Super-Twisting Sliding Mode Active Disturbance Rejection Control of Permanent Magnet Synchronous Motor Based on Vector Control

The vector control speed regulation system of the permanent magnet synchronous motor founded on traditional control has some weaknesses of poor rapidity and robustness, and it is hard to satisfy the current control requirements. A new control method combining active disturbance rejection control technology with improved super-twisting sliding mode control technology (STSM-ADRC) comes up. In active disturbance rejection control, the tracking differentiator is simplified; the extended state observer is designed to observe all internal and exterior disturbances of the system; the improved super-twisting sliding mode control law is going to take the place of the error feedback control law; and the simulation is carried out. Compared with the existing control algorithms, this new control method can overcome the influence of load disturbances and track the given speed effectively. And the responsiveness and robustness of the system are strengthened.


Introduction
The strong points which belong to permanent magnet synchronous motor (PMSM) are its easy structure, good reliability, high efficiency, and excellent control performance.The considerable attraction has converged to it in modern industry [1].
The gist of the vector control technology of PMSM is to make the synthetic vector of the motor stator always perpendicular to the rotor [2].As for the control method, the speed outer loop and current inner loop with PI controller may have problems of slow response speed, poor robustness and weak anti-interference ability [3].
Targeting the issue of the weakness of PI controllers, domestic and foreign experts and scholars put forward many excellent control methods.Cheng proposed a kind of fuzzy pi control that can simplify parameter tuning, but the fuzzy rules are difficult to establish [4].[5] also recommended adaptive control, but the adaptive laws are hard to design.[6] proposed the neural network, it has a strong antiinterference ability, but it needs the amount of data for neural network training.In addition, Han proposed the active disturbance rejection control, which takes all the motor disturbances as the whole disturbance without relying on the exact mathematical model [7].[8] introduced sliding mode control which is also a robust and fast control method, but the occurrence of chattering plays restrictions on it.In contrast with sliding mode control, Levant proposed Super-twisting sliding mode control, which is valuable for reducing chattering to a certain extent [9].
In this study, the super-twisting sliding mode control and active disturbance rejection control are combined to take into account the advantages of the two control methods.On this basis, an improved STSM-ADRC is proposed.The consequence proves the effect of this control method after simulation.

Mathematical Model
The mathematical model of PMSM in the d-q axis coordinate system is: In the equation, e  is the electric angle speed of the motor.
The motion equation of PMSM is: In the equation,  represents the mechanical speed.
The torque equation which belongs to surface mounted PMSM is: In the equation, p is the number of motor poles.
Considering that effects of disturbances, Equation ( 3) is going to be expressed as: In the equation, 0 = ; F represents the whole disturbance throughout the motor.

Traditional ADRC
Active disturbance rejection control (ADRC) represents one control method that combines unknown dynamics and external disturbances into a total disturbance for observation and compensation.ADRC is made up of tracking differentiator (TD), extended state observer (ESO) and error feedback control law (NLSEF).Figure 1 shows the traditional ADRC structure.The tracking differentiator is mainly adopted to acquire the tracking value of the reference and the differential signals of each order, which can solve the problem that the input signal may have certain random noise and the input signal are discontinuous.And it can enhance the dynamic and static control performance of the motor.In the meantime, it can play the role of filtering without a lot of noise interference.The expression of tracking differentiator (TD) is as follows: The extended state observer (ESO) mainly observes the state variables and their derivatives.Only by using all input together with the output of one motor, the total disturbance of the controlled object can be reconstructed.The core content of ESO is to perform disturbance compensation.This controlled object after disturbance compensation becomes a cascade control system that is easy to design the controller.The expression of the first-order extended state observer (ESO): In the equation, z1 observes  , and z2 represents the total disturbance which is the same as F.
The error feedback control rate replaces the linear weighting in the traditional PI control with a nonlinear combination, and it employs the observation of the error from ESO to correct the results of the nonlinear combination.The error feedback control rate (NLSEF) expresses as: ( ) The expression of the fal function in Equation ( 7) is: , , , In the equation, e represents the error,  represents the filter factor, and  represents the nonlinear factor.Since the nonlinear function fal is regarded as the control function, the nonlinear error feedback control law imports one or more errors into the fal function for nonlinear superposition, so it can overcome the overshoot problem in traditional PI control.

