Observer-based controller for position regulation of stepping motor

Observer-based controller for position regulation ofstepping motor

J. De Leon-Morales, R. Castro-Linares and O. Huerta Guevara

Abstract: The design of a controller-observer scheme for the exponential stabilisation of apermanent magnet stepper motor is proposed. The technique is based on sliding-mode techniquesand nonlinear observers. Representing the stepper motor model as a singularly perturbed nonlinearsystem, a position regulation controller is obtained. Since this controller depends on the mechanicalvariables, load torque and equilibrium point, under the assumption that the rotor position isavailable for measurement, an observer design is presented to estimate the angular speed and loadtorque. Furthermore, a stability analysis of the closed-loop system is also made to provide sufficientconditions for the exponential stability of the full-order closed-loop system when the angular speedand load torque are estimated by means of the observer. The proposed scheme is applied to themodel of a permanent-magnet stepper motor.

1 Introduction

Dynamic models obtained from theoretical considerationsare frequently so complex that they may be impractical forcontrol design purposes. Thus, several methods have beensuggested in the literature for deriving reduced-orderdynamic models from high-order models. For example, theintegral manifold approach is sometimes used systematicallyto create adequatemodels of synchronousmachines. Anothermethod used formodel reduction of large-scale systems is thesingular perturbation method (see for example [1–3] and theReferences therein). This technique has also been used tostudy robustness in the presence of parasitic or unmodelleddynamics. Some of the advantages of the singular pertur-bation method are its applicability to nonlinear systems aswell as its simplicity and good performance inmany practicalcontrol situations. When the singular perturbation method isapplied, the original system is decomposed into twosubsystems of lower dimension, both described in differenttime scales. From this decomposition, a state feedback maybe designed for each lower order subsystem combining themin a so-called composite feedback that is applied to theoriginal system. In [4] and [5], an excellent description of thesingular perturbation method is given.On the other hand, sliding-mode control techniques have

been extensively used when a robust control scheme isrequired; this is when dealing with systems that haveuncertainties due to modelling errors and disturbancesignals [6–8]. Moreover, a sliding controller is character-ised as a high-speed switching controller that provides

a robust means of controlling nonlinear systems by forcingthe trajectories to reach a sliding manifold in finite time andstay on the manifold for all time. Owing to the switchingbehaviour of the controller some theoretical and practicalproblems rise. Thus, the idea of combining the singularperturbation method and the sliding-mode control techniquerepresents a good possibility of achieving classical controlobjectives for nonlinear systems having unmodelled orparasitic dynamics and parametric uncertainties.

Generally, the controller resulting from the use ofsingular perturbation methods and sliding-mode techniquesneeds, however, to have information about the state vectorof the plant. This could be possible if adequate sensors areavailable, but in most cases such sensors, if they exist, areexpensive. The reduction of the number of sensors is animportant problem for industrial applications. The sensorscontribute to an increase in the complexity of the machineryand the cost of the installation. Therefore, it is necessary toestimate the states of the system by using state observers.The design of observers for nonlinear systems is a quiteinteresting research issue but, in general, a difficult one. Todate, several methods have been suggested for the design ofthe observers. In [9], an excellent survey is made of differentapproaches proposed for such designs.

Permanent-magnet (PM) stepper motors are electromag-netic incremental motion devices that are very useful inindustrial and research laboratory applications. Thesedevices were originally designed to provided precisepositioning control since they are open-loop stable to anystep position and no feedback is needed to control them ifthe load torque in the rotor is greater than the detent torque.However, they have a step response with overshoot andrelatively long setting time. Besides, loss of synchronyappears when steps of high frequency are given. It is thusnecessary to develop control schemes to improve theperformance of stepper motors. Feedback control methodsfor this devices are difficult to implement because they arehighly nonlinear systems and it is expensive to haveaccurate measurement of some of its variables.

This paper reports the design of an observer-basedcontroller for exponential stabilisation of a stepper motorcombining the advantages of the singular perturbationmethods and sliding-mode techniques by means of

q IEE, 2005

IEE Proceedings online no. 20045066

doi: 10.1049/ip-cta:20045066

J. De Leon-Morales and O. Huerta Guevara are with the UniversidadAutonoma de Nuevo Leon, Facultad de Ingenierıa Electrica y Mecanica,Apdo. Postal 148-F, 66451 San Nicolas de Los Garza; NL Mexico

R. Castro-Linares is with the CINVESTAV-IPN, Depto. de IngenierıaElectrica, Apdo. Postal 14-700, 07000 Mexico DF, Mexico,

E-mail: [email protected]

Paper first received 30th May 2004 and in revised form 13th January 2005.Originally published online 8th June 2005

IEE Proc.-Control Theory Appl., Vol. 152, No. 4, July 2005 465

a nonlinear observer. Also, a study of the stability propertiesof the resultant closed-loop system is presented when theslow state is replaced by its estimate. Furthermore, acomparative study is included in which the performance ofthe proposed methodology is shown. An extension of thismethodology is presented considering a sophisticated modelof stepper motor.

2 Model and problem description

Permanent-magnet stepper motors are incremental-motiondevices that convert digital pulse inputs to analogue outputmotion. The PM stepper motor basically consists of a rotorand a stator. The rotor features two axially slotted cylindersdisplaced by half a slot or ‘tooth’; one of the cylinders or‘gears’ is a permanent north magnet. The stator, which isalso slotted, has a different number of teeth than the rotorgears, so that the rotor will never be aligned with the statorteeth. Each slot in the stator is an electromagnet, which canbe alternatively made north or south (for more details see[10, 11]).

The mathematical model for a PM stepper motor is givenby the following equations (see [11] for a detailedexplanation and derivation of the model; in the Appendixa more sophisticated model is considered):

diadt

¼ 1

L½va � Ria þ Kmo sinðNryÞ�

dibdt

¼ 1

L½vb � Rib � Kmo cosðNryÞ�

dodt

¼ 1

J½�Kmia sinðNryÞ þ Kmib cosðNryÞ

� Bo� Kd sinð4NryÞ � tl�dydt

¼o

ð1Þ

where ia; ib; and va; vb are the currents and voltages inphases a and b, respectively, o is the rotor angular speedand y is the rotor angular position. L and R are the self-inductance and resistance of each phase winding, Km is themotor torque constant, Nr is the number of rotor teeth, J isthe rotor inertia, B is the viscous friction constant and tl isthe load torque. The term Kd sinð4NryÞ represents the detenttorque due to the permanent rotor magnet interacting withthe magnetic material of the stator poles. In the model (1),we neglect the slight coupling between the phases, the smallchange in L as a function of y; the variation in L due tomagnetic saturation and the detent torque (since, in general,Kd ¼ 0Nm). However, the identification procedure carriedout in previous work suggests that the model (1) is adequatefor control design (see [10] and the References therein). Fora given constant pair va ¼ v�a and vb ¼ v�b; the left-handsides of the differential equations in (1) are identically zeroat an equilibrium point ðia; ia;o; yÞ ¼ ði�a; i�a;o�; y�Þ ¼ðv�a=R; v�b=R; 0; y�Þ; where Kd ¼ 0Nm has been chosen.From 0 ¼ � Km

