Static-Errorless Deadbeat Predictive Current Control Using Second

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2017.2694019, IEEE Transactions on Power Electronics

Static-Errorless Deadbeat Predictive Current Control Using Second-Order Sliding-Mode Disturbance Observer for Induction Machine Drives Bo Wang, Student Member, IEEE, Zhen Dong, Yong Yu, Gaolin Wang, Member, IEEE, and Dianguo Xu, Fellow, IEEE  Abstract—Although the conventional deadbeat predictive current control (DPCC) can theoretically provide fast current dynamic responses for induction machine (IM) drives, it suffers from problems of steady-state current error and stability due to a detrimental model mismatch. To solve these problems, this paper proposes static-errorless DPCC using a second-order sliding-mode disturbance observer (DO). The IM parameter variation and other unmodeled dynamics are regarded as lump disturbance in the modeling of IM. Based on this model, a second-order sliding-mode DO is constructed to estimate lump disturbance, and then the estimated disturbance is applied as feed-forward compensation to correct the steady-state current error. Moreover, the designed second-order sliding-mode DO effectively alleviates the chattering effect caused by the switching control. Since the designed DO works in parallel with the DPCC, the fast dynamic responses of the DPCC is retained. Experimental results confirmed the validity of the proposed scheme. Index Terms—Deadbeat predictive current control (DPCC), second-order sliding-mode, disturbance observer (DO), induction machine (IM) drives.

I. INTRODUCTION

T

he indirect field oriented control (IFOC)-based induction machine (IM) drive has been employed in many industrial applications for its superior speed-regulation performance. The double-loop structure is typically adopted in IFOC: an outer speed loop to regulate the rotor speed; and an inner current loop to adjust the stator current, tracking its reference [1]. As the inner loop, the current loop is expected to exhibit a fast dynamic response, which can permit a high-bandwidth outer speed loop. To achieve this, varieties of control strategies have been introduced into the current loop, e.g., hysteresis control [2], proportional-integral (PI) control in stationary or

Manuscript received November 3, 2016; revised March 7, 2017; accepted April 9, 2017. This work was supported in part by the Research Fund for the National Science Foundation of China (51377032), and in part by the Research Fund for the National Science and Technology Support Program (2014BAF08B05). The authors are with the School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin 150001, China (e-mail: wangbo6222@ 126.com; [email protected]; [email protected]; WGL818@ hit.edu.cn; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

synchronous reference frame [3], [4], proportional-resonant (PR) control [5], sliding-mode control (SMC) [6], and predictive current control (PCC) [7]–[11]. Among these control schemes, it has been established that the PCC offers the fastest dynamic response [12]. As a well-known PCC, deadbeat predictive current control (DPCC) can make full use of the system model and provide the optimal actuation to force the feedback current to track its reference [7]. Thus the DPCC has recently received much attention, and has been applied for current loop in grid-connected three-phase inverter control [9], [12] as well as in machine drives [7], [8]. While the DPCC can provide a fast current response, it is often fragile due to an excessive dependence on the system model [13]. As is well known, system parameter variation and other unmodeled dynamics are inevitable in an industrial system, leading to undesired model mismatch. Accordingly, this model mismatch would cause severe drawback of a steady-state control error in the DPCC. When this error reaches a certain degree, the control system turns to unstable. To overcome this drawback, many improvements have been made to the DPCC. The presented DPCCs in [14] and [15] can stably operate under a maximum inductance mismatch of 50% nominal value or a maximal control delay of two sampling periods. To further enhance robustness and stability, modified DPCCs were presented in [13] and [16], where Luenberger-type observers have been applied to predict the next moment current for the control law. These approaches make the system insensitive to model mismatch by reducing the controller’s dependence on the system model. Although these approaches can improve the robustness and stability of the DPCC, the main drawback of the steady-state current error still exists. In [17], an adaptive robust predictive current control is proposed to correct the steady-state current error, using a parallel adaptive strategy. However, the extra error correction term increases the algorithmic complexity and adjusting difficulty. In [18] and [19], the linear disturbance observer (DO) is introduced into the PCC, thus essentially eliminating the steady-state current error for the three-phase grid-connected inverter control. However, at present, there are still many gaps and weakness in IM drive applications. In this paper, a DPCC with a second-order sliding-mode DO is proposed for IM drives. The main contributions of the proposed scheme can be summarized as: (1) retaining the

