Adaptive sliding mode current control with sliding mode disturbance observer for PMSM drives

Accepted Manuscript Adaptive sliding mode current control with sliding mode disturbance observer for PMSM drives Yong-ting Deng, Jian-li Wang, Hong-wen Li, Jing Liu, Da-peng Tian

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S0019-0578(18)30481-6 https://doi.org/10.1016/j.isatra.2018.11.039 ISATRA 2987

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ISA Transactions

Received date : 16 January 2018 Revised date : 3 August 2018 Accepted date : 27 November 2018 Please cite this article as: Y.-t. Deng, J. Wang, H.-w. Li et al. Adaptive sliding mode current control with sliding mode disturbance observer for PMSM drives. ISA Transactions (2018), https://doi.org/10.1016/j.isatra.2018.11.039 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Adaptive Sliding Mode Current Control with Sliding Mode Disturbance Observer for PMSM Drives Yong-ting Deng, Jian-li Wang*, Hong-wen Li, Jing Liu, Da-peng Tian Changchun Institute of Optics, Fine Mechanic and Physics, Chinese Academy of Science, Changchun, Jilin 130033, China *Corresponding author at: Changchun Institute of Optics, Fine Mechanic and Physics, Chinese Academy of Science, Changchun, Jilin 130033, China. E-mail addresses: [email protected] (J. Wang).

*Highlights (for review)

   

This paper proposes a current control strategy for the PMSM drives by combing the adaptive sliding mode control (ASMC) and sliding mode disturbance observer (SMDO). The ASMC is utilized to provide online compensation for disturbances caused by the PMSM model uncertainties. The SMDO is designed to estimate the external disturbances and add a corresponding feedback compensation item to the output of the ASMC current controller. The effectiveness of the proposed method is verified by experimental results.

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Adaptive Sliding Mode Current Control with Sliding Mode Disturbance Observer for PMSM Drives Abstract - This paper focuses on the current control of a permanent magnet synchronous motor (PMSM) for electric drives with model uncertainties and external disturbances. To improve the performance of the PMSM current loop in terms of the speed of response, tracking accuracy, and robustness, a hybrid control strategy is proposed by combining the adaptive sliding mode control and sliding mode disturbance observer (SMDO). An adaptive law is introduced in the sliding mode current controller to improve the dynamic response speed of the current loop and robustness of the PMSM drive system to the existing parameter variations. The SMDO is used as a compensator to restrain the external disturbances and reduce the sliding mode control gains. Experiments results demonstrate that the proposed control strategy can guarantee strong anti-disturbance capability of the PMSM drive system with improved current and speed-tracking performance. Keywords - Adaptive sliding mode control, model uncertainties, sliding mode disturbance observer, permanent magnet synchronous motor, external disturbances 1. Introduction Permanent magnet synchronous motors (PMSMs) have been extensively used in various industrial applications owing to their attractive characteristics, such as high precision, high power density, and high torque-to-current ratio [1-2]. Field-oriented control (FOC), which has features such as low torque ripple, fast dynamic response, and high control precision, has been widely utilized in PMSM drive systems. It results in a cascade control structure with a double loop, where the external speed loop influences the speed response, and the internal current loop determines the dynamic and steady performance of the entire system [3]. In industrial applications, PMSM drive systems usually face model uncertainties caused by parameter variations and unavoidable external disturbances. The model uncertainties caused by ambient temperature and magnetic saturation, vary significantly for different working conditions, resulting in significant changes to motor parameters [4]. In addition, external disturbances are inevitable for drive systems in practice. Therefore, it is challenging to derive an exact PMSM model and the corresponding motor parameters in a servo system, and it is difficult to obtain a current loop with precise steady-state control and fast dynamic response. Accordingly, several control methods, such as adaptive control [5,6], sliding mode control (SMC) [7-8], robust hybrid control [9], and predictive control [10-12], have been proposed for the current loop. These advanced control methods have improved the performance of the PMSM control system effectively. Among the existing methods, SMC is a popular and effective strategy for nonlinear systems with disturbances, and it can achieve fast dynamic response and strong robustness [13-18]. In the design of an SMC controller, a reasonable switching gain should be selected. When large parameter variations are presented in the current loop, a large switching gain should be designed for the SMC current controller to guarantee the stability and anti-disturbance capability of the system. However, a large switching gain can lead to a discontinuous control signal and serious sliding mode chattering with high-frequency oscillation [19]. Parameter variations affect the performance of the SMC controller, and it is usually not possible for them to be measured directly. The adaptive law is a simple and effective method to handle the model uncertain disturbances. In [20-24], the adaptive law was developed for unknown and bounded system uncertainties to guarantee the tracking performance. In [25-27], control methods based on the adaptive law were utilized to estimate the unknown model uncertainties caused by parameter variations, and a chattering-free SMC was designed based on the estimated values. These studies showed that the adaptive control method can achieve a robust performance against parameter perturbations. However, severe external disturbances usually exist in PMSM drive systems, which can degrade the control performance of the adaptive SMC (ASMC) for current controllers. Moreover, a large switching gain may further deteriorate the chattering phenomenon. To solve these problems, a