JR½v�a sinðNry

�Þ � v�b cosðNry�Þ� � tl

J; two

possible values for y� are obtained, i.e.

y�� ¼ 2

Nr

arctanKmv

�a �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2mðv�aÞ2 þ K2

mðv�bÞ2 � t2l R

2

q�Kmv

�b � tlR

0@

1A

By setting e ¼ L; and considering the following change ofco-ordinates of the form x ¼ colððo� o�Þ; ðy� y�ÞÞ andz ¼ colððia � i�aÞ; ðib � i�bÞÞ: In addition, Nr is a knownparameter, and we make the assignment

va � v�a ¼ � sinðNrðx2 þ y�ÞÞu;ub � v�b ¼ cosðNrðx2 þ y�ÞÞu

where u is the new scalar control input. The model (1) canbe put in the standard singular form

_xx ¼ f1ðxÞ þ F1ðxÞz;e_zz ¼ f2ðxÞ þ F2ðxÞzþ g2ðxÞu

where xðt0Þ ¼ x0; zðt0Þ ¼ z0 and

f1ðxÞ¼K4ði�b cosðaÞ� i�a sinðaÞÞ�K5x1�K7

x1

!;

F1ðxÞ¼K4

�sinðaÞ cosðaÞ

0 0

!; f2ðxÞ¼

K2x1 sinðaÞ

�K2x1 cosðaÞ

!;

F2ðxÞ¼�K1 0

0 �K1

!; g2ðxÞ¼

�sinðaÞ

cosðaÞ

!

with K1 ¼ R; K2 ¼ Km; K3 ¼ Nr; K4 ¼ Km=J; K5 ¼ B=J;K6 ¼ Kd=J ¼ 0; K7 ¼ tl=J and a ¼ K3ðx2 þ y�Þ:

Therefore, considering the above representation of thestepping motor, the goal is as follows: find a controllerbased on singular perturbation methods and slidingtechniques such that the closed-loop system consisting ofthe system and the controller is exponentially stable in adesirable equilibrium point.

In general, a controller requires the full measurement ofall variables of the system. However, only currentmeasurements and position measurements are usuallyavailable in practice. Furthermore, it would also be adequateto estimate the load torque tl: Thus, to estimate this torqueand the speed, an observer must be designed. Other types ofobservers for this kind of electromechanical devices havebeen designed and tested (see, for example [12], and theReferences therein). Then, the control problem addressed inthis paper is as follows: Assuming that the physicalparameters of the stepping motor are known and themeasurable variables are the currents ia; ib and rotor angularposition y; find a controller based on singular perturbationmethods and sliding techniques and design an observer toestimate the no measurable variables such that the overallclosed-loop system is locally exponentially stable at theequilibrium point [Note 1].

In the Appendix, a more sophisticated model of the PMstepper motor is considered and, under suitable conditions, acontrol design is obtained using the procedure proposed inthis paper.

3 Sliding-mode control

We now develop the control strategy for the stepper motor.To begin the development, the stepper motor model can berepresented by the so-called standard singularly perturbedform:

_xx ¼ f1ðxÞ þ F1ðxÞz xðt0Þ ¼ x0

e_zz ¼ f2ðxÞ þ F2ðxÞzþ g2ðxÞu zðt0Þ ¼ z0ð2Þ

where t0 � 0; x 2 Bx � Rn is the slow state, z 2 Bz � Rm

is the fast state, u 2 Rr is the control input and e 2 ½0; 1Þ; is

Note 1: This control problem can be extended to the case of constantreferences are considered to regulate the rotor position.

IEE Proc.-Control Theory Appl., Vol. 152, No. 4, July 2005466

the small perturbation parameter. f1; f2; the columns of thematrices F1; F2; and g2 are assumed to be bounded withtheir components being smooth functions of x. Bx and Bz

denote closed and bounded subsets centred at the origin. It isalso supposed that f1ð0Þ ¼ f2ð0Þ ¼ 0 and, for u ¼ 0; theorigin ðx; zÞ ¼ ð0; 0Þ is an isolated equilibrium state, andthat F2ðxÞ is nonsingular for all x 2 Bx:The slow reduced system is found by making e ¼ 0 in (2),

obtaining the nth order slow system:

_xxs ¼ f ðxsÞ þ gðxsÞusðxsÞ xsðt0Þ ¼ x0

zs ¼ hðxsÞ :¼ �F�12 ðxsÞ½ f2ðxsÞ þ g2ðxsÞus�

ð3Þ

where xs; zs and us denote the slow components of theoriginal variables x, z and u, respectively, and

f ðxsÞ ¼ f1ðxsÞ � F1ðxsÞF�12 ðxsÞf2ðxsÞ

gðxsÞ ¼ �F1ðxsÞF�12 ðxsÞg2ðxsÞ

ð4Þ

usðxsÞin the first equation of (3) denotes the slow statefeedback, which only depends on xs:The fast dynamics (or, equivalently, boundary layer

system) is obtained by transforming the (slow) time scale tto the (fast) time scale t :¼ ðt � t0Þ=e and introducing thedeviation � :¼ z� heðx; eÞ: The original system (2) thenbecomes

d~xx

dt¼ ef f1ð~xxÞ þ F1ð~xxÞ½� þ heð~xx; eÞ�g

d�

dt¼ f2ð~xxÞ þ F2ð~xxÞ½� þ heð~xx; eÞ� þ g2ð~xxÞu

� @heð~xx; eÞ@ ~xx

d~xx

dt ð5Þwhere �ð0Þ ¼ z0 � heðx0Þ; ~zzðtÞ :¼ zðetþ t0Þ; with ~zzð0Þ ¼z0; and ~xxðtÞ :¼ xðetþ t0Þ; with ~xxð0Þ ¼ x0:The so-called composite control for the original system

(2) is defined by

uðx; �; EÞ ¼ uesðx; EÞ þ uef ðx; �; EÞ ð6Þ

where ues and uef denote the slow and fast components ofthe control, respectively. If uesð~xx; EÞ and @heð~xx; EÞ=@ ~xx arebounded and ~xx remains relatively constant with respect to t;then the term E@heð~xx; EÞ=@ ~xx can be neglected for Esufficiently small. Since the second equation of (5) definesthe fast reduced subsystem, an OðEÞ approximation can beobtained for this subsystem using the first equation of (4)and setting e ¼ 0 in (5), this is

d�apxdt

¼ F2ð~xxÞ�apx þ g2ð~xxÞuf ð7Þ

where �apx; heð~xx; 0Þ ¼ hð~xxÞ and uf are OðEÞ approximationsfor �; heð~xx; eÞ and uef during the initial boundary layer and�apxð0Þ ¼ z0 � hðx0; 0Þ:

3.1 Sliding-mode control design

The sliding-mode control for the system (2) is designed intwo stages. First, the slow control is designed for the slowsubsystem (3). To do this, let us consider a ðn� rÞ-dimensional slow nonlinear switching surface defined by

ssðxsÞ ¼ colðss1ðxsÞ; . . . ; ssrðxsÞÞ ¼ 0 ð8Þ

where each function ssi : Bx ! R; i ¼ 1; . . . ; r; is a C1

function such that ssið0Þ ¼ 0: The equivalent control

method [2] is used to determine the slow reduced systemmotion restricted to the slow switching surface ssðxsÞ ¼ 0;obtaining the slow equivalent control

use ¼ � @ss@xs

gðxsÞ� ��1 @ss

@xsf ðxsÞ

� �ð9Þ

where the matrix ð@ss=@xsÞgðxsÞ is assumed to benonsingular for all xs 2 Bx:

Remark 1: The assumption that the matrix ð@ss=@xsÞgðxsÞ isnonsingular is not restrictive. The switching surface shouldbe considered to satisfy this condition. For the case thatthis matrix is not square, numerical methods can be used toobtain the pseudo-inverse of this term. Moreover, thisassumption is met for electromechanical systems such asthe induction motor, synchronous generator and severalelectrical machines. All of them depend on the switchingsurface and the control objective.

Substitution of (9) into (3) yields the slow sliding-modeequation

_xxs ¼ feðxsÞ

where feðxsÞ ¼ In � gðxsÞ @ss@xsgðxsÞ

h i�1@ss@xs

� �f ðxsÞ; with In

denoting the n� n identity matrix.

To complete the slow control design one sets [6, 7]

us ¼ use þ usN ð10Þwhere use is the slow equivalent control (9), which actswhen the slow reduced system is restricted to ssðxsÞ ¼ 0;while usN acts when ssðxsÞ 6¼ 0: In this work the control usNis selected as

usN ¼ � @ss@xs

gðxsÞ� ��1

LsðxsÞssðxsÞ

where LsðxsÞ is a positive-definite matrix of dimensionr � r; whose components are C0 bounded nonlinear realfunctions of xs; such that kLsðxsÞk � rs; for all xs 2 Bx witha constant rs>0: The equation that describes the projectionof the slow subsystem motion outside ssðxsÞ ¼ 0 is given by

_sssðxsÞ ¼ �LsðxsÞssðxsÞ ð11ÞThe stability properties of ssðxsÞ ¼ 0 in (11) can be studiedby means of the Lyapunov function candidate VðxsÞ ¼12sTs ðxsÞssðxsÞ; whose time derivative along (11) satisfies

_VVðxsÞ ¼ �sTs ðxsÞLsðxsÞssðxsÞ; for all xs 2 Bx

From the C1 properties of ssðxsÞ one also has that kssðxsÞ �ssð0Þk � ls

skxsk; 8 xs 2 Bx; where ls

sis the Lipschitz

constant of ssðxsÞ with respect to xs: Then, we obtain_VVðxsÞ � �rsa1kxsk2; where a1 ¼ l2ss : Thus, the existence ofa slow sliding mode can be concluded.

The system (3) with the control (10) yields the slowreduced closed-loop system, which is represented asfollows:

_xxs ¼ feðxsÞ þ psðxs; usNÞ ð12Þ

where psðxs; usNÞ ¼ gðxsÞusN :We now introduce the following assumption:

A1: The equilibrium xs ¼ 0 of _xxs ¼ feðxsÞ þ psðxs; usNÞ islocally exponentially stable.


By a converse theorem of Lyapunov (see [4]), Assump-tion A1 assures the existence of a Lyapunov functionVs ¼ VsðxsÞ that satisfies

c1kxsk2 � Vs � c2kxsk2;

@Vs

@xfeðxsÞ þ psðxs; usNÞ � �c3kxsk2;

@Vs

@x

�� c4kxsk

ð13Þ

for some positive constants c1; c2; c3 and c4: One may useVsðxsÞ as a Lyapunov function candidate to investigatethe stability of the origin xs ¼ 0 as an equilibrium point forthe system (12). Using Assumption A1 and (13), the timederivative of Vs along the trajectories of (13) satisfies_VVðxsÞ � �c3kxsk2; and the reduced slow system (13) isexponentially stable.

The fast control design for the subsystem (8) can beobtained in a similar way to the one used for the slowcontrol. That is, one considers an ðm� rÞ-dimensional fastswitching surface defined by sf ð�apxÞ ¼ colðsf1ð�apxÞ; . . . ;sfrð�apxÞÞ ¼ 0; where each function sfi : Bz ! R;i ¼ 1; . . . r; is also a C1 function such that sfið0Þ ¼ 0:The complete fast control takes the form

uf ¼ ufe þ ufN ð14Þwhere ufe is the fast equivalent control given by

ufeð~xx; �apxÞ ¼ �@sf@�apx

g2ð~xxÞ� ��1 @sf

@�apxF2ð~xxÞ�apx

� �ð15Þ

and

ufNð~xx; �apxÞ ¼ �@sf@�apx

g2ð~xxÞ� ��1

Lf ð�apxÞsf ð�apxÞ ð16Þ

In (15) and (16), the matrix ð@sf =@�apxÞg2ð~xxÞ is assumed tobe nonsingular, for all ð~xx; �apxÞ 2 Bx � Bz; and Lf ð�apxÞ is apositive-definite matrix of dimension r � r; whose com-ponents are C0 bounded nonlinear real functions of �apx;such that kLf ð�apxÞk � rf ; for all ð~xx; �apxÞ 2 Bx � Bz; with aconstant rf :

The projection of the fast subsystem motion outsidesf ð�apxÞ ¼ 0 is described by

dsfdt

¼@sf@�apx

d�apxdt

¼ �Lf ð�apxÞsf ð�apxÞ ð17Þ

and arguments similar to the ones used for the slowsubsystem motion can be applied to the system (17) toconclude the existence of a fast sliding mode.

When the complete fast control (14) is substituted in (7),the fast reduced closed-loop system takes the form

d�apxdt

¼ gcð~xx; �apxÞ ð18Þ

where

gcð~xx; �apxÞ ¼ F2ð~xxÞ�apx � g2ð~xxÞ@sf@�apx

g2ð~xxÞ� ��1

�@sf@�apx

F2ð~xxÞ�apx þ Lf ð�apxÞsf ð�apxÞ� �

:

The following assumption is now introduced:

A2: The equilibrium �apx ¼ 0 of d�apx=dt ¼ gcð~xx; �apxÞ islocally exponentially stable.

From Assumption A2, by a converse theorem ofLyapunov (see [4]), there exists a Lyapunov function Wf ¼Wf ð�apxÞ that satisfies

�cc1k�apxk2 � Wf � �cc2k�apxk2;

@Wf

@�apxgcð~xx; �apxÞ � ��cc3k�apxk2;

@Wf

@�apx

�� cc4k�apxk

ð19Þ

for some positive constants �cc1; �cc2; �cc3 and �cc4:One may also use Wf ð�apxÞ as a Lyapunov function

candidate to investigate the stability of the origin �apx ¼ 0 asan equilibrium point for the system (18). Using assumptionsA2 and (19), the time derivative ofWf along the trajectoriesof (18) then satisfies dWf ð�apxÞ=dt � ��cc3k�apxk2; and thereduced fast system (18) is exponentially stable.