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superiority of fast current response of the DPCC; (2) the drawback of steady-state current error is overcome by applying on-line disturbance estimation and compensation; (3) the proposed second-order sliding-mode DO alleviates the chattering effect in the conventional sliding-mode DO. This paper is organized as follows: In Section II, the mathematical model of an IM with uncertainties is presented. Then this model is improved with the inclusion of control delay. Section III analyzes the conventional DPCC and its drawback in the discrete domain. In Section IV, the conventional DPCC is combined with a second-order sliding-mode DO to solve its drawback. Then the stability and chattering suppression of the proposed DO is analyzed. The experimental results on an industrial IM test bench are provided in Section V. II. BASIC OPERATING PRINCIPLES OF IM A. IM Model with Uncertainties In the d-q synchronous reference frame, the ideal mathematical model of an IM based on IFOC [1] can be expressed as

d R L2  R L2 Lm u r  sd  isd   s r 2r m isd  e isq   Ls Lr Tr  Ls  Ls Lr  dt (1)  usq Rs L2r  Rr L2m Lm d isq  e isd     dt isq    Ls Lr r r  Ls  Ls L2r  where isd and isq are stator current d-q components; usd and usq are stator voltage d-q components; Rs and Rr are stator and rotor resistances; Ls, Lr and Lm are self and mutual inductances; λr is rotor flux; ωe is synchronous angular speed; ωr is rotor angular speed; Tr = Lr / Rr is rotor time constant; and σ = 1 – L2m / (Ls Lr) is leakage coefficient. Then we can rewrite (1) in its matrix form di  Ai  B  u  d  (2) dt where T

T

f   B1 Ai  B  u  d  d  ε   d  ε

L R 1 L , d sd   m 2 r r , d sq  m r r ,  Ls Lr Lr 1 0  0 1 I = , and J =   . 0 1  1 0 

However, this ideal model is affected by varieties of uncertainties, such as parameter (resistances and inductances) variation, and unmodeled dynamics, leading to model mismatch. Considering all these uncertainties as lump disturbance, (2) can be rewritten as di  Ai  B  u  d  f  (3) dt where Aʹ and Bʹ are the matrices using the nominal values of IM parameter; and f = [fd fq]T is the lump disturbance, which is

(4) T

where “  ” denotes parameter variation; and ε =  d  q  denotes unmodeled dynamics. Applying the forward Euler approximation in (3), we obtain the discrete form of IM model with uncertainties

i  k  1 =  I  ATs  i  k   BTs u  k   d  k   f  k  (5)

where Ts is the sampling period. B. IM Model for One-Sampling Period Control Delay In an ideal control system, sampling and ADC conversion, algorithm calculation, and PWM update are supposed to work at the same instant. However, this is impossible for an industrial microprocessor-based system due to the digital control delay. Fig. 1 shows a typical control timing sequence of an IM drive system using symmetrical PWM strategy. The sampling point is set to synchronize with the PWM interrupt signal, which can make the measured current closer to its average current without the switching noise [20]. Then, one sampling period Ts is necessary for ADC conversion and algorithm calculation. Finally, PWM update is completed in the following period. Thus, we can assume an ideal control by applying the delayed measured current iʹ(t) ≡ i(t–Ts). Considering this delay, the IM discrete model in (5) can be rewritten as i   k  =  I  ATs  i   k  (6)  BTs  u  k  1  d  k  1  f  k  1  . Ts Calculate u(k-2)

Calculate u(k-1)

Calculate u(k)

Update u(k-2)

Update u(k-1)

PWM Carrier

Update u(k-3)

T

i = isd isq  , u = usd usq  , d = dsd dsq  , R L2  Rr L2m , a2  e , A = a1 I  a2 J , B = b1 I , a1  s r  Ls L2r

b1 

given by

(k-2)Ts Sampling point

(k-1)Ts ADC Conversion

kTs Algorithm Calculation

(k+1)Ts PWM Interrupt Signal

Fig. 1. A typical control timing sequence of an IM drive system.

C. IFOC-based IM Drive System Fig. 2 depicts the block diagram of the IFOC-based IM drive using the proposed current controller. The overall control system includes an IM, a rectifier and inverter set, and a control unit. The entire control algorithm is realized in the control unit, where a commonly-used double-loop structure is applied. A PI controller is used for the outer speed loop, and the proposed current controller is used for the inner current loop. The reference of isq (representing the torque current) varies according to the speed loop output, and the reference of isd (representing the field current) remains constant under rated speed. An incremental encoder is applied to measure the IM rotor speed. The rotor flux position θ is obtained by integrating

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synchronous angular speed ωe for coordinate transformation. Rectifier

