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2 Feeback Linearization of SpeeSensorless Inuction Motor Control with Torque Compensation. Introuction Cristiane Cauuro Gastalini, Rorigo Zelir Azzolin, Rorigo Pailha Vieira 3 an Hilton Abílio Grünling 4,,3,4 Feeral University of Santa Maria Feeral University of Rio Grane 3 Feeral University of Pampa Brazil This chapter aresses the problem of controlling a threephase Inuction Motor (IM without mechanical sensor (i.e. spee, position or torque measurements. The elimination of the mechanical sensor is an important avent in the fiel of low an meium IM servomechanism; such as belt conveyors, cranes, electric vehicles, pumps, fans, etc. The absence of this sensor (spee, position or torque reuces cost an size, an increases reliability of the overall system. Furthermore, these sensors are often ifficult to install in certain applications an are susceptible to electromagnetic interference. In fact, sensorless servomechanism may substitute a measure value by an estimate one without eteriorating the rive ynamic performance especially uner uncertain loa torque. Many approaches for IM sensorless servomechanism have been propose in the literature is relate to vectorcontrolle methoologies. One of the propose nonlinear control methoologies is base on Feeback Linearization Control (FLC, as first introuce by (Marino et al., 99. FLC provies rotor spee regulation, rotor flux amplitue ecoupling an torque compensation. Although the strategy presente by (Marino et al., 99 was not a sensorless control strategy, funamental principles of FLC follow servomechanism esign of sensorless control strategies, such as (Gastalini & Grunling, 9; Marino et al., 4; Montanari et al., 7; 6. The purpose of this chapter is to present the evelopment of two FLC control strategies in the presence of torque isturbance or loa variation, especially uner low rotor spee conitions. Both control strategies are easily implemente in fixe point DSP, such as TMS3F8 use on real time experiments an can be easily reprouce in the inustry. Furthermore, an analysis comparing the implementation an the limitation of both strategies is presente. In orer to implement the control law, these algorithms mae use of only twophase IM stator currents measurement. The values of rotor spee an loa torque states use in the control algorithm are estimate using a Moel Reference Aaptive System (MRAS (Peng & Fukao, 994 an a Kalman filter (Caroso & Grünling, 9, respectively. This chapter is organize as follows: Section presents the fifthorer IM mathematical moel. Section 3 introuces the feeback linearization moelling of IM control. A simplifie 6

3 98 Electric Machines an Drives FLC control strategy is escribe in Section 4. The propose methos for spee an torque estimation, MRAS an Kalman filter algorithms, respectively, are evelope in Sections 5 an 6. State variable filter is use to obtain erivative signals necessary for implementation of the control algorithm, an this is presente in section 7. Digital implementation in fixe point DSP TMS3F8 an real time experimental results are given in Section 8. Finally, Section 9 presents the conclusions.. Inuction motor mathematical moel A threephase N pole pair inuction motor is expresse in an equivalent twophase moel in an arbitrary rotating reference frame (q, accoring to (Krause, 986 an (Leonhar, 996 accoring to the fifthorer moel, as t I qs = a I qs ω s I s a 3 a λ qr a 3 Nωλ r a 4 V qs ( t I s = a I s ω s I qs a 3 a λ r a 3 Nωλ qr a 4 V s ( t λ qr = a λ qr (ω s Nω λ r a L m I qs (3 t λ r = a λ r (ω s Nω λ qr a L m I s (4 t ω = µ (λ B r I qs λ qr I s J ω T L J T e = µ J (λ r I qs λ qr I s (5 (6 In equations ((6: I s = ( I qs, I s, λr = ( λ qr, λ r an Vs = ( V qs, V s enote stator current, rotor flux an stator voltage vectors, where subscripts an q stan for vector components in (q reference frame; ω is the rotor spee, the loa torque T L,electrictorqueT e an ω s is the stationary spee, θ is the angular position of the (q reference frame with respect to a fixe stator reference frame (αβ, where physical variables are efine. Transforme variables relate to threephase (RST system are given by x αβ = K x RST (7 Let x q = e jθ x αβ (8 [ ] with e jθ cos θ sin θ = an K = sin θ cos θ x q an x αβ stan for twoimensional voltage flux an stator current vector, respectively on (q an (αβreference frame. The relations between mechanical an electrical parameters in the above equations are a = Ls L r L m, a = R ( r Ls L r R s, a L = L m L m L r a r L s a, a 3 =, a a 4 = an µ = NL m ; a JL r a

