Speed Sensor Less Control Of Im Using Sliding Mode Observer With Variable Boundary Layer

SICE Annual Conference 2008 August 20-22, 2008, The University Electro-Communications, Japan

Speed Sensorless Control of Induction Motor Using Sliding Mode Observer with Variable Boundary Layer Min Yeong Jang, Bong Su Jang, Jun Ik Jeong, Yong Hun Park, and Young Ahn Kwon School of Electrical Engineering, Pusan National University, Busan, Korea (Tel : +82-51-510-2372; E-mail: [email protected]) Abstract: The vector control in the speed and torque controlled ac drive is typically implemented through measuring the rotor speed or position. However, speed and position sensors require the additional mounting space, reduce the reliability in harsh environments and increase the cost of a motor. Therefore, many studies have been performed for the elimination of speed and position sensors. This paper investigates an improved sliding mode observer for the speed sensorless control of an induction motor. The proposed control strategy is the sliding mode observer with a variable boundary layer for a low-chattering and fast-response control. zmulation and experimentation have been performed to verify the proposed control algorithm. Keywords: induction motor, sensorless control, sliding mode observer

1. INTRODUCTION The vector control in the speed and torque controlled ac drive is widely used for a high performance application. The vector control of an induction motor is typically implemented through measuring the rotor speed or position. However, speed and position sensors require the additional mounting space, reduce the reliability in harsh environments and increase the cost of a motor. Various control algorithms for the elimination of speed and position sensors have been proposed[1-5]: algorithms using state equations, model reference adaptive systems (MRASs), Luenberger or Kalmanfilter observers, saliency effects, sliding mode controls, artificial intelligences, direct controls of torque and flux, and the current error correction. Most sensorless algorithms are based on the flux and speed estimations which are obtained from the voltage equations, and so they are sensitive to the electrical and mechanical parameters. This paper investigates an improved sliding mode observer for the speed sensorless control of an induction motor. The sliding mode control is typically robust to the plant parameter variation and system disturbance[6,7]. However, a sliding mode control has a chattering problem due to the control discontinuity and switching action. The proposed sliding mode control in this paper is using the sliding mode observer with a variable boundary layer for a low-chattering and fast-response control. The proposed algorithm is verified through the simulation and experimentation.

ωe qs

ωr

θe θr

as

cs

Fig. 1 The real, stationary and synchronously rotating reference axes ª Ȝ abcs º ª L s « »=« T ¬Ȝ abcr ¼ ¬(L sr )

f bs

f cs ] ,

( f abcr )T = [ f ar

f br

f cr ] ,

R s = diag[ R s

L ms

L ls + L ms

L ms

− L ms

+ L ms

− L ms

L ms

Llr + L ms

L ms

− L ms

‫ڍ ڌ‬

º − L ms » » − L ms » , » » L ls + L ms » ¼ º − L ms » » − L ms » , » » L lr + L ms » ¼ ‫ڍ ڌڍ ڌ‬

− L ms

‫ڍ ڌڍ ڌ‬

Rr ] ,

‫ڍ ڌ ڍ ڌ‬

- 748 -

Rr

(2)

Rs ] ,

+ L ms

‫ڍ ڌڍ ڌ‬

ª «L ls « Ls = « − « « «¬ − ª «L lr « Lr = « − « « «¬ −

Rs

‫ڍ ڌ‬

R r = diag[ Rr

(1)

L sr º ªi abcs º »« » L r ¼ ¬i abcr ¼

where ( f abcs )T = [ f as

‫ڍ ڌڍ ڌ‬

º ªi abcs º »« » R r + pL r ¼ ¬i abcr ¼

ds

αs

Fig. 1 shows the real, stationary and synchronously rotating axes of a 3-phase symmetrical induction motor. The voltage and flux equations in the real axes may be expressed as pL sr

θ sl

βs

ar

2. MATHEMATICAL MODELING OF INDUCTION MOTOR

ª v abcs º ªR s + pL s « »=« T ¬ v abcr ¼ ¬ p(L sr )

bs

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L sr

2ʌ º 2ʌ ª cos (ș r + ) cos (ș r − ) » « cos ș r 3 3 « 2ʌ » 2ʌ = Lms «cos (ș r − ) cos ș r cos (ș r + )» 3 » 3 « 2ʌ 2ʌ » « cos ș r » «cos (ș r + 3 ) cos (ș r − 3 ) ¼ ¬

