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.