Speed Sensorless Vector Control of Induction Motors Based on Robust Adaptive Variable Structure Control Law ∗
O. Barambones∗ , A.J. Garrido∗ , F.J. Maseda∗ and P. Alkorta† Dpto. Ingenier´ıa de Sistemas y Autom´atica, Universidad del Pa´ıs Vasco. E.U.I.T.I de Bilbao, Plaza de la Casilla, 48012 Bilbao (Spain). Tel: +34 946014459; Fax: +34 946014300; E-mail:
[email protected] † E.U.I.T.I de Eibar, Avenida de Otaola 29, 20600 Eibar (Spain)
A BSTRACT A novel sensorless adaptive robust control law is proposed to improve the trajectory tracking performance of induction motors. The proposed design employs the so called vector (or field oriented) control theory for the induction motor drives and the designed control law is based on an integral slidingmode algorithm that overcomes the system uncertainties. The proposed sliding-mode control law incorporates an adaptive switching gain to avoid calculating an upper limit of the system uncertainties. The proposed design also includes a new method in order to estimate the rotor speed. In this method, the rotor speed estimation error is presented as a first order simple function based on the difference between the real stator currents and the estimated stator currents. The stability analysis of the proposed controller under parameter uncertainties and load disturbances is provided using the Lyapunov stability theory. Finally simulated results show, on the one hand that the proposed controller with the proposed rotor speed estimator provides high-performance dynamic characteristics, and on the other hand that this scheme is robust with respect to plant parameter variations and external load disturbances. I. I NTRODUCTION Field oriented control method is widely used for advanced control of induction motor drives. By providing decoupling of torque and flux control demands, the vector control can govern an induction motor drive similar to a separate excited direct current motor without sacrificing the quality of the dynamic performance. However, the field oriented control of induction motor drives presents two main problems that have been providing quite a bit research interest in the last decade. The first one relies on the uncertainties in the machine models and load torque, and the second one is the precise computation of the motor speed without using speed sensors. The decoupling characteristics of the vector control is sensitive to machine parameters variations. Moreover, the machine parameters and load characteristics are not exactly known, and may vary during motor operations. Thus the dynamic characteristics of such systems are very complex and nonlinear. Therefore, many studies have been made on the motor drives in order to preserve the performance under these
1-4244-0681-1/06/$20.00 '2006 IEEE
parameter variations and external load disturbances, such as nonlinear control, optimal control, variable structure system control, adaptive control and neural control [7], [8], [11], [12], [13]. To overcome the above system uncertainties, 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 system in the past decade [2], [3], [6], [14], [17]. The sliding-mode control can offer many good properties, such as good performance against unmodelled dynamics, insensitivity to parameter variations, external disturbance rejection and fast dynamic response [20]. These advantages of the slidingmode control may be employed in the position and speed control of an AC servo system. However the traditional sliding control schemes requires the prior knowledge of an upper bound for the system uncertainties since this bound is employed in the switching gain calculation. This upper bound should be determined as precisely as possible, because as higher is the upper bound higher value should be considered for the sliding gain, and therefore the control effort will also be high, which is undesirable in a practice. In order to surmount this drawback, in the present paper it is proposed an adaptive law to calculate the sliding gain which avoids the necessity of calculate an upper bound of the system uncertainties. Otherwise, a suitable speed control of an induction motor requires a precise speed information, therefore, a speed sensor, such a resolver and encoder, is usually adhered to the shaft of the motor to measure the motor speed. However, a speed sensor can not be mounted in some cases, such as motor drives in a adverse environments, or high-speed motor drives. Moreover, such sensors lower the system reliability and require special attention to noise. Therefore, sensorless induction motor drives are widely used in industry for their reliability and flexibility, particularly in hostile environments. Speed estimation methods using Model Reference Adaptive System MRAS are the most commonly used as they are easy to design and implement [4], [10], [21]. However, the performance of these methods is deteriorated at low speed because of the increment of nonlinear characteristics [9], [15]. In this paper the authors proposes a new robust sensorless vector control scheme consisting on the one hand of an adaptive rotor speed estimation method based on MRAS in order
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to improve the performance of a sensorless vector controller in a low speed region. In the presented method, the stator current is estimated and then it is compared with the real stator current. Next, the stator current error is used in the adaptive law to estimate the rotor speed. The proposed method can provide a fast speed estimation and improve the performance of other speed estimation methods in a low speed region and at zero-speed. In addition, the proposed variable structure control algorithm presents an adaptive sliding gain that is estimated on-line in order to compensate the present system uncertainties. Using the proposed sensorless variable structure control to govern the induction motor drive, the rotor speed becomes insensitive to variations in the motor parameters and load disturbances. Moreover, the proposed control scheme, provides a good transient response and exponential convergence of the speed trajectory tracking in spite of parameter uncertainties and load torque disturbances. This report is organized as follows. The rotor speed estimation is introduced in Section 2. In section 3, the proposed robust speed control with adaptative sliding gain is presented, then the closed loop stability of the proposed scheme is demonstrated using the Lyapunov stability theory, and the exponential convergence of the controlled speed is also provided. In the Section 4, some simulation results are presented. Finally some concluding remarks are stated in the last Section.
