Modulating functions based differentiator of the pseudo-state for a class of fractional order linear systems

Abstract

In this paper, an algebraic and robust fractional order differentiator is designed for a class of fractional order linear systems with an arbitrary differentiation order in [0,2]. It is designed to estimate the fractional derivative of the pseudo-state with an arbitrary differentiation order as well as the one of the output. In particular, it can also estimate the pseudo-state. Different from our previous works, the considered system no longer relies on the matching conditions, which makes the system model be more general. First, the considered system is transformed into a fractional differential equation from the pseudo-state space representation. Second, based on the obtained equation, a series of equations are constructed by applying different fractional derivative operators. Then, the fractional order modulating functions method is introduced to recursively give algebraic integral formulas for a set of fractional derivatives of the output and a set of fractional derivative initial values. These formulas are used to non-asymptotically and robustly estimate the fractional derivatives of the pseudo-state and the output in discrete noisy cases. Third, the required modulating functions are designed. After giving the associated estimation algorithm, numerical simulation results are finally given to illustrate the accuracy and robustness of the proposed method.