Vector Control-based Speed Sensor Less Control Of Im Using Sliding-mode Controller

Proceedings of the 6th World Congress on Intelligent Control and Automation, June 21 - 23, 2006, Dalian, China

Vector Control-Based Speed Sensorless Control of Induction Motors Using Sliding-Mode Controller* Ping Liu

Lanying Hao

Shandong Electric Power College Erhuan Nan Road Jinan, Shandong, 250002, China

School of Control Science and Engineering Shandong University Jinan, Shandong 250061, China [email protected]

[email protected] Abstract – In order to eliminate the influence of parameter uncertainties on field-oriented control for induction motor drive, a sliding-mode speed controller with integral sliding surface is proposed. In this proposed control scheme the sliding-mode control strategy is worked in conjunction with the field-oriented control method to improve the controller performance. The estimated rotor speed used in speed feedback loop is calculated by an adaptive algorithm based on measured terminal quantities. Stability analysis based on Lyapunov theory is presented to demonstrate that the rotor speed is exponential convergent. Simulation results are also presented to confirm the characteristics of the proposed overall control scheme. Index Terms - induction motor, sliding-mode control, fieldoriented, speed sensorless control

I. INTRODUCTION Up to now, indirect field-oriented technique [1] has been widely used for the control of induction motor servo drive in high-performance applications. The technique guarantees the decoupling of torque and flux control commands of the induction motor, so makes the induction motor be controlled linearly as a separated excited dc motor. But the decoupled control performance is still influenced by the uncertainties, due to the unpredictable parameter variations, external load disturbances, and unmodeled and nonlinear dynamics. To overcome these drawbacks, optimal control, sliding-mode control, adaptive control and intelligent control are proposed [2-4]. In the past decade, sliding-mode control strategy have attracted many researchers and made a great development for the speed control of the induction motor drives. It bases on advantages of sliding-mode control [5]: good performance against unmodeled dynamics, robustness to parameter variations and external disturbances, and fast dynamic response. In indirect field-oriented control of induction motors, knowledge of rotor speed is required to establish speed loop feedback. But generally, rotor speed is obtained by motor speed detecting device. Conventional motor speed detecting device usually adopts tachogenerator or digital shaft-position encoders. These sensors increase cost of the control system, exist difficulties in mounting and maintaining these speed sensors, and make the system easy be disturbed, lower the system reliability, destroy rigidity and flexibility of induction *

motor. Because of these problems, it is an important requirement to eliminate the speed sensor from the control system. Therefore, speed sensorless control of induction motor has been focused on many studies and researches in the control of induction motor drives. The core of problem is to estimate rotor speed in order to obtain and apply it to the speed feedback subsystem. From the beginning of the 1980s, there were series research works throughout the world to control induction machines without the need for speed sensors. Different methods are used for flux and speed estimation. In these schemes the speed is usually obtained by the measurement of stator voltages and rotor currents [6-8]. However, the estimation is usually heavily dependent on machine parameters. This paper proposes a new robustness control strategy. In the proposed control algorithm the rotor speed used in speed feedback loop is obtained by speed observer system based on the measurement of terminal quantities, and meanwhile, to eliminate the influence of parameter uncertainties, a robustness sliding-mode speed controller is used that will work in conjunction with the field oriented technique. Then, stability analysis using Lyapunov stability theory is given to demonstrate that the rotor speed is exponential convergent. And simulation results verify validity of the proposed overall control scheme. II. EQUATIONS OF INDUCTION MOTOR IN THE ROTOR FLUX ORIENTATION Voltage equations of a squirrel-cage induction motor in the rotor flux orientation: dψ ds (1a) − ω eψ qs , u ds = R s i ds + dt dψ qs (1b) u qs = Rs iqs + + ωeψ ds , dt dψ (1c) 0 = Rr idr + dr − (ωe − ωr )ψ qr , 0 = Rr idr +

dt dψ qr dt

+ (ωe − ωr )ψ dr ,

and the following flux equations: ψ ds = Ls ids + Lm idr ,

(2a)

ψ qs = Ls iqs + Lm iqr ,

(2b)

This work is partially supported by the department of Science & Technology of Shandong Province (Grant No.03BS089).

