1

Hybrid Control of Induction Motorsvia Sampled Closed Representations

Bernardino Castillo–Toledo, Stefano Di Gennaro, Alexander G. Loukianov, and Jorge Rivera

Abstract—In this work a controller for IMs is proposed.A continuous feedback is first applied to obtain a discretetime model in closed form. Then, on the basis of these exactsampled dynamics, a discrete time controller ensuring speedand flux modulus reference tracking is determined, makinguse of the sliding mode technique. The resulting controllerishence hybrid, in the sense that it contains both continuousand discrete time terms. It is shown how to implement sucha hybrid controller using the so–called exponential holder,which is the only device to be implemented analogically,together with an analog integrator. Moreover, a discretetime reduced–order observer is designed for rotor fluxes andload torque estimation. The performance of the proposedcontroller is finally studied by simulation and experimentaltests.

Index Terms—Induction motors, Sampled closed form,Sliding mode control, Exponential holder.

I. I NTRODUCTION

Induction Motors (IMs), due to the characteristics oftheir nonlinear dynamics, constitute an interesting test–bed for the synthesis of advanced control laws. Moreover,IMs are widely used in industrial applications. Some ofthe reasons of their success are their simple mechanicalconstruction, their low maintenance requirements and thelower cost compared to other devices, such as DC orstepper motors. Starting from the pioneering work of [6],

Manuscript received July 21, 2007. Accepted for publication May 30,2008.

Copyright c© 2007 IEEE. Personal use of this material is permit-ted. However, permission to use this material for any other purposesmust be obtained from the IEEE by sending a request to pubs–permissions@ieee.org.

Work partially supported by the Consiglio Nazionale delle Ricerche(C.N.R., Italy), by the Ministero degli Affari Esteri (M.A.E., Italy), andby the Consejo Nacional de Ciencia y Tecnologıa (Conacyt, Mexico)under project 46538. A preliminary version of the present paper hasbeen presented at the 2004 American Control Conference, Boston,Massachusetts, USA.

B. Castillo–Toledo, A. G. Loukianov are with the Centro de Inves-tigacion y de Estudios Avanzados – CINVESTAV del IPN, UnidadGuadalajara, Av. Cientifıca, Col. El Bajıo, Zapopan, 45010, Jalisco,Mexico (e–mail:toledo, louk@gdl.cinvestav.mx) .

S. Di Gennaro is with the Department of Electrical and Infor-mation Engineering and Center of Excellence DEWS, Universityof L’Aquila, Poggio di Roio, 67040 L’Aquila, Italy (correspond-ing author: +39 0862 434461; fax: +39 06 233233142; e–mail:digennar@ing.univaq.it).

J. Rivera is with the Centro Universitario de Ciencias Exactas e Inge-nierıas, Departamento de Electronica, Universidad de Guadalajara, Av.Revolucion 1500, Col. Olimpica, 44430, Guadalajara, Jalisco, Mexico(e–mail:jorge.rivera@cucei.udg.mx).

field oriented control has become a classical technique forIM control, which involves nonlinear state transformationand feedback for the asymptotic decoupling of the rotorvelocity and flux modulus [12]. More recently, variousnonlinear control design approaches have been consideredto improve the performance. Active research areas includeinput–output feedback linearization [27], passivity [19],[43], adaptive backstepping [25], [41], sliding modes [11],[24], [38], [39], [42], [44], [47], among others.

The implementation of control laws designed for thesedevices is nowadays generally made in discrete time.In fact, the recent advancements in digital microproces-sor technology have rendered cheaper, simpler and moreflexible the discrete time implementation of controllersdesigned using continuous time techniques. The mainproblem here is the degradation of the expected perfor-mance, due to the fact that the design of the control lawis not made using the sampled dynamics of the system. Itis worth noting that the determination of an appropriatesampling period and the evaluation of its influence on thecontinuous controller performance are important issues,and presently a well–recognized methodology to deal withthese aspects has not been established yet.

This problem has motivated in the last decades the workof various researchers, with the aim of improving thecontrol performance by designing the controller directly onthe basis of the digital model (see [36], [37] and referencestherein). Once the digital model has been obtained, variousimportant issues regarding the controller performance canbe faced, such as parameter variations, observer design,determination of the sampling period, modeling of theactuator’s dynamics, etc. Clearly, the quality of the so-lutions given in this way depend on the accuracy of thedigital model. In this regard, unfortunately the problemof sampling continuous time systems is not trivial. Infact, in general a sampled closed representation of thesampled dynamics does not exist: while for linear systemsa sampled model in closed form can be easily obtained [4],for nonlinear systems in general the sampled data repre-sentations are given in form of infinite series [28], [29],[33], [34], [36]. Hence, in practice, one uses truncatedmodels of desired approximation [13], [18]. This difficultymotivates the following possible solutions:

1. The design of the control law in the continuous timesetting and its implementation by means of zero

2

order holder;2. The use of much simpler discretization methods,

such as those due to Euler (backward or forward),Tustin, etc., and the design of the control law in thedigital setting.

The first solution has the drawback of possible poorperformance of the resulting sampled controller. In orderto recover this performance, a multirate control approachcan be pursued, see [31] and its application to synchronousmotors [35] or to rigid spacecraft [15]. The second solu-tion [13] has the disadvantage that the accuracy of theresulting approximate discrete time system decreases asthe sampling period increases.

An alternative to these approaches is presented in [16],[32], where an original hybrid control methodology ispresented. First, a continuous control puts the system ina finite discretizable form, and then a discrete controlsatisfies the control objectives. An advantage of obtainingan exact discrete time model is that the sampling periodto be used in the implementation of the control lawcan be high, at least theoretically. Other aspects, suchas actuators, data acquisition, intersampling behavior etc.,limit in practice its value. Clearly, this methodology can beapplied provided that the continuous control can be appliedby means of analog devices. In this sense the resultingcontroller is hybrid, since it applies a continuous controlplus a digital one. It is evident that this is the critical pointof this methodology. In fact, if the control can be appliedonly in discrete time, the continuous control has to besampled, so that eventually the system is not in a closedform.

