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Multi-Objective Robust Control ofRotor/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 IntroductionOverviewApplications
2 System DynamicsMagnetic BearingsRotordynamics
3 Robust ControlController DesignModel UncertaintyRobust Stability and PerformanceNumerical Results and Simulations
4 Multi-Objective LPV ControlLinear Parametrically Varying (LPV) SystemsMixed Performance SpecificationsNumerical Results and Simulations
I. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 2011 2 / 51
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Outline
1 IntroductionOverviewApplications
2 System DynamicsMagnetic BearingsRotordynamics
3 Robust ControlController DesignModel UncertaintyRobust Stability and PerformanceNumerical Results and Simulations
4 Multi-Objective LPV ControlLinear Parametrically Varying (LPV) SystemsMixed Performance SpecificationsNumerical Results and Simulations
I. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 2011 3 / 51
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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)
I. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 2011 4 / 51
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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 andgrinding spindles
Electric motors
Turbomolecular pumps
Blood pumps
Computer hard diskdrives, x-ray devices, ...
I. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 2011 6 / 51
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Outline
1 IntroductionOverviewApplications
2 System DynamicsMagnetic BearingsRotordynamics
3 Robust ControlController DesignModel UncertaintyRobust Stability and PerformanceNumerical Results and Simulations
4 Multi-Objective LPV ControlLinear Parametrically Varying (LPV) SystemsMixed Performance SpecificationsNumerical Results and Simulations
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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 arealmost 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 − r
)2
−(
i−s0 + r
)2)
on the rotor. The magnetic bearing constant kM is
kM :=µ0AAn2
c
4cosαM
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. Linearization in one axis around the operating point leads to
Fr∼= Fr |OP +
∂Fr
∂i
∣
∣
∣
∣
OP(ic − ic OP) +
∂Fr
∂r
∣
∣
∣
∣
OP(r − rOP) .
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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. Linearization in one axis around the operating point leads to
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 ks
defined as
ki := 4kMi0s2
0
and ks := −4kMi20s3
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
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 +
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 IAS 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 = (12 mub,sd 1
2mub,cd)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 IAS 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 = (12 mub,sd 1
2mub,cd)T .
Control objective is to stabilize the system and to minimize the rotordisplacements (whirl) with acceptable control effort.
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Outline
1 IntroductionOverviewApplications
2 System DynamicsMagnetic BearingsRotordynamics
3 Robust ControlController DesignModel UncertaintyRobust Stability and PerformanceNumerical Results and Simulations
4 Multi-Objective LPV ControlLinear Parametrically Varying (LPV) SystemsMixed Performance SpecificationsNumerical 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
Measurement(Feedback)Input
w z
u y
Manipulated
K
P
PerformanceOutputInput
Exogenous
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Controller design
Measurement(Feedback)Input
w z
u y
Manipulated
K
P
PerformanceOutputInput
Exogenous
Q: How to choose K ?
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Controller design
Measurement(Feedback)Input
w z
u y
Manipulated
K
P
PerformanceOutputInput
Exogenous
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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H2 and H∞-norms
The definitions are
‖M‖∞ := supω
σ(
M(jω)) (
Note : σ(M) :=√
λmax (M∗M))
‖M‖2 :=
√
12π
∫ ∞
−∞Trace
(
M(jω)∗M(jω))
dω
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H2 and H∞-norms
The definitions are
‖M‖∞ := supω
σ(
M(jω)) (
Note : σ(M) :=√
λmax (M∗M))
‖M‖2 :=
√
12π
∫ ∞
−∞Trace
(
M(jω)∗M(jω))
dω
For SISO LTI systems,‖M‖∞ = sup
ω|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−
log ω
|W |dB
ωc log ω
|W |dB
ωuωl log ω
|W |dB
ωc
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Model uncertainty
Uncertainty in Rotor/AMB Models
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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−1p (jω)∆(jω) W−1
q (jω))
= σ(
∆(jω))
≤ 1 ∀ω ∈ Re
∆ :=
[
δks I 00 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 + Buu
q = Cqx + Dqw w
z = Czx + Dzuu
y = Cyx + Dyw w
p = ∆q
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Robust stability and performance
w z
qp
w z
∆
M
N
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Robust stability and performance
w z
qp
w z
∆
M
N
Nominal Stability (NS) ⇔ M is internally stable
Nominal Performance ⇔ NS, and σ(
M(jω))
< γ ∀ω ∈ Re
Robust Stability (RS) ⇔ NS, andN to be stable ∀∆ : σ
(
∆(jω))
≤ 1 ∀ω ∈ Re
Robust Performance ⇔ RS, andσ(
N(jω))
< γ ∀∆ : σ(
∆(jω))
≤ 1 ∀ω ∈ Re
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Robust stability - Structured singular value
Transfer matrix of the closed-loop uncertain system in LFT form is
N = Mzw + Mzp∆(I − Mqp∆)−1Mqw .
