Volume 46, Number 2, 2005 51 Speed Sensorless DTC for Induction Motor Based on an Improved Adaptive Flux Observer S. BELKACEM, F. NACERI and A. BETTA LEB-Research Laboratory, Department of Electrical Engineering, University of Batna Rue Chahid Med El Hadi Boukhlouf, Batna 05000, Algeria E-mail: firstname.lastname@example.org Abstract – This paper presents a method for estimation of the rotor speed and stator flux based on the theory of an adaptive control and a direct torque control (DTC). A linear observer for estimation of the stator flux is synthesized, by using a Lyapunov function. The adaptive observer is associated to a direct torque control of an induction motor. To illustrate the performances and the robustness of this observer, a simulation results is presented. I. INTRODUCTION performance, state estimators or observers are usually more preferable than physical In recent years, several studies have measurements -. been developed which propose alternative In DTC, the flux is conventionally solutions to the FOC control of a PWM obtained from the stator voltage model, using inverter-fed motor drive with two objectives: the measured stator voltages and currents. first, achievement of an accurate and fast This method, utilizing open-loop pure response of flux and torque, and second, integration suffers from increased noise on reduction in the complexity of the control voltage and current and quantization errors in system. Among the various proposals, Direct the digital system, in addition to the offset, Torque and Flux Control (DTFC) also termed gain and conversions factors in the low speed Direct Torque Control (DTC), has found wide operation range, even with the correct acceptance . knowledge of the stator resistance. We Since its introduction in 1985, the direct propose to use an adaptive observer to torque control (DTC) , principle was improve the estimation of the components of widely used for IM drives with fast dynamics. the stator flux and the rotor speed. Despite its simplicity, DTC is able to produce Adaptive speed observers seem to be very fast torque and flux control, if the torque between the most promising methods thanks and the flux are correctly estimated, is robust to their good performance versus computing with respect to motor parameters and time ratio. They have the advantage to perturbations. provide both flux and mechanical speed As it is well known, speed sensors like estimates without problems of open-loop tachometers or incremental encoders increase integration. Besides, the adaptive observer the size and the cost of systems unnecessarily. has the interesting property to provide a Similar problems arise with the addition of mechanism for on-line tuning of key model search coils or Hall Effect sensors to the parameters such as the stator or rotor motor for the measurement of flux, hindering resistance and then to compensate for their functionality in terms of implementation. drift due to motor heating for example . Thus, to improve the overall system 52 ACTA ELECTROTEHNICA For the speed regulation, the saturation complex loop of return, independently of the of the manipulated variable can involve a rotor parameters. The fact that the DTC phenomenon of racing of the integral action control the switches directly, without passing during the great variations (starting of the by regulators, it clearly improves its dynamic machine), which is likely to deteriorate the performances compared to the vector control performances of the system or even to -. destabilize