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An Improved Rotor Resistance Estimator for Induction Motors
Proceedings of the 15th IFAC Symposium on System Identification Saint-Malo, France, July 6-8, 2009
An Improved Rotor Resistance Estimator for Induction Motors G. Kenn´ e ∗ T. Ahmed-Ali ∗∗ F. Lamnabhi ∗∗∗ A. Arzand´ e ∗∗∗∗ J. C. Vannier † ∗
Laboratoire d’Automatique et d’Informatique Appliqu´ ee (LAIA), D´ epartement de ´ G´ enie Electrique, Universit´ e de Dschang, B.P. 134 Bandjoun, Cameroun, Email: [email protected] ∗∗ Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen (GREYC), Universit´ e de Caen, France, Email: [email protected] ∗∗∗ Laboratoire des Signaux et Syst` emes (L2S), CNRS–SUPELEC, Universit´ e Paris XI, 3 Rue Joliot Curie, 91192 Gif-sur-Yvette, France, Email: [email protected] ∗∗∗∗ ´ ´ ´ D´ epartement Energie, Ecole Sup´ erieure d’Electricit´ e (SUPELEC), 3 Rue Joliot Curie, 91192 Gif-sur-Yvette, France, Email: [email protected] † ´ ´ ´ D´ epartement Energie, Ecole Sup´ erieure d’Electricit´ e (SUPELEC), 3 Rue Joliot Curie, 91192 Gif-sur-Yvette, France, Email: [email protected]
Abstract: In this paper, an online rotor resistance estimator for induction motor adaptive control is presented. The proposed algorithm is an improvement of the previous work Kenne et al.. The rotor resistance scheme uses the rotor speed, the stator current, voltage provided by the controller, time-derivatives of the stator current and voltage. The time derivatives of the stator current and voltage are estimated by using 2nd order sliding mode observer. The convergence of the estimated rotor resistance to the nominal value in finite time is achieved when the classical persistent excitation condition is satisfied by the input signal. Experimental results with 100% variation of the stator resistance and online variation of the rotor resistance show that the proposed algorithm gives better performance compared to the previous work. Keywords : Time-varying parameter estimation, second order sliding mode, equivalent control. 1. INTRODUCTION It is well known in literature (e.g., see Marino et al. [2000], Castaldi et al. [2005], Marino et al. [2005]) that the rotor resistance of an induction motor (IM) may vary up to 100% during operation of the IM and that due to this large variation, the online estimated value of this parameter is required to design an efficient feedback control of high performance induction motors using the rotor speed, stator current and stator voltage signals measurements. In the area of IM parameter estimation and adaptive control of IM, various approaches have been recently developed Marino et al. [2000], Bartolini et al. [2003], Castaldi et al. [2005], Marino et al. [2005], Barut et al. [2005], Picardi and Scibilia [2006], Koubaa [2006a,b], Castillo et al. [2007], Roncero-S´ anchez et al. [2007], Wang et al. [2007], Mezouar et al. [2006, 2008], but only partial and quite weak results have been obtained in term of the investigation of the online variation of these parameters in real-time experiments. Amount the above contributions, the problem of IM time-varying parameter has been studied in Bartolini et al. [2003], Barut et al. [2005], Picardi and Scibilia [2006], Mezouar et al. [2006, 2008] but there is a lack of experimental results which are the only way to verify the effectiveness of those methods. In Koubaa [2006a,b], ⋆ The main part of the experimental setup used in this work has been supported by the “D´epartement Energie, Ecole Sup´erieure d’Electricit´ e, Gif-sur-Yvette, Paris, France”.
978-3-902661-47-0/09/$20.00 © 2009 IFAC
Wang et al. [2007], interesting algorithms for IM parameter estimation are proposed using least square technique but online variation of the IM parameters was not investigated. In Roncero-S´ anchez et al. [2007], a method for rotor resistance estimation for indirect field oriented control of IM based on reactive power reference model is presented under motoring and generating modes. Sensitivity of the algorithm to errors in other machines parameters is investigated but without variation of the rotor resistance. In Castillo et al. [2007], a robust nested sliding mode regulation with application to rotor flux modulus and rotor speed of IM with unknown load torque has been introduced. The variation of the stator/rotor resistance has been investigated but the estimation of these parameters was not achieved. Therefore, the problem of combining parameter adaptation with rotor flux observer and rotor speed/flux tracking remains open in general and particularly when the unknown parameters are assumed to be time-varying. This paper revisits this problem and establishes an online rotor resistance estimator for IM adaptive control. The proposed method is an improvement of the previous work Kenne et al.. In the latter contribution, real-time speed and flux adaptive control of induction motors using unknown time-varying rotor resistance and load torque has been proposed. The rotor resistance estimation scheme used in Kenne et al. demands information about timederivatives of stator currents and voltages. This method presented many interesting properties (including high de-
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15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 coupling property) but has two main drawbacks. In this scheme the above hardly measured variables have been defined by using high gain observers and the errors of the time-derivative estimation of the stator currents were very high and this affects the rotor resistance estimation accuracy. A global convergence and stability analysis, taking into account the interconnections between the different estimators has not been investigated. In this paper, we used second-order sliding mode observers to define the hardly measured variables (time-derivatives of stator currents and voltages) and a global convergence and stability analysis is investigated. In Section 2, the description of IM model and problem statement are presented. The design procedure of the rotor resistance identifier is revisited in Section 3 using 2nd order sliding mode observer and a global convergence and stability analysis is given. Experimental results of online implementation are presented in Section 4 to verify the effectiveness of the proposed approach. Finally, some concluding remarks are given in Section 5.
