Direct Torque Tracking PI-Controller Design for Switched Reluctance Motor Drive using Singular Perturbation Method

Transcription

1 Proceeding of the 17th World Congre The International Federation of Automatic Control Direct Torque Tracking PI-Controller Deign for Switched Reluctance Motor Drive uing Singular Perturbation Method Sanjib Kumar Sahoo Sanjib Kumar Panda Jian-Xin Xu Valery D. Yurkevich Department of Electrical and Computer Engineering, National Univerity of Singapore, 4 Engineering Drive-3, Singapore, ; eleahoo@nu.edu.g Department of Electrical and Computer Engineering, National Univerity of Singapore, 4 Engineering Drive-3, Singapore, ; elekp@nu.edu.g Department of Electrical and Computer Engineering, National Univerity of Singapore, 4 Engineering Drive-3, Singapore, ; elejxx@nu.edu.g Automation Department of Novoibirk State Technical Univerity, Novoibirk, , Ruia. yurkev@ac.c.ntu.ru Abtract: The problem of torque tracking PI-controller deign for witched reluctance motor SRM drive i dicued, where SRM magnetization characteritic are highly nonlinear, and torque i a complex and coupled function of phae current and rotor poition. A ditinctive feature of the control ytem deigned i that two-time-cale motion are artificially forced in the cloed-loop ytem by the election of control law parameter. Hence, ingular perturbation method i ued to analyze the cloed-loop ytem propertie. Stability condition impoed on the fat and low mode, and a ufficiently large mode eparation rate, can enure that after fat ending of the fat-motion tranient, the torque tracking error dynamic are a deired by the control ytem deign pecification and they are inenitive to SRM nonlinearitie. An accurate polynomial model of a prototype SRM magnetization characteritic i ued for imulation tudie of the propoed controller. However, only a imple trapezoidal profile for SRM inductance i ued for calculation of the control voltage. Simulation reult for contant motor torque at different peed how that motor peak-to-peak torque ripple are minimized to about 5% of the average torque, epecially for low peed operation. Keyword: witched reluctance motor, direct torque control, robut tracking control, ingular perturbation method 1. INTRODUCTION SRM ha a robut mechanical tructure and many advantage over other electromagnetic motor. Thee include: winding only on tator pole facilitate eae of cooling, abence of permanent magnet or winding on rotor make it uitable for high peed operation, no hoot-through fault in power converter add to over all drive ytem robutne. However, exceive torque ripple due to the difficulty in torque control ha made it unpopular for high-performance drive application. Torque ripple are particularly critical at low peed operation when thee may caue peed ripple and lead to degradation in the product quality. Many reearcher have propoed SRM torque controller imilar to other drive a hown by Huain [2002], by converting torque reference into current reference and then uing current controller to realize the current in the phae winding. However, due to the highly nonlinear and complex nature of SRM magnetization characteritic, torque-to-current converion become quite complex and prone to error. Alternatively, direct torque control DTC cheme doe not need to convert the torque reference to equivalent current reference. The defining characteritic of DTC are: 1 no need for torque-to-current converion to obtain phae current reference for a given motor torque, and 2 no need for current controller. Intead, the motor torque i etimated and compared with the reference torque, and the torque error i ued for obtaining the control voltage directly. 2. NONLINEAR STATE EQUATIONS FOR DTC SCHEME In DTC cheme, phae torque i the plant output wherea phae voltage v i the control input. Motor torque T i, θ being a function of both phae current i and rotor poition θ, torque dynamic can be written a: /08/$ IFAC / KR

