deal with a nonlinear term of the IGC model, we introduce an adaptive control method into the backstepping scheme to design an IGC law such that the LOS rate is ISS with respect to target maneuvers and model uncertainties, and the stability of the overall system also can be guaranteed.…”