Chapter 6

,

Direct Control of a Permanent Magnet Synchronous Machine ¹

6.1. Introduction

Direct torque control (DTC) consists of a direct control approach, point of view in order to directly determine the converter optimal commutations from a control objective.

DTC appeared in the 1980s [TAK 85, TAK 87]. Since then, a large variety of algorithms have been proposed, relying on the heuristic refinements of the commutation choice [CAN 00], [HAS 99].

Direct control differs from conventional control and helps to obtain much faster response times. Indeed, within a scalar or vector control, the system to be controlled is defined by a continuous or discrete model; the converter being considered as a simple gain. The control algorithm then determines, at each calculation step, the voltages to be applied to the motor, which, via a PWM technique, dictates the choice of the converter commutations.

Here the focus is on direct control of permanent magnet synchronous machines (PMSM) by different approaches. After a historical review of the DTCs, some improvements to its performance time are detailed through calculation of the inverter configurations on a constant calculation horizon.

Currently, the direct predictive approaches considering the controlled process as a dynamic hybrid system (DHS), are rich in potential in power electronics applications and give promising results [MOR 08] [LIN 07].

This approach is illustrated for the PMSM.

6.2. Model of the permanent magnet synchronous machine in the d-q reference frame

Equations leading to the modeling of the PMSM in the d-q reference frame are recalled here.

Equations for the voltages:

[6.1]

[6.2]

Equations for the magnetic fluxes:

[6.3]

[6.4]

Expression for the electromagnetic torque:

[6.5]

Special case for the non-salient pole machine:

[6.6]

6.2.1. State modeling

From relations [6.2], [6.4] and [6.5] the following state equation can be determined:

[6.7]

For a non-salient pole machine L_d =L_q =L_s and by separating the part relative to the rotor flux, we obtain:

[6.8]

6.3. Conventional DTC with free switching frequency

6.3.1. General principle

DTC is based on the direct determination of the converter configuration from information on the torque evolution and the machine flux with respect to the references set by the user.

Initially proposed for asynchronous machines [TAK 85], application to the PMSM is also found in [ZOL 97, ZOL 98, CHU 98, RAH 98, RAH 03].

DTC is based on heuristics determination of the commutation state of the converter, from indicative signals coming from hysteresis controllers, measuring the gap between the torque and the reference flux and their estimated values.

From the works of Depenbrock and Takahashi [TAK 87, DEP 88], many techniques of commutation choices can be made. Here, the aim is not to present the latest evolutions of DTC, but to simply present its basic principle.

To illustrate this type of control, let us consider a two-level conventional converter for which the commutation voltages will be noted v_i j€{0...7}.

For the eight configurations of the converter, the three-phase voltages project in a fixed two-phase reference frame noted α,β¹ as follows.

Neglecting the stator resistance, the evolution of the flux during a calculation step will follow the corresponding voltage vector.

The flux estimator helps to determine the actual flux argument θ_s , and thus to define in which sector is to be found. For one sector, only four converter configurations are considered. For example, in sector 1, the ν₂ configuration will increase the flux and consequently the internal angle linked to θ_s and thus the torque, whereas the ν₅ configuration will decrease the flux and the torque respectively. For the six sectors, the configurations are given in Table 6.2.

The mimic diagram of the control is then as follows.

DTC determines the converter configuration via a simple decision algorithm associated with estimates or observers of the flux and of the torque &_em . Its construction requires two hysteresis controllers, providing pieces of information on the flux and torque tendencies in the decision table (Figure 6.3). The presence of these controllers imposes commutation durations depending on the chosen threshold and on the operating point of the motor. This variability often leads to undesirable acoustic noises. Moreover, within a digital realization, in order to detect the thresholds‘ crossings of the hysteresis controllers, it is necessary to have a shift examination, which imposes a very low calculation step and leads to calculation constraints in strict real time.

