# A New Adaptive Control for Five-Phase Fault-Tolerant Flux-Switching Permanent Magnet Motor

Authors: Hongyu Tang; Wenxiang Zhao; Chenyu Gu
Journal: International Journal of Rotating Machinery
 (2016)

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## Abstract

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.


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## Body

## 1. Introduction

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 [6].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 [7]. 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 [8]. 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 [9]. 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) [12], adaptive control [13], fuzzy control [6], neural network control [14], 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 [17]. However, its sliding mode surface suffers from chattering [18], 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 [19]. 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 [15]. The control algorithm can be designed step by step through appropriate Lyapunov functions [20]. 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 thed-q 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

Figure1 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 [24]. 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 “E”-shaped units, namely, E-cores, and a concentrated armature coil is wound around every two units, while a piece of magnet is embedded in the middle of two E-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 [25].Figure 1
Five-phase FT-FSPM motor. (a) Cross section. (b) Prototype.
(a) (b)As shown in Figure2, 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.Figure 2
Magnetic field distributions of FT-FSPM motor by armature current only.


## 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 Figure3.Figure 3
Five-phase fundamental coordinates of the FT-FSPM motor.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 as(1)ψd=Ls+2.5Lmd1id+ψm,ψq=Ls+2.5Lmq1iq,where Ls is the leakage inductance, id and iq are currents along the d and q axes respectively, ψm is the PM flux, Lmd1 and Lmq1 are the stator inductances along the d and q axes, respectively, and ψd and ψq are the flux along the d and q axes, respectively. By the 5/2 coordinate transformation, the currents along the d and q axes, that is, id and iq, are expressed as(2)id=23cos⁡θiA+cos⁡θ-25πiB+cos⁡θ-45πiC+cos⁡θ+45πiD+cos⁡θ+25πiE,iq=23-sin⁡θiA-sin⁡θ-25πiB-sin⁡θ-45πiC-sin⁡θ+45πiD-sin⁡θ+25πiE,where iA,iB,iC,iD, and iE are the five-phase currents and θ 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 as(3)ud=rsid-ωLs+52Lmq1iq+Ls+52Lmd1diddt,uq=rsiq+ωLs+52Lmd1id+ωψm+Ls+52Lmq1diqdt,Te=2.5npψmiq+Lmd1-Lmq1idiq,where np is the pairs number of rotor pole, ud and uq are the voltages along the d and q axes respectively, ω is the rotor angular speed, rs is the stator resistance, and Te is the electromagnetic torque. Although the d-axis inductance is not equal to the q-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 d-q 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 as(4)Te=Jdωdt+Bω+Tl,ω=dθdt,where J is the moment of inertia, B is the viscous friction coefficient, and Tl 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 currentsid and iq, and the rotor angular speed ω. 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 id=0 control strategy on field orientation. According to the space vector control principle, the five-phase motor stator currents generate α-β axis currents according to the 5/2 transformation. Subsequently, the d-q axis currents are generated by the Park transformation and they are sent into the controller. Since the electromagnetic torque Te is linearly proportional to the q-axis current iq, the torque output can be controlled by adjusting the current iq. The backstepping controller realizes the current closed-loop control and generates control voltages ud and uq 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 Tl according to the adaptive law.Figure 4
Block diagram of the FT-FSPM motor system.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 variableu.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 thatω∗ is the desired speed signal and ω is the actual speed signal. They are enough smooth. The speed tracking error signal is defined as(5)eω=ω∗-ω.The derivation ofeω is given by(6)e˙ω=ω˙∗-ω˙=ω˙∗+1JTl+Bω-52npψmiq-52npLmd1-Lmq1idiq.For the speed subsystem, the first Lyapunov function is defined asV1=1/2eω2, and its derivation is expressed as(7)V˙1=eωe˙ω=eωJJω˙∗+Tl+Bω-52npψmiq-52npLmd1-Lmq1idiq=-keω2+eωJJω˙∗+Tl+Bω-52npψmiq-52npLmd1-Lmq1idiq+kJeω,where k 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 id∗ and iq∗ are defined as(8)id∗=0,iq∗=25npψmJω˙∗+Tl+Bω+kJeω.Accordingly, the currentsid and iq will converge to id∗ and iq∗, so that id=id∗ and iq=iq∗. Therefore, (8) can be substituted into (7). Subsequently, (7) becomes V˙1=-keω2<0, which satisfies the speed convergence requirements. However, the load torque Tl is unknown in the actual operation and it is replaced by the estimated value of load torque T^l. According to the vector control theory, the current id is always forced to be zero, in order to orient all the linkage flux in the d-axis. Therefore, the d-q axis current tracking errors are defined as(9)ed=id∗-id,eq=iq∗-iq.When (8) and (9) are substituted into (6), the form is expressed as(10)e˙ω=1J52npψmeq+52npLmd1-Lmq1ediq-T~l-kJeω,where T~l is estimation error, T~l=T^l-Tl.