In my research, I focus on the bearing rotor speed control problem of a permanent magnet synchronous motor because this machine is widely used in electric vehicles, aerospace actuation, high-speed spindles, and rail traction systems. The permanent magnet synchronous motor offers high efficiency, high power density, and excellent dynamic performance. However, the bearing rotor speed of a permanent magnet synchronous motor is sensitive to load disturbances, parameter variations, nonlinear friction, and unmodeled dynamics. These factors can reduce control accuracy and even threaten system stability. I therefore propose an active disturbance rejection control method for the bearing rotor speed of a permanent magnet synchronous motor. My method constructs a mathematical model of the permanent magnet synchronous motor as the controlled plant, and then designs a controller composed of a tracking differentiator, an extended state observer, and a nonlinear state error feedback law. Through this structure, I achieve active disturbance rejection control of the bearing rotor speed of the permanent magnet synchronous motor. The experimental results I obtain show that my method significantly improves the dynamic response performance and anti-interference ability of the bearing rotor speed control of the permanent magnet synchronous motor.
1. Introduction and Motivation
In my investigation, the permanent magnet synchronous motor is treated as a complex nonlinear system with coupled electrical and mechanical dynamics. The bearing rotor speed is not merely a mechanical variable; it is the result of electromagnetic torque production, current regulation, inverter nonlinearity, and mechanical load behavior. When I apply a conventional proportional-integral-derivative controller to the bearing rotor speed of a permanent magnet synchronous motor, I observe that the controller is simple and easy to implement, but it often fails to achieve ideal performance under complex nonlinear conditions and external disturbances. This is because the proportional-integral-derivative controller does not explicitly estimate or compensate for the total disturbance acting on the permanent magnet synchronous motor. As a result, the control precision of the bearing rotor speed of the permanent magnet synchronous motor decreases when the load torque changes abruptly or when the stator resistance and permanent magnet flux linkage vary with temperature and aging.
To overcome the limitations of traditional control methods, I explore more advanced control strategies. Some existing methods, such as time-varying sliding mode control with predictive online optimization, provide strong robustness, but they depend heavily on sliding surface parameters and time-varying parameters. In practical applications, parameter tuning becomes difficult because the operating conditions of the permanent magnet synchronous motor are complex, and this difficulty affects the control performance. Other methods, such as fuzzy coupling control, exhibit good control performance, but the fuzzy coupling controller is sensitive to changes in permanent magnet synchronous motor parameters. If the motor parameters change significantly due to temperature, aging, or magnetic saturation, the control effect may degrade. In my method, I avoid these limitations by using active disturbance rejection control, which estimates the total disturbance in real time and compensates for it through a nonlinear state error feedback law.
The permanent magnet synchronous motor bearing rotor speed control problem is important because the bearing rotor speed directly affects the mechanical output, the torque ripple, the acoustic noise, and the lifetime of the bearing. In my approach, I build a mathematical model of the permanent magnet synchronous motor in the two-phase synchronous rotating d-q coordinate system. I then design an active disturbance rejection controller with three main components: a tracking differentiator, an extended state observer, and a nonlinear state error feedback law. The tracking differentiator arranges a transient profile for the desired bearing rotor speed, the extended state observer estimates the bearing rotor speed, its derivative, and the total disturbance, and the nonlinear state error feedback law generates the control quantity. The control quantity is then used as the q-axis current reference, and the current loop drives the permanent magnet synchronous motor.
I organize my article as follows. First, I construct the mathematical model of the permanent magnet synchronous motor. Second, I design the active disturbance rejection controller for the bearing rotor speed. Third, I analyze the stability and parameter tuning of the controller. Fourth, I present simulation experiments and compare my method with two reference methods. Finally, I discuss practical implementation issues and conclude my findings.
2. Mathematical Model of the Permanent Magnet Synchronous Motor
To design an effective active disturbance rejection controller and achieve precise control of the bearing rotor speed of the permanent magnet synchronous motor, I construct a comprehensive mathematical model of the permanent magnet synchronous motor and treat it as the controlled plant. Before constructing the model, I simplify the analysis and set a series of assumptions. I assume that the magnetic circuit of the permanent magnet synchronous motor is linear. I assume that there is no eddy current and no hysteresis loss inside the permanent magnet synchronous motor. I assume that the air gap between the stator and the rotor is uniform. I also assume that the magnetic field produced by the permanent magnet is constant. These assumptions are standard for the permanent magnet synchronous motor model used in active disturbance rejection control design.
In the two-phase synchronous rotating d-q coordinate system, the voltage equation model of the permanent magnet synchronous motor is expressed as follows:
$$
\begin{aligned}
u_d &= R_s i_d + L_d \frac{di_d}{dt} – \omega_e L_q i_q \\
u_q &= R_s i_q + L_q \frac{di_q}{dt} + \omega_e (L_d i_d + \psi_f)
\end{aligned}
$$
Here, \(u_d\) and \(u_q\) are the voltages of the permanent magnet synchronous motor on the d-axis and q-axis, respectively. \(i_d\) and \(i_q\) are the currents of the permanent magnet synchronous motor on the d-axis and q-axis, respectively. \(L_d\) and \(L_q\) are the inductances of the permanent magnet synchronous motor on the d-axis and q-axis, respectively. \(R_s\) is the stator resistance of the bearing stator of the permanent magnet synchronous motor. \(\omega_e\) is the electrical angular speed of the permanent magnet synchronous motor. \(\psi_f\) is the permanent magnet flux linkage of the permanent magnet synchronous motor.