Using the rotor flux and motor speed, the stator current is represented as: 1 ψdr + wr Tr ψqr + Tr ψ˙ dr (3) ids = Lm 1 ψqr − wr Tr ψdr + Tr ψ˙ qr (4) iqs = Lm where wr is the rotor electrical speed and Tr = Lr /Rr is the rotor time constant. From the equations (3) and (4) and using the estimated speed, the stator current is estimated as: ˆids
=
ˆiqs
=
1 Lm 1 Lm
From the stator voltage equations in the stationary frame it is obtained [5]: d Lr ˙ vds − Rs ids − σLs ids (1) ψdr = Lm dt d Lr ψ˙ qr = vqs − Rs iqs − σLs iqs (2) Lm dt where ψ is the flux linkage; L is the inductance; v is the voltage; R is the resistance; i is the current and σ = 1 − L2m /(Lr Ls ) is 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 dq-axis components in the stationary reference frame.
ψqr − w ˆr Tr ψdr + Tr ψ˙ qr
(5) (6)
Subtracting the equations of the estimated stator currents (5) and (6) from the equations of the stator currents (3) and (4) the difference in the stator current is obtained as: ids − ˆids iqs − ˆiqs
The current paper proposes a new rotor speed estimation method to improve the performance of a sensorless vector controller in the low speed region and at zero speed. 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.
ˆr Tr ψqr + Tr ψ˙ dr ψdr + w
where ˆids and ˆiqs are the estimated stator currents and w ˆr is the estimated rotor electrical speed.
II. P ROPOSED ROTOR SPEED ESTIMATOR Many schemes [1], [16], [18], 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. However, the performance of these methods is deteriorated at a low speed because of the increment of nonlinear characteristic of the system [15].
