Nonlinear robust control design without requiring any priori information on the control sign

This paper present a robust control scheme for a second order nonlinear system that have not only additive nonlinear uncertainties but also unknown multiplicative signs. These signs are called control directions since they represent effectively the direction of motion under any given control. Except for the unknown control directions, the second order system satisfies the generalised matching conditions (GMC). The GMC design is also called back-stepping because the control is designed by working backwards through two integrators. The design procedure can be generalized and applied to nonlinear systems because it basically forms a sequence of state transformations or sometimes referred to are a recursive design. The proposed robust control is continuous and guarantees global stability of uniform ultimate boundedness without a priori knowledge of the control directions nor the knowledge of nonlinear dynamics except their size bounding functions. This is achieved by online identifying control directions and by utilizing transition laws that change smoothly and accordingly the signs of robust controls. The analysis and design is done using Lyapunov direct method.