robust control of magnetic bearings

7/31/2019 Robust Control of Magnetic Bearings

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Multi-Objective Robust Control of

Rotor/Active Magnetic Bearing Systems

Ibrahim Sina Kuseyri

Ph.D. Dissertation

June 13, 2011

I. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 2011 1 / 51


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Outline

1 Introduction

Overview

Applications

2 System Dynamics

Magnetic Bearings

Rotordynamics

3 Robust ControlController Design

Model Uncertainty

Robust Stability and Performance

Numerical Results and Simulations

4 Multi-Objective LPV Control

Linear Parametrically Varying (LPV) Systems

Mixed Performance Specifications




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Outline

1 Introduction

Overview

Applications

2 System Dynamics

Magnetic Bearings

Rotordynamics


Model Uncertainty









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Overview

Radial electromagnetic bearing

50 100 150 200 250 300 350

50

100

150

200

250



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Overview

Radial electromagnetic bearing

50 100 150 200 250 300 350

50

100

150

200

250

Horizontal rotor with active magnetic bearings (AMBs)



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Advantages of rotor/AMB systems

No mechanical wear and friction.

No lubrication therefore non-polluting.

High circumferential speeds possible (more than 300 m/s).

Operation in severe and demanding environments.

Easily adjustable bearing characteristics (stiffness, damping).

Online balancing and unbalance compensation.

Online system parameter identification possible.



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Applications

Satellite flywheels

Turbomachinery

High-speed milling and

grinding spindles

Electric motors

Turbomolecular pumps

Blood pumps

Computer hard diskdrives, x-ray devices, ...



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Outline

1 Introduction

Overview

Applications

2 System Dynamics

Magnetic Bearings

Rotordynamics


Model Uncertainty









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Electromagnetic Bearings

The AMB model considered is based on the zero leakage assumption:

Magnetic flux in a high permeability magnetic structure with small airgaps is confined to the iron and gap volumes.

In the configuration above, the forces in orthogonal directions are

almost decoupled and can be calculated separately.



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Electromagnetic bearings

Two opposing electromagnets at orthogonal directions cause the force

Fr = F+ F = kM i+s0 r2

is0 + r2

on the rotor. The magnetic bearing constant kM is

kM :=0AAn

2c

4

cos M

with M denoting the angle between a pole and magnet centerline.



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The non-linearities of the magnetic force are generally reduced by

adding a high bias current i0 to the control currents

ic in each control

axis. Linearization in one axis around the operating point leads to

Fr = Fr|OP +Fri

OP

(ic ic OP) +Frr

OP

(r rOP) .



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The non-linearities of the magnetic force are generally reduced by

adding a high bias current i0 to the control currents

ic in each control

axis. Linearization in one axis around the operating point leads to

Fr = Fr|OP +Fri

OP

(ic ic OP) +Frr

OP

(r rOP) .

At ic OP = 0 and rOP = 0, the linearized magnetic bearing force of thebearing for small currents and small displacements is given by

Fr,lin = kiic ksr

with the actuator gain ki and the open loop negative stiffness ksdefined as

ki := 4kMi0

s20and ks := 4kM

i20s30



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Rotordynamics

Equations of motion for a rigid rotor may be derived from

F= P= ddt

(Mrv) , and M = H = ddt

(I) .

a b

bearing A bearing B

fa1

fa2

fa3

fa4

fb1

fb2

fb3

fb4

x,

y,

z,

mub,s

mub,c

mub,c

CGd

2



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Rotordynamics

The equations of motion for the four degrees of freedom are

x =1

Mr[fA,x + fB,x +

Mr2

g+mub,s

22dcos (t + s)] ,

y =1

Mr [fA,y + fB,y +

Mr2g+

mub,s

2 2dsin (t + s)] ,

=1

Ir[Ip + a(fA,y) + b(fB,y) + (a+ b)

2mub,c

2dsin (t + c)] ,

=

1

Ir [Ip

+

a(fA,x) +

b(

fB,x)

(a+ b)

2m

ub,c

2dcos(

t+

c)]

.


R /AMB d l i


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Rotor/AMB model in state-space

The equations of motion for the electromechanical system in thestate-space form are

xr =

0 I

AS AG()

xr + Bwr w + Bur u+ g,

where xr := (x y x y )T, u = (icA,x icA,y icB,x icB,y)

T,

w = ( 12 mub,sd12 mub,cd)

T.


