Trajectory-based optimal linearization for nonlinear autonomous vector fields

This paper deals with an optimal approximation in the least square sense of nonlinear vector fields. The optimal approximation consists of a linearization along a trajectory that approximates the nonlinear solution from the initial state to the equilibrium position. It is shown that the optimal linearization can be seen as a generalization of the classical linearization. Furthermore, the optimal linearization can approximate the derivative at the equilibrium point, and the order of the method is the same as the nonlinearity, since the approximation depends on the initial state. We also show that the method can be used to study the asymptotic stability of the equilibrium of a nonlinear vector fields, especially in the nonhyperbolic case. Simulation shows good agreement between the linearized and the nonlinear systems.