Motion control for a vertical rigid rotor rotating in electromagnetic bearings

ISSN 1064�2307, Journal of Computer and Systems Sciences International, 2011, Vol. 50, No. 5, pp. 683–697. © Pleiades Publishing, Ltd., 2011.Original Russian Text © D.V. Balandin, M.M. Kogan, 2011, published in Izvestiya Akademii Nauk. Teoriya i Sistemy Upravleniya, 2011, No. 5, pp. 3–17.

683

INTRODUCTION

The problems of levitation and motion control for bodies in a varying electromagnetic field are consid�ered in a large number of publications. Most of them study linearized models, and conclusions on a systemefficiency are made based on these models [1–4]. A large number of publications are also devoted to thestudy of nonlinear models under different simplifying assumptions [5–9]. In [6], the method of Lyapunovfunctions in the assumption of nonlinear dependence (on currents only, and not on displacements) offorces and torques acting on the rotor from electromagnets is used to substantiate the stability of a rigidrotor on two electromagnetic bearings with a proportional–differential controller.

In [7–9] the problems of current control in electromagnet coils for providing stability of levitation ofa ferromagnetic body are considered. In these papers, the central idea is the idea of feedback linearization,well developed in the control theory; this idea consists in the replacement of nonlinear functions includedin the mathematical model of an object by some new generalized control.

In this paper we consider the problems of control for a vertically rotating rigid rotor in electromagneticbearings; we use and develop the idea of feedback linearization, namely, additionally to the requirementof asymptotic stability, we use the requirements of boundedness of control currents and some phase vari�ables of the system. Moreover, we describe decentralized (depending on local measurable variables only)and centralized (depending on the whole set of measurable variables) controls. Modern methods based onthe theory of convex optimization and linear matrix inequalities [10–12] are used for solving the controlproblems.

1. FEEDBACK LINEARIZATION OF ELECTROMAGNETIC SUSPENSION

For illustrating the idea of feedback linearization, we consider the mechanical system shown in Fig. 1;here, 1 is the suspended ferromagnetic body with the mass m, and 2 and 3 are the electromagnets poweredby alternating currents that provide the body levitation. The point of origin is taken to be in the middlebetween the electromagnets; one half of the distance between them is equal to S0.

The motion equation of the object along Oz axis has the form [1]

(1.1)

where mg is the force of gravity acting on the body,

determine the forces acting on the body from the upper and lower electromagnets, I1, I2 are the currentsin the upper and lower electromagnets, respectively, and L0 is the inductance of each of the electromag�

= − + − ,�� 1 2mz mg F F

= , =

− +

2 20 0 0 01 2

1 22 20 02 2( ) ( )

L S L SI IF F

S z S z

CONTROL IN DETERMINISTIC SYSTEMS

Motion Control for a Vertical Rigid Rotor Rotating in Electromagnetic Bearings

D. V. Balandin and M. M. KoganLobachevsky State University of Nizhny Novgorod, pr. Gagarina 23, Nizhny Novgorod, 603950 Russia

Nizhny Novgorod State University of Architecture and Civil Engineering, ul. Il’inskaya 65, Nizhny Novgorod, 603950 RussiaReceived March 21, 2011; in final form, April 19, 2011

Abstract—The problems of control of a motion of a rigid rotor in electromagnetic bearings are con�sidered. The main ides of synthesis of the control laws is the approach based on feedback linearizationof the original nonlinear mathematical model of the system. The method of Lyapunov functions andmethods connected with solution of linear matrix inequalities are used in synthesis of control laws.

DOI: 10.1134/S1064230711050042

684

JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL Vol. 50 No. 5 2011

BALANDIN, KOGAN

nets. Note that if , , and the following equality is satisfied: , the bodyis in the state of unstable equilibrium, .

For stabilizing this system in a neighborhood of the equilibrium, different control laws in the feedbackform that use information on the current state (position and velocity) of the body can be applied. It shouldbe noted that in practice the current in the electromagnet circuit can be changed only by changing voltagesupplied to the inductance coil. Therefore, generally speaking, the considered motion equation should beadded by one more equation describing the process of current variation depending on voltage variation,which in turn varies according to a certain law with account of the current system state. Nonetheless, here�inafter we assume that there exists a technical possibility to implement the so called “current control” inthe following form: , with high precision and small delay. In essence, this meansthat the typical time of establishment of the required current in the electromagnet circuit is much shorterthan the typical time connected with the body motion (for some realistic electromagnetic suspensiondesigns, this assumption is quite justified). Thus, hereinafter we consider the current, rather than the volt�age, as the control variable.

Let us choose the control in the form , ( ). In this caseEq. (1.1), linearized in the neighborhood of the equilibrium takes the form

(1.2)

where , . If the parameter k1 is chosen in such a way that, and , the equilibrium state of system (1.1) is asymptotically stable in the linear approx�

imation. At the same time, the analysis of asymptotic stability in large of this nonlinear system for the cho�sen law of current variation is quite difficult [6], and as far as we know, it has not been performed yet. Thiscircumstance is a serious disadvantage of this control method. In this study, we consider another approachto choosing the law of variation of the control current based on the method known in control theory as“feedback linearization”.

