observer-based second order sliding mode control laws for stepper motors

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Control Engineering Practice 16 (2008) 429–443

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Observer-based second order sliding mode control lawsfor stepper motors

F. Nollet, T. Floquet�, W. Perruquetti

LAGIS UMR CNRS 8146, Ecole Centrale de Lille, Cite scientifique, BP 48, 59651 Villeneuve d’Ascq Cedex, France

Received 23 March 2006; accepted 31 May 2007

Available online 10 August 2007

Abstract

This paper deals with the design of an observer-based second order sliding mode control law for the stepper motor. The control

objective is to perform accurate tracking of the motor position. The motor velocity is obtained via a second order sliding mode observer.

The second order sliding mode controller is shown to be robust with respect to time-varying load torque perturbation and parametric

uncertainties. Experimental results illustrate the efficiency of the proposed approach.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Stepper motor; Flatness; Higher order sliding mode; Observer based control

1. Introduction

Stepper motors are nonlinear, electromechanical incre-mental actuators widely used as positioning devices. Theirability to provide accurate control over speed and positioncombined with their small size and relatively low cost makestepper motors a popular choice in a range of applications.In particular, permanent magnet stepper motors deliverhigher peak torque per unit weight and have a highertorque to inertia ratio than DC motors. Furthermore, theyare more reliable and, being brushless machines, requireless maintenance. However, as stated in Zribi and Chiasson(1991) and the references therein, using the stepper motorin an open-loop configuration results in low performance.Due to technological breakthroughs in digital signalprocessors, continuous time closed-loop control laws forposition regulation (or tracking) were developed in theliterature. Zribi and Chiasson (1991) considered theposition control of stepper motors by exact feedbacklinearization. Bodson, Chiasson, Novotnak, and Rekowski(1993) reported on an experimental implementation of a

e front matter r 2007 Elsevier Ltd. All rights reserved.

nengprac.2007.05.008

ing author. Tel.: +333 20 67 60 13; fax: +33 3 20 33 54 18.

esses: [email protected] (F. Nollet),

ec-lille.fr (T. Floquet),

[email protected] (W. Perruquetti).

feedback linearizing controller that guarantees positiontrajectory tracking by using field-weakening techniquesand a speed observer. Sira-Ramirez (2000) developed aflatness-based approach for the passivity-based trajectorytracking of the motor currents and position. In this work,the exact knowledge of the dynamics of the stepper motorsystem was required and the robustness issue with respectto load torque perturbations and parametric uncertaintieswas not discussed. An adaptive tracking controller for themotor position error that compensates for parametricuncertainties was designed by Speagle and Dawson (1993)when the dynamics of the stepper motor is not fully known.The stepper model is a flat system, i.e. all the state variables

and the inputs can be parameterized in terms of so-called flatoutputs (or linearizing outputs) and a finite number of theirsuccessive time derivatives (see Fliess, Levine, Martin, &Rouchon, 1992; Fliess, Levine, Martin, & Rouchon, 1995;Fliess, Levine, Martin, & Rouchon, 1999 for theoreticalbackground and Grochmal & Lynch, 2007; Horn, Bamberger,Michau, & Pindl, 2003 for some applications). Flat systemsare dynamical systems that are linearizable to controllablelinear systems by means of an endogenous feedback (i.e. thatdoes not require external variables to the system). Moreover,the flatness property considerably facilitates the off-linetrajectory planning aspects for the system. Lastly, the flatoutputs, being devoid of any zero dynamics, completely

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ARTICLE IN PRESSF. Nollet et al. / Control Engineering Practice 16 (2008) 429–443430

guarantee total internal stability of the system states andoutputs, even if those outputs are nonminimum phase.

Sliding mode methods have already been proved efficientin the field of electrical machines. Furthermore, it has beenshown (see Sira-Ramirez, 2002 or Zribi, Sira-Ramirez, &Ngai, 2001 for example) that the association of flatness andsliding modes leads to valuable and quite simple controlschemes that achieve robust tracking of reference trajec-tories. In fact, the combination of sliding mode and flatnessimplies some input/output decoupling properties and arobust dynamical linearization. Zribi et al. (2001) deve-loped a first order sliding mode control law for thetrajectory tracking of the stepper motor position. However,the sliding motion can be lost if some parameter variationsor exogenous perturbations occur.

In this paper, an observer based second order sliding modecontrol law is designed to solve the problem of accuratetrajectory tracking for the stepper motor position. It isshown, theoretically and by the means of experiments, thatsecond order sliding mode algorithms can cope with time-varying load perturbations and parametric uncertainties.Higher order sliding modes also yield less chattering andbetter convergence accuracy with respect to first order slidingmodes. In order to avoid noise on the speed measurementfrom a tachometer and to reduce the number of sensors, onlyposition and current measurements are assumed to beavailable. Thus, a second order sliding mode observer isdesigned to estimate the motor speed in finite time. Thisobserver also provides fast estimation of the unknown (time-varying) load torque perturbation. The closed-loop stabilityof the combined observer–controller scheme is proved. Someexperimental results are given and analyzed to illustrate theefficiency of the sliding mode controller.

This article is organized as follows. The stepper modeland the problem statement are given in Section 2. InSection 3, some background to higher order sliding modesare given and an observer based second order sliding modecontrol law is designed to solve the position trackingproblem. The experimental set-up (reference trajectories,benchmark description) and the implementation results aredescribed in Section 4.