it completely, the solution consists in correcting the integral action . A. Behavior of Stator Flux In the reference (α, β) related to the II. MODELLING OF THE INDUCTION stator flux, the stator flux could be obtained MOTOR by the following equation: In this section, the induction motor dΨ s Vs = Rs I s + (2) model is given by : dt ⎧ x = Ax + BU By neglecting the voltage drop due to ⎨ (1) the resistance of the stator to simplify the ⎩ y = Cx study (for high speeds), we find: where t T Ψ s ≈ Ψ s0 + ∫ Vs dt (3) x = ⎡isα ⎣ isβ ψrα ψ rβ ⎤ ⎦ 0 T T For one period of sampling, the voltage U = ⎡Vsα ⎣ Vsβ ⎤ , y = ⎡isα ⎦ ⎣ isβ ⎤ ⎦ vector applied to the asynchronous machine M remains constant, we can write: C = ⎡1 0 0 0⎤ , K = , Ψ s (k +1) ≈ Ψ s (k)+VsTe (4) ⎢0 1 0 0⎥ ⎣ ⎦ σLs Lr Rs Rr M 2 Then ΔΨ s ≈ VsTe γ= + 2 • Ψ s (k) : is the stator flux vector to the step σLs Lr σLs of current sampling; ⎡- γ 0 K ⎤ Kωr • Ψ s (k +1) : is the stator flux vector to the ⎢ Tr ⎥ ⎡ 1 ⎤ step of following sampling; ⎢ K ⎥ ⎢ σL 0 ⎥ • ΔΨ s : is the variation of the stator vector ⎢0 -γ - Kωr ⎥ A= ⎢ Tr ⎥ ⎢ s , B= ⎢ 0 1 ⎥ flux; ⎥ ⎢M 0 - 1 - ωr ⎥ ⎢ 0 σLs ⎥ • Te : is the sampling period. ⎢ Tr Tr ⎥ ⎢ 0 0 ⎢0 M 1 ⎥ ⎣ 0 ⎥⎦ For a constant period of sampling, ΔΨ s ωr - is proportional to the voltage vector applied ⎢ ⎣ Tr Tr ⎥ ⎦ to the stator of the induction motor. III. PRINCIPLE OF THE DTC B. Behavior of the Torque The DTC Technique makes it possible The electromagnetic torque is to precisely control the stator flux and the proportional to the vector product between electromagnetic torque. It is based only on the the vectors of stator and rotor flux according knowledge of the stator currents and the to the following expression: voltages and the rotor speed in steady state Ce = k (Ψ s ×Ψ r ) = k Ψ s Ψ r sin ( δ ) (5) operation. The stator voltage makes it with: possible to easily consider stator flux starting from I S, V S and ω S. The values of the flux • Ψ s : is the stator vector flux; and the torque are then calculated without • Ψ r : is the rotor vector flux; Volume 46, Number 2, 2005 53 • δ: is the angle between the stator and The symbol ^ denotes estimated values rotor flux vectors. and G is the observer gain matrix. By using an adaptation mechanism, we C. Development of the Commutation Strategy can estimate the rotor speed. The system states and the parameters can also be In order to exploit the possible estimate. The speed adaptation mechanism is sequences of operation of the inverter on two deduced by using a Lyapunov theory. The levels, the classical selection table of the DTC estimation error of the stator current and the is summarised in the table I, which shows the rotor flux represents the difference between strategy of commutation suggested by the observer and the model of the motor. The Takahashi to control the stator flux and the dynamic error is given by . electromagnetic torque of induction motor. d TABLE. I. Selection table for direct torque control. e = ( A+GC ) e+ ΔAxˆ (7) dt ΔΨs ΔCe S1 S2 S3 S4 S5 S6 where: 1 V2 V3 V4 V5 V6 V1 1 0 V7 V0 V7 V0 V7 V0 e= x - x ˆ (8) -1 V6 V1 V2 V3 V4 V5 ⎡0 0 0 p kΔωr ⎤ 1 V3 V4 V5 V6 V1 V2 ⎢ ⎥ 0 0 V0 V7 V0 V7 V0 V7 ˆ ⎢0 ΔA= A- A= ⎢ 0 -pkΔωr 0 ⎥ ⎥ (9) -1 V5 V6 V1 V2 V3 V4 ⎢0 0 0 -pΔωr ⎥ ⎢ ⎥ ⎣0 0 pΔωr 0 ⎦ The Fig. 1 gives the partition of the ˆ Δω r = ω r - ω r , (10) complex plan in six angular sectors S I = 1… 6. we consider the following Lyapunov β function, V3(010) V2(110) 1 V3(DF,AC) V2 (IF, IT) V = eT e+ ( Δωr )2 (11) λ 3 2 V5(DF,DC) V6 (AF, DT) V0,7(000) Where