This paper presents a novel non-singular fast terminal sliding mode control (NFTSMC) based on the deep flux weakening switching point tracking method in order to improve the control performance of permanent interior magnet synchronous motor (IPMSM) drive systems. The mathematical model of flux weakening (FW) control is established, and the deep flux weakening switching point is calculated accurately by analyzing the relationship between the torque curve and voltage decline curve. Next, a second-order NFTSMC is designed for the speed loop controller to ensure that the system converges to the equilibrium state in finite time. Then, an extended sliding mode disturbance observer (ESMDO) is designed to estimate the uncertainty of the system. Finally, compared with both the PI control and sliding mode control (SMC) by simulations and experiments with different working conditions, the method proposed has the merits of accelerating convergence, improving steady-state accuracy, and minimizing the current and torque pulsation.
Interior permanent magnet synchronous motor (IPMSM) has merits concering simple structure, flux density and strong coercivity of neodymium-iron-boron permanent magnet [
Common FW control strategies include the formula calculation method, lookup table (LUT), gradient descent, and negative
At present, the PI algorithm has become the mainstream method for FW control because of its simple algorithms and mature technology. The PI algorithm is a linear control method that is widely used in motor control. However, IPMSM is a nonlinear system with various uncertainties, such as internal disturbance caused by high-speed and unmodeled dynamic excitation caused by deepened coupling of the
Many advanced nonlinear control theories have been applied in IPMSM high-performance in recent years. For example, model predictive control (MPC) [
To reduce the dependence on motor parameters, sliding mode control (SMC) has become a hot topic vis-a-vis the FW control of IPMSM. An SMC controller was designed to replace the PI controller in the voltage closed-loop [
This paper presents a novel NFTSMC method based on deep flux weakening switching point tracking. It achieves fast and accurate tracking of reference speed and improves the robustness of the IPMSM drive system. The main contributions of this paper are summarized as follows:
This method that combines NFTSMC with FW control is presented to improve the speed response of the IPMSM in flux weakening region. It has the features of both NFTSMC and FW control. More specifically, while the FW control based on deep flux weakening switching point tracking expands the actual speed range, the NFTSMC based on extended sliding mode disturbance observer (ESMDO) accelerates speed convergence, improves steady-state accuracy, and minimizes the current pulsation and torque pulsation. The mathematical model of MTPV control is simplified. The deep flux weakening switching point is obtained by analyzing the relationship between the torque and the voltage drop curves. The NFTSMC speed controller is designed based on the input and the output, which reduces the dependence on motor parameters. ESMDO estimates the unknown part of external disturbances. It is added to the input of the NFTSMC speed controller as a feedforward compensation item.
Ignoring stator core saturation, eddy current, and hysteresis loss, permanent rotor magnet without winding damping winding and mutual leakage inductance between stator windings, stator voltage equation of IPMSM in the
In steady-state operation, the stator voltages can be approximately simplified as:
The motor operation should meet the following requirements:
FW control consists of three processes: maximum torque per ampere (MTPA) control, constant power control, and maximum torque per volt (MTPV) control.
When the actual motor speed is lower than the nominal speed, the current trajectory of MTPA runs along the curve OA in
Constructing Lagrange function from
The current trajectory in the shallow flux weakening region runs along with the curve AB in
The current trajectory in the deep flux weakening region runs along the curve BC in
Substituting
The expression of the
When the motor is running, the saturation of the output voltage of the inverter is judged by constructing a voltage closed-loop. When
The conventional
The current trajectory moves leftward from point A to point B. If setting point E as a switching point, the actual current cannot reach point B, resulting in a narrow speed regulation range. It can be seen from
According to
Substituting
To obtain the decline direction of voltage, a new voltage change function is designed by
Combining
According to
The
To further simplify the control algorithm, the relationship between compensation value of
To avoid integral saturation of PI and chattering problem of SMC, a second-order NFTSMC speed controller is designed to improve the performance of IPMSM, and an ESMDO is designed to estimate the unknown disturbances.
