Stability and Performance Analysis of Fractional Order Controller over Conventional Controller Design

Transcription

1 ISSN (r : ISSN (Ole) : Sably ad erformace Aaly of Fracoal Order Coroller oer Coeoal Coroller Deg aredra Kumar #,Sul Kumar Chaudhary #, Elecal Egeerg Deparme, SuRe Uery Alwar, Rajaha, Ida paredrar@gmal.com kumar_cul@yahoo.co. Abrac---I h paper, a ew comparae approach ha bee propoed for relable coroller deg. Sce ad egeer are ofe cofroed wh he aaly, deg, ad yhe of real-lfe problem. The fr ep uch ude he deelopme of a 'mahemacal model' whch ca be codered a a ubue for he real problem. The mahemacal model ued here a a pla. Fracoal egral ad derae hae foud wde applcao he corol of dyamcal yem whe he corolled yem ad he coroller are debed by a e of fracoal order dffereal equao. Here he ably of fracoal order yem checked a he dffere leel ad foud ha he ably rego large he complex plae. Th large ably rego prode he more flexbly for yem mplemeao he corol egeerg. Geerally, a aalycally or expermeally approache are ued for degg he coroller. If a fracoal order coroller deg approach ued for a ge pla he he corolled parameer ge he beer reul. Keyword---Fracoal Order Coroller, Fracoal Order Calculu, Sably, erformace Aaly, MATLAB, Fuco uder Cla, Zegler-Nchol Mehod I. INTRODUCTION The echque model order reduco ued all feld of Elecal, Chemcal, Aeropace, Mechacal ec. I he large proce corol yem ad mechacal produco houe, he model order reduco play a mpora role o ake he deco for he fal produc []-[3]. Geerally, he work wh large cale yem ery complex ad me-coumg [4]. To check he ably of he yem fr we make a mahemacal model of he pla. If he Orgal yem model doe o mach he dered performace of he mplemeg yem, he a coroller deged o fulfll he requreme of he dury. The deged coroller may be a full order or may be fracoal order. The mplemeao of he coroller deped o he pla. If a corol yem afe her ably codo by he Rouh-Hurwz ably era [5] he ay aalycal or expermeal approache are ued. O he oher had, f corol yem requre he ably rego beyod he Rouh-Hurwz era or requre more flexbly ha wha a approach ueful? So o fulfll he ably codo beyod he Rouh- Hurwz era a fracoal order approach [6] ad [7] ued here o deg he coroller. Here he comparae aaly prode he opo o op a coroller deg mehod for he ge pla. II. ID CONTROLLER TRANSFER FUNCTION The block dagram for a ID coroller how Fg. (). The ID coroller may be repreeed mahemacal form a, u( ( ) [ de k e( + e( d + T ] () T d d Fg.. ID coroller block dagram DOI:.87/je/7/93/79333 Vol 9 No 3 Ju-Jul 7 468

2 ISSN (r : ISSN (Ole) : Wh he ge block dagram u( deoe corol gal ad e( deoe he error gal of he yem. Here k repree he proporo ga adt, T d ued for he egral ad derae me coa repecely. The rafer fuco G c ( of he correpodg ID coroller ge a Gc ( k[ + + Td ] () T Equao () ca be rewre a k ( k + + k (3) G c 3 Here k ad k 3 ued for egral ga ad derae ga alue of he coroller repecely. The objece o dere a coroller uch ha he performace of he augmeed proce mache wh he dered performace of he model. I he compuaoal yem, he dered performace hould be afed by he cloed loop corol yem. To fulfll hee ere requreme a ID coroller dered form of full order ad fracoal order. III. A. The roduco o fracoal calculu. FRACTIONAL ORDER SYSTEM FUNDAMENTALS The erm fracoal-order calculu by o mea ew. I a geeralzao of ordary dffereao by o-eger derae. The heory of fracoal-order derae wa deeloped maly he 9h ceury [8- ]. I he deelopme of fracoal order calculu, here appeared dffere defo of fracoal-order dffereao ad egrao. To reduce o a geeral form fracoal calculu from egrao ad dffereao o he fracoal order fudameal operaor α D f (, where α ad are he lm ad R he drece of operao. The couou egrao dffereal operaor [] d > d αd f ( (4) < ( dτ ) α There are arou defo for fracoal egrao ad dffereao. Some of he defo pread ou drecly a of eger-order calculu. The deep-rooed depo clude he Cauchy egral formula, he Gruwald Leko (GL) defo ad Rema Loulle (RL) defo are ge [] a Defo : - Cauchy egral formula D γ Γ( γ ) f ( πj f ( τ ) γ + ( τ c dτ Where c he mooh cure ecrclg he gle alue fuco f ( Defo : - Gruwald Leko (GL) defo αd α h Lm f ( h h j j ( ) f ( jh) j Here [.] repree he eger par. Defo 3: - Rema-Loulle (RL) defo αd d f ( Γ( ) d f ( τ ) + ( τ ) α dτ (5) (6) (7) DOI:.87/je/7/93/79333 Vol 9 No 3 Ju-Jul 7 469

