Design and performance analysis of an iterative flux sliding-mode observer for the sensorless control of PMSM drives

ISA Transactions xxx (xxxx) xxx

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Research article

Design and performance analysis of an iterative flux sliding-mode observer for the sensorless control of PMSM drives Shuaichen Ye School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China

highlights • • • •

The LPF and the compensation module in the conventional SMO reduce the dynamic performance. Conventional flux SMO for the PMSM sensorless control suffers from chattering issue. An embedded LPF is designed by constructing the feedback matrix for the flux SMO. By iteratively using the flux SMO in one current control loop, chattering can be much reduced.

article

info

Article history: Received 12 December 2018 Received in revised form 7 April 2019 Accepted 12 April 2019 Available online xxxx Keywords: Sensorless control Iterative flux sliding-mode observer (IFSMO) Permanent magnet synchronous motor (PMSM)

a b s t r a c t To improve the performance of permanent magnet synchronous motor (PMSM) drives, a sensorless control scheme based on a novel iterative flux sliding-mode observer (IFSMO) is proposed in this paper. Two major drawbacks of the conventional sliding-mode observer (SMO), namely, chattering phenomena and high-order harmonics, are discussed. These drawbacks affect the estimation accuracy of the SMO and reduce the control reliability of the system. To eliminate high-order harmonics, a flux SMO is designed by expanding the PMSM state equations with the PM flux. The flux SMO estimates the rotor speed and position using the flux linkage instead of back-EMF information. Moreover, to reduce the chattering in the estimation results, the proposed flux SMO is iteratively used in one current sampling period to adaptively adjust the observer gain. An overall PMSM sensorless control system based on the proposed IFSMO is designed, and an experimental platform using the TMS320F28335 digital signal processor (DSP) controller is built. The superior chattering reduction and harmonic suppression characteristics of the proposed IFSMO are experimentally validated, and the experimental results verify the feasibility of using the proposed IFSMO-based PMSM sensorless scheme in practical applications. © 2019 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction Permanent magnet synchronous motors (PMSMs) have been widely used as servo motors in multiple highly dynamic and high-precision industrial applications, such as electric ship propellers [1] and centrifugal air compressors [2], due to their advantages of high power density and high reliability [3]. In these applications, the vector control method, also referred to as the field-oriented control (FOC) method, proposed by Blaschke [4] in 1972, is the most efficient strategy for PMSM control and drive systems. The FOC method requires precise knowledge of the realtime rotor position and speed, which are generally detected by mechanical position sensors, such as the resolver [5], encoder [6] and Hall-effect sensors [7]. However, installing sensors on the rotor shaft increases the size and inertia of the PMSM and limits E-mail address: [email protected].

the use of the system in harsh environments, such as under high-temperature and high-vibration conditions. Motivated by these problems, sensorless methods have been developed to replace the mechanical sensors used to obtain the rotor speed and position. Most sensorless control methods for PMSMs can be classified into two categories: (1) high-frequency signal injection methods and (2) back-EMF model-based methods. High-frequency signal injection methods perform well even in low-speed and standstill operations. However, the main drawbacks of such methods lie in the external noise produced by the injected signals and the fact that the methods cannot be applied to surface-mounted PMSMs. Unlike high-frequency signal injection methods, back-EMF model-based methods have no restriction regarding the structure of the PMSM and do not require external high-frequency signal sources. Back-EMF model-based methods mainly include the model reference adaptive system (MRAS) observer [8–10], the Luenberger observer [11,12], the extended Kalman filter (EKF) observer [13,14] and the sliding-mode observer (SMO) [15–19].

https://doi.org/10.1016/j.isatra.2019.04.009 0019-0578/© 2019 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: S. Ye, Design and performance analysis of an iterative flux sliding-mode observer for the sensorless control of PMSM drives. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.04.009.