4 Feeback Linearization of SpeeSensorless Inuction Motor Control with Torque Compensation 99 where R s, R r, L s an L r are the stator/rotor resistances an inuctances, L m is the magnetizing inuctance, J is the rotor inertia, B is the viscous coefficient an N is the number of pole pairs. In the control esign, the viscous coefficient of (5 is consiere to be approximately zero, i.e. B. 3. Feeback Linearization Control The feeback Linearization Control (FLC general specifications are two outputs rotor spee an rotor flux moulus, as [ y = ω λ qr λ r ] T = [ ω λr ] T (9 which is controlle by twoimensional stator voltage vector V s, on the basis of measure variables vector y = I s. The evelopment concept of this control strategy is completely escribe in (Marino et al., 99 an it will be omitte here. Following the concept of inirect fiel orientation evelope by Blaschke, (Krause, 986 an (Leonhar, 996, the purpose of FLC control is to align rotor flux vector with the axis reference frame, i.e. λ r = λ r λ qr = ( The conition expresse in ( guarantees the exact ecoupling of flux ynamics of ((4 from the spee ynamics. Once rotor flux is not irectly measure, only asymptotic fiel orientation is possible, accoring to (Marino et al., 99 an (Peresaa & Tonielli,, then lim λ r = λ r lim λ qr = ( t t It is efine y = [ ω re f λr ] T,whereωre f an λr are reference trajectories of rotor spee an rotor flux. The spee tracking, flux regulation control problem uner spee sensorless conitions is formulate consiering IM moel ((5 uner the following conitions: (a Stator currents are measurable; (b Motor parameters are known an consiere constant; (c Loa torque is estimate an it is applie after motor flux excitation; ( Initial conitions of IM state variables are known; (e λr is the flux constant reference value an estimate spee ω an reference spee ω re f are the smooth reference boune spee signals. FLC equations are evelope consiering the fifthorer IM moel uner the assumption that estimate spee tracks real spee, an therefore it is acceptable to replace measure spee with estimate spee ( i.e. ω k ω. In aition, the torque value is estimate using a Kalman filter. Fig. presents the block iagram of FLC Control. 3. Flux controller From the ecoupling properties of fiel oriente transformation (, the control objective of the flux controller is to generate a flux vector aligne with the axis to guarantee inuction motor magnetization. Then, substituting ( in (4 is (a = λr t λ r ( a L m

5 Electric Machines an Drives I s λ r λ r t SVF Flux Control SVF Is t Current Control I s i s ωk PI u s u s v s v s v qs q rst I RST rst Inution 3 Motor V RST ω ref I qs i Sαβ v Sαβ ωk t Spee Control ref SVF e T L PI i q i q SVF i qs Iqs t Current Control I qs ωk PI u qs u qs v qs Spee MRAS Estimation ω Kalman Filter ω k T L ref ωk I qs Ktn T e T m /J s B/J ω s T L Fig.. Feeback Linearization Control propose The rotor flux λ r is estimate by a moel erive from the inuction motor mathematical moel, (3 an (4, that makes use of measure stator currents ( I qs, I s an estimate spee ω variables. t λ r = a λ r j(ω s Nω λ r a L m I s (3 where the stationary spee is ω s = N ω a L m λ i qs. The igital implementation of the flux controller is mae using Euler iscretization an the erivative rotor flux signal is obtaine by a state variable filter (SVF. 3. Spee controller The spee control algorithm uses the same strategy aopte for the flux subsystem an it is compute from (5, as i q = µλ r ( T L J t ω re f To compensate for spee error between estimate spee an reference spee, e ω = ω ω re f, a proportional integral compensation is propose, as follows (4 (i.e.

