SSRG International Journal of Thermal Engineering (SSRG-IJTE) Volume 4 Issue 1 January to April 2018

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1 SSRG Inernaonal Journal of Thermal Engneerng (SSRG-IJTE) Volume 4 Iue 1 January o Aprl 18 Opmal Conrol for a Drbued Parameer Syem wh Tme-Delay, Non-Lnear Ung he Numercal Mehod. Applcaon o One- Sded Hea Conducon Syem Cong Huu Nguyen #1, Ma Trung Tha * 1 Tha Nguyen Unvery, Tha Nguyen cy, Ve Nam Tha Nguyen Unvery of Technology, Tha Nguyen cy, Ve Nam Abrac Th paper preen a oluon of an opmal conrol problem for a parabolc-ype drbued parameer yem wh me delay, non-lnear governed by a hea-conducon equaon. The yem appled o a pecfc one-ded hea-conducon yem n a heang furnace o conrol emperaure for a lab followng he mo accurae burnng andard [], [6]. The arge of problem o fnd an opmal conrol gnal o ha he error beween he drbuon of real emperaure of he objec and he dered emperaure mnmum afer a gven perod of me f [], [6], [9]. Afer olvng he problem, buldng he algorhm and eablhng he conrol program, we have proceeded o run he mulaon program on a lab of Samo and a lab of Daome o e calculng program. Keyword: opmal conrol, drbued parameer yem, delay, non-lnear, numercal mehod. I. INTRODUCTION Opmal conrol for drbued parameer yem appled n many feld uch a hea reamen, compong he magnec maeral, eel rollng, ec. In ome prevou echnologe [], [6], [7], he heang proce wa carred ou n a burnng furnace wh FO heavy ol, uch a burnng n eel rollng or n he procee of manufacure of alumnum, gla. In h cae, he ranfer funcon of he furnace he delayed nera, and he relaonhp beween he emperaure of furnace he parabolcype paral dfferenal equaon wh he boundary condon of ype 3. If we conder he opmal conrol problem for he mo accurae burnng proce, he conrol objec now drbued parameer yem wh me-delay. Wh h problem, ome auhor have been nereed n and olved by varaonal mehod, ung he Ponryagn maxmum prncple or numercal mehod a n [1], [], [6]. However, n ome oher echnologe, he furnace an elecrcal furnace, mean burned by a reor wre uch a hea reamen of mechancal par, compong he magnec maeral, ec. Hence, he ranfer funcon of reor furnace alo he fr order nera yem wh me delay n he form of: W ( ) Y ( ) k. e X ( ) ( 1) Bu, a h cae k he coeffcen dependng on he emperaure n he furnace. Acually, by denfyng a real reor furnace, k vare conderably, for example n a reor furnace wh a emperaure range of -5 C. (Th wll be demonraed laer). If he opmal conrol problem for he mo accurae burnng proce condered, hen h he opmal conrol problem for he objec wh drbued, delayed, nonlnear parameer. The nonlneary of k ha make he oluon of he problem becomng complex. Thu, n order o olve he problem, h paper dvded k no everal value, hen apple he Laplace ranform and he numercal mehod o gve an explc oluon o hea ranfer problem. II. THE PROBLEM OF OPTIMAL CONTROL A. The objec model A a ypcal drbued parameer yem, a one-dmenonal hea conducon yem condered. The proce of one-ded heang of objec, whch haped lke a reangular box n a furnace decrbed by he parabolc-ype paral dfferenal equaon, a follow n [], [6], [8], [9]: a q ( x, ) q ( x, ) x (1) where q(x,), he emperaure drbuon n he objec, he oupu needng o be conrolled, ISSN: Page 1

