Feedback Linearizing Control of Induction Motor Drive by P-I Controllers in RTDS Environment

Transcription

1 Jounal of Automation and Contol Engineeing Vol., No. 4, Decembe 03 Feedback ineaizing Contol of Induction Moto Dive by P-I Contolles in TDS Envionment Kanungo Baada Mohanty and Madhu Singh Dept. of Electical Engg., National Institute of Technology, oukela, India Abstact This pape eviews the model of the induction moto in stationay efeence fame, with stato cuent and oto flux components as state vaiables. The state feedback lineaizing and decoupling contol as applied to the induction moto dive is pesented. With this contol, the dive system is decoupled into two linea subsystems: electical and mechanical. Compehensive and systematic pocedues ae developed to detemine the gains of the Popotional Integal (P-I) contolles fo electical and mechanical subsystems. The contolled dive system is simulated in MATAB SIMUINK and expeimented in TDS envionment, and esults ae pesented. Index Tems feedback lineaization; flux estimato; stationay efeence fame; popotional-integal contolle; eal time digital simulato (TDS) I. INTODUCTION ecently the subject of nonlinea contol is occupying an inceasingly impotant place in automatic contol engineeing and has become a necessay pat of the fundamental backgound to contol engineeing []-[]. Its potential application in the aea of induction moto contol is emeging as the thust aea fo eseach wok. The induction motos ae the most pefeed fo industial application because of simplicity, eliability, low cost, uggedness, and suitability to wok in volatile envionment. It does not equie maintenance and is pollution fee. So, it is also well acceptable in automation industies. But it equies complex contol stategy, because it possesses thee inheent dawbacks as follows. It is a highe ode nonlinea dynamic system with intenal coupling of states. Some state vaiables like oto cuents and flux ae not diectly measuable. Vaiations in paametes like oto esistance due to tempeatue, and magnetizing inductance due to satuation have significant impact on the system dynamics. Many attempts have been made in past to optimize the pefomance and simplify the contol stategy of the induction moto. Out of these Field Oiented Contol o Vecto Contol poposed by Blaschke [3], and Hasse [4] has emeged successfully to achieve the high pefomance equiement. As a esult it has been aggessively accepted by the automation industies by eplacing bulky, costly DC moto dive which has commutation poblem. The vecto contol methods ae complex to implement. Because in vecto contol method, the decoupling elationship is obtained by means of a pope selection of state coodinates, unde the hypothesis that the oto flux is kept constant. The toque is only asymptotically decoupled fom the flux i.e., decoupling is obtained only in steady state, when the flux amplitude is constant. Coupling is still pesent, when flux is weakened in ode to opeate the moto at highe speed within the input voltage satuation limit o when flux is adjusted in ode to maximize powe efficiency [5], [6]. This has futhe led to intoduction nonlinea geometic contol theoy paticulaly feedback lineaization, which can achieve completely decoupled toque and flux amplitude of the induction moto [7]-[8]. But fo the satisfactoy pefomance, the moto paametes of the contolled plant must be pecisely known and accuate knowledge of the flux is equied. In the last decade, a good numbe of eseach woks has been epoted incopoating vaious contol schemes to simplify and to enhance the pefomance. The wok includes seveal methods fo accuate estimation of flux. But the contol pefomance is still influenced by the uncetainties of the plant. Theefoe, the motivation behind this wok is to design a suitable obust contol scheme to combat the uncetainties aising in pactical application. In this wok, decoupling of flux and speed in induction moto dive with state feedback lineaization technique, and design of the popotional-cum-integal (P-I) contolles fo flux and speed contol loop ae pesented. The nonlinea dynamics of the induction moto in the stationay efeence fame is pesented in section II. The nonlinea model of