**Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors**

José Luis Azcue, Alfeu J. Sguarezi Filho and Ernesto Ruppert

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48388

## **1. Introduction**

558 Fuzzy Controllers – Recent Advances in Theory and Applications

Transactions on Automatic Control 36 6, pp. 683–691.

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[5] M.S.Huq, Analysis and control of hybrid orthosis in therapeutic treadmill locomotion

[6] A. Arifin, T. Watanabe, and N. Hashimiya, (2002). A Test of fuzzy controller for cyclecycle control of FES-induced hemiplegics gait: Computer simulatilm in single-joint control: Proc. 36th Control of Japanese Soc.& Med. & Bid. Eng., Tohoku Chapter, p.30. [7] H. M. Franken, P. H. Veltink, R.Tijsmans , H. Nijmeijer, , and H. B. K. Boom, (1993). Identification of passive knee joint and shank dynamics in paraplegics using quadriceps

stimulation, IEEE Transactions on Rehabalitation Engineering, vol. 1,pp. 154-164. [8] M.S. Hatwell, B.J. Oderkerk, C.A. Sacher, and G.F. Inbar,(1991). The development of a model reference adaptive controller to control the knee joint of paraplegics. IEEE

[9] P. H. Veltink. Control of FES-induced cyclical movements of the lower leg. Med Biol

[10] Y.L. Chen, S.C. Chen, W.L. Chen, C.C. Hsiao, T.S. Kuo and J.S. Lai, Neural network and fuzzy control in FES-assisted locomotion for the hemiplegic, J Med Eng Technol 28

[11] A. Yardimci, A Survey on Use of Soft Computing Methods in Medicine, Artificial Neural Networks – ICANN 2007,Springer Berlin / Heidelberg, pp. 69-79, 2007. [12] T.C. Chin, and X. M. Qi, Genetic Algorithms for learning the rule base of fuzzy logic

[13] B.S. K. K. Ibrahim, M.O. Tokhi M.S. Huq, and S.C. Gharooni, An Approach for Dynamic Characterisation of Passive Viscoelasticity and Estimation of Anthropometric Inertia

[14] M. Ferrarin, and A. Pedotti. The relationship between electrical stimulus and joint torque: a dynamic model. IEEE Transactions on Rehabilitation Engineering, vol. 8 (3),

[15] A.K. Rzepniewski, Cycle-to-cycle control of multiple input-multiple output manufacturing processes, Ph.D. Thesis in Mechanical Engineering, Massachusetts

[16] Fahey, T.D., Harvey, M., Schroeder, R. and Ferguson, F. (1985) Influence of sex differences and knee joint position on electrical stimulation-modulated strength

[17] Bobet, J. and Stein, R.B. (1998). A simple model of force generation by skeletal muscle during dynamic isometric contractions. IEEE Trans Biomed Eng, 45(8):1010-1016. [18] Yang, J., F. Fung, J., Edamura, M. et al., (1991). H-reflex modulation during walking in spastic paretic subjects. Canadian Journal of Neurological Sciences 18(4):443–452. [19] Lance JW. Symposium synopsis. In: Feldmann RG, Young RR, Koella WP, eds.Spasticity: disordered motor control. Chicago: Year Book Medical

[20] Kralj, A. and Bajd, T, Functional Electrical Stimulation, Standing and Walking After

for paraplegia, PhD Thesis. The University of Sheffield, Sheffield, UK, 2009.

The electric drives are very common in industrial applications because they provide high dynamic performance. Nowadays exist a wide variety of schemes to control the speed, the electromagnetic torque and stator flux of three-phase induction motors. However, control remains a challenging problem for industrial applications of high dynamic performance, because the induction motors exhibit significant nonlinearities. Moreover, many of the parameters vary with the operating conditions. Although the Field Oriented Control (FOC) [16] schemes are attractive, but suffer from a major disadvantage, because they are sensitive to motor parameter variations such as the rotor time constant, and an incorrect flux estimation at low speeds. Another popular scheme for electric drives is the direct torque control (DTC) scheme [15][8], and an another DTC scheme based on space vector modulation (SVM) technique that reduces the torque ripples. This scheme does not need current regulators because its control variables are the electromagnetic torque and the stator flux. In this chapter we use the DTC-SVM scheme to analyze the performance of our proposed fuzzy controllers.

In the last decade, there was an increasing interest in combining artificial intelligent control tools with conventional control techniques. The principal motivations for such a hybrid implementation were that fuzzy logic issues such as uncertainty (or unknown variations in plant parameters and structure) can be dealt with more effectively. Hence improving the robustness of the control system. Conventional controls are very stable and allow various design objectives such as steady state and transient characteristics of a closed loop system. Several [5][6] works contributed to the design of such hybrid control schemes.

However, fuzzy controllers, unlike conventional PI controllers do not necessarily require the accurate mathematic model of the process to be controlled; instead, it uses the experience and knowledge about the controlled process to construct the fuzzy rules base. The fuzzy logic

©2012 Azcue et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

controllers are a good alternative for motor control systems since they are well known for treating with uncertainties and imprecision's. For example, in [1] the PI and fuzzy logic controllers are used to control the load angle, which simplifies the induction motor drive system. In [7], the fuzzy controllers are used to dynamically obtain the reference voltage vector in terms of torque error, stator flux error and stator flux angle. In this case, both torque and stator flux ripples are remarkably reduced. In [10], the fuzzy PI speed controller has a better response for a wide range of motor speed and in [3] a fuzzy self-tuning controller is implemented in order to substitute the unique PI controller, present in the DTC-SVM scheme. In this case, performance measures such as settling time, rise time and ITAE index are lower than the DTC-SVM scheme with PI controller.

The fuzzy inference system can be used to modulate the stator voltage vector applied to the induction motor [18]. In this case, unlike the cases mentioned above, the quantity of available vectors are arbitrarily increased, allowing better performance of the control scheme and lower levels of ripple than the classic DTC. However, it requires the stator current as an additional input, increasing the number of input variables. In this chapter we design and analyze in details three kinds of fuzzy controllers: the PI fuzzy controller (PI-F), the PI-type fuzzy controller (PIF) and the self-tuning PI-type fuzzy controller (STPIF). All of these fuzzy controllers are applied to a direct torque control scheme with space vector modulation technique for three-phase induction motor. In this DTC-SVM scheme, the fuzzy controllers generate corrective control actions based on the real torque trend only while minimizing the torque error.

#### **2. Background**

#### **2.1. The three-phase induction motor dynamical equations**

By the definitions of the fluxes, currents and voltages space vectors, the dynamical equations of the three-phase induction motor in stationary reference frame can be put into the following mathematical form [17]:

$$d\vec{u}\_s = R\_s \vec{i}\_s + \frac{d\vec{\psi}\_s}{dt} \tag{1}$$

Where *γ* is the load angle between stator and rotor flux space vector, *P* is a number of pole

Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors 561

The three-phase induction motor model was implemented in MATLAB/Simulink as is shown

In the direct torque control if the sample time is short enough, such that the stator voltage space vector is imposed to the motor keeping the stator flux constant at the reference value. The rotor flux will become constant because it changes slower than the stator flux. The electromagnetic torque (6) can be quickly changed by changing the angle *γ* in the desired direction. This angle *γ* can be easily changed when choosing the appropriate stator voltage

For simplicity, let us assume that the stator phase ohmic drop could be neglected in (1).

vector at a speed which is proportional to the magnitude of the stator voltage space vector. By selecting step-by-step the appropriate stator voltage vector, it is possible to change the stator

**3. Direct torque control scheme with space vector modulation technique** In Fig. 1, we show the block diagram for the DTC-SVM scheme [14] with a Fuzzy Controller, the fuzzy controller will be substitute for the three kind of proposed Fuzzy Controller one for time. The DTC-SVM scheme is an alternative to the classical DTC schemes [15], [8] and [9]. In this one, the load angle *γ*∗ is not prefixed but it is determinate by the Fuzzy Controller. Equation (6) shows that the angle *γ*∗ determines the electromagnetic torque which is necessary to supply the load. The three proposed Fuzzy Controllers determine the load angle using the torque error *e* and the torque error change Δ*e*. Details about these controllers will be presented in the next section. Figure 1 shows the general block diagram of the DTC-SVM scheme, the inverter, the control signals for three-phase two-level inverter is generated by the DTC-SVM

In this chapter we consider the magnitude of stator flux reference as a constant. Therefore, we can use the relation presented in equation (8) to calculate the stator flux reference vector.

> *<sup>ψ</sup>r*) + *<sup>j</sup>*<sup>|</sup> *ψ*∗

Δ

*ψs*/*dt* = *us*. During a short time Δ*t*, when the voltage space vector is applied

*ψ<sup>s</sup>* ≈ *us* · Δ*t* (7)

*ψ*∗

*ψr* in order to estimate the next angle of the stator flux reference.

