Volume 16 Number 2
April 2019
Article Contents
Saber Krim, Souflen Gdaim, Abdellatif Mtibaa and Mohamed Faouzi Mimouni. Contribution of the FPGAs for Complex Control Algorithms: Sensorless DTFC with an EKF of an Induction Motor. International Journal of Automation and Computing, vol. 16, no. 2, pp. 226-237, 2019. doi: 10.1007/s11633-016-1017-z
Cite as: Saber Krim, Souflen Gdaim, Abdellatif Mtibaa and Mohamed Faouzi Mimouni. Contribution of the FPGAs for Complex Control Algorithms: Sensorless DTFC with an EKF of an Induction Motor. International Journal of Automation and Computing, vol. 16, no. 2, pp. 226-237, 2019. doi: 10.1007/s11633-016-1017-z

Contribution of the FPGAs for Complex Control Algorithms: Sensorless DTFC with an EKF of an Induction Motor

Author Biography:
  • Soufien Gdaim received the electrical engineer degree from National School of Engineering of Sfax, Tunisia in 1998. In 2007, he received the M. Sc. degree in electronics and real-time informatics from Sousse University, and received the Ph. D. degree in electrical engineering in 2013 from ENIM, Tunisia.
    His research interests include rapid prototyping and reconfigurable architecture for real-time control applications of electrical system.
    E-mail: sgdaim@yahoo.fr

    Abdellatif Mtibaa holds a diploma in electrical engineering in 1985 and received the Ph. D. degree in electrical engineering in 2000. He is currently a professor in micro-electronics and hardware design with Electrical Department at the National School of Engineering of Monastir and head of Circuits Systems Reconfigurable ENIM-Group at Electronics and Microelectronics Laboratory. He has authored/coauthored over 150 papers in international journals and conferences. He served on the technical program committees for several international conferences. He also served as a co-organizer of several international conferences.
    His research interests include system on programmable chip, high level synthesis, rapid prototyping and reconfigurable architecture for real-time multimedia applications.
    E-mail: abdellatif.mtibaa@enim.rnu.tn

    Mohamed Faouzi Mimouni received the DEA degree of science from ENSET, Tunisia in 1986. In 1997, he obtained his doctorate degree in electrical engineering from ENSET, Tunisia. He is currently a full professor of electrical engineering with Electrical Department at the National School of Engineering of Monastir. He has authored/coauthored over 100 papers in international journals and conferences. He served on the technical program committees for several international conferences.
    His research interests include power electronics, motor drives, solar and wind power generation.
    E-mail: mfaouzi.mimouni@enim.rnu.tn

  • Corresponding author: Saber Krim received the electrical engineer degree from National School of Engineering of Monastir, Tunisia in 2011. In 2013 he received his M. Sc. degree in electrical engineering from Monastir University, Tunisia. He is currently a Ph. D. candidate with University of Monastir, Tunisia.
    His research interests include rapid prototyping and reconfigurable architecture for real-time control applications of electrical system.
    E-mail: krimsaber@hotmail.fr (Corresponding author)
    ORCID iD: 0000-0002-9294-5698
  • Received: 2015-04-17
  • Accepted: 2015-10-26
  • Published Online: 2017-10-27
  • In a conventional direct torque control (CDTC) of the induction motor drive, the electromagnetic torque and the stator flux are characterized by high ripples. In order to reduce the undesired ripples, several methods are used in the literature. Nevertheless, these methods increase the algorithm complexity and dependency on the machine parameters such as the space vector modulation (SVM). The fuzzy logic control method is utilized in this work to decrease these ripples. Moreover, to eliminate the mechanical sensor the extended kalman filter (EKF) is used, in order to reduce the cost of the system and the rate of maintenance. Furthermore, in the domain of controlling the real-time induction motor drives, two principal digital devices are used such as the hardware (FPGA) and the digital signal processing (DSP). The latter is a software solution featured by a sequential processing that increases the execution time. However, the FPGA is featured by a high processing speed because of its parallel processing. Therefore, using the FPGA it is possible to implement complex algorithms with low execution time and to enhance the control bandwidth. The large bandwidth is the key issue to increase the system performances. This paper presents the interest of utilizing the FPGAs to implement complex control algorithms of electrical systems in real time. The suggested sensorless direct torque control using the fuzzy logic (DTFC) of an induction motor is successfully designed and implemented on an FPGA Virtex 5 using xilinx system generator. The simulation and implementation results show proposed approach's performances in terms of ripples, stator current harmonic waves, execution time, and short design time.
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Contribution of the FPGAs for Complex Control Algorithms: Sensorless DTFC with an EKF of an Induction Motor

