Sliding mode fuzzy control of magnetic bearing systems

SLIDING MODE FUZZY CONTROL OF MAGNETIC BEARING SYS...

14th World Congress ofIFAC

Copyright l ' 1999 IFAC 14th Trienniai World Congress. Beijing. P.R.

J-3e-02-2

<:'~hina

SLIDING MODE FUZZY CONTROL OF MAGNETIC BEARING SYSTEMS

Takahiro Kosaki and Manabu Sano

Dept. ofComputer Science~ Hiroshima City University 3-4-1, Ozuka-higashi, Asaminami-ku, Hiroshima 73 J-3194~ Japan. Phone&Fax: +81-82-830-1669 E-mail: kosaki(~ys.cs.hiroshima-cu.ac.jp

Abstract: The stabilization of a magnetic bearing system is reduced to the t\Vo degrees-offreedom control problem about rotational motion of a rotor. In this study~ a sliding mode fuzzy controller is taken as one of robust approaches for this problem~ When the rotor rotates, it is required for the reliable operation to overcome two major obstacles: imbalance force and gyroscopic effects. For imbalance force compensation, a VSS disturbance observer is used. Fuzzy decoupling units are also attached to the control law to reduce coupling effects between the rotations. Experimental results show the stable rotation ofthe rotor is achieved using the fuzzy controller. C"opyright © /9991F4C Keywords: fuzzy control, variabJe...structure systelns, disturbance rejection.

systems~

I. INTRODUCTION

magnetic bearings, multivariable

problems about each translation, and the other is a MIMO control problem about rotational motion including interaction. While the rotor is rotating at high speed, strong interaction with rotations exists owing to gyroscopic effects. The application of various robust control methodologies have been reported by many researchers previously (e~g. Sivrioglu and Nonami, 1998; Lin and Gau~ 1997; Mohamed and Vishniac, 1995).

Active magnetic bearings are capable of supporting rotors at high speed without mechanical friction or lubrication. The advantages for this technology are due to its ability of complete contact-free suspension of rotors and very fast rotationaJ speed. Therefore~ magnetic bearings are widely used in a variety of industrial applications such as milling spindles and compressors for natural gas. The magnetic bearing systems are highly unstable and, thus, require sophjsticated controllers to achieve stable operation. A rotor suspended by the magnetic bearings has the four degrees of freedom except for the rotation on its axis. The stabilization problem ofthe system can be res.olved into the following two ones: one is two SISO control

Based on the similarity ofruJe based fuzzy controllers to variable structure system (VSS) controllers with sliding modes, luany types of fuzzy controllers to guarantee the system stability have been developed (Kawaji and Matsunaga, 1991 ~ H""rang and Lin, 1992; Wu and Liu, 1996; H",,·ang and Tomizuk~ 1994). A VSS

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Copyright 1999 IFAC

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SLIDING MODE FUZZY CONTROL OF MAGNETIC BEARING SYS...

is a control system switch ing control Iaws in different regions (Utkin, 1977)~ If once the system state is in the sliding mode~ it is robust to the parameter variations and disturbances. Meanwhile, fuzzy rule based control provides the ability of smooth and fine tuning. In this paper, we design a sliding mode fuzzy logic controller for the magnetic bearing system using the design technique of the VSS. The contro)Jer is combined with the VSS disturbance observer to compensate for imbalance forces acting on the rotor during its rotation (Tian et al., ] 996). The simple fuzzy decoupllng scheme is also appJie·d to cope with the interaction between the rotational motion of the rotor due to the gyroscopic effects (Nie, 1997). The fuzzy control system based on the VSS is realized on a magnetic bearing experimental hardware. The perfonnance ofthe developed fuzzy logic controIJer for the system is evaluated through experiments.

2. SYSTEM MODELAND DYNAMICS The basic structure of an active magnetic bearing is sho\Vll in Fig. 1. The main system components are fOUT pairs of electromagnets which control the radial and rotational motion of the rotor. The force of each pair of electromagnets is defined as ~. (i = 1, ..., 4). In particular, F 1 and F 2 are parallel to the x axis, and F} and F4 are parallel to the y axis. The coordinate system o . .~:vz is fixed on the space so that the origin 0 corresponds to the center ofgravity of the rotor in the steady-state.

