The Fuzzy Logic-Based Stator-Flux-Oriented Direct Torque Control for Three-Phase Asynchronous Motor

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Abstract

This article proposes a Takagi–Sugeno fuzzy logic-based (TS-FLB) controller applied to the stator-flux-oriented direct torque control (SFO-DTC) scheme. The conventional SFO-DTC scheme has two PI regulators to generate the reference stator voltage vector necessary to maintain the load. The objective is to reduce the quantity of these controllers, hence we propose a single TS-FLB controller to substitute both PI regulators. The rule base for the proposed TS-FLB controller is defined in function of the stator flux error and the electromagnetic torque error, and it utilizes trapezoidal and triangular membership functions (MFs) for the fuzzification of its inputs. Constant switching frequency and low torque ripple are obtained using the space vector modulation technique. The simulation and experimental results show that the TS-FLB controller obtains a decoupling torque and stator flux control and low torque ripple. These results were obtained for various operating conditions such as step change in the motor load, speed reversion with no load and the application of an arbitrary speed profile. These results show that the TS-FLB controller achieves a good performance as expected, validating the proposed scheme.

Introduction

In the last years the direct torque control (DTC) has become a popular technique for three-phase asynchronous motors (AM) drives. It provides a fast dynamic torque response without the use of current regulators (Takahashi and Noguchi 1986; Depenbrock 1988). Nowadays exist some other alternative DTC schemes to reduce the torque ripples using the space vector modulation (SVM) technique (Habetler et al. 1992; Kang and Sul 1999). Actually, the aim objective of DTC research is to improve its performance or to propose new schemes to control the AMs (Shyu et al. 2010; Zhang et al. 2010; Zaid et al. 2010; Metidji et al. 2012).

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 tool in the motor control area, e.g., in Abu-Rub et al. (2004), the PI and fuzzy logic controllers (FLC) are used to control the load angle which simplifies the IM drive system.

In Chen et al. (2005), 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 Koutsogiannis et al. (2007), 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 Maeda and Murakami (1992), He et al. (1993), Park et al. (1995), and Azcue and Ruppert (2010).

In Lin and Xu (2010), Jiang et al. (2008), and Ding et al. (2007), fuzzy systems were proposed in which outputs are a specific voltage vector numbers, similarly to the classic DTC scheme (Takahashi and Noguchi 1986). On the other hand, in Viola et al. (2006), a fuzzy inference system was proposed to modulate the stator voltage vector applied to the induction motor, but it considers the stator current as an additional input.

In Pan and Zhang (2009), two fuzzy controllers are used to generate the two components of the reference voltage vector instead of two PI controllers, similarly, in Cao et al. (2009), 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.

Unlike the schemes mentioned before, in this article it is designed a fuzzy logic-based (FLB) controller and this is the Sugeno type (Takagi and Sugeno 1985), and from now we call this FLC as Takagi–Sugeno FLB (TS-FLB) controller.

The TS-FLB controller is designed to substitute the stator flux and torque PI regulators present in conventional stator-flux-oriented direct torque control (SFO-DTC) scheme. The TS-FLB controller calculates the direct and 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 electromagnetic torque error. The MFs used for the fuzzification of the TS-FLB controller inputs have trapezoidal and triangular shapes because these functions are suitable for real-time operations (Dubois and Yager 1993).

The proposed TS-FLB controller has two different sets of rules. The first set of rules has nine rules; each rule has two coefficients in the consequence. Similarly, the second set of rules has nine rules too; each rule has the same coefficients but with the order interchanged, being not necessary other different coefficients values. It is the major contribution of this article, because only two coefficients for both outputs in each rule are needed, reducing the computational cost of its implementation. The simulation and experimental results show that the proposed TS-FLB controller for the SFO-DTC scheme has a good performance when it was tested under various operating conditions, validating the proposed scheme.

This article is organized as follows. In Sect. 2, we give the theoretical background for the dynamical equations of the three-phase AM and the DTC principles. In Sect. 3, the topology of the proposed control scheme is analyzed and in Sect. 4, the proposed TS-FLB controller is described in details mentioning different aspects of its design. Section 5 presents the simulations and experimental results of the TS-FLB controller, and in the end, the conclusion is given in Sect. 6.