2 SSRG Inernaonal Journal of Thermal Engneerng (SSRG-IJTE) Volume 4 Iue 1 January o Aprl 18 dependng on he paal coordnae x wh x L and he me wh f, a he emperaureconducng facor (m /), L he hckne of objec (m), f he allowed burnng me () The nal and boundary condon are gven n [],[6],[9]. q ( x, ) q ( x ) con q ( x, ) x q ( x, ) x () x x L q (, ) v ( ) (3) (4) wh a he hea-ranfer coeffcen beween he furnace pace and he objec (W/m. C), a he heaconducng coeffcen of maeral (W/m. C), and v() a he emperaure of he furnace repecvely ( C). The emperaure v() of he furnace conrolled by volage u(), he emperaure drbuon q(x,) n he objec conrolled by mean of he fuel flow v(), h emperaure conrolled by volage u(). Therefore, he emperaure drbuon q(x,) wll depend on volage u(). The relaonhp beween he provded volage for he furnace u() and he emperaure of he furnace v() uually he fr order nera yem wh me delay a n [1], [], [6], [9]. T. v ( ) v( ) k. u ( ) (5) where T he me conan, he me delay; k he ac ranfer coeffcen; v() he emperaure of he furnace and u() he provded volage for he furnace (conrolled funcon of he yem). However, n expreon (5), k he changng coeffcen dependng on he emperaure n he furnace, mean k a funcon of emperaure v. So, he ac ranfer coeffcen can be expreed by he equaon: k k ( v ), o k a nonlnear coeffcen. Thu, he expreon (5) can be expreed by he equaon: T. v ( ) v ( ) k. u ( ) (6) However, when he coeffcen k nonlnear, dffcul o fnd a oluon, on he oher hand, we can no o apply he Laplace ranform. Therefore, he paper wll perform he lnearzaon of coeffcen k no N value: k1, k, k 3,..., k N. In whch, we aume ha coeffcen k1, k, k 3,..., k N are conan. B. The objecve funcon The problem e ou a follow: we have o deermne a conrol funcon u() wh ( f ) n order o mnmze he emperaure dfference beween he drbuon of dered emperaure q * (x) and real emperaure of he objec q(x, f ) a me = f. I mean a he end of he heang proce o enure emperaure unformy hroughou he whole maeral: L J c q * ( x ) q ( x, f ) d x m n (7) The conraned condon of he conrol funcon : U 1 u () U (8) wh U 1,U are he lower and upper lm of he upply volage repecvely (V). Th problem called he mo accurae burnng problem. III. IDENTIFICATION OF RESISTOR FURNACE MODEL 1. Reor furnace model. Tranfer funcon of reor furnace Accordng o Zegler-Nchol, he reor furnace model can be expreed n he form of a ranfer funcon a he fr order nera yem havng delayed a follow: V ( ) k. e W ( ) U ( ) ( T 1) (9) The block dagram of he denfyng yem hown n Fg.3. Lne volage Inpu Volage U Power Converer Malab/Smulnk Reor Furnace Fg 1. Reor furnace model U() u() W() Reance Furnace Card NI USB-68 Oupu Temperaure V V() Fg. Tranfer funcon of reor furnace v() Fg 3. Dagram of daa collecng yem ISSN: Page