Control Unit

isd ,ref

d k 

r ,ref+

-



Proposed Current Controller

udq

e j

+

u

uDC

SVPWM

PI



+

slip +

r

Rr isq

 Lr isd 

Slip Calculation

 j isd , isq e i

iref ( k ) +

 BTs 

-

1

d  k  1 +

+

z

uk 

I  ATs

1

-

+

+ -

+

+

i   k  1

BTs

+

+

z 1

i  k 

I  ATs

f  k  1

BTs

Inverter

isq ,ref

d

IM discrete model

Conventional DPCC

3*380V 50Hz

I  ATs

Fig. 3. Block diagram of the conventional DPCC in the discrete domain.

abc



Conventional DPCC

IM discrete model

IM

d k 

3

Encoder

Fig. 2. Block diagram of IFOC-based IM drive using the proposed current controller.

iref  k 

Gf  z

ek 

+ -

+ +

Gc  z 

d  k  1

z 1

uk 

+

-

Gp  z 

i  k 

f  k  1

III. CONVENTIONAL DPCC AND ITS DRAWBACK The conventional DPCC has been used to achieve fast current response for IM drives [7]. The command voltage is calculated in accordance with the IM model once every sampling period. Then, this command voltage works through a fixed-switching-frequency PWM modulator to force the current to track its reference in the next sampling instant. By applying one sampling period advanced of (6) and setting iref(k) = iʹ(k+2), we can obtain the command voltage of a two-sample deadbeat as

uk  

iref  k   i   k + A  i   k +  d  k   f  k  . (7) BTs B

Since the disturbance f(k) cannot be calculated in the conventional DPCC, it is typically ignored. Applying f(k) = 0 in (7), we retain the command voltage of the conventional DPCC

uk  

iref  k   i   k + A  i   k +  d  k  . BTs B

(8)

Known as “predictive control”, the unknown variables iʹ(k+1) and d(k) in (8) need to be predicted during period kTs to (k+1)Ts. First, iʹ(k+1) can be predicted by setting f(k-1) = 0 in (6). And second, the back-EMF d(k) is predicted by assuming that its variation over the sampling periods is linear, which results in (9) d  k   2d  k  1  d  k  2 . Fig. 3 shows the block diagram of the conventional DPCC in the discrete domain. Fig. 4 is its simplification, where the controller is rearranged into two parts: the equivalent filter

G f  z    I  ATs  . 2

(10)

and the controller

Gc  z    I  ATs   zI  I  ATs  2

1

 BTs 

1

z.

(11)

The IM discrete model is reformulated as 1

Gp  z   BTs  zI   I  ATs  .

(12)

In Fig. 4, the back-EMF d(k) can be regarded as a feed forward term to the controller, cancelling d(k-1) on the machine side by considering the delay z 1 . Then the system closed-loop transfer function can be derived as

Fig. 4. Simplification of the conventional DPCC in the discrete domain.

F z 

G f Gc z 1G p I  Gc z 1G p

 z 2 .

(13)

As a two-sample deadbeat control, the DPCC can achieve fast current response; however, there is a critical drawback: the steady-state current error caused by the undesired IM model mismatch. When IM model mismatch occurs, f(k-1) ≠ 0 in Fig. 4. The error transfer function from f(k-1) to e(k) can be calculated as

Φe  z  

ez

f z



Gp  z 

1  Gc  z  z 1Gp  z 

.

(14)

According to the final-value theorem, the steady-state error caused by the disturbance is expressed as

e     lim z 1

zI  I Φe  z  f  z  . zI

(15)

Considering that the disturbance is a unit step-function signal f(t) = I(t), we obtain

f  z 

zI . zI  I

(16)

Substituting (14) and (16) into (15) yields

e  

2 BTs . I  ATs

(17)

Therefore, it contains an inevitable steady-state current error in the conventional DPCC, which seriously deteriorates the control performance. When this error increases to a certain value, control system break-down is imminent. IV. PROPOSED DPCC WITH SECOND-ORDER SLIDING-MODE DO In this section, the conventional DPCC is combined with a second-order sliding-mode DO to overcome the drawback of steady-state current error. First, the super-twisting algorithm (STA), a commonly used second-order SMC, is introduced. Second, a STA-based DO (STADO) is designed to estimate system disturbance. Finally, the overall approach of DPCC+