6 Feeback Linearization of SpeeSensorless Inuction Motor Control with Torque Compensation ( i q = k p_iq k i_iq e ω (5 s These gains values (k p_iq, k i_iq are etermine consiering an inuction motor mechanical moel. The reference quarature component stator spee current is erive from (4(5, as i qs = i q i q (6 In DSP implementation, the spee controller is iscretize using the Euler metho an the rotor spee erivative (4 is compute by a SVF. 3.3 Currents controller From ( an (, the currents controller is obtaine, as u qs = a 4 ( a iqsω s is a 3λr N (ω re f e ω t I qs an u s = ( a a is ω siqsa a 3 λ r 4 t I s (8 where proportional integral gains of the current error ( u qs = k pv k iv ĩ qs (9 s an u s = (7 ( k pv k iv ĩ s s ( in which ĩqs = I qs iqs an ĩs = I s is. These gains (k pv, k iv are etermine consiering a simplifie inuction motor electrical moel, which is obtaine by loa an locke rotor test. Hence, current controllers are expresse as v qs = u qs u qs ( v s = u s u s ( In DSP, currents controller are igitally implemente using iscretize equation (7( base on the Euler metho, an the stator current erivative is obtaine by SVF using stator currents measures. 4. Simplifie feeback linearization control In orer to reuce the number of computation requirements, a simplifie feeback linearization control scheme is propose. In this control scheme, one part of the current controller (6(7 is suppresse an only a proportional integral controller is use. This moification minimizes the influence of parameters variation in the control system. Fig. presents the block iagram of the Simplifie FLC propose. The currents controller of simplifie FLC are efine as

7 Electric Machines an Drives ( v qs = k pv k iv s v s = ( k pv k iv s ĩ qs (3 ĩ s (4 λ r SVF ω ref t λ r Flux Control i s I s PI v s v s v qs q rst I RST rst Inution 3 Motor V RST ωk e PI iq i Sαβ v Sαβ t ref Spee Control SVF TL i q i qs I qs PI v qs Spee MRAS Estimation ω Kalman Filter ref ω k TL ωk I qs Ktn T e T m /J s B/J ω s T L Fig.. Propose Simplifie Feeback Linearization Control Flux an Spee Controller are compute exactly as in the previous scheme, as ( an (4(6. 5. Spee estimation MRAS algorithm A squirrelcage threephase inuction motor moel expresse in a stationary frame can be moelle using complex stator an rotor voltage as in (Peng & Fukao, 994 for squirrelcage IM v r = v s = R s i s L s t i s L m t i r (5 = R r i r jnωl r i r jnωl m i s L r t i r L m t i s (6

8 Feeback Linearization of SpeeSensorless Inuction Motor Control with Torque Compensation 3 The voltage an the current space vectors are given as x = x α jx β, x {v s, i s, i r }, relative to the transforme variables present in (7. The inuction motor magnetizing current is expresse by i m = L r i r i s (7 L m Two inepenent observers are erive to estimate the components of the counterelectromotive vectors. ê m = L m L r i m = L m L r (ωi m Tr i m Tr i s (8 e m = v s R s i s σl s t i s (9 where σ = L m L s L r. The instantaneous reactive power maintains the magnetizing current, an its value is efine by cross prouct of the counterelectromotive an stator current vector q m = i s e m (3 Substituting (8 an (9 for e m in (3 an noting that i s i s =, which gives ( q m = i s v s σl s t i s an q m = L m L r ( (i m i s ω (i m i s T r Then, q m is the reference moel of reactive power an q m is the ajustable moel. The estimate spee is prouce by the proportional integral aaptation mechanism error of both moels, an an MRAS system can be rawn as in Fig.3 This algorithm is customary for spee estimation an simple to implement in fixe point DSP, such as in (Gastalini & Grunling, 9; OrlowskaKowalska & Dybkowski, ; Vieira et al., 9. The SVF blocks are state variable filters an are explaine in greater etail in Section 7. These filters compute erivative signals an are applie in voltage signals to avoi aition noise an phase elay among the vectors as was propose by (Martins et al., Loa torque estimation Kalman filter The reuce mechanical IM system can be represente by the following equations [ ] ω = B n [ ] J J ω J T t T L T e (33 L y = [ ][ ] ω (34 T L The Kalman Filter coul be use to provie the value of torque loa or isturbances T L. Since (5(6 is nonlinear, the Kalman filter linearizes the moel at the actual operating (3 (3

