fuzybueno

7/31/2019 fuzybueno

1/11

A self-tuning fuzzy PID-type controller design for unbalance compensationin an active magnetic bearing

Kuan-Yu Chen b , Pi-Cheng Tung a, * , Mong-Tao Tsai a , Yi-Hua Fan ba Department of Mechanical Engineering, National Central University, Jhongli City, Taoyuan County 32001, Taiwan, ROC b Department of Mechanical Engineering, Chung Yuan Christian University, Jhongli City, Taoyuan County 32023, Taiwan, ROC

a r t i c l e i n f o

Keywords:Fuzzy PID-type controllerSelf-tuning mechanismActive magnetic bearingUnbalanced force observer

a b s t r a c t

This paper presents a design for a fuzzy gain tuning mechanism dealing with the problem of unbalancedvibration problem in an active magnetic bearing (AMB) system. For the purpose of enhancing the perfor-mance of the AMB system, we replace the conventional proportional-integral-derivative (PID) controllerwith a self-tuning fuzzy PID-type controller (FPIDC). The shaft displacement and the unbalanced forces of the rotor are evaluated by model-based observation. If there are model uncertainties in the rotor systemor nonlinearities in themagnetic bearing system, this observer may not work well at any operating speed.A fuzzy gain tuner is added to adjust the actuating signal of the self-tuning FPIDC in order to overcomethe disturbances and suppress the unbalancing vibration. The experimental results show that the pro-posed scheme allows for a remarkable improvement in reducing vibration in an unbalanced AMB systemas well as demonstrate an efcient reduction in the shaft displacement of the rotor.

2008 Elsevier Ltd. All rights reserved.

1. Introduction

Interest in the industry in active magnetic bearings (AMBs) con-tinues to grow. AMBs offer some important advantages over con-ventional ball or roller bearings, such as no contact betweenbearings, and consequently, no need for lubricants, which makesthem very useful in special high temperature or vacuum environ-ments. However, it is difcult to design active controls for mag-netic bearing systems because of their high nonlinearity andopen-loop unstable electromagnetic dynamics. In recent years,several nonlinear control techniques have been proposed ( Behal,Costic, Dawson, & Fang, 2001; Lvine, Lottin, & Ponsart, 1996; Que-iroz & Dawson, 1996 ) for AMB systems including sliding mode(Torres, Sira-Ramirez, & Escobar, 1999 ), feedback linearization(Smith & Weldon, 1995 ), and hybrid control ( Al-Holou, Lahdhiri, Joo, Weaver, & Al-Abbas, 2002 ), all designed to improve their dis-turbance rejection properties and robustness in terms of unmod-eled dynamics and parameter uncertainties. The characteristic of the sliding mode control is its robustness or insensitivity to mod-eling errors and disturbances. In practical systems, however, it isdifcult to achieve the fast switching control that is generally re-quired to implement most sliding mode control designs. The draw-back of feedback linearization is that it is necessary to know thewhole state of a nonlinear system before the controller is designed.

For real systems, this is often a great hindrance, as many of thestates cannot be effectively measured. Feedback linearization isalso sensitive to modeling errors that result from the fact that anexact model of a nonlinear system is generally not available.

There is one further problem with rotor unbalancing in an AMBsystem that we must not ignore, that is it appears as synchronousrotor displacement as well as synchronous transmitted force. Var-ious methods to solve the problem of unbalanced vibration havebeen discussed. Chen and Lewis (1992) combined an accelerationestimator with a proportional-derivative (PD) controller to sup-press the vibration caused by unbalanced forces. Higuchi, Otsuka,& Mizuno (1992) proposed a periodic learning control which uti-lized the period of oscillation and the characteristics of the systemto identify the unbalancing force and reduce the vibration. Model-based controllers are also sometimes used in AMBs, although areliable model is not always known for all operating conditions(Higuchi, Mizumo, & Tsukamoto, 1990; Matsumura, Fujita, & Oka-wa, 1990 ). Methods including acceleration estimator and model-based observer designs are frequency dependent. Lum, Coppola,and Bernstein (1996) proposed an adaptive autocentering ap-proach that was frequency independent, and compensated fortransmitted forces that occurred due to unbalance in an AMB sys-tem. They showed that the adaptive autocentering control objec-tive is equivalent to the attenuation of synchronous rotorvibration caused by mass unbalance.

In this paper, we describe a fuzzy PID-type controller (FPIDC)that operates via the parameters self-tuning method, to help

0957-4174/$ - see front matter 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.eswa.2008.10.055

* Corresponding author.E-mail address: [email protected] (P.-C. Tung).