*<sup>s</sup>* <sup>|</sup> sin(*γ*<sup>∗</sup> <sup>+</sup> <sup>∠</sup>

*qs*. Therefore, the output of the fuzzy controller *γ*<sup>∗</sup> is

*<sup>s</sup>* can be decomposed in two

*ψr*) (8)

*ψs* in the direction of the stator voltage space

*<sup>m</sup>*/(*LsLr*) is the dispersion factor.

**2.2. The principle of direct torque control**

Thus the stator flux space vector moves by Δ

flux in the required direction.

**3.1. Flux reference calculation**

perpendicular components *ψ*∗

added to rotor flux angle ∠

In stationary reference frame, the stator flux reference

 *ψ*∗ *<sup>s</sup>* <sup>=</sup> <sup>|</sup> *ψ*∗

*ds* and *ψ*<sup>∗</sup>

*<sup>s</sup>* <sup>|</sup> cos(*γ*<sup>∗</sup> <sup>+</sup> <sup>∠</sup>

pairs and *<sup>σ</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>L</sup>*<sup>2</sup>

in [2].

space vector.

Therefore *d*

it has:

scheme.

$$0 = R\_r \vec{i}\_r + \frac{d\vec{\psi}\_r}{dt} - j\omega\_r \vec{\psi}\_r \tag{2}$$

$$
\vec{\psi}\_s = L\_s \vec{i}\_s + L\_m \vec{i}\_r \tag{3}
$$

$$
\vec{\psi}\_r = L\_r \vec{i}\_r + L\_m \vec{i}\_s \tag{4}
$$

Where *us* is the stator voltage space vector, *is* and *ir* are the stator and rotor current space vectors, respectively, *ψ<sup>s</sup>* and *ψr* are the stator and rotor flux space vectors, *ωr* is the rotor angular speed, *Rs* and *Rr* are the stator and rotor resistances, *Ls*, *Lr* and *Lm* are the stator, rotor and mutual inductance respectively.

The electromagnetic torque *te* is expressed in terms of the cross-vectorial product of the stator and the rotor flux space vectors.

$$t\_{\varepsilon} = \frac{3}{2} P \frac{L\_m}{L\_r L\_s \sigma} \vec{\psi}\_r \times \vec{\psi}\_s \tag{5}$$

$$t\_{\varepsilon} = \frac{3}{2} P \frac{L\_m}{L\_r L\_s \sigma} \left| \vec{\psi}\_r \right| \left| \vec{\psi}\_s \right| \sin(\gamma) \tag{6}$$

Where *γ* is the load angle between stator and rotor flux space vector, *P* is a number of pole pairs and *<sup>σ</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>L</sup>*<sup>2</sup> *<sup>m</sup>*/(*LsLr*) is the dispersion factor.

The three-phase induction motor model was implemented in MATLAB/Simulink as is shown in [2].

## **2.2. The principle of direct torque control**

2 Will-be-set-by-IN-TECH

controllers are a good alternative for motor control systems since they are well known for treating with uncertainties and imprecision's. For example, in [1] the PI and fuzzy logic controllers are used to control the load angle, which simplifies the induction motor drive system. In [7], the fuzzy controllers are used to dynamically obtain the reference voltage vector in terms of torque error, stator flux error and stator flux angle. In this case, both torque and stator flux ripples are remarkably reduced. In [10], the fuzzy PI speed controller has a better response for a wide range of motor speed and in [3] a fuzzy self-tuning controller is implemented in order to substitute the unique PI controller, present in the DTC-SVM scheme. In this case, performance measures such as settling time, rise time and ITAE index are lower

The fuzzy inference system can be used to modulate the stator voltage vector applied to the induction motor [18]. In this case, unlike the cases mentioned above, the quantity of available vectors are arbitrarily increased, allowing better performance of the control scheme and lower levels of ripple than the classic DTC. However, it requires the stator current as an additional input, increasing the number of input variables. In this chapter we design and analyze in details three kinds of fuzzy controllers: the PI fuzzy controller (PI-F), the PI-type fuzzy controller (PIF) and the self-tuning PI-type fuzzy controller (STPIF). All of these fuzzy controllers are applied to a direct torque control scheme with space vector modulation technique for three-phase induction motor. In this DTC-SVM scheme, the fuzzy controllers generate corrective control actions based on the real torque trend only while minimizing the

By the definitions of the fluxes, currents and voltages space vectors, the dynamical equations of the three-phase induction motor in stationary reference frame can be put into the following

*dt* (1)

*is* + *Lmir* (3)

*ir* + *Lmis* (4)

*<sup>ψ</sup><sup>r</sup>* <sup>×</sup> *<sup>ψ</sup> <sup>s</sup>* (5)

sin(*γ*) (6)

*ψr* are the stator and rotor flux space vectors, *ωr* is the rotor

*ψr* (2)

*is* + *d ψs*

*ir* + *d ψr dt* <sup>−</sup> *<sup>j</sup>ω<sup>r</sup>*

Where *us* is the stator voltage space vector, *is* and *ir* are the stator and rotor current space

angular speed, *Rs* and *Rr* are the stator and rotor resistances, *Ls*, *Lr* and *Lm* are the stator,

The electromagnetic torque *te* is expressed in terms of the cross-vectorial product of the stator

than the DTC-SVM scheme with PI controller.

**2.1. The three-phase induction motor dynamical equations**

*us* = *Rs*

0 = *Rr*

 *ψ<sup>s</sup>* = *Ls*

 *ψ<sup>r</sup>* = *Lr*

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

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

*ψ<sup>s</sup>* and

rotor and mutual inductance respectively.

and the rotor flux space vectors.

torque error.

**2. Background**

mathematical form [17]:

vectors, respectively,

In the direct torque control if the sample time is short enough, such that the stator voltage space vector is imposed to the motor keeping the stator flux constant at the reference value. The rotor flux will become constant because it changes slower than the stator flux. The electromagnetic torque (6) can be quickly changed by changing the angle *γ* in the desired direction. This angle *γ* can be easily changed when choosing the appropriate stator voltage space vector.

For simplicity, let us assume that the stator phase ohmic drop could be neglected in (1). Therefore *d ψs*/*dt* = *us*. During a short time Δ*t*, when the voltage space vector is applied it has:

$$
\Delta \vec{\psi}\_s \approx \vec{u}\_s \cdot \Delta t \tag{7}
$$

Thus the stator flux space vector moves by Δ *ψs* in the direction of the stator voltage space vector at a speed which is proportional to the magnitude of the stator voltage space vector. By selecting step-by-step the appropriate stator voltage vector, it is possible to change the stator flux in the required direction.

#### **3. Direct torque control scheme with space vector modulation technique**

In Fig. 1, we show the block diagram for the DTC-SVM scheme [14] with a Fuzzy Controller, the fuzzy controller will be substitute for the three kind of proposed Fuzzy Controller one for time. The DTC-SVM scheme is an alternative to the classical DTC schemes [15], [8] and [9]. In this one, the load angle *γ*∗ is not prefixed but it is determinate by the Fuzzy Controller. Equation (6) shows that the angle *γ*∗ determines the electromagnetic torque which is necessary to supply the load. The three proposed Fuzzy Controllers determine the load angle using the torque error *e* and the torque error change Δ*e*. Details about these controllers will be presented in the next section. Figure 1 shows the general block diagram of the DTC-SVM scheme, the inverter, the control signals for three-phase two-level inverter is generated by the DTC-SVM scheme.

#### **3.1. Flux reference calculation**

In stationary reference frame, the stator flux reference *ψ*∗ *<sup>s</sup>* can be decomposed in two perpendicular components *ψ*∗ *ds* and *ψ*<sup>∗</sup> *qs*. Therefore, the output of the fuzzy controller *γ*<sup>∗</sup> is added to rotor flux angle ∠ *ψr* in order to estimate the next angle of the stator flux reference.

In this chapter we consider the magnitude of stator flux reference as a constant. Therefore, we can use the relation presented in equation (8) to calculate the stator flux reference vector.

$$\vec{\psi}\_s^\* = |\vec{\psi}\_s^\*|\cos(\gamma^\* + \angle\vec{\psi}\_r) + j|\vec{\psi}\_s^\*|\sin(\gamma^\* + \angle\vec{\psi}\_r) \tag{8}$$

**Figure 1.** Direct torque control with space vector modulation scheme and fuzzy controller.

Moreover, if we consider the stator voltage *us* during a short time Δ*t*, it is possible to reproduce a flux variation Δ *ψs*. Notice that the stator flux variation is nearly proportional to the stator voltage space vector as seen in the equation (7).