2. Stepper motor model and problem statement

The model of the stepper motor can be described by thefollowing system (Bodson et al., 1993):

dia

dt¼

1

Lðva � Ria þ KO sinðNyÞÞ;

dib

dt¼

1

Lðvb � Rib � KO cosðNyÞÞ;

dOdt¼

1

JðKð�ia sinðNyÞ þ ib cosðNyÞÞ � f vO� CrÞ;

dydt¼ O;

8>>>>>>>>>><>>>>>>>>>>:where ia and ib are the coil currents in phase A and B,respectively. The state variables y and O are the angular

displacement and the angular velocity of the shaft of themotor. The input variables va and vb are the voltagesapplied on the windings of the phases A and B, respec-tively. N is the number of rotor teeth. The rotor load inertiaJ, the resistances and inductances R and L of each phasewinding, the motor torque constant K and the viscousfriction coefficient f v are assumed to be known andconstant. Cr represents the load torque perturbation. Thedirect-to-quadrature transformation (DQF) or Park trans-formation (Park, 1929) removes the trigonometric termswith the following change of variables:

½id ; iq�T ¼Mp½ia; ib�

T;

½vd ; vq�T ¼Mp½va; vb�

T

(with Mp ¼

cosNy sinNy

� sinNy cosNy

� �,

where id is the direct current, iq is the quadrature current,vd is the direct voltage and vq is the quadrature voltage. Themodel in this frame, can be written as

did

dt¼

1

Lðvd � Rid þNLOiqÞ;

diq

dt¼

1

Lðvq � Riq �NLOid � KOÞ;

dOdt¼

1

JðKiq � f vO� CrÞ;

dydt¼ O:

8>>>>>>>>>><>>>>>>>>>>:

(1)

It was shown in Sira-Ramirez (2000) and Zribi et al. (2001)that the stepper motor with outputs y1 ¼ y and y2 ¼ id is aflat system. A system is flat if there exist m differentiallyindependent functions yj ’s of the state (called the flatoutputs) and their time derivatives such that any systemvariable (states and inputs) is a function of the yj’s and afinite number of their time derivatives. Assuming thatCr ¼ 0, the variables of the stepper motor model can beparameterized in terms of the position and the directcurrent, and their time derivatives:

y ¼ y1;

O ¼ _y1;

id ¼ y2;

iq ¼1

KðJ €y1 þ f v _y1Þ;

vd ¼ L _y2 þ Ry2 �NL

K_y1ðJ €y1 þ f v _y1Þ;

vq ¼JL

Kyð3Þ1 þ

1

KðLf v þ RJÞ €y1 þ

Rf v

Kþ K þNLy2

� �_y1:

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:The main control objective is to follow a given positionreference trajectory yrðtÞ. The second flat output, id , willalso be constrained to follow a reference trajectory idrðtÞ.The off-line trajectory planning is simple using the flatnessproperty. Once the reference trajectories yr and idr havebeen chosen, one can easily obtain, without integratingany differential equations, the trajectories of all thereference variables Or, iqr, vdr and vqr which satisfy the

ARTICLE IN PRESSF. Nollet et al. / Control Engineering Practice 16 (2008) 429–443 431

motor dynamics:

didr

dt¼

1

Lðvdr � Ridr þNLOriqrÞ;

diqr

dt¼

1

Lðvqr � Riqr �NLOridr � KOrÞ;

dOr

dt¼

1

JðKiqr � f vOÞ;

dyr

dt¼ Or:

8>>>>>>>>>><>>>>>>>>>>:

(2)

In fact, one has

iqrðtÞ ¼1

KJd2yr

dt2þ f v

dyr

dt

� �;

OrðtÞ ¼dyr

dt;

vdrðtÞ ¼ Ldidr

dtþ Ridr �NLOriqr;

vqrðtÞ ¼ Ldiqr

dtþ Riqr þNLOridr þ KOr:

8>>>>>>>>>>><>>>>>>>>>>>:

(3)

Therefore, the problem is to stabilize to the origin thetracking error e ¼ ½e1; e2; e3; e4�

T ¼ ½id � idr; iq � iqr;O� Or;y� yr�

T whose dynamics is given by

_e1 ¼1

Lðvd � Re1 þNLðe3e2 þ e3iqr þ e2OrÞÞ;

_e2 ¼1

Lðvq � Re2 �NLðe3e1 þ e3idr þ e1OrÞ � Ke3Þ;

_e3 ¼1

JðKe2 � f ve3 � CrÞ;

_e4 ¼ e3;

8>>>>>>>><>>>>>>>>:

(4)

with vd ¼ vd � vdr and vq ¼ vq � vqr.Note that the flat outputs are linked to the control inputs by

_e1 ¼1

Lvd þ m1ðeÞ;

eð3Þ4 ¼

K

JLvq þ m2ðeÞ þ

f v

J2Cr �

1

J

dCr

dt;

8>><>>:where

m1ðeÞ ¼1

Lð�Re1 þNLðe3e2 þ e3iqr þ e2OrÞÞ;

m2ðeÞ ¼ �K

JLðRe2 þNLðe3e1 þ e3idr þ e1OrÞ þ Ke3Þ

�f v

J2ðKe2 � f ve3Þ:

8>>>>>><>>>>>>:

Thus, due to the flatness property, the stepper motor isequivalent to two input–output decoupled linear systems incontrollable canonical form. This greatly simplifies the designof a state feedback controller while suggesting naturally asliding mode approach. Indeed, a linearizing state feedback(that is known to be sensitive with respect to exogenousperturbations and parametric uncertainties) can be made morerobust with this technique. Furthermore, the canonical formsusually obtained for the design of sliding mode control lawsinduce a zero dynamics (Isidori, 1995) on which closed-loop

stability is dependent. Here, it is not the case since the flatoutputs are devoid of zero dynamics.

