On the robustness analysis of triangular nonlinear systems: iISS and practical stability

International audienceThis note synthesizes recent results obtained by the authors on the stability and robustness analysis of cascaded systems. It focuses on two properties of interest when dealing with perturbed systems, namely integral input-to-state stability and practical stability. We present sufficient conditions for which each of these notions is preserved under cascade interconnection. The obtained conditions are of a structural nature, which makes their use particularly easy in practice

Lyapunov stabilization of discrete-time feedforward dynamics

The paper discusses stabilization of nonlinear discrete-time dynamics in feedforward form. First it is shown how to define a Lyapunov function for the uncontrolled dynamics via the construction of a suitable cross-term. Then, stabilization is achieved in terms of u-average passivity. Several constructive cases are analyzed

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

International audienceThis 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 non- vanishing (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

Revisiting the iISS small-gain theorem through transient plus ISS small-gain regulation

International audienceRecently, 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