  • Corresponding author: Saber Krim received the electrical engineer degree from National School of Engineering of Monastir, Tunisia in 2011. In 2013 he received his M. Sc. degree in electrical engineering from Monastir University, Tunisia. He is currently a Ph. D. candidate with University of Monastir, Tunisia.
    His research interests include rapid prototyping and reconfigurable architecture for real-time control applications of electrical system.
    E-mail: krimsaber@hotmail.fr (Corresponding author)
    ORCID iD: 0000-0002-9294-5698

Abstract: In a conventional direct torque control (CDTC) of the induction motor drive, the electromagnetic torque and the stator flux are characterized by high ripples. In order to reduce the undesired ripples, several methods are used in the literature. Nevertheless, these methods increase the algorithm complexity and dependency on the machine parameters such as the space vector modulation (SVM). The fuzzy logic control method is utilized in this work to decrease these ripples. Moreover, to eliminate the mechanical sensor the extended kalman filter (EKF) is used, in order to reduce the cost of the system and the rate of maintenance. Furthermore, in the domain of controlling the real-time induction motor drives, two principal digital devices are used such as the hardware (FPGA) and the digital signal processing (DSP). The latter is a software solution featured by a sequential processing that increases the execution time. However, the FPGA is featured by a high processing speed because of its parallel processing. Therefore, using the FPGA it is possible to implement complex algorithms with low execution time and to enhance the control bandwidth. The large bandwidth is the key issue to increase the system performances. This paper presents the interest of utilizing the FPGAs to implement complex control algorithms of electrical systems in real time. The suggested sensorless direct torque control using the fuzzy logic (DTFC) of an induction motor is successfully designed and implemented on an FPGA Virtex 5 using xilinx system generator. The simulation and implementation results show proposed approach's performances in terms of ripples, stator current harmonic waves, execution time, and short design time.

Saber Krim, Souflen Gdaim, Abdellatif Mtibaa and Mohamed Faouzi Mimouni. Contribution of the FPGAs for Complex Control Algorithms: Sensorless DTFC with an EKF of an Induction Motor. International Journal of Automation and Computing, vol. 16, no. 2, pp. 226-237, 2019. doi: 10.1007/s11633-016-1017-z
Citation: Saber Krim, Souflen Gdaim, Abdellatif Mtibaa and Mohamed Faouzi Mimouni. Contribution of the FPGAs for Complex Control Algorithms: Sensorless DTFC with an EKF of an Induction Motor. International Journal of Automation and Computing, vol. 16, no. 2, pp. 226-237, 2019. doi: 10.1007/s11633-016-1017-z
  • The basic concept of the direct torque control (DTC) of an induction motor is to control both electromagnetic torque and stator flux[1]. In order to control the stator flux magnitude and the electromagnetic torque, two hysteresis comparators are utilized[1, 2]. The conventional DTC (CDTC) is characterized by outstanding dynamic response and simple structure, compared with the field-oriented control (FOC), because of the absence of pulse width modulation (PWM) block and current proportional-integral (PI) regulators used in the FOC[1-5]. Yet, the hysteresis comparators and switching table utilized in the CDTC cause high ripples in the stator flux, the electromagnetic torque, the stators current harmonic waves, the variation of the switching frequency, and the acoustic noises[6-9]. In order to address the CDTC problems, many schemes are used. Most of them utilize the space vector modulation (SVM) in the DTC which is referred as DTC-SVM, to produce the optimal voltage vectors. Thus, the switching frequency is fixed, so the stator flux and the electromagnetic torque ripples are reduced[4, 10-17]. Various methods are used based on deadbeat control[15], sliding mode control[13, 18-20], PI control[14], etc. However, the DTC-SVM algorithm is more complex than the CDTC and it is sensitive to the variation of several machine parameters, like the predictive controller[21]. To improve the CDTC performances another method is based on the multilevel inverter[22-24]; nevertheless, the hardware cost goes up. In this work, an approach of an intelligent technique using the fuzzy logic (FL), is put forward in order to overcome the previous CDTC disadvantages, especially the reduction in the electromagnetic torque and the stator flux ripples, as well as the harmonic waves of the stator current. The FL is a reasoning method that uses the language rules and does not require any mathematical model. In this case, two hysteresis comparators and a switching table are replaced by an FL block. To avoid any mechanical sensor drawbacks like cost and maintenance, an extended Kalman filter (EKF) is chosen[25-30] to estimate the rotor's speed, the stator's flux, and the stator's current. The execution time of the sensorless controller based on an EKF algorithm using the software device DSP is assessed in tens or hundreds of microseconds because of its algorithm complexity. In order to overcome the software solution limitations, the FPGA is selected due to its parallel processing. Currently, the control algorithms of electrical machines are becoming more and more complex, like the direct torque FL control (DTFC), the predictive controllers[31, 32], sensorless controllers[25-29], etc. Today, most complex algorithms are implemented with software controllers like the digital signal processing and control engineering (DSPACE).