λ is a positive constant, the derivative V4(011) 4 7 1 V1(100) of Lyapunov function is giving by : α d V = eT ⎧( A ( ωr ) +GC ) + ( A ( ωr ) +GC ) ⎫ e T ⎨ ⎬ 5 6 Secteur 1 dt ⎩ ⎭ ˆ ˆ 2 ˆ V5(001) V6(101) - 2KΔωr (e1 x4 - e2 x3 )+ Δωr ωr λ (13) Fig. 1. Partition of the complex plan in six angular With ωr is the estimated rotor speed ˆ sectors S I = 1… 6. ˆ ˆ e1 = x1 - x1 , e2 = x2 - x2 AF: Increase the Flux, DF: decrease the Flux. IT : Increase the torque, DT: decrease the torque. From equation (12), we can deduce the IF : Increase the flux, DF: decrease the flux. adaptation law for the estimation of the rotor speed by the equality between the second and IV. ADAPTIVE FLUX AND SPEED the third term, we obtain: OBSERVERS ωr = λ.K ( e1 x4 - e2 x3 ) ˆ ˆ ˆ (13) A linear state observer for the stator flux With K: is a positive constant. can then be derived as follows, by To enhance the dynamic behavior of the considering the mechanical speed as constant speed observer, we add a proportional term. parameter: The speed adaptation laws become : ˆ ˆ X = AX + BU + G (iˆ - i ) (6) s s 54 ACTA ELECTROTEHNICA t k1 : is a coefficient obtained by the pole ωr = k p ( e1 x4 - e2 x3 ) +ki ∫ (e1 x4 - e2 x3 )dt (14) ˆ ˆ ˆ ˆ ˆ placement . 0 where: k p and ki are positive gains V. SYSTEM OF SPEED REGULATION A. Synthesis of the adaptive observer Very often, the transfer of the manipulated variable is affected by one or To solve the problem related especially more nonlinear elements (hysteresis, to the estimation of stator flux and the rotor saturation). The saturation of the manipulated speed we have recourse to adaptive observer; variable can involve a phenomenon of racing the mechanism adaptive is a PI regulator. of the integral action during the great The block diagram of the observer with variations (starting of the machine), which is the adaptive mechanism is given in Fig. 2. likely to deteriorate the performances of the isα system or even to destabilize it completely. Induction motor Vsα To overcome this phenomenon, a solution + isβ Vsβ consists in correcting the integral action according to the diagram of Fig. 3. + ˆ - isα Observer kω Model + δ1 - δ ˆ isβ ˆ Φ rβ ˆ Φ rα + + + Adaptation 1/S Saturation Mechanism ˆ ωr ωref - - - Integral + ωr Fig. 2. Global adaptive observer structure. 1/ kω B. Observer gain selection Fig. 3. Structure of the anti-windup system. The observer gain matrix is chosen so The correction of the integral action is that to impose a fast dynamic for the based on the difference between the values of observer. The observer matrix is presented as ( δ ) upstream and downstream from the follow:  limiting device, balanced by the coefficient ⎡ g1 -g2 ⎤ 1 ⎢ ⎥ such as -: ⎢ g2 g1 ⎥ kω G= ⎢ ⎥ (15) ⎢ g3 -g4 ⎥ 1 ⎢ g3 ⎥ X R (k +1) = X R (k) - ( )(δ(k) - δ1 ) (16) ⎣ g4 ⎦ kω Where: g1, g2, g3, g4 are given by: The stator flux according to rotor flux is ⎛ R 1- σ 1 ⎞ given by: g1 = ( 1- k1 ) ⎜ s + + ⎟ ⎝ σLs σTr Tr ⎠ ⎧ M ⎪Ψ sα = σLsisα + Ψ rα g 2 = ( k1 -1) ωr ˆ ⎪ Lr ⎨ (17) 1- K 2 ⎛ Rs 1 - σ K ⎞ k1 - 1 ⎛ Rs 1- σ 1 ⎞ ⎪Ψ sβ = σLsisβ + M Ψ rβ g3 = ⎜ + + ⎟+ ⎜ + + ⎟ ⎪ ⎩ Lr K ⎝ σLs σTr Tr ⎠ K ⎝ σLs σTr Tr ⎠ k -1 ˆ g4 = - 1 ωr K with: k1 > 0 Volume 46, Number 2, 2005 55 Vsβ Vsα ˆ2 ˆ2 isα Ψ sα + Ψ sβ Adaptive Flux α β ˆ Observer abc Ψs ωr ˆ i sβ ˆ Ψ sβ ˆ Ψ sα ˆ ˆ isα i sβ - Sector Select ˆ + Ψs ia ib ic ˆ θs Flux Comparator. ωr + Τe PI Anti- Look up Inverter Windup Table - + - ˆ Τe ωr ˆ Torque Comparator. ˆ Ψ sα ˆ Ψ sβ IM p ⎛ Ψ sα ˆsβ - Ψ s ˆsα ⎞ ⎜ ˆ i ˆ i ⎟ ˆ isα ˆ isβ ⎝ β ⎠ Fig. 4. Speed Sensorless