A Computational Method of the Formal Linearization by the Cubic Spline Interpolation

A computational method of the formal linearization for nonlinear systems is proposed by using a cubic spline interpolation. We introduce a linearizing function that consists of the state variables, their squares, and the cubes, thus the dimension of the linearizing function is determined finitely by the number of state variables. Then we construct the differential equation with respect to the linearizing function. By approximating the nonlinear terms by the cubic spline interpolation, we obtain the linearized form with respect to the linearizing function. We contrive to determine the coefficients of the cubic spline interpolation, so that we can easily get them by simple matrix calculations. From the features of the cubic spline interpolation, a nonlinear system is converted into the piecewise linear systems, but they are continuously connected on the whole domain. A nonlinear filter is synthesized as an application of the method and is verified through numerical examples.