A New Adaptive Control for Five-Phase Fault-Tolerant Flux-Switching Permanent Magnet Motor
The five-phase fault-tolerant flux-switching permanent magnet (FT-FSPM) motor can offer high efficiency and high fault-tolerant capability. In this paper, its operation principle is presented briefly and its mathematical model is derived. Further, a new adaptive control for an FT-FSPM motor, based on the backstepping method and the sliding mode control strategy, is proposed. According to the backstepping method, the current controllers and voltage control laws are designed to track the speed and minimize the current static error, which enhance the dynamic response and the ability to suppress external disturbances. In order to overcome the influence of parameter variations, according to sliding mode control theory, the virtual control variables and the adaptive algorithm are utilized to approach uncertainty terms. Three Lyapunov functions are designed, and the stability of the closed-loop system is analyzed in detail. Finally, both simulation and experimental results are presented to verify the proposed control method.
In recent times, flux-switching permanent magnet (PM) (FSPM) motors have gained considerable attention. In the stator, these motors have PMs which are easy to cool, thus reducing the risk of PM demagnetization caused by overheating of the magnets. Moreover, FSPM motors offer a simple rotor structure and good mechanical integrity [1–5]. Hence, they are suitable for high accuracy and high speed operations .
In certain high reliability applications, continuous operation of the motor drive is important. The three-phase motors have exhibited a good performance, but they still suffer from limitations under fault conditions . Multiphase PM motors have advantages such as high efficiency, high power density, and phase number redundancy, and they are widely used in special fields such as naval propulsions, aerospace, and military equipment . In the multiphase PM motors, the phase number redundancy ensures that the motor achieves a trouble-free operation and high reliability, even if a phase or a two-phase fault occurs. Recently, by adopting multiphase and fault-tolerant teeth, a five-phase fault-tolerant FSPM (FT-FSPM) motor was proposed, exhibiting merits like high fault tolerance and high reliability.
The precise mathematical model for the FT-FSPM motors is difficult to obtain, because it is a strong nonlinear system with unmeasurable internal and external unknown disturbances . As a result, it is difficult for the linear control schemes to achieve high performance. On the other hand, a variety of advanced intelligent strategies have been introduced to improve the operational performances of the motor drive [10, 11], such as the sliding mode control (SMC) , adaptive control , fuzzy control , neural network control , and robust control [15, 16]. Among these control strategies, the SMC has been known to be a very effective way to control the nonlinear system, because it is not sensitive to external disturbances and parameter variations, displays robustness and a fast response, and is easy to implement . However, its sliding mode surface suffers from chattering , which affects the control performances. Owing to the uncertainty of parameter variations, low-pass filters and the rotor position compensating methods are used to reduce sliding mode surface chattering . The backstepping control is a control strategy developed recently for uncertain and nonlinear systems, and it can linearize the nonlinear system using the state feedback . The control algorithm can be designed step by step through appropriate Lyapunov functions . The adaptive backstepping observer and the backstepping integral were presented to improve the robustness and tracking accuracy, but there still exist static errors and larger speed overshoots [21–23]. The combination of the backstepping control and the SMC has become a major strategy in the field of study of uncertain nonlinear systems. So far, the backstepping SMC method for the FT-FSPM motor system has almost never been reported.
This paper proposes a new backstepping SMC method for the FT-FSPM motors, which can remarkably suppress sliding mode chattering and accurately track the motor speed. Intermediate virtual control variables and the axis voltage control laws are designed to achieve the desired motor speed and position control, ensuring the stability of the system. The external interferences and internal mechanical parameter variations are collectively referred to as the uncertainty terms, which are approached by the adaptive learning algorithm, access to the best state of the system.
2. FT-FSPM Motor
Figure 1 shows the cross section of a five-phase FT-FSPM motor. This motor has 10 stator slots and 18 rotor poles. Its rotor is similar to that of a switched reluctance motor . The curial merit of this topology is the introduction of fault-tolerant teeth, which interleave with the armature teeth. In contrast, the stator of a conventional FSPM motor consists of many U-shaped laminated segments. The five-phase FT-FSPM motor consists of 10 “”-shaped units, namely, -cores, and a concentrated armature coil is wound around every two units, while a piece of magnet is embedded in the middle of two -cores. Thus, the phase windings of adjacent stator poles are essentially isolated, leading to the merits of phase decoupling. Therefore, this motor has good fault-tolerant capability, which can ensure high reliability of the motor system. Besides, owing to the additional teeth, it has less mutual inductance .
As shown in Figure 2, according to the principle of minimum reluctance, the flux will flow down from the stator core yoke as much as possible. Therefore, the armature reaction has no impact on PMs.
3. Mathematical Model
Although the FT-FSPM motor has PMs in the stator, its PM flux and back EMF (Electromotive Force) appear bipolar sinusoidal under no load. Hence, its fundamental coordinate system can be described as shown in Figure 3.