are negligible compared to other dynamics of the induction motor states variables (e.g., R˙ r ∼ = 0) = 0 and ω˙ ∼ = 0, R˙ s ∼ and that the other parameters are constants. Remark 1. The stator voltage usually can be discontinuous. This is the case when the IM is fed by PWM voltagesource-inverter. Assumption (i) can be satisfied if inverter reference voltages are used in the estimation scheme instead of the measured voltages. Our goal is to design a rotor resistance estimation algorithm using 2nd order sliding mode observer when the motor is driven in closed-loop with a controller which stabilizes the rotor flux magnitude and the rotor speed to references values with adaptation of the rotor resistance and load torque. The dynamic equation of the IM used for the rotor resistance identifier is obtained by differentiating (3) combined with (2) and by taking into account assumption (iii) (for more details see Kenne et al.). This transformation gives the following differential equation d2 is = f0 + Rr f1 dt2
2. DESCRIPTION OF THE IM MODEL Assuming linear magnetic circuits, the dynamics of a balanced IM, expressed in a fixed reference frame attached to the stator are given by Leonhard [1984], Marino et al. [2000] dω Te Dω TL = − − (1) dt m m m Rr dλr Rr = (− I + np ωJ)λr + M is (2) dt Lr Lr dis M Rr (− I + np ωJ)λr =− dt σLs Lr Lr 1 M 2 Rr 1 − (Rs + vs (3) )is + 2 σLs Lr σLs 0 −1 1 0 , , J= where I = 1 0 0 1 λra vsa isa is = . , λr = , vs = λrb vsb isb In (1) (2) and (3), the state variables are rotor speed ω, rotor flux (λra , λrb ) and stator currents (isa , isb ); the control inputs are stator voltages (vsa , vsb ); the measured variables are (ω, isa , isb ) while (λra , λrb ) are not measured; the parameters are the external load torque TL , total motor and load moment of inertia m, rotor and stator winding resistances (Rr , Rs ), inductances (Lr , Ls ) and mutual inductance M ; the number of pole pairs is np and the damping gain is D. To simplify the notations in (1), (2), 2 (3), we use σ = 1− LM (leakage parameter), Te = µiTs Jλr s Lr
(4)
where f0 and f1 are given as follows dv
dis dis + ωJ β2 is + β3 vs + β1 dt dt dt dis f1 = γ2 vs − γ3 is − γ4 dt
f0 = γ1
s
− Rs
and β1 , β2 , β3 , γ1 , γ2 , γ3 , γ4 are IM parameters: β1 = n p γ1 =
1 σLs
−np np Rs β3 = σLs σLs 1 Rs 1 γ2 = γ3 = and γ4 = . σLs Lr σLs Lr σLr
β2 =
The model given by (4) is an adequate representation of the motor dynamics (Kenne et al.) if dω |ω| ≪ 2 . dt Tr
(5)
Introducing the variables x1 = is , x2 = dis /dt, u1 = vs and u2 = dvs /dt, model (4) can be rewritten as x˙ 1 = x2 , u˙ 1 = u2 ,
x˙ 2 = f (t, x1 , x2 , u1 , u2 ) + ξ(t, x1 , x2 , u1 , u2 ) 3n M (electromagnetic torque) with the constant µ = 2Lp r . y1 = x1 , Note that the parameter Rr is typically uncertain because Rr may vary up to 100% of its nominal value due to rotor y2 = u 1 (6) heating. The following assumptions will be considered unwhere the nominal part of the system is represented by til further notice: (i). The stator current and voltage are continuous and the function f and while the uncertainties are concenbounded with time-derivatives bounded piecewise-continuous. trated in the term ξ(t, x1 , x2 , u1 , u2 ). Note that the term (ii). The rotor resistance Rr ∈ ΩRr , where ΩRr is a f (t, x1 , x2 , u1 , u2 ) is expressed as function of the derivacompact set of R. tives of the stator current and voltage which cannot be (iii). It is also assumed that the rate of variation of the obtained directly using numerical differentiation due to the rotor resistance Rr , stator resistance Rs and rotor speed ω presence of noise.
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15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 3. DESIGN PROCEDURE OF THE ROTOR RESISTANCE
Remark 3. To minimize the influence of the measurement noise in (7) and (8), we approximated the stator current and voltage time-derivatives as solutions of the following equations
ESTIMATE
3.1 State observer design To estimate the unavailable stator current and voltage derivatives let us consider the following 2nd order sliding mode observer Levant [1998b] x ˆ˙ 1i = x ˆ2i + zx1i , i = a, b x ˆ˙ 2i = z 2 ,
(7)
xi
3.2 Rotor resistance adaptation algorithm
ˆ2i + zu1i , i = a, b u ˆ˙ 1i = u u ˆ˙ 2i = z 2 ,
(8)
zu2i
are given by the
zx1i ,
where the variables following expressions
ui zx2i ,
zu1i
and
x1i − x1i |1/2 sign(ˆ x1i − x1i ) zx1i = −λxi |ˆ x1i − x1i ) zx2i = −αxi sign(ˆ zu1i zu2i
1/2
u1i − u1i | = −λui |ˆ
dx ¯˙ 2i +x ¯˙ 2i = x ˆ˙ 2i , i = a, b (15) dt du ¯˙ 2i τu +u ¯˙ 2i = u ˆ˙ 2i , i = a, b (16) dt where τx and τu are suitable positive numbers which → 0. τx
In this section the method used in Kenne et al. is revisited and more details are given concerning the convergence conditions. Assuming that the unavailable states x2i and u2i are provided in finite time by the observer (7) and (8), the IM model (4) can be rewritten as
(9)
sign(ˆ u1i − u1i )
u1i − u1i ). (10) = −αui sign(ˆ The solutions of the above observer are understood in the Filippov sense Filippov [1988]. Taking ex1i = x ˆ1i − x1i , ˆ2i − x2i , the ˆ1i − u1i , and eu2i = x ˆ2i − x2i , eu1i = u ex2i = x following dynamics errors equations are obtained
x˜˙ 2i = f˜0i + Rr f˜1i , i = a, b
(17)
with x ˜2i = xˆ2i ˆ2a − Rs x f˜0a = γ1 u ˜2a − ω β2 x1b + β3 u1b + β1 x˜2b ˆ2b − Rs x ˜2b + ω β2 x1a + β3 u1a + β1 x˜2a f˜0b = γ1 u f˜1i = γ2 u1i − γ3 x1i − γ4 x˜2i .
Estimation of the parameter Rr can be achieved using (17) as the reference model of the IM. This estimation can be treated separately using a state observer combined (11) e˙ x2i = F (t, x1 , x2 , u1 , u2 ) − αxi sign(ex1i ) with a static parameter estimation law or simultaneous e˙ u1i = eu2i − λui |eu1i |1/2 sign(eu1i ) using an adaptive state observer combined with a dynamic (12) parameter estimation law. The first approach requires the e˙ u2i = −u˙ 2i − αui sign(eu1i ) inversion of the system (17). However, the results obtained with using this approach always exhibit peak values. These peak F (t, x1 , x2 , u1 , u2 ) = −f (t, x1 , x2 , u1 , u2 )−ξ(t, x1 , x2 , u1 , u2 ). values are due to the singularity of the denominators of Assuming that the states of the system is bounded as the static parameter estimation law. Therefore, it is more reported in assumption (i), then there exists positive convenient to choose a dynamic parameter estimation law combined with an adaptive state observer (2nd approach). constants µF and µu such that the inequalities In order to design the rotor resistance estimate using |F (t, x1 , x2 , u1 , u2 )| < µF (13) the 2nd approach, let us consider the following adaptive observer |u˙ 2i | < µu (14) ˆ r f˜1i + f˜0i − k ˜ sign(x ˆ˜˙ 2i = R ˆ˜2i − x hold ∀t, x1 , x2 , u1 and u2 . x ˜2i ), i = a, b. (18) e˙ x1i = ex2i − λxi |ex1i |1/2 sign(ex1i )
xi
Theorem 1. Assuming that assumptions (i) to (iii) are satisfied and conditions (5), (13) and (14) hold, then the variables of the observer (7) and (8) converge to the states of the nonlinear model of the IM (6). The proof of the convergence of the states estimates to the real states can be found in Levant [1998a]. The variables of the observer (7) and (8) converge in finite time to the states of the nonlinear model of the IM (6) in the ideal case (when the available states are not contaminated with measurement noise N (t)). Otherwise, if sup N (t) = ζ, the above observer provides estimation of x2i and u2i with accuracy proportional to (ζ)1/2 (µF )1/2 and (ζ)1/2 (µu )1/2 , respectively. Remark 2. The first attempt to choose the observer parameter can be achieved by applying the guidelines given in Levant [1998a]. e.g. αxi and λxi can be chosen as follows. αxi = a1 µF and λxi = a2 (µF )1/2 with a1 = 1.1 and a2 = 1.5.