1-4244-0332-4/06/$20.00 ©2006 IEEE

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(1d)

ψ dr = Lr idr + Lm ids , ψ qr = Lr iqr + Lm iqs ,

(2c) (2d)

where u ds , u qs are the applied voltages to phases d and q of the stator, respectively; ids , iqs , are the corresponding stator currents. idr , iqr are d- and q-axis rotor currents. The rotor and stator flux in the direct axis are given by ψ dr and ψ ds , whereas in the quadrature axis they are defined by ψ qr and

ψ qs . The rotor speed is given by ω

r

and the angular speed of

the rotor flux linkage vector by ω e . Rs , Rr are the stator and rotor resistances; Ls , Lr are the stator and rotor selfinductances; Lm is the stator-rotor mutual inductance. On the assumption that the effects of magnetic saturation, core loss and skin effect are neglected. The electrical model is augmented by the mechanical subsystem given as: 3 L Te = P m (ψ dr iqs −ψ qr ids ) , (3) 4 Lr J dω r B + ω r + TL = Te  (4) P dt P Where J and B denote the motor-load moment of inertia and the viscous friction coefficient; P is the number of pole pairs and TL is the load torque. If the vector control is fulfilled, then: ψ dr = ψ r , (5) dψ dr =0, dt

(6)

ψ qr = 0 ,

(7)

From (1c) and (1d), we obtain:

ψ dr = Lm ids ,

(8)

Then the electromagnetic torque is controlled only by q-axis stator current and becomes: 3 L Te = P m ψ dr i qs . (9) Lr 4 III. SPEED OBSERVER DESIGN From (1a) and (1b), the estimated d-q components of stator flux linkages are as follows: dψˆ ds (10a) = u ds − R s ids + ωˆ eψˆ qs , dt dψˆ qs (10b) = u qs − R s i qs − ωˆ eψˆ ds , dt where ∧ denotes estimate. From (2a)-(2d), the estimated d-q components of rotor flux linkages are: L L L − L2m ψˆ dr = r ψˆ ds − s r i ds , (11a) Lm Lm

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Lr L L − L2m ψˆ qs − s r i qs , (11b) Lm Lm Therefore, the flux observer based on the voltage model is given by [9]: dψˆ ds * (12a) = u ds − R s ids + ωˆ eψˆ qs + K I 1 (ψ dr − ψˆ dr ) , dt dψˆ qs * (12b) = u qs − Rs iqs − ωˆ eψˆ ds + K I 2 (ψ qr − ψˆ qr ) , dt L L L − L2m ψˆ dr = r ψˆ ds − s r i ds , (12c) Lm Lm

ψˆ qr =

Lr L L − L2m ψˆ qs − s r i qs . (12d) Lm Lm Where, KI1 and KI2 are correction parts, which need to be tuned in the simulation in order to obtain the desired result. The desired values of rotor flux under the rotor flux linkages oriented in the d-axis are given by: * * ψ dr = Lm ids , (13a)

ψˆ qr =

* ψ qr =0,

(13b)

Estimated synchronously rotating speed of ωˆ e is achieved using PI controller with zero command signal: ⎛ 1 ⎞ * ⎟ ψˆ qr − ψ qr ωˆ e = K ω ⎜⎜1 + , (14) ⎟ T s ω ⎠ ⎝ Where, Kω and Tω are tuning parameters of PI controller. R i qs Then the estimated rotor speed is: ωˆ r = ωˆ e − r . Lr i ds

(

)

IV. SLIDING-MODE CONTROL Under the complete field-oriented control, the mechanical equation (4) can be equivalently described as: dωˆ r + aωˆ r + f = bi qs* , (15) dt where K PTL B , (16) a= , b= T , f = J J J and K T is defined as: 3P 2 Lm * ψ dr , 4 Lr Furthermore, consider (15) with uncertainties: ωˆ r = −(a + Δa)ωˆ r − ( f + Δf ) + (b + Δb)i qs* , KT =

(17)