disturbance observer can be used to improve the system control performance because of its ability to handle external disturbances [28-30]. Many studies have shown that disturbance observers using the sliding mode technique can achieve the desired estimation performance. The sliding mode disturbance observer (SMDO) has the advantages of convenience of appliancation and strong robustness [23,31-35]. In [19,36-37], external disturbances were observed using SMDO, and thereafter a feedback compensation based on the observed value was implemented, where the switching gain should be larger than the upper bound of the disturbance compensation error. Therefore, the disturbance observer can be used to select a smaller value of the switching gain and ensure strong anti-disturbance capability of the system. To solve the current-control problem of PMSM drives, in this paper, a composite current control method is proposed by combining ASMC with SMDO. The parameter variations and external disturbances of the PMSM model are analyzed, and a novel PMSM model is derived. The derived model contains an ideal part with nominal parameters and a disturbance part with parameter mismatches. Based on the ideal part of the model, a conventional sliding mode current controller is designed. With regard to the problem caused by parameter variations (e.g., the change in stator resistance, stator inductance, and rotor flux linkage), the adaptive law is employed to compensate the uncertain disturbances. In order to suppress the influence of external disturbances and restrain the sliding mode chattering, a SMDO is designed, where the corresponding feedback compensation is implemented for the proposed ASMC current controller utilizing the estimated values. Therefore, the proposed control strategy can achieve fast dynamic response, strong anti-disturbance capability, and precise steady-state performance of the PMSM drive system with severe external disturbances and parameter variations. Experiments have been performed to demonstrate the effectiveness of the proposed control method. This paper is organized as follows. In Section 2, the PMSM model is introduced, and the model uncertainties and external disturbances are analyzed. In Section 3, the design of the proposed ASMC controller and its stability analysis are discussed. In Section 4, the details of the SMDO are presented and the stability of ASMC with SMDO compensation is analyzed. In Section 5, experiments are performed to demonstrate the effectiveness of the proposed control strategy. Section 6 concludes this paper. 2. PMSM model and model-based disturbance analysis 2.1 Mathematical model of PMSM In most applications of the variable-speed PMSM drive, the d-q axis models in the rotating reference frame are derived using the Park transformation so that the stator current control can be applied. The voltage equations of the PMSM in the synchronous rotating reference frame can be described as follows [1]: Lq  did R 1   s id  p iq  ud  dt L L L  d d d (1)  di  q   Rs i  p Ld i  p  f  1 u q d q  dt Lq Lq Lq Lq  where ud and uq are the d- and q- axis stator voltages, respectively; id and iq are the d- and qaxis stator currents, respectively; Rs is the stator resistance; Ld and Lq are the d- and q- axis stator inductances, respectively;  is the mechanical rotor angular speed;  f is the flux linkage; p is the number of pole pairs. The surface-mounted PMSM is considered, where the d- and q- axis stator inductances satisfy Ld  Lq  Ls . Under a mechanical load, the dynamic motion equation of the PMSM can be written as follows: d (2) J  Te  TL  B dt where J is the moment of inertia; Te is the electromagnetic torque; B is the viscous friction

coefficient, TL is the load torque. FOC is generally utilized to control the PMSM drive system, and the d-axis current is maintained at zero in order to maximize the output torque. Therefore, the decoupled electromagnetic torque equation is expressed as follows: 3 (3) Te  p f iq  Kt  iq 2 where K t is the PMSM torque constant. Based on such a mathematical model of PMSM, the disturbance analysis can be implemented for the PMSM drive system. 2.2 Model-based disturbance analysis For PMSM drive systems with unavoidable parameters variations and external disturbances, the stator resistance increases with the increase in the temperature, and the uncertainties caused by parameter variations increase under high-speed operation, as the variations of flux linkage and stator inductance are proportional to the product of operating speed and these parameters [4]. The test parameters of PMSM are given by:  Rs  Rs 0  Rs  (4)  Ls  Ls 0  Ls      f0 f  f where Rs 0 , Ls 0 , and f 0 are the fixed nominal parameters of the PMSM; Rs , Ls , and

 f represent the parameter variations. Considering the parameter variations, the voltage equations of the PMSM can be rewritten as follows:  ( Ls 0  Ls )iq  ( Rs 0  Rs )iq  uq  p ( Ls 0  Ls )id  p ( f 0   f ) (5)   ( Ls 0  Ls )id  ( Rs 0  Rs )id  ud  p ( Ls 0  Ls )iq Thus, the disturbances of the d-q axis voltages caused by parameter mismatches can be expressed as follows:   f q  Rs iq  pLs id  p f  Ls iq (6)    f d  Rs id  pLs iq  Ls id where f d and f q are the uncertain disturbances of the d- and q-axis voltages, respectively. Generally, it is assumed that the variations of PMSM parameters are bounded, i.e., Rs  b1 ,

Ls  b2 , and  f  b3 are satisfied, where b1, b2, and b3 are positive constants. Thus, the electrical equation (5) can be rewritten as follows:  uq  Rs 0iq  p Ls 0id  p f 0  Ls 0iq  f q (7)   ud  Rs 0id  p Ls 0iq  Ls 0id  f d In addition to the above disturbances, which are caused by parameter variations, the external disturbances owing to load-torque variations also significantly influence the control performance of the PMSM. Combing the mechanical dynamic equation (2) and electromagnetic torque equation (3) yields: K B d (t ) (8)   t iq   J J J where d (t ) represents the unknown external disturbances. In the above model, the external disturbances affect the speed control performance. Moreover, owing to the back electromotive force (back EMF) in the voltage equation, the fluctuation of speed also influences the control of the current. In order to achieve a fast transient response and

robust performance, ASMC and SMDO offer an effective approach for the precise tracking control of PMSM drive systems. 3. Design of current controller The SMC scheme is employed in the inner current-loop control of the PMSM and its advantage is that the linearization of the nonlinear voltage model at some operating points during the design process is not required. Moreover, SMC is attractive owing to its fast response and ability to deal with disturbances. In this section, a sliding mode current controller is presented; subsequently, the adaptive control is applied based on the sliding mode controller, where the adaptive law is used for the online estimation of the model uncertain disturbances caused by parameter variations. 3.1 Design of adaptive sliding mode current controller Current control algorithms should guarantee the precise tracking of the current reference value in the presence of disturbances. To achieve such a control objective, the d- and q-axis current-tracking errors are defined as follows:  eq  iqref  iq (9)  ed  idref  id where idref and iqref are the d- and q-axis current reference values, respectively, and ed and

eq are the d- and q- axis current-tracking errors, respectively. By considering the current-tracking errors ed and eq as the state variables, and the d-q axis voltages ud and uq as the control inputs, the state-space equation of the d-q axis currents can be expressed as follows:

 eq   A1 A2   eq   B1 0  uq   C1  (10) e    A A  e    0 B  u   C  4 d  2 d    2  d  3 where A1   Rs 0 Ls 0 , A2   p , A3  p , A4   Rs 0 Ls 0 , B1  1 Ls 0 , B2  1 Ls 0 , C1  E1  f q Ls 0 , C2  E2  f d Ls 0 , E1  Rs 0 Ls 0 iqref  p f 0 Ls 0 , E2   piqref , and E1 and E 2 are the disturbance terms. In the design of the sliding mode controller, the integral surface is chosen to remove the steady-state errors and ensure the precision of current control. t    sq   eq  c1 0 eq d  (11) s    t   d  ed  c2 ed d  0   where c1 and c2 are positive integral coefficients, which can guarantee the asymptotic rate of the current-error convergence. Taking the time derivative of the sliding mode surface equation (11) yields:

 sq   eq  c1eq   s   e  c e   d  d 2 d

(12)