STSM-ADRC Design
Since the error feedback tracking rate in ADRC is equivalent to PD control, the rapidity and stability are promoted by replacing the error feedback tracking rate with the super-twisting sliding mode control (STSMC).Another, tracking differentiator in ADRC is mainly to smooth the given reference signal, and it is different from the working state belonging to PMSM in this research.To reduce that difficulty for parameter adjustment, the tracking differentiator part in ADRC is omitted.The final STSM-ADRC controller structure is shown in Figure 2. , the spinning can be realized.The torque has connections with the amplitude of the stator current.For sinusoidal PMSM, the size of d i is independent of electromagnetic torque, and it has the superiority of low loss and high efficiency.The speed sensor adopts hall elements, the speed outer loop of the control system adopts STSM-ADRC control, and the PI control is applied in the inner loop.Finally, the structural diagram of the PMSM velocity control is expressed in Figure 3.

Motor vector control block diagram based on STSM-ADRC
To deal with the chattering problem brought by the discontinuity of the first-order derivative in the first-order sliding mode and the unmodeled characteristics caused by frequent switching control, the STSMC algorithm is proposed.The control input's derivative is applied to the derivative of the sliding mode surface as a new virtual control quantity, thereby weakening chattering and improving robustness.It is no longer sliding on the whole sliding surface of S = 0, but spiraling along the sliding manifold  =  ̇= 0 line.The super-twisting sliding mode control (STSMC) is derived from a secondorder sliding mode control algorithm [10] .The traditional STSMC control expression is: In the equation, the job is to design the rate parameters: 1 k and 2 k .Function sgn is a discontinuous function.In the control process, the discontinuous function will cause chattering in the system.Therefore, the hyperbolic tangent function tanh is adopted instead of the function sgn.
At the same time, in the traditional STSMC, 2 k in Equation ( 9) is a constant, and 2 k can be designed as an adaptive parameter, which simplifies the process of parameter tuning.The adaptive law is: At the same time, the total disturbance F is replaced by 2 z in Equation ( 6).

Comparative Analysis of System Simulation
According to the constructed STSM-ADRC structural diagram exhibited in Figure 3, there is a simulation model which is constructed to identify the more efficient result of the proposed strategy.
The parameters of PMSM are shown in Table 1.Given the motor's initial speed of 1400 r/min, at 0.07 s, one load with the size of 5 Nmis suddenly loaded to the motor, and the entire simulation process lasts for 0.15 s.The motor speed simulation waveform is in Figure 4. Figure 4 illustrates that after the motor starts, the motor speed under PI control overshoots, the peak value is about 15 r/min, and the speed reaches stability at 0.015 s.The motor speed under ADRC reaches 1400 r/min at 0.013 s.The motor speed under the control of STSM-ADRC reaches better without any overshoot, the velocity becomes stable at 0.01 s.At 0.07 s, the load was abruptly added to the motor, the speed under PI control decreased by 35 r/min, and reached 1400 r/min after 0.025 s.The speed under ADRC decreased by 27 r/min and reached 1400 r/min after 0.0032 s.The motor speed under STSM-ADRC control decreased by 25 r/min and reached 1400 r/min after 0.003s.Moreover, by pointing at the regulation for motor velocity change, the starting velocity is 1400 r/min, then, the velocity is reduced to 1200 r/min at 0.07 s, and the entire simulation process lasts for 0.15s.The speed simulation waveform of the motor appears in Figure 5.
Figure 5 explicates that the reference speed is reduced to 1200 r/min at 0.07 s, and the motor speed under PI control is reduced to 1121 r/min, reaching 1200 r/min after 0.007 s.The motor speed under ADRC reaches 1200 r/min after 0.005 s.The motor speed under the control of STSM-ADRC does not drop below 1200 r/min and reaches 1200 r/min after 0.003 s.
Based on simulation results, STSM-ADRC is superior to PI and ADRC in overshoot and fastness.

Conclusion
The STSM-ADRC control method is suggested in this research.The adaptive law also contributes to the parameter k2 in STSM-ADRC.It can satisfy the requirements in sudden load changes and frequent speed changes which is better than the traditional PI control and ADRC.And the research also proves that this control effect about the motor speed control is strengthened, the anti-interference ability is enhanced, and the response speed of this control is improved.

Figure 1 .
Figure 1.ADRC control system block diagram

Figure 4 .
Figure 4. Speed curve under load disturbance Figure 5. Speed curve under speed change