Comparison of stability of centralized and decentralized ... - IFToMM

12th IFToMM World Congress, Besanc¸on, June 18-21, 2007
CK-xxx
Comparison of stability of centralized and decentralized nonlinear control systems
for electromagnetic suspension
N.G. Kodochigov 1 V.S. Vostokov 2 V.S. Gorbunov 3 S.V. Lebedeva 4 I.V. Drumov 5
OKBM ОКBМ ОКBМ VGАVТ ОКBМ
Nizny Novgorod,
Russia
Nizny Novgorod,
Russia
Nizny Novgorod,
Russia
Nizny Novgorod,
Russia
Nizny Novgorod,
Russia
Abstract - The paper presents the results of performed
comprehensive analyses of full electromagnetic suspension
stability using Lyapunov method. Analyses proved that the
dynamic mechatronic system “rotor + electromagnetic
bearings control system (EMB CS)” is stable without any
additional measures needed to linearize the magnetic force
in systems with centralized control. In case of
decentralized control, overall stability cannot be proven.
However, stability of a specific mode of motion is shown.
Keyword: stability, centralized, decentralized, control, electromagnet
I. Introduction
Mode of rotor motion in EMBs is studied in a large
number of papers. Suffice it to say that international
conferences on rotor dynamics are held at regular
intervals. The majority of publications are devoted to
various methods of linearizing the magnetic force
(introduction of displacement currents [1,2] or special
nonlinearities [3], use of nonlinearity compensators, etc.)
and control techniques (use of filters [4]). Some papers
concentrate on proving dynamic system stability using the
Lyapunov method [3].
The electromagnetic suspension system has several
nonlinearities:
- quadratic dependence of magnetic force on control
current;
- hyperbolic dependence of magnetic force on the air
gap;
- saturation of radial electromagnet iron;
- supply of increased voltage to electromagnets until
the control current reaches the preset value.
1 E-mail: kodochigov@okbm.nnov.ru
2 E-mail: gorbunov@okbm.nnov.ru
3 E-mail: gorbunov@okbm.nnov.ru
4 E-mail: vip@aqua.sci-nnov.ru
5 E-mail gorbunov@okbm.nnov.ru
From the viewpoint of zero equilibrium
stability, the most important factor is nonlinear
dependence of magnetic force on the control
current because at the zero point the control current
is zero, and the derivative from the quadratic
function of magnetic force dependence on the
control current is also zero.
Lately, Lyapunov method has been used in a
number of publications [5] to prove stability of zero
equilibrium state with account of some
nonlinearities (quadratic dependence of magnetic
force on control current, and saturation of radial
electromagnet iron) and gyroscopic crosscouplings.
However, analytical investigations are
performed using a one-mass rotor model with one
EMB.
It is proven that the zero equilibrium state of
the one-mass EMB model [6] is asymptotically
stable in nonlinear approximation under centralized
control.
The statement is proven using Lyapunov
method.
In this paper we shall prove asymptotic
stability of the zero equilibrium state of full (3-
dimensional) electromagnetic suspension of
turbomachine vertical rotor under centralized
control, i.e. when two independent control signals
(based on rotor inclination and mass center
position) are generated along each EMB coordinate.
It is proven that stability of the zero
equilibrium state of an EMB CS, which functions
based on signals of rotor-stator gap variation in the
EMB area, is possible only for a certain type of
rotor motion (e.g. when rotor lateral misalignment
prevails). For other motion types, stability is not
proven.
This work can be considered a first step in
validation of the possibility to create a CS without
“pre-loading" the rotor, which helps to reduce EMB
working temperatures.

II.
Stability of zero equilibrium state in a full
electromagnetic suspension with centralized
control
A. Mathematical model of rotor electromagnetic
suspension
The object of the study is a vertical rotor in two EMBs.
Displacement current is not used; therefore only
electromagnet windings opposite to rotor displacement
direction are in use. Control of rotor axial motion is
organized conventionally and is not considered in this
paper. It is assumed that influence of vertical forces on the
horizontal constituent is weak.
The complete mathematical model of the rotor with
gyroscopic cross couplings is further considered.
The mathematical model of rotor uses dynamic
equations in coordinates of mass center and rotation
angles. The initial model of full electromagnetic
suspension with account of equations for translational and
oscillatory motion has the form [2]:
A && α + & βСΩ
= Mα
m && x = F x
(1.1)
A && β − & αСΩ
= M
m && y = F y
,
where α is angle of rotor displacement in yz plane
(Figure 1); β is angle of rotor displacement in xz plane; x
and y are mass center coordinates; А is moment of rotor
inertia relatively to each axis lying in the rotor equatorial
plane; m is rotor mass; С is moment of rotor inertia
relatively to central rotation axis; Ω is absolute angular
speed of rotation around the central axis; F , F are
forces from the radial EMB acting along axes x and y;
M M α
,
β
are moments from the radial EMB acting at
angles α and β .
B. Control system structure
β
Mathematical model of the control system does not
account for sensor response time or current in
electromagnet windings. Centralized control is considered;
that is when mass center based control does not lead to
occurrence of the moment of force, and angle based
control does not lead to displacement of the mass center.
