Global adaptive HOSM differentiators via monitoring functions and hybrid state-norm observers for output feedback

ABSTRACT

An algorithm for adaptation of the gains of higher-order sliding mode-based exact differentiators is developed for the case where the upper bound for the ρth derivative of the tracking error signal exists but it is unknown. Unlikely other publications in the literature, the developed adaptive algorithm based on monitoring functions guarantees global and exact tracking when used in closed-loop output feedback. In the closed-loop scenario, a global-exact and finite-time estimate for the variables $\dot{e}\left(t\right),\ddot{e}\left(t\right),\dots ,e^{(\rho -1)}\left(t\right)$ is applied to construct the sliding surface of the proposed sliding mode controller. The class of uncertain systems of arbitrary relative degree (ρ ≥ 1) takes into account time-varying perturbations with unknown bounds and state-dependent nonlinearities satisfying a linear growth condition with any unknown rate. The norm of the unmeasured state is majorised by using a hybrid state-norm estimator. Numerical examples and an engineering application to wing rock control are presented in order to illustrate the properties and advantages of the novel adaptation approach for sliding mode control design.

KEYWORDS:

• Uncertain nonlinear systems
• variable structure systems
• higher-order sliding mode control
• adaptive exact differentiators
• output feedback
• global stability

Acknowledgments

This work has been supported in part by CNPq, FAPERJ and CAPES, Brazil.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Let {λ_i } be the eigenvalues of A _c , the stability margin of A _c is defined by ${\gamma }_0:={\min }_{\mathrm{i}}[-\mathrm{Re}\left({\lambda }_{\mathrm{i}}\right)]$.