3 Volage (V) Temperaure (oc) SSRG Inernaonal Journal of Thermal Engneerng (SSRG-IJTE) Volume 4 Iue 1 January o Aprl 18 The yem ue NI USB-68 nerface card o oupu he volage gnal o he furnace conrol and collec he emperaure gnal n he furnace. The volage paed from he compuer o he NI USB- 68 card, hrough he DAC o he volage converer, hereby changng he volage uppled o he furnace. The emperaure n he furnace meaured by he hermocouple, afer va he andardzaon k alo aken o he NI USB-68 Card, whch conver no dgal gnal and end emperaure daa o compuer for he denfcaon of he yem model. Pu no he furnace a ep volage: u() =.1(), afer a perod of me Δ=45(), he furnace emperaure reach o v f 5( C). We obaned a he oupu he emperaure repone of he reor furnace a n Fg. 4. Temperaure (C) τ T 1 d Tme () Fg 4. Temperaure repone of he reor furnace Accordng o Zegler-Nchol, we can deermne me conan of he furnace a: T1(); 13() The ac ranfer coeffcen deermned: k v f 5, 7 U where v f he e emperaure, U he uppled volage o he furnace. Commen When he furnace emperaure reached o v f, we have k con, however, he emperaure of he furnace v() depend on he me, a he furnace emperaure change from amben emperaure v o e emperaure v f, hen k alo change dependng on emperaure, mean k con. Thu, very dffcul o deermne exacly he coeffcen k a each me. To analyze he coeffcen k, ha depend on he emperaure v(), heorecally we can perform a: Keep e emperaure v f = con, call he volage changng nerval are Δu (V), he changng nerval of furnace emperaure are Δv ( C). Pung no he furnace ha he volage change n he form of ep, afer a perod of me Δ, he volage ncreae a quany Δu o he volage u = (V), he ac ranfer coeffcen of he furnace k correpond o each me nerval Δ can be calculaed: k v / u 1,, 3,..., N (9) From Eq. (9), we ee ha n he whole burnng me from o f, for each par of value (Δu, Δv), we wll have N value k, o when applyng o fnd he oluon for he problem, he number of calculaon wll be large and exremely complex. So, he auhor ha done a follow: Do no upply drecly he volage (V) ha pu no he furnace he volage change n he form of ep, afer each me nerval Δ = 45 (), f he volage ncreae a quany Δu, he furnace emperaure wll ncreae accordngly a quany Δv o he volage u = (V), he furnace emperaure wll reach he e emperaure v f = 5 ( C), he e me = 135 (). Thu, n realy, we wll have many par of value (Δu, Δv) and wll have many value k repecvely. By denfyng he real reor furnace, we aw k need only 3 or 4 value ha can be afed (oupu emperaure wll be reached o e emperaure v f 5 ( C), o whn he e emperaure range, h paper only conder 3 value of he coeffcen k, ha k k1; k ; k 3. The analy of coeffcen k no 3 value wll make he oluon of he problem becomng mpler. The denfcaon curve hown n Fg Tme () Fg 5. Expermenal reul of reor furnace model denfcaon Commen From he expermenal characerc curve hown n Fgure 5 and expreon (9), he ac ranfer coeffcen of he furnace k no conan, depend on he emperaure n he furnace v(). To deermne he coeffcen k a hown n Fg.5, we have Fg.6. ISSN: Page 3

