**3. The proposed direct torque control scheme**

The Figure 3 shows the proposed DTC-SVM scheme, this scheme only needs sense the DC link and the two phases of the stator currents of the three-phase induction motor. In the DTC-SVM scheme the electromagnetic torque error (*Eτ*) and the stator flux error (*Eψs*) are the inputs and the stator voltage components are the outputs of the Takagi-Sugeno fuzzy controller, these outputs are represented in the stator flux reference frame. Details about this controller will be presented in the next section.

## **3.1. Stator voltage calculation**

4 Will-be-set-by-IN-TECH

*β*

*ψβ<sup>s</sup>*

*uqs* <sup>=</sup> <sup>2</sup>

For every sampled period *Ts*, the equation (8) is approximated by

*u*∗

voltage components in the stator-flux-oriented coordinates [6].

<sup>3</sup>*<sup>P</sup> Rs*

Stator-flux-oriented

*q*

**Figure 1.** Stator-flux-oriented reference frame.

error is used the PI controller, it is:

used to control the electromagnetic torque, it is:

*u*∗

component, it is:

values *u*∗

*ds*, *u*<sup>∗</sup>

reference frame

*ψαs*

*Tem ψs*

From equation (8), the stator flux is controlled with the stator voltage direct component *uds*.

When the three-phase IM operates at high speeds the term *Rsids* can be neglected and the stator voltage can to become proportional to the stator flux change with a switching frequency 1/*Ts*. However, at low speeds the term *Rsids* is not negligible and with the aim to correct this

*ds* = (*KP<sup>ψ</sup>* + *KIψ*/*s*)(*ψ*<sup>∗</sup>

*qs* = (*KPTem* + *KITem* /*s*)(*T*<sup>∗</sup>

From the equation (12), the electromagnetic torque can be controlled with the stator voltage quadrature component if the term *ωsψs* is decoupled. A simple form to decoupled it is adding the term *ωsψs* to the output of the controller as is shown in Fig. 2. Then, the PI controller is

Finally, the outputs of the PI flux and PI torque controllers can be interpreted as the stator

Where *ωs* is the angular speed of the stator flux vector. The equations (12) and (13) show that the component *uds* has influence only on the change of stator flux magnitude, and the component *uqs*, if the term *ωsψs* is decoupled, can be used for torque adjustment. Therefore, after coordinate transformation *dq*/*αβ* into the stationary reference frame, the command

*qs* are delivered to SVM. In [3] this scheme is analyzed in detail.

*em* <sup>−</sup> *<sup>T</sup>*<sup>ˆ</sup>

*ψs* = *ψds* Stationary

*d*

*ωs*

reference frame

*uds* = *Rsids* + Δ*ψs*/*Ts* (13)

*α*

+ *ωsψ<sup>s</sup>* (12)

*<sup>s</sup>* <sup>−</sup> *<sup>ψ</sup>*ˆ*s*) (14)

*em*) + *ωsψ<sup>s</sup>* (15)

The stator voltage calculation use the DC link voltage (**Udc**) and the inverter switch state (*SWa*, *SWb*, *SWc*) of the three-phase two level inverter. The stator voltage vector *us* is determined as in [5]:

$$\vec{u}\_s = \frac{2}{3} \left[ (\mathbf{S}\_{Wa} - \frac{\mathbf{S}\_{Wb} + \mathbf{S}\_{Wc}}{2}) + j\frac{\sqrt{3}}{2} (\mathbf{S}\_{Wb} - \mathbf{S}\_{Wc}) \right] \mathcal{U}\_{dc} \tag{16}$$

### **3.2. Space vector modulation technique**

In this work is used the space vector modulation (SVM) technique with the aim to reduce the torque ripple and total harmonic distortion of the current, is therefore necessary to understand the operation and fundamentals that governing their behavior. This concept was discussed in publications such as [24], [12] and [27]. For our purpose the basic ideas are summarized. In Fig. 4 and Fig. 5 are shown the three-phase two level inverter diagram, where the state of the

**Figure 3.** Takagi-Sugeno fuzzy controller in the direct torque control with space vector modulation scheme.

switches follow the following logic.

