Robust exponential arbitrary time control of nonlinear systems with input delay

In this study, we aim to present an exponential arbitrary-time control scheme for nonlinear systems subjected to input time delay and additive disturbances. To this end, a new finite-time controller was developed for certain nonlinear systems, which ensures exponential tracking in a time independent of initial conditions. Due to the presence of unknown disturbances, predictors cannot be directly used to control the system, being a challenging issue. Therefore, to utilize predictor feedback in disturbed nonlinear input delay systems, a disturbance observer was designed. The proposed disturbance observer guarantees the exponential arbitrary time estimation of unknown disturbances. Next, an adaptive exponential arbitrary time controller was designed for the system using the proposed disturbance observer. In addition, the Lyapunov theorem was utilized to prove the disturbance observer and tracking errors converged to the origin in an arbitrary time. In the end, the state tracking error was analyzed accurately. It was ensured that the upper bound of the tracking error was bounded and converged to zero in the adjusted arbitrary time. Ultimately, the main results of this study will be illustrated and validated by simulation and experimental results.