ˆ r − Rr , the dynamics ˆ˜2i − x˜2i and eRr = R Defining exi ˜ = x of the observer error can be computed using (17) as ˜ e˙ xi ˜ sign(exi ˜ ), i = a, b. ˜ = eRr f1i − kxi
(19)
To achieve the design of the rotor resistance identifier the following additive assumptions are required. Assumption (iv). It is assumed that the following rotor resistance identifiability condition holds. |f˜1i (t)| ≥ δ > 0, i = a, b.
(20)
Assumption (v). It is assumed that the gain of the adaptive observer is chosen such that ˜ kxi ˜ ≥ |eRr f1i |max , i = a, b.
(21)
Remark 4. The identifiability condition (20) can be substituted by the following persistency of excitation condition
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15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 condition Fradkov et al. [1999]. There exists α > 0, T > 0, t0 > 0 such that for all t ≥ 0 t+T
f˜1 (s)f˜1 (s)T ds ≥ αI > 0.
(22)
t
The physical circumstances for which condition (20) or (22) holds, are that the rotor currents of the IM are not zero. Taking into consideration Assumption (v), a sliding mode regime will occur on the manifold exi ˜ = 0 in finite time. Therefore the dynamic equation of the adaptive observer error can be rewritten as ˜ e˙ xi ˜ signeq (exi ˜ ) = 0, i = a, b ˜ = eRr f1i − kxi f˜T ux˜ or eRr = 1 eq (23) ||f˜1 ||2
where uTx˜eq = kx˜a signeq (ex˜a ), kx˜b signeq (ex˜b ) is the equivalent control which cannot be computed since the rotor resistance estimation error is not available. Using the average control as the approximation of the equivalent control ux˜eq Utkin [1992], we considered the following rotor resistance adaptation law (kRr > 0 is the tuning parameter). ˜T ˆ˙ r = −kRr sign f1 ux˜eq . R (24) ||f˜1 ||2 The main result can be summarized by the following theorem. Theorem 2. Given the reference model (17) and the adaptive observer (18). Suppose that the parameter of the observer (18) is selected according to (21) with assumptions (i) to (iv) satisfied, then the rotor resistance estimate ˆ r given by (24) converges in finite time to the nominal R value Rr . Proof. In order to prove Theorem 2, the following Lyapunov candidate function is considered. 1 W = e2Rr . 2
(25)
Its time-derivative is: ˙ = eRr e˙ Rr . W (26) Under assumptions (iii), (iv) and (v), (26) can be rewritten as ˙ = −kRr |eRr |. W (27) From (27), we can deduce that, the rotor resistance estimation error converges in finite time to zero (i.e., the rotor ˆ r converges to the nominal value resistance estimate R Rr ) if the identifiability condition (20) or equivalently the persistency of excitation condition (22) is satisfied. 3.3 Global convergence and stability analysis The global convergence and stability analysis taking into account the interconnections between the different estimators are based on the separation principle theorem. The finite-time convergence of the stator current and voltage time-derivatives observers and the rotor resistance estimator allows to design the observer/estimator and the control
law separately. i.e., the separation principle is satisfied (Levant [1998a], Levant and Levantovsky [1993]). The only requirement for its implementation is the boundedness of the states of the system (e.g., boundedness of fˆω , fˆλr , ˆbω and ˆbλr ) in the operational domain and the persistency of excitation condition. This condition is satisfied under normal operating conditions of the IM. Therefore, if the controller is known to stabilize the IM (as it has been proved in Kenne et al. for the non-adaptive controller), one of the admissible and recommended ways is to choose the observer dynamics fast enough to provide for the exact evaluation of the stator current and voltage timederivatives to the rotor resistance estimator which should ˆ r in an acceptable range and maintain the estimate R achieve its convergence before leaving some operational domain, where the stabilization is assured. 4. EXPERIMENTAL RESULTS A 5kW induction motor whose data are reported in Appendix has been used for the experiments. The experimental setup includes a development system DSP1103, an input/output electronics board (for analog/digital conversions) and a Personal Computer (PC). A PWM power converter with switching frequency of 10 kHz is controlled by a DSP. The external load torque TL is produced by a loaded dc generator. The motor instantaneous speed is measured by an optical incremental encoder with 1024 lines per revolution. The stator currents are measured by Hall-type sensors. All measured electrical parameters are converted by 16-b A/D converter channels with 1µ s conversion time. A DSP1103 performs data acquisition and implements in real-time within the Matlab/Simulink environment software with sampling time of 105.8µ s. In all experiments the measured stator currents used in the estimation scheme were passed through a low-pass filter with cut-off frequency of 1000 rad/s. We performed two experiments. In both experiments, the magnitude of the rotor flux is required to reach the reference value of 0.7W b in 0.75s and the unloaded motor is required to reach the speed reference value of 136.30rad/s in 3.6s. In both cases, the experiments are carried out during motor start-up and after the motor is operated with load torque for over 60 min. The parameters α and λ of the 2nd order sliding mode observer (7) and (8) have been selected according to the procedure described in Remark 2 and a finer tuning has been performed to reduce the observer errors and finally the following tuning parameters were used. αxi = 3 × 107 , λxi = 3000, αui = 108 , λui = 5000, τx = τu = 0.2ms. In both experiments, the parameters of the rotor resistance identifier (24) were chosen as follows. kx˜i = 675000, kRr = 0.65. The equivalent control has been approximated using first order low-pass filter with time-constant of 0.1ms. After the motor start-up and in both cases, the external unknown load torque TL estimated using the method described in Kenne et al. is applied. In both experiments the controller parameters and the load torque estimator parameters are the same as in Kenne et al. (i.e. cω = 200, cλr = 200, kvsd = 7, kvsq = 1, kω = 130 and kTL = 65). To verify the effectiveness of the proposed method in the “hot case” operation of the IM, both experiments were carried out using the estimate
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15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009
Control voltage component
Stator voltage observer error 10 5
(V)
(v)
200 0
0 −5
−200 2
4
6
8 10 12 Stator current component
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20
2
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(A)
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(b)
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Fig. 1. Experiment 1: Performance of the proposed method with 100% variation of the stator resistance Rs (from 0.22Ω to 0.44Ω). (a): Control voltage, stator current and applied load torque. (b): Observer/adaptive errors. i− Rotor speed
i− Stator resistance with 100% variation (from 0.22Ω to 0.44Ω)
150
0.45
100
(Ohm)
(rad/s)
0.4
50
ωref
0.35 0.3 0.25
ω: actual value 0 0
2
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0.8
1.2
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(Ohm)
(Wb)
8
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ii− Rotor resistance
0.4
RrN estimated value
0.8 0.6
0.2 0 0
Reference value Estimated value 2
4
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Time (s)
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(a)
(b)
Fig. 2. Experiment 1: Performance of the proposed method with 100% variation of the stator resistance Rs (from 0.22Ω to 0.44Ω). (a): Rotor speed and rotor flux magnitude. (b): Stator resistance and rotor resistance estimate.
