Robust Linear Parameter Varying Induction Motor Control with Polytopic Models

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1 SERBIAN JOURNAL OF ELECRICAL ENGINEERING Vol. 1, No. 2, June 213, UDK: : DOI: /SJEE D Robust Lnea Paamete Vayng Inducton Moto Contol wth Polytopc Models Khama Dalla 1, Makouf Abdessalem 1, Dd Sad 1, Lab Chf-Alaou 2 Abstact: hs pape deals wth a obust contolle fo an nducton moto whch s epesented as a lnea paamete vayng systems. o do so lnea matx nequalty (LMI) based appoach and obust Lyapunov feedback contolle ae assocated. hs new appoach s elated to the fact that the synthess of a lnea paamete vayng (LPV) feedback contolle fo the nne loop take nto account oto esstance and mechancal speed as vayng paamete. An LPV flux obseve s also syntheszed to estmate oto flux povdng efeence to cted above egulato. he nducton moto s descbed as a polytopc model because of speed and oto esstance affne dependence the values can be estmated on lne dung systems opeatons. he smulaton esults ae pesented to confm the effectveness of the poposed appoach whee obustness stablty and hgh pefomances have been acheved ove the ente opeatng ange of the nducton moto. Keywods: Inducton moto, LMI, LPV contolle, Lyapunov feedback contolle, Polytopc epesentaton. 1 Intoducton he Inducton moto s wdely used n ndusty due to the smple mechancal stuctue and easy mantenance. Howeve ths moto pesents a challengng contol poblem fo thee easons. he dynamcal system s hghly non lnea, the oto flux s not usually measuable and fnally the oto esstance value vaes consdeably wth a sgnfcant mpact on the system dynamcs. he tends n nducton moto contol system s to use effectve obust contolle desgn such as H and othe obust contol appoaches [3, 4, 5, 21]. Futhemoe the man advantage of usng feld-oented contol of voltage-contolled nducton moto s that good pefomance can be acheved va non-lnea state feedback [2]. 1 Depatment of Electcal Engneeng, Batna Unvesty, Unvesté de Batna, Avenue Chahd Mohamed Boukhlouf 5 Batna, Algea; E-mals: khama.dalla@yahoo.com; a_makouf@yahoo.f; s_dd@yahoo.f 2 I.U. de l'asne Dépatement GEll. 13 avenue Fanços Mtteand, 288 Cuffes, Laboatoe des echnologes Innovantes (L..I), Unvesté de Pcade Jules Vene, Sossons, Fance; E-mal: lab.alaou@u-pcade.f 335

2 K. Dalla, M. Abdessalem, D. Sad, L. Chf-Alaou In ths wok, the synthess of a LPV system contolle use a lnea matx nequalty (LMI) appoach [6, 7] and Lyapunov theoy [8]. LPV systems ae a specal class of systems, whch ae lnea tme nvaant (LI) system fo evey fxed value of the paamete vecto () t hs paamete can be measued on lne dung contol opeaton. Also the LPV contol technque can elmnate the tedous pocess of manually tunng contol, gan and can povde a systematc gan-schedulng method [9, 1, 11, 23]. In ou case t s assumed that only the stato cuent and the oto speed ae avalable fo measuement, the oto esstance estmaton s beyond the scope of ths wok. he contol law conssts of fast nne loop used to tack stato cuent efeence geneated by the Lyapunov theoy assocated to a sldng mode contol of the flux and the speed of the moto. hs appoach shows good obustness and hgh pefomance wth espect paamete and load toque vaaton. he nducton moto model descbed n the (, ) fame can be wtten as an LPV system wch can be tanslated n polytopc epesentaton because of affne dependence wth the oto speed and the oto esstance. hs featue wll be exploted n desgnng a self gan scheduled LPV feedback contolle fo the nne loop [12, 13, 14]. Also, the LPV moto stuctue can be used to mpove the obustness of the flux obseve and to compute the wost case flux estmaton eo n tems of H nom espectng paamete vaaton [17]. he pape s oganzed as follows. In Secton 2, the LPV modelng and contol synthess condtons ae deved fo affne paamete dependent systems. In Secton 3 the contol stuctue of the stato cuent s gven. In Secton 4 obust non lnea contols s obtaned fo the speed and flux contol. In Secton 5 the obust flux obseve synthess wth mxed senstvty stuctue s gven. Valdaton wth numecal smulatons fo all theoetcal esultng and ntepetatons ae pesented n Sectons 6 and 7. 2 Inducton Moto LPV Model 2.1 Inducton moto models he state space epesentaton of the nducton moto n the stato efeence fame s gven as follows: a12 p1 a22 p 1 a1 2 a 2 2 Vs, s a3 2 pa3 a4 a5 2 s b V (1) s s pa31 a32 a 4 a52 s b whee 336

