Start Point and Trajectory Analysis for the Minimal Time System Design Algorithm

Transcription

1 Start Pont and Trajectory Analy for the Mnmal Tme Sytem Degn Algorthm ALEXANDER ZEMLIAK, PEDRO MIRANDA Department of Phyc and Mathematc Puebla Autonomou Unverty Av San Claudo /n, Puebla, 757 MEXICO Abtract: - The dfferent degn trajectore have been analyzed n the degn pace on the ba of the new ytem degn methodology Optmal poton of the degn algorthm tart pont wa analyzed to mnmze the computer degn tme The ntal pont electon ha been done on the ba of the before dcovered acceleraton effect of the ytem degn proce The geometrcal dvdng urface wa defned and analyzed to obtan the optmal poton of the algorthm tart pont Numercal reult of both pave and actve nonlnear electronc crcut degn prove the poblty of the optmal electon of the degn algorthm tart pont Key-Word: - Tme-optmal degn algorthm, control theory applcaton, optmal tart pont electon Introducton The problem of the computer tme reducton of a large ytem degn one of the eental problem of the total qualty degn mprovement Bede the tradtonally ued dea of pare matr technque and decompoton technque []-[5] ome another way were determne to reduce the total computer degn tme The generalzed theory for the ytem degn on the ba of control theory formulaton wa elaborated n ome prevou work [6]-[8] Th approach erve for the tme-optmal degn algorthm defnton On the other hand th approach gve the poblty to analyze wth a great clearne the degn proce whle movng along the trajectory curve nto the degn pace The man concepton of the theory the ntroducton of the pecal control functon, whch, on the one hand generalze the degn proce and, on the other hand, they gve the poblty to control degn proce to acheve the optmum of the degn objectve functon for the mnmum computer tme Th poblty appear becaue practcally an nfnte number of the dfferent degn tratege that et wthn the bound of the theory, but the dfferent degn tratege have the dfferent operaton number and eecuted computer tme On the bound of th concepton, the tradtonal degn trategy only a one repreentatve of the enormou et of dfferent degn tratege A hown n [8] the potental computer tme gan that can be obtaned by the new degn problem formulaton ncreae when the ze and complety of the ytem ncreae but t realzed only n cae when we have the algorthm for the optmal trajectore real contructon We can defne the formulaton of the ntrnc properte and pecal retrcton of the optmal degn trajectory a one of the frt problem that need to be olved for the optmal algorthm contructon Problem Formulaton The degn proce for any analog ytem degn can be defned [8] a the problem of the generalzed objectve functon F ( X, U ) mnmzaton by mean of the vector equaton: X + = X wth the contrant: where ( u ) g ( X ) + t H () j j =, j =,,, M () N X R, X ( X X ) =,, K X R the vector of M X R N = K + M ), the ndependent varable and the vector the vector of dependent varable ( g j X for all j the ytem model, the ( ) teraton number, t the teraton parameter,

2 t R, H H(X,U) the drecton of the generalzed objectve functon ( X U ) F, decreang, U the vector of the pecal control functon U = ( u, u,, u m ), where u j Ω; Ω = { ; } The generalzed objectve functon F ( X, U) defned a: F ( X, U) = C( X ) + ψ( X, U) where C ( X ) the ordnary degn proce cot functon, and ψ X,U the addtonal penalty functon: ( ) M ψ ( X, U ) = u j g j ( X ) Th problem formulaton ε j= permt to redtrbute the computer tme epene between the problem () olve and the optmzaton procedure () for the functon F ( X, U) The control vector U the man tool for the redtrbuton proce n th cae Practcally an nfnte number of the dfferent degn tratege are produced becaue the vector U depend on the optmzaton current tep The problem of the optmal degn trategy earch formulated now a the typcal problem for the functonal mnmzaton of the control theory The functonal that need to mnmze the total CPU tme T of the degn proce Th functonal depend drectly on the operaton number and more generally on the degn trajectory that ha been realzed The man dffculty of th problem defnton unknown optmal dependence of all control functon u j Th problem the central for uch a type of the degn proce defnton 3 Trajectory Analy The problem of the ntal pont electon for the degn proce one of the eental problem of the tme-optmal algorthm contructon The analy of the degn proce and acceleraton effect for the mplet electronc crcut of the Fg wa provded n [9] Th the two-dmenonal cae Fg Smplet one node crcut The vector of the tate varable X ha two component X = (, ) where = R, = V The nonlnear element ha the followng dependency: R n = r + bv Ung the Law of Krchhoff we can obtan the followng functon g(x): ( X ) ( + r + b ) g (3) = The objectve functon defned by the formula C( X ) = ( k V ), where kv ha the fed value There only one control functon u n th cae becaue there only one dependent parameter The degn trajectory for th eample the curve n two-dmenonal pace, f the numercal degn algorthm appled The optmzaton procedure and the electronc ytem model, n accordance wth the new degn methodology [9], are defned by the net two equaton: ( X U ) + = + t f, ( ) g ( X ) =, =, (4) u (5) where U the vector of control varable, and the component of the movement drecton f ( X, U ) for the =, depend on the optmzaton method Thee functon, for the gradent method for eample, are gven by the formula: δ f ( X, U ) = F ( X, U ) (6) δ δ ( ) ( ) ( u ) f X, U = u F X, U + [ + η( X )] (6 ) δ t where ( X U ) F, the generalzed objectve functon, F ( X, U ) = C( X ) + ug ( X ), η ( X ) ε the mplct functon ( + = η ( X )) and t gve the value of the parameter from the equaton (5), and the operator δ for =, mean: δ δ F F δ F F = +, F = δ δ

