Self organizing neural-network-based adaptive control for linear ultrasonic motor

2006 IEEEInternational Conferenceon Systems, Man, andCybernetics

October8-11, 2006, Taipei, Taiwan

Self-Organizing Neural-Network-Based Adaptive Control

for

Linear

Ultrasonic

Motor

Chun-Fei

Hsu,

Member,

IEEE,

and

Tsu-Tian Lee,

Fellow,

IEEE

Abstract-In this paper, an self-organizing

neural-network-based adaptive control (SONNAC) system is

developed. The SONNAC system is comprised of a neural controller and acompensation controller. The neural controller

utilizes aself-organizing neural network (SONN) to mimic an ideal controller, and the compensation controller is designed to compensate for the approximation error between the neural

controller and the ideal controller. When the approximation performance ofthe SONN isnot good enough, the SONN can create new neurons in the hidden layer to decrease the

approximationerror.Moreover,theadaptive laws of controller

parameters arederived in the sense of Lyapunov, so that the

stability ofthesystem can beguaranteed. Finally,toinvestigate

theeffectiveness oftheproposed SONNAC system, the design

methodologyisappliedtocontrol a linearultrasonic motor.

I. INTRODUCTION

T HEcomputed torque orinverse dynamics technique is a l specialapplication of feedback nonlinearsystems. These

design methods are based on a good understanding of the controlled system dynamics and its environment; however, it isunpracticaltopreciselymodelacomplex nonlinear system. To tackle this problem, the neural-network-based control techniques havepresented the alternative design approaches

for the uncertain nonlinear systems [1-5]. Thesuccessful key element istheapproximationtheory, where theparameterized

neural network can approximate the unknown system

dynamics ortheideal controllerafterlearning. Someof these learning algorithms are based on the backpropagation

algorithm. However, these approaches are difficult to guarantee thestability and robustness of closed-loop system. To overcomethis difficulty, someof the learningalgorithms arebasedontheLyapunovstabilitytheorem. Thetuning laws of the neural network have been designed to guarantee the systemstabilityintheLyapunovsense.

Although the control performances are acceptable in [1-5], the learning algorithm only includes the parameter

learning, andthey havenotconsidered thestructure learning

of the neural network.If thenumber ofneuronsin thehidden layer is chosentoolarge,thecomputation loadingisheavyso

that they are not suitable for practical applications. If the

Manuscript received December25,2005.This workwassupportedinpart

bythe National ScienceCouncil of theRepublicofChina underGrantNSC 93-2213-E-155-038.

Chun-Fei Hsu is with the Department of Electrical and Control Engineering, National Chiao-Tung University, Hsinchu 300, Taiwan,

RepublicofChina(e-mail:[email protected]).

Tsu-TianLeeiswith theDepartment of ElectricalEngineering,National Taipei UniversityofTechnology, Taipei 106,Taiwan, Republic of China (e-mail:[email protected]).

numberofneurons in the hidden layer is chosen too small, the learning performance may be not good enough to achieve desired control performance. To tackle this problem, several self-constructing neural networks,consisting of structure and parameterlearning phases, have been proposed [6-8]. These two-phase learning algorithms not only decide the structure ofneural network but also adjust the parameters of neural network. Recently, some self-constructing neural networks have been applied to solve several control problems. However, some of them can not guarantee the system stability; and some of them require too complex design proceduretoachievesatisfactoryperformance.

Modern mechanical systems, such as machine tools and automatic inspection machines, often require high-speed high-accuracy linear motions. These linear motions are usually realized using the rotary motors with a mechanical transmission, such as reduction gears and lead screw. These transmission mechanisms not only significantly reduce the linearmotion speed and dynamic response, but also introduce the backlash and large friction. To tackle this problem, a linear ultrasonic motor (LUSM) is introduced to apply the linear motion without using any mechanical transmission. The LUSM has much merit, such as high precision, fast control dynamics, large driving force, smaller dimension, highholdingforce, silence andmore minimumstep size than the class electromagnetic motors, so that it can be used in many different applications [9]. However, the driving principle of the LUSM is based on the ultrasonic vibration force of piezoelectric ceramic elements and mechanical frictional force. Therefore, its mathematical model is

