63
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Chapter 7: Sub-Optimal Control
7.1. Rigid Rotor
Figure 7.1 summarizes the rotor under consideration. The reference systems is
involved in order to control sections of “A” and “B” by magnetic bearings. The body is
deformable, it greatly simplifies the discussion in the realization of the mathematical
model and study of the frequency response that would be further aggravated by the study
of flexural and torsion critical speeds.
Fig. 7.1: Schematic view of rigid rotor configuration for mathematical model.
The mathematical model that shows the radial displacement is shown in (7.1.1), [6] and
[7]:
M b qb  t     G b qb  t   f  i  t  , qb  t    B q1f g  B q1f  t 
..
.
(7.1.1)
where the used matrices are: mass matrix Mb , gyroscopic matrix G b , coordinate
transformation matrix Bq1 , actuators force f  i  t  , qb  t   , harmonic excitation f  t 
gravity force f g ;
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64
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 mb11
 0
Mb  
 mb31

 0
 0
g
G b   21
 0

 g 41
 1
 l
B q   bA
 0

 0
0
mb13
mb 22
0
0
mb33
mb 42
0
g12
0
0
g 23
g32
0
0
g 43
0
1
0
lbB
1
0
lbA
0
0 
mb 24 
0 

mb 44 
(7.1.2)
g14 
0 
g34 

0 
(7.1.3)
0 
0 
1 

lbB 
 i  i  t 
bias
control
f  i  t  , qb  t    k 

 g0  qb  t 

(7.1.4)
  ibias  i  t control
  
  g0  qb  t 
2



2



41
(7.1.5)
i  t   ibias  i  t controllo  current running in the coils

 g0  gap
q t  generic displacement of "A" and "B" sections of rotor
 b 
(7.1.6)


m cos  t 


2    J r  J a  cos  t  
f  t   


m sin  t 


    J r  J a  sin  t  
(7.1.7)
 0 
 0 

fg  
  mg 


 0 
(7.1.8)
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65
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By using Taylor’s series expansion, the actuator’s force expression (7.1.5) is linearized as
shown in (7.1.9):
f  i  t  , qb  t    K I i   t   K S qb  t 
(7.1.9)
where the matrices K I (7.1.10) and K S (7.1.11), are respectively current and stiffness
gain matrix:
 kiA
0
KI  
0

0
 k xA
 0
KS  
 0

 0
0
0
kiA
0
0
kiB
0
0
0
0
k yA
0
0
k xB
0
0
0
0 
0

kiB 
(7.1.10)
0 
0 
0 

k yB 
(7.1.11)
The mathematical model is shown in a state space form (7.1.12) in order to have the state
vector for a feedback of outputs signals:
z  t   A z  t   B i   t   B d f g  f   t  
(7.1.12)
where the used matrices are shown in (7.1.13), (7.1.14), (7.1.15) and (7.1.16)
I
 0

A   1

1
 M b K S M b G b  
 0 
B   1  
M b K I 
84
88
(7.1.13)
(7.1.14)
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66
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 0 
Bd   1 1  
Mb Bq 
C   I 0 
84
(7.1.15)
48
(7.1.16)
7.2. Sub-Optimal control law
The optimal control law is carried out by using Bryson’s Rules, [11];
max qbi  0.001m

max ui  12 A
(7.2.17)
where the matrices Q 

1

Q  diag 
 max xbA



1

R  diag 
 max icAx


,
88
e R
1
,
44
are those shown in (18) e (19), [12,13];
1
,


, 0, 0, 0, 0 
2


(7.2.18)


2


(7.2.19)
1
  max y   max x   max y 
2
2
2
bA
,
 
2
bB
1
max icAy
,
 
2
bB
1
max icBx
,
 
2
1
max icBy

and where z  t  is the state vector (7.2.20):
z  t   [qTb  t  qTb  t ]T
(7.2.20)
In this paper the optimal control law in the time-varying angular velocity was obtained by
extracting a second-order polynomial that expresses the variation of a generic element of
the control matrix in order to vary the angular velocity as shown in figure 7.2. The
variation of the element represented by the blue color is superimposed on the red using a
computer code developed in MATLAB environment which provides a second-order
polynomial expression. The control matrix is characterized by two matrix block;
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67
__________________________________________________________________________
K  K qb
K qb 
(7.2.21)
where K qb and K qb are shown in (7.2.22);
K q  A a 2  Bb   Cc
b