The objective of feedback linearization is to choose the control currents in such a way that the closed�loop system takes the form of the linear system

(1.3)

where a, b are arbitrary positive coefficients. This can be achieved in different ways by defining the currentsI1 and I2, for example, as

(1.4)

=1 01I I =2 02I I − = /2 201 02 0 02I I mgS L

= , =�0 0z z

= , �1 1( )I I z z = , �2 2( )I I z z

= − − �1 0 1 2I I k z k z =2 0I = /20 0 02I mgS L

= , =�( 0 0)z z

= − − ,�� 1 2mz K z K z

= −/ /2 21 1 0 0 0 0 0 0K k L I S L I S = /2 2 0 0 0K k L I S

> /1 0 0k I S >2 0k

= − − ,�� mz az bz

001 0

00

1 ( ) 0,

0,

az bzS z az bzI

I FS

S z az bz

⎧ | + |+ − + <⎪= ⎨

⎪ − + ≥⎩

for

for

�

�

�

0

Oz

1

3

2

S0

I1

Оz

I2

Fig. 1.


MOTION CONTROL FOR A VERTICAL RIGID ROTOR ROTATING 685

(1.5)

where , and the current I0 is determined from the condition . Indeed, by substi�tuting expressions (1.4) and (1.5) into original nonlinear differential equation (1.1), we obtain linear dif�ferential equation (1.3).

Now it is necessary to choose the parameters a, b in Eq. (1.3) according to the requirements to the con�sidered electromechanical system. The main requirement is the asymptotic stability of the system. ForEq. (1.3) this requirement is obviously satisfied if .

A more general approach to feedback linearization is to replace the right�hand side of (1.1) by somecontrol u; then the system acquires the form

(1.6)

and the currents I1, I2 can be expressed as follows:

(1.7)

(1.8)

This method of linearization makes it possible to form the control laws for a suspended body not onlywith respect to the system state, but also the measured output, i.e., for example, if information on the bodyposition only is used, and the information on its velocity is absent.

2. MATHEMATICAL MODEL OF MOTION OF A VERTICAL RIGID ROTOR ROTATING IN RADIAL BEARINGS

Let us consider a more complex mechanical system shown in Fig. 2; it describes the motion of a rigidvertically rotating rotor held by radial electromagnetic bearings (four pairs of electromagnets). It isassumed that in the direction of Oz axis the rotor is held at the center of mass by an electromagnetic sus�pension considered in the previous section. Assuming that the rotor displacement along Oz axis is small,as compared to the gap S0, we consider separately the rotor motion along Oz according to Eq. (1.1) and itsmotion in radial electromagnetic bearings described by the following differential equations:

(2.1)

where x, y are the coordinates of the center of mass of the rotor, are the angles of rotation of the rotorwith respect to x and y axes, respectively, are the distances from the center of mass to the upper andlower electromagnetic bearings, respectively, the indices u and l indicate the electromagnetic forces actingon the rotor from the upper (u) and lower (l) electromagnets, are the principal moments of inertia ofthe rotor, and is the given angular rotation frequency of the rotor with respect to z axis. The pairs of elec�tromagnetic forces in Eqs. (2.1) are determined as follows:

(2.2)

02

000

0 0,

( ) 0,

az bzI

I az bzS z az bzS

F

+ <⎧⎪

= | + |⎨ + + ≥⎪⎩

for

for

�

�

�

=0F mg = ./20 0 0(2 )L I S mg

> , >0 0a b

= ,��mz u

001 0

00

1 ( ) 0,

0,

uS z uI

I FS

S z u

⎧+ − ≥⎪

= ⎨⎪ − <⎩

for

for

02

000

0 0,

( ) 0.

uI

I uS z uS

F

≥⎧⎪

= ⎨+ <⎪

⎩

for

for

α = − − + − − ωβ,

β = − − − + ωα,

= − + − ,

= − + − ,

�

��

��

�

��

��

1 2 1 2 2 1

1 3 4 2 3 4

3 4 3 4

2 1 2 1

( ) ( )

( ) ( )

u u l lz

u u l lz

u u l l

u u l l

J l F F l F F J

J l F F l F F J

mx F F F F

my F F F F

α,β

,1 2l l

, zJ Jω

⎧ ⎫− = − ,⎨ ⎬

− +⎩ ⎭

⎧ ⎫− = − ,⎨ ⎬

− +⎩ ⎭

2 20 0 2 1

2 1 2 20 0

2 20 0 2 1

2 1 2 20 0

2 ( ) ( )

2 ( ) ( )

u u u u

u u

l l l l

l l

L S I IF F

S y S y

L S I IF F

S y S y

686


BALANDIN, KOGAN

(2.3)

where are the measurable displacements of the rotor in the upper and lower electromagneticbearings (Fig. 3). These quantities for the rigid rotor are connected with the variables by the fol�lowing kinematical relations (here, the assumption of small angles is used):

