A sensorless variable structure control of induction motor ... - CiteSeerX

Electric Power Systems Research 72 (2004) 21–32

A sensorless variable structure control of induction motor drives O. Barambones∗ , A.J. Garrido Dpto. Ingenieria de Sistemas y Automatica E.U.I.T.I Bilbao, Universidad del Pais Vasco, Plaza de la Casilla, 48012 Bilbao, Spain Received 29 September 2003; received in revised form 29 January 2004; accepted 13 February 2004 Available online 2 June 2004

Abstract In this paper, an indirect field-oriented induction motor drive with a sliding-mode controller is presented. The design includes rotor speed estimation from measured stator terminal voltages and currents. The estimated speed is used as feedback in an indirect vector control system achieving the speed control without the use of shaft mounted transducers. Stability analysis based on Lyapunov theory is also presented, to guarantee the closed loop stability. The high performance of the proposed control scheme under load disturbances and parameter uncertainties is also demonstrated via simulation examples. © 2004 Elsevier B.V. All rights reserved. Keywords: Sensorless variable structure; Sliding-mode controller; Lyapunov theory

1. Introduction Indirect field-oriented techniques microprocessors are now widely used for the control of induction motor servo drive in high-performance applications. With the field-oriented techniques [3,7,16], the decoupling of torque and flux control commands of the induction motor is guaranteed, and the induction motor can be controlled linearly as a separated excited dc motor. However, the control performance of the resulting linear system is still influenced by the uncertainties, which are usually composed of unpredictable parameter variations, external load disturbances, unmodelled and nonlinear dynamics. Therefore, many studies have been made on the motor drives in order to preserve the performance under these parameter variations and external load disturbances, such as nonlinear control, optimal control, variable structure system control, adaptive control and neural control [8–10]. In the past decade, the variable structure control strategy using the sliding-mode has been focussed on many studies and research for the control of the ac servo drive systems [2,4,11,13]. The sliding-mode control can offer many good properties, such as good performance against unmodelled

∗

Corresponding author. Tel.: +34-94-6014459; fax: +34-94-4441625. E-mail address: [email protected] (O. Barambones).

0378-7796/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2004.02.004

dynamics, insensitivity to parameter variations, external disturbance rejection and fast dynamic response [15]. These advantages of the sliding-mode control may be employed in the position and speed control of an ac servo system. On the other hand, in indirect field-oriented control of induction motors, a knowledge of rotor speed is required in order to orient the injected stator current vector and to establish speed loop feedback control. Tachogenerators or digital shaft-position encoders are usually used to detect the rotor speed of motors. These speed sensors lower the system reliability and require special attention to noise. In addition, for some special applications such as very high-speed motor drives, there exist difficulties in mounting these speed sensors. Recently, many research has been carried on the design of speed sensorless control schemes [5,6,14,12,17]. In these schemes the speed is obtained based on the measurement of stator voltages and currents. However, the estimation is usually complex and heavily dependent on machine parameters. Therefore, although sensorless vector-controlled drives are commercially available at this time, the parameter uncertainties impose a challenge in the control performance. This paper presents a new sensorless vector control scheme consisting on the one hand of a speed estimation algorithm which overcomes the necessity of the speed sensor and on the other hand of a novel variable structure control

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O. Barambones, A.J. Garrido / Electric Power Systems Research 72 (2004) 21–32

law with an integral sliding surface that compensates the uncertainties that are present in the system. The closed loop stability of the proposed scheme is demonstrated using the Lyapunov stability theory, and the exponential convergence of the controlled speed is provided. This report is organized as follows. The rotor speed estimation is introduced in Section 2. Then, the proposed variable structure robust speed control is presented in Section 3. In Section 4, some simulation results are presented. Finally some concluding remarks are stated in the last section.