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STADO is presented, successfully overcoming the drawback of steady-state current error of the conventional DPCC. A. Principle of STA The SMC is a powerful algorithm for uncertain systems due to its invariance property; however, the chattering problem limits its application [21]. The second-order SMC is considered effective for this problem. As a well-known second-order SMC, the STA has been proposed by Levant [22]. Since then, many researchers have developed this effective approach [23]. Currently, the STA has formed an intact theoretical system, and has been widely used for system control and observation. Consider an auxiliary equation (18) x  uv where x is the state variable, u is the control input, and v is the system disturbance. Applying the simplest form of STADO to ensure the convergence of the sliding-mode surface s  x  xˆ , we obtain 12   xˆ  u  vˆ   s sign  s  (19)  ˆ v    sign s     ^ where represents the estimated value; α, λ > 0 are the observer gains, and sign() is the sign function. A detailed analysis can be accessed in [24]. Applying the forward Euler approximation in (19), we retain the discrete form of STADO  xˆ  k  1  xˆ  k   T u  k   vˆ  k    s  k  1 2 sign s  k    s  .  vˆ  k  1  vˆ  k   Ts sign  s  k    (20) According to [23] and [24], the special features of the STA can be summarized as: 1) STA retains the strong robustness of the conventional SMC. 2) STA essentially alleviates the chattering problem caused by high frequency switching control in the conventional SMC. 3) STA does not need the derivate value of the sliding-mode surface, which is appropriate for a system of relative degree 1.





d  k  1 +

-

u  k  1

B

v  k  1 +

+

Ts

x  k  1 +

+

z 1

x k 

I  ATs +

Ts

+

xˆ  k  1 +

+

z 1

xˆ  k 

I  ATs -

vˆ  k 

+

z 1

+

1 s  k 

12

+ -

sk 

sign  s  k  

1sign  s  k  

-

STADO Fig. 5. Block diagram of the proposed STADO.

Step 1 xˆ  k  =  I  ATs  xˆ  k   BTs  u  k  1  d  k  1 



 Ts vˆ  k  1  1 s  k  vˆ  k  1

12

Step 2

vˆ  k   vˆ  k  1  Ts1sign  s  k  

sign  s  k  

vˆ  k 

Step 3 sk 

s  k   x  k   xˆ  k 



fˆ  k   vˆ  k  B fˆ  k 

For the next sampling period

Fig. 6. Flowchart of the proposed STADO.

where α1, λ1 > 0 are the designed observer gains, and s(k) is defined as s  k   x  k   xˆ  k  . Figs. 5 and 6 are the block diagram and flowchart of the proposed STADO, respectively.

B. STADO First, defining a variable substitution in (6)

 x  i  v   Bf

(21)

where x and v are intermediate variables. Substituting (21) into (6), we can rewrite the IM discrete model as x  k  =  I  ATs  x  k  (22)  BTs  u  k  1  d  k  1  Ts v  k  1 . Then applying the discrete STA (20) into (22), a STADO can be designed as

 xˆ  k  =  I  ATs  xˆ  k   BTs  u  k  1  d  k  1   12   Ts vˆ  k  1  1 s  k  sign  s  k      vˆ  k   vˆ  k  1  Ts1sign  s  k   (23)



IM discrete model



1) Stability analysis of STADO: The stability of the continuous and discrete STA-based observer has been discussed in [24] and [25]. As analyzed in [24], let α1 > C > 0, λ1 > 0, and defined a function Φ(α1, λ1, C) < 1. Then assuming that the observer input signal has a derivative with Lipschitz’s constant C, the sufficient conditions for the observer finite-time convergence are

1  C   2  4C 1  C .  1 1  C 

(24)

Notice that condition (24) is a very crude estimation and derived in the continuous domain. Although the discrete STA-based observer shares similar characteristics with the continuous one with small enough Ts, the condition (24) is still only sufficient condition for the discrete-time system. To ensure the finite-time convergence of the STA-based observer

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in the discrete domain, a Lyapunov function candidate

V  s 2  k  is considered first, and thus

DPCC + STADO

iref  k 

V  s 2  k + 1  s 2  k    s  k + 1  s  k    s  k + 1  s  k   .

(25)

Gf  z

+

ek  -

Gc  z 

IM discrete model

d k  + uk 

+

Multiplying (25) by sign 2  s  k   , the necessary and

+

+

d  k  1 f  k  1 -

z 1

+

fˆ  k 

where Φe(z) is the error transfer function from f(k-1) to e(k), as shown in (14). Then we can obtain the steady-state error of the DPCC+STADO, considering both fˆ  k  and f  k  1

zI  I   zI  I  e     lim  Φe  z  f  z   Φe  z  fˆ  z   . (31) z 1  zI zI  Assuming that the estimated disturbance converges to its actual value in finite sampling periods, we retain (32) f  z   z 1 fˆ  z  via considering the control delay z 1 , as shown in Fig. 7. Then substituting (30) and (32) into (31) yields

