Algebraic Control of Integrating Processes with Dead Time by Two Feedback Controllers in the Ring R MS

Transcription

1 Algebrac Control of Integratng rocee wth Dead Te by Two Feedback Controller n the ng MS Lbor ekař and oan rokop Abtract The objectve of th contrbuton to deontrate the utlzaton of algebrac controller degn n an unconventonal rng whle control ntegratng procee wth te delay. In contrat to any other ethod, the propoed approach not baed on the te delay approxaton. A control tructure cobnng a ple feedback loop and a two-degree-of-freedo control tructure condered. Th tructure can be alo conceved a a ple feedback loop wth nner tablzng loop. The control degn perfored n the rng of retarded quapolynoal (Q) eroorphc functon ( MS ) - an algebrac ethod baed on the oluton of the Bézout equaton wth the Youla-Kučera paraeterzaton preented. Fnal controller ay be of o-called anochronc type and they enure feedback loop tablty, trackng of the tep reference and load dturbance attenuaton. Aong any poble tunng ethod, the donant pole agnent ethod adopted. Th approach copared wth the conventonal polynoal LQ ethod ung an llutratve ulaton exaple.. Keyword Te delay yte, algebrac control, Bézout dentty, Youla-Kučera paraeterzaton. I I. INTODUCTION INTEATIN odel appear whle odelng a or energy accuulaton, a rotaton of achnere, etc. and they contan underable pole whch need to be hfted by utable degn of the feedback loop. A well, the great deal of technologcal and other procee, uch a dtrbuted network, long tranon lne n pneuatc yte or neural network [, to nae a few, own an nput-output te delay. The preence of a delay ental proble wth controller degn due to the fact that the delay gnfcantly nfluence the dynac properte of a feedback control yte. The cobnaton of ntegratng behavor of the yte and delay ake controller degn ore dffcult and t requre utlzaton of oe advanced procedure. There have been recently nvetgated varou prncple for Manucrpt receved Deceber 9, 8: eved veron receved... Th work wa upported by the grant of Mntry of Educaton, Youth and Sport of the Czech epublc, MSM Lbor ekař wth Toa Bata Unverty n Zlín, ná. TM 5555, 76 Zlín, Czech epublc (phone: ; fax: ; e-al: pekar@fa.utb.cz). oan rokop wth Toa Bata Unverty n Zln, ná. TM 5555, 76 Zlín, Czech epublc (e-al: prokop@fa.utb.cz). control of ntegratng procee wth te delay. A group of ethod utlze tandard I or ID controller n an effort to get a certan optzaton and robutne, ee e.g. [-[3. In [4 allowable I and D controller gan have been nvetgated. One type of a pole agnent approach n [5 wa developed. Soe dea are baed on generalzed Sth predctor, e.g. [6-[7 or even ore general control loop [8, degned anly n order to obtaned the atfactory dturbance repone. Lat but not leat, author preent predctve approache anly ncorporatng tate-pace decrpton e.g. n [9. One of the ot gnfcant approache n odern control theory a faly of algebrac ethod. Unlke oe tradtonal tate-pace odel, algebrac tool are baed on fractonal decrpton of yte where a tranfer functon can be reed a a rato of two eleent n an approprate rng. Fro the htorcal pont of vew and the natural correpondence between te-doan decrpton and the tranforaton for dcrete-te yte, tradtonal tranfer functon are repreented by polynoal fracton. Th dea wa adopted for contnuou-te yte a well. Th decrpton eployed for algebrac control trategy for ntegratng delayed yte n [ where the control tructure wth two feedback controller (Fg. ) condered. However, a tranfer functon can be wrtten a a fractonal feld of ore general algebrac tructure rng. One of bac requreent on a control yte that both a plant and a controller are proper and a control yte nternally table, whch brng the poblty of ntroducton of another frequently ued rng, S (the rng of Hurwtz table and proper ratonal functon) [, [. Algebrac control phloophy n th rng then lot Bézout dentty (Dophantne equaton) along wth the Youla-Kučera paraeterzaton to obtan table and proper controller. Neverthele, utlzaton of th rng rather retrctve whle dealng wth te delay yte nce requre ratonal approxaton of onental reng delay, uually va the frt order adé approxaton. Th contrbuton preent tranfer functon decrpton avodng any te delay approxaton. The rng of table and proper retarded quapolynoal eroorphc functon ( MS ) for th purpoe utlzed. A ter of th rng a rato of two o-called quapolynoal where the denonator Iue 4, Volue, 8 49

