MULTIDIMENSIONAL EXTENICS THEORY

Transcription

1 U.P.B. Sci. Bull., Series A, Vl. 75, Iss. 1, 213 ISSN MULTIDIMENSIONAL EXTENICS THEORY Ovidiu Ilie ŞANDRU 1, Luige VLǍDAREANU 2, Paul ŞCHIOPU 3, Victr VLǍDAREANU 4, Alexadra ŞANDRU 5 Î această lucrare vm extide teria Exteics de la cazul uidimesial cuscut î prezet la cazul -dimesial, > 1. Datritǎ imprtaţei pe care aceastǎ terie are î rezlvarea multr prbleme practice, extesia pe care realizǎm î acest articl ferǎ dmeiului aplicativ maximul de csisteţǎ tehicǎ pe care aceasta îl pate feri. This paper will exted Exteics Thery frm the e-dimesial case kw at preset t the -dimesial case, > 1. Due t the imprtace f this thery i slvig multiple practical prblems, the extesi preseted i this paper brigs a maximum f techical csistecy t the applicative field. Keywrds: Exteics thery, We idicatrs, dyamic systems with status idicatrs. 1. Itrducti Fr slvig certai paradxical prblems which are fte met i daily practice, Prf. Cai We develped a efficiet mathematical thery which he geerically amed Exteics, amely the sciece f extedig the meas f ivestigati used i classical mathematics 6. The results btaied up t the preset by Prf. Cai deal exclusively with prblems expressable by uidimesial mathematical mdels. Sice this restricts the area f applicability f a meas f ivestigati which eve uder these cditis prved t have widespread 1 Departmet f Mathematical Mdels ad Methds, Uiversity POLITEHNICA f Bucharest, 313 Splaiul Idepedeţei, 642 Bucharest, Rmaia, isadru@yah.cm. 2 Istitute f Slid Mechaics f Rmaia Academy, C-ti Mille 15, Bucharest 1, Rmaia, e- mail: luigiv@arexim.r. 3 Departmet f Electric Techlgy ad Reliability, Uiversity POLITEHNICA f Bucharest, Bd. Iuliu Maiu 1-3, Bucharest 6, Rmaia, schipu@tehfi.pub.r. 4 Departmet f Electric Techlgy ad Reliability, Uiversity POLITEHNICA f Bucharest, Bd. Iuliu Maiu 1-3, Bucharest 6, Rmaia, vladareauv@gmail.cm. 5 Departmet f Electric Techlgy ad Reliability, Uiversity POLITEHNICA f Bucharest, Bd. Iuliu Maiu 1-3, Bucharest 6, Rmaia, alexadra_sadru@yah.cm. 6 Fr further details this matter see papers [4, 9, 1, 11], where [9] represets the early wrk f prf. Cai We t which we shall refer expressly withi ur article ad [4, 1, 11] represet develpmets f the pits f view frmulated i [9] fllwed by umerus applicative examples particularly fr the egieerig field.

2 4 O. I. Şadru, L. Vlǎdareau, P. Şchipu, V. Vlǎdareau, A. Şadru applicatis 7 it is ly atural that e becmes ccered with its develpmet i a way that allws a geeralizati f the existig thery which wuld cmprise the practical prblems that eed t be mathematically expressed by multidimesial mdels 8. This paper attempts t d just that. As ca be see frm the ctet, the trasiti frm the uidimesial case t the multidimesial is either direct, as ituitive aspects d t fucti well withi the geeralizati prcess, r immediate, sice reachig the geeral cadre f the thery ecessitates a much mre sphisticated mathematical apparatus. 2. Maximal extesi f the ti f distace i Exteics Thery By d we dete the euclidea distace, i.e. 1/2 2 d( x, y) = ( yk xk), where x = ( x1 2 ) k = 1, x,.., x, ad y = ( y ) 1, y2,.., y represet tw pits frm. This thig allws us t csider the relati δ ( x, A) if d( x, y) =, y A which, as is kw, represets the distace frm pit x t the set A. I classical mathematics the ti f distace frm a give pit t a certai set f pits is sufficiet t express whether that pit belgs r t t the set csidered. I Exteics thery, the relati that ca exist betwee a pit x ad a set A is t be exteded, with the iteti f expressig mre tha the simple idea that x A r x A. I rder t btai this result we prpse replacig the idicatr δ defied earlier with the idicatr s defied as fllws: 7 See i this respect papers [1-7], [1-12]. 8 Preccupatis i this directi d t represet a tpic itrduced by us exclusively. The first iitiative f this kid belgs t prf. Freti Smaradache (see paper [8]) wh was able t develp prf Cai We s thery i tw particular directis: e referrig t the situati i which the sets f the csidered mathematical mdel have cetral symmetry ad the secd i which the prblems beig studied admit the s-called attracti pit priciple. Hwever, the thery preseted by us i this paper fudametally differs frm the apprach f prf. Smaradache ad the results that we btaied are i geeral, distict. Whe cmpared t the thery f prf. Smaradache, ur thery represets ather variat f geeralizig the thery f prf. Cai We, which leads t ther implicatis up the applicative field.

