Position Control of a PM Stepper MotorUsing Neural Networks

Gang FengSchool of Electrical Engineering, University of New South Wales

Sydney, NSW 2052, AustraliaDept. of MEEM, City University of Hong Kong

Tat Chee Ave., Kowloon, Hong KongEmail: megfeng@cityu.edu.hk

AbstractThis paper considers position control of a PM steppermotor. A new control scheme is proposed based on a kindof exact linearization controller and a neural networkbased compensating controller. This scheme takesadvantages of simplicity of the model based controlapproach and uses the neural network controller tocompensate for the motor modeling uncertainties. Theneural network is trained on line based on Lyapunovtheory and thus its convergence is guaranteed.

1. Introduction

Due to the various disadvantages of DC motors,positioning systems are more and more implemented byusing induction motors or stepper motors. Originally,stepper motors were designed to provide precisepositioning control within an integer number of stepswithout using any sensors. This is an open-loop operation.However, using the stepper motors in an open-loopconfiguration results in very low performance. Inparticular, the PM stepper motors have a step responsewith significant overshoot and a long settling time.Therefore, feedback control has been proposed for steppermotor position systems. A number of feedback controltechniques have been developed during the last few years.One of them is the so-called feedback linearizationcontroller [1-3]. The basic idea is to design a feedbackcontroller so that the nonlinear stepper motor systembecomes a linear system. Then the techniques of linearcontrol systems can be used to design the controller for thelinearized stepper motor system so that the requiredperformance can be achieved.

It is understandable that such a control technique iscapable of providing excellent positioning results if thecomplete dynamics of the stepper motor are known.However it actually results in no better performance thanthat of the conventional fixed gain controllers due to thefact that it is very hard to obtain perfect dynamic model forthe stepper motors in practice. Moreover, there also existuncertainties, as well as time-varying effects in practice.All these factors lead to the difficulty of achieving higherprecision positioning performance.

Recently, increasing attention has been paid to the use ofartificial neural networks in nonlinear control [4-6]. One

possible way to use the neural networks is to replace eitherthe entire control system or the feedforward controller withneural networks. Such research work was based on thedesire to obtain the benefits of model-based controlwithout a priori knowledge of system dynamics. However,in many cases, the nominal dynamic model of the steppermotor can be found a priori indeed. Therefore, a prioriknowledge of stepper motor dynamic model should beappropriately used rather than totally discarded.

In this paper, a new scheme for stepper motor positioningcontrol is proposed. This scheme takes advantage ofsimplicity of the conventional methods, and incorporates acompensating controller to achieve high positioningperformance. The compensating controller is based on anRBF neural network, which is trained on-line to identifythe stepper motor modeling uncertainties.

The rest of the paper is organized as follows. Section 2presents the problem formulation and Section 3 proposes anew control scheme which is followed by some concludingremarks in section 4.

2. Problem formulation

The equations describing the stepper motor can be given asbelow [7],

LNKRivdt

dirmaa

a /)]sin([ +=

LNKRivdtdi

rmbbb /)]cos([ = (1)

JJNK

GNiKNiKdtd

LrD

rbmram

//)]4sin()cos()sin([

+=

=

dtd

where ba ii , and ba vv , are the currents and voltages inphase A and B respectively, L and R are the self-inductance and resistance of each phase winding, mK isthe motor torque constant, rN is the number of rotorteeth, J is the rotor inertia, G is the viscous frictionconstant, is the rotor speed, is the motor position, Lis the load torque, and the term )4sin( rD NK represents

the detent torque due to the permanent rotor magnetinteracting with the magnetic material of the stator poles.

As shown in [1], if the modeling is perfect, with anappropriate nonlinear coordinate transformation and exactfeedback linearization, the stepper motor equations can berewritten as,

BuAxx += (2)where

=

0000000001000010

A

=

10010000

B

3321 /][:][ Kxxx = ,JKK m /:,: 3 == ,

and 4x is related with the maximum rated phase current.Then a possible linear control law for u can be chosen as,

+

=

0)(

0000

4

33

22

11

40

323130 tr

x

xx

xx

xx

a

aaau

d

d

d

(3)

where 3321 /][:][ Kxxx dddddd = , ddd ,, arethe desired position, speed and acceleration profilesrespectively, and 3/)( Ktr d= . It can be seen that withthe above controller, the closed loop system can beexpressed as,

xAx c ~~ = (4)where

[ ]Tddd xxxxxxxx 4332211:~ =

=

40

323130

000001000010

a

aaaAc .

