Pole Placement of a Nonlinear Electromagnetic System by the Receptance Method

This paper presents the problem of pole placement for the control of a nonlinear electromagnetic system using the receptance method. A pair of identical magnets and coils are mathematically modeled to create the nonlinear stiffness in the electromagnetic system. The nonlinear stiffness can be varied by adjusting the input electrical current of the coils. The transfer function of the open-loop nonlinear system is obtained at a low level of excitation, in which the system is weakly nonlinear. By doing so, the evaluation of the mass, spring, and damper matrices, which are generally required, is avoided. In further steps, to show the system's nonlinear behavior, the excitation level is raised and the open-loop receptances are measured at various levels. The nonlinear system's poles are assigned using the linear feedback control method and the Sherman-Morrison formula at various levels of excitation. The system's response is dependent on the amplitude, thus, to get the feedback gains, an iterative approach is required. At various excitation levels and positions of the magnets with respect to the coils, the performance of the nonlinear control has been investigated. When the excitation level varies, feedback control can adapt to the changes in the amplitude and the distance, and the performance of the active control system is well maintained.