3 A hown n [9] we need to elect the ntal pont of the degn proce wth the negatve coordnate In th cae the acceleraton proce can be realzed The famly of the degn curve for the crcut on Fg, whch correpond to the modfed tradtonal degn trategy (u=) and the negatve ntal value of the econd coordnate ( <) of the vector X hown n Fg for the -D phae pace Thee curve have dfferent tart pont but the ame fnal pont F The tart pont were elected on the crcle arc and have the dfferent ntal coordnate The pecal curve S-F, whch marked by thck lne, the eparatng curve Th curve eparate the trajectore that are the canddate for the acceleraton effect achevement (all curve that le under the curve S-F), and the trajectore that can not produce the acceleraton effect (curve that le over the curve S-F) It clear that the projecton of the fnal pont F to all curve of the frt group defne the wtchng pont of the optmal trajectory, whch produce the acceleraton effect All curve of the frt group (-7) approach to the fnal pont F from the left de, and all curve of the econd group (9-6) approach to the fnal pont from the rght de The comparon of the relatve computer tme for all curve of the Fg hown n Fg 3 Fg Trajectore of the modfed tradtonal trategy for the dfferent tart pont wth the negatve coordnate The eparatng curve S-F ha the mnmal computer tme among all of the trajectore At the ame tme th curve can not be ued a the ba for the tmeoptmal trajectory contructon becaue the projecton of the pont F to th curve the ame pont F, but the movement low down near th pont Only the curve that le under the curve S-F erve a the frt part of the tme-optmal trajectory wth the followng jump to the pont F The relatve computer tmeτ of the optmal trajectore wth acceleraton effect (on the ba of the curve -7, Fg ) hown n Fg 4 a the functon of the curve number n The curve 9-6 can be optmzed too but n th cae the tme reducton about -5% only take place Fg 4 how that the total computer tme ncreae when the tart pont approache to the curve S-F, and on the contrary, the more acceleraton can be obtaned f the tart pont le far from the curve S-F (from curve 7 to curve ) So, the tart pont electon wth at leat one negatve ntal coordnate of the vector X and the value of th coordnate that gve the tart pont poton under the eparatng lne are the uffcent condton for the acceleraton effect appearance More detal analy how that the negatve value of the tart pont coordnate below the eparate lne the uffcent condton for the acceleraton effect but not the neceary The phae dagram of Fg 5 nclude two type of the eparate lne The frt lne AFB eparate the trajectore that draw to the fnal pont F from the left and from the rght The econd eparate lne CTFB dvde all the phae pace to the two ubpace All the pont and trajectore that le nde th eparate lne can not produce the acceleraton effect On the other hand, all the pont that le outde the eparate lne and correpondng trajectore produce the acceleraton effect Thee geometrcal condton are the neceary and uffcent to obtan the acceleraton effect Fg 3 Relatve computer tmeτ a the functon of the curve number n Fg 4 Relatve computer tme τ of the optmal trajectore wth acceleraton effect a the functon of the curve number n

4 Fg 5 Phae dagram - for one-node crcut The N-dmenonal cae ha been analyzed below The econd eample correpond to the crcut of Fg 6 Th crcut ha fve ndependent varable a admttance y, y, y3, y4, y5 (K=5) and four dependent varable a nodal voltage V, V, V3, V4 (M=4) Non-lnear crcut element have dependence: y ( ) n = an + bn V V, y ( ) n = an + bn V3 V Non-lnearty parameter bn, bn are equal to The tate parameter vector X nclude nne component: = y, = y, 3 = y3, 4 = y4, 5 = y5, 6 = V, 7 = V, 8 = V3, 9 = V4 The ytem of the optmzaton proce nclude nne equaton and the crcut model nclude four equaton Fg 7 Phae dagram 5-9 for four-node crcut The regon outde the eparate lne nclude the pont and the trajectore that can produce the acceleraton effect In th cae, a for the frt eample, the eparate lne or more general the eparate hyper-urface defne the neceary and uffcent condton for the acceleraton effect etence Actve nonlnear crcut are analyzed below A crcut of the trantor amplfer that cont of three trantor cell hown n Fg 7 The Eber- Moll tatc model of the trantor ha been ued Fg 8 Crcut topology for three-cell trantor amplfer Fg 6 Four-node crcut topology The phae pace of the total tate parameter ha nne dmenon The eparate lne are tranformed to the eparate hyper-urface n th cae The phae projecton of the eparate hyper-urface (eparate lne one and two), whch correpond to the plane 5-9 are hown n Fg 7 The one, two and three trantor cell crcut were analyzed eparately The one trantor cell crcut wa analyzed a the frt eample In th cae we have three ndependent varable y, y, y3 a admttance (K=3) and three dependent varable V, V, V3 a nodal voltage (M=3) The tate parameter vector X nclude component: = y, = y, 3 = y3, 4 = V, 5 = V, 6 = V 3 Fg 9 correpond to the trajectory graph of the modfed tradtonal degn trategy for three above mentoned type of the trantor amplfer

