To Determine the Characteristic Polynomial Coefficients Based On the Transient Response

Transcription

1 ICCAS Jue -, KINTEX, Gyeogg-Do, Kore To Determe the Chrctertc Polyoml Coeffcet Bed O the Tret Repoe Mohmmd Her d Mohmmd Sleh Tvzoe Advced Cotrol Sytem Lb., Electrcl Egeerg Deprtmet, Shrf Uverty of Techology, Tehr, Ir (Tel : ; E-ml: her@.hrf.edu) (Tel : ; E-ml: tvzoe@mehr.hrf.edu) Abtrct: Th pper preet method to determe the chrctertc polyoml of cloed loop ll-pole ytem order to obt dered tret repoe term of the overhoot d peed (rg/ettlg tme). The method djut the overhoot by dog ome chge the chrctertc rto of the Beel-Thompo flter. The cloed loop peede the tued by utble choce of the geerlzed tme cott. Smulto reult re preeted to evlute the chevemet d mke compro wth thoe of mlr method. Keyword: Chrctertc polyoml, CDM, Tret repoe, Beel-Thompo flter. INTRODUCTION To defe the dered cloed loop tret repoe term of the chrctertc polyoml oe of the mot mportt ue the cotrol ytem deg. Almot ll the reerche the re hve dcued the mtter for the pecl form of the cloed loop trfer fucto.e. ll-pole trfer fucto [-]. The relto betwee chrctertc polyoml coeffcet d tret repoe h bee tuded, for the frt tme 9, by Grhm d h collegue who developed the Itegrl Tme Abolute Error (ITAE) tdrd form []. I 96, Keler dd ome chge the defed form to get tep repoe wth le ocllto d overhoot []. H propoed form cue 8 percet overhoot. It lo more tble (h le oclltory repoe) compro wth IATE form. The bc of the ew method determg the chrctertc polyoml coeffcet re tude tht Nl h doe the 96 [6]. Oe of the mot mportt method the ctegory the Coeffcet Dgrm Method (CDM) whch h bee troduced by Mbe [7]. Although, the CDM codered ew method, t prcple hve bee utlzed the dutry for bout yer [8]. The mot recet method the Chrctertc Rto Agmet (CRA) tht h bee troduced [9]. Th method proft by the tructure ued the deg of the Butterworth flter d t offer chrctertc polyoml by whch utble tret repoe c be obted. I th ote, we hve propoed ew method to determe chrctertc polyoml for ll-pole ytem prg the me de of the CRA method. We be our deg o the Beel-Thompo flter whch produce tep repoe wth dered overhoot d peed. The pper h bee orgzed follow. I Secto, chrctertc rto defed d djuted to obt dered peed pecfcto. Tug of the overhoot dcued Secto. Cocluo of the work derved Secto.. ADJUSTING SPEEDINESS OF THE RESPONSE To trt the dcuo, we ue Nl defto [6]. Suppoe tht p ( Hurwtz polyoml wth potve rel coeffcet () p( Chrctertc rto re defed ( < ): α () + Alo geerlzed tme cott : () Hvg, α d τ, the coeffcet of the polyoml re uquely determed : τ α α τ, () α... α Now, uppoe tht H ( d G ( re trfer fucto wth the followg form: N( m G( D( d A( m H ( B( b m m + + d + + b m m m m d b The followg theorem tted d proved [9]: Theorem: If G ( d H ( produce output y (t) d y( β t) for the me put r (t), the the coeffcet of thee two trfer fucto re relted ccordg to the followg rule: j / β for,, L, m d b / β for j d j j,, L, ; Wth due tteto to the defto gve the begg of the ecto d the bove theorem, oe coclude: Reult : If G ( tble d mmum phe the H ( wll be determed uquely wth the followg relto: A N B D α α for,, L, m, α α for,, L,, / d. b / A N (/ β ) τ, B D () (6) (/ β ) τ d

2 ICCAS N D Where α, α, A α d α B re chrctertc rto of N (, D (, A ( d B ( repectvely. Alo τ N, D A τ, τ d τ B re ther correpodg geerlzed tme cott. Exmple : Coder the followg ytem wth the chrctertc polyoml coeffcet tht were determed bed o the IATE tdrd for []. Jue -, KINTEX, Gyeogg-Do, Kore fmou flter uch Butterworth, Chebyhev, d Ellptc, t frequecy mgtude hgh frequece h le lope (Fg. ) []. Therefore, the frequecy repoe of th flter dpt wth the Chetut codto. G ( (7) The chrctertc rto d geerlzed tme cott of the ytem re: α, α, α, α ] [.68,.6,.779,.],. [ Fg. how the ut tep repoe of the ytem. A t oberved, the re tme bout.8 ec. Now uppoe tht oe wt to obt re tme of ec wthout y chge the overhoot. Wth due tteto to Reult, t uffce to keep ll of α fxed d chooe the τ : The reulted ytem wll be : G ( (8) I Fg., the tep repoe of th ytem lo depcted. A t expected, the re tme th ce ec. Fg. Beel-Thomo flter repoe for dfferet order. Chrctertc polyoml coeffcet of Beel-Thomo flter re tke from Beel polyoml whch hve bee troduced by Krll d Fk 98 []. The chrctertc polyoml defed : ( + k)! p( , k (9) k ( k)! k! From (): α k ( + )( )( + ), k () k + k + k Fg. Re tme djutmet (Exmple ).. TUNING OF THE OVERSHOOT I th prt, we put the topc of our dcuo o the b of the obted reult Chetut expermetl tude []. He dvocted tht lower (or o) reot pek well low teeper tteuto lope the hgh frequecy rego ecery to cheve mller overhoot. Here, we hve tke help from Beel-Thomo flter deg method for electo chrctertc polyoml whch reult dered tret repoe. Amog flter, th flter h bee kow mxmlly-flt tme dely []. The frequecy repoe of Beel-Thomo flter doe ot hve y reot pek (Fg. ) d compro wth the other Fg. Mgtude repoe of tdrd flter. The followg relto ext mog ( k k α k : + ) α () Relto () ely proved by mple replcg of from (). α k