2 17th IFAC World Congre IFAC'08 dt T dt = T 1 ψ ir ψ dθ + T dθ θ dt θ dt + 1 ψ v. 1 Rewriting1 in the form of general nonlinear tate equation: T = f + bu, 2 where f = T b = T ψ 1 ψ, 1 ir ψ θ dθ + T dθ dt θ dt, u = v. 3 The phae flux-linkage ψi, θ i a nonlinear function a SRM i commonly operated in deep magnetic aturation. It can be een from preceding equation that the toque dynamic are highly nonlinear and time-varying. Linear control deign method would not be able to provide highperformance. A hyterei type DTC for SRM had been reported by Inderka et al. [2002]. Hyterei controller applie full DC link voltage to the phae winding and hence require very high ampling frequency to keep the output torque within a narrow band of the reference. Any digital controller implementation require an anti-aliaing filter, which i baically a low pa filter. Thi filter add a phae lag to the feedback ignal, and hence the change in current may not how in the feedback immediately. Due to thi reaon, a bang-bang type controller will give rie to an ocillatory output. Intead of applying either V dc, +V dc or zero only, the controller hould apply a variable voltage between V dc and +V dc to track the torque reference moothly, a hown by Neuhau et al. [2006]. A pule-width-modulated PWM converter i ued to upply the variable voltage to the SRM. Neuhau et al. [2006] have propoed to ue flux-linkage a the control variable and ue predictive, dead-beat control to achieve intantaneou torque control. However, converion of phae torque to phae fluxlinkage uing a look-up table i not a direct torque control cheme a uch. Secondly, etimation of flux-linkage from phae voltage integrating the volt-econd i prone to error. Model-baed nonlinear control technique uch a feedback linearization ued by Panda et al. [1996] require an accurate plant model. A modelling of SRM magnetization characteritic i difficult and prone to error due to manufacturing tolerance, the controller hould be robut to model inaccuracie. A control deign method ha been propoed in thi paper where two-time-cale motion are artificially forced in the cloed-loop ytem by proper election of control law parameter. Hence, control ytem propertie are analyzed uing ingular perturbation method. A PI controller reult from thi method which i implemented uing a digital controller. The following ection how the propoed torque controller tructure, deign methodology and imulation reult. Fig. 1. Block diagram of peed control ytem uing an SRM: ω -peed reference; ω fb -peed feedback; T i the motor torque reference, T fb i motor torque feedback from the torque etimator, d i duty cycle from torque controller, v phae winding voltage from the converter, θ i poition feedback from the encoder Fig. 2. Four PI torque controller for the four phae winding Table 1. Specification of Prototype SRM Rated Power 1 hp No. of pole 8/6 Max peed Rated Torque 4000 rmin 1.78 N.m 3. SRM BASED MOTION CONTROL SYSTEM The block diagram of a peed control ytem uing witched reluctance motor i hown in Fig.1. The peed controller generate the motor torque reference T for the torque controller. Fig.2 how the internal detail of the torque controller for SRM. Due to limited available DC-link voltage and finite phae winding inductance; the phae current, and therefore phae torque can not be commutated intantaneouly. Firt, a torque haring function TSF ditribute the motor torque among the phae winding o that phae torque reference do not change intantaneouly. For the laboratory prototype SRM given in Table 1, Fig.3 how the TSF that ditribute the demanded torque T among two neighboring phae, Tinc and T dec ; thereby enuring a mooth and trackable growth and fall of the torque demand for each phae. The phae jut entering into the torque producing region ha to produce Tinc wherea the phae nearing the aligned poition ha to produce Tdec. A infinitely many TSF can atify thi requirement; additional contraint like efficiency and operating peed range hould be conidered for deciding the TSF. A TSF 3323