3 Robust Lnea Paamete Vayng Inducton Moto Contol wth Polytopc Models a a M a M a LR a 2 2 M M, 1, b 1, 2 1 1, 2, 3 2, s 4 2, L L LL s LL s LL s LL s Ls wth 1, 2 R, ( s, s ) ae the stato cuent components, (, ) ae the oto flux components and ( V, V ) s the stato voltage components. s s he electomagnetc toque s gven by: M p φ. (2) L e s 2.2 Polytopc nducton moto epesentaton LPV model of nducton moto s descbed by state space epesentaton of the fom x A( θ( t)) xbu G( θ ): (3) y Cx whee θ 1 2 () t R () t s a tme vayng paamete. Accodng to [22] and based on the theoy of heatng mateals, the oto esstance R can be taken as tme vayng paamete snce t can be accuately estmated on lne [2]. hus: A( θ ) A(, R) A A1RA2. (4) Specfcally fo ou poblem, the paamete vecto θ ( t) has the followng convex decomposton: θ () t, 1,, (5) whee gves the cone of polytopc paamete ange. he cone values of paamete ange ae: 11 (, mn ), 12 (, max ), 21 (, Rmn ), 22 (, Rmax ). (6) At the vetces values of the plant matx s: G( θ ) 1G( 11) 2G( 12) 3G( 21) 4G( 22), (7) wth: () t mn R() t Rmn () t max R() t Rmax 1, 3, 2, 4. R R R R max mn max mn max mn max mn

4 K. Dalla, M. Abdessalem, D. Sad, L. Chf-Alaou he stuctue of the LPV contolle of the system (7) s than gven by polytopc epesentaton as followng: 3 LPV Stato Cuent Contol 4 k( ) k( ) ( ) A θ B θ K θ 1 k( ) Dk( ). (8) C θ θ LPV stato cuent contolle s desgned n the stato fame. Its man advantage s that the nconvenences elated to the Pak tansfomaton whch could sgnfcantly affect the pefomances ae avoded [18]. 3.1 LPV contol backgound L 2 Gan pefomance Consde an open loop LPV system P descbed by x ( t) A( θ( t)) x( t) B1( θ( t)) w( t) B2( θ( t)) u( t), P: z() t C1( θ()) t x() t D11( θ()) t w() t D12( θ()) t u(), t y() t C2( θ()) t x() t D21( θ()) t w(), t whee y denote the measued output, z the contolled output, w the efeence and dstubance nputs and u the contol nputs. he matces n (9) ae affne functons of the paamete vecto that vaes n polytope Θ wth vetces θ,..., 1 θ, that s: θ( t) Θconv θ,..., θ θ,, 1. he LPV synthess poblem conssts n fndng a contolle K( θ ) descbed by: x K() t AK( θ()) t xk() t BK( θ()) t y() t K ( θ): (1) u() t CK( θ()) t xk such that the closed- loop system (11) (wth nput w and output z s ntenally stable and the nduced L 2 nom of w z s bounded by a gven numbe fo all possble paamete taectoes: ξ () t Acl ( θ( t)) Bcl ( θ( t)) ξ() t Pcl : () t cl ( ( t)) cl ( ( t)) () t. (11) z C θ D θ w he chaactezaton of obust stablty and pefomance fo the closed-loop system P cl (11) s poved by the followng theoem. (9) 338