Following this idea, in this work we will show how toimplement for IMs a continuous control which puts thesystem in a form so that a sampled closed representationdoes exist. We will show that in this continuous control thesampled values of the variables, necessary to implementthe control, appear along with an exponential term, usuallycalled exponential holder. The latter is the only item tobe implemented via an analog device, easily realizable,together with an analog integrator. In a certain sense,the exponential holder and the analog integrator representthe “minimum requisites” that have to be implementedanalogically. Once the sampled data representation inclosed form of the IM dynamics has been obtained, wedesign a digital controller necessary to impose the controlspecifications, i.e. for rotor angular velocity and rotor fluxsquare modulus tracking. A discrete time reduced–orderobserver is then designed to avoid the measurement ofload torque and rotor flux, the latter achievable by Hallsensors, quite imprecise and expensive to place within themotor.

Hence, analogously to the case of finite discretizabil-ity [16], [32], we need to determine first a continuousfeedback in order to obtain a closed representation of

the sampled dynamics. It will be shown that, in ourcase, such a closed form can be obtained by means of asimple control. The resulting system is then characterizedby the dynamics of fluxes and currents which are linearin appropriate coordinates. Using the solutions of thesedynamics, the derivation of the discrete time equations forrotor angular position and velocity is then possible.

The paper is organized as follows. In Section 2 thecontinuous time IM model is briefly reviewed, and theexact sampled dynamics of the IM are derived. In Sec-tion 3 a discrete time sliding mode control for rotorangular velocity and rotor flux square modulus trackingis designed. To remove the hypothesis of rotor flux andload torque measurements, a reduced discrete time ob-server is proposed in Section 4. Moreover, the practicalimplementation of the proposed controller by means ofan exponential holder are discussed. Section 5 presentssome simulation and experimental results. Final commentsconclude the paper.

II. SAMPLED DYNAMICS OF INDUCTION MOTORS

In the following a sampled version of the dynamics ofan IM will be determined. Under the assumptions of equalmutual inductance and linear magnetic circuit, the IMmodel in the stator reference frame(a, b) is given by [27]

φr = −αφr + pωℑφr + αLmIs

Is = αβφr − pβωℑφr − γIs +1

Lσ

u

ω = µITs ℑφr −

1

JT

θ = ω

(1)

whereθ andω are the rotor angular position and velocityrespectively,T is a bounded load torque,J is the rotormoment of inertia,

φr =

(

φr,a

φr,b

)

, Is =

(

is,a

is,b

)

, u =

(

ua

ub

)

are the rotor flux vector, the stator current vector, thecontrol input voltage vector, and

α =Rr

Lr

, β =Lm

LσLr

, γ =1

Lσ

(

L2m

L2r

Rr + Rs

)

Lσ = Ls −L2

m

Lr

, µ =3

2

pLm

JLr

, ℑ =

(

0 −11 0

)

.

Here Ls, Lr Lm are the stator, rotor and mutual in-ductances, respectively,Rs and Rr are the stator androtor resistances, andp is the number of pole pairs. It isworth noting thatℑ represents the anticlockwise rotationoperator ofπ/2.

The following assumption, acceptable whenT variesslowly with respect to the electrical dynamics [44], will beinstrumental for deriving the sampled model of the motordynamics.

3

(H1) The load torqueT can be approximated by a signalC which is piece–wise constant over the samplingperiodδ. ⋄

In order to obtain sampled dynamics of the systemdynamics (1) in closed form, in the spirit of [16], [32],let us consider first the following feedback

u = uf (t) + epθℑvk, uf = pLσωℑ(Is + βφr) (2)

with uf the continuous control which makes the systemfinite discretizable.

Remark 1: Note thatuf in (2) and the termepθℑ mustbe implemented by analog devices, while the new controlvk = v(kδ) =

(

va,k vb,k

)T, designed on the basis of

the discrete time representation of the system derived inthe following, can be implemented by a digital computerand a zero–order holder. In Section IV it will be explainedhow to determineuf from the sampled measurements, andusing analog devices given by an integrator and a so–calledexponential holder [8], [9]. The resulting controller willbehybrid, namely it will contain discrete and continuous timeterms. ⋄

With the continuous feedback (2) and under (H1) thefollowing controlled dynamics are obtained

φr = −αφr + pωℑφr + αLmIs

Is = αβφr − γIs + pωℑIs +1

Lσ

epθℑvk

ω = µITs ℑφr −

1

JC

θ = ω.

(3)

As common in the IM’s literature [27], we consider here-inafter a change of coordinates in an appropriate rotatingframe [23]. In particular, we consider a transformationwhich puts in light that the flux and current dynamicsare linear, with the current dynamics linear thanks to thefeedback (2). Hence, we consider the globally definedchange of coordinates

ΦIωθ

=

e−pθℑφr

e−pθℑIs

ωθ

, e−pθℑ =

(

cos pθ sin pθ

− sin pθ cos pθ

)

(4)with e−pθℑ an orthogonal matrix. The transformed vari-ablesΦ, I in (4) are the rotor flux and the stator currentrotated according to the electrical rotor angular positionpθ. Obviously, (4) preserves the vectors norms. Notealso that the complex notatione−jpθ(Va + jVb) could beequivalently used for the termse−pθℑV , V = φr, Is in (4).

In the new variables (4) and with the inputvk, the

dynamics (3) are expressed as follows

Φ = −αΦ + αLmI

I = αβΦ − γI +1

Lσ

vk

ω = µITℑΦ − 1

JC

θ = ω

(5)

since de−pθℑ/dt = −pωℑe−pθℑ, e−pθℑℑ = ℑe−pθℑ.Note that equations (5) are nonlinear, but the sampledclosed form is now easily obtained noting that the dy-namics forΦ and I are linear, and the controlvk willbe designed to be constant over the sampling periodδ. Denoting Φk = Φ(kδ), Ik = I(kδ), ωk = ω(kδ)and Ck = C(kδ), and considering as outputs to becontrolled the rotor angular velocity and the rotor fluxsquare modulus, some calculations given in Appendix 1yield the closed representationfor the sampled dynamicsof system (5)

Φk+1 = a11Φk + a12Ik + b1vk

Ik+1 = a21Φk + a22Ik + b2vk

ωk+1 = ωk + η1ITk ℑΦk +

(

η2Φk + η3Ik

)Tℑvk − Ck

Jδ

θk+1 = θk + ωkδ + κ1ITk ℑΦk +

(

κ2Φk + κ3Ik

)Tℑvk

− Ck

J

δ2

2(6)

wherea11, a12, a21, a22, b1, b2, ηi, κi, i = 1, 2, 3, areconstants. Note that the matrix

Ad =

(

a11II2×2 a12II2×2

a21II2×2 a22II2×2

)

= A ⊗ II2×2 (7)

with II2×2 the identity matrix2 × 2 and⊗ the Kroneckerproduct, is Schur sinceA is Shur for every choice of thesampling periodδ (see Appendix 1).