Expert Systems with Applications 36 (2009) 85608570

Contents lists available at ScienceDirect

Expert Systems with Applications

j o u rn a l homepa ge : www.e l s ev i e r. com/ loca t e / e swa
mailto:[email protected]://www.sciencedirect.com/science/journal/09574174http://www.elsevier.com/locate/eswahttp://www.elsevier.com/locate/eswahttp://www.sciencedirect.com/science/journal/09574174mailto:[email protected]

7/31/2019 fuzybueno

2/11

compensate for magnetic nonlinearity and reduce unbalancedvibration. Fuzzy logic controllers (FLCs) may be viewed as a non-conventional way to design feedback controllers in cases whereit is convenient and effective to build a control algorithm withoutrelying on a precise mathematical model of the controlled system.The fuzzy control law can be designed conveniently, after research-ing the relationship between input and output of the system. In re-cent years, there have been a number of reports on the successfuluse of FLCs for AMB systems. Hung (1995) designed a nonlinearcontroller for a dual-acting magnetic bearing. Fuzzy reasoningwas used to adjust the output of a linear proportional-integral-derivative (PID) controller only when the plant state and inputwere far from the point of equilibrium. Hong and Langari (2000)proposed a fuzzy logic control scheme for an AMB system sub- jected to harmonic disturbances, where the AMB system was rep-resented by means of a TakagiSugenoKang fuzzy model.

Fuzzy control is often viewed as a form of nonlinear PID con-trol because it provides nonlinear input/output mapping. Hence,the majority of fuzzy control applications are classied as FPIDCs.FPIDCs poses the potential to achieve better system performanceover conventional PID controllers, but in nonlinear systems theuse of only static or xed valued scaling factors (SFs) may notbe sufcient to provide optimal performance and robustness inthe face of both process disturbances and modeling errors. Mostreal processes actually consist of nonlinear high-order systemsand may have considerable dead-time. Sometimes parametersmay change randomly with time or with changes in the ambientenvironment. To overcome these drawbacks, a lot of research hasbeen focused on the tuning of the input/output SFs of FPIDCs viaon-line self-tuning schemes. Chung, Chen, & Lin (1998) developeda method for the self-tuning of both the input and output SFs of aPI-type fuzzy controller via a fuzzy tuner. Mudi and Pal (1999)proposed a robust self-tuning scheme for the output SFs of fuzzyPI- and PD-type controllers, considered equivalent to the control-ler gain. Here, the output SF is modied at each sampling time bya gain updating factor, which is dependent on the trend of theoutput of the control process. The gain updating factor is com-puted on-line using a model independent fuzzy rule base denedin terms of error ( e) and change of error ( D e). This self-tuningfuzzy PI- and PD-type controller scheme shows a remarkably im-proved performance over conventional FLCs without this gaintuning mechanism. Woo, Chung, and Lin (2000) presented an-other parameter adaptive method which used a function tuner.They dened two empirical functions with respect to the errorsignal ( e(t )). These were used to adjust the input SF ( K d) for eand the integral constant ( b ) of the FPIDC, to reduce the oscilla-tion and shorten the settling time of the system. Gzelkaya, Ek-sin, and Ye s il (2003) developed a parameter adaptive methodto adjust the K d and b of an FPIDC using an on-line fuzzy infer-ence mechanism. The fuzzy inference mechanism used to adjust

the related coefcients has two inputs, one of which is calledthe system error, the other normalized acceleration. The normal-ized acceleration gives relative rate information regarding thefastness or slowness of the system response. The mechanism thatprovides this information can be interpreted as a relative rate ob-server. The simulation results in Gzelkaya et al. (2003) clearlydemonstrate the efciency of this system compared to other re-lated a lesser number of parameters must be tuned, and theyare methods. This is because that is more robust relative to thesystem parameters or structural changes must be tuned.

In this study, we proposed an effective method for tuning thecoefcients of an FPIDC used for controlling an AMB system. A sys-tem built for experimental purposes provide evidence at the supe-rior performance over that at a conventional PID control system.

Furthermore, we use a fuzzy gain tuning mechanism to deal withthe problem of the unbalanced vibration in an AMB system. We

rst consider the uncertainties of the magnetic parameters. We de-sign a model-based observer to evaluate the shaft displacementand unbalanced forces of the rotor. The hysteresis phenomenonand magnetic leakage arising from magnetic ux density and mag-netic eld intensity make the forces acting on the magnetic bearingto be nonlinear. This means that an observer designed based on thenominal dynamic characteristics of the rotor, will not work well forthe entire operating range. Thus, we propose a model-based fuzzyunbalance estimator to overcome estimation errors caused byparameters and modelling uncertainties. The effectiveness of thenewmodel-based unbalance compensator with a fuzzy gain tuningmechanism for AMB systems is compared experimentally with thatof a self-tuning FPIDC.