3. Second order sliding mode control laws

Sliding mode control laws for nonlinear systems havebeen widely investigated and developed since they wereintroduced by Utkin (1977). The objective of this methodis, by means of a discontinuous control, to constrain thesystem to evolve and stay, after a finite time, on a slidingmanifold where the resulting behavior has some prescribeddynamics. Sliding mode control exhibits relative simplicityof design and robustness properties with respect tomatched perturbations. However, some problems existsuch as the chattering phenomenon (that is to say highfrequency oscillations along the sliding motion). Thisdrawback can be very harmful for the motor since thediscontinuous control can cause overheating of the coilsand excite some unmodeled high frequency dynamics.Here, it is supposed that the reader is familiar with theprinciples of the sliding mode theory. Further details canbe found in the following books: Edwards and Spurgeon(1998), Perruquetti and Barbot (2002) or Utkin (1992).Sliding mode control has already been applied in order

to stabilize the stepper motor position around a desiredtrajectory when no load torque is applied (Zribi et al.,2001). In this work, the sliding variables were chosen as

s1 ¼ k1e4 þ k2e3 þ1

JðKe2 � f ve3Þ; k1; k240,

s2 ¼ e1.

Then, using a discontinuous control action with largeenough control gains, the finite time convergence onto thesliding manifolds s1 ¼ 0 and s2 ¼ 0 was obtained, result-ing in the finite time stabilization of the direct current errore1 and the asymptotic stabilization of the position errore4 since, in the sliding mode ðs1 ¼ 0Þ, the equivalentdynamics is:

k1e4 þ k2 _e4 þ €e4 ¼ 0.

However, if an unknown load torque Cr appears, theequivalent dynamics becomes:

k1e4 þ k2 _e4 þ €e4 ¼ �1

JCr

and this implies that the sliding motion is not asymptoti-cally stable. This is due to the fact that the necessarycondition for a sliding mode to be insensitive to someperturbations, namely the matching condition (Drazenovic,1969), is not satisfied. One could apply some adaptivemethods in order to circumvent this difficulty but thiswould require that the load torque is constant. In thefollowing, a second order sliding mode control lawinsensitive to time-varying load torques is designed.


3.1. Generalities

The principle of higher order sliding mode control is toconstrain the system trajectories to reach and stay, after afinite time, on a given sliding manifold Sr in the state space(Emel’yanov, Korovin, & Levantovsky, 1986; Perruquetti& Barbot, 2002). Consider a system whose dynamics isgiven by

x ¼ f ðt; xÞ þ gðt;xÞw, (5)

where x 2 Rn is the system state, w 2 R is the control and f,g are sufficiently smooth vector fields. The sliding manifoldis defined by the vanishing of a corresponding sliding

variable S : Rþ � Rn ! R, and its successive time deriva-tives up to a certain order, i.e. the rth order sliding set

Sr ¼ fðt;xÞ 2 Rþ � Rn : Sðt;xÞ

¼ _Sðt;xÞ ¼ � � � ¼ Sðr�1Þðt;xÞ ¼ 0g.

A control law w leading to such a behavior will be called anrth order ideal sliding mode algorithm with respect toce:italic>S. This discontinuous control law usually acts onthe rth time derivative of the sliding variable. Thus, higherorder sliding modes can reduce, by far, the chatteringphenomenon while preserving the robustness properties.Indeed, assume that the system has relative degree p withrespect to the sliding variable. If the sliding mode order r ischosen such that ppr� 1, the discontinuous algorithmgenerating the rth order sliding mode is applied to wðr�pÞ

and thus the actual input of the system is continuous.An ideal sliding mode does not exist in practice since it

would imply that the control commutes with infinitefrequency. Because of the technological limitations suchas switching time delays and/or small time constants in theactuators, this frequency is finite. Thus, the motion onlytakes place in a neighborhood of the sliding manifold andis called a real higher order sliding mode. A motionsatisfying the following relations:

jSj ¼ OðtrÞ; j _Sj ¼ Oðtr�1Þ; . . . ; jSðr�1Þj ¼ OðtÞ,

where t is the sampling period, is said to be a rth order real

sliding mode with respect to S. From this definition, it canbe seen that the higher the order of the sliding mode is, themore accurate the convergence on S ¼ 0 is.

The problem of interest in the case of the stepper motoris to generate a second order sliding mode on anappropriately chosen sliding surface and, thus, to constrainthe trajectories system to evolve in finite time on

S2 ¼ fx 2 Rn : S ¼ _S ¼ 0g. (6)

It is obtained with a control law acting on the second timederivative of the sliding variable, which can be written inthe general form:

€S ¼ fðt;S; _SÞ þ jðt;S; _SÞW , (7)

where W ¼ w (resp. W ¼ _w) when the system has relativedegree 2 (relative degree 1) with respect to S.