The simplified mathematical model of the FT-FSPM motor can be obtained by the coordinate transformation, and the motor system can be reduced, decoupled, and linearized. Since the motor has ideal sinusoidal flux linkage, only the fundamental flux linkage is considered in this work. According to the law of the magnetic circuit, the magnet equation is described aswhere is the leakage inductance, and are currents along the and axes respectively, is the PM flux, and are the stator inductances along the and axes, respectively, and and are the flux along the and axes, respectively. By the 5/2 coordinate transformation, the currents along the and axes, that is, and , are expressed aswhere , and are the five-phase currents and is the position angle of the rotor. From (2), the five-phase currents are transferred to the two-phase currents of the rotating coordinate system. Owing to the adoption of the field orient control, it is necessary to know the relationship between the currents, voltages, and torque in the rotating coordinate system. The stator voltage and electromagnetic torque equations of the rotor coordinate system can be described aswhere is the pairs number of rotor pole, and are the voltages along the and axes respectively, is the rotor angular speed, is the stator resistance, and is the electromagnetic torque. Although the -axis inductance is not equal to the -axis inductance, the difference between them is little. As evident from (3), the electromagnetic torque of the FT-FSPM motor is mainly related to the - axis current and inductance. According to Newton’s laws of motion, without considering the elastic torsional moment coefficient and the motor mechanical movement, the motion equation is expressed aswhere is the moment of inertia, is the viscous friction coefficient, and is the load torque.
4. Controller Design
4.1. Design of Backstepping Control
It is clear that the model of a FT-FSPM motor has a high nonlinear characteristic, because there exists a coupling between the currents and , and the rotor angular speed . The block diagram of the FT-FSPM motor drive system based on the backstepping SMC is shown in Figure 4. The FT-FSPM motor drive adopts control strategy on field orientation. According to the space vector control principle, the five-phase motor stator currents generate axis currents according to the 5/2 transformation. Subsequently, the axis currents are generated by the Park transformation and they are sent into the controller. Since the electromagnetic torque is linearly proportional to the -axis current , the torque output can be controlled by adjusting the current . The backstepping controller realizes the current closed-loop control and generates control voltages and in the FT-FSPM motor drive system. Subsequently, the space vector pulse width modulation (PWM) is generated by the 2/5 coordinate transformation. The adaptive sliding mode controller is use to identify external interferences and internal parameter variations. Load torque estimation is used to estimate according to the adaptive law.
In practice, the load torque of the FT-FSPM motor usually changes and the parameter uncertainties can deteriorate the operational performance. The traditional proportional integral derivative control cannot provide a satisfactory performance because the parameters are fixed. Therefore, the speed identification system usually suffers from disturbances. When the motor operates, its internal parameters and external interferences may have fluctuations. Accordingly, the backstepping controller is used to improve the anti-interference ability and the adaptive sliding mode algorithm is adopted to estimate the uncertainty terms.
In case of the existing SMC, the chattering problem is inevitable during the process of the sliding mode switching and it affects the stability of the system. To reduce or eliminate sliding mode chattering, the backstepping control strategy is applied to the SMC algorithm. A new backstepping SMC method replaces traditional proportional integral derivative control. Compared to the existing SMC method, the backstepping SMC is not sensitive to internal parameter variations and external interferences, which can ensure that the system trajectory reaches and stays on the sliding mode surface. The adaptive sliding mode controller can automatically track parameter changes and adjust the control variable .
The design of backstepping control for the FT-FSPM motor system consists of two steps. First, the speed subsystem is designed using the voltage control law, which guarantees speed stability. Subsequently, for the current subsystem, a robust control law is designed to realize the current track. It is assumed that is the desired speed signal and is the actual speed signal. They are enough smooth. The speed tracking error signal is defined as
The derivation of is given by
For the speed subsystem, the first Lyapunov function is defined as , and its derivation is expressed aswhere is the speed feedback gain and it is a positive constant. In (7), apart from the speed term, there are current terms and the speed error term. Therefore, the desired values of the current control terms and are defined as
Accordingly, the currents and will converge to and , so that and . Therefore, (8) can be substituted into (7). Subsequently, (7) becomes , which satisfies the speed convergence requirements. However, the load torque is unknown in the actual operation and it is replaced by the estimated value of load torque . According to the vector control theory, the current is always forced to be zero, in order to orient all the linkage flux in the -axis. Therefore, the - axis current tracking errors are defined as
The core of the motor driving system is made of the five-phase currents from the inverter output and they are controlled by the space vector PWM. The current loop is a closed-loop. Therefore, the current control of and is the key for the FT-FSPM motor. According to the sliding mode theory, the derivative form of the current error is described as
Furthermore, in order to eliminate the derivative term of the -axis current error in the control algorithm, the second Lyapunov function is defined as
To guarantee the global asymptotic stability in the current loop, the axis control voltages are designed as
The load torque estimation law is described as
Further, (14) and (15) are substituted into (13), and the form is expressed aswhere and are positive constants. Subsequently, (16) becomes . It satisfies the asymptotically stable condition of the system. For the FT-FSPM motor drive system, the tracking errors , and converge toward zero. The voltage control law is a concrete way to achieve the backstepping strategy for the motor system.
