Finite-time and fixed-time input-to-state stability: Explicit and implicit approaches

Abstract

The present article gathers the analysis of non-asymptotic convergence rates (finite-time and fixed-time) with the property of input-to-state stability. Theoretical tools to determine this joint property are presented for the case where an explicit ISS Lyapunov function is known, and when it remains in implicit form (e.g. as a solution of an algebraic equation). For the case of finite-time input-to-state stability, necessary and sufficient conditions are given whereas for the fixed-time case only a sufficient condition is obtained. Academic examples and numerical simulations support the obtained results.

On Converse Lyapunov Theorem for Fixed-Time Input-to-State Stability

“…The converse Lyapunov theorem for finite-time ISS has been obtained in [12] under additional restrictions:…”

“…The equivalent Lyapunov characterizations of the finite-time stability are well known [1], while for the fixed-time stability they are quite new [11]. The ISS systems having finite-time convergence rates in the disturbance-free setting are introduced in [6] with sufficient conditions in terms of existence of Lyapunov functions, and its necessary counterpart is proposed in [12]. The sufficient characterization of fixed-time ISS is also given in [12] (see also a survey [5]).…”

“…The ISS systems having finite-time convergence rates in the disturbance-free setting are introduced in [6] with sufficient conditions in terms of existence of Lyapunov functions, and its necessary counterpart is proposed in [12]. The sufficient characterization of fixed-time ISS is also given in [12] (see also a survey [5]). This work aims on filling the gap by proposing the necessary Lyapunov conditions for (nearly) fixed-time ISS property.…”

Finite-Time Stability Tools for Control and Estimation

“…In order to define the property of accelerated ISS or iISS, conventional class-KL functions are no longer suitable (Hong et al, 2010), therefore, generalizations of these functions are usually utilized (Hong et al, 2010;Lopez-Ramirez et al, 2020), as the following one: Definition 2.6 (Lopez-Ramirez et al, 2020). A continuous function…”

“…Proof. The proof for the FT-ISS case was given in (Lopez-Ramirez et al, 2020) (we need just its mild modification related with the definition of α or ζ). The FT-iISS can be proven similarly.…”