**3. Concept of PI DTC-SVM based position control**

#### **3.1 Computing of flux reference coordinates**

been substituted by active vectors. Obtained results are illustrated in **Figure 4**. Referring to **Figure 4d**, one can notice that the demagnetization problem has been removed, while performances of the motor, for high speeds, are not affected. However, the torque ripple amplitudes rise considerably (**Figure 4c**)

with respect to the one yielded by the *Takahashi* DTC strategy, which

performances. In fact, this approach presents a low torque ripple amplitude (**Figure 5c**). Moreover, it completely eliminates the demagnetization

Further investigation of the stator flux has been achieved through the representation of the stator flux vector extremity locus in the ð Þ *α*, *β* plane. This has been

*Locus of the extremities of* Φ*s, with (a) basic Takahashi DTC strategy, (b) modified Takahashi DTC strategy*

*Average commutation frequency of the inverter power switches, (a) basic Takahashi DTC strategy,*

*and (c) DTC-SVM strategy with a constant commutation frequency.*

• **Figure 5** shows that the DTC-SVM strategy exhibits high dynamical

represents a severe drawback.

*Direct Torque Control Strategies of Electrical Machines*

phenomenon (**Figure 5d**).

**Figure 6.**

**Figure 7.**

**46**

*(b) modified Takahashi DTC strategy.*

The slip angular reference speed *ω*<sup>∗</sup> *<sup>r</sup>* , which is the output of the PI controller, will be used to calculate the argument of the stator flux reference. In the reference frame ð Þ *<sup>α</sup>*, *<sup>β</sup>* , coordinates of the reference stator flux *<sup>ϕ</sup>*<sup>∗</sup> *<sup>α</sup><sup>s</sup>* and *ϕ*<sup>∗</sup> *<sup>β</sup><sup>s</sup>* are calculated from the polar coordinates according to the following expressions:

$$\begin{cases} \phi\_{\text{at}}^{\*} = |\Phi\_{\text{s}}^{\*}| \cos \theta\_{\text{s}}^{\*} \\\\ \phi\_{\text{ft}}^{\*} = |\Phi\_{\text{s}}^{\*}| \sin \theta\_{\text{s}}^{\*} \end{cases} \tag{5}$$

#### **3.2 Computing of voltage reference coordinates**

The coordinates of references of voltage vectors *v* <sup>∗</sup> *<sup>α</sup><sup>s</sup>* and *v* <sup>∗</sup> *<sup>β</sup><sup>s</sup>* in ð Þ *α*, *β* frame are determined by the following equations:

$$\begin{cases} \begin{aligned} \boldsymbol{V}\_{\alpha s}^{\*} &= \frac{\boldsymbol{\phi}\_{\alpha s}^{\*} - \boldsymbol{\phi}\_{\alpha s}}{T\_{\epsilon}} + R\_{s} \boldsymbol{i}\_{\alpha s} \\\\ \boldsymbol{V}\_{\beta s}^{\*} &= \frac{\boldsymbol{\phi}\_{\beta s}^{\*} - \boldsymbol{\phi}\_{\beta s}}{T\_{\epsilon}} + R\_{s} \boldsymbol{i}\_{\beta s} \end{aligned} \tag{6}$$

Finally, they are introduced to the SVM block, which uses them to control the inverter switches ð Þ *Sa*, *Sb*, *Sc* .

#### **3.3 Position control loop**

The objective is the design of a suitable controller as described by **Figure 8**.

**Figure 8.** *Position control loop.*

Then, we have:

$$\begin{cases} \frac{d\theta}{dt} = \Omega\_m \\\\ \frac{d^2\theta}{dt^2} = \frac{1}{J}T\_{cm} - \frac{K\_l}{J}\sin\theta \\\\ \frac{d^3\theta}{dt^3} = \frac{1}{J}\frac{dT\_{cm}}{dt} - \frac{K\_l}{J}\frac{d\theta}{dt}\cos\theta \end{cases} \tag{7}$$

Observing this transfer function, it is clear that it contains two imaginary poles. This leads to a certain difficulty to control the system with a PID controller *C p*ð Þ:

In fact, the system does not present any stability margin. Moreover, to have an adequate dynamical behavior, the derivative time constant ð Þ *Td* should be larger

The implementation scheme of a DTC-SVM based position regulation of an induction motor is shown in **Figure 9**. The idea is based on the decoupling between

The amplitude of this vector will be imposed equal to the nominal value of the stator flux, but the argument will be calculated according to the desired performances. In fact, the error between the reference position *θ* <sup>∗</sup> and the measured one *θ* is applied to the position regulator whose output provides the slip angular reference

Coordinates of the reference stator flux in the reference frame ð Þ *α*, *β* are computed from its polar coordinates according to Eqs. (5). The coordinates of the reference

*<sup>β</sup><sup>s</sup>* are determined using Eqs. (6). Finally, the SVM block, which uses these later to generate the convenient stator

voltages inverter in each modulation period, ensuring working with a constant

*<sup>r</sup>* , which will be used to calculate the argument of the stator flux reference.

1 *Tip*

þ *Tdp*

(14)

*C p*ðÞ ¼ *Kc* 1 þ

*DTC-SVM Approaches of an Induction Motor Dedicated to Position Control Applications*

that the integral time constant ð Þ *Ti* , which is strongly not recommended.

the amplitude and the argument of the stator flux reference vector.

**3.4 DTC-SVM based position control scheme**

*DOI: http://dx.doi.org/10.5772/intechopen.94436*

*<sup>α</sup><sup>s</sup>* and *v* <sup>∗</sup>

*Induction motor position regulation based on the DTC-SVM strategy.*

speed *ω*<sup>∗</sup>

**Figure 9.**

**49**

voltage vector *v* <sup>∗</sup>

commutation frequency.

This yields:

$$\begin{split} \frac{d^3 \theta}{dt^3} &= \frac{1}{J} \left( -\frac{1}{\tau} T\_{em} + \frac{A}{\tau} \alpha\_r \right) - \frac{K\_l}{J} \frac{d\theta}{dt} \cos \theta \\\\ &= -\frac{1}{\tau} \left( \frac{d^2 \theta}{dt^2} + \frac{K\_l}{J} \sin \theta \right) + \frac{A}{J\tau} \alpha\_r - \frac{K\_l}{J} \frac{d\theta}{dt} \cos \theta \end{split} \tag{8}$$

Thus, we can write:

$$\frac{d^3\theta}{dt^3} + \frac{1}{\tau}\frac{d^2\theta}{dt^2} + \frac{K\_l}{J}\frac{d\theta}{dt} + \frac{K\_l}{J\tau}\theta + \rho\left(\theta, \frac{d\theta}{dt}\right) = \frac{A}{J\tau}\rho\_r\tag{9}$$

where:

$$\rho \left( \theta, \frac{d\theta}{dt} \right) = \frac{K\_l}{J} \frac{d\theta}{dt} (\cos \theta - 1) + \frac{K\_l}{J\tau} (\sin \theta - \theta) = \mathbf{0} \left( \theta, \ \frac{d\theta}{dt} \right)^3 \tag{10}$$

For small values of *θ*, *φ θ*, *<sup>d</sup><sup>θ</sup> dt* � � can be neglected, and then the mechanical part of the machine can be represented by a third order linear system described by the following transfer function:

$$\frac{\theta}{\alpha\_r} = \frac{\frac{A}{f\tau}}{\left(p^2 + \frac{K}{f}\right)\left(p + \frac{1}{\tau}\right)}\tag{11}$$

It is to be noted that the application of the following nonlinear feedback represents a nonlinear compensator:

$$
\rho\_{r'} = a\_r - \frac{J\tau}{K\_l} \rho \left(\theta, \frac{d\theta}{dt}\right) \tag{12}
$$

This loop realizes a feedback linearization. The transfer function between *θ* and *ω*0 *<sup>r</sup>* is expressed as:

$$\frac{\theta}{\frac{\alpha \prime\_r}{\alpha \prime\_r}} = \frac{\frac{A}{f\pi}}{\left(p^2 + \frac{K}{f}\right)\left(p + \frac{1}{\pi}\right)}\tag{13}$$

which is an exact transfer function without any approximation.

*DTC-SVM Approaches of an Induction Motor Dedicated to Position Control Applications DOI: http://dx.doi.org/10.5772/intechopen.94436*

Observing this transfer function, it is clear that it contains two imaginary poles. This leads to a certain difficulty to control the system with a PID controller *C p*ð Þ:

$$\mathbf{C}(p) \;= \mathbf{K}\_{\varepsilon} \left( \mathbf{1} + \frac{\mathbf{1}}{T\_i p} + T\_d p \right) \tag{14}$$

In fact, the system does not present any stability margin. Moreover, to have an adequate dynamical behavior, the derivative time constant ð Þ *Td* should be larger that the integral time constant ð Þ *Ti* , which is strongly not recommended.