6.3.2. Experimental application of DTC

The experimental bench is composed of a PMSM with non-salient pole 1.6 kW Leroy Sommer, associated with an incremental coder of 4,096 points for the position measurement.

6.3.2.1. The motor parameters

The motor PMSM is with non-salient poles 1.6 kW with three pole pairs (p₁ =3.) associated with an incremental coder of 4,096 points for the position measurement.

Rating 3,000 rpm, the supplied mechanical torque is near 5 Nm. In the d-q reference frame linked to the rotor, the machine parameters are the following:

−R = 2Ω;

−L = 9.15 mH;

− ψ_f = 290 mWb.

The load is constituted of an identical PMSM which, via a bridge rectifier, produces a load resistor.

The 15 kW ARCEL inverter includes three Eupec modules at IGBT. The control of the latter is made via optical fibers. The dead-time has been set at 3 μs.

The control is programmed in the C language and is implemented on a DSPACE 1104 card.

In order to spot the crossing of the hysteresis controllers, the calculation time has led to the choice of a sampling period of T_e = 28 μs.

To test this DTC, we have applied a square wave corresponding to an inversion of the nominal torque is applied. The motor speed is not overrun and thus it evolves from −1000 rpm to 1000 rpm.

We obtain a rising time of 400 (μs, i.e. about 15 calculation steps. The cogging torque ripple is 1.8 Nm.

The type of direct control leads to a variable commutation period, associated with torque rippes and undesirable acoustic noise depending on the operating point.

In order to compensate for these disadvantages, a direct torque control is developed, with a set commutation period T_e during which three converter configurations will be chosen.

6.4. DTC at a fixed switching frequency

6.4.1. Principle of the control

The torque control dictates controlling the flux vector Ψ in module and in phase at each calculation step. At a sampling instant, it is possible, via the current measurement and the knowledge of the motor parameters, to estimate or observe the flux Ψ. The torque objective can be expressed in the α, β reference frame by a Ψ^# reference. To reach this objective, it is necessary during a constant calculation step to apply voltages ensuring a flux variation ΔΨ (see Figure 6.5).

Considering, as previously, that the evolutions of the flux vector are rectilinear and in the direction of the voltage vector, the following commutation diagram can be drawn.

For each sector, it is possible to break down the vector ΔΨ on the directions of the active commutations v_j, v_j+1 and the null commutations v₀ and v₇ : i.e. T_j, T_j+1 and T_0.7 the corresponding application times. T₀₇ is chosen so that:

T_j + T_j+1 +T_0.7 = T_e

with T_e the calculation step.

For the sectors s ∈ {1 ··· 6}, by denoting it follows that:

For example, in the sector hence:

The flux variation module is limited by the circle encountering the hexagon of the vector representation in Figure 6.6. The maximum value is worth:

Noting , it follows that:

The null commutations are such that ρ₁ + ρ₂ ρ_z = 1

To accurately define the commutations, the control profiles of each leg have been characterized. For center-pulse, Table 6.4 is obtained.

Despite the fact that this type of calculation is similar to a vector modulation in the α,β reference frame of the voltages, it is indeed a direct motor control approach.

6.4.2. Development of the reference vector Ψ^#

There are several methodological approaches to determine the stator flux reference [LLO 03]. For a non-salient pole machine, the component of the stator current on the q axis is an image of the motor torque. As long as the motor speed does not require defluxing action, a null reference on the direct component of the stator torque can be considered, i.e. I_d ^# = 0 .

For a torque set-point C_em^# , relation [6.6] gives:

Relations [6.3] and [6.4] help to determine components in the d-q reference frame of the reference stator flux: Ψ_d^# = L_d .I_d ^# + Ψf and Ψ _q ^# = L_q . I_q ^# :

To obtain the suitable variation vector ΔΨ , it is necessary to express the reference flux at the instant k+1. During a calculation step T_e , the flux vector Ψ^# is considered to rotate at constant speed and is shifted by an angle ω_r T_e .

