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Abstract-The robustness properties of integral sliding-mode controllers are studied. This note shows how to select the projection matrix in such a way that the euclidean norm of the resulting perturbation is minimal. It is also shown that when the minimum is attained, the resulting perturbation is not amplified. This selection is particularly useful if integral sliding-mode control is to be combined with other methods to further robustify against unmatched perturbations.is taken as a special case. Simulations support the general analysis and show the effectiveness of this particular combination. Index Terms-, robust control, sliding-mode control (SMC), variable structure systems.

International audienceThe paper surveys mathematical tools required for stability and convergence analysis of modern sliding mode control systems. Elements of Filippov theory of differential equations with discontinuous right-hand sides and its recent extensions are discussed. Stability notions (from Lyapunov stability (1982) to fixed-time stability (2012)) are observed. Concepts of generalized derivatives and non-smooth Lyapunov functions are considered. The generalized Lyapunov theorems for stability analysis and convergence time estimation are presented and supported by examples from sliding mode control theory

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SUMMARYIn this paper, a higher-order sliding-mode observer is proposed to estimate exactly the observable states and asymptotically the unobservable ones in multi-input-multi-output nonlinear systems with unknown inputs and stable internal dynamics. In addition the unknown inputs can be identified asymptotically. Numerical examples illustrate the efficacy of the proposed observer.

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