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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

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


Outline






Overview

Radial electromagnetic bearing

50 100 150 200 250 300 350

50

100

150

200

250


Overview

Radial 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 andgrinding spindles

Electric motors

Turbomolecular pumps

Blood pumps

Computer hard diskdrives, x-ray devices, ...


Outline






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.


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.



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) .



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

·


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


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 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 .


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.


Outline






Controller design

Kym u

di

n+

+v

di

nw

+ +

z

yuP

K

v ym

ue

G

G


Controller design

Measurement(Feedback)Input

w z

u y

Manipulated

K

P

PerformanceOutputInput

Exogenous


Controller design


w z

u y

Manipulated

K

P


Exogenous

Q: How to choose K ?


Controller design


w z

u y

Manipulated

K

P


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


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


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


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


Model uncertainty

Uncertainty in Rotor/AMB Models


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−1p (jω)∆(jω) W−1

q (jω))

= σ(

∆(jω))

≤ 1 ∀ω ∈ Re

∆ :=

[

δks I 00 ΩI

]


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


Robust stability and performance

w z

qp

w z

∆

M

N


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


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 .







+for

all ∆ with σ(∆) ≤ 1 .


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 .







+for

all ∆ with σ(∆) ≤ 1 .


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 .


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


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


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)


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)



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

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


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


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


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)


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)



Simulations

Simulation Environment in SIMULINK


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)

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


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

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)


Outline






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.


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


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

)

·



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 .



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)| < ∞ .



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 ,




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 ·


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.


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.



xz1

z2

y

=

A(Ω) B1(Ω2) B2

C1 0 D12

C2 0 0C D 0

xwu


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 .



xz1

z2

y

=

A(Ω) B1(Ω2) B2

C1 0 D12

C2 0 0C D 0

xwu


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 ]


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


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

)

·




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

)

·



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


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


Simulations with Multi-objective LPV Controller

Simulations for the LPV system are made using the LFR Toolbox fromONERA for MATLAB R©-Simulink.


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


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


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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