Thecontrol problemis to force the rotor angular veloc-ity ωk and the rotor flux modulusΦT

k Φk to track somedesired referencesωr,k, Φr,k, ensuring at the same timedisturbance rejection. We assume that these references arebounded with bounded increments, where

|∆ωr,k| = |ωr,k+1 − ωr,k| ≤ ∆ωr,max

|∆Φk| = |Φr,k+1 − Φr,k| ≤ ∆Φmax.

Moreover, also the load torqueCk is bounded by a valueCmax. Furthermore, we assume that theCk has boundedincrements. Finally, we need to ensure that this incrementvanishes whenδ is zero. This can be formalized with thefollowing assumption.

(H2) The piece–wise load torqueCk is such thatCk+1 =Ck + ∆C(k, δ) with, for each integerk

|∆C(k, δ)| ≤ ∆Cmax∆(δ), limδ→0

∆(δ) = 0. ⋄

The control problem will be solved in the next section bymeans of a discrete time sliding mode control.

4

Remark 2: As last observation, it is interesting to com-pareuf in (2) with the well–known Field Oriented Control(FOC) [12]. The FOC can be determined considering the(d, q)–model

φr,dq = −αφr,dq + pωℑφr,dq + αLmIs,dq − ǫ0ℑφr,dq

=

(−αφr,d + αLmis,d

0

)

Is,dq = αβφr,dq − pβωℑφr,dq − γIs,dq

+1

Lσ

udq − ǫ0ℑIs,dq

=

(

αβφr,d − γis,d + ǫ0is,q + 1Lσ

ud

−pβωφr,d − γis,q − ǫ0is,d + 1Lσ

uq

)

ω = µITs,dqℑφr,dq −

1

JT = µφr,dis,q −

1

JT

θ = ω

(8)

obtained from (1) considering

Idq=

(

is,d

is,q

)

= e−ǫ0ℑIs, φr,dq =

(

φr,d

0

)

= e−ǫ0ℑφr

udq=

(

ud

uq

)

= e−ǫ0ℑu, ǫ0 = pω +αLm

φr,d

is,q

(9)It is not surprising that (9) resembles (4), since theidea behind the FOC is to have linear flux and currentsdynamics, as in the finite discretization. In both cases,first one determines a transformation which renders linearthe flux dynamics. Then, one looks for a control lawwhich cancels the nonlinearities in the current dynamics.Indeed, the transformation (9) is different from (4), andthe resulting models (5), (8) are different in the currentdynamics. This leads to different control expressions. Infact, the FOC is given by [12]

udq = Lσ

(

− αβφr,dq + pβωℑφr,dq + ǫ0ℑIs,dq + vdq

)

= Lσ

−αβφr,d − ǫ0is,q − αLm

φr,di2s,q + vd

pβωφr,d + ǫ0is,d + αLm

φr,dis,dis,q + vq

(10)

wherevdq =(

vd vq

)T

, with

vd = −kd,1(φr,d − φref) − ki,1

∫ t

0

(

φr,d(τ) − φref(τ))

dτ

vq = −kd,2(ωr,d − ωref) − ki,2

∫ t

0

(

ωr,d(τ) − ωref(τ))

dτ

and φref , ωref the flux and angular velocity references,kd,1, kd,2, ki,1, ki,2 > 0. One hence obtains the followingclosed–loop dynamics

φr,dq = −αφr,dq + pωℑφr,dq + αLmIs,dq − ǫ0ℑφr,dq

=

(−αφr,d + αLmis,d

0

)

Is,dq = −γIs,dq + vdq

ω = µITs,dqℑφr,dq −

1

JT = µφr,dis,q −

1

JT

θ = ω

(11)

From (10) it is clear that FOC imposes the feedbacklinearization of (8) [22]. Hence, the FOC eventually ob-tains a linear model for the flux and current dynamics,as uf in (2). In this sense, the FOC can be appliedto obtain a finite discretizable system as well, namelyfrom (11) one could obtain an exact closed form for thesampled dynamics. Indeed, it easy to see thatuf is moregeneral than FOC, since it cancels just the terms which donot allow the finite discretizability. In fact, in the(d, q)coordinatesuf can be written as

uf,dq = e−ǫ0ℑuf = Lσ

(

pβωℑφr,dq + pωℑIs,dq

)

and it is readily checked that

udq = uf,dq + Lσ

(

− αβφr,dq +αLm

φr,d

is,qℑIs,dq + vdq

)

and it is clear that the implementation of the FOC wouldimply also the implementation of further terms. ⋄

III. D ISCRETE–TIME CONTROL OF INDUCTION

MOTORS

The discrete time controlvk in (2) is designed here-inafter, in order to solve the posed control problem, fol-lowing the sliding–mode approach. For, we first derive theexpression of the tracking error dynamicse1,k = ωk−ωr,k,e2,k = ΦT

k Φk −Φr,k, which are the new outputs which wewant to force to zero. Using (6), the error dynamics canbe written as follows

e1,k+1 = ωk + η1ITk ℑΦk + λT

1,kvk − δ

JCk − ωr,k+1

e2,k+1 = a211Φ

Tk Φk + 2a11a12Φ

Tk Ik + a2

12ITk Ik

+ λT2,kvk + b2

1vTk vk − Φr,k+1

(12)where

λ1,k = −ℑ(

η2Φk + η3Ik

)

, λ2,k = 2b1

(

a11Φk + a12Ik

)

.(13)

It is interesting to note thatλ1,k represents the vectorη2Φk + η3Ik rotatedπ/2 clockwise.