2 quapolynoal table and the whole rato proper wth repect to hghet -power. The only effort to degn controller n th rng for ntegraton delayed yte n [3 where a ple DOF control tructure wa utlzed. In th paper, an algebrac approach baed on Bézout dentty and the Youla-Kučera paraeterzaton ung control yte wth two controller condered. The preented feedback yte can be coprehend and olved n double eanng; frt, one can take the yte a a whole, whch ean that the overall nput-output tranfer functon for controller degn are utlzed, and on the ba of th knowledge the approprate controller tructure are deterned; econd, the control yte can be vewed a a ple control feedback wth the nner (tablzng) feedback loop. In th cae, the nner loop olved frt and the an loop follow. Fnal controller of the o-called anochronc type enure (n both cae) feedback loop tablty, tep reference trackng and load dturbance attenuaton, and they are tuned by the pole agnent ethod decrbed e.g. n [4. Both approache are teted and verfed ung an llutratve ulaton exaple and they are copared wth the lnear quadratc (LQ) polynoal approach [, whch deontrate the uefulne and applcablty of the propoed ethod. II. SYSTEM DESCITION IN MS IN A. MS ng Algebrac control ethod are baed on nput-output yte forulaton n the for of a tranfer functon. Conventonal tranfer functon n the for of a rato of two polynoal are not drectly applcable for odel contanng delay due to onental reultng fro the Laplace tranfor of delay. In order to re the nuerator and denonator n polynoal, the frt order adé approxaton then uually utlzed; however, there t alo poble to ue another way. atonal approxaton can be avoded o that the tranfer functon can be perfored n the rng of table and proper Q-eroorphc functon, MS. Any functon n th rng a rato of two retarded quapolynoal y/x, n general, where a denonator quapolynoal Hurwtz table and the rato proper. Quapolynoal, n contrat to polynoal, are fored not only by weghted u of -power but alo by onental relatng to delay. A denonator quapolynoal x of degree n ean e.g. n [5. The nuerator y of an eleent n MS can be factorzed n the for y ( ) ), where τ > and y a retarded quapolynoal of degree l ( ) l h l y ( ) y τ j j A quapolynoal fracton called proper ff l n. j B. Integratng Delayed lant n MS MS rng can be naturally utlzed for decrpton of yte wth delay n both left and rght de of an approprate dfferental equaton. The tranfer functon of the plant or the controller then reed a a rato of two eleent n MS rng. Th contrbuton deal wth ntegratng te delay yte ncrbed wth the tranfer functon, B MS A K B A where an approprate table quapolynoal of degree one. Th quapolynoal can not be of order hgher than one becaue the tranfer functon factorzaton then would not be copre; detal about coprene for MS are n [3. The utable for of dcued n the Secton 4 where algebrac controller degn decrbed. III. CONTOL SYSTEM Up to th day, the algebrac controller degn prncple n MS (decrbed further n Secton 4) wa eployed for the ple feedback loop wth one degree of freedo (DOF) whch pctured n Fg., and for the control yte wth two degree of freedo (DOF), ee Fg.., and for the nternal odel tructure (IMC) only, [3, [5-[6. However, th contrbuton deal wth the control yte wth two controller cobnng DOF and DOF tructure, ee Fg. 3. (3) n h n xj ( x ϑ ) j j where retarded refer to the fact that the hghet -power not affected by onental. Quapolynoal table ff t own no fnte zero uch that e { },.e., a ter n MS rng analytc n the rght half coplex plane. Stablty can be verfed by the Mchalov tablty crteron, whch can be ued due to the valdty of arguent prncple, ee detal Fg. DOF control yte tructure. Iue 4, Volue, 8 5