3 ICCAS We kow tht the overhoot for the Beel-Thomo flter le th %. Hece, If we chooe α k ccordg to (), the repoe wll hve overhoot le th %. The followg lgorthm propoed to determe chrctertc polyoml tht reult djutble overhoot log wth the requred peed: Algorthm: - Select the geerlzed tme cott rbtrrly ( τ ). - Chooe pproprte degree for the chrctertc polyoml ( ). - Clculte α ccordg to (). - Adjut the geerlzed tme cott ( τ ) order to get repoe wth demded peed (Reult ). Jue -, KINTEX, Gyeogg-Do, Kore - Adjut the geerlzed tme cott ( τ ) order to get repoe wth demded peed (Reult ). Fg. Decree trfer fucto mgtude lope by decree γ. For exmple f the purpoe to rech ettlg tme of t d the reulted repoe t tge of the lgorthm h the ettg tme of t, the τ hould be determed : t τ τ t () The chrctertc polyoml whch deduced from the bove lgorthm gve u repoe wth very low overhoot. If oe decde to obt lower overhoot or repoe wthout overhoot, th lgorthm hould be modfed to tfy Chetut' crter. We try to decree the lope of the trfer fucto mgtude t hgh frequece by pplyg the followg modfcto. α replced by γα ( < γ ) d α k ( k ) re dvded by γ. Thee chge tll tfy (). I other word, we propoe the followg chrctertc rto. + α γ. + α k ( + )( )( + )( k + k + k γ ) () Decree of γ, reduce the lope of the trfer fucto mgtude the hgh frequece. Fg. how the effect for 7 d three dfferet γ. The multo reult how tht oe c rech to lmot % overhoot by decreg γ. Accordg to [], we kow tht f the curvture of the become lrger, the ytem become more tble. Decree of γ lo cree the ytem reltve tblty. Fg. how the pot for 8, 6 d three dfferet γ. Tble how bdwdth, g mrg, phe mrg, overhoot, ettlg tme, re tme d trckg error ccordg to tegrl error dce crter for three dfferet γ. A t oberved decree γ produce lrger g d phe mrg. The lgorthm the modfed follow. Modfed Algorthm: - Select the geerlzed tme cott rbtrrly ( τ ). - Chooe pproprte degree for the chrctertc polyoml ( ). - Set γ le th oe d the clculte α from (). Th electo wll reult lower overhoot. A pecfc reult c be cheved by few trl d error terto. Fg. Icree ytem reltve tblty by decree γ. Tble. Bdwdth, G d Phe mrg, Overhoot, Settlg d Rg tme d error mout for three dfferet γ γ γ. 9 γ. 8 Bdwdth.9 Hz.68 Hz.667 Hz G mrg Phe mrg Overhoot.9% Settlg tme.69 ec.687 ec.7997 ec Re tme.7 ec.877 ec.868 ec Error (IAE) Error (IATE) Error (ISE) Error (ITSE) Exmple: We deg the cotroller for the me ytem [9] ug the bove lgorthm. Our purpoe to cotrol the ytem uch tht the cloed loop tep repoe h o overhoot d % ettlg tme of ec. Trfer fucto of the ytem : ( 6( + +.) G ( () 6 d( We ue two degree of freedom cotroller tructure [6]. Fg. 6 how the block dgrm of the cloed loop ytem. Coder the followg polyoml. ( b( b b + b, f ( f () The followg umpto re codered to get zero-free cloed loop ytem.