(3)

v βs = Rs i βs + pλβs

(4)

0 = Rr iαr + pλαr + ωr λβr

(5)

0 = Rr i βr + pλβr − ωr λαr

(6)

λαs = Lls iαs + Lm (iαs + iαr ) = Ls iαs + Lm iαr

(7)

λβs = Lls i βs + Lm (iβs + iβr ) = Ls i βs + Lm iβr

(8)

λαr = Llr iαr + Lm (iαs + iαr ) = Lr iαr + Lm iαs

(9)

λβr = Llr i βr + Lm (i βs + i βr ) = Lr i βr + Lm iβs

(10)

where L m =

and TL is the load torque.

This paper proposes a novel sensorless control algorithm based on the sliding mode observer. In general the sliding mode observer for a motor control is implemented through the error between the measured and estimated currents[8-10]. In an induction motor, the error between the measured and estimated currents is used to construct sliding mode surfaces so that after sliding mode happens, the estimated fluxes are driven to converge to real ones exponentially. From (3) - (10), the state equations of the sliding mode observer in the stationary reference frame may be expressed as

(12)

0 = Rr idr + pλdr − (ωe − ωr )λqr

(13)

0 = Rr iqr + pλqr + (ωe − ωr )λdr

(14)

λds = Lls ids + Lm (ids + idr ) = Ls ids + Lm idr

(15)

λqs = Lls iqs + Lm (iqs + iqr ) = Ls iqs + Lm iqr

(17)

λdr = Llr idr + Lm (ids + idr ) = Lr idr + Lm ids

(18)

λqr = Llr iqr + Lm (iqs + iqr ) = Lr iqr + Lm iqs

(19)

(23) (24)

ªλ º ªi º ªi º ªvα s º Įr » Įs x=« s », i = « », Ȝ = « , v =« », r «λ » s « vβ s » «¬Ȝ r »¼ s «¬i ȕs »¼ ¬ ¼ ¬ ȕr ¼ ªA A º A = « 11 12 » , A = 11 «¬A 21 A 22 »¼

L ª 1/ Tr A = m « 12 σ L L ¬ − ωr s r

The electromagnetic torque in the synchronously rotating speed reference frame may be expressed as 3 P Lm (λdr iqs − λqr ids ) 2 2 Lr

ˆi = C xˆ s

where ^ means the estimated value, K is the switching gain, u is switching function, and

A

Te =

(22)

‫ڌ‬

v qs = Rs iqs + pλds + ωe λds

d xˆ = A xˆ + B v + K u ( i − ˆi ) s s s dt

Ȝˆ = D xˆ r

From (1) - (2), d - and q - axis voltage equations in the reference frame with the synchronously rotating speed of ω e may be expressed as (11)

(21)

where J is the inertia coefficient and D is the friction coefficient, ω m is the mechanical speed of the rotor,

3 dθ r L ms , ω r = . 2 dt

v ds = R s i ds + pλ ds − ω e λ qs

d ωm + Dωm + TL dt

3. IMPROVED SLIDING MODE OBSERVER FOR SENSORLESS INDUCTION MOTOR

From (1) - (2), α - and β - axis voltage and flux equations in the stationary reference frame fixed to the stator may be expressed as vαs = R s i αs + pλαs

T=J

§ R −σ · ª1 0º ¸ « », − ¨¨ s + ¸ ©σ Ls σ Tr ¹ ¬0 1¼

ωr º

L ª1 0 º , A 21 = m « » , » 1/ Tr ¼ Tr ¬0 1¼

ª− 1 / Tr − ω r º L2m L =« = − , σ 1 , Tr = r , » 22 L s Lr Rr ¬ ω r − 1 / Tr ¼

ªB º ª0 0º 1 ª 1 0º B = « 1» , B = , B =« « » », 1 2 σL ¬ 0 1 ¼ ¬0 0¼ «¬B2 »¼ s

(20)

where P is the number of poles.

