Active disturbance rejection contro

Active disturbance rejection control comprises on-line identification and compensation of the disturbances. A variety of methods to address this issue have been already published, see Chen et al. (2016), and the references therein. The disturbance identification scheme is crucial for active disturbance rejection. In particular, high order sliding mode observers (HOSMOs) have been exploited to identify disturbances due to its attractive features: robustness against uncertainties and finite time convergence. Theoretically, HOSMO converge in finite time to the exact value of the signal. Therefore, HOSMOs have been broadly used in active disturbance controller design. For instance, in Mallavalli and Fekih (2017); R´ıos et al. (2017) a HOSM identification method was used to estimate the disturbances. The estimated values of the disturbanceswere compensated through the controller. For ensuring the stability of the closed-loop system, however, the controller was turned on until the HOSM observer had converged. In practical situation the undesired peaking-phenomena effect may arise if, for instance, the controller is turned on at the same time that the observer (Khalil, 2002). Therefore a formal closed-loop stability analysis is needed. Recently, the advent of Lyapunov functions for HOSM differentiators, see Cruz-Zavala and Moreno (2016), has provided a powerful tool to analyze the observer-controller closed-loop stability since the initial time. For instance, in Chalanga et al. (2016) the closed-loop stability of an observer-controller HOSM scheme was tackled. Nevertheless, only matched perturbations were considered in that paper. Similarly, in Aguilar-Iban˜ez et al. (2017), a Lyapunov perspective was presented to analyze the closedloop stability of a system affected by matched disturbances.The goal of this work is to present an active disturbance rejection approach for a gyroscopic machine affected by matched an unmatched disturbances ensuring the stability of the closed-loop system. To this aim, first, a HOSMO is used to identify the unknown disturbances. Thus, the identified signals are injected through the control signal following a backstepping-like methodology. Finally, the closed-loop stability is investigated. Indeed, the contribution of this work is the stability analysis of the overall observer-controller scheme employing Lyapunov’s theory. Such analysis sheds light on the correct selection of the controller gains. It is shown that choosing the controller gains following the design criteria, ensures the boundedness of the state trajectories. Furthermore, the state trajectories converge asymptotically to zero once the HOSM