R t /AMB d l i t t


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Rotor/AMB model in state-space

The equations of motion for the electromechanical system in thestate-space form are

xr =

0 I

AS AG()

xr + Bwr w + Bur u+ g,

where xr := (x y x y )T, u = (icA,x icA,y icB,x icB,y)

T,

w = ( 12 mub,sd12 mub,cd)

T.

Control objective is to stabilize the system and to minimize the rotor

displacements (whirl) with acceptable control effort.


O tli


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Outline

1 Introduction

Overview

Applications

2 System Dynamics

Magnetic Bearings

Rotordynamics


Model Uncertainty








C t ll d i


18/70

Controller design

Kym u

di

n+

+ v

di

nw

+ +

z

yuP

K

{v ym

ue}

G

G


Controller design


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Controller design

Measurement(Feedback)Input

w z

u y

Manipulated

K

P

Performance

OutputInput

Exogenous


Controller design


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Controller design


w z

u y

Manipulated

K

P

Performance

OutputInput

Exogenous

Q: How to choose K?


Controller design


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Controller design


w z

u y

Manipulated

K

P

Performance

OutputInput

Exogenous

Q: How to choose K?

A: Minimize the size (e.g. H or H2-norm) of the closed-loop

transfer function M from w to z.

w zM


H and H norms


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H2 and H-norms

The definitions are

M := sup

M(j)

Note : (M) :=

max(MM)

M2 := 12

Trace

M(j)M(j)

d


H2 and H -norms


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H2 and H-norms

The definitions are

M := sup

M(j)

Note : (M) :=

max(MM)

M2 := 12

Trace

M(j)

M(j)

d

For SISO LTI systems,

M

= sup

|M(j)

|= peak of the Bode plot

M2 =

12

|M(j)|2 d area under the Bode plot


Frequency Weighting


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Frequency Weighting

Can fine-tune the solution by using frequency weights on w and z.

K +

ym

udi do

n

+

+

+

++

+

v

u

n

di do

eWr

Wu Wi Wo We

Wn

+

e

ri ri ri ymG

log

|W|dB

c log

|W|dB

ul log

|W|dB

c


Model uncertainty


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Model uncertainty

Uncertainty in Rotor/AMB Models


Model uncertainty


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Model uncertainty

Uncertainty in Rotor/AMB Models

Model Parameter Uncertainty (such as AMB stiffness ks)

Neglected High Frequency Dynamics (high frequency flexible

modes of the rotor)

Nonlinearities (such as hysteresis effects in AMB)

Neglected Dynamics (such as vibrations of rotor blades)

Setup Variations (e.g., a controller for an AMB milling spindle

should function with tools of different mass)

Changing System Dynamics (gyroscopic effects change the

location of the poles at different operating speeds)


Closed-loop rotor/AMB system with uncertainty


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Closed loop rotor/AMB system with uncertainty

K

WqWp

WzWww zw z

p q

yu

P

p q

P

W1p (j) (j) W1q (j)

=

(j)

1 Re :=

ksI 0

0 I


Closed-loop rotor/AMB system with uncertainty


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Closed loop rotor/AMB system with uncertainty

Overall system in the state-space form

K

WqWp

WzWww zw z

p q

yu

P

p q

P

x = Ax + Bpp+ Bww + Buuq = Cqx + Dqww

z = Czx + Dzuu

y = Cyx + Dyww

p = q


Robust stability and performance


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w z

qp

w z

M

N




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y p

w z

qp

w z

M

N

Nominal Stability (NS) M is internally stableNominal Performance

NS, and M(j) < ReRobust Stability (RS) NS, and

N to be stable : (j) 1 ReRobust Performance RS, andN(j) < : (j) 1 Re


Robust stability - Structured singular value


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y g

Transfer matrix of the closed-loop uncertain system in LFT form is

N=

Mzw +

Mzp(

I

Mqp)

1Mqw

.




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y g


N=

Mzw +

Mzp(

I

Mqp)

1Mqw

.

For robust stability

I Mqp(s)(s)1

should have no poles in C+

for

all with () 1 .

Meaning that = detIMqp(j) = 0, with () 1, Re.




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y g


N = Mzw

+ Mzp

(I

Mqp

)1Mqw

.


I Mqp(s)(s)1


for

all with () 1 .

Meaning that = detIMqp(j) = 0, with () 1, Re.Therefore, robust stability holds if and only if

inf

{() : det

I Mqp(j)= 0, Re} > 1 .




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y g


N = Mzw

+ Mzp

(I

Mqp

)1Mqw

.


