**1. Introduction**

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

[16] Leonhard, W. [1985]. *Control of Electrical Drives*, Springer-Verlag Berlin Heidelberg New

[17] Novotny, D. W. & Lipo, T. A. [1996]. *Vector Control and Dynamics of AC Drives*, Clarendon

[19] Ryu, J. H., Lee, K. W. & Lee, J. S. [2006]. A unified flux and torque control method for dtc-based induction-motor drives, *IEEE Trans. on Power Electronics* 21(1): 234–242. [20] Shady M. Gadoue, D. G. & Finch, J. W. [2005]. Tuning of pi speed controller in dtc of induction motor based on genetic algorithms and fuzzy logic schemes, *International*

[21] Sheu, T.-T. & Chen, T.-C. [1999]. Self-tuning control of induction motor drive using neural

[22] 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

[23] Xue, Y., Xu, X., Habetler, T. G. & Divan, D. M. [1990]. A low cost stator flux oriented voltage source variable speed drive, *Conference Record of the 1990 IEEE Industrial*

[24] Yamamura, S. [1992]. *Spiral Vector Theory of AC Circuits and Machines*, Clarendon Press

[18] Phillips, C. [2000]. *Feedback Control Systems*, Pretince Hall.

network identifier, *IEEE Transactions on Energy Conversion* 14(4).

*Conference on Technology and Automation* .

*Aplications Society Annual Meetting* 1: 410–415.

York Tokyo.

–827.

OXFORD.

Press OXFORD.

The Direct Torque Control (DTC) has become a popular technique for three-phase Induction Motor (IM) drives because it provides a fast dynamic torque response without the use of current regulators [23][9], however, nowadays exist some other alternative DTC schemes to reduce the torque ripples using the Space Vector Modulation (SVM) technique [11][14]. In general the use of fuzzy systems does not require the accurate mathematic model of the process to be controlled. Instead, it uses the experience and knowledge of the involved professionals to construct its control rule base. Fuzzy logic is powerful in the motor control area, e.g., in [1] the PI and Fuzzy Logic Controllers (FLC) are used to control the load angle which simplifies the IM drive system.

In [8] the FLC is used to obtain the reference voltage vector dynamically 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 [15] the fuzzy PI speed controller has a better response for a wide range of motor speed. Different type of adaptive FLC such as self-tuning and self-organizing controllers has also been developed and implemented in[20][4].

In [18], [13] and [10] are proposed fuzzy systems which outputs are a specific voltage vector numbers, similarly to the classic DTC scheme[23]. On the other hand, in [26] is proposed a fuzzy inference system to modulate the stator voltage vector applied to the induction motor, but it consider the stator current as an additional input.

In [19] two fuzzy controllers are used to generate the two components of the reference voltage vector instead of two PI controllers, similarly, in [7] flux and torque fuzzy controllers are designed to substitute the original flux and torque PI controllers, but these schemes use two independent fuzzy controllers, one for the flux control and another one for the torque control.

©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 Azcue et al., licensee InTech. This is a paper 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.

Unlike the schemes mentioned before, the aim of this chapter is to design a Takagi-Sugeno (T-S) Fuzzy controller to substitute flux and torque PI controllers in a conventional DTC-SVM scheme. The T-S fuzzy controller calculates the quadrature components of the stator voltage vector represented in the stator flux reference frame. The rule base for the proposed controller is defined in function of the stator flux error and the electromagnetic torque error using trapezoidal and triangular membership functions. The direct component of the stator voltage takes a linear combination of the inputs as a consequent part of the rules, however, the quadrature component of the stator voltage takes the similar linear combination used in the first output but with the coefficients interchanged, not to be necessary another different coefficients values for this output.

The simulation results shown that the proposed T-S fuzzy controller for the DTC-SVM scheme have a good performance in terms of rise time (*tr*), settling time (*ts*) and torque ripple when it was tested at different operating conditions validating the proposed scheme. The chapter is organized as follows. In section 2 the direct torque control principles of the DTC for three-phase induction motor is presented. In section 3 the topology of the proposed control scheme is analyzed and in section 4 the proposed T-S fuzzy controller is described in detail mentioning different aspects of its design. Section 5 presents the simulations results of T-S fuzzy controller, and in the end, the conclusion is given in Section 6.

## **2. Direct Torque Control principles**

### **2.1. Dynamical equations of the three-phase induction motor**

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 [25]:

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

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

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

The stator-flux-oriented direct torque control (SFO-DTC) based on space vector modulation scheme have two PI controllers as is shown in Fig. 2. This control strategy relies on a simplified description of the stator voltage components expressed in stator-flux-oriented coordinates

Therefore, in this reference frame the stator flux quadrature component is zero as is shown in Fig. 1 that means *ψ<sup>s</sup>* = *ψds* and *ψqs* = 0. Also, in this reference frame the electromagnetic

*dψs*

*uds* = *Rsids* +

*Tem* <sup>=</sup> <sup>3</sup>*<sup>P</sup>*

*iqs* <sup>=</sup> <sup>2</sup> 3*P Tem ψs*

However, if the equation (11) is substitute in the equation (9), we can obtain the expression to control the electromagnetic torque applying and appropriate stator voltage quadrature

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)

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

*ψs* in the direction of the stator voltage space

*dt* (8)

<sup>2</sup> *<sup>ψ</sup>siqs* (10)

(11)

343

*uqs* = *Rsiqs* + *ωsψ<sup>s</sup>* (9)

in [3], the code source of this implementation is shared in MATLAB CENTRAL [2].

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

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

space vector.