(18)

where Δa , Δb and Δf denote parameter uncertainties and are defined as: Δa䰉Δ ( B J ) , Δb = Δ ( K T J ) , Δf = Δ ( P TL J ) , Define the tracking speed error as: e(t ) = ω r* (t ) − ωˆ r (t ) , (19) Taking the derivative of (19) with respect to time yields: e(t ) = ω r* (t ) − ωˆ r (t ) = −ae(t ) + u (t ) + d (t ) , (20) where

* u (t ) = −bi qs + aω r* + f + ω r* ,

(21)

and * d (t ) = Δaωˆ r + Δf − Δbi qs , (22) According to (20), the sliding-surface is defined as [10]:

∫

t

S (t ) = e(t ) − (k − a)e(τ )dτ = 0 ,

(23)

0

where (k − a ) is strictly negative. When the sliding mode occurs, S (t ) = S (t ) = 0 , the dynamical behavior of the controlled system can be equivalently expressed as: e(t ) = (k − a)e(t ) , (24) From (24) it is evident that after reaching the sliding-mode surface s , the speed error e(t ) will converge to zero immediately. In other words, estimated speed ωˆ r can track the desired speed command asymptotically. Assume that: d (t ) < β , (25) Then, based on the uncertain system (18), some researches designed the sliding-mode speed controller as [5, 10]: u (t ) = ke(t ) − β sgn(s ) , (26) However, as the robustness of the SMC is not guaranteed during the hitting phase and the system response relies on the hitting time, it is expected that the hitting time should be minimized. To solve this problem, in this paper the slidingmode speed controller is designed as [11, 12]: u (t ) = ke(t ) − β sgn(s ) − rs , (27) Where k is the gain defined as above, β is the switching gain, r is the positive constant gain. From (21) and (27), we obtain the torque current command: 1 * i qs (t ) = [− ke(t ) + β sgn( s ) + rs + aω r* + f + ω r* ] . (28) b Above designed sliding-mode speed controller has a short hitting time, that is to say the method resolves the speed tracking problem for the induction motor with some uncertainties in mechanical parameters and load torque. And at the same time, a continuous term rs is used that can either shorten hitting time or reduce the chattering. Fig. 1 is the overall induction motor control scheme. V. ANALYSIS OF STABILITY By choosing a Lyapunov function candidate: V (x, t ) = 0.5s(x, t )2 , (29) the first time derivative of (29) is: V (x, t ) = s (x, t )s(x, t ) = s(x, t )[e(t ) − (k − a )e(t )] , (30) Substituting (20) into (27), then V (x, t ) = s (x, t )[− β sgn (s(x, t )) − rs (x, t ) + d (t )]

= − β s (x, t ) − r s (x, t ) + d (t ) s(x, t ) 2

≤ − β s(x, t ) − r s(x, t ) + d (t ) s (x, t ) 2

≤ − s(x, t ) (β − d (t ) ),

(33)

Assume that l = (β − d (t ) ), t 1 , t 2 are the hitting time of the SMC by (26) and (27) under the sliding conditions s (x, t )s(x, t ) = −l s ( x, t ) , (34) and s (x, t )s(x, t ) = −rs 2 (x, t ) − l s( x, t ) , (35) respectively. Solve (34) between t=0 and t=t1, (35) between t=0 and t=t2: s(t = 0) , (36) t1 = l ⎛ r s(t = 0) ⎞ ⎟ ln⎜⎜1 + ⎟ l ⎠. t2 = ⎝ (37) r while x > 0 , since x > ln(1 + x) , suppose the initial state of s(t = 0) and l for (36) and (37) are the same, then t1 > t 2 .

