1. 1 Modeling and Control of Magnetic Bearing Systems Zongli Lin Department of Electrical and Computer Engineering University of Virginia U.S.A. ICCA ’02 Xiamen University China

2. 2 Acknowledgement This talk is based on joint work with the following colleagues and students: Paul Allaire Edgar Hilton Tingshu Hu Hai Zhang under the support of: Guoxin Li Kevin Skadron Costin Untaroiu NASA Goddard Research Center AFS Trinity Power Corporation Bin Huang Marty Humphries Wei Jiang

3. 3 Outline of the Presentation  Magnetic Bearings and Their Applications u Bearing Nonlinearities  Modeling of Flexible Rotors u Robust Control Design u Control Implementation u Issues to Be Addressed

4. 4 Magnetic Bearings and Their Applications Advantages over Ball Bearings Magnetic bearings take radial or thrust loads by utilizing a magnetic field to support the shaft rather than a mechanical force as in fluid film or rolling element bearings. No wear Long life Elimination of oil supply Vibration control Very low power consumption Diagnostic capability

5. 5 Magnetic Bearings and Their Applications Energy Storage Flywheel Systems (Electromechanical Batteries) Transferring energy of various forms into rotational inertial energy of the flywheel: solar energy wind power inertial energy of a car ···

6. 6 Magnetic Bearings and Their Applications Energy Storage Flywheel Systems (Electromechanical Batteries)

7. 7 Magnetic Bearings and Their Applications

8. 8 Artificial Heart Pumps

9. 9 Magnetic Bearings and Their Applications Turbines Magnetic spindles (For high speed grinding)

10. 10 Nonlinearities in Magnetic Bearings Dynamic equation: Balance beam magnetic bearing test rig

11. 11 Nonlinearities in Magnetic Bearings A conventional current biasing strategy: Linearization at A constrained control problem: Input constraint: State constraint: Let

12. 12 Nonlinearities in Magnetic Bearings Normalization: Letand Then, Null controllable region: Maximal possible stability region:

13. 13 Nonlinearities in Magnetic Bearings Description ofand: Letandbe the eigenvalues of Denote Proposition 1. 2.

14. 14 Nonlinearities in Magnetic Bearings Dependence of on :

15. 15 Nonlinearities in Magnetic Bearings Stabilizing controllers : Proposition Under the feedback law is the stability region.

16. 16 Nonlinearities in Magnetic Bearings Performance vs : Proposition Under the feedback law 1. The convergence rate increases as is increased; 2. The disturbance is better rejected as is increased; Disadvantage of using large : Power consumption

17. 17 Nonlinearities in Magnetic Bearings Experimental results, :

18. 18 Nonlinearities in Magnetic Bearings Experimental results, :

19. 19 Nonlinearities in Magnetic Bearings Experimental results, convergence rates under :

20. 20 Modeling of Flexible Rotors Flexible rotor on magnetic bearings test rig:

21. 21 Modeling of Flexible Rotors Finite element method:

22. 22 Modeling of Flexible Rotors Bending Modes Measured Frequency(Hz) Calculated Frequency(Hz) Difference (%) 1153.0152.85-0.10 2310.5310.40-0.03 3614.0613.54-0.07 41067.51070.380.27 51455.01473.971.30 Experimental verification:

23. 23 Modeling of Flexible Rotors Model order reduction:

24. 24 Modeling of Flexible Rotors Gyroscopic effects : Uncertainty characterization:

25. 25 Robust Control Design Possible methods : control design … not successful LPV control design … computationally not implementable (yet) synthesis … treating rotor speed p as uncertainty, valid for a segment of speed range Piecewise synthesis … switching between controllers, bumpy transience, bumpless transfer?

26. 26 Robust Control Design Piecewise synthesis :

27. 27 Robust Control Design Switching between two controllers :

28. 28 Robust Control Design Bumpless transfer : Main Idea: Build an observer that estimates the off line controller state from the on line controller output Use the estimate state as the initial state at time of switching As a result,

29. 29 Robust Control Design Bumpless transfer : experimental results (12,000 rpm)

30. 30 Controller Implementation RT-Linux is used to support real-time computation. A 700 MHz Intel P III with a single 5 channel A/D card and a single 5 channel D/A card is the necessary hardware currently implemented RTic-Lab serves as the software operating platform for controller testing Current hardware and software setup :

31. 31 Parallelization, multiprocessor version of the RT-Linux Parallelization, multiprocessor version of the RT-Linux Use of SSE instructions supported in the Pentium III and IV Use of SSE instructions supported in the Pentium III and IV Use of the higher-speed commodity processors Use of the higher-speed commodity processors Controller Implementation Larger computational capacity for LPV controllers and/or higher rotor speeds :

32. 32 Issues to Be Addressed Balancing : mechanical and/or by magnetic bearing Unbalance force:

33. 33 Issues to Be Addressed Model reduction of gyroscopic systems : This will be critical for the design and implementation of LPV controllers

34. 34 Issues to Be Addressed Voltage saturation : Rotor dynamics Dynamics of the circuits

35. 35 Issues to Be Addressed Control objective: To keep the rotor centered in the presence of disturbance d. In steady state, it is required that

36. 36 Issues to Be Addressed Power loss reduction by bearing and control design Larger biasing currents in general lead to better performance Larger biasing currents in general lead to better performance Larger biasing currents incur more power loss Larger biasing currents incur more power loss Observation Approach For a given bias level, optimize the performance by appropriate control design