improved direct torque control of induction motors using

Archives of Control SciencesVolume 23(LIX), 2013No. 3, pages 361–376

Improved direct torque control of induction motorsusing adaptive observer and sliding mode control

DJILALI KOUCHIH, MOHAMED TADJINE and MOHAMED SEGHIR BOUCHERIT

This paper presents the synthesis of an adaptive observer which is used for the improvementof the direct torque control of induction motor drives. The observer detects stator flux compo-nents in two-phase stationary reference frame, rotor speed and stator resistance by measureof the stator terminal voltages and currents. The observer is adapted using a simple algorithmwhich does not imply a high computational load. Stability analysis based on Lyapunov theoryis performed in order to guarantee the closed loop stability. Simulation tests under load distur-bance and stator resistance variation are provided to evaluate the consistency and performanceof the proposed control technique in the low and high speeds.

Key words: adaptive observer, direct torque control, induction motor, sensorless, slidingmode control

1. Introduction

Direct Torque Control (DTC) is currently known as the technique being mostly em-ployed to control induction motors (IM). It requires accurate knowledge of the magni-tude and the angular position of the controlled flux. The flux is conventionally obtainedfrom the stator voltage model, using the measured stator voltages and currents [1-2].This method uses open loop pure integration and suffers from the well known problems,especially at low speed operation mode [3].

Many researchers have been involved in designing of sensorless control of the IM.Most methods are essentially based on the Model Reference Adaptive System (MRAS)or on the reactive power based reference model [4-5]. The MRAS algorithm is very sim-ple but its greatest drawback is the sensitivity to uncertainties in the motor parameters.Others methods, based on Extended Kalman Filter (EKF) or on the intelligent techniques(fuzzy logic and neural networks), have been used by many authors [6-11]. These meth-ods imply a high computational load. In addition, the variable structure techniques arealso used by many researchers [12-16].

D. Kouchih is with Electronic Department, University Saad Dahlab, B.P. 270, Blida, Algeria. E-mail: [email protected]. M. Tadjine and M.S. Boucherit are with Process Control Laboratory, Auto-matic Control Department, ENP, 10, Avenue Badi H., B.P. 182, Alger, Algeria. E-mails: [email protected],ms [email protected].

Received 14.03.2013.

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UnauthenticatedDownload Date | 1/19/18 1:34 AM

362 D. KOUCHIH, M. TADJINE, M.S. BOUCHERIT

Generally, the rotor speed estimation is affected by parameter variations, especiallythe stator resistance due to temperature, particularly at low speeds [17-18]. Therefore, toimprove the estimation of the components of the stator flux, it’s necessary to compensatethis parameter variation in sensorless control by using an online adaptation of the statorresistance.

On the other hand and for the speed control, the parametric variation modifies theperformances of the control system when we use regulator with fixed parameters. How-ever, the performances will be degraded face to internal and external disturbances [19].To offer control robustness, many strategies have been proposed in literature [20-24]. Inthis work, we are interested to the Sliding Mode Control (SMC). The SMC can offermany good properties [25]. The problem of undesirable chattering can be remedied byreplacing the switching function by a smooth continuous function [26].

2. DTC strategy

2.1. Principles

The basic idea of the DTC is to choose the optimal vector voltage which producesthe desired flux and torque. The stator flux is given by:

Φs = Φs0 +

t∫0

(V s −RsIs)dt. (1)

If the stator resistance is ignored, the stator flux can be approximated:

Φs ≈ Φs0 +

t∫0

V sdt. (2)

During one period of sampling Te, the vector voltage applied to the machine remainsconstant, we can write:

Φs(k+1)≈ Φs(k)+V sTe, (3)

so the increment is∆Φs ≈V sTe. (4)

Therefore, to increase the stator flux, we can apply a vector of voltage that is co-linearin its direction and vice-versa.

The electromagnetic torque produced by the IM can be expressed:

Ce = kT ΦsΦr sinγ (5)

where kT is a constant [18]. The electromagnetic torque depends upon the effective valueof the stator flux and rotor flux and their relative position γ. Actual torque and flux arecompared with the reference values and control signals are produced.

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IMPROVED DIRECT TORQUE CONTROL OF INDUCTION MOTORSUSING ADAPTIVE OBSERVER AND SLIDING MODE CONTROL 363

The stator voltage is being obtained using the following equation:

V s =2Udc

3

[Sa +Sbe j 2π

3 +Sce j 4π3

](6)

where Udc, Si (i = a,b,c) are DC voltage and the signals of the gats of the inverter,respectively. The choice of the stator vector voltage depends on the desired variation forthe module of stator flux, upon its rotation sense and the desired evolution for the torque.Through the components of stator flux indicated by the reference point (α,β) bound tothe stator, we can decompose the space of stator flux into six sectors [12]. The blockdiagram of an induction motor based DTC strategy is drawn in figure 1. The structure ofadaptive observer is presented in figure 2.