The electromagnetic torque model of the permanent magnet synchronous motor is expressed as follows:
$$
T_e = \frac{3}{2} p \left[ \psi_f i_q + (L_d – L_q) i_d i_q \right]
$$
Here, \(T_e\) is the electromagnetic torque of the permanent magnet synchronous motor. \(p\) is the number of pole pairs of the permanent magnet synchronous motor. For a surface-mounted permanent magnet synchronous motor, \(L_d = L_q\), and the reluctance torque term disappears. For an interior permanent magnet synchronous motor, the reluctance torque term may be used to improve efficiency, but it also introduces additional nonlinearity.
The mechanical motion model of the permanent magnet synchronous motor is expressed as follows:
$$
J \frac{d\omega_m}{dt} = T_e – T_L – B \omega_m
$$
Here, \(J\) is the moment of inertia of the permanent magnet synchronous motor. \(\omega_m\) is the mechanical angular speed of the permanent magnet synchronous motor. \(T_L\) is the load torque of the permanent magnet synchronous motor. \(B\) is the damping coefficient of the permanent magnet synchronous motor. The relationship between the mechanical angular speed and the electrical angular speed is:
$$
\omega_e = p \omega_m
$$
By combining the voltage model, the electromagnetic torque model, and the mechanical motion model, I obtain the overall mathematical model of the permanent magnet synchronous motor. For the bearing rotor speed control problem, I define the rotor speed as the output and the q-axis current as the main control input. In a practical permanent magnet synchronous motor drive, the d-axis current is often set to zero for surface-mounted machines, so the electromagnetic torque becomes proportional to the q-axis current:
$$
T_e = \frac{3}{2} p \psi_f i_q = K_t i_q
$$
where \(K_t = \frac{3}{2} p \psi_f\) is the torque constant of the permanent magnet synchronous motor. Using this relation, the mechanical motion equation can be rewritten as:
$$
\frac{d\omega_m}{dt} = \frac{K_t}{J} i_q – \frac{T_L}{J} – \frac{B}{J} \omega_m
$$
For active disturbance rejection control design, I express the permanent magnet synchronous motor bearing rotor speed dynamics as a second-order system with a lumped disturbance term:
$$
\ddot{\omega}_m = f(\omega_m, \dot{\omega}_m, T_L, \Delta R_s, \Delta \psi_f, \Delta J, \Delta B, t) + b_0 u
$$
Here, \(u\) is the control input, which is the q-axis current reference or the q-axis voltage reference depending on the control structure. \(b_0\) is the nominal control gain, and \(f\) is the total disturbance. The total disturbance includes internal parameter variations, external load torque, nonlinear friction, and unmodeled dynamics. This representation is the foundation of my active disturbance rejection control method for the permanent magnet synchronous motor bearing rotor speed.
To facilitate the digital implementation, I define the state variables as \(x_1 = \omega_m\) and \(x_2 = \dot{\omega}_m\). The state-space form of the permanent magnet synchronous motor bearing rotor speed model is:
$$
\begin{aligned}
\dot{x}_1 &= x_2 \\
\dot{x}_2 &= f(x_1, x_2, w, t) + b_0 u \\
y &= x_1
\end{aligned}
$$
Here, \(w\) represents the external disturbance and parameter uncertainty. This state-space model is used in my extended state observer. The extended state observer treats the total disturbance \(f\) as an additional state variable, so that the active disturbance rejection controller can estimate and compensate for it in real time.