Theoretical Backgrounds

Dynamical Equations of the Three-Phase Asynchronous Motor

By using the definitions of the flux, current, and voltage space vector (SV), the dynamical equations of the three-phase AM in stationary reference frame can be put into the following mathematical form (Vas 1998):

$$\begin{aligned}&\vec {u}_{\text {s}}=R_{\text {s}} \vec {i}_{\text {s}} + \frac{\text {d} \vec {\psi }_{\text {s}}}{\text {d}t} \end{aligned}$$
(1)
$$\begin{aligned}&0=R_{\text {r}} \vec {i}_{\text {r}}+\frac{\text {d}\vec {\psi }_{\text {r}}}{\text {d}t}-j\omega _{\text {r}}\vec {\psi }_{\text {r}} \end{aligned}$$
(2)
$$\begin{aligned}&\vec {\psi }_{\text {s}}=L_{\text {s}}\vec {i}_{\text {s}}+L_{\text {m}} \vec {i}_{\text {r}} \end{aligned}$$
(3)
$$\begin{aligned}&\vec {\psi }_{\text {r}}=L_{\text {r}} \vec {i}_{\text {r}}+L_{\text {m}} \vec {i}_{\text {s}} \end{aligned}$$
(4)

where \(\vec {u}_{\text {s}}\) is the stator voltage SV, \(\vec {i}_{\text {s}}\) and \(\vec {i}_{\text {r}}\) are the stator and rotor current SVs, respectively, \(\vec {\psi }_{\text {s}}\) and \(\vec {\psi }_{\text {r}}\) are the stator and rotor flux SVs, respectively, \(\omega _{\text {r}}\) is the rotor angular speed, \(R_{\text {s}}\) and \(R_{\text {r}}\) are the stator and rotor resistances, respectively, \(L_{\text {s}}, L_{\text {r}}\), and \(L_{\text {m}}\) are the stator, rotor, and mutual inductance, respectively.

The electromagnetic torque is represented by

$$\begin{aligned} t_{\text {em}}&= \frac{3}{2}P\frac{L_{\text {m}}}{L_{\text {r}} L_{\text {s}} \sigma } \vec {\psi }_{\text {r}}\times \vec {\psi }_{\text {s}} \end{aligned}$$
(5)
$$\begin{aligned} t_{\text {em}}&= \frac{3}{2}P\frac{L_{\text {m}}}{L_{\text {r}} L_{\text {s}} \sigma } \left| \vec {\psi }_{\text {r}}\right| \left| \vec {\psi }_{\text {s}}\right| \sin (\rho ) \end{aligned}$$
(6)
$$\begin{aligned} t_{\text {em}}&= \frac{3P}{2} \vec {\psi }_{\text {s}}\times \vec {i}_{\text {s}} \end{aligned}$$
(7)

where \(\rho \) is the load angle between stator and rotor flux SVs, \(P\) is a number of pole pairs, and \(\sigma =1-L^{2}_{\text {m}}/(L_{\text {s}} L_{\text {r}})\) is the dispersion factor.

Direct Torque Control

If the sample time is short enough, such that the stator voltage SV is imposed to the motor keeping the stator flux constant at the reference value. The rotor flux SV will become constant because it changes slower than the stator flux SV. The electromagnetic torque represented in Eq. (6) can be quickly changed by changing the angle \(\rho \) in the desired direction.

The angle \(\rho \) can be easily changed when choosing the appropriate stator voltage SV. For simplicity, let us assume that the stator phase ohmic drop could be neglected in \(\vec {u}_{\text {s}}=R_{\text {s}} \vec {i}_{\text {s}} + \frac{\text {d} \vec {\psi }_{\text {s}}}{\text {d}t}\). Therefore, \(\text {d} \vec {\psi }_{\text {s}} /\text {d}t=\vec {u}_{\text {s}}\). During a short time \(\Delta t\), when the stator voltage SV is applied we have:

$$\begin{aligned} \Delta \vec {\psi }_{\text {s}}&\approx \vec {u}_{\text {s}}\cdot \Delta t \end{aligned}$$
(8)

Thus, the stator flux SV moves by \(\Delta \vec {\psi }_{\text {s}}\) in the direction of the stator voltage SV at a speed which is proportional to the magnitude of the stator voltage SV. By selecting step-by-step the appropriate stator voltage SV, it is possible to change the stator flux SV in the required direction.