2. Calculation of the motor speed Many schemes [1] based on simplified motor models have been devised to sense the speed of the induction motor from measured terminal quantities for control purposes. In order to obtain an accurate dynamic representation of the motor speed, it is necessary to base the calculation on the coupled circuit equations of the motor. Since the motor voltages and currents are measured in a stationary frame of reference, it is also convenient to express these equations in that stationary frame. From the stator voltage equations in the stationary frame it is obtained [3]: ˙ dr = Lr vds − Lr Rs + σLs d ids ψ (1) Lm Lm dt Lr d Lr ˙ (2) ψqr = Rs + σLs vqs − iqs Lm Lm dt where ψ is the flux linkage; L the inductance; v the voltage; R the resistance; i the current and σ = 1 − L2m /(Lr Ls ) the motor leakage coefficient. The subscripts r and s denotes the rotor and stator values, respectively, refereed to the stator, and the subscripts d and q denote the d- and q-axis components in the stationary reference frame. The rotor flux equations in the stationary frame are [3]: ˙ dr = Lm ids − wr ψqr − 1 ψdr ψ Tr Tr

(3)

˙ qr = Lm iqs + wr ψdr − 1 ψqr ψ Tr Tr

(4)

where wr is the rotor electrical speed and Tr = Lr /Rr is the rotor time constant. ¯ r ) in relation to The angle θ e of the rotor flux vector (ψ the d-axis of the stationary frame is defined as follows: ψqr θe = arctan (5) ψdr being its derivative: θ˙ e = we =

˙ qr − ψqr ψ ˙ dr ψdr ψ 2 + ψ2 ψdr qr

(6)

Substituting the Eqs. (7) and (8) in the Eq. (6) it is obtained: ψdr iqs − ψqr ids Lm (7) = we = wr − 2 + ψ2 Tr ψdr qr Then Substituting the Eq. (6) in the Eq. (7), and finding wr we obtain: 1 Lm ˙ ˙ (ψdr iqs − ψqr ids ) , wr = 2 ψdr ψqr − ψqr ψdr + Tr ψr (8) 2 + ψ2 . where ψr2 = ψdr qr Therefore, given a complete knowledge of the motor parameters, the instantaneous speed wr can be calculated from the previous equation, where the stator measured current and voltages, and the rotor flux estimated obtained from a rotor flux observer based on Eqs. (1) and (2) are employed.

3. Variable structure robust speed control In general, the mechanical equation of an induction motor can be written as: Jw ˙ m + Bwm + TL = Te

(9)

where J and B are the inertia constant and the viscous friction coefficient of the induction motor system, respectively; TL is the external load; wm the rotor mechanical speed in angular frequency, which is related to the rotor electrical speed by wm = 2wr /p, where p is the pole numbers and Te denotes the generated torque of an induction motor, defined as [3]: Te =

3p Lm e e e e (ψ i − ψqr ids ) 4 Lr dr qs

(10)

e and ψ e are the rotor-flux linkages, with the subwhere ψdr qr script ‘e’ denoting that the quantity is referred to the synchronously rotating reference frame; ieqs and ieds are the stator currents, and p is the pole numbers. The relation between the synchronously rotating reference frame and the stationary reference frame is performed by the so-called reverse Park’s transformation:     xa cos(θe ) −sin(θe )     xd  xb  =  cos(θe − 2π/3) −sin(θe − 2π/3)  x q xc cos(θe + 2π/3) −sin(θe + 2π/3) (11)

where θ e is the angle position between the d-axis of the synchronously rotating reference frame and the a-axis of the stationary reference frame, and it is assumed that the quantities are balanced. Using the field-orientation control principle [3] the current component ieds is aligned in the direction of the rotor flux ¯ r , and the current component ieqs is aligned in the vector ψ


direction perpendicular to it. At this condition, it is satisfied that: e ψqr = 0,

e ¯ r| ψdr = |ψ

(12)

Therefore, taking into account the previous results, the equation of induction motor torque (10) is simplified to: Te =