 zI  I e      lim  Φe  z  z 1 fˆ  z  z 1  zI zI  I    z 1Φe  z  fˆ  z    0. zI 

C. DPCC with STADO The approach of DPCC+STADO is developed as shown in Fig. 7. Unlike the single DPCC approach, the proposed STADO online estimates the system disturbance caused by IM parameter variation and unmoldeled dynamics. And then the estimated disturbance is applied as feed-forward compensation, which can overcome the drawback of steady-state current error of the conventional DPCC. The command voltage uʹ(k) of DPCC+STADO is designed as (28) u  k   u  k   fˆ  k  . Substituting (8) into (28) yields

u  k  

iref  k   i   k + A  i   k +  d  k   fˆ  k  . (29) BTs B

Φe  z   

1  Gc  z  z 1G p  z 

An experimental setup was built to validate the proposed algorithm, as shown in Fig. 8. Two identical IMs are connected via flexible coupling (see Fig. 9), where one is used to test the algorithm, and the other one is used to provide the load. The IM parameters are listed in Table I. A rectifier and inverter set, which is controlled by an ARM-based control board, drives the IM. The adopted STM32F103 ARM is a 32-bit fixed-point microcontroller with 72 MHz of maximum operating frequency. Two-phase currents are measured using the sampling resistances. An incremental encoder with the resolution of 1024 P/R is applied to measure the IM mechanical speed. Rectifier

Inverter

Algorithm Test

Load Supply Encoder

+

IM

IM

3

3

Shaft

E

Load supply system SVPWM 72 MHz ARM STM32F103 SARAM

  z 1Φe  z 



V. EXPERIMENTAL RESULTS

In the following, we further analyze the steady-state current error of the proposed DPCC+STADO. According to Fig. 7, the error transfer function from fˆ  k  to e  k  can be expressed as

z 1Gp  z 

(33)

Therefore, compared to the conventional DPCC, the proposed DPCC+STADO can essentially eliminate the steady-state current error.

dc-link

Equation (27) is a discrete integrate equation. Thus, the estimated disturbance can be equivalently obtained by integrating the chattering signal, meaning that the chattering caused by the switching control is suppressed in the proposed STADO.



3*380V 50Hz

(27)

i  k 

Fig. 7. Block diagram of the DPCC+STADO.

2) Chattering Suppression Analysis: The chattering suppression technique is necessary for SMC application. For the proposed STADO, substituting the second equation of (21) into the second equation of (23), we retain

 fˆ  k   fˆ  k  1  Ts 1 sign  s  k   . B

+

Gp  z 

STA-based Disturbance Observer

sufficient conditions for the existence of the hyperplane on s(k) can be obtained as presented in [26]

   s  k +1  s  k   sign  s  k    0 (26) .  s k + 1  s k sign s k  0               From another perspective, the stability of the discrete STA can also be analyzed by using the quadratic Lyapunov functions and the linear matrix inequality technique, as shown in [25]. If the control gains are selected according to the conditions shown in [25], the trajectory of the discrete STA will be confined into a boundary layer around the sliding-mode surface, then staying inside indefinitely obeying a quasi-sliding mode behavior. Therefore, the stability of the discrete STA-based observer can be ensured by selecting the observer gains based on the inequalities (26) and the conditions presented in [25].

-

Flash

A/D Converter D/A Converter

Computer ARM-Based Control Board

(30)

Oscilloscope

Fig. 8. Block diagram of the experimental setup.

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TABLE I PARAMETERS OF THE INDUCTION MACHINE

Fig. 9. Photograph of the IMs.

The IFOC (see Fig. 1) is used as the basic control algorithm. The symmetrical SVPWM strategy is applied, and the PWM switching frequency is 6 kHz. The sampling period is set to 166.7 µs to provide sufficient execution time for the proposed algorithm. The PI controller gains of the speed loop are selected as Kps = 2.5 and Kis = 50. The STADO gains are selected as α1 = 50; λ1 =1. First, to show the superiority of the proposed algorithm in steady-state current error elimination, Figs. 10 to 13 depict the comparative experimental results of the conventional DPCC 0.5Lm

Quantity

Value

PN VN ωN np Rs Rr Lm Ls , Lr Jtot

rated power rated voltage rated speed number of pole pairs stator resistance rotor resistance mutual inductance stator, rotor inductance total inertia