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Among these methods, the SMO [15–19] has the attractive advantages of a simple algorithm, robustness to parameter variations and favorable dynamic performance; therefore, the SMO is the most widely used sensorless control method for PMSM drives [20]. However, the conventional back-EMF model-based SMO suffers from two shortcomings, high-order harmonics and the chattering issue, which considerably affect the performance of the sensorless control system. Thus, the main motivations of this research lie in eliminating high-order harmonics and reducing chattering by proposing an improved SMO method. Generally, high-order harmonic components are mixed with the estimated back-EMF signals for the following reasons [21]: (1) the flux spatial harmonics caused by the gap between two PMs and (2) the nonlinearity of the PMSM drive inverter. These harmonics reduce the accuracy of rotor position and speed estimates. Thus, many methods [22–24] have been proposed to eliminate these harmonics in back-EMF estimates. Inoue et al. [22] proposed a method that uses the error between the reference and actual voltages of the inverter to compensate for its nonlinearity, thus reducing the harmonics. However, according to [23], under light-load and no-load running conditions, the voltage error cannot be easily determined. Suk-Hwa [24] established an accurate mathematical model for calculating the PM flux spatial harmonics; however, this method increases the complexity of the system. Therefore, harmonic components are difficult to eliminate at the mechanistic level. Many researchers have also attempted to suppress harmonics using different types of filters. Wibowo [25] designed a cascade low-pass filter (LPF) to extract fundamental back-EMF signals and compared this method with the conventional SMO experimentally. However, Wang [26] verified that the LPF has hardly any effect on eliminating specific harmonics, such as the third-order harmonic; thus, an adaptive notch filter (ANF) method has been proposed to address this issue. Based on Wang’s research, Song and his partners [21] noticed that fifth-order and seventh-order harmonics still existed in the signals filtered by the ANF; thus, the researchers proposed the synchronous frequency-extract filter (SFF) to thoroughly filter out various types of harmonics. However, although the performance of filters can be greatly improved by using different structures, introducing such filters still causes a phase delay and complicates the system, thereby reducing the dynamic performance of the PMSM sensorless control. In this paper, a flux SMO method that eliminates the filter and the compensation module is proposed; this novel method expands the state equations of the conventional SMO and estimates the rotor position and speed using the filtered flux PM flux linkage instead of the back-EMF model. Although the flux SMO is effective in suppressing high-order harmonics by filtering the flux signals with a feedback matrix, the gain value of this observer is still needed to be an appropriate constant to stabilize the observer and to converge the slidingmode motion to the switching surface. According to [27], this constant value causes relatively fast convergence to the slidingmode surface when the estimation error is small, resulting in an overshoot. Thus, chattering emerges in the estimations due to these repeated overshoots in one control loop. Theoretically speaking, for an ideal sliding mode motion using a signum function, the constant observer gain issue does not cause chattering because of infinity switching frequency of the system. However, this research focuses on applying the sliding-mode theory to some practical instruments; due to the mechanical delay and the dynamic nonidealities of the system, characteristic of infinity switching frequency cannot be achieved, thus chattering appears [28]. Generally, the most widely used chattering-reduction method for sliding-mode control is to replace the discontinuous signum switching function in the conventional SMO with continuous functions, such as the saturation

function [25,29] and the sigmoid function [30]. Most recently, some high-order SMO methods have been proposed [31,32], and these methods exhibited better chattering-reduction performance compared with traditional continuous switching function methods. However, the high-order derivative in these methods may increase their implementation complexity on the digital signal processor (DSP) and increase the calculation burden. Furthermore, certain regulation modules have also been proposed and integrated into the SMO to adjust the constant observer gain [27], but the major drawback of these methods is that the additional module incorporated into the estimation process may deteriorate the dynamic performance of the system. In contrast to previous studies, this paper explores a method that iteratively utilizes the proposed flux SMO several times in one current control loop and adjusts the observer gain value in a timely manner according to the estimation error. Hence, chattering caused by the constant gain value can be effectively reduced, and no additional module is needed. In theoretical aspect, the main contributions of this paper are as follows: (1) the state equations of the conventional SMO are expanded with the PM flux, and the proposed flux SMO estimates the rotor speed using the filtered flux linkage, which suppresses the high-order harmonic components and does not increase the complexity of the system; (2) by iteratively utilizing the proposed flux SMO in one current sampling period and adaptively adjusting the observer gain, chattering in the estimates can be considerably reduced. From a practical standpoint, the contributions are as follows: (1) a sensorless control scheme based on the proposed IFSMO is designed and implemented with a TMS320F28335 DSP controller-based experimental platform; (2) the effectiveness of the proposed IFSMO and its feasibility in practical applications are validated, and the superior performance of this method is verified through comparative experiments. The remainder of this paper is organized as follows. The dynamic model of the PMSM is briefly introduced in Section 2. Section 3 proposes the flux SMO. Section 4 introduces the IFSMO and illustrates the corresponding block diagram. Section 5 constructs the sensorless control scheme based on the proposed IFSMO and describes the experimental platform. The excellent performance and effectiveness of the proposed IFSMO-based sensorless control scheme are illustrated based on experimental results in Section 6. Section 7 concludes the paper. 2. Modeling of the PMSM A nonsalient three-phase PMSM prototype is considered in this paper. The PMSM consists of a rotor with surface-mounted PMs and a stator with a Y-connected wind located every 120◦ around the circle. The three-phase motor is regarded as a nonlinear time-varying system; however, by neglecting the imaginary part in the Y-connected windings, the system can be transformed into a two-phase voltage model in the synchronous rotatingd − q coordinates, which is presented as follows