Stator-Flux-Oriented Direct Torque Control

The stator-flux-oriented DTC (SFO-DTC) has two PI regulators. The outputs of the flux and torque PI regulators can be interpreted as the stator voltage components in the stator-flux-oriented coordinates as is shown in Fig. 1 (Xue et al. 1990; Buja and Kazmierkowski 2004). The control strategy relies on a simplified description of the stator voltage components, expressed in stator-flux-oriented coordinates as:

$$\begin{aligned} u_{\text {ds}}&= R_{\text {s}}i_{\text {ds}}+\frac{\text {d}\psi _{\text {s}}}{\text {d}t} \end{aligned}$$
(9)
$$\begin{aligned} u_{\text {qs}}&= R_{\text {s}}i_{\text {qs}}+\omega _{\text {s}}\psi _{\text {s}}=k_st_{\text {em}}+\omega _{\text {s}}\psi _{\text {s}} \end{aligned}$$
(10)

where \(k_{\text {s}}=R_{\text {s}}/\psi _{\text {s}}\) and \(\omega _{\text {s}}\) is the angular speed of the stator flux SV. The above equations show that the component \(u_{\text {ds}}\) has influence only on the change of stator flux magnitude, and the component \(u_{\text {qs}}\), if the term \(\omega _{\text {s}}\psi _{\text {s}}\) is decoupled, can be used for torque adjustment. Therefore, after coordinate transformation \(\text {d}q/\alpha \beta \) into the stationary frame, the command values \(u^{*}_{\text {ds}},u^{*}_{\text {qs}}\), are delivered to SVM. In Azcue and Ruppert (2010), this scheme is analyzed in detail.

This SFO-DTC scheme requires the stator flux and electromagnetic torque estimators, which can be performed as it is proposed in Fig. 1.

Fig. 1
figure1

Stator-flux-oriented direct torque control scheme

The Proposed SFO-DTC Scheme

Figure 2 shows the proposed SFO-DTC scheme, this scheme only needs sense the DC link voltage and the two phases of the stator currents of the three-phase AM. In this scheme the TS-FLB controller has the electromagnetic torque error (\(E_{\tau }\)) and the stator flux error (\(E_{\psi _{\text {s}}}\)) as inputs, and the stator voltage components as outputs. These outputs are represented in the stator-flux-oriented reference frame. Details about the TS-FLB controller are going to be presented in the next section. In this section it is described the calculation of the stator voltage vector and the estimation of electromagnetic torque and the stator flux SV.

Fig. 2
figure2

Takagi–Sugeno fuzzy logic controller in the SFO-DTC scheme

Stator Voltage Calculation

The stator voltage calculation use the DC link voltage \(\mathbf{U}_{\mathbf{dc}}\) and the inverter switch state (\(S_{\text {a}}, S_{\text {b}}, S_{\text {c}}\)) of the three leg of the two level inverter. The stator voltage SV \(\vec {u}_{\text {s}}\) is determined by (Bertoluzzo et al. 2007):

$$\begin{aligned} \vec {u}_{\text {s}}=\frac{2}{3}\left[ \left( S_{\text {a}}-\frac{S_{\text {b}}+S_{\text {c}}}{2}\right) +j\frac{\sqrt{3}}{2}(S_{\text {b}}-S_{\text {c}})\right] U_{\text {dc}} \end{aligned}$$
(11)

Electromagnetic Torque and Stator Flux Estimation

The stator flux SV estimation depends on the back electromotive force (emf):

$$\begin{aligned} \vec {\psi }_{\text {s}}&= \int (\vec {u}_{\text {s}} - R_{\text {s}}\cdot \vec {i}_{\text {s}}){\text {d}}t \nonumber \\ \vec {\psi }_{\text {s}}&= \int (\vec {\text {emf}}) {\text {d}}t \end{aligned}$$
(12)

when the stator flux is calculated with Eq. (12) it has problems associated with a pure integrator. With the aim to solve this problem the integrator with an adaptive compensation method proposed in Hu and Wu (1998) is used. This method can be used to accurately estimate the motor flux including its magnitude and phase angle over a wide speed range.

