Analytical design of active disturbance rejection control for nonlinear uncertain systems with delay

For first-order nonlinear uncertain systems with delay, the analytical design of active disturbance rejection control (ADRC) with matched time delay is rigorously studied. The conditions of stabilizing the closed-loop system are studied in the form of parameters' explicit sets such that the tuning laws of ADRC can be directly obtained and intuitively demonstrated in engineering practice. The necessary and sufficient condition of the asymptotical stability for the closed-loop system in the case of linear time-invariant is explicitly given. The condition of stabilizing the closed-loop system with nonlinear uncertain dynamics is quantitatively presented. Moreover, the water tank experiment illustrates the proposed tuning laws.