# SPMSM Sensorless Control System Based on ASMO

Authors: Yu Cao; Chuang Liu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Aiming at the chattering problem of traditional sliding mode observer (SMO) in surface-mounted permanent magnet synchronous motor (SPMSM) control system when estimating rotor position and speed and adaptive sliding mode observer (ASMO) is designed in this paper. The observer can reduce chattering and avoid the introduction of the low-pass filter, which simplifies the system structure, considering the difficulty in determining the sliding mode gain of the existing sliding mode observer, and an adaptive law is designed to adjust the sliding mode gain with the change of the back electromotive force, to meet the operation requirements of the system and improve the control accuracy of the sensorless control system. Finally, the simulation is built through Matlab/Simulink platform. It is proved that the proposed control strategy can satisfy the system’s accuracy and simultaneously reduce the chattering and solve the problem that the gain is difficult to determine.


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## Body

## 1. Introduction

Surface-mounted permanent magnet synchronous motor has obvious advantages of high mechanical efficiency and high power factor [1], so it is widely used in wind power generation systems. Because wind turbines often operate in harsh environments, mechanical sensors’ installation and use will increase the system’s cost and reduce its reliability [2]. Therefore, speed sensorless control has become a research hotspot. At present, the commonly used algorithms include sliding mode observer, model reference adaptive control [3–7], extended Kalman filter [8–11], and artificial intelligence method [12–14]. The sliding mode observer has been widely used in speed sensorless SPMSM rotor position and speed estimation due to its insensitivity to disturbance and parameters, fast response, and other advantages [15]. However, the sliding mode control structure’s characteristics bring strong robustness and cause system chattering, which hinders its application in practical engineering. Therefore, the low-pass filter introduced in the traditional SMO algorithm filters the back EMF. Because the cutoff frequency of the traditional low-pass filter is fixed, the ripple component in the back EMF cannot be eliminated, which seriously affects the estimation accuracy of rotor position [16]. To obtain a smooth estimation of the back EMF and SMO based on a variable cutoff frequency low-pass filter and a modified back EMF, the observer is proposed in the literature [17]. The SMO can effectively suppress the high frequency and ripple components of the back EMF. However, the filter’s introduction causes the system’s phase delay to a certain extent, and the two-stage filtering structure also causes the system’s design complexity. It is not easy to implement. In literature [18], a sliding mode observer with a sinusoidal saturation function is designed to reduce chattering. However, because the saturated boundary layer is closely related to the approaching velocity and buffeting, how to choose a reasonable boundary layer is a big problem. In literature [19, 20], a sliding mode observer with improved reaching law is proposed to reduce the chattering caused by sliding mode control. Although the purpose is achieved in some cases, the method will be invalid when the control deviation changes to zero, and the sliding mode gain is also zero.To avoid introducing a low-pass filter, a new type of sliding mode observer was designed in the literature [21]. The hybrid nonsingular sliding mode terminal sliding surface was applied to the traditional linear sliding mode surface, which reduced the phase lag and chattering problems. However, the optimization of hybrid nonsingular sliding mode design is complicated, and the hardware requirements are relatively high, so it is not easy to realize. Literature [22–25] proposed a sliding mode observer based on the super-twisting algorithm. Although the scheme can effectively suppress sliding mode chattering, it is not easy to choose a good sliding mode gain. The sliding mode gain’s stability condition is related to the control function error. Literature [26] proposed a neural network estimation strategy that is used to obtain sliding mode gain. Although this scheme can weaken chattering, the designed variable parameters make the system more complex. Literature [27] proposed a sliding mode observer based on fuzzy radial basis function (RBF) neural network to adjust sliding mode gain to better adapt to external disturbances parameter changes. It needs experience in practice and is difficult to master. The super twisting sliding mode observer based on the adaptive algorithm avoids the introduction of complex neural network on the basis of solving the difficulty in determining the sliding mode gain, but only limited time to enter the sliding mode variable contains households within the neighborhood of plastering, damaging the convergence precision of the algorithm [28].To avoid the above problems, a sliding mode observer based on the adaptive algorithm is proposed in this paper. The observer has the characteristics of a first-order low-pass filter, which can effectively filter the high-frequency sliding mode noise contained in the estimated back EMF without adding a low-pass filter. Simultaneously, an adaptive gain law varying with the back EMF is designed to adjust the observer’s parameters online, which solves excessive gain and effectively suppresses the system’s chattering. Finally, the information on rotor position and speed is estimated by using the arctangent function.