Tr ψqr (wr − w ˆr ) Lm Tr = − ψdr (wr − w ˆr ) Lm =
(7) (8)
In the above equations (7) and (8), the difference of the stator current and the estimated stator current is a sinusoidal value because it is a function of the rotor flux. However, if equation (7) is multiplied by ψqr and equation (8) is multiplied by ψdr and then are added together it is obtained: (ids − ˆids )ψqr − (iqs − ˆiqs )ψdr = Tr 2 2 ψqr (wr − w ˆr ) (ψdr + ψdr ) Lm
(9)
Unlike the equations (7) and (8), equation (9) uses the rotor flux magnitude which remains constant. From equation (9) the error of the rotor speed is obtained as follows: ewr = wr − w ˆr = c (ids − ˆids )ψqr − (iqs − ˆiqs )ψdr (10) where: c=
1 Lm Lm = 2 2 Tr ψdr + ψqr Tr ψr2
Therefore, from the equation (10) the speed estimation error is calculated from the stator current and rotor flux. Using Lyapunov stability theory we can derive the following adaptation law for speed estimation:
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dw ˆr = α ewr , dt
α>0
(11)
where α is de adaptation gain that should be chosen greater than zero. To demonstrate that the previous adaptation law makes the estimated speed error drops into zero, we can define the following Lyapunov function candidate: V (t) =
1 2 e (t) 2 wr
= ewr e˙ wr = ewr (−w ˆ˙ r ) 2 = −α ewr
(12)
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 ewr (t) tends to infinity, then the equilibrium at the origin ewr (t) = 0 is globally asymptotically stable. Therefore ewr (t) tends to zero as the time t tends to infinity. Moreover, taking into account the previous Lyapunov function we can conclude that the rotor speed error converges to zero exponentially. From equation (12) we can obtain that V derivative verifies: α V˙ (t) = −α e2wr = − V (t) 2 The solution of the previous differential equation is:
3p Lm e e e e (ψ i − ψqr ids ) (15) 4 Lr dr qs e e where ψdr and ψqr are the rotor-flux linkages, with the subscript ‘e’ denoting that the quantity is refereed to the synchronously rotating reference frame; ieqs and ieds are the stator currents, and p is the pole numbers. Te =
On the basis of the fact that the velocity of outer control loop is much slower than the estimated inner loop, hence the assumption of wr approaching a constant is reasonable on deriving the following equations. Then, the time derivative of the previous Lyapunov function candidate is: V˙ (t)
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 is the rotor mechanical speed in angular frequency, which is related to the rotor electrical speed by wm = 2 wr /p where p is the pole numbers and Te denotes the generated torque of an induction motor, defined as [5]:
(13)
1 2 α e (t) = V (t0 ) exp(− t) 2 wr 2 which implies that the rotor speed error converges to zero exponentially. V (t) =
Therefore, the rotor speed wr can be calculated using the proposed speed estimator which only make use of the measured stator voltages and currents in order to estimate the rotor speed. At this point it should be noted that, due to the convergence of the proposed estimation scheme, the estimated speed, w ˆr , can be approximated to the real one, wr , up to any desirable tolerance. Therefore these signals are fully interchangeable and for notation simplicity purposes both signals will be denoted as wr in the rest of the paper. III. VARIABLE STRUCTURE ROBUST SPEED CONTROL WITH
The relation between the synchronously rotating reference frame and the stationary reference frame is performed by the so-called reverse Park’s transformation: ⎤ ⎤ ⎡ ⎡ − sin(θe ) cos(θe ) xa ⎣ xb ⎦ = ⎣ cos(θe − 2π/3) − sin(θe − 2π/3) ⎦ xd xq cos(θe + 2π/3) − sin(θe + 2π/3) xc (16) where θe is the angle position between the d-axix of the synchronously rotating and the stationary reference frames, and it is assumed that the quantities are balanced. The angular position of the rotor flux vector (ψ¯r ) related to the d-axis of the stationary reference frame may be calculated by means of the rotor flux components in this reference frame ( ψdr , ψqr ) as follows:
ψqr θe = arctan (17) ψdr where θe is the angular position of the rotor flux vector. Using the field-orientation control principle [5] the current component ieds is aligned in the direction of the rotor flux vector ψ¯r , and the current component ieqs is aligned in the direction perpendicular to it. At this condition, it is satisfied that: e e = 0, ψdr = |ψ¯r | (18) ψqr Therefore, taking into account the previous results, the equation of induction motor torque (15) is simplified to: 3p Lm e e ψ i = KT ieqs (19) 4 Lr dr qs where KT is the torque constant, and is defined as follows: Te =
KT =
J w˙ m + Bwm + TL = Te
(20)
∗
e denotes the command rotor flux. where ψdr
With the above mentioned proper field orientation, the dynamic of the rotor flux is given by [5]: e dψdr ψe Lm e + dr = i dt Tr Tr ds
ADAPTIVE SLIDING GAIN
In general, the mechanical equation of an induction motor can be written as:
3p Lm e∗ ψ 4 Lr dr
(21)
Then, the mechanical equation (14) becomes:
(14)
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w˙ m + a wm + f = b ieqs
(22)
where the parameters are defined as:
Then the sliding surface is defined as:
KT TL B , b= , f= ; (23) J J J Now, we are going to consider the previous mechanical equation (22) with uncertainties as follows:
a=
w˙ m = −(a + a)wm − (f + f ) + (b + b)ieqs
(24)
where the terms a, b and f represents the uncertainties of the terms a, b and f respectively. It should be noted that these uncertainties are unknown, and that the precise calculation of its upper bound are, in general, rather difficult to achieve. Let us define define the tracking speed error as follows: ∗ e(t) = wm (t) − wm (t)
(25)
∗ where wm is the rotor speed command.