Figure 1. DTC bloc diagram.

Figure 2. Global adaptive observer.

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2.2. Correction of the stator flux

We use a two-level comparison of hysteresis in order to maintain the extremity of thevector of stator flux in one circular crown. A Boolean variable indicates if the amplitudeof the flux must be increased (δ = 1) or (δ = 0) decreased in order to maintain:

|Φ∗s −Φs|¬ ∆Φs (7)

where Φ∗s is the reference stator flux and ∆Φs the stator flux hysteresis band.

2.3. Correction of the torque

We use a three-level comparison of hysteresis in order to control the motor in bothsenses of rotation. Variable µ indicates if the torque must be increased (µ = 1), to main-tain its constant (µ = 0) and (µ = −1) to reduce the torque. This correction assures tooperate in all four quadrants.

The stator voltage applied to the motor depends upon the position of the stator fluxin the sectors. The optimal switching logic defines the best vector and flux references.Table 1 shows the switching order.

Table 5. Switching table of the IM based DTC

Sectors 1 2 3 4 5 6

δ = 1µ =+1 110 010 011 001 101 100µ = 0 111 000 111 000 111 000

µ =−1 101 100 110 010 011 001

δ = 0µ =+1 010 011 001 101 100 110µ = 0 000 111 000 111 000 111

µ =−1 001 101 100 110 010 011

2.4. Correction of mechanical speed

To correct the mechanical speed, we use a sliding mode controller. The sliding modetechnique is developed from variable structure control to reject the disturbances, model-ing uncertainties or parameter variation. It is a technique to adjust feedback by previouslydefining surface [25]. The system which is controlled will be forced to that surface, thenthe behavior of the system slides to the desired equilibrium point.

Design of the control system will be demonstrated for a nonlinear multi-input systemof the form [25]:

X (ni)i (t) = fi(X)+

m

∑j=1

bi j(X)U j, i = 1, ...,m, j = 1, ...,m (8)

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where the vector U of components U j is the control input vector, and the state X iscomposed with Xi and their first (ni − 1) derivatives. Such systems are called squaresystems, since they have as many control inputs U j as states to be controlled Xi. We areinterested in the problem of having the state X track a desired time-varying state Xd .

Let us define a vector s of components si by:

si =

(ddt

+λi

)ni−1

ei, (9)

ei = Xdi −Xi. (10)

The purpose of the sliding mode control is to keep the system motion on the manifoldS, which is defined as:

S = X : e = 0 (11)

where e is the tracking error vector

e = Xd −X . (12)

The sliding mode control should be chosen such that the candidate Lyapunov func-tion Vi satisfies the Lyapunov stability criteria. If we let:

Vi =12

s2i (13)

thenVi = sisi. (14)

The candidate Lyapunov function should satisfy the Lyapunov stability criteria. This canbe assured for:

12

ddt

s2i =−ηi |si| ηi > 0. (15)

Equation (13) states that the squared distance to the surface, measured by s2i decreases

along all system trajectories. The control function can be expressed as follows [26]:

Ucomi =Ueq

i +Uni (16)

Where Ueqi is the equivalent control function and Un

i is the correction factor which mustbe calculated so that the Lyapunov stability condition is satisfied. The correction factorcan be expressed as [19]:

Uni = Kisgn(si). (17)

Here sgn is the sign function and Ki is the controller gain designed from the Lyapunovstability. It was well known that sliding mode technique generates undesirable chatter-ing. This problem can be remedied by replacing the switching function by a smoothcontinuous function. One possible approximation is the sigmoid-like function [26]

sat (si) =si

|si|+ψi(18)

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where ψi is a small positive scalar.For the correction of the mechanical speed, the control function will satisfy reaching

conditions in the following form

T ∗e = T eq

e +T ne (19)

where T ∗e is the reference torque, T eq

e is the equivalent control torque, T ne is the correction

factor and must be calculated so that the stability condition for the selected control issatisfied.

For n = 1, the sliding surface can be expressed by:

s(Ω) = Ω∗−Ω. (20)

During the sliding mode and in permanent regime, we have:

s(Ω) = 0, s(Ω) = 0. (21)

We haveΩ =

1J(Te−Tl − f vΩ) (22)

where J is the inertia of the rotor and the connected load, Te the electromagnetic torque,Tl the load torque, Ω is the mechanical angular speed and fv is the viscose friction coef-ficient. The equivalent control can be expressed as follows

T eqe = J

(Ω∗+Tl + fvΩ

). (23)

During the convergence mode, the condition s(Ω)s(Ω)¬ 0 must be verified. We obtain:

s(Ω) =−1J

T ne . (24)

Therefore the correction factor is given by:

T ne = K · sat (s) . (25)

To verify the system stability condition, parameter K must be positive.