⎡ [ ] ud uq

⎢Rs +

=⎣

d dt

ωe Lq

Lq

⎤ −ωe Ld ⎥ [i ] [0 ] d d ⎦ iq + ωe λf Rs + Ld

(1)

dt

where ud,q , id,q and Ld,q represent the dq-axis stator voltages, the dq-axis stator currents and the dq-axis stator inductances, respectively; Rs is the stator-winding resistance; ωe is the rotor angular speed; and λf is the flux linkage of the permanent magnets. Generally, model (1) is used for the PMSM vector controller design, whereas for the observer design, this model should be transformed into the stationary α − β coordinates using the inverse Park transformation. Moreover, because the flux linkage

Please cite this article as: S. Ye, Design and performance analysis of an iterative flux sliding-mode observer for the sensorless control of PMSM drives. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.04.009.

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Fig. 1. Relation between stationary coordinates and synchronous rotating coordinates.

3

potential application to a class of mechanical systems. Wang [34] presented a new sliding-mode control design methodology for fuzzy singularly perturbed systems (SPSs) and verified the applicability and superiority of the novel method by designing a controller for an electric circuit system. The main principle of sliding-mode control is to drive the system states to the switching surface using the equivalent control inputs. This approach ensures system robustness to parameter variations and external disturbances. In the PMSM sensorless control, the sliding-mode control method can replace the original physical sensor on the shaft to estimate the rotor position, which is also referred to as the SMO. Fig. 2 shows a basic FOC scheme for the PMSM sensorless control based on the conventional SMO. Based on the dynamic model of the PMSM and sliding-mode variable structure theory, the conventional SMO can be expressed as follows [27]. d

[

ˆiα ˆiβ

dt

] =−

[

Rs

ˆiα ˆiβ

Ls

] +

1

[ ]

Ls

1 uα − uβ Ls

[

k · sig(˜ iα )

]

k · sig(˜ iβ )

(4)

Additionally, the sliding surface can be defined as follows:

[ ] sα sβ

s=

[ =

˜iα ˜iβ

]

[ =

ˆiα − iα ˆiβ − iβ

] (5)

where ‘‘ ˆ ’’ denotes the estimate, ‘‘ ˜ ’’ denotes the error, and k is the gain of the SMO. The switching function sig(·) is defined as follows: 2

sig = Fig. 2. PMSM sensorless control scheme based on the conventional SMO.

of the PM is a constant value, the transformation between the synchronous rotating d − q coordinates and the stationary α − β coordinates can be performed directly, and the relation between these two frames is graphically presented in Fig. 1. Thus, the voltage equations based on the two-phase stationary α − β coordinates are represented as follows:

⎡ [ ] uα uβ

⎤

d

⎢Rs + dt Ls =⎣ 0

0 Rs +

d dt

Ls

[ ] [ ] eα ⎥ iα ⎦ i + e β

β

(2)

where uα , uβ , iα and iβ are the α - and β -axis stator voltages and stator currents, respectively; Ls is the stator-winding inductance (satisfying Ld = Lq = Ls for nonsalient motors); eα and eβ are the α - and β -axis back-EMF values, respectively; and θe represents the angular position of the rotor. Therefore, the stator current equations can be derived from Eq. (2) as follows: d dt

[ ] iα iβ

=−

Rs Ls

[ ]

1 iα + iβ Ls

[ ]

1 uα − uβ Ls

[ ] eα eβ

(3)

where eα = −ωe λf sinθe and eβ = ωe λf cosθe . 3. Design of the flux SMO

1+

e[−σ (ˆis −is )]

−1

(6)

where is is the column vector of the stator currents and σ is a positive real number. By combining Eqs. (3)–(5), the estimation error can be obtained. d dt

[ ] sα sβ

=−

Rs

[ ]

Ls

1 sα + sβ Ls

[ ]

k eα − eβ Ls

[

]

sig(sα ) sig(sβ )

(7)

In addition, when the observer gain k is a positive value and sufficiently large, both the existence of sliding motion and the asymptotic stability of the SMO can be guaranteed in the global scope [35]. Therefore, based on the equivalent control method, when the system reaches the sliding surface (s = s˙ = 0), the relation between the control function and the back-EMF values can be represented as follows.

[

k · sig(˜ iα )

]

[ ]

e = α eβ ˜ k · sig(iβ )

= λf ωe

[

− sin θe cos θe

] (8)

Thus, the angular position of the rotor can be directly calculated from the back-EMF estimates using the arc tangent function.

θˆe = − arctan(

eˆ α eˆ β

)

(9)

3.1. Conventional SMO

3.2. The proposed flux SMO

Sliding-mode control is a variable structure control and has been widely used in many complex nonlinear systems reported in the latest literature. For example, Li [33] investigated a fuzzy integral sliding-mode control approach to address the stabilization issue for nonlinear descriptor systems and considered its

In our previous research [36], a flux SMO was proposed, and its effectiveness was experimentally validated. In contrast to the previous approach, we use sig(·) as the switching function instead of sat(·).