#### **3.4 DTC-SVM based position control scheme**

Then, we have:

This yields:

Thus, we can write:

where:

*ω*0

**48**

*<sup>r</sup>* is expressed as:

*φ θ*,

*dθ dt* � �

following transfer function:

sents a nonlinear compensator:

For small values of *θ*, *φ θ*, *<sup>d</sup><sup>θ</sup>*

*d*3 *θ dt*<sup>3</sup> <sup>¼</sup> <sup>1</sup>

> *d*3 *θ dt*<sup>3</sup> <sup>þ</sup>

¼ *Kl J dθ* *dθ dt* <sup>¼</sup> <sup>Ω</sup>*<sup>m</sup>*

8

*Direct Torque Control Strategies of Electrical Machines*

>>>>>>>>>>><

>>>>>>>>>>>:

*<sup>J</sup>* � <sup>1</sup> *τ Tem* þ

> *d*2 *θ dt*<sup>2</sup> <sup>þ</sup>

¼ � <sup>1</sup> *τ*

> 1 *τ d*2 *θ dt*<sup>2</sup> <sup>þ</sup>

*d*2 *θ dt*<sup>2</sup> <sup>¼</sup> <sup>1</sup> *J*

*d*3 *θ dt*<sup>3</sup> <sup>¼</sup> <sup>1</sup> *J dTem dt* � *Kl J dθ dt* cos *<sup>θ</sup>*

*Tem* � *Kl*

*A <sup>τ</sup> <sup>ω</sup><sup>r</sup>*

*Kl <sup>J</sup>* sin *<sup>θ</sup>*

!

� �

*Kl J dθ dt* þ *Kl*

*dt* ð Þþ cos *<sup>θ</sup>* � <sup>1</sup>

*dt*

*θ ωr* ¼

*θ ω*0 *r* ¼

which is an exact transfer function without any approximation.

*<sup>J</sup>* sin *<sup>θ</sup>*

� *Kl J dθ dt* cos *<sup>θ</sup>*

> þ *A Jτ*

*<sup>J</sup><sup>τ</sup> <sup>θ</sup>* <sup>þ</sup> *φ θ*,

*Kl Jτ*

the machine can be represented by a third order linear system described by the

*<sup>p</sup>*<sup>2</sup> <sup>þ</sup> *<sup>K</sup> J* � �

*<sup>ω</sup><sup>r</sup>*<sup>0</sup> <sup>¼</sup> *<sup>ω</sup><sup>r</sup>* � *<sup>J</sup><sup>τ</sup>*

It is to be noted that the application of the following nonlinear feedback repre-

*Kl φ θ*,

This loop realizes a feedback linearization. The transfer function between *θ* and

*A Jτ*

> *<sup>p</sup>* <sup>þ</sup> <sup>1</sup> *τ* � �

*<sup>p</sup>*<sup>2</sup> <sup>þ</sup> *<sup>K</sup> J* � �

*A Jτ*

> *<sup>p</sup>* <sup>þ</sup> <sup>1</sup> *τ* � �

*dθ dt* � �

*<sup>ω</sup><sup>r</sup>* � *Kl J dθ dt* cos *<sup>θ</sup>*

*dθ dt* � �

ð Þ¼ sin *<sup>θ</sup>* � *<sup>θ</sup>* <sup>0</sup> *<sup>θ</sup>*, *<sup>d</sup><sup>θ</sup>*

� � can be neglected, and then the mechanical part of

¼ *A Jτ*

> *dt* � �<sup>3</sup>

(7)

(8)

(10)

(11)

(12)

(13)

*ω<sup>r</sup>* (9)

The implementation scheme of a DTC-SVM based position regulation of an induction motor is shown in **Figure 9**. The idea is based on the decoupling between the amplitude and the argument of the stator flux reference vector.

The amplitude of this vector will be imposed equal to the nominal value of the stator flux, but the argument will be calculated according to the desired performances. In fact, the error between the reference position *θ* <sup>∗</sup> and the measured one *θ* is applied to the position regulator whose output provides the slip angular reference speed *ω*<sup>∗</sup> *<sup>r</sup>* , which will be used to calculate the argument of the stator flux reference. Coordinates of the reference stator flux in the reference frame ð Þ *α*, *β* are computed from its polar coordinates according to Eqs. (5). The coordinates of the reference voltage vector *v* <sup>∗</sup> *<sup>α</sup><sup>s</sup>* and *v* <sup>∗</sup> *<sup>β</sup><sup>s</sup>* are determined using Eqs. (6).

Finally, the SVM block, which uses these later to generate the convenient stator voltages inverter in each modulation period, ensuring working with a constant commutation frequency.

**Figure 9.** *Induction motor position regulation based on the DTC-SVM strategy.*

#### **3.5 Concept of sliding mode DTC-SVM based position control**

Sliding mode (SM) controllers perform well in non nonlinear systems than PI controllers [13, 14]. Indeed, the sliding mode control is a type of variable structure systems characterized by the high simplicity and the robustness against insensitivity to parameter variations and external disturbances [14–16]. Considering a nonlinear system described by the following state equation:

$$
\dot{X} = f(X) + \mathbf{g}(X)U\tag{15}
$$

that is to say:

This leads to:

\_ *<sup>S</sup><sup>θ</sup>* <sup>¼</sup> <sup>1</sup>

That is to say:

*<sup>J</sup>* � <sup>1</sup> *τ Tem* þ

¼ 0

where:

*Ueq*,*<sup>θ</sup>* <sup>¼</sup> <sup>1</sup>

**Figure 10**.

**51**

control law becomes:

*<sup>A</sup> Tem* � <sup>2</sup>

*λ*1*T*

*<sup>S</sup><sup>θ</sup>* <sup>¼</sup> *<sup>d</sup>*<sup>2</sup>

*εθ*ðÞ¼ *<sup>t</sup>* f g *εθ*ð Þþ *<sup>t</sup>*<sup>0</sup> ½ � *<sup>ε</sup>*\_*θ*ð Þþ *<sup>t</sup>*<sup>0</sup> *<sup>λ</sup>*1*εθ*ð Þ *<sup>t</sup>*<sup>0</sup> ð Þ *<sup>t</sup>* � *<sup>t</sup>*<sup>0</sup> *<sup>e</sup>*�*λ*1ð Þ *<sup>t</sup>*�*t*<sup>0</sup> . In this case, function *h X*ð Þ is expressed as:

*DOI: http://dx.doi.org/10.5772/intechopen.94436*

*h X*ð Þ¼ <sup>1</sup> *J*

> \_ *<sup>S</sup><sup>θ</sup>* <sup>¼</sup> *<sup>d</sup>*<sup>3</sup>

þ 2*λ*<sup>1</sup>

*A τ U*

*εθ dt*<sup>3</sup> <sup>þ</sup> <sup>2</sup>*λ*<sup>1</sup>

1 *J*

*<sup>h</sup>*<sup>1</sup> *<sup>X</sup>*<sup>∗</sup> ð Þ*X*\_ <sup>∗</sup> <sup>¼</sup> <sup>Ω</sup>€ <sup>∗</sup>

the required control remaining the system on the sliding surface:

*<sup>A</sup>* <sup>ð</sup>*Tem* � *Kl* sin *<sup>θ</sup>*Þ � *<sup>J</sup><sup>τ</sup>*

Then, the slip angular reference speed *ω<sup>r</sup>* can be expressed by:

**3.6 SM controllers with adaptive parameters estimation**

*Tem* � *Kl*

*εθ dt*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*λ*<sup>1</sup>

*DTC-SVM Approaches of an Induction Motor Dedicated to Position Control Applications*

*dεθ dt* <sup>þ</sup> *<sup>λ</sup>*<sup>2</sup>

with: *εθ* <sup>¼</sup> *<sup>θ</sup>* � *<sup>θ</sup>* <sup>∗</sup> . This choice takes into account that the error decreases exponentially after reaching the sliding surface. In fact, if *S<sup>θ</sup>* ¼ 0, for *t*≥ *t*0, we have:

*Tem* <sup>þ</sup> <sup>2</sup>*λ*1Ω*<sup>m</sup>* <sup>þ</sup> *<sup>λ</sup>*<sup>2</sup>

To remain the state of the system on the sliding surface *<sup>S</sup><sup>θ</sup>* <sup>¼</sup> 0, we have: \_

*d*2 *εθ dt*<sup>2</sup> <sup>þ</sup> *<sup>λ</sup>*<sup>2</sup> 1 *dεθ*

*<sup>J</sup>* sin *<sup>θ</sup>*

*<sup>m</sup>* <sup>þ</sup> <sup>2</sup>*λ*1Ω\_ <sup>∗</sup>

Then, it is easy to express the so-called equivalent control which corresponds to

*A λ*2

The new structure of this control approach is given by the block diagram of

If system (14) depends on an unknown parameter vector *<sup>γ</sup>* <sup>¼</sup> *<sup>γ</sup>*<sup>1</sup> *<sup>γ</sup>* ½ � <sup>2</sup> … *: <sup>T</sup>*, the expression of the control depends on *γ*, that is to say: *Ueq* ¼ *Ueq*ð Þ*γ* and the applied

