Positive electrical circuits with zero transfer matrices and their discretization

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1 omptr ppcato Ectrca Egrg Vo. 6 DO.8/j Potv ctrca crct wt zro trafr matrc a tr crtzato Taz Kaczork Baytok Uvrty of Tcoogy 5 5 Baytok. Wjka 5D ma: kaczork@.pw..p Potv coto tm a crt tm ar ctrca crct wt zro trafr matrc ar ar. t ow tat tr xt a arg ca of potv ctrca crct wt zro trafr matrc. T ctrca crct ar racab obrvab a tab for a va of t rtac ctac a capactac. T crt tm ar potv ctrca crct ar troc. t ow tat: t crt tm ctrca crct aymptotcay tab for a va of t crtzato tp f a oy f t corrpog coto tm ctrca crct aymptotcay tab; t crtzato of t coto tm ctrca crct o ot cag tr racabty obrvabty a trafr matrc. KEYWODS: potv ctrca crct racabty obrvabty tabty crtzato trafr matrx. trocto yamca ytm ca potv f t trajctory tartg from ay ogatv ta tat rma forvr t potv ortat for a ogatv pt. ovrvw of tat of t art potv ytm tory gv t moograp [5 ]. Varty of mo avg potv bavor ca b fo grg coomc oca cc boogy a mc tc. T oto of cotroabty a obrvabty a t compoto of ar ytm av b troc by Kama [5 6]. T oto ar t bac cocpt of t mor cotro tory [ 7]. Ty av b ao xt to potv ar ytm [5 ]. T compoto of t par ( B a ( of t potv crt tm ar ytm a b ar [9]. T potv crct a tr racabty a b vtgat [ ] a cotroabty a obrvabty of ctrca crct [8 ]. T racabty of ar ytm coy rat to t cotroabty of t ytm. Spcay for potv ar ytm t coo for t cotroabty ar mc trogr ta for t racabty [ ]. Tt for t racabty a cotroabty of taar a potv ar ytm ar gv [ 7 ]. T potvty a racabty of fractoa coto tm ar ytm a ctrca crct av b ar [ 5 9 ].

2 T. Kaczork / Potv ctrca crct wt zro trafr matrc a tr Dcopg zro of potv crt tm ar ytm av b troc []. Stabty of fractoa ar D crt tm a coto tm ytm a b vtgat t papr [ ] a of D fractoa potv ar ytm [6]. T oto of practca tabty of potv fractoa crt tm ar ytm a b troc [6] a t potv ar ytm cotg of bytm wt ffrt fractoa orr a b aayz []. Som rct trtg rt t fractoa ytm tory a t appcato ca b fo [ ]. T racabty a obrvabty of fractoa potv coto tm ar ytm av b ar [8] a cotrctabty a obrvabty of taar a potv ctrca crct [7]. t papr t potv coto tm a crt tm ctrca crct wt zro trafr matrc w b ar. T papr orgaz a foow. cto t bac fto a torm cocrg t potvty racabty a obrvabty of ctrca crct ar rca. Potv ctrca crct wt zro trafr matrc ar prt cto. Potv crt tm ctrca crct ar aay cto. T racabty obrvabty a trafr matrc of crt tm ar ytm ar ar cto 5. ocg rmark ar gv cto 6. m T foowg otato w b : t t of ra mbr m rprt t t of m ra matrc ot t t of m matrc wt ogatv a M ta for t t of Mtzr matrc (ra matrc wt ogatv off agoa tr t tty matrx.. Potvty racabty a obrvabty of ctrca crct or ar ctrca crct compo of rtor capactor co a votag (crrt orc. t tat varab (t compot of t tat vctor x (t w coo t votag o t capactor a t crrt t co. Ug Krcoff aw w may crb t ar crct trat tat by t tat qato x ( t x( t B( t (a y ( t x( t (b wr a ( x t m ( t m B p ( ar t tat pt a otpt vctor y t p.

3 T. Kaczork / Potv ctrca crct wt zro trafr matrc a tr Dfto. [ ] T ar ctrca crct ( ca (tray p potv f t tat vctor x ( t a otpt vctor y ( t t for ay m ta coo x ( a a pt ( t t. Torm. [5 ] T ar ctrca crct potv f a oy f m p M B. ( Dfto. [5 ] T potv ctrca crct ( ca racab tm t t ] f for vry gv fa tat tr xt a pt t [ f m ( t [ t f ] x f wc tr t tat of t ctrca crct from zro ta coo x( to t fa tat x f. Dfto. [] matrx ca mooma f ac t row a ac t com cota oy o potv try a t rmag tr ar zro. Torm. [ ] T potv ctrca crct ( racab f a oy f t racabty matrx [ B B B] m ( cota a mooma matrx. Dfto. [ ] T potv ctrca crct ( ca obrvab tm t t ] f kowg t pt ( t m a t pt y ( t p for [ f t [ t f ] t pob to f t q ta coo x x(. Torm. [ ] T potv ctrca crct ( obrvab tm t [ t f ] f a oy f t matrx M agoa a t matrx O cota a mooma matrx. T trafr matrx of t potv ctrca crct ( gv by T( [ ] B p m ( (5 wr p m ( t t of p m ratoa matrc. Torm. [ ] f t par (B of t taar ctrca crct ( ot racab t om po zro cacato occr aj[ ] B. (6 t[ ] f t par ( of t taar ctrca crct ( ot obrvab t om po zro cacato occr p (