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 ofid and iq is the key for the FT-FSPM motor. According to the sliding mode theory, the derivative form of the current error is described as(11)e˙d=Rsid-npωLmq1iq-udLmd1,e˙q=2B-KJ5npψmJ5np2ψmiq+Lmd1-Lmq1idiq-Bω-T^l+Rsiq+npωLmd1id+npωψm-uqLmq1+25npψmJω¨∗+T^˙l+KJω˙∗.Furthermore, in order to eliminate the derivative term of theq-axis current error in the control algorithm, the second Lyapunov function is defined as(12)V2=V1+12ed2+12eq2+12λT~l2.Further,(13)V˙2=eωe˙ω+ede˙d+eqe˙q+T~lT^˙lλ=-keω2-kded2-kqeq2+eωe˙ω+keω+ede˙d+kded+eqe˙q+kqeq+T~lT^˙lλ.To guarantee the global asymptotic stability in the current loop, thed-q axis control voltages are designed as(14)ud=Rsid-npωLmq1iq+kdLmd1ed,uq=Rsiq+npωLmd1id+npωψm+kqLmq1eq-2Lmq15npψmJB-kJ5npψm2eq+5np2Lmd1-Lmq1ediq+Bω+T^l-25npψmJω¨∗+T^˙l+kJω˙∗.The load torque estimation law is described as(15)T^˙l=λeωJ-2B-kJ5npψmJeq.Further, (14) and (15) are substituted into (13), and the form is expressed as(16)V˙2=-keω2-kded2-kqeq2,where kd and kq are positive constants. Subsequently, (16) becomes V˙2≤0. It satisfies the asymptotically stable condition of the system. For the FT-FSPM motor drive system, the tracking errors eω,ed, and eq 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 parametersrs, B, J, and Tl 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(17)ω˙=1J52npψmiq+52npLmd1-Lmq1idiq-Tl-Bω.It is simplified as follows:(18)ω˙=Au-BJω-TlJ,where A=1/J[5/2npψmiq+5/2np(Lmd1-Lmq1)id]. Because the current iq is the key to motor control, the control term is selected as u=iq. The estimated value of the load torque T^l replaces Tl. In order to improve the control accuracy and performance of the FT-FSPM motor, the changes in the uncertainty terms are considered, and(19)ω˙=A+ΔAu-B+ΔBJ+ΔJω-T^l+ΔT^lJ+ΔJ,where ΔA contains the variations caused by the changing resistance rs(20)ω˙=Au+Bmω+F,where F is regarded as the total uncertainty term; that is, F=ΔAu-ΔB/J+ΔJω-(T^l+ΔT^l)/J+ΔJ, and  Bm=-B/J+ΔJ. Further, ΔA,ΔB, and (T^l+ΔT^l)/J+ΔJ 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, x1 is defined as the rotor position angle error; that is, x1=θ-θ∗, and the derivative form is x˙1=ω-θ˙∗. Further, x2 is a virtual control term. Subsequently, the position angle speed ω can be expressed using x1 and x2 as follows:(21)ω=x2+θ˙∗-c1x1,where x2=ω-θ˙∗+c1x1 and thus x˙1=ω-θ˙∗=x2-c1x1.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(22)S=kx1+x2.The derivative form is expressed as(23)S˙=kx˙1+x˙2.To verify the feasibility of above assumptions, the third Lyapunov function is defined as(24)V3=V2+12x12+12S2+12γF~2.The derivative form of (24) is given by(25)V˙3=V˙2+12x12+12S2+12γF~2=V˙2+x1x2-cx12+SS˙-1γF~F^˙=V˙2+x1x2-cx12+Sk1x2-c1x1+Bmx2+θ˙∗-c1x1+Au+F-θ¨∗+c1x˙1-1γF~F^˙=V˙2+x1x2-cx12+Sk1x2-c1x1+Bmx2+θ˙∗-c1x1+Au+F^-θ¨∗+c1x˙1-1γF~F^˙+γS,where γ is positive constant and F^ is the estimated value of F. Further, F~ is the estimation error and F~=F-F^. Assuming that the parameter variations and external interferences are slow, the uncertainty term can be approximated as a constant where F≤Fm, and Fm is the max of F. In this paper, the exponential approaching law is adopted and the form is -hS-usgn⁡(S). The exponential term -hS 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 usgn⁡(S) is added. When S is close to zero and the approaching speed is u instead of zero, it ensures limited arriving time. Therefore, the adaptive controller is designed as(26)u=A-1-k1x2-c1x1-Bmx2+θ˙∗-c1x1-F^+θ¨∗-c1x˙1-hS-μsgn⁡S.The adaptive law is designed as(27)F^˙=-γS.Subsequently, (26) and (27) are substituted into (25), resulting in the following:(28)V˙3=-keω2-kded2-kqeq2+x1x2-c1x12-hS2-μS,Q=c1+hk12hk1-1/2hk1-1/2h, and XT=x1x2. These equations are substituted into (28); then(29)V˙3=-keω2-kded2-kqeq2-XTQX-μS≤0.Taking into account the values of h, c1, and k1, Q is guaranteed to be a positive definite matrix. Subsequently, (29) becomes V˙3≤0. 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 ω, the position angle signal θ, and θ∗ 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.Figure 5
Block of the adaptive sliding mode controller.