Therefore *d*

it has:

**2.2. Direct Torque Control**

flux in the required direction.

torque is calculated by

Thus the stator flux space vector moves by Δ

*2.2.1. Stator-flux-oriented direct torque control*

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

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

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

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 is expressed in terms of the cross-vectorial product of the stator and the rotor flux space vectors.

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

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

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 [3], the code source of this implementation is shared in MATLAB CENTRAL [2].

### **2.2. Direct Torque Control**

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

Unlike the schemes mentioned before, the aim of this chapter is to design a Takagi-Sugeno (T-S) Fuzzy controller to substitute flux and torque PI controllers in a conventional DTC-SVM scheme. The T-S fuzzy controller calculates the quadrature components of the stator voltage vector represented in the stator flux reference frame. The rule base for the proposed controller is defined in function of the stator flux error and the electromagnetic torque error using trapezoidal and triangular membership functions. The direct component of the stator voltage takes a linear combination of the inputs as a consequent part of the rules, however, the quadrature component of the stator voltage takes the similar linear combination used in the first output but with the coefficients interchanged, not to be necessary another different

The simulation results shown that the proposed T-S fuzzy controller for the DTC-SVM scheme have a good performance in terms of rise time (*tr*), settling time (*ts*) and torque ripple when it was tested at different operating conditions validating the proposed scheme. The chapter is organized as follows. In section 2 the direct torque control principles of the DTC for three-phase induction motor is presented. In section 3 the topology of the proposed control scheme is analyzed and in section 4 the proposed T-S fuzzy controller is described in detail mentioning different aspects of its design. Section 5 presents the simulations results of T-S

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 is expressed in terms of the cross-vectorial product of the stator

fuzzy controller, and in the end, the conclusion is given in Section 6.

**2.1. Dynamical equations of the three-phase induction motor**

*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.

coefficients values for this output.

**2. Direct Torque Control principles**

mathematical form [25]:

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.

### *2.2.1. Stator-flux-oriented direct torque control*

The stator-flux-oriented direct torque control (SFO-DTC) based on space vector modulation scheme have two PI controllers as is shown in Fig. 2. This control strategy relies on a simplified description of the stator voltage components expressed in stator-flux-oriented coordinates

$$
\mu\_{ds} = R\_s i\_{ds} + \frac{d\psi\_s}{dt} \tag{8}
$$

$$
\mu\_{q\text{s}} = R\_{\text{s}} i\_{q\text{s}} + \omega\_{\text{s}} \psi\_{\text{s}} \tag{9}
$$

Therefore, in this reference frame the stator flux quadrature component is zero as is shown in Fig. 1 that means *ψ<sup>s</sup>* = *ψds* and *ψqs* = 0. Also, in this reference frame the electromagnetic torque is calculated by

$$T\_{em} = \frac{\Im P}{2} \psi\_s i\_{qs} \tag{10}$$

$$i\_{\mathfrak{g}s} = \frac{2}{3P} \frac{T\_{\mathfrak{em}}}{\psi\_{\mathfrak{s}}} \tag{11}$$

However, if the equation (11) is substitute in the equation (9), we can obtain the expression to control the electromagnetic torque applying and appropriate stator voltage quadrature

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

component, it is:

$$
\mu\_{qs} = \frac{2}{3P} R\_s \frac{T\_{em}}{\psi\_s} + \omega\_s \psi\_s \tag{12}
$$

*ψ*∗ *s*

+

*T*∗

−

*Eψ<sup>s</sup>*

*em* <sup>+</sup> + +

PI

*<sup>E</sup><sup>τ</sup> <sup>u</sup>*<sup>∗</sup>

*ψ*ˆ*s T*ˆ *em*

implement the T-S fuzzy controller proposed.

presented in the next section.

in [5]:

**3.1. Stator voltage calculation**

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

**3.2. Space vector modulation technique**

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

*ωsψs*

*qs*

*dq*

(*θψs*)

*us*

The SFO-DTC based on space vector modulation scheme requires the flux and the torque estimators, which can be performed as it is proposed in this chapter, this scheme is used to

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

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

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

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

√ 3

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

*Udc* (16)

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

*is*

*αβ*

SVM

Voltage Flux angle calculation *αβ*

*ABC*

PWM signals

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

*Udc*

*ia*

*ib*

*ψds ψqs*

Flux and torque estimator

**Figure 2.** Conventional stator-flux-oriented direct torque control scheme.

*u*∗ *ds*

−

PI

Im

*Udc*

345

From equation (8), the stator flux is controlled with the stator voltage direct component *uds*. For every sampled period *Ts*, the equation (8) is approximated by

$$
\mu\_{\rm ds} = R\_{\rm s} i\_{\rm ds} + \Delta \psi\_{\rm s} / T\_{\rm s} \tag{13}
$$

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 error is used the PI controller, it is:

$$
\mu\_{ds}^{\*} = (K\_{P\psi} + K\_{I\psi}/s)(\psi\_s^{\*} - \hat{\psi}\_s) \tag{14}
$$

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 used to control the electromagnetic torque, it is:

$$
\mu\_{qs}^\* = (K\_{PT\_{em}} + K\_{IT\_{em}}/s)(T\_{em}^\* - \hat{T}\_{em}) + \omega\_s \psi\_s \tag{15}
$$

Finally, the outputs of the PI flux and PI torque controllers can be interpreted as the stator voltage components in the stator-flux-oriented coordinates [6].

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 values *u*∗ *ds*, *u*<sup>∗</sup> *qs* are delivered to SVM. In [3] this scheme is analyzed in detail.

**Figure 2.** Conventional stator-flux-oriented direct torque control scheme.

The SFO-DTC based on space vector modulation scheme requires the flux and the torque estimators, which can be performed as it is proposed in this chapter, this scheme is used to implement the T-S fuzzy controller proposed.