3 17th IFAC World Congre IFAC' Torque feedback Fig. 3. Cubic torque haring function with the phae torque hare; θ on i the tarting point of the overlap region; θ v the width of the overlapping region when two nearby phae actually hare the total motor torque with a cubic egment a defined in 5-6 ha been ued in thi work a cubic polynomial are the implet function to provide mooth phae torque reference. T inc = fθ T dec = T T inc. 4 fθ = A + Bθ θ on + Cθ θ on 2 + Dθ θ on 3 5 where, A = 0 ; B = 0 ; C = 3T ; D = 2T, 6 with θ on i the tarting point for phae current conduction and θ v i the width of the overlapping region of conduction between two neighboring phae. The next tak i to realize the phae torque reference by applying uitable control technique. An accurate cloedloop torque controller need the actual motor torque a feedback. The phae current and rotor poition are given a feedback to the torque etimator for etimation of motor torque. The following ection dicue the torque etimator ued for torque feedback. θ 2 v θ 3 v Commonly, a train-gauge type of torque tranducer i ued for meaurement of torque. However, thi i bulky, expenive and need pace for intallation. An accurate torque etimator in term of phae current and rotor poition i developed from a flux-linkage model ψi, θ. Firt, the flux-linkage had been meaured at different rotor poition from and phae current value from 1 A 9 A. Thi range of rotor poition and phae current were divided into four region: 1 for mall current near unaligned poition, 2 for mall current near aligned poition, 3 for large current near unaligned rotor poition and, 4 for large current near aligned poition. Fifth order polynomial are ued for capturing the variation of flux-linkage with rotor poition with current being contant. Then, variation of the polynomial coefficient for different current i again captured by another fifth order polynomial in current. Finally, the torque etimator i derived from flux-linkage model uing co-energy principle i.e. T i, θ = i θ ψι, θdι. The torque etimator i given 0 a: T i, θ = N 1,k,1,m kθ k 1 for 0 i i and 0 θ θ h = N 2,k,1,m kθ k 1 for 0 i i and θ h θ θ a = T 1 i, θ + N 1,k,2,m for i i i max and 0 θ θ h = T 2 i, θ + N 2,k,2,m for i i i max and θ h θ θ a, kθ k 1 kθ k 1 where T 1 i, θ = T 2 i, θ = N 1,k,1,m N 2,k,1,m kθ k 1, kθ k 1. 7 Fig. 4. Matching of meaured tatic torque data with the torque predicted by the torque etimator+ derived from the polynomial flux-linkage model. The etimator wa verified by comparing with the phae torque meaured under locked rotor condition. A can be een in Fig.4, the matching between the etimated torque and the meaured torque i quite accurate. Hence, thi torque etimator ha been ued for accurate torque feedback. However, the propoed two-time cale DTC cheme can be ued for any other method of torque feedback a well. 3324

4 17th IFAC World Congre IFAC'08 4. TORQUE TRACKING PI-CONTROLLER DESIGN 4.1 Continuou-time PI-Controller Let the tracking error be defined a e = T ref T, where the torque behavior i governed by 2, and the torque dynamic i given by 1. The controller i being deigned o that the deired tracking error behavior given by hold, where λ > 0. ė = λe 8 In accordance with the deign methodology Yurkevich [2004], let u conider the torque tracking error controller given by µ u = k[λe + ė], 9 where µ i a mall poitive parameter, k i a gain elected in accordance with the induced fat-motion tability requirement a hown below. Note that the control law 9 may be expreed in term of tranfer function, of a conventional PI-controller [ kλ u = µ + k ] e. µ 4.2 Two-time-cale motion analyi In accordance with 2 and 9, conider the equation of the cloed-loop ytem given by ė = T ref f bu, 10a µ u =k[λe + ė]. 10b By ubtituting of 10a into 10b, the ytem 10 can be rewritten a ė = T ref f bu, 11a µ u =k[λe + T ref f bu]. 11b Since µ i the mall poitive parameter, the cloed-loop ytem equation 11 ha the tandard ingular perturbation form and, accordingly, the ingular perturbation method ee Tikhonov [1952], Kokotović et al. [1999], Naidu [2002] may be ued to analyze the cloed-loop ytem propertie. Hence, from 11, we obtain the fat-motion ubytem FMS given by µ u = kbu + k[λe + T ref f], 12 where e, f, b and T ref are treated a the frozen variable during the tranient in 12 that aumption can be provided by µ 0, that lead to increae of time-cale eparation degree between fat and low mode in the cloed-loop ytem. Let the gain k i elected uch that kb > 0, then the FMS i table. Hence, after the rapid decay of tranient in 12, we have the teady tate more preciely, quaiteady tate for the FMS 12, where ut = u t and u = u id = b 1 [λe + T ref f], 13 where u id i the nonlinear invere dynamic olution. Subtitution of 13 into 10a yield the low-motion ubytem SMS equation in the form 