DTC System Using Adaptive Flux Observer. VI. PROPOSED SENSORLESS VII. SIMULATION RESULTS INDUCTION MOTOR DRIVE The proposed sensorless IM drive of The proposed sensorless IM drive block Fig. 4 was verified using simulations. During diagram is shown in Fig. 4. The drive the simulations, the torque set value is limited operates at constant stator flux uses DTC to to 25 (N.m). provide torque control. The speed controller In order to show the performances and is a (PI anti-windup) regulator that generates the robustness of the adaptive observer, we the reference torque. Measurements of two simulated different cases, which are presented line currents and the average voltage over a thereafter. The static and dynamic PWM switching period are used in the performances of the observer are analyzed adaptive flux observer and the direct torque according to the simulation of the following control. The IM stator flux is estimated by the transients: adaptive observer and used in the DTC control. The estimated speed is used in the PI A. Speed Inversion anti-windup speed regulator and for flux To test the robustness of the System, we weakening. The IM under study is a 4-kW vary a reference from 157 rad/sec to –157 two-pole squirrel-cage motor, the parameters rad/sec at time t = 2s. Fig. 5 presents the of which are listed in Table II. actual and the estimated speeds ωr , ωr ˆ TABLE II. INDUCTION MOTOR PARAMETERS respectively and the estimation error of ωr . P = 4 KW f = 50 Hz Wr = 1440 rpm Vr = 380 V Fig. 6 presents actual flux Ψ s , the estimated Rs = 1.2 Ω Rr = 1.8 Ω ˆ flux Ψ s and estimation error Ψ s . The Ls = 0.1554 H Lr = 0.1568 H estimation algorithm is robust because the Tr = 0.0871 s M = 0.15 H variation of the speed is important and the J = 0.07 kgm2 P=2 estimated speed follows the real speed when the motor starts and at the moment of speed 56 ACTA ELECTROTEHNICA er (rad/sec) Wr (rad/sec) Wr (rad/sec) ||Ψs| (Wb) |Ψs| (Wb) ^ ^ er (Wb) Fig. 5. The real, estimated speed and estimation error. Fig. 6. The real, estimated stator flux magnitude and estimation error. Wr (rad/sec) Wr (rad/sec) |Ψs| (Wb) ||Ψs| (Wb) ^ ^ er (rad/sec) er (Wb) Fig. 7. The real, estimated speed and estimation error. Fig. 8. Evolution of the stator magnitude. inversion. Fig. 9. illustrates the response of adaptive observer, the simulation was stator current and the electromagnetic torque. established in low speed. The Fig. 7 and Fig. 8 illustrate the simulation results of the B. Operation at low speed process of speed estimation with a speed To test the speed estimation by using an reference equal to ±50 rad/sec. We can see that the speed follow perfectly the speed Ψsβ (Wb) Ψsα (Wb) Fig. 9. Responses of electromagnetic torque and stator current. Fig. 10. Trajectory of the estimated stator flux. Volume 46, Number 2, 2005 57 Wr (rad/sec) Wr (rad/sec) ^ er (rad/sec) Fig.11.Variation test of the load torque Fig.12. Response of the stator voltage and electromagnetic torque reference. The deviation detected in Fig. 10 is Fig.13.a, 13.b illustrates the evolution of the caused by the instantaneous reversal of the estimated and the real magnitude of the stator speed of the load torque at the zero crossing flux with an increase of 100 % of the stator of the speed; however, it is important to note resistance at 0.25s. We note that according to that the control system demonstrates a good these results that the observer corrects well performance even under those variations. the magnitude of the stator flux and its reference does