3p Lm e e ψ i = KT ieqs 4 Lr dr qs

(13)

where Kt is the torque constant, and is defined as follows: KT =

3p Lm e∗ ψ 4 Lr dr

(14)

e∗ denotes the command rotor flux. where ψdr With the above mentioned proper field orientation, the dynamic of the rotor flux is given by [3]: e ψe dψdr Lm e + dr = i dt Te Lr ds

(15)

Then, the mechanical Eq. (9) becomes: w ˙ m + awm + f = bieqs

(16)

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In order to obtain the speed trajectory tracking, the following assumption should be formulated:(A1) The gain k must be chosen so that the term (k − a) is strictly negative, therefore k < 0. Then the sliding surface is defined as:

t (k − a)e(τ) dτ = 0 (24) S(t) = e(t) − 0

The variable structure speed controller is designed as: u(t) = ke(t) − βsgn(S)

(25)

where the k is the gain defined previously, β the switching gain, S the sliding variable defined in Eq. (23) and sgn(·) is the sign function. In order to obtain the speed trajectory tracking, the following assumption should be formulated:(A2) The gain β must be chosen so that β ≥ |d(t)| for all time. Theorem 1. Consider the induction motor given by Eq. (18). Then, if assumptions (A1) and (A2) are verified, the control law (25) leads the rotor mechanical speed wm (t) so that the speed tracking error e(t) = wm (t) − w∗m (t) tends to zero as the time tends to infinity.

where the parameter are defined as: a=

B , J

b=

KT , J

f =

TL ; J

(17)

Now, we are going to consider the previous mechanical Eq. (16) with uncertainties as follows: w ˙ m = −(a + a)wm + (f + f) + (b + b)ieqs

(18)

where the terms a, b and f represents the uncertainties of the terms a, b and f, respectively. Let us define the tracking speed error as follows: e(t) =

wm (t) − w∗m (t)

(19)

where w∗m is the rotor speed command. Taking the derivative of the previous equation with respect to time yields: e˙ (t) = w ˙m −w ˙ ∗m = −ae(t) + u(t) + d(t)

(20)

where the following terms have been collected in the signal u(t), u(t) = bieqs (t) − aw∗m (t) − f(t) − w ˙ ∗m (t)

(21)

and the uncertainty terms have been collected in the signal d(t), d(t) = −awm (t) − f(t) + bieqs (t)

(22)

Now, we are going to define the sliding variable S(t) with an integral component as:

t S(t) = e(t) − (k − a)e(τ) dτ (23) 0

where k is a constant gain.

The proof of this theorem will be carried out using the Lyapunov stability theory. Proof: Define the Lyapunov function candidate: 1 V(t) = S(t) S(t) (26) 2 Its time derivative is calculated as: ˙ = S[˙e − (k − a)e] V˙ (t) = S(t) S(t) = S[(−ae + u + d) − (ke − ae)] = S[u + d − ke] = S[ke − β sgn(S) + d − ke] = S[d − β sgn(S)] ≤ − β − |d| |S| ≤ 0

(27)

It should be noted that the Eqs. (23), (20) and (25), and the assumption (A2) have been used in the proof. Using the Lyapunov’s direct method, since V(t) is clearly positive-definite, V˙ (t) is negative definite and V(t) tends to infinity as S(t) tends to infinity, then the equilibrium at the origin S(t) = 0 is globally asymptotically stable. Therefore S(t) tends to zero as the time t tends to infinity. Moreover, all trajectories starting off the sliding surface S = 0 must reach it in finite time and then will remain on this surface. This system’s behavior once on the sliding surface is usually called sliding mode [15]. When the sliding mode occurs on the sliding surface (24), ˙ = 0, and therefore, the dynamic behavior then S(t) = S(t) of the tracking problem (20) is equivalently governed by the following equation: ˙ = 0 ⇒ e˙ (t) = (k − a)e(t) S(t) (28) Then, under assumption (A1), the tracking error e(t) converges to zero exponentially.