The original slow and fast state variables are now used toconstruct the composite control (6), i.e. uðx; �Þ ¼ usðxÞ þuf ðx; �Þ; where (10, 14)

us ¼ � @ss@x

gðxÞ� ��1 @ss

@xf ðxÞ þ LsðxsÞssðxsÞ

� �

uf ¼ �@sf@�

g2ðxÞ� ��1 @sf

@�F2ðxÞ� þ Lf ð�Þsf ð�Þ

� �

When the composite control (6,10,14) is substituted in (2),one obtains the closed-loop nonlinear singularly perturbedsystem

_xx ¼ fcðx; �Þ

e _�� ¼ gcðx; �Þ � E@h

@x½ fcðx; �Þ�

ð20Þ

where � ¼ z� hðxÞ; xðtoÞ ¼ xo; zðtoÞ ¼ zo and

fcðx; �Þ ¼ f ðxÞ þ F1ðxÞ� � gðxÞ

� @ss@x

gðxÞ� ��1 @ss

@xf ðxÞ þ LsðxÞssðxÞ

� �:

In the present work the Lyapunov function candidates Vs

and Wf are instrumental to investigate the stability proper-ties of the closed-loop system obtained when the compositecontrol u ¼ us þ uf is used and an observer is introduced toestimate the state of the original system.

4 Nonlinear estimator

Since the above control design depends on measurable andnon-measurable variables, it is necessary to estimate thosenon-measurable variables in order to implement thiscontroller. Hence, an observer design is presented toestimate the non-measurable variables.

It is clear that there is no systematic method to design anobserver for a given nonlinear control system. However,several designs are available according to the specificcharacteristics of the nonlinear system considered.

In this paper, let us now consider the class of nonlinearsingularly perturbed systems described by (2) together withan output variable y 2 R; such that y ¼ qðxÞ; where q is acontinuously differentiable function of Bx and depends onthe slow state [Note 2]. In addition, it is assumed that

Note 2: This observer design can be extended to the multi-output case.


the fast state z is an input vector to the slow subsystem (2)and that it is completely measurable.Consider the nonlinear system

_xx ¼ f1ðxÞ þ F1ðxÞz y ¼ qðxÞ ð21Þ

where z is an input for the system. If there exists a mappingT : Rn ! Rn;which is a diffeomorphism fromBx ontoTðBxÞ;such that (21) can be written in the new co-ordinates as

_zz ¼ Azþ Gðz; zÞy ¼ Cz

ð22Þ

where z ¼ colðz1; z2; . . . ; zmÞ;Gðz; zÞ is a n� m matrix withGðz; zÞ ¼ colðg1ðz1; zÞ; g2ðz1; z2; zÞ; . . . ; gnðz1; z2; . . . ; zn�1;zn; zÞÞ; and the pair (A,C) is in the canonical observable form,that is

A ¼

0 1 0

..

. ... . .

. ...

0 0 1

0 0 0

0BB@

1CCA

and C ¼ ð 1 0 0 Þ: Then, we say that system (21) isuniformly observable for any input (see [13] for more details).One now assumes that the mappings gj : R

i ! R; for i ¼1; . . . ; n are globally Lipschitz.Next, we will design an observer for systems (21) by

exploiting this triangular structure. This property of thenonlinearity is important because it ensures the uniformobservability of the system.We can establish the following.

Theorem 1: Suppose system (21) is uniformly observable.Let also K be a n� 1 constant column vector such thatRelfðA� KCÞg 2 C�: Then, system

dzzdt

¼ Azzþ Gðzz; zÞ � R�1ð‘ÞKðy� yyÞ ð23Þ

where Rð‘Þ ¼ diagð1; 1=‘; . . . ; 1=‘n�1Þ; with ‘ a positiveconstant such that ‘>0; is an exponential observer forsystem (22) whose dynamics can be arbitrarily fast.

Proof: Consider the change of co-ordinates � ¼ Rð‘Þz: Inthese new co-ordinates, and due to the form of the matrix Aand the column vector C, one has that Rð‘ÞAR�1ð‘Þ ¼‘A; CR�1ð‘Þ ¼ C; and the dynamics of system (22) can bewritten as

_�� ¼ ‘A� þ Rð‘ÞGðR�1ð‘Þ�; zÞy ¼ C�

In a similar way, the observer system (23) can be expressedas

d��

dt¼ ‘A�� þ Rð‘ÞGðR�1ð‘Þ��; zÞ � ‘KCð� � ��Þ

yy ¼ C��

If we now define the estimation error as e� ¼ � � ��; itsdynamics are given by

_ee� ¼ ‘ðA� KCÞe� þ Rð‘ÞGð�; ��; zÞ ð24Þ

where Gð�; ��; zÞ ¼ GðR�1ð‘Þ�; zÞ � GðR�1ðyÞ��; zÞ: On theother hand, since, by assumption, the autonomous system_ee� ¼ ‘ðA� KCÞe� is asymptotically stable, there exists aLyapunov function Ve�

¼ eT� Pee� with Pe a symmetricpositive definite matrix, which is the unique solutionof the Lyapunov equation (see [4]) ðA� KCÞTPe þ Pe

ðA� KCÞ ¼ �Qe; where Qe is an arbitrary symmetricpositive definite matrix such that

c�1ke�k2 � Ve�� c�2ke�k2;

@Ve�

@e�ðA� KCÞe� � �c�3ke�k2;

@Ve�

@e�

�� c�4ke�k

where c�1 ¼ lminðPeÞ; c�2 ¼ lmaxðPeÞ; c�3 ¼ lminðQeÞ andc�4 ¼ 2lmaxðPeÞ: Taking the time derivative of Ve�along the trajectories of the estimation error dynamics(24) one has

_VVe�¼ �‘c�3ke�k2 þ 2eT� PeRð‘ÞGð�; ��; zÞ� �‘c�3ke�k2 þ c�4ke�kkRð‘ÞGð�; ��; zÞk

Since the mappings gis are globally Lipschitz, one also hasthat kRð‘ÞGð�; ��; zÞk � rke�k; where r is a Lipschitz

constant. Thus _VVe�� ake�k2; where a ¼ ð‘c�3 � c�4rÞ>0:

Ifr is small enough and satisfying the bound r � �rr<‘c�3=c�4;

the observer system (23) is exponentially stable with aconvergence rate that can be made arbitrarily fast, that is_VVe�

� � ac�2Ve�

:

5 Closed-loop stability

Suppose that a composite control (6, 10, 14) has beendesigned such that the nonlinear singularly perturbedsystem (20) is uniformly bounded, and that an observer(23), with exponential rate of convergence, is alsodesigned. The fundamental question of knowing if thestability of the closed-loop system is preserved, when thestate is replaced by its estimate in the control law, is nowaddressed. The purpose of this Section is to give sufficientconditions that ensure the stability of the closed-loopsystem with observer.