3 ( ) ( ) Q, Q (7) n whch, Q and are fro MS and ( ) A( ) ( ) B( ) [ ( ) Q( ) M (8) Fg. DOF control yte tructure. correpond to the charactertc (qua)polynoal of the cloed loop. Both external nput, W and D, are condered to be tep functon,.e. W w d HW w H D d, D (9) F F W D w d Fg. 3 ropoed control tructure wth two controller. where w ( ) and ( ) d are arbtrary table polynoal of degree one and H W, H D, F W, F D MS. In chee, W the reference gnal, D the load dturbance, E refer to the control error, U the controller output, U the plant nput, and Y ean the plant output (controlled value) n the Laplace tranfor. The plant tranfer functon depcted a Q, next, Q ( ) and are feedback controller tranfer functon for DOF and DOF tructure and for the control yte wth two controller, repectvely, and or repreent feedforward controller n the approprate chee. Ung th decrpton, the followng correpondence between the tructure can be wrtten: For DOF chee hold:, Q Q (4) For DOF chee hold:, (5) Q Q The advantage of the tructure wth two controller ret n the poblty to atfy decouplng of reference trackng and load dturbance rejecton. The followng tranfer functon can be derved n the control yte n general: IV. DIECT ALEBAIC CONTOLLE DESIN IN MS IN The algebrac controller degn preented n th contrbuton uppoe that all tranfer functon and gnal n the control yte are n the for of rato of eleent n MS ; thu, a feld of fracton aocated wth the MS rng ntroduced. The control yte chee pctured n Fg. 3 can be graped ether a the whole yte correpondng to tranfer functon (6) or a an nner feedback loop wth controller Q and outer loop wth controller. Let u now decrbe the forer, ay drect, approach. Uual requreent on the control yte are thee: cloed-loop tablty, ayptotcal reference trackng and load dturbance attenuaton. A. Control Syte Internal Stablzaton It natural to requre that all gnal n the control loop avod pule ode, whch brng the noton of nternal tablty, ee e.g. [, [7. Conder plant (3) where A and B are copre eleent n MS. If there ext functon, T where MS ( ) ( ) Q ( ) T atfyng the Dophantne equaton WY WE E W Y B, W E DE D where controller are DY M B Q M B M A Y D B ( ) M (6) ( ) ( ) B( ) T ( ) A then the et of all controller that nternally tablze the control loop deduced fro the paraeterzaton of the partcular oluton Iue 4, Volue, 8 5

4 T B Z T A Z T A Z Z MS, p q ( ) p ( ) p[ q q [ (8) The proof of the prevou tateent can be done analogouly a, e.g. n [3, [7, appled to control yte hown n Fg. 3. A free paraeter Z can be choen properly to fulfll other control degn requreent. eultant controller are gven by (7) wth repect to the dtrbuton of the oluton,. Concretely, the partcular oluton of for plant (3) are fro the oluton of the followng equaton T (3) Wthout lo of generalty, let T α and, and the reanng tak to fnd a utable table quapolynoal. Hence, (3) reult n α (4) The requreent α to be real; therefore the plet ha to be of the for αk (5) The eental feature of retarded quapolynoal t tablty whch can be tuded e.g. ung the Mchajlov crteron, [8-[9. Va a coputatonal procedure analogou to the one decrbed for untable yte n [6, one can derve the tablty condton a π α (6) A Kτ where A > the gan argn ( A correpondence to the tablty border). B. eference Trackng and Dturbance ejecton A wa entoned above, the convenent opton of Z n the paraeterzaton enable to fnd the oluton of, o that requreent of reference trackng and dturbance rejecton are accoplhed. If both nput are condered a tep functon (9), t are fro tranfer functon (6) that nuerator of and Q ut have dervatve pattern. In other word, untable (zero) pole of F W and F D ut be canceled by zero pole of and Q,.e. ther nuerator ut be ether of the for p p, q q (7) or, eventually for quapolynoal Both nuerator (7) and (8) of and Q enure at leat one zero root. For the plant (3) and the partcular oluton (3)-(5), the choce Z α n paraeterzaton yeld T αk α[ αk αk (9) Obvouly, and T are fro MS and the for of enure reference trackng and dturbance rejecton. C. araeterzaton of T The oluton of the Bézout dentty wth paraeterzaton gve and T; however, the controller tranfer functon nvolve Q and and thee eleent are obtaned by paraeterzaton of T accordng to. Hence, the lat tep n controller degn the dtrbuton of T onto and wth repect to deand on the for of Q, ee (7) and (8). Thu, functon T n (9), wth repect to, (7) and (8) and takng a dtrbuton paraeter γ, can be forulated a T α [ αk αk γ α α αk Q whch reult n controller Q ( ) ( γ ) α αk γ α Kα Q ( γ ) α Hence, the proportonal and generalzed (delayed) proportonal-ntegratve controller are obtaned. D. Alternatve Choce of Z The choce of a electable eleent Z α a wa propoed above, not the only poblty how to fnd controller atfyng ultaneou nternal tablty, reference trackng and dturbance rejecton. To deontrate th feature, let Z αk λ λ K Iue 4, Volue, 8 5