    On the other hand, in some industrial applications for controlling of induction machines, where a high control quality and a large bandwidth are required, the software solutions are not recommended due to its sequential treatment that decreases the processing speed. More the algorithm is complex; more the execution time is longer. As a consequence the calculation delays introduced in the control closed loop affects the control bandwidth. The complexity of these algorithms necessitates powerful digital devices in terms of processing speed, such as the hardware field programmable gate array (FPGA). In this paper, the hardware DTFC implementation on the FPGA with an EKF is used to get high control performances. The FPGA is utilized to overcome the DSPACE limitations in terms of execution time, which is assessed in tens or hundreds of microseconds. However, the execution time using the FPGA does not exceed 5 s. Thanks to their inherent parallel processing; the FPGAs are outperforming the software solutions. In addition, the FPGA is cheaper relative to the DSPACE, which reduces the system cost[33]. In order to implement control algorithms on the FPGAs, two principal methods are used, like the programming very-high-speed integrated circuits hardware description language (VHDL)[34-37], and the Xilinx system generator (XSG). The XSG offers the possibility to generate the VHDL code of the proposed control algorithm. In this work the XSG is chosen, thanks to its simplicity, portability and marketing rapidity. The proposed algorithm has been designed and encoded utilizing the XSG, used in several areas[38, 39]. This paper is organized as follows. In Section 2, the induction motor model and the DTC principle are presented. The FL system is described in Section 3. In Section 4, an overview of the EKF principle is presented. In Section 5, the performances of the FPGAs related to the DSPs and the XSG design flow are given and discussed. The simulation results using the XSG and the implementation results using the FPGA Virtex 5 are shown and discussed in Section 6. The conclusion is presented in Section 7.

  • The state model of the induction motor (IM) in concordia reference is expressed by following equations.

    $ \begin{align} \begin{array}{l} {\mathop{i}\limits^{\centerdot } _{S\alpha } = \dfrac{1}{\sigma } \left(\dfrac{R_{S} }{L_{S} } +\dfrac{R_{r} }{L_{r} } \right)\mathop{i}\nolimits_{S\alpha } -\omega \mathop{i}\nolimits_{S\beta } +\dfrac{R_{r} }{\sigma L_{r} L_{S} } \mathop{\varphi }\limits^{} _{S\alpha } } +\\ \qquad {\dfrac{\omega }{\sigma L_{S} } \mathop{\varphi } \limits^{} _{S\beta } +\dfrac{1}{\sigma L_{S} } \mathop{v}\nolimits_{S\alpha } } \end{array} \end{align} $

    (1)

    $ \begin{align} \begin{array}{l} {\mathop{i}\limits^{\centerdot } _{S\alpha } = \dfrac{1}{\sigma } \left(\dfrac{R_{S} }{L_{S} } +\dfrac{R_{r} }{L_{r} } \right)\mathop{i}\nolimits_{S\beta } -\omega \mathop{i}\nolimits_{S\alpha } -\dfrac{\omega }{\sigma L_{S} } \mathop{\varphi }\limits^{} _{S\alpha } } +\\ \qquad\; {\dfrac{R_{r} }{\sigma L_{r} L_{S} } \mathop{\varphi }\limits^{} _{S\beta } +\dfrac{1}{\sigma *L_{S} } \mathop{v}\nolimits_{S\beta } } \end{array} \end{align} $

    (2)