Thus it appears that there are no coordinate cross
couplings between equations (1.1). Equations for a
proportional-differential controller are considered. For a
rigid rotor with centralized control, position of mass center
x
y
and rotation angles are calculated based on the
measured rotor displacement in EMBs relative to
the stator central axis. It follows that:
x aα
α вu
= x bα
α l
= −
where u stands for “upper”, l stands for “lower”, а
is the distance between the upper bearing and the
rotor mass center; b is the distance between the
lower bearing and the rotor mass center; x αu is
calculated rotor displacement (relative to stator
central axis) in the upper bearing, with the fixed
mass center; x αl s calculated rotor displacement
(relative to stator central axis) in the lower bearing,
with the fixed mass center.
Equations for y β and β are of a similar form.
Equations for forces and moments from
electromagnets will be of the form:
F
x
= −2k1I
m.
c.
x
I
m.
c.
x
М = −k1(
Iα
u
Iα
u
a − Iα
l
Iα
b)
, (1.2)
α
l
where m.c. stands for “mass center”; k 1 =L 0 /2x 0 ; L 0
is inductivity of electromagnet winding with the
rotor in central position; I m.c.x is mass center based
control current; I αl , I αu is angle based control
current; x 0 is nominal air gap in the electromagnet.
Equation (1.2) is written in the assumption that
2 2
( x0 ± ∆x)
≈ x0
(this assumption holds because
out of two electromagnets the one that is active is
the one with the air gap larger than x 0
). Here ∆x is
displacement of the bearing rotor part relative to the
stator central axis
Equations for F у and М β are of a similar form.
Let us pass on from equations (1.1) in variables
α,β to new equations in variables x αu , y βu according
to correlations α= x αu /а; β= y βu /а:
& x
y
αв
βв
A + &
СΩ = −k
а a
& y
βв
xα
в
A − & СΩ = −k
I
а a
&&
1
( Iα
в
Iα
в
а − Iα
н
Iα
н
b)
I
а
1
(
βв
βв
−
βн
βн
)
m x = −2k1I m . c.
x
I
m . c.
x
I
I
= k
x + k
x&
I
I
3 3
a + b
= −k1
I
2
a
3 3
a + b
b = −k1
I
2
a
m . c.
x рx dx
(1.3)
α u
I
β u
= k
= k
рα
рβ
x
αu
y
βu
+ k
+ k
dα
dβ
x&
αu
y&
βu
I
αв
αв
I
βв
βв

where k px is proportional coefficient of mass center based
controller; k dx is differential coefficient of mass center
based controller; k pα, k pβ is proportional coefficient of α
and β angle based controller; k dα, k dβ is differential
coefficient of α and β angle based controller.
All coefficients k p and k d (with indices) are positive.
Equations for y,I m.c.y are of a similar form.
Thus, in assumption of centralized control, the
equations become independent, which permits us to use
earlier obtained results to write one second-degree
equation [6].
C. System stability
Control is independent of rotor inclination angle and
mass center deviation from the central axis. It can be easily
shown that system (1.3) can be transformed into two
equations for control current:
2k1
d
2
m I&
m. c.
xI
&
m.
c.
x
+ k
px
( I
m.
c.
x
I
m.
c.
xI
m.
c.
x
) = −4k1kdx
⋅ I
m.
c.
x
( I&
m.
c.
x
)
3 dt
and
3 3
A
a b
2
[ I&&
u
I&
u
I&&
uI
& +
⋅
α α
+
β βu
] = − k1
⋅{[
k ( ) 2 ( ) ]
2
p
I
u
I
uI
&
α α α αu
+ kd
α
Iα
u
I&
αu
+
a
a
2
+ [ k ( I I I&
) + 2k
I ( I&
) ]}.
pβ
βв
βв
βв
dβ
βв
βв
So, the sought variables are control currents I m.c.x,
I m.c.y, I αu, I βu.
The Lyapunov function has the form (1.4):
2
2
I&
I&
m.
c.
x m.
c.
y 2k1
2
2
v = m + m + ⋅{
kpx
Im . c.
x
Im . c.
x
+ kpy
Im . c.
y
Im . c.
y}
+
2 2 3
(1.4)
2 2 3 3
I&
I&
αu
βu
k1
a + b
2
2
A + A + ⋅{
k + } > 0
2 pα
Iαu
Iαu
kpβ
Iβu
Iβu
2а
2а
3 a
and accordingly, its derivative taken with account of
original equations is:
3 3
2
2 a + b
2
2
− 4k1 ⋅{
kdx
Im.
c.
x
I&
m.
c.
x
+ kdy
Im.
c.
y
I&
m.
c.
y
} − 2k1
⋅{
k + } < 0
2 d
I
u
I&
α α αu
kdβ
I
βu
I&
βu
a
At the same time,
d
dt
( I I)
= 2 I I&
.
d
dt
( I II)
= 3 I II&
, and
Based on the above, we can conclude that equilibrium
state
= = = = = =
x
y
I
I
I
=
m. c.
x m.
c.