8. Hence, after the damping of fat tranient, the deired tracking error behavior i fulfilled depite that f and g are unknown complex and coupled function of phae current and rotor poition. The control law parameter T, µ, k, and λ hould be elected uch that it would provide: firt, the FMS tability; econd, the deired degree of time-cale eparation between fat and low motion in the cloed-loop ytem 11. Note that the degree of time-cale eparation η can be etimated by the ratio of SMS time contant to the FMS time contant, that i η = kb/λµ. 4.3 Digital PI-Controller For implementation of the torque tracking error controller, let u conider a Z-tranform of 9 preceded by a ZOH. Hence, the pule-tranfer function of the digital PI-controller H u/e z = k [ 1 + λt ] µ z 1 follow, where T i a ampling period. Note that for mall T the ZOH can be approximated by a time delay τ = T /2 ee Åtröm and B. Wittenmark [1997], then in order to conider the effect of controller dicretization on the cloed-loop ytem propertie, intead of 10, a nonlinear peudo-continuou-time model ėt = T ref t ft btut τ, 14a µ ut =k[λet + ėt]. 14b with the delay τ = T /2 taken into account ee Yurkevich [2004], Yurkevich et al. [1997]. Then, from 14, we obtain the FMS given by µ ut = kbtut τ+k[λet+ T ref t ft], 15 where et, ft, bt and T ref t are treated a the contant value during the tranient in 15. From 15, the correponding tranfer function of the openloop FMS with pure time delay G o F MS = kb exp T /2, 16 µ reult. Then the croover frequency ω c and the phae margin P M on the Nyquit plot of 16 are given by ω c = kb/µ and P M = [π ω c T ]/2, accordingly. It i clear to ee, an increae of b lead to decreae of P M given that the ampling period T i fixed. Hence, in order to maintain the contant value of the fat-motion ubytem phae margin P M, take k = ˆb 1, where ˆb i an etimate of the parameter b. Aume that the ampling period T i fixed and T = Take, for example, the deired phae margin P M d given by P M d = 1 rad, and the deired degree of timecale eparation given by η d = 60. Then, from the above, the parameter µ and λ T µ = 2π/2 P M d , λ = η d µ 17 reult given that the condition kb = 1 hold. 3325

5 17th IFAC World Congre IFAC'08 Value of ˆb i obtained uing a trapezoidal profile for phae inductance i.e. auming that inductance i directly proportional to the overlap between tator and rotor pole. The analytical expreion for the nominal model i ψi, θ = Lθi and Lθ i given by Lθ = L u, for 0 0 < θ θ 1 = L u + K θ ; θ = θ θ 1, for θ 1 < θ θ 2 = L a, for θ 2 < θ 30 0, 18 where L u i the phae inductance at the unaligned rotor poition, K = dl dθ i the lope of the aumed phae inductance variation v. rotor poition, θ = 0 i the unaligned rotor poition and θ = 30 0 i the aligned rotor poition. For the prototype motor, L u = 0.01 H, L a = 0.04 H, θ 1 = 7 0, θ 2 = 27 0, K = 0.09 Nm/A 2. Then, uing T = 1 2 Ki2, ψ = L u + Kθ and T = Ki. Finally, ˆb i obtained a: Ki ˆb = 19 L u + Kθ The election of controller parameter in accordance with the deired phae margin P M d of the fat-motion ubytem 15 provide a guarantee of FMS tability on controller dicretization. Thee control parameter are ued in imulation of the controller. The imulation reult and their analyi are provided in the next ection. the fact that the controller deign i quite imple. Fig.9 how the torque reference and correponding etimated torque for one phae winding. The phae torque tracking error i hown to be within ±0.15 N m. Correponding phae current and phae voltage are hown in Fig.10, which how that phae current tay controlled for the propoed DTC cheme. The phae voltage i high during the increaing and decreaing part of the phae torque reference, a expected. It can alo be noted that the phae torque tracking error and motor torque ripple increae with increaing peed. Thi i explained by the fact that bandwidth of the controller λ i fixed, wherea the bandwidth of the phae torque reference increae with motor peed. 5. SIMULATION RESULTS The controller performance ha been verified through numerical imulation uing MATLAB-SIMULINK. Simulation i carried out for full motor torque of 1.8 Nm, at two peed 40 rpm and 240 rpm. Other important parameter were on-angle θ on = 5 0, overlap angle θ v = 5 0, ampling time T = 200 µ and DC-link voltage V dc = 200V. The ampling time i choen baed on the typical execution time on our laboratory prototype controller. Firt, a bang-bang type DTC cheme ha been imulated. The hyterei bandwidth wa et at 0.1 Nm. Due to the limited ampling frequency, the phae torque doe not tay within the hyterei band. Thi reult in exceive torque ripple, almot up to 50%, a can be een in Fig.5. Next, the bang-bang type controller had been imulated with V dc = 48V typical voltage for electric vehicle ytem and T = 50µ. Fig.6 how that torque ripple are much le at about 5 %. Thee two reult indicate that bang-bang type controller can be ued for low DC-link voltage and if the ampling time of the controller can be kept low. In SRM controller, the main computation involve that of the torque haring function TSF and etimation of phae torque. The ampling time can be kept low with a fater proceor, but that olution will be expenive. However, the propoed PI controller give equivalent performance for the available DC-link voltage and for the ame ampling time. Fig.7 how the motor torque and two phae torque with motor operating at 1.8 Nm and 40 rpm. It can be een that torque tracking ripple are negligible. Next, imulation are carried out for 240 rpm. Fig.8 how the motor torque with two phae torque. The peak-to-peak ripple have increaed to about 0.1 N m, which i about 5% of the average torque. Thi reult i very promiing conidering Fig. 5. Simulation reult for a bang-bang type DTC cheme: actual motor torque howing the torque ripple and two phae torque, with motor peed 40 rpm, motor torque at 1.8 Nm, with V dc = 200V and T = 200µ Fig. 6. Simulation reult for a bang-bang type DTC cheme: actual motor torque howing the torque ripple and two phae torque, with motor peed 40 rpm, motor torque at 1.8 Nm, with V dc = 48V and T = 50µ 3326

6 17th IFAC World Congre IFAC'08 Fig. 7. Simulation reult for propoed PI baed DTC: actual motor torque howing the torque ripple and two phae torque, with motor peed 40 rpm, motor torque at 1.8 Nm Fig. 9. Simulation reult for propoed PI baed DTC: phae1 torque reference, actual torque and tracking error in Nm; with motor peed 240 rpm, motor torque at 1.8 Nm Fig. 8. Simulation reult for propoed PI baed DTC: actual motor torque howing the torque ripple and two phae torque, with motor peed 240 rpm, motor torque at 1.8 Nm REFERENCES I. Huain. Minimization of torque ripple in SRM drive. IEEE Tran. Ind. Electron., volume 49, no. 1, page 28 39, Feb R. B. Inderka, and R. W. De Doncker. DITC-direct intantaneou torque control of witched reluctance drive. IEEE IAS Annual Meeting, volume 3, page , C. R. Neuhau, N. H. Fuengwarodakul, and R. W. De Doncker. Predictive PWM-baed Direct Intantaneou Torque Control of Switched Reluctance Drive. 37th IEEE Power Electronic Specialit Conference, page , June, 2006, Jeju, Korea. S. K. Panda, P. K. Dah. Application of nonlinear control to witched reluctance motor: a feedback lineariation approach. IEE Proc., Electric Power Application, volume 143, no. 5, page , Sept V. D. Yurkevich. Deign of nonlinear control ytem with the highet derivative in feedback, Serie on Stability, Vibration and Control of Sytem, Serie A - volume 16, Fig. 10. Simulation reult for propoed PI baed DTC: phae1 current and voltage; with motor peed 240 rpm, motor torque at 1.8 Nm World Scientific, 2004, 352 p. P. V. Kokotović, H. K. Khalil, and J.O Reilly. Singular Perturbation Method in Control: Analyi and Deign, Philadelphia, PA: SIAM, A.N. Tikhonov. Sytem of differential equation containing a mall parameter multiplying the derivative. Mathematical Sb., Mocow, volume 31, no. 3, page , D. S. Naidu. Singular Perturbation and Time Scale in Control Theory and Application: An Overview. Dynamic of Continuou, Dicrete & Impulive Sytem DCDIS Serie B: Application & Algorithm, volume 9, no. 2, page , June K.J. Åtröm, B. Wittenmark. Computer-controlled ytem: theory and deign, 3rd ed., Prentice-Hall information and ytem cience erie, Upper Saddle River, N.J. : Prentice Hall, 1997, 557 p. V. D. Yurkevich, M. J. B lachuta, and K. Wojciechowki. Deign of digital controller for MIMO non-linear timevarying ytem baed on dynamic contraction method. Proc. of European Control Conf. ECC 97, Bruel, Belgium,