Decreased vibration control for centrifuges: A new adaptive hybrid control technique

ControlEng. Practice,Vol. 4, No. 12, pp. 1693-1700, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0967-0661/96 $15.00 + 0.00

Pergamon PII:S0967-0661(96)00186-4

DECREASED VIBRATION CONTROL FOR CENTRIFUGES: A NEW ADAPTIVE HYBRID CONTROL TECHNIQUE Wen Yu*, Tianyou Chai* and Yi Yuan** *Research Center of Automation, Northeastern University, Shenyang, 110006, P.R. China **Faculty of Mechanical Engineering, Northeastern University, Shenyang, 110006, P.R. China

(Received August 1995; in final form October 1996) Abstract: This p a p e r presents a new control approach which can restrain the vibration of centrifuges. In t h e new scheme, changed s u p p o r t stiffness and self-tuning feedforward PID control of the decreased vibration force are effectively combined. Some advanced techniques, (such as adaptive modification for u n s t a b l e vibration regions, two-level c o m p u t e r control, curve-tracking control), are used to make the centrifuge s y s t e m r u n in a m i n i m u m - v i b r a t i o n state. A horizontal centrifugal exp e r i m e n t is used to test the new control method. Experimental results show the effectiveness of the p r o p o s e d scheme. KeFword: tion force

self-tuning PID, feedforward compensation, changing s u p p o r t stiffness, decreased vibra-

1. I N T R O D U C T I O N

can prevent serf-vibration.

There are many

ways of changing the natural frequency. A typical and effective method is to change the sup-

Manufacturing bias, filling material or a rotat-

port stiffness of the rotor bearing (Palazzolo

ing frequency nearing the natural frequency may

and Jagannathan, 1993).

cause centrifuge eccentricity, and this will cause centrifuge self-vibration.

2. Use of active control of the decreased vibration

The unstable vibration

(Viderman and Porat, 1987). The centrifuge's

becomes more and more serious when the rotating

vibration can be decreased by exerting a force on the rotors.

frequency is increased, especially for large rotatory machines. Almost all centrifuges have some kind of eccentricity, and self-vibration during starting

As it is very difficult to describe the vibration of a

and stopping is inevitable, so vibration control is

centrifuge, and there are currently no effective con-

very i m p o r t a n t for the stable operation of rotatory

trol methods or actuators, active control for the vi-

machines. However, as most of the studies on the

bration of centrifuges is still at the level of theoret-

vibration of r o t a t o r y machines were based on an

ical preparation and simulation experiments (Tzou

analysis of the unstable vibration characteristics of the centrifuge, they usually only change the op-

more difficult than industrial process control, an

erating mode to prevent vibration. These control techniques are passive.

effective and realizable controller needs to be designed for rotatory machines.

Recent a t t e m p t s to design active controllers for centrifuges have made great progress in two directions:

Simple and effective experimental equipment for changing the support stiffness has been made by

1. Changing the natural frequency of the centrifuge's rotor. Moving the natural frequency

and Gade, 1988).

As vibration control is much

the authors of this paper. Based on this device a new control approach is proposed here, which uses many advanced and effective techniques, such as

1693

Wen Yu et

1694

al.

hybrid decreased vibration control, two-level com-

lem, and difficult to illustrate.

From a simpli-

puter control, self-tuning P I D control, feedforward

fied dynamic model of the centrifuge, which uses

compensation, prediction of the unstable vibration

three Newton equations, three Euler equations, six

region, and curve-tracking control. Compared with

derivation equations and four Navior-Stakes equa-

other decreased vibration controllers, this method

tions (Wolf, 1968), a liquid-solid coupling vibration

has the following properties:

equation is given as follows (Yuan, 1995):

• The control algorithm is simple, and the actuator is realizable and rapid.

J

ail

• The demands for observation of the running

a12

[o

[ a21

state are low. • The controller is adaptable to changes in the

+

parameters of the centrifuge. • It is suitable for real-time control, and not only eliminates self-vibration, but also restrains vi-

where

bration in the stable operation mode.