Generally, the basic sliding motion of the SMC can be divided into two steps. In the first step, the system trajectory is forced to tend to the sliding mode surface from the random initial status within a finite time. This step is defined as the reaching process. In the second step, the system trajectory should stay on the sliding surface under the parameter variations and external disturbances once it reaches the sliding surface. This step is defined as the sliding process. The equation ss  0 solely ensures the asymptotical stability and convergence without providing solutions for achieving the trajectory. The reaching law method can deal with the reaching process directly, and can improve the dynamic performance of the system. Therefore, the

regular exponential reaching law is used based on a reasonable choice of switching gains and exponential terms, which should be compatible with the variations of the sliding mode surface and system states. The reaching law is expressed as follows:

 sq   g * ( sq )  (kq sign( sq )  ktq sq )  *  sd   g ( sd )  (kd sign( sd )  ktd sd )

(13)

where xq  eq and xd  ed are the current-error variables of the d- and q- axis, respectively,

kq  0 and k d  0 are the switching gains of the reaching law, ktq  0 and ktd  0 are the exponential coefficients of the reaching law, and sign( s) is the sign function, which is defined as follows:  1 if ( s  0)  sign( s )   0 if ( s  0) (14) 1 if ( s  0)  The discontinuous sign function sign( s) results in serious chattering phenomenon and undesired responses. Several techniques that inhibit the chattering in the control signal exit [27]. Among these techniques, the boundary layer method is commonly used, and is considered effective because it is based on a continuous saturation function. The saturation function is expressed as follows:

if ( s   )  1  sat ( s)   s /  if ( s   ) (15)   1 if ( s   )  where   0 is the layer thickness. A comparison of equations between (14) and (15) shows that the boundary layer method is an effective tradeoff between the performance and control discontinuity. Thus, the reaching law is rewritten as follows:

 sq   g (sq )  (kq sat (sq )  ktq sq )  sd   g (sd )  (kd sat (sd )  ktd sd )

(16)

By combining (10), (12), and (16), the control law for the sliding mode current controller is derived as follows:  *   fq R   kq sat ( sq )  ktq sq   uq  Ls 0   c1  s 0  eq  p ed   Ls 0  Ls 0    (17)    Rs 0  fd  * ud  Ls 0   c2  L  ed  p eq  L  kd sat ( sd )  ktd sd  s0  s0    Equation (17) shows that the derived SMC law contains the model uncertain disturbances. However, it is difficult to obtain the exact values of parameter variations. If the SMC control law does not add the model uncertain terms, the SMC needs to select large control gains to guarantee its robustness against parameter variations. Large switching gains lead to the well-known chattering phenomenon, which can excite the high-frequency modes of the system [26]. To solve these problems, the adaptive law is employed to estimate the parameter variations online. The adaptive control law is expressed as follows: 1 ˆ  f q   sq  1 (18)  1  fˆ  s  d  2 d where fˆq and fˆd are the estimated values of f q and f d , respectively; 1 and  2 are positive constants used to determine the estimated speed of the adaptive law. Thus, the ASMC for the current loops can be expressed as follows:

   fˆ R   uq*  Ls 0   c1  s 0  eq  ped  q  kq sat (sq )  ktq sq    Ls 0  Ls 0        * Rs 0  fˆd  kd sat ( sd )  ktd sd   ed  peq  ud  Ls 0   c2   Ls 0  Ls 0   

(19)

3.2 Stability analysis of adaptive sliding mode current controller In order to analyze the stability of the ASMC in equation (19), the following Lyapunov functions are defined: 1 2 1  2  Vq  2 sq  2 L 1 f q  s0 (20)  V  1 s 2  1  f 2  d 2 d 2 Ls 0 2 d where f  fˆ  f and f  fˆ  f are the estimation errors of the disturbance. q

q

q

d

d

d

Differentiating equation (20) with respect to time t yields:

1  Vq  sq sq  L f q f q s0   uq* fq Rs 0  )eq   p ed  E1  )  1 ( fˆq  f q ) f q   sq ((c1  Ls 0 Ls 0 Ls 0 Ls 0   V  s s   2 f f d d d d  d Ls 0  Rs 0 ud* fd 2 ˆ    sd ((c2  L )ed  L  p eq  E2  L )  L ( f d  f d ) f d s0 s0 s0 s0  Sequentially, substituting (18) and (19) into (21) yields:  f q  fˆq   E1  g ( sq ))  1 ( fˆq  f q ) fˆq Vq  sq ( Ls 0 Ls 0      sq f q  s ( E  g ( s ))  1 f sq q 1 q q  Ls 0 Ls 0 1    sq ( E1  g ( sq ))   f d  fˆd   E2  g ( sd ))  2 ( fˆd  f d ) fˆd Vd  sd ( Ls 0 Ls 0   s f s     d d  sd ( E2  g ( sd ))  2 f d d Ls 0 Ls 0  2    s ( E  g ( s )) d 2 d  From equation (16), it follows that: Vq  sq ( E1  g ( sq ))    sq ( E1  kq sat ( sq )  ktq sq )  2   sq ( E1  kq sat ( sq ))  ktq sq  Vd  sd ( E2  g ( sd ))    sd ( E2  kd sat ( sd )  ktd sd )   s ( E  k sat ( s ))  k s 2 d 2 d d td d 

