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

Abstract

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.