у u βu
αl
βl
=
α
is
asymptotically stable. Thus, asymptotic stability of the
nonlinear system of full electromagnetic suspension is
proven.
It should be noted that stability is proven in variables
or:
I
I
I
0
Im. c.
x;
Iα u;
I βu
; but it follows from
I = k x + k x&
that
α u
рα
αu
αu
+ kdα
xαu
dα
αu
k
р
x &
α → 0 . So it is evident that
x α u
→ 0 if t → ∞ . Thus, we can assert that
stability in variables α , & α is also proven. The same
holds for x , x& , y,
y&
,β.
III.
System stability with account of inertia
of control current build-up in
electromagnet winding
In order to simplify things, we will consider the
case of rotor holdup in one coordinate (e.g. along x
axis).
The equation for control current with account
of current inertia in EMB winding is written as
follows:
k
p
x + kd
x& = τ I&
+ I, (2.1)
where k p is controller proportional coefficient;
k d is controller differential coefficient;
τ is time constant of electromagnet winding
L 0
τ = ; R is EMB winding resistance.
R
If we introduce dimensionless quantity
~ t
t = ,
τ
equation (2.1) will be transformed into:
dI kd dx
I k
px
dt
~ + = +
τ dt
~ . (2.2)
Equations (1.1) are transformed in the same
way:
2
m d x L0
I I
2 2
dt
~ = − .
τ 2x0
2
2
d x L0τ
Substituting
I I
dt
~ = −
into (2.2),
2x0m
we receive:
2
d I dI kdτL0
dx
I I k
p
dt
~ +
dt
~ = − +
2
2x
m dt
~ . (2.3)
0
If we define expressions to the left of I I and
( I I)
dt
~
∫
as c and h, we obtain:
2
d I dI
( )
~
~ + ~ + c I I + h = 0
2 ∫ I I dt
,
dt dt

2
d I dI d
h I I dt
~
h I I dt
~
~ + ~ + ~ ( ) ) ( ) ( c h)
I I
dt dt dt
( + = − − .(2.4)
2 ∫ ∫
Multiplying the left and right sides of equation (2.4) by
dI h I I)
dt
~
dt
~ + ∫ ( , we obtain:
dI ~ 2
( ~ + h∫
( I I)
dt )
d
( ) ~
~ [ dt
c − h c − h
2
+ I II + ( h ( ) ) ]
2
3 2
∫ I I dt =
dt
h
dI
( ( )
~ 2
= − ~ + h∫
I I dt ) ,
dt
or:
dI
2
(
~
~ + h∫(
I I)
dt )
c −h
( c −h)
~ 2
v =
dt
+ I II + ( h ( ) )
2 3 2
∫ I I dt
, (2.5)
b
d dI
( )
~
~ ν = −(
~ + h∫ I I dt
)
2 < 0 .
dt dt
2
kdτL
k
0
pτ
L0
If c = and h = , stability condition
2x0m
2x0m
c>h transforms into inequality k d
> k p
τ .
The obtained result can be used in studies of stability
of full rotor suspension described above. Thus, as there are
no cross-couplings between the equations, stability of full
electromagnetic suspension with centralized CS and with
account of current inertia in EMB winding can be
considered proven
.
IV. Stability of full electromagnetic suspension of
vertical rotor with a decentralized CS
Let us consider one coordinate, neglecting gyroscopic
forces, since it was shown above that they do not affect the
result but only make calculations bulkier.
Original equations have the form
mX
1"
= −F(
X
1)
+ F(
X
2
)
,
mX
2"
= −F(
X
2
) + F(
X
1)
where Х 1 and Х 2 are coordinates of the upper and lower
bearings.
After transformations similar to those described in
Section 1, we receive cross terms I I 1 2
' and I I ' 2 1
that do
not permit to conclude whether the Lyapunov function or
its derivative are sign-constant.
Next, we replace control currents I1 and I2 and their
derivatives with angle and mass center based control
currents (analogous to replacement of variables). After
addition of the cross terms we receive only eigenproducts
I I α α
' and I ' c. m.
I
c.
m.
, but of different signs.
If we introduce them under the derivative sign,
we receive additional terms in the Lyapunov’s
2 2
−
function that have the form of
I + I α m . c . and
are identically equal to zero. Thus, on the one hand,
stability cannot be concluded as the resulting
function is not sign-constant; but we can assume
that if one component is always greater than the
other (e.g. if parallel rotor motion prevails), then
stability will be ensured. Numerical calculations on
more detailed nonlinear models and experiments
with vertical rotors confirm that such assumption
can be true.
V. Conclusions
A. Analyses of full electromagnetic suspension
stability performed using Lyapunov method
proved that the dynamic mechatronic system
“rotor + CS” is stable without any additional
measures to linearize the magnetic force.
B. Preference should be given to centralized
control systems.
C. The demonstrated result of performed analyses
is of practical value as it permits to abandon use
of displacement currents that linearize the
system but lead to overheat of electromagnet
windings.
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2000. P. 23-25.
[4] Fujiwara H., Matsushita O., Okubo H. Stability
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