(21)

(22)

(23)

Similar to the stability analysis of the second-order system studied in [26, 38], for the asymptotic stability of the closed-loop system under the Lyapunov function equation (23), the following inequality should be satisfied: 2   sq ( E1  kq sign( sq ))  ktq sq  0 (24)  2   sd ( E2  kd sign( sd ))  ktd sd  0 From the above equation, it follows that:   ( E1  kq ) / ktq 1   s  0 s k  E  k s      q q tq 1 q q     s  0  s k  E  k d d td 2 d    ( E2  kd ) / ktd  1  2  ktq sq  sq ( E1  kq sign( sq )) sd   (25)    2  ( E1  kq ) / ktq ktd sd  sd ( E2  kd sign( sd ))  1   s  0  s k  E  k sq    q q tq 1 q     sd  0  sd ktd  E2  kd  ( E2  kd ) / ktd  1   sd   If (| E1 |  | kq |) / ktq and (| E2 |  | kd |) / ktd are bounded, (| E1 |  | kq |) / ktq  q   and

(| E2 |  | kd |) / ktd   d   . Thus the equation (25) can be satisfied for the entire state space except for a small neighborhood of the sliding surface where | sq |  q and | sd |  d , if  q   q and  d   d . Thus, we can conclude that there exist parameters kq , ktq , kd and ktd for which the condition of asymptotic stability of the closed-loop system is satisfied for the entire state space, except for a small neighborhood of the sliding surface. Thus, the closed-loop system can be brought to any small neighborhood of the sliding surface by choosing a proper set of control parameters satisfying the following inequality:

 kq  ktq q | E1 |   kd  ktd  d | E2 | where  q and  d are the layer thicknesses of the sliding surface.

(26)

Equation (26) guarantees that Vq  0 and Vd  0 for all s   . Therefore, Vq and Vd are bounded, and the system trajectories approach  near the sliding mode surface within a finite time using the ASMC law in equation (19). When the system trajectory moves into the sliding mode surface, the switching gains in equation (26) play significant roles in the control effort. Therefore, when the system is at the steady state, the switching gains of the ASMC controller should increase with the increase of the external load disturbances. Owing to the switching function, large switching gains can lead to discontinuous control signal and serious chattering with high-frequency dynamics. In order to improve the robustness of the ASMC controller to the external load disturbances and to further suppress the system chattering, the SMDO technique is introduced as a compensation method in Section 4. 4. Design of disturbance observer 4.1 Design of sliding-mode disturbance observer In this section, the SMDO is introduced as a disturbance observer technique for the PMSM control system. The external load disturbances are estimated online, and are subsequently utilized as feedback to the ASMC current controller. The detailed principle of the SMDO is described as follows.

In a practical PMSM drive system, the system disturbances are considered to vary very slowly compared with the system state in every sampling period of the speed loop. Thus, the derivative of d (t ) in (8) with respect to time t can be regarded as d (t )  0 . By considering the mechanical rotor angular speed  and the external load disturbance d (t ) as the state variables, the electromagnetic torque Te as the control input, and  as the system output, the state-space equation of the system can be expressed as follows:

      

    B / J 1 / J    1 / J   Te   0   d   0  d   0   y  1 0   d 

(27)

By considering the mechanical rotor angular speed  and the external load disturbance

d (t ) as the estimated values, the SMDO equation can be represented as follows: ˆ    B / J 1/ J  ˆ  1/ J  1 (28)   Te    (e )  ˆ     0  d   0   dˆ   0 l  where l is the observer gain and  (e ) is the SMC function with the speed observation error

e    ˆ . Based on equations (27) and (28), the equation of the observation error can be written as follows: B 1  e   e  edis   (e ) (29) J J  edis  l  (e ) where e  d  dˆ is the observation error of the external load disturbance. dis

The sliding mode surface is given as follows:

s  e    e d t

0

(30)

where  is a positive integral coefficient. To reduce the chattering caused by SMDO, the exponential reaching law based on the boundary layer method is adopted: s  (k sat ( s )  kt s ) (31) where k  0 is the switching gain and kt  0 is the exponential coefficient of the reaching law. Considering edis / J as the disturbance term, and substituting equation (31) into equation (29) yields: B (32)  (e )  (  )  e  k sat (s )  kt s J Based on the above sliding mode observer, when the observer trajectory reaches the sliding surface s  0 within a finite time and remains on it, the following condition should be satisfied:

 s  s  0  e  e  0

(33)

Combining equations (29) and (33) yields:

edis   J  (e )   edis  l  (e )

(34)

Thus, the estimation error of the external load disturbance with respect to time t is expressed as follows:

(35) edis  cd  el / J t where cd is a constant. To ensure that the observed error of the external load disturbance converges to zero, the observer gain should satisfy l  0 . Moreover, the observer gain determines the rate of convergence of the estimation error. To analyze the stability of the SMDO, the Lyapunov function is defined as follows: 1 (36) V  s 2 2 Differentiating V with respect to time t and applying the control law in equation (32) yields:

V  s  s B 1 e  edis   (e )   e ) J J 1  s ( edis  k sat ( s )  kt s ) J 1  s ( edis  k sat ( s ))  kt s2 J  s (

(37)

Similar to the stability analysis of the ASMC controller, to satisfy the finite-time Lyapunov stability theorem V  0 , the control gains are chosen as follows: 1 (38) k  kt    edis J where   is the layer thickness of the sliding surface. From equation (38), we can conclude that the SMDO is asymptotically stable with a suitable observer gain, and the observer trajectory moves toward and reaches the sliding mode surface within a finite time. 4.2 Stability analysis of ASMC with SMDO compensation The observed value of external load disturbance is used as feedback to compensate the output of the d-q axis current loops. Considering the compensation terms, the ASMC law in equation (19) can be rewritten as follows:  * fˆq R  g ( sq )  kcq dˆ )  ucq  Ls 0 ((c1  s 0 )eq  ped  Ls 0 Ls 0  (39)  ˆ Rs 0 fd  * ˆ ucd  Ls 0 ((c2  L )ed  peq  L  g ( sd )  kcd d ) s0 s0  where kcq and kcd are the compensation coefficients of the d- and q-axis controllers, respectively. According to equation (7), the compensation terms kcq dˆ and kcd dˆ correspond to the disturbances pid  p f 0 / Ls 0 and  piq , respectively, and the signs of kcq dˆ and

kcd dˆ depend on the PMSM rotation direction and currents, respectively. In order to analyze the stability of the ASMC controller with disturbance compensation in equation (39), the Lyapunov functions (20) can be modified as follows:

1 2 1 1  2 2  Vcq  2 sq  2 L 1 f q  2 | Jl | edis  s0  1 1 1 2 V  s 2  2 fd 2  edis cd d  2 2 Ls 0 2 | Jl | The stability analysis of the optimized ASMC law is as follows:

(40)

1  Vcq  sq sq  L f q f q  edis edis s0  *  ucq fq Rs 0  )eq   p ed  E1  )  1 ( fˆq  f q ) f q  edis edis   sq ((c1  Ls 0 Ls 0 Ls 0 Ls 0   V  s s   2 f f  e e d d d d dis dis  cd Ls 0  * Rs 0 ucd f    s (( c  ) e   p eq  E2  d )  2 ( fˆd  f d ) f d  edis edis d 2 d  Ls 0 Ls 0 Ls 0 Ls 0 

(41)

Based on the finite-time Lyapunov stability theorem, and substituting equations (39) and (40) into equation (41), the following equation is obtained:  sq f q   sq ( E1  kcq dˆ  kq sat ( sq )  ktq sq )  1 fˆq f q  |  (e ) |2 Vcq   Ls 0 Ls 0   2 ˆ   sq ( E1  kcq d  kq sat ( sq )  ktq sq ) |  (e ) |    sq ( E1  kcq dˆ  kq sat ( sq )  ktq sq )  0 (42)  sd f d   2 2 ˆ ˆ Vcd   L  sd ( E2  kcd d  kd sat ( sd )  ktd sd )  L f d f d  |  (e ) | s 0 s 0    s ( E  k dˆ  k sat ( s )  k s ) |  (e ) |2 d 2 cd d d td d   ˆ   sq ( E1  kcq d  kq sat ( sq )  ktq sq )  0  The stability analysis of the ASMC controller with disturbance compensation is similar to that of the ASMC controller. To satisfy Vcq  0 and Vcd  0 , the control gains should be selected as follows:  kq  ktq  cq  E1  kcq dˆ  (43)   kd  ktd  cd  E2  kcd dˆ  where  cq and  cd are the modified layer thicknesses of the sliding surface after the SMDO compensation. By comparing equation (26) with equation (43), it can be concluded that the ASMC controller with SMDO compensation does not require larger switching gains when a sudden external load disturbance is imposed on the PMSM drive system. Once the SMDO is employed, the external load disturbances can be estimated and compensated simultaneously, and the chattering of the ASMC current controller is further reduced. The composite control scheme is shown in Fig. 1, where the controller is composed of the ASMC current controller and the SMDO based feedback part. The composite scheme improves the robustness of the PMSM drives. Moreover, it can suppress the sliding mode chattering phenomenon.

2

1

Adaptive Control Law(18)

ASMC Current Controller

fˆq kd ktd

d / dt

i 0 i * d * q

ed

eq

-

sd

Sliding Mode s q Surface (11)

-

kq ktq

d / dt

fˆd Park 1

* ucd

Sliding Mode Controller (39)

u

d, q

* cq

,

Sliding Mode Observer (28)

SV

d, q ,



d / dt



s Sliding Mode Surface (30)

e

-

3-phase inverter

u PWM

i

id iq dˆ

Vdc u

i

,

a, b

u a ub u c ia ib

Clark

Park

PMS M

d / dt

ˆ

Sliding Mode Distrubance Observer

Fig. 1 Block diagram of the ASMC+SMDO method. 5. Experimental results and discussion Experiments based on a PMSM drive system are presented in this section to demonstrate the effectiveness of the proposed ASMC+SMDO scheme. Fig. 2 shows the overall structure diagram of the PMSM drive system, where the ASMC+SMDO strategy and FOC method are utilized to control the PMSM. A sketch of the experimental setup is shown in Fig. 3, and the corresponding experimental platform is shown in Fig. 4. The proposed control scheme is realized based on the DSP-TMS320F28335 and FPGA-EP3C40F324-based drive setup. The sampling frequency of the speed loop is 1 kHz, and the counterpart of the current loop is 15 kHz. The PMSM parameters used in this study are listed in Table 1. Experiments were performed to evaluate the effectiveness of the proposed ASMC+SMDO scheme compared with the traditional proportional integral (PI) current controller under various operating conditions. According to the aforementioned analysis, the experimental coefficients are selected as follows. For a fair comparison, the PI controller is applied in the speed loop throughout the experiments, with kP  0.012 and kI  0.003 . The parameters of the PI current controllers in the d- and q- axis are the same, and they are selected as kP  5.0 and kI  0.15 . The parameters of the ASMC current controllers in the d- and q- axis are the same, and they are selected as c1  3 , c2  3 , kq  100 , ktq  60 , kd  100 , ktd  60 , 1  500 ,  2  500 , and   0.1 . The parameters of the SMDO kt  0.01 , l  20 , kcq  135 , and kcd  -105 .

are

selected

as   0.5 , k  0.05 ,

DC

ref  

iqref

Speed controller

ASMC Current Controller

 iq

idref  0  id

uq*

* ucq



ud*

u



* cd

Parkˉ¹

dq  

kcq dˆ

kcd dˆ

u

SVPWM

u

PWM1 PWM2 PWM3 PWM4 PWM5 PWM6

Threephase inverter

e

SMDO

A B C

  dq

i i

ab  

ia

ia

ib

ib

PMSM Encoder

Clark

Park



External disturbance

Position and Speed detection

Fig. 2 Diagram of structure of the PMSM servo system based on the ASMC+SMDO scheme. Intelligent Power Module (IPM)