Please cite this article as: S. Ye, Design and performance analysis of an iterative flux sliding-mode observer for the sensorless control of PMSM drives. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.04.009.

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Based on [36], the proposed flux SMO can be expressed as follows: d

[

ˆi λˆ

dt

]

[

][

A = 11 0

A12 · A22

ˆi λˆ

] [ ][ ] [ ] [ ] B uα I −k · sig(sα ) + + · 0 uβ L −k · sig(sβ ) (10)

]T

]T

ˆiα ˆiβ , λˆ = λˆ α λˆ β , and L ∈ R 2×2 is the where ˆi = feedback matrix. The estimation error can be subsequently yielded as follows:

[

d

[ ]

[

s

A = 11 0 λ dt ˜

[

] [ ]

[ ] [

]

s A12 I −k · sig(sα ) · ˜ + · A22 L −k · sig(sβ ) λ

(11)

]

]T

[

V = 0.5s s = 0.5(sα + sβ ) T

2

2

(12)

Differentiating Eq. (12) with respect to time and substituting Eq. (11) into this equation yields the following formula: V˙ = −

R Ls

(s2α + s2β ) +

ωe Ls

sα (˜ λβ − ˜ λα ) − k sα · sig(sα ) + sβ · sig(sβ )

[

]

(13) To guarantee the asymptotic stability of the SMO in the global scope, the condition V˙ < 0 should be satisfied. Thus, the scope of the observer gain can be set as follows.

⏐ ⏐ ⏐} {⏐ ⏐ ωe ⏐ ⏐ ωe ⏐ ⏐ ⏐ ⏐ ˜ ˜ k > max ⏐ λα ⏐ , ⏐ λβ ⏐⏐ Ls Ls

(14)

3.3. Design of the feedback matrix Compared with the conventional SMO, the proposed flux SMO exhibits major advancement in that a feedback matrix L can be constructed by extending the state equations with the flux linkage. In the conventional SMO, the equivalent control was used to approximate the back-EMF and estimate the rotor position directly. However, in the proposed flux SMO, the estimated flux linkage is used to calculate the rotor position and speed instead of the back-EMF, and the equivalent control signal is first multiplied by the feedback matrix before being added to the flux linkage estimates. Thus, the feedback matrix can serve as a filter or a correction process that extracts the fundamental flux linkage signal and reduces the estimated error. According to the analyses above, the design of the feedback matrix is essential to the flux SMO. This section discusses the design of the feedback matrix based on the disturbance input caused by the speed estimation errors. Considering the speed estimation errors, Eq. (10) can be rewritten as follows: d

[

dt

ˆi λˆ

]

[

A = 11 0

][

A12 · A22

ˆi λˆ

] [ ] [ ][ ] B I −k · sig(sα ) + u+ · +H 0 L −k · sig(sβ ) (15)

where H is the disturbance input caused by the speed estimation errors:

[ H =

H1 H2

]

[ ] ∆A12 ˆ = λ ∆A22

[ ]

[ A11 = 0 λ dt ˜ d

(16)

where ∆A12 = ˜ A12 − A12 = (−1/Ls )∆ωe J , ∆A22 = ˜ A22 − A22 = ∆ωe J and ∆ωe = ˜ ωe − ωe is the speed estimation error.

] [ ]

s

[ ] [

]

[

s A12 I −k · sig(sα ) H1 · ˜ + · + A22 L −k · sig(sβ ) H2 λ

] (17)

After sliding-mode motion occurs, s˙α = sα = 0 and [s˙β = s] β = ueq−α 0. The equivalent control is defined as follows: ueq = = ueq−β

[

]

k · sig(sα ) , which can be obtained from the first two equations k · sig(sβ )

of Eq. (17):

[

is the column where ˜ λ = ˜ λβ = λˆ α − λα λˆ β − λβ λα ˜ vector of the flux estimation error. To design the observer gain k, which guarantees the stability and reachability of the sliding mode, Lyapunov theory is employed, and the Lyapunov function candidate is defined as follows.

