Simultaneous Arrival to Origin Convergence: Sliding-Mode Control Through the Norm-Normalized Sign Function

In this article, simultaneous arrival to origin (SATO) convergence is defined - all state elements arriving at the origin at the same time. Accordingly, a relevant sufficient condition is proposed for SATO convergence. Based on this formulation of SATO convergence, the classical and norm-normalized (NN) sign functions are revisited. Their differences are studied with applications in sliding-mode control design. Both functions (expectedly when properly invoked) contribute to system stability, while the NN sign function enables the system to accomplish SATO convergence. This finding shows the distinctive merit of the NN sign function in achieving more than finite-time stability for a sliding-mode control system. Extensions to the scenario with a networked system are studied, where, using the NN sign function, the networked system (now with the SATO convergence property) drives all the agents to reach consensus simultaneously. Additionally, for double integrator systems and Euler-Lagrange systems, singularity-free sliding-mode control laws are designed and demonstrated to achieve SATO convergence.