14th World Congress ofIFAC

The equations of motion of the rotor with four degrees of freedom are described by Inx == Fj + F2 + Fd:r (1 ) my = F'.", + F4 + Fdy , (2) 'J

]rB+ J aoxP = Ft - F2 + Fd8 , J riP - J awiJ =- F3 - f 4 + Fdf/>'

(3) (4) where m is the mass of the rotor, J a is the moment of inertia in the axial direction, and J r is the mOlnent of

inertia in the radial direction. The rotor is assumed to rotate at the constant angular speed 0). F fix' F dy~ F dfP and F d? are disturbance forces. The simplified equation for each electromagnetic force ~ is given with respect to the air gap x j and the coil current if by - (i j +JO )2 (i}-lo)2 F, = k k 'l.i =. 1, ... , 4~ ) {Xj-X o )2 (xj+XoY-' where k is the proportional constant associated with the characteristics of electromagnets, /0 is the bias current, and X o is the nominal gap length. Using Taylor series expansion for small values of i and x, it becomes F j == kIf j + k 2 X j , (5) where

6,

5 J.

k1 == 4kl o I X k2 = 4k1 I X Using (5)., (J)-(4) can be described by ..

x

2k.,

=--~

m

..

2k') m

k) (.

x+m

l(

k1 m

.)

+ F dx,

.)

+

+1 2

-

y=--~ y+-(l3+ l 4

..

2k,,12

kIl..

F

d~',

-

.

(}~JB+T(l) -l2)-caN/J+FdB , r 2

(6) (7) (8)

r

k 1/ ( . ;; 2k 2 1 ., Li F Y'=---fjJ+- ll-l4)+COJu+ d~

(9) Jr' ~ where c = J(/J" and 1 is half the length of the rotor. The relationship between the voltage ej and the coil current ~ is given by di, e·=L-J+Ri· J·=1 .. ··· .. 4,

Jr

Prior to develop the mathematical model ofthe magnetic bearing system, it is assumed that a) The rotor is a rigid body and symmetrical with respect to its center of gravity. b) AB radial bearings have the same characteristics. c) There are no hysteresis, flux leakage, and saturation effects.

] dt J' '" where R is the coil resistance, L is the coil inductance. If R and L are assumed to be the same in all coils~ then _L d (i 1 +12) + RC'I +12 .) ex 1 dt ' 3 +i4 ) L d(i RC' .) e,. + ll,~+l4 dt ., ,T

en -- L dei,dt-iz) + R("II fJ

- L d(i3 -i4 ) dt

et/J -

.)

-l')

- ,

+ R(·l]. -14.) .

The control problem of the radial system (6)-(9) can be divided into two main probJems: one is two SISO control problems about the translations along x and y~ and the other is a MIMO control problem about the rotations including the interaction owing to the gyroscopic effects. Hereafter? the latter problem with

Fig. 1. A magnetic bearing system model. 5106

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ISBN: 0 08 043248 4

14th World Congress of IFAC

SLIDING MODE FUZZY CONTROL OF MAGNETIC BEARING SYS...

respect to the rotational motion is discussed. The statespace representation is

xCt) = yet)

Ax(t) + Bu(t) + Fa

,

= Cx(t) ,

(lQ)

E=[O -ltJ] 0 .

where x(t) =

OJ

[Xl,X2,X3,X4tX5,X6]T

=[6,t/J,8t~,il -i2 ,ij -i4 ]T,

Combining (13) and (14) can be rewritten as

xd (t) = AdXd (t) + Bdu(t) ,

T

u(t);;::;[ee,e.p-J ,

Yd (t) == CdXd (t) ,

0 0

2k 2 l 2

i A==

B=[: C-[~

0

0

0

0

0

0

0

- cw

~

0

CO)

0

0

0

1

0

0

r

2k 2 l

0

i

Jr

2

i

0

0

0

0

R -L

0

0

0

0

0

0

0

-

0

0

0

0

0

0

0

0

0

Ad=[Z ~].

0

Bd

=[:].

