Semi-analytical methods for optimal energy transfer in RC ladder networks

Transkrypt

1 J. Baranowski, W. Mikowski AGH Universiy of Science and Tecnology, Deparmen of Auomaics Semi-analyical meods for opimal energy ransfer in RC ladder neworks Absrac. Tis paper presens a semi-analyical soluion o e opimal conrol problem arising wi energy ransfer roug ransmission line. Disribued parameer RC line is approimaed wi a RC ladder nework. Formulas for opimal conrol are deermined up o a momen wen mari eponenial as o be compued. Tis final elemen as o be compued numerically. Paper ends wi resuls of simulaions and a discussion of numerical issues. Sreszczenie. Praca przedsawia semi-analiyczne rozwiazanie problemu serowania opymalnego wysępujacego przy przesyle energii poprzez linię długa. Linia długa RC jes aproksymowana za pomoca układu łancucowego ypu RC. Wzory okreslaj ace serowanie opymalne sa wyznaczone do momenu koniecznosci wyliczenia eksponeny macierzy. Ta częsc obliczen musi byc wykonana numerycznie. Praca konczy się wynikami symulacji i dyskusja zagadnien numerycznyc. Meody semi-analiyczne w opymalnym przesyle energii w układac drabinkowyc RC) Keywords: energy ransfer, RC ladder nework, maimum principle, opimisaion Słowa kluczowe: przesył energii, układ drabinkowy RC, zasada maksimum, opymalizacja Inroducion Analysis of elecrical ladder neworks is an imporan aspecs of conrol eory and eoreical elecrical engineering. Teir imporance comes from connecion o e models of infinie dimensional processes especially ermal), were ey can be used as efficien and inuiive finie dimensional approimaions. Auors recen works considered especially sabilisaion of neworks, among e oers undamped LC ype [, and nonlinear RC ype [3. Also under consideraion were problems of conrol in ladder neworks were considered in [ and [3 and applicaions in modeling of ouseold buildings [9,. Tis work is a coninuaion of resuls presened in [, 5,. Opimal conrol problems arise in many real life siuaions. One ineresing problem is eaing wi an elecric curren see for eample [). We consider an elecric nework sown in e figure. Te resisance of e volage source R and e oupu resisance R H are given. Assuming a i w ) e energy producing ea on e oupu resisance is given by E y)i w )d e ransmission line is described by e equaion, z) rc, z) z z l. Le z i, l/n, i,,..., n and, k )/) k ), k,,..., n.weave, z) z, z + ), z) ), z), z ) for z k )/ and k,,..., n. Ten e RC ransmission line can be approimaed by e RC ladder nework sown in e figure, were R rl and C cl see for eample [). I is desired o ransfer e required energy roug e ransmission medium a e same ime minimising e energy delivered o e nework. Fig.. Approimaion of a RC ransmission line wi a RC ladder nework Fig.. Scemaic represenaion of a ransmission line In order o find suc inpu volage u) a would realise e desired goal one needs o formulae e maemaical model of e problem. Consider a omogeneous long elecric RC ransmission line, i.e. one were e parameers per e uni leng resisance r and capaciy c) are consan and independen of e spaial variable z. An infiniesimal par of Opimal conrol problem We consider e elecric RC ladder nework sown in e figure. Is parameers R, R,R H and C are known. Te sysem can be described by e following sysem of equaions see for eample [,, ) ) ẋ) A)+Bu), ) [ ) )... n ) T, y) W) PRZEGLAD ELEKTROTECHNICZNY Elecrical Review), ISSN 33-97, R. NR 9a/ 5