3 Multidimesial exteics thery 5 (, ), ( ) δ x A x A s ( xa, ) =, (1) δ x, A, x A where by A we deted the abslute cmplemet f A, i.e. A = \ A. We shall w shw that this ew idicatr keeps all the prperties f the idicatr ρ ( x, [ a, b] ) a+ b b a = x, (2) 2 2 itrduced by Cai We i [9] fr the particular case where x ad set A is a iterval f the real umbers set f the frm [ a, b ], with a< b, which it als geeralizes, i the sese that the restricti applied t ur idicatr, t the case studied by Prf. Cai, cicides with ρ. Ideed, the idicatr s verifies the prperties detailed belw: Prpsiti 1. Fr ay pit x ad ay set A, if x A, where with A is deted the iterir f the set A i tplgy iduced by the metric d fixed earlier the space s xa<,, ad reciprcally., the ( ) The prf f this setece results directly frm the defiiti f the idicatr s. Prpsiti 2. Fr ay pit ( ) x ad ay set A, we have x A s x, A >, where with A is ted the clsure f set A i tplgy iduced by metric d space. As earlier the prf f this setece results directly frm the defiiti f idicatr s. Prpsiti 3. Fr ay pit s xa, > s xb,, x. ( ) ( ) x ad ay sets A ad B i Prf. We eed t aalyze three distict cases., if A B the

4 6 O. I. Şadru, L. Vlǎdareau, P. Şchipu, V. Vlǎdareau, A. Şadru Case l. x A. Let x '' B s that δ ( x, B) = d( x, x'' ) ad x ' A s that the pits x, x', x '' be clliear, ad x ' be betwee x ad x '', see Figure 1 9. Uder δ x, A d x, x' < d x, x'' = δ x, B. It fllws these cditis ( ) ( ) ( ) ( ) s( x, A) > s ( x, B). Nte. The symbls A ad B used abve dete the budaries f sets A ad B respectively, meaig A = A\ A, respectively, B= B\ B. Case 2. x B\ A Fig. 1.. I this case we have δ ( x, A) δ ( x, B) ( x, A) > ( x, B) s s. See Fig. 2. >, r equivaletly, Fig Please te that bth figure 1 ad ther figures, t which we shall refer t hereiafter, d t exhaustively cver the multitude f all situatis evisaged by the demstrati (fr example sets A ad B must t ecessarily be f dmai type). The purpse f these figures is limited t ly prvidig a ituitive visual framewrk meat t help fix the ideas withi that demstrati.

5 Multidimesial exteics thery 7 Case 3. x B. Let x '' A s that d( x, x'' ) δ ( x, A) = ad x ' [ xx, ''] B, where [ x, x ''] represets the set f all pist f the straight lie determied by pits x ad x '' situated betwee x ad x ''. See Fig. 3. With this preamble we ca write δ ( x, A) = d( x, x'' ) > d( x, x' ) δ ( x, B), which meas s x, A > s x, B. ( ) ( ) Fig. 3. Prpsiti 4. I the particular case x, A [ a, b] idicatrs ρ ( x, A) ad s ( x, A) cicide. The prf t this affirmati results directly frm calculati. 3. Maximal extesi f the Cai We idicatr =, a, b, a < b, the With the help f the s idicatr we defied i the previus paragraph it is pssible t build a ew vital idicatr fr Exteics Thery, amely s( xa, ) S ( xab,, ) =, (3) s x, B s x, A ( ) ( ) defied fr ay x ad fr ay sets A ad B i fr which A B. This idicatr geeralizes the idicatr ρ ( xa, ) K( x, A, B) =, (4) ρ x, B ρ x, A ( ) ( )

