# An improved pseudo-state estimator for a class of commensurate fractional order linear systems based on fractional order modulating functions

Fellow

Systems & Control Letters Volume 118, August 2018, Pages 29-34

Yan-Qiao Wei a;b, Da-Yan Liu a;b, Driss Boutat b, Yi-Ming Chen a;c;d

a School of Sciences, Yanshan University, Qinhuangdao, Hebei, P.R.China, 066004
b INSA Centre Val de Loire, Université d'Orléans, PRISME EA 4229, Bourges Cedex 18022, France
c LE STUDIUM Loire Valley Institute for Advanced Studies, 1 rue Dupanloup 45000 Orléans, France
PRISME (INSA-Institut National des Sciences Appliquées)-88, Boulvevard Lahitolle, 18000 Bourges, France

## Abstract

In this paper, a non-asymptotic pseudo-state estimator for a class of commensurate fractional order linear systems is designed
in noisy environment. Different from existing modulating functions methods, the proposed method is based on the system
model with fractional sequential derivatives by introducing fractional order modulating functions. By applying the fractional
order integration by parts formula and thanks to the properties of the fractional order modulating functions, a set of fractional
derivatives and fractional order initial values of the output are analogously obtained by algebraic integral formulas. Then,
an explicit formula of the pseudo-state is accomplished by using the fractional sequential derivatives of the output computed
based on the previous results. This formula does not contain any source of errors in continuous noise-free case, and can be used
to non-asymptotically estimate the pseudo-state in discrete noisy case. The construction of the fractional order modulating
functions is also shown, which is independent of the time. Finally, simulations and comparison results demonstrate the efficiency
and robustness of the proposed method

## Keywords

Non-asymptotic method
Fractional order modulating functions
Robust estimation
Pseudo-state estimator
Fractional derivative initial values