#### **3.2. Stator voltage calculation**

The stator voltage calculation uses 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 [4]:

$$\vec{u}\_s = \frac{2}{3} \left[ (\mathbf{S}\_{Wa} - \frac{\mathbf{S}\_{Wb} + \mathbf{S}\_{Wc}}{2}) + j\frac{\sqrt{3}}{2} (\mathbf{S}\_{Wb} - \mathbf{S}\_{Wc}) \right] \mathcal{U}\_{dc} \tag{9}$$

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

As drawn by Fig. 1 the electromagnetic torque and the stator flux estimation depend on the stator voltage and the stator current space vectors,

$$
\vec{\psi}\_s = \int (\vec{u}\_s - R\_s \cdot \vec{i}\_s) dt \tag{10}
$$

Notice that *Tr* = *Lr*/*Rr* is the rotor time constant, and *ψrq* = 0. Substituting this expression in

1 + *sTr*

Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors 563

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

*Lr*

*<sup>r</sup>* is the rotor flux according to the equation (12). Since the voltage model is based on

*isd* (12)

*is* <sup>−</sup> *<sup>U</sup> comp*) (14)

*is* (13)

*<sup>s</sup>*) (15)

*is* (16)

*<sup>ψ</sup><sup>r</sup>* <sup>×</sup> *<sup>ψ</sup> <sup>s</sup>* (17)

*U comp*

<sup>−</sup> PI

−

*us*

+

+

Eq. 17 *te*

 *ψr*

 *ψs*

*<sup>ψ</sup>rd* <sup>=</sup> *Lm*

(*vs* − *Rs*

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 [13]. The rotor

The estimator scheme shown in the Fig. 2 works with a good performance in the wide range

*is*

*<sup>r</sup>*

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

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

 *ψr*( *ψs*)

Where LPF means low pass filter. In the other hand, when equations (14) and (16) are replaced

 *ψs*

−

+

*ψi s* *Lm*

*m*

*Rs*

the equation (11) yields:

Where *ψi*

flux

of speeds.

*is*

In the current model the stator flux is represented by:

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

equation (1), the stator flux in the stationary reference frame is given by

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

measurement, the voltage model is adapted through the PI controller.

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

*ψrdq*

*e*−*jθ<sup>r</sup> ejθ<sup>r</sup>*

*isdq*

LPF

*θψ <sup>r</sup> θψ <sup>r</sup>*

arctan(*ψrq*/*ψrd*)

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

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

 *ψs*( *ψr*)

 *ψr*

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

**Figure 2.** Stator and rotor flux estimator, and electromagnetic torque estimator.

in (5) we can estimated the electromagnetic torque *te* as:

*ψi*

The problem with this kind of estimation is that for 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 [13]. The rotor flux *ψrdq* represented in the rotor flux reference frame is given by:

$$
\vec{\psi}\_{rdq} = \frac{L\_m}{1 + sT\_r} \vec{i}\_{sdq} - j \frac{(\omega\_{\Psi\_r} - \omega\_r)T\_r}{1 + sT\_r} \vec{\psi}\_{rdq} \tag{11}
$$

562 Fuzzy Controllers – Recent Advances in Theory and Applications Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors <sup>5</sup> Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors 563

> Notice that *Tr* = *Lr*/*Rr* is the rotor time constant, and *ψrq* = 0. Substituting this expression in the equation (11) yields:

$$
\psi\_{rd} = \frac{L\_m}{1 + sT\_r} i\_{sd} \tag{12}
$$

In the current model the stator flux is represented by:

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

*t* ∗ *e*

+ +

Fuzzy Controller

a flux variation Δ

in [4]:


− −

voltage space vector as seen in the equation (7).

*us* <sup>=</sup> <sup>2</sup> 3 

stator voltage and the stator current space vectors,

**3.2. Stator voltage calculation**

flux reference frame is given by:

Flux reference calculation  *ψ*∗ *s*

∠ *ψr*

*te*

Us

*is*

Torque and Flux Estimator

**Figure 1.** Direct torque control with space vector modulation scheme and fuzzy controller.

(*SWa* <sup>−</sup> *SWb* <sup>+</sup> *SWc*

 *ψ<sup>s</sup>* = 

model to improve the flux estimation as in [13]. The rotor flux

*<sup>ψ</sup>rdq* <sup>=</sup> *Lm*

1 + *sTr*

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

Moreover, if we consider the stator voltage *us* during a short time Δ*t*, it is possible to reproduce

The stator voltage calculation uses 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

<sup>2</sup> ) + *<sup>j</sup>*

As drawn by Fig. 1 the electromagnetic torque and the stator flux estimation depend on the

The problem with this kind of estimation is that for low speeds the back electromotive force (emf) depends strongly of the stator resistance, to resolve this problem is used the current

*isdq* <sup>−</sup> *<sup>j</sup>*

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

*ψs*. Notice that the stator flux variation is nearly proportional to the stator

√ 3

<sup>2</sup> (*SWb* <sup>−</sup> *SWc*)

(*us* <sup>−</sup> *Rs* ·*is*)*dt* (10)

 *ψs*

Δ *ψ<sup>s</sup>* 1 Δ*t u*∗ *s*

> Voltage calculation

Control Signals

SVM

*α* − *β* ABC

Im

*Udc* (9)

*ψrdq* represented in the rotor

*ψrdq* (11)

*Udc*

*Ia*

*Udc*

*Ib*

$$
\vec{\psi}\_s^i = \frac{L\_m}{L\_r}\vec{\psi}\_r^i + \frac{L\_s L\_r - L\_m^2}{L\_r}\vec{i}\_s \tag{13}
$$

Where *ψi <sup>r</sup>* is the rotor flux according to the equation (12). Since the voltage model is based on equation (1), the stator flux in the stationary reference frame is given by

$$
\vec{\psi}\_s = \frac{1}{s} (\vec{v}\_s - R\_s \vec{i}\_s - \vec{\mathcal{U}}\_{comp}) \tag{14}
$$

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

$$\vec{\mathcal{U}}\_{comp} = (K\_p + K\_l \frac{1}{s})(\vec{\psi}\_s - \vec{\psi}\_s^i) \tag{15}$$

The *Kp* and *Ki* coefficients are calculated with the recommendation proposed in [13]. The rotor flux *ψr* in the stationary reference frame is calculated as:

$$
\vec{\psi}\_r = \frac{L\_r}{L\_m}\vec{\psi}\_s - \frac{L\_s L\_r - L\_m^2}{L\_m}\vec{i}\_s\tag{16}
$$

The estimator scheme shown in the Fig. 2 works with a good performance in the wide range of speeds.

**Figure 2.** Stator and rotor flux estimator, and electromagnetic torque estimator.

Where LPF means low pass filter. In the other hand, when equations (14) and (16) are replaced in (5) we can estimated the electromagnetic torque *te* as:

$$t\_{\varepsilon} = \frac{3}{2} P \frac{L\_m}{L\_r L\_s \sigma} \vec{\psi}\_r \times \vec{\psi}\_s \tag{17}$$

## **4. Design of fuzzy controllers**

#### **4.1. The PI fuzzy controller (PI-F)**

The PI fuzzy controller combines two simple fuzzy controllers and a conventional PI controller. Note that fuzzy controllers are responsible for generating the PI parameters dynamically while considering only the torque error variations. The PI-F block diagram is shown in Fig. 3, this controller is composed of two scale factors *Ge*, *G*Δ*<sup>e</sup>* at the input. The input for fuzzy controllers are the error (*eN*) and error change (*e*Δ*N*), and theirs outputs represent the proportional gain *Kp* and the integral time *Ti* respectively. These parameters *Kp* and *Ti* are adjusted in real time by the fuzzy controllers. The gain *KP* is limited to the interval [*Kp*,*min*, *Kp*,*max*], which we determined by simulations. For convenience, *KP* is normalized in the range between zero and one through the following linear transformation.

$$K\_p' = \frac{K\_p - K\_{p, \text{min}}}{K\_{p, \text{max}} - K\_{p, \text{min}}} \tag{18}$$

*4.1.1. Membership Functions (MF)*

0 0.2 0.4 0.6 0.8 1

**Figure 4.** Membership functions for the inputs *eN*, Δ*eN*.

center of mass method for defuzzification [11].

0 0.2 0.4 0.6 0.8 1

**Figure 5.** Membership functions for the first fuzzy controller output *Kp*.

Grado de pertenencia

Degree of membership

In the Fig. 3, the first fuzzy controller receives as inputs the errors *eN*, Δ*eN*, each of them has three fuzzy sets that are defined similarly, being only necessary to describe the fuzzy sets of the first input. The first input *eN* has three fuzzy sets whose linguistic terms are N-Negative, ZE-Zero and P-Positive. Each fuzzy set has a membership function associated with it. In our particular case of, these fuzzy sets have trapezoidal and triangular shapes as shown in Fig.4.

Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors 565

**N ZE P**

−1.5 −1 −0.5 0 0.5 1 1.5

The output has two fuzzy sets whose linguistic values are associated with them are S-small and B-Big, respectively. Both have trapezoidal membership functions as shown in Fig.5. The universe of discourse of the fuzzy sets is defined over the closed interval [−0.5, 1.5]. The fuzzy controller uses: triangular norm, Mamdani implication, max-min aggregation method and the

**S B**

−0.5 0 0.5 1 1.5

Similarly, the second fuzzy controller has the same fuzzy sets for its two inputs, however, its output is defined by three constant values defined as 1.5, 2 and 3 which linguistic values associated with them are S-Small, M-Medium and B-Big. This controller uses the zero-order Takagi-Sugeno model which simplifies the hardware design and is easy to

The universe of discourse of these sets is defined over the closed interval [−1.5, 1.5].

Then, considering that the fuzzy controller output is a normalized value *K*� *<sup>p</sup>*, we obtain *Kp* by:

$$K\_p = (K\_{p, \text{min}} - K\_{p, \text{max}})K\_p' + K\_{p, \text{min}} \tag{19}$$

However, for different reference values the range for the proportional gain values is chosen as [0, *Kp*,*max*],

$$K\_p = K\_{p,max} K\_p'\tag{20}$$

Due to nonlinearities of the system and in order to avoid overshoots for large reference torque *r*, it is necessary to reduce the proportional gain. We use a gain coefficient *ρ* = 1/(1 + 0002 ∗ *r*) that depends on the reference values. In order to achieves real time adjustment for the *Kp* values. Therefore, *Kp*,*max* = *ρKp*,*max*<sup>0</sup> where the value *Kp*,*max*<sup>0</sup> = 1.24 was obtained through various simulations. Note that both *ρ* and *Kp*,*max* decreases as the reference value increases. Consequently, the gain *Kp* decreases. The PI-F controller receives as input the torque error *e* and as output the motor load angle *γ*∗.

**Figure 3.** PI Fuzzy Controller block diagram.

564 Fuzzy Controllers – Recent Advances in Theory and Applications Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors <sup>7</sup> Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors 565

## *4.1.1. Membership Functions (MF)*

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

The PI fuzzy controller combines two simple fuzzy controllers and a conventional PI controller. Note that fuzzy controllers are responsible for generating the PI parameters dynamically while considering only the torque error variations. The PI-F block diagram is shown in Fig. 3, this controller is composed of two scale factors *Ge*, *G*Δ*<sup>e</sup>* at the input. The input for fuzzy controllers are the error (*eN*) and error change (*e*Δ*N*), and theirs outputs represent the proportional gain *Kp* and the integral time *Ti* respectively. These parameters *Kp* and *Ti* are adjusted in real time by the fuzzy controllers. The gain *KP* is limited to the interval [*Kp*,*min*, *Kp*,*max*], which we determined by simulations. For convenience, *KP* is normalized in

> *<sup>p</sup>* <sup>=</sup> *Kp* <sup>−</sup> *Kp*,*min Kp*,*max* − *Kp*,*min*

However, for different reference values the range for the proportional gain values is chosen as

*Kp* = *Kp*,*maxK*�

Due to nonlinearities of the system and in order to avoid overshoots for large reference torque *r*, it is necessary to reduce the proportional gain. We use a gain coefficient *ρ* = 1/(1 + 0002 ∗ *r*) that depends on the reference values. In order to achieves real time adjustment for the *Kp* values. Therefore, *Kp*,*max* = *ρKp*,*max*<sup>0</sup> where the value *Kp*,*max*<sup>0</sup> = 1.24 was obtained through various simulations. Note that both *ρ* and *Kp*,*max* decreases as the reference value increases. Consequently, the gain *Kp* decreases. The PI-F controller receives as input the torque error *e*

x

*Kp*

*Ti*

*Kp*,*max*

PI Controller

*p*

(18)

*<sup>p</sup>*, we obtain *Kp* by:

*γ*∗

*<sup>p</sup>* + *Kp*,*min* (19)

*<sup>p</sup>* (20)

the range between zero and one through the following linear transformation.

*K*�

Then, considering that the fuzzy controller output is a normalized value *K*�

*<sup>G</sup>*Δ*<sup>e</sup> <sup>K</sup>*�

Fuzzy controller (I)

> Fuzzy controller (II)

*Kp* = (*Kp*,*min* <sup>−</sup> *Kp*,*max*)*K*�

**4. Design of fuzzy controllers**

**4.1. The PI fuzzy controller (PI-F)**

and as output the motor load angle *γ*∗.

<sup>−</sup> <sup>Δ</sup>*<sup>e</sup>* <sup>Δ</sup>*eN*

*e eN*

**Figure 3.** PI Fuzzy Controller block diagram.

*Ge*

+

[0, *Kp*,*max*],

*Z*−<sup>1</sup>

In the Fig. 3, the first fuzzy controller receives as inputs the errors *eN*, Δ*eN*, each of them has three fuzzy sets that are defined similarly, being only necessary to describe the fuzzy sets of the first input. The first input *eN* has three fuzzy sets whose linguistic terms are N-Negative, ZE-Zero and P-Positive. Each fuzzy set has a membership function associated with it. In our particular case of, these fuzzy sets have trapezoidal and triangular shapes as shown in Fig.4. The universe of discourse of these sets is defined over the closed interval [−1.5, 1.5].

**Figure 4.** Membership functions for the inputs *eN*, Δ*eN*.

The output has two fuzzy sets whose linguistic values are associated with them are S-small and B-Big, respectively. Both have trapezoidal membership functions as shown in Fig.5. The universe of discourse of the fuzzy sets is defined over the closed interval [−0.5, 1.5]. The fuzzy controller uses: triangular norm, Mamdani implication, max-min aggregation method and the center of mass method for defuzzification [11].

**Figure 5.** Membership functions for the first fuzzy controller output *Kp*.

Similarly, the second fuzzy controller has the same fuzzy sets for its two inputs, however, its output is defined by three constant values defined as 1.5, 2 and 3 which linguistic values associated with them are S-Small, M-Medium and B-Big. This controller uses the zero-order Takagi-Sugeno model which simplifies the hardware design and is easy to introduce programmability [19]. The defuzzification method used for this controller is the weighted sum.

### *4.1.2. Scaling Factors (SF)*

The PI-F controller has two scaling factors, *Ge* and *G*Δ*<sup>e</sup>* as inputs, while the fuzzy controllers outputs are the gain *K*� *<sup>p</sup>* and the integral time *TI* respectively. From these values we can calculate the parameter *KI* = *Kp*/*TI*.

The scale factor ensures that both inputs are within the universe of discourse previously defined. The parameters *Kp* and *KI* are the tuned parameters of the PI controller. The inputs are normalized, by:

$$e\_N = G\_\ell \cdot e$$

**eN** / **ΔeN N ZE P N** BBB **ZE** SBS **P** BBB

Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors 567

*p*

**eN** / **ΔeN N ZE P N** SSS **ZE** BMB **P** SSS

The PI-type fuzzy controller (PIF) is a fuzzy controller inspired by a digital PI controller, which is depicted in Fig. 8. It is composed by two input scale factors "*Ge*, *G*Δ*e*" and one output scale

This controller has a single input variable, which is the torque error "*e*" and one output variable

In (23), *k* is the sampling time and Δ*γ*∗(*k*) represents the incremental change of the controller output. We wish to emphasize here that this accumulation (23) of the controller output takes

Fig.9 shows the self-tuning PI-type fuzzy controller (STPIF) block diagram, its main difference

The MF for PIF controller are shown in Fig. 10(a). This MF for input variables "*eN*, Δ*eN*" and

The MF's for GTF controller are shown in Fig. 10(a) and in Fig. 10(b) for input and output variables respectively. Input variables "*eN*, Δ*eN*" are defined in the closed interval [-1,1] and

Most of the MF variables have triangular shape [Fig. 10] with 50% overlapping neighbor functions, except the extremes which are trapezoidal. The linguistic variables are referred to as: NL-Negative Large, NM-Negative Medium, NS-Negative Small, ZE-Zero, VS-Very Small,

The two inputs SF "*G*Δ*e*, *Ge*" and the output SF "*Gγ*<sup>∗</sup> " can be adjusted dynamically through updating the scaling factor "*α*". "*α*" is computed on-line, using a independent fuzzy rule model

*<sup>N</sup>*" are normalized in the closed interval [-1,1].

place out of the fuzzy part of the controller and it does not influence the fuzzy rules.

with the PIF controller is the gain tuning fuzzy controller (GTF) block.

the output variable "*α*" is defined in the closed interval [0,1].

S-Small, SL-Small Large and so on as shown in Table 3 and in Table 4.