5 ntead of Z α. A electable potve real paraeter λ enure that Z and brng an addtonal degree of freedo. Then T MS [ λ λ αk ( αk λ) K( λ) (3) The controller degn procedure preented n Secton 4 wa baed on the control yte decrpton n the for of (the whole) cloed loop tranfer functon (6). However, one can conceve the chee n Fg. 3 a a control yte wth an nner pre-tablzaton loop contanng controller Q and outer loop wth controller whch provde dturbance rejecton and etpont trackng. To avod the preence of nput dturbance n the nner feedback for controller degn, let the control yte chee be rearranged a n Fg. 5. Obvouly, all tranfer functon (6) tll hold; neverthele, controller degn for the nner loop exclude the aupton of the nput dturbance. The dea that nner feedback pre-tablze the controlled proce,.e. zero pole to be oved to the left, and the outer feedback controller enure already entoned requreent for pretablzed yte. The dtrbuton of T onto Q and n the for λ γ α αλ K λ λ ( γ ) α K Q λ Hence, the reultng et of controller the followng (4) Fg. 4 The tructure of anochronc controller n (5). Q K λ[ Q λ γ α αλ λ ( γ ) α K λ[ (5) A can be een, denonator of fnal controller (5) are quapolynoal, and th feature refer to o-called anochronc for of the controller. However, thee type of controller are a eay to pleent ether on C or LC, ee [, a the tradtonal ID controller; whch can be ealy deducted fro the Matlab-Sulnk ulaton block chee of e.g. dplayed n Fg. 4. V. CONTOLLE DESIN WITH THE E-STABILIZIN INNE LOO Fg. 5 econfgured control chee. A. Inner Loop re-tablzaton Let an ntegratng delayed plant (3) be pre-tablzed ung a proportonal controller Q q. The condton for cloedloop tablty, ee e.g. [7, then gven by the Dophantne equaton q (6) The natural tak to fnd a utable table retarded quapolynoal ( ) whch provde a copre factorzaton of the plant tranfer functon. The oluton of (6) gve ( ) q (7) The requreent q to be real; therefore, larly a n (4) and (5), the plet ha to be of the for ( ) q (8) where the tablty of condtoned agan by (6) n whch q placed ntead of α. Thu, the tranfer functon of the nner pre-tablzed feedback loop Iue 4, Volue, 8 53

6 (9) q B. Syte Stablzaton Now the tak to control the pre-tablzed loop ung a ple feedback wth controller. However, the nuerator and denonator n (9) are not fro MS and thu the tranfer functon ut be factorzed a ( τ ) qk (3) where a table (qua)polynoal. In order to have a ple a poble, let λ (3) where λ > a electable real paraeter whch brng an addtonal degree of freedo. Naturally one can take another ; an exaple deontratng t preented n Secton 5D. The outer cloed-loop tablty property enured by a oluton of the Dophantne equaton qk λ λ whoe a partcular oluton read (3) λ, (33) q Th oluton can be further paraeterzed accordng to ung the Youla-Kučera paraeterzaton a A Z B Z λ q qk λ λ Z Z ; Z MS (34) to fulfll other control requreent,.e. reference trackng and dturbance rejecton, va an approprate choce of the free eleent, Z. C. eference Trackng and Dturbance ejecton araeterzaton (34) enable to fnd the oluton of (3), o that requreent of reference trackng and dturbance rejecton wll be accoplhed. The requreent that both denonator of Laplace for of external nput, F W and F D, dvde. If both nput, w(t) and d(t), are condered a tep functon (9), ut contan at leat one zero pole. Let Z λ K q ( λ) then the outer feedback controller read λ [ qk K qk λ[ qk λ[ qk K[ λ[ (35) (36) Obvouly, th controller of anochronc tructure a n (5) agan and t tructure lar to the one pctured n Fg. 4. ecall that the nner-feedback controller proportonal, Q q ; however, n ter of algebrac phloophy t can be wrtten alo n factorzaton a Q q Q [ λ[ qk λ[ q D. An Alternatve Soluton (37) A wa entoned n Secton 5B, table (qua)polynoal can be choen unlke n (3). Another natural choce ( ) ( ) q (38) whch agree wth the denonator of the non-factorzed nner-feedback tranfer functon. Thu the factorzed one read qk qk qk (39) q In th cae, the tablzng Dophantne equaton q ha one of partcular oluton (4) Iue 4, Volue, 8 54