    $ \begin{align} \mathop{\varphi }\limits^{\centerdot } _{S\alpha } = -R_{S} \mathop{i}\nolimits_{S\alpha } +\mathop{v}\nolimits_{S\alpha }\qquad\qquad\qquad\qquad\qquad\;\;\, \end{align} $

    (3)

    $ \begin{align} \mathop{\varphi }\limits^{\centerdot } _{S\beta } = -R_{S} \mathop{i}\nolimits_{S\beta } +\mathop{v}\nolimits_{S\beta }\qquad\qquad\qquad\qquad\qquad\;\;\, \end{align} $

    (4)

    $ \begin{align} T_{e} = \frac{3}{2} p\left(\varphi _{s\alpha } i_{s\beta } -\varphi _{s\beta } i_{s\alpha } \right)\qquad\qquad\qquad\qquad \end{align} $

    (5)

    $ \begin{align} J\frac{{\rm d}\Omega }{{\rm d}t} = T_{e} -T_{L} -f\Omega\qquad\qquad\qquad\qquad\qquad\; \end{align} $

    (6)

    where

    $ \sigma = 1-\frac{M^{2} }{L_{S} L_{r} }. $

    $ R_{S} $: The stator resistance.

    $ R_{r} $: The rotor resistance.

    $ L_{S} $: The stator inductance.

    $ L_{r} $: The rotor inductance.

    $ M $: Mutual inductance.

    $ T_{L} $: Load torque.

    $ J $: Rotor inertia.

    $ f $: Viscous friction coefficient.

    $ p $: Pole pairs number.

    $ \Omega $: Mechanical angular speed.

    $ \omega = p{\kern 1pt} {\kern 1pt} \Omega $: Electrical angular speed.

    $ \left(\varphi _{S\alpha } \quad \varphi _{S\beta } \right)^{\rm T} $: $ \alpha $-$ \rm axis $ and $ \beta $-$ \rm axis $ stator flux.

    $ \left(v_{S\alpha } \quad v_{S\beta } \right)^{\rm T} $ : $ \alpha $-$ \rm axis $ and $ \beta $-$ \rm axis $ stator voltage.

    $ \left(i_{S\alpha } \quad i_{S\beta } \right)^{\rm T} $ : $ \alpha $-$ \rm axis $ and $ \beta $-$ \rm axis $ stator current.

  • The CDTC diagram of the induction motor drives, based on the speed PI controller, is given by Fig. 1. Accordingly, the CDTC is a control technique based on the stator flux orientation. It is also based on stator flux and electromagnetic torque estimators, two hysteresis controllers and a switching table. The CDTC's basic idea is to control the electromagnetic torque and the stator flux module[1]. For every sampling time, the voltage vector is selected according to the electromagnetic torque error and the stator's flux error, and its position in the circular trajectory.

    Figure 1.  Block diagram of the CDTC

    The $ \alpha $$ \beta $ components of the stator current vector are calculated using the system of (7).

    $ \begin{align} \left\{\begin{array}{l} {i_{s\alpha } = i_{sa} } \\ {i_{s\beta } = \dfrac{i_{sa} +2i_{sb} }{\sqrt{3} } } .\end{array}\right. \end{align} $

    (7)

    The $ \alpha $$ \beta $ components of the stator voltage vector are calculated using the system of (8).

    $ \begin{align} \left\{\begin{array}{l} {v_{s\alpha } = \dfrac{2}{3} {E}\left(S_{a} -\dfrac{S_{b} -S_{c} }{2} \right)} \\ {v_{s\beta } = \dfrac{2}{3} {E}\left( \dfrac{S_{b} -S_{c} }{\sqrt{3}} \right) } \end{array}\right. \end{align} $

    (8)

    where $ S{}_{a} $, $ S{}_{b} $ and $ S{}_{c} $ are the inverter switching states. If $ S{}_{i} $ = 1 ($ i = a, b, c $) the upper switch of the inverter is on, if $ S = 0 $ the upper switch of the inverter is off.

    The stator flux is given by (9), which is integrated from (3) and (4).

