As a new strategy for magnetic levitation envisioned in 1990s, the Inductrack system with permanent magnets (PMs) aligned in Halbach arrays has been intensively studied and applied in many projects. Due to the nonlinear, time-varying electro-magneto-mechanical coupling in such a system, the dynamic behaviors are complicated with transient responses, which in most cases can hardly be predicted with fidelity by a steady-state Inductrack model. Presented in this paper is a benchmark 2-DOF transient Inductrack model, which is derived from the first laws of nature, without any assumed steady-state quantities. It is shown that the dynamic response of the Inductrack dynamic system is governed by a set of nonlinear integro-differential equations. As demonstrated in numerical simulations with the transient model, unstable vibrations in the levitation direction occur when the traveling speed of the vehicle exceeds a threshold. To resolve this instability issue, feedback control is implemented in the Inductrack system.
In the development, an assembly of Halbach arrays and active coils that are wound on the PMs is proposed to achieve a controllable source magnetic field. In this preliminary investigation, the proposed control system design process takes two main steps. First, a PID controller is set and tuned based on a simple lumped-mass dynamic system. Second, the nonlinear force-current correlation is obtained from a lookup table that is pre-calculated by steady-state truncation of the full transient Inductrack model. With the implemented feedback control algorithms, numerical examples display that the motion of the vehicle in levitation direction can be effectively stabilized at different traveling speeds. Although only a 2-DOF transient model is used here, the modeling technique and the controller design approach developed in this work are potentially applicable to more complicated models of Inductrack Maglev systems.