ˆ r given by (24) in the adaptive controller Kenne et al. R and over-estimated initial values have been chosen for the rotor resistance estimation dynamics. In the first experiment, the robustness property with respect to stator resistance uncertainty is tested with 100% variation of the stator resistance (from 0.22Ω up to 0.44Ω). The results obtained in this case are reported in Fig. 1 and Fig. 2. In the second experiment, the performance of the algorithm to track the variation of the rotor resistance has been investigated. In this case, the online variation of the rotor resistance has been carried out using a 3−phase variable rheostat for rotor wound induction motors. The experimental results are shown in Fig. 3. From Fig. 3, one may conclude that the proposed estimator is slow. From the fact that the value of the additional resistance (0.47Ω) has been introduced progressively/manually and not instantaneously by using an external 3−phase variable rheostat for rotor wound induction motor, the convergence time is less than 6 sec. The convergence time can be reduced by increasing the value of the rotor resistance estimation gain kRr (see equation (24) and the proof of theorem 2). The above results confirm as in Kenne et al. that there is no significant effect on the rotor resistance estimate and the adaptive control algorithm for a wide range of variation of the stator resistance (up to 100%). This robustness
property can be justified theoretically by the fact that the rotor resistance estimator (24) is based on the sliding mode theory which is a known technique for improving robustness under parameter uncertainties. Note that the residual parameter estimation error is due to the measurement noise, mismatching between the motor and the model parameters, ohmic heating during experiments and unmodeled dynamics. We can also note that in ˆ˜1 − both experiments, the adaptive observer error ex˜1 = x x ˜1 of the proposed method is particularly slower than the value obtained in the previous work Kenne et al.. This can justify the accuracy of the proposed rotor resistance estimation scheme and the good tracking performance of the adaptive controller. The very good behaviour of the proposed algorithm to track the variation of the parameter with very good accuracy demonstrates that this scheme can be used to detect and estimate the rotor resistance fault if its value is outside some acceptable known bounds. 5. CONCLUSION An improved rotor resistance estimator for induction motor adaptive control has been investigated. The time derivatives of the stator current and voltage are estimated using 2nd order sliding mode observer. The finite-time convergence of the estimated rotor resistance in finite time
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15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009
i− Rotor speed (rad/s)
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Fig. 3. Experiment 2: Tracking performance of the proposed method with respect to online variation of the rotor resistance.
to the nominal value has been achieved under feasible persistency of excitation condition. A global convergence and stability analysis has also been investigated. Experimental results with 100% variation of the stator resistance and online variation of the rotor resistance show that the proposed algorithm gives better performance compared to the previous work Kenne et al.. Important advantages of the proposed methodology include that it is an online method (e.g, the value of Rr can be continuously updated) in order to make the controller more beneficial for motor efficiency since very high tracking performance of the rotor resistance variation is achieved. APPENDIX: INDUCTION MOTOR DATA
Rated power Rated speed Rated torque Rated frequency Rated current Stator resistance Rotor resistance Stator inductance Rotor inductance Mutual inductance Number of pole pairs Motor-load inertia
5 kW. 1500 rpm. 32 Nm. 50 Hz. 22.9 A. RsN = 0.22 Ω. RrN = 0.52 Ω. LsN = 0.052 H. LrN = 0.0516 H. MN = 0.0495 H. np = 2. m = 0.12 kg m2 .
REFERENCES Giorgio Bartolini, Alessandro Pisano, and Pierluigi Pisu. Simplified exponentially convergent rotor resistance estimation for induction motors. IEEE Transactions on Automatic Control, 48:325–330, 2003. Murat Barut, Seta Bogosyan, and Metin Gokasan. Speed sensorless direct torque control of induction motors with rotor resistance estimation. Energy Conversion and Management, 46:335–349, 2005. P. Castaldi, W. Geri, M. Montanari, and A. Tilli. A new adaptive approach for on-line parameter and state estimation of induction motors. Control Engineering Practice, 13:81–94, 2005. B. Castillo, S. D. Gennaro, A. Loukianov, and J. Rivera. Robust nested sliding mode regulation with application
to induction motors. Proceedings of the 2007 American Control conference, New York City, USA, pages 5242– 5247, 2007. A. F. Filippov. Differential equations with discontinuous right-hand sides. Dordrecht, The netherlands: Kluwer, 1988. A. L. Fradkov, L. V. Miroshnik, and V. O. Nikiforov. Nonlinear and adaptive control of complex systems. Kluwer Academic Publishers, 1999. G. Kenne, T. Ahmed-Ali, F. Lamnabhi-Lagarrigue, and A. Arzand´e. Real-time speed and flux adaptive control of induction motors using unknown time-varying rotor resistance and load torque. to appear in IEEE Transaction on Energy Conversion. Yassine Koubaa. Application of least-squares techniques for induction motor parameters estimation. Mathematical and Computer Modelling of Dynamical Systems, 12: 363–375, 2006a. Yassine Koubaa. Asynchronous machine parameters estimation using recursive method. Simulation Modelling Practice and Theory, 14:1010–1021, 2006b. W. Leonhard. Control of electric drives. Springer Verlag, 1984. A. Levant. Robust exact differentiation via sliding mode technique. Automatica, 34:379–384, 1998a. A. Levant. Robust exact differentiation via sliding mode technique. Automatica, 34:379–384, 1998b. A. Levant and L. V. Levantovsky. Sliding order and sliding accuracy in sliding mode control. International Journal of Control, 58:1247–1263, 1993. R. Marino, S. Peresada, and P. Tomei. On-line stator and rotor resistance estimation for induction motors. IEEE Trans. On Contr. Syst. Technology, 8:570–579, 2000. R. Marino, S. Peresada, and C. M. Verrelli. Adaptive control for speed-sensorless induction motors with uncertain load torque and rotor resistance. International Journal of Adaptive Control and Signal Processing, 19: 661–685, 2005. A. Mezouar, M. K. Fellah, S. Hadjeri, and Y. Sahali. Adaptive speed sensorless vector control of induction motor using singularly perturbed sliding mode observer. IEEE Industrial Electronics-IECON 2006, pages 932– 939, 2006. A. Mezouar, M. K. Fellah, and S. Hadjeri. Adaptive sliding-mode-observer for sensorless induction motor drive using two-time-scale approach. Simulation Modelling Practice and Theory, 2008. C. Picardi and F. Scibilia. Sliding-mode observer with resistances or speed adaptation for field-oriented induction motor drives. IEEE Industrial Electronics-IECON 2006, pages 1481–1486, 2006. P. Roncero-S´ anchez, A. Garc´ia-Cerraba, and V. FeliuBatlle. Rotor resistance estimation for induction machines with indirect field-orientation. Control Engineering Practice, 15:1119–1133, 2007. V. I. Utkin. Sliding modes in optimization and control. Springer-Verlag, 1992. Kaiyu Wang, John Chiasson, Marc Bodson, and Leon M. Tolbert. An online rotor time constant estimator for the induction machine. IEEE Transactions on Control Systems Technology, 15:339–348, 2007.