induction moto is lineaized by state feedback lineaization and decoupling technique in section III. The induction moto is thus decoupled into two linea subsystems: electical subsystem and mechanical subsystem. In section IV, systematic pocedues ae used to select the gains of the P-I contolles fo each subsystem. In section V, closed loop contol technique based on state feedback lineaization algoithm is simulated using P-I contolles and esults ae discussed. The contol scheme is simulated in MATAB SIMUINK envionment and veified in TDS envionment. Manuscipt eceived Decembe, 0; evised Apil 8, Engineeing and Technology Publishing doi: 0.70/joace

2 Jounal of Automation and Contol Engineeing Vol., No. 4, Decembe 03 II. MODEING OF INDUCTION MOTO The dynamic equations epesenting induction moto in the stato fixed α-β efeence fame ae as: p V () m m m s i s s i s s s s s p V () m m m s i s s i s s s s s p i (3) m s p i (4) m s whee, B ( Te Tl ) (5) J J m ( ) s is the leakage coefficient; (i αs, i βs ), (ψ α, ψ β ), (V αs,v βs ) ae espectively the α-β component of the stato cuent, oto flux and stato voltage, ( s, s ), (, ) ae stato and oto paametes (esistance and inductance), m is magnetizing inductance and ω is the moto speed. The electomagnetic toque developed is given by T K ( i i ) (6) e T s s whee, 3p KT p is the numbe of pole pais. III. FEEDBACK INEAIZATION Feedback lineaization is an appoach to nonlinea contol design which has attacted geat deal of eseach inteest in ecent yeas. The cental idea of the appoach is to algebaically tansfom a nonlinea system dynamics into a fully o patially linea one so that linea contol technique can be applied. This diffes entiely fom conventional lineaization techniques. Feedback lineaization is achieved by exact state tansfomation. Theefoe, it uses a nonlinea tansfomation on system vaiables expessing them in a new suitable coodinate system which enables the intoduction of a feedback, so that an input-output o state lineaization in new coodinates is achieved. The theoetical foundation and systematic appoach can be found in [], [7]-[8]. In ode to contol the induction moto in field oientation schemes to get a dc moto like pefomance, the oto speed and oto flux must be decoupled. Theefoe, outputs to be contolled ae chosen as the oto speed, ω and oto flux, ψ. The magnitude of oto flux is given as m (7) The time deivative of oto flux linkage is (8) Substituting fom (3) and (4) into (8) m m p i s p i (9) s Simplifying (9), the oto flux dynamic equation is m ( i s i s ) (0) Fom (5) and (6), oto speed dynamic equation is B KT ( i s i s ) T () l J J J Equations (0) and () descibe flux and mechanical system, which has i αs, and i βs as two contol inputs. Thus, it epesents a coupled system. Theefoe, the nonlinea feedback theoy [] is used to eliminate this coupling elationship between the contol inputs i αs, i βs and the system outputs ψ and ω. et u and u be taken as two new contol input which convets coupled system into decoupled one [7]. Equations (0) and () with new contol input can be ewitten as: m u () B Tl Ku T (3) J J J Fom (0), (), () and (3) the expession fo contol inputs can be witten as [7]. u i s si s (4) u i i (5) s s Above equations ae ewitten in (6) and (7) fo deivation of i s andi s in tems of u and u i s u u (6) i s u u (7) Equations (6) and (7) epesent a feedback lineaization decoupling contolle. The block diagam of feedback lineaizing contolle is shown in Fig.. The tansfomed model of induction moto as given in () and (3) is linea and decoupled. The developed toque and the oto flux ae independently contolled. The induction moto model is now decoupled into two linea subsystems: Electical subsystem and Mechanical subsystem, shown in Fig. and Fig. 3, espectively. The electical subsystem is given by () and mechanical subsystem is given by (3). The oto flux and moto speed can be contolled by P-I contolles [8]. P-I Contolles ae developed using linea contol theoy in [8], to obtained desied steady state and tansient pefomance. The pefomance of the dive system 03 Engineeing and Technology Publishing 307