## 2. Traditional SMO

In theαβ two-phase stationary coordinate system, the mathematical model of SPMSM can be expressed as [17](1)i˙α=−RsLiα+uαL−zαLi˙β=−RsLiβ+uβL−zβL,where uα, uβiα, iβ are the stator currents and voltages of α-β axis, respectively; Rs is the stator resistance; L is the stator inductance; zα, zβ are the back EMFs of the α-β axis; and the expression is(2)zα=−ψfωesinθezβ=ωeψfcosθe,where ψf is the rotor flux linkage, θe is the rotor position angle, and ωe is the electric angular velocity of the rotor.According to (1), the SPMSM sliding mode observer can be constructed.(3)i^˙α=−RsLi^α+uαL−hαLi^˙β=−RsLi^β+uβL−hβL,where i^α, i^β are the α-β axis estimation values of the stator current.The stator current error system can be obtained by subtracting (1) from (3).(4)i¯˙α=−RsLi¯α−zα−hαLi¯˙β=−RsLi¯β−zβ−hβL,where i¯α=i^α−iα and i¯β=i^β−iβ are the errors between the observed current value and the actual current value. hα, hβ are the SMO control laws, and the expression is(5)hα=ksigni¯αhβ=ksigni¯β,where k is the sliding mode gain, and k>maxzα,zβ.When the system state reaches the sliding mode surface and enters the sliding mode, the system state remains on the sliding mode surface and satisfiesi¯α=0,i¯β=0, and according to (5), it can be obtained.(6)zα=hα=ksigni¯αzβ=hβ=ksigni¯β.Equation (6) contains a nonlinear switching function, which will cause high-frequency chattering of the system, and is not conducive to extracting continuous estimation of the back EMFs. Therefore, the first-order LPFs should be introduced for filtering processing, namely:(7)z^α=wcS+wczαz^β=wcS+wczα,where z^α, z^β is the estimated value of the back EMFs; S is a complex frequency; wc is the cutoff frequency of LPFs. Due to the introduction of a first-order low-pass filter, and to obtain a better filtering effect, a lower cutoff frequency should be selected, but this will lead to a large phase lag in the estimation of the back electromotive force, which requires phase compensation for the rotor position. The estimated value of rotor position and speed after compensation is(8)θ^e=−arctanz^αz^β+arctanω^ewcω^e=z^2α+z^2β1/2ψf.


## 3. Design of the ASMO

To accurately estimate the SPMSM sensorless control system of the rotor position and speed, and to avoid the introduction of the first-order low-pass filter, this paper proposes a sliding mode observer based on the adaptive algorithm, which weakens the chattering phenomenon of the system and reduces LPFs estimation error.


