Research on improved sliding mode control of permanent magnet synchronous motor with disturbance compensation

In order to address the shortcomings of traditional sliding mode variable structure control in permanent magnet synchronous motor vector control, such as the state quantity’s inability to converge to 0 in limited time, too large chattering, too large load disturbance, poor dynamic performance, a non-singular fast terminal sliding mode control algorithm in speed loop is proposed. At the same time, a new approach rate is designed by comparing the advantages and disadvantages of different approach rates, which can restrain the natural chattering of sliding mode and improve the approach rate. On this basis, in order to increase response time, lessen chattering, and enhance the observer’s capacity for adaptation, a load torque observer is constructed, load disturbance is added to the speed loop as feedforward compensation, and a new approach rate is added to the observer. The simulation results manifest that the design successfully tracks load torque, enhances the system’s dynamic performance, and lessens the impact of sliding mode chattering.


INTRODUCTION
At present, PMSM is applied in electric vehicles, intelligent robots, aerospace, and other fields for its high power density and factor, superior control performance, and strong reliability [1] [2].Sliding mode variable structure control is concerned because of its low requirement on model precision and strong robustness.However, the chattering of traditional SMC is large, and the robustness and chattering of the motor control system are further affected because the load torque cannot be measured in the actual control.
In literature [3], the integrated sliding mode variable structure controller is intended to prevent chattering, suppress high-frequency disturbance, and eliminate the difference of velocity signal.The improvement of sliding mode reaching law [4][5] can also reduce chattering problems.In literature [6], both the velocity loop and the current loop are subjected to sliding mode control, and simulation results demonstrate that both the resilience and the speed of the system are greatly increased.However, the design is complicated and parameter selection is difficult.SMC is roughly linear sliding mode surface, which makes state quantity cannot converge to 0 in a finite time.In literature [7] [8], terminal sliding mode (TSM) is used, which can make the state variables have a better effect of zero convergence and also has strong robustness for model errors as well as an external disturbance.However, the singular phenomenon easily generated is also solved in literature [9].In addition, unknown load disturbance will greatly reduce the stability and dynamic performance.Therefore, an extended state observer is built [10], which effectively tracks load disturbance.
In this paper, a non-singular fast terminal sliding mode surface is used.A new reaching rate is devised, which greatly improves the arrival rate and reduces the jitter.An improved load torque observer is devised to introduce the load disturbance into the speed loop as feedforward compensation, which can accelerate the response speed and reduce chattering.The results of the simulation show that the design can effectively reduce chattering, improve dynamic performance, enhance robustness, and effectively lower the effect of load disturbance and variation on motor control.

Sliding surface design
The traditional linear sliding mode surface S(x)=Cx is relatively simple in design, however, the system state quantity can't converge to zero in finite time.In order to attain higher management performance, after the lookup and scan of a number of one-of-a-kind sliding mode surfaces, a non-singular fast terminal sliding mode surface (NFTSM) is chosen for plan and research.This sliding mode surface solves the trouble that the state quantity can't converge to 0 in a restrained time, avoids the occurrence of non-singular phenomena, and increases the response speed at the same time.The design of NFTSM is shown below: Where x is the state variable, the reference speed in PMSM is defined as ref  , and the actual speed is m  .
Then the speed error is: The rate of change of error is: p, q, u, and v are all odd numbers greater than 0, and 1 < p/q < 2, 1 < u/v < 2;  and  numbers are greater than zero.Formula (1) can converge to zero in a finite moment by designing appropriate parameters and sliding mode control law.When S=0, it is known that the rate of change of velocity error is: According to the above formula, the new / uv x  introduced by NFTSM can make the system converge faster than NTSM.When the system error moves towards the sliding mode plane, the convergence rate of the error is mainly influenced by exponential terms , and the convergence rate is close to exponential change.When the velocity error is small, i.e., when the state variable approaches the sliding mode plane, the rate of convergence of errors is mainly influenced by the linear term -x.Combined with each other, the errors can be guaranteed to converge quickly in different stages of convergence.

New approach rate design of NFTSM
The sliding mode approach rate can improve the dynamic performance quality and make the system move into the sliding mode quickly and effectively.The common approach law mainly includes the general approach rate, constant speed approach rate, power approach rate, etc.According to the advantages and disadvantages of the above approach rates, a new approach rate is designed on the basis of the power approach rate, which can reduce chattering and greatly increase the speed of the approach motion.
The design of the new approach rate is as follows: sgn( ), ( ) Where, k>0,  is the power term exponent; a and b are the maximum and minimum values that  can achieve, and a>1,0<b<1.
The main idea of the new approach law designed in this paper is to change the traditional power approach rate into the variable exponential power approach rate and add the system state variable x into the power term index to achieve this goal.By analyzing the approach law, we can see that: When the system running track keeps relatively away from the sliding mode surface, x is big, x e  approaches 0, and then is approximately equal to a(a>1).It closes to the sliding-mode surface at a fast speed, which effectively solves the problem that the low velocity and the long motion time when the traditional power-reaching law keeps away from the sliding-mode surface.As it approaches and reaches the surface, x inch by inch decreases to close to 0, then x e  gradually approaches 1, and  is approximately equal to b(0<b<1), which greatly reduces the reaching velocity when the system approaches the sliding-mode surface and is conducive to weakening chattering.
Next, to verify the stability of the new approach rate, the Lyapunov function is selected as: The derivative of the above equation can be obtained: Since k is greater than 0,V  less than 0. Therefore, the above equation satisfies the Lyapunov stability theorem, which shows that the overall stability is feasible