Notice that the system (12) depends on quadratic termsof the control signalvk sinceb1 > 0 for all sampling timeδ > 0. This complicates the control design procedure. Inorder to simplify the design of a control law, one considersas new control

Vk =

(Va,k

Vb,k

)

= Tkvk, Tk =

(

λT1,k

λT2,k

)

(14)

vk = T −1k Vk. It can be verified that the transformationTk

is invertible for allk, with

det Tk = dk = λT2,kℑλ1,k = 2b1

(

Φk

Ik

)T

R

(

Φk

Ik

)

6= 0

5

(b1 > 0) and

R=

(

a11η2II2×212(a12η2 + a11η3)II2×2

12(a12η2 + a11η3)II2×2 a12η3II2×2

)

(15)

nonsingular for allδ (see Appendix 1). Note that

T −1k =

1

dk

ℑ(

−λ2,k λ1,k

)

and

(

T −1k

)T(

T −1k

)

=1

d2k

(

λT2,kλ2,k −λT

1,kλ2,k

−λT1,kλ2,k λT

1,kλ1,k

)

.

With such a control transformation,e1,k+1, e2,k+1 de-pend only onVa,k, Vb,k, respectively, and equations (12)become

e1,k+1 = ωk + η1ITk ℑΦk − δ

JCk − ωr,k+1 + Va,k

e2,k+1 = akV2b,k + bkVb,k + ck

(16)

where

ak =b21

d2k

λT1,kλ1,k ≥ 0, bk = 1 − 2b2

1

d2k

λT1,kλ2,kVa,k

ck =1

4b21

λT2,kλ2,k +

b21

d2k

λT2,kλ2,kV2

a,k − Φr,k+1

=

(

1

4b21

+b21

d2k

V2a,k

)

λT2,kλ2,k − Φr,k+1

=

(

Φk

Ik

)T(

a211 a11a12

a11a12 a212

)

(

Φk

Ik

)

+b21

d2k

λT2,kλ2,kV2

a,k − Φr,k+1.

It is worth noting thatak is zero whenΦk = −η3Ik/η2.A discrete time sliding–mode control will be designed

hereinafter, to force to zero the error dynamics. A naturalchoice as sliding functions are the errorse1,k, e2,k.

Since the control objective is to design a controller forangular velocity tracking and disturbance rejection, con-sideringe1,k as sliding surface, one can impose that thistracking error be zeroed in one step, namelye1,k+1 = 0.Considering the first of (16), this is achieved considering

Va,k = −ωk − η1ITk ℑΦk +

δ

JCk + ωr,k+1. (17)

Usually, such a control is called equivalent control, andis such that the sliding manifolde1,k = 0 is renderedinvariant.

Analogously, we determineVb,k imposing that the track-ing error e2,k is zeroed in one or more steps, or tendsasymptotically to zero. The determination of the equiva-lent control follows from simple considerations, reportedhereinafter.

1. If ak > 0 and ∆k = b2k − 4akck ≥ 0, considering

the second of (16), one imposesakV2b,k + bkVb,k +

ck = 0. Hence, the control

Vb,k = − bk

2ak

±√

∆k

2ak

straightforwardly imposese2,k+1 = 0, i.e. ensuresthat e2,k is zeroed in one step.

2. If ak > 0 and∆k < 0, thene2,k can not be zeroedin one step. In this case we can only determineVb,k so thate2,k decreases at each time step. Thiscan be achieved as follows. For a certainr > 0,representing the duration of the decrease requiredon e2,k, one iteratively imposes

e2,k+j+1 = ak+jV2b,k+j + bk+jVb,k+j + ck+j = pj

j = 0, · · · , r, with pj chosen so that|e2,k+j+1| <|e2,k+j |, and such that the new discriminant

∆k,j = b2k+j − 4ak+j(ck+j − pj)

is equal to a fixed minimum value∆min,j ≥ 0.Hence, pj = ck+j + (∆min,j − b2

k+j)/(4ak+j).Therefore, one easily works out the values of thecontrolVβ ensuring this decrease

Vb,k+j =−bk+j ±

√

∆min,j

2ak+j

, j = 0, · · · , r.

With this choicee2,k tends asymptotically to zero.Clearly, the fastest decrease ofe2,k is obtained for∆min,j = 0, i.e. with Vb,k+j = bk+j/(2ak+j), j =0, · · · , r, namely imposingVb,k = bk/(2ak) whenak > 0 and ∆k < 0. Clearly if, afterr steps,∆k

become nonnegative,Vb,k will be chosen as in thepoint 1. (or 3.) so obtaining zero error inr+1 steps.

3. If ak = 0, then it is possible to impose straight-forwardly e2,k+1 = 0 in one step with the controlVb,k = −ck/bk.

We can resume this discussion saying that with thefollowing equivalent control

Vb,k =

− bk

2ak±

√∆k

2akif ak > 0 and∆k ≥ 0

− bk

2akif ak > 0 and∆k < 0

−ck

bkif ak = 0

(18)

with ∆k = b2k − 4akck, the flux tracking error is zeroed

in one or more steps, or tends asymptotically to zero.We have shown thate1,k, e2,k are forced to zero asymp-

totically. One should ensure thatIk does not diverge toinfinity. This aspect is correlated with the stability of whatare called the zero dynamics which, roughly speaking, arethose dynamics obtained when the output of the systemis zero (see [22] for the continuous time setting and [30]for the discrete time case). We do not carry out a formal

6

study of this aspect (but in Appendix C we give some moredetails), and we just mention thatIk remains bounded (thiscan be also seen from the experimental results given inthe following), while the flux angular positionθΦ,k andthe rotor angular positionθk in practice take value in[−π, π). This prevents the controlvk from saturating, andthe suggested scheme can be applied to solve the posedcontrol problem.

IV. D ISCRETE–TIME CONTROL FROM

SAMPLED MEASUREMENTS

In this section we consider and fix the drawbacks ofthe proposed control (2), (14). The first problem is themeasurability of the rotor flux and of the load torque. Thisproblem is solved by a simple discrete time reduced–orderobserver, as shown in Section IV-A. The second problemconcerns how the continuous feedback (2) is implemented,i.e. the availability of the signalsθ(t), Is(t), φr(t). Thisaspect is considered in Section IV-B.

A. Flux and Load Torque Observers

In practical cases, the rotor flux and load torque arenot measured, and this is a crucial aspect in the realimplementations. The literature presents various works onthis subject, in the case of continuous time see [1], [3],[5], [14], [21], [24], [38], [40], [45] among the others. Lessnumerous are the results in the discrete time setting [2],[7], [10]. In the following we derive flux and torquediscrete time observers for the discrete model (6). The fluxobserver is proposed to have the following (quite obvious)form

Φk+1 = a11Φk + a12Ik + b1vk. (19)

Then, the dynamical error equation becomes

eΦ,k+1 = a11eΦ,k, eΦ,k = Φk − Φk.

