A non-linear backstepping control of Permanent Magnet Synchronous Motor (PMSM)

. in the industry, the permanent magnet synchronous motors are one of the most widely used motors and have superior performance compared to the other types of motors. The principal objective of this paper is to ameliorate the performance of PMSMs by implementing a robust non-linear backstepping control. The first part deals with the vector control of mechanical sensors using PI controller. In the second part, we shed the light on the non-linear backstepping control, using Lyapunov function, of PMSM from the point of view of stability and robustness. The results are validated by MALTAB/SIMULINK. The results obtained show the good stability and good dynamic of PMSM’s control by backstepping controller, also the PI is very sensitive to parameter’s machine variation compared with the backstepping controller.


Introduction
Due to its simplicity and performance, the permanent magnet synchronous machine among the most used machines in different fields and industry. For this reason, the current scientific research converges on the control of the asynchronous machine. The classical PI controller is the most used technique because it is applicable and simple to implement, but this technique is not satisfactory because of the non-linearity of the PMSM [1]. This is why we have decided to develop different non-linear techniques more advanced that ensures the stability and robustness of the system. Among these methods we quote backstepping, artificial intelligent and sliding mode control [2,3,4]. The present work highlighted the application of the PI controller and the backstepping control to PMSM. The latter controller presents a promising alternative to control methods for nonlinear systems, to ensure the control by the backstepping, it is necessary to combine the choice of the function of Lyapunov with that of the control laws [5]. this allows us to guarantee the global stability of the system at any time, and thus to ensure a good regulation and setpoint tracking [6].

Mathematical model of PMSM
The mathematical model of the Permanent Magnet Synchronous Motor in the d-q reference frame is described as the next form [7].

The PMSM's field-oriented control
We get the PMSM with a surface mounted PMSM, = , in the field vector control we set the current to zero in order to have an optimal linearization of the torque, so: The electromagnetic torque is given by the follows form:

The PMSM's PI controller law
The law of the PI controller is: We can define the regulator parameters for and by = 2 ; = . (5) For the speed we have: 5 Backstepping design

current control
The control law of backstepping for current is:

Speed control
The control law of backstepping for the speed current is:

current Control
The control law of backstepping for current is:

Discussion
The figure 1.a shows that an overshoot in speed tracking for PMSM's PI controller. when we applied the load, the speed decreased, and we observe a high value for id current. The electromechanically torque compensates the applied load with an exceed value of the torque (1.c). For the speed inversion the figure (1.b) shows that in the moment of the inversion of speed the iq current and the torque make a high drop. When the parameter's machine varies effects on the iq current and on the torque. The figure 2 (a, b et c) illustrate that for the backstepping controller, the speed reaches the speed tracking with a fast response time and without the exceeding, the same when we inverse the speed. The backstepping controller assure a good speed dynamics and disturbance's rejection. an instantaneous increase in the torque to compensate for the load applied at that moment. The id and iq currents stator's characteristics at the start-up the machine lead to a high current afterwards then we observe a decrease as the machine has the normal operating regime. The decoupling that we introduce in this controller technique (id=0) applied of PMSM illustrated by the iq and id currents.

Conclusion
In this work we present the PMSM's vector control based by the classical PI controller. The analysis of the robustness of this controller shows that is not robust to the parameter's machine variation and to disturbances. To have a satisfactory result for PI controller, it is necessary to ensure the elimination of disturbances and the variation of the machine parameters.
The Backstepping controller, is a nonlinear controller that current research is converging on. It based on a recent methodology using the Lyapunov function, proposed in this paper to improve the PMSM's control. According the results obtained we conclude that this controller has a good robustness to parameter's variation and disturbances compared with a PI controller. The synthesis of this nonlinear controller led to the globally asymptotically stability.