4 SSRG Inernaonal Journal of Thermal Engneerng (SSRG-IJTE) Volume 4 Iue 1 January o Aprl 18 Temperaure (C) Δv 1 Δv Δv Tme () Fg 6. Temperaure repone o deermne Δv. From Fg.5 and Fg.6, we can deermne he emperaure changng range Δv and he volage changng range Δu, he ac ranfer coeffcen of he furnace k correpondng o each me nerval Δ are calculaed a n Table 1. Table 1. Table of coeffcen k N Δv ( C) Δu (V) k v u , , Commen From Fg.6 and Table 1, he ac ranfer coeffcen of he furnace wll ncreae a he furnace emperaure ncreae over me. When he furnace emperaure range change from amben emperaure o abou 5 C, he value of k change que a lo. IV. THE SOLUTION OF PROBLEM The paper propoe a mehod o olve he opmal problem for he above yem a follow: we dvded he heang me from o f no 3 equal me perod Δ 1 = Δ = Δ 3 and call: - Δ 1 = 1 correpondng o Δv 1 = v v 1, we have k 1 - Δ = 1 correpondng o Δv = v 1 v, we have k - Δ 3 = f correpondng o Δv 3 = v v f, we have k 3 wh 1 / 3; f f / 3; f a he allowed burnng me; v a amben emperaure, v f a he e emperaure. The proce of fndng he opmal oluon nclude ep: - Sep 1: Fnd he relaonhp beween q(x,) and he conrol gnal u(). Namely, we have o olve he equaon of hea ranfer (relaonhp beween v() and q(x,)) wh boundary condon ype-3 combned wh ordnary dfferenal equaon wh me delay (relaonhp beween u() and v()) - Sep : Fnd he opmal conrol gnal u * () by ubung q(x,) found n he fr ep no he funcon (7), afer ha fndng opmal oluon u * (). We conder n he fr perod of me Δ 1 = 1 a follow: A. Fnd he relaonhp beween q 1 (x,) and he conrol gnal u 1 () To olve he paral dfferenal equaon (1) wh he nal and he boundary condon (), (3), (4), we apply he Laplace ranformaon mehod wh he me parameer. On applyng he ranform wh repec o, he paral dfferenal equaon reduced o an ordnary dfferenal equaon of varable x. The general oluon of he ordnary dfferenal equaon fed o he boundary condon, and he fnal oluon obaned by he applcaon of he nvere ranformaon. Tranformng Laplace (1), we obaned: Q1 ( x, ) a Q 1 ( x, ) x (1) where: Q1( x, ) L q1( x, ) Afer ranformng he boundary condon (3), (4), we have: Q1 ( x, ) x x Q1 ( x, ) x x L Q 1(, ) V1( ) (11) (1) From Eq. (6), aumng he delayed objec afy he condon: 6 T/ < 1 n [5], [9]. To olve h problem, he fr order nera yem wh me delay replaced by fr-order Pade appxmaon (Pade 1) Tranformng Laplace (6), we obaned: ( T 1) V1 ( ) k1. U 1 ( ). e k1. U 1 ( ) (13) where: V () L v ( ) ; U ( ) L u ( ) 1 1 (14) The general oluon of (1) : 1 1. x. x a a Q1 ( x, ) A1 ( ). e B1 ( ). e (15) where: A 1 (); B 1 () are he parameer need o be fnd. Afer ranformng, we have he funcon: ISSN: Page 4

5 SSRG Inernaonal Journal of Thermal Engneerng (SSRG-IJTE) Volume 4 Iue 1 January o Aprl 18 Q1 ( x, ).( L x ).( L x ) a a U 1( ). k1. e e. L. L. L. L a a a a a T 1 e e.. e e Pung G1 ( x, ).( L x ).( L x ) a a k1. e e (16). L. L. L. L a a a a a T 1 e e.. e e (17) We have: Q 1 (x,) = G 1 (x,).u() (18) From (18), accordng o he convoluon heorem, he nvere ranformaon of (18) gven by q 1 (x,) = g 1 (x,)* u 1 () We can wre (19) q1 ( x, ) g 1 ( x, ). u1 ( ) d or q1 ( x, ) g 1 ( x, ). u1 ( ) d () where g ( x, ) L 1 G ( x, ) (1) 1 1 Therefore, f we know he funcon g 1 (x,), we wll be able o calculae he emperaure drbuon q 1 (x,) from conrol funcon u 1 (). To fnd q 1 (x,) n (), we need o fnd he funcon (1). Ung he nvere Laplace ranformaon of funcon G 1 (x,) we have he followng reul: k k1. k k. co ( L x ) a k g 1 ( x, ). e k L k k L k co n a a a k1 k1. k1. c o L x a k1. e k1l k1 k1l 