3 ISSN (r : ISSN (Ole) : The followg fuco ge below obaed by Laplace Traform of he GL ad RL fracoal dffereal-egral. The zero al codo ad order ge he followg reul ± D ± [ α f ( ; ] F( B. Fracoal order yem The fracoal-order yem he exeo form of he radoal eger order yem. Fracoal order yem gaed from he fracoal-order dffereal equao. A clac -erm lear fracoal order dffereal equao (FODE) aumed by α D y( α D y( + α D y( Le coderg he corol fuco o whch pu gal appled o FODE yem Eq. (9) a follow: α y( αd y( + α D y( u( D Afer Laplace raform of Eq. (), we ge α Y ( α Y ( + α Y ( U ( From Eq. (), we ca oba a fracoal-order rafer fuco a Y ( G( U ( α + α α () I broad, for a dyamc yem wh gle arable ad fracoal order rafer fuco of a yem ca be defed a γ b G( a γ + b + a bm a y m Here b (,... m), a (,... ) are coa ad γ (,... m), (,... ) are radom real or raoal umber ad whou lackg geeraly, ca be prebed aγ > γ >...γ m m ad m > m >... The commeurable fracoal order yem Eq. (3) ca alo be expreed commeurable form by he mul-alued rafer fuco b + b bm H (, ( > ). a + a a m Noe ha eery fracoal order yem may be repreeed he form of Eq. (4) ad doma of H( meag a Rema hee. IV. STABILITY OF FRACTIONAL ORDER SYSTEM Sably oe of he mo freque erm ued leraure whe we deal wh he dyamcal yem ad her behaor. I mahemacal ocabulary, ably heory addree he coergece clarfcao of dffereal or dfferece equao. A yem (LTI) ad o be able f he roo of characerc polyomal are had egae real par. I he cae of fracoal order yem (LTI), he ably o ame a of eger oe. Impora po ha, for a fracoal order yem, he roo may le o he rgh half of complex plae Fg. (). (8) (9) () () (3) (4) DOI:.87/je/7/93/79333 Vol 9 No 3 Ju-Jul 7 47

4 ISSN (r : ISSN (Ole) : Fg.. Sable ad uable rego of LTI fracoal order yem Theorem: - Accordg o Mago ably heorem he fracoal order rafer fuco π q able f ad oly f arg( σ ) q, whereσ, ( < q < ) wh σ C, If, a gle roo of D(, he yem cao be able. h roo of D ( σ ). N( G ( D( Aboe heorem ably rego how Fg. (), Idcae he whole plae where q. I how he Rouh-Hurwz ably ad q ed o he egae real ax for q. A we kow ha oly he pole play a mpora role he ably of a yem. So he ably aeme doe by deomaor oly ad umeraor doe o affec he ably of a FOTF. The ably of fracoal order yem ca be aalyzed aoher way alo. Le coderg here, he characerc equao of a geeral fracoal order yem a: α + α α α (5) For, we ca raform he Eq. (5) o he σ-plae. α o Here α σ k σ m ad m he lea commo mulple of ѵ. (6) For a ge α, f he abolue phae of all roo of raform Eq. (6) φ σ arg(σ ), we ca cloe he followg po for he ably of fracoal order yem. π π. The ably codo a < arg( σ ) <. m m π. The ocllao codo a arg( σ ). m If ay lear me ara (LTI) fracoal order yem afy he aboe wo po he he yem able oherwe uable. V. FRACTIONAL ORDER CONTROLLER DESIGN Maxmum of he work fracoal order corol yem are hypohecal aure. Coroller deg ad applcao ru-hrough ery mall. I h paper, he core objece o pread o he fracoal order corol (FOC) o exame he yem corol performace. The fracoal-order I λ D μ coroller wa propoed a a broad ew of he ID coroller wh egraor of real order λ ad dffereaor of real order μ. The DOI:.87/je/7/93/79333 Vol 9 No 3 Ju-Jul 7 47