$$\mathcal{S}\_{Wi} = \begin{cases} 1, \text{the switch } \mathcal{S}\_{Wi} \text{ is ON and the switch } \bar{\mathcal{S}}\_{Wi} \text{ is OFF} \\ 0, \text{ the switch } \mathcal{S}\_{Wi} \text{ is OFF and the switch } \bar{\mathcal{S}}\_{Wi} \text{ is ON} \end{cases} \tag{17}$$

1 *Z Z Z*

1 *Z Z Z*

In Fig. 6 the hexagon is divided in six sectors, and any reference voltage vector is represented as combination of adjacent active and zero voltage vectors, e.g. the voltage vector *U* <sup>∗</sup> is localized in sector I between active vectors *U* <sup>1</sup> and *U* 2, as is shown in Fig. 8, and considering

*<sup>U</sup>* <sup>∗</sup> · *Tz* <sup>=</sup> *<sup>U</sup>* <sup>1</sup> · *<sup>T</sup>*<sup>1</sup> <sup>+</sup> *<sup>U</sup>* <sup>2</sup> · *<sup>T</sup>*<sup>2</sup>

*T*1 *Tz* + *U* <sup>2</sup> *T*2 *Tz*

(19)

347

*U* <sup>∗</sup> = *U* <sup>1</sup>

*SWa SWb SWc*

*Ia Ib Ic*

*n*

*a b c*

*Uan Ubn Ucn*

*SWb*

*S*¯ *Wb*

*P*

*Udc*

*N*

**Figure 4.** Three-phase two level inverter with load

*P*

*Udc*

*N*

**Figure 5.** Simplified three-phase two level inverter with load

a enough short switching period, it is:

*UaN*

*SWa*

*S*¯ *Wa*

*UnN*

*UaN*

*UnN*

*Ia Ib Ic*

*n*

*a b c*

*S*¯ *Wc*

*SWc*

The Takagi-Sugeno Fuzzy Controller Based Direct Torque Control with Space Vector Modulation for Three-Phase Induction Motor

*Uan Ubn Ucn*

Where i=a,b,c and considering that the switch *S*¯ *Wi* is the complement of *SWi* is possible to resume all the combinations only considering the top switches as is shown in Table 1.


**Table 1.** Switching vectors

Where *S*0, *S*1, *S*2, *S*3, *S*4, *S*5, *S*<sup>6</sup> and *S*<sup>7</sup> are switching vectors. These switching vectors generate six active voltage vectors (*U* 1, *U* 2, *U* 3, *U* 4, *U* <sup>5</sup> and *U* 6) and two zero voltage vectors (*U* <sup>0</sup> and *U* 7) as are shown in the Figures 6 and 7. The generalized expression to calculate the active and zero voltage vectors is:

$$\mathbf{J}\_n = \begin{cases} \frac{2}{3} \sqrt{3} \mathbf{J}\_{d\mathbf{c}} \cdot e^{j(2\mathbf{n}-1)\frac{\pi}{6}}, n = 1, \ldots, 6\\ 0 & , n = 0, 7 \end{cases} \tag{18}$$

Where *Udc* is the DC link voltage.

**Figure 4.** Three-phase two level inverter with load

6 Will-be-set-by-IN-TECH

*us*

**Figure 3.** Takagi-Sugeno fuzzy controller in the direct torque control with space vector modulation

1, the switch *SWi* is ON and the switch *S*¯

resume all the combinations only considering the top switches as is shown in Table 1.

*S*<sup>6</sup> and

*U <sup>n</sup>* =

 2 3

0, the switch *SWi* is OFF and the switch *S*¯

Vector *SWa SWb SWc*

*S*<sup>0</sup> 000

*S*<sup>1</sup> 100

*S*<sup>2</sup> 110

*S*<sup>3</sup> 010

*S*<sup>4</sup> 011

*S*<sup>5</sup> 001

*S*<sup>6</sup> 101

*S*<sup>7</sup> 111

six active voltage vectors (*U* 1, *U* 2, *U* 3, *U* 4, *U* <sup>5</sup> and *U* 6) and two zero voltage vectors (*U* <sup>0</sup> and *U* 7) as are shown in the Figures 6 and 7. The generalized expression to calculate the active and

<sup>√</sup>3*Udc* · *<sup>e</sup>j*(2*n*−1) *<sup>π</sup>*

*is*

SVM

Voltage Flux angle calculation *<sup>α</sup>* <sup>−</sup> *<sup>β</sup>*

*ABC*

*Wi* is OFF

*Wi* is the complement of *SWi* is possible to

*S*<sup>7</sup> are switching vectors. These switching vectors generate

0 , *<sup>n</sup>* <sup>=</sup> 0, 7 (18)

<sup>6</sup> , *n* = 1, ..., 6

*Wi* is ON (17)

PWM

*Udc*

*ia*

*ib*

*ψ*∗ *s*

scheme.