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Robust stability - Structured singular value
Transfer matrix of the closed-loop uncertain system in LFT form is
N = Mzw + Mzp∆(I − Mqp∆)−1Mqw .
For robust stability(
I − Mqp(s)∆(s))−1 should have no poles in C
+for
all ∆ with σ(∆) ≤ 1 .
Meaning that =⇒ det(
I−Mqp(jω)∆)
6= 0, ∀∆ with σ(∆) ≤ 1, ∀ω ∈ Re .
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Robust stability - Structured singular value
Transfer matrix of the closed-loop uncertain system in LFT form is
N = Mzw + Mzp∆(I − Mqp∆)−1Mqw .
For robust stability(
I − Mqp(s)∆(s))−1 should have no poles in C
+for
all ∆ with σ(∆) ≤ 1 .
Meaning that =⇒ det(
I−Mqp(jω)∆)
6= 0, ∀∆ with σ(∆) ≤ 1, ∀ω ∈ Re .
Therefore, robust stability holds if and only if
inf∆σ(∆) : det
(
I − Mqp(jω)∆)
= 0, ∀ω ∈ Re > 1 .
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Robust stability - Structured singular value
Transfer matrix of the closed-loop uncertain system in LFT form is
N = Mzw + Mzp∆(I − Mqp∆)−1Mqw .
For robust stability(
I − Mqp(s)∆(s))−1 should have no poles in C
+for
all ∆ with σ(∆) ≤ 1 .
Meaning that =⇒ det(
I−Mqp(jω)∆)
6= 0, ∀∆ with σ(∆) ≤ 1, ∀ω ∈ Re .
Therefore, robust stability holds if and only if
inf∆σ(∆) : det
(
I − Mqp(jω)∆)
= 0, ∀ω ∈ Re > 1 .
Inversion leads to the definition
µ∆(M) :=1
inf∆ σ(∆) : det(
I − Mqp(jω)∆)
= 0 < 1 ∀ω ∈ Re .
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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 b
sB
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 =
(
38s + 1200
s + 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]
Gai
n [d
B]
Wu
10−2
100
102
104
106
0
2
4
6
8
10
12
Frequency [rad/s]
Gai
n [d
B]
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
alue
s (d
B)
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
alue
s (d
B)
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
alue
s (d
B)
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
alue
s (d
B)
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
Imag
inar
y A
xis
−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
sedl
oop
Pha
sesh
ift fo
r jo
urna
l dis
plac
emen
ts(u
nbal
ance
cha
nnel
)
XAYAXBYB
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
sedl
oop
Pha
sesh
ift fo
r jo
urna
l dis
plac
emen
ts(u
nbal
ance
cha
nnel
)
XAYAXBYB
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
gula
r V
alue
s (d
B)
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
gula
r V
alue
s (d
B)
Closed−loop SVs with K2r
10−2
10−1
100
101
102
103
104
−200
−150
−100
−50
0
50
Frequency[rad/s]Clo
sedl
oop
Pha
sesh
ift fo
r jo
urna
l dis
plac
emen
ts(u
nbal
ance
cha
nnel
)
XAYAXBYB
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
sedl
oop
Pha
sesh
ift fo
r jo
urna
l dis
plac
emen
ts(u
nbal
ance
cha
nnel
)
XAYAXBYB
185
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Robust stability of the uncertain closed-loop system
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 the operation speed constant (3000 rpm), robuststability 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
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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 are shown below.
H∞ performance γ of the system for Ωmax = 4085 rpm is 47.86.