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An Improved Rotor Resistance Estimator for Induction Motors G. Kenn´ e ∗ T. Ahmed-Ali ∗∗ F. Lamnabhi ∗∗∗ A. Arzand´ e ∗∗∗∗ J. C. Vannier † ∗
Laboratoire d’Automatique et d’Informatique Appliqu´ ee (LAIA), D´ epartement de ´ G´ enie Electrique, Universit´ e de Dschang, B.P. 134 Bandjoun, Cameroun, Email: [email protected] ∗∗ Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen (GREYC), Universit´ e de Caen, France, Email: [email protected] ∗∗∗ Laboratoire des Signaux et Syst` emes (L2S), CNRS–SUPELEC, Universit´ e Paris XI, 3 Rue Joliot Curie, 91192 Gif-sur-Yvette, France, Email: [email protected] ∗∗∗∗ ´ ´ ´ D´ epartement Energie, Ecole Sup´ erieure d’Electricit´ e (SUPELEC), 3 Rue Joliot Curie, 91192 Gif-sur-Yvette, France, Email: [email protected] † ´ ´ ´ D´ epartement Energie, Ecole Sup´ erieure d’Electricit´ e (SUPELEC), 3 Rue Joliot Curie, 91192 Gif-sur-Yvette, France, Email: [email protected]
Abstract: In this paper, an online rotor resistance estimator for induction motor adaptive control is presented. The proposed algorithm is an improvement of the previous work Kenne et al.. The rotor resistance scheme uses the rotor speed, the stator current, voltage provided by the controller, time-derivatives of the stator current and voltage. The time derivatives of the stator current and voltage are estimated by using 2nd order sliding mode observer. The convergence of the estimated rotor resistance to the nominal value in finite time is achieved when the classical persistent excitation condition is satisfied by the input signal. Experimental results with 100% variation of the stator resistance and online variation of the rotor resistance show that the proposed algorithm gives better performance compared to the previous work. Keywords : Time-varying parameter estimation, second order sliding mode, equivalent control. 1. INTRODUCTION It is well known in literature (e.g., see Marino et al. [2000], Castaldi et al. [2005], Marino et al. [2005]) that the rotor resistance of an induction motor (IM) may vary up to 100% during operation of the IM and that due to this large variation, the online estimated value of this parameter is required to design an efficient feedback control of high performance induction motors using the rotor speed, stator current and stator voltage signals measurements. In the area of IM parameter estimation and adaptive control of IM, various approaches have been recently developed Marino et al. [2000], Bartolini et al. [2003], Castaldi et al. [2005], Marino et al. [2005], Barut et al. [2005], Picardi and Scibilia [2006], Koubaa [2006a,b], Castillo et al. [2007], Roncero-S´ anchez et al. [2007], Wang et al. [2007], Mezouar et al. [2006, 2008], but only partial and quite weak results have been obtained in term of the investigation of the online variation of these parameters in real-time experiments. Amount the above contributions, the problem of IM time-varying parameter has been studied in Bartolini et al. [2003], Barut et al. [2005], Picardi and Scibilia [2006], Mezouar et al. [2006, 2008] but there is a lack of experimental results which are the only way to verify the effectiveness of those methods. In Koubaa [2006a,b], ⋆ The main part of the experimental setup used in this work has been supported by the “D´epartement Energie, Ecole Sup´erieure d’Electricit´ e, Gif-sur-Yvette, Paris, France”.
978-3-902661-47-0/09/$20.00 © 2009 IFAC
Wang et al. [2007], interesting algorithms for IM parameter estimation are proposed using least square technique but online variation of the IM parameters was not investigated. In Roncero-S´ anchez et al. [2007], a method for rotor resistance estimation for indirect field oriented control of IM based on reactive power reference model is presented under motoring and generating modes. Sensitivity of the algorithm to errors in other machines parameters is investigated but without variation of the rotor resistance. In Castillo et al. [2007], a robust nested sliding mode regulation with application to rotor flux modulus and rotor speed of IM with unknown load torque has been introduced. The variation of the stator/rotor resistance has been investigated but the estimation of these parameters was not achieved. Therefore, the problem of combining parameter adaptation with rotor flux observer and rotor speed/flux tracking remains open in general and particularly when the unknown parameters are assumed to be time-varying. This paper revisits this problem and establishes an online rotor resistance estimator for IM adaptive control. The proposed method is an improvement of the previous work Kenne et al.. In the latter contribution, real-time speed and flux adaptive control of induction motors using unknown time-varying rotor resistance and load torque has been proposed. The rotor resistance estimation scheme used in Kenne et al. demands information about timederivatives of stator currents and voltages. This method presented many interesting properties (including high de-
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15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 coupling property) but has two main drawbacks. In this scheme the above hardly measured variables have been defined by using high gain observers and the errors of the time-derivative estimation of the stator currents were very high and this affects the rotor resistance estimation accuracy. A global convergence and stability analysis, taking into account the interconnections between the different estimators has not been investigated. In this paper, we used second-order sliding mode observers to define the hardly measured variables (time-derivatives of stator currents and voltages) and a global convergence and stability analysis is investigated. In Section 2, the description of IM model and problem statement are presented. The design procedure of the rotor resistance identifier is revisited in Section 3 using 2nd order sliding mode observer and a global convergence and stability analysis is given. Experimental results of online implementation are presented in Section 4 to verify the effectiveness of the proposed approach. Finally, some concluding remarks are given in Section 5.
