Bounded-Energy-Input Convergent-State Property of Dissipative Nonlinear Systems: An iISS Approach

Abstract

For a class of dissipative nonlinear systems, it is shown that an iISS gain can be computed directly from the corresponding supply function. The result is used to prove the convergence to zero of the state whenever the input signal has bounded energy, where the energy functional is determined by the supply function.

On the robustness of hysteretic second-order systems with PID: iISS approach

“…This together with the converse Lyapunov function for A-GAS system, we can obtain the A-iISS Lyapunov function similar to the proof of [7,Theorem 3.1].…”

“…It is shown in [2] that it is integral input-to-state stable (iISS) if the system is (a) 0-GAS and (b) dissipative with supply function σ. In [7] it is shown that for a class of dissipative systems, the iISS gain is equal to the supply function σ. Here, we explore the robustness of system (12) by applying the modified concept of iISS, where instead of discussing the iISS with respect to the origin, we are interested in the iISS property with respect to a set A (A-iISS).…”

“…Proof: The proof of Theorem 3.5 follows the same arguments as in the proof of Theorem 3.1 in [7]. Firstly, the arguments in [17] which use the converse Lyapunov theorem for GAS system are replaced by the similar arguments using the converse Lyapunov theorem for A-GAS system as discussed in [18].…”

“…By replacing the compact set K ⊂ R n in [7, Assumption (A)] and in the rest of the proof in [7] by B A l , we can obtain the same lemmas as given in [7, Lemma 3.2, Lemma 3.3 and Lemma 3.4]. This together with the converse Lyapunov function for A-GAS system, we can obtain the A-iISS Lyapunov function similar to the proof of [7,Theorem 3.1].…”

On the new notion of input-to-state safety

“…For instance, a well-known nonlinear small-gain theorem in [9] is based on the use of β and γ. The study of convergence input convergence state property as in [7] is based on the use of ISS Lyapunov function. However, as mentioned in the Introduction, existing results on robustness have focused on the systems' stability and there is not many attention on the robustness analysis on systems' safety.…”