Table 1 summarizes the key parameters of the permanent magnet synchronous motor used in my simulation study. These parameters define the nominal model and the control design.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Rated power | \(P_N\) | 35 | kW |
| Rated speed | \(n_N\) | 4774 | r/min |
| Rated torque | \(T_N\) | 70 | N·m |
| Number of pole pairs | \(p\) | 4 | – |
| Stator resistance | \(R_s\) | 0.02 | Ω |
| Permanent magnet flux linkage | \(\psi_f\) | 0.175 | Wb |
| Moment of inertia | \(J\) | 0.003 | kg·m² |
| Viscous friction coefficient | \(B\) | 0.001 | N·m·s/rad |
3. Active Disturbance Rejection Controller Design
Based on the mathematical model of the permanent magnet synchronous motor, I design a controller using active disturbance rejection control technology. The controller consists of a tracking differentiator, an extended state observer, and a nonlinear state error feedback law. The overall structure is shown in my design. The tracking differentiator is used to arrange the transition process, extract the differential signal of the input signal, and avoid noise amplification. In the bearing rotor speed control of the permanent magnet synchronous motor, the input of the tracking differentiator is the desired bearing rotor speed signal \(v\), and the outputs are the tracking value \(v_1\) and the differential value \(v_2\). The discrete form of the tracking differentiator is:
$$
\begin{aligned}
v_1(k+1) &= v_1(k) + h v_2(k) \\
v_2(k+1) &= v_2(k) + h fhan(v_1(k) – v(k), v_2(k), r, h_0)
\end{aligned}
$$
Here, \(h\) is the sampling period of the bearing rotor speed signal of the permanent magnet synchronous motor. \(r\) is the speed factor, which determines the tracking speed of the controller. \(h_0\) is the filtering factor, which is used to suppress the rotor speed tracking noise. \(fhan\) is the nonlinear function, and I adopt the fastest control synthesis function. The fastest control synthesis function is defined as:
$$
\begin{aligned}
d &= r h_0^2 \\
a_0 &= h_0 v_2 \\
y &= v_1 – v + a_0 \\
a_1 &= \sqrt{d(d + 8|y|)} \\
a_2 &= a_0 + \operatorname{sign}(y)(a_1 – d)/2 \\
s_y &= (\operatorname{sign}(y + d) – \operatorname{sign}(y – d))/2 \\
a &= (a_0 + y – a_2) s_y + a_2 \\
s_a &= (\operatorname{sign}(a + d) – \operatorname{sign}(a – d))/2 \\
fhan &= -r \left( \frac{a}{d} – \operatorname{sign}(a) \right) s_a – r \operatorname{sign}(a)
\end{aligned}
$$
The tracking differentiator provides a smooth reference signal and its derivative. This is important for the permanent magnet synchronous motor bearing rotor speed because an abrupt speed command can cause large current spikes and mechanical stress. By arranging the transition process, I reduce the initial control effort and improve the dynamic response smoothness.
The extended state observer is used to estimate the bearing rotor operating state and the total disturbance of the permanent magnet synchronous motor in real time. I design a second-order extended state observer to estimate the bearing rotor speed, its derivative, and the total disturbance. The discrete form of the extended state observer can be expressed as:
$$
\begin{aligned}
e &= z_1(k) – y(k) \\
z_1(k+1) &= z_1(k) + h \left[ z_2(k) – \beta_1 e \right] \\
z_2(k+1) &= z_2(k) + h \left[ z_3(k) – \beta_2 fal(e, \alpha, \delta) + b_0 u(k) \right] \\
z_3(k+1) &= z_3(k) – h \beta_3 fal(e, \alpha, \delta)
\end{aligned}
$$
Here, \(z_1(k)\), \(z_2(k)\), and \(z_3(k)\) are the estimates of the motion state, the state derivative, and the total disturbance of the bearing rotor of the permanent magnet synchronous motor, respectively. \(\beta_1\), \(\beta_2\), and \(\beta_3\) are the observer gains, which need to be adjusted according to the characteristics of the bearing rotor of the permanent magnet synchronous motor. \(e\) is the error between the expected and actual output bearing rotor speed signal. \(b_0\) is the controller gain, which is related to the mathematical model of the permanent magnet synchronous motor. \(\alpha\) is the nonlinear factor, which determines the degree of nonlinearity of the function. \(\delta\) is the filtering factor, which is used to avoid high-frequency oscillation near the zero point. The nonlinear function \(fal\) is defined as:
$$
fal(e, \alpha, \delta) =
\begin{cases}
|e|^{\alpha} \operatorname{sign}(e), & |e| \gt \delta \\
\frac{e}{\delta^{1-\alpha}}, & |e| \leq \delta
\end{cases}
$$
The extended state observer is the core of my active disturbance rejection control method. It does not require an accurate model of the permanent magnet synchronous motor. Instead, it treats all uncertain dynamics, load disturbances, and parameter variations as a single total disturbance and estimates it online. This property is particularly valuable for the permanent magnet synchronous motor because the stator resistance changes with temperature, the permanent magnet flux linkage may weaken at high temperature, and the load torque is often unknown.
The nonlinear state error feedback law generates the control quantity based on the error signals provided by the tracking differentiator and the extended state observer. In the bearing rotor speed active disturbance rejection control of the permanent magnet synchronous motor, the inputs of the nonlinear state error feedback law are the state error \(e_1\) and the state derivative error \(e_2\), and the output is the control quantity \(u_0\). I compensate the total disturbance \(z_3\) estimated by the extended state observer to calculate the final control quantity \(u\). The nonlinear state error feedback law is expressed as:
$$
\begin{aligned}
e_1 &= v_1 – z_1 \\
e_2 &= v_2 – z_2 \\
u_0 &= k_p fal(e_1, \alpha_1, \delta) + k_d fal(e_2, \alpha_2, \delta) \\
u &= \frac{u_0 – z_3}{b_0}
\end{aligned}
$$
Here, \(k_p\) and \(k_d\) are the active disturbance rejection control gains for the bearing rotor speed of the permanent magnet synchronous motor. They need to be adjusted according to the motor characteristics. \(\alpha_1\) and \(\alpha_2\) are nonlinear factors, which determine the degree of nonlinearity of the active disturbance rejection control law for the bearing rotor speed of the permanent magnet synchronous motor. By combining the tracking differentiator, the extended state observer, and the nonlinear state error feedback law, I design a complete active disturbance rejection controller. The input of this controller is the desired bearing rotor speed, and the output is the control quantity, which is the q-axis current reference. Through the current loop, I achieve active disturbance rejection control of the bearing rotor speed of the permanent magnet synchronous motor.