Order of the controller K3 (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
gula
r V
alue
s (d
B)
Singular values of controller K3
10−2
100
102
104
106
−1000
−800
−600
−400
−200
0
200
Singular Values
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
Closed−loop SVs with K3
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Simulations
Simulation Environment in SIMULINK
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Simulations
Simulation Environment in SIMULINK (Rotor/AMB)
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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)
Unb
alan
ce F
orce
(N
ewto
ns)
0 100 200 300 400 500 600−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
Time (msec)
Vol
ts
Sensor Noise
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Simulations
0 0.1 0.2 0.3 0.4 0.5−2
−1
0
1
2
3
4
5
6
Time (sec)
XA (
Vol
ts)
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 (
Vol
ts)
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 (A
mpe
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 (A
mpe
res)
Control current for Bearing A (y−axis)
Rotor position and control currents during start-upI. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 2011 35 / 51
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Simulations
Mass center displacement (eccentricity) due to unbalance of the rotoris 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. Therefore, the H∞ controller K2 reducesthe unbalance whirl amplitude of the rotor more than 95%.
0 0.1 0.2 0.3 0.4 0.5−2
−1.5
−1
−0.5
0
0.5
Time (sec)
XA (
Vol
ts)
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 (
Vol
ts)
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 (A
mpe
res)
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 (A
mpe
res)
Control current for Bearing A (y−axis)
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Outline
1 IntroductionOverviewApplications
2 System DynamicsMagnetic BearingsRotordynamics
3 Robust ControlController DesignModel UncertaintyRobust Stability and PerformanceNumerical Results and Simulations
4 Multi-Objective LPV ControlLinear Parametrically Varying (LPV) SystemsMixed Performance SpecificationsNumerical Results and Simulations
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LPV Systems
xzy
=
A(ρ) Bw(ρ) Bu(ρ)Cz(ρ) Dzw (ρ) Dzu(ρ)Cy(ρ) Dyw (ρ) Dyu(ρ)
xwu
Parameters ρ(t) are measured in real-time with sensors for control.
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LPV Systems
xzy
=
A(ρ) Bw(ρ) Bu(ρ)Cz(ρ) Dzw (ρ) Dzu(ρ)Cy(ρ) Dyw (ρ) Dyu(ρ)
xwu
Parameters ρ(t) are measured in real-time with sensors for control.
Hence controller is also parameter-dependent, using the availablereal-time information of the parameter variation.
u y
w z
Pρ
Kρ
ρ
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Mixed Performance Specifications
Suppose a specific control task leads to the generalized LPV plant
xz1
z2
y
=
A(ρ) B1(ρ) B2(ρ)C1(ρ) D11(ρ) D12(ρ)C2(ρ) D21(ρ) D22(ρ)C(ρ) D(ρ) 0
xwu
Using an LPV controller, K (ρ, ρ), the closed-loop system can bedescribed in the form
xcl
z1
z2
=
A BC1 D1
C2 D2
(
xcl
w
)
·
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Mixed Performance Specifications
xcl
z1
z2
=
A BC1 D1
C2 D2
(
xcl
w
)
L2 gain of the w → z1 channel is defined as
αopt := infK∈K
sup‖w‖2 6=0
‖z1‖2
‖w‖2
where K := set of all stabilizing controllers .
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Mixed Performance Specifications
xcl
z1
z2
=
A BC1 D1
C2 D2
(
xcl
w
)
L2 gain of the w → z1 channel is defined as
αopt := infK∈K
sup‖w‖2 6=0
‖z1‖2
‖w‖2
where K := set of all stabilizing controllers .
To quantify the gain of the channel w → z2 we use the induced norm
βopt := infK∈K
sup‖w‖2 6=0
‖z2‖∞‖w‖2
·
Remark: ‖z‖2 :=√
∫
∞
−∞z(t)T z(t) dt < ∞ , ‖z‖∞ := ess supt∈R
|z(t)| < ∞ .
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Mixed Performance Specifications
We can use a single Lyapunov function to achieve both of the controlobjectives (though conservatively) and the problem can be defined asminimizing an upper bound βm under the constraint α < αm .
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Mixed Performance Specifications
We can use a single Lyapunov function to achieve both of the controlobjectives (though conservatively) and the problem can be defined asminimizing an upper bound βm under the constraint α < αm .