It becomes:

[6.9]

However, during a calculation time τ, the estimated flux Ψ of the machine is shifted by an angle ???, which will affect the control.

To compensate for this sensitivity of the method at the calculation time, we will apply the control at the instant (k +1) is applied so as to lead to the flux reference Ψ ^# at the instant (k + 2). We will thus have:

6.4.2.1. Flux estimation

From the calculation of the components on the d-q axes, relations [6.3] and [6.4] give: Ψ_d = L_d .I_d (k) + Ψ_f, Ψ_q=L_q .I_q (k):

[6.10]

Knowing the rotor position, relations [6.9] and [6.10] ensure the calculation of the flux variation vector ΔΨ , which will help in the calculation of the cyclic ratios inside the inverter leg (see Tables 6.1 and 6.2)

6.4.3. Experimental results of DTC on a period of fixed calculation

With the same experimental configuration as the one taken to test the conventional DTC, a set-point change is carried out, corresponding to the inversion of the nominal torque.

Although the calculation times are short (about 70 μs ), we have chosen a long calculation period ( T_e = 200 μs ), to show the control robustness.

The rising time is about 600 μs , i.e. three calculation steps and the torque ripple is 0.5 Nm. With respect to conventional DTC, the calculation step is increased by a ratio of 7, while reducing the ripple on the torque by about a ratio of 4.

This improvement comes from the fact that it is better to generate a control based on an analytical model, rather than a control based on heuristic laws built on tendency evolutions.

Regarding the calculation constraints in real time inherent in conventional DTC, it is clear that with the use of hysteresis controllers – if they are of interest during an analogical realization in the current case using DSP and microcontrollers – the obligation to have a very short calculation step increases the cost of the realization. However, to get round these difficulties, the use of dedicated circuits (FPGA, ASIC etc.) can be an alternative in some cases.

6.5. Predictive direct control

6.5.1. Introductions

DTC has several variants. The previously presented method helps to obtain synchronous modulation. These direct control techniques are a new methodological approach, where the control of the continuous magnitudes, such as the flux and the torque, leads to the direct piloting of the cells‘ switches.

A more general approach to the direct control, by considering that the control magnitudes are no longer supply voltages of the motor, but are configurations that can be selected form the inverter. To clarify this methodology, the set converter/machine is considered as a dynamic hybrid system (DHS).

This type of system consists of an energy modulator with m configurations (hereeight for one two-level inverter) and one continuous process with n state variables(here two for the PMSM).

From the references consisting of electric magnitudes (here the two stator currents in the d-q reference frame), the best commutation state(s) of the converter and their application times are determined. To reach this objective, a formal representation of the converter machine behavior is developed for a direct predictive control, ensuring in the state space, the pursuit of the references fixed by the user.

6.5.2. General principle of predictive direct control

At a calculation instant k, an initial point is defined in the state space of dented dimension n of the system to control. For the j different configurations of the converter, the evolution of the characteristic point of the state vector will be done in j different directions. The control objective can be represented by a point in this state space that is pursued.

Strict calculation constraints lead to the use of a rustic model for real timecalculation. The model is recalculated at each calculation step to determine the converter configurations. The model is only valid locally in the step time frame.

The general approach, in order to establish a predictive control algorithm of a dynamic hybrid system, is as follows:

1. Obtainment of a general model of the behavior of the set energy modulator continuous process. During this phase, one or several non-linear models can be established. We thus obtain one or several local models, valid on the decision horizon and the required state space. The control can take m different configurations. If we note j the corresponding index (j∈{1:m}), knowing the measured state x(t), these m models at the instant (t + τ) can be written:

[6.11]

τ representing here the prediction horizon;

2.  search for the linearity domain. The principle of our approach presupposes that, for all the possible motor configurations, the trajectories in the state space are rectilinear. This constraint implies the determination of the maximum application time τ_max of a control and of the limits in the state space satisfying this condition;

3.  determination of a local model. For the previously defined limits (decision horizon limited by τ_max and validity domain in the state space), a rustic model is calculated at each decision instance;

4.  development of one or several choice strategies; from these local models and from the operating point marked in the state space at an instant t, it is easy to determine all the directions in the d_j state space corresponding to the different states of the u_j control.