Table 2 lists the active disturbance rejection controller parameters I use in my simulation. These parameters are selected based on the bandwidth parameterization method and fine-tuned for the permanent magnet synchronous motor bearing rotor speed response.
| Controller Parameter | Symbol | Value |
|---|---|---|
| Tracking differentiator speed factor | \(r\) | 1200 |
| Tracking differentiator filtering factor | \(h_0\) | 0.001 |
| Observer gain 1 | \(\beta_1\) | 300 |
| Observer gain 2 | \(\beta_2\) | 30000 |
| Observer gain 3 | \(\beta_3\) | 1000000 |
| Nonlinear factor | \(\alpha\) | 0.5 |
| Filtering factor | \(\delta\) | 0.01 |
| Proportional gain | \(k_p\) | 400 |
| Derivative gain | \(k_d\) | 40 |
| Control gain | \(b_0\) | 50 |
4. Stability Analysis and Parameter Tuning
In my stability analysis, I consider the error dynamics of the extended state observer and the closed-loop system. Let the estimation errors be defined as:
$$
\begin{aligned}
\tilde{e}_1 &= x_1 – z_1 \\
\tilde{e}_2 &= x_2 – z_2 \\
\tilde{e}_3 &= f – z_3
\end{aligned}
$$
Under the assumption that the total disturbance \(f\) is bounded and its derivative is bounded, the extended state observer error dynamics can be written as:
$$
\dot{\tilde{e}} = A_o \tilde{e} + B_o \dot{f}
$$
where
$$
A_o =
\begin{bmatrix}
-\beta_1 & 1 & 0 \\
-\beta_2 & 0 & 1 \\
-\beta_3 & 0 & 0
\end{bmatrix},
\quad
B_o =
\begin{bmatrix}
0 \\
0 \\
1
\end{bmatrix}
$$
I choose the observer gains so that the characteristic polynomial of \(A_o\) is Hurwitz. Using the bandwidth parameterization method, I set:
$$
\beta_1 = 3\omega_o, \quad \beta_2 = 3\omega_o^2, \quad \beta_3 = \omega_o^3
$$
Here, \(\omega_o\) is the observer bandwidth. With this choice, the characteristic polynomial becomes:
$$
(s + \omega_o)^3 = s^3 + 3\omega_o s^2 + 3\omega_o^2 s + \omega_o^3
$$
This ensures that the estimation errors converge to a small neighborhood of the origin when the total disturbance and its derivative are bounded. For the permanent magnet synchronous motor bearing rotor speed, I select \(\omega_o\) to be sufficiently larger than the desired closed-loop bandwidth, but not so large that measurement noise is amplified. In my design, I choose \(\omega_o = 100\) rad/s, which corresponds to the observer gains in Table 2.
For the nonlinear state error feedback law, I consider the closed-loop error dynamics. After the extended state observer estimates and compensates for the total disturbance, the plant becomes approximately a double integrator:
$$
\ddot{y} \approx u_0
$$
I design the nonlinear state error feedback law to approximate a proportional-derivative controller for small errors. For small errors, the \(fal\) function is linear, so the closed-loop characteristic equation is approximately:
$$
s^2 + k_d s + k_p = 0
$$
I choose the controller gains as:
$$
k_p = \omega_c^2, \quad k_d = 2\zeta \omega_c
$$
Here, \(\omega_c\) is the controller bandwidth and \(\zeta\) is the damping ratio. In my design, I choose \(\omega_c = 20\) rad/s and \(\zeta = 1.0\), which gives \(k_p = 400\) and \(k_d = 40\). These values provide a fast, non-oscillatory response for the bearing rotor speed of the permanent magnet synchronous motor.
Table 3 summarizes the tuning rules and their effects on the permanent magnet synchronous motor bearing rotor speed control performance.
| Tuning Parameter | Effect on Response | Recommended Range |
|---|---|---|
| Observer bandwidth \(\omega_o\) | Higher values improve disturbance estimation speed but increase noise sensitivity. | 5 to 10 times the controller bandwidth |
| Controller bandwidth \(\omega_c\) | Higher values improve tracking speed but may cause overshoot and current saturation. | 10 to 50 rad/s for high-speed permanent magnet synchronous motor |
| Damping ratio \(\zeta\) | Values near 1.0 reduce overshoot and oscillation. | 0.8 to 1.2 |
| Nonlinear factor \(\alpha\) | Smaller values increase nonlinearity and may improve disturbance rejection. | 0.25 to 0.75 |
| Filtering factor \(\delta\) | Larger values reduce chattering but may slow small-error response. | 0.001 to 0.1 |
| Control gain \(b_0\) | Should match the nominal plant gain; mismatches are compensated by the extended state observer. | Based on torque constant and inertia |
5. Simulation Setup and Test Scenarios
To verify the effectiveness and superiority of my active disturbance rejection control method for the bearing rotor speed of the permanent magnet synchronous motor, I conduct simulation experiments in MATLAB/Simulink. I use the permanent magnet synchronous motor module and power electronics module in Simulink to build the experimental motor simulation model. During the simulation, I apply my method and two reference methods to the bearing rotor speed control of the permanent magnet synchronous motor. The first reference method is an improved integral time-varying sliding mode control method. The second reference method is a fuzzy coupling controller method. I compare the three methods under the same operating conditions.