ª1 0 0 0º ª0 0 1 0º C=« , D=« » ». ¬0 1 0 0¼ ¬0 0 0 1¼

The mechanical equation of a motor may be expressed as

The proposed sliding mode control in this paper is using the sliding mode observer with a variable

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boundary layer for a low-chattering and fast-response control. Fig. 2 shows the variable boundary layer of the proposed sliding mode observer.

Fig. 4 (a), (b) and (c) show the speed responses in the speed commands of 20rpm, 50rpm and 800rpm and in the no load.

φ

φ

30

50rpm

200rpm

Speed [rpm]

u

u 200rpm

Variable Boundary layer

20

10

0

u 50rpm

30

Speed [rpm]

s

Real Speed 0

1

2

3

0

1

2

3

4

5

20

10

0

Estimated Speed 4

5

Time [sec]

Fig. 2 Control input with a variable boundary layer The rotor speed is estimated from the currents and fluxes obtained in (22) as follows: (25)

iqs*

PI

νqs θe

iqs

νβs

Space Vector PWM

Proposed SMO

3

0

1

2

3

4

5

Estimated Speed 4

5

G

(b)

Vdc

iαs

800

Vector Rotation

ids ωr

Vector Rotation

2

Time [sec]

Speed [rpm]

PI

PI

1

25

0

iβs

400

0

IM



Real Speed 0

1

2

3

0

1

2

3

4

5

800

Speed [rpm]

ids* ω r*

να s

Real Speed 0

50

The overall system of the proposed sensorless control algorithm is shown in Fig. 3. νds

25

0

Speed [rpm]

+ K i ³ (k 1 sgn(iˆαs − iαs )λˆ βr − k 2 sgn(iˆβs − i βs )λˆαr ) dt

50

Speed [rpm]

ω r = K p (k 1 sgn(iˆαs − iαs )λˆβr − k 2 sgn(iˆβs − i βs )λˆαr )

G

(a)

Fig. 3 Configuration of the overall system

400

0

4. SIMULATION

Estimated Speed 4

5

Time [sec]

(c)

The simulation has been performed to verify the proposed control algorithm applied to a sensorless induction motor. Table 1 shows the specification of the induction motor used in the simulation and experimentation.

Fig. 4 Speed responses in the speed command of (a) 20 rpm (b) 50 rpm (c) 800 rpm As shown in Fig. 4, the proposed sensorless control algorithm has good speed responses in the low and high speeds. Fig. 5(a) and Fig. 5(b) are the simulation results obtained for the comparison with the sliding mode control algorithm without variable boundary layer in case of considering the parameter variation. Fig. 5(a) and Fig. 5(b) show the speed responses in case the rotor winding resistance is increased by 30% of the nominal value and the load torque of 6Nm is applied in the middle of the operation of 200rpm. As shown in the

Table 1 Motor specification Rated Power

3 hp

Rr

1.5 Ω

Rated Voltage

220 V

Ls

245 mH

Pole Numbers

4

Lr

247 mH

Rs

2.7 Ω

Lm

236 mH

G

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figures the proposed sensorless control algorithm has an improved and robust performance. Speed [rpm]

50

100 300

Speed [rpm]

25

200

0

4

6

8

10

12

14

1

2

3

0

1

2

3

0

6

8

10

12

5

Estimated Speed 4

5

Time [sec]

Estimated Speed 4

4

25

200

100

Real Speed 0

50

Real Speed

Speed [rpm]

Speed [rpm]

300

G

(b)G

14

Time [sec]

G

(a)

Speed [rpm]

800

100 300

Speed [rpm]

400

200

0

Real Speed 4

6

8

10

12

14

0

3

0

1

2

3

6

8

10

12

G

5. EXPERIMENTS AND DISCUSSIONS The experimentation has been performed to verify the proposed algorithm applied to a sensorless induction motor. The Intel-Pentium microprocessor system is used for the digital processing of the proposed algorithm. Fig. 6 (a), (b) and (c) show the experimental speed responses in the speed commands of 20rpm, 50rpm and 800rpm and in the no load.

