# Speed Sensor-Less Control System of Surface-Mounted Permanent Magnet Synchronous Motor Based on Adaptive Feedback Gain Supertwisting Sliding Mode Observer

Authors: Lei Zhang; Jing Bai; Jing Wu
Journal: Journal of Sensors
 (2021)

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## Abstract

Aiming at the problems of severe chattering and difficulty in low-speed operation of the surface-mounted permanent magnet synchronous motor (SPMSM) sensor-less speed control system based on the traditional sliding mode observer (SMO), this paper proposes a sensor-less control strategy of supertwisting sliding mode observer based on adaptive feedback gain (AFG-STA-SMO). This strategy combines the supertwisting algorithm (STA) with the equivalent feedback principle and designs an adaptive law to compensate for the rotor position error by adjusting the feedback gain coefficient online. Secondly, considering the ripple component in the back electromotive force (back-EMF), the Kalman filter gets a smoother back-EMF signal, further improving the rotor position estimation accuracy. The stability of the system is proved by using the Lyapunov function. Finally, the feasibility and effectiveness of the proposed control strategy are verified by MATLAB/Simulink simulation.


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## Body

## 1. Introduction

Surface-mounted permanent magnet synchronous motor has a small size, high power density, high efficiency, low rotor loss, and solid environmental adaptability [1–4]. It is widely used in the wind power generation system and industrial transmission field. In order to realize high-performance control of permanent magnet synchronous motor, it is necessary to install speed and position sensors such as a photoelectric encoder to get accurate rotor position and speed information. However, installing a photoelectric encoder will increase the system’s cost and require a high operating environment [5]. As a result, various sensor-less techniques have been developed to estimate rotor position and speed over the past few decades.At present, speed sensor-less control algorithms are mainly divided into two categories: the high-frequency signal injection method, which depends on the motor’s salient spatial characteristics. Accurate rotor position information can be obtained at all speeds, including zero speed, but it is only applicable to the built-in motor with much noise. The second type uses the motor speed parameters (such as the back-EMF) in the motor fundamental excitation mathematical model to estimate the rotor position and speed, which has significant dynamic performance but is only suitable for the operation range of medium and high-speed. The commonly used fundamental back-EMF observation algorithms include SMO [6–9], model reference adaptive [10–11], and extended Kalman filter [12]. SMO has been widely studied because of its simple implementation, insensitivity to parameter changes, and external interference [13]. However, the chattering caused by the sign switching function is an inevitable problem in SMO. Therefore, the low-pass filter (LPF) was introduced to filter the back-EMF. Since the traditional LPF cut-off frequency is fixed, the back-EMF’s ripple component cannot be eliminated, seriously affecting the rotor position’s estimation accuracy [14]. In literature [15], an adaptive filter is proposed to filter out the sliding mode noise, but the causes of the system’s phase delay to a certain extent. The literature [16–18] proposed an SMO based on the supertwisting algorithm to observe back-EMF to solve this problem. This method can effectively suppress the chattering phenomenon caused by the switching function, and a good control effect can be obtained in the medium and high-speed range. However, with the decrease of motor speed, the back-EMF gradually decreases, and the observation accuracy also decreases. In order to improve the rotor estimation when the motor runs at low speed, an SMO is based on equivalent feedback proposed in the literature [19]. The observer can feed back on the estimated back-EMF to the stator current observation and calculation to achieve the motor’s low-speed operation. However, the feedback gain coefficient is only selected according to the speed, so the algorithm lacks adaptability. Based on the equivalent feedback sliding mode observer, a feedback gain adaptive algorithm is proposed in the literature [20, 21] to realize the rotor angle compensation at different speeds. However, this method still needs to introduce an LPF and compensate for the rotor position delay, which increases the complexity of the system.In this paper, an STA-SMO with adaptive feedback gain was proposed by combining the equivalent feedback principle with the supertwisting algorithm. The observer feeds back on the estimated back-EMF to the stator current observation and calculation and designs an adaptive law to realize rotor angle compensation at different speeds, which improves the observer’s low-speed performance and simplifies the selection of sliding mode gain. Simultaneously, the Kalman filter is used to filter out the ripple component in the back-EMF and further improve the sensor-less control precision; the Lyapunov function analyzes the system’s stability.


## 2. Design of SMO

### 2.1. Mathematical Model of SPMSM

The mathematical model of surface-mounted PMSM, theα-βstationary coordinate system, is [22]
(1)UαUβ=R+ddtL00R+ddtLiαiβ+EαEβ,where Uαβ, iαβ, Eαβ, R, and L are α-β axis voltages, α-β axis currents, α-β axis back-EMFs, stator resistance, and stator inductance, respectively. And Eαβ satisfies
(2)EαEβ=ωeψf−sinθecosθe,where ψf is the flux linkage, θe is the rotor position angle, and ωe is the rotational speed.The derivative of equation (2) is obtained:
(3)ddtEαEβ=ωe‐ψfωe‐ψfωecosθesinθe=ωe−EβEα.As shown in equation (3), the back-EMF contains the rotor position and speed information.


