Optimal control of a magnetic bearing without bias and saturating voltages

Conventional Active Magnetic Bearings (AMB) are operated using a bias current (or flux) to achieve greater linearity and dynamic capability. Bias, however, results in undesirable rotating losses and consequent rotor heating. While control without bias flux is an attractive alternative, it is considerably more complex due to both force slew rate limitations and actuator nonlinearity. In this paper, optimal control of a magnetic bearing without bias is investigated. A single degree-of-freedom system consisting of a mass and two opposing electromagnets is considered. The optimal control problem is examined for a cost function that penalizes both poor regulation and rotational energy lost. Though a standard optimization procedure does not directly yield an analytical solution, it does show that the optimal control is always bang-bang including possibly a singular arc. First, the minimum time problem is solved for a simple switching law in three dimensional state space. A non-standard, physics-based approach is then employed to obtain an optimal solution for the general problem. The final result is an optimal variable structure feedback controller. This result provides a benchmark which can be used for evaluation of the performance of a practical feedback controller designed via other methods. The practical controller will be designed to support a flexible rotor and achieve robustness and optimally reject disturbance. This result may also be applied to many other applications which contain opposing quadratic actuators. KEY WORDS: active magnetic bearing; opposing quadratic actuators; nonlinear; constrained control; time-state optimization; bang-bang; singular arc