Robust Control of Rotor/AMB Systems

Robust Control of
Rotor/Active Magnetic Bearing Systems
˙Ibrahim Sina Kuseyri
Bo˘gazic¸i University
15/03/2011
˙I. Sina Kuseyri (Bo˘gazic¸i University) Robust Control of Rotor/AMB Systems 15/03/2011 1 / 34

Outline
1 Introduction
Overview
Applications
2 System Dynamics
Magnetic Bearings
Rotordynamics
3 Robust Control
Controller Design
Model Uncertainty and Robustness
4 Numerical Results and Simulations

Outline
1 Introduction
Overview
Applications
2 System Dynamics
Magnetic Bearings
Rotordynamics
3 Robust Control
Controller Design

Overview
Radial magnetic (electromagnetic) bearing
50 100 150 200 250 300 350
50
100
150
200
250

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 ﬂywheels
Turbomachinery
High-speed milling and grinding spindles
Electric motors
Turbomolecular pumps
Blood pumps
Computer hard disk drives, x-ray devices, ...

Outline
1 Introduction
Overview
Applications
2 System Dynamics
Magnetic Bearings
Rotordynamics
3 Robust Control
Controller Design

Electromagnetic Bearings
The AMB model considered is based on the zero leakage assumption
which says that magnetic ﬂux in a high permeability magnetic structure
with small air gaps is conﬁned to the iron and gap volumes.

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
c
4
cos αM
with αM denoting the angle between a pole and magnet centerline.

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. Hence electromagnetic force in one axis can be linearized
around the operating point as
Fr
∼= Fr |OP +
∂Fr
∂i OP
(ic − ic OP) +
∂Fr
∂r OP
(r − rOP) ·

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. Hence electromagnetic force in one axis can be linearized
around 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 the
bearing for small currents and small displacements is given by
Fr,lin = kiic − ksr
with the actuator gain ki and the open loop negative stiffness ks
deﬁned as
ki := 4kM
i0
s2
0
and ks := −4kM
i2
0
s3
0
·

Rotordynamics
Equations of motion for a rigid rotor may be derived from
F = ˙P =
d
dt
(Mr v) , and M = ˙H =
d
dt
(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
CG d
2

Rotordynamics
The equations of motion for the four degrees of freedom are
¨x =
1
Mr
[fA,x + fB,x +
Mr
√
2
g +
mub,s
2
Ω2
d cos (Ωt + ϕs)] ,
¨y =
1
Mr
[fA,y + fB,y +
Mr
√
2
g +
mub,s
2
Ω2
d sin (Ωt + ϕs)] ,
¨ψ =
1
Ir
[−ΩIp
˙θ + a(−fA,y ) + b(fB,y ) +
(a + b)
2
mub,c Ω2
d sin (Ωt + ϕc)] ,
¨θ =
1
Ir
[ΩIp
˙ψ + a(fA,x ) + b(−fB,x ) −
(a + b)
2
mub,c Ω2
d cos (Ωt + ϕc)] .

Rotor/AMB model in state-space
The equations of motion for the electromechanical system in the
state-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 = (1
2 mub,sd 1
2 mub,cd)T .

Rotor/AMB model in state-space
The equations of motion for the electromechanical system in the
state-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 = (1
2 mub,sd 1
2 mub,cd)T .
Control objective is to stabilize the system and to minimize the rotor
displacements (whirl) with moderate control effort.

Outline
1 Introduction
Overview
Applications
2 System Dynamics
Magnetic Bearings
Rotordynamics
3 Robust Control
Controller Design

Controller design
K
ym u
di
n+
+
v
di
n
w
+ +
z
yu
P
K
{
v ym
u
e}
G
G

Controller design
Controlled
Output
Input and/or
Disturbance
Measurement
(Feedback)Input
w z
u y
Manipulated
P
K
(Controller)
(Generalized Plant)

Controller design
Controlled
Output
Input and/or
Disturbance
Measurement
(Feedback)Input
w z
u y
Manipulated
P
K
(Controller)
(Generalized Plant)
Q: How to choose K?

Controller design
Controlled
Output
Input and/or
Disturbance
Measurement
(Feedback)Input
w z
u y
Manipulated
P
K
(Controller)
(Generalized Plant)
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 ∞ := sup
ω
¯σ M(jω) Note : ¯σ(M) := λmax (M∗M)
M 2 :=
1
2π
∞
−∞
Trace M(jω)∗M(jω) dω

H2 and H∞-norms
The deﬁnitions are
M ∞ := sup
ω
¯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 = 1
2π
∞
−∞
|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
u
di do
n
+
+
+
++
+
v
˜u
˜n
˜di ˜do
eWr
Wu Wi Wo We
Wn
+
−
˜e
˜ri ri ri − ym
G−
log ω
|W|dB
ωc log ω
|W|dB
ωuωl log ω
|W|dB
ωc

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
K
WqWp
WzWw˜w ˜zw z
p q
yu
˜∆
˜P
˜p ˜q
¯P
σ W−1
p (jω) ∆(jω) W−1
q (jω) = σ ˜∆(jω) ≤ 1 ∀ ω ∈ Re
∆ :=
δksI 0
0 ΩI

Closed-loop rotor/AMB system with uncertainty
Overall system in the state-space form
K
WqWp
WzWw˜w ˜zw z
p q
yu
˜∆
˜P
˜p ˜q
¯P
˙x = Ax + Bp˜p + Bw ˜w + Buu
˜q = Cqx + Dqw ˜w
˜z = Czx + Dzuu
y = Cy x + Dyw ˜w
p = ∆q

Outline
1 Introduction
Overview
Applications
2 System Dynamics
Magnetic Bearings
Rotordynamics
3 Robust Control
Controller Design