0

xd(k+l) = (])dxd(k)+rdu(k) (16) The switching gain matrix M(e(k)) with a boundary Jayer w

R -L

is represented by rtlHCde(k)

M(e(k»=

Ir

~HCde(k)11 p,

~]. =[ Fd

p is a constant. If ~HCde(k)11 < y, ~HCde(k)11

= y. r is a positive real number and works to avoid chattering. H is found such that HC d ::::: r~PL' and here it is taken as a unit matrix. PI is the solution of the following Lyapunov matrix equation:

dO

F F

]

d4J

.


3. 1 VSS Observer

G==(R+CdPC~)-]CdP
In the above magnetic bearing system (10), the measurable state variables are Xl and x l The VSS observer is constructed to estimate the other state variables (Tian et al., 1996). Furthermore, in the real equipment, imbalance in the rotor mass often causes periodic oscillation with the same frequency as the rotational speed Cd while the rotor is rotating~ The VSS observer also estimates the periodic disturbance forces F d8 and F d , in (10) simultaneously. The imbaJance forces can be expressed by the following sinusoidal disturbance

3,2 Switching Manifold and Sliding Mode Controller For the discrete-time system (16)j the switching manifold is defined as a(k)=Sxd(k).

forces: ===

(1- c)-rm cos(m t + 13)

,

= Ax(t)+ Bu(t)+ Dw(t) , wet) Ew(t) , ::=

(19)

After the state trajectory xjk) has reached the sliding manifold (19), from the condition of O'(k) ; : : O'(k+ 1) the control input can be obtained by u(k) =-(Srd )-lS(0 d -I)xd{k). (20) The state equation of the system in the sliding mode is given by

(J I)

Fdr; = (1- C)TW2 sin(a> t+ /1) , (12) where ris the angle betvleen the geometrical and inertial axis, i.e., amount of dynamic imbalance. P is an initial value. Substituting (11) and (12) into (10) gives x(t)

(17)

such that 4»0 ~ ~d - GC d is a stable matrix. P is the solution of the algebraic matrix Riccati equation and R >0.

<

2

(18)

where e(k):;;:: xd(k) -xd (k) is the estimation error and

3. CONTROL SYS
Fd8

Cd-[C OJ.

Then the VSS observer is attached to the following discrete-time system derived from (15):

r

1 L

0

1 0

Xd(t):;;:: [x, WJT ,

5J!..

r

(15)

where

(13)

xJ (k

(14)

+ I) =

{4> - r(Sr)-l 8(4) - I) }xd (k) . (21)

The matrix S is detennined so as to stabilize the control

where 5107

Copyright 1999 IFAC

ISBN: 0 08 043248 4

14th World Congress of IFAC

SLIDING MODE FUZZY CONTROL OF MAGNETIC BEARING SYS...

system (2 I), which is found by ST = (R 2 + r~P2rd )-l r~P2«Ja ' where P2 is the positive defmite solution of the Riccati equation and R 2 > O~Then a candidate of the Lyapunov function is selected as V(k) ==

2 !-u (k) 2 '

(22)

If the fo))owing reaching condition is satisfied, the existence of sliding mode is guaranteed and the system is driven into the sliding mode: V(k+ 1) < V(k) . (23) From (19), (20), (22) and (23), the sliding mode control law is given by u(k) ~ -(Sr d ) -I S(<1J d - I)x d (k)

- Kclfa(k)lt) sgn(o(k)) .

(24)

o(k) = O. In the vicinity of the switching manifold, the transfer characteristic of ~Jk) is nonlinear. The control rule (25) can be refonnulated as follows: Rule n; if am (k) is In then ufm is Un. where n = 1, ... , 9 and m = 1, 2. The center of gravity method is used for defuzzification 9

9

= k~ W n U 'J.,. I k~ l1l'2 1

U.f

1"1

n=l

IP=I

.

where M'n = f.12."r1 (0 m(k» is the value of the membership function of the nth fuzzy set of C'rm(k). As shown in Fig. 2, the triangle type· of membership functions is chosen.