7 ( qk τ, (4) q and by decon Z q n the Youla-Kučera paraeterzaton, the fnal outer-feedback controller tructure enurng reference trackng and dturbance rejecton then q [ q ) (4) whch a delayed I controller of the ae tructure a n ; however, one can notce that a proportonal and an ntegral coeffcent cannot be ultaneouly the ae a thoe n. The nner-feedback controller Q q agan. VI. TUNIN OF CONTOLLES The fnal et of controller,, (5), (36) and (4), tll contan unknown paraeter that have to be et properly. There are naturally plenty of approache olvng the proble of controller tunng. In th contrbuton, the well applcable and relatvely ple tunng ethod called drect pole placeent, whch wa decrbed e.g. n [4, utlzed. Th ethod enable to precrbe the dered et of donant pole of the cloed loop, the axu nuber of whch gven by the nuber, k, of unknown paraeter n the charactertc quapolynoal. If the donant pole are denoted aσ,... k, the charactertc equaton a, and a vector of r unknown paraeter a v, then the followng yte of k lnear equaton obtaned wth unknown paraeter q (or α whch ha the ae eanng) and λ. Thu, let the frt quapolynoal n (45) be taken. Snce there a ngle paraeter to be found, q, the only ultple real donant root or a conjugate par of coplex root can be precrbed. Moreover, tablty condton (6) cannot be otted. In any real applcaton, ocllatory ode n the output gnal are underable; therefore the optal choce of precrbed pole n the for of the leftot donant real root uggeted. We wll propoe here two way how to derve the condton for the precrpton of thee optal pole. The frt deducton are fro the obervaton of the value of q coputed fro (43) and (45) for a precrbed table real pole. Startng the pole poton n the left neghborhood of the tablty border (.e. σ ) and contnung toward to negatve nfnty, the value of q ntally re up untl t axu value reached and conequently lope down behnd th pont. A concrete exaple of th behavor can be een n Fg. 6. Thu, there ext two dtnct value of the choen pole for the ae value of the controller paraeter, except the axu pont. Th ean that whenever a pole to the left of the axu pont choen, there ut ext another pole to the rght whch donant. Therefore, the tak of choong the leftot donant pole converted to the earchng the axu of q, whch ndcate the poton of the optal real pole. (, v), k σ... (43) For coplex pole, one root fro each coplex conjugate par taken and (43) dvded nto two equaton of the for e I { ( σ, v) } { ( σ, v) } (44) The gnfcant feature that et (43) and (44) are lnear wth repect to unknown paraeter, whch ake the oluton eay to fnd. If r > k, the equaton are olved ung Moore-enroe peudo-nveron of non-quared atrx [; on the other hand, f r k, the et full rank and can be olved ung a a coon et of algebrac lnear equaton,.e. peudo-nveron becoe an nveron. In the partcular cae of delayed ntegrator, all et of fnal controller reult n two dparate charactertc quapolynoal [ q [ q ( λ) (45) q Fg. 6 Dependence of on a precrbed donant pole σ ; K, τ 5. The pole placeent condton cobnng (43) and (45) read ( σ, q ) [ σ q ( τσ ) (46) The objectve to axze the functon q (σ ), σ <, whch gven by (46) a Iue 4, Volue, 8 55

8 σ q (47) ( τσ ) The well-known procedure of analytc earchng the axu of contnuou functon yeld the optal donant pole choce σ,ot τ (48) whch gve the optal controller paraeter q,ot (49) Kτe The nteretng feature of th reult ret n the fact that the derved cloed-loop pole trplcate, whch evdent fro the followng tateent d dq ( σ, q ) σ σ,ot q q,ot (5) The econd poblty how to derve the optal controller paraeter choce to obtan the leftot real donant pole baed on the oluton of cobnaton of (44) and (45). If σ α jω, the followng et of equaton hold α qk ω q ( α τ ) co( τω ) ( α τ ) n( τω ) (5) Conder one conjugate par of precrbed pole,.e., then the oluton of (5) by cancellaton of the factor q K read ω arctan( ω τ ) (5) α whch decrbe the dependence of a real and an agnary part of the root of (45). Calculaton of lt l ω [ ω arctan( ω τ ) τ (53) lead to the ae reult a n (48). The polynoal factor n the econd charactertc quapolynoal n (45) lnear, thu, an addtonal table root, λ, can be precrbed. Obvouly, f λ <τ, then th pole becoe donant and the nfluence of the quapolynoal factor q on the yte dynac uppreed. In [, the uggeton for the choce of λ λ (54) τ whch, n coparon wth (48), doe not allow for the donant pole. A can be een fro above entence, root of the charactertc quapolynoal (real or coplex) ut be choen carefully becaue oe attept to place the excevely to the left n the coplex plane can lead nto the followng tuaton. Due to the fact that the quapolynoal (45) ha the nfnty nuber of root, the chan of coplex root can ove to the rght near to the tablty border (agnary ax) and thu thee root can take over the role of donant pole of the yte. Coeffcent γ n and (5) nfluence the feedback yte behavor a well becaue t appear n the nuerator of cloed loop tranfer functon; however, t doe not pact the pectru of the yte. In the llutratve exaple below, varou value of γ are et randoly. There are naturally other poblte how to et the unknown paraeter of controller, e.g. n ore coputatonal way va artfcal ntellgence approache baed on genetc algorth, [. VII. LQ OLYNOMIAL METHOD The ethodology preented above n th contrbuton ought to be copared wth another approach to deontrate t uablty. Take a ethod preented n [ whch baed on a ratonal approxaton of onental n a plant tranfer functon followed by optal pole-placeent va nzaton of a quadratc cot functon. Look at a bref yet ore detaled decrpton of the LQ ethod. The control yte condered a n Fg. 3 agan. The plant tranfer functon (3) approxated ung the frt order adé approxaton.5τ ( τ) (55).5τ.e. a ratonal fracton decrpton obtaned. The rng of polynoal ntead of MS thu ued. In th rng, tablzng equaton ( ) ( ) B( ) T ( ) M ( ) A (56) olved where M the charactertc polynoal and dtrbuton hold. araeterzaton not ued; on the other hand, et drectly n the for enurng the ayptotc trackng and load dturbance attenuaton,.e. a n (7), wherea the oluton T dtrbuted accordng to T T ( ) ( ) Q( ) n n t, r, Q t ; r q γ t t ( γ ) t r for γ,,,... n n q (57) Iue 4, Volue, 8 56