(2.4)

3. FEEDBACK LINEARIZATION OF RADIAL ELECTROMAGNETIC BEARINGS

Let us choose the control currents in such a way that the forces with which the electromagnets act onthe rotor linearly depend on and their derivatives, i.e.,

(3.1)

⎧ ⎫− = − ,⎨ ⎬

− +⎩ ⎭

⎧ ⎫− = − ,⎨ ⎬

− +⎩ ⎭

2 20 0 3 4

3 4 2 20 0

2 20 0 3 4

3 4 2 20 0

2 ( ) ( )

2 ( ) ( )

u u u u

u u

l l l l

l l

L S I IF F

S x S x

L S I IF F

S x S x

, , ,u u l lx y x y, ,α,βx y

α,β

= + β , = − β , = − α , = + α .1 2 1 2u l u lx x l x x l y y l y y l

, , ,u l u lx x y y

− = − + ,

− = − + ,

− = − + ,

− = − + .

�

�

�

�

2 1 1 1

2 1 2 2

3 4 3 3

3 4 4 4

( )

( )

( )

( )

u uu u

l ll l

u uu u

l ll l

F F a y b y

F F a y b y

F F a x b x

F F a x b x

I1n

z

I1v

I2n

I3n

I4n

I2v

I3v

I4v

0

ω

αβ

y

x

Fig. 2.



For this purpose, we represent the expressions for all currents as follows:

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

where

1 1

1 1 10 1 1

01

0 0,

( ) 0,

u u

u u uu u u

a y b y

I a y b yS y a y b y

F

+ <⎧⎪

= +⎨+ + ≥⎪

⎩

for

| |for

�

�

�

1 10 1 1

2 01

1 1

( ) 0,

0 0,

u uu u u

u

u u

a y b yS y a y b y

I F

a y b y

⎧ +− + <⎪

= ⎨⎪ + ≥⎩

| |for

for

�

�

�

2 2

1 2 20 2 2

01

0 0,

( ) 0,

l l

l l ll l l

a y b y

I a y b yS y a y b y

F

+ <⎧⎪

= +⎨+ + ≥⎪

⎩

for

| |for

�

�

�

2 20 2 2

2 01

2 2

( ) 0,

0 0,

l ll l l

l

l l

a y b yS y a y b y

I F

a y b y

⎧ +− + <⎪

= ⎨⎪ + ≥⎩

| |for

for

�

�

�

3 30 3 3

3 01

3 3

( ) 0,

0 0,

u uu u u

u

u u

a x b x S x a x b xI F

a x b x

⎧ +− + <⎪

= ⎨⎪ + ≥⎩

| |for

for

�

�

�

3 3

4 3 30 3 3

01

0 0,

( ) 0,

u u

u u uu u u

a x b xI a x b x S x a x b x

F

+ <⎧⎪

= +⎨+ + ≥⎪⎩

for

| |for

�

�

�

4 40 4 4

3 01

4 4

( ) 0,

0 0,

l ll l l

l

l l

a x b x S x a x b xI F

a x b x

⎧ +− + <⎪

= ⎨⎪ + ≥⎩

| |for

for

�

�

�

4 4

4 4 40 4 4

01

0 0,

( ) 0,

l l

l l ll l l

a x b xI a x b x S x a x b x

F

+ <⎧⎪

= +⎨+ + ≥⎪⎩

for

| |for

�

�

�

= .

0 001

2

L SF

yv

xv

x

y

Fig. 3.

688


BALANDIN, KOGAN

The above control is called the decentralized control, thus underlining the fact that each control cur�rent depends on its local variables only. For example, the current is determined by the values of themeasurable variables yu and only and is independent of other variables; all other control currents aredetermined in a similar way.

After the expressions for the currents are substituted into original system (2.1) with account of kine�matical relations (2.4), we obtain the linear system of differential equations in the matrix–vector form,

(3.10)

where , and the matrices are determined as follows:

In system (3.10) the matrix M determines inertial properties of the system, D, the dissipative proper�ties, C, the rigidity properties, G, gyroscopic forces, and F, the circulation forces. Note that the matricesM, D, C are symmetric, and G, F, skew�symmetric.

4. CHOICE OF LINEARIZING FEEDBACK PARAMETERS

Now the problem consists in choosing the parameters in system of equations (3.10) in such a waythat the technical requirements to the considered object are satisfied. The first and most important of theserequirements is obviously the requirement of asymptotic stability of linear system (3.10). Omitting com�plete and detailed analysis of the possible choice of parameters, we try to simplify this system. For exam�ple, we choose the parameters ai in such a way that the matrix F vanishes, and the matrices D, C are positivedefinite. In this case, it can easily be shown that the system is asymptotically stable. Indeed, we choose thefollowing positive definite quadratic form as the Lyapunov function:

then its derivative, taking into account the system, has the form

Therefore, according to the Barbashin–Krasovskii theorem, the trivial solution to this system is asymp�totically stable.