complex, andthemotorparametersaretime-varying because of increasing temperature and changes in motor drive operating conditions. For control system designs, the conventional controltechnologiesarealways basedonagood

understanding of the controlled system; however,theLUSM

dynamic model is difficult to obtain. Therefore, it is very difficulttocontrol theLUSMusing the conventional control theory. Totackle thisproblem,somedesign techniques have beenadopted for the LUSMcontrol [10-12]; however, these

design procedures are overly complex or may cause large

chatteringinthe controlefforts which will wear the bearing mechanismand exciteunmodelleddynamics.

The motivation ofthispaperis todesigna self-organizing neural-network-based adaptive control (SONNAC) system for the LUSM. The SONNACsystemiscomprisedofaneural controller and a compensation controller. The neural controller uses a self-organizing neural network(SONN) to

isdesignedto recoverthe residual of theapproximationerror.

Thelearning phase of SONNAC includesa structurelearning

andaparameterlearning. Inthestructure learning phase,the developed SONN can on-line create new neurons in the hidden layer asthe approximation performance isnot good

enough, thus the learning capability and flexibility can be upgraded. In the parameter learning phase, the controller parameters can be on-line tuned based on the Lyapunov

function, sothatthestability oftheclosed-loop systemcanbe guaranteed. Finally, theexperiment results areperformedto

demonstrate the effectiveness of the proposed SONNAC design method for LUSM.

II. MODELINGOF LINEARULTRASONIC MOTOR The structure of the LUSM is a large face ofa relatively

thin rectangular piezoelectric ceramic device. The driving

principles of the LUSMarebasedonthe ultrasonic vibration force of piezoelectric ceramic element and mechanical frictional force. Figure 1 shows theprincipalstructureofthe LUSM considered in this study [9]. The stator vibrator is fitted withbending and longitudinalpiezoelectric actuators.

They are driven by two electrical sources of identical frequency, but with a phase difference that is carefully

controlled. Atthe vibration tip, an elliptical motion is thus created, resultant of the elliptical and longitudinal motion. The bending actuators convert a large electrical power to

mechanicaloutput andthelongitudinal actuatordynamically

changes the force along thepre-load directionto adjust the frictional force betweenthe stator and therotor. Friction is inevitableintheLUSM. Itisahighly complicatedprocessto

attempt to build an explicit mathematical friction model for the LUSM because friction plays a dual role: it does not

simply contributetothenonlineardynamics (e.g.,deadzone)

ofthe LUSM, but it also serves as thedriving force for the movingpart. Therefore, theLUSMdynamic equationisvery complicated and theparameters ofthe elements arenoteasy

toknow.

For developing the control law, the LUSM can be describedas a second-order nonlinear dynamic equation by the Newton's Lawas

[M+m]x=F(x)+G(x)u (

where M is themass ofthemoving table; m is themassof the payload; x=[x

x]4

represents the position and velocity of themoving table; F(x) is the nonlineardynamic function including friction, ripple force and external disturbance; G(x) is thegainofthe LUSMresonantinverter;and u is the inputforcetothe LUSM.Rewriting(1),thedynamic equation of the LUSMcanbe obtained as

F(x) + G(x) u

M+m M+m

=

f

(x)

+

g(x)u

(2)

F(x) G__x_

where f(x) and

g(x)(x)

x) Since these

M+m M+m

LUSM poses an interesting and challenging dilemma to the control problem. longitudinal piezoelectric actuator 'vibration tip - bending piezoelectric actuator

Fig. 1 Structureofthepiezoelectric-type linear ultrasonic motor.

III. IDEALCONTROL

The tracking control problem ofthe LUSM system is to find a control law so that the state trajectory x can track a reference command xc.Thetrackingerroris definedas

e=x -x (3)

andasliding surface is definedas

s-e +ke+k2 |e(r)dr (4)

where

k,

and k2 are non-zero positive constants. If the system dynamics areexactly known, anideal controller can bedesignedas

eq hU

wheretheequivalent controller ueq isrepresentedas

1

Ueq g(=x

(-f

(x)+?K +kIe+k,e)

andthehitting controller uhi isdesignedas Uh

=-I

sgn(s)

g(x)

(5)

(6)

(7)

where q is a positive constant and

sgn(.)

is the sign function. Substituting(5), (6) and (7) into (2),yields

s=-isgn(s). (8)

Inordertodrive the s -X 0, consider theLyapunovfunction candidate in thefollowing form

I1.

