Analysis and control of coexisting attractor transitions in a class of rigid vibro-impact systems

In order to solve the problem that coexisting attractor transition control is not easy to achieve in rigid vibro-impact systems, the coexisting attractors and the transition control of a single-degree-of-freedom vibro-impact system with clearance is explored in this paper. Firstly, the Poincaré map of system and its Jacobi matrix are obtained by constructing the local map. Secondly, the coexistence and transition characteristics between stable and unstable periodic motions are studied by the cell mapping method, the shooting method, and parameter continuation algorithms. Finally, with the boundary point being whether a collision occurs between the controlled system and the target system, a piecewise controller based on Lyapunov stability theory and the backstepping method is proposed to adapt to the rigid vibro-impact system with clearance and multistability coexistence. The controller employs the impact occurrence between the controlled system and the target system as the boundary point. Numerical simulation results demonstrate that the designed controller can effectively control the transition of the attractors.