6 8 O. I. Şadru, L. Vlǎdareau, P. Şchipu, V. Vlǎdareau, A. Şadru itrduced by Prf. Cai i [9] fr the case x, x, A = [ a, b ], B ( a, b) =, a, b, a, b, a< a < b < b. The idicatr S has all the prperties detailed belw. Prpsiti 5. Fr ay tw sets A ad B i fr which A B we have S x, AB, < 1, if x B 1 S xab,, <, if x B\ A S xab,,, if ( ) x A, ad reciprcally. ; ( ) ; ( ) Prf. Usig the prperties f the idicatr s detailed i the previus seteces it is easy t deduce that: I the first case where x B, we have s ( xa, ) > ad s ( xb, ) >. It s x, B + s x, A < s x, A. Takig it accut that fllws that ( ) ( ) ( ) ( xb, ) ( xa, ) < S ( xab,, ) < 1; I the secd case where x B\ A, we have s ( xb, ) ad ( ) Csequetly, ( x, B) + ( x, A) ( x, A) s( xb, ) s ( xa, ) <, we ca usually deduce that 1 S ( xab,, ) <. I the last case where x A, we have ( xa, ) s( xb, ) s ( xa, ) <, it ca be see immediately that S ( xab,, ). s s, (see Prpsiti 3) we btai the iequality s xa>,. s s s. Sice additially Prpsiti 6. I the particular case x, A [ a, b] ( a, b) < < <, the idicatrs K ( xab,, ) ad ( x, AB, ) a a b b s. Sice =, a, b, a, b, S cicide. Prf. I the hyptheses abve the idicatrs ρ ad s cicide (see Prpsiti 4). 4. Aplicati May f the state f the art techical istallatis such as thse which emit high itesity radiati brig abut areas which are dagerus fr humas. We shall assume that we wish t secure such a istallati thrugh a cetralized system usig electric sesrs which versees the dager areas ad depedig the gravity f the situati ca sed warig sigals fr users r eve stp the system. Fr this we shall te with X the area i the surrudig space

7 Multidimesial exteics thery 9 (mathematically mdeled thrugh 3 ) iside which the radiati is ver the admitted safety level ad with X, ( X X ) that area iside X i which the level f radiati is uacceptable (fatal) fr humas. The sesrs muted i the areas 3 \ X, X \ X ad X sed t the cetral mitrig ad ctrl uity the spatial crdiates f all perss implicated i the activity. The sftware applicati which iterprets the data received frm the sesrs must accmplish the fllwig fuctis: 1) Whe the spatial crdiates x = ( x1, x2 x3) f a user belg t the area 3 \ X, the istallati is allwed t fucti uimpeded; 2) Whe the spatial crdiates x = ( x1, x2 x3) f a user belg t the area X \ X, the verseer system must sed warig sigals; 3) Whe the spatial crdiates x = ( x1, x2 x3) f a user belg t the area X, the verseer system must stp the istallati. Prducig such a sftware applicati is much simplified by usig the idicatr S ( x, X, X) which was defied earlier i relati (3) i which A = X ad B X S x t, X, X < 1, fr ay =. Ideed, accrdig t Prpsiti 5, if ( a ( ) ) a A, where by A we have deted the set f emplyees servig the istallati which we refer t ad by xa ( t ) the spatial crdiates f the emplyee a at the mmet t, the mitrig ad ctrl system will t sed ut a warig sigal the istallati is left t fucti at rmal capacity; if S ( xa ( t), X, X) [ 1,) fr at least a a A, the the mitrig ad ctrl system seds ut warig sigals fr the istallati users, but the istallati is allwed t ctiue fuctiig; if S ( xa ( t), X, X) fr at least a a A, the the cetralized cmmad system stps the istallati ucditially. The prblem aalyzed previusly referred t the simple case f a techlgical istallati with ly e risk factr. The sluti preseted ca be exteded t the geeral case f istallatis ctaiig multiple risk factrs. T exemplify this we shall aalyze the case f a similar istallati, but with > 1 surces f radiati. The iteded gal is the same as i the case previusly studied: desigig a mitrig system which will war f apprach it dagerus areas ad if ecessary iterrupt the use f thse radiati surces which culd edager humas. I rder t fix the ideas we shall suppse X1, X2,.., X t be the dager areas distributed fr each f the surces f the istallati which emit radiati ad X1, X2,.., X ( X1 X1, X2 X 2,.., X X ) t be the areas i their immediate veciity f their surces, where the dager fr persel