4.2. Adaptive Sliding Mode Controller Design
Notably, the controller design does not consider changes from the parameters of the motor itself. In the actual operation, the parameters , , , and can change with the speed variation and the external interferences. These factors will affect the stability of the FT-FSPM motor drive system and lower the system control accuracy. Although the upper bound of the uncertainty term is difficult to determine, effective estimations of the uncertainty term can be achieved using the adaptive sliding mode algorithm. The motion equation is rewritten as
It is simplified as follows:where . Because the current is the key to motor control, the control term is selected as . The estimated value of the load torque replaces . In order to improve the control accuracy and performance of the FT-FSPM motor, the changes in the uncertainty terms are considered, andwhere contains the variations caused by the changing resistance where is regarded as the total uncertainty term; that is, , and . Further, , and are uncertainty terms of the motor parameter variations and external load interferences. In order to facilitate analysis, the rotor position angle of the motor is introduced to analyze the uncertainty terms. Further, is defined as the rotor position angle error; that is, , and the derivative form is . Further, is a virtual control term. Subsequently, the position angle speed can be expressed using and as follows:where and thus .
To overcome the interferences and parameter variations, the sliding mode term is introduced. The selection of sliding mode switching function is very important. For easy operation and less sliding mode chattering, taking into account the prior backstepping control analysis, the sliding mode surface is selected as
The derivative form is expressed as
To verify the feasibility of above assumptions, the third Lyapunov function is defined as
The derivative form of (24) is given bywhere is positive constant and is the estimated value of . Further, is the estimation error and . Assuming that the parameter variations and external interferences are slow, the uncertainty term can be approximated as a constant where , and is the max of . In this paper, the exponential approaching law is adopted and the form is . The exponential term can guarantee a greater speed to approach the sliding surface away from the switching surface. However, a simple exponential term cannot guarantee limited arriving time and the constant speed term is added. When is close to zero and the approaching speed is instead of zero, it ensures limited arriving time. Therefore, the adaptive controller is designed asThe adaptive law is designed asSubsequently, (26) and (27) are substituted into (25), resulting in the following:, and . These equations are substituted into (28); thenTaking into account the values of , , and , is guaranteed to be a positive definite matrix. Subsequently, (29) becomes . According to the Lyapunov stability theory, the FT-FSPM motor system is stable. The block of the adaptive sliding mode controller is shown in Figure 5. The rotor angle speed signal , the position angle signal , and are the inputs of the controller. In this regard, the purpose of increasing the virtual control terms is to design the adaptive sliding mode algorithms to approach the total uncertainty term.
5. Simulation and Experimental Results
5.1. Simulation Results
To verify the effectiveness of the backstepping SMC algorithm, a simulation model is built using MATLAB/Simulink. The five-phase FT-FSPM motor parameters are as follows: the rated power kW, the phase voltage = 200 V, the rated speed = 600 r/min, the rated torque = 7.8 N·m, the stator resistance = 2.56 Ω, the winding inductances = 36 mH and = 35 mH, the moment of inertia = 0.00062 kg·m2, the friction coefficient = 0.00031 N·m·s, and the permanent magnet flux = 0.183 Wb.
The parameters of the backstepping SMC are selected as
To test the robustness of the controller against changes in the load, the simulations are performed under two different cases. The reference speed is = 600 r/min. In the first case, at s, the load torque is changed from 5 N·m to 7.5 N·m, and the result is shown in Figure 6. In the other case, at s, the load torque is changed from 5 N·m to 2.5 N·m.
As seen from Figures 6 and 7, at the start, the simulation results of the proposed SMC show that the speed response can quickly track the reference speed within 0.01 s, but the existing SMC lags by 0.05 s. Notably, the speed overshoot in the case of the proposed SMC can be reduced from 10% to 7.5%, compared to the existing SMC. Moreover, the existing SMC causes the currents and to produce large chattering initially, but the proposed SMC can minimize instant chattering due to the effectiveness of the voltage () control laws and the sliding mode control algorithm. It can be seen that the -axis current is well decoupled from the motor speed and is regulated to zero. is proportional to the -axis current and is almost simultaneous to the changing current . When the motor starts, because of the FT-FSPM motor location torque, there are large electromagnetic torque and current spikes. However, the electromagnetic torque ripple is small under normal operation. The response trajectories of the five-phase currents are consistent with the results of the theoretical analysis and tend to steady within 0.01 s. They present sine waves and corresponding changes in the amplitude. In the instant of the load torque mutations, the speed, current, and torque fluctuations of the proposed SMC are smaller than those of the existing SMC, although the response rapidity is almost same. Therefore, the proposed SMC method promises better implementation as compared to the existing SMC method for the FT-FSPM motor system.
