Adaptive Backstepping Nonsingular Terminal Sliding Mode Control of Servo System Based on New Sliding Mode and Reaching Law

: Aiming at the problem that the control accuracy of permanent magnet synchronous motor (PMSM) servo system is easily affected by parameter uncertainty, an adaptive backstepping nonlinear nonsingular terminal sliding mode control (ABNNTSMC) method is proposed in this paper. Based on the existing fast nonsingular terminal sliding mode, a new piecewise nonlinear nonsingular terminal sliding mode is proposed to improve the convergence speed and ensure the steady-state accuracy. At the same time, a new reaching law with attenuation term is proposed to reduce the vibration gradually. Finally, the adaptive law is used to estimate the moment of inertia and viscous friction coefficient of the system to improve the robustness of the system. Simulation results show that ABNNTSMC has higher steady-state accuracy and smaller vibration compared with several existing results.

changes and strong robustness to disturbances. The common sliding surfaces are linear sliding surface [9], integral sliding surface [10], fractional sliding surface [11] and terminal sliding surface. Among them, the biggest advantage of the terminal sliding surface is that it can strictly prove the convergence time of the sliding surface, which provides a mathematical guarantee for the parameter tuning of the sliding surface. Then, with the continuous improvement of many scholars, the singular point problem of terminal sliding surface after derivation is solved, and further developed to fast nonsingular terminal sliding surface [12][13][14].
Besides the improvement of sliding mode surface, reaching law is also an important part of sliding mode control design. The design of reaching law not only determines the speed of the servo system converging to the sliding surface, but also affects the chattering. Reference [6] showed that a new reaching law combines exponential reaching law and power reaching law, which synthesizes the advantages of the two reaching laws. But these reaching laws all have a common problem, that is, the value is only related to s, so that the amplitude of buffeting does not decrease with the decrease of state error. Therefore, the reaching law with attenuation term has become an important research direction in recent years. Reference [15] showed that an improved adaptive variable rating exponential reaching law is proposed, which can shorten the time to reach the equilibrium point and reduce the chattering at the same time.
The advantage of backstepping control is that it can effectively deal with the nonlinear control problem of the system, adjust fewer parameters, and is easy to realize in engineering. Therefore, backstepping control has been widely valued by academia. Reference [16] showed that an adaptive backstepping control for nonlinear switched systems is proposed based on equivalent mapping.
Reference [17] showed that the backstepping control is further combined with the recurrent Hermite polynomial neural network to provide adaptive error estimation for nonlinear backstepping control.
Finally, sliding mode control is combined with backstepping control in many literatures. References [1,2] showed that adaptive sliding mode backstepping control is introduced to electro-hydraulic servo system, which overcomes the nonlinear problem in complex system. Reference [18] showed that adaptive radial basis function network based on sliding mode backstepping controller is introduced, which can overcome the uncertainty of the system. Among the uncertainties of the system, the moment of inertia and viscous friction coefficient are the key factors to determine the control performance of the system. Among the traditional parameter identification methods, the common ones are sine signal integration method [19,20], steady-state direct method [21,22], least square method with forgetting factor [23]. There is a significant problem in these methods, that is, they need to have specific conditions for speed and load torque. However, it is impossible for the system to keep in a specific motion condition all the time, so the adaptive identification algorithm will have more advantages. Reference [24] showed that genetic algorithm is used to optimize adaptive parameters to estimate unknown disturbances in perturbed systems. On the basis of backstepping control, Lyapunov function method is used to design adaptive estimation of load variables [25,26]. Therefore, the adaptive algorithm is introduced into the identification of moment of inertia and viscous friction coefficient to ensure that the estimated value can track the actual value in real time.
The contribution of this paper is that based on the traditional nonsingular terminal sliding surface, a nonsingular terminal sliding surface with nonlinear switching term is proposed. The sliding surface can ensure faster speed at start-up and higher steady-state accuracy when the response curve is close to the given signal. Then a reaching law with attenuation term is designed, which can further reduce the chattering as the error decreases. Finally, this paper combines sliding mode control with backstepping control, and introduces an adaptive identification algorithm for moment of inertia and viscous friction coefficient to enhance the robustness of the control system.