It can be verified that|a11| < 1 (see expression inAppendix A). Hence,Φk asymptotically converges toΦk.

As far as the load torque estimation is concerned,we have assumed that(H2) holds. Then, the followingobserver can be used

ωk+1 = ωk + η1ITk ℑΦk +

(

η2ΦTk + η3I

Tk

)

ℑvk

− δ

JCk + l1(ωk − ωk)

Ck+1 = Ck + l2(ωk − ωk).

(20)

Note that the estimationωk is necessary only to determineCk. Setting eω,k = ωk − ωk and eC,k = Ck − Ck asrotor angular velocity and load torque estimate errors,respectively, the dynamical error equations become

(

eω,k+1

eC,k+1

)

=

(

−l1 − δJ

−l2 1

)

(

eω,k

eC,k

)

+

(

η1ITk − η2v

Tk

0

)

ℑeΦ,k +

(

0

1

)

∆C(k, δ).

(21)

Note that eΦ,k tends asymptotically to zero andη1ITk

andη2vTk are bounded terms. Due to the presence of the

term ∆C(k, δ), which acts as a perturbation, choosingl1 and l2 such that the state matrix in (21) is Schur,then ωk and Ck converge asymptotically toωk and Ck

in a practical sense [17], namely the trajectory of theestimation error belongs to a neighborhood of radiusr ofthe origin. Note that, on the basis of the assumption(H2),the perturbation term∆C(k, δ) can be made small byreducing the sampling periodδ, namely

|∆C(k, δ)| ≤ ∆Cmax∆(δ) ≤ ∆Cmax∆(δ0)

for δ ≤ δ0, with δ0 sufficiently small. In this way oneobtains a smaller value for the radiusr and, hence, a betterobserver performance.

B. Implementation of (2)

The main issue concerning the implementation of thecontrol law designed here is the realization of the contin-uous part of control (2), where the stator currentIs(t),the rotor fluxφr(t) = epθ(t)ℑΦ(t) and the rotor angularposition and velocityθ(t), ω(t) are required. For, weshow here that this continuous control can be written asa combination of the sampled valuesθk, ωk, Φk. Thecoefficients of this combination depend on a time varyingterm, usually called exponential holder. This exponentialholder is the only term to be implemented via an analogdevice, together with an analog integrator.

The stator currentIs can be measured by analog de-vices, while the rotor angular position and velocityθ, ωare commonly measured by a digital device, such as anencoder, meaning that their valuesθk, ωk are availableat sampling instants. Furthermore,φr is estimated by thediscrete time observer (19), (20), and only the valuesφr,k = epθkℑΦk are available at sampling instants.

Therefore, we need to reproduce the continuous timesignalsθ(t), ω(t), φr(t) from knowledge of the valuesθk, ωk, Φk. This can be done as follows. Referring toAppendix A and considering equations (24), (25), settingτ = t − kδ, τ ∈ [0, δ), and consideringΦk in theplace ofΦk, for Φ(t) andω(t) one obtains the followingexpressions

Φ(t) = Φ(τ + kδ) = ϕ1(τ)Φk + ϕ2(τ)Ik + ζ1(τ)vk

ω(t) = ω(τ + kδ) = ωk + ν1(τ)ITk ℑΦk

+ ν2(τ)ΦTk ℑvk + ν3(τ)IT

k ℑvk − Ck

Jτ.

(22)

Note that the expressions ofϕ1(τ), ϕ2(τ), ζ1(τ), νi(τ),i = 1, 2, 3, contain terms of the formeςτ , ς = ±ρor ς = ±ω0, see Appendix A. In fact, the hyperbolictrigonometric functions can be represented with exponen-tial terms. The termeςτ is usually calledexponential

7

holder, and is physically realizable by an analog de-vice [8], [9]. In particular, in [9] an electronic realizationof such device is proposed and a circuit realization isgiven for sinusoidal signals. In some sense, the exponentialholder represents the “minimum requisite” that has to beimplemented analogically to reproduceΦ(t), ω(t).

Therefore, (22) allows us to reconstruct the values ofΦ(t), ω(t), for all t. Onceω(t) has been reconstructed, itcan be integrated in order to obtainθ(t) by means of ananalog integrator device.

V. SIMULATION AND EXPERIMENTAL RESULTS

In this section we first compare the proposed controlscheme with different control laws, by means of simula-tions, and then we describe the experimental results.

A. Simulation Results

We considered an IM with the following plate data:three–phase, two pole, 0.14 KW, 0.74 p.f., 1600 rpm, 220V, 1.3 A, and withRr = 10.1, Rs = 14 Ω, Ls = 400,Lr = 412.8, Lm = 377 mH, J = 0.01 Kg m2.

Hence, α = 24.467 Ω/H, β = 16.398 1/H, γ =402.62 Ω/H, µ = 2.7398 1/(Kg m2), Lσ = 5.5695×10−2

H. Moreover,ω0 = 198.6 Ω/H, ρ = 213.5 Ω/H, whereΩ / H = Hz. The sampling period isδ = 230 µs, whichis a typical value in applications, see for instance [5],and which is in accordance with the value used in theexperimental section below. It is worth noting that thechoice of the sampling period is an important aspect indigital implementation of control laws, and one popularway to check the influence of the sampling periodδon the controller performance is by simulation. In theintroduction we have already noted that an advantage ofobtaining an exact discrete time model is that the samplingperiod can be theoretically high, although other aspectslimit in practice its value.

The gain control parameters have been chosen equalto l1 = 0.0824, l2 = −0.8244. With these values, it ispossible to check that matrix (7) is Schur. A load torqueT of 0.2 Nm, with a variation to 1 Nm att = 4.6 s, hasbeen considered in the simulations.