VI. SIMULATION RESULTS In this section the proposed overall closed-loop control system including the speed observer has been investigated by simulation. The parameters of induction motor used in this drive system are: J=0.00035Kg/m2, P=1, Rs=24.6Ω, Rr=16.1Ω, Ls=1.493H, Lr=1.493H, Lm=1.46H, The ω r∗ is changed from 0 to 100 rad/sec, and the ψ dr∗ is set to 2.58 Wb. Fig. 2 shows the rotor speed tracking performance with load torque T L = 0 . It can be seen that the proposed SMC control scheme has a fast response time, a good tracking performance, and torque has a fast convergence speed. Even in the transient, only a 4.5% speed error appears. This means that even during the speed transients the scheme can track the speed command accurately. ∗ ψ dr

∗ i ds

1/Lm

uds IM

ids SMC

ω r∗

PI ∗ i qs

uqs

Speed Observer

(31) Fig. 1 Overall induction motor control scheme

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(d-q)

iqs ωˆ r

2

= −r s(x, t ) − s (x, t ) (β − d (t ) ) ,

From (25), we know that (31) is not positive: V (x, t ) = s (x, t )s(x, t ) ≤ 0 . (32) According to equation (32), the sliding-mode reaching condition is satisfied, that is to say the uncertain system (20) is asymptotically stable. In the same way, if we design the sliding-mode speed controller as u (t ) = ke(t ) − β sgn(s ) , then V (x, t ) = s (x, t )[− β sgn (s(x, t )) + d (t )]

* ψ dr

If we choose the sliding-mode speed controller as u (t ) = ke(t ) − β sgn(s ) , other conditions are the same as above, we can obtain speed tracking performance shown as Fig. (3-1). From it, the rotor speed response time is about 0.45 sec. However, in the proposed sliding-mode control algorithm the speed response time is about 0.02 sec as shown in Fig. (32), it is shorter than above case. In order to test the robustness of the controllers with rotor resistance uncertainty, the rotor resistance is stepped to Rr = 2 Rrn , other conditions are the same as above. From Fig. 4 it can be seen that at the steady state there is a little speed error not more than 0.2%, and a 5.3% speed error during the transient. The result shows that the proposed scheme is robust to rotor resistance uncertainty. 6

60 40

reference value

20 0 0

4

10 100

2 0 -2 -4

2

time 4 (s)

6

-6 0

8

2

time 4(s)

(2-1)

6

speed error (rad/s)

actual value

rotor speed (rad/s)

80

speed error (rad/s)

rotor speed (rad/s)

100

In the control scheme, make the torque command change from 0Nm to 1.0Nm. The Fig. 5 shows the corresponding speed response. When the load torque changed, the control scheme has a good robustness to external load torque variation with maximum speed error 8% during the transient, no steady state error. The results of Fig. 4 and Fig. 5 show that the proposed scheme is robust to rotor resistance uncertainty and external load variation. Fig. 6 is Error performance of the rotor speeds and fluxs. Fig. (6-2) is speed error between estimated and actual values, Fig. (6-4) is speed error between estimated and reference values. From Fig. 6 we can see that the estimated rotor speed and flux approach the actual values. That means the observer system has high estimation accuracy and satisfies our requests completely.

80 60 40

5

0

-5

20

8 0 0

(2-2)

2

time 4 (s)

6

8

-10 0

2

time 4 (s)

6

8

(5-1) (5-2) Fig. 5 Speed response when the torque command is changed

0.4

12

0.2

100 rotor speed (rad/s)

0.1 0 -0.1 0

2

time 4 (s)

6

8

(2-3) Fig. 2 Speed tracking performance when

80 60

estimated value

40 20 0

TL = 0

speed error (rad/s)

torque (Nm)

0.3

-20 0

10

actual value 2

8 6 4 2 0

time 4 (s)

6

-2 0

8

2

(6-1)

98

reference speed

97

actual speed

96

95 0.4

95 0.4

(s)

0.8

1

actual speed

97

96 time 0.6

100

99 98

0.45 time 0.5 (s) 0.55

0.6

d-axis rotor flux (W b)

speed error (rad/s)

rotor speed (rad/s)

refererce speed

40 20 0 0

actual speed

4

2

time 4

(s)

6

8

estimated speed 20

time 4

(s)

8 6 4 2 0

-20 0

-2 0

2

time 4 (s)

6

8

2

2.5

6

8

4 estimated value

reference value

x 10

8

2

6

8

-3

estimated value

0 -2 reference value -4

2

time 4 (s)

6

8

-8 0

2

time 4 (s)