1uI�uy

ζ = − ζ − ζ + ζ + ζ,�� M D C G F

ζ = α βT( )x y , , , ,M D C G F

⎛ ⎞⎜ ⎟⎜ ⎟= ,⎜ ⎟⎜ ⎟⎝ ⎠

0 0 0

0 0 0

0 0 0

0 0 0

J

JM

m

m⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

+

+= ,

+

+

2 21 1 2 2

2 23 1 4 2

3 4

1 2

0 0 0

0 0 0

0 0 0

0 0 0

b l b l

b l b lDb b

b b⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

+

+= ,

+

+

2 21 1 2 2

2 23 1 4 2

3 4

1 2

0 0 0

0 0 0

0 0 0

0 0 0

a l a l

a l a lCa a

a a

− ω −⎛ ⎞⎜ ⎟ω − −⎜ ⎟= ,

−⎜ ⎟⎜ ⎟− −⎝ ⎠

1 1 2 2

3 1 4 2

3 1 4 2

1 1 2 2

0 0

0 ( ) 0

0 0 0

( ) 0 0 0

z

z

J b l b l

J b l b lG

b l b l

b l b l

−⎛ ⎞⎜ ⎟− −⎜ ⎟= .

−⎜ ⎟⎜ ⎟− −⎝ ⎠

1 1 2 2

3 1 4 2

3 1 4 2

1 1 2 2

0 0 0

0 0 ( ) 0

0 0 0

( ) 0 0 0

a l a l

a l a lF

a l a l

a l a l

,i ia b

= ζ + ζ ζ,ζ

��

T TV M C

= − ζ ≤ .ζ

��

T2 0V D



For further simplification, we can consider just two parameters instead of eight ones ( ) if it isassumed that

(4.1)

In this case, the equations of system (3.10) take the simplest form,

(4.2)

It is clear that this system is asymptotically stable if .Along with the requirement of asymptotic stability, the rotor control system should satisfy other

requirements, for example, such as the boundedness of control currents and given or maximum degree ofstability of the system. The first requirement is determined by natural physical constraints, and the secondone, by the best in some sense response of the system to perturbations.

Let us consider one variant of statement of the problem on the choice of the sought parameters of thelinearizing feedback for radial electromagnets. Let the following relations be satisfied at the initial timeinstant of the rotor motion:

and let the velocities be equal to zero. Then if

(4.3)

taking into account (3.2)–(3.9), and (4.1) during some initial time interval the currents in the electromag�nets are bounded by I

∗. It should be noted, however, that the boundedness of currents by the value of I

∗

at any subsequent time interval, generally speaking, is not guaranteed. At the same time, taking intoaccount the asymptotic stability of the system, it is quite likely that after some transient period the systemstate turns out to be in the domain in which the constraints on the currents are satisfied.

For the maximum value of the parameter a0, according to inequality (4.3), we formulate the problemof maximization with respect to the parameter b0 of the degree of stability of system (4.2). In the mathe�matical sense, the problem consists in finding eight roots of the characteristic equation of system (4.2),determination of the degree of stability for the fixed parameter b0 (i.e., determination of the minimumwith respect to the absolute value real part among the eight roots of the characteristic equation), and find�ing the parameter b0 maximizing the degree of stability.

Let us consider this problem in more detail. For analyzing system (4.2), we use the dimensionless vari�ables and parameters. Assuming that

we introduce the new dimensionless variables as follows:

where

,0 0a b ,i ia b

= = , = =

+ +

= = , = = .

++

2 11 3 0 2 4 0

1 2 1 2

122 4 01 3 0

1 21 2

,l l

a a a a a al l l l

llb b b b b bl ll l

α = − α + α − ωβ

β = − β + β + ωα

= − +

= − + .

�

��

��

�

��

��

1 2 0 0

1 2 0 0

0 0

0 0

( ) ,

( ) ,

( ),

( )

z

z

J l l a b J

J l l a b J

mx a x b x

my a y b y

> , >0 00 0a b

+ ≤ , + ≤ ,2 2 2 2

0 0u u l lx y S x y S

, , ,u u l lx y x y

≤ ,

20

0 20

*8

L Ia

S

= = ,

20

0 20

**8

L Ia a

S

, ,' ' 'x y t

= , = , = ,0 0' ' 'x S x y S y t Tt

= .

*mTa

690


BALANDIN, KOGAN

In terms of the new dimensionless variables (after omitting strokes), we obtain the following system:

(4.4)

where

Let us consider an example with particular parameter values

For these parameters, we have

The choice of the parameter from the condition of maximization of the degree of stability of the sys�tem yields , and the degree of stability in this case is equal to 0.86. Thus, the sought value of b0 is

Therefore, taking into account (4.1), the sought parameters in this case are as follows:

Note also that for these parameter values the control currents, at least during some initial time interval,do not exceed I* = 1.4 A.