V,

(s)= --S(9)

2

Differentiating (9) withrespect totimeandusing(8),yields

V,(s)=ss

=-7IsI

<O. (10)

In summary, the ideal control system presented in (5) can guarantee the system stabilityin the sense oftheLyapunov. However, if the system dynamics

f(x)

and g(x) are unknown or perturbed, the ideal control cannot be implemented. Thus, a model-free control technology, which is termed as self-organizing neural-network-based adaptive control (SONNAC), is proposed in the following section to achieve desiredtrackingperformance.

IV. SELF-ORGANIZINGNEURAL-NETWORK-BASED ADAPTIVE CONTROL

The proposed SONNAC system is comprised of a neural controller and a compensation controller. The neural controller uses a SONN, in which the neurons inthehidden layer can on-line split up as theapproximation performance is not good enough, to mimic an ideal controller. The compensation controller is designed to compensate for the approximation error between the neural controller and the ideal controller.

A. DescriptionofSelf-OrganizingNeuralNetwork

A single-hidden-layer SONN is shown in Fig. 2. The outputof this SONN takes the form

yn

y= wI-a(viS)

i=l

(11)

variation of the weights. As shown in Fig. 3, the k-th neuron satisfying (14) is divided into two neurons, and the newly created neuron is denoted by k'-thneuron. The new weights connected to the k'-thneuron aredecided as follows

Vk,(N+1)= Vk(N) (15)

wk(N+1)=aWk(N) (16)

where a is a positiveconstant. The weights connected to the k-th neuron aredetermined as follows

Vk(N+1)=Vk(N) (17)

Wk

(N

+

1)

=

(1-ca)

wk

(N).

(18)

This method is induced from the facts that the weights connected to the newly created neuron will share the large variations of theweights.

where s and y are the input and output of the SONN, respectively, a represents the hidden-layer activation function, v; is the interconnection weight between the input and hidden layers, and Wj is the interconnection weight between the hidden and outputlayers. Theseweights will be on-line adjusted in the following derivation. The activation function in this paper isconsideredas asigmoid function

a

(I)

+e)(12)

Bycollecting all theweights of the SONN, equation (1 1) can beexpressedina vectorformas

y=

WT((Vs)

(13)

where V=

[vl,

v2 ,...,v,E R and W=

[wl,

w2

,...,w]E

R' A main property of neural network regarding feedback control purpose is the universal function approximation property. Ingeneral, the approximation error decreases as the net size m increases. In general, the number of neurons in hidden layer should be determined by trial-and-error to achieve favorableapproximation.

To tackle this problem, this paper proposed a simple

disjunctionalgorithm such that the k-thneuronin the hidden layer splits up at the N-th sampling time, if the following condition is satisfied

nk

I|

+IWI

01

k=1,2,...,m

i=l

(14)

where 0 denotes a disjunction threshold value. When the approximated nonlinear functions are too complex, the disjunction threshold value should be chosenas asmall value sothattheneurons canbeeasilycreated. Thetuning laws v; and i4 willbe derivedin thenextsubsection. The proposed

disjunction algorithmis derivedfrom the observation that if the left hand side of (14) is larger than the disjunction

threshold value, which implies theprecise approximation is hard to capturebecause theupdating of theweightvalues is relativelylarge. Forthatreason, ifcondition(14)is satisfied,

then a new neuron is created to spread the relatively large

s

Fig.2.Thestructure ofself-organizing neural network.

s Vk kh Wk

Ifcondition (14) issatisfied

t O(1 a)wk.