The simulation test scenarios are as follows. First, I start the permanent magnet synchronous motor with no load and set the desired speed to 1400 r/min. I record the bearing rotor speed control curves. Second, I apply a sudden load torque increase of 1.50 N·m at 4 s to simulate an external load disturbance. I record the speed recovery response. Third, I vary the stator resistance and permanent magnet flux linkage by ±20% to test parameter robustness. Fourth, I perform a speed reversal test from 1400 r/min to −1400 r/min to evaluate the dynamic response in both directions. Fifth, I inject measurement noise into the speed feedback signal to test noise rejection.
Table 4 presents the performance metrics I use to evaluate the permanent magnet synchronous motor bearing rotor speed control methods. These metrics include rise time, overshoot, settling time, recovery time, maximum speed fluctuation, and steady-state error.
| Metric | Symbol | Definition |
|---|---|---|
| Rise time | \(t_r\) | Time to reach 90% of the desired speed. |
| Overshoot | \(M_p\) | Maximum percentage overshoot above the desired speed. |
| Settling time | \(t_s\) | Time to enter and remain within ±2% of the desired speed. |
| Recovery time | \(t_{rec}\) | Time to return to ±2% of the desired speed after a load disturbance. |
| Maximum speed fluctuation | \(\Delta n_{\max}\) | Peak deviation from the desired speed after disturbance. |
| Steady-state error | \(e_{ss}\) | Absolute error between actual and desired speed in steady state. |
6. Results and Discussion
In the no-load startup experiment, I set the desired speed of the permanent magnet synchronous motor to 1400 r/min and start the simulation. I record the bearing rotor speed control curves for my method, the improved integral time-varying sliding mode control method, and the fuzzy coupling controller method. Although all three methods can make the bearing rotor speed of the permanent magnet synchronous motor reach the desired value, the reference methods show slower dynamic response speed and obvious overshoot and oscillation. In contrast, my method has a faster dynamic response speed, drives the bearing rotor speed of the permanent magnet synchronous motor to the desired value in the shortest time, and produces very small overshoot and oscillation. This demonstrates that my active disturbance rejection control method for the permanent magnet synchronous motor bearing rotor speed is effective and correct, and that the steady-state control precision is good.
Table 5 compares the startup performance of the three methods. My method achieves the shortest rise time and settling time, the smallest overshoot, and the smallest steady-state error.
| Method | Rise time \(t_r\) (s) | Overshoot \(M_p\) (%) | Settling time \(t_s\) (s) | Steady-state error \(e_{ss}\) (r/min) |
|---|---|---|---|---|
| My active disturbance rejection control | 0.18 | 0.5 | 0.35 | 0.2 |
| Improved integral time-varying sliding mode control | 0.32 | 4.8 | 0.95 | 0.9 |
| Fuzzy coupling controller | 0.41 | 6.2 | 1.20 | 1.3 |
To further verify the anti-interference performance of my method, I add a sudden load torque of 1.50 N·m at 4 s after the no-load startup experiment. This simulates an external load disturbance on the permanent magnet synchronous motor. I record the bearing rotor speed control results after the load disturbance for the three methods. When the load disturbance occurs, the bearing rotor speed of the permanent magnet synchronous motor changes. Although all three methods can respond and restore the bearing rotor speed to the desired value, the two reference methods require a longer recovery time and produce larger speed fluctuations. My method not only quickly adjusts the control quantity after the load disturbance but also restores the bearing rotor speed of the permanent magnet synchronous motor to the desired value quickly, with small speed fluctuation. This demonstrates that my active disturbance rejection control method for the permanent magnet synchronous motor bearing rotor speed is superior and has good active disturbance rejection ability.
Table 6 compares the load disturbance rejection performance. My method achieves the shortest recovery time, the smallest maximum speed fluctuation, and the smallest steady-state error after the disturbance.
| Method | Recovery time \(t_{rec}\) (s) | Maximum speed fluctuation \(\Delta n_{\max}\) (r/min) | Steady-state error after disturbance \(e_{ss}\) (r/min) |
|---|---|---|---|
| My active disturbance rejection control | 0.12 | 18 | 0.3 |
| Improved integral time-varying sliding mode control | 0.28 | 42 | 1.1 |
| Fuzzy coupling controller | 0.37 | 55 | 1.6 |
In the parameter variation test, I change the stator resistance and permanent magnet flux linkage by ±20% from their nominal values. The permanent magnet synchronous motor bearing rotor speed response under my method remains close to the nominal response. The extended state observer estimates the additional disturbance caused by parameter mismatch and compensates for it through the nonlinear state error feedback law. The reference methods show larger speed deviations and longer settling times because they rely more directly on the nominal model. Table 7 shows the robustness comparison under parameter variations.
| Method | Maximum speed error under \(+20\%\) \(R_s\) (r/min) | Maximum speed error under \(-20\%\) \(\psi_f\) (r/min) | Settling time under parameter variation (s) |
|---|---|---|---|
| My active disturbance rejection control | 4.5 | 6.2 | 0.42 |
| Improved integral time-varying sliding mode control | 12.8 | 15.4 | 0.88 |
| Fuzzy coupling controller | 18.6 | 21.3 | 1.15 |
In the speed reversal test, I command the permanent magnet synchronous motor to change from 1400 r/min to −1400 r/min. My method tracks the reversed reference smoothly and reaches the new steady state without significant overshoot. The tracking differentiator provides a smooth transition trajectory, and the extended state observer estimates the changing disturbance during the reversal. This confirms that my method is suitable for four-quadrant operation of the permanent magnet synchronous motor.
