Rational approximations to design controllers for unstable processes, including dead-time

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1 X Iteratioal Symoium o Idutrial Electroic INDEL 04 Baa Luka November Ratioal aroximatio to deig cotroller for utable rocee icludig dead-time arko Bošković tudet Faculty of Electrical Egieerig Uiverity of Eat Saraevo Eat Saraevo Boia ad Herzegovia bokovic.marko.etf@gmail.com Tomilav B. Šekara Faculty of Electrical Egieerig Uiverity of Belgrade Belgrade Serbia tomi@etf.r ila R. Raaić Bori Jakovlević Faculty of Techical Sciece Uiverity of Novi Sad Novi Sad Serbia raaa@u.ac.r bakov@u.ac.r Abtract Thi aer addree the roblem of deigig comlex cotroller for utable idutrial rocee with traort delay uder cotrait o robute ad erformace. The olutio to the cotrol deig roblem i obtaied i a o-ratioal form which i ratioalized by uig variou method. The aer alo reet a comarative aalyi of differet aroximatio techique. By mea of umerical imulatio it ha bee how that the rooed method lead to adequate erformace ad robute idice. Keyword Ratioal aroximatio Comlex cotroller Robute Frequecy domai. I. INTRODUCTION The aer coider the roblem of deigig comlex cotroller of utable idutrial rocee icludig traort delay obervig the retrictio o erformace ad robute. Whe deigig comlex cotroller for utable rocee with traort delay a time-delay may alo aear withi the trafer fuctio of the cotroller C(). Thu after the deig hae the cotroller trafer fuctio i ot ratioal ad oe ofte ecouter iteral itability. Thi iteral itability i the cotroller aear a oe or more utable diole i.e. air of ole ad zero at the ame oit at the right half-lae of the -lae. Elimiatio of the itral itability by mea of uitable ratioal aroximatio of the cotroller trafer fuctio i a itriic te i olvig the cotrol roblem uder coideratio ad thu rereet a key motivator for the reet aer. Several method of ratioal aroximatio of C() are coidered i the reet aer. It i well kow that early 94% of feedback i idutry i realized by PI/PID cotroller [] while thi ercetage i over 97% i etrochemical idutry [ 3]. Owig to thi high igificace of PI/PID the efficiet ad imle rocedure for tuig of arameter of idutrial cotroller have bee develoed [4 5] a well a the otimizatio rocedure [6-9] for deigig PI/PID with miimum IAE obervig the retrictio o robute which meet the criterio reeted i []. I additio to the reviouly metioed method there are method for deigig PID cotroller derived from IC cotroller [3-5]. For IC method of deigig cotroller there i oe adutable arameter λ which for a arrow cla of rocee directly ifluece the time cotat of the cloed loo ytem. Reoe to a Heaviide diturbace of the roce cotrolled by a cotroller obtaied by IC method i deedet o the domiat dyamic of the roce. E.g. if the roce ha domiat ocillatory dyamic reoce to ay diturbace will be ocillatory. The roblem of cotrol of comlex rocee (multile itabilitie multile atatim domiat time delay) ca ot be olved adequately by alyig PID cotroller the baic reao for develoig method of deigig comlex cotroller. For the uroe of accomlihig adequate idice of robute ad erformace for a wide cla of table ad utable rocee ew method [6-9] have bee develoed for determiatio of comlex cotroller baed o modified IC tructure. However the rule of deig of comlex cotroller by alyig thee method have ot bee defied for the geeral form of roce trafer fuctio () but they are defied for ecific clae of rocee () [6-9]. Cotroller C() ad it ratioal aroximatio defied i [3] i deiged for the geeral form of the roce trafer fuctio ()H()ex(-τ)/Q() uder