3.4 Fuzzy Decoupling Scheme With the increase of the rotational speed (J}or the rotor and ~ ~ the IDToscopic effects cause angular velocity coupling bet,"'een (1 and f/J. One simple approach to reduce the coupling effects is the fuzzy decQupIing scheme proposed by Nie (1997)~ The following cross feedback tenns are combined as the decoupling units to the control law (24):

e

3.3 Sliding Mode Fuzzy Logic Controller

Based on the similarity of fuzzy logic controHers to sliding mode controllers, we derive the sliding mode fuzzy logic controller (Palm et al., 1997) by replacing KCllo(k)IDsgn(O'(k») in (24) by the corresponding control law U f == K f (uek») . (25) The sliding mode fuzzy logic controller changes the magnitude utk) depending on the signed distance a(k) betvveen the state vector xJ..k) and the switching manifold

udi

ud2

= kd1 (w~t/J)f/J ' == kdZ((J)tB)B ,

where i d1 and kd2 are on-line tuned by the fuzzy inference according to (1), Band
crmax cr

Fig~

Fig. 2. Membership functions for the sliding mode fuzz~y controller.

3. Membership functions for the fuzzy decoupling components.

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SLIDING MODE FUZZY CONTROL OF MAGNETIC BEARING SYS...

the simple structure. The fuzzy inference system consists of a set of six fuzzy linguistic if-then rules expressed in

Rule n: if lJJ is Qa and 8 is e b then k dJ is Kt.: ~ nand (9 are the fuzzy sets characterized by the fuzzy membership functions. K denotes the fuzzy set in the consequent part. The c.orresponding membership functions are shown in Fig. 3. The weighted average of each rule's output is represented as A

kdm

6

6

It:::: I

n=l

= L lV'nk;m I L

,",In,

where the degree of match on the premise part li-'n is detennined by W n = /lQeJ (OJ) l\j1,giJ (6) and A is minoperator. #Qa (m) and J.'eb (6) are the membership grades of (J) and B in the fuzzy sets. The case of
4. EXPERIMENTAL RESULTS

The perfonnance of the sliding mode fuzzy logic controller is tested by using the magnetic bearing experimental system. Three fuzzy logic controllers are employed at the same time: one is the MIMO fuzzy controller described in this paper and the others are two SISO fuzzy controllers to control the displacement x andy designed by the same procedure~ Table I gives the parameters in the experimental system (Magnetic

value

parameter k

2.80x tO~7[Nm2/A2]

m

O~263

L

8.80x I0-4 [H]

~

1~59xlO~3

Ja

1.OlxIO- [kgm2]

R T

4.0 [0]

J

IQ

0.50 [AJ

Xo

4.0xIO-4 [m]

[kgm 2 ]

Fig. 6 shows the transient responses of the system in all sensor positions when the sliding mode controller (dotted line) and the sliding mode fuzzy logic controller

(solid )ine) are used respectively. In comparison with O.nr------r,----,,-------......,..--~ ......... 0.4

e

.,-."'11

£

p

~

-

-

p

---- -

~

f-- ----------~ -------- --- -~ -~ ----------r---~ -~- -----

P

........ - . . - - - .. _....... ....,~ - - -_ ...... _ .. - - . . -~ .......... - . . -- - - ....... ~. - - ... - - - ....... --~ -- ......... _. ----

OT2

1\'

0,

I

~

:

I/~.

.;; -0.2 -0--4

•

,

,

,

----+-.~-.~____i:~---~-l.

-

I

r{'- -------- -~ --- -~ --- ----~ -~ -----" ----~ ------- ~ ----:. -- p-- - - - - --

Jj------.---t-- --~

-
p-

~

---

-~ ~_. -

1

o

-

value

0.1 J I [m] [kg]

Moments, 1997). In the experimental hardware, the endpoint displacement of the rotor from a baseline is detected by a Hall effect sensor. The total movable range of translational motion in the x and y directions is approximately ± 0.6 mm. The rotor has a diameter of 12.4 mm. The magnetic bearing system consists of eight electromagnets altogether. A single electromagnet has a horseshoe shaped coil core. The flow diagram of control signals is shown schematicaJly in Fig. 5. The sliding mode fuzzy logic controller is realized in C language. and implemented on a Pentiwn-based personal c·omputer at a sampling rate of I kHz. The control comlnands are send to the experimental system through AID and D/A interfaces. The rotor is actuated for the rotation by an air turbine drive. The rotational speed is controlled by changing the supply air to it using a pneumatic valve and sensed by an optical encoder.