5. GENERAL APPROACH TO SYNTHESIS OF CONTROL LAWS

Let us consider original system (2.1) and introduce the following notation:

(5.1)

Substituting these relations into system (2.1), we obtain the control system

(5.2)

α = −μ α + α − ρβ

β = −μ β + β + ρα

= − −

= − − ,

��

��

��

�

�

��

�

��

( ) ,

( ) ,

,

b

b

x x bx

y y by

μ = , ρ = ω , = .�

* *1 2

01zJml l m b b

J J a ma

0

2 21 2

30

* 1 4 0 07 13 7

0 74 0 017 0 3 0 2

0 5 10 314 (3000 min)

z

I L m

J J l l

S −

= . , = . , = . ,

= . , = . , = . , = . ,

= . × , ω = .

A H kg

kg m kg m m m

m rad/s turns/

4 2* 7 10 1 4 10

1 11 0 1

a T −

= × , = . × ,

μ = . , ρ = . .

N/m s

�b

=� 1.8b

30 1 76 10b = . × .N s/m

,i ia b

4 41 3 2 4

3 31 3 2 4

2 8 10 4 2 10

0 70 10 1 06 10

a a a a

b b b b

= = . × , = = . ×

= = . × , = = . × .

N/m N/m

N s/m N s/m

− = ,

− = ,

− = ,

− =

2 1 1

2 1 2

3 4 3

3 4 4

u u

l l

u u

l l

F F u

F F u

F F u

F F u

α = − + − ωβ,

β = − + ωα,

= + ,

= + .

�

��

��

�

��

��

1 1 2 2

1 3 2 4

3 4

1 2

z

z

J l u l u J

J l u l u J

mx u u

my u u



Now, different problems of synthesis of the controls can be considered, using the theory andmethods developed for linear control systems. After the control laws are determined in the feedback form,the control currents can be represented as

(5.3)

(5.4)

Indeed, for this choice of the control currents

(5.5)

The other currents are determined in a similar way.Unlike the decentralized control considered in the previous section, this control is called centralized,

since each control component can depend on the whole set of measurable variables.According to the technical requirements to the control system, different statements of the problem of

synthesis of the control laws in system (5.2) are possible. Let us present one of these statements of the

problem: it is necessary to find the control laws in the form of a linear state feedback providingthe given degree of stability of system (5.2), the boundedness of the controls

(5.6)

along any trajectory of the system starting from zero initial velocities from the domain of values of with the “maximum size” belonging to the set of possible positions of the rotor,

(5.7)

These inequalities should be satisfied along any trajectory of the system; taking into account kinematicalrelations (2.4), they have the form

(5.8)

Note that if such controls ui are found, each of the control currents with account of (3.2)–(3.9) defi�nitely does not exceed

The formulated problem can be solved using linear matrix inequalities [10].

6. SYNTHESIS OF CONTROL LAWS USING LINEAR MATRIX INEQUALITIES

Let us introduce the new (dimensionless) variables and parameters as follows:

where , . Then the equations of system (5.2) after omitting primes take the form

(6.1)

, , ,1 2 3 4u u u u

1

1 10 1

01

0 0,

( ) 0,uu

u

I uS y u

F

≥⎧⎪

= ⎨+ <⎪⎩

for

| |for

10 1

2 01

1

( ) 0,

0 0.

uu

uS y u

I F

u

⎧− ≥⎪

= ⎨⎪ <⎩

| |for

for

⎧ ⎫− = − = .⎨ ⎬

− +⎩ ⎭

2 20 0 2 1

2 1 12 20 02 ( ) ( )

u u u u

u u

L S I IF F u

S y S y

= ,, 1 4,iu i

≤ γ, = ,| | 1 4,iu i

α ,β , ,0 0 0 0x y

+ ≤ , + ≤2 2 2 2 2 2

0 0 .u u l lx y S x y S

+ β + − α ≤ , − β + + α ≤ .2 2 2 2 2 2

1 1 0 2 2 0( ) ( ) ( ) ( )x l y l S x l y l S

γ= .

0

0

8*

SI

L

= , = , = , = γ , α = α , β = β ,/ /0 0 0 0 0 0' ' ' ' ' 'x S x y S y t Tt u u S l S l

= γ/0T mS = + /0 1 2( ) 2l l l

1 1 2 2 0

1 3 2 4 0

3 4

1 2

,

,

,

u u

u u

x u u

y u u

α = −μ + μ − ρ β

β = μ − μ + ρ α

= +

= + ,

�

��

��

�

��

��

692


BALANDIN, KOGAN

where , and the constraints for the phase and control variables arerepresented as

(6.2)

where .

