LMI-Based Robust Position Control of an Impacting Oscillator with Double-Side Asymmetric Rigid Constraints

This work proposes the synthesis of a linear state-feedback control law to robustly stabilize a non-autonomous impacting hybrid system, namely the one-degree-of-freedom (1-DoF) mechanical oscillator with two asymmetric rigid unilateral constraints. This mechanical system with impulse effects is exited with a sinusoidal forcing signal, which is considered as an external persistent disturbance. In order to compute the controller gain, we use the Linear Matrix Inequality (LMI) method. This approach is based mainly on the use of the S-procedure lemma for the linear differential equation during the oscillation phase and also the Finsler lemma for the algebraic equation during the impact phase. We show that the design methodology gives rise to stability conditions expressed in terms of Bilinear Matrix Inequalities (BMIs). Unfortunately, the search for a feasible solution of a BMI constraint is an NP-hard problem in general. Then, to obtain traceable stability conditions, these BMIs are transformed into LMIs by applying different mathematical tools. As a result, we show via numerical simulations that the mass of the impacting oscillator is stabilized around the desired position, the zero-equilibrium point.