5 Robust Lnea Paamete Vayng Inducton Moto Contol wth Polytopc Models heoem: he LPV system (11) has a quadatc stablty and L 2 gan level f thee exsts a matx X such that: Acl ( θ) X XAcl ( θ) XBcl ( θ) Ccl ( θ) Bcl ( θ) X I Dcl ( θ). (12) cl ( ) cl ( ) C θ D θ I hs mples fo synthess nequaltes (12) that, wthout loss of genealty, we can eplace the seach ove the polytope Θ by the seach ove the vetces of ths set Consequently, condton (12) can be educed to a fnte set of lnea matx nequaltes (LMI). 3.2 Computaton of self-scheduled LPV contolle We assume that paamete dependence of the plant P s affne and Θ s polytope wth vetces θ, 1,2,...,. Accodng to the esult n [6, 7], one LPV contolle K ( θ ) can be computed though the followng steps: *Compute the vetex contolles K ( AK, B,,), (1 ) K C K as follows: Solve the set of LMIs (13) and (14) XA ˆ BK C2 ˆ A ˆ K A AY B2 C K, (13) ( ˆ XB1 B 21 ) K D B 1 I ˆ C1 C 1 Y D12 C K D 11 I X I I Y. (14) whee ( ) denotes tems whose expessons follow the equement that the matx s self-adont. hs step gves ( Aˆ, ˆ, ˆ K B ) K C K and symmetc matces X and Y. Compute A K, B K and C K by: 1 ( ˆ ˆ ˆ A N A XAY B C Y XB C ) M, K K K 2 2 K 1 ˆ K K B N B, ˆ C C M, whee N and M ae matces such that I XY NM. K K 339

6 K. Dalla, M. Abdessalem, D. Sad, L. Chf-Alaou Fnally the state space matces of the LPV polytopc contolle K (.) as a convex combnaton of the vetex contolles s gven by: A B A B θ. (15) K K K K ( ) CK 1 CK 3.3 Loop shappng-mxed senstvty stuctue o each obectves n tems of pefomances and obustness of system contol we have to ntoduce weghtng functons actng as fequency fltes on the I/O sgnals of the systems [15, 16]. It can be shown that obust stablty, efeence tackng, dstubance and nose attenuaton can be defned wth senstvty functon S ( I GK ) 1 complementay senstvty functon I S and the closed loop tansfe functon KS. hus, H mxed WS s senstvty cteon espect followng nequalty: 1, whee W must WKS be a hgh-pass flte functon to nsue obustness aganst neglect dynamcs and W a low-pass flte to guaantee good tackng accuacy. s 3.4 LPV cuent contolle desgn he K(, R ), as t s shown n Fg. 1, s a cuent LPV feedback contolle allowng to tacks the set pont efeence sef. he nput of contolle s the dffeence between sef and s obtaned fom G(, R ) epesentng the nducton moto. Fg. 1 Mxed senstvty stuctue of H takng. he cuent feedback contolle s obtaned usng polytopc epesentaton of nducton moto gven by (1). he measued output s consdeed as 34

7 Robust Lnea Paamete Vayng Inducton Moto Contol wth Polytopc Models y w. s s and the extenal nputs s of efeence cuent sef sef he contolle outputs ae the stato voltage components u Vs V s. he obust multvaable LPV contolle has to povde satsfactoy pefomance ove the whole opeatng ange of the moto. he LPV contolle n polytopc epesentaton wth fou vetces s computed usng the LMI toolbox. Each vetex can be consdeed as an LI contolle wth eght states. he L 2 gan bound guaanteeng the closed loop system pefomance and stablty s equal fo ou case to ( = 1.2). In ode to obtan the optmal contolle the weghtng functon used ae as followng: 4 Speed and Flux Contolle 2 2 Ws dag,, s.2 s W dag,. s.2 s.2 (16) he Lyapunov theoy assocate to sldng mode contol technque s used to desgn speed and flux contolle. hs contol system allows obust contol of all tansent electomagnetc phenomena n a moto. o smplfy the synthess pocedue of contolle the oto flux s oented on the d axs as t s gven by followng elatons:,. (17) d q he nducton moto model can be expessed n the synchonous fame and specfcally the dynamcs of flux and speed ae gven by the followng equaton: R R d d sd d, (18) L L dt d e l J f, (19) dt whee l s the load toque and e the electomagnetc toque gven by M e p dsq. (2) L he equatons descbed by (18) and (19) can be ewtten as 341