I Mqp(s)(s)1


for

all with () 1 .

Meaning that = detIMqp(j) = 0, with () 1, Re.Therefore, robust stability holds if and only if

inf

{() : det

I Mqp(j)= 0, Re} > 1 .

Inversion leads to the definition

(M) :=1

inf {() : detI Mqp(j)

= 0} < 1 Re.


Numerical Results - System Data


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A

A

bearing A bearing B

touch-down bearing A touch-down bearing B

displacement sensors

magneticmagnetic

sA

a b

sB

LD

LS

dDSection A-A

dS

g

Symbol Value Unit Symbol Value Unit Symbol Value Unit

MS 85.90 kg LS 1.50 m s0 2.0 103 m

MD 77.10 kg LD 0.05 m s1 0.5 103 m

Ir 17.28 kgm2 dS 0.10 m i0 3.0 A

Ip 2.41 kgm2 dD 0.50 m kM 7.8455 105 Nm2/A2

a 0.58 m sA 0.73 m ks 3.5305 105 N/m

b 0.58 m sB 0.73 m ki 235.4 N/A


Numerical Results - Weighting functions


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Wu =

38s+ 1200

s+ 50000

I4 We =s+ 0.05

s+ 0.01

I4

102

100

102

104

106

5

0

5

10

15

20

25

30

35

Frequency [rad/s]

Gain[dB]

Wu

102

100

102

104

106

0

2

4

6

8

10

12

Frequency [rad/s]

Gain[dB]

We


Results with the H controllers for the nominal system


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Maximum operation speed = 3000 rpm ( 314.2 rad/s)

102

100

102

104

106

100

80

60

40

20

0

20

Singular Values

Frequency (rad/sec)

SingularValues(dB)

Singular values of controller K1

102

100

102

104

106

100

80

60

40

20

0

20

Singular Values

Frequency (rad/sec)

SingularValues(dB)


102

100

102

104

106

140

120

100

80

60

40

20

0

20

40

Singular Values

Frequency (rad/sec)

SingularValues(dB)

Closedloop SVs with K1

102

100

102

104

106

140

120

100

80

60

40

20

0

20

40

Singular Values

Frequency (rad/sec)

SingularValues(dB)



Results with the H controllers for the nominal system


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Table: H performance with K1 for different design parameters

Maximum speed (rpm) Maximum mass center displacement (m) 1500 0.25103 70.96

3000 0.25103 97.06

6000 0.25103 99.81

1500 0.50103 89.57

3000 0.50103 99.24

6000 0.5010

3

100.07

Table: H performance with K2 for different design parameters

Maximum speed (rpm) Maximum mass center displacement (m)

1500 0.25103 11.41

3000 0.25103 15.42

6000 0.25103 31.77

1500 0.50103 12.62

3000 0.50103 21.05

6000 0.50103 52.01


Critical speeds (eigenfrequencies)


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PoleZero Map

Real Axis

ImaginaryAxis

250 200 150 100 50 0 50 100 150 200 250

60

40

20

0

20

40

60

x: Openloop eigenfrequencies at standstill (rad/s)

117(x2)

117(x2)

65.8(x2)

65.8(x2)

100

101

102

103

104

200

150

100

50

0

50

100

Frequency (Speed) [rad/s]ClosedloopPhaseshiftforjournaldispla

cements(unbalancechannel)

XA

YA

XB

YB

120

Phase shift with K1

100

101

102

103

104

200

180

160

140

120

100

80

60

40

20

0

Frequency[rad/s]ClosedloopPhaseshiftforjournaldispla

cements(unbalancechannel)

XA

YA

XB

YB

150


Results with the reduced order H controllers


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The H norm of the closed-loop system at 3000 rpm with the reduced

ordered controllers K1r and K2r (4 states are eliminated) increases

from 99.24 to 529.55 and from 21.05 to 62.07 respectively.

102

100

102

104

106

140

120

100

80

60

40

20

0

20

40

60

Singular Values

Frequency (rad/sec)

SingularValues(dB)

Closedloop SVs with K1r

102

100

102

104

106

140

120

100

80

60

40

20

0

20

40

Singular Values

Frequency (rad/sec)

SingularValues(dB)

Closedloop SVs with K2r

102

101

100

101

102

103

104

200

150

100

50

0

50

Frequency[rad/s]ClosedloopPhaseshiftforjournaldisplace

ments(unbalancechannel)

XA

YA

XB

YB

170

102

101

100

101

102

103

104

200

180

160

140

120

100

80

60

40

20

0

Frequency[rad/s]ClosedloopPhaseshiftforjournaldisplace

ments(unbalancechannel)

XA

YA

XB

YB

185


Robust stability of the uncertain closed-loop system


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Keeping the uncertainty on the bearing stiffness constant (25%),robust stability of the closed-loop system is tested for several

maximum operating speeds with -analysis.