Let us consider the augmented closed-loop nonlinearsingularly perturbed system described by

_xx ¼ f ðxÞ þ F1ðxÞ� þ gðxÞusðxxÞ; xðt0Þ ¼ x0

e _�� ¼ F2ðxÞ� þ g2ðxÞuf ðxx; �Þ þ e@h

@��

� _ee�

� e@h

@xþ @h

@��

@g@x

� f ðxÞ þ F1ðxÞ� þ gðxÞusðxxÞ

�_ee� ¼ ‘ðA� KCÞe� þ Rð‘ÞGð�; ��; � þ hðxÞÞ; e�ðt0Þ ¼ e�0y ¼ C� ð25Þ

where �ðt0Þ ¼ z0 � hðx0Þ and ‘; K are selected as in Section3. Note that the composite control now depends on theestimate xx; where x ¼ T�1ðR�1ð‘Þ�Þ ¼ g�1ð�Þ: The system(25) can be rewritten as


_xx ¼ fcðx; �Þ þ gðxÞDusðx; xxÞ; xðt0Þ ¼ x0

e _�� ¼ gcðx; �Þ þ g2ðxÞDuf ðx; �; xxÞ þ e@h

@��

� _ee�

� e@h

@xþ @h

@��

@g@x

� nfcðx; �Þ þ gðxÞDusðx; xxÞ

o_ee� ¼ ‘ðA� KCÞe� þ Rð‘ÞGð�; ��; � þ hðxÞÞ; e�ðt0Þ ¼ e�0y ¼ C�

ð26Þ

where �ðt0Þ ¼ z0 � hðx0Þ and fc; gc are defined as inSection 2, and

Dusðx; xxÞ ¼ usðxxÞ � usðxÞ;Duf ðx; xx; �Þ ¼ uf ðxx; �Þ � uf ðx; �Þ

ð27Þ

From the properties of the functions involved in us and uf ;one has that Dus and Duf satisfy the local Lipschitzconditions

kDusðx; xxÞk � mske�k; kDuf ðx; xx; �Þk � mf ke�k

for all ðx; xx; �Þ 2 Bx � Bx � Bz; where ms and mf are theLipschitz constants of usðxÞ and uf ðx; �Þ with respect to xand ðx; �Þ; respectively. From the fact that the columns ofg(x) and g2ðxÞ are bounded, one has:

kgðxÞDusðx; xxÞk � m0mske�k;kg2ðxÞDuf ðx; xx; �Þk � m2mf ke�k

ð28Þ

for all ðx; xx; �Þ 2 Bx � Bx � Bz; and e� 2 Bx: m0 and m2 aresome positive constants.

In view of the properties of all the functions involved infcðx; �Þ; this satisfies the local Lipschitz condition

k fcðx; �Þ � fcðx; 0Þk ¼ kF1ðxÞ�k � lf �k�k;8ðx; �Þ 2 Bx � Bz

ð29Þ

where lf� is the Lipschitz constant of fcðx; �Þ; with respect tothe fast variable �: Furthermore, fcð0; 0Þ ¼ 0; thus

k fcðx; 0Þk � lfx1kxk 8x 2 Bx ð30Þ

where lfx1 denotes the Lipschitz constant of fcðx; 0Þ withrespect to x. Also, from the continuous differentiability ofh(x) it follows that

dh

dx

�� lhx ;

dh

d��

�� lh�� ;

dgdx

�� lgx ; 8x 2 Bx; �� 2 B��

ð31Þ

where lhx ; lh�� and lgx are positive constants. Now, set a1 ¼c3;a2 ¼ �cc3

e� �cc4lf�ðlhx þ lh�� lgxÞ

� ; b1¼ c4lf �þ �cc4ðlhx þ lh�� lgxÞ

lfx1 ;b2¼ c4m0ms; b3¼ �cc4ðlhx þ lh�� lgxÞm0msþ 1e�cc4m2mf þ �cc4lh��ð‘aMþrÞ:

To study the stability properties of the closed-loopsystem, consider the following Lyapunov function

Lðx; �; e�Þ ¼ VsðxÞ þWf ð�Þ þ Ve�ðe�Þ:

where VsðxÞ;Wf ð�Þ; and Ve�ðe�Þ are the Lyapunov

functions of the slow subsystem, the fast subsystem andthe estimation error dynamics, respectively.

The following result gives sufficient conditions toassure the local exponential stability of the overallclosed-loop nonlinear singularly perturbed system (26):

Theorem 2: Consider a nonlinear singularly perturbedsystem (2) in closed loop with the control (6, 10, 14)using an observer (23) for estimating the unmeasurablestates, whose estimation error dynamics convergesexponentially to zero. Thus, if there exist some numbers0< di < 1; i ¼ 1; 2; 3; 4; such that mco ¼ minfa0; b0; c0g>0;where a0 ¼ a1 � ðb1d1=2Þ � ðb2d2=2Þ; b0 ¼ a2 � b1=2d1�b3d3=2; c

0 ¼ a� b2=2d2 � b3=2d3; for sufficientlysmall E; then the augmented closed-loop nonlinearsingularly perturbed system (26) is locally exponentiallystable.

Sketch of proof:We proceed to compute the time derivativeof L along the trajectories of each subsystem. Taking thenorm of all terms and substituting the correspondinginequalities, after some computations it follows finally that

_LLðxðtÞ; �ðtÞ; e�ðtÞÞ � �a0kxk2 � b0k�k2 � c0kek2

� �mcoLðxðtÞ; �ðtÞ; e�ðtÞÞ:

This last inequality implies that

LðxðtÞ; �ðtÞ; e�ðtÞÞ � Lðx0; �0; e�0Þe�mcoðt�t0Þ

Then the states x, �; and e� are locally exponentially stablefor all t � t0:

6 Application to PM stepper motor

In this Section, we present the procedure for designing theobserver-based controller using the aforementioned method.

6.1 Control law design

When e ¼ 0; one obtains a unique root zs ¼ hðxsÞ (3) givenby

z1ðsÞ ¼ h1ðxsÞ ¼K2xs1 sinðasÞ � sinðasÞusðxsÞ

K1

z2ðsÞ ¼ h2ðxsÞ ¼�K2xs1 cosðasÞ þ cosðasÞusðxsÞ

K1

and the slow reduced subsystem is given by (4) with

f ðxsÞ ¼�O xs1 þ

K4

K1½v�b cosðasÞ � v�a sinðasÞ� � K7

xs1

!;

gðxsÞ ¼K4

K1

0

!

whereO ¼ ðK4K2=K1Þ þ K5 and as ¼ K3ðxs2 þ y�Þ: Since itis desired that the rotor position tracks the reference signal,which in this case is the equilibrium point [Note 3], a slownonlinear switching function is chosen as:

ssðxsÞ ¼ s1xs1 þ s2xs2

where s1 and s2 are constant real coefficients [Note 4]. Thischoice leads, in accordance with Section 2, to the slowcontrol

Note 3: In this analysis, we do not consider time-varying reference signals.However, it is possible to include this kind of signal as long as someadditional assumptions are imposed to guarantee the local exponentialstability.