9 where coeffcent γ dvde a weght between and Q. The polynoal M condered a a product of two table polynoal n the for M M M (58) where M a onc for of the polynoal obtaned by the pectral factorzaton M A ϕ ) [ [ A B B M ( ) M ( } (59) where φ the weghtng coeffcent and the aterk denote the pectral factor of an approprate polynoal. Condton (59) hold for the nzaton of the quadratc cot functon J { e t u& t ϕ dt (6) Fnally, the econd polynoal n (58) uggeted a M (6) τ whch agree wth the choce (54). A can be een, there one electable real paraeter, φ >, that nfluence the cloed loop pole locaton. The fnal controller for a plant (3) have the followng for of tranfer functon J ISTE t { e t u& t ϕ } dt (63) The gnfcant feature of th crteron that t handcap latter gnal value,.e. t hgher value ndcate lower control repone ettleent. Let K and τ 5. The reference gnal w(t) for t < and w(t) for t < 3. The tep nput dturbance d(t) -. enter at te t ; hence, the proce of retoraton of zero control error due to the nput dturbance nfluence ISTE crteron gnfcantly. Sulaton reult for every ngle fnal controller are dcued ndependently. A. eult for Drect Controller Degn Frt aue controller whch copre two adjutable paraeter: α, γ. The optal value of α gven by (49) a α. 736 whch enure the trple leftot real donant yte pole ( σ,,3. ). For the ake of coparon, chooe two coplex conjugate donant pole, accordng to (5), a σ 4,5.8 ±.j and σ 6,7.3 ±.j whch gve α. 835 and α. 5, repectvely. Two dfferent dtrbuton paraeter γ are evenly choen a γ. 5 and γ. 75. The ulaton reult are n Fg. 7 - Fg.. A quanttatve reult collaton n the for of value of ISE and ISTE crteron are n Table I. Q r q q p r r p (6) VIII. ILLUSTATIVE EXAMLE Th ulaton exaple copoed n the Matlab-Sulnk envronent deontrate the uablty of the propoed controller degn ethod n the MS rng and t preent a benchark tet by coparon the ulaton reult wth the polynoal LQ ethod. Two crtera a ntruent for a quanttatve coparon are pleented. The frt one follow crteron (6) and thu t ected that the optal LQ ethod hould perfor the bet reult. Let u call the crteron ply a ISE (Integrated Squared Error) crteron. A electon of a potve real paraeter φ enable to deterne the pact of control gnal dervaton,.e. the hgher value of φ reult n a oother coure of u&(t) functon, and let φ 5 choen. The econd ISTE (Integrated Squared Te Error) crteron forulated a Fg. 7 Step etpont and load dturbance repone of u(t) ung controller ; K, τ 5, γ.5, d -.. Iue 4, Volue, 8 57

