IDENTIFICATION OF TIME-VARYING ROTOR AND STATOR RESISTANCES OF INDUCTION MOTORS

Transcription

1 IDENTIFICATION OF TIME-VARYING ROTOR AND STATOR RESISTANCES OF INDUCTION MOTORS G. Kenné T. Ahmed-Ali H. Nkwawo and F. Lamnabhi-Lagarrigue L2S C.N.R.S SUPELEC Université Paris XI 3 Rue Joliot Curie 99 Gif-sur-Yvette France fax: +33 () {lamnabhikenne}@lss.supelec.fr Département de Génie Electrique IUT-FV Bandjoun Université de Dschang Cameroun Département GEII IUT Université Paris XIII 99 Avenue Jean Baptiste Clément 9343 France homere.nkwawo@iutv.univ-paris3.fr ENSIETA 2 Rue François Verny 2986 Brest Cedex 9 France ahmedali@ensieta.fr Keywords: time-varying parameter induction motor identification sliding mode observer. Abstract This paper deals with the identification of the stator and rotor resistances. The originality of this work is that it considers both rotor and stator resistances as time-varying parameters. Two schemes for the rotor resistance identification and one scheme for both stator and rotor resistances identification are outlined. The first and the third schemes use the measured stator current voltage command and rotor speed while the second scheme uses only stator current and voltage commands. All schemes are based on the sliding mode observer and the convergence of the estimates to their true values does not required the persistence of excitation conditions to be satisfied by the input signal. Introduction This paper deals with the identification of stator and rotor resistances. The originality of this work is that both rotor and stator resistances are considered as time-varying parameters. It is well known in the literature that the rotor resistance is a parameter which largely varies during operation and it is crucial in the design of high performance induction motor control algorithms when flux measurements are not available. Rotor resistance may vary up to % and stator resistance up to 5% due to rotor and stator heating and can be hardly recovered using thermal models temperature sensors. Several works in this area exist in the literature one can cite the work of [] using a ninth order estimation algorithm which provides on-line exponentially convergent estimates of both rotor and stator resistances when persistence of excitation conditions are satisfied and the stator currents integrals are bounded on the basis of rotor speed stator voltages and stator currents measurements. Rotor flux is also asymptotically recovered. But the proposed algorithm always make an assumption that both stator and rotor resistances are constant during the estimation process. Moreover the identification schemes use the nominal values of stator and rotor resistances. Another result was proposed in [2] the rotor resistance estimation scheme is outlined using least mean square and the adaptive algorithms in the transient state under the speed sensorless control of induction motor. Moreover [3] used least squares identification techniques for the estimation of both electrical and mechanical parameters of an induction motor under slowly varying rotor speed. Recently [4] proposed algorithms for the rotor and stator resistances estimation based on the model reference adaptive system (MRAS) approach. In this work the fact that these parameters are timevarying is not investigated. The contribution of this paper is the extension of this work to the case of time-varying parameters by using the work of [7] and [8] which use sliding mode observer in the parametric identification schemes. In fact it has been shown in [8] and [9] that parametric identification scheme based on the variable structure provides better result than least square estimation technique. The remainder of the