### 2.2. Traditional SMO

In order to obtain the back-EMF, equation (1) is rewritten as follows:
(4)ddtiαiβ=‐RLiαiβ+1LUαUβ−1LEαEβ.According to equation (4), the traditional SMO can be designed as
(5)ddti^αi^β=‐RLi^αi^β+1LUαUβ−1Lsvαvβ,where i^αβ is the observed value of stator current, Uαβ is the control input of the observer, and vαβ is the estimated value of the back electromotive force.
(6)vαvβ=ksigni^α−iαksigni^β−iβ,where sign is the signum function and k is the sliding mode gain coefficient.From equation (6), it can be seen that there is a discontinuous function sign in the estimated back-EMF value, which makes the back-EMF contain a large number of sliding mode noise and harmonic components, so it needs to introduce a LPF to filter it. Since the phase and amplitude errors are introduced into the LPF, it must be compensated, which increases the complexity of the system.In order to further improve the low-speed performance of speed sensor-less control, a SMO based on equivalent feedback electromotive force was proposed in literature [19], as shown in the following equation:
(7)ddti^αi^β=‐RLi^αi^β+1LUαUβ−1Lsvα+εvαeqvβ+εvβeq,where ε is the feedback gain coefficient and vαβeq are the equivalent feedback electromotive force values.
(8)vαeqvβeq=ωcτs+ωcvαωcτs+ωcvβ,where ωc is the cut-off frequency of the LPF and τ is the time constant of the LPF.When the motor is running at low speed, the SMO based on the equivalent feedback EMF can improve the value of the equivalent feedback EMF by reasonably designing parameterε, so as to solve the problem that the response EMF is too small and the rotor position cannot be estimated effectively, and improve the control precision of the system. However, the method still needs to introduce a low-pass filter and conduct behavior compensation.In order to retain the advantages of the equivalent feedback SMO and avoid using a LPF, a new sliding mode speed sensor-less control method was designed in literature [23]. In this method, a new state observer is designed to observe the equivalent feedback signal vαβeq, thus avoiding the use of a LPF. However, the design and debugging of the state observer are complicated.


## 2.1. Mathematical Model of SPMSM

The mathematical model of surface-mounted PMSM, theα-βstationary coordinate system, is [22]
(1)UαUβ=R+ddtL00R+ddtLiαiβ+EαEβ,where Uαβ, iαβ, Eαβ, R, and L are α-β axis voltages, α-β axis currents, α-β axis back-EMFs, stator resistance, and stator inductance, respectively. And Eαβ satisfies
(2)EαEβ=ωeψf−sinθecosθe,where ψf is the flux linkage, θe is the rotor position angle, and ωe is the rotational speed.The derivative of equation (2) is obtained:
(3)ddtEαEβ=ωe‐ψfωe‐ψfωecosθesinθe=ωe−EβEα.As shown in equation (3), the back-EMF contains the rotor position and speed information.


## 2.2. Traditional SMO

In order to obtain the back-EMF, equation (1) is rewritten as follows:
(4)ddtiαiβ=‐RLiαiβ+1LUαUβ−1LEαEβ.According to equation (4), the traditional SMO can be designed as
(5)ddti^αi^β=‐RLi^αi^β+1LUαUβ−1Lsvαvβ,where i^αβ is the observed value of stator current, Uαβ is the control input of the observer, and vαβ is the estimated value of the back electromotive force.
(6)vαvβ=ksigni^α−iαksigni^β−iβ,where sign is the signum function and k is the sliding mode gain coefficient.From equation (6), it can be seen that there is a discontinuous function sign in the estimated back-EMF value, which makes the back-EMF contain a large number of sliding mode noise and harmonic components, so it needs to introduce a LPF to filter it. Since the phase and amplitude errors are introduced into the LPF, it must be compensated, which increases the complexity of the system.In order to further improve the low-speed performance of speed sensor-less control, a SMO based on equivalent feedback electromotive force was proposed in literature [19], as shown in the following equation:
(7)ddti^αi^β=‐RLi^αi^β+1LUαUβ−1Lsvα+εvαeqvβ+εvβeq,where ε is the feedback gain coefficient and vαβeq are the equivalent feedback electromotive force values.
(8)vαeqvβeq=ωcτs+ωcvαωcτs+ωcvβ,where ωc is the cut-off frequency of the LPF and τ is the time constant of the LPF.When the motor is running at low speed, the SMO based on the equivalent feedback EMF can improve the value of the equivalent feedback EMF by reasonably designing parameterε, so as to solve the problem that the response EMF is too small and the rotor position cannot be estimated effectively, and improve the control precision of the system. However, the method still needs to introduce a low-pass filter and conduct behavior compensation.In order to retain the advantages of the equivalent feedback SMO and avoid using a LPF, a new sliding mode speed sensor-less control method was designed in literature [23]. In this method, a new state observer is designed to observe the equivalent feedback signal vαβeq, thus avoiding the use of a LPF. However, the design and debugging of the state observer are complicated.


## 3. Design of AFG-STA-SMO

### 3.1. Supertwisting Algorithm

The general form of the supertwisting algorithm [17] is as follows:
(9)dx^1dt=‐k1x~11/2signx~1+x2+ρ1,dx^2dt=‐k2signx~1+ρ2,where xi, x^i, x~i, ki, ρi, and sign are state variables, estimation of state variables, the error between estimated and actual state variables, sliding mode gain coefficient, perturbation term, and signum function, respectively.


### 3.2. AFG-STA-SMO

Since SPMSM high-precision vector control requires accurate motor speed and rotor position information, traditional SMO has parameter perturbation and chattering. This paper combines the principle of equivalent feedback with STA-SMO, and the design is as follows:(10)ddti^αi^β=‐RLi^αi^β+1LUαUβ−1Lvα+l⋅vαeqvβ+l⋅vβeq−k1⋅i~α1/2⋅signi~αk1⋅i~β1/2⋅signi~β,ddtvαvβ=k2⋅signi~αk2⋅signi~β,vαeqvβeq=vαvβ,where l is the feedback gain coefficient, i^αβ are the observation value of the stator currents, Uαβ are α-β axis voltages, ‐R/Liα+1/LUα and ‐R/Liβ+1/LUβ are regarded as the perturbation terms, vα=∫k2⋅signi~αdt and vβ=∫k2⋅signi~βdt are the estimated value of back-EMFs, and vαeq and vβeq are the equivalent feedback electromotive force values.The difference between equations (4) and (10) can be obtained:
(11)ddti~αi~β=‐RsLsi~αi~β−1Lse~αe~β−k1⋅i~α1/2⋅signi~αk1⋅i~β1/2⋅signi~β,where i~α=i^α−iα, i~β=i^β−iβ, e~α=vα+lvαeq−Eα, and e~β=vβ+lvβeq−Eβ.After the state variable of the observer reaches the sliding mode surfacei~α=0 and i~β=0, according to the sliding mode variable structure equivalent principle, we can get
(12)EαEβ=vα+lvαeqvβ+lvβeq=1+l⋅∫k2⋅signi~αdt1+l⋅∫k2⋅signi~βdt.From equation (12), the integral term vαβ filters out high-frequency sliding mode noise, so introducing low-pass filters and phase compensation is avoided, and the system algorithm is simplified. It can be seen from equation (12) that the sliding mode gain k2 and feedback gain l are proportional to the speed. When the motor is running at high speed, the sliding-mode gain k2 and the feedback gain l are larger. When the motor runs at low speed, the sliding-mode gain k2 and the feedback gain l are smaller.