In order to design second order sliding mode algorithms,it is necessary to establish some assumptions (see forexample Bartolini, Ferrara, Levant, & Usai, 1999) to allowthe reachability of the sliding surface and the bound of thevariable €S. Particularly, it is assumed that there existpositive constants S0, km, KM , C0 such that 8x 2 Rn andjSðt;xÞjoS0, the system satisfies the following conditions:

0okmpjjðt;S; _SÞjpKM and jfðt;S; _SÞjoC0.

Different kinds of second order sliding mode algorithmscan be found in the literature (see Bartolini et al., 1999;Emel’yanov et al., 1986; Fridman & Levant, 2002):Twisting, Sampled Twisting, Super Twisting, Sub-Optimal,et. Hereafter, two algorithms that induce a second ordersliding mode for the system (5) and that will be applied tothe stepper motor are described.

Sampled twisting algorithm: This algorithm providesgood robustness properties and can be applied when therelative degree is 2. It does not require the knowledge of thetime derivative of the sliding variable and takes intoaccount some practical constraints such as the sampling ofthe measurement and the control. The sampling period t isassumed to be the same for both the control law and themeasurement. The sampled twisting algorithm can bewritten as follows:

w9wteðS;DSÞ ¼�lMsignðSÞ if SDS40;

�lmsignðSÞ if SDSp0;

(

with

DS90; k ¼ 0;

ðSðktÞ � Sððk � 1ÞtÞÞ; kX1:

(

Under sufficient conditions:

lm44KM

S0;

lm4C0

km

;

lM4KMlm

km

þ 2C0

km

;

8>>>>>>><>>>>>>>:

(8)

it can be shown that the system trajectories converge ontothe real second order sliding set

S2real ¼ fjSj ¼ Oðt2Þ; j _Sj ¼ OðtÞg.

Super twisting algorithm: This algorithm has been devel-oped for systems with relative degree 1 to avoid thechattering phenomena. The control law comprises twocontinuous terms that, again, do not depend upon the firsttime derivative of the sliding variable. The discontinuityonly appears in the control input time derivative:

w9wstðSÞ ¼ v1 þ v2 with

_v1 ¼ �bsignðSÞ;

v2 ¼ �ajSj1=2signðSÞ;

a;b40:

8><>: (9)


If the control gains satisfy the sufficient conditions (Levant,1993),

b4C0

km

and aX4C0

k2m

ðKMbþ C0Þ

ðkmb� C0Þ(10)

one can obtain the convergence in finite time on the slidingsurface (6).

3.2. Full-state feedback controller

In this part, it is assumed that all the state variables areavailable for control and that a load torque is applied tothe motor. In order to achieve the control objective forsystem (4), i.e. the tracking of the position and directcurrent, second order sliding mode control laws will bedesigned.

Position tracking: Let us define the following slidingvariable:

S1 ¼ ke4 þ _e4 ¼ ke4 þ e3,

with k40. Note that this sliding variable does not dependon €e4 and is completely independent of the motorparameters. The system has relative degree 2 with respectto S1 and the successive time derivatives of S1 are

_S1 ¼ k _e4 þ €e4 ¼ ke3 þ €e4 (11)

and

€S1 ¼ k _e3 þ eð3Þ4

¼k

JðKe2 � f ve3Þ þ

K

JLvq þ m2ðeÞ

�k

J�

f v

J2

� �Cr �

1

J

dCr

dt. ð12Þ

A solution lies in the design of a second order sliding modealgorithm that only requires the knowledge of S1 (that is tosay only the tracking error variables). One could use forinstance the sub-optimal algorithm described by Bartoliniet al. (1999). This algorithm, after an initialization time,needs in real time the exact knowledge of the singular valueof the sliding variable, that is to say, the correspondingvalue when _S1ðtÞ ¼ 0. Here, the sampled twisting algorithmwill be applied to the stepper motor.1 For this, set

K

JLvq ¼ �

k

JðKe2 � f ve3Þ � m2ðeÞ þ wteðS1;DS1

Þ. (13)

One obtains

€S1 ¼ f1ðtÞ þ wteðS1;DS1Þ, (14)

where

f1ðtÞ ¼ �k

J�

f v

J2

� �Cr �

1

J

dCr

dt.

1Note that both algorithms were experimentally tested. It appeared that

the sampled twisting algorithm was more efficient and easier to implement.

Assume that the load torque and its first time derivative arebounded2 such that

suptjCrðtÞjoP0,

supt

dCr

dt

��oP1.

The second time derivative of S1 has a form similar to (7),with f ¼ f1 and j ¼ 1. Thus, the system trajectoriesevolve, after a finite time, onto the second order slidingset S2

real if lm and lM satisfy the convergence conditions(8), i.e.:

lm4k

J�

f v

J2

��P0 þ

1

JP1 and

lM4lm þ 2k

J�

f v

J2

��P0 þ

1

JP1

� �.

When S1 ¼ 0, one gets exponential convergence of e4 tozero since, in the sliding mode, the equivalent dynamics are

ke4 þ _e4 ¼ 0.