*γ*∗(*k*) = *γ*∗(*k* + 1) + Δ*γ*∗(*k*) (23)

**4.2. The PI-type fuzzy controller (PIF) and The self-tuning PI-type fuzzy**

**Table 1.** Rule base for fuzzy controller (I), output *K*�

**Table 2.** Rule base for fuzzy controller (II), output *TI*

which is the motor load angle "*γ*∗" given by:

*4.2.1. Membership Functions (MF)*

output variable "Δ*γ*∗

*4.2.2. Scaling factors*

factor "*Gγ*<sup>∗</sup> ". Finally it uses saturation block to limit the output.

**controller (STPIF)**

$$
\Delta e\_N = \mathbb{G}\_{\Delta \varepsilon} \cdot \Delta e \tag{22}
$$

#### *4.1.3. The rule bases*

The rules are based on simulation that we conducted of various control schemes. Fig.6 shows an example for one possible response system. Initially, the error is positive around the point *a*, and the error change is negative, then is imposed a large control signal in order to obtain a small rise time.

To produce a large signal control, the PI controller should have a large gain *Kp* and a large integral gain *KI* (small integral time *TI*), therefore,

**Figure 6.** Response system.

The rule base for the first fuzzy controller is in Table 1, also, the rule base for the second fuzzy controller is in Table 2.

Fig. 7 shows the control surface for the first and second fuzzy controllers. This figure clearly shows the non-linear relationship between (*e*, Δ*e*, *Kp*) and (*e*, Δ*e*, *TI*)



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

introduce programmability [19]. The defuzzification method used for this controller is the

The PI-F controller has two scaling factors, *Ge* and *G*Δ*<sup>e</sup>* as inputs, while the fuzzy controllers

The scale factor ensures that both inputs are within the universe of discourse previously defined. The parameters *Kp* and *KI* are the tuned parameters of the PI controller. The inputs

The rules are based on simulation that we conducted of various control schemes. Fig.6 shows an example for one possible response system. Initially, the error is positive around the point *a*, and the error change is negative, then is imposed a large control signal in order to obtain a

To produce a large signal control, the PI controller should have a large gain *Kp* and a large

*Rx* : if *eN* is *S* and Δ*eN* is *N* then *Kp* is *G*

The rule base for the first fuzzy controller is in Table 1, also, the rule base for the second fuzzy

Fig. 7 shows the control surface for the first and second fuzzy controllers. This figure clearly

shows the non-linear relationship between (*e*, Δ*e*, *Kp*) and (*e*, Δ*e*, *TI*)

*<sup>p</sup>* and the integral time *TI* respectively. From these values we can

*eN* = *Ge* · *e* (21) Δ*eN* = *G*Δ*<sup>e</sup>* · Δ*e* (22)

weighted sum.

*4.1.2. Scaling Factors (SF)*

outputs are the gain *K*�

are normalized, by:

*4.1.3. The rule bases*

small rise time.

**Figure 6.** Response system.

controller is in Table 2.

calculate the parameter *KI* = *Kp*/*TI*.

integral gain *KI* (small integral time *TI*), therefore,


**Table 2.** Rule base for fuzzy controller (II), output *TI*

## **4.2. The PI-type fuzzy controller (PIF) and The self-tuning PI-type fuzzy controller (STPIF)**

The PI-type fuzzy controller (PIF) is a fuzzy controller inspired by a digital PI controller, which is depicted in Fig. 8. It is composed by two input scale factors "*Ge*, *G*Δ*e*" and one output scale factor "*Gγ*<sup>∗</sup> ". Finally it uses saturation block to limit the output.

This controller has a single input variable, which is the torque error "*e*" and one output variable which is the motor load angle "*γ*∗" given by:

$$
\gamma^\*(k) = \gamma^\*(k+1) + \Delta\gamma^\*(k)\tag{23}
$$

In (23), *k* is the sampling time and Δ*γ*∗(*k*) represents the incremental change of the controller output. We wish to emphasize here that this accumulation (23) of the controller output takes place out of the fuzzy part of the controller and it does not influence the fuzzy rules.

Fig.9 shows the self-tuning PI-type fuzzy controller (STPIF) block diagram, its main difference with the PIF controller is the gain tuning fuzzy controller (GTF) block.

### *4.2.1. Membership Functions (MF)*

The MF for PIF controller are shown in Fig. 10(a). This MF for input variables "*eN*, Δ*eN*" and output variable "Δ*γ*∗ *<sup>N</sup>*" are normalized in the closed interval [-1,1].

The MF's for GTF controller are shown in Fig. 10(a) and in Fig. 10(b) for input and output variables respectively. Input variables "*eN*, Δ*eN*" are defined in the closed interval [-1,1] and the output variable "*α*" is defined in the closed interval [0,1].

Most of the MF variables have triangular shape [Fig. 10] with 50% overlapping neighbor functions, except the extremes which are trapezoidal. The linguistic variables are referred to as: NL-Negative Large, NM-Negative Medium, NS-Negative Small, ZE-Zero, VS-Very Small, S-Small, SL-Small Large and so on as shown in Table 3 and in Table 4.

#### *4.2.2. Scaling factors*

The two inputs SF "*G*Δ*e*, *Ge*" and the output SF "*Gγ*<sup>∗</sup> " can be adjusted dynamically through updating the scaling factor "*α*". "*α*" is computed on-line, using a independent fuzzy rule model

10 Will-be-set-by-IN-TECH 568 Fuzzy Controllers – Recent Advances in Theory and Applications Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors <sup>11</sup>

*Z*−<sup>1</sup>

*Z*−<sup>1</sup>

+ +

*<sup>N</sup>* Δ*γ*<sup>∗</sup> *γ*<sup>∗</sup>

*<sup>N</sup>* Δ*γ*<sup>∗</sup> *γ*<sup>∗</sup>

*N*.

Δ*γ*∗

Δ*γ*∗

*α*

*<sup>G</sup>γ*<sup>∗</sup> PI-type

Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors 569

Fuzzy (PIF)

+ + +

Gain tuning Fuzzy (GTF)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

ZE VS S SL ML L VL

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

gain factor (α)

*<sup>N</sup>* of the PIF controller. (b) the gain updating factor *α*

(b) Membership function for *α* for GTF controller output

**Figure 10.** Membership functions for: (a) error *eN* and change of error Δ*eN* is the same for PIF and GTF

(a) Membership function for *eN*, Δ*eN* and Δ*γ*<sup>∗</sup>

NL NM NS ZE PS PM PL

*<sup>G</sup>*Δ*<sup>e</sup> <sup>G</sup>γ*<sup>∗</sup> PI-type Fuzzy (PIF)

*Z*−<sup>1</sup>

**Figure 8.** PI-type fuzzy controller.

*Z*−<sup>1</sup>

+

<sup>−</sup> <sup>Δ</sup>*<sup>e</sup>* <sup>Δ</sup>*eN*

*G*Δ*<sup>e</sup>*

*e eN*

*Ge*

<sup>−</sup> <sup>Δ</sup>*<sup>e</sup>* <sup>Δ</sup>*eN*

*e eN*

**Figure 9.** Self-tuning PI-type fuzzy (STPIF) controller.

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

Degree of membership

controllers as well as for the output Δ*γ*∗

Degree of membership

*Ge*

(a) Control surface for fuzzy controller (I), output *K*� *p*.

(b) Control surface for fuzzy controller (II), output *TI*.

**Figure 7.** Control surface for: (a) fuzzy controller (I) and (b) fuzzy controller (II)

defined in terms of "*e*, Δ*e*". The relationship between the SF and the input/output variables of the STPIF controller are shown bellow:

$$
\varepsilon\_N = \mathbb{G}\_\ell \cdot \varepsilon \tag{24}
$$

$$
\Delta \mathbf{e}\_N = \mathbf{G}\_{\Delta \mathbf{e}} \cdot \Delta \mathbf{e} \tag{25}
$$

$$
\Delta \gamma^\* = (\mathfrak{a} \cdot \mathbb{G}\_{\gamma^\*}) \cdot \Delta \gamma\_N^\* \tag{26}
$$

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

−1.5 −1 −0.5 0 0.5 1 1.5

(a) Control surface for fuzzy controller (I), output *K*�

−1.5 −1 −0.5 0 0.5 1 1.5

(b) Control surface for fuzzy controller (II), output *TI*.

defined in terms of "*e*, Δ*e*". The relationship between the SF and the input/output variables of

Δ*γ*<sup>∗</sup> = (*α* · *Gγ*<sup>∗</sup> ) · Δ*γ*<sup>∗</sup>

error change (Δe) error (e)

−1.5 −1 −0.5 0 0.5 1 1.5 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

**Figure 7.** Control surface for: (a) fuzzy controller (I) and (b) fuzzy controller (II)

*p*.

