A robust control design technique for nonlinear uncertain systems with unknown time varying control direction

In this paper, a continuous robust control design scheme is proposed for first-order nonlinear systems that have both nonlinear uncertainties and unknown time varying control directions. The so-called control direction is the multiplier of the control term that effectively represents the direction of motion under any given control. Nonlinear robust control is designed to online and continuously identify sign changes of the unknown control direction and to guarantee stability of uniform ultimate boundedness. Robust control requires only three conditions: the nominal system is stable, the control direction is smooth, and the uncertainties are bounded by known functions. Continuity of the proposed robust control is achieved by using a so-called shifting law that changes smoothly the sign of robust control and tracks the change of the control direction. The analysis and design is proceeded using the Lyapunov´s direct method