Design of NFTSM speed controller
The vector control with id=0 is adopted.The input velocity error of the speed controller is used to track the given mechanical angular velocity.The output is the reference value of the current of the q-axis.Take the derivative of Equation (3) again, and plug it into the electromagnetic torque equation： 3 2 By derivation of Equation (1), we can obtain: This is obtained by combining the above two formulas According to Equation (5), the speed control law can be written as:

DESIGN OF LOAD TORQUE OBSERVER
The variation of load torque and other parameters affects the robustness of the system.In this section, a sliding mode extended observer is built to observe the load disturbance changes, and the observed value is fed forward to the speed loop for adjustment, so as to realize the rapid suppression of the load disturbance.
Based on the PMSM electromagnetic torque and motion equation, because of the high switching frequency of the controller, the load torque can be considered to be a constant value in the control cycle.The PMSM equation of state is as follows: Establishing an extended sliding mode observer:  is the feedback gain, ˆm  is the evaluated value of the electrical angular velocity, and ˆL T is the evaluated value of the load torque.In the equation 2 ˆŝgn( ) ,  is consistent with the above, which is different from the traditional sliding mode extended observer.

ŝgn( )
, it speeds up the response, improves the tracking ability for unknown disturbances, and decreases the chattering effect.The observation error equation is obtained by subtracting Equation (13) from Equation (12) as follows：

SIMULATION
To verify the correctness and superiority of this design, Matlab/Simulink software is used for simulation research.
The given speed of the motor is 1000 r/min.The system starts with no load and runs with a 5N• m load at 0.2s.To verify the control effect of the new approach rate NFTSM speed control strategy, the SMC speed control strategy, traditional approach rate NFTSM speed control strategy, and the new approach rate NFTSM were compared and simulated respectively, and the simulation results were compared and analyzed.
The following figure compares the speed response of the new approaching rate NFTSM (NNFTSM) with that of the traditional SMC.As shown in Figure 1, under the control of NNFTSM, the SMC overshoot is significantly greater than that of NNFTSM, about 40 r/min, and only 20 r/min.Under the action of NNFTSM control, velocity enters the steady state at 0.013s, while under the action of SMC control, velocity enters the steady state at 0.015 s.It can be seen that the NNFTSM control can approach the given speed faster than the SMC control.When the load is abruptly added, the speed drops by 6r/min under the control of traditional SMC, and it is difficult to reach the given speed value.NNFTSM avoids the problem that SMC cannot rapidly converge to 0 within a limited time.After the load is abruptly added, the speed drops by 4r/min and quickly rises to the given speed, which has stronger robustness.
The following figure 2 compares the speed response of the new approaching rate NFTSM (NNFTSM) with that of the NFTSM with the general reach rate.As can be seen from the local magnifying diagram, under the NNFTSM speed controller, the response time is fast and the steady state is reached faster, and the given speed is reached at about 0.013s.The overshoot is smaller, about 20r/min, and the steady-state precision is high with almost no fluctuation.Under NFTSM control with a general reaching rate, overshoot is as high as 50r/min, with large fluctuations and obvious chattering, and the specified speed is reached stably at about 0.021s.When the load is disturbed by 5N at 0.2s, the NFTSM speed generally decreases by 7r/min, accounting for about 0.7% of the rated speed.The NFTSM speed can quickly recover to the given speed, and the required time is only 0.003 s.However, obvious chattering can be observed.The rotation speed decreased by 4r/min, but quickly recovered to the specified rotation speed after 0.003s, greatly reducing chattering and small fluctuation.The following figure respectively shows the electromagnetic torque response waveform under three kinds of control.We can see from Figure 3 to 5, under the control of NNFTSM, the electromagnetic fluctuation is smaller, and the fluctuation range is about 0.3 after the steady state, which indicates higher steady-state accuracy and better stability.
To verify the effect of the load torque observer designed, when PMSM runs stably, we change the load torque.The motor starts at 0s with 5 N• m, and the load torque becomes 10N• m at 0.2s.It can be obtained from Figure 6, the observed value has a lag of about 0.015 seconds compared with the actual value, so it has good tracking ability, and the chattering of the observed waveform is small, with high accuracy.The following figure shows the comparison simulation of the rotational speed response of NNFTSM with disturbance compensation ignored and disturbance compensation introduced:  7, the speed without disturbance compensation dropped about 10r/min when the load was suddenly added, and it took a long time to recover to the specified speed.When the load disturbance compensation was introduced, the speed dropped 4r/min and reached the specified speed after 0.004 seconds, which further improved the disturbance immunity and better dynamic performance.

CONCLUSIONS
This paper is based totally on the lookup on the sliding mode management of everlasting magnet synchronous motor.By way of the use of NFTSM in a speed loop, the error can't converge to zero in restrained time and the non-singular phenomenon is avoided.Then by using enhancing the achieving rate of the speed controller, the reaching rate is increased, and the effect of chattering is reduced.This approach rate is introduced into the load torque observer to reduce the chattering and achieve the tracking of the load disturbance.Through simulation analysis, the format in this paper successfully suppressed the chattering of the system and increased the response pace and anti-interference potential of the system, suggesting that the dynamic overall performance is superior.
the velocity evaluated 2 ˆLL e T T is the load torque estimation error, and the sliding mode surface is defined as 1 ˆmm se    .

Figure 1 .
Figure 1.Speed response of NNFTSM control and SMC control

Figure 2 .
Figure 2. NNFTSM control and NFTSM control speed response

Figure 7 .
Figure 7.Comparison of observer speed with or without load disturbance