.q_

0

b22

c21 all

:

~

(1) 0

--2C

= 1-

+

(I +

• Eccentricity is compensated for by feedforward a21 = i + 2D

control. In order to evaluate the adaptive hybrid control,

bll =

25[-~ -

1

a horizontal centrifugal experiment is used to test several control methods.

Experimental results

show that the approach presented in this paper is b21 = 2 c - 2ill(1 - 2#)

more effective t h a n the others. This paper is organized as follows. The dynamic model of the centrifuge, support stiffness control

b22 = 4# c12

=

-f12(1 - l / f 2 ) ( 1 + 2/~}

and the decreased vibration force control model are introduced in Section 2. In Section 3, the con-

c2, =

trol system structure and control algorithms are discussed. An experimental evaluation is given in Section 4. Section 5 concludes the paper.

Dynamic

model

of the centrifuge

A typical horizontal centrifuge rotor is shown in Fig. 1

u x -- i U y

i=

j-:-~,

e - i c o s e) f = ~b , f l --

o,o

y is the displacement of the centrifuge vibration fl is an intermediate variable

d e c r e a s e d v i b r a t i o n force filled l i q u i d motor

(I + 2/~)f}2

1

q = ~(sin

system

2iflc -

P = ~--~o u :

2. P L A N T D E S C R I P T I O N 2.1.

k(t)-

_N~__ ~ ..................

c is a dumped coefficient, e n t r a n c e of material

# is the density of filled liquid 1 - f is the filling rate

rotor bearing rotor

Fig. 1. The structure of horizontal centrifuges

a is the radius of the rotor a - b is the width of the filled liquid (see Fig.l) w is the critical rotation speed

The vibration problem of a centrifuge belongs to the dynamic category of a rigid body filled with .liquid. It is a complex liquid-solid coupling prob-

u~, uy are projections of the decreased vibration force in the horizontal and vertical directions

Decreased Vibration Control for Centrifuges

1695

2.2. Support stiffness control

is the angular displacement of the rotor k(t) is the support stiffness

Support stiffness is controlled by the oil pressure, and is regulated by the control voltage of the oil

e is the eccentricity distance

valve. Because the oil pressure changes the stiffness of the rubber spring in the rotor bearing, the

w0 is the natural frequency rn is the mass.

support stiffness of the rotor bearing is changed. It is very difficult to express this nonlinear process by means of equationsi especially the relation between

In order to analyze unstable vibration, let

the rubber spring and the support stiffness of the

y

(2)

(Xo - iyo)e -ivt

=

where Xo, Yo are constants, and v is the angle speed of rotor. From the analysis of Yuan (1995), the rotation of a centrifuge can also be described as:

rotor bearing. An experiment has been designed to test this relation, where the input is the control voltage q(t) of the oil valve, and the output is the natural vibration frequency of the centrifuge. The support stiffness k(t) is calculated from the natural Vibration frequency. The experimental result is shown in Fig.3.

(r ÷ / ~ ) v 4 + 2 [2icr + 12(r - 1)1 v3+ [12 (r - 5 -

+ 2ica(r

k(t)

- 2) - r]

(3)

+212 [1 - 3ic12 - 2122(p + 1)] v W122 [1 - 2ic12 - 122(1 +/~)] ----0

,xlo6

2.3

j

support 2.2-

components of v are positive, t ~ m ly(t)l = co and this is defined as unstable vibration. Let 1 - f and fl be variable. Eq.3 can then be solved using Newton-Raphson methods (Burden and Faired, 1985). The relation between unstable y(t) and 1 - f, and 12 is shown in Fig.2.

_

Y

siffness 2.1"

where r = ~_-~y. If r,/z, 12 are constants, v determines the state of the rotor. When the imaginary

_

2.01.9"

J 0.6

[~ 0,8

I 1.0

1.2

1.4

I

'

~

voltage(V}

1.6

Fig. 3. Relation between the support stiffness and the oil valve

The natural frequency of the centrifuge is changed by the support stiffness, so 12 changes, followed by regulation of the oil valve. This can move the un-

I-1 1.0

stable operating point A to stable points B or C in

0.8

Fig.2. However unsuitable control can also change B or C to A, so the determination of the unstable region is very important for vibration control.