+

*T*∗ *em*

−

+

−

*Eτ*

switches follow the following logic.

**Table 1.** Switching vectors

zero voltage vectors is:

Where *Udc* is the DC link voltage.

*S*0, *S*1, *S*2, *S*3, *S*4, *S*5,

Where

*SWi* =

Where i=a,b,c and considering that the switch *S*¯

*ψ*ˆ*s*

*T*ˆ*em*

TS Fuzzy Controller

*u*∗ *qs* *d* − *q*

(*θψs*)

*α* − *β*

signals *<sup>E</sup>ψ<sup>s</sup>*

*ψds ψqs*

Flux and torque estimator

*u*∗ *ds*

Im

*Udc*

**Figure 5.** Simplified three-phase two level inverter with load

In Fig. 6 the hexagon is divided in six sectors, and any reference voltage vector is represented as combination of adjacent active and zero voltage vectors, e.g. the voltage vector *U* <sup>∗</sup> is localized in sector I between active vectors *U* <sup>1</sup> and *U* 2, as is shown in Fig. 8, and considering a enough short switching period, it is:

$$\begin{aligned} \vec{U}^\* \cdot T\_z &= \vec{U}\_1 \cdot T\_1 + \vec{U}\_2 \cdot T\_2\\ \vec{U}^\* &= \vec{U}\_1 \frac{T\_1}{T\_z} + \vec{U}\_2 \frac{T\_2}{T\_z} \end{aligned} \tag{19}$$

**Figure 6.** Switching and voltage vectors

**Figure 7.** Linear region to work without overmodulation

The times *T*<sup>1</sup> and *T*<sup>2</sup> are calculated using trigonometric projections as is shown in Fig. 8, it is:

$$T\_1 = \frac{|\vec{U}^\*|}{|\vec{U}\_1|} \cdot T\_2 \frac{\sin(\frac{\pi}{3} - \phi)}{\sin(\frac{2\pi}{3})} \tag{20}$$

*U* 1 *T*1 *Tz*

*U* 1

*U* 1

*S*1(100)

*α*

*S*1(100)

*α*

*φ*

*U* 7

**Figure 8.** Voltage vector *U* <sup>∗</sup> and its components in sector *I*

shown in Fig. 10.

 *S*0(000) *S*7(111)

 *S*0(000) *S*7(111)

*S*3(010)

*S*3(010)

*Ubc*

*U* <sup>2</sup> *U* <sup>3</sup>

*β*

II

*Ubc*

*U* <sup>2</sup> *U* <sup>3</sup>

*β*

II

*S*2(110)

*<sup>S</sup>*2(110)

*U* 0

**Figure 9.** Switching sequence for odd sector.

*U* 0

**Figure 10.** Switching sequence for even sector.

*<sup>U</sup>* <sup>7</sup>

*<sup>U</sup>* <sup>7</sup>

I

I

*U* 2 *T*2 *Tz*

Where *T*<sup>0</sup> and *T*<sup>7</sup> are the times of applications of zero vectors in a switching period. Once calculated the times of applications of each adjacent voltage vectors the next step is to follow a specific switching sequence for the symmetrical space vector modulation technique, this one depends if the reference vector is localized in an even or odd sector, e.g. in Fig. 9 is

*S*7), however for even sector the switching sequence is contrary to the case for odd sector as is

*T*0 4

*T*0 4

*T*1 2

*T*2 2

*T*1 2

*T*2 2

*π* <sup>3</sup> − *φ*

*U* 2

*U* 1

The Takagi-Sugeno Fuzzy Controller Based Direct Torque Control with Space Vector Modulation for Three-Phase Induction Motor

