1 Basic Control Theory and Its Application in AMB Systems Zongli Lin University of Virginia

2 Representation of a Plant

3

4 Example 4

5 Principles of Feedback Tracking Requirement

6 Disturbance Rejection Disturbance Rejection Requirement (change in load, aero or mechanical forces, etc)

7 Sensitivity Sensitivity to plant uncertainties

8 High Gain  Instability High-Gain Causes Instability k=256

9 Limitations of Constant Feedback Constant feedback is often insufficient

10 Integral Action Use of integral action for zero steady state error (r(t)=1(t), e.g., raise rotor vertical position) K(s) = kK(s) = k/s Observation: Large k causes actuator saturation

11 Differential Action Use of differential action for closed loop stability K(s)=kK(s)=k P +k D s In general: PID control

12 Example Decentralized PI/PD position control of active magnetic bearings Above: cross section of the studied active magnetic bearing system Right: cross section of the studied radial magnetic bearing Reference: B. Polajzer, J. Ritonja, G. Stumberger, D. Dolinar, and J. P. Lecointe, “Decentralized PI/PD position control for active magnetic bearings”, Electrical Engineering, vol. 89, pp , 2006.

13 Stabilization Stabilization: PD control

14 Steady State Error Steady state error reduction: PI Control (e.g., to avoid mechanical contact) PI/PD control configuration

15 PD/PD vs PID Control PD: Choice of K d and T d is ad hoc. PI: Choice of K i and T i is ad hoc. PID: Choice of 3 parameters even harder

16 Experimental Results

17 Experimental Results

18 Lag and Lead Compensation Compensation objectives: Increased PM Increased steady state accuracy

19 Lag and Lead Compensation Phase lag compensator

20 Lag and Lead Compensation Phase lead compensator

21 Two mass symmetric model of the rotor in an LP centrifugal compressor An LP Centrifugal Compressor (ISO )

22 Mathematical model An LP Centrifugal Compressor (ISO )

23 Transfer functions: An LP Centrifugal Compressor (ISO )

24 Decentralized control design Stabilization requires a large increase in the phase An LP Centrifugal Compressor (ISO ) Open-loop poles:

25 Three PD controllers (to approximate 3 lead compensators) An LP Centrifugal Compressor (ISO )

26 Compensated bode plots An LP Centrifugal Compressor (ISO ) Closed-loop poles:

27 An LP Centrifugal Compressor (ISO ) Closed-loop poles in the presence of aero cross coupling:

28 Challenges in Control of AMB Systems PID Control and lead/lag compensators Choice of parameters Coupling between channels Flexible rotor leads to higher order plant model

29 A More Realistic Design Example

30 A More Realistic Design Example Parameter varying (gyroscopic effects): Potential approach: LPV (Linear Parameter Varying) Approach – Based on gain scheduling – Conservative in performance

31 A More Realistic Design Example Piecewise  Design several  controllers at different speeds; Each controllers robust in a speed range; Switch between controllers as the speed varies; Bumpless switching is the key

32 A More Realistic Design Example Control Switching Conditions for a Bumpless Transfer:

33 A More Realistic Design Example Bumpless Transfer Build an observer that estimates the off line controller state from the on line controller output Use the estimated state as the initial state at time of switching As a result,

34 A More Realistic Design Example Piecewise  controller design Controller 1 at nominal speed 10,000 rpm Controller 2 at nominal speed 15,000 rpm Each covers +/ rpm 48 th order controllers

35 A More Realistic Design Example Transfer at 12,000 rpm (upper bearing, x direction)

36 Transfer at 12,000 rpm (upper bearing, y direction) A More Realistic Design Example

37 Nonlinearity of the AMB input-output characteristics More Challenges in Control of AMB Systems