Design of sliding mode controller for PMSM
The derivation of sliding mode control algorithm is based on the mathematical model of permanent magnet synchronous motor [26]. The d-q models of motor in synchronous rotating coordinate system are expressed as follows: where,  is the rotation angle,  is the rotation speed, n P is the pole pair number, f  is Based on the above mathematical model, the step input signal is defined as d  firstly, then the tracking error e and the differential of tracking error e are defined as follows: Reference [14] defined a nonsingular terminal sliding mode as follow: This sliding mode has high convergence rate and can guarantee the convergence in finite time.
However, when the system state is close to the equilibrium state, the convergence rate will decrease and the convergence accuracy is insufficient. In this paper, the sliding mode is improved as follow: where, c is the coefficient of linear part of sliding mode, 0 c  .
The comparison of the response curves of the two sliding modes is shown in Fig. 1:

Fig. 1 Comparison of two sliding modes
It can be seen from Fig. 1 that the rising speed of 1 s is faster than that of 2 s , but the convergence accuracy of 2 s is higher than that of 1 s . Therefore, a new nonlinear switching sliding mode is proposed by combining the two sliding modes. When the error is large and the system state is far away from the equilibrium state, the 1 s is adopted; when the error is small and the system state is close to the equilibrium state, 2 s is adopted. 11 where,  is the error threshold of sliding mode switching.
After the sliding mode is designed, a new reaching law is improved based on the reaching law in Ref. [14] as follows, In the reaching law, ks  can accelerate the reaching speed when the system state is far away from the sliding mode, and reduce the reaching speed when the system state is close to the sliding mode. When the system is in the start-up phase, e is larger; When the system response is close to the given signal, the error e and e are smaller. Therefore, the es  can ensure that the system has a larger reaching speed when it starts up. And In practical application, the uncertainty of system parameters will have an important impact on the control effect. In particular, the moment of inertia and friction viscosity coefficient will inevitably change in the long-term work. Therefore, this paper uses the adaptive algorithm to estimate the moment of inertia and viscous friction coefficient, and estimates J and B with Ĵ and B , to provide accurate parameters for the control variables. At this time, the control quantity q u is as follows, Errors of estimate Ĵ and B are defined as follows, Adaptive laws of Ĵ and B are defined as follows, The Lyapunov function 2 V is defined as follows, 22 According to Lyapunov stability theorem, the estimated value of system parameters will converge asymptotically, and the state of the system will converge to sliding mode surface. That is, even if the motor parameters are uncertain due to long-term operation, the state of the system can still ensure stable convergence and has strong robustness.

Simulations
The simulation model in this paper borrows the parameter data in Ref. [26], as shown in Table 1:

Table 1 Model parameters and controller parameters
According to the parameters in Table 1, the simulation model is built, as shown in Fig. 2: Parameter of backstepping controller q k = d k =100 Parameter of adaptive controller 1  = 2  =100 [14], adaptive linear sliding mode control (ALSMC) [5] and global sliding mode control (GSMC) [10].
The simulation results are shown in Fig. 3-Fig. 7. The comparison of simulation data of various sliding mode control is shown in Table 2: The frequency is high and the amplitude decreases slightly It can be seen from Fig. 3 that ABNNTSMC has the strongest tracking performance, and it can track the input signal in 0.008s. And there is no overshoot at all because ABNNTSMC will switch to the sliding mode surface with smaller steady-state accuracy when it is close to the stable state. However, the tracking performance of the other three methods is worse than ABNNTSMC, and GSMC has obvious overshoot. Therefore, ABNNTSMC is the best in tracking performance.
It can be seen from Fig. (a) of Fig. 4 that ABNNTSMC and ANFTSMC have the highest steady-state accuracy. Both ALSMC and GSMC have obvious vibration, which indicates that their steady-state accuracy is insufficient. It can be seen from Fig. (b) that ABNNTSMC and ANFTSMC can reach the steady-state accuracy of 10 -4 within 0.1s, but the convergence accuracy of ABNNTSMC is higher than that of ANFTSMC. This shows that ABNNTSMC has the highest steady-state accuracy.
It can be seen from Fig. 5 that ABNNTSMC has the maximum rising speed, which can reach 769.2 rad/s. Moreover, when ABNNTSMC completes the tracking of the input signal, the speed can quickly drop to 0 rad/s, and the state is very stable. However, the rising speed of the other three methods is too slow, so it takes a longer time to complete the tracking of the input signal, resulting in response lag. This shows that ABNNTSMC can not only achieve high tracking speed, but also stabilize rapidly after tracking.
It can be seen from Fig. 6 that the GSMC belongs to the integral sliding surface, so the system state is on the sliding surface from the beginning. In addition to the other three kinds of sliding mode control, ABNNTSMC only needs 0.005s to reach the sliding mode surface without overshoot.
ANFTSMC can also approach the sliding mode surface at 0.06s, but the later approaching speed is too slow. Although ALSMC has a fast approaching speed, it has serious overshoot and does not meet the requirements of high precision. This shows that ABNNTSMC has a fast approaching speed, and there is no overshoot. The reaching law proposed in this paper can meet the higher tracking requirements.
It can be seen from Fig. (a) of Fig. 7 that the vibration frequency of ABNNTSMC is very low, and the vibration amplitude is gradually decreasing, from 10 -6 to 10 -8 , and then to lower. It can be seen from

Conclusion
For PMSM servo system, in order to improve the previous sliding mode control design, an adaptive backstepping nonlinear nonsingular terminal sliding mode control (ABNNTSMC) is proposed.