8 1 O. I. Şadru, L. Vlǎdareau, P. Şchipu, V. Vlǎdareau, A. Şadru is maximum. Oce these zes are established, we may the csider, fr each f the alarms systems f the istallati, a idicatr ( ) s( xx, k ) ( xx, ) s( xx, ) S x, Xk, Xk =, k = 1,2,..,, (5) s k f the frm csidered earlier i the case f a e-surce istallati. The alarm systems are desiged t fucti accrdig t the same priciple as i the case f ly e alarm system, but idepedetly frm e ather, thus, if at a certai amut f time t all idicatrs S ( xa( t), Xk, Xk), k = 1, 2,..,, are strictly smaller tha 1 fr ay a A, where A is the set f persel, ad xa () t are the spatial crdiates f emplyee a at time t, the the techlgical prcess is left uimpeded; if at a certai time t there is a A fr which e r mre f the idicatrs S ( xa( t), Xk, Xk), k = 1,2,..,, have values betwee 1, iclusively, ad, exclusively, the the mitrig system r systems i that area will sed ut warig sigals; ad if at a certai time t there is a A fr which e r mre f the idicatrs S ( xa( t), Xk, Xk), k = 1,2,..,, have values greater r equal t the the mitrig ad ctrl system r systems i thse areas will autmatically stp this r these surces f radiati emissi. I the case f the system with mre degrees f freedm i regards t the pssibility f prducig a uwated icidet, situatis may arise which are yet mre cmplicated t mitr. Fr example, i a istallati with, > 1 risk factrs, if stppig a certai subsystem f the istallati is impssible (fr techical reass r thse relatig t disaster preveti) uless the etire istallati is stpped, the the mitrig ad ctrl systems f the istallati, which supervise (each f them) the surces f ptetial dager, will have t dispse f a relative idepedece f acti ly, a cetral uit fr ctrl ad mitrig beig required t sythesize the ifrmati frm the lcal mitrig systems. Mrever, this cetral uit must have the pwer t verride the lcal systems whe the situati requires a verall iterrupti f the istallati. I rder t realize such a mitrig system we wuld prpse that the lcal mitrig ad ctrl systems are equipped with a idicatr f the frm (5), ad the cetral uit with a tempral idicatr f the frm: () ( () ) k { a k k k, a } S t = sup S x t, X, X 1 A. Parameter t appearig abve desigates the mmet i time at which the mitrig is made.

9 Multidimesial exteics thery 11 The fuctiig priciple f the prtecti system: As lg as idicatr S() t is strictly egative, the lcal mitrig systems are allwed t act idepedetly (meaig they will sigal, wheever ecessary, the presece f persel i the mderate risk areas Xk \ Xk, k = 1,2,.., ). Hwever, whe a critical situati arises, where the idicatr S( t ) wuld have a value greater r equal t, the cetralized safety system wuld cmmad the iterrupti f the etire istallati. 5. Cmmets The applicable examples f the idicatrs ( x, X, X) ( xa() t X k Xk) k, a { } S ad S() t = = sup S,, 1 A respectively, give i the previus paragraph suggest the itrducti f a ew ccept i the thery f dyamic systems, that f dyamic system with status idicatr 1. By this ti a imprtat class f systems ca be delimitated withi dyamic systems, which, by the status idicatrs they are edwed with, ca beefit frm special methds t slve a great umber f specific issues, such as thse related t the real-time quality ctrl prcess regardig their w fuctiig. R E F E R E N C E S [1] He Bi, Zhu Xuefeg, Hybrid Extesi ad Adaptive Ctrl, Ctrl Thery & Applicatis,25,22 (2), [2] Zhi Che, Ygqua Yu, T Fid the Key Matter-Elemet Research f Extesi Detectig, It. Cf. Cmputer, Cmmuicati ad Ctrl Techlgies (CCCT), Flrida, USA, 23, 7. [3] Wg Chigchag,Che Jeyag, Adaptive Extesi Ctrller Desig fr Nliear Systems, Egieerig Sciece, 21,3(7), [4] Yag Chuya, Cai We, Extesi Egieerig, Sciece Press, Beijig, 22. [5] Yag Chuya, Li Weihua, Li Xiamei, Recet Research Prgress i Theries ad Methds fr the Itelliget Dispsal f Ctradictry Prblems, Jural f Guagdg Uiversity f Techlgy,211, 28 (1), [6] Gua Feg-Xu, Wag Ke-Ju, Study Extesi Ctrl Strategy f Pedulum System, Jural f Harbi Istitute f Techlgy, 26,38 (7), I a very abstract maer the ti f dyamic system edwed with status idicatr assumes, by defiiti, the existece f a set ( ), Σ I made up f a dyamic system Σ ad a idicatr I f the states which the system Σ passes thrugh durig its fuctiig.

10 12 O. I. Şadru, L. Vlǎdareau, P. Şchipu, V. Vlǎdareau, A. Şadru [7] Che Jeyag, Wg Chigchag, Extesi Ctrller Desig via Slidig Mde Ctrl, Egieerig Sciece, 21,3 (7), [8] Flreti Smaradache, Geeralizatis f the Distace ad Depedet Fucti i Exteics t 2D, 3D, ad -D, vixra.rg, [9] Cai We, Extesi Set ad N-Cmpatible Prblems, Advaces i Applied Mathematics ad Mechaics i Chia, Pekig: Iteratial Academic Publishers, 199,1-21. [1] Cai We, Extesi Thery ad Its Applicati, Chiese Sciece Bulleti,1999, 44 (17), [11] Cai We, Shi Yg, Exteics, its Sigificace i Sciece ad Prspects i Applicati, Jural Of Harbi Istitute f Techlgy, 26, 38 (7): [12] Zha Yawei, Study f Cceptual Desig f Extesi fr Mechaical Prducts, Egieerig Sciece,21, 18 (6),