Instructor: Prof. Masayuki Fujita (S5-303B)

Transcription

1 Robust Control Instructor: Prof. Masayuki Fujita (S5-303B) 1/4/2016 T: Magnetic Bearing: Robust Performance Reference: M. Fujita, K. Hatake, F. Matsumura and K. Uchida An Experimental Evaluation and Comparison of Control for a Magnetic Bearing 12th IFAC World Congress, Sydney, Australia, July 18-23,

2 Real Physical System Magnetic Bearing Figure Magnetic Bearing 2

3 Assumptions The rotor is rigid and has no unbalance All Electromagnets are identical Attractive force of an electromagnet is in proportion to the square of the ratio of the electric current to the gap length The resistance and the inductance of the electromagnet coil are constant and independent of the gap length Small deviations from the equilibrium point are treated 3

4 Nominal Model State-Space Representation g: deviations from the steady gap lengths between the electromagnets and the rotor i: deviations from the steady currents of the electromagnets e: deviations from the steady voltages of the electromagnets The subscripts v and h stand for the vertical motion and horizontal motion of the magnetic bearing. 4

5 Mathematical Model 5

6 Nominal Model Gyroscopic effect : If (ignore gyroscopic effect) - (v) Vertical plant - (h) Horizontal plant Nominal model 6

7 Model Uncertainty Perturbation (gyro effect ) Uncertainty weight Robust stability 7

8 Performance Performance weight Nominal performance 8

9 Loop Shaping For frequencies: For frequencies: Loop shaping : : 9

10 Loop Shaping Design Procedure [Step 1] Loop Shaping Selecting shaping functions and, the singular values of the nominal plant are shaped to have a desired open loop shape. Let represent this shaped plant, and should be selected such that has no hidden unstable modes. 10

11 Loop Shaping Design Procedure [Step 2] Robust Stabilization The maximum stability margin is calculated. If, return to Step 1 and and are reselected. Otherwise, is appropriately selected as, and an controller is synthesized for. [Step 3] Final Controller The final controller can be obtained by the combination of and as 11

12 Loop Shaping Design Procedure Normalized left coprime factorization Uncertainties Robust stabilizing problem 12

13 Loop Shaping Design Procedure 13

14 Design for vertical motion Design for horizontal motion 14

15 Loop Transfer Function 15

16 Nominal Performance and Robust Stability NP Test RS Test 16

17 Interconnection Structure 0 0 u y K Fig. Feedback Structure 17

18 Mixed Sensitivity Design u W T (s) y WT y g y MATLAB Command systemnames = 'G WT'; inputvar = '[ u(4) ]'; outputvar = '[ WT; G ]'; input_to_g = '[ u ]'; input_to_wt = '[ G ]'; GWT = sysic; w u K (s) W T (s) z 1 W S (s) z 2 y MATLAB Command % Generalized Plant systemnames = 'WP GWT'; inputvar = '[ w(4); u(4) ]'; outputvar = '[ WP; GWT(1:4); GWT(5:8)+w ]'; input_to_wp = '[ GWT(5:8)+w ]'; input_to_gwt = '[ u ]'; Gmix = sysic; nmeas = 4; ncon = 4; [Kmix, CLmix, gammix, infomix] = hinfsyn(gmix, nmeas, ncon, 'tolgam', 0.1); 18

19 -Controller Set of Plants Robust Stability Robust Performance 19

20 -Controller: Structured Singular Value Linear Fractional Transformation Block Structure Structured Singular Value Robust Performance Test 20

21 -Controller: D-K Iteration D-K Iteration : 1st 2nd order fit for the D-scaling 21

22 -Controller: D-K Iteration D-K Iteration : 2nd 22

23 -Controller: Analysis 23

24 -Controller: Analysis 24

25 Digital Implementation and Experiments Sampling Time : : : Discretization : Tustin Transform MATLAB Command G = tf(num, den); Gd = c2d(g, T, tustin ); 25

26 Digital Implementation and Experiments Additional weight is about 3.3 kg Experiment 1 Sine-wave type signal of only one cycle The frequency of the sine-wave is 10 Hz The peak value is 6 V for the vertical case and 4.5 V for the horizontal case Experiment 2 Step type signal Applied voltage is 5 V 26

27 Results of Experiment 1 : Nominal 27

28 Results of Experiment 1 : Perturbed 28

29 Results of Experiment 2 : Nominal 29

30 Results of Experiment 2: Perturbed 30

31 Unbalance Control Free parameter Condition 31

32 Unbalance Control Design for vertical motion Design for horizontal motion Rotational speed (1200 rpm) 32

33 Unbalance Control: Experimental Results 33

34 Unbalance Control: Experimental Results 34

35 Unbalance Control: Experimental Results 35

36 Unbalance Control: Experimental Results 36

37 Unbalance Control: Experimental Results 37