0 0 0 1 1 0 0 0

*Tz*

0 0 1 1 1 1 0 0

0 0 0 1 1 0 0 0

*Tz*

*T*7 4 *T*7 4

0 1 1 11 1 1 0

*T*7 4 *T*7 4

*T*2 2

*T*2 2

0 0 1 1 1 1 0 0

0 1 1 11 1 1 0

*S*0, *S*1, *S*<sup>2</sup> and 349

*T*0 4

*T*0 4

*T*1 2

*T*1 2

*U* <sup>∗</sup>

2*π* 3

observed the optimum switching sequence and the pulse pattern for odd sector (

$$T\_2 = \frac{|\vec{U}^\*|}{|\vec{U}\_2|} \cdot T\_z \frac{\sin(\phi)}{\sin(\frac{2\pi}{3})}\tag{21}$$

Where *T*<sup>1</sup> and *T*<sup>2</sup> are the times of application of the active vectors in a switching period, *TZ* is the switching period and *φ* is the angle between the reference voltage vector and the adjacent active vector (*U* 1). If the sum of times *T*<sup>1</sup> and *T*<sup>2</sup> is minor of the switching period, the rest of the time is apply the zero vectors, it is:

$$T\_0 = T\_7 = T\_Z - T\_1 - T\_2 \tag{22}$$

348 Induction Motors – Modelling and Control The Takagi-Sugeno Fuzzy Controller Based Direct Torque Control with Space Vector Modulation for Three-Phase Induction Motor <sup>9</sup> 349 The Takagi-Sugeno Fuzzy Controller Based Direct Torque Control with Space Vector Modulation for Three-Phase Induction Motor

**Figure 8.** Voltage vector *U* <sup>∗</sup> and its components in sector *I*

8 Will-be-set-by-IN-TECH

*Ubc*

*U* <sup>2</sup> *U* <sup>3</sup>

*β*

II

*S*2(110)

*S*3(010)

 *S*0(000) *S*7(111)

IV

III

*U* 0

*<sup>U</sup>* <sup>7</sup>

I

VI

*S*6(101)

*U* <sup>∗</sup> <sup>|</sup>*<sup>U</sup>* ∗| <sup>=</sup> *Udc* <sup>|</sup>*<sup>U</sup>* <sup>1</sup><sup>|</sup> <sup>=</sup> <sup>√</sup><sup>2</sup> 3 *Udc*

*U* <sup>5</sup> *U* <sup>6</sup>

V

*S*5(001)

*Ubc*

*U* <sup>2</sup> *U* <sup>3</sup>

*β*

*U* 0 *U* 7

*<sup>T</sup>*<sup>1</sup> <sup>=</sup> <sup>|</sup>*<sup>U</sup>* ∗| <sup>|</sup>*<sup>U</sup>* <sup>1</sup><sup>|</sup>

*<sup>T</sup>*<sup>2</sup> <sup>=</sup> <sup>|</sup>*<sup>U</sup>* ∗| <sup>|</sup>*<sup>U</sup>* <sup>2</sup><sup>|</sup>

*U* <sup>5</sup> *U* <sup>6</sup>

The times *T*<sup>1</sup> and *T*<sup>2</sup> are calculated using trigonometric projections as is shown in Fig. 8, it is:

· *Tz*

· *Tz*

Where *T*<sup>1</sup> and *T*<sup>2</sup> are the times of application of the active vectors in a switching period, *TZ* is the switching period and *φ* is the angle between the reference voltage vector and the adjacent active vector (*U* 1). If the sum of times *T*<sup>1</sup> and *T*<sup>2</sup> is minor of the switching period, the rest of

sin( *<sup>π</sup>*

sin(*φ*) sin( <sup>2</sup>*<sup>π</sup>*

<sup>3</sup> − *φ*) sin( <sup>2</sup>*<sup>π</sup>*

*Uca*

*Uca*

**Figure 7.** Linear region to work without overmodulation

the time is apply the zero vectors, it is:

*U* 4

*LinearRegion*

*U* 4

 *S*4(011)

**Figure 6.** Switching and voltage vectors

*Uab*

*Uab*

*T*<sup>0</sup> = *T*<sup>7</sup> = *TZ* − *T*<sup>1</sup> − *T*<sup>2</sup> (22)

*U* 1

*α*

<sup>3</sup> ) (20)

<sup>3</sup> ) (21)

*U* 1

*S*1(100)

*α*

Where *T*<sup>0</sup> and *T*<sup>7</sup> are the times of applications of zero vectors in a switching period. Once calculated the times of applications of each adjacent voltage vectors the next step is to follow a specific switching sequence for the symmetrical space vector modulation technique, this one depends if the reference vector is localized in an even or odd sector, e.g. in Fig. 9 is observed the optimum switching sequence and the pulse pattern for odd sector ( *S*0, *S*1, *S*<sup>2</sup> and *S*7), however for even sector the switching sequence is contrary to the case for odd sector as is shown in Fig. 10.