retrictio o robute ad eitivity to meaurmet oie. Adutable arameter of a comlex cotroller C() are time cotat λ ad factor of relative damig ζ of domiat ole of the roce i cloed loo with cotroller C() a i [6]. Adutig of arameter ζ allow achievig comromie betwee idice of robute ad erformace which i ot oible for comlex cotroller deiged by IC [3-5] or by modified IC [6-9]. Through a erie of imulatio of a wide cla of idutrial rocee a comario of differet method of ratioal aroximatio of C() i order to achieve a adequate idex erformace/robute ad iteral tability of the cotroller ha bee obtaied. II. COPARATIVE ANALYSIS OF SEVERAL ETHODS OF RATIONAL APPROXIATION OF INTERNALLY UNSTABLE CONTROLLERS The cotrol tructure with C() cotroller i reeted i Fig.. ff () decribe the feed forward from the et oit y to cotrol igal u ad will ot be coidered here. For a wide cla of trafer fuctio of idutrial rocee i τ () H() e / Q() where Q() ad H() are olyomial 00

2 y ff () u d () C () Figure. Cotrol tructure with cotroller C() of the order deg Q ( ) deg H ( ) 0 ad H(0) 0. Comlemetary eitivity fuctio of the cotrolled roce () of Fig. i give by relatio T () L ()/( L ()) [3] the fuctio of feedback trafer beig of the form L () C () (). Let the deired comlemetary eitivity fuctio T() i give by where: N() τ N() e T T() () P () η P () ( λ ζλ ) ζ O() N ad adutable arameteri λ > 0 η R determied o the bai of deired eerformace of the cloed loo ytem. From relatio () cotroller C() for roce trafer fuctio () for maximum atteuatio of diturbace d or i defied a T( ) N( ) Q( ) C () () () T() H() F() τ where F() P() e N() [30]. I geeral arameter η η are determied to achieve cacellig of ole of roce () ad zero of fuctio F() [9] where for utable rocee a iteral itability arie i comlex cotroller (). Free arameter of comlex cotroller () are time cotat λ > 0 ad factor of relative damig ζ > 0 of the cloed loo ytem a i [630]. Damig factor itroduced i the comlex cotroller deig lay a igificat role i accomlihig a comromie betwee the erformace ad robute idece. It hould be metioed that arameter ζ affect eitivity to the meauremet oie at high frequecie lim C(i ω) (3) ω C (i ω ) (i ω ) I order to accomlih a comromie betwee the deired erformace IAE ad max /(L(iω)) time cotat λ ω hould fullfil coditio ωλ max /( C(i ω) (i ω)). (4) ive ζ ad (4) time cotat λ i determied by olvig two oliear algebraic equatio y Ci ( ω) ( iω) / 0 (5) ( C(i ω) (i ω) )/ ω 0 (6) a i [6 30]. The iitial choice of arameter ζ hould be ζ ad arameter λ from the viciity of the etimated traort delay. By determiig time cotat λ for differet value of arameter ζ oe achieve comromie betwee value IAE ad max ω L(iω)/(L(iω)) uder coditio that i the cae of a utable roce () the utable diole i cotroller C() i removed. I order to remove the utable diole if it exit i the cotroller everal method foud i the literature reduce to alicatio of Padé aroximatio. A. Padé aroximatio of cotroller C() Padé aroximatio i oe od the mot frequet ratioal aroximatio met i the cotrol ytem ad wider. It ca be calculated a follow [63] C () Cˆ () N d 0 N N i c i 0 i (7) where N i the uer-ecified cotroller order to achieve the deirable erformace ecificatio for the load diturbace reectio ad c i ad d are determied by the followig two liear matrix equatio. d b c d b b0 0 0 c (8) d b b b b c N N N N N b b b b c b N N N N bn bn bn b 3 c b N (9) b b b b c b N N3 N4 N N N where () i bi f (0)/ i! i N are the aclauri coefficiet if ci 0 of f() C() ad c0 hould be take a c 0. if ci < 0 I thi way uder certai coditio o arameter λ ζ ad N the iteral itability of cotroller C() ca be removed. However the obtaied cotroller Cˆ N() i ot alway a adequate aroximatio of C() thu idex robute/erformace could be imaired. For thi reao i [9 30] other method of ratioal aroximatio are met. The method of ratioal aroximatio of C() alied i [9] i eece alo make ue of Padé aroximatio i the 0