    $ \begin{align} \varphi _{s} = \int \left(v_{s} -R_{s} i_{s} \right) \, {\rm d}t. \end{align} $

    (9)

    The electromagnetic torque is calculated by

    $ \begin{align} T_{e} = \frac{3}{2} p\left(\varphi _{s\alpha } i_{s\beta } -\varphi _{s\beta } i_{s\alpha } \right). \end{align} $

    (10)

    The stator flux components are estimated by

    $ \begin{align} \left\{\begin{array}{l} {\varphi _{s\alpha } = \int \left(v_{s\alpha } -R_{s} i_{s\alpha } \right) \, {\rm d}t} \\ {\varphi _{s\beta } = \int \left(v_{s\beta } -R_{s} i_{s\beta } \right) \, {\rm d}t}. \end{array}\right. \end{align} $

    (11)

    The expression of the stator flux magnitude is given as follows:

    $ \begin{align} \left|\varphi _{s} \right| = \sqrt{\varphi _{s\alpha }^{2} +\varphi _{s\beta }^{2} }. \end{align} $

    (12)

    The stator flux and electromagnetic torque errors are given by (1) and (14).

    $ \begin{align} \Delta \varphi = \left|\varphi _{s} \right|^{*} -\varphi _{s} \end{align} $

    (13)

    $ \begin{align} \Delta T_{em} = T_{em}^{*} -T_{em}. \end{align} $

    (14)

    For each sampling time, the voltage vector is chosen using the electromagnetic torque error, the stator flux error, and the stator flux vector position in the circular trajectory, which is given by Fig. 2.

    Figure 2.  Sector flux region and voltage vector

    When the flux is located in a zone $ i\, (i = 1, \cdots, 6) $, the stator flux and the electromagnetic torque can be controlled by selecting one of the following eight voltage vectors.

    1) If $ v_{i+1} $ is selected, the stator flux and the electromagnetic torque increase.

    2) If $ v_{i-1} $ is selected, the stator flux increases and the electromagnetic torque decreases.

    3) If $ v_{i+2} $ is selected, the stator flux decreases and the electromagnetic torque increases.

    4) If $ v_{i-2} $ is selected, the stator flux and the electromagnetic torque decrease.

    5) If $ v_{0} $ or $ v_{7} $ is selected, the electromagnetic torque decreases and the electromagnetic torque remains unchanged.

    Finally, as shown in Fig. 1, the two outputs of the hysteresis controllers and the number of the sector where the stator flux is located, present the inputs of the switching table (Table 1)[40].

    Table 1.  Selection of the voltage vector of the CDTC

  • The FL controller is used in this paper to improve the performances of the induction motor which is controlled by the CDTC strategy. Fig. 3 schematically depicts the CDTC of an induction motor based on the FL. In this figure, the hysteresis controllers and the switching table are replaced by the FL. It can be noticed that the stator flux linkage vector angle, and the errors of stator flux module and the electromagnetic torque present the inputs; whereas, the selected voltage vectors are the outputs of the FL controller.

    Figure 3.  Fuzzy direct torque control (DTFC) scheme

    Generally, in an FL controller system three main parts can be distinguished: Fuzzification, fuzzy reasoning, and defuzzification[20, 41]. Fuzzification is based on the membership functions, as shown in Fig. 4.

    Figure 4.  Diagram of the fuzzy logic controller

    The membership function shape affects the function of each fuzzy rule. The fuzzy system has three inputs in the form of fuzzy sets. The stator flux error has two fuzzy sets (positive: P and negative: N). The electromagnetic torque error has three fuzzy sets (positive large: PL, ZEro: ZE, and negative large: NL). The stator flux linkage vector angle has seven fuzzy sets. In fact, there are $ 2\times3\times7 = 42 $ rules, as shown in Table 2. Several defuzzification methods are utilized in the literature, such as the method of the center of gravity and the method of the Maximum. In this paper, the maximum method is used thanks to its simplicity. The membership functions of the stator flux's error $ \delta\varphi $, electromagnetic torque's error $ \delta{T}_{em} $, the stator flux's linkage vector angle $ {\theta}_{S} $, and the chosen voltage vector are illustrated by Figs. 5-8, respectively.

    Table 2.  Fuzzy control rule table

    Figure 5.  Fuzzy membership functions of flux error $\Delta\varphi$

    Figure 6.  Fuzzy membership functions of torque error $\Delta{T}_{em}$

    Figure 7.  Stator flux linkage vector angle ${\theta}_{s}$

    Figure 8.  Fuzzy membership functions of output

    Each control rule can be described using the three input variables of the stator flux error $ \Delta\varphi $, the electromagnetic torque error $ \Delta{T}_{em} $ and the angle $ {\theta}_{S} $, and the output variable $ {V}_{i} $. To obtain $ {V}_{6} $, the following rule is used.