Figure 3 shows a block diagram of this method. The main idea of this method is the fact that the motor stator flux SV is orthogonal to its back emf. The quadrature detector detect if the orthogonality between the estimated stator flux SV and bemf is maintained.

Fig. 3
figure3

Block diagram of the adaptive compensation method

The operating principle of this method is explained by using a vector diagram as shown in Fig. 4. The estimated stator flux SV is a sum of two vectors, a feedforward vector \(\vec {\psi }_1\) which is the output of the low pass (LP) filters (\(\psi _{\text {d} 1}\) and \(\psi _{{\text{ q }} 1}\)) and a feedback vector \(\vec {\psi }_2\) which is composed of \(\psi _{\text {d} 2}\) and \(\psi _{{\text{ q }} 2}\). Ideally, the stator flux SV \(\vec {\psi }_{\text {s}}\) should be orthogonal to the \(\vec {\text {emf}}\), and the output of the quadrature detector is zero. When an initial value or dc drift is introduced to the integrator, the above orthogonal relation is lost, and the phase angle between the flux and emf vectors is no longer 90°, which yields an error signal defined by

$$\begin{aligned} \Delta \vec {e}&= \vec {\psi }_{\text {s}}\cdot \vec {\text {emf}}/\left| \vec {\psi }_{\text {s}}\right| =(\psi _{\text {qs}}\cdot \text {emf}_{\text {q}}+\psi _{\text {ds}}\cdot \text {emf}_{\text {d}})/\left| \vec {\psi }_{\text {s}}\right| \nonumber \\ \Delta \vec {e}&= \left| \text {emf}\right| \cos (\gamma ) \end{aligned}$$
(13)

Assuming that the magnitude of the feedback vector \(\vec {\psi }_2\) is increased to \(\vec {\psi }^{'}_2\) as shown in Fig. 4 due to a dc offset or initial value problem, the phase angle \(\gamma \) will be greater that 90°. The quadrature detector will generate a negative error signal. The output of the PI regulator \(\psi _{\text {cmp}}\) is reduced and so is the feedback vector. As a result, the stator flux SV \(\vec {\psi }^{'}_{\text {s}}\) moves toward the original position of 90° until the orthogonal relationship between \(\vec {\psi }_{\text {s}}\) and \(\vec {\text {emf}}\) is reestablished. If \(\gamma \) is less than 90° for some reason, an opposite process will occur, which brings \(\gamma \) back to 90°. Therefore, the modified integrator with the adaptive control can adjust the flux compensation level \(\psi _{\text {cmp}}\) automatically to an optimal value such that the initial value and dc drift problems are essentially eliminated.

Fig. 4
figure4

Vector diagram showing the emf and flux linkage relationship

The stator flux SV angle \(\theta _{\psi _{\text {s}}}\) necessary to orient the system is obtained by

$$\begin{aligned} \theta _{\psi _{\text {s}}}=\tan ^{-1}\left( \frac{\psi _{\text {qs}}}{\psi _{\text {ds}}}\right) \end{aligned}$$
(14)

with the stator flux obtained with the adaptive compensation method and with the measured stator current it is possible to estimate the electromagnetic torque through the Eq. (7).

Design of Takagi–Sugeno Fuzzy Logic-Based Controller

The Takagi–Sugeno fuzzy logic controller is based on a suitable set of fuzzy rules, carried out from both the knowledge of the experimental behavior and the internal structure of the controlled system. In order to design a TS-FLB controller, the following steps must be performed:

  1. (1)

    development of a suitable rule set;

  2. (2)

    selection of input/output variables and their quantization in fuzzy sets;

  3. (3)

    definition of MFs to be associated with the input variables; and

  4. (4)

    selection of the defuzzification method.