Taking the derivative of the previous equation with respect to time yields: ∗ e(t) ˙ = w˙ m − w˙ m = −a e(t) + u(t) + d(t)
(26)
where the following terms have been collected in the signal u(t), ∗ ∗ (t) − f (t) − w˙ m (t) (27) u(t) = b ieqs (t) − a wm
S(t) = e(t) +
Now, we are going to propose the sliding variable S(t) with an integral component as: t (a + k)e(τ ) dτ (29) S(t) = e(t) +
(30)
Then, we can design a variable structure speed controller, that incorporates an adaptive sliding gain, in order to control the AC motor drive, as follows: ˆ u(t) = −k e(t) − β(t)γ sgn(S)
(31)
where the k is the gain defined previously, βˆ is the estimated switching gain, γ is a positive constant, S is the sliding variable defined in eqn. (29) and sgn(·) is the signum function. The switching gain βˆ is adapted according to the following updating law: ˙ βˆ = γ |S|
ˆ β(0) =0
(32)
where γ is a positive constant that let us choose the adaptation speed for the sliding gain. In order to obtain the speed trajectory tracking, the following assumptions should be formulated: (A 1) The gain k must be chosen so that the term (a + k) is strictly positive. Therefore the constant k should be k > 0. (A 2) There exits an unknown finite non-negative switching gain β such that β > dmax + η where dmax ≥ |d(t)| constant.
η>0
∀ t and η is a positive
Note that this condition only implies that the uncertainties of the system are bounded magnitudes. (A 3) The constant γ must be chosen so that γ ≥ 1. Theorem 1: Consider the induction motor given by equation (24). Then, if assumptions (A 1), (A 2) and (A 3) are verified, the control law (31) leads the rotor mechanical speed ∗ (t) wm (t) so that the speed tracking error e(t) = wm (t) − wm tends to zero as the time tends to infinity. The proof of this theorem will be carried out using the Lyapunov stability theory. Proof : Define the Lyapunov function candidate: V (t) =
0
where k is a constant gain, and a is a parameter that was already defined in equation (23).
(a + k)e(τ ) dτ = 0
0
and the uncertainty terms have been collected in the signal d(t), d(t) = −a wm (t) − f (t) + b ieqs (t) (28) In order to compensate for the above described uncertainties that are presented in the system, it is proposed a sliding adaptive control scheme. In the sliding control theory, the switching gain must be constructed so as to attain the sliding condition [19], [20]. In order to meet this condition a suitable choice of the sliding gain should be made to compensate for the uncertainties. To select the sliding gain vector, an upper bound of the parameter variations, unmodelled dynamics, noise magnitudes, etc, should be known, although in practical applications there are situations in which these bounds are unknown, or at least difficult to calculate. A solution could be to choose a sufficiently high value for the sliding gain, but this approach could cause a to high control signal, or at least more activity control than necessary in order to achieve the control objective. One possible way to overcome this difficulty is to estimate the gain and to update it by some adaptation law, so that the sliding condition is achieved.