are negligible compared to other dynamics of the induction motor states variables (e.g., R˙ r ∼ = 0) = 0 and ω˙ ∼ = 0, R˙ s ∼ and that the other parameters are constants. Remark 1. The stator voltage usually can be discontinuous. This is the case when the IM is fed by PWM voltagesource-inverter. Assumption (i) can be satisfied if inverter reference voltages are used in the estimation scheme instead of the measured voltages. Our goal is to design a rotor resistance estimation algorithm using 2nd order sliding mode observer when the motor is driven in closed-loop with a controller which stabilizes the rotor flux magnitude and the rotor speed to references values with adaptation of the rotor resistance and load torque. The dynamic equation of the IM used for the rotor resistance identifier is obtained by differentiating (3) combined with (2) and by taking into account assumption (iii) (for more details see Kenne et al.). This transformation gives the following differential equation d2 is = f0 + Rr f1 dt2
2. DESCRIPTION OF THE IM MODEL Assuming linear magnetic circuits, the dynamics of a balanced IM, expressed in a fixed reference frame attached to the stator are given by Leonhard [1984], Marino et al. [2000] dω Te Dω TL = − − (1) dt m m m Rr dλr Rr = (− I + np ωJ)λr + M is (2) dt Lr Lr dis M Rr (− I + np ωJ)λr =− dt σLs Lr Lr 1 M 2 Rr 1 − (Rs + vs (3) )is + 2 σLs Lr σLs 0 −1 1 0 , , J= where I = 1 0 0 1 λra vsa isa is = . , λr = , vs = λrb vsb isb In (1) (2) and (3), the state variables are rotor speed ω, rotor flux (λra , λrb ) and stator currents (isa , isb ); the control inputs are stator voltages (vsa , vsb ); the measured variables are (ω, isa , isb ) while (λra , λrb ) are not measured; the parameters are the external load torque TL , total motor and load moment of inertia m, rotor and stator winding resistances (Rr , Rs ), inductances (Lr , Ls ) and mutual inductance M ; the number of pole pairs is np and the damping gain is D. To simplify the notations in (1), (2), 2 (3), we use σ = 1− LM (leakage parameter), Te = µiTs Jλr s Lr
(4)
where f0 and f1 are given as follows dv
dis dis + ωJ β2 is + β3 vs + β1 dt dt dt dis f1 = γ2 vs − γ3 is − γ4 dt
f0 = γ1
s
− Rs
and β1 , β2 , β3 , γ1 , γ2 , γ3 , γ4 are IM parameters: β1 = n p γ1 =
1 σLs
−np np Rs β3 = σLs σLs 1 Rs 1 γ2 = γ3 = and γ4 = . σLs Lr σLs Lr σLr
β2 =
The model given by (4) is an adequate representation of the motor dynamics (Kenne et al.) if dω |ω| ≪ 2 . dt Tr
(5)
Introducing the variables x1 = is , x2 = dis /dt, u1 = vs and u2 = dvs /dt, model (4) can be rewritten as x˙ 1 = x2 , u˙ 1 = u2 ,
x˙ 2 = f (t, x1 , x2 , u1 , u2 ) + ξ(t, x1 , x2 , u1 , u2 ) 3n M (electromagnetic torque) with the constant µ = 2Lp r . y1 = x1 , Note that the parameter Rr is typically uncertain because Rr may vary up to 100% of its nominal value due to rotor y2 = u 1 (6) heating. The following assumptions will be considered unwhere the nominal part of the system is represented by til further notice: (i). The stator current and voltage are continuous and the function f and while the uncertainties are concenbounded with time-derivatives bounded piecewise-continuous. trated in the term ξ(t, x1 , x2 , u1 , u2 ). Note that the term (ii). The rotor resistance Rr ∈ ΩRr , where ΩRr is a f (t, x1 , x2 , u1 , u2 ) is expressed as function of the derivacompact set of R. tives of the stator current and voltage which cannot be (iii). It is also assumed that the rate of variation of the obtained directly using numerical differentiation due to the rotor resistance Rr , stator resistance Rs and rotor speed ω presence of noise.
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Remark 3. To minimize the influence of the measurement noise in (7) and (8), we approximated the stator current and voltage time-derivatives as solutions of the following equations
ESTIMATE
3.1 State observer design To estimate the unavailable stator current and voltage derivatives let us consider the following 2nd order sliding mode observer Levant [1998b] x ˆ˙ 1i = x ˆ2i + zx1i , i = a, b x ˆ˙ 2i = z 2 ,
(7)
xi
3.2 Rotor resistance adaptation algorithm
ˆ2i + zu1i , i = a, b u ˆ˙ 1i = u u ˆ˙ 2i = z 2 ,
(8)
zu2i
are given by the
zx1i ,
where the variables following expressions
ui zx2i ,
zu1i
and
x1i − x1i |1/2 sign(ˆ x1i − x1i ) zx1i = −λxi |ˆ x1i − x1i ) zx2i = −αxi sign(ˆ zu1i zu2i
1/2
u1i − u1i | = −λui |ˆ
dx ¯˙ 2i +x ¯˙ 2i = x ˆ˙ 2i , i = a, b (15) dt du ¯˙ 2i τu +u ¯˙ 2i = u ˆ˙ 2i , i = a, b (16) dt where τx and τu are suitable positive numbers which → 0. τx
In this section the method used in Kenne et al. is revisited and more details are given concerning the convergence conditions. Assuming that the unavailable states x2i and u2i are provided in finite time by the observer (7) and (8), the IM model (4) can be rewritten as
(9)
sign(ˆ u1i − u1i )
u1i − u1i ). (10) = −αui sign(ˆ The solutions of the above observer are understood in the Filippov sense Filippov [1988]. Taking ex1i = x ˆ1i − x1i , ˆ2i − x2i , the ˆ1i − u1i , and eu2i = x ˆ2i − x2i , eu1i = u ex2i = x following dynamics errors equations are obtained
x˜˙ 2i = f˜0i + Rr f˜1i , i = a, b
(17)
with x ˜2i = xˆ2i ˆ2a − Rs x f˜0a = γ1 u ˜2a − ω β2 x1b + β3 u1b + β1 x˜2b ˆ2b − Rs x ˜2b + ω β2 x1a + β3 u1a + β1 x˜2a f˜0b = γ1 u f˜1i = γ2 u1i − γ3 x1i − γ4 x˜2i .
Estimation of the parameter Rr can be achieved using (17) as the reference model of the IM. This estimation can be treated separately using a state observer combined (11) e˙ x2i = F (t, x1 , x2 , u1 , u2 ) − αxi sign(ex1i ) with a static parameter estimation law or simultaneous e˙ u1i = eu2i − λui |eu1i |1/2 sign(eu1i ) using an adaptive state observer combined with a dynamic (12) parameter estimation law. The first approach requires the e˙ u2i = −u˙ 2i − αui sign(eu1i ) inversion of the system (17). However, the results obtained with using this approach always exhibit peak values. These peak F (t, x1 , x2 , u1 , u2 ) = −f (t, x1 , x2 , u1 , u2 )−ξ(t, x1 , x2 , u1 , u2 ). values are due to the singularity of the denominators of Assuming that the states of the system is bounded as the static parameter estimation law. Therefore, it is more reported in assumption (i), then there exists positive convenient to choose a dynamic parameter estimation law combined with an adaptive state observer (2nd approach). constants µF and µu such that the inequalities In order to design the rotor resistance estimate using |F (t, x1 , x2 , u1 , u2 )| < µF (13) the 2nd approach, let us consider the following adaptive observer |u˙ 2i | < µu (14) ˆ r f˜1i + f˜0i − k ˜ sign(x ˆ˜˙ 2i = R ˆ˜2i − x hold ∀t, x1 , x2 , u1 and u2 . x ˜2i ), i = a, b. (18) e˙ x1i = ex2i − λxi |ex1i |1/2 sign(ex1i )
xi
Theorem 1. Assuming that assumptions (i) to (iii) are satisfied and conditions (5), (13) and (14) hold, then the variables of the observer (7) and (8) converge to the states of the nonlinear model of the IM (6). The proof of the convergence of the states estimates to the real states can be found in Levant [1998a]. The variables of the observer (7) and (8) converge in finite time to the states of the nonlinear model of the IM (6) in the ideal case (when the available states are not contaminated with measurement noise N (t)). Otherwise, if sup N (t) = ζ, the above observer provides estimation of x2i and u2i with accuracy proportional to (ζ)1/2 (µF )1/2 and (ζ)1/2 (µu )1/2 , respectively. Remark 2. The first attempt to choose the observer parameter can be achieved by applying the guidelines given in Levant [1998a]. e.g. αxi and λxi can be chosen as follows. αxi = a1 µF and λxi = a2 (µF )1/2 with a1 = 1.1 and a2 = 1.5.