<sup>1</sup>*<sup>θ</sup>* � *Kl*

<sup>þ</sup> *<sup>λ</sup>*<sup>2</sup>

*<sup>m</sup>* <sup>þ</sup> *<sup>λ</sup>*<sup>2</sup> 1 \_ *θ* <sup>∗</sup> (28)

<sup>1</sup>Ω*<sup>m</sup>* <sup>þ</sup> *<sup>τ</sup>Kl*

*ω<sup>r</sup>* ¼ *Ueq*,*<sup>θ</sup>* � *U*0,*<sup>θ</sup>*sign ð Þ *S<sup>θ</sup>* (30)

*U* ¼ *Ueq* þ Δ*U* (31)

<sup>1</sup> � *Kl*

*<sup>J</sup>* cos *<sup>θ</sup>*

*<sup>A</sup>* <sup>Ω</sup>*<sup>m</sup>* cos *<sup>θ</sup>* <sup>þ</sup> *<sup>J</sup><sup>τ</sup>*

<sup>1</sup>*εθ* ¼ 0 (24)

*<sup>J</sup>* sin *<sup>θ</sup>* (25)

*dt* <sup>¼</sup> <sup>0</sup> (26)

<sup>Ω</sup>*<sup>m</sup>* � *<sup>h</sup>*<sup>1</sup> *<sup>X</sup>*<sup>∗</sup> ð Þ*X*\_ <sup>∗</sup>

*S<sup>θ</sup>* ¼ 0.

(27)

*<sup>A</sup> <sup>h</sup>*<sup>1</sup> *<sup>X</sup>*<sup>∗</sup> ð Þ*X*\_ <sup>∗</sup>

(29)

A choice of the sliding surface *S X*ð Þ can be given by:

$$\mathbf{S}(X) = h(X) - h(\mathbf{X}^\*) \tag{16}$$

with *X*<sup>∗</sup> is a reference trajectory.

In order to decide a system trajectory, the equivalent control *Ueq* represents the required control to reach and to remain on the sliding surface. The corrected term Δ*U* is required to guarantee the remaining on the surface *S X*ð Þ¼ 0.

Thus, one can choose for the controller the following expression:

$$U = U\_{eq} + \Delta U \tag{17}$$

The equivalent control can be designed as follows: when the system remains on the sliding surface, we have *S X*ð Þ¼ 0, then \_ *S X*ð Þ¼ 0. Since:

$$\dot{\mathbf{S}}(\mathbf{X}) = h\_1(\mathbf{X})[f(\mathbf{X}) + \mathbf{g}(\mathbf{X})U] - h\_1(\mathbf{X}^\*)\dot{\mathbf{X}}^\* = \mathcal{F}(\mathbf{X}, \mathbf{X}^\*) + \mathcal{G}(\mathbf{X})U \tag{18}$$

where *<sup>h</sup>*1ð Þ¼ *<sup>X</sup> dh dX*.

This yields the following expression of the equivalent control:

$$\boldsymbol{U}\_{eq} = \left[ h\_1(\boldsymbol{X}) \mathbf{g}(\boldsymbol{X}) \right]^{-1} \left[ h\_1(\boldsymbol{X}^\*) \dot{\boldsymbol{X}}^\* - h\_1(\boldsymbol{X}) \boldsymbol{f}(\boldsymbol{X}) \right] = - \left[ \boldsymbol{G}(\boldsymbol{X}) \right]^{-1} \boldsymbol{\mathcal{F}}(\boldsymbol{X}, \boldsymbol{X}^\*) \tag{19}$$

under the regularity of matrix *G*ð Þ¼ *X* ½ � *h*1ð Þ *X g X*ð Þ . The term Δ*U* can be expressed as:

$$
\Delta U = -U\_0 \operatorname{sign} \left[ \mathcal{G}^T(X) \mathcal{S}(X) \right] \tag{20}
$$

In fact, if we consider the Lyapunov function:

$$V(X) = \mathbb{S}^T \mathbb{S} > \mathbf{0} \tag{21}$$

Its differential with respect to time is expressed as:

$$\dot{V} = \mathbf{S}^T \dot{\mathbf{S}} = \mathbf{S}^T \mathbf{G}(\mathbf{X}) \Delta U = -U\_0 \mathbf{S}^T \mathbf{G}(\mathbf{X}) \text{sign} \left[ \mathbf{G}^T(\mathbf{X}) \mathbf{S}(\mathbf{X}) \right] = -U\_0 \left\| \left[ \mathbf{G}^T(\mathbf{X}) \mathbf{S}(\mathbf{X}) \right] \right\|\_1 \le \mathbf{0} \quad \text{(22)}$$

This yields that the closed loop system is stable.

#### *3.5.1 Position sliding mode controller*

The sliding surface is expressed as:

$$S\_{\theta} = \left(\frac{d}{dt} + \lambda\_1\right)^2 \varepsilon\_{\theta} = \mathbf{0} \tag{23}$$

*DTC-SVM Approaches of an Induction Motor Dedicated to Position Control Applications DOI: http://dx.doi.org/10.5772/intechopen.94436*

that is to say:

**3.5 Concept of sliding mode DTC-SVM based position control**

system described by the following state equation:

*Direct Torque Control Strategies of Electrical Machines*

with *X*<sup>∗</sup> is a reference trajectory.

the sliding surface, we have *S X*ð Þ¼ 0, then \_

*Ueq* <sup>¼</sup> ½ � *<sup>h</sup>*1ð Þ *<sup>X</sup> g X*ð Þ �<sup>1</sup> *<sup>h</sup>*<sup>1</sup> *<sup>X</sup>*<sup>∗</sup> ð Þ*X*\_ <sup>∗</sup> � *<sup>h</sup>*1ð Þ *<sup>X</sup> f X*ð Þ

In fact, if we consider the Lyapunov function:

*<sup>V</sup>*\_ <sup>¼</sup> *<sup>S</sup>TS*\_ <sup>¼</sup> *STG*ð Þ *<sup>X</sup>* <sup>Δ</sup>*<sup>U</sup>* ¼ �*U*0*S<sup>T</sup>G*ð Þ *<sup>X</sup>* sign *<sup>G</sup><sup>T</sup>*ð Þ *<sup>X</sup> S X*ð Þ

This yields that the closed loop system is stable.

*<sup>S</sup><sup>θ</sup>* <sup>¼</sup> *<sup>d</sup>*

*dt* <sup>þ</sup> *<sup>λ</sup>*<sup>1</sup> � �<sup>2</sup>

Its differential with respect to time is expressed as:

under the regularity of matrix *G*ð Þ¼ *X* ½ � *h*1ð Þ *X g X*ð Þ .

*dX*.

The term Δ*U* can be expressed as:

*3.5.1 Position sliding mode controller*

**50**

The sliding surface is expressed as:

\_

where *<sup>h</sup>*1ð Þ¼ *<sup>X</sup> dh*

A choice of the sliding surface *S X*ð Þ can be given by:

Δ*U* is required to guarantee the remaining on the surface *S X*ð Þ¼ 0. Thus, one can choose for the controller the following expression:

This yields the following expression of the equivalent control:

h i

<sup>Δ</sup>*<sup>U</sup>* ¼ �*U*<sup>0</sup> sign *<sup>G</sup><sup>T</sup>*ð Þ *<sup>X</sup> S X*ð Þ

Sliding mode (SM) controllers perform well in non nonlinear systems than PI controllers [13, 14]. Indeed, the sliding mode control is a type of variable structure systems characterized by the high simplicity and the robustness against insensitivity to parameter variations and external disturbances [14–16]. Considering a nonlinear

In order to decide a system trajectory, the equivalent control *Ueq* represents the required control to reach and to remain on the sliding surface. The corrected term

The equivalent control can be designed as follows: when the system remains on

*S X*ð Þ¼ *<sup>h</sup>*1ð Þ *<sup>X</sup>* <sup>½</sup> *f X*ð Þþ *g X*ð Þ*U*� � *<sup>h</sup>*<sup>1</sup> *<sup>X</sup>*<sup>∗</sup> ð Þ*X*\_ <sup>∗</sup> <sup>¼</sup> *<sup>F</sup> <sup>X</sup>*, *<sup>X</sup>*<sup>∗</sup> ð Þþ *<sup>G</sup>*ð Þ *<sup>X</sup> <sup>U</sup>* (18)

*<sup>X</sup>*\_ <sup>¼</sup> *f X*ð Þþ *g X*ð Þ*<sup>U</sup>* (15)

*S X*ð Þ¼ *h X*ð Þ� *h X*<sup>∗</sup> ð Þ (16)

*U* ¼ *Ueq* þ Δ*U* (17)

¼ �½ � *<sup>G</sup>*ð Þ *<sup>X</sup>* �<sup>1</sup>

*V X*ð Þ¼ *<sup>S</sup>TS*><sup>0</sup> (21)