C

0.6

o.4~ 0.2-

iF I

I

I

I

I

[

1

1.2

1.4

1.6

1.8

2.0

I

I

b

f]

2.3. Controlled model of decreased vibration force Using the Laplace transformation on Eq.1 and eliminating fl, the relation between u and y is

Fig. 2. Unstable vibration region y{s) =

The shaded part in the figure is the unstable zone,

kp go + g l s + g2s 2 + g3s 3 u(s)

(4)

where gcr. . . . . b1~c21, g l : c21a12 + b12b21, g2 = a12b21azl -b22b11, g3 = a12a~l-b22a11, kp : pb22.

where the whirl frequency is less than the rotation frequency. This is also called "sub-resonance vibration" .

Support stiffness k(t) , i.e. c21, corresponds with the parameters go and gl of system (4). If the

Wen Yu et al.

1696

sample period of the experiment is 500m8 and a

as a supervisor and for calculation of the optimum.

zero-order-holder is used, the transfer function of

The control periods are:

the impulse response of Eq.4 is e(z_l)_

B(z -1) _ y(t)

A(z

~ti~solacement? f o r c e ¢ |ff.etection II control [

u(t)"

~ ~

~

~ ](':'-':'.'" [Icontroll

The influence of eccentricity e(t), which is the last term on the right-hand side of Eq.1, can be calculated from the vibration displacement, so eccen-

1. curve-tracking of changing stiffness ~ ] 12. closed-loop velocity control ] | 13. decrease vibration force output ~ 4. data processing ~ [

| ] /

tricity is regarded as a measurable bounded distur|

bance, and can be compensated for by feedforward control.

Regardless of the system noise, the dis-

crete form of the decreased vibration force control model is

A(z-Xjy(t) = B ( z - 1 ) u ( t -

1) + qe(t).

I I 1. ~2. |3. [4.

Optimum Computer(IBM-~86)

calculation of the unstable vibration region management of the changing stiffness tracking curv( decreased vibration force adaptive control man-machine interface

(5) Fig. 4. The structure of the centrifuge's computer control system

3. DESIGN OF T H E D E C R E A S E D

• Decreased vibration force: 500ms

VIBRATION CONTROLLER

• Changing support stiffness:80ms

Some special techniques in designing the structure

• Velocity control: lOOms

of the control system are used in order to satisfy

A standard PID algorithm is used to control the

the following requirements:

motor's velocity. The block diagram of the control

• The unstable vibration region must be crossed

system is shown in Fig.5.

rapidly during starting and stopping • If a change in the operating conditions causes the system to enter the unstable region, it should move rapidly into the stable region. Velocity Control Loop

• The amplitude of the vibration must be minimized in the stable operating mode.

3.1. The structure of the control system The length of the experimental device is 0.4m, the

Changing Stiffness Control Loop

inner radius is 0.044m, the weight is 9.7kg, the

eccentricity

filled liquid is NaC1 with a density of 1.2g/ml, the ~self--

feedftrward I ] cont]rolGy J tu n i n g ~ ~ ~ - ~ _ ~

[ [

II ..... ~J Idispla . . . . . t~ ~ ..... t . . . . t , o n ~ se . . . . ["

filling rate is 1 - f = 0.6, f =

b_ (see Fig.l)

and without any control the natural frequency is 25.6Hz. In order to satisfy two conflicting demands (complex calculations and rapid response), a two-level

i

Decreased Vibration Force Control Loop

computer controlled system is used as shown in Fig.4. Each subsystem is divided into several func-

Fig. 5. Block diagram of the control system

tional modules, and the two subsystems are conEach

The velocity sensor is a Hull component. The two

module is characterized by an operating mode, an

displacement sensors are mounted with horizontal

operating strategy and an operator.

