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

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