Note 4: Evidently, other switching surfaces can be considered. The choicedepends on the control objective.


usðxsÞ ¼ useðxsÞ þ usN ðxsÞ ð32Þ

with

use ¼K1

K4

K4K2

K1

þ K5 �s2s1

� xs1 þ K7

� �� ½v�b cosðasÞ � v�a sinðasÞ�

usN ¼� K1LsðxsÞs1K4

ðs1xs1 þ s2xs2Þ

ð33Þ

where LsðxsÞ ¼ ls>0: On the other hand, (11) takes theform _sssðxsÞ ¼ lsssðxsÞ; thus there exists a slow slidingmode. Substitution of the slow control (32, 33) into the slowreduced subsystem yields the slow reduced closed-loopsystem:

_xxs ¼ feðxsÞ þ psðxs; usNÞ¼ Asxs

where As ¼ � s2s1þ ls

� �s2ls

s1

1 0

!and psðxs; usNÞ ¼ 0:

By choosing s1>0 and s2>0; one guarantees the exponen-tial stability of system _xxs ¼ Asxs:Since in the OðEÞ approximation of the exact fast

subsystem, given by (18), the constant matrix F2 has itstwo eigenvalues at �K1 ¼ �R; there is no need for fastcontrol. That is, one sets uf ¼ 0; and the fast reduced closed-loop system is given by

d�apxdt

¼ gcð~xx; �apxÞ ¼ F2�apx

Then, using the Lyapunov function candidate Wf ð�apxÞ ¼�TapxPf �apx; where Pf is a symmetric positive definite matrix,it is easy to prove that the equilibrium of this system isexponentially stable.Finally, the composite control becomes

uðxÞ ¼ usðxÞ

6.2 Speed and torque estimation

The controller designed above requires mechanical vari-ables, load torque and equilibrium point. Then, for controlimplementation purposes, we must measure all of them.However, a reduction in the number of sensors reduces thecost of the overall control system. Assuming that the rotorposition is usually available in practice using encoders andthe load torque is unknown, an observer is designed toestimate the angular speed and load torque.Let x3 ¼ tl=J so that now one has the assignment x ¼

colððo� o�Þ; ðy� y�Þ; tl=JÞ for the slow state. This yieldsto the new augmented slow subsystem

_xx ¼ �ff 1ðxÞ þ �FF1ðxÞzy ¼ qðxÞ

ð34Þ

where

�ff 1ðxÞ ¼K4i

�b cosðaÞ � K4i

�a sinðaÞ � K5x1 � x3

x1

0

0B@

1CA;

�FF1ðxÞ ¼�K4 sinðaÞ K4 cosðaÞ

0 0

0 0

0B@

1CA

with output qðxÞ ¼ x2; and z as the input of the system.

Using the observability condition given in [14] (Theorem73, Section 7.3.2, p. 418), one can verify that the system(34) is locally observable; then an observer can be designedfor this systems.

Remark 2: It is clear that other observers can be consideredin this case. For instance, in [15] a position and velocitysensorless control for a brushless DC motor using anadaptive sliding mode observer is presented. Some con-ditions are required to determine the gain of the observer.However, when the angular speed is small and the motorload is large, these conditions are not satisfied. Then, therotor position should be sensed. On the other hand, in [16] asensorless observer for induction motors is proposed. Thiskind of observer can be considered in this work providedthat one finds a change of coordinates to transform thestepper motor model to the suitable representation required.

Considering the following change of co-ordinates z1 ¼ x2;z2 ¼ x1; z3 ¼ K4i

�b cosðaÞ � K4i

�a sinðaÞ � K5x1 � x3; the

system (34) can be written in the form (22) with

A ¼0 1 0

0 0 1

0 0 0

0@

1A; Gðz; zÞ ¼

0

C1

C2

0@

1A;

where C1 ¼ �K4 sinðaÞz1 þ K4 cosðaÞz2; C2 ¼ K5C1þz2K3K4½i�b sinðaÞ þ i�a cosðaÞ� � K5z3 and a ¼ K3ðx2 þ y�Þ:Then, the observer for system (34) is given by (23) with theobserver gain K selected in such a way that ðA� KCÞ isHurwitz.

7 Simulation results

The stepper motor described by (1) was simulated togetherwith the controller and observers designed above, using thefollowing nominal values of motor parameters, whichwere chosen as in [17]: R ¼ 10O; Km ¼ 0:113N � m=A;Nr ¼ 50; B ¼ 0:001Nm=rad=s; Kd ¼ 0Nm; tl ¼ 0:05Nm;J ¼ 5:7� 10�6Kgm2 and L ¼ 0:0011H: Also, for a voltagepair v�a ¼ 2:1621V; v�b ¼ 5:4054V; the following equili-brium point was i�a ¼ 0:21621A; i�b ¼ 0:54054A; o� ¼ 0rad=s; y� ¼ 0:0065385 rad: Furthermore, the control andobserver parameters were chosen as follows s1 ¼ 1; s2 ¼500; s3 ¼ 5000; ls ¼ 10000; ‘ ¼ 10 and K ¼ colð1; 1; 1Þ:

In the following studies, the initial conditions of the motorvariables and the estimates were fixed as iað0Þ ¼ 0:21621A;ibð0Þ ¼ 0:54054A; oð0Þ ¼ 0 rad=s; yð0Þ ¼ 0:031416 rad is

Fig. 1 Rotor Position


the resolution limit of the device [17], ooð0Þ ¼ 0:001 rad=s;yy ¼ 0 rad and ttlð0Þ ¼ 0:045Nm:

Furthermore, for comparison the exact linearisationcontroller proposed by [17] was also employed to showthe performance with respect to the proposed controlscheme. The time open-loop as well as the two closed-loop responses of the rotor angular speed and the rotor

angular position are shown in Figs. 1 and 2, respectively.We can observe that the exact linearisation controller showsmore oscillations than the proposed controller, which has afaster response with no oscillations.

In this work, the simulations were performed using thehigh-gain observer for implementing the proposed con-troller. In Figs. 3 and 4 the phase currents are shown.

Fig. 2 Angular speed

Fig. 3 Phase current a

Fig. 4 Phase current b

Fig. 5 Rotor position (Kd variation)

Fig. 6 Angular speed (Kd variation)

Fig. 7 K7 (Kd variation)


The value of the load torque tl and the parameter Kd werenext changed to show the performance of the proposedmethodology under parametric perturbations. In this case, tlwas changed from 0.05 to 0.06Nm and Kd from 0 to0.0043Nm, at t ¼ 0:02 s and t ¼ 0:055 s:Furthermore, unlike in [17], an observer for estimating tl

and o was considered for implementing the exact

linearisation controller. The simulations were performedusing the same high-gain observer considered in Section 4.

