Finite time stabilization of a perturbed double integrator with unilateral constraints

A discontinuous second order sliding mode (twisting) controller is utilized in a full state feedback setting for the finite time stabilization of a perturbed double integrator in the presence of both a unilateral constraint and uniformly bounded persisting disturbances. The unilateral constraint involves rigid body inelastic impacts causing jumps in one of the state variables. Firstly, a non-smooth state transformation is employed to transform the unilaterally constrained system into a jump-free system. The transformed system is shown to be a switched homogeneous system with negative homogeneity degree where the solutions are well-defined. Secondly, a non-smooth Lyapunov function is identified to establish uniform asymptotic stability of the transformed system. The global, uniform, finite time stability is then proved by utilizing the homogeneity principle of switched systems. The novelty lies in achieving finite time stabilization in the presence of jumps in one of the states without the need to analyze the Lyapunov function at the jump instants. The proposed results are of theoretical significance as they bridge non-smooth Lyapunov analysis, quasi-homogeneity and finite time stability for a class of impact mechanical systems.