µ-analysis OF INDIRECT SELF CONTROL OF AN INDUCTION MACHINE Henrik Mosskull

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1 -ANALYSIS OF INDIRECT SELF CONTROL OF AN INDUCTION MACHINE Henrik Mokull Bombardier Tranportation, SE-7 7 Väterå, Sweden S, Automatic Control, KTH, SE- Stockholm, Sweden Abtract: Robut tability and performance of an induction machine controlled with Indirect Self Control (ISC) are examined through -analyi. The analyi indicate poor robut performance at higher rotor peed. Uing reaonable parameter deviation between plant and controller, the predicted bad performance i however hard to verify through imulation. The teted parameter variation eem not to generate the wort cae model error. In imulation, the ISC therefore outperform linear controller optimized with repect to the robut performance criterion (at dipene of nominal performance). The large predicted enitivity to model error however explain the large impact on performance due to eemingly mall difference in the implementation of the ISC control. Copyright IFAC Keyword: Motor control, control application, tability analyi, robutne, H-infinity control.. INTRODUCTION Field-oriented controller (FOC) are frequently ued non-linear controller for induction machine. The idea with field-orientation wa originally to mimic control of DC-machine for induction motor. It ha later been hown that the claical field-oriented controller actually perform aymptotic linearization and decoupling (Marino, et al., 99). Stability of FOC ha been invetigated for example in (De Wit et al., 99) and (Bazanella and Reginatto, ), where focu i on robutne regarding error in the etimate of the rotor reitance. A different cla of controller for induction machine i formed by the o called Direct Torque Control (DTC) technique; ee (Buja and Kazmierkowki, ) for an overview. One method within thi family, which i run at contant witching frequency, i the Indirect Self Control (ISC). With ISC, torque and tator flux magnitude are controlled by PI controller in cloed-loop, ee (Jänecke et al., 989) and (Maichak, 99). The linearized cloedloop dynamic of an induction motor drive uing ISC wa examined in (Mokull, ), where alo tuning rule for the controller parameter were given to achieve deired bandwidth and tability margin. The robutne analyi wa, however, performed under the implifying aumption of perfect knowledge of parameter, for example ued for tator flux etimation. One way to include flux etimation in the robutne analyi of the cloed loop drive i to apply -analyi, a in done for FOC in (Thoma, 99). A the analyi i till performed on linear model (a oppoed to the work in (De Wit et al., 99) and (Bazanella and Reginatto, )), the reult may only be valid for mall perturbation around an operating point. On the other hand, not only robutne in term of tability but alo robut performance can be conidered. To further evaluate the ISC, -ynthei controller are deigned for ome fixed operating point and the performance i compared to that of the ISC.. INDUCTION MACHINE The induction machine can be decribed by the following et of complex-valued pace vector

2 equation, where pace vector are indicated by upercript, (Steimel, ): ψ () t = Ri () t + u () t () Rr Rr ψr() t ψ () t = + jnpωm() t ψr() t () L L Here ψ (t) and ψ r (t) repreent the tator and rotor fluxe, repectively, and ω m (t) i the mechanical rotor peed. The number of pole pair of the induction machine i denoted by n p, wherea R and R r tand for the reitance in the tator and rotor winding. Finally, the machine inductance are denoted L and L. Thi repreentation, where all leakage inductance i put in the rotor meh, i called the Γ-model and can be viualized by the equivalent circuit diagram (ECD) hown in Fig.. i R L R r. LINEAR PROCESS MODEL The induction machine i decribed by equation (), () and (). Input are the three tator voltage, repreented by the magnitude and frequency of the tator voltage pace vector u (t), and output are the control variable torque and tator flux magnitude. To obtain a linear model of the proce, thee equation are preferably rewritten in the form of nonlinear tate pace equation in the following way R m = R + m + mr coδ + mu coδ (7) u L L L m r = m coδ mr (8) T T R mr mu δu = inδ inδu + ωu (9) L m m u Ψ L i.. Ψ r jn p ω m Ψ r Fig.. Induction machine Γ-ECD. Uing a polar notation with magnitude and angle for the fluxe, i.e. ψ () () () () jχ () t jχr () t, ψr r t = m t e t = m t e () the electrical torque can be expreed a, (Steimel, ), np T = m () t mr () t in ( χ () t χr () t ) () L. INPUT CONSTRAINTS Modern drive often ue voltage ource inverter to feed the induction machine. The inverter convert a DC-voltage, denoted U d, to three AC tator voltage with variable