    $ {R}_{5} $: If ($ \Delta $$ \varphi $ is P) & ($ \Delta $$ \varphi $ is PL) & ($ {\theta}_{S} $ is $ {\theta}_{5} $) then ($ {V}_{i} $ is $ {V}_{6} $).

  • Two forms of the Kalman filter (KF) are used in the literature, like the basic KF and the EKF. The basic KF is a linear observer (deterministic type). For nonlinear systems, such as the induction machine, the EKF is more suitable to overcome the linear observer limitations. In this work, the EKF is utilized to estimate the unmeasured states, like the stator flux and the rotor speed, using the stator currents that are easily measurable by current sensors and the voltage vector that is easily estimated using the inverter control sequences ($ {S}_{a} $, $ {S}_{b} $, $ {S}_{c} $) and (8).

  • The EKF is based on the discret model of the induction motor. The discretization of the induction motor model is obtained using the Euler approximation, as follows[42]:

    $ \begin{align} X\left( {k + 1} \right) = A_d X\left( k \right) + B_d U\left( k \right) \end{align} $

    (15)

    where

    $ \begin{align} \left\{ \begin{array}{l} A_d = {\rm e}^{AT_e } \approx I + AT_e \\ B_d = \int_0^{T_e } {{\rm e}^{A\tau } Bd\tau \approx BT_e }. \\ \end{array} \right. \end{align} $

    (16)

    Using (16), (15) becomes:

    $ \begin{align} X\left(k+1\right) = \left(I+AT_{e} \right)X\left(k\right)+BT_{e} U\left(k\right). \end{align} $

    (17)

    The digital model of the induction motor is then represented by the following system of equations:

    $ \begin{align} \left\{\begin{array}{l} {X\left(k+1\right) = f\left(X\left(k\right), U\left(k\right), k\right)} \\ {Y\left(k\right) = h\left(X\left(k\right), k\right).} \end{array}\right. \end{align} $

    (18)
  • The induction motor cannot be represented by a perfectly deterministic model, so it is necessary to introduce the various sources of uncertainties that are modeled by noises. The stochastic model of the induction motor is as follows:

    $ \begin{align} \left\{\begin{array}{l} {X\left(k+1\right) = f\left(X\left(k\right), U\left(k\right), k\right)+w\left(k\right)} \\ {Y\left(k\right) = h\left(X\left(k\right), k\right)+V\left(k\right)} \end{array}\right. \end{align} $

    (19)

    where $ w(k) $ represents the disturbances vector applied to the system and $ V(k) $ represents the measurement noises.

    $ f\left({X\left(k \right), U\left(k \right), k} \right) = \\ \left[\!\! {\begin{array}{*{20}c} {\left({1-\gamma T_e } \right)i_{s\alpha }+\dfrac{{\, m_{sr} R_r }}{{\sigma \, L_s \, L_r^2 }}\, T_e \varphi _{r\alpha }+\dfrac{{\, \, m_{sr} }}{{\sigma \, L_s \, L_r }}\, \omega T_e \varphi _{r\beta }+ \dfrac{{T_e }}{{\sigma L_S }}v_{s\alpha } } \\ {\left({1-\gamma T_e } \right)i_{s\beta}-\dfrac{{\, \, m_{sr} }}{{\sigma \, L_s \, L_r }}\, \omega T_e \varphi _{r\alpha }+\, \dfrac{{\, m_{sr} R_r }}{{\sigma \, L_s \, L_r^2 }}T_e \varphi _{r\beta }+ \dfrac{{T_e }}{{\sigma L_S }}v_{s\beta } } \\ {\dfrac{{\, R_r m_{sr} }}{{L_r }}T_e i_{s\alpha }+\left({1-\dfrac{{R_r }}{{L_r }}T_e } \right)\varphi _{r\alpha }-\omega T_e \, \varphi _{r\beta } \, } \\ {\dfrac{{\, R_r m_{sr} }}{{L_r }}T_e i_{s\beta }+ \omega \, T_e \, \varphi _{r\alpha } + \left({1-\dfrac{{R_r }}{{L_r }}T_e } \right)\varphi _{r\beta } \, } \\ {\omega _r } \\ \end{array}}\!\! \right] $

    where $ \gamma = \dfrac{{R_s L_r^2 + R_r m_{sr}^2 }}{{\sigma L_s L_r^2 }} $.