In order to compare different control solutions, we haveconsidered four control schemes

a) The sampling of the classical FOC, see equa-tion (10), with the state observer proposed in [46].

b) Digital control and observer determined on the basisof the Euler approximation [26]

φr,k+1 = φr,k + δ(

− Ψkφr,k + αLmIs,k

)

Is,k+1 = Is,k + δ(

βΨkφr,k − γIs,k +1

Lσ

uk

)

ωk+1 = ωk + δ(

µITs,kℑφr,k − 1

JTk

)

θk+1 = θk + δωk

of the dynamics (1), withΨk = αId −pωkℑ, andId

the identity matrix.c) Control law given by (14), (19), (20), and the

sampling ofuf in (2), namely

uk = pLσωkℑ(

Is,k + βφr,k

)

+ epθkℑvk (23)

where(

φr,k

Is,k

)

=

(

epθkℑΦk

epθkℑIk

)

.

d) Control law given by (14), (19), (20), and (2)implemented as explained in Section IV.B.

e) Control law as in d), but with a doubled samplingperiodδ = 460 µs.

In Figures 1,?? the simulation results for these fiveschemes are compared. The rotor angular velocity ref-erence increases from 0 to 100 rad/s in the first 20 s,and then is piece–wise constant and equal to 70 rad/sfor t ∈ [30, 40), and to 40 rad/s fort ∈ [40, 50]. Theflux modulus reference has a constant value of 0.2 Wb2.The other variables have not been reported due to spacelimitations. Note that the behaviors of the controllers c)and d) are practically coincident, at least for the chosensampling rate. This means that the simple sampling of (2)is an effective solution, which avoids the use of analogdevices, at least for sufficiently small sampling periodsδ. In fact, it is obvious that asδ → 0 one recoversthe continuous behavior. Anyway, on the basis of thecomments already done, there is no theoretical assurancein the case c) that the digital representation is still ac-curate asδ increases. It is also worth noting that thecontroller d) performs better with respect to the sampledFOC or a digital controller determined on the basis ofsimpler discrete time representations, such as the Eulerapproximation. The controller e) shows a behavior similarto that in d), since the exact continuous reproduction isensured at the sampling time by the sampled closed model,in accordance with the theory. Note that nothing can beensured for the intersamplings.

As final consideration, since the implementation ofthese control laws requires the knowledge of the angularvelocity, whose measurement is usually quite poor inpractice, a degeneration of the controller performance isexpected due to measurement noise. In order to evaluatethis important aspect, in Figures??, ??we have consideredthe previous controllers with measurement noise. In thiscase, the performance of the controllers c) and d) arepractically the same. But, the degradation in performanceof controller b) and the sensitivity of controller a) withrespect to load torque variations suggest the superiority ofthe proposed controller d).

B. Experimental Results

The experimental setup, shown in Figure??, consistsof an IM with the same characteristics given in the

8

simulation section. The power module is a three–phaseinverter Semikron IGBT Power Electronics Teaching Sys-tem, which incorporates a three–phase rectifier fed by athree–phase AC variator. The control and PWM generationare implemented with a DSP board (Dspace 1104). Thesignals from the phase currents are acquired by analog–to–digital converters included in the DSP board.

The motor is loaded by a dc generator connected to avariable–power resistor. In the tests, we reproduced a loadtorqueT of 0.2 Nm, with a variation to 1 Nm att = 4.6 s,as in the simulations.

The motor position and speed are measured by anoptical encoder with 1024 ppr (pulse per revolution) ofresolution and filtered by a first–order Butterworth low–pass filter having a 8 rad/s edge frequency, to attenuatethe measurement noise. The stator phase currents aremeasured by Hall–type sensors.

On the base of the time constraints of the imple-mentation, the sampling period has been fixed equal toδ = 230 µs, which is a typical value in applications, asalready observed. However, we recall that an advantage ofobtaining an exact discrete time model is that the samplingperiod to be used in the implementation of the control lawcan be high, at least theoretically. Other aspects, such asactuators, data acquisition, etc., limit in practice its value.

Figure ?? shows the output tracking results. The ref-erences have been chosen as in the simulation section.The results are similar to those in the simulation section,showing slight deviations corresponding to step variations.These differences can be attributed to the high–frequencyeffects of the unmodelled dynamics of the actuator, notconsidered in the mathematical model. Finally, Figure??gives the experimental behaviors of the componentsis,a,k,is,b,k of the stator current vectorIs,k, and those of theinput voltagesua,k, ub,k.

VI. CONCLUSIONS

In this work, an exact sampled data representationof the IM has been obtained. This sampled data modelhas been determined by means of a particular controllerwhich induces a sampled closed representation for theoriginal continuous time model. On the basis of the exactsampled data model, an observer–based controller, whichguarantees asymptotic reference tracking for the velocityand flux in presence of an unknown load torque, has beendesigned. Some interesting issues, such as the robustnessof the controller with respect to parameter variations,determination of the sampling period, modeling of theactuator’s dynamics, are currently under study.

ACKNOWLEDGEMENTS

The authors thank the anonymous referees for theirvaluable comments which improved the paper.

APPENDIX ASAMPLED DYNAMICS OF SYSTEM (8)

Integrating equations (5) on the time interval[kδ, t),t < (k + 1)δ, one obtains

(

Φ(t)

I(t)

)

=

(

ϕ1(t − kδ)II2×2 ϕ2(t − kδ)II2×2

ϕ3(t − kδ)II2×2 ϕ4(t − kδ)II2×2

)(

Φk

Ik

)

+

(

ζ1(t − kδ)II2×2

ζ2(t − kδ)II2×2

)

vk

ω(t) = ωk + µ

∫ t

kδ

ITℑΦ dτ − Ck

J(t − kδ)

θ(t) = θk +

∫ t

kδ

ω(τ) dτ

(24)

where, denoting “sinh”, “cosh” by “sinh”, “ cosh”, respec-tively

ϕ1(t) = e−ρt

[

γ − α

2ω0

sinhω0t + coshω0t

]

ϕ2(t) =αLm

ω0e−ρt sinhω0t

ϕ3(t) =αβ

ω0e−ρt sinhω0t

ϕ4(t) = e−ρt

[

− γ − α

2ω0

sinhω0t + coshω0t

]

ζ1(t) =Lm

Rs

[

1 − e−ρt

[

ρ

ω0sinhω0t + coshω0t

]]

ζ2(t) =1

Rs

[

1 − e−ρt

[

1

ω0

(

ρ − Rs

Lσ

)

sinhω0t + coshω0t

]]

with ρ = (α + γ)/2 = (LsRr/Lr + Rs)/(2Lσ), and

ω0 =√

(γ − α)2 + 4α2βLm/2 =√

ρ2 − αRs/Lσ

γ − α =(

Rs + (2L2m − LsLr)Rr/L2

r

)

/Lσ.