2 Were A R n n is a ridiagonal Jacobi mari [, suc a A [a ij a ij a ij, for i j > a ii n, for i, 3,...,n, RC a + rr )) n RC, a nn + rr H )) n RC, a i+,i n, for i,,...,n, RC a i,i+ a i+,i, for i,,...,n. rv) R nv + R, B n rr ) RC e, W n rr H )R H e T R n, e [... T R n, e n [... T R n. Formulaion of e problem Le T and E be fied, e goal is o find a conrol signal a in e ime orizon T delivers an amoun of energy E o e receiver R H delivering minimal amoun of e overall energy o e RC nework see for eample [). Formal definiion of e problem is o find suc conrol u U d suc a: Ju) Ju ) u U d, were Ju) is e performance inde given by ) Ju) u)i)d n nr + R u)[u) )d, wi given by ) and ). U d is e se of admissible conrols { T U d u : y) d R H 3) n } R T H nr H + R/) n ) d E I sould be noed a e se U d. To see a, eamine for eample u) conssuc a: R H y) d n R H nr H + R/) n ) d E. Now, we consider e space L p,t), p [, ) wi e norms f p [ f) p d /p. From e Hölder inequaliy see for eample [5) we ave u) )d u u. Te sysem ) is asympoically sable, conrollable and observable e pair A, B) is conrollable [ and W, A) is observable, see also [3). Consequenly: R + R/n)Ju) u u) )d u u / see ) and 3)),for every u U d e norm finie and Ju) as u. Tus ere eiss e opimal conrol u, cf. ). We can noice a Ju) J u). Applicaion of Maimum Principle I is possible o consruc an algorim for deerminaion of opimal conrol wi use of e Maimum Principle [7,. One needs o observe a e problem is an opimal conrol problem wi equaliy consrains on erminal sae. To do so new sae variables are defined: ) ẋ n+ ) n ), n+ ), ẋ n+ ) u)u) )), n+ ), one can see a because n+ T ) corresponds o e consrain given by e admissible se 3) and n+ T ) is used o represen an inegral performance inde we ave n+ T ) nr H + R/) n R H E, n+ T ) nr + R Ju). n Le us inroduce e adjoin funcion ψ: R R n and inroduce e following noaion and ) [) T n+ ) n+ ) T ψ) [ψ) T ψ n+ ) ψ n+ ) T. Ten we obain e Hamilonian in e form: 5) H ψ), ),u)) ψ) T [A)+Bu) + ψ n+ ) n ) + + ψ n+ )u)[u) ). Using ransversaliy condiions we ave ψ n+ ) ρ cons, ψ n+ ), ψt ) R n and e adjoin funcion ψ is a soluion o e following sysem of equaions ) ψ) A T ψ) e u) ρe n n ) 5 PRZEGLAD ELEKTROTECHNICZNY Elecrical Review), ISSN 33-97, R. NR 9a/

3 Using e Maimum Principle see for eample [7,, ), from 5) we ge: 7) u) [BT ψ)+ ) Te conrol 7) depends on e real number ρ and is called e eremal conrol. We will searc for e opimal conrol u among e eremal conrols 7). From ), ) and 7), we obain e canonical sysem in e following form: ẋ) ) Z ψ) ψ) ) ) ψt ), were [ A + Z Be 9) e e T ρe n e T n e Z Φ ) Φ ) ) Φ 3 ) Φ ) Ten from ) and ) we ave ) Φ )ψ) ) ψ) Φ )ψ). BBT A e B T If E, en ), cf. 3). Tus from ) we ge ψ). Since ψt ), cf. ), from ) we ave: ) de Φ T ). Obaining e opimal conrol Te following algorim can be used for e deerminaion of opimal conrol. I is a varian of algorim proposed in [.. Deermine e parameer ρ using equaion ).. Because rankφ T ) n using gaussian eliminaion deermine e dependence of soluion of Φ T )ψ) on a single parameer γ in paricular α α ψ). 3). γ αγ α n 3. From 3), ) and 3) calculae γ and subsequenly ψ). Tis will be discussed below.. From 7) and ) deermine ) u) [BT ψ)+ ) [ B T Φ )+e T Φ ) ψ), As one can easilly see seps and deermine e eremal soluions, wile e opimal one is cosen in sep 3 by fulfilling e equaliy consrain 3). Sep 3 requires special discussion. Le us consider e inegral in 3). n ) d, le us denoe ) ψ) e Z T en e T n e Z [ ) ψ) e ZT d [ ) ψ) [ en e T n ) e Z d ψ) Q en e T n. We reformulae e inegral e ZT Qe Z d e ZTT e ZT T e ZT Qe Z d e ZT T e ZT T ) Qe Z d We denoe Ψ) e Z, and formulae a sysem of equaions Ψ) Z Ψ) 5) Ẋ) Q Z T, X) Ψ) In n X) en One can see a if en and e ZT T n n e ZT T ) Qe Z d ΨT ) T XT ) Z G Q Z T e e G Z Ξ ) Ξ ) ΨT ) T XT )e ZTT Ξ T ) Finally denoing e ZTT P P Ξ T ) P 3 P we ge n ) d γ α T P α and from 3) we ave e formula on γ E ) γ ± α T P α d PRZEGLAD ELEKTROTECHNICZNY Elecrical Review), ISSN 33-97, R. NR 9a/ 5