3. Adaptive observer

The objective is to determine the adaptation mechanism of the speed and the statorresistance. The structure of the observer is based on the induction motor model in stator

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reference frame. The state equations of the induction motor can be expressed as follows[27]:

diαs

dt=− 1

σLs

(Rs +Rr

L2m

L2r

)iαs +

1σLs

RrLm

L2r

Φαr +1

σLsω

Lm

LrΦβr +

1σLs

vαs

diβs

dt=− 1

σLs

(Rs +Rr

L2m

L2r

)iβs −

1σLs

ωLm

LrΦαr +

1σLs

RrLm

L2r

Φβr +1

σLsvβs

dΦαr

dt=

RrLm

Lriαs −

Rr

LrΦαr −ωΦβr

dΦβr

dt=

RrLm

Lriβs −

Rr

LrΦβr +ωΦαr

(26)

where vαs, vβs are the components of stator voltage vector, ids, iqs are the componentsof stator current vector, Φαr, Φβr are the components of rotor flux vector, σ is the leak-age factor, Rs and Rr are stator and rotor resistance, Ls and Lr represent the stator androtor cyclic inductances and Lm is the stator-rotor cyclic mutual inductance. ω is themechanical pulsation.

The previous state system can be expressed in the form:dXdt

= AX +BU

Y =CX(27)

where

XT =(iαs iβs Φαr Φβr

), Y =

(iαs

iβs

), U =

(vαs

vβs

).

The matrices are defined by:

A =

−a 0RrLm

Lrbω

Lm

b

0 −a −ωLm

bRrLm

LrbRrLm

Lr0 −Rr

Lr−ω

0RrLm

Lr+ω −Rr

Lr

B =

1σLs

0

01

σLs0 00 0

, C =

(1 0 0 00 1 0 0

),

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a =1

σLs

(Rs +Rr

L2m

L2r

), b = σLsLr, σ = 1− L2

m

LsLr

A linear state observer can be then derived by considering the mechanical speedas a constant parameter since its variation is very slow in comparison to the electricalvariables. The model of the observer is defined as follows [28]:

dXdt

= AX +BU +G(Y −Y

)Y =CX .

(28)

The machine parameters are assumed to be perfectly known, the mechanical pulsationand the stator resistance are unknown. Let define:

δω = ω−ωδRs = Rs −Rs.

(29)

The symbol „^” denotes estimated values and G is the observer gain matrix.We will determine the differential system describing the evolution of the error:

e = X − X . (30)

The state matrix of the observer can be written as:

A = A+δA (31)

where

δA =

− 1σLs

δRs 0 0Lm

bδω

0 − 1σLs

δRs −Lm

bδω 0

0 0 0 −δω0 0 +δω 0

. (32)

Then, we can write:dXdt

= AX +BU +G(Y −Y

)(33)

ordXdt

= AX +BU +GCe. (34)

From (27), (28) and (30), we get:

dedt

= AX − AX +GCe, (35)

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thusdedt

= (A+GC)e−δAX . (36)

Let define the Lyapunov function as:

V = eT e+(δω)2

λ1+

(δRs)2

λ2(37)

where λ1 and λ2 are positive scalars. This function should contain terms of the differ-ences δω and δRs to obtain mechanism adaptation.

The stability of the observer is guaranteed for the condition [25]:

dVdt

< 0. (38)

The derivative of the Lyapunov function is as follows

dVdt

= 2eT dedt

+2δωλ1

dδωdt

+2δRs

λ2

dδRs

dt. (39)

First element of (39) can easily calculated

2eT dedt

= 2eT (A+GC)e−2eT δAX . (40)

The rotor flux components can not be measured. The flux dynamic is faster than themachine parameters dynamic. Therefore, to simplify (40), we can accept that

Φαr = Φαr

Φβr = Φβr.(41)

ThuseT δAX =

Lm

bδω(Φβreiαs − Φαreiβs

)− 1

σLsδRs(iαseiαs + iβseiβs

). (42)

For the second and third terms of (39), we can write

2δωλ

dδωdt

= 2δωλ1

ddt

ω−2δωλ1

ddt

ω

2δRs

λ2

dδRs

dt= 2

δRs

λ2

ddt

Rs −2δRs

λ2

ddt

Rs.

(43)

We consider the hypothesis of a slowly varying regime for the machine parameters, thus:

dωdt

≈ 0

dRs

dt≈ 0,

(44)

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thusdωdt

=dδωdt

dRs

dt=

dδRs

dt.