Let us write system (6.1) in the canonical form of a control system,

(6.3)

where , , and is the so called target output of the system usedhereinafter for convenient representation of the constraints for the control and phase variables. The matri�ces A, B included in this system of differential equations can be conveniently written in the block form,

where I is the fourth order identity matrix,

The matrices have the form

and the matrices P1, P2 are as follows:

The solution of these problems using linear matrix inequalities is described in detail in [13, 14]. In thispaper, we briefly present the main idea of this solution. Let the sought control in the form of a linear feed�back be

Substituting this control into Eqs. (6.3), we obtain the closed�loop system

(6.4)

μ = , μ = , ρ = ω/ / /1 1 0 2 2 0 0 zml l J ml l J J T J

+ βσ + − ασ ≤ ,

− βσ + + ασ ≤ ,

≤ , = , ,| |

2 21 1

2 22 2

( ) ( ) 1

( ) ( ) 1

1 1 4i

x y

x y

u i

σ = , σ =/ /1 1 0 2 2 0l l l l

ξ = ξ + ,

= ξ + , = , ,

�

1 6i i i

A Bu

z C D u i

ξ = αβ αβ��

T( )x y x y =

T1 2 3 4( )u u u u u = ,, 1 6,iz i

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

⎛ ⎞= , = ,⎜ ⎟Γ⎝ ⎠ 1

0 0

0

IA B

B

−ρ −μ μ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ρ μ −μ⎜ ⎟ ⎜ ⎟Γ = , = .⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

0 1 2

0 1 21

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 1 1

0 0 0 0 1 1 0 0

B

,i iC D

1 1 2 2

3 4 5 6

( 0) ( 0) 0 3 6

0 1 2

(10 00) (0100) (0010) (00 01)

i

i

C P C P C i

D i

D D D D

= , = , = = , ,

= = , ,

= , = , = , = ,

for

for

⎛ ⎞ ⎛ ⎞σ −σ σ σ⎜ ⎟ ⎜ ⎟

σ σ σ −σ⎜ ⎟ ⎜ ⎟= , = .⎜ ⎟ ⎜ ⎟σ −σ⎜ ⎟ ⎜ ⎟−σ σ⎝ ⎠ ⎝ ⎠

2 21 1 2 2

2 21 21 2

1 2

1 2

1 2

0 0 0 0

0 0 0 0

0 1 0 0 1 0

0 0 1 0 0 1

P P

= Θξ.u

ξ = + Θ ξ,

= + Θ ξ, = , .

� ( )

( ) 1 6i i i

A B

z C D i



Let us consider the quadratic form ; this form will be used hereinafter as the Lyapunovfunction for system (6.4). The condition of asymptotic stability (with the given degree of stability r) for thissystem can be represented as the following condition for quadratic forms to be definite [10]:

(6.5)

Let us write the inequalities determining the constraints for the phase and control variables:

, or, in more detail,

(6.6)

Let us consider the set of initial states of the system in the form of an ellipsoid determined by the ine�

quality . If inequalities (6.5) are satisfied, any trajectory of the system starting from the initialstate belonging to this ellipsoid does not leave it. Let us formulate the following problem: it is necessary tofind an ellipsoid of the maximum size (in other words, to find a symmetric positive definite matrix Y) andthe feedback matrix such that inequalities (6.5) are satisfied, and the sought ellipsoid is embedded inthe set determined by inequalities (6.6).

In terms of linear matrix inequalities, the stated problem has the following form [13, 14]: it is necessary

to find the matrix , the matrix Z, and the minimum value of the parameter satisfying thematrix inequalities

(6.7)

where , and is the block matrix. The latter inequality with the minimum possible valueof makes it possible to find the only ellipsoid for which the domain of initial values of the coordinates

is “maximal” (for zero initial velocities), and the initial coordinate values are such that the

phase trajectory of the system, starting from any point of the set , where

does not leave the e phase space domain determined by the constraints on the phase and control variables.

Actually, this inequality means that the ellipsoid contains a sphere with the maximum possible

radius equal to . Note that taking into account Schur’s lemma [10], the last inequality in (6.7) can bewritten in the form of the linear matrix inequality with respect to Y and λ

Thus, we obtain the system of linear matrix inequalities with respect to the variables Y, Z, λ

(6.8)

If the solution to this system for the smallest value of is obtained, then the feedback matrix can be

determined using the formula .

