State-feedback control via LMI approach of a 1-DOF disturbed impacting mechanical oscillator under double-side rigid constraints

This paper proposes a state-feedback controller using Linear Matrix Inequality (LMI) method for the position control of a single-degree-of-freedom, periodically forced, impact mechanical oscillator subject to asymmetric double-side unilateral rigid constraints. The main idea in the synthesis of the controller lies in the use of the S-procedure Lemma to solve the stabilization problem by considering only the region inside which the oscillator's state evolves. The periodic forcing input is considered as an external disturbance for which we desire compensating it via control. As a result, the synthesis problem of the control law is transformed into a solving problem of a set of Bilinear Matrix Inequalities (BMIs). To convert these BMIs into LMIs, a linearization procedure was employed mainly through the Schur complement Lemma and the matrix inversion Lemma. We show the effectiveness of the developed state-feedback controller in the control of the position of the impact oscillator even in the presence of the periodic forcing input as a persistent exogenous disturbance.