1 Tk1 c o n a a a. k. c o ( L x ) 1 a. e L. L L. L 1 T. n co. a a a a ( ) q 3 (x,) wh n he range Δ 3 = f, we alo ranform he ame a he cae of fndng q 1 (x,), we fnally ge he followng reul: k k. k k. c o ( L x ) a k g ( x, ). e k L k k L k co n a a a k1 k. k1. co L x a k1. e k1l k1 k1l 1 Tk1 c o n a a a. k. co ( L x ) a. e L. L L. L 1 T. n co. a a a a k k 3. k k. co ( L x ) a k g 3 ( x, ). e k L k k L k co n a a a k1 k 3. k1. c o L x a k1. e k1l k1 k1l 1 Tk1 c o n a a a. k. co ( L x ) 3 a. e L. L L. L 1 T. n co. a a a a k 1 / T ; k1 / In he cae, e replaced by Taylor approxmaon n [], [6]. Eq. (6) become: U 1 ( ) ( T 1) V1 ( ) k1. U 1 ( ). e k1 1 ( 3 ) ( 4 ) (5) In general cae, n order o fnd he funcon g μ (x,) (μ=1,,3), we alo have he ame ranformaon a n he cae of Pade 1, fnally we obaned he funcon g μ (x,) accordng o he Taylor expanon a follow: To fnd he emperaure drbuon q (x,) wh n he range Δ = 1 and he emperaure drbuon ISSN: Page 5

6 SSRG Inernaonal Journal of Thermal Engneerng (SSRG-IJTE) Volume 4 Iue 1 January o Aprl 18 k k. k. c o ( L x ) a k g ( x, ). e k L k k L 1 k co n a a a k1 k. k1. c o L x a k1. e k1l k1 k1l 1 T. k1 c o n a a a k c o ( L x ) a. e L. L L. L 1 T 1. n co. a a a a k 1 / T ; k1 1/ In Eq.(3), Eq.(4), Eq.(5) and Eq. (6): calculaed from he formula: a / L he oluon of he equaon:. g L / B B he coeffcen BIO of he maeral. he hea-ranfer facor (W/m. C) he hea-conducng facor of objec (W/m. C) L he hckne of objec (m), a he emperaure-conducng facor (m /) he me delay of he furnace () T he me conan of he furnace () k 1 ; k ; 3 ( 6 ) k are he ac ranfer coeffcen of he furnace correpondng o perod of me Δ 1 ; Δ ; Δ 3 repecvely. Concluon: We have olved a yem of parabolc-ype paral dfferenal equaon wh boundary condon of ype-3 (he relaonhp beween v 1 () and q 1 (x,)) combned wh he ordnary dfferenal equaon wh me delay (he relaonhp beween u 1 () and v 1 ()). Thu, f we are no nereed n he opmal problem, we can calculae he emperaure feld n he objec when we know he uppled volage for he furnace (The problem know he hell o fnd he core), a follow: The relaonhp beween he uppled volage for he furnace u() and he emperaure feld drbuon n he objec q(x,): q ( x, ) g ( x, ) * u ( ) g ( x, ). u ( ) d (7) wh =1 3; = f ; f he allowed burnng me (). B. Fnd he opmal conrol gnal u * () by ung numercal mehod To fnd he u * (), we have o mnmze he objecve funcon (7), mean: where L J c q * ( x ) q ( x, f ) d x m n (8) 1 q ( x, f ) g1 ( x, ). u1 ( ) d g ( x, ). u ( ) d g 3 ( x, ). u 3 ( ) d f 1 (9) and q * (x) he dered emperaure drbuon; q(x, f ) he real emperaure drbuon of he objec a me = f. A calculaed n [], [6], [9] he negral numeral mehod ued by applyng Smon formula o he rgh-hand de of he objecve funcon (8). The L, he hckne of he objec, dvded no n equal lengh ( n an even number). Thu, he objecve funcon expreed a n [], [6]. n J c [ u *] L q * ( x ) q ( x, f ) (3) where are he wegh agned o he value of negrand a he pon x. The value of x and he wegh are known for each negraon formula. If he Smpon compoe formula ued, he value of x and are gven by n [], [6]. x L / n ; (,1,..., n ) n 1 / 3 n 1 3 n 1 4 / 3 n 