ˆ r − Rr , the dynamics ˆ˜2i − x˜2i and eRr = R Defining exi ˜ = x of the observer error can be computed using (17) as ˜ e˙ xi ˜ sign(exi ˜ ), i = a, b. ˜ = eRr f1i − kxi
(19)
To achieve the design of the rotor resistance identifier the following additive assumptions are required. Assumption (iv). It is assumed that the following rotor resistance identifiability condition holds. |f˜1i (t)| ≥ δ > 0, i = a, b.
(20)
Assumption (v). It is assumed that the gain of the adaptive observer is chosen such that ˜ kxi ˜ ≥ |eRr f1i |max , i = a, b.
(21)
Remark 4. The identifiability condition (20) can be substituted by the following persistency of excitation condition
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15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 condition Fradkov et al. [1999]. There exists α > 0, T > 0, t0 > 0 such that for all t ≥ 0 t+T
f˜1 (s)f˜1 (s)T ds ≥ αI > 0.
(22)
t
The physical circumstances for which condition (20) or (22) holds, are that the rotor currents of the IM are not zero. Taking into consideration Assumption (v), a sliding mode regime will occur on the manifold exi ˜ = 0 in finite time. Therefore the dynamic equation of the adaptive observer error can be rewritten as ˜ e˙ xi ˜ signeq (exi ˜ ) = 0, i = a, b ˜ = eRr f1i − kxi f˜T ux˜ or eRr = 1 eq (23) ||f˜1 ||2
where uTx˜eq = kx˜a signeq (ex˜a ), kx˜b signeq (ex˜b ) is the equivalent control which cannot be computed since the rotor resistance estimation error is not available. Using the average control as the approximation of the equivalent control ux˜eq Utkin [1992], we considered the following rotor resistance adaptation law (kRr > 0 is the tuning parameter). ˜T ˆ˙ r = −kRr sign f1 ux˜eq . R (24) ||f˜1 ||2 The main result can be summarized by the following theorem. Theorem 2. Given the reference model (17) and the adaptive observer (18). Suppose that the parameter of the observer (18) is selected according to (21) with assumptions (i) to (iv) satisfied, then the rotor resistance estimate ˆ r given by (24) converges in finite time to the nominal R value Rr . Proof. In order to prove Theorem 2, the following Lyapunov candidate function is considered. 1 W = e2Rr . 2
(25)
Its time-derivative is: ˙ = eRr e˙ Rr . W (26) Under assumptions (iii), (iv) and (v), (26) can be rewritten as ˙ = −kRr |eRr |. W (27) From (27), we can deduce that, the rotor resistance estimation error converges in finite time to zero (i.e., the rotor ˆ r converges to the nominal value resistance estimate R Rr ) if the identifiability condition (20) or equivalently the persistency of excitation condition (22) is satisfied. 3.3 Global convergence and stability analysis The global convergence and stability analysis taking into account the interconnections between the different estimators are based on the separation principle theorem. The finite-time convergence of the stator current and voltage time-derivatives observers and the rotor resistance estimator allows to design the observer/estimator and the control
law separately. i.e., the separation principle is satisfied (Levant [1998a], Levant and Levantovsky [1993]). The only requirement for its implementation is the boundedness of the states of the system (e.g., boundedness of fˆω , fˆλr , ˆbω and ˆbλr ) in the operational domain and the persistency of excitation condition. This condition is satisfied under normal operating conditions of the IM. Therefore, if the controller is known to stabilize the IM (as it has been proved in Kenne et al. for the non-adaptive controller), one of the admissible and recommended ways is to choose the observer dynamics fast enough to provide for the exact evaluation of the stator current and voltage timederivatives to the rotor resistance estimator which should ˆ r in an acceptable range and maintain the estimate R achieve its convergence before leaving some operational domain, where the stabilization is assured. 4. EXPERIMENTAL RESULTS A 5kW induction motor whose data are reported in Appendix has been used for the experiments. The experimental setup includes a development system DSP1103, an input/output electronics board (for analog/digital conversions) and a Personal Computer (PC). A PWM power converter with switching frequency of 10 kHz is controlled by a DSP. The external load torque TL is produced by a loaded dc generator. The motor instantaneous speed is measured by an optical incremental encoder with 1024 lines per revolution. The stator currents are measured by Hall-type sensors. All measured electrical parameters are converted by 16-b A/D converter channels with 1µ s conversion time. A DSP1103 performs data acquisition and implements in real-time within the Matlab/Simulink environment software with sampling time of 105.8µ s. In all experiments the measured stator currents used in the estimation scheme were passed through a low-pass filter with cut-off frequency of 1000 rad/s. We performed two experiments. In both experiments, the magnitude of the rotor flux is required to reach the reference value of 0.7W b in 0.75s and the unloaded motor is required to reach the speed reference value of 136.30rad/s in 3.6s. In both cases, the experiments are carried out during motor start-up and after the motor is operated with load torque for over 60 min. The parameters α and λ of the 2nd order sliding mode observer (7) and (8) have been selected according to the procedure described in Remark 2 and a finer tuning has been performed to reduce the observer errors and finally the following tuning parameters were used. αxi = 3 × 107 , λxi = 3000, αui = 108 , λui = 5000, τx = τu = 0.2ms. In both experiments, the parameters of the rotor resistance identifier (24) were chosen as follows. kx˜i = 675000, kRr = 0.65. The equivalent control has been approximated using first order low-pass filter with time-constant of 0.1ms. After the motor start-up and in both cases, the external unknown load torque TL estimated using the method described in Kenne et al. is applied. In both experiments the controller parameters and the load torque estimator parameters are the same as in Kenne et al. (i.e. cω = 200, cλr = 200, kvsd = 7, kvsq = 1, kω = 130 and kTL = 65). To verify the effectiveness of the proposed method in the “hot case” operation of the IM, both experiments were carried out using the estimate