3.7 kW 380 V 1500 r/min 2 1.142 Ω 0.825 Ω 118.9 mH 124.4 mH 0.0256 kg·m2

and the proposed DPCC+STADO under machine parameter variation. As the major disturbance, the mutual inductance Lm is selected as the tested parameter in this section [17], [19], and Lm ranges from 50% to 200% of Lm ( Lm represents the nominal value of the machine mutual inductance). The waveforms of isd, isq, and their references isd,ref, isq,ref, and the phase current iu are given per figure. In addition, the waveforms of the estimated disturbance fˆ and fˆ are also provided in Figs. 11 and 13. sq

sd

2 Lm

isd

0.5Lm

isd ,ref

4A

0

Symbol

2 Lm

isd

isd ,ref

4A

0

isq

isq 0

isq ,ref

4A

isq ,ref

4A

0

iu

iu 0

0

0

8A 0.1

0.2

0.3

0.4

0.5 Time(s)

0.6

0.7

0.8

0.9

1

0

8A 0.1

0.2

0.3

(a) 300 rpm, no load

0.4

0.5 Time(s)

0.6

0.7

0.8

0.9

1

(b) 300 rpm, rated load

Fig. 10. Response of the conventional DPCC under Lm mismatch at 300 rpm.

0.5Lm

2 Lm

isd

isd ,ref

4A

0

0.5Lm

2 Lm

isd

isd ,ref

4A

0

isq

isq 0

isq ,ref

isq ,ref

4A

0

fˆd

0

4A

fˆd

0

fˆq

0.2p.u.

fˆq

0.2p.u.

iu

iu 0

0

0

8A 0.1

0.2

0.3

0.4

0.5 Time(s)

0.6

(a) 300 rpm, no load

0.7

0.8

0.9

1

0

8A 0.1

0.2

0.3

0.4

0.5 Time(s)

0.6

0.7

0.8

0.9

1

(b) 300 rpm, rated load

Fig. 11. Response of the DPCC+STADO under Lm mismatch at 300 rpm.

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2 Lm

0.5Lm

isd ,ref

isd

4A

0

2 Lm

0.5Lm

isd

4A

0

isd ,ref isq

isq 0

isq ,ref

4A

isq ,ref

4A

0

iu

iu 0

0

0

8A 0.1

0.2

0.3

0.4

0.5 Time(s)

0.6

0.7

0.8

0.9

1

0

8A 0.1

0.2

(a) 1500 rpm, no load

0.3

0.4

0.5 Time(s)

0.6

0.7

0.8

0.9

1

(b) 1500 rpm, rated load

Fig. 12. Response of the conventional DPCC under Lm mismatch at 1500 rpm.

0.5Lm

2 Lm

isd

isd ,ref

4A

0

0.5Lm

2 Lm

isd

isd ,ref

4A

0

isq

isq 0

isq ,ref

isq ,ref

4A

4A

0

fˆd

fˆd

0

0

fˆq

0.4p.u.

fˆq

0.4p.u.

iu

iu 0

0

0

8A 0.1

0.2

0.3

0.4

0.5 Time(s)

0.6

0.7

0.8

0.9

1

(a) 1500 rpm, no load

0

8A 0.1

0.2

0.3

0.4

0.5 Time(s)

0.6

0.7

0.8

0.9

1

(b) 1500 rpm, rated load

Fig. 13. Response of the DPCC+STADO under Lm mismatch at 1500 rpm.

Figs. 10 and 11 are the responses of the two controllers under Lm mismatch at 300 rpm, respectively. Fig. 10 reveals the existence of significant steady-state current error (especially in isq), using the conventional DPCC under both no-load and rated-load conditions. The steady-state current error increases with the increment of Lm mismatch. E.g., in Fig. 10 (a), there’s almost no isq steady-state error with accurate Lm, while this error increases to 1.0 A with 50% Lm and 1.4 A with 200% Lm . By contrast in Fig. 11, isd and isq accurately track their references using the DPCC+STADO, indicating that the estimated fˆsd and fˆ can adjust the controller to remove the steady-state current sq

error caused by Lm mismatch. Figs. 12 and 13 present the responses of the two controllers under Lm mismatch at 1500 rpm, respectively. In Fig. 12, the problem of steady-state current error becomes more severe compared to Fig. 10. When Lm increases to 200% Lm , isd and isq suffer current oscillations under both no-load and rated-load

conditions. In Fig. 13, when the DPCC+STADO is applied, isd and isq can accurately track their references during the entire operation, even under 200% Lm . Second, to further verify the dynamic performance of the DPCC+STADO under machine parameter variation, Fig. 14 shows the response against step changes of isd and isq with Lm  50%Lm . The amplitudes of isd,ref and isq,ref abruptly change from 2 A to 6 A at 0.3 s and from 0 A to 6 A at 0.7 s, respectively. As expected, isd and isq can quickly converge to their references without overshoot. The system disturbance caused by Lm mismatch is estimated by STADO, and then the estimated disturbance fˆsd and fˆsq is used for control compensation, eliminating the steady-state current error. Moreover, to show the overall performance of the DPCC+STADO, Figs. 15 and 16 depict the experimental results of speed increase and speed reversal with Lm  50%Lm , respectively. The waveforms of the rotor speed ωr, the rotor

