Robust Control of Rotor/AMB Systems

Robust Control ofRotor/Active Magnetic Bearing Systems

Ibrahim Sina KuseyriBogazici University

15/03/2011

I. Sina Kuseyri (Bogazici University) Robust Control of Rotor/AMB Systems 15/03/2011 1 / 34

Outline

1 IntroductionOverviewApplications

2 System DynamicsMagnetic BearingsRotordynamics

3 Robust ControlController DesignModel Uncertainty and Robustness

4 Numerical Results and Simulations


Outline






Overview

Radial magnetic (electromagnetic) bearing

50 100 150 200 250 300 350

50

100

150

200

250

Horizontal rotor with active magnetic bearings (AMBs)


Overview

Radial magnetic (electromagnetic) bearing

50 100 150 200 250 300 350

50

100

150

200

250

Horizontal rotor with active magnetic bearings (AMBs)


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.


Applications

Satellite flywheelsTurbomachineryHigh-speed milling and grinding spindlesElectric motorsTurbomolecular pumpsBlood pumpsComputer hard disk drives, x-ray devices, ...


Outline






Electromagnetic Bearings

The AMB model considered is based on the zero leakage assumptionwhich says that magnetic flux in a high permeability magnetic structurewith small air gaps is confined to the iron and gap volumes.

If the electromagnets in a radial bearing are arranged as shown above,the forces in orthogonal directions are almost decoupled and can becalculated separately.


Electromagnetic bearings

Two opposing electromagnets at orthogonal directions cause the force

Fr = F+ − F− = kM

((i+

s0 − r

)2

−(

i−s0 + r

)2)

on the rotor. The magnetic bearing constant kM is

kM :=µ0AAn2

c4

cosαM

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



The non-linearities of the magnetic force are generally reduced byadding a high bias current i0 to the control currents ∓ic in each controlaxis. Hence electromagnetic force in one axis can be linearizedaround the operating point as

Fr ∼= Fr |OP +∂Fr

∂i

∣∣∣∣OP

(ic − ic OP) +∂Fr

∂r

∣∣∣∣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 = ki ic − ksr

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

ki := 4kMi0s2

0and ks := −4kM

i20s3

0·



The non-linearities of the magnetic force are generally reduced byadding a high bias current i0 to the control currents ∓ic in each controlaxis. Hence electromagnetic force in one axis can be linearizedaround the operating point as

Fr ∼= Fr |OP +∂Fr

∂i

∣∣∣∣OP

(ic − ic OP) +∂Fr

∂r

∣∣∣∣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 = ki ic − ksr

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

ki := 4kMi0s2

0and ks := −4kM

i20s3

0·


Rotordynamics

Equations of motion for a rigid rotor may be derived from

F = P =ddt

(Mr v) , and M = H =ddt

(Iω) .

θ

a b

bearing A bearing B

φ

ψ

fa1

fa2

fa3

fa4

fb1

fb2

fb3

fb4

x, ζ

y, η

z, ξ

mub,s

mub,c

mub,c

CGd2


Rotordynamics

The equations of motion for the four degrees of freedom are

x =1

Mr[fA,x + fB,x +

Mr√2

g +mub,s

2Ω2d cos (Ωt + ϕs)] ,

y =1

Mr[fA,y + fB,y +

Mr√2

g +mub,s

2Ω2d sin (Ωt + ϕs)] ,

ψ =1Ir

[−ΩIpθ + a(−fA,y ) + b(fB,y ) +(a + b)

2mub,c Ω2d sin (Ωt + ϕc)] ,

θ =1Ir

[ΩIpψ + a(fA,x ) + b(−fB,x )− (a + b)

2mub,c Ω2d cos (Ωt + ϕc)] .


Rotor/AMB model in state-space

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

xr =

(0 I

AS AG(Ω)

)xr + Bwr w + Bur u + g ,

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

w = (12mub,sd 1

2mub,cd)T .

Control objective is to stabilize the system and to minimize the rotordisplacements (whirl) with moderate control effort.


Rotor/AMB model in state-space

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

xr =

(0 I

AS AG(Ω)

)xr + Bwr w + Bur u + g ,

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

w = (12mub,sd 1

2mub,cd)T .

Control objective is to stabilize the system and to minimize the rotordisplacements (whirl) with moderate control effort.


Outline






Controller design

Kym u

di

n+

+v

di

nw

+ +

z

yuP

K

v ym

ue

G

G


Controller design

ControlledOutput

Input and/orDisturbance

Measurement(Feedback)Input

w z

u y

Manipulated

P

K(Controller)

(Generalized P lant)

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


Controller design

ControlledOutput



w z

u y

Manipulated

P

K(Controller)


Q: How to choose K ?