5

G

300

Speed [rpm]

30

20

10

Real Speed 1

2

3

4

300

5

Speed [rpm]

0

10

Estimated Speed 0

1

2

3

Time [sec]

(a)

4

200

100

20

0

4

Fig. 6 Experimental speed responses in the speed command of (a)20 rpm (b) 50 rpm (c) 800 rpm G The proposed sensorless control algorithm has good speed responses in the low and high speeds same as the simulation result. Fig. 7 is the experimental result obtained for the comparison with the sliding mode control without variable boundary layer in case that the rotor resistance is increased by 30% of the nominal value, and the load torque 6Nm is applied in the middle of the operation of the speed command 200rpm. As shown in the experimental results, the proposed sensorless control algorithm has an improved and robust performance.G

by 30% with the load variation (200rpm, 0ൺ6Nm) (a) without variable boundary layer (b) with variable boundary layer

30

5

(c)G

14

Fig. 5 Speed response in the rotor resistance increased

0

4

Estimated Speed

Estimated Speed 4

(b)G

Speed [rpm]

2

Time [sec]

Time [sec]

Speed [rpm]

1

400

200

100

Real Speed 0

800

Speed [rpm]

Speed [rpm]

300

6

8

10

4

6

8

10

12

Estimated Speed 12

Time [sec]

G

14

200

100 5

Real Speed 4

(a)

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14

G

the proposed algorithm shows a better performance in the parameter variation compared to the conventional algorithm.

Speed [rpm]

300

200

100

Speed [rpm]

300

REFERENCES Real Speed 4

6

8

10

4

6

8

10

12

[1] Edited by K. Rajashekara, A. Kawamura, and K. Matsuse, Sensorless Control of AC Motor Drives, IEEE Press, 1996. [2] J. Holtz, " Sensorless control of induction motor drives," Proc. IEEE, vol.90, pp.1359–1394, Aug. 2002. [3] P. Vas, Sensorless Vector and Direct Torque Control, Oxford Univ. Press, 1998. [4] Y. A. Kwon and S. H. Kim, “New scheme for speed-sensorless control of induction motor‚” IEEE Trans. Ind. Electr., vol.51, pp.545-550, June 2004. [5] Z. Yan and V. Utkin, “Sliding mode observers for electric machines- an overview,” IEEE Proc IECON, pp.1842-1847, 2002. [6] J. J. Slotine, "Sliding Controller Design for Nonlinear Systems," Int. J. Contr., Vol. 40, No. 2, pp.421-434, 1984. [7] V. Utkin, J. Guldner and J. Shi, Sliding Mode Control in Electromechanical Systems, Taylor and Francis, 1999. [8] Z. M. A. Peixoto, P. F. Seixas, B. R. Menezes, and P. C. Cortizo, "Speed control of permanent magnet motors using sliding mode observers for induced EMF position and speed estimation," IEEE Proc IECON, pp.1023-1028, 1995. [9] F. Parasiliti, R. Petrella, and M. Tursini, “Adaptive sliding mode observer for speed sensorless control of induction motors,” IEEE IAS Annual Meeting, pp. 2277-2283, 1999. [10] J. Li, L. Xu, and Z. Zhang, “An adaptive sliding mode observer for induction motor sensorless speed control,” IEEE IAS Annual Meeting, pp.1329-1334, 2004.

14

200

100

Estimated Speed 12

14

Time [sec]

(b)G

G

Fig. 7 Experimental speed response in the rotor resistance increased by 30% with the load variation (200rpm, 0ൺ6Nm) (a) without variable boundary layer (b) with variable boundary layer

6. CONCLUSIONS This paper proposed a novel speed sensorless control algorithm of an induction motor based on the sliding mode observer. The sliding mode observer is implemented through the error between the measured and estimated currents. The error between the measured and estimated currents is used to construct sliding mode surfaces so that after sliding mode happens, the estimated fluxes are driven to converge to real ones exponentially. The proposed sliding mode control in this paper is using the sliding mode observer with a variable boundary layer for a low-chattering and fast-response control. The simulation and experimental results indicate that the proposed algorithm shows good speed responses in the low and high speeds, and shows robust speed responses in the rotor resistance variation. Especially,

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