The single-

and vertical orientations, as shown in Fig.6. Ec-

chip controller is used for rapid response and con-

centricity detection is achieved using an indirect

trol output, while the "optimum computer" is used

method. It is calculated from the two displacement

nected by means of serial communication.

Decreased Vibration Control for Centrifuges

1697

values.

/

~

computer]

,iiiiiiiii 3.5

......

:

'

~

i ¸¸

3 25 2

'.~,

Fig. 6. Vibration displacement detection

o.s

.... i ............. .........

i i

•

i ~

i .............

:

.............

heuristic t i m e t

3.2. Prediction of unstable vibration The unstable vibration region is judged from the amplitude of vibration and resonance analysis created by the F F T board. The centrifuges may reenter the unstable region when changed stiffness or a decreased vibration force are used, so it is very important to predict whether a new operating mode will lie in an unstable region. As Eq.3 and Fig.2 can only give approximate calculation under the ideal conditions, they are difficult to use directly. Eq.2 and Eq.3 can also be written as an experimental equation:

y(t) = kl(i ) × f ( r , n , # , c , v , e , u )

(6)

where f(.) is determined off-line by experimental data, and kl(i ) is the ith modification coefficient, modified on-line by a heuristic algorithm.

Fig. 7. The experimental result of modification coefficient k(t) (1/ Y × a / V ~ ) , a = ~-&T[Ey 2 1 ~ Yi, V/is the degree of accuracy, Fi is a weighting term of the degree of accuracy, and Fa is the approach gain. Fa and F/ are constants, determined experimentally.

where Vi =

-

In the experiment, n = 8. The model is modified after n values are detected. The experimental result of this algorithm are shown in Fig.7. When i = 20 and i = 45, the operating mode is changed. This shows that the heuristic algorithm has good convergence properties. It can predict the unstable vibration effectively.

The unstable region can be defined as ly(t)] >_ M

3.3. Curve-tracking control of changing stiffness

where M is the upper limit of the vibration amplitude in the stable region. According to the calcu-

A method of changing the stiffness is used to move

lated value y' = k! × f(-) and the detected vibration displacement y(t), k! is modified adaptively. There are several parameter-estimation algorithms that can be used (Goodwin and Sin, 1987), but they often involve some assumptions such as strict passivity real (SPR} and persistent excitation (BE}. In this paper only a scalar kl(i } needs to be estimated, not a parameter vector, so a modified stochastic gradient algorithm is used. This method is similar to that of SIEMENS Ltd. (1987}. The following adaptive algorithm is used to modify k I

k (i + 1) =

y{0 + va[y-; -

(7)

where Ira is the theoretical approach gain, k I (0) = 1. Let

vo --- ro(1 - ¢ , a }

{8}

the centrifuge rapidly into the stable region. Because neither the relationship between the oil valve and the stiffness, nor that between the stiffness and the stable region can be known, Fig.2 shows that different 1 - f and fl correspond to different unstable regions. So the control voltage of the oil valve will need to have different velocity and acceleration respectively to make the centrifuge leave the unstable vibration region. From the experimental results and Fig.2, the different curves for the different operating modes are given in Fig.8. These curves can be selected by the "optimum computer", or by an operator in a different operating mode. The selected curve is sent to the single-chip controller by serial communication, and the changhag stiffness control follows this curve directly. Before the control voltage is added to the valve, the unstable region prediction module is used to

Wen Yu et al.

1698 conditiov

curve ~voltage

~,

7 C ~ a

f0.8

f<0.4

voltage

.f > 0.1

q~

sxpe~ence

f>0.4

/

~u(t) = klan(t) + h~(t)+ ka[A~(t) - A~(t - 1)] + ale(t )

(11)

where ~(t) = - y ( t ) , A~(t) : ~ ( t ) - ~ ( t - 1), G / ( z -1) = - A i E ( z -1) + S ( z - 1 ) ] , kp : - ) k ( g I --}-

~, voltage

voltage

f<0.8

A feedforward PID controller (see Fig.5) is:

curve

condition

/ fl

292), h = A(g0 + gl + g2), kd = Ag2. The closed-loop equation is:

(QA~ + P B , ) y ( t ) = (qQ + BrS)e(t) +(Q + B r E ) M ( t ) .