+ Fuse

Vdc

WP

VP

UP

R

Load W V U

S

Voltage Sensor

WN

VN

UN

PMSM

Encoder

_

PC

Current Sensor



IPM Fault Signal

To A/D Interface

Reference Speed

*

Speed Controller Algorithm

Current Sensors

Proposed Current Controller Algorithm

Optocoupler

ia ib

SVPWM Generator and Fault Protection

A/D Interface

 DSP-TMS320F28335

FPGA-EP3C40F324

Fig. 3 Configuration of the experimental setup. TABLE 1 Parameters of PMSM. Symbol Quantity Stator phase resistance Rs

Value and Unit 15.42 

Ls

d,q axis inductances

30.08 mH

Kt

Torque constant

0.41 N·m/A

J

Rotational inertia Number of pole pairs

0.138 kg·cm² 4

p DC power

Magnetic powder brake

Oscilloscope

PMSM and Encoder

PMSM driver

PC

Fig. 4 Photograph of the experimental platform.

Encoder Interface

5.1 Current-tracking performance in the speed open-loop system In order to verify the current control performance, experiments were carried out in a speed open-loop system, where the reference current is a step constant using the conventional PI and ASMC methods. The reference currents of the d- and q- axis are given as idref  0 and iqref  0.2  0.35 A , respectively. Then, the d-axis reference value is set to idref  0.2 A . The experimental results of the PI and ASMC current controllers are shown in Figs. 5 and 6. Fig. 5 shows a comparison of the results of the two controllers, with the step change of the q-axis reference value ranging from 0.2 A to 0.35 A without a load. The phase current responses show that the PMSM begins to speed up when the q-axis current increases. The settling time of the ASMC controller is shorter than that of the PI controller. Therefore, the ASMC controller has a faster dynamic response than the PI controller. 0.5

0.5

0.4

0.4 0.3

iq

Current(A)

Current(A)

0.3 0.2 0.1

4

5

6 Time(s)

7

8

-0.1

9

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 -0.1

-0.3

-0.3

5.98

6 Time(s)

6.02

6.04

6.06

5

6 Time(s)

7

8

9

5.96

5.98

6 Time(s)

6.02

6.04

6.06

-0.1 -0.2

5.96

4

0

-0.2

-0.4

id

0

Current(A)

Ia(A)

0.2 0.1

id

0 -0.1

iq

-0.4

(a) (b) Fig. 5 Experimental results under step q-axis current: (a) Experimental results of the PI method (b) and experimental results of the ASMC method. Fig. 6 shows a comparison of the results of the two controllers, with the step change of the daxis reference value in the range 0–0.2 A without a load. It shows that the settling time of the ASMC is shorter than that of the PI, which is similar to the q-axis current response. Therefore, the results further illustrate the superiority of the ASMC current controller.

0.5

0.5

iq

0.4

0.4

0.3

0.3 Current(A)

Current(A)

iq

0.2 0.1

0.1

id

4

5

6 Time(s)

7

8

-0.1

9

4

5

6 Time(s)

7

8

9

5.96

5.98

6 Time(s)

6.02

6.04

6.06

0.5

Ia(A)

Ia(A)

0.5

0

-0.5

id

0

0 -0.1

0.2

5.96

5.98

6 Time(s)

6.02

6.04

6.06

0

-0.5

(a) (b) Fig. 6 Experimental results under step d-axis current: (a) Experimental results of the PI method and (b) experimental results of the ASMC method 5.2 Robustness to model uncertain disturbances In order to evaluate the performance of the proposed ASMC scheme under parameter variations, experiments were carried out at 400 r/min and 800 r/min without a load. The experimental results of the conventional PI and ASMC are shown in Figs. 7-12. The tested PMSM electric parameters were the inductance Ls , stator resistance Rs , and permanent magnetic flux linkage f . As the rated parameters of the PMSM cannot be changed freely, the sensitivities of the three parameters are tested by changing their values in the ASMC controller, e.g., the stator inductance of the ASMC is set to Ls1  60.16mH , whereas the nominal value is Ls 0  30.08mH . This indicates that the stator inductance is reduced to half of its rated value. The three parameters are changed online according to the following three steps, 0.5Ls 0  Ls 0  2Ls 0 , 0.5Rs 0  Rs 0  2Rs 0 , and 0.5 f 0   f 0  2 f 0 . Figs. 7 and 8 respectively show a comparison of the current results obtained using the PI and ASMC methods, respectively, under inductance mismatch conditions. They show that the real d-axis current cannot track the reference current value accurately under the PI control, and the current-tracking error increases with the speed owing to the inductance mismatch. Moreover, the q-axis current is affected slightly, owing to the existence of the coupling in the PMSM model. The results are consistent with the theoretical analysis of the novel model (7) in Section 2, where the inductance variation mainly generates the d-axis disturbance. With respect to the ASMC method, the model uncertain disturbance caused by a variation in inductance is estimated using the adaptive law, and is designed as a feedback item to the SMC current controller. Therefore, the steady-state current-tracking error is effectively eliminated. Figs. 9 and 10 show the comparison current results of the PI and ASMC methods under stator resistant mismatch, respectively. It is observed that the real q-axis current cannot track the reference current value accurately under PI control, and the current-tracking error increases with

speed owing to the stator resistant mismatch. Moreover, the d-axis current is also slightly affected because of the existence of the coupling. As shown in the novel model (7), the resistance variation mainly generates q-axis disturbance. In the ASMC method, the adaptive law estimates the model uncertain disturbance caused by the stator resistant variation, and it generates a compensation part for the SMC current controller. Therefore, the ASMC controller can realize a system robust to parameter variation, and the current-tracking error of the steady state is eliminated effectively. Figs. 11 and 12 show a comparison of the current results obtained using the PI and ASMC methods, respectively, under permanent-magnetic flux linkage mismatch. The experimental results show that the real q-axis current has evident errors compared with the reference current value, and the tracking error increases with the speed owing to the flux mismatch. The flux variation mainly causes the q-axis disturbance according to the novel model (7), and the d-axis is affected slightly. The tracking responses of the d- and q- axis currents confirm the results of the analysis. With respect to the ASMC method, the disturbance caused by parameter variation is also compensated by the adaptive law, and this guarantees a better tracking performance of the current control. According to the above experimental results, the effectiveness of the proposed ASMC+SMDO scheme in restraining the model uncertain disturbances is verified under different parameter variations. Ls  2Ls 0