Therefore, the error dynamics of the positioning controlsystem can be well designed by suitably choosing thecoefficients 323130 ,, aaa and 40a .

However, if the modeling is not perfect or there existuncertainties which is always the case in practice, then theabove design might lead to the significant degrade of theperformance. In this case, we propose a new controlscheme which includes a compensating controller. It issupposed that the nominal model of the stepper motor isknown a priori. Then the feedback linearization controldiscussed above can be designed based on this nominalmodel. However, in this case, the stepper motor controlsystem equation will not be expressed as in eqn.(2).Instead, there will exist an uncertainty term f(x) in theequation, which will be the unknown nonlinear function of

the motor state variables. It is supposed that the uncertaintyterm satisfies the matching condition, that is, the term is inthe range of the control signal u(t). In such a case, thesystem can be expressed as,

)]([ xfuBAxx ++= (5)

It can be clearly seen that the performance of the steppermotor position control system will degrade due to thisuncertainty term f(x). Our objective is to design acompensating controller to improve the stepper motorpositioning performance. If the nonlinear function f(x)were known a priori, then a modified controller

co uuu += (6)

+

=

0)(

0000

4

33

22

11

40

323130 tr

x

xx

xx

xx

a

aaau

d

d

d

o (7)

)(xfuc = (8)would lead to the closed loop system expressed as ineqn.(4), i.e., the known modeling uncertainties could bewell compensated.

Unfortunately, the non-linear function f(x) is unknown apriori in practice. Therefore the above modified controllercould not be implemented. However, this controllersuggests indeed that a well estimated function )( xf of thenon-linear function f(x) could be used to improve thestepper motor positioning performance. It is noted thatthere exist another nonlinear function )~(xh such that

)~()( xhxf = by taking notice of the definition of the statevariable x~ . With the same control law (6), (7) and a newcompensating control law, which will be discussedsubsequently, the closed loop stepper motor control systemcan be expressed as, )]~([~~ xhuBxAx cc ++= (9)

Due to their great approximation capability, artificialneural networks will be used in this paper to identify thisnon-linear function [8-10]. For this we make the followingassumptions.

Assumption 1: The closed loop stepper motor controlsystem, whose controller is designed based on the nominalstepper motor model, is stable, and the tracking errorvector x~ belongs to a compact set.

Assumption 2:(i) Given a positive constant 0 and a continuous

function RCh : , where rmRC is acompact set, there exists a weight vector = *such that the output ),~( xh of the neuralnetwork architecture with n* nodes satisfies

0*

~

|)~(),~(|max

xhxhCx

,

where n* may depend on precision parameter0 and the function h.

(ii) The output ),~( xh of the neural networkarchitecture is continuous with respect to itsarguments for all finite ( ,~x ).

Remark 1: This paper is mainly concerned with thecompensating controller design for positioningperformance, thus it is reasonable to assume the controlsystem based on the nominal model is stable in theassumption 1. Assumption 2 is also reasonable due to theuniversal function approximation capability of the neuralnetworks.

3. A new control schemeSuppose the unknown nonlinear function h( x~ ) beparameterized by a static RBF neural network with output

),~( xh , where *nR is the adjustable weight, and ndenotes the number of weights in the neural networkapproximation. Then the eqn.(2) can be rewritten as

))],~()~((),~([~~ ** xhxhxhuBxAx cc +++= (10)where * denotes the optimal weight values in theapproximation for x~ belonging to a compact set

nx RMC

2)( , that is }||~||:~{:)( xx MxxMC = . Ingeneral, the "optimal" weight * in eqn.(10) could takearbitrarily large values. However, in order to avoid anynumerical problems that may arise due to too large weightsand to prevent the weights from drifting to infinity, we areonly concerned with weights that belong to a largecompact set )( MB , where M is a design constant, and

}||||:{:)( MMB = denotes a ball of radius M . Inthe design of adaptive law, we also restrict the estimate of* to the compact set )( MB through the use of aprojection approach. In this way, the optimal weight * isdefined as the element in )( MB that minimizes thefunction |),~()~(|| xhxh | for Cx ~ , i.e.

||}),~()~(||sup{minarg:)(~)(

*

xhxhxMCxMB

=

Now eqn.(10) can be expressed as)]~(),~([~~ * xxhuBxAx cc +++= (11)

where denotes the error due to the use of the neuralnetwork,

),~()~(:)~( * xhxhx = (12)The error is bounded by a finite constant vector 0 ,where

||),~()~(||sup: *~

0 xhxhCx

=

According to the properties of the RBF neural networks,the function ),~( *xh can be expressed in the form

)~(),~( ** xxh T = (13)where * is a matrix representing the optimal weightvalues subject to the constraints ||*|| M, the vector field

*)~( nRx which is refereed to regressor, is Gaussian typeof functions defined element-wise as

)|~|

exp()~( 22

i

icxx

= , i = 1, 2, .., n*.