t
1˜ ˜ 1 S(t)S(t) + β(t) β(t) 2 2
(33)
where S(t) is the sliding variable defined previously and ˜ = β(t) ˆ − β. β(t)
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Its time derivative is calculated as: ˜ β(t) ˜˙ ˙ + β(t) V˙ (t) = S(t)S(t) ˜ β(t) ˆ˙ = S · [e˙ + (a + k)e] + β(t)
When the sliding mode occurs on the sliding surface (30), ˙ then S(t) = S(t) = 0, and therefore the dynamic behavior of the tracking problem (26) is equivalently governed by the following equation:
= S · [(−a e + u + d) + (k e + a e)] + β˜ γ|S| = S · [u + d + k e] + (βˆ − β)γ|S| ˆ sgn(S) + d + k e + (βˆ − β)γ|S| = S · −k e − βγ ˆ sgn(S) + βγ|S| ˆ = S · d − βγ − βγ|S| ˆ ˆ = d S − βγ|S| + βγ|S| − βγ|S|
˙ S(t) =0
then
(34)
(35) V˙ (t) ≤ 0
e(t) ˙ = −(a + k)e(t)
(40)
Then, under assumption (A 1), the tracking error e(t) converges to zero exponentially.
≤ |d||S| − βγ|S| ≤ |d||S| − (dmax + η)γ|S| = |d||S| − dmax γ|S| − η γ|S| ≤ −η γ|S|
⇒
(36)
It should be noted that in the proof have been used the equations (29), (26), (31) and (32), and the assumptions (A 2) and (A 3).
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 (40), where the tracking error tends to zero. Finally, the torque current command, ie∗ qs (t), may be obtained directly substituting eqn. (31) in eqn. (27): 1 ∗ ∗ ie∗ + w˙ m +f (41) k e − βˆ γ sgn(S) + a wm qs (t) = b
Using the Lyapunov’s direct method, since V (t) is clearly positive-definite, V˙ (t) is negative semidefinite and V (t) tends ˜ tends to infinity, then the equilibto infinity as S(t) and β(t) ˜ rium at the origin [S(t), β(t)] = [0, 0] is globally stable, and ˜ are bounded. Since S(t) therefore the variables S(t) and β(t) is bounded then it is deduced that e(t) is bounded.
Therefore, the proposed variable structure speed control with adaptive sliding gain resolves the speed tracking problem for the induction motor, with some uncertainties in mechanical parameters and load torque.
On the other hand, making the derivative of equation (29) it is obtained,
In this section we will study the speed regulation performance of the proposed sensorless adaptive sliding-mode field oriented control versus reference and load torque variations by means of simulation examples. The block diagram of the proposed robust control scheme is presented in figure 1, where the block ‘VSC Controller’ represent the proposed adaptive sliding-mode controller, and it is implemented by equations (29), (41), and (32). 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 ‘dq e → abc’ makes the conversion between the synchronously rotating and stationary reference frames, and is implemented by equation (16). The block ‘Current Controller’ consists of a three hysteresis-band current PWM control, 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. e∗ The block ‘ie∗ ds Calculation’ provides the current reference ids from the rotor flux reference through the equation (21). The block ‘wr and Flux Calculation’ represent the proposed rotor speed estimator and flux calculator, and is implemented by the equations (11), (1) and (2) respectively and the block ‘IM’ represents the induction motor.
˙ S(t) = e(t) ˙ + (a + k)e(t)
(37)
then, substituting the equation (26) in the above equation, ˙ S(t) = −ae(t) + u(t) + d(t) + (a + k)e(t) = ke(t) + d(t) + u(t)
(38)
˙ From equation (38) we can conclude that S(t) is bounded because e(t), u(t) and d(t) are bounded. Now, from equation (34) it is deduced that ˙ − β γ d |S(t)| V¨ (t) = d S(t) dt ˙ which is a bounded quantity because S(t) is bounded.
(39)
Under these conditions, since V¨ is bounded, V˙ is a uniformly continuous function, so Barbalat’s lemma let us conclude that V˙ → 0 as t → ∞, which implies that S(t) → 0 as t → ∞. 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 [20].
IV. S IMULATION R ESULTS
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wr∗ +
e-
−6
VSC Controller
ie∗ qs -
ie∗ qs-
dq e → abc ie∗ ds-
Limiter
6 w ˆr-
θe
Field Weakening
-
e∗ ψdr
ie∗ ds Calculation w ˆr
-
Current Controller
6
θe Calculation ψ 66 ψ dr
i∗abc -
qr
wr and Flux Calculation
Pulses ? PWM Inverter
iabc vabc ? IM
Fig. 1.