### 3.1. ASMO

For the SPMSM model (1) and (2), the SMO is constructed [29, 30].(9)i^˙α=−RsLiα+uαL−z^αL−Mi¯α1/2signi¯αi^˙β=−RsLiβ+uβL−z^βL−Mi¯β1/2signi¯βz^α=∫Nsigni¯αdtz^β=∫Nsigni¯βdt,where i^α, i^β are the α-β axis estimated currents; z^α, z^β are the α-β axis estimated back EMFs; M, N are the sliding mode gains.The stator current error system can be obtained by subtracting (1) from (9).(10)i¯˙α=−RsLi¯α−zα−z^αL−M⋅i¯α1/2⋅signi¯αi¯˙β=−RsLi¯β−zβ−z^βL−M⋅i¯α1/2⋅signi¯α,where z¯α=z^α−zα and zβ=z^β−zβ are the errors between the observed voltage value and the actual voltage value.When the system state reaches the sliding mode surface and enters the sliding mode, the system state remains on the sliding mode surface and satisfiesi¯α=0,i¯β=0, and according to (9), it can be obtained.(11)z^αz^β=zαzβ=∫Nsigni¯αdt∫Nsigni¯βdt.From (11), it can be seen that the estimated back EMFs is a continuous function for the partial derivative of time because of the integral term. Therefore, it is unnecessary to introduce first-order LPF for filtering, and the phase delay of the estimated value of the back EMFs is not caused, which further simplifies the system structure.To prove the stability of STA-SMO, the Lyapunov function is(12)G1=12i¯α2+i¯β2.The derivative of (12) can be obtained.(13)G˙1=−RsLi¯2α+i¯2β−1Lz¯αi¯α+z¯βi¯β−Mi¯α3/2+i¯β3/2.WhenG˙1<0, the observed current of SMO converges in finite time. So M should be designed to be [30].(14)M>sup−RsL⋅i¯2α+i¯2βi¯α3/2+i¯β3/2−1L⋅z¯αi¯α+z¯βi¯βi¯α3/2+i¯β3/2,where sup represents the upper bound.When the current of the system converges in finite time, namely,i¯˙α=i¯α=i¯˙β=i¯β=0, from (10) that(15)signi¯α=−z¯αM⋅L⋅i¯α1/2,signi¯β=−z¯βM⋅L⋅i¯β1/2.By combining (11) and (15), we get(16)z^˙α=−N⋅z¯αM⋅L⋅i¯α1/2,z^˙β=−N⋅z¯βM⋅L⋅i¯β1/2.By deriving (2), we get(17)z˙α=−ψfωecosθe=−zβωez˙β=−ωeψfsinθe==zαωe.(16) and (17) can be subtracted:(18)z¯˙α=−N⋅z¯αM⋅L⋅i¯α1/2+ωe⋅zβ,z¯˙β=−N⋅z¯βM⋅L⋅i¯β1/2−ωe⋅zα.Similarly, the Lyapunov function can define as(19)G2=12z¯α2+z¯β2.By deriving (19), we get(20)G˙2=−Nz¯2αM⋅L⋅i¯α1/2+z¯2βM⋅L⋅i¯β1/2+ωe⋅zβ⋅z¯α−ωe⋅zα⋅z¯β.To ensure the convergence of SMO, namely,G2<0, N should be designed as(21)N>supM⋅L⋅ωe⋅zβ⋅z¯α−zα⋅z¯βz¯2α/i¯α1/2+z¯2β/i¯β1/2.


### 3.2. Design and Analysis of Adaptive Law

Compared with the traditional sliding mode control, the algorithm’s control structure is simple and easy to implement. However, it can be seen from (9) and (15) that the observer’s stability condition requires that the sliding mode gain is more significant than the supremum of a function about the control error. However, the control error cannot be determined in the existing system, so the system’s stability condition cannot directly determine the sliding mode’s selection range. So, in reality, we often try to choose more extensive parameters, which leads to the increase of system chattering, and even the system cannot usually run. When the sliding mode variable is near the sliding mode surface, the chattering of the system is increased due to the overestimation of the control gain, thus affecting the stability of the system. At the same time, in order to prevent the sliding mode variable away from the sliding mode surface, because the control gain estimate is too small, the system rapidity is affected.Therefore, in this paper,M and N are designed as sliding mode gains that change with the change of the back EMF, namely,(22)M=λ1+ζ1z^α+z^β,N=λ2+ζ2z^α+z^β,where λ1 and λ2 are both small numbers greater than 0; ζ1 and ζ2 are adaptive coefficients more significant than 0, and it is shown in Figure 1.Figure 1
Block diagram of ASMO.Stability analysis is given as follows.From (13), −Rs/Li¯2α+i¯2β and −Mi¯α3/2+i¯β3/2 are terms that are always less than 0.Therefore, a sufficient condition for the existence of (15) is constructed as follows:(23)−RsLi¯2α+i¯2β−1Lz¯αi¯α+z¯βi¯β−Mi¯α3/2+i¯β3/2<0,i¯α>0,i¯β>0,−RsLi¯2α+i¯2β−1L−z¯αi¯α−z¯βi¯β−Mi¯α3/2+i¯β3/2<0,i¯α<0,i¯β<0.Since−Rs/Li¯2α+i¯2β<0, −Mi¯α3/2+i¯β3/2<0, it can be deduced that the condition satisfying G˙1<0 is as follows:(24)N>maxzα,zβ=ωeψf=z^α+z^β.Therefore, N can be designed as(25)N=λ2+ζ2z^α+z^β.From (21), we can know N>supM. According to (21), M can be designed as(26)M=λ1+ζ1z^α+z^β,where λ1 is less than λ2; ζ1 is less than ζ2.


