B. Research on coupled control of MPMSMs based on NAISMC and SMDO

In synchronous coupling control, multi-motor system is prone to problems such as unstable operation of the motor system and reduced control performance. In view of this phenomenon, SMC and disturbance observer are used to optimize the whole motor system. SMC is also named variable structure control and its essence is a kind of nonlinear control. In the dynamic process, it can make intentional changes with the current state of the system [19]. It is not affected by object parameters and disturbances, and has the characteristics of fast response, insensitivity to disturbances, and simple implementation.

SMC has a very important role in MPSMS control, which can provide better multi-motor control results. SMC is a sliding mode-based control method that tracks the desired control target by constantly changing the form of the current model. This control method has several advantages. First is to better control stability. By changing the form of the current model, SMC can enhance the stability of MPMSM system. Secondly, it can improve the dynamic performance of the multi-motor system. By switching different forms of sliding mode, the SMC can improve the dynamic performance of the system, so that the system can still maintain the best state under the changing external conditions. Next, it can reduce the system control error. By using sliding mode, SMC can reduce the error and thus improve the accuracy of the system. Finally, flexible control between multiple motors can be achieved. By switching different sliding mode forms, SMC can achieve flexible control effects to meet different drive requirements.

Formula (6) is the definition of sliding surface function. S is the sliding surface function. x refers to the system status parameter. t indicates the time, and R^m denotes the real number range.

Formula (7) is the solution of SMC function. μ is the control variable. μ⁺(x,t)≠μ⁻(x,t) expresses the sliding mode in the sliding state.

Because PMSM is a strongly coupled nonlinear system, it responds quickly to the changes of parameters and is vulnerable to external interference. It is difficult to make the MMSC stable. This paper studies the optimization of traditional SMC by using NAISMC. Taking the surface-mounted PMSM as the research object, its mathematical model expression is shown in Formula (8) [20].

In Formula (8), U_x and U_y denote the shaft voltage on the x and y axes respectively. φ_x and φ_y are stator flux chains. ω_e is the electrical angular velocity. Ri_x and Ri_y represent the stator resistance of the motor on the x and y axes.

Formula (9) is the calculation of stator flux linkage [21, 22]. In Formula (9), L_x and L_y are the inductance components on the x and y axes, respectively. For the convenience of calculation, the two values should be the same. φ_f is the permanent magnet magnetic chain.

Formula (10) shows the mechanical motion expression of the motor. J indicates the moment of inertia. T_e stands for electromagnetic torque. T_L indicates the load torque. B means the damping coefficient. ω_m expresses the mechanical angular velocity [23, 24].

The NAISMC is mainly composed of two parts: reaching and maintaining motion. On the basis of rapidly approaching the sliding surface, it is also necessary to maintain the stability of the sliding surface under the action of the control law. According to Formulas (6) to (10), NAISMC is constructed.

Formula (11) shows the mathematical expression of NAISMC [24]. Where, s˙ denotes the sliding surface function under the optimization of reaching law. k indicates the switching item coefficient, k>0. sat(s) is the introduced saturation function. ka+(1−a)e−δ|s|2+λ|s|β means a toggle item. When |s| is large, the switch item becomes ka+λ|s|β. This can make the whole system converge to stability faster and make the error variable reach the sliding surface quickly. When |s| is small, the switching item becomes the coefficient k. At this time, the error variables only switch in the boundary layer, and the system is more stable. 0<a<1, λ>0, δ>0, k₁>0, 1<β<2. All of the above are motor controller parameters. e is a mathematical constant.

At present, there are many researches on the design of disturbance observer, for example, the design of load torque observer which can resist the change of load disturbance by generating electromagnetic torque [25]. The observer can effectively improve the load capacity of the system, but it has the defects of slow response speed, long convergence time, and excessive compensation current. In addition, some scholars have introduced adaptive control algorithm into the disturbance observer to accurately measure the variation of load disturbance. It is compensated to speed loop regulator to reduce load disturbance. The anti-disturbance effect of the observer is better than that of the traditional observer. However, it still has the defects of instability and easy overshoot, so there is still room for optimization. The improved reaching law is used to optimize the SMC. A SMDO based on the improved reaching law is proposed. It aims to optimize the observation indexes of motor load disturbance. For PMSM, due to the limitation of the rated load of the motor, its load effect changes little, so the following expression can be satisfied [26].

Formula (12) is the expression for the integration of the load torque. Let the mechanical angular velocity ω_m be the output quantity, d be the total perturbation, and T be the internal parameter uptake. The mathematical model of the sliding mode perturber is obtained using the integral sliding mode surface as shown in Formula (13)

In Formula (13), e_ω indicates the control law when the speed difference is the input. λ_ω denotes integral sliding mode gain. The approach law algorithm is optimized, so that the final sliding mode perturber meets the requirements as shown in Formula (14).

In Formula (14), X indicates the independent variable error. q is the approach law coefficient. h(X) meets the limiting conditions shown in Formula (15) [27].

In Formula (15), k₂ is the approaching law coefficient. η, τ are the independent variable boundary layer, which satisfies η>0 and 0<τ<1. Lyapunov function can prove the stability of the disturbance observer.

The improved SMC and SMDO are used in the improved DCC of MPMMS. The control structure schematic diagram is shown in Fig 4. To test the results of improved DCC of MPMMS with NAISMC and SMDO, a simulation experiment will be designed on the basis of this control structure schematic diagram to analyze the motor performance under traditional and improved SMC.