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2A

position θ, the estimated disturbance fˆsd and fˆsq , and the

isd

0

2A

isd ,ref

isd

0.145

0.155

0

2A

isq ,ref

isq

phase current iu are separately given in each figure. Fig. 15 is the response during speed increase from 0 to the rated value under no-load. Fig. 16 shows the response during speed reversal at rated speed under rated-load. It reveals that the overall transition process is definitely stable and smooth. Thus, the proposed DPCC+ STADO can achieve satisfactory performance during speed increase and speed reversal.

isd ,ref

2A

0

isq ,ref

isq

0.345

0.355

0

0.2p.u.

fˆq

0

VI. CONCLUSIONS

fˆd 8A

iu

0

0

0.05

0.1

0.15

0.2

0.25 0.3 Time(s)

0.35

0.4

0.45

0.5

Fig. 14. Response of the DPCC+STADO against step changes of isd and isq under Lm  50%Lm . 1000rpm

r

0 0.5p.u. 0

REFERENCES

 0.4p.u.

[1]

fˆq

0

fˆd 8A

iu

0

0

0.1

0.2

0.3

0.4

0.5 Time(s)

0.6

0.7

0.8

0.9

1

Fig. 15. Response of the DPCC+STADO during speed increase from 0 to rated speed under Lm  50%Lm , no load. 1000rpm

r

0



0.5p.u.

0 0.4p.u.

fˆq

0

fˆd 8A

iu

0

0

1

2

3

4

5 Time(s)

6

The presented DPCC+STADO can effectively overcome the intrinsic drawback of the steady-state current error of the conventional DPCC. The designed STADO online estimates the system disturbance that is caused by parameter variation and other unmodeled dynamics, and then this estimated disturbance is used for control compensation. The appropriate selection of observer gains ensures the stability and finite-time convergence of the designed STADO. By applying the STADO in parallel with the DPCC algorithm, the inherent advantage of fast current response of the conventional DPCC is retained. As verified by experimental tests, the proposed scheme can achieve static-errorless current control under Lm mismatch ranging from 50% to 200% of its nominal value.

7

8

9

10

Fig. 16. Response of the DPCC+STADO during speed reversal at rated speed under Lm  50%Lm , rated load.

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[13] J. C. Moreno, J. M. Espi Huerta, R. G. Gil, and S. A. Gonzalez, “A robust predictive current control for three-phase grid-connected inverters,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1993–2004, Jun. 2009. [14] Y.-R. Mohamed and E. El-Saadany, “An improved deadbeat current control scheme with a novel adaptive self-tuning load model for a three-phase PWM voltage-source inverter,” IEEE Trans. Ind. Electron., vol. 54, no. 2, pp. 747–759, Apr. 2007. [15] Q. Zeng and L. Chang, “An advanced SVPWM-based predictive current controller for three-phase inverters in distributed generation systems,” IEEE Trans. Ind. Electron., vol. 55, no. 3, pp. 1235–1246, Mar. 2008. [16] J. R. Fischer, S. A. Gonzalez, M. A. Herran, M. G. Judewicz, and D. O. Carrica, “Calculation-delay tolerant predictive current controller for three-phase inverters,” IEEE Trans. Ind. Inf., vol. 10, no. 1, pp. 233–242, Feb. 2014. [17] J. M. Espi, J. Castello, R. García-Gil, G. Garcera, and E. Figueres, “An adaptive robust predictive current control for three-phase grid-connected inverters,” IEEE Trans. Ind. Electron., vol. 58, no. 8, pp. 3537–3546, Aug. 2011. [18] K. J. Lee, B. G. Park, R. Y. Kim, and D. S. Hyun, “Robust predictive current controller based on a disturbance estimator in a three-phase gridconnected inverter,” IEEE Trans. Power Electron., vol. 27, no. 1, pp. 276– 283, Jan. 2012. [19] C. Xia, M. Wang, Z. Song, and T. Liu, “Robust model predictive current control of three-phase voltage source PWM rectifier with online disturbance observation,” IEEE Trans. Ind. Informat., vol. 8, no. 3, pp. 459–471, Jul. 2012. [20] V. Blasko, V. Kaura, and W. Niewiadomski, “Sampling of discontinuous voltage and current signals in electrical drives: A system approach,” IEEE Trans. Ind. Appl., vol. 34, no. 5, pp. 1123–1130, Sep./Oct. 1998. [21] K. D. Young, V. I. Utkin, and U. Ozguner, “A control engineer’s guide to sliding mode control,” IEEE Trans. Control Syst. Technol., vol. 7, pp. 328–342, Mar. 1999. [22] A. Levant, “Sliding order and sliding accuracy in sliding mode control,” Int. J. Control, vol. 58, pp. 1247–1263, 1993. [23] I. Boiko, L. Fridman, A. Pisano, and E. Usai, “Analysis of chattering in systems with second-order sliding modes,” IEEE Trans. Automat. Contr., vol. 52, no. 11, pp. 2085–2102, 2012. [24] A. Levant, “Robust exact differentiation via sliding mode technique,” Automatica, vol. 34, no. 3, pp. 379–384, Mar. 1998. [25] I. Salgado, I. Chairez, and B. Bandyopadhyay, “Discrete-time non-linear state observer based on a super twisting-like algorithm,” IET Control Theory Appl., vol. 8, no. 10, pp. 803–812, Jul. 2014. [26] S. Z. Sarpturk, Y. Istefanopulos, and O. Kaynak, “On the stability of discrete-time sliding mode control systems,” IEEE Trans. Autom. Control, vol. AC-32, no. 10, pp. 930–932, Oct. 1987.

