comment vous trouver ce travail: Abstract
Direct torque control (DTC) is a widely used control method for induction motors, offering rapid dynamic response and relatively simple implementation. However, it presents high torque and flux ripples and variable switching frequencies. To overcome these limitations, the Double Power Super Twisting Algorithm (DPSTA) was proposed, combining the advantages of the Super Twisting Algorithm (STA) to reduce chattering and those of the Double Power Convergence Law (DPRL) to improve system speed and dynamic quality. However, optimal tuning of the DPSTA controller's sliding mode gains presents a major challenge. In order to solve this problem, we propose an extension of the DPSTA algorithm by integrating a single-neuron adaptive approach. Simulations were carried out on Matlab Simulink to test the performance of the proposed controller. The results show that SN-DPSTC enables dynamic and real-time adjustment of the controller gains, while offering better performance in terms of chattering reduction, improved dynamic response and robustness to load disturbances.

Introduction:
Induction motors have enjoyed great success in industry and transport applications due to their affordability, robustness and low maintenance costs[1], [2], [3]. However, these motors present particular challenges in terms of control due to their non-linear dynamics and sensitivity to parameter variations[4], [5]. Their control requires advanced techniques to guarantee optimum performance.
Direct torque control (DTC) is a control technique widely used in induction motor (IM) control applications, as it offers rapid dynamic response and ease of implementation[6], [7], [8]. Nevertheless, conventional DTC has significant limitations, including high torque and flux ripples, as well as variable switching frequencies[9]. These problems adversely affect system performance, generate mechanical vibrations and lead to increased component resistance.
To overcome these limitations, the Double Power Super Twisting Algorithm (DPSTA) has been proposed[10], combining the advantages of the Super Twisting Algorithm (STA), which reduces the chattering phenomenon[11], [12], [13], and those of the Double Power Convergence Law (DPRL), which improves the reaching speed as well as the dynamic quality of the system[14], [15]. As described in our previous work [10], the performance of the proposed controller is evaluated in the context of DTC control of IM driving electric vehicle, and compared with that of PI, fuzzy logic and traditional STA controllers. Numerical results show that DPSTA improves the robustness of the control system and significantly reduces chattering, while maintaining high dynamic performance.
However, since the operating conditions of induction motors vary considerably due to various factors such as load changes, temperature variations and component ageing, determining the optimum sliding-mode gains of the DPSTA controller is a major challenge in maintaining optimum performance. Incorrect setting of these parameters can lead to degraded dynamic performance and even system instability, which is unacceptable in critical industrial environments.
To overcome these challenges, we propose in this paper an extension of the DPSTA algorithm that integrates an adaptive single-neuron approach. The adaptive single neuron method offers an efficient solution for dynamically adjusting controller parameters in real time. This approach exploits the learning and adaptation capabilities of neural networks, even in a simplified form with a single neuron, to optimize controller parameters in response to variations in operating conditions without over-complexifying the controller[16], [17], [18]. Compared with traditional multi-layer neural networks, a single neuron reduces computational complexity and facilitates practical implementation, while retaining effective adaptability[19], [20].
Recent studies have shown that the use of artificial intelligence techniques, such as neural networks and adaptive algorithms, can lead to significant improvements in dynamic response and stability. In [21], [22], [23], [24], the authors applied neural adaptive control to tune the three PID control parameters, namely the proportional, integral and differential coefficients, in order to effectively solve the problem of difficult gain parameterization of the conventional PID controller. The results showed that neural adaptive control improves the performance of conventional PID control, offering high robustness, stability and dynamic performance.
With the integration of a single adaptive neuron into the DPSTA algorithm, we aim to improve robustness, torque and flux ripples, as well as dynamic response. This article is structured as follows: In the second section, we outline the concept of the single adaptive neuron and its integration into the DPSTA controller for the DTC method. The third section presents the system modeling and simulation method. We then present the simulation results and a detailed performance analysis of the proposed system. At the end of the study, we conclude by summarizing the main contributions.
Single-neuron Double Power Super Twisting Controller Design:
Double power super twisting controller
The DPST Algorithm has been been designed to enhance the performance of the STA algorithm by exploiting the properties of the DPSMRL \cite{mencou2024direct}. The main idea is to replace the switching function of the STA with that of DPSMRL. The basic form of this algorithm is as follows:
\begin{equation} \label{e1}
    {\left\{\begin{split} &\frac{d x_1}{d t}=-k_1 \phi_1\left(x_1\right)+x_2+\varphi_1\left(x_1, t\right) \\&
    \frac{d x_2}{d t}=-k_2 \phi_2\left(x_1\right)+\varphi_2\left(x_2, t\right)\end{split}\right.}
\end{equation}
with:
\begin{equation*} \label{e1}
        {\left\{\begin{split} &\phi_1(x)=\vert x\vert^{1/2}\;sign(x)+\lambda\vert x\vert^{3/2}\;sign(x)\\& \phi_2(x) = \phi_1^2(x)^{\prime}=2 \phi_1(x) \cdot \phi_1(x)^{\prime}  =\operatorname{sign}(x)+4 \lambda|x| \operatorname{sign}(x)+\frac{3}{2} \lambda^2|x|^2 \operatorname{sign}(x)\end{split}\right. \;\;\; ;\;  \lambda\geq0}
\end{equation*}
where $k_1$ and $k_2$ are sliding mode gains coefficients.