In order to test the speed tracking performance of the FT-FSPM motor under changing speed, within 0.25 s, the speed is increased from 600 r/min to 800 r/min and the speed is decreased from 600 r/min to 400 r/min, respectively. As shown in Figures 8 and 9, by using the proposed SMC algorithm, the response speed can quickly track the reference speed signal and the speed tracking error is very small. As can be seen from the simulation of the rotor position angle, the response of the rotor position angle can reflect the changes in speed.
With regard to the uncertainty terms, the simulations of the rotor position angle are designed to validate the feasibility of the algorithm and the performance of the position tracking. A sinusoidal signal is taken as the input signal. Figure 10 shows the tracking of the motor position angle and the trajectories of the controller input when a load of 0.2 N·m is applied to the system at the rotor position angle = 90°. Figure 11 shows the tracking of the motor position angle and the trajectories of the controller input when the interference is applied to the system at the rotor position angle = 270°. In spite of the abrupt application of load, the rotor can track the ideal position after a transient fluctuation. Although mutation changes caused by the uncertainty terms are present, the time taken for adjustment of the position angle is very short and the fluctuation is small. The control input has a very quick response and the total uncertainty term is supposed to be a sinusoidal signal. Figure 12 shows that can track adaptively and realize the estimation of uncertainty terms using the adaptive sliding mode algorithm.
5.2. Experimental Results
To verify the feasibility of the proposed SMC further, an experimental platform for the FT-FSPM motor control was built in the laboratory, as shown in Figure 13. The experimental platform devices include a DSP2812 chip, a Mitsubishi IPM, a 2048 line photoelectric encoder, a power module, Hall sensors, a torque sensor, and power and driver modules. A DC motor is used as the load. The experimental results are shown as follows.
The experimental results under a speed of 200 r/min are shown in Figures 14 and 15. Figure 14 shows that the currents have a big transient fluctuation when the load increases from 4 N·m to 6 N·m. It can be observed that the torque and the speed fluctuations of the proposed SMC are substantially smaller than those of the existing SMC. The speed response of the proposed SMC is approximately 0.3 s faster than that of the existing SMC. After changing the load, the current changed significantly. Figure 15 shows that the current and torque ripples have been reduced significantly under the proposed SMC method, when the load decreases from 6 N·m to 4 N·m. The speed response of the proposed SMC is approximately 0.4 s faster than that of the existing SMC. After 1–1.5 s, the speed, the currents, and the torque tend to be steady. Obviously, the fluctuation of the proposed SMC method is significantly smaller, indicating that the proposed SMC can absorb the sudden load changes better and has an anti-interference capability. Compared to the existing SMC, the proposed SMC can improve the robustness and the rapidity of the FT-FSPM motor drive system.
The experimental waveforms of currents and are shown in Figure 16. Accordingly, because of the adopted = 0 control strategy, the current is almost a straight line at zero, while the current also changes with the changing load. Owing to the proposed SMC method, the rapidity of response of current is improved and the speed response time decreases by approximately 0.1 s. At the instant the load changes, there exist small fluctuations and almost no overshoot. When the motor operates normally, the current ripple is also significantly reduced, by approximately 10 to 15%. These results indicate that this method is superior to the existing SMC, in terms of both the rapidity and the robustness.
The five-phase FT-FSPM motor has a simple rotor structure and high fault-tolerant capability, which is effective to the PM fever and faulty problems. To reduce the influences of the system parameter uncertainties and external interferences, a nonlinear adaptive SMC algorithm has been proposed for the FT-FSPM motor, based on the backstepping control theory. The introduction of the virtual control terms, the speed tracking error, and the estimated errors of the d-q axis currents are the control terms, and the voltage control laws are designed using the backstepping method. These can improve the system robustness, and the dynamic and static performances under time-varying load torques, owing to the adaptive SMC algorithm for the uncertainty terms. The proposed SMC can effectively achieve speed tracking and reduce the chattering, and the system stability under this method has been demonstrated. It ensures that the five-phase FT-FSPM motor has better transient performance and the stability of the motor system has been secured. The simulations and the experimental results confirm the validity of the proposed method. In view of the characteristics of the FT-FSPM motor, the next step of the research group will be to continue to study the medium-high speed performance of the motor.
The authors declare that there is no conflict of interest regarding the publication of this paper.
This work was supported in part by the National Natural Science Foundation of China (Project no. 51422702).
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