Instability analysis for semi-active control systems with semi-active inerters

In this paper, the instability of semi-active control systems caused by the utilization of semi-active inerters is analyzed. The purpose of this study is to demonstrate the possibility of inducing instability by applying semi-active inerters in vibration control systems, a phenomenon which is essential for the applications of semi-active inerters but has not been drawn much attention in the existing results. A constructive method is proposed to prove the instable effect caused by switching-state semi-active inerters based on a general multi-degree-of-freedom (DOF) mass-chain system. The underlying idea in the proof is to construct a multi-DOF unstable semi-active control system with semi-active inerters based on its 1-DOF counterparts. First of all, it is theoretically proved that for a 1-DOF system, both undamped and damped cases, instability can be induced by semi-active inerters under certain switching laws. Then, following the idea of modal analysis, unstable n-DOF semi-active inerter-based system with proportional damping is constructed via coordinate transformation. Moreover, for general damping case, phase portraits are depicted to show the instability caused by semi-active inerters. In this way, the conclusion is reached that semi-active inerter can induce instability if the inertance improperly controlled, and such a finding is still valid when the inerter nonlinearities are considered. Hence, the stability issue should not be neglected in the practical application of semi-active inerters.