[

Similarly to Eq. (11), the estimation error equations can be obtained as follows:

ueq

u = eq−α ueq−β

]

= A12˜ λ + H1

(18)

By substituting Eq. (18) into the last two equations of Eq. (17), d dt

˜ λ = A22˜ λ + L · (−ueq ) + H 2 ˜ = A22 λ + L · (−A12˜ λ − H 1) + H 2 = (A22 − LA12 )˜ λ + (H 2 − LH 1 )

(19)

Then, by substituting Eq. (16) into Eq. (19), d ˜ λ = ωe Q ˜ λ + Q ∆ωe λˆ (20) dt where Q = 1/Ls LJ + J . ˆ to the The transfer function of the disturbance input ∆ωe λ ˜ flux linkage estimated error λ can be obtained as follows: T (s) = (sI − ωe Q )−1 Q

(21)

The major design purpose of the feedback matrix is to ensure that the flux SMO has good robustness against the external disturbance signals. In this paper, the H-infinite norm is applied to measure robustness, which means that the purpose of the feedback matrix is to minimize the H-infinite norm of the transfer function Eq. (21). We assume that the feedback matrix has the following form: L = g1 I + g2 J

(22)

where g1 and g2 are constant values, which need to be determined. Thus, the Q matrix can be rewritten as follows: g1 g2 )J = −α I + β J (23) Q = (− )I + (1 + Ls Ls where α = g2 /Ls and β = 1 + g1 /Ls The H-infinite norm of the transfer function T (s) is

∥T (s)∥∞ = supσmax [T (jω)] ω √ = sup λmax [T H (jω)T (jω)] ω

= sup √ ω

(24)

∥Q ∥2 (ω−|βωe |)2 +(αωe )2

where σmax represents the maximum singular value of the matrix, λmax represents the maximum eigenvalue of the matrix and ∥ · ∥2 represents the induced norm of the matrix. By substituting Eq. (23) into the last equation of Eq. (24), under the frequency condition ω = |βωe |, the H-infinite norm of the transfer function T (s) can be simplified as follows:

∥T (s)∥∞

⏐ ⏐√ ⏐1⏐ β = ⏐⏐ ⏐⏐ 1 + ( )2 ωe α

(25)

It is obvious that the H-infinite norm of T (s) has a minimum value when parameter β equals zero:

⏐

⏐

⏐1⏐ min ∥T (s)∥∞ = ⏐⏐ ⏐⏐ ωe

(26)

Please cite this article as: S. Ye, Design and performance analysis of an iterative flux sliding-mode observer for the sensorless control of PMSM drives. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.04.009.

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5

Fig. 3. The concept of the IFSMO.

Thus, the feedback matrix can be obtained as follows:

[

−Ls L= α Ls

−α Ls − Ls

] (27)

Substituting β = 0 into Eq. (23) and Eq. (20), the equation of the flux linkage estimated error can be obtained as follows: d dt

˜ λ + αωe I˜ λ = −α ∆ωe I λˆ

Fig. 4. Block diagram of the IFSMO.

(28)

Eq. (28) clearly shows that the pole of the flux observer is

(−αωe , 0) (when α > 0, the observer is asymptotically stable), the bandwidth of the observer is αωe and the estimates of λα and λβ are decoupled. Eq. (28) and simulation results indicate that with increasing α , the response speed and the bandwidth of the system increase, while the robustness and the chattering reduction characteristic degrade; conversely, with decreasing α , the response speed and the bandwidth of the system decrease, while the robustness and the chattering reduction characteristic upgrade. Thus, in practical applications, when choosing the α value, there is a tradeoff between the response speed and the chattering reduction performance. In this paper, to obtain the most favorable performance of the system, the value of α is chosen to be 3. Moreover, Eq. (28) can be regarded as an LPF of the flux linkage estimates, which would filter out high-order harmonics and external noise in the flux linkage estimates. Thus, by extending the state equations with the flux linkage and constructing the feedback matrix properly, the proposed flux SMO can possess an embedded LPF that replaces the external LPF and phase compensation module in the conventional SMO. 4. The proposed IFSMO According to the foregoing analysis, under a given operating condition of the PMSM, the observer gain should be a sufficiently large constant to guarantee the stability and reachability of the sliding mode. This large gain leads to the sliding-mode motion rapidly approaching the switching surface when the estimation error is large, which improves the control efficiency of the system. However, as the estimation error decreases, a smaller gain value is needed because the fast-approaching characteristic of the original large gain would cause overshoots to the switching surface for the sliding-mode motion. These overshoots reduce the performance of the estimation and cause chattering in the results. Therefore, it is necessary to adaptively adjust the observer gain according to the estimation error and reduce chatter. Thus, the IFSMO is designed by iteratively using the flux SMOs with different observer gains in the estimation and control processes. Fig. 3 illustrates the concept of the proposed IFSMO, which divides one stator current control period into four parts and adds three virtual current sampling steps between two actual sampling steps. The observer gain is adjusted before each virtual sampling step according to the real-time estimation error. The block diagram of the IFSMO for the sensorless control of a PMSM is shown in Fig. 4. In the estimation process, the flux SMO process is repeated four times, and its observer gain is gradually

Fig. 5. The PMSM sensorless control scheme based on the conventional SMO with the LPF and compensation module.