For the sake of tractable analysis, ic and i are chosen apriori and kept fixed. The local training techniquespresented in [11] could be used for appropriately choosingthe centres and widths of the RBF neural network. Or thecentres can be chosen as the mesh points equallydistributed inside the compact set. In such case, the onlyadjustable parameters appear linearly with respect to theknown nonlinearity (x).

Now, eqn.(11) can be rewritten as)]~()~([~~ * xxuBxAx Tcc +++= (14)

Next, we will show that an adaptive compensatingcontroller can be added to achieve improved positioningperformance. The new compensating control law is asfollows:

)~( xu Tc = (15)where is the estimated parameter for *.

This control law leads to the closed loop system expressedas

)]~()~()~([~~ * xxxBxAx TTc ++= = )]~()~(~[~ xxBxA Tc ++ (16)

where *:~ = denotes the parameter estimation error.

Choosing a Lyapunov function candidate2||~||

21

~~

21

FT xPxV

+= > 0 (17)

where 2|||| F denotes the Frobenius matrix norm, definedas = ij ijF rR 22 ||:|||| , and the matrix P is positive definiteand satisfies the following Lyapunov equation

QPAPA Tcc =+ (18)with Q 0.

Taking the time derivative of V along the trajectories ofeqn.(16), we have

)~~(1]~~~~[21

TTT trxPxxPxV ++=

)~(1~])~(~[]~)(~[21

TTTTTccT trxPBxxPAPAx ++++=

)~(1~~~)~(]~~[21

TTTTTT trxPBxPBxxQx ++=

(19)Noting that

)~)~(~(~~)~( TTTT xxPBtrxPBx =

if we choose the parameter update law as

)~(~

~)~( 20 MxPBx

cPBxxT

T=

, (20)

>==

otherwisePBxandMif

cTTT

00~||||1

0 (21)

It can be easily verified that if the initial parameters arechosen to be inside the ball, i.e., || )0( ||:={tr[ )0( T )0( ]}1/2 M, then we have || )( t || M forall t 0. Then we have

xPBxxPBtrxQxV TTTTTT ~)~~)~(~(1~~21

+++=

xPBPBxxtrxQx TTTTT ~]~)~)~([1~~21

+++=

)~~

(~~~21

20

TTT

TTT

MPBx

trcxPBxQx += (22)

Now let's have a look at the last term in the aboveequation,

0

)(~

)~(~

)~~

(

*

20

20

20

=

=

TTTT

TTT

TTT

trMPBx

c

trMPBx

c

MPBx

trc

Actually, it is noted that when || || < M or || || = M and0~ TT PBx , then c0 = 0, the above inequality is trivial;

when || || = M and 0~ > TT PBx , then, due to the factthat ||*|| M, we can have 0)( * TT , therefore,the above inequality is also true. In other words, theprojection will not make the derivative of the Lyapunovtype function more positive. Therefore we have

xPBxQxV TTT ~~~21 +

)](22||~||)([||~||21

||~||||||)(||~||)(21

max0min

max02

min

PxQx

xBPxQ

=

+(23)

where max(P) and mix(Q) denote the maximumeigenvalue of matrix P and the minimum eigenvalue ofmatrix Q respectively.

Consequently, it can be concluded from the eqn.(23) thatthe system is convergent and tracking error x~ will belong

to a residue of radius R0 = 0 with )()(22

:min

max

QP

= .

This implies that the positioning error will also belong tothe residue.

4. Conclusions

A new stepper motor control scheme is developed in thispaper. The proposed scheme consists of a conventionalfeedback linearization controller, which is based on theknown nominal stepper motor dynamics model, and acompensating controller, which is based on the RBF neuralnetwork. The compensating controller is used to improvethe stepper motor positioning performance. The neuralnetwork is trained on-line based on Lyapunov theory andlearning convergence is thus guaranteed.

References

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