9 4 Electric Machines an Drives Reference Moel i Sαβ v Sαβ SVF SVF e v R i L i t m Sαβ s Sαβ s Sαβ q i e m Sαβ m q m q m im k im im i e L Sαβ m k i i i t TR T T R R TR m m m Sαβ Aaptative Moel q i e m Sαβ m k Kalman Filter PI q m Fig. 3. Reactive Power MRAS Spee Estimation point (Aström & Wittenmark, 997. In aition, this filter takes into account the signal noise, which coul be generate as pulse with moulation rivers. Assuming the efinitions x k = [ ] T, ω k T L Am = B n J J, B m = J, C m = [ ] an y k = ω. Then, the recursive equation for the iscrete time Kalman Filter (De Campos et al., is escribe by K(k=P(kC T m ( C m P(kC T m R (35 where K(k is the Kalman gain. The covariance matrix P(k is given by P(k=(I A m t s (P(k K(kC m P(k(I A m t s T (B m t s Q(B m t s T (36 Therefore, the estimate torque T L is one observe state of the Kalman filter x k (k=(i A m t s x k (kb m t s u(k(i A m t s K(k( ω C m x k (k (37 giving ω ω k an x k (k= [ ω k T L ] T. The matrices R an Q are efine accoring to noise elements of preicte state variables, taking into account the measurement noise covariance R an the plant noise covariance Q. 7. State variable filter The state variable filter (SVF is use to mathematically evaluate ifferentiation signals. This filter is necessary in the implementation of FLC an MRAS algorithms. The transfer function of SVF is of secon orer as it is necessary to obtain the first orer erivative. G sv f = ω sv f (sω sv f (38

10 Feeback Linearization of SpeeSensorless Inuction Motor Control with Torque Compensation 5 where ω sv f is the filter banwith efine at aroun 5 to times the input frequency signal u sv f. The iscretize transfer function, using the Euler metho, can be performe in statespace as x svf (k = A svf x svf (kb svf u sv f (k (39 [ ] [ ] [ ] x where A svf = ωsv, B f ω svf = sv f ωsvf an x svf =. x The state variables x an x represent the input filtere signal an input erivative signal. 8. Experimental results Sensorless control schemes were implemente in DSP base platform using TMS 3F8. Experimental results were carrie out on a motor with specifications:.5cv, 38V,.56A, 6 Hz, R s = 3.4Ω, R r = 4.96Ω, L r = 44.8mH, L s = 4.4mH, L m = 388.5mH, N = an nominal spee of 88 ra/s. The experimental analyses are carrie out with the following operational sequence: The motor is excite (uring s to s using a smooth flux reference trajectory. Starting from zero initial value, the rotor spee reference grows linearly until it reaches the reference value. Thus, the reference rotor spee value is kept constant. 3 During stanstate, a step constant loa torque is applie. In orer to generate loa variation for torque isturbance analyses, the DC motor is connecte to an inuction motor rivingshaft. Then, the loa shaft varies in accorance with DC motor fiel voltage an inserting a resistance on its armature. Fig. 4 an Fig. 5 epict performance of both control schemes: FLC control an simplifie FLC control with rotor spee reference of 8 ra/s. In these figures measure spee, estimate spee, stator (q currents an estimate torque are illustrate. Fig.6 an Fig. 7 present experimental results with 36 ra/s rotor spee reference. Fig. 8 an Fig. 9 show FLC control an Simplifie FLC with 45 ra/s rotor spee reference. The above figures present experimental results for low rotor spee range of FLC control an Simplifie FLC control applying loa torque. In accorance with the figures above, both control schemes present similar performances in steay state. It is verifie that both schemes respon to compensate torque variations. With respect to Simplifie FLC, it is necessary to carrefully select fixe gains in orer to guarantee the alignment of the rotor flux on the axis. 9. Conclusion Two ifferent sensorless IM control schemes were propose an evelope base on nonlinear control FLC Control an Simplifie FLC Control. These control schemes are compose of a fluxspee controller, which is erive from a fifthorer IM moel. In the implementation of feeback linearization control (FLC, the control algorithm presents a large number of computational requirements. In the simplifie FLC scheme, a substitution of FLC currents controllers by two PI controllers is propose to generate the stator rive voltage. In orer to provie the rotor spee for both control schemes, a MRAS algorithm base on reactive power is applie. To correctly evaluate whether this Simplifie FLC oes not affect control performance, a comparative experimental analysis of a FLC control an a simplifie FLC control is presente. Experimental results in DSP TMS 3F8 platform show the performance of both systems

