A globally Mittag-Leffler bounded high-gain observer for systems with unknown dynamics and noisy measurements

In this work, we present a globally Mittag-Leffler bounded high-gain observer for fractional order nonlinear systems with unmodeled dynamics and additive measurement noise at the output. Our proposal starts from an alternative representation of the fractional order system, whose output does not depend on the additive measurement noise and in which the original system's output is treated as an additional state variable. This representation allows us two things: 1) to simultaneously estimate the state variables and the uncertain term and 2) to incorporate into the design scheme a fractional integral-type contribution, which is useful to give robustness against the measurement noise and the unmodeled dynamics, as well as to attenuate the noise amplification, typical of any high-gain observer. Through the corresponding mathematical analysis, we prove that the estimation error of the proposed observer is uniformly bounded and converges asymptotically to a globally Mittag-Leffler compact attractive set, this is, the proposed observer is globally Mittag-Leffler bounded. Additionally, we show that under certain conditions, such as an integer derivation order or the absence of measurement noise, the proposed observer exhibits some particular properties. Finally, we consider a continuously stirred biochemical reactor to exemplify our design methodology. The numerical results confirm that the observer is able to accurately estimate the state variables as well as the uncertainty term of the fractional model. In other words, the globally Mittag-Leffler bounded high-gain observer is robust against measurement noise and uncertainties.