robust control of rotor/amb systems
TRANSCRIPT
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Robust Control ofRotor/Active Magnetic Bearing Systems
Ibrahim Sina KuseyriBogazici University
15/03/2011
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Outline
1 IntroductionOverviewApplications
2 System DynamicsMagnetic BearingsRotordynamics
3 Robust ControlController DesignModel Uncertainty and Robustness
4 Numerical Results and Simulations
I. Sina Kuseyri (Bogazici University) Robust Control of Rotor/AMB Systems 15/03/2011 2 / 34
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Outline
1 IntroductionOverviewApplications
2 System DynamicsMagnetic BearingsRotordynamics
3 Robust ControlController DesignModel Uncertainty and Robustness
4 Numerical Results and Simulations
I. Sina Kuseyri (Bogazici University) Robust Control of Rotor/AMB Systems 15/03/2011 3 / 34
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Overview
Radial magnetic (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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Overview
Radial magnetic (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 flywheelsTurbomachineryHigh-speed milling and grinding spindlesElectric motorsTurbomolecular pumpsBlood pumpsComputer hard disk drives, x-ray devices, ...
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Outline
1 IntroductionOverviewApplications
2 System DynamicsMagnetic BearingsRotordynamics
3 Robust ControlController DesignModel Uncertainty and Robustness
4 Numerical Results and Simulations
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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.
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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.
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Electromagnetic bearings
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·
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Electromagnetic bearings
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·
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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
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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)] .
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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(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.
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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(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.
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Outline
1 IntroductionOverviewApplications
2 System DynamicsMagnetic BearingsRotordynamics
3 Robust ControlController DesignModel Uncertainty and Robustness
4 Numerical Results and Simulations
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Controller design
Kym u
di
n+
+v
di
nw
+ +
z
yuP
K
v ym
ue
G
G
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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
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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-looptransfer function M from w to z.
w zM
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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
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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
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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
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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 − ym G−
logω
|W |dB
ωc logω
|W |dB
ωuωl logω
|W |dB
ωc
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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 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)
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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
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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 + Bw w + Buuq = Cqx + Dqw wz = Czx + Dzuuy = Cyx + Dyw wp = ∆q
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Outline
1 IntroductionOverviewApplications
2 System DynamicsMagnetic BearingsRotordynamics
3 Robust ControlController DesignModel Uncertainty and Robustness
4 Numerical Results and Simulations
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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
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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
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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
)
Singular values of controller K2
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
)
Closed−loop SVs with K2
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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
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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
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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
I. Sina Kuseyri (Bogazici University) Robust Control of Rotor/AMB Systems 15/03/2011 28 / 34
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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
I. Sina Kuseyri (Bogazici University) Robust Control of Rotor/AMB Systems 15/03/2011 29 / 34
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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
)
Singular values of controller K3
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
)
Closed−loop SVs with K3
I. Sina Kuseyri (Bogazici University) Robust Control of Rotor/AMB Systems 15/03/2011 30 / 34
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Simulations
Simulation Environment in SIMULINK (Rotor/AMB)
I. Sina Kuseyri (Bogazici University) Robust Control of Rotor/AMB Systems 15/03/2011 31 / 34
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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
I. Sina Kuseyri (Bogazici University) Robust Control of Rotor/AMB Systems 15/03/2011 32 / 34
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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
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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)
I. Sina Kuseyri (Bogazici University) Robust Control of Rotor/AMB Systems 15/03/2011 34 / 34