4 T. Kaczork / Potv ctrca crct wt zro trafr matrc a tr aj[ ] (7 t[ ] wr aj[ ] ot t ajot matrx of [ ]. Torm 5. f for t taar ctrca crct ( T( [ ] B (8 t O (9 wr O a ar f by ( a ( rpctvy. Proof. Proof gv []. Torm 6. t for t taar ctrca crct ( t coo (8 b atf. T t par (B racab f t par ( obrvab f B. Proof. Proof gv [].. ar ctrca crct wt zro trafr matrc Examp. or t ctrca crct ow Fg. wt gv rtac ctac capactac a votag orc. Fg.. Ectrca crct of Examp Ug Krcoff aw w may wrt t qato (. t otpt y w coo y. ( T qato ( a ( ca b rwrtt t form

5 T. Kaczork / Potv ctrca crct wt zro trafr matrc a tr B y (a wr B [ ]. (b By Torm t ctrca crct potv for a va of a c from (b w av M B. ( T trafr fcto of t ctrca crct T ( [ ] [ ] B ( for a va of a. Not tat t[ ] (5 a t ctrca crct tab for a va of a. By Torm a t potv ctrca crct wt (b racab a obrvab c t matrc [ B B] O (6 av oy o mooma com a o mooma row rpctvy. From (6 w av O. (7 gra ca t ca of potv ctrca crct wt zro trafr matrx ca b prt t form ow Fg.. Torm 7. T ca of ctrca crct ow Fg. ar potv ctrca crct wt zro trafr fcto f a oy f tr commo part ar potv ctrca crct. xamp of potv ctrca crct wt zro trafr matrx for gv t foowg xamp. 5

6 T. Kaczork / Potv ctrca crct wt zro trafr matrc a tr Examp. or t ctrca crct ow Fg. wt gv rtac k k... 9 ctac... capactac a votag orc j j. Fg.. Potv ctrca crct wt zro trafr matrx Fg.. Potv ctrca crct Ug Krcoff aw w may wrt t qato 6

7 T. Kaczork / Potv ctrca crct wt zro trafr matrc a tr 7. ( ( (8a t otpt w coo y. (8b T qato (8 ca b wrtt t form y B (9a wr

8 T. Kaczork / Potv ctrca crct wt zro trafr matrc a tr 8. B (9b Not tat t ctrca crct potv for a va of t rtac ctac a capactac c 6 M 6 B a 6. t ay to cck tat t trafr matrx of t potv ctrca crct ] [ ( 6 B T (

9 T. Kaczork / Potv ctrca crct wt zro trafr matrc a tr a t crct tab for a va of t rtac ctac a capactac c t matrx a two zro row. gra ca w av t foowg coco. oco. T mbr of zro row of t matrx qa to t mbr of otpt of t ytm.. Dcrtzato a potv crt tm ctrca crct or t ar ctrca crct crb by (. ppyg t approxmato x( t x( t x x x( t Z {...} ( to ( w obta t corrpog crt tm ctrca crct crb by t qato x x B (a x( t y x (b wr x x( t x( x (t y y(t a B B. (c Dfto 5. [] T crt tm ctrca crct ( ca (tray p potv f x y Z for ay ta coo a a m Z. Torm 8. T crt tm ctrca crct ( potv f a oy f m p B. ( Proof. Proof gv [ ]. Torm 9. T crt tm ctrca crct ( potv f a oy f t coto tm ctrca crct ( potv a ( max a wr a... ar t agoa tr of t matrx M. Proof. From (c t foow tat f a oy f M coo ( atf. t w kow tat t gva z... of t matrx rat wt t gva... of t matrx by z... x a t. (5 Torm. T crt tm ctrca crct ( aymptotcay tab for f a oy f t coto tm ctrca crct ( aymptotcay tab. ar 9