The simulations results under Kd variation are shown inFigs. 5–9, and the plots under tl variation are shown inFigs. 10–14. These changes correspond to a variation in thetorque of 20% and to a typical value of Kd; i.e. between 5and 10% of the value of Kmio; where io is the rated current.

Fig. 8 Control action va

Fig. 9 Control action vb

Fig. 10 Rotor position (tl variation)

Fig. 11 Angular speed (tl variation)

Fig. 12 K7 and its variation

Fig. 13 Control va action (tl variation)


Finally, both parameters were changed simultaneously andtheirs dynamic responses are given in Figs. 15–18.

The real and the estimate constant K7 ¼ tl=J underperturbations is given in Fig. 19. We can see that theobserver performs well under parametric variations, withoutdeterioration of the responses.

From the Figures we can say that the exact linearisationpresents some degradation in the quality of the responses,i.e. a large overshoot, more oscillations and steady-stateerrors. On the other hand, the proposed observer-basedcontrol strategy exhibits the best transient behaviour andworks well for different parameter and operating conditions.

Fig. 14 Control vb action (tl variation)

Fig. 15 Rotor position (tl and Kd variations)

Fig. 16 Angular speed (tl and Kd variations)

Fig. 17 Phase current a

Fig. 18 Phase current b

Fig. 19 K7 and its variation


8 Conclusions

In this paper, a controller-observer scheme has beenpresented and studied for a class of nonlinear singularlyperturbed systems, where the dynamics are jointly linear inthe fast variables and the control inputs, but nonlinear in theslow state variables. Assuming that the state of the system isavailable, a composite control is first designed using thesliding-mode technique in such a way that the closed-loopsystem is locally exponentially stable.Considering that the system’s output is a function of the

slow state variable, the fast state variable is available andthe slow system is observable, an observer has beendesigned to estimate the slow variable exponentially. In thepaper, a set of sufficient conditions is given under which theexponential stability of the closed-loop system, togetherwith the estimation error dynamics of the observer, can beguaranteed. These conditions are expressed in terms ofthose that assure the exponential stability of the closed-loopsystem without an observer and the exponential stability ofthe observer. Using the permanent-magnet stepper motormodel with uncertainties in the load torque and detenttorque, the controller–observer design has been illustratedand a comparative study with the exact linearisation controlwas carried out. Better results were obtained with theproposed scheme. An extension of the methodologyproposed here would be to study a more general class ofnonlinear singularly perturbed systems where the wholestate is not available.

9 Acknowledgment

This work is supported by PAICYT, Mexico, under grantCA-866-04.

10 References

1 Saberi, A., and Khalil, H.K.: ‘Quadratic-type Lyapunov functions forsingular perturbed systems’, IEEE Trans. Autom. Control, 1984, 29, (6),pp. 542–550

2 Wilkelman, J.R., Chow, J.H., Allemong, J.J., and Kokotovic, P.V.:‘Multi-time-scale analysis of a power systems’, Automatica, 1980, 16,pp. 35–43

3 Xu, X., Mathur, R.M., Jiang, J., Rogers, G.J., and Kundur, P.:‘Modeling of generators and theirs controls in power systemssimulations using singular perturbations’, IEEE Trans. Power Syst.,1998, 13, (1), pp. 109–114

4 Khalil, H.K.: ‘Nonlinear systems’ (Mcmillan Publishing Company,1996, 2nd edn.)

5 Kokotovic, P.V., Khalil, H.K., and O’Reilly, J.: ‘Singular perturbationmethods in control: analysis and design’ (Academic Press, London,1986)

6 Decarlo, R.A., Zak, S.H., and Matthews, G.P.: ‘Variable structurecontrol of nonlinear multivariable systems: a tutorial’, Proc. IEEE,1988, 76, (3), pp. 212–233

7 Utkin, V.I.: ‘Sliding modes in control and optimization’ (Springer-Verlag, Berlin, Germany, 1992)

8 Sira-Ramirez, H., Spurgeron, S.K., and Zinober, A.S.I.: ‘Robustobserver-controller design for linear systems’, Lect. Notes ControlInf. Sci., 1994, 193, pp. 161–180

9 Bornard, G., Celle-Couenne, F., and Gilles, G.: ‘Observability andobservers’, in Fossard, A.J., and Normand-Cyrot, D. (Eds.): ‘Systemesnon lineaires’ (Masson, Paris, 1993), (1), pp. 177–221

10 Bodson, B., Chiason, N., Novotnak, R.T., and Rokowski, R.B.: ‘High-performance non-linear feedback control of a permanent magnetstepper motor’, IEEE Trans. Control Syst. Tech., 1993, 1, (1), pp. 5–14

11 Kenjo, T.: ‘Stepping motors and their microprocessor controls’(Clarendon, Oxford, UK, 1984)

12 Chiasson, J.N., and Novotnak, R.T.: ‘Nonlinear speed observer for thePM stepper motor’, IEEE Trans. Autom. Control., 1993, 38, (10),pp. 1584–1588

13 Gauthier, J.P., Hammouri, H., and Othman, S.: ‘A simple observer fornonlinear systems: application to bioreactors’, IEEE Trans. Autom.Control, 1992, 37, pp. 875–880

14 Vidyasagar, M.: ‘Nonlinear systems analysis’ (Prentice Hall, Engle-wood Cliffs, NJ, USA, 1993, 2nd edn. )

15 Furuhashi, T., Sangwongwanchi, S., and Okuma, S.: ‘A position-and-velocity sensorless control for brushless DC motors using adaptivesliding mode observer’, IEEE Trans. Ind. Electron., 1992, 39, (2),pp. 89–95

16 Ghanes, M., De Leon, J., and Glumineau, A.: ‘Validation of aninterconnected high gain observer for sensorless induction motor onlow frequencies benchmark: application to an experimental set-up’, IEEProc. Control Theory Appl., 1999, 152, (4)

17 Zribi, M., and Chiasson, J.: ‘Position control of a PM stepper motor byexact linearization’, IEEE Trans. Autom. Control, 1991, 36, (5),pp. 620–625

11 Appendix

We consider a more sophisticated model of the PM steppermotor for which we develop a control strategy based onsingular perturbation methods and sliding mode techniques.The equations describing the dynamical behaviour of thePM stepper motor model are given by (see [17])

va ¼ Ria þdLAdt

þ dfA

dt

vb ¼ Rib þdLBdt

þ dfB

dt

Jdodt

¼ ia þdfA

dyþ ib

dfB

dy� Kd sinð4NryÞ � Bo� tL

dydt

¼ o

where fA ¼ fAðyÞ is the flux in phase a due to thepermanent magnet rotor, fB ¼ fBðyÞ is the flux in phase bdue to the permanent magnet rotor, LA ¼ LAðia; ib; yÞ is theflux in phase a due to ia; ib; LB ¼ LBðia; ib; yÞ is the flux inphase b due to ia; ib:

On the other hand, assuming that the matrix

N ¼

@LA@ia

@LA@ib

@LB@ia

@LB@ib

0BB@

1CCA

is nonsingular, then the above model can be represented asfollows

dodtdydt

0BB@

1CCA¼ �Kd

Jsinð4NryÞ�

B

Jo� tL

Jo

!