Block diagram of the proposed adaptive sliding-mode control
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 Ω, Rr = 0.228 Ω, Ls = 35.5 mH, Lr = 35.5 mH, and Lm = 34.7 mH. The system has the following mechanical parameters: J = 1.357 kg.m2 and B = 0.05 N.m.s. It is assumed that there are an uncertainty around 20 % in the system parameters, that will be overcome by the proposed sliding control. In addition the following values have been chosen for the ˆ controller parameters: α = 150, k = 25, γ = 30 and β(0) = 0. In this 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 60 rad/s. Then, the rotor speed is constant until t = 0.9 s, and finally decelerates down to 0 rad/s. The system starts with an initial load torque TL = 0 N.m, and at time t = 0.6 s the load torque steps from TL = 0 N.m to TL = 225 N.m, and as before, it is assumed that there is an uncertainty around 20 % in the load torque. Figures 2 and 3 show the desired rotor speed (dashed dotted line), the real rotor speed (dashed line) and the estimated rotor speed (solid line ). As it may be observed, after a transitory time in which the sliding gain is adapted, the rotor speed tracks the desired speed in spite of system uncertainties. Nevertheless, at time t = 0.6 s a little speed error can be observed. This error appears because there is a torque increment at this time, and then the control system lost the so called ‘sliding mode’ because the actual sliding gain is too small for the new uncertainty introduced in the system due to the load torque increment. But, after a small time, the sliding gain is adapted so that this gain can compensate for the system
uncertainties and then the rotor speed error is eliminated. In this figure, it can also be seen that the proposed estimator performs very well in a low speed region. Figure 3 shows, the first instants enlargement of the figure 2. In this figure it may be seen that the proposed rotor speed adaptation law estimates the real rotor speed with quite a bit precision and rapidity. Figure 4 presents the time evolution of the estimated sliding gain. The sliding gain starts from zero and then it is increased until its value is high enough in order to compensate for the system uncertainties. Then, after t = 0.3 s, the sliding gain is remained constant because the system uncertainties remain constant as well. Later at time 0.6 s, there is an increment in the system uncertainties caused by the rise in the load torque. Therefore the sliding gain is adapted once again in order to overcome the new system uncertainties. As it can be observed in the figure after the sliding gain is adapted, it remains constant again, since the system uncertainties remains constant as well. It should be noted that the adaptive sliding gain allows to employ a smaller sliding gain. In this way, it is not necessary to choose the gain value high enough to compensate for all the possible system uncertainties, because with the proposed adaptive scheme the sliding gain is adapted (if necessary) when a new uncertainty appears in the system in order to surmount this uncertainty. Figure 5 shows the time evolution of the sliding variable. In this figure it can be seen that the system reach the sliding condition (S(t) = 0) at time t = 0.3 s, but the system lost this condition at time t = 0.6 s due to the torque increment which produces an increment in the system uncertainties that could not be compensated by the actual value of the sliding gain.
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Figure 6 shows the d-component of estimated stator current (solid line), and the real stator current (dashed line). In this figure it may be observed that the stator current is estimated very fast. Furthermore, 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. Next, in the constant speed region the current is lower because the motor torque only has to compensate the friction and the load torque. Then, at time t = 0.6 s the current increases because the load torque has been increased, and at time t = 0.9 s the current decreases in the deceleration zone.
25
Rotor Speed (rsd/s
20
10
5
0
Figure 7 shows the motor torque. As in the case of the current (fig. 6), the motor torque has a high initial value in the speed acceleration zone, then the value decreases in a constant region, later at time t = 0.6 s the current increases due to the load torque increment, and at time t = 0.9 s the torque decreases in the deceleration zone. This figure shows that the so-called chattering phenomenon appears in the motor torque. Although this high frequency changes in the torque will be reduced by the mechanical system inertia, they could cause undesirable vibrations in the rotor, which may be a problem for certain systems. However, for the systems that do not support this chattering, it may be eliminated substituting the sign function by the saturation function in the control signal [19].