-

-

-.

--~ ~~

,

-

_.- -. -

p

-- -

-~

I

,

---

0.1

0.2

0.3

0,4

I I

I l

• 1

• I

v:

:

:

:

0.3

0.4

.~-

-

-

-

_.-

0.5

I ~: a~:~:~:~~ r:: :::::::r:::::::::::f::::::::::::t::::::::::::

Table 1 System parameters

parameter

14th World Congress of IFAC

o~

~:: r":":::::::: r":::-:::"::(::::"::::: :~::::::::: :::;:::::::::::: o

0.1

0.2

0.5

r[sj

J.Ox 10- 3 (s]

Position senser outpUlS c.

...(j,6

XI

--'-

L....-~~--.L-

o

,

Y2

0.4 ~.- - - - - - • - - - - - ~ - - - -

...J.--_ _

0.2

0.1

-

-

-

-

-

-

-

0.5

I

~

p

_ _- . J

-----J.~

0.3

~ ~~ ~ -

-

-

-

-

-

-.

~~

•

-

-

-

-

-

p

-

~ ~ -

-

-

}. -

-

-

-

_. -

•

-

-

•

1_::: ~~t~~:::t::::~:::::::_::::::::_)_:: _ :-::--t:::::_:::. -0.4

- -- -- - - - ~ - - -~ - -- - - - - - - - --~ -- - - - -- - -- - ~t

-0.6 - . -_ _-----"---

o

0.1

~

_

l.------_ _- - ' - -_ _~ _ i . . _ . ~ _ ~

0.2

0.3

0.4

()5

I[s]

Fig. 6. Transient responses ofthe rotor ends; (a) sliding mode fuzzy logic controller (solid), (b) sliding mode controller (dot).

Fig. 5. Configuration of the experimental system. 5109

Copyright 1999 IFAC

ISBN: 0 08 043248 4

SLIDING MODE FUZZY CONTROL OF MAGNETIC BEARING SYS...

14th World Congress ofIFAC

0.('1 r------'-----l,r-------~----.,-----,.., - - - - - . - ,- - - (},4

~

- - - - - - - - - - -

-~ -

~-

- - - - - - - - - - -

~-

- - - - - - - - - - -

- - - -- - - - - -

-~ -

0.: ------------;---....-----~---- ..----' -t------------;---.--------

~ -0.2

- --- --- -- - - -~ --- - - - -- - ~ ~-:----_.- - - - - - ~-- -- - - - - -- --~ -- - - - - - - - ---

-
- - - - - - - - - - - - ~ - - - • - - - - - - - - ~ - - - - - - - - - ~ - - ~ - - - - - - - - - - - -~ -. - - - - - - - - - -

-O.b 0

0.'1

~-i3 ~---------"O!L-4~--

0:2

ACKNOWLEDGMENT

- - - - - - - - . --

This study is supported in part by the Electro-Mechanic Technology Advancing Foundation.

0.5

I/sJ

I

::~ ::::::::: ::~::::::::::::t::: :::::::::~ ::::::::::::f::::::::::1

;:'-'1.2 -OA

-.- - - -- - - - --~ --- -- - - - - - - -~ -- -- - - - - - - --~- - - - - - - -- - --~ - ----- -- - - --

---------·--r------------~-·---------+---------- --r------·- -

-

-0.6 O~--~O_':-l---O~.2----0-.i.....3--~-O..i.-.4---.......J0.5 lIs) I

ilo

0.4

E

g

~

•

0.2

_

I

-- _......... - - _ .. -.. -- - - .. - _... ----., - - - - -- -:- - .... - -- _.- ... - ------ -------"i------- ------f-----. ------~-- ------ ----:-,' -----------_... - - - ---- _. -' -

_.-.

......