The remainder of this paper can be summarized as follows. InSection 2 we discuss the two-input FPIDC structures and parame-ter adaptive mechanisms. In Section 3 we discuss the modeling of the AMB system, the basic analysis of the proposed model-basedobserver, and the structure of the fuzzy gain tuning mechanism. Fi-nally, the experimental setup and results are discussed, and someconclusions are provided in Sections 4 and 5.

2. Self-tuning fuzzy PID-type controller

2.1. Fuzzy PID-type controller with a parallel structure

Qiao and Mizumoto (1996) described a FPIDC structure wherethe fuzzy PD- and PI-type controllers are simply connected to-gether in parallel, as shown in Fig. 1 a. They utilized the productsum inference method, center of gravity defuzzication method,and triangular uniformly distributed membership functions (MFs)for the inputs and a crisp output to design the FPIDC. The outputof the FPIDC can be given by

u uPD uPI aU bR Udt a A PK ee DK d _e bR

A PK ee DK d _edt a A b At aK eP bK dDe bK eP

R edt aK dD_e;

1

where aK eP + bK dD, bK eP , and aK dD are the equivalent proportional,integral, and derivative gains, respectively. The relation betweenthe input and output variables of the FPIDC in (1) is given byU = A + PE + DD E , where E = K ee and D E = K de . Here K e and K d aretheSFs for the input variables e and e , respectively, and b is the inte-gral constant for the output variable U .

0.50-0.5 1

NB1

-1

NM PM PBZE

FLC

Derivative

Estimator

++

+-

FPIDC

++

a

b

Fig. 1. (a) Closed-loop control structure of the FPIDC; (b) MFs for E and D E .

K.-Y. Chen et al./ Expert Systems with Applications 36 (2009) 85608570 8561

7/31/2019 fuzybueno

3/11

The MFs for the scaled error E and derivative of D E error areshown in Fig. 1 b, where the fuzzy variables labeled: NB, NM,ZE, PM, and PB, represent negative big, negative medium,zero, positive medium, and positive big, respectively. Thefuzzy control rules for computing U are given in Table 1 .

2.2. Self-tuning method for the FPIDC

Gzelkaya et al. (2003) developed a fuzzy parameter regulatorfor tuning the input and out put SFs K d and b of the FPIDC. The fuz-zy parameter regulator has two inputs: one of which is the abso-lute value of error ( jej) and the other one is the normalizedacceleration ( r m). The output variable of the fuzzy parameter regu-lator is designated c. The normalizedacceleration r m(k) is dened as

r mk K r mdek dek 1

de K r m

ddekde

; 2

where de (k) is the incremental change in error given byde (k) = e(k) e(k 1), dde (k) is the acceleration in error given bydde (k) = de (k) de (k 1), and K r m is the SF for r m(k). In (2) , de () isthe maximum change between de (k) and the previous valuede (k 1), designated as follows:

de dek; jdekjP jdek 1 jdek 1 jdekj< jdek 1 j

3

Here K d and b are adjusted by multiplying and dividing its predeter-

mined value by c, as shown below:K d K ds K fd K f c; 4

and

b bs

K f c05

where K ds and b s are the initial values of K d and b , K f isthe SF for theoutput of the fuzzy parameter regulator, and K fd is the additionalparameter that affects only K d. A block diagram of the controllerstructure is shown in Fig. 2 a. The MFs for the input variables r mand jej, and the output variable c chosen are symmetrical triangularuniformly distributed functions, as shown in Fig. 2 b and c, wherethe fuzzy variables for jejand c are S, SM, M, and L, represent

small, small medium, medium, and large, respectively, andthe fuzzy variables for r mare S, M, and F, representing slow,moderate, and fast, respectively. Table 2 shows the fuzzy con-trol rules for the computation of the output variable c.

3. Modeling of the AMB system with unbalance compensation

3.1. Modeling of the AMB system

The basic structure of the horizontal rotor system used in thisstudy is shown in Fig. 3 . The rotor is suspended by an electric mag-netic bearing (EMB) on the free end. The other end is connected toan inductive motor by a exible coupling. Two position sensors tomeasure the shaft displacement in both the horizontal and vertical

directions are installed at the free end of the rotor.The relations between the position of the coupling, magnetic

bearing, and the center of mass of the rotor are shown in Fig. 4 ,where O is the mass center of the rotor; and F 1F 4 denote the four

Table 1

TakagiSugeno type fuzzy rule base for the computation of U .