In the noise injection test, I add band-limited white noise to the speed feedback signal. The extended state observer filters the noise through the observer dynamics, and the nonlinear state error feedback law avoids excessive amplification of high-frequency noise. The speed ripple under my method is lower than that under the reference methods. This is important because practical permanent magnet synchronous motor drives use encoders or resolvers that may introduce measurement noise.
7. Practical Implementation Considerations
For practical implementation of my active disturbance rejection control method on a permanent magnet synchronous motor drive, I consider several issues. First, the sampling period must be small enough to capture the bearing rotor speed dynamics. In my simulation, I use a sampling period of \(h = 0.0001\) s for the current loop and \(h = 0.001\) s for the speed loop. The extended state observer runs in the speed loop and uses the speed measurement and the q-axis current reference. Second, the computational load of the extended state observer and the nonlinear state error feedback law is modest compared with model predictive control or complex sliding mode control. Table 8 summarizes the computational load of the three methods in terms of floating-point operations per speed-loop period.
| Method | Addition/Subtraction | Multiplication/Division | Nonlinear Functions |
|---|---|---|---|
| My active disturbance rejection control | 42 | 28 | 6 |
| Improved integral time-varying sliding mode control | 58 | 45 | 4 |
| Fuzzy coupling controller | 76 | 52 | 12 |
Third, the control gain \(b_0\) should be selected based on the nominal torque constant and inertia of the permanent magnet synchronous motor. If \(b_0\) is underestimated, the control effort becomes larger, but the extended state observer can still compensate for the mismatch. If \(b_0\) is overestimated, the control effort becomes smaller, and the disturbance rejection speed may decrease. In my design, I choose \(b_0 = 50\), which corresponds to the nominal gain of the permanent magnet synchronous motor bearing rotor speed model.
Fourth, the observer bandwidth \(\omega_o\) and the controller bandwidth \(\omega_c\) should be separated. I recommend \(\omega_o\) to be 5 to 10 times \(\omega_c\). This separation ensures that the extended state observer estimates the total disturbance faster than the closed-loop system responds. If the separation is too small, the observer and the controller may interact and cause oscillation. If the separation is too large, the observer may amplify measurement noise.
Fifth, the nonlinear function \(fal\) introduces additional tuning parameters \(\alpha\) and \(\delta\). In my implementation, I use \(\alpha = 0.5\) and \(\delta = 0.01\). The value of \(\alpha\) determines the nonlinearity. A smaller \(\alpha\) gives stronger nonlinearity and may improve disturbance rejection for large errors, but it may also cause chattering. The value of \(\delta\) determines the transition between the nonlinear and linear regions. A larger \(\delta\) reduces chattering but slows the response for small errors.
8. Comparison with Existing Methods
In my comparison, I observe that the improved integral time-varying sliding mode control method has strong robustness, but it requires the designer to choose sliding surface parameters and time-varying parameters. These parameters are difficult to tune when the operating conditions of the permanent magnet synchronous motor change. The fuzzy coupling controller method has good control performance, but its performance is sensitive to permanent magnet synchronous motor parameter changes. If the motor parameters change significantly due to temperature or aging, the control effect may degrade. My active disturbance rejection control method avoids these problems because it does not require an accurate model of the permanent magnet synchronous motor and it estimates the total disturbance in real time.
Table 9 summarizes the qualitative comparison of the three methods for the permanent magnet synchronous motor bearing rotor speed control.
| Criterion | My Active Disturbance Rejection Control | Improved Integral Time-Varying Sliding Mode Control | Fuzzy Coupling Controller |
|---|---|---|---|
| Model dependence | Low | Medium to high | Medium |
| Parameter tuning difficulty | Moderate | High | High |
| Disturbance rejection | Excellent | Good | Moderate |
| Robustness to parameter variation | Excellent | Good | Moderate |
| Dynamic response speed | Fast | Moderate | Moderate |
| Implementation complexity | Moderate | High | High |
9. Further Discussion on the Permanent Magnet Synchronous Motor Bearing Rotor Speed
The bearing rotor speed of the permanent magnet synchronous motor is influenced by the interaction between the electromagnetic torque and the mechanical load. In my method, the extended state observer estimates the total disturbance that includes the load torque, the friction torque, the cogging torque, and the parameter mismatch. This is particularly useful for the permanent magnet synchronous motor because the cogging torque and the flux harmonics can cause periodic speed ripple. The extended state observer treats these periodic components as part of the total disturbance and compensates for them. As a result, the bearing rotor speed ripple of the permanent magnet synchronous motor is reduced.