I~

• ~

•

I

)

•

0ft"'f~~~~rl_\:+~~f._4::_~_I__f!,r__f_:;~~~L+_~_/_..J.._.l

:; -0.2 -- ---. - - -- -0.4

----. - - - - --

-~---- - -------L-----. ----- -.. -- --- --- ----~ ----- -----~~

-t-----------+------------~ -------_.---~ ---.-_..----

-0.6 ~---:-1~----a..------'---~----'-----.-l o O.t 0.2 0.3 0.4 0.5

-----,--l-----r:--~______....__:---

O.61r---~: r--"

I

0.4

------~ - - ---~-

0.:2

------------~------------:.------------~------------~--------~---

0 ~: ..().4 -0.2

- - -- - - -- - - -;- ------ ------:- ----- - -- -- --:------- -- ----

~~~~:~~~~:~~~~:~..,..,..::::::~~::::::...-:....=-.:-~~ ~ r------------f~ -----.. ------:-~ --_. --------:---------,--

-. ----------.. . -----.. ---.. -r-- -------- "- . ------ - -- ---.. --.. . ---~~

~

~

Hwang, G-C. and S.-C. Lin (1992). A stability approach to fuzzy control design for nonlinear systems. FuzZ)/ Sets and Systems~ 48,279-287. Hwang, Y.-R. and M. Tomizuka (1994). Fuzzy smoothing algorithms for variable structure systems. IEEE Trans. on Fuzzy Systems~ 2-4, 277284. Kawaji, S. and N. Matsunaga (1991). Fuzzy control of VSS type and its robustness. Proc. IFSA World Congress, vol. Eng., 81-84. Lin., L.-C. and T.-B. Gau (1997). Feedback linearization and fuzzy control for conical magnetic bearings. IEEE Trans. on Control Systems Technology, 5-4,

417-426.

- - - - - - - - - - - -~

~

REFERENCES

~

~.6 ()~-----:-"'O~-l~--O..J..~2----0....l..:3----0........ ~4---....Jf).5 l[sJ

Fig. 7. Responses in the steady-state at a rotor speed of 2,000 rpnl.

the result of the conventional sliding mode contra 1~ less oscillation appears in the fuzzy control case~ On the other han~ Fig. 7 shows the steady-state responses in the case with the fuzzy decQupling units at the Totational speed of 2,000 rpm. Even when the rotor rotates at high speed., th e control system provides stable performance with small coupling effects.

5. CONCLUSIONS The sliding mode fuzzy logic controller combined with the VSS disturbance observer and the fuzzy decoupling units has been developed to stabilize the magnetic bearing system based on the VSS design methodology. The fuzzy controller has been implemented on the experimental system and verified its performance. In the experimental results, the influences of the imbalance disturbance forces and the interactions caused by the coupling effects are suppressed while the rotor is rotating at high speed.

Magnetic Moments LLC (1997). MBC500 Magnetic Bearing System Operating Instructions. Magnetic Moments LLC. Mohamed~ A.M. and L Busch-Vishniac (1995). Imbalance compensation and automation balancing in lnagnetic beaTing systems using the Qparameterization theory. IEEE Trans. on. Control Systems Technofogy-:; 3, 202-2 11. Nie, J. (1991). Fuzzy control ofmulti variable nonlinear servomechanisms with explicit decoupling scheme. IEEE Trans. on Fuzzy Systems, 5-2, 304-311 ~ Palm, R.) D. Driankov and H. Hel1endoom (1997). Model Based Fuzzy Control. Springer. SivriogJu, S. and K. Nonami (1998). Sliding mode control with time-varying hyperplane for AMB Systems. JEEEIASME Trans. on i\.1echatronics, 3], 51-59. Tian, H., K. Nonami, Y. Shi, and A. Shimada (1996). Sliding mode servo controller design for sensorless flexible rotor-magnetic bearing systems with VSS disturbance observer. Trans_ ofS1CE., 32-8, 12421251. Utkin, V.l. (1977). Variable structure systems vlith sliding modes. IEEE Tram. on Automatic Control, AC-22-2'l 212-222.

Wu, J. C. and T. S. Liu (1996). A sliding-lnode approach to fuzzy control design. IEEE Tral1s~ on Control LC;ystems Technology, 4-2,141-151.

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ISBN: 0 08 043248 4