### 3.3. Optimization of AFG-STA-SMO

The derivative of equation (12) is obtained:
(13)ddtEα=‐ωe⋅Eβ=1+l⋅k2⋅signi~α,ddtEβ=ωe⋅Eα=1+l⋅k2⋅signi~β.Although the harmonic component is effectively filtered by integral operation, the back-EMF’s observed value still contains a ripple component, which will lead to error if used to estimate the rotor position directly. Therefore, the Kalman filter is used in this paper to obtain a smoother back-EMF signal, which further improves the rotor position estimation accuracy.According to equation (13), the state equation of the Kalman filter is designed as [24]
(14)ddtE^αddtE^βddtω^e=‐E^βω^e−E~αλE^αω^e−E~βλE^βE~α−E^αE~β,where λ is the adaptive gain, ω^e is the estimated rotational speed, E^α and E^β are the final observed values of back-EMF, E~α=E^α−vα−lvαeq, and E~β=E^β−vβ−lvβeq.Considering that the adaptive gain coefficient which is too large will cause the system to be unstable and too small will cause the system to converge too slowly, so this paper designs the adaptive gain according to the actual speed of the motor as(15)λ=λa+κωe,where λa and κ are both normal numbers and ωe represents the actual speed of the motor.Stability analysis(16)V=12E~2α+E~2β+ω~2e.where ω~e=ω^e−ωe.Since the mechanical time constant is much larger than the electrical time constant, it is considered that the speed does not change within an estimation period. which can be obtained from equation (3) and equation (14):
(17)ddtE~αddtE~βddtω~e=‐E^βω~e−E~βω^e−E~αλE^αω~e+E~αω^e−E~βλE^βE~α−E^αE~β.Substitute equation (17) into equation (16) and derive
(18)ddtV=E~αddtE~α+E~βddtE~β+ω~eddtω~e=‐E~αE^βω~e−E~αE~βω^e−λE~2α+E^αE~βω~e+E~αE~βω^e−λE~2β+E~αE^βω~e−E^αE~βω~e=‐λE~2α+E~2β≤0.Therefore, according to the Lyapunov stability criterion, the system with the Kalman filter is stable. The structure schematic diagram of AFG-STA-SMO is shown in Figure1.Figure 1
Block diagram of AFG-STA-SMO.


### 3.4. System Stability Analysis

The Lyapunov function can define as(19)V1=12i~α2+i~β2.From equations (19) and (11), it can be obtained that
(20)ddtV1=‐RLi~2α+i~2β−1Le~αi~α+e~βi~β−k1i~α3/2+i~β3/2<0.In order to ensure the convergence of the current-sliding mode observer,d/dtV1<0 must satisfy; then, the sliding mode gain k1 can be obtained as follows:
(21)k1>sup‐RL⋅i~2α+i~2βi~α3/2+i~β3/2−1L⋅e~αi~α+e~βi~βi~α3/2+i~β3/2,where sup represents the upper bound.When the current-sliding mode observer converges,d/dti~α=i~α=d/dti~β=i~β=0, from equation (11) that
(22)signi~α=‐e~αk1⋅L⋅i~α1/2,signi~β=‐e~βk1⋅L⋅i~β1/2.Substitute equation (22) into equation (10) and get
(23)ddtvα+lvαeq=‐1+lk2⋅e~αk1⋅Ls⋅i~α1/2,ddtvβ+lvβeq=‐1+lk2⋅e~βk1⋅Ls⋅i~β1/2.It can be obtained from equations (23) and (3) that
(24)ddte~α=‐1+lk2⋅e~αk1⋅Ls⋅i~α1/2+ωe⋅Eβ,ddte~β=‐1+lk2⋅e~βk1⋅Ls⋅i~β1/2−ωe⋅Eα.Similarly, the Lyapunov function can be defined as(25)V2=12e~α2+e~β2.From equations (25) and (24), the following equation can be obtained:
(26)ddtV2=‐1+lk2e~2αk1⋅L⋅i~α1/2+e~2βk1⋅L⋅i~β1/2+ωe⋅Eβ⋅e~α−ωe⋅Eα⋅e~β<0.To ensure the convergence of the current-sliding mode observer,d/dtV2<0 must satisfy; then, the sliding mode gain k2 can be obtained as follows:
(27)1+lk2>supk1⋅Ls⋅ωe⋅Eβ⋅e~α−Eα⋅e~βe~2α/i~α1/2+e~2β/i~β1/2.Equations (21) and (27) give the stability conditions of the established system.


### 3.5. Adaptive Law Analysis of Feedback Gain Coefficient

In equation (20), ‐R/Li~2α+i~2β and −k1i~α3/2+i~β3/2 is always less than 0. A sufficient condition for the existence of equation (20) is constructed as follows:
(28)‐RLi~2α+i~2β−1Le~αi~α+e~βi~β−k1i~α3/2+i~β3/2<0,i~α>0,i~β>0,‐RLi~2α+i~2β−1L‐e~αi~α−e~βi~β−k1i~α3/2+i~β3/2<0,i~α<0,i~β<0.Since‐R/Li~2α+i~2β<0 and −k1i~α3/2+i~β3/2<0, it can be deduced that the condition satisfying d/dtV1<0 is as follows:
(29)1+lk2>maxEα,Eβ=ωeψf,where ωe is the actual speed and ψf is the flux linkage.From equation (29), it can be seen that the selection of l affects the size of the sliding mode gain of k2. The larger the l is, the lower the limit of the effective range of k2 is. Conversely, the smaller the l is, the higher the lower limit of the effective range of k2 is.According to equation (29),
(30)l>ωeψfk2−1,where k2>ψf.According to equation (30), 1>ψf/k2>0 is known, so the feedback gain coefficient l can be designed as follows:
(31)l=δωe−1,where δ is a normal number, which is related to the gain of sliding mode k2. So k2 meet the following conditions:
(32)k2>ψfδ.In order to selectδ conveniently, 0<δ<1 is set in this paper.