If for constants \ensuremath{\delta }\ensuremath{_{1}} \ensuremath{\geq } 0 and $\xi_1=\frac1{{{\phi_1^{'}}_{\;}\left(x_1\right)}^{\;}} $:
\begin{equation} \label{e2}
{\left\vert \varphi_1\right\vert\leq\delta_1\left\vert \zeta_1\right\vert\;\;and\;\;\varphi_2=0\;\;;\;\forall t\geq0}
\end{equation}

The system converges to the equilibrium point $x(0,0)$ in finite time if the sliding mode gains $k_1$ and $k_2$ satisfy the conditions :

\begin{equation} \label{e3}
{\left\{\begin{array}{l}k_1>2\delta_1\\k_2>k_1\frac{5\delta_1k_1+4\delta_1^{2}}{2\left(k_1-2\delta_1\right)}\end{array}\right.}
\end{equation}

The proof of finite-time stability and the estimation of convergence time are well presented in our previous paper \cite{mencou2024direct}.

As detailed in \cite{mencou2024direct}, the DPST control command is expressed as follows:
\begin{equation} \label{e4}
u(t)=-k_1\left(\left(|s(t)|^{\frac{1}{2}}+\lambda|s(t)|^{\frac{3}{2}}\right) \operatorname{sign}(s(t))\right)-\int k_2\left(\left(1+4 \lambda|s(t)|+\frac{3}{2} \lambda^2|s(t)|^2\right) \operatorname{sign}(s(t))\right)
\end{equation}

with $s(t)$ the sliding variable.

Although the controller ensures system stability as long as the gains satisfy the stability conditions (\ref{e3}), these conditions allow a wide range of values, which makes it difficult to calculate the specific gain values $k_1$ and $k_2$. This causes problems for the design of optimal controller parameters.

To solve this problem, we designed a Single Neuron Adaptive Double-Power-Super-Twisting (SNA\_ DPST) controller for adjusting the gain of the DPST controller in real-time. This approach merges the benefits of the SN adaptive control method, known for its simplicity and adaptability, with the principles of DPSTA control theory. The dspecific steps involved in designing this control algorithm is described below.
Single-Neuron Double Power Super Twisting Controller design:
Figure 1 shows the overall block diagram of the single neuron Double Power Super Twisting Sliding controller (SN_DPSTC) approach proposed in this study.


Since the single neuron controller operates using a numerical control, it is essential to discretise the DPST control command described in (\ref{e4}). We used the direct finite difference method for this discretisation. This approach allowed us to transform the continuous controller into an incremental controller, adapted for a numerical environment. 