This leads to defining the mixed objective functional
I(
K (X ))
:= inf βm | ∃ a functionX (ρ) satisfying α < αm and β < βmfrom the solution of the following infinite dimensional LMIs for all (ρ, ρ):
X = X T ≻ 0 ,
X + ATX + XA XB CT1
BTX −I DT1
C1 D1 −α2mI
≺ 0 , C2X−1C2 ≺ βmI ,
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Mixed Performance Specifications
We can use a single Lyapunov function to achieve both of the controlobjectives (though conservatively) and the problem can be defined asminimizing an upper bound βm under the constraint α < αm .
This leads to defining the mixed objective functional
I(
K (X ))
:= inf βm | ∃ a functionX (ρ) satisfying α < αm and β < βmfrom the solution of the following infinite dimensional LMIs for all (ρ, ρ):
X = X T ≻ 0 ,
X + ATX + XA XB CT1
BTX −I DT1
C1 D1 −α2mI
≺ 0 , C2X−1C2 ≺ βmI ,
where X is defined to be
X :=
m∑
i=1
∂X∂ρi
ρi ·
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Controller Synthesis
A full order controller K which satisfying the mixed objective functionalI(
K (X ))
can be constructed, if there exist parameter-dependentfunctions X (ρ), Y (ρ) with X ≻ 0 , Y ≻ 0 , and E(ρ),F (ρ),G(ρ) withG(ρ) = DK (ρ), such that
X + AT X + XA + FC + (FC)T XB1 + FD (C1 + D12GC)T
(XB1 + FD)T −I (D11 + D12GD)T
C1 + D12GC D11 + D12GD −α2mI
≺ 0 ,
−Y + AY + YAT + B2E + (B2E)T B1 + B2GD (C1Y + D12E)T
(B1 + B2GD)T −I (D11 + D12GD)T
C1Y + D12E D11 + D12GD −α2mI
≺ 0 ,
βmI C2Y + D22E C2 + D22GC(C2Y + D22E)T Y I(C2 + D22GC)T I X
≻ 0 .
Inequalities above consist of convex but infinite-dimensionaloptimization problem.
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LPV Model for Rotor/AMB Systems
xz1
z2
y
=
A(Ω) B1(Ω2) B2
C1 0 D12
C2 0 0C D 0
xwu
System has parameter dependence to Ω(t) due to gyroscopic effectsand to Ω2(t) due to unbalance forces.
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LPV Model for Rotor/AMB Systems
xz1
z2
y
=
A(Ω) B1(Ω2) B2
C1 0 D12
C2 0 0C D 0
xwu
System has parameter dependence to Ω(t) due to gyroscopic effectsand to Ω2(t) due to unbalance forces.
Letting all of the parameter dependent functions to have an affinestructure, (such as X (Ω) = X0 + ΩX1) infinite-dimensional inequalitiesfor controller synthesis become a series of LMIs with lineardependence on Ω and linear/quadratic/cubic dependence on Ω .
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LPV Model for Rotor/AMB Systems
xz1
z2
y
=
A(Ω) B1(Ω2) B2
C1 0 D12
C2 0 0C D 0
xwu
System has parameter dependence to Ω(t) due to gyroscopic effectsand to Ω2(t) due to unbalance forces.
Letting all of the parameter dependent functions to have an affinestructure, (such as X (Ω) = X0 + ΩX1) infinite-dimensional inequalitiesfor controller synthesis become a series of LMIs with lineardependence on Ω and linear/quadratic/cubic dependence on Ω .
Hence one only needs to check these matrix inequalities at thevertices of the polytope defined by P = [Ωmin,Ωmax ] × [Ωmin, Ωmax ]
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Numerical Results with LPV (L2) Controllers
LPV controller for the parameter (rotor speed) dependent rotor/AMBsystem can be designed via semidefinite programming satisfyingseveral 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:
10−2
100
102
104
106
108
−300
−250
−200
−150
−100
−50
0
50
Singular Values
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
LPV Closed−loopSVs at 3000 RPM
10−2
100
102
104
106
108
−250
−200
−150
−100
−50
0
50
Singular Values
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
LPV Closed−loopSVs at 6000 RPM
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Results with LPV (L2) Controllers
Controller to robustly stabilize the system with L2 performance issynthesized inside a four-dimensional convex hull with the rotor speedrange from 0 rad/s to 614 rad/s (6000 rpm), and angular accelerationrange from -15 rad/s2 to 15 rad/s2.
