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To reduce the chattering issue and improve the dynamic tracking performance of the speed regulation of the permanent magnet synchronous motor (PMSM) system, the sliding mode control (SMC) with an enhanced exponential reaching law (EERL) is proposed in this paper. Compared to the SMC with conventional reaching law (CRL), the proposed SMC based on the EERL makes the convergence rate be associated with the change of the system state variable. In order to implement the minimum resource utilization of the embedded processor while guaranteeing the performance of the system, the event triggered strategy is developed for SMC, which possesses the excellent robustness and saves the energy cost. The stability of the event triggered SMC control law is proved by the Lyapunov function and the minimum control execution time interval is also derived. Finally, the effectiveness of the proposed EERL and the satisfactory performance of the event triggered SMC are demonstrated by the simulation results.

The PMSM is widely applied to industries due to its excellent characteristics, such as simple structure, small size, high power density, low noise, and friendly maintenance. However, the practical PMSM system is multivariable, strongly coupled, and nonlinear, which is easily affected by unmodeled dynamics, parameter variations, and load disturbance [

To decrease the influence of these aforementioned problems, various control theories have been proposed, such as fuzzy control [

However, the SMC is still troubled by the chattering phenomenon presently. The chattering phenomenon is that the output of the control signal is mixed with finite amplitude and high-frequency oscillation signal, which is damage to the mechanical system. In order to suppress the chattering issue, different approaches have been proposed, such as the integral sliding surface method [

Consequently, the EERL is proposed in this paper, which is based on the choice of an exponential term. In this method, the system state (error signal) is relative to the gain of the switching function. If the motion trajectories of the system state variables go far away from the sliding mode surface, the gain of the switching function will increase. On the contrary, if the motion trajectories of the system state variables are approaching the sliding mode surface, the gain of the sign function tends to zero. Hence, it can adapt to variations of the system state and sliding mode surface. Compared to the conventional SMC, the reaching law can reduce the reaching time and suppresses the chattering phenomenon.

In real work, the controllers are implemented on the digital platform. Therefore, the time triggered control scheme is commonly used, which is the periodic control method. In this strategy, the control law is updated at every sampling instants. Although this method is easy to implement, the source utilization is not economical. The event triggered strategy can avoid the unnecessary energy consumption under the premise of satisfactory control performances. The event triggered control loop is visualized in Figure

The event triggered control loop.

This paper is organized as follows. In Section

Assuming that the distribution of the permanent magnetic field of the rotor is sinusoidal in the air gap space, induction electromotive force in armature winding is also sinusoidal. Considering that the saturation of stator core is neglected and magnetic circuit is linear, the inductance parameters are constant and the eddy current and hysteresis are overlooked. There is also no damping winding on the rotor. The mathematical model of surface mounted PMSM on

Actually, even though the parameters will change, the variable of the motor is bounded. Thus,

The vector form of (

The vector control is adapted to the PMSM system in Figure

The control loop of PMSM.

In order to illustrate the proposed EERLSMC theory, the second-order nonlinear model is considered as

Substituting (

It is clear that the control law includes the discontinuous term. Increasing the coefficient

The reaching time

It can be seen that

In the speed control law (

According to the Lyapunov stability criterion, the Lyapunov function

Equation (

In practical implementation, the time triggered execution which also called Riemann sampling is generally adopted. The control law is only updated at a specific sampling instant via this method. Assuming the sampling instants

In this paper, instead of relying on time triggered execution, the event triggered method is adopted, in which the control law is updated when the triggering rule is satisfied. Thus, the time interval

Now the stability of the system is analyzed. It is defined that

Considering system (

Let the Lyapunov function

Once

Considering system given by (

where

The time derivate of (

with the initial condition

Theorem

In this paper, the vector control

Parameters of PMSM.

Characteristic | Symbol | Value |
---|---|---|

Stator phase resistance | | 2.875 Ω |

d- and q-axes inductances | | 8.5 mH |

Number of pole pairs | | 4 |

viscous damping | | 0.008 N. m. s |

Torque Constant | | 1.05 N. m |

Rotational inertia | | 0.003 kg.m2 |

In the simulation, the disturbances are not considered. So let

Comparison of the trajectory of system state and the controller output of the two methods. (a) The trajectory of sliding mode control of the two methods. (b) The controller output of the two methods.

Comparison of the speed/torque response of the two methods. (a) Speed curve of the two methods. (b) Torque curve of the two methods.

Comparison of current response of the two methods. (a) The three-phase stator current based on EERLSMC method. (b) The three-phase stator current based on CRLSMC method.

Figure

Figures

Figure

Section

(a) Speed regulation at 1000 RPM. (b) System state variable

The triggering instants

Figure

In this paper, an enhanced reaching law based on the exponential term is proposed and the stability of the EERLSMC has proved by the Lyapunov function. The proposed reaching law is applied to the speed regulation of the PMSM system. Then, the event triggered SMC is devised for the PMSM control system, by which the control law is updated when the triggering rule is satisfied. From the simulations and the comparisons, the following conclusions can be drawn: (1) The proposed reaching law can effectively suppress the chattering phenomenon and improve the reaching time and make the motor system more stable. (2) Compared to the SMC with conventional reaching law, the proposed SMC based on the EERL makes the convergence rate is associated with the change of the system state variable, which has a fast calculation. (3) Through the simulations, the effectiveness of the proposed event triggered SMC has been verified. The stability of the system is guaranteed. The interval time between two triggering instants is greater than the time triggered. Due to the bigger sampling interval, the controller executes as little as possible. Thus, the energy consumption which is owing to the communication is minimized. Even if the energy is not a concern, the control tasks are executed infrequently, so that the processor will have more time to complete other important tasks.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The work described in this paper was supported in part by Natural Science Foundation of China under Grant NSFC51677020, in part by the Provincial Key Laboratory of Power Electronics Energy Saving Technology and Equipment (no. szjj2016-093), in part by China Postdoctoral Science Foundation under Grant 2015M572457, and in part by Fundamental Research Funds for the Central Universities (no. ZYGX2016J209).