2
K qb  Dd   Ee   F f
(7.2.22)
Fig. 7.2: Polyfit of k11 varying with angular speed.
Each single square block matrix is represented by a polynomial of second degree as a
function of angular velocity, where the matrices A a , Bb , Cc , Dd , Ee and F f
are the
coefficients of each polynomial matrix. By reworking the expression of the control signal:
i   t    K qb
K qb  qTb  t  qTb  t 
T
(7.2.23)
so that the feedback signals is sketched in other two different form (7.2.24) and (7.2.25):
i Δ (t )    Aa 2  Bb  Cc
Dd 2  Ee  F f  z (t )
(7.2.24)
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68
_________________________________________________________________________


i   qb  t  , qb  t  ,    2  Aa qb  t   Dd qb  t      Bbqb  t   Eeqb t    Cc qb t   Ff qb t   (7.2.25)


which shows that the generic control signal is a linear combination of matrix components
and control components of the state vector and speed that are traveling:

ii qbj  t  , qbj  t  ,      2




 k11 0
0 k
22
K0  
 k31 0

 0 k42




a q
4
ij
bj
j 1
k13
0


 

t    d q t       b q t    e q t      c q t    f q t    (7.2.26)
4
ij
bj
j 1
k15
0
k17
0
k24
0
k26
0
k33
0
k35
0
k37
0
k44
0
k46
0


4
4
ij
bj
j 1
ij
bj
j 1
 
4
4
ij
bj
ij
j 1
j 1
0 
k28 
0 

k48 
bj

(7.2.27)
In reality, the physical system does not reflect the mathematical model, since only one
sensor positioned along an axis captures a single signal and speed. So in this work it was
decided to eliminate from the matrix control, set at zero angular velocity of the elements
placed outside the main diagonal of each block Cc 
44
and Ff 
44
in order to obtain
the control matrix (7.2.28):
K d   diag k11 , k22 , k33 , k44  diag k15 , k26 , k37 , k48
(7.2.28)
7.3. Results and Discussion
The values of maximum eccentricity and angle of deviation of the main axis of
inertia were derived on the basis of the relations present in [10] and shown in table VI.
Table VI
Properties of Rotor
Symbol
Description
S.I.
m
mass of rotor
20.75 kg
Jr
radial inertia
1.85 kg  m2
Ja
axial inertia
0.0902 kg  m2