The discrimination of $u(t)$ is given by:

\begin{equation} \label{e5}
{\begin{split}
u(k)=k_1^{\prime}|s(k)|^{\frac{1}{2}} \operatorname{sgn}(s(k))+k_2^{\prime}|s(k)|^{\frac{3}{2}} \operatorname{sign}(s(k))+k_3^{\prime} \sum_{i=0}^k \operatorname{sign}(s(i))  \\ +k_4^{\prime} \sum_{i=0}^k|s(i)| \operatorname{sgn}(s(i))+k_5^{\prime} \sum_{i=0}^k|s(i)|^2 \operatorname{sgn}(s(i))
  \end{split} }
\end{equation}

with :

\begin{equation} \label{e6}
{\left\{\begin{array}{c}
k_1^{\prime}=k_1 \\
k_2^{\prime}=k_1 \lambda \\
k_3^{\prime}=k_2 \\
k_4^{\prime}=4 k_2 \lambda \\
k_5^{\prime}=\frac{3}{2} k_2 \lambda^2
\end{array}\right.}
\end{equation}

The incremental controller is given by:

\begin{equation} \label{e7}
  {\begin{array}{l}\begin{split}
      \dot \Delta u(k)&=u(k)-u(k-1) 
      \\&=k_1^{\prime}\left(|s(k)|^{\frac{1}{2}} \operatorname{sgn}(s(k))-|s(k-1)|^{\frac{1}{2}} \operatorname{sgn}(s(k-1))\right) 
      \\& +k_2^{\prime}\left(|s(k)|^{\frac{3}{2}} \operatorname{sign}(s(k))-|s(k-1)|^{\frac{3}{2}} \operatorname{sign}(s(k-1))\right)
      \\& +k_3^{\prime} \operatorname{sign}(s(k))+k_4^{\prime}|s(k)| \operatorname{sgn}(s(k))+k_5^{\prime}|s(k)|^2 \operatorname{sign}(s(k)) 
  \end{split}\end{array} }  
\end{equation}

we define the set variables as:

\begin{equation} \label{e8}
{\left\{\begin{array}{c}
x_1(k)=|s(k)|^{\frac{1}{2}} \operatorname{sgn}(s(k))-|s(k-1)|^{\frac{1}{2}} \operatorname{sgn}(s(k-1)) \\
x_2(k)=|s(k)|^{\frac{3}{2}} \operatorname{sign}(s(k))-|s(k-1)|^{\frac{3}{2}} \operatorname{sign}(s(k-1)) \\
x_3(k)=\operatorname{sign}(s(k)) \\
x_4(k)=|s(k)| \operatorname{sgn}(s(k)) \\
x_5(k)=|s(k)|^2 \operatorname{sign}(s(k))
\end{array}\right.}
\end{equation}

These state variables are the inputs to our single neuron, as shown in \autoref{fig1}. To each input variable $x_i(k)$, we assign a weighting factor $\omega_i(k)$. 

Based on the input variables and the weighting factors, the neuron calculates the incremental controller:

\begin{equation} \label{e9}
{\Delta u(k)=K \sum_{i=1}^5 \omega_i(k) x_i(k) }
\end{equation}

where $K$ is the neuron gain coefficient $K>0$.

Therefore:

\begin{equation} \label{e10}
{\left\{\begin{array}{c}
k_1^{\prime}=K \omega_1(k) \\
k_2^{\prime}=K \omega_2(k) \\
k_3^{\prime}=K \omega_3(k) \\
k_4^{\prime}=K \omega_4(k) \\
k_5^{\prime}=K \omega_5(k)
\end{array}\right.}
\end{equation}

Adaptive learning was chosen for the adjustment of the weighting factors because it allows the weights to be modified dynamically in response to variations in the error. This approach guarantees faster and more stable convergence of the model. We define the objective function $J$ as a measure of the squared error $J=\frac{s(k)^2}{2}$:

\begin{equation} \label{e11}
{\Delta \omega_i(k)=-\eta_i \frac{\partial J}{\partial \omega_i}}
\end{equation}

Using improved Hebb supervised learning, we obtain:

\begin{equation} \label{e12}
{\Delta \omega_i(k)=\eta_i s(k) u(k-1)(2 s(k)-s(k-1))}
\end{equation}

To avoid uncontrolled weight gain, the output is obtained after normalizing the weighting factors as follows:

\begin{equation} \label{e13}
{\Delta u(k)=K \sum_{i=1}^5 \omega_i^{\prime}(k) x_i(k) \quad \text { with: 
       } \omega_i^{\prime}(k)=\frac{\omega_i(k)}{\sum_{i=1}^5 \omega_i(k)}}
\end{equation}