Direct current tracking: For the problem of the directcurrent trajectory tracking, define the following slidingvariable:

S2 ¼ e1. (15)

The system has relative degree one with respect to thissliding variable, since:

_S2 ¼1

Lvd þ m1ðeÞ. (16)

One could apply a first order sliding mode control law (seeNollet, Floquet, & Perruquetti, 2003 or Zribi et al., 2001).However, this would induce chattering phenomena. In orderto reduce the chattering, a first order dynamic sliding modecontrol scheme was designed in Zribi et al. (2001). Never-theless, this method requires the knowledge of further timederivatives of the state. Here, the chattering phenomenon isovercome by applying the super-twisting algorithm (thediscontinuous part will act on the second time derivative ofthe direct current). Furthermore, this algorithm only requiresthe knowledge of e1. For this, let us define

1

Lvd ¼ �m1ðeÞ þ wstðS2Þ. (17)

One gets

€S2 ¼ _wstðS2Þ ¼ �b signðS2Þ �1

2ajS2j

�1=2 _S2. (18)

Again, one has a form similar to (7), with f ¼ 0 and j ¼ 1.Thus, choosing a and b sufficiently large, e1 tends to zero infinite time.Let us now briefly show that the designed control laws

are insensitive with respect to parametric uncertainties.For this purpose, rewrite the tracking error model (4)

2The algorithm can cope with perturbations whose first time derivative

has countable unbounded discontinuities. This is the case for the square

wave form of load torque disturbance.


as follows:

_e1 ¼1

Lþ d1

� �vd �

R

Lþ d2

� �e1 þNðe3e2 þ e3iqr þ e2OrÞ;

_e2 ¼1

Lþ d1

� �vq �

R

Lþ d2

� �e2 �Nðe3e1 þ e3idr þ e1OrÞ

�K

Lþ d3

� �e3;

_e3 ¼K

Jþ d4

� �e2 �

f v

Jþ d5

� �e3 �

1

J þ d6Cr;

_e4 ¼ e3;

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

(19)

d1 ¼ dð1LÞ, d2 ¼ dðR

LÞ, d3 ¼ dðK

LÞ, d4 ¼ dðK

JÞ, d5 ¼ dðf v

JÞ, d6 ¼

dðJÞ represent the parametric uncertainties that areassumed to be constant. Then, the second time derivativeof S1 can be written as

d2S1

dt2¼

K

JLþ Z1

� �vq þ

k

JðKe2 � f ve3Þ þ m2ðeÞ

þ Z2ðeÞ þ DðCrÞ,

with

Z1 ¼ d41

Lþ d1

� �þ d1

K

J,

DðCrÞ ¼f v þ d5JJðJ þ d6Þ

�k

J þ d6

� �Cr �

1

J þ d6

dCr

dt

and where Z2ðeÞ is a function of the tracking error state andthe uncertainties. Applying the control law (13) leads to

d2S1

dt2¼ 1þ

JL

KZ1

� �wte S1;DS1

� �þ Z2ðeÞ þ DðCrÞ, (20)

where Z2ðeÞ is also a function of the tracking error state andthe uncertainties. One can still recognize a form similar to(7), with

fðtÞ ¼ Z2ðeÞ þ DðCrÞ,

jðtÞ ¼ 1þJL

KZ1.

Thus with a suitable choice of the control gains lm and lM ,the system trajectories evolve after a finite time on S1 ¼ 0,i.e. ke4 þ e3 ¼ 0 and the tracking position error isasymptotically stabilized. Similarly, it could be shown thatthe finite time stabilization of the direct current error e1 canbe obtained in spite of the uncertainties by increasing thegains a and l.

3.3. Speed observer

Motivated by the practical consideration that positionmeasurement by optical encoders is much more reliablethan the noisy speed measurement by tachometers, onlyposition measurement is used. The objective is to recon-struct the unmeasured signal O that is needed in the controldesign. A second order sliding mode observer is designed

as follows:

dOdt¼

1

JðKiq � f vOÞ þ rðy� yÞ � wðy� yÞ;

dydt¼ O;

8>>><>>>:where w is a discontinuous output injection and r is apositive scalar. The observation error is defined as �y ¼y� y and �O ¼ O� O. Its dynamics is given by

_�y ¼ �O;

€�y ¼ _�O ¼ �f v

J�O � r�y þ wð�yÞ �

1

JCr:

8<: (21)

Consider �y ¼ 0 as a sliding manifold and define thediscontinuous action wð�yÞ as a real twisting algorithm:

wð�yÞ ¼ wteð�y;D�yÞ ¼�lo

M signð�yÞ if �yD�y40;

�lom signð�yÞ if �yD�yp0:

(

One has

€�y ¼ �1

JCr �

f v

J�O � r�y þ wteð�y;D�y Þ.

Again, one can recognize a form similar to (7) with f ¼� 1

JCr and j ¼ 1. The term �ðf v=JÞ�O � r�y is an additive

stabilizing linear part and it can be shown (Floquet, Barbot,& Perruquetti, 2003) that, under the conditions (8), i.e.:

lom4

P0

Jand lo

M4lom þ 2

P0

J, (22)

the trajectories of the system (21) still evolve, after a finitetime, on the second order sliding set f�y ¼ �O ¼ 0g. Thus, Oconverges in finite time towards O. This type of observer isof interest for the following reasons. It provides a finite timeestimation of the motor velocity whatever the control law is.Furthermore, one can also get an estimation Cr of the loadtorque. Indeed, according to the equivalent control method,while sliding mode occurs on �y ¼ �O ¼ 0, one gets thefollowing equivalent dynamics (obtained by writing €�y ¼ 0):

weq �1

JCr ¼ 0, (23)

where weq is the equivalent information injection andrepresents the mean value of the signum function in slidingmode. It can be obtained in finite time via a low pass filter(Utkin, 1992) or by some continuous approximation of thesignum function (Edwards & Spurgeon, 1998). Thus after afinite time, one gets an estimation of the load torque:

Cr ¼ Jweq � Cr. (24)

3.4. Closed-loop stability

It is well known that the separation principle does nothold for nonlinear systems. This means that the observerand the controller cannot be designed separately (which iscontrary to the linear case). Thus, let us analyze thecombined observer–controller stability, when the control


law is designed on the basis of the speed observer. For this,consider the whole dynamics of the closed-loop system,given by Eqs. (4), (21):

_e1 ¼1

Lðvd � Re1 þNLðe3e2 þ e3iqr þ e2OrÞÞ;

_e2 ¼1

Lðvq � Re2 �NLðe3e1 þ e3idr þ e1OrÞ � Ke3Þ;

_e3 ¼1


_e4 ¼ e3;

_�O ¼ �f v

J�O �

1

JCr � r�y þ wð�yÞ;

_�y ¼ �O:

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

(25)

The control laws are based on the output of the observer O.Thus, vd and vq are henceforth given by

vq ¼JL

K�

k

JðKe2 � f v�3Þ � m02ðeÞ þ wteðS1;DS1

Þ

� �;

vd ¼ Lð�m01ðeÞ þ wstðS2ÞÞ;

8><>:

(26)

with

m01ðeÞ ¼1

Lð�Re1 þNLð�3e2 þ �3iqr þ e2OrÞÞ;

m02ðeÞ ¼ �K

JLðRe2 þNLð�3e1 þ �3idr þ e1OrÞ þ K�3Þ

�f v

J2ðKe2 � f v�3Þ;

�3 ¼ O� Or ¼ e3 � �O:

8>>>>>>>>><>>>>>>>>>:The second order sliding mode speed observer gives anobservation error which converges to zero in finite time.Then, O converges in finite time towards O whatever thecontrol law is. After this transient time, the control lawbehaves as in Section 3.2 and the control objective isfulfilled. Let us show that during the transient time of theconvergence of the observer, the variables of the wholesystem (25) remain bounded. After substituting (26) into(25), one has

_e1 ¼ wstðS2Þ þNðe2 þ iqrÞ�O;

_e2 ¼J

KwteðS1;DS1

Þ � ge2 þ gf v

Ke3

� Nðe1 þ idrÞ þK

Lþ g

f v

K

� ��O;

_e3 ¼1


_e4 ¼ e3;

_�O ¼ �f v

J�O �

1

JCr � r�y þ wð�yÞ;

_�y ¼ �O;

8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>:where g ¼ k � f v=J.

Denoting X ¼ ½e1; e2; e3; e4; �O; �y�T, the system can be

written as _X ¼ f ðX Þ þ g where

f ðX Þ ¼

Niq�O

�ge2 þ gf v

Ke3 � Nid þ

K

Lþ g

f v

K

� ��O

1

JðKe2 � f ve3Þ

e3

�f v

J�O � r�y

�O

26666666666666664

37777777777777775

and

g ¼

wstðS2Þ

J

KwteðS1;DS1

Þ

�1

JCr

0

�1

JCr þ wð�yÞ

0

26666666666666664

37777777777777775

.

Under the practical assumptions that the direct andquadrature currents are saturated (jid jpidmax , jiqjpiqmax

),and since g is a bounded function such that kgkpg, onecan write

k _XkpKkXk þ g, (27)

where K is a positive constant. Integrating (27) leads to

kX ðtÞkpkX ð0Þk þZ t

0

ðKkX ðtÞk þ gÞdt.

Applying the Gronwall Lemma (Khalil, 1992), one has

kX ðtÞkpkX ð0Þk expðKtÞ þg

Kexp½ðKtÞ � 1�

and, this inequality implies that the variables of the wholesystem (25)–(26) are bounded in finite time.

4. Implementation of the controllers

4.1. Reference trajectories

Position: The control objective is to bring the motorposition from yrðtiÞ ¼ yi ¼ 0 to yrðtf Þ ¼ yf ¼ 6 rad fastenough (from the initial time instant ti to the final time tf )and with the least amount of overshoot. In order to havesufficiently smooth dynamics (velocity and acceleration),without discontinuity, electrical peak, or torque undula-tions, the following additional constraints are chosen:

_yrðtiÞ ¼_yrðtf Þ ¼

€yrðtiÞ ¼€yrðtf Þ ¼ 0.

Thus, the reference trajectory can be chosen as a fifth orderpolynomial equation:

yrðtÞ ¼ yi þ ðyf � yiÞð10D3t � 15D4

t þ 6D5t Þ,

ARTICLE IN PRESS

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

2

4

6

Reference position

(rad)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

10

20

30

Reference speed

(rad/s

)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0

100

-100

-200

200

Reference acceleration

(rad/s

2)

Time (s)

Fig. 1. Reference trajectories.

F. Nollet et al. / Control Engineering Practice 16 (2008) 429–443436

with

Dt ¼ðt� tiÞ

ðtf � tiÞ; ti ¼ 0 s.

The reference trajectories for tf ¼ 0:5 s are given in Fig. 1.Direct current: From the mechanical equation,

dOdt¼

1

JðKiq � f vO� CrÞ

note that the motor torque is a function of iqðtÞ. In order toobtain the maximum torque, that is say to iqrðtÞ maximal, itis advisable to choose a direct current reference trajectorysuch as idrðtÞ ¼ 0 or idrðtÞo0. Moreover, in order tominimize3 the Joule losses:

J ¼

Z tf

0

Rði2d þ i2qÞdt

one can choose idrðtÞ ¼ 0.