    Fig. 9 shows the structure of the EKF of an induction motor:

    Figure 9.  EKF diagram

    The EKF algorithm is given by the following steps (see Fig. 10):

    Figure 10.  EKF algorithm

    where $ Q $ is the covariance matrix of the system noise which is $ {[5\times 5]}$. $ R $ is the covariance matrix of the measurement noise which is $ {[2\times 2]}$. $ Q $ and $ R $ are two diagonal.

    $ \begin{align} F\left(k\right) = \frac{\partial f\left(X\left(k\right), U\left(k\right)\right)}{\partial X\left(k\right)} \left|_{X\left(k\right) = \mathop{X}\limits^{\wedge } \left(\frac{k}{k}\right)} \right. \end{align} $

    (20)

    $ \begin{align} H\left(k\right) = \frac{\partial h\left(X\left(k\right)\right)}{\partial X\left(k\right)} \left|_{X\left(k\right) = \mathop{X}\limits^{\wedge } \left(k\right)}. \right. \end{align} $

    (21)

    The sensorless direct torque control of an induction motor based on the fuzzy logic and an extended Kalman filter is given by the Fig. 11.

    Figure 11.  Sensorless DTFC-EKF diagram

  • To reduce the execution time, several software solutions are used; where two of them are given in Table 3. The first solution is based on reducing the algorithm complexity that downgrades the algorithm performances. The second one is based on using the multi-DSP, which is possible, but it decreases the system's integration and reliability. Finally, the hardware FPGA-based implementation is an alternative solution that can be used to overcome the software solution limitations.

    Table 3.  Different solutions to reduce the execution time

    Nowadays, the hardware solutions that are based on the FPGAs are outperforming the software ones based on the DSPs, because of the inherent parallelism of the FPGAs. Utilizing the hardware solutions, the designer may develop and implement more and more complex algorithms, having a negligible execution time compared to the one obtained with DSPs. A comparison between the software solutions and the FPGA Virtex 5, in terms of execution time, is presented by the implementation results, and then discussed.

  • The XSG software is a toolbox that was developed by Xilinx. It can be integrated into a Matlab/Simulink environment and it can let the user to create parallel FPGA systems. The created models can be displayed as blocks, and connected to other blocks of the Matlab/Simulink-like. Once the system is developed from the XSG, a VHDL code can be generated, exactly reproducing the behavior observed in Matlab. The design flow using the XSG is given in Fig. 12.

    Figure 12.  Xilinx system generator design flow

  • The DTFC-EKF is an algorithm with a high complexity, composed of several blocks. In this section some blocks are presented. Using (10), the design of the electromagnetic torque in the XSG is given by Fig. 13. The design of the stator flux components in the XSG is given by Fig. 14.

    Figure 13.  Design of electromagnetic torque estimator in XSG

    Figure 14.  Design of stator flux components from XSG

    When the stator flux is located, the FL block propagates three inputs: the stator flux error, the electromagnetic error, and the angle $ {\theta}_{S} $. The electromagnetic torque is presented by three fuzzy sets: positive large (PL), zero (ZE) and negative large (NL). In this study, the design of the fuzzy PL set of the electromagnetic torque error in the XSG is presented, which is modeled by (22) and given by Fig. 15.

    $ \begin{align} \left\{\begin{array}{l} {\rm if}\; \Delta T_{em} >a_{1} \; {\rm then}\; \mu (\Delta T_{em} ) = 1 \\ {\rm if}\; \Delta T_{em} <0\, \, {\rm then}\; \mu (\Delta T_{em} ) = 0 \\ {\rm else}\; \mu (\Delta T_{em} ) = \dfrac{1}{a_{1} } \Delta T_{em}. \end{array}\right. \end{align} $

    (22)

    Figure 15.  Design of membership function of the PL fuzzy set of the electromagnetic torque error from XSG

    Mamdani's inference method is used in this work based on min-max decision. The output of each fuzzy rule is given by the minimum function between the three membership function inputs. The design of the minimum function in the XSG is illustrated by Fig. 16.

    Figure 16.  Design of minimum function in XSG

    For the interconnection between all the rules, the "OR" operator is used, which is carried out by the maximum function. The design of the maximum function between two inputs is given as follows by Fig. 17.