4 Numerical eperimens and discussion Below we presen resuls of numerical compuaions for n,, 3,. Figure 3 presens ow conrol canges wi e rise of n. In figures - 7 e sae variables for differen n are presened. As one can easily see e conrols for differen n are smoo and similar o one anoer. Also similar rends are visible also in sae rajecories. 7 n n n3 n 5 u Fig. 5. Evoluion of sae variables for n Fig. 3. Conrols for differen n Fig.. Evoluion of sae variables for n 3 Fig.. Evoluion of for n Some discussion is owever needed regarding e numerical compuaion. For n all compuaions can be performed analyically. Namely equaion ) becomes see [) gωρ)t ) ωρ) Z were ωρ) Z + ρ ) Z, R + R H +3R/ Z CR + R/)R H + R/), Z C R + R/), Fig. 7. Evoluion of sae variables for n is equaion as infinie number of soluions, owever i can 53 PRZEGLAD ELEKTROTECHNICZNY Elecrical Review), ISSN 33-97, R. NR 9a/

5 be sown afer iresome compuaion, a ψ) ± R H + R/)ω Z R H T E + were ω is e smalles posiive soluion of ω Z ) ) ) ) Z + ω T Z gωt) ω Z and opimal conrol is given by e following formula: u ) [BΦ )+Φ ) ψ ), R B R + R)RC Φ ) Z ω sin ω Φ ) cosω Z ω sin ω Opimal rajecory is given by ) Φ )ψ ) Te case of n is owever e only one were analyical soluion can be obained compleely wiou e use of compuers. For iger orders some numerical compuaion is needed. For n, 3 equaion ) can be solved direcly owever problem becomes increasingly badly scalled. Eigenvalues of mari Z increase eir absolue values by e mulipliciy of n. And because ey are evenly spread among posiive and negaive ones e mari eponenial e Z becomes ill condiioned. For n deerminan becomes useless for compuaion and problem can be solved only by reducion of e smalles eigenvalue o zero. For n>5prob- lem is so badly condiioned a e algorim is unable o solve i. Tere can be cerain aemps o improve e condiioning, owever in auors opinion direc opimisaion meods could lead o beer resuls. Conclusions Te problem of opimal energy ransfer was considered and a semi analyical soluion was presened. Applicaion of classical resuls of conrol eory leads o poenially effecive formulas and is an alernaive o non deerminisic meods suc as evoluionary algorims [. Work financed from NCN-Naional Science Cenre funds no. N N5. BIBLIOGRAPHY [ M. Aans and P. Falb. Opimal Conrol. An Inroducion o e Teory and Is Applicaions. Serowanie opymalne. Wsęp do eorii i jej zasosowania). Wydawnicwa Naukowo Tecniczne, Warszawa, 99. in Polis. [ J. Baranowski, M. Długosz, M. Ganobis, W. Mikowski, A. Obraczka, and P. Skruc. Modelowanie i serowanie wybranyc procesów energeycznyc z wykorzysaniem układów łancucowyc modeling and conrol of seleced energy processes using ladder neworks. In R. Dindorf, edior, KKA : XVII Krajowa Konferencja Auomayki : Kielce Cedzyna, 9.. r. : sreszczenia referaów, pages 9 7. Wydawnicwo Poliecniki Święokrzyskiej,. [3 J. Baranowski, M. Długosz, M. Ganobis, P. Skruc, and W. Mikowski. Applicaions of maemaics in seleced conrol and decision processes. Maemayka Sosowana : pismo Polskiego Towarzyswa Maemaycznego, /53nr spec.):5 