**Figure 10.** Switching sequence for even sector.

 *S*7(111) *<sup>U</sup>* <sup>7</sup>

*S*1(100)

*α*

The details about the implementation of SVM algorithm in MATLAB/Simulink are presented in [3] in page 97.

### **3.3. Electromagnetic torque and stator flux estimation**

The Figure 3 shows that the electromagnetic torque and the stator flux estimation depends of the stator voltage and the stator current space vectors, therefore:

$$
\vec{\psi}\_s = \int (\vec{u}\_s - \mathsf{R}\_s \cdot \vec{i}\_s) dt \tag{23}
$$

*e*−*jθ<sup>r</sup> ejθ<sup>r</sup>*

*isdq*

*is*

(*u*∗

LPF

*θψ <sup>r</sup> θψ <sup>r</sup>*

 *ψs*( *ψr*)

*is*

*<sup>r</sup>*

 *ψr*

*te* <sup>=</sup> <sup>3</sup> 2 *<sup>P</sup> Lm LrLsσ* 

**Figure 11.** Stator and rotor flux estimator, and electromagnetic torque estimator.

replaced in (5) it is estimated the electromagnetic torque.

**4. Design of Takagi-Sugeno fuzzy controller**

interchanged how is shown in the Figure 12.

**Figure 12.** The structure of a fuzzy logic controller.

**4.1. Membership functions**

is know as subjective approach [22].

*Eψ<sup>s</sup> Eτ*

*ψi*

*ψrdq*

 *ψr*( *ψs*)

Where LPF means low pass filter. In the other hand, when the equations (27) and (29) are

The Takagi-Sugeno Fuzzy controller takes as inputs the stator flux error *Eψ<sup>s</sup>* and the electromagnetic torque error *Eτ*, and as outputs the quadrature components of the stator

linear combination of the inputs as a consequent part of the rules, similarly, the second output

*Fuzzi f ication De f uzzi f ication Fuzzy In f erence Engine*

> for *u*∗ *ds* for *u*∗ *qs*

The Membership Functions (MF) for T-S fuzzy controller are shown in Figure 13 and in Figure 14, for the stator flux error and the electromagnetic torque error, respectively. These MF's shape and parameters was found through trial and error method with multiple simulations and with the knowing of the induction motor response for every test. This method

*qs* = −*bEψ<sup>s</sup>* + *aE<sup>τ</sup>*

*ds* = *aEψ<sup>s</sup>* + *bE<sup>τ</sup>*

*qs*) takes the similar linear combination used in the first output but with the coefficients

voltage vector, represented in the stator flux reference frame. The first output (*u*∗

If *Eψ<sup>s</sup>* is FE and *E<sup>τ</sup>* is TE then *u*<sup>∗</sup>

If *Eψ<sup>s</sup>* is FE and *E<sup>τ</sup>* is TE then *u*<sup>∗</sup>

 *ψs*

−

+

*ψi s*

arctan(*ψrq*/*ψrd*)

Eq. 30 *te*

 *ψr*

*ds*) takes a

 *ψs* 351

*<sup>ψ</sup><sup>r</sup>* <sup>×</sup> *<sup>ψ</sup> <sup>s</sup>* (30)

*u*∗ *ds u*∗ *qs*

*U comp*

<sup>−</sup> PI

*Rs*

−

The Takagi-Sugeno Fuzzy Controller Based Direct Torque Control with Space Vector Modulation for Three-Phase Induction Motor

*us*

+

+

The problem in this type of estimation is when in low speeds the back electromotive force (emf) depends strongly of the stator resistance, to resolve this problem is used the current model to improve the flux estimation as in [17]. The rotor flux represented in the rotor flux reference frame is:

$$
\vec{\psi}\_{rdq} = \frac{L\_m}{1 + sT\_r} \vec{i}\_{sdq} - j \frac{(\omega\_{\Psi\_r} - \omega\_r)T\_r}{1 + sT\_r} \vec{\psi}\_{rdq} \tag{24}
$$