paper is organized as follow. In the second Section we describe the model of the induction motor. In the third one we present a new scheme of the rotor resistance identification. We study two cases. In the first case the rotor speed is supposed to be available while in the second case it is neither measured nor estimated. In both cases the stator current and voltage are measurable and the rotor resistance is time-varying. Another scheme for the stator and rotor resistances identification is presented in Section 4. This scheme used the measured stator current voltage commands and rotor speed. Both stator and rotor resistances are time-varying. All designs are based on the sliding mode observer and the convergence of the estimates to their true values does not required the persistence of excitation conditions to be satisfied by the input signal. Section 5 is dedicated to the simulation results and section 6 concludes the paper. 2 Model description The dynamic model of an induction motor in stator reference frame is given by [5] d Ω = 3 n p M i T s Jλ r α 2 ml r m Ω τ l () m d λ r = ( R r I + n p ΩJ)λ r + R r Mi s (2) L r L r

2 d i s = M L r ( R r L r I + n p ΩJ)λ r (R s + M 2 R r L 2 )i s + v s (3) r [ ] [ ] I = J = ( ) ( ) ( ) isd vsd λrd i s = v i s = λ sq v r = sq λ rq Ωλ r R r and L r are respectively rotor angular velocity flux resistance and inductance v s i s R s and L s are respectively stator command voltage current resistance and inductance n p is the number of poles pair σ = M 2 L sl r is the leakage parameter M is the mutual inductance between stator and rotor winding m is the moment of inertia of the rotor α is the damping gain and τ L is the external load torque. In order to eliminate the unobservable flux equations (2) and (3) are transformed by using an assumption which considers that the rotor speed changes significantly slower relative to the rotor flux and is considered as a constant parameter. By differentiating (3) and using (2) we obtain d 2 2 i s = (α I +Ωβ J) d i s +(α 2 I +Ωβ 2 J)i s +(α 3 I +Ωβ 3 J)v s + α 4 d v s (4) α = ( R s + R r σl r ) α 2 = R sr r L r α 3 = R r L r β = n p β 2 = n pr s β 3 = n p α 4 = Equation (4) ([4]) can be transformed to di s = a +Ωb + ɛ(t) (5) the functions a b and ɛ(t) are defined as follows a = (c + α )i s + α 2 i s + α 3 v s + α 4 v s b = J(β i s + β 2 i s + β 3 v s ) ɛ(t) = ɛ(t )exp( ct) and i s = s + c i s i s = s s + c i s v s = s + c v s v s = s s + c v s s is the operator d and c is a strictly positive constant. In the following analysis it is assume that the stator current i s voltage command v s rotor speed Ω and their time-derivatives are continuous and bounded. 3 Rotor resistance identification In this section we identify only the rotor resistance by using an estimator based on the sliding mode observer ([2]). We consider that the stator current and voltage are measurable and the rotor resistance is time-varying R r = R rn + R r with Ṙr = R r µ µ is a known positive constant. We also consider the both cases the rotor speed is available or not. We can easily see that by separating the terms containing R r in equation (5) we obtain di s = f + R r f 2 +ΩJf 3 + ɛ (6) f = (c + ρ )i s + ρ 2 v s f 2 = k i s + k 2 i s + k 3 v s f 3 = β i s + β 2 i s + β 3 v s and ρ = R s ρ 2 = β = n p β 2 = n pr s β 3 = n p k = σl r k 2 = ρ L r k 3 = ρ 2 L r. (7) 3. Identification with the rotor speed available Let us consider the following sliding observer dî s = f + ˆR r f 2 +ΩJf 3 + u is (8) u is = K is sign(î s i s ) is the input of the identifier î s is the adaptative observer and ˆR r is the estimated value of the resistance R r. By defining e is = î s i s the observer error and e r = ˆR r R r the parameter estimation