adjusted from a relatively large value to a relatively small value in each virtual current sampling step (i.e., k1 > k2 > k3 > k4 ) to ensure the reachability of the sliding mode and to avoid the chattering caused by overshoot issues. The computational speed of this system is highly dependent on the working frequency of the DSP microprocessor. Experimental results verified that four flux SMO repetitions is the maximum allowable number in one control cycle using the TMS320F28335 DSP-based controller. Thus, choosing this number of repetitions not only ensures the maximum observer gain adjustment range but also does not increase the calculation complexity. Compared to the conventional SMO method, the control hardware for the proposed IFSMO is not altered, and only an iteration loop for the flux SMO is added to the algorithm; thus, there is hardly any implementation complexity or cost increase. After the IFSMO process, the rotor position and speed estimations are obtained from the flux estimates using a basic phaselocked loop (PLL) [37], which includes the flux normalization module and the proportional integral (PI) regulator. 5. The proposed IFSMO-based PMSM FOC scheme and experimental setup According to Ref. [25], the PMSM sensorless FOC scheme based on the conventional SMO with the LPF and compensation module is shown in Fig. 5. In this paper, to verify the validity of the IFSMO, a PMSM sensorless FOC scheme based on the proposed method is designed, and an experimental platform is built. The

Please cite this article as: S. Ye, Design and performance analysis of an iterative flux sliding-mode observer for the sensorless control of PMSM drives. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.04.009.

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sampling periods for the current control cycle and the speed control cycle were both set to 0.1 ms. Additionally, the parameters of the PI controllers were set to kp = 0.03 and ki = 3 for the speed controller and to kp = 150 and ki = 300 for the current controller. The observer gains for one current control period were set to k1 = 130, k2 = 100, k3 = 60, and k4 = 30, and the parameters of the feedback matrix were set to g1 = 0.0054 and g2 = 0.0162. 6. Experimental results

Fig. 6. The PMSM sensorless control scheme based on the proposed IFSMO.

To verify the superior performance of the PMSM sensorless control scheme based on the proposed IFSMO method, several groups of comparative experiments were performed. In this section, the proposed IFSMO are mainly compared with two existing methods, the conventional SMO method and the flux SMO method. The conventional SMO method represents the PMSM sensorless control scheme based on a traditional SMO, which is presented in Section 3.1, with the LPF and the compensation module; the flux SMO method represents the case in which the PMSM sensorless control scheme uses the flux SMO with a determined constant gain value for only one time in every current control loop. First, comparisons among these three methods are presented under steady motor operating conditions. The purpose of the steady-state experiments is to validate the chattering suppression performance of the proposed method. Then, comparisons are performed under varying speed, load and motor parameter conditions to prove that the proposed method yields a faster dynamic response and displays higher disturbance robustness than the existing methods do. 6.1. The steady-state experiments

Fig. 7. The experimental platform. Table 1 Parameters of the PMSM prototype. Parameters

Value

Parameters

Value

Rated voltage (V) Rated power (kW) Rated speed (r/min) Rated torque (N m) Pole number

36 0.4 3000 3.8 4

Stator-winding resistance () Stator-winding inductance (H) Flux linkage (Wb) Inertia (kg m2 )

0.667 0.0054 0.18 0.0012

overall block diagram of the novel control scheme is shown in Fig. 6. In this scheme, the three-phase stator current should be transformed into two-phase synchronous coordinates using the Clark transform for vector control. Then, as a result of the vector control process, a reference current is generated from the inverter and passed to the PMSM. Proportional–integral (PI) controllers are implemented for both current and speed control using the errors between the command and estimated values. Compared with the conventional scheme in Fig. 5, the novel scheme replaces the conventional SMO and the LPF with the proposed IFSMO. In addition, an encoder module is integrated to compare the actual motor speed value with the estimated value. A photograph of the experimental platform is shown in Fig. 7, which consists of the following components: a nonsalient threephase PMSM prototype with specifications listed in Table 1, which is equipped with a 2500-line incremental encoder; a DSP controller board based on the TMS320F28335 provided by the TI Corporation; an air switch is used to protect the circuit; and 36-V and 15-V DC voltage sources are used to power the PMSM and the motor drive board, respectively. In the experiment, the switching frequency of the PMSM drive was set to 10 kHz with a switching dead time of 2 µs, and the