(45)

Finally, we obtain:

dVdt

= 2eT (A+GC)e− 2Lm


)(46)

+2

σLsδRs(iαseiαs + iβseiβs

)+2

δωλ1

ddt

ω+2δRs

λ2

ddt

Rs.

If the term dVdt = 2eT (A+GC)e is negative, the condition dV

dt < 0 is verified for:

2σLs

δRs(iαseiαs + iβseiβs

)− 2Lm


)(47)

+2δωλ1

ddt

ω+2δRs

λ2

ddt

Rs = 0.

This condition can be verified if:2

δωλ1

ddt

ω =2Lm


)2

δRs

λ2

ddt

Rs =− 2σLs

δRs(iαseiαs + iβseiβs

).

(48)

We obtain the adaptation mechanism in the form:ω =

∫λ1

Lm

b

(Φβreiαs − Φαreiβs

)dt

Rs =∫

−λ21

σLs

(iαseiαs + iβseiβs

)dt.

(49)

The matrix of gain G is selected such as the eigenvalues of the matrix A+GC are inthe left plane half of the complex plan and that the real part of the eigenvalues is largerin absolute value than the real part of the eigenvalues of the state matrix A [28].

The estimated electromagnetic torque is expressed as: We obtain the adaptationmechanism in the form:

Ce =32

pLm

Lr

(Φαr iβs − Φβr iαs

). (50)

The reference of the stator flux Φsn is deduced using the equation of the induction motorsteady-state model.

Vsn = ΦsnRs

Ls

√√√√√√(

1−σ LsLrRsRr

ωslωs

)2+(

LrRr

ωsl +LsRs

ωs

)2

1+(

σωslLrRr

)2 (51)

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where ωsl is the rotor variables pulsation in nominal mode and Vsn is the nominal valueof the stator phase voltage.

4. Simulation results

The technique presented in the previous sections, has been implemented in the MAT-LAB environment. The IM parameters has been as follows: 3 [kW], 220/380 [V], 50[Hz], Rs = Rr = 1.84[Ω], Ls = Lr = 0.17 [H], Lm = 0.16 [H], J = 0.0145 [kgm2],fv = 0.0038 [Nms/rd].

4.1. Low speeds

To illustrate performances of the proposed control, we simulated a load-less startingup mode and nominal torque applied at time t = 0.3 [sec]. The synthesized observerallows us to reconstruct all the state variables. For the DTC simulation, torque and fluxhysteresis bands are 0.2 [Nm] and 0.01 [Wb] respectively. The load torque is fixed to 20[Nm]. The speed of reference equals to ±50 [rpm]. Figures 3-8 summarizes the controlsystem performance.

Figure 3. Observed and reference rotor speed. Figure 4. Observed electromagnetic torque.

4.2. High speed mode

The high speed mode depends on the machine and load mechanical possibilities.The load torque is an important factor for the choice of the acceptable high speed. Thehigh speed mode is important, particularly in the case of electrical vehicles, wherein, weadopt the weakening of stator flux – see figure 9.

We simulated a load-less starting up mode with a speed of reference equals to 1415[rpm]. Then at t = 0.5 [sec], we imposed a high speed of 2000 [rpm]. The simulationresults are shown in figures 10-12.

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Figure 5. Effective value of the stator flux. Figure 6. The hodograph of the stator flux vector.

Figure 7. Direct and quadrature stator currents errors.

Figure 8. Observed stator resistance.

From the simulation results, we can estimate the machine variables in the differentworking from low to high speeds. It appears that the load torque and stator resistancevariations do not allocate the performances of the proposed control. The flux tracks thereference value and it is insensitive to parameters variation. The speed response also

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Figure 9. Reference stator flux.

Figure 10. Observed and referenced rotor speed.

Figure 11. Effective value of the stator. Figure 12. The hodograph of the stator flux vector.

stays insensitive to parameters variation. The global control scheme introduces high per-formances of robustness, stability and precision, particularly, under uncertainties causedby parameter variation.

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5. Conclusion

In this paper a new adaptive observer design method has been presented. The globalcontrol scheme introduces high performances of robustness. Stability and precision, par-ticularly, under uncertainties caused by load and stator resistance variation is also pre-served. Furthermore, this observation method presents a simple algorithm that has theadvantage to be easily implementable.

The adaptive observer uses an adaptive mechanism for the speed and the stator re-sistance estimation. This approach relies on the improvement of an estimation of thecomponents of the stator flux.

We can note that the estimation of the stator flux by the adaptive observer makes theIM based on DTC more robust if stability is concerned.

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improved direct torque control of induction motors using

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