*eN* = *Ge* · *e* (24) Δ*eN* = *G*Δ*<sup>e</sup>* · Δ*e* (25)

*<sup>N</sup>* (26)

error change (Δe) error (e)

−1.5 −1 −0.5 0 0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

output K '<sup>p</sup>

Output TI

the STPIF controller are shown bellow:

**Figure 9.** Self-tuning PI-type fuzzy (STPIF) controller.

**Figure 10.** Membership functions for: (a) error *eN* and change of error Δ*eN* is the same for PIF and GTF controllers as well as for the output Δ*γ*∗ *<sup>N</sup>* of the PIF controller. (b) the gain updating factor *α*

#### *4.2.3. The rule bases*

The incremental change in the controller output Δ*γ*∗ *<sup>N</sup>* to PIF controller is defined as,

$$R\_X \text{ : if } e\_N \text{ is } E \text{ and } \Delta e\_N \text{ is } \Delta E \text{ then } \Delta \gamma\_N^\* \text{ is } \Delta \Gamma\_N^\*.$$

"*te*" is approaching to the torque reference "*t*

is "*PM*" and "Δ*e*" is "*PM*" then "*α*" is "*VL*".

"*te*" is moving away from the torque reference "*t*∗

The nonlinear relationship between "*e*, Δ*e*, Δ*γ*∗

∗

On the other hand, when "*e*" and "Δ*e*" have the same sign, it means that the torque estimate

Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors 571

large value by "*α*" in order to avoid that the torque depart from the torque reference, e.g., if "*e*"

−1

(a) Surface of PIF controller output (Δ*γ*∗

−1

(b) Surface of GTF controller output (*α*).

The inference method used in PIF and GTF controllers is the Mamdani's implication based on

−1

−0.5

max-min aggregation. We use the center of area method for defuzzification.

0

error (e)

0.5

**Figure 11.** Surface of: (a) PIF controller and (b) GTF controller

1 0.2 0.4 0.6 0.8

gain updating factor (α)

−0.5

0

Change of error (Δe)

0.5

1

Change of error (Δe) error (e)

−1

−0.5 0

0.5

1

−0.5

output controller (Δγ\*

N)

0

0.5

−0.5

0

*<sup>N</sup>*).

0.5

1

small value by "*α*", for instance, if "*e*" is "*PM*" and "Δ*e*" is "*NM*" then "*α*" is "*S*".

*<sup>e</sup>* ", then the output SF "*Gγ*<sup>∗</sup> " must be reduced to a

*<sup>e</sup>* ", the output SF "*Gγ*<sup>∗</sup> " must be increased to a

*<sup>N</sup>*" and "*e*, Δ*e*, *α*" are shown in Fig. 11.

Where *E* = Δ*E* = ΔΓ∗ *<sup>N</sup>* = {*NL*, *NM*, *NS*, *ZE*, *PS*, *PM*, *PL*}. The output *α* of the GTF controller is determined by the following rules:

$$R\_{\mathcal{X}} \text{ : if } e\_N \text{ is } E \text{ and } \Delta e\_N \text{ is } \Delta E \text{ then } \alpha \text{ is } \mathcal{X}$$

Where *E* = Δ*E* = {*NL*, *NM*, *NS*, *ZE*, *PS*, *PM*, *PL*} and *χ* = {*ZE*, *VS*, *S*, *SL*, *ML*, *L*, *VL*}. The rule base for Δ*γ*∗ *<sup>N</sup>* and *α* are shown in Tab. 3 and Tab. 4 respectively.


**Table 3.** Fuzzy rules for the computation of Δ*γ*∗ *N*


**Table 4.** Fuzzy rules for the computation of *α*

#### *4.2.4. Gain tuning fuzzy*

The purpose of the GTF controller is update continuous the value of *α* in every sample time. The output *α* is necessary to control the percentage of the output SF "*Gγ*<sup>∗</sup> ", and therefore for calculating new "Δ*γ*∗",

$$
\Delta \gamma^\* = (\mathfrak{a} \cdot \mathbb{G}\_{\gamma^\*}) \cdot \Delta \gamma\_N^\* \tag{27}
$$

The GTF controller rule base is based on knowledge about the three-phase IM control, using a DTC type control according to the scheme proposed in [14], in order to avoid large overshoot and undershoot, e.g., when "*e*" and "Δ*e*" have different signs, it means that the estimate torque "*te*" is approaching to the torque reference "*t* ∗ *<sup>e</sup>* ", then the output SF "*Gγ*<sup>∗</sup> " must be reduced to a small value by "*α*", for instance, if "*e*" is "*PM*" and "Δ*e*" is "*NM*" then "*α*" is "*S*".

On the other hand, when "*e*" and "Δ*e*" have the same sign, it means that the torque estimate "*te*" is moving away from the torque reference "*t*∗ *<sup>e</sup>* ", the output SF "*Gγ*<sup>∗</sup> " must be increased to a large value by "*α*" in order to avoid that the torque depart from the torque reference, e.g., if "*e*" is "*PM*" and "Δ*e*" is "*PM*" then "*α*" is "*VL*".

The nonlinear relationship between "*e*, Δ*e*, Δ*γ*∗ *<sup>N</sup>*" and "*e*, Δ*e*, *α*" are shown in Fig. 11.

12 Will-be-set-by-IN-TECH

*Rx* : if *eN* is *E* and Δ*eN* is Δ*E* then *α* is *χ*

Where *E* = Δ*E* = {*NL*, *NM*, *NS*, *ZE*, *PS*, *PM*, *PL*} and *χ* = {*ZE*, *VS*, *S*, *SL*, *ML*, *L*, *VL*}. The

**ΔeN** / **eN NL NM NS ZE PS PM PL NL** NL NL NL NM NS NS ZE **NM** NL NM NM NM NS ZE PS **NS** NL NM NS NS ZE PS PM **ZE** NL NM NS ZE PS PM PL **PS** NM NS ZE PS PS PM PL **PM** NS ZE PS PM PM PM PL **PL** ZE PS PS PM PL PL PL

*Rx* : if *eN* is *E* and Δ*eN* is Δ*E* then Δ*γ*<sup>∗</sup>

*<sup>N</sup>* and *α* are shown in Tab. 3 and Tab. 4 respectively.

*N*

**ΔeN** / **eN NL NM NS ZE PS PM PL NL** VL VL VL L SL S ZE **NM** VL VL L L ML S VS **NS** VL ML L VL VS S VS **ZE** S SL ML ZE ML SL S **PS** VS S VS VL L ML VL **PM** VS S ML L L VL VL **PL** ZE S SL L VL VL VL

The purpose of the GTF controller is update continuous the value of *α* in every sample time. The output *α* is necessary to control the percentage of the output SF "*Gγ*<sup>∗</sup> ", and therefore for

Δ*γ*<sup>∗</sup> = (*α* · *Gγ*<sup>∗</sup> ) · Δ*γ*<sup>∗</sup>

The GTF controller rule base is based on knowledge about the three-phase IM control, using a DTC type control according to the scheme proposed in [14], in order to avoid large overshoot and undershoot, e.g., when "*e*" and "Δ*e*" have different signs, it means that the estimate torque

*<sup>N</sup>* (27)

*<sup>N</sup>* to PIF controller is defined as,

*<sup>N</sup>* is ΔΓ<sup>∗</sup> *N*

*<sup>N</sup>* = {*NL*, *NM*, *NS*, *ZE*, *PS*, *PM*, *PL*}. The output *α* of the GTF controller

*4.2.3. The rule bases*

Where *E* = Δ*E* = ΔΓ∗

rule base for Δ*γ*∗

is determined by the following rules:

**Table 3.** Fuzzy rules for the computation of Δ*γ*∗

**Table 4.** Fuzzy rules for the computation of *α*

*4.2.4. Gain tuning fuzzy*

calculating new "Δ*γ*∗",

The incremental change in the controller output Δ*γ*∗

(a) Surface of PIF controller output (Δ*γ*∗ *<sup>N</sup>*).

(b) Surface of GTF controller output (*α*).

**Figure 11.** Surface of: (a) PIF controller and (b) GTF controller

The inference method used in PIF and GTF controllers is the Mamdani's implication based on max-min aggregation. We use the center of area method for defuzzification.

14 Will-be-set-by-IN-TECH 572 Fuzzy Controllers – Recent Advances in Theory and Applications Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors <sup>15</sup>

## **5. Simulation results**

We have conducted our simulation with MATLAB simulation package, which include Simulink block sets and fuzzy logic toolbox. The switching frequency of the pulse width modulation (PWM) inverter was set to be 10kHz, the stator reference flux considered was 0.47 Wb. In order to investigate the effectiveness of the three proposed fuzzy controllers applied in the DTC-SVM scheme we performed several tests.