3 Jounal of Automation and Contol Engineeing Vol., No. 4, Decembe 03 lagely depends upon the choice of the contolle. Design of contolles is discussed in section IV. u u Figue. Feedback lineaizing and decoupling contolle Figue. Block diagam of the open loop electical subsystem Figue 3. Block diagam of the open loop mechanical subsystem IV. Feedback ineaizing and Decoupling Contolle, Equations (6)-(7) Ψ αs u m s DESIGN OF P-I CONTOES The electical and mechanical subsystems obtained above ae type zeo systems as the integal tem is absent in both. So, a step input leads to a steady state eo. Hence, contolles with integating action ae equied fo both the subsystems. In this scheme two Popotionalcum-Integal (P-I) contolles ae used, one fo the electical subsystem and anothe fo the mechanical subsystem. To have a feel of the influence of contolle gains on the chaacteistic of the dive system in geneal and the factos affecting the gains of the P-I contolles in paticula, the tansfe function of the electical and mechanical subsystem ae analyzed. Dive system with designed contolles is simulated and it is veified that simulation esponse with the designed contolles is satisfactoy. P-I contolles ae designed using Modulus optimum method. The contolle is designed so as to make the modulus of the closed loop tansfe function, unity ove a wide fequency ange, stating fom zeo. A. P-I Contolle fo Electical Subsystem Using the induction moto paametes given in Section V, =0.5H, m =0.5H and =0.64Ω, the open loop tansfe function fo the electical subsystem shown in Fig. can be expessed as m () s 5.4 u( s) s s This is a fist ode tansfe function. Time constant of flux is: Ψ βs i αs i βs 0.09 sec The time constant of the open system esponse is 0.09 s. Fo a unit step input, 90% ise time of flux is.3 times the electical time constant, i.e.,.3x 0.09= 0.6 s, theeby giving sluggish esponse. In ode to tack the * efeence flux ψ and to impove the system esponse one P-I contolle is used as shown in Fig. 4. The tansfe function G 0 (s) of the fowad path including that of the contolle is G 0 (s) = G (s). G (s) K s K p i ( s K ) p i. s s0.846 ss ( 0.846) Then, the oveall closed loop tansfe function of the unity feedback flux contol loop 5.4( K s K ) p i () s G () s 0 ss ( 0.846) * ( s) G ( s) 5.4( K s K ) 0 p i ss ( 0.846) 5.4( K s K ) p i s ( K ) s 5.4K p i The chaacteistic polynomial of the closed loop tansfe function: s ( K ) s 5.4K is p i of second ode and compaed with the standad fom: ( s ns n ), whee, ω n is the natual fequency of oscillation and ξ is the damping facto. Assuming the natual fequency of oscillation, n to be 75 ad/s, 5.4K 565 i 565 Ki In ode to make the system citically damped (i.e.,=), equating coefficients of s, popotional gain K p is obtained K p = n = K 5.67 p 5.4 Figue 4. Block diagam of the closed loop electical subsystem with P-I contolle With these values of K p and K i, the contol law fo the electical subsystem is: u t * * 5.67( ) ( ) Thus closed loop poles ae at -75, fathe away fom the oigin of complex s-plane, than the open loop pole at ise time fo unit step speed input n 0 dt 03 Engineeing and Technology Publishing 308

4 Jounal of Automation and Contol Engineeing Vol., No. 4, Decembe 03 is s. This makes the closed loop 75 system s dynamic esponse faste and moe stable. The dive system is simulated taking diffeent value of n and it is obseved fom the simulation studies that n = 75 ad/s is moe acceptable. B. P-I Contolle fo Mechanical Subsystem Using the induction moto paametes given in Section V, =0.5H, m =0.5H and =0.64Ω, J=0.6 kg.m and B=0.035 kg.m /s, the open loop speed tansfe function is: () s KT 8 u( s) Js B s 0. Figue 5. Block diagam of the closed loop mechanical subsystem with a P-I contolle This is a fist ode tansfe function. Time constant of speed is 4.55sec It s ise time is s and pole at -0. gives sluggish esponse. In ode to tack the efeence speed and to impove the tansient pefomance a P-I contolle is used as shown in Fig. 5. The tansfe function of the fowad path including that of the contolle is G ( s) G ( s). G ( s) K ps Ki K 8( K T ps Ki). s ( Js B ) ss ( 0.) Then, the oveall closed loop tansfe of using feedback speed contol loop can be expessed as: () s 8( K ps Ki) * () s s (0. 