## 3.1. Supertwisting Algorithm

The general form of the supertwisting algorithm [17] is as follows:
(9)dx^1dt=‐k1x~11/2signx~1+x2+ρ1,dx^2dt=‐k2signx~1+ρ2,where xi, x^i, x~i, ki, ρi, and sign are state variables, estimation of state variables, the error between estimated and actual state variables, sliding mode gain coefficient, perturbation term, and signum function, respectively.


## 3.2. AFG-STA-SMO

Since SPMSM high-precision vector control requires accurate motor speed and rotor position information, traditional SMO has parameter perturbation and chattering. This paper combines the principle of equivalent feedback with STA-SMO, and the design is as follows:(10)ddti^αi^β=‐RLi^αi^β+1LUαUβ−1Lvα+l⋅vαeqvβ+l⋅vβeq−k1⋅i~α1/2⋅signi~αk1⋅i~β1/2⋅signi~β,ddtvαvβ=k2⋅signi~αk2⋅signi~β,vαeqvβeq=vαvβ,where l is the feedback gain coefficient, i^αβ are the observation value of the stator currents, Uαβ are α-β axis voltages, ‐R/Liα+1/LUα and ‐R/Liβ+1/LUβ are regarded as the perturbation terms, vα=∫k2⋅signi~αdt and vβ=∫k2⋅signi~βdt are the estimated value of back-EMFs, and vαeq and vβeq are the equivalent feedback electromotive force values.The difference between equations (4) and (10) can be obtained:
(11)ddti~αi~β=‐RsLsi~αi~β−1Lse~αe~β−k1⋅i~α1/2⋅signi~αk1⋅i~β1/2⋅signi~β,where i~α=i^α−iα, i~β=i^β−iβ, e~α=vα+lvαeq−Eα, and e~β=vβ+lvβeq−Eβ.After the state variable of the observer reaches the sliding mode surfacei~α=0 and i~β=0, according to the sliding mode variable structure equivalent principle, we can get
(12)EαEβ=vα+lvαeqvβ+lvβeq=1+l⋅∫k2⋅signi~αdt1+l⋅∫k2⋅signi~βdt.From equation (12), the integral term vαβ filters out high-frequency sliding mode noise, so introducing low-pass filters and phase compensation is avoided, and the system algorithm is simplified. It can be seen from equation (12) that the sliding mode gain k2 and feedback gain l are proportional to the speed. When the motor is running at high speed, the sliding-mode gain k2 and the feedback gain l are larger. When the motor runs at low speed, the sliding-mode gain k2 and the feedback gain l are smaller.


## 3.3. Optimization of AFG-STA-SMO

The derivative of equation (12) is obtained:
(13)ddtEα=‐ωe⋅Eβ=1+l⋅k2⋅signi~α,ddtEβ=ωe⋅Eα=1+l⋅k2⋅signi~β.Although the harmonic component is effectively filtered by integral operation, the back-EMF’s observed value still contains a ripple component, which will lead to error if used to estimate the rotor position directly. Therefore, the Kalman filter is used in this paper to obtain a smoother back-EMF signal, which further improves the rotor position estimation accuracy.According to equation (13), the state equation of the Kalman filter is designed as [24]
(14)ddtE^αddtE^βddtω^e=‐E^βω^e−E~αλE^αω^e−E~βλE^βE~α−E^αE~β,where λ is the adaptive gain, ω^e is the estimated rotational speed, E^α and E^β are the final observed values of back-EMF, E~α=E^α−vα−lvαeq, and E~β=E^β−vβ−lvβeq.Considering that the adaptive gain coefficient which is too large will cause the system to be unstable and too small will cause the system to converge too slowly, so this paper designs the adaptive gain according to the actual speed of the motor as(15)λ=λa+κωe,where λa and κ are both normal numbers and ωe represents the actual speed of the motor.Stability analysis(16)V=12E~2α+E~2β+ω~2e.where ω~e=ω^e−ωe.Since the mechanical time constant is much larger than the electrical time constant, it is considered that the speed does not change within an estimation period. which can be obtained from equation (3) and equation (14):
(17)ddtE~αddtE~βddtω~e=‐E^βω~e−E~βω^e−E~αλE^αω~e+E~αω^e−E~βλE^βE~α−E^αE~β.Substitute equation (17) into equation (16) and derive
(18)ddtV=E~αddtE~α+E~βddtE~β+ω~eddtω~e=‐E~αE^βω~e−E~αE~βω^e−λE~2α+E^αE~βω~e+E~αE~βω^e−λE~2β+E~αE^βω~e−E^αE~βω~e=‐λE~2α+E~2β≤0.Therefore, according to the Lyapunov stability criterion, the system with the Kalman filter is stable. The structure schematic diagram of AFG-STA-SMO is shown in Figure1.Figure 1
Block diagram of AFG-STA-SMO.