## 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 currentsid and iq, and the rotor angular speed ω. 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 id=0 control strategy on field orientation. According to the space vector control principle, the five-phase motor stator currents generate α-β axis currents according to the 5/2 transformation. Subsequently, the d-q axis currents are generated by the Park transformation and they are sent into the controller. Since the electromagnetic torque Te is linearly proportional to the q-axis current iq, the torque output can be controlled by adjusting the current iq. The backstepping controller realizes the current closed-loop control and generates control voltages ud and uq 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 Tl according to the adaptive law.Figure 4
Block diagram of the FT-FSPM motor system.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 variableu.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 thatω∗ is the desired speed signal and ω is the actual speed signal. They are enough smooth. The speed tracking error signal is defined as(5)eω=ω∗-ω.The derivation ofeω is given by(6)e˙ω=ω˙∗-ω˙=ω˙∗+1JTl+Bω-52npψmiq-52npLmd1-Lmq1idiq.For the speed subsystem, the first Lyapunov function is defined asV1=1/2eω2, and its derivation is expressed as(7)V˙1=eωe˙ω=eωJJω˙∗+Tl+Bω-52npψmiq-52npLmd1-Lmq1idiq=-keω2+eωJJω˙∗+Tl+Bω-52npψmiq-52npLmd1-Lmq1idiq+kJeω,where k 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 id∗ and iq∗ are defined as(8)id∗=0,iq∗=25npψmJω˙∗+Tl+Bω+kJeω.Accordingly, the currentsid and iq will converge to id∗ and iq∗, so that id=id∗ and iq=iq∗. Therefore, (8) can be substituted into (7). Subsequently, (7) becomes V˙1=-keω2<0, which satisfies the speed convergence requirements. However, the load torque Tl is unknown in the actual operation and it is replaced by the estimated value of load torque T^l. According to the vector control theory, the current id is always forced to be zero, in order to orient all the linkage flux in the d-axis. Therefore, the d-q axis current tracking errors are defined as(9)ed=id∗-id,eq=iq∗-iq.When (8) and (9) are substituted into (6), the form is expressed as(10)e˙ω=1J52npψmeq+52npLmd1-Lmq1ediq-T~l-kJeω,where T~l is estimation error, T~l=T^l-Tl.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 ofid and iq is the key for the FT-FSPM motor. According to the sliding mode theory, the derivative form of the current error is described as(11)e˙d=Rsid-npωLmq1iq-udLmd1,e˙q=2B-KJ5npψmJ5np2ψmiq+Lmd1-Lmq1idiq-Bω-T^l+Rsiq+npωLmd1id+npωψm-uqLmq1+25npψmJω¨∗+T^˙l+KJω˙∗.Furthermore, in order to eliminate the derivative term of theq-axis current error in the control algorithm, the second Lyapunov function is defined as(12)V2=V1+12ed2+12eq2+12λT~l2.Further,(13)V˙2=eωe˙ω+ede˙d+eqe˙q+T~lT^˙lλ=-keω2-kded2-kqeq2+eωe˙ω+keω+ede˙d+kded+eqe˙q+kqeq+T~lT^˙lλ.To guarantee the global asymptotic stability in the current loop, thed-q axis control voltages are designed as(14)ud=Rsid-npωLmq1iq+kdLmd1ed,uq=Rsiq+npωLmd1id+npωψm+kqLmq1eq-2Lmq15npψmJB-kJ5npψm2eq+5np2Lmd1-Lmq1ediq+Bω+T^l-25npψmJω¨∗+T^˙l+kJω˙∗.The load torque estimation law is described as(15)T^˙l=λeωJ-2B-kJ5npψmJeq.Further, (14) and (15) are substituted into (13), and the form is expressed as(16)V˙2=-keω2-kded2-kqeq2,where kd and kq are positive constants. Subsequently, (16) becomes V˙2≤0. It satisfies the asymptotically stable condition of the system. For the FT-FSPM motor drive system, the tracking errors eω,ed, and eq 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 parametersrs, B, J, and Tl 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(17)ω˙=1J52npψmiq+52npLmd1-Lmq1idiq-Tl-Bω.It is simplified as follows:(18)ω˙=Au-BJω-TlJ,where A=1/J[5/2npψmiq+5/2np(Lmd1-Lmq1)id]. Because the current iq is the key to motor control, the control term is selected as u=iq. The estimated value of the load torque T^l replaces Tl. In order to improve the control accuracy and performance of the FT-FSPM motor, the changes in the uncertainty terms are considered, and(19)ω˙=A+ΔAu-B+ΔBJ+ΔJω-T^l+ΔT^lJ+ΔJ,where ΔA contains the variations caused by the changing resistance rs(20)ω˙=Au+Bmω+F,where F is regarded as the total uncertainty term; that is, F=ΔAu-ΔB/J+ΔJω-(T^l+ΔT^l)/J+ΔJ, and  Bm=-B/J+ΔJ. Further, ΔA,ΔB, and (T^l+ΔT^l)/J+ΔJ 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, x1 is defined as the rotor position angle error; that is, x1=θ-θ∗, and the derivative form is x˙1=ω-θ˙∗. Further, x2 is a virtual control term. Subsequently, the position angle speed ω can be expressed using x1 and x2 as follows:(21)ω=x2+θ˙∗-c1x1,where x2=ω-θ˙∗+c1x1 and thus x˙1=ω-θ˙∗=x2-c1x1.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(22)S=kx1+x2.The derivative form is expressed as(23)S˙=kx˙1+x˙2.To verify the feasibility of above assumptions, the third Lyapunov function is defined as(24)V3=V2+12x12+12S2+12γF~2.The derivative form of (24) is given by(25)V˙3=V˙2+12x12+12S2+12γF~2=V˙2+x1x2-cx12+SS˙-1γF~F^˙=V˙2+x1x2-cx12+Sk1x2-c1x1+Bmx2+θ˙∗-c1x1+Au+F-θ¨∗+c1x˙1-1γF~F^˙=V˙2+x1x2-cx12+Sk1x2-c1x1+Bmx2+θ˙∗-c1x1+Au+F^-θ¨∗+c1x˙1-1γF~F^˙+γS,where γ is positive constant and F^ is the estimated value of F. Further, F~ is the estimation error and F~=F-F^. Assuming that the parameter variations and external interferences are slow, the uncertainty term can be approximated as a constant where F≤Fm, and Fm is the max of F. In this paper, the exponential approaching law is adopted and the form is -hS-usgn⁡(S). The exponential term -hS 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 usgn⁡(S) is added. When S is close to zero and the approaching speed is u instead of zero, it ensures limited arriving time. Therefore, the adaptive controller is designed as(26)u=A-1-k1x2-c1x1-Bmx2+θ˙∗-c1x1-F^+θ¨∗-c1x˙1-hS-μsgn⁡S.The adaptive law is designed as(27)F^˙=-γS.Subsequently, (26) and (27) are substituted into (25), resulting in the following:(28)V˙3=-keω2-kded2-kqeq2+x1x2-c1x12-hS2-μS,Q=c1+hk12hk1-1/2hk1-1/2h, and XT=x1x2. These equations are substituted into (28); then(29)V˙3=-keω2-kded2-kqeq2-XTQX-μS≤0.Taking into account the values of h, c1, and k1, Q is guaranteed to be a positive definite matrix. Subsequently, (29) becomes V˙3≤0. 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 ω, the position angle signal θ, and θ∗ 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.Figure 5
Block of the adaptive sliding mode controller.