4 n / 3 n n an even number Smlarly, appled o he rgh-hand de of he equaon (9). The perod of me f devded no hree equal nerval m 1, m m 3 ha m 1, m m 3 are an even number, oo. where m 1 me nerval from o 1 m me nerval from 1 o m 3 me nerval from o f Therefore, q(x, f ) caculaed: m 1 q ( x, ) g ( x, ). u ( ) f 1 j1 1 j1 1 j1 j1 (31) he value of by n [], [6]. m ( ) g ( x, ). u ( ) 1 j j j j m 3 ( ) g ( x, ). u ( ) f j3 3 j3 3 j3 j3 j 1 ; j ; j 3 and j 1 ; j ; j 3 are gven ISSN: Page 6

7 SSRG Inernaonal Journal of Thermal Engneerng (SSRG-IJTE) Volume 4 Iue 1 January o Aprl 18 j j 1 11 / m 1 m 1 / 3 m m / 3 m 1 4 m 1 / 3 m 1 j j / m m 1 / 3 m 1 3 m 1 4 / 3 m 4 m / 3 m j j 3 3 f / m 3 m 1 / 3 m m / 3 m 3 4 m 3 / 3 m 3 Pung 1j 1 j1 1 j1 ( j 1,1,,..., m 1 ) j m1, m1 1,.., m j3 m, m 1,.., m 3 c.. g ( x, ); u1( j ) u 1 j1 c ( ).. g ( x, ); u( j ) u j j 1 j j 3j f j3 3 j3 (3) c ( ).. g ( x, ); u3( j ) u 3 j3 u u u u * ; q * ( x ) q j1 j j j 3 (33) Subung (31), (3) and (33) no (3), we obaned: n m1 m m 3 * * J [ u ] L... c q c u c u c u 1 j j j j 3 j j j1 j j3 (34) The conraned condon of he conrol funcon (Lm of he uppled volage for he furnace) are decrbed a follow: (35) (j = m 1 +m +m 3 ) U 1 u j U The performance ndex (34) a quadrac funcon of he varable u j wh conran (35) are lnear, he problem become a quadrac programmng problem. Th problem can be obaned by ung numercal mehod afer a fne number of eraon of compuaon. Alhough a oluon of he quadrac programmng problem obaned afer a fne number of eraon of compuaon, bu algorhm more complcaed han ha of he mplex mehod for lnear programmng. If he performance ndex aken a L (36) J c q * ( x ) q ( x, f ) d x nead of (8), he lnear programmng echnque can be ued drecly. On applyng he ame procedure a menoned above, he approxmae performance ndex correpondng o (36) wren a J... _ n m m m 1 3 * J L c c q c u c u c u 1 j j j j 3 j j j1 j j3 (37) The problem of mnmzng (37) under he conran (35) can be pu no a lnear programmng form by ung known echnque [], [6]. By nroducng (n+1) non-negave auxlary and (=,1, n), he mnmzaon of (37) equvalen o he mnmzaon of wh he conran _ n c ' ( ) (38) J L m 1 m m 3 * c j u j q j, and U 1 u j U (j = m 1 +m +m 3 ) (39),1,..., n Thu, wh any u j, he mnmum value of (38) aaned by eng = f m 3 * c ju j q non-negave and = f j m 3 * c ju j q negave. Then, clearly j mn _ J c = mn _ J c ' Thu, we can replace he oluon of (34) wh he conran (35) by mnmzng he problem (38) wh he conran (39). By ung he mple mehod n [], [4], he opmal oluon of (38), (39) can be reached afer a fne number of eraon. C. Calculae he emperaure of he furnace v() and he emperaure drbuon n he objec q(x,) (4) or 1) Calculae he emperaure of he furnace v() We know ha v() and u() have he relaon: T. v ( ) v ( ) k. u ( ) v ( ) k. u ( ) v ( ) f (v, u ) T (41) Baed on mproved Euler formula, we have: v ( j 1) v ( j ) ( l1 l l3 ). f ( u, v ( j)) ISSN: Page 7