Fig. 8. Tracking curve of changing stiffness

decide if q(t) makes y(t) unstable, k(t) is determined by the control voltage from Fig.3, and y(t) is calculated from Eq.6. If y(t) is unstable, the control output is halted and a message is sent to the optimum computer. The optimum computer will modify the tracking curve.

(12)

S is a feedforward weighting term, and a proper choice of S can eliminate the influence of e(t). To eliminate the steady-state error caused by e(t), let

S=qE(1).

(13)

An indirect robust self-tuning algorithm is adopted. Eq.5 can be written as

3.4. Self-tuning PID feedforward control of de-

creased vibration force

1)T0 + M(t).

(14)

The normalization signal n(t) is used to ensure a

If the operating mode is in a stable region, the decreased vibration force adaptive control module is used. As stiffness and eccentricity often change, the parameters of the system in Eq.5 will be changed. In order to design a PID structure controller, the model for adaptive control is selected as a secondorder model; Eq.5 is written as

A~(z-1)y(t) = B ~ ( z - 1 ) u ( u - 1)+ M ( t ) + qe(t)

y(t) : X ( t -

(9)

where Ar(z -1) = 1 + alz -1 + a2z -2, Br(z -1) =

bo + blz - 1 , A m = A - A r , B m = B - B~, M(t) = - A , ~ y ( t ) + B m u ( t - 1) is the unmodeled dynamics. Decreased vibration control is a regulation problem, so let the reference input w(t) = 0. The aim of the self-tuning controller is that of minimizing

bound on the unmodeled dynamic M(t)

o<,<,_~{x,(,),c},

.(t) ---- m a x

c > 0.

Xt Let X " ( t ) = x_~, e"(t) = y"(t) - X~(t)O~(t - 1),

M " ( t ) = M___~. Assume that the upper bound of M " ( t ) is defined as V,,ax > {Mn(t){. Use the leastsquares algorithm with relative dead zone Or(t) = Or(t - 1) + K(t)en(t)

(15)

K(t) = a ( t ) P ( t - 2)X~"(t - 1) [1+ X n ( t - 1 ) P ( t - 2 ) X n ( t - 1)] -1

(16)

P(t- I) P(t- 2) -g(Ox~(t- 1)P(t- 2)

(17)

--

where

J = [Py(* + 1) + Qu(t) + se(t)l 2 (x =

where P, Q, S are weighting terms. Along the same hnes as that of Yu and Chai (1994), the PID control law can be obtained from:

Q(z -1) : A-1(1 - z -1) - EBr P = EAr + z - l G G

:-

gO -t- yl Z - 1 + g2z - 2 .

(lO)

o

I e'(t) [~_ /~Vmax

e,~(t) e~(t)+aVm.x

Era(t ) < __f~Vma x

= [p +

1/(1

- ~)11/~

; > 0

From Eq.14-Eq.17 the lower model ,4r, Br are estimated, and u(t) is calculated from Eq.10 and Eq.11. The stability and convergency can be seen in the paper by Yu and ChM (1994).

Decreased Vibration Control for Centrifuges

1699

4. E X P E R I M E N T A L EVALUATION

The experimental equipment is shown in Fig.4. The centrifuge is accelerated to 2300r/rain. T w o kinds of graph will be used: 3D graph of vibration displacement relatedto COo and CO, and 2D graph of the time history of vibration displacement. In the 3D graph, the z orientation is the vibration displacement y(t), the z orientation is the rotation frequency CO, and the 9 orientation is the natural frequency co0. In the 2D graph, the z orientation is time t, and the ~ orientation is the vibration displacement

4 o (.0

(11 z )

al

¢~ --~--, •

~

' ;

:

-:

:,

,'

,

6 4

y(t). y (ram)

Fig.9 shows the results without any active control. The sub-resonance vibration appears at 63.7273.17Hz. This is the unstable vibration region. Using only changing stiffness control (Palaszolo and Jagannathan, 19931, the 3D graph during the starting period and the time history of the displacement at operating rotation speed are shown in Fig.10. The sub-resonance vibration almost disappears, but the first-order frequency vibration is very large, and the displacement in the operating rotation speed decreases to half that of Fig.9.

dl. 1~

a~,oO

-4.1~

i

-0.~6

0.625

1.250

1.615

.