9,. [ J. Baranowski and W. Mikowski. Sabilisaion of e second order sysem wi a ime delay conroller. In Papers. IFIP TC7 Conference on Sysem Modelling and OpimizaionOpimal glucose level regulaion for Inensive Care uni paiens: CSE- DOM approac, pages 9, Palais Rouge, Buenos Aires, Argenina, July 7 - July 3s 9. Organized by: Herramienas Gerenciales. [5 J. Baranowski and W. Mikowski. Analyical meods for opimal energy ransfer in rc ladder neworks. In Maeriały XIV sympozjum Podsawowe Problemy Energoelekroniki, Elekromecaniki i Mecaroniki, pages 53, Wisła, 9- grudnia. [ J. Baranowski and W. Mikowski. Sabilisaion of lc ladder nework wi e elp of delayed oupu feedback. Conrol and Cyberneics, ),. Acceped for publicaion. [7 W. Bołianski. Meody maemayczne serowania opymalnego. Wydawnicwa Naukowo-Tecniczne, Warszawa, 97. [ A. G. Bukowskii. Teory of Opimal Conrol of Disribued Parameer Sysems. Elsevier, New York, 99. [9 M. Długosz, W. Mikowski, J. Baranowski, P. Piaek, and P. Skruc. Układy obwodowe w modelowaniu procesów cieplnyc budynków. In Maeriały XIV sympozjum Podsawowe Problemy Energoelekroniki, Elekromecaniki i Mecaroniki, pages 9 5, Wisła, 9- grudnia. [ M. Długosz, P. Piaek, J. Baranowski, and P. Skruc. Algorymy serowania i zarzadzania budynkami mieszkalnymi. Pomiary Auomayka Roboyka,. Złożone do druku. [ M. Gajer. Zasosowanie ecniki obliczen ewolucyjnyc w obszarze eorii obwodów elekrycznyc. Przeglad Elekroecniczny, 7):5 53,. [ H. Górecki. Opymalizacja i serowanie sysemów dynamicznyc. Uczelniane Wydawnicwa Naukowo Dydakyczne AGH, Kraków,. [3 T. Kaczorek. Serowalnosc i obserwowalnosc liniowyc obwodów elekrycznyc. Przeglad Elekroecniczny, 79a): 5,. [ J. Klamka. Conrollabiliy of Dynamical Sysems. PWN, Warszawa, 99. in Polis, englis ediion: Kluwer Academic Publisers, 99). [5 J. Kudrewicz. Analiza funkcjonalna dla auomayków i elekroników. PWN, Warszawa, 97. [ W. Mikowski. Synesis of rc-ladder nework. Bull.Acad.Pol. Sci., Tec.Sci., ):33 37, 99. [7 W. Mikowski. Tridiagonal marices and eir applicaions in circuis eory. Macierze rójprzekaniowe i ic zasosowania w eorii obwodów. In Proc. of Seminar on Elecrical Engineering "Beskidy 9"., volume, pages 9 3, Isebna-Pieraszonka, 3- Ocober 99. Conference Arcives PTETiS. [ W. Mikowski. Analysis of ladder and ring rc-neworks. Bull. Acad. Pol. Sci., Tec. Sci., 53):5 5, 997. [9 W. Mikowski. Remarks on sabiliy of posiive linear sysems. Conrol and Cyberneics, 9):95 3,. [ W. Mikowski. Remarks abou energy ransfer in an rc ladder nework. In. J. Appl. Ma. Compu. Sci.,, 3):93 9, 3. [ W. Mikowski. Wybrane problemy nagrzewania pradem elekrycznym. In D. Szeliga, M. Pierzyk, and J. Kusiak, ediors, KomPlasTec : informayka w ecnologii meali : maeriały XIII konferencji : Szczawnica 5 sycznia, pages 9, Kraków,. Wydawnicwo Naukowe Akapi. [ L. Ponriagin, W. Bołianski, R. Gamkrelidze, and E. Miszczenko. Maemaiczeskaja ieoria opimalnyc processow. Nauka, Moskwa, ediion, 93. [3 P. Skruc, J. Baranowski, and W. Mikowski. Dynamic feedback sabilizaion of nonlinear rc ladder nework - sabilizacja nieliniowyc obwodów drabinkowyc rc za pomoca dynamicznego sprzężenia zwronego. Elekryka, 5):9 33,. Auors: P.D. Jerzy Baranowski, Prof. Wojciec Mikowski AGH Universiy of Science and Tecnology, Faculy of Elecrical Engineering, Auomaics, Compuer Science and Elecronics, Deparmen of Auomaics, Al. Mickiewicza 3/B, 3-59 Krakow, Poland, jb@ag.edu.pl, wojciec.mikowski@ag.edu.pl PRZEGLAD ELEKTROTECHNICZNY Elecrical Review), ISSN 33-97, R. NR 9a/ 5