6.5.3. Restriction of the short-circuit current of permanent magnet machines

For the PMSM, the flux and torque control determine the references to obtain on the components of the stator currents in the d-q reference frame (I_d ^# and I_q ^#).

In order to express the model of the inverter/motor set, the eight configurations chosen by the converter are expressed with the PMSM state model given by [6.8].

The discrete controls u_A, u_B ,u_C , are associated with the three converter legs. For example, the state of the inverter leg A can be represented by a discrete variable u_A such, that if u_A = 1 , the upper transistor of leg A is on; if u_A = 0 , the lower transistor of leg A is the on transistor.

These three discrete variables lead to eight inverter configurations for these. The stator voltage components are computed in the d-q reference frame.

[6.12]

Among the eight possible configurations, six lead to non-null voltages at the motor terminals and two give null voltages. Only seven different configurations are retained for the converter.

During the decision horizon, our work hypotheses suppose evolutions of the currents I_d and I_q , rectilinear in the state space and whose norm is proportional to the application time τ of the control.

On the one hand, the decision horizon time being about a hundred microseconds, the current evolutions in the inductances can be considered as rectilinear. On the other hand, the electric and mechanical time-constants being well separated, the matrices A and B₂ of [6.8] are considered to be constant, and the property of the linear trajectories will be true in the all the state space.

With these hypotheses, it is possible to establish a simplified behavior model according to:

with:

The state vector is expressed for a horizon time corresponding to the calculation step, by discretization of equation [6.8] in the 1^st order. It follows that:

[6.13]

Expressing the components in the d-q reference frame at the instant k+1 with relation [6.12] obtains:

[6.14]

From relation [6.14], it is possible to determine in the I_d,I_q plane, the seven possible directions plane dj (j ∈ {0: 6}), coming from the point O, representative of the initial state at the k instant.

In this same plane, the torque and flux objective determine an objective point to be reached.

The chosen approach is a multi-step strategy with three decisions, for which we apply three converter configurations; two corresponding to the non-null voltages at the motor terminals and the third corresponding to the null voltages.

Let us note τ₁ and τ₂ the application times corresponding to the two states of the converter, supplying the non-null voltages to the motor and τ_z the application time of the two configurations giving the null voltages.

The application times of these configurations are calculated so that their sum is equal to a constant time corresponding to the calculation period T_e .

With this constraint, the reachable state space points are included in a triangle defined by the extremities of the vectors corresponding to the three chosen directions [MOR 08]. The vector modules correspond to the maximum application time, i.e. here T_e (Figure 6.9).

6.5.3.1. Determination of the reference state vector

In the case of a PMSM, the electromagnetic torque is proportional to the I_q current. The Joule losses minimization, when the poles are non-salient, thus leads to maintaining the I_d current at a null value.

In the state space, the point defining the objective to be pursued is characterized by the two current references I_d^# = 0 and I_q^# , image of the electromagnetic torque, calculated by a conventional control algorithm independent of the hybrid control (speed and/or position loop).

Thus, the space reachable in the I_d , I_q plane will be constituted by the polygon linking the six extremities of the active directions (voltages different from zero).

If the coordinates of the point D ( I_d^# I_q^# ) are in the polygon, it will always be possible to find three states of the converter reaching this point.

In the opposite case, a set-point is defined at the intersection point of the OD direction with the corresponding side of the polygon.

For a set-point represented by point D, it is easy, at the expense of very few tries, to select the directions d _j and d _j+1 which, help to reach the objective D with the d_z direction.

Determination of the application times τ_i ,τ_i+1,τ_z of the converter with the directions d_i , d_i+1, d_z corresponds to the resolution of the two projection equations and of the condition on the sum of the application times.

