Nonasymptotic Pseudo-State Estimation for a Class of Fractional Order Linear Systems

This paper aims at designing a nonasymptotic and robust pseudo-state estimator for a class of fractional order linear systems which can be transformed into the Brunovsky's observable canonical form of pseudo-state space representation with unknown initial conditions. First, this form is expressed by a fractional order linear differential equation involving the initial values of the fractional sequential derivatives of the output, based on which the modulating functions method is applied. Then, the former initial values and the fractional derivatives of the output are exactly given by algebraic integral formulae using a recursive way, which are used to nonasymptotically estimate the pseudo-state of the system in noisy environment. Second, the pseudo-state estimator is studied in discrete noisy case, which contains the numerical error due to a used numerical integration method, and the noise error contribution due to a class of stochastic processes. Then, the noise error contribution is analyzed, where an error bound useful for the selection of design parameter is provided. Finally, numerical examples illustrate the efficiency of the proposed pseudo-state estimator, where some comparisons with the fractional order Luenberger-like observer and a new fractional order $\mathcal{H}_{\infty}$-like observer are given.