Ls  2Ls 0

Ls  Ls 0

Ls  Ls 0 Ls  0.5Ls 0

Ls  0.5Ls 0 0.4

0.4 Idref Id

0.2

0.2

0.1

0.1

0

0

-0.1

-0.1

-0.2

-0.2

-0.3

2

4

6

8 Time(s)

10

12

14

Idref Id

0.3

Id(A)

Id(A)

0.3

-0.3

16

0.8

2

4

6

8 Time(s)

10

12

14

0.8 Iqref Iq

Iqref Iq

0.6 Iq(A)

Iq(A)

0.6

0.4

0.4

0.2

0.2

0

16

2

4

6

8

10 Time(s)

12

14

16

0

2

4

6

8

10

12

14

16

Time(s)

Fig. 7 Experimental results of the PI method with Ls mismatch under various speed conditions: (a) Experimental results at a speed of 400 r/min and (b) experimental results at a speed of 800 r/min.

Ls  2Ls 0

Ls  2Ls 0

Ls  Ls 0

Ls  Ls 0

Ls  0.5Ls 0

Ls  0.5Ls 0

0.4

0.4 Idref Id

0.2

0.1

0.1

0 -0.1

0 -0.1

-0.2 -0.3

Idref Id

0.3

0.2

Id(A)

Id(A)

0.3

-0.2 2

4

6

8

10

12

14

-0.3

16

2

4

6

8

Time(s)

0.8

12

14

16

0.8 Iqref Iq

Iqref Iq

0.6 Id(A)

0.6 Iq(A)

10 Time(s)

0.4

0.2

0.4

0.2

0 2

4

6

8

10 Time(s)

12

14

0

16

2

4

6

8

10 Time(s)

12

14

16

Fig. 8 Experimental results of the ASMC method with Ls mismatch under various speed conditions: (a) Experimental results at a speed of 400 r/min and (b) experimental results at a speed of 800 r/min. Rs  2 Rs 0

Rs  2 Rs 0

Rs  Rs 0

Rs  Rs 0

Rs  0.5Rs 0

Rs  0.5Rs 0 0.4

0.4 Idref Id

0.2

0.2

0.1

0.1

0

0

-0.1

-0.1

-0.2

-0.2

-0.3

-0.3 2

4

6

8

10

12

14

Idref Id

0.3

Id(A)

Id(A)

0.3

16

2

4

6

8

0.8

14

16

Iqref Iq 0.6 Iq(A)

Iq(A)

12

0.8

Iqref Iq

0.6

0.4

0.2

0

10 Time(s)

Time(s)

0.4

0.2

2

4

6

8 10 Time(s)

12

14

16

0

2

4

6

8

10

12

14

16

Time(s)

Fig. 9 Experimental results of the PI method with Rs mismatch under various speed conditions: (a) Experimental results at a speed of 400 r/min and (b) experimental results at a speed of 800 r/min.

Rs  2 Rs 0

Rs  2 Rs 0 Rs  Rs 0

Rs  Rs 0 Rs  0.5Rs 0

Rs  0.5Rs 0 0.4

0.4

Idref Id

0.2

0.2

0.1

0.1

0

0

-0.1

-0.1

-0.2

-0.2

-0.3

2

4

6

8

10

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14

Idref Id

0.3

Id(A)

Id(A)

0.3

-0.3

16

2

4

6

8

Time(s)

10

12

14

0.8

0.8 Iqref Iq

Iqref Iq

0.6

Iq(A)

Iq(A)

0.6 0.4

0.4

0.2 0

16

Time(s)

0.2

2

4

6

8 10 Time(s)

12

14

0

16

2

4

6

8 10 Time(s)

12

14

16

Fig. 10 Experimental results of the ASMC method with Rs mismatch under various speed conditions: (a) Experimental results at a speed of 400 r/min and (b) experimental results at a speed of 800 r/min.  f  2 f 0  f  2 f 0

 f  f 0

 f  f 0

 f  0.5 f 0

 f  0.5 f 0 0.4 Idref Id

0.2 0.1

Id(A)

0.3 0.2 Id(A)

Idref Id

0.3

0.4

0.1

0 -0.1

0

-0.2

-0.1

-0.3

-0.2 -0.3 2

4

6

8

10

12

14

2

4

6

8 10 Time(s)

12

14

16

16

Time(s) 0.8

0.8

Iqref Iq

0.4 0.2 0

Iqref Id

0.6

Iq(A)

Iq(A)

0.6

0.4

0.2

2

4

6

8

10 Time(s)

12

14

16

0

2

4

6

8

10 Time(s)

12

14

16

Fig. 11 Experimental results of the PI method with f mismatch under various speed conditions: (a) Experimental results at a speed of 400 r/min and (b) experimental results at a speed of 800 r/min.

 f  2 f 0

 f  2 f 0

 f  f 0

 f  f 0

 f  0.5 f 0

 f  0.5 f 0 0.4

0.4

Idref Id

0.3

0.2

0.1 0

0 -0.1

-0.2

-0.2

2

4

6

8 10 Time(s)

12

14

-0.3

16

Iqref Iq

4

6

8

10 Time(s)

12

14

16

Iqref Iq

0.6

Iq(A)

0.6 0.4

0.4 0.2

0.2 0

2

0.8

0.8

Iq(A)

0.1

-0.1

-0.3

Idref Id

0.3

Id(A)

Id(A)