E

NB NM ZE PM PB

D E NB 1 0.7 0.5 0.3 0NM 0.7 0.4 0.2 0 0.3ZE 0.5 0.2 0 0.2 0.5PM 0.3 0 0.2 0.4 0.7PB 0 0.3 0.5 0.7 1

0.6670.333 10

SM M LS1

0 1

S1

-1

FM

FLC++

+-

FPIDC

++

Relative rate observer

+-

FPR

Fuzzy tuner

+ -

a

b

c

Fig. 2. (a) Block diagram of the relative rate observer FPIDC; (b) MFs for r m; (c) MFsfor jej and c.

Table 2

Fuzzy rule base for the computation of c.

r y

S M F

jej S M M L SM SM M L M S SM ML S S SM

2

1

4

3Motor

Platform

Magnetic bearing

MassSensor

Back-up bearingFlexible coupling

Fig. 3. Basic structure of the AMB system.

8562 K.-Y. Chen et al./ Expert Systems with Applications 36 (2009) 85608570

7/31/2019 fuzybueno

4/11

attractive magnetic forces in the x and y directions. The rotor is as-sumed to be rigid and symmetrical with uniform mass unbalance.

The dynamic equations describing the movement at the rotor bear-ing system about the mass center are

F x m x m pr X2 cos X t h F x1 F 1 F 3 ;

F y m y m pr X2 sin X t h mg F y1 F 2 F 4 ;

M y I h y X I p _h x dm pr X2 cos X t h aF x1 bF 1 F 3 ;

M x I h x X I p _h y dm pr X2 sin X t h aF y1 bF 2 F 4 ;

6

where m is the mass of the rotating shaft; m p is the mass unbalanceat the rotating disk; h is the initial angle of the unbalanced massmeasured from the X axis; X is the speed of rotation around thespinning Z axis; x, y, h

x, and h

yare the radial and rotating displace-

ments of the mass center, respectively; I and I p are the transverseand polar mass moments of inertia of the rotor; F x1 and F y1 arethe coupling forces; x1 and y1 are the shaft displacements corre-sponding to the X and Y axes at the exible coupling; and x2 and y2 are the shaft displacements at the magnetic bearing.

According to Tadeoand Cavalca (2003) and neglecting the effectof rotation, the exible coupling forces corresponding to the X andY axes can be expressed as follows:

F x1 c T _ x1 kT x1 ;F y1 c T _ y1 kT y1 ;

7

where c T is the equivalent damping; and kT is the equivalent stiff-ness of the coupling.

The magnetic forces provided by the EMB are functions of thewidth of the magnetic gap and the current driving the electromag-nets ( Hsiao, Fan, Chieng, & Lee, 1996 ). Thus, the four magneticforces can be written as a function of the driving currents and vari-ations in the magnetic gap,

F n f 0 kiin kddn f in ; dn; n 1 ; 2 ; . . . ; 4 ; 8

where f 0 is the static magnetic force when the driving current isin = 0 and the magnetic gap variation is dn = 0; kd and ki are theforcedisplacement stiffness factor and the forcecurrent stiffnessfactor; and f (in,dn) is a high-order term for the magnetic forcesdue to the coil currents and magnetic gap variations.

Let the four electromagnets have the same static magnetic force

f 0 and the same coefcients ka and ki; the four magnetic forces F 1F 4 can now be rewritten as

F 1 F 3 kii1 kd x2 f i1 ; x2 ;F 2 F 4 kii2 kd y2 f i2 ; y2 :

9

For simplicity, the system equations for the designed controllerindicate displacements in the locations of the exible couplingand magnetic bearing. Since the rotor is assumed to be rigid andthe displacement from the desired position is assumed to be small,

the relationships between the shaft positions ( x1 , x2 , y1 , y2 ) and themass center ( x, y,h x,h y) can be shown as

x bx1 ax2

a b

bx1 ax2L

;

y by1 ay 2

a b

by1 ay 2L

;

h x y1 y2a b

y1 y2

L;

h y x2 x1a b

x2 x1

L ;

10

where a is the distance between the exible coupling and the masscenter; b is the distance between the magnetic bearing and themass center; and L= a + b.

The dynamics of the systemcan be rearranged in matrix form asfollows:

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

2666437775

x1 x2 y1 y2

2666437775

c T b 3 0 a 1 a 1c T b 2 0 a 2 a 2a 1 a 1 c T b 3 0a 2 a 2 c T b 2 0

2666437775

_ x1_ x2_ y1_ y2

2666437775

kT b 3 kdb 2 0 0

kT b2 kdb 1 0 00 0 kT b3 kdb 20 0 kT b2 kdb 1

2666437775

x1 x2 y1 y2

2666437775

kib 2 0

kib 1 00 kib 20 kib 1

2666437775

i1i2

c2 0 b 2 0

c1 0 b 1 00 c2 0 b 20 c1 0 b 1

2666437775

m pr X2 cos X t h

m pr X2 sin X t h

f i1 ; x2 f i2 ; y2

266664377775

0

0

1

1

2666437775 g ;

11a

where a 1 I pX aIL , a 2

I p X bIL , b 1

b21

1m, b2

ab1

1m, b3

a21

1m, c1

bd1

1m; and c2 ad1 1m. Eq. (11a) can be expressed as

M x C _ x Kx Bu Ew Dg ; 11b

where x = [ x1 x2 y1 y2]T is the state vector; u = [ i1 i2]T is the input

vector; and w = [m pr X 2cos( X t + h) m pr X 2sin( X t + h) f (i1 , x2) f (i2 , x2)]T is the vector of the disturbance forces.