## 3.4. System Stability Analysis

The Lyapunov function can define as(19)V1=12i~α2+i~β2.From equations (19) and (11), it can be obtained that
(20)ddtV1=‐RLi~2α+i~2β−1Le~αi~α+e~βi~β−k1i~α3/2+i~β3/2<0.In order to ensure the convergence of the current-sliding mode observer,d/dtV1<0 must satisfy; then, the sliding mode gain k1 can be obtained as follows:
(21)k1>sup‐RL⋅i~2α+i~2βi~α3/2+i~β3/2−1L⋅e~αi~α+e~βi~βi~α3/2+i~β3/2,where sup represents the upper bound.When the current-sliding mode observer converges,d/dti~α=i~α=d/dti~β=i~β=0, from equation (11) that
(22)signi~α=‐e~αk1⋅L⋅i~α1/2,signi~β=‐e~βk1⋅L⋅i~β1/2.Substitute equation (22) into equation (10) and get
(23)ddtvα+lvαeq=‐1+lk2⋅e~αk1⋅Ls⋅i~α1/2,ddtvβ+lvβeq=‐1+lk2⋅e~βk1⋅Ls⋅i~β1/2.It can be obtained from equations (23) and (3) that
(24)ddte~α=‐1+lk2⋅e~αk1⋅Ls⋅i~α1/2+ωe⋅Eβ,ddte~β=‐1+lk2⋅e~βk1⋅Ls⋅i~β1/2−ωe⋅Eα.Similarly, the Lyapunov function can be defined as(25)V2=12e~α2+e~β2.From equations (25) and (24), the following equation can be obtained:
(26)ddtV2=‐1+lk2e~2αk1⋅L⋅i~α1/2+e~2βk1⋅L⋅i~β1/2+ωe⋅Eβ⋅e~α−ωe⋅Eα⋅e~β<0.To ensure the convergence of the current-sliding mode observer,d/dtV2<0 must satisfy; then, the sliding mode gain k2 can be obtained as follows:
(27)1+lk2>supk1⋅Ls⋅ωe⋅Eβ⋅e~α−Eα⋅e~βe~2α/i~α1/2+e~2β/i~β1/2.Equations (21) and (27) give the stability conditions of the established system.


## 3.5. Adaptive Law Analysis of Feedback Gain Coefficient

In equation (20), ‐R/Li~2α+i~2β and −k1i~α3/2+i~β3/2 is always less than 0. A sufficient condition for the existence of equation (20) is constructed as follows:
(28)‐RLi~2α+i~2β−1Le~αi~α+e~βi~β−k1i~α3/2+i~β3/2<0,i~α>0,i~β>0,‐RLi~2α+i~2β−1L‐e~αi~α−e~βi~β−k1i~α3/2+i~β3/2<0,i~α<0,i~β<0.Since‐R/Li~2α+i~2β<0 and −k1i~α3/2+i~β3/2<0, it can be deduced that the condition satisfying d/dtV1<0 is as follows:
(29)1+lk2>maxEα,Eβ=ωeψf,where ωe is the actual speed and ψf is the flux linkage.From equation (29), it can be seen that the selection of l affects the size of the sliding mode gain of k2. The larger the l is, the lower the limit of the effective range of k2 is. Conversely, the smaller the l is, the higher the lower limit of the effective range of k2 is.According to equation (29),
(30)l>ωeψfk2−1,where k2>ψf.According to equation (30), 1>ψf/k2>0 is known, so the feedback gain coefficient l can be designed as follows:
(31)l=δωe−1,where δ is a normal number, which is related to the gain of sliding mode k2. So k2 meet the following conditions:
(32)k2>ψfδ.In order to selectδ conveniently, 0<δ<1 is set in this paper.


## 4. Rotor Position Estimation

Since the sliding mode control is accompanied by high-frequency chattering in the sliding mode, the high-frequency chattering phenomenon will estimate in the back-EMF. The rotor position estimation method based on the arctangent function will introduce this chattering into the division operation; significantly, when the observed value of back-EMF exceeds zero, the rotor error will amplify. Therefore, PLL [25] is adopted in this paper to replace the arctangent function to extract the rotor’s position information, as shown in Figure 2.Figure 2
Schematic diagrams of PLL.Hypothesisη=Ld−Lqωeid−piq+ω^eζf, when θe−θ^e<π/6 and sinθe−θ^e=θe−θ^e is established, according to Figure 2, can get the following relationship:
(33)ΔE=‐E^αcosθ^e−E^βsinθ^e=ηsinθecosθ^e−ηsinθ^ecosθe=ηsinθe−θ^e≈ηθe−θ^e.At this time, the equivalent block diagram of Figure2 is shown in Figure 3.Figure 3
Equivalent schematic diagrams of PLL.According to Figure3, the transfer function of PLL can be obtained, namely,
(34)Gs=θ^eθe=2ξωn+ω2ns2+2ξωns+ω2n,where ωn=Kp/2η/Ki, ξ=ηKi, and ωn determine the bandwidth of the PI regulator.


## 5. Simulation

According to Figure4, MATLAB is used to build a simulation model, and the control scheme adopts vector control based on id=0. The motor parameters in the simulation experiment are shown in Table 1.Figure 4
SPMSM speed sensor-less control block diagram of AFG-STA-SMO.Table 1
SPMSM parameters.
ParametersValuesStator resistanceRs2.875ΩPole-pair number4Magnetic fluxψf0.175 WbD-axis induction Ld8.5e-3HQ-axis induction Lq8.5e-3HDamping factorB0DC bus voltageUdc310 VPole-pair number4Magnetic fluxψf0.175 WbRotational inertiaJ0.001 kg·m2The control system parameters of SPMSM based on the traditional sliding mode observer are as follows:k=200. The control system parameters of SPMSM based on traditional STA-SMO are as follows: k1=5000 and k2=200000. The control system parameters of STA-SMO PMSM based on fixed feedback gain are k1=5000, k2=700, and l=500. The control system parameters of PMSM based on AFG-STA-SMO are as follows: k1=5000, k2=700, and δ=0.5.


### 5.1. Improved AFG-STA-SMO Simulation

When the SPMSM is under sensor-less control, the rotor position and speed signals are hidden in the back-EMF observation value. This paper mainly uses a super spiral sliding mode observer to extract rotor position and speed information from the back-EMF observations. Although this strategy can avoid introducing low-pass filters, it cannot effectively filter the back-EMF ripple component. In order to obtain a smooth back-EMF signal, this article introduces a Kalman filter to filter the back-EMF signal again. The simulation result is shown in Figure5. To verify the effectiveness of the strategy proposed in this paper, the motor adopts a no-load starting mode with a given speed of 100 r/min. At 0.2 s, the speed is increased to 300 r/min; at 0.4 s, the speed is increased to 1100 r/min. Figure 5 shows the back-EMF response waveforms under two different strategies.Figure 5
Back-EMF response diagram.Table2 shows that the back-EMF fluctuations of the improved strategy proposed in this paper at three different speed stages are 50%, 8%, and 2.5% of the back-EMF fluctuations of the unimproved strategy, respectively.(Ea1: AFG-STA-SMO without Kalman filter; Ea2: AFG-STA-SMO with Kalman filter). By analyzing Figure 5 and the above results, it can be seen that the introduction of the Kalman filter can effectively suppress the ripple component in the back-EMF observation value, thereby obtaining a smoother back-EMF observation signal and further improving the accuracy of the rotor position estimation.Table 2
Back-EMF fluctuation.
ResultEa1Ea2UnitBack-EMF fluctuation (a)7.34~7.327.325~7.315VBack-EMF fluctuation (b)22.05~21.92522.025~22.015VBack-EMF fluctuation (c)80.9 ~ 80.580.7 ~ 80.69V