þ 1

J

dfA

dy1

J

dfB

dy00

� ia

ib

�

diadtdibdt

0B@

1CA¼

@LA@ia

@LA@ib

@LB@ia

@LB@ib

0BB@

1CCA

�1

va �Ria �@LA@y

þ dfA

dy

� o

vb �Rib �@LB@y

þ dfB

dy

� o

0BBB@

1CCCA

Furthermore, in order to represent this system in a singularperturbed form, we assume that the components of the

matrix N are given by @LA@ia

¼ ej1;@LA@ib

¼ ej2;@LB@ia

¼ ej3;@LB@ib

¼ej4; where e is a small parameter, such that the secondequation can be written as

e

diadtdibdt

0B@

1CA¼M�1

va �Ria �@LA@y

þ dfA

dy

� o

vb �Rib �@LB@y

þ dfB

dy

� o

0BBB@

1CCCA

where M ¼ j1 j2

j3 j4

� : Defining x¼ colðo�o�;y� y�Þ;

z¼ colðia � i�a; ib � i�bÞ and making the assignment


va � v�a ¼� sinðNrðyþ y�ÞÞu;vb � v�b ¼ cosðNrðyþ y�ÞÞu;where u is the new scalar control input, the system can beexpressed as

_xx¼ f1ðxÞþF1ðxÞze_zz¼Gðx; z;uÞ

ð35Þ

where

f1ðxÞ

¼ �Kd

Jsinð4Nrðx2þy�ÞÞ�B

Jx1�

tLJþ1

J

dfA

dyi�aþ

1

J

dfB

dyi�b

x1

!;

F1ðxÞ¼1

J

dfA

dy1

J

dfB

dy0 0

0@

1A;

Gðx;z;uÞ

¼M�1

�Rz1�@LA@y

þdfA

dy

� x1�sinðNrðx2þy�ÞÞu

�Rz2�@LB@y

þdfB

dy

� x1þcosðNrðx2þy�ÞÞu

0BBB@

1CCCA

Remark 3: It is clear that this structure is nonlinear anddifferent from those considered in this paper. However, thenonlinearities are found in the fast subsystem. In fact, as wewill see, this difficulty can be overcome in the controldesign.

Now, to determine the slow subsystem, take e ¼ 0; then itfollows that the roots of Gðx; z; uÞ ¼ 0 depend on the terms@LA@x2

¼ @LAðz1;z2;x2Þ@x2

and @LB@x2

¼ @LBðz1;z2;x2Þ@x2

: Assuming that there

exists a unique root and following the same ideas as in [5],the unique roots of Gðx; z; uÞ ¼ 0 are of the form z ¼hðx; uÞ; i.e.

z1 ¼ h1ðx1; x2; uÞ ¼ H1ðx1; x2Þ þ H2ðx1; x2Þuz2 ¼ h2ðx1; x2; uÞ ¼ H3ðx1; x2Þ þ H4ðx1; x2Þu

Replacing the roots of z in (35), it follows that

dx1dt

dx2dt

0BB@

1CCA

¼ �Kd

Jsinð4Nrðx2 þ y�ÞÞ�B

Jx1 �

tLJþ 1

J

dfAðx2Þdx2

i�a þ1

J

dfB

dyi�b þC1

x1

0@

1A;

þC2

0

� u

where C1 ¼ 1J

dfAðx2Þdx2

H1ðx1;x2Þþ dfBðx2Þdx2

H3ðx1;x2Þ�

and

C2 ¼ 1J

dfAðx2Þdx2

H2ðx1;x2Þ þ dfBðx2Þdx2

H4ðx1;x2Þ�

u:

On the other hand, introducing the deviation � :¼z� hðx; uÞ and transforming the time scale t :¼ ðt � t0Þ=e;since h1ðx1; x2; uÞ and h2ðx1; x2; uÞ are the roots ofGðx; z; uÞ ¼ 0; it follows that the fast subsystem is given by

d�1dtd�2dt

0B@

1CA ¼ M�1 � sinðNrðx2 þ y�ÞÞuf � R�1

cosðNrðx2 þ y�ÞÞuf � R�2

�

Taking uf ¼ 0; the resulting system has the form

d�

dt¼ �RM�1�

To prove the stability of such a system, let Vð�Þ ¼ �TM� bea candidate Lyapunov function. The time derivative alongthe trajectories of the system is given by _VVð�Þ ¼ �2�T� ��mVð�Þ< 0: Hence, the fast subsystem is exponentiallystable.

The slow control can be obtained using the followingslow nonlinear switching function:

ssðxsÞ ¼ s1xs1 þ s2xs2

This choice leads, in accordance with Section 2, to the slowcontrol

usðxsÞ ¼ useðxsÞ þ usN ðxsÞ

with

use ¼�1

C2

(�Kd

Jsinð4Nrðx2 þ y�ÞÞ � B

Jx1 �

tLJþ s2s1x1

þ 1

J

dfAðx2Þdx2

i�a þ1

J

dfBðx2Þdx2

i�b þC1

)

usN ¼ � Lss1C2

ðs1oþ s2ðx2 þ y�ÞÞ

To compare with the controller obtained with the simplifiedmodel, we consider that

dfA

dy¼ �Km sinðNrðx2 þ y�ÞÞ;

dfB

dy¼ Km cosðNrðx2 þ y�ÞÞ; @LA

@ia¼ L;

@LB@ib

¼ L;

@LA@ib

¼ 0;@LB@ia

¼ 0;dLAdy

¼ 0;dLBdy

¼ 0;

then, it follows that

H1ðx1; x2Þ ¼Km

RsinðNrðx2 þ y�ÞÞx1;

H2ðx1; x2Þ ¼ � 1

RsinðNrðx2 þ y�ÞÞ;

H3ðx1; x2Þ ¼ �Km

RcosðNrðx2 þ y�ÞÞx1;

H4ðx1; x2Þ ¼1

RcosðNrðx2 þ y�ÞÞ:

and

C1 ¼ � 1

J

K2m

Rx1; C2 ¼

1

J

Km

R:

Finally, using these expressions, the controller for thesimplified system is given by

use ¼K1

K4

K4K2

K1

þ K5 �s2s1

� xs1 þ K7

� �� ½v�b cosðaÞ � v�a sinðaÞ�

usN ¼ �K1Lss1K4

ðs1xs1 þ s2xs2Þ

where K1 ¼ R; K2 ¼ Km; K3 ¼ Nr; K4 ¼ Km=J;K5 ¼ B=J;K6 ¼ Kd=J;K7 ¼ tl=J; a ¼ K3ðx2 þ y�Þ:


Documents

Observer-based controller for position regulation of stepping motor