3.2. Model-based unbalanced forces observer

Here, we describe a decentralized force estimator for compen-sating disturbance forces. Considering Eq. (11a) , the dynamics of the suspended magnetic part can be rearranged as follows:

1 0

0 1 x2 y2 kdb 1 00 kdb 1 x2 y2 kib 1 00 kib 1 i1i2 dx2dy2 ;

12

where dx2 and dy2 are the sum of the disturbance forces, including

unbalanced forces, force of gravity, coupling forces, and gyroscopicforces for both the x and y directions.

2

3

4

1

Fig. 4. Geometry relationship of the rotor system.


7/31/2019 fuzybueno

5/11

Self-TuningFPIDC

Magnetic BearingSystem + -

Unbalanced ForceObserver

FuzzyInferenceSystem erivative

EstimatorFuzzy Gain Tuning M

Model-Based Unbalance Compensator

Fig. 5. Control structure of the AMB system with the model-based unbalance compensator.

0.50-0.5 1

NB1

-1

NS PS PBZE

Fig. 6. Membership functions of Y 2 and D Y 2 .

air gap, g

electromagnet

+i 0 i y

i 0 x+ i

f

x x

i i-0 y

i - i f y

shaftrotor

0 x

y

a

b

Fig. 7. (a) End view of the magnetic bearing; (b) photograph of the experimental setup.

Table 3

TakagiSugeno type fuzzy rule base for the computation of a .

Y 2

NB NS ZE PS PBD Y 2 NB 1 0.7 0.5 0.3 0

NS 0.7 0.4 0.2 0 0.3ZE 0.5 0.2 0 0.2 0.5PS 0.3 0 0.2 0.4 0.7PB 0 0.3 0.5 0.7 1


7/31/2019 fuzybueno

6/11

The sum of disturbance forces can be expressed as

dx2 c1 m pr X2 cos X t h c T b 2 _ x1 a 2 _ y1 kT b 2 x1 a 2 _ y2 ;

dy2 c1 m pr X2 sin X t h a 2 _ x1 c T b 2 _ y1 kT b 2 y1 a 2 _ y2 g :

13 It is obvious from Eq. (12) that thesystemcan be separated into twosimilar sub-systems. Hence, we design a force estimator y2 for that

is also suitable for x2 .

The dynamic equation of y2 can be expressed as

_ y2 y2" # 0 1kdb 1 0" #y2_ y2" #

0

kib1" #i2 0

1" #dy2 Ay2 Bi2 Ddy2 ; y y2 g s 0

y2_ y2" # Cy2 ;

14

Fig. 8. Shaft displacements on the Y axis and orbits of the rotor center for the PID controller: (a) 10 Hz; (b) 20Hz; (c) 30 Hz; (d) 40Hz; (e) 50Hz; (f) 60 Hz; (g) 70 Hz; (h)80 Hz.


7/31/2019 fuzybueno

7/11

where y2 y2_

y2T and g s is the gain of the position sensor. We

can calculate the rank of the observable matrix by

rank V rankC

C A" # ! rank g s 00 g s 2 : 15 The system is thus a fully observable system. The Luenberger state

estimator can be written as

_ y2

y2" #0 1

kdb 1 0" #^ y2_ y2" #

0

kib 1" #i2 l1

l2" # y y2 ^ y y2 A ^ y2 Bi2 L y y2 y y2 ;

^ y y2 g s 0 ^ y2_ y2" # C y2 :

16

Fig. 9. Shaft displacements on the Y axis and orbits of the rotor center obtainedusing the self-tuning FPIDC:(a) 10Hz; (b) 20Hz; (c) 30Hz; (d) 40Hz; (e) 50Hz; (f) 60Hz; (g)70 Hz; (h) 80Hz.