Moreover, keeping the operation speed constant (3000 rpm), robust

stability is tested for uncertainty in bearing stiffness.

3000 3500 4000 4500 5000 5500 60000.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Maximum rotor speed (RPM)

mu

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Uncertainty in bearing stiffness (%)

mu


Results with the robust H controller


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Singular values of the controller and the closed-loop system for a

maximum operating speed of 4085 rpm are shown below.

H performance of the system for max = 4085 rpm is 47.86.

Order of the controller K3 (twelve) can not be reduced since it leads to

the instability of the closed-loop system.

102

100

102

104

106

100

80

60

40

20

0

20

Singular Values

Frequency (rad/sec)

SingularValues(dB)


102

100

102

104

106

1000

800

600

400

200

0

200

Singular Values

Frequency (rad/sec)

SingularValues(dB)



Simulations


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Simulation Environment in SIMULINK


Simulations


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Simulation Environment in SIMULINK (Rotor/AMB)


Simulations


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We analyze the H performance of the closed-loop system using the

controller K2 in the simulations. Disturbance acting on the system, i.e.,unbalance force and sensor/electronic noise, are shown below.

0 0.1 0.2 0.3 0.4 0.5100

80

60

40

20

0

20

40

60

80

100

Time (sec)

UnbalanceForce(Newtons)

0 100 200 300 400 500 6000.03

0.02

0.01

0

0.01

0.02

0.03

0.04

Time (msec)

Volts

Sensor Noise


Simulations


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0 0.1 0.2 0.3 0.4 0.52

1

0

1

2

3

4

5

6

Time (sec)

XA

(Volts)

Rotor displacement in Bearing A(xdirection)

0 0.1 0.2 0.3 0.4 0.56

5

4

3

2

1

0

1

2

Time (sec)

YA

(Volts)

Rotor displacement in Bearing A(ydirection)

0 0.1 0.2 0.3 0.4 0.54

3

2

1

0

1

2

Time (sec)

ic

,Ax(Amperes)

Control current for Bearing A(xaxis)

0 0.1 0.2 0.3 0.4 0.52

1

0

1

2

3

4

Time (sec)

ic,Ay(Amperes)

Control current for Bearing A(yaxis)

Rotor position and control currents during start-up


Simulations


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Mass center displacement (eccentricity) due to unbalance of the rotor

is assumed to be 0.25 103 m in the simulations.

Peak value of the vibration (except the transient) is less than 0.1 V,corresponding to 14 106 m. Therefore, the H controller K2 reducesthe unbalance whirl amplitude of the rotor more than 95%.

0 0.1 0.2 0.3 0.4 0.52

1.5

1

0.5

0

0.5

Time (sec)

XA

(Volts)

Rotor displacement in Bearing A (xdirection)

0 0.1 0.2 0.3 0.4 0.52

1.5

1

0.5

0

0.5

1

Time (sec)

YA

(Volts)

Rotor displacement in Bearing A (ydirection)

0 0.1 0.2 0.3 0.4 0.53

2

1

0

1

2

3

4

Time (sec)

ic,Ax(Amperes)

Control current for Bearing A (xaxis)

0 0.1 0.2 0.3 0.4 0.53

2

1

0

1

2

3

4

Time (sec)

ic,Ay(Amperes)

Control current for Bearing A (yaxis)


Outline


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1 Introduction

Overview

Applications2 System Dynamics

Magnetic Bearings

Rotordynamics


Model Uncertainty








LPV Systems


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x

z

y =

A() Bw() Bu()Cz() Dzw() Dzu()

Cy() Dyw() Dyu()

x

w

u

Parameters (t) are measured in real-time with sensors for control.


LPV Systems


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x

z

y =

A() Bw() Bu()Cz() Dzw() Dzu()

Cy() Dyw() Dyu()

x

w

u

Parameters (t) are measured in real-time with sensors for control.