8 SSRG Inernaonal Journal of Thermal Engneerng (SSRG-IJTE) Volume 4 Iue 1 January o Aprl 18. ( ) ( ) 1 f k u j v j v ( j 1) v ( j ). m m m T 1 3 where (4) l1 1 / m1; l ( 1 ) / m ; l3 ( f ) / m 3 ; 1 f / 3; f / 3 ; f he allowed burnng me (); k can ge: k k1 k k 3 /3 wh m 1, m, m 3 are he number of me nerval correpondng o he me nerval 1,, 3 T he me conan of he furnace. Afer ranformng, we ge v ( j 1) or v( j) k.( l1 l l3 ). u ( j ) v ( j )[ T ( l1 l l3 )] T (43) k.( l1 l l3 ). u ( j 1) v ( j 1).[ T ( l1 l l3 )] T (44) wh j = m 1 + m +m 3 So, when knowng u * () we can calculae v() from Eq. (44). ) Calculae he emperaure drbuon n he objec q(x,) To calculae q(x,) when knowng u * (), we ue he prvou calculaed reul. Here alo ue he numercal mehod [], [3], [4], [6]. From Eq. (31), we have 1 q ( x, j ) g 1( x, j ). u1( ) d 1 g ( x, j ). u ( ) d f n ; f g 3 ( x, j ). u 3 ( ) d (45) ; n he number of layer of pace. Accordng o rapezodal formula [3], [4]. Afer calculang, we obaned j1 q ( x, ) j 1 l 1. g 1 ( x, j 1l1 ). u 1 ( ) j jl. g ( x, j l ). u ( ) j3 j3l3. g 3 ( x, j3l3 ). u 3 ( ) (46) V. SIMULATION RESULTS Afer buldng he algorhm and eablhng he conrol program, we have proceeded o run he mulaon program o e calculang program. A. The mulaon for a lab of Samo The phycal parameer of he objec = 6 (W/m. C); =.955 (W/m. C) a= 4.86*1-7 (m /); L =.3 (m) The parameer of he furnace T = 1 (); = 13 (); k1 1.8 ; k 3.3 ; k 3 5 The dered emperaure drbuon q * = 3 C The perod of heang me f = 4 () Lm he emperaure of furnace u() 5 C Lm he emperaure of lab urface: q(, ) 35 C Lm under volage: U 1 =15 (V) Lm upper volage: U =5 (V) Wh hee parameer, he coeffcen B calculaed a follow:.l/ 6., 3 /.955 1, 9 B Thu, he lab of Samo a hck objec becaue he coeffcen B greaer han.5. Havng 6 T/ <1 To calculae he opmal heang proce, we chooe n = 6 and m 1 =m =m 3 =m=4. Afer he mulaon, we have reul lke n Fgure 7 and Fgure 8. ISSN: Page 8

9 SSRG Inernaonal Journal of Thermal Engneerng (SSRG-IJTE) Volume 4 Iue 1 January o Aprl 18 u*() v() q(x,) q* q(x,) Fg 7. The opmal heang proce for a lab of Samo (followng Taylor Approxmaon, wh e = 3.77e -6 ) Fg 8. The opmal heang proce for a lab of Samo (followng Pade Approxmaon, wh e = e -7 ) ISSN: Page 9

10 SSRG Inernaonal Journal of Thermal Engneerng (SSRG-IJTE) Volume 4 Iue 1 January o Aprl 18 B. The mulaon for a lab of Daome The phycal parameer of he objec = 6 (w/m C); =. (w/m. C) a= 3.6*1-7 (m /); L =.4 (m) The parameer of he furnace T = 1 (); = 13 () k1 1.8 ; k 3.3 ; k 3 5 The dered emperaure drbuon q * = 4 C The perod of heang me f = 44 () Lm he emperaure of furnace u() 75 C Lm he emperaure of lab urface q(, ) 55 C Lm under, upper volage: U 1 =15 (V);U = (V) Wh hee parameer, he coeffcen B calculaed a follow: B.L/ 6., 4 /. 1 Thu, he lab of Daome a very hck objec becaue he coeffcen B greaer han.5. Havng 6 T/ <1 To calculae he opmal heang proce, we chooe n = 1 and m 1 =m =m 3 =m=1. Afer he mulaon, we have reul lke n Fgure 9 and Fgure 1. Fg 9. The opmal heang proce for a lab of Daome (followng Taylor Approxmaon, wh e = 1.319e -6 ) Fg 1. The opmal heang proce for a lab of Daome (followng Pade Approxmaon, wh e = 5.943e -9 ) ISSN: Page 1