## 3.1. ASMO

For the SPMSM model (1) and (2), the SMO is constructed [29, 30].(9)i^˙α=−RsLiα+uαL−z^αL−Mi¯α1/2signi¯αi^˙β=−RsLiβ+uβL−z^βL−Mi¯β1/2signi¯βz^α=∫Nsigni¯αdtz^β=∫Nsigni¯βdt,where i^α, i^β are the α-β axis estimated currents; z^α, z^β are the α-β axis estimated back EMFs; M, N are the sliding mode gains.The stator current error system can be obtained by subtracting (1) from (9).(10)i¯˙α=−RsLi¯α−zα−z^αL−M⋅i¯α1/2⋅signi¯αi¯˙β=−RsLi¯β−zβ−z^βL−M⋅i¯α1/2⋅signi¯α,where z¯α=z^α−zα and zβ=z^β−zβ are the errors between the observed voltage value and the actual voltage value.When the system state reaches the sliding mode surface and enters the sliding mode, the system state remains on the sliding mode surface and satisfiesi¯α=0,i¯β=0, and according to (9), it can be obtained.(11)z^αz^β=zαzβ=∫Nsigni¯αdt∫Nsigni¯βdt.From (11), it can be seen that the estimated back EMFs is a continuous function for the partial derivative of time because of the integral term. Therefore, it is unnecessary to introduce first-order LPF for filtering, and the phase delay of the estimated value of the back EMFs is not caused, which further simplifies the system structure.To prove the stability of STA-SMO, the Lyapunov function is(12)G1=12i¯α2+i¯β2.The derivative of (12) can be obtained.(13)G˙1=−RsLi¯2α+i¯2β−1Lz¯αi¯α+z¯βi¯β−Mi¯α3/2+i¯β3/2.WhenG˙1<0, the observed current of SMO converges in finite time. So M should be designed to be [30].(14)M>sup−RsL⋅i¯2α+i¯2βi¯α3/2+i¯β3/2−1L⋅z¯αi¯α+z¯βi¯βi¯α3/2+i¯β3/2,where sup represents the upper bound.When the current of the system converges in finite time, namely,i¯˙α=i¯α=i¯˙β=i¯β=0, from (10) that(15)signi¯α=−z¯αM⋅L⋅i¯α1/2,signi¯β=−z¯βM⋅L⋅i¯β1/2.By combining (11) and (15), we get(16)z^˙α=−N⋅z¯αM⋅L⋅i¯α1/2,z^˙β=−N⋅z¯βM⋅L⋅i¯β1/2.By deriving (2), we get(17)z˙α=−ψfωecosθe=−zβωez˙β=−ωeψfsinθe==zαωe.(16) and (17) can be subtracted:(18)z¯˙α=−N⋅z¯αM⋅L⋅i¯α1/2+ωe⋅zβ,z¯˙β=−N⋅z¯βM⋅L⋅i¯β1/2−ωe⋅zα.Similarly, the Lyapunov function can define as(19)G2=12z¯α2+z¯β2.By deriving (19), we get(20)G˙2=−Nz¯2αM⋅L⋅i¯α1/2+z¯2βM⋅L⋅i¯β1/2+ωe⋅zβ⋅z¯α−ωe⋅zα⋅z¯β.To ensure the convergence of SMO, namely,G2<0, N should be designed as(21)N>supM⋅L⋅ωe⋅zβ⋅z¯α−zα⋅z¯βz¯2α/i¯α1/2+z¯2β/i¯β1/2.