The TS-FLB controller has as inputs the stator flux error \(E_{\psi _{\text {s}}}\) and the electromagnetic torque error \(E_{\tau }\), and it has as outputs the direct and the quadrature components of the stator voltage SV. These components are represented in stator-flux-oriented reference frame.

The TS-FLB controller has two different sets of rules. The first set of rules, for the output (\(u^{*}_{\text {ds}}\)), have a linear combination of the inputs as a consequent part of the rules. Similarly, the second set of rules, for the output (\(u^{*}_{\text {qs}}\)), have the similar linear combination used in the first set of rules but with the interchanged coefficients order as shown in Fig. 5.

Fig. 5
figure5

The structure of TS-FLB controller

Membership Functions

The MF for TS-FLB controller are shown in Figs. 6 and 7 for the stator flux error and the electromagnetic torque error, respectively. These MFs are used in fuzzification process to convert numerical variables into fuzzy variables. The shape and the universe of discourse were adjusted through the simulation process using trial and error method. 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 Fig. 6. 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 Fig. 7, 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.

Fig. 6
figure6

Membership function for stator flux error input

Fig. 7
figure7

Membership function for electromagnetic torque error input

The Fuzzy Rule Base

A set of rules for the direct component of the stator voltage \(u^{*}_{\text {ds}}\) are defined by rules of the following form:

$$\begin{aligned}&R^{1}_{V_{ds}}:\quad \hbox { if } E_{\psi _s} \hbox { is N} \hbox { and } E_{\tau } \hbox { is N} \\&\quad \quad \qquad \,\, \hbox { then } V^{R_1}_{ds}=a\cdot E_{\psi _s} + b\cdot E_{\tau } \nonumber \end{aligned}$$
(15)

However, the set of rules for the quadrature component of the stator voltage \(u^{*}_{\text {qs}}\) are defined by rules of the following form:

$$\begin{aligned}&R^{1}_{V_{qs}}:\quad \hbox { if } E_{\psi _s} \hbox { is N} \hbox { and } E_{\tau } \hbox { is N} \\&\quad \quad \qquad \,\, \hbox { then } V^{R_1}_{qs}=-b\cdot E_{\psi _s} + a\cdot E_{\tau } \end{aligned}$$

The a and b constants are the coefficients of the first-order polynomial function typically present in the consequent part of the firs-order Takagi–Sugeno fuzzy controllers. Observe that these coefficients for \(V^{{\text {R}}_1}_{\text {ds}}\) and \(V^{{\text {R}}_1}_{\text {qs}}\) are the same but with interchanged order, being not necessary other different coefficients for each one, as they are repeated for all the rules. The complete rule base to calculate \(u^{*}_{\text {ds}}\) and \(u^{*}_{\text {qs}}\) is shown in Table 1, where (\(a=5\); \(b=0.1\)), (\(c=6.5\); \(d=0.2\)), (\(e=8\); \(f=0.1\)).

Table 1 Rule base table for computation of \(u^{*}_{\text {ds}}\) and \(u^{*}_{\text {qs}}\)

Inference Method

In general, operators on fuzzy sets use triangular norms, which may be divided into T-norms (AND operators) and S-norms (OR operators) (Gupta and Qi 1991a, b). T-norms perform an intersection operation on fuzzy sets and have a particular importance in fuzzy logic control. T-norm is usually denoted as T (a,b). The T-norms used in the proposed TS-FLB controller is defined as:

$$\begin{aligned}&\mu ^{{\text {R}}_i}=T(\mu ^{i}_{{\text {E}}_{\psi _{\text {s}}}},\mu ^{i}_{{\text {E}}_{\tau }}) = \mu ^{i}_{{\text {E}}_{\psi _{\text {s}}}} \cdot \mu ^{i}_{{\text {E}}_{\tau }}\nonumber \\&\text {for}\quad i=1,\ldots ,n; \,\, n=9 \end{aligned}$$
(16)

where \(\mu ^{i}_{{\text {E}}_{\psi _{\text {s}}}} \) and \(\mu ^{i}_{{\text {E}}_{\tau }}\) are MFs degrees of the first and second TS-FLB controller inputs, respectively, and \(\mu ^{{\text {R}}_i}\) is the truth value of the preposition.