    Figure 17.  Design of maximum function in XSG

  • In order to validate the theoretical study of the DTFC with the EKF, simulation results are presented in what follows using the toolbox XSG. The stator flux reference is 0.91 Wb. At $ t = 0.3 $ s, a load torque of 10 N$ \cdot $m is applied. The induction motor parameters are listed in the appendix.

  • Fig. 18 shows that the rotor speed has quickly reached its reference value at a start time of 0.05 s, thanks to the fast dynamics of the DTC. Figs. 19-23 show the evolution of the stator flux module, the stator flux trajectory, the electromagnetic torque, the stator current module, and the three-phase stator current, respectively (Color versions of Figs. 1923 in this paper are available online). All these figures present the DTFC performances in terms of ripples, which are dramatically decreased. Table 4 represents a comparative study between the CDTC and the DTFC in terms or ripples.

    Figure 18.  Evolution of rotor speed: (a) CDTC, (b) DTFC

    Figure 19.  Evolution of stator flux: (a) CDTC, (b) DTFCl

    Figure 20.  Evolution of stator flux vector trajectory: (a) CDTC, (b) DTFC

    Figure 21.  Evolution of electromagnetic torque: (a) CDTC, (b) DTFC

    Figure 22.  Evolution of stator current module: (a) CDTC, (b) DTFC

    Figure 23.  Evolution of stator current: (a) CDTC, (b) DTFC

    Table 4.  FDTC performances

  • The sensorless control technique, given by Fig. 11, is evaluated by simulation results, which show that the convergence between real and estimated variables is obtained. Fig. 24 illustrates the evolution of the real and estimated speed. It can be seen that the estimated speed has quickly reached the real speed. In Figs. 25 and 26, it is noticed that the estimated stator flux follows the real stator flux. In Figs. 27 and 28, it is clear that the estimated stator current follows the real stator current. In order to test the robustness and the performances of the EKF algorithm, simulation testing with different speed references and at a low speed are performed as shown in Fig. 29 (Color version of Fig. 29 in this paper is available online). It is noticeable that the estimated rotor speed follows the real rotor speed.

    Figure 24.  Evolution of real and estimated rotor speed

    Figure 25.  Evolution of stator flux components: $\varphi _{s\alpha } $ and ${\mathop{\varphi }\limits^{\wedge }}_{s\alpha } $

    Figure 26.  Stator flux components: $\varphi _{s\beta } $ and ${\mathop{\varphi }\limits^{\wedge }}_{s\beta } $

    Figure 27.  Stator current components: $i_{s\alpha } $ and ${\mathop{i}\limits^{\wedge }}_{s\alpha } $

    Figure 28.  Stator current components: $i_{s\beta } $ and ${\mathop{i}\limits^{\wedge }}_{s\beta } $

    Figure 29.  Rotor speed

  • The obtained synthesis and timing results are archived in Table 5. It is noted that the FPGA is characterized by a high processing speed thanks to its parallel treatment. In this paper, the execution time is $ 2.57 $ s using the Xilinx Virtex-5 FPGA with an xc5vfx70t-3ff1136 package. In papers [25-29] the total execution time has been evaluated from 90 to 500 s, due to the serial treatment of the software solutions.

    Table 5.  Synthesis results for FPGA based sensorless DTFC-EKF

    Once the system model is designed, with the XSG, the VHDL code is easily generated. Then the logic synthesis and the routing steps are performed [43] using Xilinx integrated synthesis environment (ISE). The register transfer level schematic of the sensorless DTFC-EKF, using the Xilinx ISE 12.4, is given by Fig. 30.

    Figure 30.  Synthesis result of SDTFC-EKF using Xilinx ISE 12.4

  • In this paper an FPGA implementation of a sensorless DTFC of an induction motor using an EKF has been presented. The DTC and the FL are, at first, analyzed. Then an overview of the EKF is presented. After that, the XSG principle and the design of some blocks of the sensorless DTFC-EKF in the XSG are presented. The simulation results show the performances of the intelligent technique, which reduces the torque and the flux ripples, and the stator current distortion. The simulation results of the DTFC-EKF show the convergence of the estimated magnitudes to the actual ones. In the case of the hardware implementation on the FPGA the obtained execution time is evaluated to 2.57, which is negligible relative to the one obtained with the software implementation on the DSP controller.

  • Table 6.  Induction machine parameters

Reference (43)

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