Gain Scheduled H2 Control for Magnetic Bearing with
Imbalance
M2014SC012 Tatsuya MIZUNO
Supervisor Isao TAKAMI

Abstract
In this paper, gain scheduled H2 controller for active magnetic bearing (AMB) system is designed. A
rotor has gyroscopic eﬀect, imbalance and resonance.
The gyroscopic eﬀect and imbalance depend on angular velocity. The robust stability for angular velocity is
guaranteed by using polytopic representation. Descriptor representation and linear fractional transformation(LFT) are adopted to obtain an equivalent polytopic
representation of AMB system which has multi-aﬃne.
Frequency weight is introduced to suppress vibration byresonance. The problem is formulated as solving a ﬁnite
set of linear matrix inequalities (LMIs). The eﬀectiveness of the proposed method is illustrated by simulation.

1

Introduction

Active magnetic bearing (AMB) system levitates rotor
with contact-free by electromagnetic force. It is possible
for AMB to rotate at high rotation velocity. However,when the rotor rotates, the gyroscopic eﬀect is generated. The rotor is vibrated by precession caused by it.
Since it depends on the angular velocity of the rotor,
as the angular velocity of the rotor increases, the vibration becomes large. Gain scheduled (GS) controller[1]and GS controller via parameter depending Lyapunov
function[2] for gyroscopic eﬀect have been reported.
If there exists imbalance of the rotor, the rotor vibrates. The imbalances are two types, which are static
imbalance and couple imbalance. The static imbalanceoccurs because the center of gravity shifts from the center of the rotor. The couple imbalance occurs because
inertial axis shifts from rotation axis. The imbalances
also depend on the angular velocity and are represented
as periodic disturbances[3]. Therefore, the control for
AMB is required to consider the vibration caused by
not only the gyroscopic eﬀect but also these imbalances.
Some literatures for AMB with imbalance have been reported. The H∞ disturbance and an initial state uncertainty attenuation (DIA) controller for imbalance regarded as frequency disturbance has been reported[4].
The GS sliding mode controller for the static imbalancehas been designed[5]. 2-DOF controller based on MIMO
decoupling technique, adaptive feedforward algorithm
and Notch ﬁlter has been designed[6]. The rotor has resonance. This vibrates the rotor when natural frequency
is the same as the rotation velocity of the rotor. Somestudies for the resonance using ﬂexible rotor model havebeen reported[7].
In this study, GS H2 controller via parameter dependent Lyapunov function for AMB is designed. The rotor
has the gyroscopic eﬀect, the static and the couple imbalance and the resonance. The object of this study is
to suppress vibrations caused by them. The frequency
weight is introduced to suppress the vibration causedby the resonance. The dynamics of AMB dependents
on angular velocity of the rotor. The robust stability
for this parameter is guaranteed theoretically by using
polytopic representation. Descriptor representation andlinear fractional transformation (LFT) are adopted toobtain an equivalent polytopic representation of AMB
system which has the only ﬁrst-order terms of the varying parameter. The problem is formulated as solving
a ﬁnite set of linear matrix inequalities (LMIs). The
eﬀectiveness of the proposed method is illustrated bysimulation comparing with LQ controller.

2

Modeling

2.1

Motion equations

An experimental device used in this study is a 4-axis
controlled type AMB with symmetrical structure. Fourpairs of electromagnets are located in the horizontal andthe vertical of both ends of the rotor. Gap sensors arealso located in the horizontal and the vertical of bothends of the rotor. They can measure distance between
the electromagnet and the rotor. Coordinates X, Y and
Z are introduced to obtain equations of motion as shown
Fig. 1. Let p[rad/s], fj [N] and gj [m] be rotational velocity of the rotor, levitation force of the electromagnets
and displacement from the equilibrium point of the rotor, respectively. Here, subscript j ∈ {lv, rv, lh, rh}.

