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In this work, we study finite-time stability of switched and hybrid systemsin the presence of unstable modes. We present sufficient conditions in terms ofmultiple Lyapunov functions for the origin of the system to be finite timestable. More specifically, we show that even if the value of the Lyapunovfunction increases in between two switches, i.e., if there are unstable modesin the system, finite-time stability can still be guaranteed if the finite timeconvergent mode is active long enough. In contrast to earlier work where theLyapunov functions are required to be decreasing during the continuous flowsand non-increasing at the discrete jumps, we allow the Lyapunov functions toincrease \emph{both} during the continuous flows and the discrete jumps. Asthus, the derived stability results are less conservative compared to theearlier results in the related literature, and in effect allow the hybridsystem to have unstable modes. Then, we illustrate how the proposed finite-timestability conditions specialize for a class of switched systems, and present amethod on the synthesis of a finite-time stabilizing switching signal forswitched linear systems. As a case study, we design a finite-time stable outputfeedback controller for a linear switched system, in which only one of themodes is controllable and observable. Numerical example demonstrates theefficacy of the proposed methods.

Finite-Time Stability of Switched and Hybrid Systems with Unstable Modes