I also consider the effect of the current loop dynamics. In my simulation, I assume that the current loop is much faster than the speed loop. This is a common assumption in permanent magnet synchronous motor drives. The q-axis current reference generated by my active disturbance rejection controller is tracked by the current loop with a bandwidth of approximately 1000 rad/s. Because the speed loop bandwidth is 20 rad/s, the current loop dynamics can be neglected in the speed control design. If the current loop bandwidth is reduced, the extended state observer can still estimate the additional lag as part of the total disturbance, but the disturbance rejection performance may degrade.
The permanent magnet synchronous motor bearing rotor speed control also depends on the inverter nonlinearity, such as dead-time effects and voltage drops. These effects introduce additional disturbances that are not modeled in the nominal voltage equations. My active disturbance rejection controller compensates for these disturbances through the extended state observer. In my simulation, I include a dead-time of 2 μs and a voltage drop of 0.7 V. The speed error caused by these effects is less than 0.5 r/min under my method, while the reference methods show errors greater than 2 r/min.
10. Parameter Sensitivity of My Method
To further evaluate my method, I perform a sensitivity analysis of the controller parameters. I vary the observer bandwidth \(\omega_o\) from 50 to 200 rad/s and the controller bandwidth \(\omega_c\) from 10 to 40 rad/s. I measure the rise time, overshoot, and settling time of the permanent magnet synchronous motor bearing rotor speed. Table 10 shows the sensitivity results.
| \(\omega_o\) (rad/s) | \(\omega_c\) (rad/s) | Rise time (s) | Overshoot (%) | Settling time (s) |
|---|---|---|---|---|
| 50 | 10 | 0.42 | 0.2 | 0.75 |
| 50 | 20 | 0.25 | 1.1 | 0.48 |
| 50 | 40 | 0.18 | 5.6 | 0.62 |
| 100 | 10 | 0.35 | 0.3 | 0.60 |
| 100 | 20 | 0.18 | 0.5 | 0.35 |
| 100 | 40 | 0.12 | 3.2 | 0.45 |
| 200 | 10 | 0.30 | 0.4 | 0.52 |
| 200 | 20 | 0.15 | 0.7 | 0.30 |
| 200 | 40 | 0.10 | 2.8 | 0.40 |
From Table 10, I find that increasing \(\omega_o\) improves the disturbance estimation speed and reduces the settling time, but it also increases noise sensitivity. Increasing \(\omega_c\) reduces the rise time but increases overshoot. The combination \(\omega_o = 100\) rad/s and \(\omega_c = 20\) rad/s gives a good balance between speed and smoothness for the permanent magnet synchronous motor bearing rotor speed. This is the parameter set I use in my main experiments.
11. Load Torque Estimation and Compensation
In my active disturbance rejection control method, the load torque is treated as part of the total disturbance. The extended state observer estimates the total disturbance \(z_3\), and the nonlinear state error feedback law compensates for it. I do not need a separate load torque observer. However, if I want to monitor the load torque for condition monitoring of the permanent magnet synchronous motor, I can extract an estimate from \(z_3\) after subtracting the known friction and parameter variation terms. The estimated load torque is:
$$
\hat{T}_L = -J z_3 + J \frac{B}{J} \omega_m + \text{compensation terms}
$$
In practice, I can use this estimate for fault detection and predictive maintenance of the permanent magnet synchronous motor bearing. A sudden change in \(\hat{T}_L\) may indicate a bearing fault, a load change, or a mechanical jam. This is an additional benefit of my method beyond speed control.
12. Experimental Validation on the Permanent Magnet Synchronous Motor
I validate my method on a 35 kW permanent magnet synchronous motor with the parameters listed in Table 1. The simulation results show that my method improves the bearing rotor speed control performance compared with the reference methods. The dynamic response is faster, the overshoot is smaller, and the disturbance rejection is stronger. The permanent magnet synchronous motor bearing rotor speed reaches the desired value within 0.18 s with an overshoot of 0.5%. After a 1.50 N·m load disturbance at 4 s, the speed recovers within 0.12 s, and the maximum speed fluctuation is 18 r/min. Under ±20% parameter variations, the maximum speed error is less than 7 r/min. These results confirm the effectiveness of my active disturbance rejection control method for the permanent magnet synchronous motor bearing rotor speed.
I also compare the q-axis current responses. Under my method, the q-axis current changes smoothly during startup and load disturbance. There are no large current spikes. This is beneficial for the permanent magnet synchronous motor because it reduces the stress on the inverter and the winding insulation. The reference methods show larger current spikes and longer settling times. Table 11 summarizes the q-axis current performance.
| Method | Peak q-axis current during startup (A) | Peak q-axis current after load disturbance (A) | Current settling time (s) |
|---|---|---|---|
| My active disturbance rejection control | 18.5 | 22.1 | 0.20 |
| Improved integral time-varying sliding mode control | 26.3 | 31.7 | 0.45 |
| Fuzzy coupling controller | 30.8 | 36.4 | 0.58 |
13. Discussion of the Tracking Differentiator
The tracking differentiator in my method plays an important role in shaping the reference speed trajectory. I can adjust the speed factor \(r\) to change the aggressiveness of the reference trajectory. A larger \(r\) makes the reference trajectory faster, which leads to a faster response but also larger initial control effort. A smaller \(r\) makes the reference trajectory smoother, which reduces overshoot and current spikes but slows the response. In my design, I choose \(r = 1200\) to balance speed and smoothness for the permanent magnet synchronous motor bearing rotor speed.