8 K. Dalla, M. Abdessalem, D. Sad, L. Chf-Alaou dd 1 f1 sd dt (21) d f2 e dt R whee f1 d, f2 l f. L Pactcally, ae f non lnea functons and stongly affected by tempeatue, satuaton skn effects and dffeent nonlneates nduced by hamonc polluton due to convetes of fequences and nose measuements. he obectves s to detemne a contol law makng possble to mantan flux oentaton, tackng efeence speed and flux even n the pesence of paamete vaatons and measuement noses [2]. o do so we can wte the followng expesson f f f, whee f ˆ s the tue non lnea feedback functon (NLFF), f s the effectve NLFF and f s the NLFF vaaton aound. f. he f can be geneated fom the vaatons of paametes as ndcated above. We assume that all of the f, whee the ae known bounds. Knowledge of the s not dffcult obtan, snce one can use a suffcently lage numbe to satsfy the constant f : dd ˆ 1 f1 f1 sd, dt d J fˆ 2 f2 e. dt (22) Poposton: In the case of all flux state model, f the flux oentaton constants s satsfed the followng contol laws ae used ˆ sd f1 d K1( d ) K11sgn( d ), Lm (23) J K ( ) K sgn( ), e ef 2 ef 22 ef whee K and K fo 1,...,3. Poof: Let the Lyapunov functon elated to the flux and speed dynamcs defned by 1 1 V ( d ) J( ef ). (24)

9 Robust Lnea Paamete Vayng Inducton Moto Contol wth Polytopc Models hs functon s globally postve-defnte ove the whole state space. Its devatve s gven by V ( )( ) J( )( ). (25) Substtutng (22) n (25), t esults: d d ef ef ( )( ˆ L V f f ) m d 1 1 sd J( )( fˆ f J ). ef e 2 2 ef (26) Let us eplace the contol law (24) n (26) we obtan V ( d ) ( f1 d K11sgn( d ) (27) ( ef ) ( f2 K22sgn( ef ) V 1, 2 V 1 K1 ( ) d K2 ef, (28) the tem ( d )( f1 K11sgn( d ) ( ef ) f2 K22sgn( e f ), ( d ), ( ef ) and l then V V 1. All vaaton f can be absobed by K f. he equaton (29) s satsfed snce K and f K. he functon gven n (28) s globally negatve-defnte. Hence, usng Lyapunov's theoem we conclude that: lm( ), lm( ). (29) t 5 Flux Obseve Desgn d ef n he flux obseve has been pefomed usng standad poblem stuctue whee the contolle s n fact the obseve and the same optmzaton mechansm s used to acheve the synthess [13, 19]. he nputs and outputs ae as t s ndcate by Fg. 2 and obustness s mpoved by tackng nto account oto esstance and speed vaatons. he desgn conssts of fndng u Gobs y to mnmze, closed- loop H LPV nom fom w to z accodng to the small gan theoem. he flux obseve can be bult up usng polytopc epesentaton of nducton moto wth mxed senstvty stuctue and t s computed unde LMI convex optmzaton usng the LMI tool box. Descbed by 2 2 = 4LI cone the obseve have 6 state. 343

10 K. Dalla, M. Abdessalem, D. Sad, L. Chf-Alaou In Fg. 2 w Vs V s m consttute the exogenous nputs, z e e the outputs, y s s the measuements and u [ ˆ ˆ s s ] the contol nput. he tackng eos of oto flux components ae gven as e ˆ and ˆ. he obust quadatc stablty and pefomance s acheved e fo.86 usng followng shapng flte: W.6.6 dag, s7.31 s Fg. 2 Mxed senstvty stuctue fo flux obseve. Fg. 3 Geneal block dagam of the suggested IM contol scheme. 344