−

= ξ ξ

T 1V Y

−

− − − − −

= ξ ξ > , ξ ≠ ,

= ξ + + + Θ + Θ ξ < ,ξ ≠ .�

T

T T T T

1

1 1 1 1 1

0 0

( 2 ) 0 0

V Y

V Y A A Y rY Y B B Y

≤ , = ,T 1 1 6i iz z i

ξ + Θ + Θ ξ ≤ , = , .T T( ) ( ) 1 1 6i i i iC D C D i

−

ξ ξ ≤T 10 0 1Y

Θ

= >

T 0Y Y λ > 0

−

+ + + + < ,

⎛ ⎞+≥ , = ,⎜ ⎟

+⎝ ⎠

≤ λ ,

T T T

T T T

T10 0

2 0

0 1 6,i i

i i

AY YA rY BZ Z B

Y YC Z Di

CY D Z I

J Y J I

= ΘZ Y =0 ( 0)J Iλ

α ,β , ,0 0 0 0x y−

ξ ξ ≤T 10 0 1Y

0 0 0 0 0( 0 0 0 0)x yξ = α βT

−

ξ ξ ≤T 10 0 1Y

− /

λ1 2

⎛ ⎞≥ .⎜ ⎟

λ⎝ ⎠

T0

0

0Y J

J I

+ + + + < ,

⎛ ⎞+≥ , = ,⎜ ⎟

+⎝ ⎠

⎛ ⎞≥ .⎜ ⎟

λ⎝ ⎠

T T T

T T T

T0

0

2 0

0 1 6,

0

i i

i i

AY YA rY BZ Z B

Y YC Z Di

CY D Z I

Y J

J I

λ

−

Θ =1ZY

694


BALANDIN, KOGAN

For the above object parameters, the dimensionless parameters used in (6.1) are

According to Section 4, we determine the value of the parameter characterizing the degree of stabilityas follows: . The numerical solution of the problem of minimization of the parameter takinginto account the constraints determined by linear matrix inequalities (6.8) using Matlab yields the follow�

ing results: , ,

Therefore, for the control in system (6.1) the degree of stability is not smaller than the given r

and along any trajectory starting from the initial state belonging to the ellipsoid , constraints (6.2)for the phase and control variables are satisfied.

7. SYNTHESIS OF CONTROL LAWS WITH “SATURATION”

The control u linear with respect to the phase variables constructed in the previous section is boundedby the given value only in some phase space domain. Let us consider another type of control, the so�calledcontrol with saturation; this control is bounded in the whole phase space. Let us again consider system (6.1)and introduce the notation for the new control variables,

(7.1)

Let us also write the expressions for the variables ui with respect to vi,

(7.2)

Taking into account the above notation, system (6.1) takes the form

(7.3)

Now we should synthesize the controls ui instead of the controls vi. Similar to the previous section, thefollowing requirements for the control system are formulated: the condition of asymptotic stability and

μ = . , μ = . , ρ = . , σ = . , σ = . .1 2 0 1 21 3 0 93 0 1 1 2 0 8

= .0 86r λ

−

= λ

T10 0 0 0J Y J I λ = .0 3 43

0

0 48 0 0 0 0 0 1 0 42 0 08 0 08 0 14

0 0 0 48 0 1 0 0 0 08 0 42 0 14 0 08

0 0 0 1 0 54 0 0 0 09 0 06 0 49 0 1

0 1 0 0 0 0 0 54 0 06 0 09 0 1 0 49

0 42 0 08 0 09 0 06 1 15 0 0 0 01 0 04

0 08 0 42 0 06 0 09 0 0 1 15 0 04 0 01

0 0

Y

. . . . − . . − . − .

. . − . . − . − . . − .

. − . . . . . − . .

. . . . − . . − . − .=

− . − . . − . . . . .

. − . . . . . − . .

− . 8 0 14 0 49 0 1 0 01 0 04 1 00 0 0

0 14 0 08 0 1 0 49 0 04 0 01 0 0 1 00

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ,⎜ ⎟⎜ ⎟⎜ ⎟

. − . − . . − . . .⎜ ⎟⎜ ⎟− . − . . − . . . . .⎝ ⎠

0

1 23 0 03 0 01 0 96 0 77 0 0 0 0 0 71

0 94 0 08 0 02 0 89 0 83 0 02 0 02 1 02

0 03 1 23 0 96 0 01 0 0 0 77 0 71 0 0

0 08 0 94 0 89 0 02 0 02 0 83 1 02 0 02

. . . − . . . . − .⎛ ⎞⎜ ⎟− . − . − . − . − . − . − . − .⎜ ⎟Θ = .

. − . − . − . . . − . .⎜ ⎟⎜ ⎟− . . − . . − . . − . .⎝ ⎠

= Θ ξ0u−

ξ ξ ≤T 1

0 1Y

= −µ + µ , = µ − µ ,

= + , = + .

1 1 1 2 2 2 1 3 2 4

3 3 4 4 1 2

u u u u

u u u u

v v

v v

− µ + µ= − , = ,

µ + µ µ + µ

+ µ − µ= , = − .

µ + µ µ + µ

1 2 4 1 1 41 2

1 2 1 2

2 2 3 2 1 33 4

1 2 1 2

u u

u u

v v v v

v v v v

α = − ρ β,

β = + ρ α,

= ,

= .

�

��

��

�

��

��

1 0

2 0

3

4

x

y

v

v

v

v



limited control currents, which results in the bounded variables ui ( ), or, taking into account (7.2),the inequalities

(7.4)

for the new control variables vi. Note that it is rather difficult to solve synthesis problems with these con�trol constraints; therefore, let us replace constraints (7.4) by the constraints of the form

(7.5)

where

For such constraints on the control variables vi the constraints are definitely satisfied, and there�fore, the constraints on the control currents are also satisfied. At the same time, it should be noted that fora simpler form of constraints (7.5) that makes it possible to synthesize a bounded control, the control“resource” is partially lost, since there exists a domain of values of vi not satisfying (7.5) for which the con�straints are satisfied.