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Fig. 1. Block diagram of the proposed sliding-mode field oriented control.

It should be noted that, a typical motion under sliding mode control consists of a reaching phase during which trajectories starting off the sliding surface S = 0 move toward it and reach it in finite time, followed by sliding phase during which the motion will be confined to this surface and the system tracking error will be represented by the reduced-order model (28), where the tracking error tends to zero. Finally, the torque current command, i∗qs (t), can be obtained directly substituting Eq. (25) in Eq. (21): i∗qs (t) =

1 [ke − β sgn(S) + aw∗m + w ˙ ∗m + f ] b

(29)

Therefore, the proposed variable structure speed control resolves the speed tracking problem for the induction motor, with some uncertainties in mechanical parameters and load torque.

4. Simulation results In this section, we will study the speed regulation performance of the proposed sliding-mode field oriented control versus reference and load torque variations by means of two different simulation examples. The block diagram of the proposed robust control scheme is presented in Fig. 1. The block ‘VSC Controller’ represents the proposed sliding-mode controller, and it is implemented by Eqs. (23) and (29). The block ‘limiter’ limits the current applied to the motor windings so that it remains within the limit value, and it is implemented by a saturation function. The block ‘dqe → abc makes the conversion between the synchronously rotating and stationary reference frames, and is implemented by Eq. (11). The block ‘Current Controller’ consists of a three hysteresis-band current PWM control,

Fig. 2. Reference and real rotor speed signals (rad/s).


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Fig. 3. Stator current isa (A).

which is basically an instantaneous feedback current control method of PWM where the actual current (iabc ) continually tracks the command current (i∗abc ) within a hysteresis band. The block ‘PWM Inverter’ is a six IGBT-diode bridge inverter with 780 V DC voltage source. The block ‘Field Weakening’ gives the flux command based on rotor speed, so that the PWM controller does not saturate. The block e∗ ‘ie∗ ds Calculation’ provides the current reference ids from the rotor flux reference through the Eq. (15).

The block ‘wr and we Estimator’ represent the proposed rotor speed and synchronous speed estimator, and is implemented by the Eqs. (8) and (6) respectively. The block ‘IM’ represents the induction motor. The induction motor used in this case study is a 50 HP, 460 V, four pole, 60 Hz motor having the following parameters: Rs = 0.087 ', Rr = 0.228 ', Ls = 35.5 mH, Lr = 35.5 mH, and Lm = 34.7 mH.

Fig. 4. Motor torque (Nm).

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Fig. 5. Stator current ieds (A).

The system has the following mechanical parameters: J = 1.662 kgm2 and B = 0.1 Nms. It is assumed that there is an uncertainty around 20% in the system parameters, that will be overcome by the proposed sliding control. The following values have been chosen for the controller parameters, k = −100, β = 30. First example: In this first example, the motor starts from a standstill state and we want the rotor speed to follow a

speed command that starts from zero and accelerates until the rotor speed is 90 rad/s. The system starts with an initial load torque TL = 50 Nm, and at time t = 1 s the load torque steps from Tl = 50 Nm to TL = 100 Nm. Fig. 2 shows the desired rotor speed (dashed line) and the real rotor speed (solid line). As may be observed, the rotor speed tracks the desired speed in spite of system uncertainties. Moreover, the speed tracking is not affected by the load

Fig. 6. Stator current ieqs (A).


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Fig. 7. Rotor flux ψdr (Wb).

torque change at the time t = 1 s, because when the sliding surface is reached (sliding mode) the system becomes insensitive to the boundary external disturbances. Fig. 3 shows the current of one stator winding. This figure shows that in the initial state, the current signal presents a high value because it is necessary a high torque to increment the rotor speed. In the constant speed region, the motor torque only has to compensate the friction and the load

torque and so, the current is lower. Finally, at time t = 1 s the current increases because the load torque has been increased. Fig. 4 shows the motor torque. As in the case of the current (Fig. 3), the motor torque has a high initial value speed in the acceleration zone, then the value decreases in a constant region and finally increases due to the load torque increment. In this figure, it may be seen that in the motor torque

Fig. 8. Stator current ψqr (Wb).

