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

In this note, simultaneous-arrival-to-origin (SATO) convergence is defined --- all the elements of the system state 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 sign function and the norm-normalized sign function are revisited. We investigate their differences with applications in sliding-mode control design, paying special attention to their convergence performance. It is found that both functions (expectedly when properly invoked) contribute to system stability, while the norm-normalized sign function additionally enables the system to achieve SATO convergence. This finding shows the distinctive merit of the norm-normalized 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 with the utilization of the norm-normalized 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-fr