## 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 powerP=1.8kW, the phase voltage U = 200 V, the rated speed n = 600 r/min, the rated torque Te = 7.8 N·m, the stator resistance rs = 2.56 Ω, the winding inductances Lmd1 = 36 mH and Lmq1 = 35 mH, the moment of inertia J = 0.00062 kg·m2, the friction coefficient B = 0.00031 N·m·s, and the permanent magnet flux ψm = 0.183 Wb.The parameters of the backstepping SMC are selected as(30)η=0.3,β=1.5,γ=12,μ=0.1,k1=15,c1=10,h=20.To test the robustness of the controller against changes in the load, the simulations are performed under two different cases. The reference speed isω∗ = 600 r/min. In the first case, at t=0.2s, the load torque Tl is changed from 5 N·m to 7.5 N·m, and the result is shown in Figure 6. In the other case, at t=0.3s, the load torque Tl is changed from 5 N·m to 2.5 N·m.Figure 6
Simulation results when increasing load from 5 N⋅m to 7.5 N⋅m. (a) Speed response. (b) Current waveformsid and iq under the existing SMC. (c) Current waveforms id and iq under the proposed SMC. (d) Waveforms of the five-phase currents. (e) Torque response.
(a) (b) (c) (d) (e)As seen from Figures6 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 id and iq to produce large chattering initially, but the proposed SMC can minimize instant chattering due to the effectiveness of the voltage (ud,uq) control laws and the sliding mode control algorithm. It can be seen that the d-axis current id is well decoupled from the motor speed and is regulated to zero. Te is proportional to the q-axis current and is almost simultaneous to the changing current iq. 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.Figure 7
Simulation results when decreasing load from 5 N⋅m to 2.5 N⋅m. (a) Speed response. (b) Current waveformsid and iq under the existing SMC. (c) Current waveforms id and iq under the proposed SMC. (d) Waveforms of the five-phase currents. (e) Torque response.
(a) (b) (c) (d) (e)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 Figures8 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.Figure 8
Speed tracking under speed changing. (a) Speed increasing. (b) Speed decreasing.
(a) (b)Figure 9
Rotor position angle tracking under changing speed. (a) Speed increasing. (b) Speed decreasing.
(a) (b)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. Figure10 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 θ = 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 θ = 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 F^ can track F adaptively and realize the estimation of uncertainty terms using the adaptive sliding mode algorithm.Figure 10
Simulation results of the backstepping SMC when the load is applied. (a) Position tracking. (b) Control input.
(a) (b)Figure 11
Simulation results of the backstepping SMC when the interference is applied. (a) Position tracking. (b) Control input.
(a) (b)Figure 12
Adaptive tracking trajectory ofF.


