Robust control design for nonlinear uncertain systems with an unknown time-varying control direction

In this paper, a continuous robust control design approach is proposed for first-order nonlinear systems whose dynamics contain both nonlinear uncertainty and an unknown time-varying control direction, The so-called control direction is the multiplier of the control term in the dynamic equation, and it effectively represents the direction of motion under any given control. A nonlinear robust control is designed to on-line and continuously identify sign changes of the unknown control direction and to guarantee stability of uniform ultimate boundedness. The proposed robust control design requires only three conditions: the nominal system is stable, the control direction is smooth, and the uncertainty in the system is bounded by a known function. The necessity of these conditions is established in this paper. Continuity of the proposed robust control is achieved by using a so-called shifting lan that changes smoothly the sign of robust control and tracks the change of the unknown control direction, Analysis and design is shown by using the Lyapunov\u27s direct method