7/31/2019 fuzybueno

8/11

If there is no disturbance term dy2 , the Luenberger observer makesthe observed error decay to zero. In other words, we can stabilizethe estimated system and make the observed displacement andvelocity of therotor approach that of a real system via suitablegainsof L. Hence, the disturbance term exists all the time. The observederror is affected by the disturbance term of dy2 , leading to variationin the estimated output. In other words, the variation in the ob-served output is a measurement criterion for the disturbance forcedy2 . Thus if we set the error integral term d

^

y2 to be

d^ y2 g sld Z y2 ^ y2 dt ; 17 then insert this into the Luenberger observer. The observer can nowbe rewritten as

_ y2 y2

d _ y2

26643775

0 1 1

kdb 1 0 10 0 0

264375

^ y2_ y2

d y2264

375

0

kib 10

264375

i2 l1l2l3

264375

y y2 ^ y y2 ;

^ y y2 g s 0 0

^ y2_ y2

d^ y2264

375

:

18 The dynamic equation for the error can now be expressed as

_e g sl1 1 0

g sl2 kdb1 0 1

g sld 0 0264

375e

0

0

1264

375d _ y2 ; 19

where_

e e y2_

e y2 dy2 d^

y2T and e y2 = y2 y2 .

The characteristic equation for Eq. (19) is

s3 g sl1 s2 g sl2 kdb 1 s g sld 0 : 20

According to the RouthHurwitz stability criterion, the stabilityconditions are

g sl1 > 0 ; g sl2 kdb 1 > 0 ; g sl1 g sl2 kdb 1 > g sld > 0 :

21

Hence, suitable gains can be selected utilizing Eq. (21) to make theobserver stable. We can also design an observer for x2 that makesboth observers stable and provides feedback to the system.

However, if thereare some model uncertainties in the rotor sys-

tem or nonlinearities in the magnetic bearing system, this observerwill not work well at any operating speed. The lag and the ampli-tude of the phase of the estimated force signal may differ for exactdisturbances. Theestimated forces of disturbance increase with therotating speed. Hence, in this study, the fuzzy gain tuning mecha-nism is used to help compensate for magnetic nonlinearities andimprove system performance.

3.3. Fuzzy tuning method for the unbalanced forces observer

Unlike conventional types of control, which need a precisemathematical model of the plant, fuzzy control design is based

Fig. 10. Observed and measured shaft displacements on the Y axis obtained using the self-tuning FPIDC: (a) 10 Hz; (b) 20 Hz; (c) 30 Hz; (d) 40Hz; (e) 50 Hz; (f) 60 Hz; (g)70 Hz; (h) 80 Hz.

Table 4

Initial values of parameters used in the experiments.

PID controller K p = 1.2, K I = 2,K D = 0.002, i0 = 1.5

Self-tuning FPIDC K e = 0.6, K ds = 1,a = 3.4, b s = 0.001,i0 = 1.5, K ae = 1.2,K r m= 0.2, K f = 2.4,K fd = 2.4

Unbalanced force observer kd = 12,000, ki = 35,b 1 = 2.6, l2 = 600,ld = 45, g s = 4000

Fuzzy gain tuner SF e = 4, SF _e 1 :2,SFa = 2.4


7/31/2019 fuzybueno

9/11

on an expert linguistic description of the system behavior. In otherwords, fuzzy logic control is a knowledge-based system. In thissection, we describe the proposed fuzzy tuning mechanism, whereinput signals obtained from the model-based force observer areused to adjust the output signal of the self-tuning FPIDC of theAMB system, thereby overcoming disturbances and suppressingunbalanced vibration.

A block diagram of our AMB system with the model-basedunbalance compensator is shown in Fig. 5 . A self-tuning FPIDC isused to calculate the control current driving the electromagnetssuspending the rotor. Furthermore, a model-based unbalance com-pensator, constructed with a model-based unbalanced forces ob-server using the fuzzy gain tuning method, is used to suppress

Fig. 11. Shaft displacements on the Y axis and orbits of the rotor center obtained using the self-tuning FPIDC with fuzzy gain tuning mechanism: (a) 10 Hz; (b) 20 Hz; (c)30 Hz; (d) 40 Hz; (e) 50 Hz; (f) 60 Hz; (g) 70 Hz; (h) 80 Hz.


7/31/2019 fuzybueno

10/11

unbalanced vibration and improve the control performance of theAMB system.

The fuzzy rules of the gain tuning method are designed basedon variances in the magnitude and the rate of change of the vibra-tion. These naturally increase as the force of the disturbance in-creases. The vibration signal is periodic and symmetric. We thususe the signal obtained from the observer and its derivative to de-sign the fuzzy rules. Various types of fuzzy inference methods thathave been used to design the fuzzy logic controllers are describedin the literature ( Chung et al., 1998; Gzelkaya et al., 2003; Mudi &Pal, 1999; Qiao & Mizumoto, 1996; Woo et al., 2000 ). It is well-known that the TakagiSugeno inference method enhances theefciency of the defuzzication process, because it greatly simpli-es the computation required by the Mamdani method. Hence, weuse the TakagiSugeno method to design the fuzzy tuner.