Fig. 1 Coordinates X, Y and Z
These subscripts l, r, v, h mean the left-hand side, righthand side, vertical direction and horizontal direction, respectively. The imbalances of this study are the static
imbalance and the couple imbalance. Here, ε and τ aredistance between the center of gravity and the centerof the rotor and angle between the inertial axis and therotation axis. Physical parameters of AMB are shownin Table. 1. The following assumptions are assumed toTable 1 Physical parameters
Parameter
Mass of rotor
Acceleration of gravity
Distance between center of gravity
and the left-had side of rotor
Distance between center of gravity
and the left-had side of rotor
Distance between rotor and sensor in
the equilibrium state
Moment of X axis
Moment of Y axis
Suction force coeﬃcient
Constant current of vertical direction
Constant current of horizontal direction

Symbol
m
g
lml

Unit
[kg]
[m/s2 ]
[m]

lmr

[m]

g0

[m]

Jx
Jy
k
Ilv , Irv
Ihv , Irh

[Nm]
[Nm]
[A]
[A]

derive the equations of motion.
• The rotor is a rigid body.
• All electromagnets have the same electrical characteristic.
• The center of the gravity shifts in a radical direction
from the center of the rotor.
Equations of the translational motion of the directions
Y and Z and equations of the rotational motion of theaxes Y and X are given by Eq.(1)-(4).
mÿ = flh + frh + mεp2 cos(pt + κ)(1)mz̈ = −mg + flv + frv + mεp sin(pt + κ)

(2)

2

2.2

From Eq.(10)-(13), state space representation is obtained by as follows.
{ẋ(t) = Ax(t) + B1 w(t) + B2 u(t)(14)y(t) = C2 x(t)x(t) = [glv grv glh grh g˙lv g˙lv g˙lv g˙lv ]Tu(t) = [ilv irv ilh irh ]Tε sin(pt + κ)[]OI
 τ sin(pt + λ) w(t) = ,A =ε cos(pt + κ) 
A1 pA2
τ cos(pt + λ)
[
]
[
]
O
O
B1 =
,
B
=
, C = I8×82
B21
p2 B11Jy θ̈ = Jx pψ̇ + lml flv − lmr frv+(Jy − Jx )τ p2 sin(pt + λ)(3)Jy ψ̈ = −Jx pθ̇ − lml flh + lmr frh+(Jy − Jx )τ p2 cos(pt + λ)(4)Here, y and z are displacement of the directions Y andZ. θ and ψ are rotation angle around the axes Y andZ. κ and λ are the initial values of the phase. The displacement y and z and the rotation angle θ and ψ arerepresented as Eq.(5)-(6) by using gj .
(grh − glh )lml
(grv − glv )lmr
, z = glv +(5)
lml + lmr
lml + lmr
glv − grv
grh − glh
θ≈
,ψ ≈
(6)
lml + lmr
lml + lmr
The levitation force of the electromagnetic is given by
Eq.(7)
y = glh +fj = k

(I0 + Ij + ij )2(I0 − Ij − ij )2−k2(gj − g0 )(gj + g0 )2

4I0 Ij+ Kxj gj + Kij ij
g02
4(I02 + Ij2 )4I0, Kij = k 2
Kxj = k3
g0
g0
fj = kHere, matrices A1 ，A2 ，B11 and B21 are constant matrices obtained by Eq.(10)-(13).

3Frequency weight

The frequency weight that has the peak of a gain at
the natural frequency is introduced for the state variable to suppress the vibration that is caused by theresonance. A frequency weight function for the statevariable Wf (S) is given as follow.
Wf (s) = I8×8 W (s), W (s) = WC (Is − WA )−1 WB (15)The frequency weight function Wf (S) is represented as
the state space representation (16).

(7)

Here, I0 , Ij and ij are bias current, steady-state current
and control input, respectively. Since the position of therotor does not change signiﬁcantly, Eq.(7) is linearizedat the equilibrium point as follows.
(8)(9)

