Safe Sliding Mode Control in Position for Double Integrator Systems

We address the problem of robust safety control design for double integrator systems. We show that, when the constraints are defined only on position states, it is possible to construct a safe sliding domain from the dynamic of a simple integrator that is already safe. On this domain, the closed-loop trajectories remain robust and safe against uncertainties and disturbances. Furthermore, we design a controller gain that guarantees convergence to the safe sliding domain while avoiding the given unsafe set. The concept is initially developed for first-order sliding mode and is subsequently generalized to an adaptive framework, ensuring that trajectories remain confined to a predefined vicinity of the sliding domain, outside the unsafe region.