Finite time stabilization of perturbed double integrator with jumps in velocity

In this paper, finite time stabilization of a perturbed double integrator is considered, incorporating jumps in the velocity at the unstable equilibrium. Rigid body inelastic impacts are considered. A robust control synthesis is presented in the presence of uniformly bounded persistent disturbances. The second order sliding mode (twisting) controller is utilized. Firstly, a non-smooth state transformation is employed to transform the original system into a jump-free system. The transformed system is shown to be a switched homogeneous system with negative homogeneity degree whose solutions are well-defined. Secondly, a non-smooth Lyapunov function is identified to establish uniform asymptotic stability of the transformed system. The global finite time stability then follows from the homogeneity principle of switched systems. Thus, using a single Lyapunov function, the global finite time stability of the origin of the system with velocity jumps is established without having to analyze the Lyapunov function at the jump instants. A finite upper bound on the settling time is also computed.