error and by taking into account equations (6) and (8) one can obtain the following dynamic equation of the observer error ė is = e r f 2 + u is ɛ. (9) If the gain K is of the input u is is choosing such that K is > e r f 2 ɛ max () with f 2 ˆRr and ɛ bounded a sliding regime occurs on the manifolds ė is = e is =([]). From this one can write u iseq = e r f 2 + ɛ () u iseq is the equivalent control. Remark: From a practical point of view it is not possible to implement u iseq. We considered the average control as the approximation of u iseq ([3]). This means that τ d ūis +ū is = u is or ū is = +τs u is (2)

3 τ is a strictly positive constant that tends to zero. In others words u iseq ū is. By taking into account () (2) and if the operation points of the motor are such that f 2 2 t then one can deduce the following expression for the parameter estimation error e r = f T 2 u iseq f ɛ (3) ɛ = f 2 T ɛ f 2 2. Now let us consider the following rotor identifier ˆR r = K r sign( f T 2 u iseq f 2 2 ) (4) which can be rewritten as ˆR r = K r sign(e r ɛ). (5) Therefore the dynamic equation of the parameter error is ė r = K r sign(e r ɛ) Ṙr (6) Proof of the convergence In order to proof the convergence of the identifier (4) let us consider the following Lyapunov candidate function V = 2 e r ɛ 2. Then its time derivative along the solution of (6) is V = (e r ɛ)(ė r ɛ) = (e r ɛ)( ˆR r Ṙr ɛ) = ( K r sign(e r ɛ) Ṙr ɛ)(e r ɛ) = K r e r ɛ (e r ɛ)( R r + ɛ) From the fact that Ṙr µ we deduce that V K r e r ɛ + µ e r ɛ + ɛ e r ɛ or V (Kr µ ɛ ) e r ɛ Furthermore since the stator and voltage commands are continuous and bounded their filtered values and hence f 2 are bounded. Moreover if the operating points of the motor are such that f 2 (t) 2 t with assumption () satisfied then ɛ and hence ɛ converge exponentially to zero (from the fact that ɛ(t) C and converges exponentially to zero). Therefore by choosing K r >µ we ensure that e r ɛ and e r converge asymptotically to zero and ˆR r R r. 3.2 Identification with the rotor speed not available This scheme is particularly interesting both economically and technically in speed sensorless control because measurement noise of the sensor can be avoided and the cost of the controller will be reduced. In this case we also use equation (6) and we consider that the rotor speed is not available. The term containing the rotor speed is eliminated by using the following property of the matrix operator J x T Jx = for any given column vector x. If we multiply the two parts of equation (6) from the left by we obtain f T 3 f3 T di s = f3 T f + R r f3 T f 2 + f3 T ɛ (7) Now let us consider the following change of variable Its time derivative is dλ Λ=f T 3 2 β i s 2 = g + R r h + ɛ (8) g = (β 3 v s +(β 2 cβ )i s ) T i s + f T 3 f h = f3 T f 2 ɛ = f3 T ɛ In order to identify the rotor resistance we introduce the following sliding observer dˆλ = g + ˆR r2 h + u Λ (9) u Λ = K Λ sign(ˆλ Λ) is the control input of the identifier. By defining e Λ = ˆΛ Λ the observer error and e r2 = ˆR r2 R r the parameter estimation error and by taking into account (8) and (9) one can derive the dynamic equation of the observer error as follow ė Λ = e r2 h + u Λ ɛ (2) From this we can say that if K Λ is choosing such that K Λ > e r2 h ɛ max (2) with h and ɛ bounded the sliding regime occurs on the manifolds ė Λ = e Λ =in finite time. Then we can write e r2 h = u Λeq + ɛ (22) u Λeq is the equivalent control approximated here as in subsection (3.). If the operating points of the motor are such that h t then the parameter estimation error is e r2 = h ( u Λeq + ɛ). (23) Now let us consider the following rotor identifier ˆR r2 = K r2 sign( h u Λeq) = K r2 sign(e r2 ˆɛ).(24) ˆɛ = h ɛ.