### 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 Figure13. 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.Figure 13
Prototype motor and experimental platform.The experimental results under a speed of 200 r/min are shown in Figures14 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.Figure 14
Response when the load suddenly increases (current (1 A/div), speed (200 r/min/div), torque (2 N·m/div), and time (1 s/div)). (a) The existing SMC control strategy. (b) The proposed SMC control strategy.
(a) (b)Figure 15
Response when the load suddenly decreases (current (1 A/div), speed (200 r/min/div), torque (2 N·m/div), and time (1 s/div)). (a) The existing SMC control strategy. (b) The proposed SMC control strategy.
(a) (b)The experimental waveforms of currentsid and iq are shown in Figure 16. Accordingly, because of the adopted id = 0 control strategy, the current id is almost a straight line at zero, while the current iq also changes with the changing load. Owing to the proposed SMC method, the rapidity of response of current iq 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.Figure 16
Measured waveforms of currentsid and iq. (a) With increasing load. (b) With decreasing load.
(a) (b)


## 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 powerP=1.8kW, the phase voltage U = 200 V, the rated speed n = 600 r/min, the rated torque Te = 7.8 N·m, the stator resistance rs = 2.56 Ω, the winding inductances Lmd1 = 36 mH and Lmq1 = 35 mH, the moment of inertia J = 0.00062 kg·m2, the friction coefficient B = 0.00031 N·m·s, and the permanent magnet flux ψm = 0.183 Wb.The parameters of the backstepping SMC are selected as(30)η=0.3,β=1.5,γ=12,μ=0.1,k1=15,c1=10,h=20.To test the robustness of the controller against changes in the load, the simulations are performed under two different cases. The reference speed isω∗ = 600 r/min. In the first case, at t=0.2s, the load torque Tl is changed from 5 N·m to 7.5 N·m, and the result is shown in Figure 6. In the other case, at t=0.3s, the load torque Tl is changed from 5 N·m to 2.5 N·m.Figure 6
Simulation results when increasing load from 5 N⋅m to 7.5 N⋅m. (a) Speed response. (b) Current waveformsid and iq under the existing SMC. (c) Current waveforms id and iq under the proposed SMC. (d) Waveforms of the five-phase currents. (e) Torque response.
(a) (b) (c) (d) (e)As seen from Figures6 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 id and iq to produce large chattering initially, but the proposed SMC can minimize instant chattering due to the effectiveness of the voltage (ud,uq) control laws and the sliding mode control algorithm. It can be seen that the d-axis current id is well decoupled from the motor speed and is regulated to zero. Te is proportional to the q-axis current and is almost simultaneous to the changing current iq. 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.Figure 7
Simulation results when decreasing load from 5 N⋅m to 2.5 N⋅m. (a) Speed response. (b) Current waveformsid and iq under the existing SMC. (c) Current waveforms id and iq under the proposed SMC. (d) Waveforms of the five-phase currents. (e) Torque response.
(a) (b) (c) (d) (e)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 Figures8 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.Figure 8
Speed tracking under speed changing. (a) Speed increasing. (b) Speed decreasing.
(a) (b)Figure 9
Rotor position angle tracking under changing speed. (a) Speed increasing. (b) Speed decreasing.
(a) (b)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. Figure10 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 θ = 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 θ = 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 F^ can track F adaptively and realize the estimation of uncertainty terms using the adaptive sliding mode algorithm.Figure 10
Simulation results of the backstepping SMC when the load is applied. (a) Position tracking. (b) Control input.
(a) (b)Figure 11
Simulation results of the backstepping SMC when the interference is applied. (a) Position tracking. (b) Control input.
(a) (b)Figure 12
Adaptive tracking trajectory ofF.


## 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 Figure13. 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.Figure 13
Prototype motor and experimental platform.The experimental results under a speed of 200 r/min are shown in Figures14 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.Figure 14
Response when the load suddenly increases (current (1 A/div), speed (200 r/min/div), torque (2 N·m/div), and time (1 s/div)). (a) The existing SMC control strategy. (b) The proposed SMC control strategy.
(a) (b)Figure 15
Response when the load suddenly decreases (current (1 A/div), speed (200 r/min/div), torque (2 N·m/div), and time (1 s/div)). (a) The existing SMC control strategy. (b) The proposed SMC control strategy.
(a) (b)The experimental waveforms of currentsid and iq are shown in Figure 16. Accordingly, because of the adopted id = 0 control strategy, the current id is almost a straight line at zero, while the current iq also changes with the changing load. Owing to the proposed SMC method, the rapidity of response of current iq 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.Figure 16
Measured waveforms of currentsid and iq. (a) With increasing load. (b) With decreasing load.
(a) (b)


## 6. Conclusion

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 thed-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.



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*Source: 3792645-2016-12-29.xml*