frequency and amplitude. The fundamental of the tator voltage are upper limited by, (Kovac, 98), mu () t u () t Ud () π Thi i a contraint on the tator voltage amplitude, but one hould alo put retriction on the tator voltage frequency to prevent the teady tate lip frequency, ω lip, from exceeding the pull-out lip frequency (Steimel, ), i.e. ω = ω ω (6) lip u np m T Here the tator frequency, ω u (t), i the time derivative of the angle of the tator voltage pace vector u (t). Thi angle will be denoted χ u (t), cf. (). Further, T = L /R r i the rotor tray time contant. i r R mr m mu δ = + inδ + inδu npω m L m T m r m () where δu () t = χu () t χ () t, δ () t = χ () t χr () t () From () and (7)-(), a linear model G from tator voltage magnitude and frequency to torque and tator flux magnitude can now be derived, i.e. T mu = G m Gu ω () u. SCALING To facilitate the controller analyi, the ytem i caled a propoed in (Skogetad and Potlethwaite, 996). Introducing diagonal caling matrice D u and D e, with the maximum expected value of the input and control error along the diagonal repectively, the induction machine tranfer function change to - G=D ˆ e GD u () Here the original tranfer function i denoted with a hat. The limit on the tator voltage magnitude follow from (), and (6) will be ued to cale the tator voltage frequency, although it only repreent the maximum teady tate frequency. Hence, by introducing the tator frequency ω a the time derivative of χ, we get (only conidering upper magnitude limitation) umax max Ud m ω,, umax = () π T Here the approximation m u m ω valid for all but very low tator frequencie wa ued. The matrix D e i determined by the maximum control error et to % of maximum torque and rated flux (ee the appendix).

3 6. INDIRECT SELF CONTROL (ISC) The continuou time ISC control law i given by ( ψ ) ψ () () () () () u t = w t + jw t t + R i t () ref T where w Ψ (t)= F Ψ e Ψ i the output of a PI controller for the tator flux magnitude and w T (t) i the output of a torque controller, given by Rr wt () t = npωm() t + T ref () t + nm p r () t (6) Rr + FT ( p) e T () t nm t p r () Here F T i a PI-controller and e T i the error in torque. The factor in front of the torque reference i the teady tate converion to lip frequency. From the analyi in (Mokull, ) it follow that, for zero torque, the linearized ISC control law can be repreented a - u() t = G ( Ce() t + C fwtref () t ) (7) where e(t) = (e T (t) e Ψ (t)) T and ( ) FT T + C( ) =, C ( ) T fw = + (8) m F ( ) ψ For the -analyi we will focu on the feedback part of the controller, which will be denoted K, i.e. - K=G C (9) Due to pule width modulation (PWM), the tator voltage in (7) will be delayed by T d. 7. CONTROL REQUIREMENTS The requirement on the cloed loop ytem will be et in term of the tranfer function KS and S, the mixed-enitivity approach (Skogetad and Potlethwaite, 996). The enitivity function i given by S = (I+GK) Input Requirement - KS. The magnitude of the tator voltage i retricted by () wherea the tator frequency i only retricted in teady tate. Neglecting the teady tate contraint on the frequency therefore motivate the following requirement (note that the abolute value of the caled input are limited by one): ( WP ( jω) K( jω) S ( jω) ) <, ω () where W P ( jω ) = (). 7. Performance Requirement - S Performance requirement on the cloed loop ytem are often given in term of torque tep repone time, or equivalently, on the torque control bandwidth, ω B. One way of auring uch a requirement i to put a contraint on the enitivity function. Thu, we demand that ( S ( jω) ) <, ω () wp ( jω ) where jω + ω B wp ( jω ) = M () jω+ ωbq The parameter Q define tolerable value for the enitivity function at teady tate and M et the limit at high frequencie. In thi work the parameter are et to M =, Q = e-6, ω B = rad/. () 7. Uncertainty In practice there i alway a time delay connected to digital controller. The controller time delay will be modeled a an uncertainty at the plant input. That i j T G e ω d = G( I+ W I I) () where W I = diag(w I, w I ) and jωtd max wi =, I < (6) ωt d max j + The maximum time delay, T dmax, i et to m, which i reaonable in for example traction application. 8. MODEL FOR CONTROLLER ANALYSIS The ytem can now be put on the tandard form ued for -analyi/ynthei, ee Fig.. d u u G I P K W I - y W P z Fig.. Plant model for -analyi. The ytem P i given by WI Wp P = (7) WpG Wp WpG -G -I -G where the following notation wa introduced W P = diag(w P, w P ). W P v z