¼ �*U*<sup>0</sup> *<sup>G</sup><sup>T</sup>*ð Þ *<sup>X</sup> S X*ð Þ � � �

*εθ* ¼ 0 (23)

*<sup>F</sup> <sup>X</sup>*,*X*<sup>∗</sup> ð Þ (19)

�

<sup>1</sup> ≤0 (22)

(20)

*S X*ð Þ¼ 0. Since:

h i

h i

$$S\_{\theta} = \frac{d^2 \varepsilon\_{\theta}}{dt^2} + 2\lambda\_1 \frac{d\varepsilon\_{\theta}}{dt} + \lambda\_1^2 \varepsilon\_{\theta} = \mathbf{0} \tag{24}$$

with: *εθ* <sup>¼</sup> *<sup>θ</sup>* � *<sup>θ</sup>* <sup>∗</sup> . This choice takes into account that the error decreases exponentially after reaching the sliding surface. In fact, if *S<sup>θ</sup>* ¼ 0, for *t*≥ *t*0, we have: *εθ*ðÞ¼ *<sup>t</sup>* f g *εθ*ð Þþ *<sup>t</sup>*<sup>0</sup> ½ � *<sup>ε</sup>*\_*θ*ð Þþ *<sup>t</sup>*<sup>0</sup> *<sup>λ</sup>*1*εθ*ð Þ *<sup>t</sup>*<sup>0</sup> ð Þ *<sup>t</sup>* � *<sup>t</sup>*<sup>0</sup> *<sup>e</sup>*�*λ*1ð Þ *<sup>t</sup>*�*t*<sup>0</sup> .

In this case, function *h X*ð Þ is expressed as:

$$h(X) = \frac{1}{J} T\_{em} + 2\lambda\_1 \Omega\_m + \lambda\_1^2 \theta - \frac{K\_l}{J} \sin \theta \tag{25}$$

To remain the state of the system on the sliding surface *<sup>S</sup><sup>θ</sup>* <sup>¼</sup> 0, we have: \_ *S<sup>θ</sup>* ¼ 0. This leads to:

$$\dot{S}\_{\theta} = \frac{d^3 \varepsilon\_{\theta}}{dt^3} + 2\lambda\_1 \frac{d^2 \varepsilon\_{\theta}}{dt^2} + \lambda\_1^2 \frac{d \varepsilon\_{\theta}}{dt} = \mathbf{0} \tag{26}$$

That is to say:

$$\begin{split} \dot{S}\_{\theta} &= \frac{1}{J} \Big( -\frac{1}{\tau} T\_{em} + \frac{A}{\tau} U \Big) + 2\dot{\lambda}\_1 \Big( \frac{1}{J} T\_{em} - \frac{K\_l}{J} \sin \theta \Big) + \left( \dot{\lambda}\_1^2 - \frac{K\_l}{J} \cos \theta \right) \Omega\_m - h\_1(X^\*) \dot{X}^\* \\ &= 0 \end{split} \tag{27}$$

where:

$$h\_1(X^\*)\dot{X}^\* = \left(\ddot{\Omega}\_m^\* + 2\dot{\lambda}\_1 \dot{\Omega}\_m^\* + \dot{\lambda}\_1^2 \dot{\theta}^\*\right) \tag{28}$$

Then, it is easy to express the so-called equivalent control which corresponds to the required control remaining the system on the sliding surface:

$$U\_{eq, \theta} = \frac{1}{A} T\_{em} - 2 \frac{\lambda\_1 \tau}{A} (T\_{em} - K\_l \sin \theta) - \frac{J\tau}{A} \lambda\_1^2 \Omega\_m + \frac{\tau K\_l}{A} \Omega\_m \cos \theta + \frac{J\tau}{A} h\_1(X^\*) \dot{X}^\* \tag{29}$$

Then, the slip angular reference speed *ω<sup>r</sup>* can be expressed by:

$$
\rho\_r = U\_{eq, \theta} - U\_{0, \theta} \text{sign} \quad (\mathbb{S}\_{\theta}) \tag{30}
$$

The new structure of this control approach is given by the block diagram of **Figure 10**.

#### **3.6 SM controllers with adaptive parameters estimation**

If system (14) depends on an unknown parameter vector *<sup>γ</sup>* <sup>¼</sup> *<sup>γ</sup>*<sup>1</sup> *<sup>γ</sup>* ½ � <sup>2</sup> … *: <sup>T</sup>*, the expression of the control depends on *γ*, that is to say: *Ueq* ¼ *Ueq*ð Þ*γ* and the applied control law becomes:

$$
\overline{U} = \overline{U}\_{eq} + \Delta U \tag{31}
$$

• Theorem

adaptive laws:

• Proof

*<sup>V</sup>*\_ <sup>¼</sup> *<sup>s</sup> T* \_ *S* þ Σ*<sup>i</sup>* 1 *ηi* Δ*γi*Δ*γ*\_*<sup>i</sup>*

> þΣ*<sup>i</sup>* 1 *ηi*

¼ � *<sup>G</sup><sup>τ</sup>*

� � � *γ*\_

*DOI: http://dx.doi.org/10.5772/intechopen.94436*

*<sup>i</sup>* ¼ �*ηiS<sup>T</sup> <sup>G</sup>*ð Þ *<sup>X</sup>*

*<sup>V</sup>* <sup>¼</sup> <sup>1</sup> 2 *<sup>S</sup>TS* <sup>þ</sup> 1 2 X *i*

Let us consider the following Lyapunov function:

respect to time: *γ*\_ ≃0. Then, we can write: Δ*γ*\_ ≃*γ*\_.

2

<sup>¼</sup> *ST <sup>G</sup>*ð Þ *<sup>X</sup>* <sup>Δ</sup>*<sup>U</sup>* <sup>þ</sup> <sup>Σ</sup>*<sup>i</sup> <sup>G</sup>*ð Þ *<sup>X</sup>*

Δ*γi*Δ*γ*\_*<sup>i</sup>* þ *o*ð Þ Δ*γ*

<sup>¼</sup> *STG*ð Þ *<sup>X</sup>* <sup>Δ</sup>*<sup>U</sup>* <sup>þ</sup> <sup>Σ</sup>*<sup>i</sup> ST <sup>G</sup>*ð Þ *<sup>X</sup>*

<sup>¼</sup> *STG*ð Þ *<sup>X</sup>* <sup>Δ</sup>*<sup>U</sup>* <sup>þ</sup> *<sup>o</sup>*ð Þ <sup>Δ</sup>*<sup>γ</sup>*

ð Þ *X S X*ð Þ

values of its components.

*rotor resistance*

*Lyapunov* theorem [17]. It is easy to show that:

**53**

� � � 1

þ *o*ð Þ Δ*γ*

Control laws (17), (19) and (20) stabilize system (15) with the following

*DTC-SVM Approaches of an Induction Motor Dedicated to Position Control Applications*

*∂Ueq ∂γi*

In the following, it assumed that vector *γ* is constant or it has slow variations

with respect to time, in such away that we can neglect its differential with.

The differential with respect to time of function *V* is expressed as:

ð Þþ *γ*

� �

� �Δ*γ<sup>i</sup>* � �

ð Þþ *γ*

� �

� �


where k k� <sup>1</sup> is the norm "1" of a vector which corresponds to the sum of absolute

*3.6.1 Position adaptive SM controller with variations on the mutual inductance and the*

The sensitivity of the DTC-SVM to (i) variations on the magnetic permeability of the stator and rotor cores, and (ii) variations on the rotor resistance, which can vary with time and operating conditions, can be removed by an online estimation of the mutual inductance and the rotor resistance. The adaptive SM of the speed can be derived based on the mutual inductance and rotor resistance estimations using the

*<sup>∂</sup>g X*ð Þ *∂γi*

> *<sup>∂</sup>G*ð Þ *<sup>X</sup> ∂γi*

Δ*U*

Δ*U*

þ 1 *ηi* Δ*γ*\_*<sup>i</sup>*

ð Þ *<sup>X</sup> S X*ð Þ h i <sup>þ</sup> *<sup>o</sup>*ð Þ <sup>Δ</sup>*<sup>γ</sup>*

*U<sup>θ</sup>* ¼ *Ueq*,*<sup>θ</sup>* � *U*0,*<sup>θ</sup>* sign ð Þ *S*<sup>Ω</sup> (36)

þ *o*ð Þ Δ*γ* 2

*∂Ueq ∂γi*

> *∂Ueq ∂γi*

<sup>2</sup> ¼ �*STG*ð Þ *<sup>X</sup>* sign *<sup>G</sup><sup>T</sup>*

<sup>2</sup> ≤0

ð Þþ *γ*

1 *ηi* Δ*γ*<sup>2</sup>

*<sup>∂</sup>G*ð Þ *<sup>X</sup> ∂γi*

Δ*U* � � (33)

*<sup>i</sup>* (34)

Δ*γ<sup>i</sup>* þ *o*ð Þ Δ*γ*

2

2

(35)