4.2. Benchmark description

The experimental benchmark, developed in LAGISlaboratory has the following structure:

�

3

crit

per

The stepper motor is a Turbo Disc P850 from thePortescap company. Its technology is based on a disc

The purpose of the paper is not the optimization of some energy

erion. However, the trajectories are chosen in order to illustrate the

formance of the control laws while being physically sensible.

magnet, whose advantages are a thin rotor disc with lowinertia and a lighter weight than other types of steppermotor. This allows for small stepping angles and fastacceleration, hence, excellent dynamic response. Afteridentification, the motor characteristics, with coils inseries, are� Torque constant: K ¼ 0:3Nm=A� Inertia: Jt ¼ 5:10�4 kgm2

� Holding torque: Cm ¼ 780:10�3 Nm� Viscous friction: f v ¼ 10�3 Nms=rad� One motor step: ffi 1; 8 (200 steps)� Teeth number: N ¼ 50� Resistance: R ¼ 2; 6O� Inductance: L ¼ 8mH

Jt is the whole inertia of the kinematic chain elements (i.e.the sum of motor inertia, encoder inertia and joint inertia).
� The computer is equipped with Matlab and Simulink
and an interface dSPACE card (1104) with theControlDesk software. The sampling period ist ¼ 10�4 s. It has been chosen to be smaller than theelectrical time constant of the motor L=R ¼ 0:003 s.
� The position is measured with an absolute optical
encoder, mounted on the motor shaft (13 bits–8192points). The absolute optical encoder accuracy is 7:67�10�4 rad (40:96 times higher than the stepper motoraccuracy).
� The load torque is generated by a powder brake whose
input ðVbrakeÞ is a square signal.


�

(ra

d)

(ra

d)

(rad/s

)(r

ad/s

)

The input voltages va and vb of each motor coil aredelivered by the dSPACE card and amplified by twodriver circuits.
� Current transducers provide the measurement of the coil
currents ia and ib. The precision of the sensors is 1% ofthe nominal current Ipn ¼ 3A.
� The limits of the voltages and the currents provided by
the power supply of the benchmark are 30V–3A.

0 0.05 0.1 0.15 0.2

0

1

2

3

4

5

6

Motor and ref

0 0.05 0.1 0.15 0.2

0

0.005

-0.005

0.01

-0.01

Posi

Ti

Fig. 2. Positions—

0 0.05 0.1 0.15 0.2

0

10

-10

20

30

40

50

Estimated an

0 0.05 0.1 0.15 0.2

0

0.2

-0.2

-0.4

-0.6

-0.8

0.4

0.6

Spe

T

Fig. 3. Speeds—

4.3. Experimental results

All the experimental results reported hereafter arerealized with the following parameters:

lm ¼ 2; lM ¼ 10; k ¼ 50,

a ¼ 8; b ¼ 1,

lom ¼ 3500; lo

M ¼ 5000.

0.25 0.3 0.35 0.4 0.45 0.5

erence positions

0.25 0.3 0.35 0.4 0.45 0.5

tion error

me (s)

Cr ¼ 0Nm.

0.25 0.3 0.35 0.4 0.45 0.5

d reference speeds

0.25 0.3 0.35 0.4 0.45 0.5

ed error

ime (s)

Cr ¼ 0Nm.

ARTICLE IN PRESS

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0

0.2

-0.2

-0.4

0.4

0.6

Motor and reference Id

(A)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0

20

-20

40

-40

Voltage Va

(V)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0

1

-1

-2

2

Current Ia

(A)

Time (s)

Fig. 4. Voltages and currents—Cr ¼ 0Nm.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

1

2

3

4

5

6

Motor and reference positions

(rad)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

1

-1

-2

-3

-4

-5

2

3

4

x 10-3 Position error

(rad)

Time (s)

Fig. 5. Positions—Cr ¼ 0Nm.

F. Nollet et al. / Control Engineering Practice 16 (2008) 429–443438


4.3.1. Experiments without perturbations

Two experiments are performed assuming there is noload torque and that the motor parameters are well known.In the first one, the final time instant is chosen astf ¼ 0:23 s, in order to have the voltages and the currentsclose to their limit values imposed by the power supply of

0 0.1 0.2 0.3

0

5

-5

10

15

20

25

Estimated and

(rad/s

)

0 0.1 0.2 0.3

0

0.1

-0.1

-0.2

-0.3

-0.4

0.2

0.3

Spee

(rad/s

)

Tim

Fig. 6. Speeds—

0 0.1 0.2 0.3

0

0.05

-0.05

0.1

-0.1

Motor and

(A)

0 0.1 0.2 0.3

0

10

-10

20

-20

Volta

(V)

0 0.1 0.2 0.3

0

0.5

-0.5

1

-1

Curr

(A)

Tim

Fig. 7. Voltages and cu

the benchmark (see Fig. 4). The resulting maximum rotorspeed is about 50 rad s�1 (see Fig. 3). It can be seen in Fig. 2that the motor position converges rapidly towards thedesired trajectory and that the error position (reference-motor position) is always less than 10�2 rad. It can benoted that, once the motor has been stabilized around the

0.4 0.5 0.6 0.7 0.8

reference speeds

0.4 0.5 0.6 0.7 0.8

d error

e (s)

Cr ¼ 0Nm.

