Sensor Less Adaptive Back Stepping Speed Control Of Im

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Proceedings of the 45th IEEE Conference on Decision & Control Manchester Grand Hyatt Hotel San Diego, CA, USA, December 13-15, 2006

SENSORLESS ADAPTIVE BACKSTEPPING SPEED CONTROL OF INDUCTION MOTOR Hou-Tsan Lee", Li-Chen Fu2'3, Feng-Li Lian2

1. Takming College, Taipei, Taiwan 2. Department ofElectrical Engineering, National Taiwan University, Taipei, Taiwan 3.Department of Computer Science and Information Engineering National Taiwan University, Taipei, Taiwan i *E-mail: h Abstract: This paper proposes an adaptive speed tracking control scheme for an induction motor subject to unknown load torque via backstepping analysis. The controller is developed under a special nonlinear coordinate transform such that speed control objective can be fulfilled. The underlying design concept is to endow the closed-loop system while lacking the knowledge of some key system parameters, such as the rotor resistance, motor inertia, and motor damping coefficient. The proposed control scheme comes along with a thorough proof derived based on Lyapunov stability theory. Experimental results are also given to validate the effectiveness of the proposed control scheme. Keywords: Adaptive control, induction motors, backstepping analysis, Lyapunov stability, 1. INTRODUCTION The induction motor control is an important issue in both motion control and servo control applications because the induction motor can operate in a wide-range of both torque and speed. And, their efficiency and robustness are useful features in industry. The control schemes based on indirect FOC are much more popular due to the advantages in applications (Marino, et al., 1999; Tajima and Hori, 1993; Espinosa, et al., 1998). In general applications, indirect vector control of induction motor is widely applied, where the rotor flux is estimated rather than being measured. This requires a priori knowledge of the machine parameters, which makes the indirect vector control scheme machine dependent. Given the fact that parameters may change significantly with temperature and there are some states that are not easily acquired, design of appropriate observers becomes crucially important to the success of the control (Krishnan and Bharadwaj, 1990; Shin, et al., 2000). Recently, the sensorless field oriented control scheme gradually appears as a popular control method for induction motor (Marino, et al., 1996; Lin and Fu, 2000). Even in particular applications, sensorless FOC shows its advantages. (Kwon, et al., 2005; Hinkkanen, et al., 2005) On the other hand, the load torque structure is also a very important knowledge for controller design to achieve high performance control. There have been many research results in the literature about the torque control of induction motor so far (Ryu, et al., 2006; Lee and Fu, 2001; Lascu, et al., 2000; Ortega and Espinosa, 1993).

This research was supported in part by the National Science Council, Taiwan, ROC, under the grant NSC 94-2213-E-002-022, NSC 95-2218-E-002-031 and Takming College, Taiwan, ROC.

1-4244-0171-2/06/$20.00 ©2006 IEEE.

Given the above observation, we propose a speed tracking control scheme based on the indirect FOC via the backstepping design (Kanellakopoulos, et al., 1992). Moreover, the proposed control scheme handles the problems with both uncertainties of rotor resistance and load torque, respectively. The system parameters of the induction motor, except its rotor resistance, and mechanical parameters, are known as mentioned previously. For rigorousness, the developed control scheme is thoroughly analyzed via Lyapunov stability theory, and the asymptotic convergence property is soundly proved. Experimental results are given to validate the performances. 2. PRELIMINARIES In this section, we will first review the mathematical description of the operational principle of an induction motor in the following sections. Before we continue the control of speed of the induction motors, we first make some basic assumptions as shown below: (A.1) The induction motor is assumed without saturation, hysteresis, eddy currents, and spatial flux harmonics. (A.2) All the states are measurable except the rotor flux. Parameters including rotor resistance, rotor inertia, damping coefficient, and the payload coefficient are assumed unknown. Proposition 1. Under the assumptions (A. 1), if the input voltages in d-q frame of the voltage-fed induction motor are defined as -Aqr v Adr V cVq cVd,

,and

qr

dr

qr

+dr

(1)

then the power transferred to the rotor of induction motor is maximal subject to the constraint

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2005; Lee, et.al., 2002; Lin, and Fu, 2000; Tajima, et al.,1993), the observer which we adopted is designed by the authors in previous literatures and the detailed discussion is omitted here. 3.2 Speed Tracking Controller Before we introduce the design of the controller, in order to avoid dealing with the discontinuous function sgn(x), we approximate it by the so-called sigmoid function smod(x) defined below:

(Vq ± ]ds) =(V c)2 at any time (Lee and Fu, 2001; Lee, et al., 2000a; Lee et al., 2000b). Of course, V does not have to be a constant. Instead, it offers one DOF (degree of freedom) control to the system. For general mechanical systems, the load torque is a function of the rotor speed o), which normally has the form TL = JL Zr + sgn(ct), )bo + b, c) + b2 sgn(ct), nrJ0 This assumption is more realistic than a constant load torque. Therefore, the mechanical load in the form aforementioned can be rearranged as TL = H)W with the constant parameter vector ® = [JL bo b, b2,T

and

the

known

function

sgn(x) smod(x) = -

vector

following: fe5

(3)

=

_a5 (al + a4 )X4 a5 pX3X5 + a5 jV OT

(5)

and then further simplify the dynamical model of an induction motor with Proposition 1, as shown in (Ortega and Espinosa, 1993; Lin and Fu, 2000), which becomes: 2hx 2axl + 2a2X3 + 4 V,

which clearly relates the voltage input V to the tracking error e5 , and lays down a ground for constructing an adaptive controller. However, there remain two difficulties before the controller design. One is to establish a backstepping designed transfer function from the parametric error term to the tracking error e5, and another is to decompose the uncertainty terms into a product of unknown parametric vector and known function vector. If we design stabilizing function for

= -2a4x2+2a3x3,

x3 = a3x +a2x2 -(al + a4)x3 + px5x4,

x~4

ex

where y > 1 determines the slope of the function. By taking such an approximation, we will be able to differentiate the payload function fL (r) for the subsequent purpose of controller design. To proceed with the design, we first write down the speed tracking error equation from (2) as: (4) JQ5 =a5x4 -OE)TW, 0 =[J bo bl b2]T , and where e5 = X5 -Ctd After W = [dsmod(xs ) X5 smod(X5 )Xs2]T differentiating both sides of (4), we obtain the

f K9r sgn(w9)W9rgn(n(C)wJ. In the sequel, we will assume that 0 is unknown. On the other hand, we would like to show that, given a desired speed command Wd , there exists a proper input signal V such that the steady state of the system exactly achieves the purpose of speed tracking, i.e., Wr=od , and the objective of maximal power transfer (Proposition 1). To this end, we first introduce a reasonable result (A.3) X2= A + A2 >n

x~2

ey _x e- 7x e ±e -7X

as

X

= -px5x3 -(a1 +a4)x4 +±WV,

a, = -c3Je5 +-TW a5

with Lyapunov

function Vb =-Je5 + (x4 a1)2.

(2) JX5 = a5x4 fL(X5) where the parameters above are defined in the literature (Lee, et.al., 2002).

2 2 Its time derivative is rendered

negative

definite

Vb -c3a5Je5 _C4(X4 _ C )2 by the control

3. OBSERVER AND CONTROLLER To proceed with the controller design, we first introduce the observers to estimate the unmeasurable rotor flux, and the unknown rotor resistance. 3.1 Observer Design Due to Assumption (A.2), we have to build an observer and a parameter estimator to estimate the rotor flux as well as the rotor resistance. There exist various types of flux observers and parameter estimators in the literature, which have been described in (Kwon, et al.,

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V=

where 3 W3

[g3 (x)±+ )W

-

c4(X4aI-,1) - a5e5]

g3 (x) =px3x5 + (al+ a4)x4 -c3a5x4

C3E)T W +1Ia5

) W

=

I

C3E)TW±1+ a5

T

and W

,

the parameter vector 0' as well as the known function vector W' in the term E)TW , i.e.,

45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006

®T14r

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0'W',

and c3, c4 > 0 and then design the input V is described as follow:

(X)±+)3

= [g3

- a5z,] , where

-2aF 0

]2 =0

(6)

X3_

Q3 =0 3 F4z2W3 (7) as the adaptive law, with some positive adaptation gains. Such fact is stated in the following theorem. Theorem 1. Consider an induction motor whose

=

dr ±,2

,{2l qr

Aqr

ds

2dr

=

v

,2 ± 2 C + dr

Aqr

e5, z2 =X4 - a1, where a, a5

C3TW+® I1

a5

® W

C

TW+IJiT WI

a5

thus e5

0 as t -X o

from Barbalat's Lemma..

are listed as follows: Rs = 0.83 Q

in the closed-loop and convergence of the rotor speed tracking, i.e., e5 -* 0 as t - (Do

Since z1

-

Q.E.D.