5 ISSN (r : ISSN (Ole) : rafer fuco of uch kd he coroller Laplace doma ha form K I μ C( K + + K, ( λ, μ > ) λ D Here K he proporoal ga coa K I he egral ga coa ad K D he derae ga coa. If λad μ, we obaed a clacal ID coroller. If λ ad μ, we obaed a D μ ad I λ coroller repecely. Thee ere coroller are he cae of I λ D μ coroller, whch prode flexbly wh a opporuy o adju he dyamc propery of fracoal order corol yem. Two ep are ued here o deg uch coroller. Sep : - Deg of K Oerhoo perceage [r], elg me ecod [T] ad ac error perceage [E] belog o roporoal ga K. I geeral, K ca be obaed by K E Here roporoal ga K eleced for mmum ac error. Sep : - Deg of K D, μ, K I ad λ To deerme hee alue for Fracoal-Order coroller deg, he followg yhe cheme ued here. Le he coroller rafer fuco C(, la rafer fuco G( ad a uy feedback appled o he yem. hae marg of corolled yem [3 ad 4] Φ m arg[ C( jω g ) G( jω g )] + π (9) Here j ωg he ooer frequecy. hae marg a depede or coa phae. Th ca be accomplhed by coroller of he form k + C( k, k, k τ K pla () Here K pla he ga of pla ad τ he me coa for he pla. Now from he Eq. (9) ad Eq. () φ m arg [ C( jω ) G( jω )] kk pla arg + π (+ ) jω (+ ) arg [( jω) ] + π π π ( + ) g g + π Here for a ge pla, we fx he ga marg. u he ga alue Eq. () oe ca fd ou he alue of. he oher dered alue k ad k are obaed from Eq. (). Now ug hee coa Eq. (), we ca oba a fracoal I λ D μ coroller, whch a parcular cae of I λ D μ coroller ha he form ( ) k + k ; C( k K k k adk k D I If he alue of K ge he he full rafer fuco of fracoal order coroller C( K + K ( ) D + K I If do a comparo wh Eq. (7), we ca ay μ ( ) adλ. VI. CONTROLLER DESIGN USING ZIEGLER-NICHOLS SECOND METHOD I h mehod, we fr e T ad T d. By ue he proporoal corol aco eae K from o a cal alue K a whch he oupu exhb uaed ocllao he yem Fg (3). (7) (8) () () (3) DOI:.87/je/7/93/79333 Vol 9 No 3 Ju-Jul 7 47

6 ISSN (r : ISSN (Ole) : Fg. 3. Cloed loop yem for proporoal coroller Thu, he cal ga K c ad he correpodg perod are deermed by experme. Accordg o Zegler-Nchol mehod he alue of he parameer K, T adt d ca be obaed by he formula how Table. Type of coroller K p.5 K p TABLE I For Crcal Ga ad Crcal erod T T d I.45 K /. ID.6 K.5.5 The ID coroller ued by he ecod mehod of Zegler-Nchol rule ge [5]. Gc( K p ( + + Td (4) T.6K ( ( + ).75K (5) 4 Equao (5) how ha he ID coroller ha a pole a he org ad double zero a. VII. EXAMLES Coder he corol yem [6] ge below fgure, whch a ID coroller ued o corol he yem Fg. (4). Here he objece o aaly he ably of he yem ad deg he coroller by propoed approach. Fg. 4. ID corol ad wo degree of freedom corol A. Sably Check ad Coroller Deg by Zegler-Nchol Mehod The fr requreme o fd ou he arg po for coderg he K p oly. Here he cloed loop repoe K p ad double zero. Le ar he ug wh DOI:.87/je/7/93/79333 Vol 9 No 3 Ju-Jul 7 473