4 8. Evaluation Criteria Controller evaluation i done by cloing the loop and introducing a fictitiou performance uncertainty P, ee Fig.. thee controller are hown in Fig., Fig. 7 and Fig. 9. The -ynthei deign how that it i poible to reduce the peak of the RP ingular value at dipene of nominal performance. Note that - ynthei i with repect to RP. u I N P y.6. ISC: Structured Singular Value - NP OP Fig.. Plant model for controller evaluation. Now four criteria can be defined in term of the tructured ingular value, (Skogetad and Potlethwaite, 996): Nominal Stability (NS): N internally table Nominal Performance (NP): ( N ( jω) ) < P, ω(and NS) (8) Robut Stability (RS): ( N ( jω) ) < I, ω (and NS) (9) Robut Performance (RP): ( ( j )), N ω < ω (and NS) () where = diag( I, P ). 9. CONTROLLER EVALUATION Thi ection how the evaluation criteria (8) () when ISC i applied to an induction machine with the data given in the appendix. The controller i tuned according to recommendation in (Mokull, ) to give a bandwidth of rad/. The analyi i done for three operating point with nominal flux and zero torque, where the tator frequency i et to % (OP), % () and 9% () of bae peed ω. The ampling time i. m for all operating point and the reult are preented in Fig., Fig. 6 and Fig. 8. Under the aumption of perfect parameter, the analyi in (Mokull, ) indicate no difference in behavior between the different operating point. Thi alo follow from Fig.. The reaon that the nominal performance at operating point three deviate from the firt two operating point i that the maximum ingular value of KS get cloe to the limit. It however tay le than the limit. Note that thi i poible a ( ) T =, which i larger than one. The robut tability and robut performance criteria for the ISC all how peak at frequencie correponding to the operating point tator frequency, ee Fig. 6 and Fig. 8. Thi wa to be expected a a linearized model of the induction motor ha large relative gain array (RGA) element at higher tator frequencie, ee (Mokull, ). Invere-baed controller, uch a (9), are uually not recommended for plant with large RGA element, (Skogetad and Potlethwaite, 996). To further evaluate the ISC, linear -ynthei controller were deigned for the three operating point. The reulting tructured ingular value with Angular Frequency [rad/] Fig.. Nominal performance criterion: ISC Synthei: Structured Singular Value - NP OP. - Angular Frequency [rad/] Fig.. Nominal performance criterion: -ynthei ISC: Structured Singular Value - RS OP - Angular Frequency [rad/] Fig. 6. Robut tability criterion: ISC Synthei: Structured Singular Value - RS OP - Angular Frequency [rad/] Fig. 7. Robut tability criterion: -ynthei.

5 ... ISC: Structured Singular Value - RP OP - Angular Frequency [rad/] Fig. 8. Robut performance criterion: ISC... -Synthei: Structured Singular Value - RP OP. - Angular Frequency [rad/] Fig. 9. Robut performance criterion: -ynthei.. SIMULATIONS In thi ection ome imulation reult are hown to check the theoretical reult obtained above. A dicrete-time ISC i compared with the deigned - ynthei controller. The -ynthei controller were dicretized through zero-order hold on the input. The imulation conit of torque tep between and Nm a flux tep from % to 9% of nominal flux at time. Step repone with the ISC are hown in Fig. when the controller ue correct motor parameter. The tep repone are caled according to (), i.e. one in the plot correpond to % of maximum torque and rated flux, repectively. The repone are more or le independent of the operating point and the cro-coupling i very mall. There i no ign of the problem predicted by peak in the tructured ingular value in Fig. 8, although the imulation loop contain time delay. The correponding reult with the -ynthei controller are hown in Fig.. We ee a wore nominal performance a i indicated by Fig., where epecially the cro-coupling from torque reference to flux i large. To ee the effect of the peak of the tructured ingular value, model error are introduced by changing the parameter of the motor while keeping the parameter ued by the controller fixed. The wort combination eem to be to increae the tator reitance and to decreae the leakage induction of the motor. In Fig. tep repone at are hown where the tator reitance of the real machine i increaed by a factor four and the leakage inductance i decreaed by a factor four. Although thee are unrealitically large model error, the reult are till not too bad with the ISC. A a matter of fact, the correponding imulation with the - ynthei controller i untable. Scaled Torque and Fluxe Step Repone: ISC OP...6 Time [] Fig.. Step repone with ISC. Scaled Torque and Fluxe Step Repone: -Synthei Controller OP...6 Time [] Fig.. Step repone with the -ynthei controller. Scaled Torque and Fluxe - - Step Repone.. Time [] Fig.. Step repone with the ISC where the tator reitance i four time larger and the leakage inductance four time maller compared to the value ued by the controller.

