Exponential Stability with RISE Controllers for Uncertain Nonlinear Systems with Unknown Time-Varying State Delays

Robust Integral of the Sign of the Error (RISE) controllers have gained popularity in recent years due to their ability to achieve asymptotic tracking error convergence using a continuous control input for uncertain nonlinear systems. In a recent breakthrough, it was shown that the tracking error convergence with RISE controllers is also exponential, whereas previously it was thought to be only asymptotic. However, it remains an open question whether this exponential stability result also holds for systems containing unknown time-varying state delays. In this paper, a novel strict candidate Lyapunov function is developed to prove exponential stability with a RISE controller for uncertain nonlinear systems involving unknown time-varying state delays. Additionally, a simulation example is provided to demonstrate the performance of a RISE controller in the presence of unknown time-varying state delays. The results indicate a higher sensitivity of tracking error and control effort to the delay frequency as compared to the delay magnitude.