A: Minimize the “size” (e.g. H∞ or H2-norm) of the closed-looptransfer function M from w to z.

w zM


Controller design

ControlledOutput



w z

u y

Manipulated

P

K(Controller)


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


H2 and H∞-norms

The definitions are

‖M‖∞ := supω

σ(M(jω)

) (Note : σ(M) :=

√λmax (M∗M)

)‖M‖2 :=

√1

2π

∫ ∞−∞

Trace(M(jω)∗M(jω)

)dω

For SISO LTI systems,‖M‖∞ = supω |M(jω)| = peak of the Bode plot

‖M‖2 =√

12π

∫∞−∞ |M(jω)|2 dω ∼ area under the Bode plot


H2 and H∞-norms

The definitions are

‖M‖∞ := supω

σ(M(jω)

) (Note : σ(M) :=

√λmax (M∗M)

)‖M‖2 :=

√1

2π

∫ ∞−∞

Trace(M(jω)∗M(jω)

)dω

For SISO LTI systems,‖M‖∞ = supω |M(jω)| = peak of the Bode plot

‖M‖2 =√

12π

∫∞−∞ |M(jω)|2 dω ∼ area under the Bode plot


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 − ym G−

logω

|W |dB

ωc logω

|W |dB

ωuωl logω

|W |dB

ωc


Model uncertainty

Uncertainty in Rotor/AMB Models

Model Parameter Uncertainty (such as AMB stiffness ks)Neglected High Frequency Dynamics (high frequency flexiblemodes 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 spindleshould function with tools of different mass)Changing System Dynamics (gyroscopic effects change thelocation of the poles at different operating speeds)


Closed-loop rotor/AMB system with uncertainty

K

WqWp

WzWww zw z

p q

yu

∆

P

p q

P

σ(W−1

p (jω) ∆(jω) W−1q (jω)

)= σ

(∆(jω)

)≤ 1 ∀ω ∈ Re

∆ :=

[δksI 0

0 ΩI

]I. Sina Kuseyri (Bogazici University) Robust Control of Rotor/AMB Systems 15/03/2011 20 / 34

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 + Bw w + Buuq = Cqx + Dqw wz = Czx + Dzuuy = Cyx + Dyw wp = ∆q


Outline






Numerical Results - System Data

A

A

bearing A bearing B

touch-down bearing A touch-down bearing B

displacement sensors

magneticmagnetic

sA

a bsB

LD

LS

dDSection A-A dS

g

Symbol Value Unit Symbol Value Unit Symbol Value UnitMS 85.90 kg LS 1.50 m s0 2.0 · 10−3 mMD 77.10 kg LD 0.05 m s1 0.5 · 10−3 mIr 17.28 kg·m2 dS 0.10 m i0 3.0 AIp 2.41 kg·m2 dD 0.50 m kM 7.8455 · 10−5 N·m2/A2

a 0.58 m sA 0.73 m ks −3.5305 · 105 N/mb 0.58 m sB 0.73 m ki 235.4 N/A


Numerical Results - Weighting functions

Wu =

(38

s + 1200s + 50000

)I4 We =

(s + 0.05s + 0.01

)I4

10−2

100

102

104

106

−5

0

5

10

15

20

25

30

35

Frequency [rad/s]

Ga

in [

dB

]

Wu

10−2

100

102

104

106

0

2

4

6

8

10

12

Frequency [rad/s]

Ga

in [

dB

]

We


Results with the H∞ controllers for the nominal system

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

10−2

100

102

104

106

−100

−80

−60

−40

−20

0

20

Singular Values

Frequency (rad/sec)

Sin

gula

r V

alu

es (

dB

)

Singular values of controller K1

10−2

100

102

104

106

−100

−80

−60

−40

−20

0

20

Singular Values

Frequency (rad/sec)

Sin

gula

r V

alu

es (

dB

)


10−2

100

102

104

106

−140

−120

−100

−80

−60

−40

−20

0

20

40

Singular Values

Frequency (rad/sec)

Sin

gula

r V

alu

es (

dB

)

Closed−loop SVs with K1

10−2

100

102

104

106

−140

−120

−100

−80

−60

−40

−20

0

20

40

Singular Values

Frequency (rad/sec)

Sin

gula

r V

alu

es (

dB

)



Results with the H∞ controllers for the nominal system

Table : H∞ performance with K1 for different design parameters

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

1500 0.25·10−3 70.963000 0.25·10−3 97.066000 0.25·10−3 99.811500 0.50·10−3 89.573000 0.50·10−3 99.246000 0.50·10−3 100.07

Table : H∞ performance with K2 for different design parameters

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

1500 0.25·10−3 11.413000 0.25·10−3 15.426000 0.25·10−3 31.771500 0.50·10−3 12.623000 0.50·10−3 21.056000 0.50·10−3 52.01


Critical speeds (eigenfrequencies)Pole−Zero Map

Real Axis

Imagin

ary

Axis

−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]

Clo

sedlo

op P

haseshift fo

r jo

urn

al dis

pla

cem

ents

(unbala

nce c

hannel)

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]Clo

sedlo

op P

haseshift fo

r jo

urn

al dis

pla

cem

ents

(unbala

nce c

hannel)

XA

YA

XB

YB

150


Results with the reduced order H∞ controllers

The H∞ norm of the closed-loop system at 3000 rpm with the reducedordered controllers K1r and K2r (4 states are eliminated) increasesfrom 99.24 to 529.55 and from 21.05 to 62.07 respectively.