Aggregation

The final output value \(u^{*}_{\text {ds}}\) inferred from \(n=9\) implications is aggregated as the average of all \(V^{{\text {R}}_i}_{\text {ds}}\) with the weights \(\mu ^{{\text {R}}_i}\):

$$\begin{aligned} u^{*}_{\text {ds}}=\frac{\sum ^{n}_{i=1}\mu ^{{\text {R}}_i}V^{{\text {R}}_i}_{\text {ds}}}{\sum ^{n}_{i=1}V^{{\text {R}}_i}_{\text {ds}}} \end{aligned}$$
(17)

and the final output value \(u^{*}_{\text {qs}}\) inferred from \(n=9\) implications is aggregated as the average of all \(V^{{\text {R}}_i}_{\text {qs}}\) with the weights \(\mu ^{{\text {R}}_i}\):

$$\begin{aligned} u^{*}_{\text {qs}}=\frac{\sum ^{n}_{i=1}\mu ^{{\text {R}}_i}V^{{\text {R}}_i}_{\text {qs}}}{\sum ^{n}_{i=1}V^{{\text {R}}_i}_{\text {qs}}} \end{aligned}$$
(18)

In TS-FLB controller, that is Sugeno type, it is not necessary the defuzzification interface (Driankov et al. 1996; Sandri and Correa 1999), as its common use in Mamdani’s (Mamdani 1974) type controllers. This is so because, in Takagi–Sugeno fuzzy controllers, each rule is already crisp and the total result is determined by the weighted sum of each rule, as shown in Eqs. (17) and (18). The TS-FLB controller was programmed in C programming language for the simulation because it facilitates its implementation in the digital signal processor TMS320F28335 from the Texas Instrument.

Simulation and Experimental Results

The simulations activities were performed using the MATLAB R2011b simulation package together with the Simulink block sets and fuzzy logic toolbox. The switching frequency of the three-phase two level inverter was set to be 10 kHz, the reference stator flux considered was set to be the rated value 0.82 Wb.

The experimental activities were realized with electronic circuits and three-phase AM. Figure 8 shows the experimental set-up built during the development stage.

Fig. 8
figure8

Experimental set-up showing the induction motor, the Foucault break, and the driven set

The experimental set-up consists of a DSP (Texas Instruments TMS320F28335) connected to a three-phase squirrel cage AM, driven by a 12 kVA Semikron three-phase inverter. A Foucault braking system imposes the mechanical load. Conditioning signal boards are necessary to acquire the motor currents and the Dc link voltage from a high level of voltage and current to an appropriate voltage level to be sampled and converted by the internal AD converter.

Initially, with the objective to calculate the processing time was implemented two cases. In the first case the fuzzy controller has only two coefficient for calculate the two partial outputs of each rule (for example \(R^{1}_{V_{\text {ds}}}, R^{1}_{V_{\text {qs}}}\)) and in the second case the fuzzy controller has four coefficients for calculate the two partial outputs of each rule (for example \(R^{1}_{V_{\text {ds}}}, R^{1}_{V_{\text {qs}}}\)). The first case include the proposed TS-FLB controller. In Table 2 is shown the number of coefficients to calculate the two partial outputs in each rule, the total number of coefficients, the processing time necessary to execute the control algorithm, the switching frequency, and the switching period of the inverter. The switching period is the maximum available time to execute the full control algorithm. The algorithm with the proposed TS-FLB controller is executed 6.5 % faster than case two and with 50 % minus coefficients.

Table 2 Computational cost and quantity of coefficients

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 motor load (from 0 to 1.0 pu) at 50 % of rated speed, step change in speed reference with no load (from 0.5 to \(-0.5\) pu), and the application of an arbitrary speed profile. The induction motor parameters and its rated values are given in Table 3.

Table 3 Rated values and parameters of motor

Figure 9a, b shows the simulation and experimental results of the rotor angular speed \(\omega _{\text {r}}\), electromagnetic torque, and phase a stator current when a step change in motor load it is applied (from 0 to 1.0 pu) with 50 % of rated speed. The electromagnetic torque and stator current are increased to maintain the motor load. Also, it is possible to observe a small undershoot and overshoot of the speed when the load is applied and retreated. This behavior is as expected.