respectively, will guarantee boundedness of all signals

=

=

e5 is bounded, and fL (X5) is also

bounded that provides ±1

=

e5 also bounded. Hence,

X4 cannot be unbounded (otherwise, x5 will be

Rm 0.53 Q

Ls

0.08601(H) ' Lr = 0.08601(H) ' Lm 0.08259(H),

4 poles, rated current 8.6 A, 220 V, 60 Hz, AC. Jm and Bm are assumed unknown. The mechanical load torque is of the form JLr±+bOs&T(W))+b1

Proof:

PX5X4

4. EXPERIMENTAL RESULTS To validate the performances of the proposed controller, we hold a series of experiments whose setup had been demonstrated in (Lee, et al., 2002). Whose parameters

and W3 are defined as. a1 -c3Jzl +-EOW TW3

A

=AX+ u

result, u is apparently bounded, and hence X will be bounded. But this fact leads to contradition to the hypothesis in the second case. This then proves the boundedness of all the states. For convergence of tracking error e5, we note that e5 is bounded, and

1 'T

and

+

no slower than x3 if x3 does grow unbounded. As a

where V is given by equation (6), subject to adaptive law (7), with some positive adaptation gains F4, and the new variables z1

2a3 ) X2 -(al +Ua4)-X3

and V, respectively, from (1) and (6), we found that the first entry of u will be bounded because x2 grows

Ctd with Od,Wd and 60d being all bounded, then the stator voltage design as (maximal power transfer) qs

a2

2hx4 V

x1 VK

where A can be shown to be Hurwitz as a result of the matrix operation. After reviewing definitions of x3

dynamics are governed by System (2) with unknown load torque under Assumptions (Al) and (A2). Given a twice-differentiable smooth desired speed trajectory

V

a3

-2a4

2a2

± b2sgn(w)w,r

In

the experiments, there is no load applied on the induction motor. The gains are[ ' 4]=[500, 100] in nonlinear adaptive backstepping controller and the

adaptation gain

unbounded). As we know, we can rearrange the dynamical equations from System (2) as shown below:

F4

is set to unity of the controller

design. Accompany with kdl kd2 =kql =kq2 0.003

ko = 0.01, kR = 200

of the observers' gains. The

experimental results of the speed tracking are

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45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006

demonstrated in the following cases: Case 1. The speed commands are smooth functions, Nd

=

50(1-exp(-27zt))+10sin(2ztt)

and

30sin(2 zt)

which meet the requirements of speed command in Theorem 1. As shown in both theoretical analysis and the results of Figure 1, the performances are satisfactory. Case 2. The speed command is a sort of step-type command (50 rad/sec) to validate the proposed controller. Figure 2 shows the boundeness of estimated parameters and also the performances of the proposed controller with a step speed command. The estimated values of the unknown parameters are truly bounded. And, the errors of both speed tracking and the deviation of rotor resistance are rapidly converged. The control objective is achieved even with the crucial condition. Case 3. The speed tracking controller is operated in a critical situation of benchmark commands (rapidly changes as 30 - 0 - -30 - 0 - 10 rads/sec). Figure 3 shows the satisfactory performances of the speed tracking. We can see that the actual speed follows the speed command. And, all the estimated parameters are bounded. Although the speed command rapidly changes, the actual speed output of the induction motor is closely follows up. On the other hand, the current input is zero as the motor stops. All the experiments are conducted without the information of the deviation of rotor resistance, the motor inertia and the damping coefficient of the induction motor. And, the parameters of load torque are unknown, either.

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novel dynamical model of an induction motor and designed under the backstepping control technique. The proposed speed tracking controller of an induction motor, which copes with the unknown motor parameters (Jm and Bin), the uncertainty of rotor resistance, and unknown load torque TL, is a Lyapunov-based design. Although the speed command is assumed being a twice differentiable trajectory, but the step type command can be directly applied to the proposed controller with satisfactory performances, i.e., the assumption can be relaxed in reality. REFERENCES 1. Espinosa, G., R. Ortega, and P. J. Nicklasson (1998). "Torque and Flux Tracking of Induction Motors." Int. J. Robust & Nonlinear Contr., Vol. 8, pp. 1-9. 2. Krishnan, R., and A. S. Bharadwaj (1990)."A Review of Parameter Sensitivity and Adaptation in Indirect Vector Controlled Induction Motor Drive Systems." IEEE Trans. Power Electron., Vol. 6, No.4, pp. 434-440. 3. Lascu, C., I. Boldea, and F. Blaabjerg (2000). "A

Modified Direct Torque Control for nduction Motor Sensorless Drive." IEEE Trans. Indust. Appl., Vol. 36, No.1, pp. 124-130. 4. Kwon, Tae-Sung, Myoung-Ho Shin and Dong-Seok Hyun (2005), "Speed sensorless stator flux-oriented control of induction motor in the field weakening region using Luenberger observer", IEEE Transactions on Power Electronics Vol. 20, no. 4, pp.864- 869. 5. Hinkkanen, M., Leppanen, V.-M., and Luomi, J. (2005), "Flux observer enhanced with low-frequency signal injection allowing sensorless zero-frequency operation of induction motors", IEEE Transactions on Industr Applications, Vol. 41, no. 1, pp.52 - 59.

