An LMI-based design of a robust state-feedback control for the master-slave tracking of an impact mechanical oscillator with double-side rigid constraints and subject to bounded-parametric uncertainty

• Published in: Communications in nonlinear science & numerical simulation 2020-03, Vol.82, p.105020, Article 105020
• Main Authors: Turki, Firas, Gritli, Hassène, Belghith, Safya
• Format: Article
• Language: English
• Subjects: BMI; Control systems; Feedback control; Feedback control systems; Hybrid systems; Impact mechanical oscillator; Impulsive hybrid non-autonomous system; Linear matrix inequalities; LMI; Mathematical analysis; Matrix; Mechanical oscillators; Oscillators; Reference systems; Robust control; Robust tracking; Simulation; Stability; State constraints; Tracking control; Tracking errors; Uncertainty
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2019.105020

•A robust feedback control of a 1-DoF impact mechanical oscillator subject to double-side rigid constraints and under bounded-parametric uncertainty is considered.•The control is achieved such that the impact mechanical oscillator, as a slave, tracks a master impact oscillator adopted as a reference model.•The dynamics of the master-slave tracking error is described by a non-autonomous system with impulsive effects.•We use the S-procedure Lemma and the Finsler Lemma to only consider the regions within which the system state evolves and hence to develop BMI stability conditions.•We use the Schur complement Lemma and the Matrix Inversion Lemma to transform these BMIs into LMIs.•A portfolio of numerical simulations is presented to illustrate the master-slave tracking. In this paper, a Linear Matrix Inequality- (an LMI-) based approach for designing a robust state-feedback controller for a 1-DoF, periodically forced, impact mechanical oscillator subject to double-side rigid barriers/constraints and under bounded parametric uncertainties to track another impact oscillator, as a master or reference system, is proposed. The dynamics of such impact oscillator is defined by a hybrid non-autonomous system with impulsive effects, for which the impulsive event occurs when the system state encounters the two barriers and the oscillation motion is limited between them. The main idea in the synthesis of the stability conditions lies in the use of the S-procedure Lemma and the Finsler Lemma in order to only consider the regions inside which the master-slave tracking error evolves. We show that the stability conditions of the tracking error are reformulated by a set of Bilinear Matrix Inequalities (BMIs). Via the Schur complement Lemma and the Matrix Inversion Lemma, a linearization procedure is realized to transform these BMIs into LMIs where the admissible maximum bounds of the parametric uncertainties are maximized. The effectiveness of the proposed feedback controller towards uncertainties is illustrated through simulation results.