Let us note the coordinates of the d_i direction. The yields.

The times τ_j ,τ_j+1,τ_z are calculated first. If centered pulses for each inverter leg are selected, Table 6.4 lists the calculation of the cyclic ratios.

6.5.4. Experimental results

The experimental results presented in Figures 6.11 and 6.12 are carried out with the same experimental test bench as previously, and correspond to an inversion of the nominal torque leading to an inversion of the rotation direction at 1,250 rpm.

The torque shows a very low ripple in steady state (of about 0.2 Nm). The inversion of the nominal torque is obtained within three calculation steps, i.e. 600 µs.

By comparison, on the same test bench, the conventional vector control with the same modulation period ( 200μs ) leads to an inversion of the nominal torque in 20 ms [MOR 07a].

This direct predictive control monitors the current evolution in the d-q reference frame. We can verify in Figure 6.12 shows that, during the inversion of the rotation direction, the phase currents have constant peak values.

The obtained results are similar to those with fixed step DTC, as previously presented. However, the approach proposed here is general and gives excellent results for any system with an energy modulator (static converters, electropneumatic motor controlled by a distributor, etc.).

In the research phase of a decision strategy (see point 4, section 6.4.2), during the development of the choice of one or several configurations of the power modulator, it is necessary to explore all the commutation possibilities of the power converter.

Eight cases arise, including two identical ones, and the calculation constraint is not too strict in real time. However, for other structures (matrix converter, multi-cell converter, etc.) the number of configurations to be calculated can be an obstacle to the proposed method.

In order to overcome this problem, an alternative is to seek for the control by inversion of the dynamic hybrid system model, representing the studied process. Generally, the problem is undersized and it is necessary to attach conditions relative to the control profiles of the cell switches in order to raise the undetermination.

To illustrate this other direct control approach, we will now present it as an alternate to the approach previously presented.

6.5.5. Predictive direct control by model inversion

An alternate to the previous approach is presented to illustrate this other direct control approach.

To develop this direct control presented in [MOR 07b], we will use model [6.14] for which the mean values of the inverter duty ratio are considered:

[6.15]

The objective to be reach in the d-q reference frame is represented by . The search for ρ(k) control will satisfy the previous relation for which .

Relation [6.15] obtains:

Getting the ρ(k) control, requires solving the following equation:

i.e. with development of C₂₃ :

[6.16]

The second member of this equation is vector λ (k) with two components:

[6.17]

From the previous equation [6.17], the three cyclic ratios may not be calculated. It is necessary to attach an additional condition to raise the undetermination.

The λ(k) vector can be expressed by a single voltage vector V_s in the α,β reference frame (see Figure 6.13). For this voltage vector V_s, there is an endless number of combinations of the cyclic ratios of the inverter legs, all leading to the same results at the instant (k+1).

The fact that an endless number of commutation choices is possible within a calculation step may easily be understood. Indeed, from the X(k) initial state, in order to reach the X^# set-point, we need two active configurations are taken in {ν ₁, ν₂, ν ₃, ν ₄, ν₅, ν ₆} and one null in {ν₀,ν₇}. For a given objective, there is only one triplet, (ν _i, ν_i+1, ν_z), but an there are an endless number of commutation sequences giving a calculation step of the different trajectories, all leading to the set- point.

Two types of sequences, usually used in electrotechnics are presented, to show this variability and raise undetermination.

In the first sequence, at each calculation step, one leg does not switch, which reduces the commutation losses of the converter. The second sequence encourages the decrease of voltage harmonics and corresponds to a centered distribution of the conduction times.

6.5.5.1. Profile with an idle leg

The commutation profiles of the three legs of the inverter vary by circular permutation, depending on the sector in which the V_s voltage vector is found.

Thus, if sectors 1 or 2 are addressed, there is no conduction on the leg C (see Figure 6.14.).

In these two latest sectors, ρ_C = 0 is the required additional condition. Table 6.5 displays the properties for the six sectors.