1 1.5 2 2.5 3 3.5

Rotor currents

1 1.5 2 2.5 3 3.5

time (s)

Electromagnetic torque

Torque

0.98 0.99 <sup>1</sup> 1.01 1.02 1.03 1.04 −5

time (s)

Stator currents

0.98 0.99 1 1.01 1.02 1.03 1.04

Rotor currents

0.98 0.99 1 1.01 1.02 1.03 1.04

time (s)

**Figure 15.** Stator and rotor currents for sudden torque change for DTC-SVM with PI-F controller.

**Figure 13.** Stator and rotor current for speed reversion for DTC-SVM with PI-F controller.

reference

Stator currents

Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors 573

−30 −20 −10 0 10 20 30

−30 −20 −10 0 10 20 30

0

−30 −20 −10 0 10 20 30

−30 −20 −10 0 10 20 30

Irabc (A)

Isabc (A)

**Figure 14.** Sudden torque change for DTC-SVM whit PI-F controller

5

T

em (N.m)

10

15

Irabc (A)

Isabc (A)

We used different dynamic operating conditions such as: step change in the motor load (from 0 to 1.0 pu) at 90 percent of rated speed, no-load speed reversion (from 0.5 pu to -0.5 pu) and the application of a specific load torque profile at 90 percent of rated speed. The motor parameters used in the tests are given in Table 5.

Fig. 12, shows the response of the speed and electromagnetic torque when speed reversion for DTC-SVM with PI-F controller is applied. Here, the rotor speed changes its direction at about 1.8 seconds. Fig. 13 shows the stator and rotor current sinusoidal behavior when applying reversion.

**Figure 12.** Speed reversion for DTC-SVM with PI-F controller.

Fig. 14 and Fig. 15 show the torque and currents responses respecively, when step change is applied in the motor load for DTC-SVM with the PI-F controller. This speed test was established at 90 percent of rated speed.

In Fig. 16, we demonstrate the speed response when applying a speed reversion for DTC-SVM with PIF controller. In this case the speed of the rotor changes its direction at about 1.4 seconds. Fig. 17 shows the electromagnetic torque behavior when the reversion is applied.

Fig. 18 and Fig. 19 show the response of the electromagnetic torque and phase *a* stator current respectively, when applying a step change in the motor load for DTC-SVM whit PIF controller. In this test the speed of the motor was set to 90 percent of rated speed.

Fig. 20 shows the behaviors of the electromagnetic torque, phase *a* stator current and the motor speed, when applying speed reversion from 0.5 pu to -0.5 pu in the DTC-SVM scheme with STPIF controller. The sinusoidal waveform of the current shows that this control technique also leads to a good current control.

572 Fuzzy Controllers – Recent Advances in Theory and Applications Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors <sup>15</sup> Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors 573

**Figure 13.** Stator and rotor current for speed reversion for DTC-SVM with PI-F controller.

**Figure 14.** Sudden torque change for DTC-SVM whit PI-F controller

14 Will-be-set-by-IN-TECH

We have conducted our simulation with MATLAB simulation package, which include Simulink block sets and fuzzy logic toolbox. The switching frequency of the pulse width modulation (PWM) inverter was set to be 10kHz, the stator reference flux considered was 0.47 Wb. In order to investigate the effectiveness of the three proposed fuzzy controllers applied

We used different dynamic operating conditions such as: step change in the motor load (from 0 to 1.0 pu) at 90 percent of rated speed, no-load speed reversion (from 0.5 pu to -0.5 pu) and the application of a specific load torque profile at 90 percent of rated speed. The motor

Fig. 12, shows the response of the speed and electromagnetic torque when speed reversion for DTC-SVM with PI-F controller is applied. Here, the rotor speed changes its direction at about 1.8 seconds. Fig. 13 shows the stator and rotor current sinusoidal behavior when applying

Rotor speed

reference

1 1.5 2 2.5 3 3.5

speed

Electromagnetic torque

Torque

1 1.5 2 2.5 3 3.5

time (s)

Fig. 14 and Fig. 15 show the torque and currents responses respecively, when step change is applied in the motor load for DTC-SVM with the PI-F controller. This speed test was

In Fig. 16, we demonstrate the speed response when applying a speed reversion for DTC-SVM with PIF controller. In this case the speed of the rotor changes its direction at about 1.4 seconds.

Fig. 18 and Fig. 19 show the response of the electromagnetic torque and phase *a* stator current respectively, when applying a step change in the motor load for DTC-SVM whit PIF controller.

Fig. 20 shows the behaviors of the electromagnetic torque, phase *a* stator current and the motor speed, when applying speed reversion from 0.5 pu to -0.5 pu in the DTC-SVM scheme with STPIF controller. The sinusoidal waveform of the current shows that this control technique

Fig. 17 shows the electromagnetic torque behavior when the reversion is applied.

In this test the speed of the motor was set to 90 percent of rated speed.

**5. Simulation results**

reversion.

in the DTC-SVM scheme we performed several tests.

parameters used in the tests are given in Table 5.

−1500 −1000 −500 0 500 1000 1500

> −20 −10 0 10 20

**Figure 12.** Speed reversion for DTC-SVM with PI-F controller.

T

established at 90 percent of rated speed.

also leads to a good current control.

em (N.m)

ωm (rpm)

**Figure 15.** Stator and rotor currents for sudden torque change for DTC-SVM with PI-F controller.

0.9 0.92 0.94 0.96 0.98 <sup>1</sup> 1.02 1.04 1.06 1.08 1.1 −60

**Figure 19.** Phase *a* stator current for sudden torque change for DTC-SVM with PIF controller.

torque

reference torque rated

time (s)

1 1.5 2 2.5 3 3.5

speed

reference

1 1.5 2 2.5 3 3.5

current Ia

**Figure 20.** Speed, torque and phase *a* stator current for speed reversion for DTC-SVM with STPIF

1 1.5 2 2.5 3 3.5

time (s)

STPIF

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

50% torque rated

time(s)

Stator Currents

Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors 575

−100 0 100 speed (rad/s)

> −20 0 20

> −20 0 20

**Figure 21.** Torque profile for DTC-SVM whit STPIF

T

em (Nm)

T

I

controller.

a(A)

em(N.m)

Isabc (A)

**Figure 16.** Speed reversion for DTC-SVM with PIF controller.

**Figure 17.** Torque bahavior for speed reversion for DTC-SVM with PIF controller.

**Figure 18.** Sudden torque change for DTC-SVM whit PIF controller.

Fig. 21 presents the results when a specific torque profile is imposed to DTC-SVM scheme with STPIF controller. In this case the electromagnetic torque follow the reference.

Fig. 22 illustrates the response of the electromagnetic torque for the DTC-SVM scheme with STPIF controller, when applying step change in the motor load. In this test we used the rise time *tr* = 5.49*mS*, the settling time *ts* = 12*mS* and the integral-of-time multiplied by the absolute magnitude of the error index *ITAE* = 199.5.

**Figure 19.** Phase *a* stator current for sudden torque change for DTC-SVM with PIF controller.

**Figure 20.** Speed, torque and phase *a* stator current for speed reversion for DTC-SVM with STPIF controller.

**Figure 21.** Torque profile for DTC-SVM whit STPIF

16 Will-be-set-by-IN-TECH

Rotor Speed

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 −200

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 −50

**Figure 17.** Torque bahavior for speed reversion for DTC-SVM with PIF controller.

time (s)

Electromagnetic Torque

0.9 0.92 0.94 0.96 0.98 <sup>1</sup> 1.02 1.04 1.06 1.08 1.1 −5

Fig. 21 presents the results when a specific torque profile is imposed to DTC-SVM scheme

Fig. 22 illustrates the response of the electromagnetic torque for the DTC-SVM scheme with STPIF controller, when applying step change in the motor load. In this test we used the rise time *tr* = 5.49*mS*, the settling time *ts* = 12*mS* and the integral-of-time multiplied by the

with STPIF controller. In this case the electromagnetic torque follow the reference.

time (s)

time (s)

Electromagnetic Torque

> −40 −30 −20 −10 0 10 20 30 40 50

> > 0

**Figure 18.** Sudden torque change for DTC-SVM whit PIF controller.

absolute magnitude of the error index *ITAE* = 199.5.

5

T

em (N.m)

10

15

20

T

em (N.m)

**Figure 16.** Speed reversion for DTC-SVM with PIF controller.

ωm (rad/s)

Alfeu J. Sguarezi Filho

*on* 51(1): 5 – 14.

pp. 185 –190.

–200.