eccentricity
4  106 m

deviation inertia axis
6 106
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69
__________________________________________________________________________
The characteristic of radial active magnetic bearings have been extrapolated using a
computer code developed in MATLAB based on [1] and are shown in table II;
Table VII
Electro-mechanic properties of radial active magnetic bearing
Symbol
Description
S.I.
kxA  k yA
stiffness gain of AMB “A”
1.25 MN / m
kxB  k yB
stiffness gain of AMB “B”
0.46 MN / m
kixA  kiyA
current gain of AMB “A”
400 N / A
kixB  kiyB
current gain of AMB “B”
147 N / A
I max A  I max B
maximum value of current in the coil
12 A
g0 A  g0 B
gap
2 103 m
The simulations were carried out in the event that the sensors are ideal, meaning that they
pick up the order of movements till 1034 m . The law of variation of the angular velocity is
assumed to be linear and reaches its maximum value at the end of the simulation with a
value of 1400 rad / s at t  10 s . Figures 7.3 and 7.4 respectively show the displacement
and control currents by law K, when at t  0.3s an excitation acts on the rotor in abruptly
way equals to 12 times the gravity force in order to simulate an exogenous excitation along
y direction. By looking to the displacement along y direction the upper electromagnet is
crossing by a value of current (blue colored) that is higher than current running in the
lower magnet (green colored). The figure 7.5 shows a picture of radial active magnetic
bearing build-up with its electromagnets and coil winding:
-5
1
x 10
-5
Spostamenti della sezione A in direzione "x"
5
x 10
Spostamenti della sezione A in direzione "y"
0
ybA [m]
xbA [m]
0
-1
-5
-2
-3
0
2
4
6
8
10
-10
0
2
Tempo [s]
-5
3
x 10
0
6
8
10
x 10
Spostamenti della sezione B in direzione "y"
-1
ybB [m]
xbB [m]
2
1
0
-1
0
4
Tempo [s]
-4
Spostamenti della sezione B in direzione "x"
-2
-3
2
4
Tempo [s]
6
8
10
-4
0
2
4
6
8
10
Tempo [s]
Fig. 7.3: Displacement of rotor’s section A and B along x and y direction.
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70
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Correnti delle bobine del cuscinetto A in direzione "x"
Correnti delle bobine del cuscinetto A in direzione "y"
6.5
12
6.4
10
icyA [A]
icxA [A]
8
6.3
6.2
6
4
6.1
6
0
2
2
4
6
8
0
0
10
2
Tempo [s]
4
6
8
10
Tempo [s]
Correnti delle bobine del cuscinetto B in direzione "x"
Correnti delle bobine del cuscinetto B in direzione "y"
6.6
12
10
6.4
icyB [A]
icxB [A]
8
6.2
6
4
6
2
0
2
4
Tempo [s]
6
8
10
0
0
2
4
6
8
10
Tempo [s]
Fig. 7.4: Current injected into the coil of active magnetic bearing A and B
Fig. 7.5: Radial active magnetic bearing.
Figures 7.6 and 7.7 show the same simulations when the control law is represented by suboptimal control matrix. In the same figures we see that the displacements in the x direction
of the part “A” and “B” belonging to the rotor and controlled by the law are time-varying
and larger by two orders of magnitude for the “A” and one order of magnitude for the
section “B”. The currents running in coils, arranged in the same direction, exhibit a
variation in their coordinated and differentiated in their Power Width Modulation (PWM)
as shown by the figures 7.4 and 7.7. From the same figures is shown that using control law
K, the final value of the currents of the coils placed in the direction of the x-axis are greater
than 10A compared to those produced by the matrix K d . The displacements in y direction
are maintained in the same order of magnitude, and comply with the condition of Brayson.
Brayson’s rule is also respected for the current injected into the control coils placed along
the axis of maximum stress, y-axis, in fact, they do not reach the value of 12A and exhibit
the same variation observed in the coils arranged in the direction x-axis. In both cases, the
sections of the rotor oscillate by increasing the angular speed.
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71
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-7
6
x 10
-4
Spostamenti della sezione A in direzione "x"
0
Spostamenti della sezione A in direzione "y"
x 10
4
-0.5
ybA [m]
xbA [m]
2
0
-2
-1
-4
-6
0
2
4
6
8
-1.5
0
10
2
4
Tempo [s]
-6
3
x 10
6
8
10
Tempo [m]
-4
Spostamenti della sezione B in direzione "x"
0
Spostamenti della sezione B in direzione "y"
x 10
2
-1
ybB [m]
xbB [m]
1
0
-2
-1
-3
-2
-3
0
2
4
6
8
-4
0
10
2
4
Tempo [s]
6
8
10
Tempo [s]
Fig. 7.6: Displacement of rotor’s section A and B along x and y direction.
Correnti delle bobine del cuscinetto A in direzione "x"
Correnti delle bobine del cuscinetto A in direzione "y"
6.4
12
6.35
10
icyA [A]
icxA [A]
8
6.3
6.25
6
4
6.2
6.15
0
2
2
4
6
8
0
0
10
2
Tempo [s]
4
6
8
10
Tempo [s]
Correnti delle bobine del cuscinetto B in direzione "x"
Correnti delle bobine del cuscinetto B in direzione "y"
6.5
12
10
6.4
icyB [A]
icxB [A]
8
6.3
6.2
6
4
6.1
6
0
2
2
4
Tempo [s]
6
8
10
0
0
2
4
6
8
10
Tempo [s]
Fig. 7.7: Current injected into the coil of active magnetic bearing A and B.
Fig. 7.8: Frequency response of AMB A varying Ω.
Fig. 7.9: Frequency response of AMB B varying Ω.
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72
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Fig. 7.10: Frequency response of AMB A varying Ω with K0.
Fig. 7.11: Frequency response of AMB A varying Ω with K0.
Figures 7.8 and 7.9 show the frequency response of the two bearings A and B, by using the
optimal control matrix K. We note that, in this case, one axis has a peak of resonance at
  0 rad / s that is verified at the frequency of 90 rad / s ; by increasing the angular speed
this peak of resonance disappears.
Figures 7.10 and 7.11 show the frequency response of the bearing “A” and “B” for the suboptimal control law K d . This produces a steady behavior in the frequency response
because it does not depend on the angular velocity  and does not give rise to resonance
peaks, while the angular velocity increases the stability characteristics remain unchanged.
In the figure 7.12 is shown an analysis based on comparison of response time and settling
time as a function of angular velocity in the presence of the two control laws
aforementioned. From figures 7.12a and 7.12b we see that the response times obtained by
sub-optimal control are lower than those obtained by time-varying law which suggests that
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73
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the sub-optimal control system exhibits a greater responsiveness but a high settling times
and thus less dynamic precision of answer than the dynamic time-varying control, figures
7.12c and 7.12d:
Fig. 7.12: Time response and settling time varying the angular speed and control matrix.
7.4. Experimental Optimal control law
By considering the data shown in tables VIII and IX a set of experiments are
performed in order to compare the substantial differences between the implemented
controllers.
The experimental test rig consists of a 304.8mm long steel shaft, 12.7mm in diameter
driven by a 48 V DC electric motor via a flexible coupling. There are two radial AMB
rotors and one balance disk mounted on a shaft.;
Table VIII
Shaft parameters
Symbol
Description
S.I.
m
mass of rotor
1.35 kg
Jr
radial inertia
4.09 104 kg  m2
Ja
axial inertia
7.22 103 kg  m2