Where *Tr* = *Lr*/*Rr* is the rotor time constant. In this reference frame *ψrq* = 0 and substituting this expression in the equation (24), it is:

$$
\psi\_{rd} = \frac{L\_m}{1 + sT\_r} i\_{sd} \tag{25}
$$

In the current model the stator flux is represented as:

$$
\vec{\psi}\_s^i = \frac{L\_m}{L\_r}\vec{\psi}\_r^i + \frac{L\_s L\_r - L\_m^2}{L\_r}\vec{i}\_s\tag{26}
$$

Where *ψi <sup>r</sup>* is the rotor flux estimated in the equation (25). The voltage model is based in the equation (1) and from there the stator flux in the stationary reference frame is:

$$
\vec{\psi}\_s = \frac{1}{s} (\vec{v}\_s - R\_s \vec{i}\_s - \vec{\mathcal{U}}\_{comp}) \tag{27}
$$

With the aim to correct the errors associated with the pure integration and the stator resistance variations with temperature, the voltage model is adapted through the PI controller.

$$
\vec{\mathcal{U}}\_{comp} = (\mathcal{K}\_p + \mathcal{K}\_i \frac{1}{\mathbf{s}}) (\vec{\psi}\_{\mathbf{s}} - \vec{\psi}\_{\mathbf{s}}^i) \tag{28}
$$

The *Kp* and *Ki* coefficients are calculated with the recommendation proposed in [17]. The rotor flux *ψr* in the stationary reference frame is calculated as:

$$
\vec{\psi}\_r = \frac{L\_r}{L\_m}\vec{\psi}\_s - \frac{L\_s L\_r - L\_m^2}{L\_m}\vec{i}\_s\tag{29}
$$

The estimator scheme shown in the Figure 11 works with a good performance in the wide range of speeds.

350 Induction Motors – Modelling and Control The Takagi-Sugeno Fuzzy Controller Based Direct Torque Control with Space Vector Modulation for Three-Phase Induction Motor <sup>11</sup> 351 The Takagi-Sugeno Fuzzy Controller Based Direct Torque Control with Space Vector Modulation for Three-Phase Induction Motor

**Figure 11.** Stator and rotor flux estimator, and electromagnetic torque estimator.

Where LPF means low pass filter. In the other hand, when the equations (27) and (29) are replaced in (5) it is estimated the electromagnetic torque.

$$t\_{\varepsilon} = \frac{3}{2} P \frac{L\_m}{L\_r L\_s \sigma} \vec{\psi}\_r \times \vec{\psi}\_s \tag{30}$$

### **4. Design of Takagi-Sugeno fuzzy controller**

The Takagi-Sugeno Fuzzy controller takes as inputs the stator flux error *Eψ<sup>s</sup>* and the electromagnetic torque error *Eτ*, and as outputs the quadrature components of the stator voltage vector, represented in the stator flux reference frame. The first output (*u*∗ *ds*) takes a linear combination of the inputs as a consequent part of the rules, similarly, the second output (*u*∗ *qs*) takes the similar linear combination used in the first output but with the coefficients interchanged how is shown in the Figure 12.

**Figure 12.** The structure of a fuzzy logic controller.

### **4.1. Membership functions**

10 Will-be-set-by-IN-TECH

The details about the implementation of SVM algorithm in MATLAB/Simulink are presented

The Figure 3 shows that the electromagnetic torque and the stator flux estimation depends of

The problem in this type of estimation is when in low speeds the back electromotive force (emf) depends strongly of the stator resistance, to resolve this problem is used the current model to improve the flux estimation as in [17]. The rotor flux represented in the rotor flux

*isdq* <sup>−</sup> *<sup>j</sup>*

*<sup>ψ</sup>rd* <sup>=</sup> *Lm*

(*vs* − *Rs*

variations with temperature, the voltage model is adapted through the PI controller.