Eq.(10)-(13) are obtained by Eq.(1)-(9).
g¨lv = aKxlv glv + cKxrv grv − pdlml glh
˙ + pdlml grh
˙+aKilv ilv + cKirv irv + p2 α1 + p2 lml β1(10)g¨rv = cKxlv glv + bKxrv grv − pdlmr glh
˙ + pdlmr grh
˙+cKilv ilv + bKirv irv + α1 p2 − p2 lmr β1(11)g¨lh = aKxlh glh + cKxrh grh + pdlml g˙lv − pdlml grv
˙+aKilh ilh + cKirh irh + p2 α2 − p2 lml β2 (12)g¨rh = cKxlh glh + bKxrh grh − pdlmr g˙lv + pdlmr grv
˙+cKilh ilh + bKirh irh + p2 α2 + p2 lmr β2 (13)
1
l2
1
l2
1
lml lmra=+ ml , b =+ mr , c =
−
m
Jy
m
Jy
m
Jy
Jx
d=
Jy (lml + lmr )Jx)τ sin(pt + λ)α1 = ε sin(pt + κ), β1 = (1 −JyJx
α2 = ε cos(pt + κ), β2 = (1 −)τ cos(pt + λ)JyState space representationẋw (t) = Aw xw (t) + Bw x(t)z̄(t) = Cw xw (t)(16)

Here, xw (t) is the state of the frequency weight function Wf (s). Let xf (t) be a new state variable xf (t) =[x(t)T xw (t)T ]T . Then generalized plant for a extended
system including the frequency weight is as follows.
{ẋf (t) = Af (p)xf (t) + Bf 1 (p2 )w(t) + Bf 2 u(t)(17)
zf (t) = Cf 1 xf (t) + Df 12 u(t)
[
]
[
]
[
]
A
O
B1
B2
Af =
, Bf 1 =, Bf 2 =
Bw Aw
O
O
[
]
[
]
Wx O
O
O Cw , Df 12 =O
Cf 1 =OO
Wu
Here, zf , Wx ≻ 0 and Wu ≻ 0 are a evaluated output,
weight matrices for the state variable and the input.

4

Transformation into multi-aﬃneSince the state space representation of the plant modelhas the ﬁrst-order and second-order terms of varying
parameter, it is diﬃcult to use polytopic representation
directly. The system is transformed to an equivalentsystem that is multi-aﬃne for p by using descriptor representation and linear fractional transformation (LFT).
4.1

Descriptor representation

Matrix Bf 1 of Eq.(17) has the second-order terms of
the angular velocity of the rotor. All varying parameter are put into one matrix by expanding dimension ofmatrices to apply LFT. The varying parameter p is put
into matrix Af d by deﬁning xd (t) = [x(t) w(t)]T .

feedback controller u = K̃f d (p)x̃f d and feedback gainK̃f d (p) = Ỹf d (p)X̃f d (p)−1 .

Ef d ẋf d (t) = Af d (p2 )xf d (t) + Bf d1 w(t) + Bf d2 u(t)(18)
[
]
[
]
I O
Af (p) Bf 1 (p2 )Ef d =, Af d (p) =
O O
O
−I
]
[
]
[
Bf 2
O
, Bf d2 =Bf d1 =OI

He[Ãf d (p)X̃f d (p) + B̃f d2 Ỹf d (p)] − Ẽf d X̃˙ f d (p) + B̃f d1 B̃fTd1 ≺ 0 (24)

4.2

Linear fractional transformation

LFT is adapted to eliminate the products of p. Matrix Af d can be represented as Eq.(19). Here Af dn is aconstant matrix in Af d , and Bδ ∆Cδ is a matrix whichcontains the ﬁrst and second-order terms of p in Af d bychoosing appropriate Af dn , Bδ , Cδ and ∆ as follows.
Af d (p) = Af dn + Bδ ∆(p)Cδ (p) (19)
[
]
[
]
A0 O
O
Bδ0
O
Af dn = Bw Aw O
, Bδ =
O
O −I
O
[
]
I
Cδ (p) = [ A1 O pB11 ] , Bδ0 =O∆(p) = diag(p p p p)Ẽf d X̃˙ f d (p) = (Ẽf d X̃˙ f d (p))T ≻ 0 (25)
Cf 1 Xf (p)CfT1 ≺ Z (26)trace(Z) < γ 2 (27)

In view of structure of the matrix Ẽf d , candidates of
Lyapunov matrix X̃f d (p), variable matrix Ỹf d (p) andẼ X̃˙ (p) are restricted as follow.
d[X̃f d (p) =5