4 ICCAS Jue -, KINTEX, Gyeogg-Do, Kore propoed method wth the oe clled CRA d w troduced [9]. Aume tht the dered repoe h % ettlg tme of ec d overhoot le th.%. Fg. 6 Two degree of freedom cotrol cofgurto. ( ( ( + +.), ( ( ( + +.) (6) The cloed loop trfer fucto c be wrtte follow: Y ( f ( ( T ( (7) R( p( Where p ( the chrctertc polyoml. p( ( ( d( + b( ( )( p( ( + +.) + +.) (8) p ( of order 9 d determed bed o the propoed lgorthm. Frt, we et. The derve the tep repoe for two vlue of γ ( γ d γ. 7 ). Whe γ the repoe h overhoot of.7% d whe γ. 7 the overhoot lmot zero (Fg. 7). The ettlg tme of o overhootg repoe equl to.96 ec. Therefore we djut τ :.96.9 Fg. 7 Step repoe of the dered cloed loop ytem wth vrou γ (Exmple ). Fg. 8 how the repoe for the fl cloed loop ytem. The cotroller polyoml re gve follow: - ( (9-) b( (9-) f ( (9-) Exmple : Through th exmple we try to compre our Fg. 8 Step repoe of the dered cloed loop ytem wth vrou τ d γ.7 (Exmple ). Tble how bdwdth, re tme d trckg error ccordg to tegrl error dce crter. A t oberved the propoed lgorthm poee lrger bdwdth d lower re tme th the CRA method. However, the CRA method produce le error. Now, we tudy the etvty of the repoe chrctertc (overhoot d rg tme) wth repect to vrto the coeffcet. Overhoot d re tme chge cued by ± % vrto d re gve Tble. We oberve tht the CRA method dcte le etvty to coeffcet vrto ome ce. However, the dfferece re eglgble. Tble. Bdwdth, re tme d error mout for preeted lgorthm d CRA lgorthm New Algorthm CRA Algorthm Bdwdth. Hz. Hz Re tme. ec.9 ec Error (IAE).98. Error (IATE).9.7 Error (ISE).8.96 Error (ITSE) Tble. Overhoot d re tme vrto wth repect to % chge the coeffcet Coeffcet Vrto Method +% Vrto -% Vrto New Algorthm O.S t.6 ec O.S.7% t 6. ec CRA Algorthm New Algorthm CRA Algorthm O.S t.6 ec O.S.6% t. ec O.S.% t. ec. CONCLUSION O.S9.6% t. ec O.S<.% t.9 ec O.S<.% t.98 ec I th pper, ew lgorthm troduced whch bed o the Beel-Thompo flter deg method. By pplyg th lgorthm, chrctertc polyoml whch poee the dered overhoot d peede c be determed. If

5 ICCAS ope loop ytem mmum phe, we c ue the cotroller deg method w ued Exmple - for cloed loop ytem to get the dered chrctertc polyoml. Jue -, KINTEX, Gyeogg-Do, Kore ACKNOWLEDGMENTS Fcl upport from Shrf Uverty of Techology grtefully ckowledged. REFERENCES [] S. Jyury d J. W. Sog, O the ythe of competor for o overhootg tep repoe, Proc. Amer. Cotrol Cof, pp , 99.G.-D. Hog, Ler cotrollble ytem, Nture, Vol., pp. 8-7, 99. [] R.H. Mddleto d S. F. Grebe, Slow tble ope-loop pole: To ccel or ot to ccel, Automtc,, pp , 999. [] G.C. Goodw, A. R. Woodytt, R. H. Mddleto d J. Shm, Fudmetl lmtto due to x zero SISO ytem, Automtc,, 87-86, 999. [] D. Grhm, d R.C. Lthrop, The ythe of optmum tret repoe: crter d tdrd form, AIEE Trcto, 7(II), pp. 7-88, 9. [] C. Keler, E Betrg zur Theore Mehrchlefger Regulto, Regelugt, 8(8), pp. 6-66, 96. [6] P. Nl, Eetl of optml cotrol. MA: Boto Techcl Publher, Cmbrdge, 969. [7] S. Mbe, Coeffcet dgrm method, I Proc. th IFAC Symp. Automtc Cotrol Aeropce, Seoul, Kore, pp. 99-, 998. [8] S. Mbe d Y.C. Km, Recet developmet of Coeffcet Dgrm Method, ASSC rd A Cotrol Coferece, Shgh,. [9] Y.C. Km, L.H. Keel d S.P. Bhttchryy, Tret repoe cotrol v chrctertc rto gmet, IEEE Tr. Automt. Cotr., 8, pp. 8-,. [] R.C. Dorf d R.H. Bhop, Moder Cotrol Sytem, 8 th Ed. Addo Weley Logm, Melo Prk, CA, 998. [] H. Chetut d R.W. Myer, Servomechm d Regultg Sytem Deg, I, New York, Wley, 99. [] L.P. Huelm, Actve d Pve Alog Flter Deg, McGrw-Hll, 99. [] S. Wder, Alog d Dgtl Flter Deg, d Ed., Newe,. [] H.L. Krll d O. Fk, A New Cl of Orthogol Polyoml: The Beel Polyoml, Tr. Amer. Mth Soc., 6, pp. -, 98. [] S. Mbe, "Bref Tutorl d Survey of Coeffcet Dgrm Method", th A Cotrol Coferece, Sgpore,. [6] C.T. Che, Alog d Dgtl Cotrol Sytem Deg: Trfer Fucto, Stte Spce d Algebrc Method. Ft. Worth, TX: Suder College Publhg, 99.