The equation of electromagnetic torque
The mechanical motion equation of IPMSM is:
Substituting
To accelerate speed transient response and improve steady-state control accuracy, an NFTSMC speed controller is designed. From
Substituting
Define the state error of the controller as:
Then, the state variable is defined as
Selecting a second-order NFTSMC surface as below [
Choose the exponential reaching law as:
From
The derivative of
Due to
It completes the proof.
Substituting
The ESMDO can estimate the uncertainty of the system by measuring the input and output of the actual system. It provides not only practical feasibility for the technical realization and practical applications in motor control. The variable is defined as
Combining
Select the speed error as the sliding mode surface:
Choose the exponential reaching law as:
The control law of the sliding mode observer can be obtained from
The derivative of
Chosen
It completes the proof.
Substituting
Substituting
To weaken the chattering of the system due to the sgn function, the continuous saturation function
To verify the feasibility and effectiveness of the proposed method, this section demonstrates the comparative analysis results of simulations and experiments.
MATLAB/Simulink simulation is used to build a motor model compared with PI, conventional SMC, and NFTSMC.
Parameters | Unit | Values |
---|---|---|
Dc voltage/ |
V | 600 |
Rated speed/ |
r/min | 1900 |
Stator resistance/ |
2.75 | |
Pole number/ |
pairs | 2 |
H | 0.004 | |
H | 0.009 | |
Magnetic flux/ |
Wb | 0.12 |
Inertia/ |
0.029 |
PI | SMC1 | NFTSMC |
---|---|---|
/ | ||
/ | / | |
/ | / | |
/ | / | |
/ | / | |
/ | / | |
/ | / |
Note: 1The sliding mode surface and reaching law of SMC:
The load torque is set as
0 | 0.4 | 0.8 | 1.6 | |
---|---|---|---|---|
1000 | 2000 | 4000 | 6000 |
The simulation comparison results are summarized in
Convergence time/s | Speed relative error rate1/% | |
---|---|---|
1000 | 0.09/0.09/0.09 | 0.48/0.48/0.10 |
2000 | 0.49/0.49/0.49 | 0.27/0.22/0.05 |
4000 | 1.00/1.00/0.98 | 0.15/0.02/0.02 |
6000 | 1.96/1.96/1.90 | 0.10/0.08/0.01 |
Note: 1Relative speed error rate represents the ratio of steady speed error amplitude of PI/SMC/NFTSMC to reference speed.
From
Setting the reference speed as a slope function to simulate uniform acceleration condition.
Convergence time/s | Speed relative error rate/% |
---|---|
1.50/1.50/1.50 | 0.03/0.018/0.001 |
From
From
To further test the effectiveness of the proposed method, the hardware-in-the-loop simulation (HILS) of IPMSM is carried out with DSP-TMS320F2812 and RT-Lab (OP5600) HILS platform.
Compared with both
Performance indicators | PI | SMC | NFTSMC |
---|---|---|---|
Torque error /Nm | |||
Torque ripple1/% | 20.0 | 15.0 | 10.7 |
Average speed error rate2/% | 0.2 | 0.16 | 0.04 |
THD value of phase current/% | 9.27 | 6.38 | 2.21 |
Notes: Torque ripple1: The ratio of torque error amplitude to the reference torque. Average speed error rate2: The average speed relative error rate with different working conditions.
This paper presents a novel NFTSMC based on deep flux weakening switching point tracking of FW control for IPMSM drive system. The deep flux weakening switching point is accurately obtained, and the NFTSMC speed controller is designed based on the input and the output to accelerate speed convergence. The unknown part of external disturbances is estimated quickly by ESMDO, which is fed back to the NFTSMC speed controller. The comparative analysis results of simulations and experiments indicate that the method proposed has excellent speed response. In addition, it can effectively minimize the current and torque pulsation, and the IPMSM drive system has better transient steady-state performance.