Fig. 9
figure9

Simulation (a) and experimental (b) results of \(\omega _{\text {r}}, t_{\text {em}}\), and \(i_{\text {as}}\) stator current when the step change in the motor load is applied with 50 % of rated speed (C2: 0.5 pu/div; C3: 5 Nm/div; C4: 10 A/div; 5 s/div)

Figure 10a, b shows the simulation and experimental results of the rotor angular speed \(\omega _{\text {r}}\), electromagnetic torque, and phase a stator current waveform when a step change in speed reference in no load conditions it is applied (from 0.5 to \(-0.5\) pu). In these figures are observed that the rotor angular speed reaches the speed reference at approximately 1.2 s. The sinusoidal waveform of the stator current shows that this control technique has a good current control; because it is inherent to the algorithm control proposed in this article.

Fig. 10
figure10

Simulation (a) and experimental (b) results of \(\omega _{\text {r}}, t_{\text {em}}\), and \(i_{\text {as}}\) stator current when a step change in speed reference is applied with no load (C1: 10 Nm/div; C2, C3: 0.5 pu/div; C4: 10 A/div; 5 s/div)

Figure 11a, b shows the simulation and experimental results of the stator flux, electromagnetic torque, and phase a stator current when a step change in motor load it is applied (from 0 to 1.0 pu). In this test is possible to observe the decoupled behavior of electromagnetic torque and stator flux. It means that if a load is applied, the stator flux is maintained constant in its rated value.

Fig. 11
figure11

Simulation (a) and Experimental (b) results of \(|\vec {\psi }_{\text {s}}|, t_{\text {em}}\), and \(i_{\text {as}}\) stator current when the step change in the motor load is applied with 30 % of rated speed (C1: 0.5 pu/div; C2: 10 Nm/div; C3: 0.5 Wb/div; C4: 10 A/div; 5 s/div)

Figure 12a, b shows the simulation and experimental results of the rotor angular speed and phase a stator current when speed profile is applied. It is the rotor angular speed tracks the reference as is expected. Also, the stator current decrease when the motor needs to work a minor speed. All the test results showed the good performance of the proposed SFO-DTC scheme with TS-FLB controller.

Fig. 12
figure12

Simulation (a) and Experimental (b) results of \(\omega _{\text {r}}\) and \(i_{\text {as}}\) stator current when an arbitrary speed profile is applied (C1, C2: 0.1 pu/div; C4: 5 A/div; 5 s/div)

Conclusion

In this article, we presented the SFO-DTC scheme with TS-FLB controller for the three-phase AM. The conventional SFO-DTC scheme has two PI regulators to generate the reference stator voltage space vector. The objective is to reduce the quantity of these controllers, hence we proposed a single TS-FLB controller to substitutes both PI regulators. The rule base for the TS-FLB controller was defined in function of the stator flux error and the electromagnetic torque error. The TS-FLB controller has two different sets of rules. The first set of rules has a linear combination of the TS-FLB controller inputs as a consequent part of the rules. Similarly, the second set of rules has the similar linear combination but with the interchanged coefficients being it not necessary other different coefficients. Constant switching frequency and low torque ripple were obtained using SVM technique. The numerical simulations and experimental results were obtained for various operating conditions such as step change in the motor load, speed reversion with no load, and the application of an arbitrary speed profile. These results show that the TS-FLB controller achieves a decoupling torque and stator flux control and low torque ripple with a good performance as expected, validating the proposed scheme.

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Acknowledgments

The authors are grateful to CAPES and FAPESP for the financial support for this research.

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Correspondence to José L. Azcue-Puma.

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Azcue-Puma, J.L., Sguarezi Filho, A.J. & Ruppert, E. The Fuzzy Logic-Based Stator-Flux-Oriented Direct Torque Control for Three-Phase Asynchronous Motor. J Control Autom Electr Syst 25, 46–54 (2014). https://doi.org/10.1007/s40313-013-0091-5

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Keywords

  • Direct torque control
  • Fuzzy controller
  • Three-phase asynchronous motor