Pole Placement of a Nonlinear Electromagnetic System by the Receptance Method

This paper addresses the challenge of pole placement for controlling a nonlinear electromagnetic system using the receptance method. The system, consisting of a pair of identical magnets and coils, is mathematically modeled to introduce nonlinear (cubic) stiffness. This nonlinear stiffness can be adjusted by varying the input electrical current to the coils and the displacement between the magnets and coils.

The transfer function of the open-loop nonlinear system is obtained under low excitation levels, where the system exhibits weak nonlinearity. Using this method, the evaluation of the mass, stiffness, and damping matrices, which are generally required, is avoided. To demonstrate the system’s nonlinear behavior, the excitation level is subsequently increased, and open-loop receptances are measured at various levels. The poles of the nonlinear system are assigned using linear feedback control and the Sherman–Morrison formula across different excitation levels. Given the system’s response dependency on amplitude, an iterative approach is employed to determine the feedback gains.

The performance of the nonlinear control is examined under varying excitation levels and different positions of the magnets relative to the coils. The feedback control adapts to changes in amplitude and displacement, ensuring the active control system's performance remains effective. Experiments were carried out to demonstrate the successful assignment of the poles.

The results demonstrate that feedback control performs well with varying excitation levels, even with small changes in the amplitude or the distance between the magnets and coils.