*U comp* = (*Kp* + *Ki*

With the aim to correct the errors associated with the pure integration and the stator resistance

The *Kp* and *Ki* coefficients are calculated with the recommendation proposed in [17]. The rotor

The estimator scheme shown in the Figure 11 works with a good performance in the wide

1 *s*

*<sup>ψ</sup><sup>s</sup>* <sup>−</sup> *LsLr* <sup>−</sup> *<sup>L</sup>*<sup>2</sup>

*Lm*

)(*<sup>ψ</sup> <sup>s</sup>* <sup>−</sup> *<sup>ψ</sup> <sup>i</sup>*

*m*

Where *Tr* = *Lr*/*Rr* is the rotor time constant. In this reference frame *ψrq* = 0 and substituting

1 + *sTr*

*LsLr* <sup>−</sup> *<sup>L</sup>*<sup>2</sup> *m*

*Lr*

*<sup>r</sup>* is the rotor flux estimated in the equation (25). The voltage model is based in the

(*ωψ<sup>r</sup>* − *ωr*)*Tr* 1 + *sTr*

(*us* <sup>−</sup> *Rs* ·*is*)*dt* (23)

*ψrdq* (24)

*isd* (25)

*is* (26)

*<sup>s</sup>*) (28)

*is* (29)

*is* <sup>−</sup> *<sup>U</sup> comp*) (27)

**3.3. Electromagnetic torque and stator flux estimation**

In the current model the stator flux is represented as:

this expression in the equation (24), it is:

*<sup>ψ</sup>rdq* <sup>=</sup> *Lm*

 *ψi <sup>s</sup>* <sup>=</sup> *Lm Lr ψi <sup>r</sup>* +

 *<sup>ψ</sup><sup>s</sup>* <sup>=</sup> <sup>1</sup> *s*

*ψr* in the stationary reference frame is calculated as:

 *<sup>ψ</sup><sup>r</sup>* <sup>=</sup> *Lr Lm* 

equation (1) and from there the stator flux in the stationary reference frame is:

1 + *sTr*

the stator voltage and the stator current space vectors, therefore:

 *ψ<sup>s</sup>* = 

in [3] in page 97.

reference frame is:

Where *ψi*

flux

range of speeds.

The Membership Functions (MF) for T-S fuzzy controller are shown in Figure 13 and in Figure 14, for the stator flux error and the electromagnetic torque error, respectively. These MF's shape and parameters was found through trial and error method with multiple simulations and with the knowing of the induction motor response for every test. This method is know as subjective approach [22].

### 12 Will-be-set-by-IN-TECH 352 Induction Motors – Modelling and Control The Takagi-Sugeno Fuzzy Controller Based Direct Torque Control with Space Vector Modulation for Three-Phase Induction Motor <sup>13</sup>

The universe of discourse for the stator flux error input is defined in the closed interval [-0.5, 0.5]. The extreme MFs have trapezoidal shapes but the middle one takes triangular shape as is shown in Figure 13. However, the universe of discourse for electromagnetic torque error input is defined in the closed interval [-20, 20] but with the objective to see the shape of the MFs only is shown the interval [-5, 5] in Figure 14, the shapes of these MF are similar to the first input. For both inputs the linguistic labels N, Ze and P means Negative, Zero and Positive, respectively.

For instance, when the consequent function of the rule *Ri* is a real number, the consequent function is a zero-order polynomial and we have a zero-order controller. If the consequent

*Ri* : if *X* is *Ai* and *Y* is *Bi* then *z* = *fi*(*X*, *Y*)

operator and the weighted average (**wtaver**) is the defuzzification method used to set the

*ds* = *aFe* + *bTe*

*ds* = *aFe* + *bTe*

*ds* = *aFe* + *bTe*

*qs* = −*bFe* + *aTe*

*qs* = −*bFe* + *aTe*

*qs* = −*bFe* + *aTe*

*u*∗

*u*∗

*u*∗

*u*∗

*u*∗

*u*∗

The simulations were performed using MATLAB simulation package which include Simulink block sets and fuzzy logic toolbox. The switching frequency of PWM inverter was set to be 10*kHz*, the stator reference flux considered was 0.47 Wb and the coefficients considered were *a* = 90 and *b* = 2. In order to investigate the effectiveness of the proposed control system and in order to check the closed-loop stability of the complete system, we performed several tests. We used different dynamic operating conditions such as: step change in the motor load (from 0 to 1.0 pu) at fifty percent of rated speed, no-load sudden change in the speed reference (from 0.5 pu to -0.5 pu), and the application of an arbitrary load torque profile at fifty percent of