**Figure 10.** *Induction motor position regulation based on the DTC-SVM with sliding mode controllers.*

where *Ueq* ¼ *Ueq*ð Þ*γ* . *γ* is the estimated vector of *γ*. Referring to Eq. (17), the differential of *S* is expressed as:

$$\begin{split} \dot{\mathcal{G}} &= \mathcal{F}(\mathcal{X}, \mathcal{X}^\*) + \mathcal{G}(\mathcal{X})\overline{\mathcal{U}} = \mathcal{F}(\mathcal{X}, \mathcal{X}^\*) + \mathcal{G}(\mathcal{X}) \left( \overline{\mathcal{U}}\_{eq} + \Delta \mathcal{U} \right) \\ &= \underbrace{\mathcal{F}(\mathcal{X}, \mathcal{X}^\*) + \mathcal{G}(\mathcal{X})U\_{eq}}\_{=0} + \mathcal{G}(\mathcal{X})\Delta U + \mathcal{G}(\mathcal{X}) \left( \overline{\mathcal{U}}\_{eq} - \mathcal{U}\_{eq} \right)}\_{=0} \\ &= \mathcal{G}(\mathcal{X})\Delta U + \mathcal{G}(\mathcal{X}) \left( \overline{\mathcal{U}}\_{eq} - U\_{eq} \right) \\ &= \mathcal{G}(\mathcal{X})\Delta U + \mathcal{G}(\mathcal{X}) \left( \overline{\mathcal{U}}\_{eq} - U\_{eq} \right) + \underbrace{\left[ \mathcal{G}(\mathcal{X}) - \overline{\mathcal{G}}(\mathcal{X}) \right] \underbrace{\left( \overline{\mathcal{U}}\_{eq} - U\_{eq} \right)}\_{=\circ(\Delta \mathcal{Y})}}\_{=\circ(\Delta \mathcal{Y})} \\ &= \underbrace{\left( \mathcal{G}(\mathcal{X})\Delta U - U\_{eq} \right)}\_{=\circ(\Delta \mathcal{Y})} \end{split} \tag{32}$$

$$\begin{split} \mathcal{G} &= \mathcal{G}(X)\Delta U + \overline{\mathcal{G}}(X) \left( \Sigma\_{i} \frac{\partial U\_{eq}}{\partial \gamma\_{i}}(\overline{\gamma}) \Delta \gamma\_{i} \right) + o(\Delta \gamma)^{2} \\ &= \overline{\mathcal{G}}(X)\Delta U + \overline{\mathcal{G}}(X) \left( \Sigma\_{i} \frac{\partial U\_{eq}}{\partial \gamma\_{i}}(\overline{\gamma}) \Delta \gamma\_{i} \right) + \left[ \overline{\mathcal{G}}(X) - \overline{\mathcal{G}}(X) \right] \Delta U + o(\Delta \gamma)^{2} \\ &= \overline{\mathcal{G}}(X)\Delta U + \overline{\mathcal{G}}(X) \left( \Sigma\_{i} \frac{\partial U\_{eq}}{\partial \gamma\_{i}}(\overline{\gamma}) \Delta \gamma\_{i} \right) + \left[ \overline{\mathcal{G}}(X) - \overline{\mathcal{G}}(X) \right] \Delta U + o(\Delta \gamma)^{2} \\ &= \overline{\mathcal{G}}(X)\Delta U + \left( \sum\_{i} \left[ \overline{\mathcal{G}}(X) \frac{\partial U\_{eq}}{\partial \gamma\_{i}}(\overline{\gamma}) + \frac{\partial \mathcal{G}(X)}{\partial \gamma\_{i}} \Delta U \right] \Delta \gamma\_{i} \right) + o(\Delta \gamma)^{2} \end{split}$$

with Δ*γ* ¼ *γ* � *γ* and *G*ð Þ X is the expression of *G*ð Þ X for *γ* ¼ *γ*.

*DTC-SVM Approaches of an Induction Motor Dedicated to Position Control Applications DOI: http://dx.doi.org/10.5772/intechopen.94436*

• Theorem

Control laws (17), (19) and (20) stabilize system (15) with the following adaptive laws:

$$\dot{\overline{\gamma}}\_i = -\eta\_i \mathbf{S}^T \left( \overline{\mathcal{G}}(\mathbf{X}) \frac{\partial U\_{eq}}{\partial \chi\_i}(\overline{\gamma}) + \frac{\partial \mathcal{G}(\mathbf{X})}{\partial \chi\_i} \Delta U \right) \tag{33}$$

• Proof

where *Ueq* ¼ *Ueq*ð Þ*γ* . *γ* is the estimated vector of *γ*. Referring to Eq. (17), the differential of *S* is expressed as:

*Direct Torque Control Strategies of Electrical Machines*

<sup>¼</sup> *<sup>F</sup> <sup>X</sup>*,*X*<sup>∗</sup> ð Þþ *<sup>G</sup>*ð Þ *<sup>X</sup> Ueq* |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} <sup>¼</sup><sup>0</sup>

¼ *G*ð Þ *X* Δ*U* þ *G*ð Þ *X* Σ*<sup>i</sup>*

¼ *G*ð Þ *X* Δ*U* þ *G*ð Þ *X* Σ*<sup>i</sup>*

¼ *G*ð Þ *X* Δ*U* þ *G*ð Þ *X* Σ*<sup>i</sup>*

<sup>¼</sup> *<sup>G</sup>*ð Þ *<sup>X</sup>* <sup>Δ</sup>*<sup>U</sup>* <sup>þ</sup> <sup>X</sup>

**52**

¼ *G*ð Þ *X* Δ*U* þ *G*ð Þ *X Ueq* � *Ueq*

¼ *G*ð Þ *X* Δ*U* þ *G*ð Þ *X Ueq* � *Ueq*

*<sup>S</sup>* <sup>¼</sup> *<sup>F</sup> <sup>X</sup>*,*X*<sup>∗</sup> ð Þþ *<sup>G</sup>*ð Þ *<sup>X</sup> <sup>U</sup>* <sup>¼</sup> *<sup>F</sup> <sup>X</sup>*, *<sup>X</sup>*<sup>∗</sup> ð Þþ *<sup>G</sup>*ð Þ *<sup>X</sup> Ueq* <sup>þ</sup> <sup>Δ</sup>*<sup>U</sup>* � �

*Induction motor position regulation based on the DTC-SVM with sliding mode controllers.*

� �

*∂Ueq ∂γi*

*∂Ueq ∂γi*

*∂Ueq ∂γi*

*<sup>i</sup> G*ð Þ *X*

þ *G*ð Þ *X* Δ*U* þ *G*ð Þ *X Ueq* � *Ueq*

h i


þ *o*ð Þ Δ*γ* 2

*<sup>∂</sup>G*ð Þ *<sup>X</sup> ∂γi*

� � <sup>þ</sup> *<sup>G</sup>*ð Þ� *<sup>X</sup> <sup>G</sup>*ð Þ *<sup>X</sup>*

ð Þ*γ* Δ*γ<sup>i</sup>* � �

ð Þ*γ* Δ*γ<sup>i</sup>* � �

ð Þ*γ* Δ*γ<sup>i</sup>* � �

ð Þþ *γ*

� �

� �

*∂Ueq ∂γi*

with Δ*γ* ¼ *γ* � *γ* and *G*ð Þ X is the expression of *G*ð Þ X for *γ* ¼ *γ*.

� �


> þ *G*ð Þ� *X G*ð Þ *X* h i

> þ *G*ð Þ� *X G*ð Þ *X* h i

> > Δ*U*

Δ*γ<sup>i</sup>*

*Ueq* � *Ueq* � � |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} <sup>¼</sup>*o*ð Þ <sup>Δ</sup>*<sup>γ</sup>*

Δ*U* þ *o*ð Þ Δ*γ*

Δ*U* þ *o*ð Þ Δ*γ*

þ *o*ð Þ Δ*γ* 2 2

(32)

2

\_

**Figure 10.**

Let us consider the following Lyapunov function:

$$V = \frac{1}{2} \mathbf{S}^T \mathbf{S} + \frac{1}{2} \sum\_{i} \frac{1}{\eta\_i} \Delta \boldsymbol{\gamma}\_i^2 \tag{34}$$

In the following, it assumed that vector *γ* is constant or it has slow variations with respect to time, in such away that we can neglect its differential with.

respect to time: *γ*\_ ≃0. Then, we can write: Δ*γ*\_ ≃*γ*\_.