11 SSRG Inernaonal Journal of Thermal Engneerng (SSRG-IJTE) Volume 4 Iue 1 January o Aprl 18 The mulaon reul are hown on Fg.7 o Fg.1. where u*() he opmal conrol gnal (opmal volage) of he furnace; v() he emperaure of he furnace; q(x,) emperaure drbuon of he lab, ncludng he emperaure of he wo urface and he emperaure of he nner layer of he lab. C. Commen In Fg. 7 and Fg. 8, we can ee ha a he me = f =4(), he emperaure drbuon of he layer n a lab of Samo q(x, f ) are all approxmaely equal q * =3 C. Therefore, he opmal oluon ha been efed. Taylor approxmaon ha e = 3.77e -6, wherea Pade 1 ha e = e -7 (wh e a he error of objecve funcon J c ). In Fg. 9 and Fg. 1, we alo can ee ha a he me = f =44(), he emperaure drbuon of he layer n a lab of Daome q(x, f ) are all approxmaely equal q * =4 C. Thu, he opmal oluon ha been efed. Taylor approxmaon ha e = 1.319e -6, Pade 1 ha e = 5.943e -9. I mean ha, f delayed objec afe he condon 6 T/ < 1 [5], [9], he fr-order Pade approxmaon wll have hgher accuracy. VI. CONCLUSIONS The paper ha olved a yem conng of parabolc-ype paral dfferenal equaon wh boundary condon ype-3 combned wh a medelayed ordnary dfferenal equaon. Namely, we have proceeded o denfy a real reor furnace n order o deermne exacly ranfer funcon of furnace, a well a have analyzed he change of ac ranfer coeffcen dependng on emperaure n he furnace. An opmal oluon for a drbued parameer yem wh me-delay, nonlnear ha been defned by ung a numercal mehod. Algorhm and opmal calculang program have been accuracy. Then, we have proceeded o run he mulaon on a lab of Samo and a lab of Daome n order o e he algorhm once agan. However, n h paper, we only mulae on Malab ofware o verfy he oluon and he reul of he calculaon. In he nex paper, we wll conduc expermen on pecfc pecmen n order o e he mulaon reul. ACKNOWLEDGMENT Th paper funded by The Scence and Technology Program for he uanable developmen of Norhern-We area, Venam Naonal Unvery, Hano. REFERENCES [1] P.K.C.Wang, "Opmum conrol of drbued parameer yem", Preened a he Jon Auomac Conrol Conference, Mnneapol, Mnn.June, pp [] Y. Sakawa, Soluon of an opmal conrol problem n a drbued parameer yem, Tran. IEEE, 1964, AC-9, pp [3] E. Celk and M. Bayram, On he numercal oluon of dfferenal algebrac equaon by Pade ere, Appled mahemac and compuaon, 137, pp , 3. [4] J. H. Mahew and K. K. Fnk, Numercal Mehod Ung Malab, 4 h Edon, Upper addle Rver, New Jerey, 4 [5] N..H.Cong, A reearch replace a delayed objec by approprae model, 6 h Ve Nam Inernaonal Conference on Auomaon, (VICA6-5) [6] N.H.Cong, N.H.Nam, Opmal conrol for a drbued parameer yem wh me delay baed on he numercal mehod, 1 h Inernaonal Conference on Conrol, Auomaon, Roboc and Von, IEEE Conference, pp , 8. [7] M. Suba, Opmal Conrol of Hea Source n a Hea Conducvy Problem, Opmzaon Mehod and Sofware, Vol 17, pp. 39-5, Turkey, 1. [8] B. Talae, H. Xu and S. Jagannahan, Neural nework dynamc progammng conraned conrol of drbued parameer yem governed by parabolc paral dfferenal equaon wh applcaon o dffuon-reacon procee, IEEE Conference Publcaon, pp , 14. [9] Ma.T.Tha, Nguyen.H.Cong, Nguyen.V.Ch, Vu.V.Dam, Applyng pade approxmaon model n opmal conrol problem for a drbued parameer yem wh me delay, Inernaonal Journal of Compung and Opmzaon, Hkar Ld, vol 4, no.1, pp.19-3, 17. ISSN: Page 11