The proposed fuzzy tuner has two-input variables and a signaloutput. The MFs for the input variables: the observed shaft dis-placement

^

y2 and the derivative of the observed shaft displacement_

^

y2 , are shown in Fig. 6 , where the universe of discourse for each in-put is normalized over the interval [ 1,1]; the fuzzy variables arelabeled NB, NS, ZE, PS, and PB and represent negativebig, negative small, zero, positive small, and positive big,respectively. Symmetrical triangular uniformly distributed MFsare assigned for the two-input variables.

The rule base for the computation of the output variable a isshown in Table 3 . This is a commonly used two-dimensional phaseplane rule base. To make the self-tuning FPIDC produce a smallunbalanced vibration in the AMB system, the controller gain isset to a large value, while

^

y2 the and_

^

y2 are simultaneously positivebig or negative big. In other words, the gain should be set to a smallvalue to maintain dynamic equilibrium when the unbalancedvibration of the AMB systemis small, such as at rest or when rotat-ing at a lower speed.

4. Experimental results

4.1. Experimental setup

The experimental setup used in this study consists of a two-axiscontrolled horizontal shaft magnetic bearing symmetric in struc-ture. The magnetic bearing has four identical electromagnetsequally spaced radially around a rotor disk which is made of lam-inated stainless steel, as shown in Fig. 7 a. Each electromagnet in-cludes a coil and a laminated core made of silicon steel ( Fuh &Tung, 1997 ). The system is driven by an AC motor through a ex-ible coupling. This helps to isolate the vibration originating fromthe motor. A pair of eddy current type proximity probes is placedoutside the shaft near the electromagnets for measuring the hori-zontal and vertical displacements at the geometric center of theshaft. A photograph of the experimental setup is shown in Fig. 7 b.

4.2. Results

There are two pairs of electromagnets in this AMB system: onthe X and Y axes. The two pairs of electromagnets are controlledsimultaneously by two PID controllers or two self-tuning FPIDCs.Diagrams (a1), (b1)(h1) in Figs. 8 and 9 show the shaft displace-ment on the Y axis; diagrams (a2), (b2)(h2) show the orbits aboutthe rotor center when only a conventional PIDcontroller and a self-tuning FPIDC are used at rotating speeds from 10 to 80 Hz. In gen-eral, the shaft displacement (of the rotor center) is smaller in thehorizontal direction than that in the vertical direction because of the effects of gravity. Hence, we only show the shaft displacementon the Y axis here. A comparison at the differences between Figs. 8and 9 show that, the control performance achieved via the self-

tuning FPIDC is better than that with the conventional PIDcontroller.

To evaluate the performance andcharacteristics of the proposedfuzzy gain tuning mechanism, we look at an AMB system con-trolled by a self-tuning FPIDC without feed-forward unbalancedforce compensation in the rst 10 s. The fuzzy updating gain isbrought into effect after 10 s. Fig. 10 shows the observed and mea-

sured shaft displacement on the Y axis obtained using a self-tuningFPIDC, at rotational speeds from 10 to 80 Hz. The initial values of all experimental parameters are shown in Table 4 . From the resultsin Fig. 10 we can see that the observed shaft displacement (ob-tained from the model-based observer) is very close to the mea-sured shaft displacement obtained from the position sensors.Fig. 11 shows the shaft displacement on the Y axis, and the orbitsabout the rotor center obtained using the self-tuning FPIDC withthe proposed fuzzy gain tuning mechanism, for rotating speedsfrom 10 to 80 Hz. It can be seen that the proposed scheme cannoticeably reduce the shaft displacement. A comparison at Figs.11 to 9 shows that the orbits around the rotor center have becomeobviously smaller.

5. Conclusion

In this study, a fuzzy gain tuning mechanism is proposed tosuppress unbalanced vibrations in an AMB system. First, a mod-el-based unbalanced forces estimator for the observation of unbal-anced forces is described. The experimental results show that theobserved shaft displacements obtained with the observer are veryclose to the measured shaft displacements observed by the posi-tion sensors. We then designed a fuzzy gain tuner to adjust theactuating signal of the self-tuning FPIDC. The experimental resultsclearly show that this scheme improves the performance of a self-tuning FPIDC for an AMB system.

We can conclude that the model-based unbalanced force obser-ver fuzzy gain tuning mechanism is indeed more efcient at sup-pressing unbalanced vibration in an AMB system, and is alsomore robust in terms of system uncertainties and nonlinearities,compared to the self-tuning FPIDC.

Acknowledgement

This project was supported by the National Science Council inTaiwan, Republic of China, under Project No. NSC 96-2628-E-008-075-MY3.