Vk kt

S

Fig. 3. Schematic illustration of thedisjunctionalgorithm. B. DesignofSONNAC

The developed SONNAC system is shown in Fig. 4; the controller is comprised of a neural controller and a

compensation controller

U=Un, + Uc,p. (19)

Theneural controller unn is utilizedtoapproximatethe ideal controller and thecompensationcontroller

u1,t

isdesignedto compensate for theapproximation errorbetween the neural controller and the ideal controller. Substituting (19) into(1),

yields

Usi= (5)ad (20u) , +ueqai).

(20)

S=

g(x)(u

-

ut7tZ

-

U,p

-q

sgn(s))

.(21

)

By the universal approximation theorem, an optimal

approximator can be designed to approximate the ideal controller, suchthat

u =u +A=

WaTJ(V*S)

+A =W 6 +A (22)

where V* and W* aretheoptimal vectors of SONN and A

isthe approximationerror.Letthenumber ofoptimalneurons

be

nW

and theneuronsbedivided intotwoparts.The first part contains n neurons which are the activated part and the secondary part contains

nW

-n neurons which do not exist yet. Thus, the optimal weights of NN

V*

and W* are

classified intwopartssuchas

V

=-[v" v.*]

and W* = a

(23)

where

V.

E R" and

W.'

E R are activated parts, and

V*

E R( -n) and

W*

E R(n* are inactivated parts,

respectively. Since theoptimalNN

weights

thatareneededto

best approximate the ideal controller are

unobtainable,

an

estimation of

unn

isgivenby

uM

=W (VaS)=Waa (24) where Va and Wa are the estimated values of the

optimal

weights V and W ,

respectively.

Definetheestimatederror

u as

U=WU-U(F *(F

= W 6+ W 6. -W 6a +A

=

WT>

₌ a+Wai+WTf +

W:*T(+A

(25)

a6a + a6a + a6a +W6i +i(5

where 6a =u(V x) , x=J(VsX) X a=aa

WV

=W*

-W and V =Va

^V

. The

Taylor

series

expansionof

(Y'

with respectto Vas canbe derivedas

(0 =( +

60V

s+h

(26)

where

(o'

is the Jacobian and h is a vector of

higher-order

terms. Therefore,

a

=(,'V>

S+h. (27) Substituting (27)into(25), yields

U =WT

OSa

+

W"

'o'V S+E

(28)

where £=

WTia +W.aT[

+

W:h+A

is assumed to be bounded by

lel

<E where E is a positive constant. However, the bound ofapproximation error Eis difficult to measure in practical applications. Thus, using the approximation error equation (28), equation (29) can be rewritten as

s(t)

-

g(x)(WfaTa+

sVa

aW +

-uC

-1

sgn(s))

.

(29)

Inthisderivation, W

ToaVa

s=[WTa'Va s]T is used since it is

Theorem 1. Consider a linear ultrasonic motor system expressed by (1). If theself-organizingneural-network-based adaptive control system isdesignedas(19). Aself-organizing

neural network splits up the neurons in the hidden layer if condition (14) is satisfied. The adaptation laws of activated parts Va and Wa aredesignedas

V =_-V =77,S2g(X)(T Wa Wa= -Wa =77,s

9(x)ka

(30) (31) where q, and

q7,

are the positive learning-rates. The compensation controller isdesignedas

uCP

=Esgn(s) (32)

where E is the estimation of theapproximation errorbound with the estimation lawgivenas

E=-E=

rEs

g(x)l

(33)

where

qE

isapositivelearning-rate. Then,thestabilityofthe

adaptive self-structuring neural network control system can beguaranteed.

Proof:

Define a Lyapunov function candidate in the following

form

V,(S, VW,E)=-2+ VTV + WTW + 1 E' .(34)