In each configuration given in Table 6.5, the undetermination of relation [6.17] can be raised. Thus, in the case where ρ_C = 0 , it follows that:

which leads to the explicit calculation of the vector ρ(k) of the cyclic ratios of the converter legs:

[6.18]

For the sectors 3, 4, 5 and 6, ρ_A = 0 and ρ_B = 0 , are solved by modifying the corresponding line of the C _ρ matrix. The three possible cases should be evaluated and only the vector ρ(k) should be kept, giving positive cyclic ratios as the sector is unknown.

Knowing that the ν₀ (ρ(k) = 0) and ν₇ (ρ(k) = 1) configurations lead to null voltages, any addition of the same magnitude on the cyclic ratios ρ_A, ρ_B, ρ _C will not modify the application times of the non-null converter configurations.

Thus, if sector 1 in Figure 6.13 is selected, adding a constant value to ρa , ρ_B, ρ _C will not modify ρ ₁ and ρ₂ since they are coming from the relative values of the leg cyclic ratios.

This property helps us to apply relation [6.18], whatever the sector where the voltage vector is. For example, if we are in sector 3 and if relation [6.18] is applied, ρ_A < 0 , ρ_B > 0 and obviously ρ _C = 0 are obtained. The values of ρ_B and ρ _C are positively shifted from ρ_A as ρ_A = 0 in this sector.

In order to reduce the calculation time, relation [6.18] is used and vector ρ(k) is calculated as follows:

[6.19]

The calculation of this direct control will be made with the help of relations [6.16], [6.18] and of selection [6.19]. This method has the advantage of being independent of the position of the considered voltage vector and has a reduced algorithmic complexity.

This direct control is derived for the reverse model for centered cyclic ratios.

6.5.5.2. Centered pulses

This commutation strategy helps to reduce the current harmonics in the load.

The control profiles of each converter leg are represented in Figure 6.15.

The configurations of null voltages ({ν₀,ν₇}), here characterized by the cyclic ratio ρ_z, are distributed on the weakest cyclic ratio (min(ρ_A,ρ_B,ρ _C)) on both sides of the largest cyclic ratio (max (ρ_A ,ρ_B,ρ _C)).

For sector 1: .

For sector 2: .

For the other sectors, the properties are given in the following table.

In those three cases, an additional condition to  relation [6.17] ensures the calculation of the cyclic ratios vector ρ(k).

For example, for sectors 1 and 4, the condition is  ρ_A + ρ _C = 1 and it follows that:

which gives:

[6.20]

As previously, it is useless to do this calculation in the three considered cases. Indeed, if we study the configurations in the six sectors:

which gives:

[6.21]

Thus, as for the previous commutation profiles, the determination of these three cyclic ratios consists of adding the same value in accordance with relations [6.22].

[6.22]

Here with centered pulses, the direct control is ensured by [6.20] on which the selections [6.22] ensure the control calculation.

The obtained results are similar to those given in Figures 6.11 and 6.12.

The value of this approach by inversion of the studied hybrid system model is that it avoids, in order to choose the control, the exploration of all the converter configurations. However, the model inversion is only possible by attaching heuristic laws on the commutation profiles of the converter legs. With this two-level converter, it is shown that a simple solution appeared. For more complex structures, such as multi-level converters, the difficulty significantly increases.

6.6. Conclusion

Conventional DT and two direct control strategies relying on calculated approaches are presented.

The direct predictive control developed in the first section relies on a simplified model for which the trajectories in the state space are considered as rectilinear with a module proportional to the application time of the control.

This control approach, relying on a dynamic hybrid system, can be considered for any physical system with commutations fulfilling these conditions.

The obtained rusticity of the model significantly impacts the possible duration of calculation in real time. This technique applied to the synchronous and asynchronous machines with an inverter helps to control the machine torque directly at the level of the switches. Very high torque dynamics could be obtained. Monitoring the currents in transient regime authorizes the realization of a speed control by adding only one regulator to this predictive control.