References

Al-Holou, N., Lahdhiri, T., Joo, D. S., Weaver, J., & Al-Abbas, F. (2002). Sliding modeneural network inference fuzzylogic control for active suspension systems. IEEE Transactions on Fuzzy Systems, 10 (2), 234246.

Behal, A., Costic, B. T., Dawson, D. M., & Fang, Y. (2001). Nonlinear control of magnetic bearing in the presence of sinusoidal disturbance. In Proceedings of the American control conference, Arlington, VA, USA (pp. 36363641).

Chen, H. M., & Lewis, P. (1992). Rule-based damping control for magnetic bearings.In Proceedings of the 3rd international symposium on magnetic bearing, Alexandria,VA, USA(pp. 2534).

Chung, H. Y.,Chen, B. C.,& Lin, J. J. (1998).A PI-type fuzzy controller with self-tuningscaling factors. Fuzzy Sets and Systems, 93 , 2328.

Fuh, C. C., & Tung, P. C. (1997). Robust stability analysis of fuzzy control systems.Fuzzy Sets and Systems, 88 , 289298.

Gzelkaya, M., Eksin,_

I., & Yes il, E. (2003). Self-tuning of PID-type fuzzy logiccontroller coefcients via relative rate observer. Engineering Applications of Articial Intelligence, 16 , 227236.

Higuchi, T., Mizumo, T., & Tsukamoto, M. (1990). Digital control system formagnetic bearings with automatic balancing. In Proceedings of the 2ndinternational symposium on magnetic bearing, Tokyo, Japan (pp. 2732).

Higuchi, T., Otsuka, M., & Mizuno, T. (1992). Identication of rotor unbalance andreduction of housing vibration by periodic learning control in magneticbearings. In Proceedings of the 3rd international symposium on magnetic bearing, Alexandria, VA, USA (pp. 571579).

K.-Y. Chen et al. / Expert Systems with Applications 36 (2009) 85608570 8569

7/31/2019 fuzybueno

11/11

Hong, S. K., & Langari, R. (2000). Robust fuzzy control of a magnetic bearing systemsubject to harmonic disturbances. IEEE Transactions on Control SystemsTechnology, 8 , 366371.

Hsiao, F. Z., Fan, C. C., Chieng, W. H., & Lee, A. C. (1996). Optimum magnetic bearingdesign considering performance limitations. JSME International Journal Series C, 39(3), 586596.

Hung, J. Y. (1995). Magnetic bearing control using fuzzy logic. IEEE Transactions onIndustry Applications, 31 (6), 14921497.

Lvine, J., Lottin, J., & Ponsart, J. C. (1996). A nonlinear approach to the control of magnetic bearings. IEEE Transactions on Control Systems Technology, 4 (5),524544.

Lum, K. Y., Coppola, V. T., & Bernstein, D. S. (1996). Adaptive autocentering controlfor an active magnetic bearing supporting a rotor with unknown massimbalance. IEEE Transactions on Control Systems Technology, 4 (5), 587597.

Matsumura, F., Fujita, M., & Okawa, K. (1990). Modeling and control of magneticbearing system achieving a rotation around the axis of inertia. In Proceedings of the 2nd international symposium on magnetic bearing, Tokyo, Japan (pp. 273280).

Mudi, R. K., & Pal, N. R. (1999). A robust self-tuning scheme for PI- and PD-typefuzzy controllers. IEEE Transactions on Fuzzy Systems, 7 , 216.

Qiao, W. Z., & Mizumoto, M. (1996). PID type fuzzy controller and parametersadaptive method. Fuzzy Sets and Systems, 78 , 2335.

Queiroz, M. S., & Dawson, D. M. (1996). Nonlinear control of active magneticbearings: A backstepping approach. IEEE Transactions on Control SystemsTechnology, 4 (5), 545552.

Smith, R. D., & Weldon, W. F. (1995). Nonlinear control of a rigid rotor magneticbearing system: Modeling and simulation with full state feedback. IEEE Transactions on Magnetics, 31 (2), 973980.

Tadeo, A. T., & Cavalca, K. L. (2003). A comparison of exible coupling models forupdating in rotating machinery response. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 25 (3), 235246.

Torres, M., Sira-Ramirez, H., & Escobar, G. (1999). Sliding mode nonlinear control of magnetic bearings. In Proceedings of the IEEE international conference controlapplications, Kohala Coast-Island, Hawaii, USA (pp. 743748).

Woo, Z. W., Chung, H. Y., & Lin, J. J. (2000). A PID type fuzzy controller with self-tuning scaling factor. Fuzzy Sets and Systems, 115 , 321326.


Documents

fuzybueno