8 K ) s 8K * p i The closed loop tansfe function is of second ode. The chaacteistic polynomial is in the fom s s. Assuming ω n to be 4 ad/s, and n n compaing the tems: 8 K i = 6; 6 Ki Assuming the system to citically damped (i.e., =) K p = n = x K p.43 8 With these gain values the speed contol law is given by: u * * 0.43( ) 0.88 ( ) The closed loop pole is at -4, fathe to the left of the oigin of complex s-plane than the open loop pole -0.. ise time fo unit step speed input is s. The dynamic esponse is faste 7.5 and moe stable. Theefoe, above gains ae acceptable. The dive system is simulated by taking diffeent value of n and it is obseved fom the simulation studies that n = 4 ad/s is a good choice. C. System Desciption The schematic block diagam of the poposed system is shown in Fig. 6. The scheme consists of two P-I contolles with feedback lineaizing algoithm, one flux estimato, and one cuent contolled PWM voltage souce invete. Two P-I contolles ae egulating flux and speed loop. Voltage model [6] is used fo flux estimation. Output of flux and speed egulato and also estimated flux ae the inputs to the decoupling contolle and its output goes to the cuent contolle. Output of the cuent contolle is utilized to geneate gate dive signal fo PWM voltage souce invete (VSI), which foces efeence cuent in the moto to develop equied toque. Figue 6. Schematic diagam of a lineaized induction moto with P-I speed and flux contolle 03 Engineeing and Technology Publishing 309

5 Jounal of Automation and Contol Engineeing Vol., No. 4, Decembe 03 V. SIMUATION ESUTS AND DISCUSSIONS The poposed contol scheme is simulated in SIMUINK and expeimented with the eal-time digital simulato (TDS). The specifications and paametes of the induction moto ae as follows. Thee Phase Squiel Cage Induction Moto :5 HP (3.7 kw), 4 pole, - connected, 45 V, 445pm, s = 7.34Ω, ls =0.0H, m =0.5H, = 5.64Ω, l = 0.0H, J=0.6 kg-m, B=0.035 kg-m /s. Simulation esults coesponding to speed esponse, electomagnetic toque esponse, stato cuent and oto flux ae pesented in Fig. 7 fo SIMUINK simulation and in Fig. 8 fo TDS. oto speed, developed electomagnetic toque, stato phase-a cuent, α-β components of oto flux ae shown unde diffeent dynamic conditions such as stating acceleation and load petubation (load application and load emoval). Fom the obtained esult the following salient featues ae obseved. Stating Dynamics: The thee phase squiel cage induction moto is fed fom a contolled voltage and fequency souce. The efeence speed is set at 500 pm with a cuent limit set at the ated value. Theefoe, the stating cuent is limited to the ated cuent when the moto builds up the equied stating toque to each the set speed. The moto eaches its set speed in 0.43s. The theoetical value is obtained in section IV as 0.306s. When the speed eo becomes zeo pm the winding cuent also educes to no load value and the developed toque becomes equal to no load toque as obseved in the stato cuent esponse shown in Fig. 7. The oto flux amplitude emains constant thoughout. oad Petubation: As shown in Fig. 7, when the moto is unning at a steady state speed of 500 pm, a load toque equal to the 0 N.m is applied at t = s and emoved at t =.5s. Application of the load esults in incease of stato cuent to coesponding value. VI. CONCUSIONS The feedback lineaization contol fo induction moto dive has been pesented to decouple speed and flux. Systematic pocedue is adopted to design P-I contolles fo electical and mechanical subsystems. The complete scheme is simulated in MATAB SIMUINK envionment. The pefomance of the system is obseved in tems of speed esponse, toque esponse, moto cuent and flux. The esults obtained establish that decoupling of flux and speed is obtained at all stages though the poposed contol algoithm. The scheme is implemented in TDS. Thee is eduction in the toque ipples and bette oto flux esponse. (a) Speed esponse (c) Stato cuent esponse (d) Alpha-Beta components of oto flux linkage (b) Electomagnetic toque esponse Figue 7. Simulation esults of dive system with P-I contolles (a) Speed esponse (b) Electomagnetic toque esponse 03 Engineeing and Technology Publishing 30