Where:

\begin{equation} \label{e14}
{\omega_i(k)= \omega_i(k-1) - \eta_i s(k) u(k-1)(2 s(k)-s(k-1))}
\end{equation}

The SNA\_DPST control command will be expressed as:

\begin{equation} \label{e15}
{u(k)= u(k-1) + K \sum_{i=1}^5 \omega_i^{\prime}(k) x_i(k) }
\end{equation}
System modelling and simulation methodology
Integration the Single Neuron Controller in DTC Control
As part of our previous studies and in order to verify the accuracy and efficiency of the proposed Single-Neuron Double Dower Duper Twisting controller, we tested it in the context of Direct Torque Control of an induction motor. Figure 2 shows the block diagram of the DTC control of an induction motor based on a Single-neuron double power super twisting controller. This control algorithm is integrated into the closed-loop speed control.
The SN DPSTC calculates the control command u(k) based on the speed error. The reference torque Tem∗ is calculated according to the following form, as explained in [10]:

\begin{equation} \label{e16}
{T_{e m}^*=f_v \omega_m^*+J \omega_m^*+T_L-J u}
\end{equation}
System modelling and simulation methodology
To compare the double-power super-twisting controller (DPSTC) and the single-neuron double-power super-twist controller (SN_DPSTC) in detail, simulations were carried out using Matlab/Simulink. The DTC simulation model based on DPSTC and NN_DPSTC, developed in Matlab, is shown in Figure 9. The detailed modelling of the DTC control is shown in our previous paper [1].

Simulations and Résultats
In this section, we present the simulation results obtained to evaluate and test the efficiency of the proposed SN-DPST controller. These performances are compared with those of the DPST and PI controllers present in [10]. For the simulation, we chose a 4KW induction motor with the parameters shown in table 1. The gain coefficient of the neuron was K = 25 and the initial value of the weighting factor was 0.1. 
Firstly, we analyze the controller responses to a reference speed setpoint of 75rad/s applied at 0.05s and a load torque of 14Nm introduced at 0.3s.
Simulation results are presented in figure 1, 2 and 3.
The speed response of the three controllers, illustrated in figure 2, shows that, under the action of the PI control, the motor speed follows the reference value with an overshoot of 5.84% and a steady-state error of 0.12rad/s in the absence of load and 0.43 rad/s when load is applied. On the other hand, under the control of the DPSTC controller, the motor operates without overshoot, but has a relatively longer regulation time: 0.15 s for speed variation and 0.03 s for torque change. In contrast, the SN_DPSTC proposed controller offers the best performance in terms of response time and steady-state error. Under the control of the SN_DPSTC controller, the motor accurately follows the setpoint signal without displacement, and with a very fast response time of 12 ms for speed variation and only 2 ms for torque change adjustment. 
Variations of the controller parameters k1, k2, k3, k4 and k5 are shown in Fig. 3. The results clearly demonstrate the ability of the proposed single-neuron adaptive algorithm to dynamically and optimally adjust these parameters according to system operating conditions.
Figure 4, which illustrates the variation of the objective function J, shows that the latter converges to 0 in steady state. This shows that the proposed algorithm succeeds in efficiently minimizing the error.
To evaluate the performance of the proposed controller more thoroughly, we test its robustness and stability under more demanding operating conditions. We apply a reference speed setpoint varying between 0rad/s and 157rad/s over a period of 22s, with a rectangular load torque signal of 14 Nm amplitude applied between 5 s and 17 s.
The simulation results presented in figure 2 illustrate that, under the action of the SN-DPSTA controller, the motor speed follows the reference signal quickly and efficiently, without overshoot, while maintaining overall system stability. Unlike the PI and DPST controllers, the adaptive neural network controller showed robustness to load disturbances and the ability to adapt dynamically to changing operating conditions.
In addition, a comparative study was carried out in terms of minimization criteria: Integral Square Error (ISE) and Integral Absolute Error (IAE). These two statistical parameters are commonly used in control systems to evaluate and compare closed-loop system performance.
	PI	DPSTC	SN-DPSTC
ISE	0.410	0.367	0.021
IAE	2.340	0.798	0.177