a

a 2 277 a a 2q7 a a 77

~~~~~~

A

where E=E-F. Differentiating(34)with respecttotime

andusing (29),and(30)-(33), gives

VTV WTW EE

V(s,Va,

Wa,

E)=ss+ a+ a a

+_

-

qg(x)lsl

+esg(x)-

Elsg(x)l

<

-(E

-

ej)jsg(x)l

<

0 .

(35)

Ifthe system dynamic g(x) is unavailable, the g(x) in the adaptive algorithms canbe reorganized as

lg(x)|sgn(g(x))

in practical applications. Therefore, the learning algorithms

for the SONNAC shown in (30), (31) and (33) can be reconstructedasfollows: (36) Va =)/1S SOT'IW a aa E=

-E

=

qmIsI

(37) (38) in which hi = 7

1g(x)l,

m2

='

wg(x)l

and

i73

=

'7E

lg(x)l

The

)1,,

172

and

q3

canbe takenasnewlearning-rates.

a scale. Therefore, thefollowing theorem can be stated and

Theexperimental results offixed-structured neural network controlareshowninFig.6. Thetrackingresponsesareshown

inFigs. 6(a)and 6(c); and the torque input areshown in Figs.

6(b) and 6(d) for 7 and 20 hidden neurons, respectively. Experimental results show that the robust tracking performance has beenachieved for different hidden neurons. If the number ofhidden neurons is chosen too small, the convergence ofthe tracking error is slow. On the other hand, if the number of hidden neurons is chosen too large, the computation loading is heavy.

(a)

1,

ControlEffort'

Fig. 4. SONNAC for linear ultrasonicmotorsystem.

A. V. EXPERIMENTAL RESULTS

The computercontrol experimental system for the LUSM is shown in Fig. 5. A servo control card is installed in the control computer, which included multi-channels of D/A,

A/D, PIO and encoder interface circuits. Thepositionof the

movingtable is feedback usingalinear scale. Theproposed

SONNAC system isrealized in the Pentiumusingthe"Turbo C"language. The control interval ofthe control systemare set as2ms. Thecontrolobjective is tocontrol the movingtable to follows a 0.035m periodic step command. Moreover, a second-order transfer function 64 ischosenasthe

s2+16s+64 referencemodel for the step command.

I5A _Isec (b) Trajectory Command (c) t o. . 'Control Effort'.: OA ... I5A ; lsec:

LIre

oerectric

CeramicMotor

Fig.5Computer-controlledlinearultrasonicmotorsystem. To illustrate the effectiveness of the proposed design

method, a comparison between a fixed-structured neural network control in [5] and the proposedSONNAC is made.

(d)

Fig. 6Experimental results of fixed-structured neural networkcontrol.

Then, the developed SONNAC is applied to controlthe LUSM. Theparameters intheproposed SONNC systemare selected as

k,

=2 , k2 =I 5,i =50, q2 =50 , 3 =0.1 v a=0.3, and = 0.5. Theexperimentalresultsof SONNAC

areshowninFig. 7. Thetracking response is shown inFigs.

7(a); the torqueinputis shown inFigs. 7(b), and the number ofthe hidden node is shown in Fig. 7(c). It shows thatthe

favorable

tracking performance

can be achieved for the

proposed

SONNACafter thestructureandparameter

learning

phases.

Moreover,

the SONNAC can on-line create new neurons in the hidden

layer

as the

approximation

ofneural network isnot

good enough.

le

Isec

are:

1)

the successful

development

ofan

SONNAC,

inwhich the

Lyapunov

stability

theorem is usedto derive the on-line

learning

algorithms.

2)

the

self-structuring

neural network has been created with easy

split-up algorithm

of hidden

neurons to achieve favorable

learning

performance. 3)

the successful

application

ofthe SONNAC to control a linear

ultrasonicmotor.

AcKNoWLEDGMENT

The authors

appreciate

the

partial

financial support from National Science Council of the

Republic

of China under Grant NSC 93-22

13-E-i155-038.

Control Effort' A. (b) Number:of Neuro Isec - 11 .0 _{I10 Isec} (c)

Fig.

7

Experimnental

results of SONNAC.

VI. CONCLUSIONS

This paper

developed

a

self-organizing

neural-network-based

adaptive

control

(SONNAC)

system,

which is

comprised

ofaneuralcontroller anda

compensation

controller. In the neural controller

design,

a

self-organizing

neuralnetwork is utilizedtomimic anidealcontroller. Inthe

compensation

controller

design,

an error estimation

mechanism is

investigated

to estimate the bound of

approximation

error. The

major

contributions ofthis paper

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