**7. References**

*CECS/UFABC, Santo André - SP, Brazil*

[1] Abu-Rub, H., Guzinski, J., Krzeminski, Z. & Toliyat, H. [2004]. Advanced control of induction motor based on load angle estimation, *Industrial Electronics, IEEE Transactions*

Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors 577

[2] Azcue P., J. L. [2010]. *Three-phase induction motor direct torque control using self-tuning pi*

[3] Azcue P., J. & Ruppert, E. [2010]. Three-phase induction motor dtc-svm scheme with self-tuning pi-type fuzzy controller, *Fuzzy Systems and Knowledge Discovery (FSKD), 2010*

[4] Bertoluzzo, M., Buja, G. & Menis, R. [2007]. A direct torque control scheme for induction motor drives using the current model flux estimation, *Diagnostics for Electric Machines, Power Electronics and Drives, 2007. SDEMPED 2007. IEEE International Symposium on*

[5] Cao, S.-G., Rees, N. & Feng, G. [1999]. Analysis and design of fuzzy control systems using dynamic fuzzy-state space models, *Fuzzy Systems, IEEE Transactions on* 7(2): 192

[6] Chen, C.-L. & Chang, M.-H. [1998]. Optimal design of fuzzy sliding-mode control: A

[7] Chen, S., Kai, T., Tsuji, M., Hamasaki, S. & Yamada, E. [2005]. Improvement of dynamic characteristic for sensorless vector-controlled induction motor system with adaptive pi mechanism, *Electrical Machines and Systems, 2005. ICEMS 2005. Proceedings of the Eighth*

[8] Depenbrock, M. [1988]. Direct self-control (dsc) of inverter-fed induction machine, *Power*

[9] Habetler, T., Profumo, F., Pastorelli, M. & Tolbert, L. [1992]. Direct torque control of induction machines using space vector modulation, *Industry Applications, IEEE*

[10] Koutsogiannis, Z., Adamidis, G. & Fyntanakis, A. [2007]. Direct torque control using space vector modulation and dynamic performance of the drive, via a fuzzy logic controller for speed regulation, *Power Electronics and Applications, 2007 European*

[11] Kovacic, Z. & Bogdan, S. [2006]. *Fuzzy Controller Design: Theory and Applications*, CRC

[12] Krause, P. C., Wasynczuk, O. & Sudhoff, S. D. [2002]. *Analysis of Electric Machinery and*

[13] Lascu, C., Boldea, I. & Blaabjerg, F. [2000]. A modified direct torque control for induction motor sensorless drive, *Industry Applications, IEEE Transactions on* 36(1): 122–130. [14] Rodriguez, J., Pontt, J., Silva, C., Kouro, S. & Miranda, H. [2004]. A novel direct torque control scheme for induction machines with space vector modulation, *Power Electronics Specialists Conference, 2004. PESC 04. 2004 IEEE 35th Annual*, Vol. 2, pp. 1392 – 1397 Vol.2.

*type fuzzy controller*, Master's thesis, University of Campinas (UNICAMP).

*Seventh International Conference on*, Vol. 2, pp. 757 –762.

comparative study, *Fuzzy Sets and Systems* 93(1): 37 – 48.

*International Conference on*, Vol. 3, pp. 1877 –1881 Vol. 3.

*Electronics, IEEE Transactions on* 3(4): 420–429.

*Transactions on* 28(5): 1045 –1053.

*Conference on* pp. 1 –10.

*Drive Systems*, IEEE Press.

Press.

URL: *http://www.sciencedirect.com/science/article/pii/S0165011496002217*

**Figure 22.** Step change in torque for DTC-SVM scheme with STPIF


**Table 5.** Induction Motor Parameters [12]

## **6. Conclusion**

In this chapter we have presented the DTC-SVM scheme that controls a three-phase IM using three different kinds of fuzzy controllers. These fuzzy controllers were used in order to determinate dynamically and on-line the load angle between stator and rotor flux vectors. Therefore, we determine the electromagnetic torque necessary to supply the motor load. We have conducted simulations with different operating conditions. Our simulation results show that the all proposed fuzzy controllers work appropriately and according to the schemes reported in the literature. However, the STPIF controller achieves a fast torque response and low torque ripple in a wide range of operating conditions such as: sudden change in the command speed and step change of the load.

## **Acknowledgements**

The authors are grateful to FAPESP and CAPES for partially financial support.

## **Author details**

José Luis Azcue and Ernesto Ruppert *School of Electrical and Computer Engineering of University of Campinas, UNICAMP, Department of Energy Control and Systems, Campinas-SP, Brazil*

576 Fuzzy Controllers – Recent Advances in Theory and Applications Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors <sup>19</sup> Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors 577

> Alfeu J. Sguarezi Filho *CECS/UFABC, Santo André - SP, Brazil*

## **7. References**

18 Will-be-set-by-IN-TECH

STPIF

reference

0.98 0.985 0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 1.03

time(s)

Rated voltage (V) 220/60Hz Rated Power (W) 2238 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*2) 0.089 P 2

In this chapter we have presented the DTC-SVM scheme that controls a three-phase IM using three different kinds of fuzzy controllers. These fuzzy controllers were used in order to determinate dynamically and on-line the load angle between stator and rotor flux vectors. Therefore, we determine the electromagnetic torque necessary to supply the motor load. We have conducted simulations with different operating conditions. Our simulation results show that the all proposed fuzzy controllers work appropriately and according to the schemes reported in the literature. However, the STPIF controller achieves a fast torque response and low torque ripple in a wide range of operating conditions such as: sudden change in the

*School of Electrical and Computer Engineering of University of Campinas, UNICAMP, Department*

The authors are grateful to FAPESP and CAPES for partially financial support.

**Table 5.** Induction Motor Parameters [12]

command speed and step change of the load.

José Luis Azcue and Ernesto Ruppert

*of Energy Control and Systems, Campinas-SP, Brazil*

**6. Conclusion**

**Acknowledgements**

**Author details**

**Figure 22.** Step change in torque for DTC-SVM scheme with STPIF

T

em (N.m)

	- [15] Takahashi, I. & Noguchi, T. [1986]. A new quick-response and high-efficiency control strategy of an induction motor, *Industry Applications, IEEE Transactions on* IA-22(5): 820 –827.
	- [16] Trzynadlowski, A. M. [1993]. *The Field Orientation Principle in Control of Induction Motors*, springer.
	- [17] Vas, P. [1998]. *Sensorless Vector and Direct Torque Control*, Oxford University Press. ISBN 0198564651.
	- [18] Viola, J., Restrepo, J., Guzman, V. & Gimenez, M. [2006]. Direct torque control of induction motors using a fuzzy inference system for reduced ripple torque and current limitation, *Power Electronics and Motion Control Conference, 2006. EPE-PEMC 2006. 12th International*, pp. 1161 –1166.
	- [19] Yamakawa, T. [1993]. A fuzzy inference engine in nonlinear analog mode and its application to a fuzzy logic control, *Neural Networks, IEEE Transactions on* 4(3): 496 –522.

20 Will-be-set-by-IN-TECH

[15] Takahashi, I. & Noguchi, T. [1986]. A new quick-response and high-efficiency control strategy of an induction motor, *Industry Applications, IEEE Transactions on* IA-22(5): 820

[16] Trzynadlowski, A. M. [1993]. *The Field Orientation Principle in Control of Induction Motors*,

[17] Vas, P. [1998]. *Sensorless Vector and Direct Torque Control*, Oxford University Press. ISBN

[18] Viola, J., Restrepo, J., Guzman, V. & Gimenez, M. [2006]. Direct torque control of induction motors using a fuzzy inference system for reduced ripple torque and current limitation, *Power Electronics and Motion Control Conference, 2006. EPE-PEMC 2006. 12th*

[19] Yamakawa, T. [1993]. A fuzzy inference engine in nonlinear analog mode and its application to a fuzzy logic control, *Neural Networks, IEEE Transactions on* 4(3): 496 –522.

–827.

springer.

0198564651.

*International*, pp. 1161 –1166.

578 Fuzzy Controllers – Recent Advances in Theory and Applications

*Edited by Sohail Iqbal, Nora Boumella and Juan Carlos Figueroa Garcia*

Fuzzy control theory is an emerging area of research. At the core of many engineering problems is the problem of control of different systems. These systems range all the way from classical inverted pendulum to auto-focusing system of a digital camera. Fuzzy control systems have demonstrated their enhanced performance in all these areas. Progress in this domain is very fast and there was critical need of a book that captures all the recent advances both in theory and in applications. Serving this purpose, this book is conceived. This book will provide you a very clear picture of current status of fuzzy control research. This book is intended for researchers, engineers, and postgraduate students specializing in fuzzy systems, control engineering, and robotics.

Fuzzy Controllers - Recent Advances in Theory and Applications

Fuzzy Controllers

Recent Advances in Theory and Applications

*Edited by Sohail Iqbal, Nora Boumella* 

*and Juan Carlos Figueroa Garcia*

Photo by Karimpard / iStock