−5
0
0.02
0.04
0.06
0.08
0.1
Time (s)
Fig. 3.
Reference and real rotor speed signals (enlargement) (rad/s)
45 40 35
Sliding Gain
30 25 20 15
70 60
w−
10
wr
5
r
w*
50
r
Rotor Speed (rad/s)
15
0
0
0.2
0.4
40
0.6 0.8 Time (s)
1
1.2
1.4
30
Fig. 4.
Sliding Gain
20 10 0 −10
0
Fig. 2.
0.2
0.4
0.6 0.8 Time (s)
1
1.2
1.4
Reference and real rotor speed signals (rad/s)
V. C ONCLUSIONS In this paper a sensorless adaptive sliding mode vector control has been presented. The rotor speed adaptation law is based on stator current equations and rotor flux equations in the stationary reference frame, and using simulation examples it is demonstrated that this adaptation law performs well in a low speed region. It is proposed a new adaptive 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 error or by changes
in the load torque. Moreover, the proposed variable structure control incorporates an adaptive algorithm to calculate the sliding gain value. The adaptation of the sliding gain, on the one hand avoids the necessity of computing the upper bound of the system uncertainties, and on the other hand allows to employ as smaller sliding gain as possible in order to overcome the actual system uncertainties. Therefore, the control signal of our proposed variable structure control schemes will be smaller that the control signals of the traditional variable structure control schemes, because in the last one the sliding gain value should be chosen high enough to overcome all the possible uncertainties that could appear in the system along the time. The closed loop stability of the presented design has been proved thought Lyapunov stability theory. Finally, by means of simulation examples, it has been shown that the proposed control scheme presents a good performance in practice, and that the speed tracking objective is achieved under uncertainties in the parameters and load torque.
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0.2
300
0
250
−0.2
200 Motor Torque (N*m)
Sliding Variable
−0.4 −0.6 −0.8 −1
50
−50
−1.4
0
0.2
0.4
0.6 0.8 Time (s)
Fig. 5.
1
1.2
−100
1.4
Sliding Variable
300
−− ds
i i
ds
200
100
0
−100
−200
0
0.2
0.4
Fig. 6.
0.6 0.8 Time (s)
1
0
0.2
0.4
Fig. 7.
400
−300
100
0
−1.2
−1.6
150
1.2
1.4
Stator Current isa (A)
ACKNOWLEDGMENTS The authors are grateful to the Basque Country University for the support of this work through the research project 1/UPV 00146.363-E-16001/2004. R EFERENCES [1] A BBONDANTI , A. AND B RENNEN , M.B., 1975, Variable speeed induction motor drives use electronic slip calculator based on motor voltages and currents, IEEE Trans. Industry Applications, vol.IA-11, pp.483-488. [2] BARAMBONES , O. AND G ARRIDO , A.J., 2004, A sensorless variable structure control of induction motor drives, Electric Power Systems Research, 72, 21-32. [3] B ENCHAIB , A. AND E DWARDS , C., 2000, Nonlinear sliding mode control of an induction motor, Int. J. of Adaptive Control and Signal Procesing, 14, 201-221. [4] B OSE , B.K., 1993, Power electronics and motion control-technology status and recent trends IEEE Trans. on Ind. Appl., vol.29, pp.902-909. [5] B OSE , B.K., 2001, Modern Power Electronics and AC Drives., Prentice Hall, New Jersey. [6] C HERN , T.L., C HANG , J. AND T SAI , K.L.,1998, Integral variable structure control based adaptive speed estimator and resistance identifier for an induction motor. Int. J. of Control, 69, 31-47. [7] H UANG , S.J., H UANG , C.L. AND L IN , Y.S., 1998, Sensorless speed identification of vector controlled induction drives via neural network based estimation., Electric Power System Research, 48, 1-10.
0.6 0.8 Time (s)
1
1.2
1.4
Motor torque (N.m)
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