Self-scheduled H1 Controllers for Magnetic Bearings

In this paper we present a design procedure for disturbance attenuation with internal stability of a rotor supported by magnetic bearings, over the whole interval of the rotor operating speeds. The nonlinear gyroscopic equations can be simpliied to a set of linear, time-varying diierential equations, owing to the linear dependence of the rotor speed in the plant dynamics. For rotors operating in high speeds, even small unbalanced masses can create large synchronous disturbances. To alleviate the forces produced due to imbalance the controller automatically balances the rotor for each operating speed. The proposed methodology addresses both the gyroscopic compensation problem due to change of the rotor speed as well as the automatic balancing problem. The approach is based on the recently developed theory of self-scheduled H1 controllers. Owing to the linear dependence of the equations on the rotor speed, the computation of the controller can be done very eeciently using convex optimization and Linear Matrix Inequalities (LMIs). NOMENCLATURE A area of each pole Co convex hull E0 nominal voltage ~ F external force in body frame F0 nominal force G0 nominal gap Ja axial moment of inertia Jr radial moment of inertia M mass of the rotor N number of coil turns P parameter polytope R coil resistance IR n n-dimensional vector space S system matrix polytope ~ T external torque in body frame ~ V velocity of center of mass in body frame ej voltage across the jth coil fij magnetic forces gj gap length of jth coil h pole width`half the length of the shaft p rotor speed 0 nominal airgap disturbance attenuation level angular displacement about Y axis 0 permeability of free space (4 10 ?7 H=m) angular displacement about Z axis j airgap magnetic ux of jth coil angular displacement about X axis ! angular velocity in body frame INTRODUCTION Magnetic bearings (MB) increasingly become the choice for high-speed, high-performance rotating machinery because of their frictionless characteristics. They utilize a magnetic eld generated by radially or axially placed electromagnets to generate the forces necessary to suspend and support a shaft without any contact with its environment. Thus, magnetic bearings are particularly useful in very high or very low temperature conditions where a lubrication-free environment is necessary. The advantages of magnetic bearings are primarily their very low power consumption (an order of magnitude lower than oil lm bearings) and their very long, maintenance-free life. Some applications where …