T: Magnetic Bearing: Reference: Robust Control Instructor: Prof. Masayuki Fujita (S5-303B) 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, 1993. Robust Performance 1 1/4/2016
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Fujita Laboratory - T: Magnetic Bearing: Robust Performance · 2016. 4. 1. · Instructor: Prof. Masayuki Fujita (S5-303B) M. Fujita, K. Hatake, F. Matsumura and K. Uchida An Experimental
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T: Magnetic Bearing:
Reference:
Robust ControlInstructor: Prof. Masayuki Fujita (S5-303B)
M. Fujita, K. Hatake, F. Matsumura and K. Uchida An Experimental Evaluation and Comparison of Control for a Magnetic Bearing12th IFAC World Congress, Sydney, Australia, July 18-23, 1993.
Robust Performance
1
1/4/2016
2Figure Magnetic Bearing
Real Physical System
Magnetic Bearing
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
4
Nominal ModelState-Space Representation
• The subscripts “v” and “h” stand for the vertical motion andhorizontal motion of the magnetic bearing.
g: deviations from the steady gap lengths between the electromagnetsand the rotor
i: deviations from the steady currents of the electromagnetse: deviations from the steady voltages of the electromagnets
5
Mathematical Model
6
Nominal Model• Gyroscopic effect :
• If (ignore gyroscopic effect)
- (v) Vertical plant
- (h) Horizontal plant
• Nominal model
Model Uncertainty• Perturbation (gyro effect )
• Uncertainty weight
• Robust stability
7
Performance• Performance weight
• Nominal performance
8
9
Loop Shaping
• For frequencies:
• For frequencies:
• Loop shaping:
:
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.
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
12
Loop Shaping Design Procedure
• Normalized left coprime factorization
• Uncertainties
• Robust stabilizing problem
13
Loop Shaping Design Procedure
14
Design for vertical motion
Design for horizontal motion
15
Loop Transfer Function
16
Nominal Performance and Robust Stability
NP Test RS Test
17Fig. Feedback Structure
K
y
00
u
Interconnection Structure
18
Mixed Sensitivity Design
)(sWT
u yyg
WTy systemnames = 'G WT'; inputvar = '[ u(4) ]';outputvar = '[ WT; G ]'; input_to_G = '[ u ]'; input_to_WT = '[ G ]';GWT = sysic;