The steady-state experiments include the following: back-EMF and flux linkage estimation experiments at a speed of 50 r/s; rotor position estimation experiments using different methods at a speed of 50 r/s; and rotor speed estimation experiments using different methods at speeds of 50 r/s and 10 r/s, respectively. Fig. 8(a) shows the back-EMF estimation results obtained using the conventional SMO method at 50 r/s, and Fig. 8(b) and (c) shows the flux linkage estimation results obtained using the flux SMO and the IFSMO, respectively, at 50 r/s. The results show that large vibrations emerge in the back-EMF estimations using the conventional SMO and the flux linkage estimation results using the flux SMO. Although these two methods both filter out the harmonics in the estimations, their observer gains remain constant, which produces chattering in the estimation process and causes vibrations in their estimation profiles. By applying the IFSMO and regulating the observer gain adaptively several times in one current control loop, chattering can be considerably reduced, and the vibrations are eliminated in the estimations of the flux linkage (refer to Fig. 8(c)). Fig. 9(a), (b) and (c) illustrates the position estimation results at 50 r/s using different methods and there estimation errors. Fig. 9(a) and (b) shows that serious chattering emerges in the position estimations using the conventional SMO method and the flux SMO method; the maximum steady estimation error for these two methods are 9.3◦ and 7.9◦ , respectively. It is worth noting that, although the flux SMO method eliminates the LPF and the compensation module in the conventional SMO, the chattering caused by the constant observer gain value still exists, and the estimation error is hardly reduced compared with that of the conventional SMO; the large errors associated with these two methods reduce the estimation accuracy and deteriorate the system control performance. By introducing the IFSMO with variable observer gain, chattering caused by the repeated overshoots

Please cite this article as: S. Ye, Design and performance analysis of an iterative flux sliding-mode observer for the sensorless control of PMSM drives. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.04.009.

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of the chattering issue. However, by introducing the proposed IFSMO method, the maximum steady estimation error decreases to only 1.7 r/s (refer to Fig. 10(c)). Similarly, Fig. 10(d), (e) and (f) illustrates that under the low-speed running condition of 10 r/s, the maximum estimation error decreases from 5.4 r/s for the conventional SMO method and 4.3 r/s for the flux SMO method to 1.73 r/s for the proposed IFSMO method. In particular, Fig. 10(a) and (d) indicates that the maximum steady error for the conventional SMO increases from 4.98 r/s to 5.4 r/s as the speed decreases because under the low-speed condition, the amplitude of the back-EMF is relatively small, and this small value reduces the estimated accuracy and magnifies chatter. This deficiency represents one of the drawbacks of using the back-EMF modelbased method. By introducing the flux linkage estimation method in this paper, the maximum estimated error under the low-speed condition is the same as that under the high-speed condition (refer to Fig. 10(c) and (f)), and this drawback can be eliminated. Therefore, it can be concluded that the speed estimation error is considerably reduced and the chattering is effectively suppressed by applying the proposed IFSMO method, regardless of whether the system operates under high-speed or low-speed conditions. 6.2. The dynamic experiments

Fig. 8. (a) Back-EMF estimation results obtained using the conventional SMO at 50 r/s. (b) Flux linkage estimation results obtained using the flux SMO at 50 r/s. (c) Flux linkage estimation results obtained using the proposed IFSMO at 50 r/s.

can be considerably reduced, and the maximum estimation error decreases to only 1.6◦ (refer to Fig. 9(c)). Fig. 10 shows the speed estimation results of the conventional SMO method, the flux SMO method and the proposed IFSMO method at speeds of 50 r/s and 10 r/s. Fig. 10(a) and (b) clearly shows that maximum estimation errors of 4.98 r/s and 4.15 r/s exist in the conventional SMO method and the flux SMO method, respectively, under high-speed operating condition. Similar to the position estimation results, large vibrations and ripples emerge in the estimation profiles of these two existing methods because

To prove that by eliminating the LPF and the compensation module in the conventional SMO, the proposed IFSMO yields better dynamic performance, experiments under varying speed, load and motor parameter conditions were separately performed. Fig. 11(a), (b) and (c) shows the speed estimation results obtained with a stepped speed response varying from 10 r/s to 50 r/s using the conventional SMO, the flux SMO and the proposed IFSMO, respectively. Fig. 11(a) illustrates that serious chattering emerges in the estimations using the conventional SMO method, which is in accord with the aforementioned steadystate experimental results. Moreover, a distinct speed estimation error of 7.6 r/s can be observed in the initial period at 50 r/s, and an adjustment time of 0.54 s is needed for the PMSM to converge to its steady operating condition. The estimation results in Fig. 11(b) demonstrate that by displacing the external LPF and the compensation module with the embedded feedback matrix in the flux SMO, the estimated speed exhibits fast convergence to its steady state, and no distinct adjustment period emerges after varying the speed. However, although the dynamic performance is improved using the flux SMO, a large steady-state error of 4.8 r/s still exists because of the chattering, which is also in accord with the steady-state results. Furthermore, by introducing the proposed IFSMO method in Fig. 11(c), the estimation error decreases to less than 1.7 r/s over all speed ranges, and no distinct unstable running state emerges as the speed increases. Thus, in addition to the chattering reduction performance, which has

Fig. 9. Rotor position estimation results obtained at 50 r/s using (a) the conventional SMO method and the corresponding estimation error; (b) the flux SMO method and the corresponding estimation error; and (c) the proposed IFSMO method and the corresponding estimation error.