## 3.2. Design and Analysis of Adaptive Law

Compared with the traditional sliding mode control, the algorithm’s control structure is simple and easy to implement. However, it can be seen from (9) and (15) that the observer’s stability condition requires that the sliding mode gain is more significant than the supremum of a function about the control error. However, the control error cannot be determined in the existing system, so the system’s stability condition cannot directly determine the sliding mode’s selection range. So, in reality, we often try to choose more extensive parameters, which leads to the increase of system chattering, and even the system cannot usually run. When the sliding mode variable is near the sliding mode surface, the chattering of the system is increased due to the overestimation of the control gain, thus affecting the stability of the system. At the same time, in order to prevent the sliding mode variable away from the sliding mode surface, because the control gain estimate is too small, the system rapidity is affected.Therefore, in this paper,M and N are designed as sliding mode gains that change with the change of the back EMF, namely,(22)M=λ1+ζ1z^α+z^β,N=λ2+ζ2z^α+z^β,where λ1 and λ2 are both small numbers greater than 0; ζ1 and ζ2 are adaptive coefficients more significant than 0, and it is shown in Figure 1.Figure 1
Block diagram of ASMO.Stability analysis is given as follows.From (13), −Rs/Li¯2α+i¯2β and −Mi¯α3/2+i¯β3/2 are terms that are always less than 0.Therefore, a sufficient condition for the existence of (15) is constructed as follows:(23)−RsLi¯2α+i¯2β−1Lz¯αi¯α+z¯βi¯β−Mi¯α3/2+i¯β3/2<0,i¯α>0,i¯β>0,−RsLi¯2α+i¯2β−1L−z¯αi¯α−z¯βi¯β−Mi¯α3/2+i¯β3/2<0,i¯α<0,i¯β<0.Since−Rs/Li¯2α+i¯2β<0, −Mi¯α3/2+i¯β3/2<0, it can be deduced that the condition satisfying G˙1<0 is as follows:(24)N>maxzα,zβ=ωeψf=z^α+z^β.Therefore, N can be designed as(25)N=λ2+ζ2z^α+z^β.From (21), we can know N>supM. According to (21), M can be designed as(26)M=λ1+ζ1z^α+z^β,where λ1 is less than λ2; ζ1 is less than ζ2.