> Rated voltage (V) 220/60Hz Rated Power (HP) 3 Rated Torque (Nm) 11.9 Rated Speed (rad/s) 179 *Rs*, *Rr*(Ω) 0.435, 0.816 *Lls*, *Llr* (H) 0.002, 0.002 *Lm* (H) 0.0693 *J*(*Kgm*<sup>2</sup>) 0.089 P (pole pairs) 2

The Figure 15 illustrates the torque response of the DTC-SVM scheme with T-S fuzzy controller when the step change in the motor load is apply. The electromagnetic torque tracked the reference torque and in this test is obtained the following good performance measures: rise time *tr* = 1.1*ms*, settling time *ts* = 2.2*ms* and torque ripple *ripple* = 2.93%. Also is observed

*ds* and *u*<sup>∗</sup> *qs*

*qs* is shown in Table 2. The **product** is the conjunction

The Takagi-Sugeno Fuzzy Controller Based Direct Torque Control with Space Vector Modulation for Three-Phase Induction Motor

*u*∗

*u*∗

*u*∗

*u*∗

*u*∗

*u*∗

*ds* = *aFe* + *bTe*

*ds* = *aFe* + *bTe*

*ds* = *aFe* + *bTe*

*qs* = −*bFe* + *aTe*

353

*qs* = −*bFe* + *aTe*

*qs* = −*bFe* + *aTe*

function is a linear we have first-order controller [21].

*ds* = *aFe* + *bTe*

*ds* = *aFe* + *bTe*

*ds* = *aFe* + *bTe*

*qs* = −*bFe* + *aTe*

*qs* = −*bFe* + *aTe*

*qs* = −*bFe* + *aTe*

rated speed. The motor parameters are given in Table 3.

*ds* and *u*<sup>∗</sup>

**E<sup>s</sup>** / **Eø N ZE P**

The rule base to calculate *u*∗

controller in the MATLAB fuzzy editor.

*u*∗

*u*∗

*u*∗

**Table 2.** Fuzzy rules for computation of *u*∗

**Table 3.** Induction Motor Parameters [16]

that the behavior of the stator current is sinusoidal.

**N** *u*∗

**ZE** *u*∗

**P** *u*∗

**5. Simulation results**

**Figure 13.** Membership function for stator flux error input (*Eψs*)

**Figure 14.** Membership function for electromagnetic torque error input (*Eτ*)

## **4.2. The fuzzy rule base**

The direct component of the stator voltage *u*∗ *ds* is determined by the rules of the form:

$$R\_{\mathcal{X}} \text{ : if } E\_{\psi\_s} \text{ is } FE \text{ and } E\_{\mathbb{T}} \text{ is } TE \text{ then } \mu\_{ds}^\* = aE\_{\psi\_s} + bE\_{\mathbb{T}}.$$

However, the quadrature component of the stator voltage *u*∗ *qs* is determined by the rules of the form:

$$R\_{\tilde{y}} \; : \text{ if } E\_{\psi\_s} \text{ is } FE \text{ and } E\_{\mathbb{T}} \text{ is } TE \text{ then } \mu\_{qs}^\* = -bE\_{\psi\_s} + aE\_{\mathbb{T}}.$$

Where *FE* = *TE* = {*N*, *ZE*, *P*} are the fuzzy sets of the inputs and, **a** and **b** are coefficients of the first-order polynomial function typically present in the consequent part of the firs-order Takagi-Sugeno fuzzy controllers.

For instance, when the consequent function of the rule *Ri* is a real number, the consequent function is a zero-order polynomial and we have a zero-order controller. If the consequent function is a linear we have first-order controller [21].

$$\mathcal{R}\_{\dot{l}} \text{ : if } X \text{ is } A\_{\dot{l}} \text{ and } Y \text{ is } B\_{\dot{l}} \text{ then } z = f\_{\dot{l}}(X, Y)$$

The rule base to calculate *u*∗ *ds* and *u*<sup>∗</sup> *qs* is shown in Table 2. The **product** is the conjunction operator and the weighted average (**wtaver**) is the defuzzification method used to set the controller in the MATLAB fuzzy editor.


**Table 2.** Fuzzy rules for computation of *u*∗ *ds* and *u*<sup>∗</sup> *qs*