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Results with LPV (L2) Controllers
Controller to robustly stabilize the system with L2 performance issynthesized inside a four-dimensional convex hull with the rotor speedrange from 0 rad/s to 614 rad/s (6000 rpm), and angular accelerationrange from -15 rad/s2 to 15 rad/s2.
L2 performance α of the closed-loop LPV system at the instantaneousspeed 6000 RPM is 56.31. Note that this performance is achieved witha controller of the form
(
xK
u
)
=
(
AK (Ω, Ω) BK (Ω)CK (Ω) DK (Ω)
)(
xK
y
)
·
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Results with LPV (L2) Controllers
Controller to robustly stabilize the system with L2 performance issynthesized inside a four-dimensional convex hull with the rotor speedrange from 0 rad/s to 614 rad/s (6000 rpm), and angular accelerationrange from -15 rad/s2 to 15 rad/s2.
L2 performance α of the closed-loop LPV system at the instantaneousspeed 6000 RPM is 56.31. Note that this performance is achieved witha controller of the form
(
xK
u
)
=
(
AK (Ω, Ω) BK (Ω)CK (Ω) DK (Ω)
)(
xK
y
)
·
If the matrix function X used for the stabilization of the closed-loopsystem is assumed to be constant (time-invariant), then the controllermatrices will not depend on the angular acceleration of the rotor, andthe controller will be of the form
(
xK
u
)
=
(
AK (Ω) BK (Ω)CK (Ω) DK (Ω)
)(
xK
y
)
·
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Results with LPV (L2) Controllers
Comparing the L2 performance of the controllers, it can be said thatthere 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 RPM
Structure of X and Y α Controller FormX = X0 + ΩX1 Y = Y0 + ΩY1 15.92 Acceleration FeedbackX = X0 Y = Y0 + ΩY1 19.13 No Acc. FeedbackX = X0 Y = Y0 27.56 No Acc. Feedback
Table: L2 performance of LPV closed-loop systems at 6000 RPM
Structure of X and Y α Controller FormX = X0 + ΩX1 Y = Y0 + ΩY1 56.31 Acceleration FeedbackX = X0 Y = Y0 + ΩY1 65.42 No Acc. FeedbackX = X0 Y = Y0 102.29 No Acc. Feedback
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Numerical Results with Multi-objective LPV Controller
A multi-objective LPV controller with mixed performance specificationis synthesized within the same convex hull as the single objective LPVcontroller for a maximum operating speed of 6000 rpm.
Generalized L2 → L∞ performance βm of the multi-objective LPVcontroller is found to be 364.4, with L2 performance level αm of 72.12at 6000 rpm.
10−2
100
102
104
106
−100
−80
−60
−40
−20
0
20
40
Singular Values
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
SVs of Multi−objectiveController at 6000 RPM
10−2
100
102
104
106
108
−250
−200
−150
−100
−50
0
50
Singular Values
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
Closed−loop SVs ofMulti−objective LPVSystem at 6000 RPM
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Simulations with Multi-objective LPV Controller
Simulations for the LPV system are made using the LFR Toolbox fromONERA for MATLAB R©-Simulink.
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Simulations with Multi-objective LPV Controller
A pulse signal with 1 V amplitude and 0.025 seconds duration and isinjected into the loop at 0.2 seconds of simulation time at the input ofthe controller. Control current and rotor position at bearing A in y-axisfor LPV control with L2 performance and with mixed performance isshown in the figures.
0 0.1 0.2 0.3 0.4 0.5−4
−3
−2
−1
0
1
2
3
4
Time (sec)
ic,A
y (A
mpe
res)
0 0.1 0.2 0.3 0.4 0.5−1.2
−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
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0 0.1 0.2 0.3 0.4 0.5−4
−3
−2
−1
0
1
2
3
4
Time (sec)
ic,A
y (A
mpe
res)
0 0.1 0.2 0.3 0.4 0.5−1.2
−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
0 0.1 0.2 0.3 0.4 0.5−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
Time (sec)
ic,A
y
0 0.1 0.2 0.3 0.4 0.5−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
Time (sec)
YA
Figure: Control current and rotor displacement with LPV “mixed” control
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Conclusion
Comparing the results, it is clear that the peak values of both thecontrol current and rotor position are suppressed in the closed-loopsystem with the multi-objective controller. Hence mixed controlprovides additional flexibility with respect to transients.
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