Stabilization of nonaffine systems: a constructive method for polynomial systems

This note focuses on the stabilization of nonaffine systems described by continuous nonlinear ordinary differential equations. First, conditions of stabilization using control Lyapunov function give theoretical but nonconstructive and restrictive results. Second, some particular nonaffine systems are considered: This is polynomial system in the control variable of order two and three (a method is also given for high-order systems). The main result is a method of construction of feedbacks for this class of polynomial systems. In the end, the polynomial example of the levitation system is stabilized using an extension of this method to discontinuous feedback.