The differential with respect to time of function *V* is expressed as:

*<sup>V</sup>*\_ <sup>¼</sup> *<sup>s</sup> T* \_ *S* þ Σ*<sup>i</sup>* 1 *ηi* Δ*γi*Δ*γ*\_*<sup>i</sup>* <sup>¼</sup> *ST <sup>G</sup>*ð Þ *<sup>X</sup>* <sup>Δ</sup>*<sup>U</sup>* <sup>þ</sup> <sup>Σ</sup>*<sup>i</sup> <sup>G</sup>*ð Þ *<sup>X</sup> ∂Ueq ∂γi* ð Þþ *γ <sup>∂</sup>g X*ð Þ *∂γi* Δ*U* � �Δ*γ<sup>i</sup>* � � þ *o*ð Þ Δ*γ* 2 � � þΣ*<sup>i</sup>* 1 *ηi* Δ*γi*Δ*γ*\_*<sup>i</sup>* þ *o*ð Þ Δ*γ* 2 <sup>¼</sup> *STG*ð Þ *<sup>X</sup>* <sup>Δ</sup>*<sup>U</sup>* <sup>þ</sup> <sup>Σ</sup>*<sup>i</sup> ST <sup>G</sup>*ð Þ *<sup>X</sup> ∂Ueq ∂γi* ð Þþ *γ <sup>∂</sup>G*ð Þ *<sup>X</sup> ∂γi* Δ*U* � � þ 1 *ηi* Δ*γ*\_*<sup>i</sup>* � � |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} <sup>¼</sup><sup>0</sup> Δ*γ<sup>i</sup>* þ *o*ð Þ Δ*γ* 2 <sup>¼</sup> *STG*ð Þ *<sup>X</sup>* <sup>Δ</sup>*<sup>U</sup>* <sup>þ</sup> *<sup>o</sup>*ð Þ <sup>Δ</sup>*<sup>γ</sup>* <sup>2</sup> ¼ �*STG*ð Þ *<sup>X</sup>* sign *<sup>G</sup><sup>T</sup>* ð Þ *<sup>X</sup> S X*ð Þ h i <sup>þ</sup> *<sup>o</sup>*ð Þ <sup>Δ</sup>*<sup>γ</sup>* 2 ¼ � *<sup>G</sup><sup>τ</sup>* ð Þ *X S X*ð Þ � � � � � � 1 þ *o*ð Þ Δ*γ* <sup>2</sup> ≤0 (35)

where k k� <sup>1</sup> is the norm "1" of a vector which corresponds to the sum of absolute values of its components.

#### *3.6.1 Position adaptive SM controller with variations on the mutual inductance and the rotor resistance*

The sensitivity of the DTC-SVM to (i) variations on the magnetic permeability of the stator and rotor cores, and (ii) variations on the rotor resistance, which can vary with time and operating conditions, can be removed by an online estimation of the mutual inductance and the rotor resistance. The adaptive SM of the speed can be derived based on the mutual inductance and rotor resistance estimations using the *Lyapunov* theorem [17].

It is easy to show that:

$$U\_{\theta} = \overline{U}\_{eq,\theta} - U\_{0,\theta} \text{ sign } (\mathbb{S}\_{\Omega}) \tag{36}$$

where:

$$\overline{U}\_{eq,\theta} = -\left[\overline{\mathcal{G}}(X)\right]^{-1}\overline{\mathcal{F}}(X, X^\*)\tag{37}$$

Thereby, Eq. (42) gives:

*DOI: http://dx.doi.org/10.5772/intechopen.94436*

**control**

*φ θ*, *<sup>d</sup><sup>θ</sup>*

**55**

• a reference stator flux Φ<sup>∗</sup>

approaches under study,

speed Ω*<sup>m</sup>* and the torque *Cem*, leading to:

• constant value of *θ* equal to 30o from 1 s to 1.5 s.

• constant value of *θ* equal to 60o from 2.5 s to 4 s.

The analysis of simulation results leads to the following items:

currents are smallest for results given by SM controllers.

*<sup>V</sup>*\_ *<sup>θ</sup>* ¼ �*U*0,*<sup>θ</sup>G*ð Þ *<sup>X</sup>* <sup>∣</sup>*S*Ω<sup>∣</sup> <sup>þ</sup> <sup>Δ</sup>*<sup>M</sup> <sup>G</sup> <sup>∂</sup>Ueq*,*<sup>θ</sup>*

þ 1 *ηθ*2 Δ*R*\_ *<sup>r</sup>*

*DTC-SVM Approaches of an Induction Motor Dedicated to Position Control Applications*

¼ �*U*0,*<sup>θ</sup>G*ð Þ *<sup>X</sup>* <sup>∣</sup>*S*Ω<sup>∣</sup> <sup>þ</sup> *<sup>o</sup>*ð Þ <sup>Δ</sup>*M*, <sup>Δ</sup>*Rr* <sup>2</sup> <sup>≤</sup><sup>0</sup>

**3.7 Simulation results investigated SM DTC-SVM approach based position**

<sup>s</sup> equal to 1 *Wb*,

• The modulation period has been fixed to *Tmod* ¼ 150*μs* in all DTC-SVM

• Constants involved in the position SM controller are: *U*0,*<sup>θ</sup>* ¼ 50 and *λθ* ¼ 100. The constants involved in the flux SM controller are: *U*0,*<sup>ϕ</sup>*, ¼ 150 and *λϕ* ¼ 2. The desired trajectory is defined by smooth variations of the position *θ*, the

• variations of *<sup>θ</sup>* form �60° (morning panel position) to 30o from 0 s to 1 s,

• variations of *θ* form 30o to 60o (afternoon panel position) from 1.5 s to 2.5 s,

• **Figures 11** and **12** present evolutions of the position *θ*, the speed Ω*m*, the torque *Cem*, the flux ∣Φ*s*∣ and the current *ias*, using PID controllers (figures indexed by 1), PID controllers with a nonlinear compensator (figures indexed by 2), and SM controllers (figures indexed by 3). It is well obvious that a good tracking has been realized by these control approaches. It is also obvious, that there is no significant difference between results yielded by PID controllers and PID controllers with a nonlinear compensator. This justifies that the nonlinear term

*dt* � � can be neglected. Moreover, ripples of the torque, the flux and stator

Simulation works have been carried out in order to investigate performances of the position control of the induction motor drive under the above-presented DTC-SVM strategies, using PID, PID with a nonlinear compensator and SM controllers. For the sake of comparison, both strategies have been considered in the same induction motor drive using the same implementation conditions, such that:

� �


*∂Rr* � �

Since *<sup>G</sup>*ð Þ *<sup>X</sup>* <sup>&</sup>gt;0,*V*\_ *<sup>θ</sup>* is negative. Then, the system is stable.

<sup>þ</sup>Δ*Rr <sup>G</sup> <sup>∂</sup>Ueq*,*<sup>θ</sup>*

*∂M* � �

þ 1 *ηθ*1 Δ*M*\_

(45)

� �


<sup>þ</sup> *<sup>o</sup>*ð Þ <sup>Δ</sup>*M*, <sup>Δ</sup>*Rr* <sup>2</sup>

Then:

$$\begin{aligned} \dot{S}\_{\theta} &= \mathcal{F}(\mathbf{X}, \mathbf{X}^\*) + \mathcal{G}(\mathbf{X}) \overline{\mathcal{U}}\_{\theta} = \mathcal{F}(\mathbf{X}, \mathbf{X}^\*) + \mathcal{G}(\mathbf{X}) \left( \overline{\mathcal{U}}\_{eq,\theta} + \Delta U\_{\theta} \right) \\\\ &= \mathcal{F}(\mathbf{X}, \mathbf{X}^\*) + \mathcal{G}(\mathbf{X}) U\_{eq,\theta} + \mathcal{G}(\mathbf{X}) \Delta U\_{\theta} + \mathcal{G}(\mathbf{X}) \left( \overline{\mathcal{U}}\_{eq,\theta} - U\_{eq,\theta} \right) \\\\ &= \mathcal{G}(\mathbf{X}) \Delta U\_{\theta} + \mathcal{G}(\mathbf{X}) \left( \overline{\mathcal{U}}\_{eq,\theta} - U\_{eq,\theta} \right) \end{aligned} \tag{38}$$

• Corollary

The following slip angular reference speed control law stabilizes the speed loop:

$$
\rho\_r = \overline{U}\_{eq, \theta} - U\_{0, \theta} \text{ sign} \quad (\text{S}\_\Omega) \tag{39}
$$

where *Ueq*,*<sup>θ</sup>* ¼ *Ueq*,*<sup>θ</sup> M*, *Rr* � �, *M* and *Rr* are estimator values of the mutual inductance and the rotor resistance given by the following updating laws:

$$\begin{aligned} \dot{\overline{M}} &= -\eta\_{\theta\_1} \overline{\mathcal{G}} \mathcal{S}\_{\theta} \left( \frac{\partial U\_{eq,\theta}}{\partial \mathcal{M}} \right) \\\\ \dot{\overline{R}}\_r &= -\eta\_{\theta\_2} \overline{\mathcal{G}} \mathcal{S}\_{\theta} \left( \frac{\partial U\_{eq,\theta}}{\partial \mathcal{R}} \right) \end{aligned} \tag{40}$$

with: *ηθ*<sup>1</sup> and *ηθ*<sup>2</sup> positive scalars, *G* ¼ *G*ð Þ *M*, *Rr* and *G* ¼ *G M*, *Rr* � � defined in Eq. (17).