According to the values of the evaluation parameters shown in Table 2, the PI controller showed the least satisfactory behavior compared to the other controllers, showing high values for IAE (0.410) and ISE (2.340). In contrast, the DPSTC controller showed superior performance, displaying lower values for the statistical parameters (IAE: 0.367 and ISE: 0.798). However, the proposed SN-DPSTC controller showed the most promising results compared with the other controllers, with an IAE of 0.021 and an ISE of 0.177. The use of an adaptive single neuron controller showed exceptional ability to minimize error over time.
These results underline the improvements brought by the integration of the single-neuron adaptive algorithm into the DPSTC controller in terms of real-time self-adjustment of controller parameters, reduced chattering, improved dynamic response and robustness to variations in load system parameters. This guarantees more efficient and reliable control, even under variable operating conditions, making it an ideal solution for applications requiring high precision and increased robustness, such as motor control in electric vehicles or other demanding industrial systems.

Conclusion
This paper presents an extension of the DPSTA algorithm by integrating a single-neuron adaptive algorithm, aimed to overcome the challenge of optimal controller gain adjustment. Numerical simulation results have shown that the proposed SN-DPSTC controller enables dynamic and optimal adjustment of control parameters in real time. The proposed SN-DPSTC controller outperformed the DPSTC and PI controllers by offering fast and accurate response to setpoint and load variations, a significant reduction in the chattering phenomenon, as well as increased robustness to disturbances and variations in operating conditions. In addition, the simplicity of the single-neuron algorithm facilitates practical implementation of the controller, while reducing computational complexity compared with traditional multi-layer neural networks.
These improvements make the SN-DPSTC controller particularly suitable for applications requiring high precision and robustness, such as motor control in electric vehicles and demanding industrial systems.


 