(6-6)

Fig. 6 Observer system performance

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6

-6

(6-5)

(4-1) (4-2) Fig. 4 Rotor speed response when Rr=2Rrn

time 4 (s)

(6-4)

2.6

2.4 0 2

10

0

2.45

1 0

2

40

2.55

3

-1 0

60

2.65

5

60

refrence speed

2.7

6

80

80

(6-3)

(3-1) (3-2) Fig. 3 speed response time performance

100

8

12

reference speed

speed error (rad/s)

99

6

(6-2)

q -ax is ro to r flu x (Wb )

100

rotor speed (rad/s)

100

rotor speed (rad/s)

101

rotor spe ed (ra d/s)

101

time 4 (s)

VII. CONCLUSIONS The indirect field-oriented control is sensitive to some uncertain parameters. Therefore, in this paper the slidingmode control strategy worked in conjunction with the fieldoriented method to improve the controller performance, eliminate the problem caused by parameter uncertainties, and reject the interference disturbances. Meanwhile, the steady state relation in the vector control was used in the observer system to calculate rotor speed. Although the construction of observer system is simple, the rotor speed was estimated exactly. Simulation results were presented and confirmed the characteristics of the proposed overall control scheme. The overall control algorithm has good performance against unmodeled dynamics, robustness to parameter variations and external disturbances, and fast dynamic response. REFERENCES [1]

F. Blaschke, “The Principle of Field Orientation as Applied to the New Transvektor Closed-Loop Control System for Rotating Machines,” Siemens Review, No.34, pp.217-220, 1972. [2] Toshiaki Murata, Takeshi Tsuchiya, and Ikuo Takeda, “Vector Control for Induction Machine on the Application of Optimal Control Theory,” IEEE Transactions on Industrial Electronics, Vol. 37, No. 4, August 1990. [3] S. K. Biswas, H. Sendaula, and T. Caro, “Self Tuning Vector Control for Induction Motors,” IEEE IAS Ann. Mtg. Conf. Rec., 1989, pp. 276280. [4] V. I. Utkin, “Sliding Mode Control Design Principles and Applications to Electric Drives”, IEEE Trans. Ind. Electron, 1993, 40, (3), pp. 23-36. [5] Kuo Kai Shyu, Faa Jeng Lin, Hsin Jang Shieh, and Bor-Sen Juang, “Robust Variable Structure Speed Control for Induction Motor Drive,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 35, No. 1, January 1999. [6] R. Joetten, and G. Maeder, “Control Methods for Good Dynamic Performance Induction Motor Drives Based on Current and Voltages as Measured Quantities,” IEEE Trans. Ind. Applicat., Vol. 1A-19, pp. 356363, May/June 1983. [7] Geng Yang, and Tung Hai Chin, “Adaptive-Speed Identification Scheme for a Vector-Controlled Speed Sensorless Inverter-Induction Motor Drive,” IEEE Transactions on Industry Applications, Vol. 29, No. 4, July/August 1993. [8] Jingchuan Li, Longya Xu, and Zheng Zhang, “An Adaptive SlidingMode Observer for Induction Motor Sensorless Speed Control,” IEEE Transactions on Industry Applications, Vol. 41, No. 4, July/August 2005. [9] Haithem Abu-Rub, Jaroslaw Guzinski, Zbigniew Krzeminski, and Hamid A. Toliyat, “Speed Observer System for Advanced Sensorless Control of Induction Motor,” IEEE Transactions on Energy Conversion, Vol. 18, No. 2, June 2003. [10] O. Barambones, and A. J. Garrido, “A Sensorless Variable Structure Control of Induction Motor Drives,” Electric Power Systems Research 72 (2004) 21-32. [11] D.Q.Zhang and S.K.Panda, “Chattering-free and fast-responses sliding mode controller,” IEE proc.-Control Therory Appl., Vol. 146, No. 2, March 1999. [12] F. Chen and M.W. Dunnigan, “Sliding-Mode Torque and Flux Control of an Induction Machine,” IEEE Proc.-Electr. Power Appl., Vol. 150, No. 2, March 2003.

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