7 ISSN (r : ISSN (Ole) : Gcl ZN C( K R( ( + )( + 5) + K By he help of Rouh-Hurwz era, he alue of K for uaed ocllao K 3. So afer pug h alue characerc equao he alue of cal frequecy ω 5. Now wh he help of able oher alue are K T T π K 8 d Afer pug he alue GcZN ( K p ( + + Td we fd ou he coroller rafer T 6.33( +.435) fuco Gc ZN (, here he double zero ex o So here he al alue are obaed. A he requreme of dury, we ca e he alue of maxmum oerhoo programmg. Geerally, accordg o he beer eablhme of he yem, he oerhoo hould be bewee% o 4%. Ug he MATLAB program, we ary he ga 9 o 3.5 wh ep ze -. ad zero a 7 o.3 wh ep ze -.. Fe ug ge he followg reul Fg. (5) ad Fg. (6). Ga (K) ad Zero (a).4 Maxmum oerhoo (m).995 The fal cloe-loop rafer fuco of he yem Gcl ZN ( (7) (6) ole-zero Map.5 Imagary Ax Real Ax Fg. 5. ole locao o ole-zero Map DOI:.87/je/7/93/79333 Vol 9 No 3 Ju-Jul 7 474

8 ISSN (r : ISSN (Ole) : U-Sep Repoe..8 Oupu.6.4. K a Sec (ec) Fg. 6. U ep repoe of he cloed loop yem B. Sably Check ad Coroller Deg by Fracoal-Order Mehod Coder he rafer fuco model of pla ge example G la ( + )( + 5) (8) Here -5 o-doma pole ad doe o affec he pla ably o we ca elmae. Afer remog he doma pole he rafer fuco of he pla G. la ( + ) (9) We are ug here he echque propoed eco 4 for fraco order coroller deg. Accordg o h Sep : - To deg he K For mmum ac error he alue of proporoal ga K, from Eq. (9) Sep : - Deg of K D, μ, K I ad λ The alue of a me coa τ ad ga of la K la. repecely Eq. (9). If we fx o ga marg ϕm > 6 for he ge corol yem. The we fd ou he alue of.3 by Eq. (). The oher dered alue k 5 ad k obaed from Eq. (). Now pug hee alue Eq. (), we go C FO ( 5 + (3) Now addg he alue of K from ep o Eq. (3), we go fal rafer fuco of fracoal order coroller a C FO ( (3) The ope loop corol yem for coroller ad pla G ol ( (3) The cloe-loop rafer fuco of ge corol yem wh uy feedback obaed a C( Gla ( Gcl ( + C( Gla ( Or DOI:.87/je/7/93/79333 Vol 9 No 3 Ju-Jul 7 475

9 ISSN (r : ISSN (Ole) : G cl ( (33) The fuco able checked he deomaor of Gcl(, ad foud ha K, dcae he yem able. Here Fg. (7) Show ha yem corolled by fracoal order coroller ha more ably rego ad Fg. (8) Idcae ha he complee deged yem able. Fgure (9) repree he comparo bewee boh ype coroller deg..5 ole-zero Map Imagary Ax able rego uable rego Real Ax.4 Fg. 7. ole poo of a cloed loop yem u-ep repoe. Oupu me(ec) Fg. 8. Sep repoe of cloed loop yem.4. U-Sep Repoe Gcl(Zegler-Nchol Gcl(Fracoal-Order).8 Oupu.6.4. K a Sec (ec) Fg. 9. Smulaeouly u ep repoe of boh yem DOI:.87/je/7/93/79333 Vol 9 No 3 Ju-Jul 7 476