0.4 0.5 0.6 0.7 0.8

reference Id

0.4 0.5 0.6 0.7 0.8

ge Va

0.4 0.5 0.6 0.7 0.8

ent Ia

e (s)

rrents—Cr ¼ 0Nm.


desired position yf , the error is 7� 10�4 rad, which is theoptical encoder accuracy. Thus, the accuracy of theposition tracking is limited by the accuracy of the chosenposition encoder. However, it should be stressed that, insimulation, the position error was 10�8 rad as expectedtheoretically (whereas a first order sliding mode control law

0 0.2 0.4 0.6 0.8

0

2

4

6

Motor and re

(rad)

0 0.2 0.4 0.6 0.8

0

0.01

-0.01

0.02

-0.02

-0.03

Posi

(rad)

0 0.2 0.4 0.6 0.8

0

0.2

-0.2

0.4

0.6

Estimated and measur

(Nm

)

Ti

Fig. 8. Positions—C

0 0.2 0.4 0.6 0.8

0

5

-5

10

15

20

25

Estimated and

(ra

d/s

)

0 0.2 0.4 0.6 0.8

0

0.5

-0.5

-1

1

1.5

Spe

(rad/s

)

T

Fig. 9. Speeds—Cr

exhibited a 10�4 rad accuracy). The control voltage canstabilize the position according to any step fraction even ifthe position reference value does not match with a wholenumber of steps at any time instant. The position trajectoryis quite smooth because the discontinuous control law actson the third time derivative of y.

1 1.2 1.4 1.6 1.8 2

ference positions

1 1.2 1.4 1.6 1.8 2

tion error

1 1.2 1.4 1.6 1.8 2

ed load torque perturbation

me (s)

rmax � 0:55Nm.

1 1.2 1.4 1.6 1.8 2

reference speeds

1 1.2 1.4 1.6 1.8 2

ed error

ime (s)

max � 0:55Nm.

ARTICLE IN PRESS

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.1

-0.1

-0.2

0.2

0.3

Motor and reference Id

(A)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

20

-20

-40

40

Voltage Va

(V)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

2

-2

-4

4Current Ia

(A)

Time (s)

Fig. 10. Voltages and currents—Crmax � 0:55Nm.

0 0.5 1 1.5 2

0

2

4

6

Motor and reference positions

(rad)

0 0.5 1 1.5 2

0

0.01

-0.01

-0.02

-0.03

0.02Position error

(rad)

0 0.5 1 1.5 2

0

0.2

-0.2

0.4

0.6

Estimated and measured load torque perturbation

(Nm

)

Time (s)

Fig. 11. Positions—Cr � Rþ 25%, Lþ 25%.

F. Nollet et al. / Control Engineering Practice 16 (2008) 429–443 441


Fig. 3 shows that the observer provides a goodestimation of the angular velocity of the motor. In Fig. 4,one can see the good tracking of the direct currenttrajectory. The error magnitude corresponds to theaccuracy of the current (ia and ib) sensors. The phasevoltage and current of the motor are also given.

0 0.5

0

5

-5

10

15

20

25

Estimated and

(ra

d/s

)

0 0.5

0

0.5

-0.5

-1

1

1.5

Spe

(ra

d/s

)

Ti

Fig. 12. Speeds—Cr � R

0

0.5

-0.5

Motor and

(A)

0

20

-20

-40

40

Volt

(V)

0 0.5

0 0.5

0 0.5

0

2

-2

-4

4

Cur

(A)

T

Fig. 13. Voltages and currents—

A second test is realized with tf ¼ 0:5 s, and thus areduced maximum rotor speed. This reference will be usedin the case of parametric uncertainties and load torqueperturbation in order to stay in the admissible voltage andcurrent limits. The results, given in Figs. 5–7, are stillsatisfactory.

1 1.5 2

reference speeds

1 1.5 2

ed error

me (s)

þ 25%, Lþ 25%.

reference Id

age Va

1 1.5 2

1 1.5 2

1 1.5 2

rent Ia

ime (s)

Cr � Rþ 25%, Lþ 25%.


4.3.2. With a load torque

The same control law has also been tested with asignificant time-varying load torque applied (see Fig. 8). Itsmaximum value is Crmax � 0:55Nm, that is to say 70% ofthe holding torque. Figs. 8–10 show that the controlobjective is still fulfilled, and that the estimate of the speedobserver stays good in spite of the influence of the loadtorque.

While the transient position error is slightly increased,the encoder accuracy is still obtained when the position isstabilized around the desired value. The load torqueestimation is quite similar to the signal delivered by theoutput of the load torque sensor. In Fig. 10, the evolutionof the voltages and currents that compensate for the loadtorque perturbation is given.

4.3.3. With a load torque and parametric uncertainties

Another experiment for testing the robustness issue withrespect to parameter variations has been realized byconsidering a 25% variation of both the resistance R andthe inductance L. The results shown in Figs. 11–13 indicatethat the control objective is still fulfilled.

5. Conclusion

In this paper, a flatness based second order sliding modecontrol combined with an angular velocity second ordersliding mode observer has been designed for the steppermotor. The use of the flatness property provides an easymethod of off-line trajectory planning while second ordersliding mode control has good robustness and trackingaccuracy properties. The second order sliding modeobserver achieves good real time estimation of the motorvelocity and the unknown time-varying load torqueperturbation. Experiments illustrated the efficiency of thesliding mode observer based controller for trajectorytracking and its capability to compensate for unknownload torques and parameter variations.

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