Design of a state observer to approximate signals by using the concept of fractional variable-order derivative

Abstract

This article proposes a state observer to find a model for a given signal, i.e. to approximate a treated signal. The design of the state observer is based on a dynamical system of equations which is generated from the increasing-order differentiation of a n-th order Fourier series. This dynamical system is set in state space representation by considering that the Fourier series is the first state and the rest of the states are the successive derivatives of the series. The purpose of the state observer is the recursive estimation of the states in order to recover the coefficients from them. This set of coefficients produces the best fit between the dynamical system and the signal. The dynamical system used for the observer conception shall be, together with the estimated coefficients, the model that will describe the signal behavior. The special feature of the proposed observer is the order of the differential equations of the model on which it is based, dα(t)ν(t)/dtα(t), which can take integer and non-integer values, i.e. α(t)∈(0,1]. Even more important, α(t) can be a smooth function such that α(t)∈(0,1] in the interval t∈[0,T]. The procedure to design the state observer of variable-order as well as some examples of its use in engineering applications are presented.