Note that for our case,γ − α > 0.For the expression ofω(t) andθ(t) one works out

ω(t) = ωk + ν1(t − kδ)ITk ℑΦk + ν2(t − kδ)ΦT

k ℑvk

+ ν3(t − kδ)ITk ℑvk − Ck

J(t − kδ)

θ(t) = θk + (t − kδ)ωk + ν4(t − kδ)ITk ℑΦk

+ ν5(t − kδ)ΦTk ℑvk + ν6(t − kδ)IT

k ℑvk

− 1

2(t − kδ)2

Ck

J

(25)

where

ν1(t) =µ

ρe−ρt sinh ρt

9

ν2(t) =µ

αRs

e−ρt

[

γ − α

2

(

1

ω0sinhω0t −

1

ρsinh ρt

)

+ coshω0t − cosh ρt

]

ν3(t) =µLm

Rs

e−ρt

[

1

ω0sinhω0t −

1

ρsinh ρt

]

ν4(t) =µ2

α + γ

[

t − e−ρt

ρsinh ρt

]

ν5(t) =µ2Lσ

α2R2s

[

γ − e−ρt

[

(γ − α)γ + 2α2βLm

2ω0

sinhω0t

+γ coshω0t

]]

− µ2

(α + γ)Rs

[

e−ρt

ρsinh ρt +

γ

αt

]

ν6(t) =µ2LmLσ

αR2s

[

1 − e−ρt

[

α + γ

ω0sinhω0t

+ coshω0t

]]

− µ2Lm

Rs

e−ρt

ρ2sinh ρt.

For t = (k + 1)δ we finally obtain the expressions in (6)with

a11= ϕ1(δ), a12= ϕ2(δ), a21 = ϕ3(δ), a22 = ϕ4(δ)

b1= ζ1(δ), b2 = ζ2(δ), η1 = ν1(δ), η2 = ν2(δ)

η3= ν3(δ), κ1 = ν4(δ), κ2 = ν5(δ), κ3 = ν6(δ).

Note that the coefficientb1 is always positive for allδ > 0,as can be verified by plottingb1 as a function ofδ. Notealso that the matrixAd in (7) can be easily transformedinto the matrix diagA, A, where

A =

(

a11 a12

a21 a22

)

.

A is a nonsingular matrix with determinant|A| = e−ρδ,and with characteristic polynomialp(λ) = λ2 + a1λ+ a0,a1 = −2e−ρδ coshω0δ, a0 = |A| = e−ρδ. Applying Jury’scriterion [4], one gets that all its roots are inside the unitcircle if and only if a0 = e−ρδ < 1, a0 ± a1 + 1 =e−ρδ ± 2e−ρδ coshω0δ + 1 > 0. Sinceρ > 0 ande−ρδ ±2e−ρδ coshω0δ+1 is always positive, one easily concludesthat Ad is Schur for every choice of the sampling periodδ > 0.

Finally, considering the matrixR given by (15), oneworks out that the determinant is

detR = (a11η3 − a12η2)4/16

where

a11η3 − a12η2 =µLm

ω0Rs

e−2ρδ(

sinhω0δ cosh ρδ

− ω0

ρcoshω0δ sinh ρδ

)

=µLm

ω0Rs

dR,δ

(

tanhω0δ −ω0

ρtanh ρδ

)

withdR,δ = e−2ρδ coshω0δ cosh ρδ.

Here,tanhω0δ/ tanhρδ − ω0/ρ is a positive, increasingfunction of δ, anddR,δ is different from zero forδ 6= 0.

APPENDIX BSAMPLING OF THE (a, b) ROTOR FLUX VECTOR

ANGULAR POSITION

In the stator reference frame (a, b), the rotor flux at timekδ can be obtained from (4)

φr,k =

(

φr,a,k

φr,b,k

)

= epθkℑΦk, Φk =

(

φa,k

φb,k

)

=

(

cos pθk − sin pθk

sin pθk cos pθk

)(

φa,k

φb,k

)

=

(

φa,k cos pθk − φb,k sin pθk

φa,k sin pθk + φb,k cos pθk

)

.

Hence, the angular position ofφr,k is given by

θΦ,k = arctanφr,b,k

φr,a,k

=φa,k sin pθk + φb,k cos pθk

φa,k cos pθk − φb,k sin pθk

= pθk + arctanφb,k

φa,k

(26)

where some trigonometric manipulations have been per-formed.

Moreover, considering that

φr,k =√

ΦTk Φk

(

cos θΦ,k

sin θΦ,k

)

one gets

Φk = e−pθkℑφr,k

=

(

cos pθk sin pθk

− sin pθk cos pθk

)

√

ΦTk Φk

(

cos θΦ,k

sin θΦ,k

)

=√

e2,k + Φr,k ϑk

with ΦTk Φk = e2,k + Φr,k, and where we have defined

ϑk =

(

cos(θΦ,k − pθk)

sin(θΦ,k − pθk)

)

.

APPENDIX CZERO DYNAMICS

The difficulties in studing the stability of the whole con-trolled system are due to the fact that the strict feedbackform [20], as that of the induction motor dynamics, is lostunder sampling. To the best of the authors’ knowledge,there are not general results on the stability of suchsampled systems. In the following we analyze the socalled zero dynamics and we give some results. Thesystem dynamics can be rewritten using the new state

10

variables(

e1,k e2,k ITk θΦ,k θk

)T, where the first

two variables are exactly the outputs of the system, andwhere the “complement” of this state transformation isgiven by the currentIk, the rotor flux angular positionθΦ,k (see (26)) and the rotor angular positionθk. The zerodynamics are the dynamics of the complement variableswhen the outputs are zero.

The current dynamics (the second of (6))

Ik+1 = a22Ik + Wk

Wk = a21

√

e2,k + Φr,k ϑk + b2T −1k Vk

have eigenvalues inside the unit circle (a22 ∈ (0, 1) forall δ > 0 – see Appendix A), so that these dynamicsare bounded input – bounded output. Unfortunately, it isnot evident how to show that the “input”Wk is bounded.In fact, while ϑk is given by bounded (trigonometric)functions,Φr,k is bounded ande2,k tends to zero, it iseasy to check thatvk = T −1

k Vk depends explicitely onIk. A possible solution to circumvent this difficulty isto show the existence of a steady state functionIr,k forIk. For, we first get rid ofuk between the flux andcurrent equations (first two equations of (6)) consideringthe variableIk = b2Φk − b1Ik. One easily obtains

Ik+1 = λIk + bΦk, Φk =√

e2,k + Φr,k ϑk

where

λ = a22 − a12b2/b1

b = (a11 − a22)b2 − a21b1 + a12b22/b1.