## 4. Simulation

The vector control strategy id = 0 is adopted to establish the system simulation model based on MATLAB/Simulink to realize the sensorless control of SPMSM. Its principal block diagram is shown in Figure2. Rated motor parameters used in the simulation are shown in Table 1.Figure 2
SPMSM speed sensor-less control block diagram of ASMO.Table 1
Parameters of the SPMSM.
QuantityName of unitSymbolRs2.875Ωψf0.175WbL8.5e−3HDC bus voltage311VRotational inertia0.003kg m2Rated power2.3kWRated speed1000r/minRated torque15N·mPole pairs numbers4The parameters of SMO arek = 200. The STA-SMO parameter configuration is M = 5000, N = 35000. The ASMO parameter configuration is λ1=235, ζ1=65, λ2=3800, ζ2=450.The initial speed of the system is 600 r/min, and no load is started. At 0.05 s, 5 N m load torque is applied; at 0.1 s, the speed changes to 1000 r/min; in the first variable speed stable operation stage (0.15 s), the load torque changes to 10 N m; at 0.2 s, the speed changes to 300 r/min; in the second variable speed stable operation stage (0.25 s), the load torque changes to 15 N·m.From Figures3 and 4 the response speed of STA-SMO and ASMO is faster than that of SMO, no matter in the motor start-up stage, speed shift stage, and load torque mutation stage. Especially at 0.2 s, when the speed decreases from 1000 r/min to 300 r/min, the SMO speed waveform has an obvious lag phenomenon due to introducing a low-pass filter. At 0.25 s, the torque load suddenly increases to 15 N·m, and the phenomenon of the rapid drop of SMO is apparent.Figure 3
Speed response diagram.Figure 4
Enlargement of local speed.
(a)(b)(c)(d)(e)(f)It can be seen that SMO is not as good as the previous two control strategies when dealing with external disturbances.From Figures3–5 and Table 2, compared with traditional SMO, the speed errors of ASMO at different speed stages are 2.65%, 3.51%, and 3.5% of SMO, respectively. Compared with STA-SMO, ASMO’s speed errors at different speeds are 72.581%, 92.857%, and 59.231% of STA-SMO, respectively.Figure 5
Speed error response diagram.Table 2
Speed error.
ResultsSMOSTA-SMOASMOSymbolError (a)170.620.45r/minError (b)18.50.70.65r/minError (c)110.650.385r/minFrom the above results and the simulation figures analysis, it can be seen that the speed error of the three control strategies is proportional to the speed. SMO has the largest chattering and speed error, and the response speed is slow. Compared with SMO, STA-SMO, because of the function of integral function, can well suppress sliding mode chattering, reduce the rotation speed error, avoid the phenomenon of rotation speed waveform lag caused by the introduction of the low-pass filter, and enhance the response speed of the system. However, because the sliding mode gain is a fixed value, it cannot meet high precision speed regulation requirements. The sliding mode gain of the ASMO can be adjusted online with speed, and the speed error and the speed waveform chattering can be minimized. The problem of excessive gain can be solved well, and the robustness of the system is enhanced.From Table3, the rotor position errors of ASMO at different speed segments are 29.13%, 4.69%, and 4% of SMO, respectively. Compared with STA-SMO, ASMO’s rotor position errors at different speeds are 75%, 96.83%, and 55.56% of STA-SMO, respectively.Table 3
Rotor position error.
ResultsSMOSTA-SMOASMOSymbolError (a)0.002060.00080.0006DegError (b)0.0130.000630.00061DegError (c)0.0250.00180.001DegFrom Figures6–9 and the analysis of the above results, the estimated rotor position waveform of SMO has obvious chattering. When the speed drops to 300 r/min, the chattering is further aggravated, and the rotor position error is the largest, which seriously affects the system control accuracy. Compared with SMO, the rotor position error of STA-SMO is small, and the chattering phenomenon is weakened, but because its sliding mode gain is a fixed value when the speed is 300 r/min, the rotor position error increases, which is not conducive to the system high precision control needs. Compared with the above two control strategies, the ASMO rotor position error is the smallest, and under different disturbances, the estimated rotor position is the same as the actual rotor position. It satisfies the high precision control requirements of the system.Figure 6
SMO rotor position response diagram.Figure 7
STA-SMO rotor position response diagram.Figure 8
ASMO rotor position response diagram.Figure 9
Rotor position error response diagram.


## 5. Conclusion

This paper presents a speed sensorless control technology of surface-mount permanent magnet synchronous motor based on ASMO. The proposed observer can effectively filter the high-frequency sliding mode noise when estimating the back EMF and avoid the lag caused by introducing the low-pass filter, thus reducing the system’s complexity.Secondly, to solve the difficulty of the observer’s sliding mode gain, an adaptive law is designed that changes with the back EMF. By adjusting the sliding mode gain online, the control precision of the system is further enhanced.Finally, by comparing with the simulation experiments of traditional SMO and STA-SMO control strategies, it can be seen that the proposed control strategy can effectively suppress sliding mode chattering, improve the accuracy of rotor position estimation, and better meet the requirements of the system speed regulation.



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*Source: 6196321-2022-03-30.xml*