Abstract
In this work, we study finite-time stability of hybrid systems with unstable modes. We present sufficient conditions in terms of multiple Lyapunov functions for the origin of a class of hybrid systems to be finite-time stable. More specifically, we show that even if the value of the Lyapunov function increases during continuous flow, i.e., if the unstable modes in the system are active for some time, finite-time stability can be guaranteed if the finite-time convergent mode is active for a sufficient amount of cumulative time. This is the first work on finite-time stability of hybrid systems using multiple Lyapunov functions. Prior work uses a common Lyapunov function approach, and requires the Lyapunov function to be decreasing during the continuous flows and non-increasing at the discrete jumps, thereby, restricting the hybrid system to only have stable modes, or to only evolve along the stable modes. In contrast, we allow Lyapunov functions to increasebothduring the continuous flows and the discrete jumps. As thus, the derived stability results are less conservative compared to the earlier results in the related literature, and in effect allow the hybrid system to have unstable modes.
Highlights
Stability of the equilibrium point or equilibrium set of switched and hybrid systems has been studied extensively in the literature
We develop sufficient conditions for FTS of a class of hybrid systems in terms of multiple Lyapunov functions
The switches in the continuous flows occur after 0.2 sec, i.e., |Tik| 0.2 sec, k ∈ Z+, and discrete jumps occur after 0.1 s for all i ∈{1, 2, . . . , 5}
Summary
Stability of the equilibrium point or equilibrium set of switched and hybrid systems has been studied extensively in the literature. Inspired by the results in (Branicky, 1998; Zhao and Hill, 2008), we study conditions for finite-time stability (FTS) of a class of hybrid systems, using multiple generalized Lyapunov functions. We relax the requirement in (Zhao and Hill, 2008; Li and Sanfelice, 2019) that each Lyapunov function is nonincreasing at the discrete jumps, and strictly decreasing during the continuous flow; instead, we allow the Lyapunov functions to increase both during the continuous flow and at the discrete jumps, and require that these increments are bounded In this respect, we allow the hybrid system to have unstable modes while still guaranteeing FTS. For the applications where the FTS mode cannot be kept active for all times, or the switching signal is not under user’s control, it is essential to study FTS under switching laws that allow the FTS mode to become inactive, and unstable modes to become active
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