10 Fg. 8 Step etpont and load dturbance repone of y(t) ung controller ; K, τ 5, γ.5, d -.. The reult n Table I ndcate that donant pole wth approprately all (but not zero) rato of ther agnary and real part can prove both crteron, epecally for lower γ value. However, a can be een fro Fg. 7 - Fg., coplex conjugate pole naturally caue the propenty to ocllaton and overhoot, whch are underable n any applcaton. Fg. 9 and Fg. n contrat to Fg. 7 and Fg. 8 how alo that ncreang of the dtrbuton paraeter γ (.e. paraeter n the nuerator of re) lead to fater change of control gnal u(t) and ore apparent overhoot; on the other hand, control repone are fater. Conder now alternatve controller (5) coprng an addtonal paraeter λ whch et by (54) a λ. 4. Control yte repone are n Fg. Fg. 4 and the nuercal coparon n Table II. In contrat to controller, Table II together wth the fgure above, evdently how that the alternatve anochronc controller (5) are uch convenent to control the plant (3). Both ISE and ISTE crteron are notably le, repone are fater and alo underhoot caued by the nput dturbance are reduced. Hgher value of γ reult n fater change of control gnal u(t) and ore apparent overhoot agan, whch pol the qualty crteron; on the contrary, t reduce underhoot a lttle. Fg. 9 Step etpont and load dturbance repone of u(t) ung controller ; K, τ 5, γ.75, d -.. Fg. Step etpont and load dturbance repone of u(t) ung controller (5); K, τ 5, λ.4, γ.5, d -.. Fg. Step etpont and load dturbance repone of y(t) ung controller ; K, τ 5, γ.75, d -.. Table I: Value of ISE and ISTE crteron when ung controller ; K, τ 5 γ α ISE ISTE Iue 4, Volue, 8 58

11 Fg. Step etpont and load dturbance repone of y(t) ung controller (5); K, τ 5, λ.4, γ.5, d -.. B. eult for Controller Degn wth the Inner Loop Controller degn utlzng the pre-tablzng nner feedback loop and the outer loop gve e.g. controller (36) and (37), and (4). Conder the forer et of controller frt, whch contan two electable paraeter: q and λ. The double donant pole gven by q a for α n the prevou ecton, ee Fg. 5 - Fg. 8. The nfluence of a change of λ deontrated n Fg. 9 and Fg.. The ISE and ISTE crteron valuaton preent Table III. Fg. 5 Fg. 8 clearly how that any change n q doe not affect the etpont repone. Approprately choen paraeter q correpondng to conjugate donant pole prove partcularly ISTE crteron, whch reveal fro Table III; however, there a tendency to overhoot after the dturbance enter. Fg. 9 and Fg. dcloe that hgher value of λ caue fater change of the control gnal yeldng a deteroraton of the ISE crteron. Fg. 3 Step etpont and load dturbance repone of u(t) ung controller (5); K, τ 5, λ.4, γ.75, d -.. Fg. 5 Step etpont and load dturbance repone of u(t) ung controller (36) and (37); K, τ 5, λ., d -.. Fg. 4 Step etpont and load dturbance repone of y(t) ung controller (5); K, τ 5, λ.4, γ.75, d -.. Table II: Value of ISE and ISTE crteron when ung controller (5); K, τ 5, λ.4 γ α ISE ISTE Iue 4, Volue, 8 59

12 Fg. 6 Step etpont and load dturbance repone of y(t) ung controller (36) and (37); K, τ 5, λ., d -.. Fg. 9 Step etpont and load dturbance repone of u(t) ung controller (36) and (37); K, τ 5, q.835, d -.. Fg. 7 Step etpont and load dturbance repone of u(t) ung controller (36) and (37); K, τ 5, λ.4, d -.. Fg. Step etpont and load dturbance repone of y(t) ung controller (36) and (37); K, τ 5, q.835, d -.. Table III: Value of ISE and ISTE crteron when ung controller (36) and (37); K, τ 5. Fg. 8 Step etpont and load dturbance repone of y(t) ung controller (36) and (37); K, τ 5, λ.4, d -.. λ q ISE ISTE Look at the alternatve oluton (4) for whch the paraeter q et a for controller (36) and (37). The correpondng reult are preented n Fg., Fg. and Table IV. Iue 4, Volue, 8 6

