A tool for the design of a self-excited oscillation of a desired amplitude and frequency in linear plants by means of the variable structure control is proposed. An approximate approach based on the describing function method is given, which requires that the mechanical plant should be a low-pass filter-the hypothesis that usually holds when the oscillations are relatively fast. The proposed approach is demonstrated via the controller design and experiments on the Furuta pendulum.Index Terms-Frequency domain methods, periodic solution, underactuated systems, variable structure systems.

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Summary The problem of output control in multiple‐input–multiple‐output nonlinear systems is addressed. A high‐order sliding‐mode observer is used to estimate the states of the system and identify the discrepancy between the nominal model and the real plant. The exact and finite‐time estimation may be tackled as long as the system presents the algebraic strong observability property. Thus, a continuous robust input‐output linearization strategy can be obtained with respect to a prescribed output. As a consequence, the closed‐loop dynamics performs robustly to uncertainties/perturbations. To illustrate the advantages of the proposed method, we introduce a study case that demands a robust linear system behavior: the self‐oscillations induced in an underactuated mechanical system through a two‐relay controller. Experiments with an inertial wheel pendulum illustrate the feasibility of the proposed approach.

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Summary This paper presents a control design algorithm that combines backstepping and high‐order sliding modes. It is known that backstepping can achieve asymptotic stability for nonlinear systems in strict‐feedback form in spite of parametric uncertainties. Nevertheless, when external perturbations are also present, only practical stability can be ensured. For the same aforementioned perturbed conditions, the combined design presented in this paper can achieve finite‐time exact tracking/regulation. At the same time, the semi‐global or global stability obtained through backstepping is preserved, and the gains of the high‐order sliding modes controller can be reduced with respect to its direct application. The design is based on recently reported combined designs that are based on the idea of virtual controls, which can contain terms based on high‐order sliding modes algorithms. The proposal also extends previous results to the multiple‐input–multiple‐output case. Copyright © 2016 John Wiley & Sons, Ltd.