Please cite this article as: S. Ye, Design and performance analysis of an iterative flux sliding-mode observer for the sensorless control of PMSM drives. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.04.009.

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Fig. 10. Rotor speed estimation results obtained using the conventional SMO method at (a) 50 r/s and (d) 10 r/s; using the flux SMO method at (b) 50 r/s and (e) 10 r/s; and using the proposed IFSMO method at (c) 50 r/s and (f) 10 r/s.

Fig. 11. Speed estimation results obtained under a stepped speed response of 10 r/s to 50 r/s using (a) the conventional SMO method, (b) the flux SMO method, and (c) the proposed IFSMO method.

Fig. 12. Speed estimation results obtained under varying loads using (a) the conventional SMO method, (b) the flux SMO method, and (c) the proposed IFSMO method.

also been verified by steady-state experimental results, the superior dynamic performance of the proposed IFSMO under varying speeds is validated. Fig. 12(a), (b) and (c) presents the speed estimation results obtained under varying loads using the conventional SMO, the flux SMO and the proposed IFSMO, respectively. In the experiment, the PMSM ran at a speed of 50 r/s, and a load of 3 Nm, which was simulated by a magnetic powder brake connected to the motor shaft, was added at 2 s. Fig. 12(a) shows that the conventional SMO method yields relatively slow recovery to the steady operating condition, and an adjustment time of 0.98 s is needed. In Fig. 12(b) and (c), by eliminating the LPF and the compensation module, the adjustment time decreases to 0.48

s for the flux SMO method and 0.42 s for the IFSMO method. Moreover, compared with the result of the flux SMO presented in Fig. 12(b), the results shown in Fig. 12(c) demonstrate that the vibration caused by the chattering and the estimation error are greatly reduced by using the IFSMO. It can be concluded that the IFSMO exhibits the best performance among these three methods because it not only reduces chattering with a variable observer gain but also yields fast speed convergence under varying loads. To verify the robustness of the proposed IFSMO method to variations in the motor parameters, experiments were performed with varying stator resistance. Fig. 13 shows the stator resistance estimates obtained using different methods with Rs varied from 0.3  to 0.6  at 2 s. It can be observed that an adjustment time of

Please cite this article as: S. Ye, Design and performance analysis of an iterative flux sliding-mode observer for the sensorless control of PMSM drives. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.04.009.

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Acknowledgments The author would like to thank the Editor and anonymous referees for their useful suggestions, which helped in improving the quality of the paper. Funding The author received no financial support for the research, authorship, and/or publication of this article. Declarations of interest The author declares no conflict of interest. References

Fig. 13. Stator resistance profiles: (a) reference variation profile, (b) estimation results of the conventional SMO-based PMSM sensorless control scheme, and (c) estimation results of the proposed IFSMO-based PMSM sensorless control scheme.

0.9 s is needed for the conventional SMO method to converge the resistance to its actual value, and the required adjustment time decreases to only 0.38 s when using the proposed IFSMO method. Thus, in addition to the superior anti-load-varying performance of the proposed method, this approach also provides higher robustness against internal disturbances than the conventional SMO method does. 7. Conclusions In this paper, a novel IFSMO-based sensorless control method is proposed for PMSM speed and position tracking. In the proposed flux SMO, a feedback matrix, which serves as an embedded LPF for the flux linkage estimations, is constructed to reduce the harmonics. Furthermore, the flux SMO is iteratively used several times in one stator current control loop to adjust the observer gain and reduce chattering. A PMSM sensorless control scheme based on the novel IFSMO is designed, and an experimental platform is built. The experimental results confirm that relative to the conventional SMO method, the proposed IFSMO offers the following advantages: (1) chattering and estimation errors are greatly reduced and (2) because the LPF and compensation module are eliminated, the response speed and dynamic performance of the proposed method are greatly improved. Therefore, the effectiveness and feasibility of the proposed IFSMO in practical engineering are validated. Future work in this context will be performed along two directions: (1) in the case in which parameters are lacking, online parameter-verification methods should be developed and integrated to eliminate the parameter-dependent characteristic of the proposed method; (2) some time-saving methods should be used to adjust the observer gain online to displace the iterative utilization of the flux SMO; therefore, the processing speed of the DSP can be greatly enhanced.

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Please cite this article as: S. Ye, Design and performance analysis of an iterative flux sliding-mode observer for the sensorless control of PMSM drives. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.04.009.