5 (a) =, =,,,K (K=3) (b) =, =,,,K (K=3) (c) =, =,,,K (K=5) (d) =3, =,,,K (K=5) (e) =, =,,,K (K=7) (f) =, =,,,K (K=7) Fg 9 Famly of the curve that correpond to the modfed tradtonal degn trategy and eparate lne for: (a), (b) one-cell; (c), (d) two-cell; and (e), (f) three-cell trantor amplfer

6 Fg 9 (a), (b) how the behavor of the trajectory projecton n the plane 3-6 Fg 9 (a) correpond to the ntal coordnate value =, and Fg 9 (b) to the value = for =,,3 There a great dfference between the actve and the pave crcut The eparate lne and (the projecton of the correpondng eparate hyper urface) have a very trong confguraton for =, that eplan the preence or the abence of the acceleraton effect On the contrary, the eparate hyper urface projecton dappear n the plane 3-6 for the ntal value = It mean that the acceleraton effect oberved alway, for any value of the coordnate 6 becaue all trajectore nclude the poblty to fnh pont jump It very nteretng that the crcut complcaton brng to the further epanon of the acceleraton effect regon We can ee th property from Fg 9 (c), (d) and (e), (f) Fg 9 (c), (d) correpond to the two-cell trantor amplfer and Fg 9 (e), (f) to the three cell amplfer There a gnfcant reducton of the regon of the acceleraton effect abence for two cell amplfer, Fg 9 (c) The projecton of the eparate hyper urface (eparate lne and ) n the plane 5 - have the ame behavor and very narrow regon of the acceleraton effect abence for =, =,,3,4,5 The acceleraton effect alway et for =3 a we can ee n Fg 9 (d) The eparate hyper urface dappear completely for three cell trantor amplfer (Fg 9 (e), (f)) and we can realze acceleraton effect practcally for all tart pont and for all trajectore 4 Concluon The ntal pont electon permt obtan acceleraton effect wth a great probablty The trajectory analy of varou degn tratege how that the concepton of the eparate lne or the eparate hyper urface n general cae very helpful to undertand and defne the neceary and uffcent condton for the degn proce acceleraton effect etence The eparate hyper urface defne the tart pont and the trajectore that can produce the acceleraton effect and can be ued for the optmal degn trajectory contructon The electon of the ntal pont outde of the eparate hyper urface the neceary and uffcent condton for the acceleraton effect etence The eparate hyper urface ha the comple tructure n general cae However, the tuaton mplfed for the actve nonlnear crcut becaue a dappearance of the eparate hyper urface for more complcated crcut It mean that the acceleraton effect can be realzed alway for the comple actve crcut Th effect reduce the total computer tme addtonally and erve a the ba for the optmal or qua-optmal algorthm contructon Acknowledgment Th work wa upported by the Unverdad Autónoma de Puebla, under project VIEPIII5G Reference: [] JR Bunch and DJ Roe, (Ed), Spare Matr Computaton, Acad Pre, NY, 976 [] O Oterby and Z Zlatev, Drect Method for Spare Matrce, Sprnger-Verlag, NY, 983 [3] A George, On Block Elmnaton for Spare Lnear Sytem, SIAM J Numer Anal Vol, No3, 984, pp [4] FF Wu, Soluton of Large-Scale Network by Tearng, IEEE Tran Crcut Syt, Vol CAS- 3, No, 976, pp [5] A Sangovann-Vncentell, LK Chen and LO Chua, An Effcent Cluter Algorthm for Tearng Large-Scale Network, IEEE Tran Crcut Syt, Vol CAS-4, No, 977, pp [6] A Zemlak, One Approach to Analog Sytem Degn Problem Formulaton, Proc IEEE Int Sym on Qualty Electronc Degn ISQED, San Joe, CA, USA, March, pp [7] A Zemlak, Sytem Degn Problem Formulaton by Control Theory, Proc IEEE Int Sym on Crcut and Sytem ISCAS, Sydney, Autrala,,Vol 5, pp 5-8 [8] AM Zemlak, Analog Sytem Degn Problem Formulaton by Optmum Control Theory, IEICE Tran on Fundamental of Electronc, Communcaton and Computer, Vol E84-A, No 8,, pp 9-4 [9] AM Zemlak, Acceleraton Effect of Sytem Degn Proce, IEICE Tran on Fundamental of Electronc, Communcaton and Computer, Vol E85-A, No 7,, pp