11 Robust Lnea Paamete Vayng Inducton Moto Contol wth Polytopc Models Fg. 3 gves a geneal descpton of the dagam block suggestng an nducton moto contol scheme. As t s shown we note that flux gven by an LPV obseve and speed ae nonlnea feedback-contolled. he stato cuent components ae tansfomed nto (, ) fame and then contolled by an LPV contolle. 6 Smulaton Results he pefomances of contolles ae nvestgated by smulaton on nducton moto whch paametes values ae gven n appendx. Full non-lnea smulatons wee caed out fo the speed, flux step demand and fo paamete vaaton see Fgs. 4 and R oque Speed (d/s) me (s) Fg. 4 Roto esstance and load toque me (s) Fg. 5 Speed tackng. Fg. 6 epesents esponse of the oto speed followng the specfed efeence. At 1.1 s a evesal speed test fom157 ad/s to 157 ad/s was pefomed wth loaded machne (14 Nm) at.5 s oque (mn) 2-2 Stato cuents (A) me (s) Fg. 6 oque tme vaaton me (s) Fg. 7 Cuent tackn g esponse. 345

12 K. Dalla, M. Abdessalem, D. Sad, L. Chf-Alaou Fg. 7 and Fg. 8 show good cuent efeence tackng wthout any effect of paamete vaaton. We can note futhemoe that the cuent peak stays wthn the admssble lmts. Howeve the tackng of the flux gven n Fg. 9 depends entely on the flux estmaton accuacy. 1 Refeence stato cuent Stato cuents (A) 5-5 Flux (wb) me (s) Fg. 8 Cuent eo zoom me (s) Fg. 9 Flux tackng efeence. 7 Concluson he man obectves of ths pape ae to show the hgh pefomances poveded by the obust non lnea contolle ove the ente opeatng ange. An LMI based appoach has been poposed to desgn a quadatcally stable flux obseve and an output Lyapunov feedback egulato to tack the stato cuent. In both cases we have obtaned a scheduled tme vayng system (LPV) whch ensues a fnte L2 attenuaton fo a gven closed-loop tansfe functon whch epesent the desgn equement. It s clealy tuned out that wth the use of the LPV technques assocated wth Lyapunov feedback contolle the obustness and stablty of the whole dve was demonstate. he man advantage of usng LPV methods s that they povde a systematc way of desgnng an H flux obseve fo the nducton moto assumng that the oto speed and esstance ae avalable. Stablty of the flux estmato was demonstated usng small-gan based analyss. he smulaton esults demonstate clealy hgh pefomances of the nducton moto contol accodng to the pofle defned above. 8 Appendx he machne paametes ae as follows: Resstance of the oto: R 4, Resstance of the stato: RS 8, Inductance of the oto: L.47H, 346