Bo Wang (S’16) was born in Shandong Province, China, in 1987. He received the B.S. degree in electrical engineering from Northwestern Polytechnical University, Xi’an, China, in 2011, and the M.S. degree in electrical engineering from Harbin Institute of Technology, Harbin, China, in 2013, where he is currently working toward the Ph.D. degree in electrical engineering. His research interests include induction machine drives and nonlinear control theories and applications.

Zhen Dong was born in Jiangsu Province, China, in 1994. He received the M.S. degree in electrical engineering from Harbin Institute of Technology, Harbin, China, in 2016, where he is currently working toward the M.S. degree in electrical engineering. His research interests include induction machine drives and robust control theories.

Yong Yu was born in Jilin Province, China, in 1974. He received the B.S. degree in electromagnetic measurement and instrumentation from the Harbin Institute of Technology (HIT), where he also received the M.S. and Ph.D. degrees in electrical engineering in 1997 and 2003, respectively. From 2004 to 2014, he was an Associate Professor in the Department of Electrical Engineering, HIT, where he has been a Professor of Electrical Engineering since 2014. His current research interests include electrical motor drives, power quality mitigation, and fault diagnosis and tolerant control of inverter.

Gaolin Wang (M’13) received the B.S., M.S. and Ph.D. degrees in Electrical Engineering from Harbin Institute of Technology, Harbin, China, in 2002, 2004 and 2008 respectively. In 2009, he joined the Department of Electrical Engineering, Harbin Institute of Technology as a Lecturer, where he has been a Professor of Electrical Engineering since 2014. From 2009 to 2012, he was a Postdoctoral Fellow in Shanghai STEP Electric Corporation. He has authored more than 30 technical papers published in journals and conference proceedings. He is the holder of seven Chinese patents. His current major research interests include permanent magnet synchronous motor drives, high performance direct-drive for traction system, position sensorless control of AC motors and efficiency optimization control of Interior PMSM.

Dianguo Xu (M’97, SM’12, F’16) received the B.S. degree in Control Engineering from Harbin Engineering University, Harbin, China, in 1982, and the M.S. and Ph.D. degrees in Electrical Engineering from Harbin Institute of Technology (HIT), Harbin, China, in 1984 and 1989 respectively. In 1984, he joined the Department of Electrical Engineering, HIT as an Assistant Professor. Since 1994, he has been a Professor in the Department of Electrical Engineering, HIT. He was the Dean of School of Electrical Engineering and Automation, HIT from 2000 to 2010. He is now the Vice President of HIT. His research interests include renewable energy generation technology, power quality mitigation, sensorless vector controlled motor drives, high performance PMSM servo system. He published over 600 technical papers. Dr. Xu is a senior member of IEEE, an Associate Editor of the IEEE Transactions on Industrial Electronics and IEEE Journal of Emerging and Selected Topics in Power Electronics. He serves as Chairman of IEEE Harbin Section.

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