### 5.2. Verification of Adaptive Feedback Gain Adjustment

In order to verify the validity of the adaptive law proposed in this article, the motor adopts a no-load starting mode with a given speed of 100 r/min. At 0.2 s, the speed is increased to 300 r/min; at 0.4 s, the speed is increased to 1100 r/min.From Figures6 and 7 and Table 3, it is shown that the speed errors of AFG-STA-SMO at three different speed stages are 54.26%, 35.45%, and 100% of STA-SMO speed errors with fixed feedback gain (FFG-STA-SMO), respectively.Figure 6
Speed response diagram.Figure 7
Speed error response diagram.Table 3
Back-EMF fluctuation.
ResultFFG-STA-SMOAFG-STA-SMOUnitError (a)0.32250.175RpmError (b)0.2750.0975RpmError (c)0.270.27RpmPosition error (a)0.00430.0007DegPosition error (b)0.000370.00009DegPosition error (c)0.000160.00016DegFigures8–10 and Table 3 show that the rotor position errors of AFG-STA-SMO at three different speed stages are 16.28%, 24.32%, and 100% of FFG-STA-SMO rotor position errors, respectively.Figure 8
Rotor position change with fixed feedback gain.Figure 9
Rotor position change with adaptive feedback gain.Figure 10
Rotor position error diagram.From the above results and the simulation diagram analysis, it can be seen that under the action of the fixed feedback gain, when the motor is running at 1000 r/min, the speed error and rotor position error are minimal. As the speed continues to decrease, the speed error and rotor position error are gradually increasing. After introducing adaptive control, the feedback gain is proportional to the speed. During the entire speed change process, the speed errors are less than the speed errors of FFG-STA-SMO; the estimated rotor position and the actual rotor position are the same, enhancing the system stability.


### 5.3. Medium and High-Speed Verification Analysis

In order to verify the advantages of the control strategy proposed in this paper in medium and high-speed operation, the motor adopts a no-load starting mode, and the given speed is 600 r/min. At 0.1 s, load torque of 4.5 N·m is applied; at 0.2 s, the speed rises to 1000 r/min; at the first variable speed stable operation stage (0.3 s), the load torque changes to 5 N·m; at 0.4 s, the speed is increased to 2000 r/min.From Figure11, the application of 4.5 N·m and 5 N·m of load torque in 0.1 s and 0.3 s, respectively, causes a slight speed drop in the actual speed. At this time, the estimated speed is still tracking the actual speed, and there is also a slight speed drop that has proved the feasibility and effectiveness of these three control strategies.Figure 11
Speed response diagram.Figure12 and Table 4 show that the speed errors of AFG-STA-SMO at three different speed stages are 0.429%, 0.556%, and 1.5% of SMO speed errors, respectively. The speed errors of AFG-STA-SMO at three different speed stages are 5%, 8.3%, and 20% of STA-SMO speed errors, respectively.Figure 12
Speed error response diagram.Table 4
Speed Error.
ResultSMOSTA-SMOAFG-STA-SMOUnitError (a)3530.15RpmError (b)4530.25RpmError (c)4030.6RpmFrom the above results and simulation waveform analysis, it can be seen that the speed error of SMO increases with the continuous increase of the speed; the sliding mode chattering is serious, which is not conducive to the system’s stable operation of the system. There is an obvious lag phenomenon in the speed rise of 2000 r/min. Compared with SMO, STA-SMO effectively suppresses sliding mode chattering due to the characteristics of high-order sliding mode and integral operation function and reduces the speed error, thereby avoiding the lag phenomenon caused by introducing low-pass filters. Compared with STA-SMO, the proposed control strategy is based on STA-SMO, and the Kalman filter and adaptive control are introduced to suppress further sliding mode chattering so that the speed error is minimized, and the change of speed error is small when coping with external disturbance.Table5 shows that the back-EMF fluctuations of AFG-STA-SMO at three different speed stages are 0.75%, 2.78%, and 0.5% of the back-EMF fluctuations of SMO, respectively. The back-EMF fluctuations of AFG-STA-SMO at three different speed stages are 12.5%, 6.25%, and 4% of the back-EMF fluctuations of STA-SMO, respectively.Table 5
THD and back-EMF fluctuation.
ResultSMOSTA-SMOAFG-STA-SMOUnitTHD (a)8.928.628.68%THD (b)1.841.271.24%THD (c)1.504.283.84%Back-EMF fluctuation (a)-31.8~-33.725-32.92~-33-32.91~-32.92VBack-EMF fluctuation (b)-72~-72.36-72.6~-72.76-72.685~-72.695VBack-EMF fluctuation (c)143~145146.35~146.646.22~146.23VFigure13, Table 5, and the above analysis of the results show that the SMO’s back-EMF has a lag due to the low-pass filter’s introduction, which affects the system’s response speed. However, STA-SMO avoids this phenomenon due to integral operation. Although these two control strategies can effectively filter out the harmonic components, there are many ripple components in the back-EMF value. If they are directly used to estimate the rotor position, they will inevitably cause larger errors. Therefore, the Kalman filter is introduced to perform filtering processing to effectively suppress the back-EMF ripple component, thereby obtaining a smoother back-EMF signal and improving control accuracy.Figure 13
Response diagram of back-EMF.