13 benchark for the ethod utlzng the rng MS. There are three electable paraeter, γ, λ, ϕ. To obtan content reult wth thoe preented n Secton 8A and Secton 8B, let λ.4 n all cae, γ γ. 5 and γ γ.75 for the coparon. The weghtng factor, ϕ, ha three varou value, ϕ, 5, 9, to tudy t nfluence agan. raphc reult are dplayed n Fg. 3 Fg. 6, and ISE and ISTE crteron are evaluated n Table V. Fg. Step etpont and load dturbance repone of u(t) ung controller (4); K, τ 5, d -.. Fg. 3 Step etpont and load dturbance repone of u(t) ung controller (6); K, τ 5, d -., γ γ.5 Fg. Step etpont and load dturbance repone of y(t) ung controller (4); K, τ 5, d -.. Table IV: Value of ISE and ISTE crteron when ung controller (4); K, τ 5. q ISE ISTE Fg. 4 Step etpont and load dturbance repone of y(t) ung controller (6); K, τ 5, d -., γ γ.5. Obvouly, thee reult gve the wort ISE and ISTE crteron, whch te fro very low control repone. On the other hand, only one controller paraeter, q, to be et and the change of control gnal u(t) are low, whch contrbute to a long workng lfe of actuator. C. eult for LQ Controller The ethodology propoed n th paper further copared wth the polynoal LQ approach whch erve a a Iue 4, Volue, 8 6

14 approxated plant tranfer functon ntead of an orgnal one. Lookng at Table V, t can be affred that the algebrac ethod utlzng the rng MS gve reult reconclable wth the optal polynoal LQ ethod for both, the drect oluton and alo for ucceve degn of the nner and the outer controller. Fg. 5 Step etpont and load dturbance repone of u(t) ung controller (6); K, τ 5, d -., γ γ.75. Fg. 6 Step etpont and load dturbance repone of y(t) ung controller (6); K, τ 5, d -., γ γ.75. Table V: Value of ISE and ISTE crteron when ung controller (6); K, τ 5. γ φ ISE ISTE Snce the ISE crteron (6) calculated wth φ 5, one would ect that the LQ ethod ung th opton gve the bet reult. However, th doe not hold a t are fro Table V. Th becaue of th ethod ue the lnear approxaton and thu the optzaton (6) ade for an IX. CONCLUSION Th paper developed the proble of algebrac control degn n the rng of table and proper Q eroorphc functon for ntegratng te delay procee. The propoed ethod doe not nvolve the delay approxaton a t cutoary; however, t utlze tranfer functon paraeterzaton wthout any lo of nforaton. The controller tructure wa derved through the oluton of the Bézout equaton together wth the Youla-Kučera paraeterzaton. The ethodology enable to fnd varou controller that atfy requreent on the cloed loop tablty, tep reference trackng and tep load dturbance attenuaton. The novel cobnaton of th algebrac ethodology and the control yte tructure cobnng conventonal DOF and DOF chee wa propoed. The control tructure conceved n double eanng: ether a a whole (one) yte or a nner (pre-tablzng) feedback loop plu the outer one. The fnal controller were tuned ung the donant pole agnent ethod where the optal ettng yeldng the leftot donant real pole wa derved. The effcency and uablty of the propoed ethodology wa verfed on a ulaton exaple and copared wth the polynoal LQ ethod. EFEENCES [ C. Jeong, J. Moon, and.. ark, Oberver-baed control of tedelayed yte, n roceedng of the 7 th WSEAS Internatonal Conference on obotc, Control & Manufacturng Technology, Hanghzou, Chna, 7, pp [ W. D. Zhang and X. M. Xu, Quanttatve perforance degn for ntegratng procee wth te delay, Autoatca, vol. 35, pp , 999. [3 L. Wang and W.. Cluett, Tunng of ID controller for ntegratng procee, IEE roc. Control Theory Appl., vol. 44, pp , 997. [4 D. Ütebay, H. Özbay, and N. ünde, A new I and ID control degn ethod for ntegratng yte, n roceedng of the 6 th WSEAS Internatonal Conference on Sgnal roceng, obotc & Autoaton, Corfu Iland, reece, 7, pp [5 K. Žáková, Contraned pole agnent controller for delayed double ntegrator yte, n roceedng of the 6 th WSEAS Internatonal Conference on Syte Scence and Sulaton n Engneerng, Vence, Italy, 7, pp. -7. [6 S. Majh and D.. Atherton, Modfed Sth predctor and controller for procee wth te delay, IEE roc. Control Theory Appl., vol. 46, pp , 999. [7 Y. Tan, F. ao, M. O. Tadé, and J. Tang, Control of ntegrator procee wth long dead-te, n roceedng of 4th World Congre of IFAC, Beng,.. Chna, 999, pp [8 W. D. Zhang, H. Wang, W. Wang, and X. Xu, Novel controller for ntegratng and table procee wth long te delay, n roceedng of 5th World Congre of IFAC, Barcelona, Span,, pp Iue 4, Volue, 8 6

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