5. CONCLUSION

Some concluding remarks are summarized as follows. The proposed control scheme is developed based on a

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for Speed Sensorless Stator Flux Orientation Control of Induction Motors." IEEE Trans. Power Electron., Vol. 22, No.2, pp. 312-318. 15. Tajima, H., and Y. Hori (1993). "Speed Sensorless Field-Orientation Control of Induction Machine." IEEE Trans. Indust. Appl., Vol. 29, No. 1, pp. 185-180. 16. Kanellakopoulos, I., P. V. Kokotovic and A. S. Morse(1 992), "A Toolkit for Nonlinear Feedback Design", System & Control Letters, pp. 83-92. 17. Ryu, J.H., Lee, K.W., and Lee, J.S.(2006), "A Unified Flux and Torque Control Method for DTC-Based Induction-Motor Drives," IEEE Transactions on Power Electronics, Vol. 21, no. 1, pp. 234 -242.

6. Lee, H. T., L. C. Fu, and H. S. Huang (2002). "Speed Tracking Control with Maximum Power Transfer of Induction Motor." IEEE Trans. Ind Electr., pp. 911-924, Vol.49, No.4,2002. 7. Lee, H. T., L. C. Fu, and H. S. Huang (2000a). "Speed Tracking Control with Maximal Power Transfer of Induction Motor." Proc. IEEE 39th Conf OnDecision and Control, pp.925-930. 8. Lee, H. T., J. S Chang, and L. C. Fu (2000b). "Exponential Stable Nonlinear Control for Speed Regulation of Induction Motor with Field Oriented PI-Controller." Int. J. Adap. Contr. & Sign.Proc., Vol. 21, No.23, pp. 297-321. 9. Lee, H. T., and L. C. Fu (2001). "Nonlinear Control of Induction Motor with Unknown Rotor Resistance and Load Adaptation." Proc. American Control Conference, pp. 55-59. 10. Lin, Y. C., and L. C. Fu (2000). "Nonlinear Sensorless Indirect Adaptive Speed Control of Induction Motors with Unknown Rotor Resistance and Load." Int. J. Adap. Contr. & Sign. Proc., Vol. 21, No.23. 11. Marion, R., S. Peresada, and P. Tomei (1996). "Adaptive Observer-Based Control of Induction Motor with Uncertainty of rotor resistance." Int. J. Adapt.Contr. & Signal Proc., Vol. 10, pp. 345-363. 12. Marino, R., S. Peresada, and P. Tomei (1999). "Global Adaptive Output Feedback Control of Induction Motors with Uncertain Rotor Resistance." IEEE Trans. Automat. Contr., Vol. 44, No. 5, pp. 968-983, 1999. 13. Ortega, R., and G. Espinosa (1993). "Torque Regulation of Induction Motors." Automatica, Vol. 29, No.3, pp. 623-633. 14. Shin, M. H., D. S. Hyun, S. B. Cho, and S. Y. Choe(2000). "An Improved Stator Flux Estimation

(al )kualSped

(bi )1ualSped

40

o40 0

2

2

4

6

8

10 d

r2) Slaor Curred

0

2

4

6

8

10

4

6

8

10

0

2

4

6

8

10

(a2) 5'gor Curre

Fig 1. Experimental results of a sinusoidal speed tracking (Left: 50(1-exp(-2gzt))+10sin(2zt); Right: 30 sin(2zt) rad/sec).

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45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006

(1i AcuaI Speed

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(1 Adua1,Sped

P)Speed Error

P)Speed Error

0

0

4

0

O

2

3

(5) Es'im3'ei J

4

5

0

02

2

3

2

3

(6) Edimaled bO

4

5

4

5

5

0.15 E

2

-0.1

0.05

0O

2

3

4

5

0.

0

Fig 2. Experimental results of a step-type speed tracking (50 rad/sec).

Fig. 3. Experimental results of speed tracking (benchmark, 30 - 0 -30 - 0 - 10 rads/sec) -

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