### 5.4. Low-Speed Verification Analysis

In order to verify the advantages of the sensor-less algorithm-controlled speed control system in low-speed operation situations, simulation experiments are carried out in low-speed areas. The motor adopts a no-load start mode, and the given speed is 100 r/min. At 0.2 s, the speed suddenly changes to 15 r/min.Figure14, Figure 15, and Table 6 show that the speed errors of AFG-STA-SMO at two different speed stages are 0.6534% and 1.1423% of the SMO speed errors, respectively. The speed errors of AFG-STA-SMO at two different speed stages are 5.3125% and 12.5% of STA-SMO speed errors, respectively.Figure 14
Response diagram of speed.Figure 15
Response diagram of speed error.Table 6
Speed error and rotor position error.
ResultSMOSTA-SMOAFG-STA-SMOUnitError (a)263.20.17RpmError (b)353.20.4RpmPosition error (a)0.30.030.0016DegPosition error (b)0.80.20.006DegFigures16–19 and Table 6 show that the rotor position errors of AFG-STA-SMO at two different speed stages are 0.53% and 0.75% of SMO rotor position errors, respectively. The rotor position errors of AFG-STA-SMO at two different speed stages are 5.33% and 3% of STA-SMO rotor position errors, respectively.Figure 16
The rotor position change of traditional SMO.Figure 17
The rotor position change of traditional STA-SMO.Figure 18
The rotor position change of AFG-STA-SMO.Figure 19
Rotor position error of the response diagram.From the above results and simulation diagram analysis, it can be seen that when the motor runs at low speed, the chattering of SMO speed waveform and rotor estimated position waveform is serious, and the speed error and rotor position error are the largest. When the speed is reduced to 15 r/min, the system almost cannot operate normally. Compared with SMO, STA-SMO suppresses sliding mode chattering to a certain extent due to the characteristics of high-order sliding mode and the function of the integrator. However, the back-EMF is too small when the motor runs at low speed, resulting in rotor position error and speed error deviation. It is not conducive to the stable operation of the system. Compared with STA-SMO, the estimated rotor position of AFG-STA-SMO is basically the same as the actual rotor position due to the introduction of equivalent feedback gain, and the speed error and rotor position error are minimal, thus avoiding the situation that the rotor position error is too large due to the small back-EMF.


## 5.1. Improved AFG-STA-SMO Simulation

When the SPMSM is under sensor-less control, the rotor position and speed signals are hidden in the back-EMF observation value. This paper mainly uses a super spiral sliding mode observer to extract rotor position and speed information from the back-EMF observations. Although this strategy can avoid introducing low-pass filters, it cannot effectively filter the back-EMF ripple component. In order to obtain a smooth back-EMF signal, this article introduces a Kalman filter to filter the back-EMF signal again. The simulation result is shown in Figure5. To verify the effectiveness of the strategy proposed in this paper, the motor adopts a no-load starting mode with a given speed of 100 r/min. At 0.2 s, the speed is increased to 300 r/min; at 0.4 s, the speed is increased to 1100 r/min. Figure 5 shows the back-EMF response waveforms under two different strategies.Figure 5
Back-EMF response diagram.Table2 shows that the back-EMF fluctuations of the improved strategy proposed in this paper at three different speed stages are 50%, 8%, and 2.5% of the back-EMF fluctuations of the unimproved strategy, respectively.(Ea1: AFG-STA-SMO without Kalman filter; Ea2: AFG-STA-SMO with Kalman filter). By analyzing Figure 5 and the above results, it can be seen that the introduction of the Kalman filter can effectively suppress the ripple component in the back-EMF observation value, thereby obtaining a smoother back-EMF observation signal and further improving the accuracy of the rotor position estimation.Table 2
Back-EMF fluctuation.
ResultEa1Ea2UnitBack-EMF fluctuation (a)7.34~7.327.325~7.315VBack-EMF fluctuation (b)22.05~21.92522.025~22.015VBack-EMF fluctuation (c)80.9 ~ 80.580.7 ~ 80.69V


## 5.2. Verification of Adaptive Feedback Gain Adjustment

In order to verify the validity of the adaptive law proposed in this article, the motor adopts a no-load starting mode with a given speed of 100 r/min. At 0.2 s, the speed is increased to 300 r/min; at 0.4 s, the speed is increased to 1100 r/min.From Figures6 and 7 and Table 3, it is shown that the speed errors of AFG-STA-SMO at three different speed stages are 54.26%, 35.45%, and 100% of STA-SMO speed errors with fixed feedback gain (FFG-STA-SMO), respectively.Figure 6
Speed response diagram.Figure 7
Speed error response diagram.Table 3
Back-EMF fluctuation.
ResultFFG-STA-SMOAFG-STA-SMOUnitError (a)0.32250.175RpmError (b)0.2750.0975RpmError (c)0.270.27RpmPosition error (a)0.00430.0007DegPosition error (b)0.000370.00009DegPosition error (c)0.000160.00016DegFigures8–10 and Table 3 show that the rotor position errors of AFG-STA-SMO at three different speed stages are 16.28%, 24.32%, and 100% of FFG-STA-SMO rotor position errors, respectively.Figure 8
Rotor position change with fixed feedback gain.Figure 9
Rotor position change with adaptive feedback gain.Figure 10
Rotor position error diagram.From the above results and the simulation diagram analysis, it can be seen that under the action of the fixed feedback gain, when the motor is running at 1000 r/min, the speed error and rotor position error are minimal. As the speed continues to decrease, the speed error and rotor position error are gradually increasing. After introducing adaptive control, the feedback gain is proportional to the speed. During the entire speed change process, the speed errors are less than the speed errors of FFG-STA-SMO; the estimated rotor position and the actual rotor position are the same, enhancing the system stability.