#### • Proof

Considering the following function:

$$V\_{\theta} = \frac{1}{2} \mathcal{S}\_{\theta}^{2} + \frac{1}{2\eta\_{\theta\_{1}}} \Delta \mathcal{M}^{2} + \frac{1}{2\eta\_{\theta\_{2}}} \Delta R\_{r}^{2} \tag{41}$$

with Δ*M* ¼ *M* � *M* and Δ*Rr* ¼ *Rr* � *Rr*. The time derivative of the *Lyapunov* function can be expressed as:

$$
\dot{\mathcal{V}}\_{\theta} = \mathcal{S}\_{\theta}\dot{\mathcal{S}}\_{\theta} + \frac{1}{\eta\_{\theta\_1}}\Delta M \Delta \dot{M} + \frac{1}{\eta\_{\theta\_2}}\Delta R\_r \Delta \dot{R}\_r \tag{42}
$$

However:

$$\overline{\mathbf{U}}\_{eq,\theta} - \mathbf{U}\_{eq,\theta} = \left(\overline{\mathbf{M}} - \mathbf{M}\right) \left(\frac{\partial U\_{eq,\theta}}{\partial \mathbf{M}}\right) + \left(\overline{R}\_r - R\_r\right) \left(\frac{\partial U\_{eq,\theta}}{\partial R\_r}\right) + o(\Delta \mathbf{M}, \ \Delta R\_r)^2 \tag{43}$$

Moreover:

$$\mathcal{G}\left(\overline{U}\_{eq,\theta} - U\_{eq,\theta}\right) = \overline{\mathcal{G}} \left(\overline{U}\_{eq,\theta} - U\_{eq,\theta}\right) + o(\Delta \mathcal{M}, \ \Delta \mathcal{R}\_r)^2\tag{44}$$

*DTC-SVM Approaches of an Induction Motor Dedicated to Position Control Applications DOI: http://dx.doi.org/10.5772/intechopen.94436*

Thereby, Eq. (42) gives:

where:

Then:

\_

• Corollary

Eq. (17).

• Proof

However:

Moreover:

**54**

where *Ueq*,*<sup>θ</sup>* ¼ *Ueq*,*<sup>θ</sup> M*, *Rr*

Considering the following function:

with Δ*M* ¼ *M* � *M* and Δ*Rr* ¼ *Rr* � *Rr*.

*Ueq*,*<sup>θ</sup>* � *Ueq*,*<sup>θ</sup>* <sup>¼</sup> *<sup>M</sup>* � *<sup>M</sup>* � � *<sup>∂</sup>Ueq*,*<sup>θ</sup>*

*G Ueq*,*<sup>θ</sup>* � *Ueq*,*<sup>θ</sup>*

*Ueq*,*<sup>θ</sup>* ¼ � *G*ð Þ *X*

*<sup>S</sup><sup>θ</sup>* <sup>¼</sup> *<sup>F</sup> <sup>X</sup>*,*X*<sup>∗</sup> ð Þþ *<sup>G</sup>*ð Þ *<sup>X</sup> <sup>U</sup><sup>θ</sup>* <sup>¼</sup> *<sup>F</sup> <sup>X</sup>*, *<sup>X</sup>*<sup>∗</sup> ð Þþ *<sup>G</sup>*ð Þ *<sup>X</sup> Ueq*,*<sup>θ</sup>* <sup>þ</sup> <sup>Δ</sup>*U<sup>θ</sup>*

� �

inductance and the rotor resistance given by the following updating laws:

*<sup>M</sup>*\_ ¼ �*ηθ*<sup>1</sup>

with: *ηθ*<sup>1</sup> and *ηθ*<sup>2</sup> positive scalars, *G* ¼ *G*ð Þ *M*, *Rr* and *G* ¼ *G M*, *Rr*

The time derivative of the *Lyapunov* function can be expressed as:

*S<sup>θ</sup>* þ 1 *ηθ*1

*∂M* � �

� � <sup>¼</sup> *<sup>G</sup> Ueq*,*<sup>θ</sup>* � *Ueq*,*<sup>θ</sup>*

\_ *Rr* ¼ �*ηθ*<sup>2</sup>

*<sup>V</sup><sup>θ</sup>* <sup>¼</sup> <sup>1</sup> 2 *S*2 *<sup>θ</sup>* þ 1 2*ηθ*<sup>1</sup>

*<sup>V</sup>*\_ *<sup>θ</sup>* <sup>¼</sup> *<sup>S</sup><sup>θ</sup>* \_

¼ *G*ð Þ *X* Δ*U<sup>θ</sup>* þ *G*ð Þ *X Ueq*,*<sup>θ</sup>* � *Ueq*,*<sup>θ</sup>*

*Direct Torque Control Strategies of Electrical Machines*

<sup>¼</sup> *<sup>F</sup> <sup>X</sup>*,*X*<sup>∗</sup> ð Þþ *<sup>G</sup>*ð Þ *<sup>X</sup> Ueq*,*<sup>θ</sup>* <sup>þ</sup> *<sup>G</sup>*ð Þ *<sup>X</sup>* <sup>Δ</sup>*U<sup>θ</sup>* <sup>þ</sup> *<sup>G</sup>*ð Þ *<sup>X</sup> Ueq*,*<sup>θ</sup>* � *Ueq*,*<sup>θ</sup>*

The following slip angular reference speed control law stabilizes the speed loop:

*G S<sup>θ</sup>*

*G S<sup>θ</sup>*

h i�<sup>1</sup>

*<sup>F</sup> <sup>X</sup>*,*X*<sup>∗</sup> ð Þ (37)

� �

*ω<sup>r</sup>* ¼ *Ueq*,*<sup>θ</sup>* � *U*0,*<sup>θ</sup>* sign ð Þ *S*<sup>Ω</sup> (39)

� �, *M* and *Rr* are estimator values of the mutual

*∂Ueq*,*<sup>θ</sup> ∂M* � �

*∂Ueq*,*<sup>θ</sup> ∂R* � �

<sup>Δ</sup>*M*<sup>2</sup> <sup>þ</sup>

<sup>Δ</sup>*M*Δ*M*\_ <sup>þ</sup>

þ *Rr* � *Rr*

� � *<sup>∂</sup>Ueq*,*<sup>θ</sup>*

1 2*ηθ*<sup>2</sup>

1 *ηθ*2

*∂Rr* � �

� � <sup>þ</sup> *<sup>o</sup>*ð Þ <sup>Δ</sup>*M*, <sup>Δ</sup>*Rr* <sup>2</sup> (44)

Δ*R*<sup>2</sup>

� �

(38)

(40)

� � defined in

*<sup>r</sup>* (41)

Δ*Rr*Δ*R*\_ *<sup>r</sup>* (42)

<sup>þ</sup> *<sup>o</sup>*ð Þ <sup>Δ</sup>*M*, <sup>Δ</sup>*Rr* <sup>2</sup> (43)

$$\begin{split} \dot{V}\_{\theta} &= -U\_{0,\theta} \mathcal{G}(\mathcal{X}) |\mathcal{S}\_{\Omega}| + \Delta \mathcal{M} \underbrace{\left[ \overline{\mathcal{G}} \left( \frac{\partial U\_{eq,\theta}}{\partial \mathcal{M}} \right) + \frac{1}{\eta\_{\theta\_{1}}} \Delta \dot{\mathcal{M}} \right]}\_{=0} \\ &+ \Delta R\_{r} \underbrace{\left[ \overline{\mathcal{G}} \left( \frac{\partial U\_{eq,\theta}}{\partial R\_{r}} \right) + \frac{1}{\eta\_{\theta\_{2}}} \Delta \dot{R}\_{r} \right]}\_{=0} + o(\Delta \mathcal{M}, \Delta R\_{r})^{2} \\ &= -U\_{0,\theta} \mathcal{G}(\mathcal{X}) |\mathcal{S}\underline{\alpha}| + o(\Delta \mathcal{M}, \ \Delta R\_{r})^{2} \le 0 \end{split} \tag{45}$$

Since *<sup>G</sup>*ð Þ *<sup>X</sup>* <sup>&</sup>gt;0,*V*\_ *<sup>θ</sup>* is negative. Then, the system is stable.

#### **3.7 Simulation results investigated SM DTC-SVM approach based position control**

Simulation works have been carried out in order to investigate performances of the position control of the induction motor drive under the above-presented DTC-SVM strategies, using PID, PID with a nonlinear compensator and SM controllers. For the sake of comparison, both strategies have been considered in the same induction motor drive using the same implementation conditions, such that:


The desired trajectory is defined by smooth variations of the position *θ*, the speed Ω*<sup>m</sup>* and the torque *Cem*, leading to:


The analysis of simulation results leads to the following items:

• **Figures 11** and **12** present evolutions of the position *θ*, the speed Ω*m*, the torque *Cem*, the flux ∣Φ*s*∣ and the current *ias*, using PID controllers (figures indexed by 1), PID controllers with a nonlinear compensator (figures indexed by 2), and SM controllers (figures indexed by 3). It is well obvious that a good tracking has been realized by these control approaches. It is also obvious, that there is no significant difference between results yielded by PID controllers and PID controllers with a nonlinear compensator. This justifies that the nonlinear term *φ θ*, *<sup>d</sup><sup>θ</sup> dt* � � can be neglected. Moreover, ripples of the torque, the flux and stator currents are smallest for results given by SM controllers.