10 ISSN (r : ISSN (Ole) : TABLE II Comparo for erformace Specfcao of Deged Coroller Coroller deg Zegler-Nchol Mehod Fracoal-Order Mehod Model Full Model Full Model Specfcao G ( G ( Re me (ec) Selg me (ec) eak amplude Oerhoo (%) A me (ec).83.6 VIII. CONCLUSIONS O behalf of he reul how he able, ome mpora po may be debed for ug of he coroller. All bac dea of fracoal calculu, he ably of fracoal order yem ad MATLAB fuco are preeed here. The ma purpoe of he paper o draw aeo o fracoal order yem ably ad aaly oer a coeoal way. Here a eger order pla corolled by full order coroller ad fracoal order coroller. I coclude here ha he fracoal order yem ha a large rego for ably whch mproe he performace of he yem. Here all rae parameer of fracoal order coroller deg yem ha beer performace oer coeoal coroller deg excep oerhoo. Here he maxmum oerhoo 9. % comparo o 9.95%, bu o a problem becaue due o h, he repoe fa ad ll 4 % he oerhoo acceped he geeral yem deg. We belee ha he comparae approach ued h paper ueful for elecg he mehod of coroller deg. REFERENCES [] er-olof, G. Carl Fredrk, Maerfel ad. Molader, Corbuo o he model reduco problem, IEEE Traaco o Auomac Corol, Vol. AC-7, No., Aprl 98, pp [] B. Badyopadhyay, A. Rao ad H. Sgh, O ade approxmao for mularable yem, IEEE Tra. Crcu ad Syem, Vol. 36, No. 4, pp , 989. [3] Y. Smamah Trucao mehod of reduco, A able alerae eleoc leer d Jauary, 98, Vol. 7, No., pp [4] M. Jamhd, A oerew o he aggregao of large cale yem, 98, pp [5] Y. Shamah, Model reduco ug he Rouh ably ero ad he ade approxmao echque, I. J. Corol, Vol., No. 3, pp , 975. [6] Axell M., Be E. M., Fracoal calculu applcao corol yem, I roceedg of he IEEE 99 Naoal Aeropace ad Eleoc Coferece, New York, 99, pp [7] Dorcak L., Numercal model for mulao he fracoal-order corol yem, UEF SAV, The Academy of Scece Iue of Expermeal hyc, Koce, Sloak Republc, 994, pp [8] Oldham K. B. ad Spaer J., The Fracoal Calculu, New York, Academc re, 974 [9] odluby I., Fracoal Dffereal Equao, Sa Dego, Academc re, 999 [] Mller K. S., Ro B., A Iroduco o he Fracoal Calculu ad Fracoal Dffereal Equao, New York, Wley, 993 [] Samko S. G., Klba A. A. ad Marche O. I., Fracoal Iegral ad Derae ad Some of Ther Applcao, Mk: Nauka Techka, 987 [] Y. Q. Che, Io era ad Dgyu Xue, Fracoal Order Corol, A Tuoral, Amerca Corol coferece, Jue -, 9 [3] Bda., commade dffue due mache elec roduco, ESAIM proceedg, ol. 5, Frace, 998, pp [4] VINAGRE, B. M. odluby, I. Dorcak, L. Felu, V., O Fracoal ID Coroller: A Frequecy Doma Approach, roceedg of he IFAC Workhop o Dgal Corol ID, Terraa, Spa,, pp [5] C.C. Hag, K. J. Arom ad W. K. Ho, Refeme of he Zegler-Nchol ug formula, IEEE roceedg a Corol Theory ad Applcao, Vol. 38, No., pp. -8, 99. [6] K. Ogaa, Moder Corol Egeerg, 4h ed., Eglewood Clff, NJ: rece Hall,. AUTHOR ROFILE aredra Kumar workg a a Aa rofeor Elecal ad Compuer Egeerg Deparme a Wolaa Sodo Uery, Ehopa. Auhor purug HD Elecal Egeerg Deparme, SuRe Uery Alwar, Rajaha, Ida. H reearch area clude he feld of model order reduco, Fracoal order model ad coroller deg. He compleed M. Tech () from IIT ROORKEE Syem ad Corol ream uder Elecal egeerg Deparme ad o Graduae Dploma (8) Embedded Syem from DOI:.87/je/7/93/79333 Vol 9 No 3 Ju-Jul 7 477

11 ISSN (r : ISSN (Ole) : C-DAC Hyderabad. He receed B.Tech (7) degree from Uar radeh Techcal Uery Luckow he ream of Eleoc ad Commucao. He publhed more ha fe aoal ad eraoal paper he feld of model order reduco ad coroller deg Sul Kumar Chaudhary graduaed from Nagpur Uery 996 eleoc, compleed h M.E. ower Eleoc from Maharh Dayaad Uery, Haryaa he year 6 ad h.d. elecal egeerg, from JamaMlaIlama, New Delh he year 4. H employme experece clude Lgaya Iue of Maageme ad Techology (LIMAT) Fardabad, Haryaa, MaaRacha college of Egg (MRCE), Fardabad, Haryaa, IIMT college of Egeerg Greaer Noda, U. ad preely workg a rofeor Elecal Egg. Deparme SuRe Uery Alwar, Rajaha. He ha more ha paper Ieraoal Joural / Coferece of repue. H area of ere Applcao of of compug echque o ower Eleoc ad Corol Syem. DOI:.87/je/7/93/79333 Vol 9 No 3 Ju-Jul 7 478