It is possible to check that|λ| < 1 (see Appendix A).These linear dynamics are bounded input – boundedoutput, forced by a bounded inputΦk (e2,k goes to zero,Φr,k is bounded by assumption,ϑk has trigonometric –sin and cosine – components). Hence,Ik is bounded andtherefore also

Ik =b2

b1

Φk − 1

b1

Ik

is bounded. Moreover, there exists a steady state

Ir,k =

k−1∑

j=0

λk−j−1 b√

Φr,j ϑj

solution of the linear equation

Ir,k+1 = λIr,k + b√

Φr,k ϑk.

It is easy to see thatIk converges toIr,k. In fact,

Ik+1 − Ir,k+1 = λ(Ik − Ir,k)

+ b(√

e2,k + Φr,k −√

Φr,k)ϑk

|λ| < 1, and Φk converges to√

Φr,k ϑk (since e2,k

tends to zero). Therefore,Ik converges asymptotically to

Ir,k =b2

b1

√

Φr,k ϑk − 1

b1

Ir,k

=b2

b1

√

Φr,k ϑk − b

b1

k−1∑

j=0

λk−j−1√

Φr,j ϑj .

Finally, the dynamics ofθΦ,k+1 and θk may be unstable,namelyθΦ,k+1, θk may go to infinity. This is physicallyadmissible since they refer to the flux and rotor angularpositions (in practice they take values in[−π, π)), and ismathematically acceptable sinceθΦ,k, θk enter in the otherequations (those ofIk) throughϑk, which is bounded.

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Fig. 1. Rotor angular velocityωk (solid) and referenceωr,k (dashed);a) FOC with observer; b) Digital control and observer (Eulerapprox-imation); c) Control law (14), (19), (20), and (2) sampled; d) Controllaw (14), (19), (20), and (2) implemented with the exponential holder;e) Control law as in d), but withδ = 460 µs

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Bernardino Castillo–Toledo received the B.Sc. degree in Electrical Engineering from theNational Polytechnic Institute (IPN), the M.Sc. Degree from the Center of Research andAdvanced Studies (CINVESTAV–IPN) and thePh. D. degree from the University of Rome“La Sapienza”, Italy, in 1981, 1985 and 1992respectively. He worked as a lecturer at theSchool of Electrical and Mechanics Engineer-ing of the IPN from 1985 to 1989. From 1985 to1995 he was at the Automatic Control Section

of the Department of Electrical Engineering of the CINVESTAV-IPN, andsince 1995, at CINVESTAV–IPN Campus Guadalajara. He held severalresearch stages at University of Rome “La Sapienza”, University ofL’Aquila and was a visiting Professor at the Laboratoire d’Automatiqueet d’Analyse des Systemes (LAAS) of the French Council for ScientificResearch (CNR) during 1999–2000 and at University of Compi`egneduring the first semester of 2002. His main research interests includenonlinear control design, the robust regulation problem and applicationof artificial neural networks and fuzzy logic techniques to control andfault diagnosis of dynamical systems.

Stefano Di Gennaro obtained the degree inNuclear Engineering in 1987 (summa cumlaude), and the Ph.D. degree in System Engi-neering in 1992, both from the University ofRome “La Sapienza”, Rome, Italy. In October1990 he joined the Department of Electrical En-gineering, University of L’Aquila, as AssistantProfessor of Automatic Control. Since 2001,he has been Associate Professor of AutomaticControl at the University of L’Aquila. He holdscourses on Automatic Control and Nonlinear

Control. In 1986 he was visiting scientist at the Nuclear Research CenterENEA – Casaccia. He has been visiting professor at the Laboratoire desSignaux et Systemes, CNRS–Paris, of the Department of Electrical Engi-neering of the Princeton University and of the Department ofElectricalEngineering and Computer Science of the University of California atBerkeley, USA, and of the Centro de Investigacion y EstudiosAvanzadosdel IPN, Unidad Ciudad de Mexico and Unidad Guadalajara, M´exico.He is working in the area of hybrid systems, regulation theory, andapplications of nonlinear control.

Alexander G. Loukianov was born in 1946,in Moscow, Russia. He graduated from Poly-technic Institute, Moscow, Russia in 1975, andreceived Ph. D. degree in Automatic Controlfrom Institute of Control Sciences of RussianAcademy of Sciences, Moscow, Russia in 1985.He was with Institute of Control Sciences since1978, and was Head of the Discontinuous Con-trol Systems Laboratory from 1994 to 1995.In 1995-1997 he has held visiting position inthe University of East London, UK, and since

April 1997, he has been with CINVESTAV–IPN, (Advanced Studiesand Research Center of the National Polytechnic Institute), GuadalajaraCampus, Mexico, as a Professor of Electrical Engineering graduateprograms. His research interests include nonlinear systems robust con-trol and Variable Structure Systems with sliding modes as applied todynamical plants with delay, electric drives and power systems control,robotics, spice and automotive control. In 1992-1995 he wasin chargeof an industrial project between his Institute and largest Russian carplant, also several international projects supported by INTAS and INCO–COPERNICUS, Brussels. He has published more than 100 technicalpapers in international journals, books and conferences, and has servedas reviewer for different international journals and conferences.

Jorge Rivera was born in El Rosario, Sinaloa,Mexico in 1975. He received the B.Sc. de-gree from Technological Institute of the Sea(ITMAR) campus Mazatlan, Mexico, in 1999,and the M.Sc. and Ph.D. degree in ElectricalEngineering from the Advanced Studies andResearch Center of the National PolytechnicInstitute (CINVESTAV IPN) campus Guadala-jara, Mexico in 2001 and 2005 respectively.Since 2006, he has been with the University ofGuadalajara Mexico, as a full–time Professor

at the Electronic Department. His research interest focus on Regulatortheory, Sliding mode control, Discrete-time nonlinear control systemsand their applications to electrical machines.