Estimators for a Class of Commensurate Fractional-Order Systems with Caputo Derivative

Abstract

Using some concepts of observability, based on algebraic properties of fractional-order systems, we can synthesize observers for nonlinear fractional-order systems. The technique used in this chapter is based on Rafael Martínez-Guerra et al., Fractional Algebraic Observability property Synchronization of Integral and Fractional Order Chaotic Systems: A Differential Algebraic and Differential Geometric Approach With Selected Applications in Real-Time, Springer 2015, [1], Rafael Martínez-Guerra and Christopher D. Cruz-Ancona. Algorithms of Estimation for Nonlinear Systems: A Differential and Algebraic Viewpoint, Springer 2017, [2]. The former verifies whether a given state of a system can be estimated from a function that depends on the output, input and their finite number of fractional-order derivatives, i.e., if state is reconstructible from output and input measurements. The methodology proposed consist in finding a canonical form for the original system, and this is obtained through a mapping given by the output of the system and its successive fractional-order derivatives. Then, we can design an observer to estimate the state of the transformed system, so-called Fractional Generalized Observability Canonical Form Rafael Martínez-Guerra et al., Fractional Algebraic Observability property Synchronization of Integral and Fractional-Order Chaotic Systems: A Differential Algebraic and Differential Geometric Approach With Selected Applications in Real-Time, Springer 2015, [1]. Finally, from an inverse mapping, obtain the estimates of the original state.