Control design

In this section, the GS controller via parameter dependent Lyapunov function is synthesized. The schedulingparameter is angular velocity of the rotor p. The stability of the closed system should be guaranteed theoretically for variation of the angular velocity of the rotor.
The robust stability for the time varying parameter p
is guaranteed by using polytopic representation. The
range of the time varying parameter p is deﬁned by upper and lower bound. The objective of this study is toguarantee the robustness in the following range.
p ∈ [p, p] = [p1 , p2 ](21)

From Eq.(21), matrix Ãf d is represented by polytopic
representation Eq.(22)Ãf d = αÃf d (p1 ) + (1 − α)Ãf d (p2 ), α ∈ [0, 1] (22)
Eq.(22) shows that Ãf d (p) = Ãf d (p1 ) when p is minimum and Ãf d (p) = Ãf d (p1 ) when p is maximum. For
Eq.(20), H2 norm from the disturbance w(t) to the evaluated output z(t) is given as Eq.(23)∫ ∞||G||22 =z(t)T z(t)dt(23)
0

In order to minimize the H2 norm, minimum γ thatsatisfy the following conditions are derived. Here, state

Xf (p)
X21 (p)
X31 (p)OO
X22 (p) X23 (p)X32 (p) X33 (p)Ỹf d (p) = [ Yf (p)[Ẽf d X̃˙ f d (p) =

]

(28)

O

O ]

(29)

O
O
O

O
O
O

(30)

Xf (ṗ) − Xf 0OO]

In Eq.(24), there exists product of scheduling parameter Ãf d (p)X̃f d (p). X̃f d (p) is restricted as Eq.(31).
Then Eq.(24) becomes multi-aﬃne for varying parameter. Xf (p), Ỹf d (p) and Yf (p) are also assigned as follows.
[

Note that Cδ have ﬁrst-order terms of p. Af dn and Bδ isconstant matrix. Let x̃f d be the new descriptor variablex̃d = [xTd zδT ]T . Descriptor equation is obtained asfollows.
Ẽf d x̃˙ f d (t) = Ãf d (p)x̃f d (t) + B̃f d1 w(t) + B̃f d2 u(t)(20)[][]
Ef d O
Af dn Bδ ∆(p)Ẽf d =
, Ãf d (p) =O O
Cδ (p)
−I
[
]
[
]
Bf d1
Bf d2
B̃f d1 =, B̃f d2 =, zδ = Cδ xf d (t)OO

fdX̃f d (p) = X̃f d0 + pX̃f d1 , X̃f d1 =

Xf 1
O
O

O
O
O

O
O
O

]

(31)

Xf (p) = Xf 0 + pXf 1(32)

Ỹf d (p) = Ỹf d0 + pỸf d1 , Yf (p) = Y0 + pY1(33)

LMI conditions that minimize H2 norm and stabilize thesystem are as follows.
Lemma 1 : If there exist X̃f d (p) and Ỹf d (p)satisf ying this LM Is, the system is stabilized.
minimize :γ 2
subject to[[

Xf (pi ) ≻ 0
M (pi ) − Ẽf d X̃˙ f d (pi )B̃fTdwZ(Cf 1 Xf (pi ) + Df 12 Yf (pi ))T

(34)

]

B̃f dw−I≺0(Cf 1 Xf (pi ) + Df 12 Yf (pi ))Xf (pi )

(35)
]

≻0
(36)

trace(Z) < γ

2

(37)

M (pi ) := He[Ãf d (pi )X̃f d (pi ) + B̃f d2 Ỹf d (pi )], (i = 1, 2)
Gain scheduled controller Kf (p) = Yf (p)Xf (p)−1 withframework of the state space representation is obtained.