4 By considering the Lyapunov candidate function V = 2 e r2 ˆɛ 2 and if the operating points of the motor are such that h t with assumptions (2) satisfied then the proof of the asymptotic convergence of ˆR r2 R r is similar to the above subsection. 4 Rotor and stator resistances identification In this section we assume that the stator current and voltage command are measurable. We also consider that the rotor speed is available. We start by separating the terms containing R r and R s in equation (5). This leads to di s = f + R s f2 + R r f3 R sr r L r i s + ɛ (25) f = ci s + v s + n p ΩJ(i s v s ) f 2 = (n p ΩJi s i s ) f 3 = v s i s L r σl r In order to identify the stator and rotor resistances we use the following change of variables λ = i T s Ji s λ 2 = 2 it s i s By choosing the gains K and K 2 such that K > g + R s h + R r h 2 + ɛ max K 2 > g 2 + R s h 2 + R r h 22 + R s R r h 23 + ɛ 2 max (29) a sliding regime occurs on the manifolds ė λi = e λi = i = 2 in finite time. This leads to dˆλ = K sign(e λ ) eq = u eq dˆλ 2 = K 2 sign(e λ2 ) eq = u 2eq (3) u eq and u 2eq are the equivalent control inputs also approximated as in subsection (3.). From the fact that ɛ i converge to zero exponentially we consider that the estimates of R s and R r verify the following equations u eq g = ˆR s h + ˆR r h 2 u 2eq g 2 = ˆR s h 2 + ˆR r h 22 + ˆR s ˆRr h 23 (3) We suppose that system (3) is identifiable in the sense of the work of ([]). This means that there exits a unique solution ˆR s and ˆR r satisfying system (3) for a given input-output behaviour t. Therefore the resolution of this system leads to the following identification laws for ˆR s and ˆR r ˆR s = B ± 2A ˆR r = (u eq g ) ˆR s h h 2 for ˆλi = λ i i= 2 (32) This leads to dλ = g + R s h + R r h 2 + ɛ dλ 2 = g 2 + R s h 2 + R r h 22 + R s R r h 23 + ɛ 2 (26) g = i T s J f g 2 = i T s f h = i T sj f 2 h 2 = i T sj f 3 h 2 = i T s f 2 h 22 = i T s f 3 h 23 = L r i T s i s ɛ = i T sjɛ ɛ 2 = i T s ɛ. Now let us consider the following sliding observer dˆλ = K sign(ˆλ λ )= K sign(e λ ) dˆλ 2 = K 2 sign(ˆλ 2 λ 2 )= K 2 sign(e λ2 ) (27) From this the dynamic equation of the observer errors will be as follows A=h h 23 B = h h 22 h 2 h 2 (u eq g )h 23 C=(u 2eq g 2 )h 2 (u eq g )h 22 and =B 2 4AC are the coefficients of the second order equation obtained and its discriminant. Equations (32) are the parameters identification laws of winding rotor induction motor. From this we can have two cases: First case = The solution will be straightforward. Second case > In this case we can obtain two solutions. If one of them is negative we consider the positive one. If both solutions are positive we make an assumption which considers that the stator resistance bounds or nominal value should be known. This assumption is generally true because it is possible to measure the nominal value of the stator resistance independently by applying a dc voltage to the stator winding and then deriving the voltage to current ratio. A choice of the correct solution of ˆR s is therefore straightforward. 5 Simulation results Efficiency of the proposed identifiers has been verified by si- e λ = K sign(e λ ) g R s h R r h 2 ɛ mulation in MATLAB/SIMULINK environments. Motor parameters nominal values ([4]) used in the simulation are given e λ2 = K 2 sign(e λ2 ) g 2 R s h 2 R r h 22 R s R r h 23 ɛ 2 (28) as

5 follow. R sn =.Ω R rn =.87Ω M =.84H L r =.H L s =.H n p =6τ L =4m=.5α=.7. We consider that R rs = R {rs} + at if t t R rs = R {rs}n + f(t) if t t R {rs} is the initial value of the resistance a is a positive constant and f(t) represents square chirp or sinusoidal signals. The magnitudes of the variations of the reference time-varying rotor and stator resistances are respectively % and 2% of the nominal values while initials values are 2% of the nominal values. All the schemes work well in both cases of slowly and constant rotor speed. Fig. and Fig. 2 show the results of the identification of the time-varying parameter R r with respectively zero and.3 initial values and function f(t) being square signal. Fig. 3 and Fig. 4 show the results of the identification of the time-varying parameters R r with initial values equal respectively to zero and.3 and function f(t) being chirp signal. Fig. 5 shows the identification of the time-varying parameter R r in the case the rotor speed is not available with function f(t) being chirp signal. Fig. 6 and Fig. 7 show the identification of both stator and rotor resistances by considering the function f(t) as sinusoidal signal. The parameters of the identifier of the rotor resistance in the