.

2.50O

,LI25

3,750

4.37S

t

5.0DQ

(minute)

Fig. 9.3D graph and time history of displacement without any control

Using changing stiffness control during the starting period, and self-tuning feedforward PID control at operating rotation speed, the 3D graph and the time history of the displacement are shown in Fig.11. The sub-resonance vibration disappears and the first-order frequency vibration is very small. The displacement at operating rotation speed is smaller than in Fig.10.

'~0

{I/z)

d

:-~2--_--&

~

~

•

The results above demonstrate that the control scheme described in this paper is effective.

(J.,J

ao.o

40.o

~-;,- -. /

•

/,"'

¢,o.o

/

oo.o

"

,"

mo.o

/:~

14

CO ( l l z )

y (---)

5. C O N C L U S I O N For engineering mechanisms, decreased vibration control is very significant, especially for high-speed rotation equipment. This paper gives a new vibration control method. The main contribution of the paper is that a new hybrid active control scheme is proposed. As changing stiffness and decreased vibration force are combined, and some advanced techniques are effectively used, the vibration of centrifuges can be decreased using a selftuning PID. Theoretical analysis and experimental results demonstrate that this scheme is effective.

H,a~

4.L~

-u.a6 o

;.12s '~.~o

~75

~.500 ~.~2~ ;;,so

7.67G !

Fig. I0. Controlled with changing stiffness

(mjntlte)

9

Wen Yu et al.

1700 ( O 0 ( l ! z)

o o

~,.u

,*o.o

60.0

um,n

mo.o

(d ('llz)

y (mm) n .2a

I

-4. lo

-e,2~

.... o

1.150

3.~.90

5.Z50

t.I)Otl

8.150

1 tl.SgO

12.25

t

(minute)

Fig. 11. Controlled with changing stiffness and decreased vibration force

6. ACKNOWLEDGEMENT This work has been supported by the Natural Science Key Foundation of China. The authors wish to thank all the reviewers for their helpful comments and suggestions.

7. REFERENCES Burden, R.L. and J.D. Faired {1985), Numerical Analysis (third edition), Prindle, Webwe & Schmidt Publishers Goodwin, G.C. and K.S. Sin (1984), Adaptive Filtering Prediction and Control, Englewood Cliffs Palazzolo, A.B. and S. Jagannathan (1993}, Hybrid Active Control of Rotor bearing Systems Using Piezoelectric Actuators,Journal of Vibration and Acoustics, V o l . l l 5 , 111-119 SIEMENS Ltd. (1990), Process Control in Cold Rolling Mills with Mathematical Models, Shanghai Baoshan Iron and Steel Group Tech. Rep. Tzou, H.S. and M. Gade (1988}, Active Vibration Isolation by Polymeric Piezoelectric with Variable Feedback Gains, AIAA Journal, Vol.26, 1014-1017 Viderman, Z. and I. Porat {1987), An Optimal Control Method for Passage of a Flexible Rotor Through Resonances, Journal of Dynamic Systems, Measurement and Control, Vol.109, 216223 Wolf, J.A. (1968), Whirl Dynamics of a Rotor Partially Filled with Liquid,Journal of Applied Mechanics, Vol.35, 166-175 Yuan, Y. (1995), Study on Centrifuges Rotor Vibration, Ph.D. thesis, Northeastern University Yu, W. and T.Y. Chai (1994), Self-tuning Feedforward PID Controller, Proc. 33th IEEE Conf. Decision and Control, USA