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ˆ r given by (24) in the adaptive controller Kenne et al. R and over-estimated initial values have been chosen for the rotor resistance estimation dynamics. In the first experiment, the robustness property with respect to stator resistance uncertainty is tested with 100% variation of the stator resistance (from 0.22Ω up to 0.44Ω). The results obtained in this case are reported in Fig. 1 and Fig. 2. In the second experiment, the performance of the algorithm to track the variation of the rotor resistance has been investigated. In this case, the online variation of the rotor resistance has been carried out using a 3−phase variable rheostat for rotor wound induction motors. The experimental results are shown in Fig. 3. From Fig. 3, one may conclude that the proposed estimator is slow. From the fact that the value of the additional resistance (0.47Ω) has been introduced progressively/manually and not instantaneously by using an external 3−phase variable rheostat for rotor wound induction motor, the convergence time is less than 6 sec. The convergence time can be reduced by increasing the value of the rotor resistance estimation gain kRr (see equation (24) and the proof of theorem 2). The above results confirm as in Kenne et al. that there is no significant effect on the rotor resistance estimate and the adaptive control algorithm for a wide range of variation of the stator resistance (up to 100%). This robustness
property can be justified theoretically by the fact that the rotor resistance estimator (24) is based on the sliding mode theory which is a known technique for improving robustness under parameter uncertainties. Note that the residual parameter estimation error is due to the measurement noise, mismatching between the motor and the model parameters, ohmic heating during experiments and unmodeled dynamics. We can also note that in ˆ˜1 − both experiments, the adaptive observer error ex˜1 = x x ˜1 of the proposed method is particularly slower than the value obtained in the previous work Kenne et al.. This can justify the accuracy of the proposed rotor resistance estimation scheme and the good tracking performance of the adaptive controller. The very good behaviour of the proposed algorithm to track the variation of the parameter with very good accuracy demonstrates that this scheme can be used to detect and estimate the rotor resistance fault if its value is outside some acceptable known bounds. 5. CONCLUSION An improved rotor resistance estimator for induction motor adaptive control has been investigated. The time derivatives of the stator current and voltage are estimated using 2nd order sliding mode observer. The finite-time convergence of the estimated rotor resistance in finite time
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Fig. 3. Experiment 2: Tracking performance of the proposed method with respect to online variation of the rotor resistance.
to the nominal value has been achieved under feasible persistency of excitation condition. A global convergence and stability analysis has also been investigated. Experimental results with 100% variation of the stator resistance and online variation of the rotor resistance show that the proposed algorithm gives better performance compared to the previous work Kenne et al.. Important advantages of the proposed methodology include that it is an online method (e.g, the value of Rr can be continuously updated) in order to make the controller more beneficial for motor efficiency since very high tracking performance of the rotor resistance variation is achieved. APPENDIX: INDUCTION MOTOR DATA
Rated power Rated speed Rated torque Rated frequency Rated current Stator resistance Rotor resistance Stator inductance Rotor inductance Mutual inductance Number of pole pairs Motor-load inertia
5 kW. 1500 rpm. 32 Nm. 50 Hz. 22.9 A. RsN = 0.22 Ω. RrN = 0.52 Ω. LsN = 0.052 H. LrN = 0.0516 H. MN = 0.0495 H. np = 2. m = 0.12 kg m2 .
REFERENCES Giorgio Bartolini, Alessandro Pisano, and Pierluigi Pisu. Simplified exponentially convergent rotor resistance estimation for induction motors. IEEE Transactions on Automatic Control, 48:325–330, 2003. Murat Barut, Seta Bogosyan, and Metin Gokasan. Speed sensorless direct torque control of induction motors with rotor resistance estimation. Energy Conversion and Management, 46:335–349, 2005. P. Castaldi, W. Geri, M. Montanari, and A. Tilli. A new adaptive approach for on-line parameter and state estimation of induction motors. Control Engineering Practice, 13:81–94, 2005. B. Castillo, S. D. Gennaro, A. Loukianov, and J. Rivera. Robust nested sliding mode regulation with application
to induction motors. Proceedings of the 2007 American Control conference, New York City, USA, pages 5242– 5247, 2007. A. F. Filippov. Differential equations with discontinuous right-hand sides. Dordrecht, The netherlands: Kluwer, 1988. A. L. Fradkov, L. V. Miroshnik, and V. O. Nikiforov. Nonlinear and adaptive control of complex systems. Kluwer Academic Publishers, 1999. G. Kenne, T. Ahmed-Ali, F. Lamnabhi-Lagarrigue, and A. Arzand´e. Real-time speed and flux adaptive control of induction motors using unknown time-varying rotor resistance and load torque. to appear in IEEE Transaction on Energy Conversion. Yassine Koubaa. Application of least-squares techniques for induction motor parameters estimation. Mathematical and Computer Modelling of Dynamical Systems, 12: 363–375, 2006a. Yassine Koubaa. Asynchronous machine parameters estimation using recursive method. Simulation Modelling Practice and Theory, 14:1010–1021, 2006b. W. Leonhard. Control of electric drives. Springer Verlag, 1984. A. Levant. Robust exact differentiation via sliding mode technique. Automatica, 34:379–384, 1998a. A. Levant. Robust exact differentiation via sliding mode technique. Automatica, 34:379–384, 1998b. A. Levant and L. V. Levantovsky. Sliding order and sliding accuracy in sliding mode control. International Journal of Control, 58:1247–1263, 1993. R. Marino, S. Peresada, and P. Tomei. On-line stator and rotor resistance estimation for induction motors. IEEE Trans. On Contr. Syst. Technology, 8:570–579, 2000. R. Marino, S. Peresada, and C. M. Verrelli. Adaptive control for speed-sensorless induction motors with uncertain load torque and rotor resistance. International Journal of Adaptive Control and Signal Processing, 19: 661–685, 2005. A. Mezouar, M. K. Fellah, S. Hadjeri, and Y. Sahali. Adaptive speed sensorless vector control of induction motor using singularly perturbed sliding mode observer. IEEE Industrial Electronics-IECON 2006, pages 932– 939, 2006. A. Mezouar, M. K. Fellah, and S. Hadjeri. Adaptive sliding-mode-observer for sensorless induction motor drive using two-time-scale approach. Simulation Modelling Practice and Theory, 2008. C. Picardi and F. Scibilia. Sliding-mode observer with resistances or speed adaptation for field-oriented induction motor drives. IEEE Industrial Electronics-IECON 2006, pages 1481–1486, 2006. P. Roncero-S´ anchez, A. Garc´ia-Cerraba, and V. FeliuBatlle. Rotor resistance estimation for induction machines with indirect field-orientation. Control Engineering Practice, 15:1119–1133, 2007. V. I. Utkin. Sliding modes in optimization and control. Springer-Verlag, 1992. Kaiyu Wang, John Chiasson, Marc Bodson, and Leon M. Tolbert. An online rotor time constant estimator for the induction machine. IEEE Transactions on Control Systems Technology, 15:339–348, 2007.
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