11 6 Electric Machines an Drives Current I (A (a IM Stator Current I s Current Iq (A (b IM Stator Current I qs Rotor Spee (ra/s ω ˆω k (c Rotor Spee Estimate an Encoer Measurement Loa T orque (N.m ( Estimate Loa Torque Fig. 4. FLC control with 8 ra/s rotor spee reference

12 Feeback Linearization of SpeeSensorless Inuction Motor Control with Torque Compensation 7 Current I (A (a IM Stator Current I s Current Iq (A (b IM Stator Current I qs Rotor Spee (ra/s ω ˆω k (c Rotor Spee Estimate an Encoer Measurement Loa T orque (N.m ( Estimate Loa Torque Fig. 5. Simplifie FLC control with 8 ra/s rotor spee reference

13 8 Electric Machines an Drives.5 Current I (A.5 Current Iq (A (a IM Stator Current I s (b IM Stator Current I qs 4 Rotor Spee (ra/s (c Rotor Spee Estimate an Encoer Measurement ω ˆω k Loa T orque (N.m ( Estimate Loa Torque Fig. 6. FLC control with 36 ra/s rotor spee reference

14 Feeback Linearization of SpeeSensorless Inuction Motor Control with Torque Compensation 9.5 Current I (A (a IM Stator Current I s Current Iq (A (b IM Stator Current I qs 4 Rotor Spee (ra/s (c Rotor Spee Estimate an Encoer Measurement ω ˆω k Loa T orque (N.m ( Estimate Loa Torque Fig. 7. Simplifie FLC control with 36 ra/s rotor spee reference

15 Electric Machines an Drives Current I (A (a IM Stator Current I s Current Iq (A (b IM Stator Current I qs 5 Rotor Spee (ra/s ω ˆω k (c Rotor Spee Estimate an Encoer Measurement Loa T orque (N.m ( Estimate Loa Torque Fig. 8. FLC control with 45 ra/s rotor spee reference

16 Feeback Linearization of SpeeSensorless Inuction Motor Control with Torque Compensation.5 Current I (A (a IM Stator Current I s Current Iq (A (b IM Stator Current I qs 5 Rotor Spee (ra/s ω ˆω k (c Rotor Spee Estimate an Encoer Measurement Loa T orque (N.m ( Estimate Loa Torque Fig. 9. Simplifie FLC control with 45 ra/s rotor spee reference