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Fig. 9. Reference and real rotor speed signals (rad/s).

appears the so-called chattering phenomenon, however, this high frequency changes in the torque will be filtered by the mechanical system inertia. Figs. 5 and 6 shows the stator currents in the rotating reference frame. As may be observed in the figures, both currents present a initial peak at the beginning, as it is usual in the starting of motors. Then the current, ieds , corresponding to the field component, remains constant. On the other hand, the current ieqs , corresponding to the torque compo-

nent, varies with the torque; that is, presents a high initial value in the acceleration zone, then the value decreases in a constant region and finally increases due to the load torque increment. Figs. 7 and 8 shows the estimated rotor flux in the stationary reference frame. As may be observed the rotor flux starts from zero and increases until the nominal value. Second example: In this second example, the final state of the previous example has been used as initial condition. That

Fig. 10. Stator current isa (A).


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Fig. 11. Motor torque (Nm) (A).

is, the motor and the reference speeds are initially 90 rad/s with a load torque of TL = 100 Mm, then at time t = 0.3 s the speed reference changes from this value to 120 rad/s. Fig. 9 shows the desired rotor speed (dashed line) and the real rotor speed (solid line). As it may be observed, the rotor speed tracks the desired speed in spite of system uncertainties as in the previous example. Nevertheless, when the speed reference steps to its final value at time t = 0.3 s,

the motor can not follow this reference instantaneously due to the physical limitations of the system. However, after a transitory time in which the motor accelerates until the final speed the trajectory tracking is obtained. Fig. 10 shows the current of one of the stator windings. This figure shows that in the initial state, the current signal is constant because the speed is constant, then the current increased because the motor is accelerating, and finally

Fig. 12. Stator current ieds .

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Fig. 13. Stator current ieqs (A).

remains constant again because the speed is constant. At this point it should be noted that the final current value is similar to the initial one. This is reasonable because the only difference in the torque of both initial (wm = 90 rad/s) and final (wm = 120 rad/s) states relies in the friction term that presents a small variation. Fig. 11 shows the motor torque. As in the case of the current (Fig. 10), the motor torque has constant value in the

initial state, then increases due to the acceleration of the motor, and finally remains constant again. Figs. 12 and 13 shows the stator currents in the rotating reference frame. As may be observed in the figures, the current ieds , corresponding to the field component, remains constant because the flux is maintained constant, and the current ieqs , corresponding to the torque component, varies with the torque.

Fig. 14. Rotor flux ψdr (Wb).


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Fig. 15. Stator current ψqr (Wb).

Finally, Figs. 14 and 15 shows the estimated rotor flux in the stationary reference frame. As may be observed the rotor flux remains constant at the nominal value.

5. Conclusions In this paper, a sensorless sliding mode vector control has been presented. The rotor speed estimator is based on stator voltage equations and rotor flux equations in the stationary reference frame. It is proposed a variable structure control which has an integral sliding surface to relax the requirement of the acceleration signal, that is usual in conventional sliding mode speed control techniques. Due to the nature of the sliding control this control scheme is robust under uncertainties caused by parameter errors or by changes in the load torque. The closed loop stability of the presented design has been proved through Lyapunov stability theory. Finally, by means of simulation examples, it has been shown that the proposed control scheme performs reasonably well in practice, and that the speed tracking objective is achieved under uncertainties in the parameters and load torque.

Acknowledgements The authors are grateful to the Basque Country University for partial support of this work through the research Project 1/UPV 00146.363-E-13992/2001.

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