6 Jounal of Automation and Contol Engineeing Vol., No. 4, Decembe 03 (c) Stato cuent esponse (d) Alpha-Beta components of oto flux linkage Figue 8. eal time simulation esults of the system with TDS EFEENCES [] J. J. E. Slotine and W. i, Applied Nonlinea Contol, Englewood Cliffs, Pentice Hall Intenational, 99, ch. 4. [] A. Isidoi, Nonlinea contol Systems, Belin, Spinge-Velag, 995, ch. 3. [3] F. Blaschke, The pinciple of field oientation as applied to the new TANSVEKTO closed- loop contol system fo otatingfield machines, Siemens eview, vol. 39, pp. 7-0, May 97. [4] K. Hasse, On the dynamic behavio of induction machine diven by vaiable fequency and voltage souce, ETZ Ach., Bd. 89, H.4, pp. 77-8, 968. [5] W. eonad, Contol of Electical Dives, Belin, Spinge-Velag, 990, ch.. [6] B. K. Bose, Moden Powe Electonics and AC Dives, New Delhi, Pentice Hall of India, 008, ch. 8. [7] C. H. Tsai and H. C. u, Sensoless decoupling contol of induction motos based on feedback lineaization, in Poc. 0 TH IEEE INES Int. Conf. on Intelligent Engineeing Systems, ondon, Sept 006, pp [8] K. B. Mohanty and N. K. De, Nonlinea contolle fo induction moto dive, in Poc. IEEE ICIT, Goa, Jan 000, pp [9] A. Isidoi, A. J. Kene, C. Goi-Giogi, and S. Monaco, Nonlinea decoupling via feedback: A diffeential-geometic appoach, IEEE Tans. Automatic Contol, vol. 6, pp , 98. [0] Z. Kezminski, Nonlinea contol of induction moto, in Poc. 0 th IFAC Wold Congess on Automatic Contol, Munich, 987, vol. 3, pp []. Maino, S. Peesada, and P. Valigi, Adaptive input-output lineaizing contol of induction motos, IEEE Tans. Automatic Contol, vol. 38, no., pp.08-, 993. [] J. Chiasson, Dynamic feedback lineaization of the induction moto, IEEE Tans. Automatic Contol, vol. 38, no. 0, pp , Oct 993. [3] A. Sabanovic and D. B. Izosimov, Application of sliding modes to induction moto contol, IEEE Tans. Ind. App., vol. 7, no., pp. 4-49, Jan 98. [4] J. Chiasson, A. Chaudhai, and M. Bodoson, Nonlinea contolles fo the induction moto, in Poc. IFAC Nonlinea Contol System Design Symp., Bodeaux Fance, June 99, pp [5] M. Wlas, Z. Kzemniski, J. Guzinski, and H. Abu-ub, Atificialneual- netwok-based sensoless nonlinea contol of induction motos, IEEE Tans. on Enegy Convetion, vol. 0, no. 3, pp , Sept 005. [6]. J. Wai and W. K. iu. Nonlinea decoupling contol fo induction moto sevo-dive using sliding mode technique, IEEE Poc. Contol Theoy Application, vol. 48, no. 3, pp. 7-3, May 00. [7] J. Soltani and G.. A. Makadeh, A cuent-based output feedback mode contol fo speed sensoless induction machine dive using adaptive sliding mode flux obseve, in Poc. Fifth Int. conf. Powe Electonics and Dive Systems, Singapoe, 003, vol., pp [8] J. Soltani and M. A. Abbasian, obust nonlinea contol of linea induction moto taking into account the pimay end effect, in Poc. IEEE 5th Int. Powe Electonics and Motion Contol Conf., Shanghai, Aug 006. Kanungo Baada Mohanty has eceived B.E. degee fom Sambalpu Univ., M.Tech. and Ph.D. degees fom Indian Institute of Technology, Khaagpu in the yeas 989, 99 and 00 espectively, in Electical Engineeing. He is a faculty membe of Electical Engg. Dept, National Institute of Technology, oukela since 99, and cuently seving as Associate Pofesso. He has published 7 jounal papes and moe than 40 confeence papes. His eseach inteests include contol and estimation in induction machine specifically vecto contol, DTC and wind enegy convesion systems. D. Mohanty is a Senio Membe of the IEEE, Fellow of the IE (India), Fellow of IETE, and ife Membe of Sola Enegy Society of India, and System Society of India. kbmohanty@nitkl.ac.in Madhu Singh has eceived B.Sc. Engg.(Electical) degee fom N.I.T Patna, MBA(Pesonnel Management) degee fom NM Institute of Economic Development and Social Change Magad Univ, M.Sc.Engg.(PoweElectonics)degee fom N.I.T Jamshedpu in the yeas 990,993,and 007 espectively. She is a faculty membe of Electical Engg. Dept, National Institute of Technology, Jamshedpu since 999. He eseach inteest aea includes powe electonics and dives, nonlinea contol and atificial intelligent contol application in moto sevo dives. Ms. Singh is a Membe of the IEEE. Fellow of IETE. madhu_nitjs@ediffmail.com.in 03 Engineeing and Technology Publishing 3