13 Robust Lnea Paamete Vayng Inducton Moto Contol wth Polytopc Models Inductance of the oto: LS.47 H, Mutual nductance: M.44H, 2 Ineta: J.4kgm, Numbe of poles pas: p 2. 9 Refeences [1] C. Attaanese, C. omasso: H Contol of Inducton Moto Dves, IEE Poceedngs Electc Powe Applcatons, Vol. 148, No. 3, May 21, pp [2] C.P. Bottua, M.F.S. Neto, S.A.A. Flho: Robust Sspeed Contol of an Inducton Moto: An H Contol heoy Appoach wth Feld Oentaton and µ-analyss, IEEE ansacton on Powe Electoncs, Vol. 15, No. 5, Sept. 2, pp [3] A. Makouf, M.E.H. Benbouzd, D. Dallo., N.E. Bouguechal: Inducton Moto Robust Contol: An H Contol Appoach wth Feld Oentaton and Input-output Lneasng, he 27 th Annual Confeence of the IEEE Industal Electoncs Socety, Denve, CO, USA, 29 Nov. 2 Dec. 21, Vol. 2, pp [4] J.C. Baslo J.A. Slva, L.G.B. Rolm, M.V. Moea: H Desgn of Roto Flux-oented Cuent-contolled Inducton Moto Dves: Speed Contol, Nose Attenuaton and Stablty Robustness, IE Contol heoy and Applcatons, Vol. 4, No. 11, Nov. 21, pp [5] D.I. Km, I.J. Ha, M.S. Ko: Contol of Inducton Motos va Feedback Lneazaton wth Input-output Decouplng, Intenatonal Jounal of Contol, Vol. 51, No. 4, 199 pp [6] P. Apkaan, P. Gahnet: A Convex Chaactezaton of Gan-scheduled H Contolles, IEEE ansacton on Automatc Contol, Vol. 4, No. 5, May 1995, pp [7] P. Apkaan, P. Gahnet, G. Becke: Self-scheduled H Contol of Lnea Paamete-vayng Systems: A Desgn Exemple, Automatca, Vol. 31, No. 9, Sept. 1995, pp [8] H.K. Khall: Nonlnea Systems, Pentce Hall, Englewood Clffs, NJ, USA, [9] B. Lu, H. Cho, D. Gegoy, K. amm: Lnea Paamete-vayng echnque fo Contol of a Magnetc Beang System, Contol Engneeng Pactce, Vol. 16, No. 1, Oct. 28, pp [1] B. Pamans, W. Symens, H. Van Bussel, J. Sweves: A Gan-schedulng-contol echnque fo Mechatonc Systems wth Poston-dependent Dynamcs, Amecan Contol Confeence, Mnneapols, MN, USA, June 26. [11] C. Wang: Contol, Stablty Analyss and Gd Integaton of Wnd ubnes, PhD hess, Unvesty College, London, UK, Mach 28. [12] D. Khama, A. Makouf, S. Dd: Contol of Inducton Moto usng Polytopc LPV Models, Intenatonal Confeence on Communcaton, Computng and Contol Applcaton, Hammamet, unsa, 3 5 Mach 211. [13] D. Fodo, R. oh: Speed Sensoless Lnea Paamete Vaant H Contol of the Inducton Moto, 43 d IEEE Confeence on Decson and Contol, Paadse Island, Bahamas, Dec. 24 Vol. 4, pp [14] M. Hlaet, C. Daengosse, F. Auge, P. Chevel: Synthess and Analyss of Robust Flux Obseves fo Inducton Machnes, IFAC Symposum on Robust Contol Desgn, Pague, Czech Republc, June,

14 K. Dalla, M. Abdessalem, D. Sad, L. Chf-Alaou [15] E. Pempan, I. Postlethwate, A. Benchab: A Lnea Paamete Vaant H Contol Desgn fo an Inducton Moto, Contol Engneeng Pactce, Vol. 1, No. 6, June 22, pp [16] S. Dd, M. adne, M.S. Nat-Sad: Robust Backsteppng Vecto Contol fo the Doubly Feed Inducton Moto, IE Contol heoy and Applcatons, Vol. 1, No. 4, July 27, pp [17] V. Uay: Electotechncs, Muszak Konyvkado, Budapest, Hungay, 197. (In Hungaan). [18] K. Wang, J. Chason, M. Bodson, L.M. olbet: An Onlne Roto me Constant Estmato fo the Inducton Machne, IEEE ansacton on Contol Systems echnology, Vol. 15, No. 2, Mach 27, pp [19] A. Chelouah: Compaason des Statéges de Commandes Nonlneae Applquées à un Moteu Asynchone Modelsés dans les Refeentels (d, q) et (α, β) Aspect Echantllonnées et Robustesse, Maste hess, he Laboatoy of Sgnals and Systems, Natonal Cente fo Scentfc Reseach, Supelec and Pas-Sud Unvesty, (In Fench). [2] D. McFalane, K. Glove: A Loop Shapng Desgn Pocedue usng H Synthess, IEEE ansacton on Contol, Vol. 37, No. 6, June 26, pp [21] E. Pempan, I. Postlethwate: L 2 and H 2 Pefomance Analyss and Gan-schedulng Synthess fo Paamete-dependent Systems, Automatca, Vol. 44, No. 8, Aug. 28, pp