Hence controller is also parameter-dependent, using the available

real-time information of the parameter variation.

u y

w z

P

K




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Suppose a specific control task leads to the generalized LPV plant

x

z1z2y

=

A() B1() B2()C1() D11() D12()C2() D21() D22()C() D() 0

xw

u

Using an LPV controller, K(, ), the closed-loop system can bedescribed in the form

xclz1z2

=A BC1 D1C2 D2

xclw




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xclz1z2 =

A B

C1

D1

C2 D2xcl

w

L2 gain of the w z1 channel is defined as

opt := infKK supw2=0 z1

2

w2where K := {set of all stabilizing controllers} .




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xclz1z2 =

A B

C1

D1

C2 D2xcl

w

L2 gain of the w z1 channel is defined as

opt := infKK supw2=0 z1

2

w2where K := {set of all stabilizing controllers} .To quantify the gain of the channel w z2 we use the induced norm

opt := infKK

supw2=0

z2w2

Remark: z2 :=

z(t)Tz(t) dt


54/70

We can use a single Lyapunov function to achieve both of the control

objectives (though conservatively) and the problem can be defined as

minimizing an upper bound m under the constraint < m.





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This leads to defining the mixed objective functional

I

K(X) := inf {m | a function X() satisfying < m and < m}from the solution of the following infinite dimensional LMIs for all (, ):

X= XT 0 ,X+ ATX+ XA XB CT1BTX I DT1

C1 D1 2mI

0 , C2X1C2 mI,





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This leads to defining the mixed objective functional

I

K(X) := inf {m | a function X() satisfying < m and < m}from the solution of the following infinite dimensional LMIs for all (, ):

X= XT 0 ,X+ ATX+ XA XB CT1BTX I DT1

C1 D1 2mI

0 , C2X1C2 mI,

whereX is defined to be

X:=m

i=1

Xi

i


Controller Synthesis

A full order controller K which satisfying the mixed objective functional


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A full order controller K which satisfying the mixed objective functionalI

K(X)

can be constructed, if there exist parameter-dependent

functions X(), Y() with X

0 , Y

0 , and E(), F(), G() with

G() = DK(), such thatX + ATX + XA + FC+ (FC)T XB1 + FD (C1 + D12GC)T(XB1 + FD)T I (D11 + D12GD)T

C1 + D12GC D11 + D12GD 2mI

0 ,

Y + AY + YAT + B2E + (B2E)T B1 + B2GD (C1Y + D12E)T(B1 + B2GD)T I (D11 + D12GD)T

C1Y + D12E D11 + D12GD 2mI

0 ,

mI C2Y + D22E C2 + D22GC

(C2Y + D22E)T Y I

(C2 + D22GC)T I X 0 .

Inequalities above consist of convex but infinite-dimensional

optimization problem.


LPV Model for Rotor/AMB Systems


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x

z1z2y

=

A() B1(2) B2

C1 0 D12C2 0 0

C D 0

xw

u

System has parameter dependence to (t) due to gyroscopic effects

and to 2(t) due to unbalance forces.




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x

z1z2y

=

A() B1(2) B2

C1 0 D12C2 0 0

C D 0

xw

u



Letting all of the parameter dependent functions to have an affine

structure, (such as X() = X0 + X1) infinite-dimensional inequalitiesfor controller synthesis become a series of LMIs with linear

dependence on and linear/quadratic/cubic dependence on .




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x

z1z2y

= A() B1(

2) B2

C1 0 D12C2 0 0

C D 0

xwu



Letting all of the parameter dependent functions to have an affine

structure, (such as X() = X0 + X1) infinite-dimensional inequalitiesfor controller synthesis become a series of LMIs with linear

dependence on and linear/quadratic/cubic dependence on .Hence one only needs to check these matrix inequalities at the

vertices of the polytope defined by P = [min, max] [min, max]


Numerical Results with LPV (L2) ControllersLPV controller for the parameter (rotor speed) dependent rotor/AMB


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LPV controller for the parameter (rotor speed) dependent rotor/AMB

system can be designed via semidefinite programming satisfying

several LMIs at all the vertices of the convex hull.

Singular values of the closed-loop system at two different speeds;

3000 and 6000 rpm are shown below:

102

100

102

104

106

108

300

250

200

150

100

50

0

50

Singular Values

Frequency (rad/sec)

S

ingularValues(dB)

LPV ClosedloopSVs at 3000 RPM

102

100

102

104

106

108

250

200

150

100

50

0

50

Singular Values

Frequency (rad/sec)

S

ingularValues(dB)

LPV ClosedloopSVs at 6000 RPM


Results with LPV (L2) ControllersController to robustly stabilize the system with L2 performance is


62/70

Controller to robustly stabilize the system with L2 performance issynthesized inside a four-dimensional convex hull with the rotor speed

range from 0 rad/s to 614 rad/s (6000 rpm), and angular acceleration

range from -15 rad/s2 to 15 rad/s2.