Numerical Results - System Data
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 · 10−3 m
MD 77.10 kg LD 0.05 m s1 0.5 · 10−3 m
Ir 17.28 kg·m2 dS 0.10 m i0 3.0 A
Ip 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/m
b 0.58 m sB 0.73 m ki 235.4 N/A

Numerical Results - Weighting functions
Wu = 38
s + 1200
s + 50000
I4 We =
s + 0.05
s + 0.01
I4
10
−2
10
0
10
2
10
4
10
6
−5
0
5
10
15
20
25
30
35
Frequency [rad/s]
Gain[dB]
Wu
10
−2
10
0
10
2
10
4
10
6
0
2
4
6
8
10
12
Frequency [rad/s]
Gain[dB]
We

Results with the H∞ controllers for the nominal system
Maximum operation speed = 3000 rpm (≈ 314.2 rad/s)
10
−2
10
0
10
2
10
4
10
6
−100
−80
−60
−40
−20
0
20
Singular Values
Frequency (rad/sec)
SingularValues(dB)
Singular values of controller K1
10
−2
10
0
10
2
10
4
10
6
−100
−80
−60
−40
−20
0
20
Singular Values
Frequency (rad/sec)
SingularValues(dB)
10
−2
10
0
10
2
10
4
10
6
−140
−120
−100
−80
−60
−40
−20
0
20
40
Singular Values
Frequency (rad/sec)
SingularValues(dB)
Closed−loop SVs with K1
10
−2
10
0
10
2
10
4
10
6
−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
Table : H∞ performance with K1 for different design parameters
Maximum speed (rpm) Maximum mass center displacement (m) γ
1500 0.25·10−3 70.96
3000 0.25·10−3 97.06
6000 0.25·10−3 99.81
1500 0.50·10−3 89.57
3000 0.50·10−3 99.24
6000 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 11.41
3000 0.25·10−3 15.42
6000 0.25·10−3 31.77
1500 0.50·10−3 12.62
3000 0.50·10−3 21.05
6000 0.50·10−3 52.01

Critical speeds (eigenfrequencies)
Pole−Zero 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)
10
0
10
1
10
2
10
3
10
4
−200
−150
−100
−50
0
50
100
Frequency (Speed) [rad/s]
ClosedloopPhaseshiftforjournaldisplacements(unbalancechannel)
XA
YA
XB
YB
120
Phase shift with K1
10
0
10
1
10
2
10
3
10
4
−200
−180
−160
−140
−120
−100
−80
−60
−40
−20
0
Frequency[rad/s]
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 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.
10
−2
10
0
10
2
10
4
10
6
−140
−120
−100
−80
−60
−40
−20
0
20
40
60
Singular Values
Frequency (rad/sec)
SingularValues(dB)
Closed−loop SVs with K1r
10
−2
10
0
10
2
10
4
10
6
−140
−120
−100
−80
−60
−40
−20
0
20
40
Singular Values
Frequency (rad/sec)
SingularValues(dB)
Closed−loop SVs with K2r
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
−200
−150
−100
−50
0
50
Frequency[rad/s]
XA
YA
XB
YB
170
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
−200
−180
−160
−140
−120
−100
−80
−60
−40
−20
0
Frequency[rad/s]
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 speed
of operation. Keeping the uncertainty on the bearing stiffness constant
(25%), robust stability of the closed-loop system is tested for several
maximum operating speeds with µ-analysis. Moreover, keeping the
operation speed constant (3000 rpm), robust stability is tested for
uncertainty in bearing stiffness.
3000 3500 4000 4500 5000 5500 6000
0.8
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 25
0
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 a
maximum operating speed of 4085 rpm (∼= 418 rad/s) are shown
below. H∞ performance γ of the closed-loop system for Ωmax = 4085
rpm is 47.86. Note that the closed-loop system with K3 have
twenty-nine inputs and thirty-two outputs, whereas the closed-loop
systems with K1 and K2 have ﬁve inputs and eight outputs. Order of
the controller K3 (which is twelve) can not be reduced since it leads to
the instability of the closed-loop system.
10
−2
10
0
10
2
10
4
10
6
−100
−80
−60
−40
−20
0
20
Singular Values
Frequency (rad/sec)
SingularValues(dB)
10
−2
10
0
10
2
10
4
10
6
−1000
−800
−600
−400
−200
0
200
Singular Values
Frequency (rad/sec)
SingularValues(dB)

Simulations
Simulation Environment in SIMULINK (Rotor/AMB)

Simulations
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 0.5
−100
−80
−60
−40
−20
0
20
40
60
80
100
Time (sec)
UnbalanceForce(Newtons)
0 100 200 300 400 500 600
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
Time (msec)
Volts
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)
X
A
(Volts)
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)
Y
A
(Volts)
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,Ax(Amperes)
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,Ay(Amperes)
Control current for Bearing A
(y−axis)
Rotor position and control currents during start-up

Simulation Results
Mass center displacement (eccentricity) due to unbalance of the rotor
is assumed to be 0.25 · 10−3 m in the simulations. Peak value of the
vibration (except the transient) is less than 0.1 V, corresponding to
14 · 10−6 m. Hence, the H∞ controller K2 reduces the unbalance whirl
amplitude 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)
X
A
(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,Ax(Amperes)
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,Ay(Amperes)
Control current for Bearing A (y−axis)

Robust Control of Rotor/AMB Systems

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Viewers also liked

Viewers also liked (20)

Similar to Robust Control of Rotor/AMB Systems

Similar to Robust Control of Rotor/AMB Systems (20)

Recently uploaded

Recently uploaded (20)

Robust Control of Rotor/AMB Systems