6 Actually, the dicrete time ISC propoed by (Maichak, 99) ue the reference magnitude for the tator flux explicitly hown in () intead of the actual magnitude. The imulation a well a the analyi performed above have conidered the cae with the actual flux magnitude. By trictly following (Maichak, 99), the imulation reult in the ideal cae with perfect motor parameter are hown in Fig.. Compared to Fig. we ee a very large diturbance in torque at a tep in the flux reference. Although the deviation in flux i mall (note only magnitude deviation), the error in torque i large and the error increae with the operating point rotor peed. Thi fit with the reult of Fig. 8. In thi cae however the error i not caued by erroneou parameter but caued by an approximation in the control law, which give an error in the magnitude of the tator flux. Scaled Torque and Fluxe Step Repone: ISC OP...6 Time [] Fig.. Step repone with ISC a propoed in (Maichak, 99).. CONCLUSION In an attempt to properly analyze robutne of Indirect Self Control (ISC) of an induction machine, -analyi wa applied to linearized model of the ytem. The analyi indicate decreaing robut tability a well a robut performance with increaing rotor peed. Problem are to be expected for frequencie around the operating point tator frequency. At imulation with different parameter of the induction machine and the controller, the ISC till performed urpriingly well. Even with large model error it outperformed -ynthei controller deigned for optimal robut performance. It wa concluded that the wort cae model error, giving large tructured ingular value during analyi, were not obtained by the tried variation of motor parameter. However, by lightly modifying the ISC control law, to ue the reference tator flux magnitude intead of the actual value, mall tep in the flux reference were hown to have large impact on the torque. Compared to the analyi for claical field-oriented controller, given in (Thoma, 99), the tructured ingular value for ISC how harper peak. Otherwie, control method relying on tator flux etimation, uch a DTC technique, are conidered more robut to parameter variation compared to field-oriented controller orienting to rotor flux. Under ideal condition, tator flux etimation only depend on one motor parameter, the tator reitance; wherea rotor flux etimation involve more motor parameter, ee e.g. (Xu et al., 99). APPENDIX: Γ MODEL MOTOR DATA Stator reitance R = 8.mΩ Rotor reitance R r = 7.mΩ Stator inductance L = 6.mH Leakage inductance L =.79mH Number of pole pair p = Rated DC-link voltage U = 7V Bae peed ω = 8rad/ Rated flux Ψ =.9V Maximum torque T max = Nm REFERENCES Bazanella, A. S. and R. Reginatto (). Intability Mechanim in Indirect Field-Oriented Control Drive: Theory and Experimental Reult. In: IFAC World Congre. Barcelona, Spain. Buja, G.S. and M.P. Kazmierkowki (). Direct Torque Control of PWM Inverter-fed AC Motor - a Survey. In: IEEE Tranaction on Indutrial Electronic,, no., De Wit, P.A.S., R. Ortega and I. Mareel (99). Indirect Field-Oriented Control of Induction Motor i Robutly Globally Stable. In: Conference on Deciion & Control. New Orlean. Jänecke, M., R. Kremer and G. Steuerwald (989). Direct Self Control, A Novel Method Of Controlling Aynchronou Machine In Traction Application. In: Record of the EPE-Conference, Aachen. Kovac, K. P. (98). Tranient Phenomena in Electrical Machine, Elevier. Marino, R., S. Pereada and P. Valiga (99). Adaptive input-output linearizing control of induction motor. In: IEEE Tranaction on Automatic Control, 8, no., 8-. Maichak, D. (99). Schnelle Drehmomentregelung im geamten Drehzahlbereich eine hochaugenutzten Drehfeldantrieb, VDI Verlag. Mokull, H (). Tuning of Field-Oriented Controller for Induction Machine. In: IEE-PEMD. Bath, UK. Mokull, H. (). Controllability Analyi of an Inverter Fed Induction Machine. In: American Control Conference, Boton, Ma. Skogetad, S. and I. Potlethwaite (996). Multivariable Feedback Control, John Wiley & Son. Steimel, A. (). Stator-Flux-Oriented High Performance Control in Traction. In: th Annual Meeting IEEE IAS Indutry Application Society. Rome. Thoma, J.L. (99). -Analyi Applied to Field Oriented Control of Induction Motor. In: th European Conference on Power Electronic and Application. Brighton, Great Britain. Xu, X, R. De Doncker and D. Novotny (988). A tator flux oriented induction machine drive, In: IEEE Power Electronic Specialit Conference, Kyoto, Japan.