A Universal Formula for Asymptotic Stabilization with Bounded Controls

Motivated by Artstein and Sontag universal formula, this brief paper presents an explicit proof of the universal formula for asymptotic stabilization and asymptotic disturbance rejection of a nonlinear system with mismatched uncertainties and time varying disturbances. We prove the stability via Lyapunov stability criteria. We also prove that the control law satisfies small control property such that the magnitude of the control signal can be bounded without the catastropic effect to the closed loop stability. For clarity, we benchmark the proposed approach with other method namely a Lyapunov redesign with nonlinear damping function. We give a numerical example to verify the results.