summary
Based on the nonlinear dynamic mathematical model of a permanent magnet synchronous motor (PMSM) , a direct feedback linearization control method is adopted to achieve input-output linearization of the PMSM system. The speed tracking control designed using this method improves the speed tracking speed while ensuring the stability of the entire closed-loop system. Furthermore, a load torque disturbance observer is employed to reduce the impact of load disturbances on the speed. Simulation results show that the feedback linearization control design is simple and has excellent speed tracking performance.
Keywords: Permanent magnet synchronous motor, direct feedback linearized torque disturbance estimation
Abstract According to nonlinear dynamic mathematical model of permanent magnet synchronous motor, the PMSM system is input-output linearized with the method of direct feedback linearization. Through this linearization model, the speed tracking control law is designed, and under the condition which can guarantee the stability of the whole closed loop system, it improves the celerity of speed tracking. Formore, the observer of disturbance torque is applied, and reduce the effect of torque disturbance to speed. The simulation results show that the designed method is simpleness which has better speed tracking performances.
Keywords: PMSM, Direct feedback linearization, Disturbance Torque Estimation
1 Introduction
Based on the nonlinear dynamic mathematical model of a permanent magnet synchronous motor, a back-propagation control method is adopted. This method overcomes the shortcomings of traditional PID control in handling nonlinear control. The speed tracking control designed using this method can improve the speed tracking speed while ensuring the stability of the entire closed-loop system. Furthermore, a load torque disturbance observer is employed to reduce the impact of load disturbances on speed. Simulation results show that the back-propagation control design is simple and has excellent speed tracking performance.
With the development of permanent magnet materials, semiconductor power devices, and control theory, permanent magnet synchronous motors (PMSMs) are playing an increasingly important role in current medium- and low-power motion control. They possess advantages such as compact structure, high power density, high air gap flux, and high torque-to-inertia ratio. Therefore, they are increasingly widely used in servo systems. However, a permanent magnet synchronous motor is a nonlinear system containing a product of angular velocity *w* and current * id* or * q* . Therefore, to achieve precise control performance, angular velocity and current must be decoupled. For high-precision speed tracking control problems, load disturbances can affect speed fluctuations. Therefore, it is necessary to estimate load disturbances to reduce their impact.
Therefore, conventional linear control methods are not ideal. To address this control problem, current nonlinear control methods mainly include variable structure control, feedback linearization, and passive control. However, the design methods for these nonlinear controls are complex and difficult to understand. This paper proposes a Backstepping control strategy based on the coordinate transformation method of vector control. This strategy not only achieves complete decoupling of the permanent magnet synchronous motor system and has a simple design method, but also demonstrates significantly superior control performance compared to traditional PID control. Furthermore, a load torque disturbance observer is designed to reduce the impact of load disturbances on speed fluctuations.
2. Thrust-back control of permanent magnet synchronous motors
2.1 Mathematical Model
The surface-mount permanent magnet synchronous motor has the following d-q model based on the synchronously rotating rotor coordinates:
Where: u <sub>d</sub> and u<sub>q </sub> are the d-axis and q-axis stator voltages; i <sub>d </sub> and i<sub> q</sub> are the d-axis and q-axis stator currents; R is the stator resistance; L is the stator inductance; TL is the constant load torque; J is the moment of inertia; B is the viscous friction system; P is the number of pole pairs; ω is the rotor mechanical angular velocity; and φ<sub> f </sub> is the permanent magnet flux.
Equations (1), (2), and (3) can be simplified as follows:
2.2 Coordinate Transformation
To achieve system decoupling and avoid the zero-dynamic system problem, ω and φf are chosen as the system outputs, and new output variables are defined as follows:
Since the system is a three-input, three-output system, and its relative orders are {1, 1, 1}, meaning their sum equals the system order, the system can be linearized with feedback and does not exhibit zero-dynamic problems. Let the control input be:
Thus, state feedback control can be designed according to the pole placement theory of linear systems as follows:
Substituting equations (1), (2), and (3) into equations (7) and (8), we obtain the actual control quantities uq and ud :
3. Load Disturbance Observer Design
In some high-precision servo systems, load disturbances can cause speed fluctuations, leading to a decrease in system servo performance. Therefore, in high-precision speed tracking control, it is necessary to estimate load disturbances and compensate for them online in real time.
Since load disturbances are not easily measured directly, they can be observed using the obtained iq and ω. Considering that the measurement of iq and ω will produce noise errors, a filter is added to the output of the TL observer to eliminate the above-mentioned effects. Taking the Laplace transform of equation (14), we get:
Let, taking the inverse Laplace transform, we get:
The designed load disturbance observer is shown in Figure 1.
Figure 1 Load disturbance observer
4 System Example Simulation
The block diagram of the direct feedback linearized control of the permanent magnet synchronous motor based on torque disturbance estimation is shown in Figure 2. The system is adjusted to reach a satisfactory configuration point. The parameters of the permanent magnet synchronous motor are shown in Table 1.
Table 1 Parameters of Permanent Magnet Synchronous Motor
The block diagram of the thrust-back control for a permanent magnet synchronous motor based on torque disturbance estimation is shown in Figure 2. The system is adjusted to reach a satisfactory configuration point. The parameters of the permanent magnet synchronous motor are shown in the attached table. Assuming a reference speed of 500 r/min, and a sudden load of 20 Nm applied at 0.2 s, the thrust-back control parameters are: k1 = 50000, k2 = 300, k3 = 20, T0 = 0.01. The simulation results are shown in Figure 3. A partial magnification of the circle in Figure 3 is shown in Figure 4. Curve 1 in Figure 4 is the speed tracking curve under thrust-back control, and curve 2 is the speed tracking curve under thrust-back control with torque disturbance estimation. The simulation results show that thrust-back control enables the system to achieve rapid speed tracking while ensuring good dynamic performance. Furthermore, thrust-back control with torque disturbance estimation further accelerates the system's tracking speed and reduces the impact of disturbances on speed fluctuations.
Figure 2 System control block diagram
To implement the back-thrust control method based on load disturbance estimation, the TMS320LF2810 DSP chip, dedicated to motor control, was selected as the digital controller, and corresponding software was developed for implementation. Figure 5 shows the implementation of the back-thrust control strategy and the generation of SVPWM via a timer interrupt subroutine. This paper applies the back-thrust control based on torque disturbance estimation to speed tracking of a permanent magnet synchronous motor. This design method reduces the number of adjustment parameters and simplifies the system's control design. Matlab simulations demonstrate that the system has excellent tracking performance, verifying the effectiveness and feasibility of the system design. Furthermore, this control strategy has been applied to the key project of Zhejiang Province: "All-Digital AC General-Purpose Servo Drive System." It shows that the number of adjustment parameters is relatively reduced compared to PID control. Parameter tuning is easier, reducing programming work, and the system has achieved good results.
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About the Author
Dr. Liu Dongliang's main research areas are AC servo drive systems and nonlinear control strategies.
Senior Engineer Yan Weican mainly engages in research on motor control.