10−2

100

102

104

106

−140

−120

−100

−80

−60

−40

−20

0

20

40

60

Singular Values

Frequency (rad/sec)

Sin

gu

lar

Va

lue

s (

dB

)

Closed−loop SVs with K1r

10−2

100

102

104

106

−140

−120

−100

−80

−60

−40

−20

0

20

40

Singular Values

Frequency (rad/sec)

Sin

gu

lar

Va

lue

s (

dB

)

Closed−loop SVs with K2r

10−2

10−1

100

101

102

103

104

−200

−150

−100

−50

0

50

Frequency[rad/s]Clo

se

dlo

op

Ph

ase

sh

ift

for

jou

rna

l d

isp

lace

me

nts

(un

ba

lan

ce

ch

an

ne

l)

XA

YA

XB

YB

170

10−2

10−1

100

101

102

103

104

−200

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

Frequency[rad/s]Clo

se

dlo

op

Ph

ase

sh

ift

for

jou

rna

l d

isp

lace

me

nts

(un

ba

lan

ce

ch

an

ne

l)

XA

YA

XB

YB

185


Robust stability of the uncertain closed-loop system

Using the model incorporating parametric uncertainty structure,nominal bearing stiffness is set to the nominal value with 25%uncertainty. Nominal speed is selected as half of the maximum speedof operation. Keeping the uncertainty on the bearing stiffness constant(25%), robust stability of the closed-loop system is tested for severalmaximum operating speeds with µ-analysis. Moreover, keeping theoperation speed constant (3000 rpm), robust stability is tested foruncertainty 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

Singular values of the controller and the closed-loop system for amaximum operating speed of 4085 rpm (∼= 418 rad/s) are shownbelow. H∞ performance γ of the closed-loop system for Ωmax = 4085rpm is 47.86. Note that the closed-loop system with K3 havetwenty-nine inputs and thirty-two outputs, whereas the closed-loopsystems with K1 and K2 have five inputs and eight outputs. Order ofthe controller K3 (which is twelve) can not be reduced since it leads tothe instability of the closed-loop system.

10−2

100

102

104

106

−100

−80

−60

−40

−20

0

20

Singular Values

Frequency (rad/sec)

Sin

gu

lar

Va

lue

s (

dB

)


10−2

100

102

104

106

−1000

−800

−600

−400

−200

0

200

Singular Values

Frequency (rad/sec)

Sin

gu

lar

Va

lue

s (

dB

)



Simulations

Simulation Environment in SIMULINK (Rotor/AMB)


Simulations

We analyze the H∞ performance of the closed-loop system using thecontroller 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 0.5−100

−80

−60

−40

−20

0

20

40

60

80

100

Time (sec)

Un

ba

lan

ce

Fo

rce

(N

ew

ton

s)

0 100 200 300 400 500 600−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Time (msec)

Vo

lts

Sensor Noise


Simulation Results

0 0.1 0.2 0.3 0.4 0.5−2

−1

0

1

2

3

4

5

6

Time (sec)

XA (

Vo

lts)

Rotor displacement in Bearing A (x−direction)

0 0.1 0.2 0.3 0.4 0.5−6

−5

−4

−3

−2

−1

0

1

2

Time (sec)

YA (

Vo

lts)

Rotor displacement in Bearing A (y−direction)

0 0.1 0.2 0.3 0.4 0.5−4

−3

−2

−1

0

1

2

Time (sec)

ic,A

x (

Am

pe

res)

Control current for Bearing A (x−axis)

0 0.1 0.2 0.3 0.4 0.5−2

−1

0

1

2

3

4

Time (sec)

ic,A

y (

Am

pe

res)

Control current for Bearing A (y−axis)

Rotor position and control currents during start-upI. Sina Kuseyri (Bogazici University) Robust Control of Rotor/AMB Systems 15/03/2011 33 / 34

Simulation Results

Mass center displacement (eccentricity) due to unbalance of the rotoris assumed to be 0.25 · 10−3 m in the simulations. Peak value of thevibration (except the transient) is less than 0.1 V, corresponding to14 · 10−6 m. Hence, the H∞ controller K2 reduces the unbalance whirlamplitude of the rotor more than 95%.

0 0.1 0.2 0.3 0.4 0.5−2

−1.5

−1

−0.5

0

0.5

Time (sec)

XA (

Volts)

Rotor displacement in Bearing A (x−direction)

0 0.1 0.2 0.3 0.4 0.5−2

−1.5

−1

−0.5

0

0.5

1

Time (sec)

YA (

Volts)

Rotor displacement in Bearing A (y−direction)

0 0.1 0.2 0.3 0.4 0.5−3

−2

−1

0

1

2

3

4

Time (sec)

ic,A

x (

Am

pere

s)

Control current for Bearing A (x−axis)

0 0.1 0.2 0.3 0.4 0.5−3

−2

−1

0

1

2

3

4

Time (sec)

ic,A

y (

Am

pere

s)

Control current for Bearing A (y−axis)


Engineering

Robust Control of Rotor/AMB Systems