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y y 2 p

synthesized inside a four-dimensional convex hull with the rotor speed



L2 performance of the closed-loop LPV system at the instantaneousspeed 6000 RPM is 56.31. Note that this performance is achieved with

a controller of the formxKu

=

AK(, ) BK()CK() DK()

xKy




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y y 2 p

synthesized inside a four-dimensional convex hull with the rotor speed



L2 performance of the closed-loop LPV system at the instantaneousspeed 6000 RPM is 56.31. Note that this performance is achieved with

a controller of the formxKu

=

AK(, ) BK()CK() DK()

xKy

If the matrix function X used for the stabilization of the closed-loop

system is assumed to be constant (time-invariant), then the controller

matrices will not depend on the angular acceleration of the rotor, andthe controller will be of the form

xKu

=

AK() BK()CK() DK()

xKy


Results with LPV (L2) ControllersComparing the L2 performance of the controllers, it can be said that


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there is virtually no loss of performance if the controller is constructedwithout the information on angular acceleration of the rotor.

Table: L2 performance of LPV closed-loop systems at 3000 RPMStructure of X and Y Controller Form

X = X0 + X1 Y = Y0 + Y1 15.92 Acceleration Feedback

X = X0 Y = Y0 + Y1 19.13 No Acc. Feedback

X = X0 Y = Y0 27.56 No Acc. Feedback

Table: L2 performance of LPV closed-loop systems at 6000 RPMStructure of X and Y Controller Form

X = X0 + X1 Y = Y0 + Y1 56.31 Acceleration Feedback

X = X0 Y = Y0 + Y1 65.42 No Acc. Feedback

X = X0 Y = Y0 102.29 No Acc. Feedback


Numerical Results with Multi-objective LPV Controller

A multi-objective LPV controller with mixed performance specification


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is synthesized within the same convex hull as the single objective LPV

controller for a maximum operating speed of 6000 rpm.

Generalized L2 L performance m of the multi-objective LPVcontroller is found to be 364.4, with L2 performance level m of 72.12at 6000 rpm.

102

100

102

104

106

100

80

60

40

20

0

20

40

Singular Values

Frequency (rad/sec)

SingularValues(dB)

SVs of MultiobjectiveController at 6000 RPM

102

100

102

104

106

108

250

200

150

100

50

0

50

Singular Values

Frequency (rad/sec)

SingularValues(dB)

Closedloop SVs ofMultiobjective LPVSystem at 6000 RPM


Simulations with Multi-objective LPV Controller

Simulations for the LPV system are made using the LFR Toolbox from


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ONERA for MATLAB R-Simulink.


Simulations with Multi-objective LPV Controller

A pulse signal with 1 V amplitude and 0.025 seconds duration and is


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injected into the loop at 0.2 seconds of simulation time at the input of

the controller. Control current and rotor position at bearing A in y-axis

for LPV control with L2 performance and with mixed performance isshown in the figures.

0 0.1 0.2 0.3 0.4 0.54

3

2

1

0

1

2

3

4

Time (sec)

ic,Ay(Amperes)

0 0.1 0.2 0.3 0.4 0.51.2

1

0.8

0.6

0.4

0.2

0

0.2

0.4

Time (sec)

YA

Figure: Control current and rotor displacement with LPV L2 control


1

2

3

4

eres) 0.2

0

0.2

0.4


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0 0.1 0.2 0.3 0.4 0.54

3

2

1

0

Time (sec)

ic,Ay(Ampe

0 0.1 0.2 0.3 0.4 0.51.2

1

0.8

0.6

0.4

Time (sec)

YA

Figure: Control current and rotor displacement with LPV L2 control

0 0.1 0.2 0.3 0.4 0.52.5

2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

Time (sec)

ic,Ay

0 0.1 0.2 0.3 0.4 0.51

0.8

0.6

0.4

0.2

0

0.2

0.4

Time (sec)

YA

Figure: Control current and rotor displacement with LPV mixed control


Conclusion


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Comparing the results, it is clear that the peak values of both the

control current and rotor position are suppressed in the closed-loop

system with the multi-objective controller. Hence mixed controlprovides additional flexibility with respect to transients.


robust control of magnetic bearings

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