[1]	P. Dinolova, V. Ruseva, and O. Dinolov, “Energy Efficiency of Induction Motor Drives: State of the Art, Analysis and Recommendations,” 2023. doi: 10.3390/en16207136.
[2]	M. L. De Klerk and A. K. Saha, “A Comprehensive Review of Advanced Traction Motor Control Techniques Suitable for Electric Vehicle Applications,” IEEE Access, vol. 9, 2021, doi: 10.1109/ACCESS.2021.3110736.
[3]	O. E. M. Youssef, M. G. Hussien, and A. El-Wahab Hassan, “A Robust Regenerative-Braking Control of Induction Motors for EVs Applications,” International Transactions on Electrical Energy Systems, vol. 2024, 2024, doi: 10.1155/2024/5526545.
[4]	P. Thirugnanam, “Advances, New Perspective and Applications in Induction Motors,” in Induction Motors - Recent Advances, New Perspectives and Applications, 2023. doi: 10.5772/intechopen.1001583.
[5]	M. V. Sarin, A. Chitra, P. Sanjeevikumar, and A. Venkadesan, “Induction Motor Control Schemes for Hybrid Electric Vehicles/Electric Vehicles,” in Artificial Intelligent Techniques for Electric and Hybrid Electric Vehicles, 2020. doi: 10.1002/9781119682035.ch9.
[6]	M. Aktas, K. Awaili, M. Ehsani, and A. Arisoy, “Direct torque control versus indirect field-oriented control of induction motors for electric vehicle applications,” Engineering Science and Technology, an International Journal, vol. 23, no. 5, 2020, doi: 10.1016/j.jestch.2020.04.002.
[7]	R. H. Kumar, A. Iqbal, and N. C. Lenin, “Review of recent advancements of direct torque control in induction motor drives - A decade of progress,” 2018. doi: 10.1049/iet-pel.2017.0252.
[8]	T. Sutikno, N. R. N. Idris, and A. Jidin, “A review of direct torque control of induction motors for sustainable reliability and energy efficient drives,” 2014. doi: 10.1016/j.rser.2014.01.040.
[9]	S. Mencou, M. Ben Yakhlef, and E. B. Tazi, “Advanced Torque and Speed Control Techniques for Induction Motor Drives: A Review,” in 2022 2nd International Conference on Innovative Research in Applied Science, Engineering and Technology, IRASET 2022, 2022. doi: 10.1109/IRASET52964.2022.9738368.
[10]	S. Mencou, M. Ben Yakhelf, and E. Tazi, “Direct torque control of induction motor based on double-power-super-twisting sliding mode speed control for electric vehicle applications,” International Journal of Power Electronics and Drive Systems, vol. 15, no. 3, pp. 1399–1409, Sep. 2024, doi: 10.11591/ijpeds.v15.i3.pp1399-1409.
[11]	R. I. Verdés, A. Ferreira de Loza, L. T. Aguilar, I. Castillo, and L. Freidovich, Emerging Trends in Sliding Mode Control, Theory and Application, vol. 318. 2021.
[12]	A. Levant, “Sliding order and sliding accuracy in sliding mode control,” Int J Control, vol. 58, no. 6, 1993, doi: 10.1080/00207179308923053.
[13]	A. Ammar, “Second-order sliding mode-direct torque control and load torque estimation for sensorless model reference adaptive system–based induction machine,” Proceedings of the Institution of Mechanical Engineers. Part I: Journal of Systems and Control Engineering, vol. 235, no. 1, 2021, doi: 10.1177/0959651820935694.
[14]	M. Tao, Q. Chen, X. He, and M. Sun, “Adaptive fixed-time fault-tolerant control for rigid spacecraft using a double power reaching law,” International Journal of Robust and Nonlinear Control, vol. 29, no. 12, 2019, doi: 10.1002/rnc.4593.
[15]	H. J. Li and Y. L. Cai, “Sliding mode control with double power reaching law,” Kongzhi yu Juece/Control and Decision, vol. 31, no. 3, 2016, doi: 10.13195/j.kzyjc.2014.1908.
[16]	L. Cheng, Z. Wang, F. Jiang, and J. Li, “Adaptive neural network control of nonlinear systems with unknown dynamics,” Advances in Space Research, vol. 67, no. 3, 2021, doi: 10.1016/j.asr.2020.10.052.
[17]	W. Tang, L. Wang, J. Gu, and Y. Gu, “Single neural adaptive PID control for small UAV micro-turbojet engine,” Sensors (Switzerland), vol. 20, no. 2, 2020, doi: 10.3390/s20020345.
[18]	S. Morris, P. K. Dash, and K. P. Basu, “A neuro-sliding mode controller for STATCOM,” Electric Power Components and Systems, vol. 32, no. 2, 2004, doi: 10.1080/15325000490195853.
[19]	C. Napole, O. Barambones, I. Calvo, and J. Velasco, “Feedforward compensation analysis of piezoelectric actuators using artificial neural networks with conventional PID controller and single-neuron PID based on hebb learning rules,” Energies (Basel), vol. 13, no. 15, 2020, doi: 10.3390/en13153929.
[20]	Y. Qin and H. Duan, “Single-neuron adaptive hysteresis compensation of piezoelectric actuator based on hebb learning rules,” Micromachines (Basel), vol. 11, no. 1, 2020, doi: 10.3390/mi11010084.
[21]	X. Wang, X. Wang, and K. Liu, “Adaptive PID Control of Coal Feeder Based on Single Neuron,” 2019. doi: 10.2991/masta-19.2019.58.
[22]	J. Hernandez-Barragan, J. D. Rios, A. Y. Alanis, C. Lopez-Franco, J. Gomez-Avila, and N. Arana-Daniel, “Adaptive single neuron anti-windup PID controller based on the extended Kalman filter algorithm,” Electronics (Switzerland), vol. 9, no. 4, 2020, doi: 10.3390/electronics9040636.
[23]	J. G. Lim, S. Fei, and L. Yu, “An improved single neuron self-adaptive PID control scheme of superheated steam temperature control system,” International Journal of System Control and Information Processing, vol. 2, no. 1, 2017, doi: 10.1504/ijscip.2017.10005169.
[24]	Y. Zhang, P. Li, and W. J. Wu, “Single neuron PID sliding mode parallel compound control for alternating current servo system,” in Procedia Engineering, 2012. doi: 10.1016/j.proeng.2012.01.261.