not deviate from its reference C. Variation of the load torques in established mode follows. Fig. 11 and Fig. 12 illustrate the VIII. CONCLUSION simulation results after the introduction of load torque between 1.5 s and 3s after a In this paper, we use a method for the leadless starting. We can see the insensibility direct torque control with sensorless of the control algorithm to load torque induction motor using an adaptive observer to variation. estimate the rotor speed and the stator flux. The adaptive observer uses an adaptation D. Effect of the Stator Resistance Variation mechanism for the speed estimation when the To show the effect of stator resistance, load torques change. This approach relies on we carried out to vary stator resistance. the improvement of an estimation of the ||Ψs| (Wb) |Ψs| (Wb) ^ Fig. 13.a. Evolution of a real stator magnitude Fig. 13.b. Evolution of an estimate stator magnitude developed by the induction motor. developed by the adaptive observer. 58 ACTA ELECTROTEHNICA components of the stator flux and the rotor 4. G. Bottiglieri, G. Scelba, G. Scarcella, A. Testa, speed. A. Consoli, "Sensorless speed estimation in induction motor drives," IEEE Electric Machines, A PI (anti-windup) regulator has been IEMDC'03., Vol 1, pp. 624 – 630, June.2003. used to replace the PI controller in the speed 5. Y. Cao, Z. Lin, and D.G. Ward, "An anti-windup control of a direct torque control. In approach to enlarging domain of attraction for conclusion, it seems that the PI (anti-windup) linear systems subject to actuator saturation," controller outperforms the classical PI IEEE Trans. Automat. Control, vol. 47, pp. 140– 145, Jan. 2002. controller in speed control of high 6. L. Zaccarian and A. Teel, "Nonlinear scheduled performance DTC motor drive. anti-windup design for linear systems," IEEE We have given a general vision on Trans. on Automat. Control, Vol.49, pp. 2055- association adaptive Observer-DTC, we can 2061, November. 2004. note that the estimation of the components of 7. C. Canudas de Wit, Modélisation contrôle vectoriel et DTC, Edition HARMES science the stator flux by the adaptive observer has Europe, Ltd: 2000. well compensated the variation of the stator 8. D. Casadei, F. Profumo, G. Serra, and A. Tani resistance and has made more robust and "FOC and DTC: two viable schemes for more stable the induction motor based DTC. induction motors torque control," IEEE Trans. Industrial Appl, vol. 17, pp.779-786, September. 2002. IX. REFERENCES 9. F. Khoucha, K. Marouani, K. Aliouane, A. Kheloui "Experimental performance analysis 1. I. Takahashi and T. Noguchi, "A new quick- of adaptive flux and speed observers for direct response and high efficiency control strategy of torque control of sensorless induction motor an induction machine," IEEE Trans. Industrial drives," IEEE Power Electronics Specialists Appl, vol. IA-22, pp. 820- 827, Sep/Oct. 1986. Conference Aachen, Germany, pp. 2678-2683, 2. C. Lascu, I. Boldea, F. Blaabjerg, "A modified 2004. direct torque control for induction motor 10. D. Casadei, G. Serra, A. Tani, "Performance sensorless drive," IEEE Trans .Industrial analysis of a speed-sensorless induction motor Appl,vol.36,pp. 122-130, Jan/Feb 2000. drive based on a constant-switching-frequency 3. J. Maes, J. Melkebeek, "Speed-sensorless direct DTC scheme," IEEE Trans. Industrial App, vol. torque control of induction motors using an 39, March/April. 2003. adaptive flux observer," Proc. of IEEE Trans .Industrial Appl, vol.36, pp. 778-785, May/June 2000.