## 5.3. Medium and High-Speed Verification Analysis

In order to verify the advantages of the control strategy proposed in this paper in medium and high-speed operation, the motor adopts a no-load starting mode, and the given speed is 600 r/min. At 0.1 s, load torque of 4.5 N·m is applied; at 0.2 s, the speed rises to 1000 r/min; at the first variable speed stable operation stage (0.3 s), the load torque changes to 5 N·m; at 0.4 s, the speed is increased to 2000 r/min.From Figure11, the application of 4.5 N·m and 5 N·m of load torque in 0.1 s and 0.3 s, respectively, causes a slight speed drop in the actual speed. At this time, the estimated speed is still tracking the actual speed, and there is also a slight speed drop that has proved the feasibility and effectiveness of these three control strategies.Figure 11
Speed response diagram.Figure12 and Table 4 show that the speed errors of AFG-STA-SMO at three different speed stages are 0.429%, 0.556%, and 1.5% of SMO speed errors, respectively. The speed errors of AFG-STA-SMO at three different speed stages are 5%, 8.3%, and 20% of STA-SMO speed errors, respectively.Figure 12
Speed error response diagram.Table 4
Speed Error.
ResultSMOSTA-SMOAFG-STA-SMOUnitError (a)3530.15RpmError (b)4530.25RpmError (c)4030.6RpmFrom the above results and simulation waveform analysis, it can be seen that the speed error of SMO increases with the continuous increase of the speed; the sliding mode chattering is serious, which is not conducive to the system’s stable operation of the system. There is an obvious lag phenomenon in the speed rise of 2000 r/min. Compared with SMO, STA-SMO effectively suppresses sliding mode chattering due to the characteristics of high-order sliding mode and integral operation function and reduces the speed error, thereby avoiding the lag phenomenon caused by introducing low-pass filters. Compared with STA-SMO, the proposed control strategy is based on STA-SMO, and the Kalman filter and adaptive control are introduced to suppress further sliding mode chattering so that the speed error is minimized, and the change of speed error is small when coping with external disturbance.Table5 shows that the back-EMF fluctuations of AFG-STA-SMO at three different speed stages are 0.75%, 2.78%, and 0.5% of the back-EMF fluctuations of SMO, respectively. The back-EMF fluctuations of AFG-STA-SMO at three different speed stages are 12.5%, 6.25%, and 4% of the back-EMF fluctuations of STA-SMO, respectively.Table 5
THD and back-EMF fluctuation.
ResultSMOSTA-SMOAFG-STA-SMOUnitTHD (a)8.928.628.68%THD (b)1.841.271.24%THD (c)1.504.283.84%Back-EMF fluctuation (a)-31.8~-33.725-32.92~-33-32.91~-32.92VBack-EMF fluctuation (b)-72~-72.36-72.6~-72.76-72.685~-72.695VBack-EMF fluctuation (c)143~145146.35~146.646.22~146.23VFigure13, Table 5, and the above analysis of the results show that the SMO’s back-EMF has a lag due to the low-pass filter’s introduction, which affects the system’s response speed. However, STA-SMO avoids this phenomenon due to integral operation. Although these two control strategies can effectively filter out the harmonic components, there are many ripple components in the back-EMF value. If they are directly used to estimate the rotor position, they will inevitably cause larger errors. Therefore, the Kalman filter is introduced to perform filtering processing to effectively suppress the back-EMF ripple component, thereby obtaining a smoother back-EMF signal and improving control accuracy.Figure 13
Response diagram of back-EMF.


## 5.4. Low-Speed Verification Analysis

In order to verify the advantages of the sensor-less algorithm-controlled speed control system in low-speed operation situations, simulation experiments are carried out in low-speed areas. The motor adopts a no-load start mode, and the given speed is 100 r/min. At 0.2 s, the speed suddenly changes to 15 r/min.Figure14, Figure 15, and Table 6 show that the speed errors of AFG-STA-SMO at two different speed stages are 0.6534% and 1.1423% of the SMO speed errors, respectively. The speed errors of AFG-STA-SMO at two different speed stages are 5.3125% and 12.5% of STA-SMO speed errors, respectively.Figure 14
Response diagram of speed.Figure 15
Response diagram of speed error.Table 6
Speed error and rotor position error.
ResultSMOSTA-SMOAFG-STA-SMOUnitError (a)263.20.17RpmError (b)353.20.4RpmPosition error (a)0.30.030.0016DegPosition error (b)0.80.20.006DegFigures16–19 and Table 6 show that the rotor position errors of AFG-STA-SMO at two different speed stages are 0.53% and 0.75% of SMO rotor position errors, respectively. The rotor position errors of AFG-STA-SMO at two different speed stages are 5.33% and 3% of STA-SMO rotor position errors, respectively.Figure 16
The rotor position change of traditional SMO.Figure 17
The rotor position change of traditional STA-SMO.Figure 18
The rotor position change of AFG-STA-SMO.Figure 19
Rotor position error of the response diagram.From the above results and simulation diagram analysis, it can be seen that when the motor runs at low speed, the chattering of SMO speed waveform and rotor estimated position waveform is serious, and the speed error and rotor position error are the largest. When the speed is reduced to 15 r/min, the system almost cannot operate normally. Compared with SMO, STA-SMO suppresses sliding mode chattering to a certain extent due to the characteristics of high-order sliding mode and the function of the integrator. However, the back-EMF is too small when the motor runs at low speed, resulting in rotor position error and speed error deviation. It is not conducive to the stable operation of the system. Compared with STA-SMO, the estimated rotor position of AFG-STA-SMO is basically the same as the actual rotor position due to the introduction of equivalent feedback gain, and the speed error and rotor position error are minimal, thus avoiding the situation that the rotor position error is too large due to the small back-EMF.


## 6. Conclusion

This paper presents a sensor-less control strategy for STA-SMO with adaptive feedback gain. The adaptive law, which compensates for rotor angle estimation error by adjusting the feedback gain coefficient online, effectively solves the motor’s problems of low-speed operation difficulty and serious sliding mode chattering.Secondly, the Kalman filter is used to reduce the ripple component in the back-EMF. The smoother back-EMF signal is obtained, and the control precision of the algorithm is further improved.The whole system realizes the full speed domain operation; the dynamic response is fast and has the strong disturbance resistance when dealing with the external disturbance. In the next step, the author intends to implement the proposed control strategy with high-performance DSP chip in order to improve the operation time extension caused by the increasing complexity of the control system.



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*Source: 8301359-2021-10-20.xml*