0.2

0

2

4

6

8 10 Time(s)

12

14

16

2

4

6

8 10 Time(s)

12

14

16

Fig. 12 Experimental results of the ASMC method with f mismatch under various speed conditions: (a) Experimental results at a speed of 400 r/min and (b) experimental results at a speed of 800 r/min. 5.3 Robustness to external disturbances In order to evaluate the performance of the proposed ASMC+SMDO scheme under an external disturbance, experiments were carried out using sudden-load and sudden-unload disturbances for the rotating speeds of 400 r/min and 800 r/min, respectively. The sudden-load and sudden-unload disturbances were realized using a magnetic powder brake. Considering the robustness of the PMSM control system, a comparison of the experiment results of the ASMC and ASMC+SMDO schemes is shown in Figs. 13-15. Fig. 13 shows the dynamic responses of the speed, phase current I a , and q -axis current I q when the motor is running at a speed of 400 r/min with the ASMC and ASMC+SMDO methods, respectively. The external disturbance load torque is set as 0.52 N·m. Fig. 14 shows the arrangement in the same sequence with the motor running at a speed of 800 r/min, and the external disturbance load torque is 0.35 N·m. The estimated results of the SMDO are shown in Fig. 15. It can be observed that step-disturbance load torque of 0.52 N·m and 0.35 N·m is detected by the observer. The SMDO can estimate the variation of the external disturbance torque precisely and rapidly. When the motor is operating at the speed of 400 r/min and a sudden load disturbance of 0.52 N·m is applied, the maximum speed fluctuation under the ASMC method is 55 r/min, whereas the ASMC+SMDO method reduces the maximum speed fluctuation to 28 r/min. The maximum speed fluctuation is reduced by 6.75%. Moreover, the adjustment time required for the speed to return to its original value decreases to the small value of 0.32 s with the ASMC+SMDO method. When the motor is operating at the speed of 800 r/min and a sudden load disturbance of 0.35 N·m is applied, the ASMC method has a maximum speed fluctuation of 30 r/min, whereas the ASMC+SMDO achieves a smaller fluctuation of 22 r/min. The maximum speed fluctuation is therefore reduced by 1%, and a much smaller adjustment time of 0.24 s is required for the speed to return to the original value.

The results show that the ASMC+SMDO method achieves a smaller fluctuation in the speed and I a responses and a small overshoot in I q irrespective of the operating conditions. In

500

Speed(r/min)

Speed(r/min)

addition, it is guaranteed that the speed, I a and I q are restored to their original values much faster than with the ASMC method, therefore, exhibits a satisfactory robustness performance compared with the ASMC method. Based on the above experimental results, the effectiveness of the proposed ASMC+SMDO scheme with respect to the rejection of external disturbances has been verified under various operating conditions. 0.48s 400 345r/min 300

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(a) (b) Fig. 13 Comparison of experimental the results of the ASMC and ASMC+SMDO methods at a speed of 400 r/min. (a) Experimental results of the ASMC method and (b) experimental results of the ASMC+SMDO method. 0.39s 800 770r/min 750

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(a) (b) Fig. 14 Comparison of experimental the results of the ASMC and ASMC+SMDO methods at a speed of 800 r/min. (a) Experimental results of the ASMC method and (b) experimental results of the ASMC+SMDO method.

reference value observed value

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(a) (b) Fig. 15 Experimental results of the SMDO under various operating conditions. (a) Experimental results at a speed of 400 r/min, and (b) experimental results at a speed of 800 r/min. 6. Conclusion To improve the anti-disturbance capability of a PMSM drive system, a strategy was proposed by combining the ASMC and SMDO. In addition, a novel model was derived for the PMSM system with parameter variations and external disturbances. In the proposed strategy, the ASMC+SMDO methods were applied, where the adaptive law was employed to estimate the model uncertain disturbances and the SMDO was utilized to estimate the external disturbances. The estimated values were designed as feedback items to the ASMC controller. This proposed ASMC+SMDO method could achieve the desired control of the PMSM system with parameter variations and external disturbances. The effectiveness of the proposed method was verified by comparing the experimental results with those derived using the traditional method. The proposed method has the potential for application to PMSM systems with extreme parameter variations, particularly for the cases where the load disturbance of the motor drive systems should be estimated online. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant no. 11603024, the Third Phase of Innovation Project of Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Science (Grant no. 065X32CN60). References [1] Y. Da, X. Shi, M. Krishnamurthy, A Novel Universal Sensor Concept for Survivable PMSM Drives, IEEE Trans. Power Electron. 28(12) (2013) 5630 - 5638. [2] S.-Y. Jung, K. Nam, PMSM Control Based on Edge-Field Hall Sensor Signals Through ANF-PLL Processing, IEEE Trans. Ind. Electron. 58(11) (2011) 5121-5129. [3] E.J. Fuentes, C.A. Silva, J.I. Yuz, Predictive Speed Control of a Two-Mass System Driven by a Permanent Magnet Synchronous Motor, IEEE Trans. Ind. Electron. 59(7) (2012) 2840-2848. [4] A. Rabiei, T. Thiringer, M. Alatalo, Improved Maximum-torque Per-ampere Algorithm Accounting for Core Saturation, Cross-coupling Effect, and Temperature for a PMSM Intended for Vehicular Applications, IEEE Trans. Transp. Electr. 2(2) (2016) 150 - 159. [5] Y.A.-R.I. Mohamed, Design and Implementation of a Robust Current-Control Scheme for a PMSM Vector Drive with a Simple Adaptive Disturbance Observer, IEEE Trans. Ind. Electron. 54(4) (2007) 1981-1988. [6] K.-H. Kim, Model Reference Adaptive Control-based Adaptive Current Control Scheme of a PM Synchronous Motor with an Improved Servo Performance, IET Electr. Power Appl. 3(1) (2009) 8–18. [7] M. Comanescu, An Induction-Motor Speed Estimator Based on Integral Sliding-Mode Current Control, IEEE Trans. Ind. Electron. 56(9) (2009) 3414-3423.

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