Robust finite-time input-to-state stability via impulsive hybrid control for uncertain dynamical systems with disturbances

Abstract

This paper studies the robust finite-time input-to-state stability (R-FT-ISS) including robust finite-time stability (R-FTS) via impulsive hybrid control (IHC) for uncertain dynamical systems (UDS) with disturbances. The notions of robust $\mathrm{GKL}$-stability, R-FT-ISS, and R-FTS are proposed. Time-based IHC (T-IHC) and state-based IHC (S-IHC) are proposed, respectively. And based on the Hamilton–Jacobi inequalities of Lyapunov-like functions, less restrictive R-FT-ISS and R-FTS criteria are established for UDS under IHC (including T-IHC and S-IHC). And the event-triggered S-IHC schemes for R-FT-ISS and R-FTS are designed. Correspondingly, the estimates of settling time for R-FT-ISS and R-FTS are also obtained, respectively. Theoretical results and numerical simulations show that both T-IHC and S-IHC can achieve not only R-FT-ISS but also R-FTS for unstable systems with structural disturbances and external disturbances. Therefore, IHC (including T-IHC and S-IHC) can eliminate the impact on stability from disturbances and thus both T-IHC and S-IHC are anti-disturbance and robust, which improves the stabilization only to ISS (not to asymptotic stability) in the presence of disturbance in the literature. It is also shown that R-FT-ISS and R-FTS can be achieved by pure impulsive control, which improves the FTS results of impulsive systems in the literature. Moreover, compared with T-IHC, S-IHC has less number of impulses and lower cost than T-IHC while T-IHC has smaller settling time.