For synthesizing the control laws, we use the notation applied in Section 3, namely, ζ = .Then system of equations (7.3) takes the form

(7.6)

We consider the following controls as the sought ones:

(7.7)

Let us show that for system (7.6) closed by control (7.7) is asymptotically stable in the large.For proving this fact, we use the method of Lyapunov functions and choose the following function:

(7.8)

where

Calculating the derivative of the Lyapunov function, due to system (7.6) with account of (7.7), we obtain

(7.9)

and

≤| | 1iu

− μ + μ≤ , ≤ ,

μ + μ μ + μ

+ μ − μ≤ , ≤

μ + μ μ + μ

1 1 4| | | |

| | | |

1 2 4

1 2 1 2

2 2 3 2 1 3

1 2 1 2

1 1

1 1

v v v v

v v v v

≤ , = , ,| | 0 1 4i U iv

−

⎧ ⎫+ μ + μ= μ , μ = , .⎨ ⎬μ + μ μ + μ⎩ ⎭

1 1 20 0 0

1 2 1 2

1 1maxU

≤| | 1iu

≤| | 1iu

α,β, ,T( )x y

ζ = − ρ ζ , ζ = + ρ ζ ,

ζ = , ζ = .

��

��

1 1 0 2 2 2 0 1

3 3 4 4

v v

v v

1 2 1 2 0

1 2 00 1 2

,

( ) .i i i i

ii ii i

c c c c U

U c c c c U

⎧⎪⎨⎪⎩

− ζ − ζ ζ + ζ ≤

=

− ζ + ζ ζ + ζ >

v

for | |

sign for | |

� �

� �

> , >1 20 0c c

=

ζ,ζ = + ζ ,ζ∑�� /

42

1

( ) [ 2 ( )]ii

i

V f

21 1 0

21 00 0 1

2 ,( )

(2 ) .i i

iii

c c Uf

U U c c U

⎧⎪⎨⎪⎩

ζ ζ ≤ζ =

|ζ | − / ζ >

/ for | |

for | |

ζ

=

= ζ + ζ ,∑ ��

4

1

'[ ( )]i i i

i

V fv

1 1 0

1 00

,' ( )

( ) .

i ii

ii

c c Uf

U c U

⎧⎪⎨ζ⎪⎩

ζ ζ ≤ζ =

ζ ζ >

for | |

sign for | |

696


BALANDIN, KOGAN

Let us introduce the following notation:

and analyze the sign of this function. We use Fig. 4 for convenience; this figure shows the domains

. The following inequalities are satisfied in these domains: , in S1;

, in the domains S2, S3; , , in the domains S4, S5, and

, in the domains S6, S7.

Then we obtain

and therefore, the function for all . Thus, according to the Barbashin–Krasovskii theorem,system (7.6) closed by control (7.7) is asymptotically stable in large.

CONCLUSIONS

Different control laws for the motion of a vertically rotating rotor in electromagnetic bearings, includ�ing decentralized and centralized controls, and centralized control with saturation, were synthesized. Theadvantages and disadvantages of the synthesized control laws providing asymptotic stability of the closed�loop system and constraints for the phase and control variables were indicated. The promising tasks forfurther study are the problem of synthesis of output control, i.e., control based on measurement of therotor displacement and dynamic estimation of the rate of variation of these displacements, as well as syn�thesis of control laws in the conditions of perturbations and inaccurate determination of some objectparameters.

ζζ,ζ = ζ + ζ� '( ) [ ( )]i i i ig fv

, = ,1 7iS i ζ + ζ ≤�| |1 2 0i ic c U ζ ≤| |1 0ic U

ζ + ζ ≤�| |1 2 0i ic c U ζ >| |1 0ic U ζ + ζ >�| |1 2 0i ic c U ζ ≤| |1 0ic U

ζ + ζ >�| |1 2 0i ic c U ζ >| |1 0ic U

22 1

2 30 1 2

4 50 1 2 1

6 7

( ) ,

[ ( ) ] ( ) ,( )

[ ( ) ] ( ) ,

0 ( )

i ii

i ii i i ii

i ii i i i

i i

c S

U c c S Sg

U c c c S S

S S

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

− ζ ,ζ ∈ζ

ζ ζ − ζ − ζ ζ ,ζ ∈

ζ,ζ =

ζ − ζ + ζ + ζ ζ ,ζ ∈

ζ ,ζ ∈

for

sign for

sign for

for

��

� � �

�

� � �

�

∪

∪

∪

ζ,ζ ≤�( ) 0ig ζ ,ζ�i i

4

4

6

2

1

0

1

3

5

5

7ζi

ζi

.

Fig. 4.



ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research (project nos. 10�01�00514,11�01�00215, and 11�01�97022r�povolzh’e) Federal Targeted Program “Scientific and Scientific�Peda�gogical Personnel of the Innovative Russia”.

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Documents

Motion control for a vertical rigid rotor rotating in electromagnetic bearings