Nowadays, the predictive approach, using a dynamic hybrid model, opens a promising research domain, arousing the interest of many research teams. The results obtained on other types of machines and other types of converter structures (multi-cell, matrix, etc.) seem to be very promising.

6.7. Bibliography

[CAN 00] CANUDAS C., Modelisation, contrôle vectoriel et DTC: commande des moteurs asynchrones 1, Hermès, Paris, 2000.

[CHU 98] CHUNG S. K., Kim H. S., Kim C. G., Youm M. J., “A new instantaneous torque control of PM synchronous motor for high performance direct drive applications”, IEEE Transactions on Power Electronics, vol. 13, no. 3, 1998, p. 380-400

[DEP 88] DEPENBROCK M., “Direct self control (DSC) of Inverter-fed induction machines”, IEEE Trans. Power Electronics, vol. PE-3, no.4, p. 420-429, 1988.

[HAS 99] EL HASSAN I., Commande Haute Performance d‘un Moteur Asynchrone sans Capteur de Vitesse par Contrôle Direct du Couple, PhD Thesis, Institut National Polytechnique of Toulouse, 1999.

[LIN 07] Lin-Shi X., RÉtif J.M., Brun X., Morel F., Valentin C., Smaoui M., “Commande des systèmes hybrides rapides: applications aux systèmes mécatroniques”, Journal Européen des Systèmes Automatisés, vol. 41, no. 7-8, p. 963-990, 2007.

[LLO 03] LLOR A. M., Control directo de par a frecuencia de modulacion constante de motores sincronos de imanes permanentes, PhD Thesis, University of Madrid Carlos III, Spain, 2003.

[MOR 07a] MOREL F., RÉRTIF J.M., LIN-SHI X., VALENTIN C., “Permanent magnet synchronous machine hybrid torque control”, IEEE Transactions on Industrial Electronics, vol 55, no. 2, p. 501-511, February 2008.

[MOR 07b] MOREL F., Commandes directes appliquées à une machine synchrone à aimants permanents alimentée par un onduleur triphase à deux niveaux ou par un convertisseur matriciel triphase, PhD Thesis, INSA, Lyon, 2007.

[MOR 08] MOREL F., RÉTIF J.M., LIN-SHI X., VALENTIN C., “Permanent magnet synchronous machine hybrid torque control”, IEEE Transactions on Industrial Electronics, vol 55, no. 2, p. 501-511, February 2008.

[RAH 98] Rahman M. F., ZHONG L., Lim K. W., “A direct torque controlled interior permanent magnet synchronous motor drive incorporating field weakening”, IEEE Transactions on Industry Applications, vol. 34, no. 6, p. 1246-1253, November- December, 1998.

[RAH 03] RAHMAN M. F., “A direct torque controlled interior permanent magnet synchronous motor drive without a speed sensor”, IEEE Transactions on Energy Conversion, vol. 18, no. 1, p. 17-22, March 2003.

[TAK 85] TAKAHASHI I., NOGUCHI T., “A new quick response and high efficiency control strategy of an induction motor”, Rec. IEEE IAS, 1985, p. 495-502.

[TAK 87] TAKAHASHI I., Asakawa S., “Ultra-wide speed control of induction motor covered 10A6 range”,,IEEE Transaction Industrial Application, IA-25:227-232, 1987.

[ZOL 97] ZOLGHADRI M. R., Contrôle direct du couple des actionneurs synchrones, PhD Thesis, Institut National Polytechnique of Grenoble, France, 1997.

[ZOL 98] ZOLGHADRI, M.R.. GUIRAUD, J., DAVOINE, J., ROYE D, “A DSP based direct torque controller for permanent magnet synchronous motor drives”, PESC 98 Record., 29^th Annual IEEE, vol. 2, 17–22 May, p. 2055–2061, 1998.

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1 Chapter written by Jean-Marie RÉTIF.

1. Here, the Concordia transformation is performed with