#### **Figure 11.**

*Induction motor position regulation considering (subscript "1") DTC-SVM approach using PI controller, (subscript "2") DTC-SVM approach using PI controller with a nonlinear compensator and (subscript "3") DTC-SVM approach using sliding mode controllers. Legend: (a) evolution of the position and its reference and (b) the speed of the motor and its reference.*

**Figure 12.**

**Figure 13.**

**57**

*and (b) the speed of the motor and its reference.*

*Induction motor position regulation considering (1) DTC-SVM approach using PI controller, (2) DTC-SVM approach using PI controller with a nonlinear compensator and (3) DTC-SVM approach using sliding mode controllers. Legend: (a) evolution of the electromagnetic torque, (b) the stator flux and (c) the stator current of phase a.*

*DTC-SVM Approaches of an Induction Motor Dedicated to Position Control Applications*

*DOI: http://dx.doi.org/10.5772/intechopen.94436*

*Induction motor position regulation, considering +100% variations on the stator resistance, (1) DTC-SVM approach using PI controller, (2) DTC-SVM approach using PI controller with a nonlinear compensator and (3) DTC-SVM approach using sliding mode controllers. Legend: (a) evolution of the position and its reference*


#### **3.8 Performance criteria**

Considering the same simulation, we propose to use performance criteria defined in the appendix.

In the following, the steady state operating point is defined by a desired position *θ* equal to 60<sup>o</sup> for the time larger than 2.5 s.

• Total Harmonic Distortion (THD)

The first criterion is the average total harmonic distortion (THD) of the stator current which is defined in the Appendix.

*DTC-SVM Approaches of an Induction Motor Dedicated to Position Control Applications DOI: http://dx.doi.org/10.5772/intechopen.94436*

**Figure 12.**

• **Figures 13** and **14** present the same variable evolutions for variations of

*Induction motor position regulation considering (subscript "1") DTC-SVM approach using PI controller, (subscript "2") DTC-SVM approach using PI controller with a nonlinear compensator and (subscript "3") DTC-SVM approach using sliding mode controllers. Legend: (a) evolution of the position and its reference and*

• Thus, the implementation of the DTC-SVM using sliding mode controllers strategies highlights high dynamical performances obtained with the lowest

Considering the same simulation, we propose to use performance criteria

In the following, the steady state operating point is defined by a desired position

The first criterion is the average total harmonic distortion (THD) of the stator

torque ripple, the lowest flux ripple and the lowest current ripple.

parameters are known and do not vary.

*Direct Torque Control Strategies of Electrical Machines*

*θ* equal to 60<sup>o</sup> for the time larger than 2.5 s.

• Total Harmonic Distortion (THD)

current which is defined in the Appendix.

**3.8 Performance criteria**

*(b) the speed of the motor and its reference.*

**Figure 11.**

defined in the appendix.

**56**

machine parameters as: þ100% variations on the stator resistance *Rs*, þ 100% variations on the rotor resistance *Rr* and � 50% variations on the mutual inductance *M*. It is clear that results, yielded from PID controllers without and with a nonlinear compensator, present important oscillations. However, SM controllers with parameter's updating give same results as in the case where

*Induction motor position regulation considering (1) DTC-SVM approach using PI controller, (2) DTC-SVM approach using PI controller with a nonlinear compensator and (3) DTC-SVM approach using sliding mode controllers. Legend: (a) evolution of the electromagnetic torque, (b) the stator flux and (c) the stator current of phase a.*

**Figure 13.**

*Induction motor position regulation, considering +100% variations on the stator resistance, (1) DTC-SVM approach using PI controller, (2) DTC-SVM approach using PI controller with a nonlinear compensator and (3) DTC-SVM approach using sliding mode controllers. Legend: (a) evolution of the position and its reference and (b) the speed of the motor and its reference.*

In this context, the frequency spectrum of the stator current *ias* has been analyzed by the observation of amplitudes of all its harmonics frequencies. **Figure 15** shows the evolution of one period of *ias* between 3 s and 4 s, its spectrum (only 20 harmonics has been presented). It is obvious that SM controllers give less ripples of

THD (%) 3.07 3.07 1.23

*DTC-SVM Approaches of an Induction Motor Dedicated to Position Control Applications*

**PI without a NL compensator PI with a NL Compensator Sliding Mode Controllers**

The total harmonic distorsion criterion of the stator current *ias* is given by

The second comparison criterion translates the torque and the flux ripples

*Zoomed shapes of (a) electromagnetic torque and (b) stator flux. In the case of (subscript "1") DTC-SVM approach using PI controller, (subscript "2") DTC-SVM approach using PI controller with a nonlinear*

*compensator and (subscript "3") DTC-SVM approach using sliding mode controllers.*

**Figure 16** presents the evolution of the torque *Tem* and the flux ∣Φ*s*∣ from 3 s to 4 s. Computations of flux ripple criteria are given by **Table 5**, and computations of torque ripple criteria are given by **Table 6**. These tables confirm that the PID controllers without a nonlinear compensator and PID controllers with a nonlinear compensator give same results. However, SM controllers give less ripples of the flux

6.

**Table 4** which shows that SM controllers give the lowest criterion.

around their steady state values <sup>∣</sup>Φ*s*<sup>∣</sup> <sup>¼</sup> 1 and *Tem*,*mean* <sup>¼</sup> *Kl* sin *<sup>π</sup>*

the stator current.

**Table 4.**

and the torque.

**Figure 16.**

**59**

• Ratio of torque and flux ripples.

*Total harmonic distortion of the stator current ias.*

*DOI: http://dx.doi.org/10.5772/intechopen.94436*

#### **Figure 14.**

*Induction motor position regulation, considering +100% variations on the stator resistance, considering (1) DTC-SVM approach using PI controller, (2) DTC-SVM approach using PI controller with a nonlinear compensator and (3) DTC-SVM approach using sliding mode controllers. Legend: (a) evolution of the electromagnetic torque, (b) the stator flux and (c) the stator current of phase a.*

#### **Figure 15.**

*Spectrum of the current ias. (a) Normalized spectrum, (b) higher harmonics of the spectrum current (c) one period of the current ias. (subscript "1") DTC-SVM approach using PI controller, (subscript "2") DTC-SVM approach using PI controller with a nonlinear compensator and (subscript "3") DTC-SVM approach using sliding mode controllers.*


**Table 4.**

**Figure 14.**

**Figure 15.**

**58**

*sliding mode controllers.*

*Induction motor position regulation, considering +100% variations on the stator resistance, considering (1) DTC-SVM approach using PI controller, (2) DTC-SVM approach using PI controller with a nonlinear compensator and (3) DTC-SVM approach using sliding mode controllers. Legend: (a) evolution of the*

*Spectrum of the current ias. (a) Normalized spectrum, (b) higher harmonics of the spectrum current (c) one period of the current ias. (subscript "1") DTC-SVM approach using PI controller, (subscript "2") DTC-SVM approach using PI controller with a nonlinear compensator and (subscript "3") DTC-SVM approach using*

*electromagnetic torque, (b) the stator flux and (c) the stator current of phase a.*

*Direct Torque Control Strategies of Electrical Machines*

*Total harmonic distortion of the stator current ias.*

In this context, the frequency spectrum of the stator current *ias* has been analyzed by the observation of amplitudes of all its harmonics frequencies. **Figure 15** shows the evolution of one period of *ias* between 3 s and 4 s, its spectrum (only 20 harmonics has been presented). It is obvious that SM controllers give less ripples of the stator current.

The total harmonic distorsion criterion of the stator current *ias* is given by **Table 4** which shows that SM controllers give the lowest criterion.

• Ratio of torque and flux ripples.

The second comparison criterion translates the torque and the flux ripples around their steady state values <sup>∣</sup>Φ*s*<sup>∣</sup> <sup>¼</sup> 1 and *Tem*,*mean* <sup>¼</sup> *Kl* sin *<sup>π</sup>* 6.

**Figure 16** presents the evolution of the torque *Tem* and the flux ∣Φ*s*∣ from 3 s to 4 s. Computations of flux ripple criteria are given by **Table 5**, and computations of torque ripple criteria are given by **Table 6**. These tables confirm that the PID controllers without a nonlinear compensator and PID controllers with a nonlinear compensator give same results. However, SM controllers give less ripples of the flux and the torque.

#### **Figure 16.**

*Zoomed shapes of (a) electromagnetic torque and (b) stator flux. In the case of (subscript "1") DTC-SVM approach using PI controller, (subscript "2") DTC-SVM approach using PI controller with a nonlinear compensator and (subscript "3") DTC-SVM approach using sliding mode controllers.*


**Table 5.**

*Flux ripple criteria.*


**Table 6.**

*Torque ripple criteria.*
