A Nonlinear Observer via Augmented Linear System based on Formal Linearization using Discrete Fourier Expansion

 Abstract—In order to expand a formal linearization method on designing an observer for nonlinear dynamic and measurement systems, we exploit the discrete Fourier expansion that can reduce computational complexity. A nonlinear multidimensional dynamic system is considered by a nonlinear ordinary differential equation, and a measurement equation is done by a nonlinear equation. Defining a linearization function which consists of the trigonometric functions considered up to the higher-order, the nonlinear dynamic system is transformed into an augmented linear one with respect to this linearization function by using the discrete Fourier expansion. Introducing an augmented measurement vector which consists of polynomials of measurement data, the measurement equation is transformed into an augmented linear one with respect to the linearization function in the same way. To these augmented linearized systems, a linear estimation theory is applied to design a nonlinear observer. Index Terms—nonlinear system, nonlinear observer, formal linearization, discrete Fourier expansion