3 followig way. The obtaied cotroller C() from () i firt decomoed i the form NQ () () Q() C () H() P() N() e P () τ (0) where: Q () ad Q () are olyomial whoe root rereet ole of roce () i the right ad left half lae of the - lae reectively. The Padé aroximatio i alied oly to a art of the receedig exreio τ Ne () D () Q()/ takig care that the obtaied P () cotroller i caual ie. N() Q () () () H() P() C Dˆ N b 0 i a i 0 i Dˆ (). () Similarly i [30] i order to avoid calculatio of Padé aroximatio of comlex fuctio the ue i made of the kow Padé aroximatio of fuctio e τ i cotroller C() followed by factorizatio ad elimiatio of utable diole to remove iteral itability of the cotroller. Sice Padé aroximatio of fuctio e τ i the viciity of 0 i give i the form Proce. ( N k)!! ( ) k 0 B () τ τ ( N)! k!( k)! e A () N ( N k)! N! N k ( τ ) k 0 ( N)! k!( N k)! ad o the bai of relatio () it follow factorizatio ad elimiatio of utable diole k () ˆ AN () N() Q() CL/ L() (3) H ()( P() AN() N() B()) therefore i the obtaied cotroller Cˆ L/ L() there are o utabile diole. B. Simulatio aalyi Comario of the rooed method for removal of iteral itability i cotroller C() () will be aalyzed through four rereetative tyical dyamic characteritic of utable idutrial rocee icludig traort delay: 3. 4e 4 e ( )(0.5 ) 0. e ( ) The obtaied cotroller are: e. (3)( ).0488( )(.3)( 0.07) ( )( 8.777) (.7760)( 0.07)( ) ( 0.99)( ) L 9.609(.68)( 0.388)( 0.385)( )( 0.07) 6/6 Proce. ( 0.387) ( 68.06)( ) ( 0.054)( )( ) ( )( 7.776) 9.837(.9890)( 0.055)( ) ( 79.8)( ) L ( )( )( )( )( 0.055) 3 Proce ( 0)( ) 3/3 Proce 4. ( 0.588) ( )( ) ( ) ( )( ) ( )( ) ˆ ( ) C/ () ( ) ˆ () oe L CN / N ˆ ( )(.8684) C3/3 () ( ) L 49.86( )( )( )( ) 4 (.0408) ( ) O the bai of Table ad trafer fuctio of the obtaied cotroller it ca be cocluded that cotroller C () are of higher order comared to thoe obtaied by other method all havig the ame. I additio i the third examle by alyig L method it wa ot oible to determie the cotroller for the give. Cotroller of tye C () ad obtaied for the ecod roce cotai double utable ole but i cotrol ytem it i ot recommedable that the cotroller itelf i utable ice thi imair robute of the cotrol loo. Cotroller of tye 0

4 ˆ () C give better reult for rocee havig domiat traort delay comared to other two cotroller tye. I deig of the cotroller it i oible to iclude arameter ζ a i [30] i order to achieve better idice of eformace ad robute. TABLE I. THE RESULTS OBTAINED BY THE PROPOSED ETHODS OF RATIONAL APPROXIATION OF CONTROLLER C(S) FOR THE SAE VALUE OF S AND ζ. Proce ethod λ IAE () () () L Cˆ 6/6() () () () L Cˆ () Cˆ 3/3 () () Cˆ / () C () () () Cˆ 3/3 () C () III. CONCLUSIONS Deig of cotroller of utable idutrial rocee icludig traort delay with retrictio o erformace ad robute i of excetioal igificace from the oit of view of idutry. The roblem of cotrol of comlex rocee (multile itabiltie multile atatim domiat time delay) ca ot be adequatly olved by uig PID cotroller which i the baic reao for develomet of the method for deig of comlex cotroller. The coditio for a comlex cotroller are that it i table of relativly lower order ad of adequate tructure for ractical realizatio. I thi work three method of ratioal aroximatio i the deig of comlex cotroller have be aalyzed. The reeted comarative aalyi ad imulatio gave the exected reult. ACKNOWLEDENT The author would like to ackowledge fiacial uort of the Serbia iitry of Sciece ad Educatio uder grat TR3300 (T.B.