17 Electric Machines an Drives in the 8ra/s, 36 ra/s an 45 ra/s rotor spee range. Both control schemes present similar performance in steaystate. Hence, the propose moification of the FLC control allows a simplification of the control algorithm without eterioration in control performance. However, it may necessary to carefully evaluate the gain selection in the simplifie FLC control, to guarantee rotor flux alignment on the axis, as well as, to guarantee speeflux ecoupling. Both control schemes inicate sensitivity with moel parameter variation, an one way to overcome this woul be the is evelopment of an aaptive FLC control laws on FLC control.. References Aström, K. & Wittenmark, B. (997. ComputerControlle Systems: Theory an Design, PrenticeHall. Caroso, R. & Grünling, H. A. (9. Gri synchronization an voltage analysis base on the kalman filter, in V. M. Moreno & A. Pigazo (es, Kalman Filter Recent Avances an Applications, InTech, Croatia, pp De Campos, M., Caratti, E. & Grunling, H. (. Design of a position servo with inuction motor using selftuning regulator an kalman filter, Conference Recor of the IEEE Inustry Applications Conference,. Gastalini, C. & Grunling, H. (9. Speesensorless inuction motor control with torque compensation, 3th European Conference on Power Electronics an Applications, EPE 9, pp. 8. Krause, P. C. (986. Analysis of electric machinery, McGrawHill. Leonhar, W. (996. Control of Electrical Drives, SpringerVerlag. Marino, R., Peresaa, S. & Valigi, P. (99. Aaptive partial feeback linearization of inuction motors, Proceeings of the 9th IEEE Conference on Decision an Control, 99, pp vol.6. Marino, R., Tomei, P. & Verrelli, C. M. (4. A global tracking control for speesensorless inuction motors, Automatica 4(6: Martins, O., Camara, H. & Grunling, H. (6. Comparison between mrls an mras applie to a spee sensorless inuction motor rive, 37th IEEE Power Electronics Specialists Conference, PESC 6., pp. 6. Montanari, M., Peresaa, S., Rossi, C. & Tilli, A. (7. Spee sensorless control of inuction motors base on a reuceorer aaptive observer, IEEE Transactions on Control Systems Technology 5(6: Montanari, M., Peresaa, S. & Tilli, A. (6. A speesensorless inirect fieloriente control for inuction motors base on high gain spee estimation, Automatica 4(: OrlowskaKowalska, T. & Dybkowski, M. (. Statorcurrentbase mras estimator for a wie range speesensorless inuctionmotor rive, IEEE Transactions on Inustrial Electronics 57(4: Peng, F.Z. & Fukao, T. (994. Robust spee ientification for speesensorless vector control of inuction motors, IEEE Transactions on Inustry Applications 3(5: Peresaa, S. & Tonielli, A. (. Highperformance robust speeflux tracking controller for inuction motor, International Journal of Aaptive Control an Signal Processing,. Vieira, R., Azzolin, R. & Grunling, H. (9. A sensorless singlephase inuction motor rive with a mrac controller, 35th Annual Conference of IEEE Inustrial Electronics,IECON 9., pp. 3 8.

18 Electric Machines an Drives Eite by Dr. Miroslav Chomat ISBN Har cover, 6 pages Publisher InTech Publishe online 8, February, Publishe in print eition February, The subject of this book is an important an iverse fiel of electric machines an rives. The twelve chapters of the book written by renowne authors, both acaemics an practitioners, cover a large part of the fiel of electric machines an rives. Various types of electric machines, incluing threephase an singlephase inuction machines or oubly fe machines, are aresse. Most of the chapters focus on moern control methos of inuctionmachine rives, such as vector an irect torque control. Among others, the book aresses sensorless control techniques, moulation strategies, parameter ientification, artificial intelligence, operation uner harsh or failure conitions, an moelling of electric or magnetic quantities in electric machines. Several chapters give an insight into the problem of minimizing losses in electric machines an increasing the overall energy efficiency of electric rives. How to reference In orer to correctly reference this scholarly work, feel free to copy an paste the following: Cristiane Cauuro Gastalini, Rorigo Zelir Azzolin, Rorigo Pailha Vieira an Hilton Abílio Grünling (. Feeback Linearization of SpeeSensorless Inuction Motor Control with Torque Compensation, Electric Machines an Drives, Dr. Miroslav Chomat (E., ISBN: , InTech, Available from: InTech Europe University Campus STeP Ri Slavka Krautzeka 83/A 5 Rijeka, Croatia Phone: 385 ( Fax: 385 ( InTech China Unit 45, Office Block, Hotel Equatorial Shanghai No.65, Yan An Roa (West, Shanghai, 4, China Phone: Fax:

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