eccentricity
4  106 m

deviation inertia axis
6 106
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74
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Table IX
Electromechanical Parameters
Symbol
Description
S.I.
kxA  k yA
stiffness gain of AMB “A”
1.44 105 N / m
kxB  k yB
stiffness gain of AMB “B”
1.44 105 N / m
kixA  kiyA
current gain of AMB “A”
38 N / A
kixB  kiyB
current gain of AMB “B”
38 N / A
I max A  I max B
maximum value of current in the coil
3A
g0 A  g0 B
gap
3.81104 m
The bearing span is 209.55mm and the disk is located in the middle span from the
outboard bearing. The total rotor assembly has a mass of 1.35Kg and is supported on two
radial AMB’s and each of these have a static load capacity of 53 N and saturation current
of 3A . Axial support is provided by stiffness of the flexible motor coupling. The AMB’s
are controlled with a dSPACE DS1103 via a SKF MB340g4-ERX for current
amplification. The digital controller is programmed using MATLab Simulink and samples
at 10kHz . Figure 7.13 a photograph of the experimental test rig.
Fig.7.13: Experimental test rig.
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75
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The speed sensor is based on a revolution key phasor. The experimental rig is levitated and
after the system reaches steady state, rotation is started. The initial jump in angular speed
due to a minimum speed restriction is followed by a 1Hz per speed ramp. When the rotor
reaches 65Hz rotation, it is allowed to rotate at constant speed for approximately 15s . The
same duty cycle was applied to the rotor levitated with each of the three controllers. In the
figure 7.14 is depicted the time history of a single inboard AMB position sensor for each of
the controllers and the corresponding rotor speed. Note that the transient at the beginning
of rotation are in arbitrary directions depending on the initial phase of unbalance:
Fig.7.14: Experimental position response at inboard AMB sensor and speed signal for sub-optimal control
law.
In order to compare the results with the conventional methods a set of experiments are
performed by different control system such PID and optimal Gaussian control in both
configuration or rather variant and invariant with the speed such in the figure 9, 10 and 11:
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76
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Fig.7.15: Experimental position response at inboard AMB sensor and speed signal for PID control law.
Fig.7.16: Experimental position response at inboard AMB sensor and speed signal for speed invariant optimal
Gaussian.
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77
__________________________________________________________________________
Fig.7.17: Experimental position response at inboard AMB sensor and speed signal for speed varying optimal
Gaussian control law.
In the figure 7.15 the PID controller exhibits higher displacements than optimal Gaussian
control shown in figures 7.16 and 7.17. In the figures 10 and 11 we can see that
displacements produced in presence of optimal control law, as a speed varying solution of
algebraic Riccati’s equation, are similar in magnitude and dynamic behavior during the
ramp-up and ramp-down of speed signal injected by motors. The test shows that
contactless motion between stator and rotor is guaranteed during the rump up of angular
speed in presence of acting loads. All the involved variables stay within their maximum
design values. The experimental comparison between the proposed method and the control
systems developed on line with speed variation has shown that the polynomial
approximation of each coefficient belonging to control matrix provide a useful tool to
perform an experimental application. The advantage of this method is obviously the
reduced computational burden of calculator that in case of speed variant solution it must
solve a set of equations to carry out the stabilizing solution of Riccati’s equation.
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78
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