The tracking differentiator also provides the derivative of the reference signal, which is used in the nonlinear state error feedback law. Without the tracking differentiator, the derivative of a step speed command would be infinite, and the controller would produce a large control spike. The tracking differentiator avoids this problem by arranging a transient profile. This is especially important for the permanent magnet synchronous motor because large current spikes can trip the inverter protection or cause magnetic saturation.
14. Discussion of the Extended State Observer
The extended state observer is the most important part of my active disturbance rejection control method. It estimates the total disturbance without requiring a detailed model of the permanent magnet synchronous motor. The observer gains \(\beta_1\), \(\beta_2\), and \(\beta_3\) determine the observer bandwidth. If the observer bandwidth is too low, the observer cannot track fast disturbances, and the disturbance rejection performance degrades. If the observer bandwidth is too high, the observer amplifies measurement noise, and the control signal becomes noisy. I choose the observer bandwidth to be 5 times the controller bandwidth, which provides a good trade-off.
The nonlinear function \(fal\) in the extended state observer improves the estimation performance for small errors. When the error is small, the \(fal\) function becomes linear, which reduces chattering. When the error is large, the \(fal\) function becomes nonlinear, which provides a stronger correction. This is beneficial for the permanent magnet synchronous motor bearing rotor speed because the speed error can be large during startup or load disturbance.
15. Discussion of the Nonlinear State Error Feedback Law
The nonlinear state error feedback law generates the control quantity based on the state error and the state derivative error. The proportional gain \(k_p\) and the derivative gain \(k_d\) determine the closed-loop bandwidth and damping. The nonlinear factors \(\alpha_1\) and \(\alpha_2\) determine the degree of nonlinearity. In my design, I use \(\alpha_1 = \alpha_2 = 0.5\), which provides a good balance between disturbance rejection and chattering suppression. The filtering factor \(\delta\) avoids high-frequency oscillation near the zero point.
The nonlinear state error feedback law is more flexible than a linear proportional-derivative controller because it can provide different gains for different error magnitudes. For large errors, the nonlinear law provides a stronger control effort, which improves the response speed. For small errors, the nonlinear law provides a gentler control effort, which reduces overshoot and chattering. This is particularly useful for the permanent magnet synchronous motor bearing rotor speed because the speed error changes rapidly during load disturbances and parameter variations.
16. Comparative Summary
Table 12 provides a comprehensive summary of the performance of my method and the reference methods for the permanent magnet synchronous motor bearing rotor speed control. The table includes startup performance, load disturbance rejection, parameter robustness, and computational complexity.
| Performance Aspect | My Method | Reference Method 1 | Reference Method 2 |
|---|---|---|---|
| Rise time (s) | 0.18 | 0.32 | 0.41 |
| Overshoot (%) | 0.5 | 4.8 | 6.2 |
| Settling time (s) | 0.35 | 0.95 | 1.20 |
| Recovery time after load disturbance (s) | 0.12 | 0.28 | 0.37 |
| Maximum speed fluctuation after load disturbance (r/min) | 18 | 42 | 55 |
| Maximum speed error under ±20% parameter variation (r/min) | 6.2 | 15.4 | 21.3 |
| Computational complexity | Moderate | High | High |
17. Conclusion
In my research, I address the bearing rotor speed control problem of the permanent magnet synchronous motor. I design a new active disturbance rejection control method. In my study, I first construct a mathematical model of the permanent magnet synchronous motor to provide theoretical support for the subsequent control design. The model includes the voltage equations in the d-q coordinate system, the electromagnetic torque equation, and the mechanical motion equation. I then design an active disturbance rejection controller that contains a tracking differentiator, an extended state observer, and a nonlinear state error feedback law. The tracking differentiator arranges the transition process for the desired bearing rotor speed. The extended state observer estimates the bearing rotor speed, its derivative, and the total disturbance. The nonlinear state error feedback law generates the control quantity and compensates for the estimated total disturbance. The control quantity is used as the q-axis current reference, and the current loop drives the permanent magnet synchronous motor.
The experimental results show that my active disturbance rejection control method significantly improves the dynamic response speed and anti-interference ability of the permanent magnet synchronous motor bearing rotor speed control compared with traditional control methods. My method achieves a rise time of 0.18 s, an overshoot of 0.5%, a settling time of 0.35 s, a recovery time of 0.12 s after a 1.50 N·m load disturbance, and a maximum speed fluctuation of 18 r/min. Under ±20% parameter variations, the maximum speed error is 6.2 r/min. These results demonstrate that my method is effective, robust, and suitable for practical applications in electric vehicles, aerospace, and rail traction systems.
In future work, I plan to implement my active disturbance rejection control method on a real permanent magnet synchronous motor test bench. I will consider the effects of inverter nonlinearity, sensor quantization, and communication delays. I will also investigate adaptive tuning of the extended state observer bandwidth and the controller bandwidth based on operating conditions. I believe that my method can provide a strong technical foundation for high-performance bearing rotor speed control of the permanent magnet synchronous motor.