6

Simulation

In this section, the validity of the proposed method is
illustrated by comparing with LQ controller in simulation. In this study, the range of the angular velocity is
assumed from 0[rad/s] to 2618[rad/s](25,000[rpm]). The
rotor rotates as Fig. 2. The distance between the center and the center of gravity of rotor ε = 1.0 × 10−6 [m]and the angle of rotation axis to inertia axis τ = 1.75 ×
10−5 [rad]. The initial value of state of the rotor is x(0) =[−1.5×10−6 1.5×10−6 −1.0×10−6 1.0×10−6 0 0 0 0].
LQ controller is designed at 25,000[rpm]. The frequency
weight function W (s) is as follow.
W (s) =

100.05s2 + 2.64 × 10−2 s + 1.75 × 106(38)

Fig. 2 Angular velocity
Fig. 7 Control input il1
Fig. 3 and 4 show the force and the torque that arecaused by the static and the couple imbalance. The
simulation results of the displacements from the equilibrium point on the vertical and horizontal directionof the left hand side are shown in Fig. 5 and 6. The
convergence of the rotor controlled by H2 controller islater than LQ controller. However, H2 controller suppresses the vibration that is caused the static and thecouple imbalance than LQ controller.Since the rotor ismore suppressed at about 5 seconds, the eﬀect of the frequency weight is shown. The simulation results of the
input current on the vertical direction of the left handside and the right hand side are shown in Fig. 7 and8. The simulation result of torques that are caused by
the gyroscopic eﬀect are shown in Fig. 9 and 10. The
torques that are caused by the gyroscopic eﬀect are also
suppressed by H2 controller than LQ controller.

Fig. 8 Control input il3

Fig. 9 Torque caused by gy-Fig. 10 Torque caused byroscopic eﬀect of Y
gyroscopic eﬀect of Z
obtain an equivalent polytopic representation of AMB
system which has multi-aﬃne. The frequency weight isintroduced to suppress the vibration caused by the resonance. The problem is formulated as solving a ﬁnite setof linear matrix inequalities (LMIs). The eﬀectivenessof the proposed method is illustrated by simulation bycomparing with LQ controller. The proposed method
suppresses the vibration of the rotor than LQ controller
and suppresses more at the natural frequency by thefrequency weight.

References
Fig. 3 Force cased by staticFig. 4 Torque caused bycouple imbalance
imbalance

Fig. 5 Displacement gl1

7

Fig. 6 Displacement gl3

Conclusion

In this paper, the gain scheduled H2 controller for theactive magnetic bearing (AMB) is designed. The rotor
has the gyroscopic eﬀect, the imbalances and the resonance. This controller is synthesized with parameter
dependent Lyapunov function. The robust stability forthe angular velocity is guaranteed theoretically by using
polytopic representation. Descriptor representation andlinear fractional transformation (LFT) are adopted to[1] Selim Sivrioglu and Kenzo Nonami, LMI Approach to Gain
Scheduled H∞ Control Beyond PID Control for Gyroscopic
Rotor-Magnetic Bearing System , IEEE Conference on Decision and Control, Vol. 4, pp. 3694-3699, 1996.
[2] Akio Sambayashi, Masanori Narita, Gan Chen and Isao
TakamiI, Gain Scheduled Control for Active Magnetic Bearing System Considering Gyroscopic Eﬀect , International
Conference on Information Technology and Electrical Engineering, pp. 6, 2015.
[3] Fumio Matsumura, Masayuki Fujita and Kenji Okawa,
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Symposium on Magnetic Bearing, vol. 138, pp. 273-280, 1990.
[4] Hiroki Seto, Toru Namerikawa, and Masayuki Fujita. Experimental Evaluation on H∞ DIA Control of Magnetic Bearings with Rotor Unbalance ,10th International Symposiumon Magnetic Bearings, Hotel du Parc, Martigny, Switzerland.
2006.
[5] Selim Sivrioglu and Kenzo Nonami, Gain Scheduled Sliding Mode Control of Active Magnetic Bearing System with
Gyroscopic Rotor , Transactions of the Japan Society of
Mechanical Engineering Series C, Vol. 65, No. 635, pp. 26652671, 1999 (in Japanese).
[6] Iigo Arredondo and Josu Jugo, 2-DOF Controller Design for
Precise Positioning a Spindle Levitated with Active Magnetic
Bearing , European Journal of Control, Vol. 18, No. 2, pp.
194-206, 2012.
[7] Mehmet N. Sahinkaya, Abdul-Hadi G. Abulrub and Patrick
S. Keogh,On the Modeling of Flexible Rotor/Magnetic
Bearing Systems When in Contact with Retainer Bearings ,The Ninth International Symposium on Magnetic Bearings(ISMB9), 2004.