first case are c =K r =.3K is = 5. In the second case c =K r2 =4K Λ = 2. For the identification of both stator and rotor resistances the parameters of the simulations are c =K = K 2 = 3. Moreover the approximation of the equivalent control has been performed using first order low-pass filter with τ =ms and the functions sign(x) are replaced by a saturation function X X +. ([6]) From the simulations results one can easily see that the proposed identifiers is suitable for time-varying parameters as rotor and stator resistances. In all cases the estimates values of the resistances converge to the true values within a very short time Figure 2: R r and its estimate ˆR r with measured rotor speed initial condition.3 and f(t) being square signal Figure 3: R r and its estimate ˆR r with measured rotor speed zero initial condition and f(t) being chirp signal Figure 4: R r and its estimate ˆR r with measured rotor speed initial condition.3 and f(t) being chirp signal Figure : R r and its estimate ˆR r with measured rotor speed zero initial condition and f(t) being square signal Figure 5: R r and its estimate ˆR r with rotor speed neither measured nor estimated and f(t) being chirp signal

6 Rs and hat Rs References [] R. Marino S. Peresada and P. Tomei On-line Stator and Rotor Resistance Estimation for Induction Motors IEEE Trans. On Control Systems Technology Vol. 8 N. 3 (May 2) Figure 6: R s and its estimate ˆR s with measured rotor speed and f(t) being sinusoidal signal [2] K. Akatsu and A. Kawamura On-line Rotor Resistance Estimation Using the Transient State Under the Speed Sensorless Control of Induction Motor IEEE Trans. On Power ElectronicsVol.5 N.3 (May 2). [3] J. Stephan M. Bodson and John Chiasson Real- Time Estimation of the Parameters and Fluxes of Induction Motors IEEE Trans. On Industry Applications Vol. 3 N. 3 (May/Jun 994) [4] A. V. Pavlov and A. T. Zaremba Real-Time Rotor and Stator Resistances Estimation on an Induction Motor Proc. of NOLCOS- St-Petersbourg (2). [5] Leonhard W. Control of electric drives Springer Verlag (984) Figure 7: R r and its estimate ˆR r with measured rotor speed and f(t) being sinusoidal signal 6 Conclusion In this paper two schemes for the rotor resistance identification and one scheme for both stator and rotor resistances identification has been designed. The first and the third schemes use the measured stator current voltage commands and rotor speed while the second scheme uses only stator current and voltage commands. The designs are based on the sliding mode observer and the convergence of the estimates to their true values does not required the persistence of excitation conditions to be satisfied by the input signal. The originality of this work is that it considers both rotor and stator resistances as time-varying and that the schemes proposed are easily implementable. This feature distinguishes the proposed identifier from the known ones. The simplified dynamic model of the induction motor used in this work depends on the parameter c. Further work is under way to derive the validity condition of this model and hence the choice of this parameter c. Further more real-time implementations deserves to be realized in order to verify the effectiveness of the proposed schemes with respect to the sensor noise discretization effects and modeling inaccuracies. Acknowledgment [6] J. J. E Slotine Sliding controller design for nonlinear systems International Journal of Control Vol. 4 N. 3 pp. 42 (984). [7] T. Ahmed-Ali F. Floret and F. Lamnabhi-Lagarrigue Robust identification and control with time-varying parameters perturbations Proc. of Amer. Contr. Conf.- ACC3 Denver Colorado USA 4-6 Jun (23). [8] F. Floret-Pontet and F. Lamnabhi-Lagarrigue Parametric identification methodology using sliding modes observer International Journal of Control Vol. 74 N. 8 pp (2). [9] Fabienne Floret Méthodes d identification pour des systèmes non-linénaires en temps continu Thèse de Doctorat Université Paris XI Sud Orsay France Novembre (22). [] Y. Lecourtier F. Lamnabhi-Lagarrigue and E. Walter Volterra and generating power series approches to identifiability testing E. Walter Editor Identifiability of parametric models Pergamon Press PP (987). [] J. J. E Slotine and W. Li Applied Nonlinear Control Prentice-Hall International Editions Englewood Cliffs (99). [2] UtkinV. I. Principles of identification using sliding regimes Soviet Physiks Doklady 26 pp (98). [3] Utkin V. I. Sliding modes in Optimization and Control Springer-Verlag (992). The authors would like to thank the anonymous reviewers for their constructive comments.