Š) TR308 ad TR3303 (.R.R). REFERENCES [] S. Yamamoto ad I. Hahimoto "Preet tatu ad future eed: the view from Jaaee idutry" I Arku ad Ray Ed. Chemical Proce Cotrol-CPCIV. Proc. 4 th Iter. Cof. o Chemical Proce Cotrol TX 99. [] L. Debourough R. iller "Icreaig cutomer value of idutrial cotrol erformace moitorig Hoeywell exeriece" i: Sixth Iteratioal Coferece o Chemical Proce Cotrol AIChE Symoium Serie Number 36 Vol [3] K.J. Ǻtröm T. Hägglud Reviitig the Ziegler-Nichol te reoe method for PID cotrol Joural of Proce Cotrol vol Setember 004. [4] S. Skogetad "Simle aalytic rule for model reductio ad PID cotroller tuig" Joural of Proce Cotrol Vol [5]. Shamuzzoha. Lee IC-PID cotroller deig for imroved diturbace reectio of time-delayed rocee Idutrial & Egieerig Chemitry Reearch vol. 46 o [6].R. ataušek T.B. Šekara PID cotroller frequecy-domai tuig for table itegratig ad utable rocee icludig deadtime J. Proce Cotrol vol [7] T.B. Šekara.R. ataušek Claificatio of dyamic rocee ad PID cotroller tuig i a arameter lae J. Proce Cotrol vol. Iue [8] H. Paagooulo K.J. Ǻtröm ad T. Hägglud Deig of PID cotroller baed o cotraied otimizatio IEE Proceedig- Cotrol Theory ad Alicatio vol Jauary 00. [9] A. Wallé K.J. Ǻtröm ad T. Hägglud Loo-haig deig of PID cotroller with cotat T i/t d ratio Aia Joural of Cotrol vol December 00. [0] C. Hwag ad C-Y. Hiao Solutio of o-covex otimizatio ariig i PI/PID cotrol deig Automatica vol November 00. [] B. Kritiao ad B. Learto Evaluatio ad imle tuig of PID cotroller with high-frequecy robute Joural of Proce Cotrol vol February 006. [] B. Kritiao ad B. Learto Robut tuig of PI ad PID cotroller: uig derivative actio deite eor oie IEEE Cotrol Sytem agazie February 006. [3] A.J. Iako ad S.F. raebe Derivative filter i a itegral art of PID deig IEE Proceedig-Cotrol Theory ad Alicatio vol Jauary 00. [4] A. Karimi. Kuze ad R. Logcham Robut cotroller deig by liear rogrammig with alicatio to a double-axi oitioig ytem Cotrol Egieerig Practice vol February 007. [5] T. B. Šekara. R. ataušek "A imle effective method to obtai a well-tued PID cotroller" Proceedig of the 5t Coferece o. ETRAN AU (i Serbia). [6] T.B. Šekara.R. ataušek Otimal tuig of a PI/PID cotroller for rocee defied by a ratioal trafer fuctio INFOTEH Vol. 6 Paer A Jahoria arch 007 (i Serbia). [7] T.B. Šekara ad.r. ataušek Otimizatio of PID cotroller baed o maximizatio of the roortioal gai uder cotrait o robute ad eitivity to meauremet oie IEEE Tra. Automatic Cotrol vol. 54 o Ja [8] T.B. Šekara ad.r. ataušek Reviitig the Ziegler-Nichol roce dyamic characterizatio J. Proce Cotrol vol [9] T. B. Šekara. R. ataušek "A four-arameter otimizatio of a PID cotroller" Proceedig of 5. Cof. ETRAN Vol. Palic Jue 008 (i Serbia). [0] T.B. Šekara.R. ataušek Otimal tuig of a PID cotroller i frequecy domai INFOTEH Paer A Boia ad Herzegovia Jahoria arch 009 (i Serbia). [] T.B. Šekara.B. Trifuovic Otimal tuig of a PID cotroller havig a differetial comeator coected i erie i frequecy domaie Proceedig of INDEL Baa Luka 4-6 November 00 (i Serbia). [] F.. Shikey How good are our cotroller i abolute erformace ad robute? eauremet ad Cotrol vol ay 990. [3] Chie I. L. IC-PID cotroller deig-a exteio IFAC Proceedig Serie vol. 6 (. 47-5) 988. [4] Fruehauf P. S. Chie I. L. & Laurite. D. Simlified IC- PID tuig rule. ISA Traactio 33()

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