10 T. Kaczork / Potv ctrca crct wt zro trafr matrc a tr Proof. t j... t t crt tm ctrca crct ( aymptotcay tab f a oy f or a z ( ( ( ( (6 ( (7.... (8 From (8 t foow tat f a oy f..... t ctrca crct ( aymptotcay tab. 5. acabty obrvabty a trafr matrc of crt tm ctrca crct t cto t w b ow tat t crtzato of t coto tm ctrca crct o ot cag tr racabty a obrvabty. Dfto 6. [] T crt tm ctrca crct ( ca racab q tp f for vry gv fa tat... q m x f tr xt a pt qc wc tr t tat from zro ta tat x to t fa tat x f. Torm. T crt tm ctrca crct ( racab q tp f a oy f q rak rak[ B B B. (9 q ] Proof. Proof mar to t proof gv []. Dfto 7. [] T crt tm ctrca crct ( ca obrvab q tp f kowg t pt qc... q a otpt qc y y... yq t pob to f t q ta coo x. Torm. T crt tm ctrca crct ( obrvab q tp f a oy f rak O rak. ( q Proof. Proof gv []. Torm. T crt tm ctrca crct ( racab q tp f a oy f t coto tm ctrca crct ( racab q tp. q

11 T. Kaczork / Potv ctrca crct wt zro trafr matrc a tr Proof. By Torm t crt tm ctrca crct ( racab q tp f a oy f q rak[ B B B ]. ( Sbttto of (c to ( y rak[ B ( B ( q B] rak [ B B q B] ( q ( q rak[ B c t matrx B q B] ( q ( ogar for ay. Torm. T crt tm ctrca crct ( obrvab q tp f a oy f t coto tm ctrca crct ( obrvab q tp. Proof. Proof mar to t proof of Torm. Torm 5. T trafr matrx T (z of t crt tm ctrca crct ( zro f a oy f t trafr matrx T ( of t coto tm ctrca crct ( zro. Proof. T trafr matrx (z of ( gv by T T ( z [ z ] B. ( Sbtttg (c a (5 to ( w obta T ( z [ ] B [ ( ] B (5 [ ] B T(. Trfor T (z f a oy f T (. T corato ca b ay xt to t potv ctrca crct. 6. ocg rmark Potv coto tm a crt tm ar ytm wt zro trafr matrc av b ar. t a b ow tat tr xt a arg ca of potv ctrca crct wt zro trafr matrc (Torm 7. T ctrca crct ar racab obrvab a tab for a va of q

12 T. Kaczork / Potv ctrca crct wt zro trafr matrc a tr t rtac ctac a capactac (Torm 5 a 6. T crt tm ar potv ctrca crct av b troc a tr potvty a aymptotc tabty av b vtgat (Torm 9. t a b ow tat: t crt tm ctrca crct aymptotcay tab f a oy f t coto tm ctrca crct aymptotcay tab (Torm ; t crt tm ctrca crct racab (obrvab f a oy f t coto tm f a oy f t coto tm ctrca crct racab (obrvab (Torm a ; t trafr matrx of crt tm ctrca crct zro f a oy f t trafr matrx of coto tm ctrca crct zro (Torm 5. T corato ca b xt to t fractoa potv ctrca crct wt zro trafr matrc. frc [] tak E. Mc. ar Sytm Brkar Boto 6. [] tm fractoa orr ytm wt ay of t rtar typ B. Po. ca. Sc. Tc. vo. 56 o. pp [] orr tat pac ytm Jora of Vbrato a otro vo. o. 9 pp [] amtr tfcato ba o fractoa orr mo Proc. Eropa otro ofrc Bapt Hgary 9. [5] Fara. a S. Potv ar Sytm: Tory a ppcato J. Wy & So Nw York. [6] Kaczork T. ymptotc tabty of potv fractoa D ar ytm B. Po. ca. Sc. Tc. vo. 57 o. pp [7] Kaczork T. otrctabty a obrvabty of taar a potv ctrca crct Ectrca vw vo. 89 o. 7 pp. 6. [8] Kaczork T. otroabty a obrvabty of ar ctrca crct Ectrca vw vo. 87 o. 9a pp [9] Kaczork T. Dcompoto of t par (B a ( of t potv crt tm ar ytm rcv of otro Scc vo. o. pp. 6. [] Kaczork T. Dcopg zro of potv crt tm ar ytm rct a Sytm vo. pp. 8. [] Kaczork T. Fractoa potv coto tm ar ytm a tr racabty t. J. pp. Mat. ompt. Sc. vo. 8 o. pp [] Kaczork T. Potv D a D Sytm oo UK: Sprgr Vrag. [] Kaczork T. Potv ctrca crct a tr racabty rcv of Ectrca Egrg vo. 6 o. pp. 8.

13 T. Kaczork / Potv ctrca crct wt zro trafr matrc a tr [] Kaczork T. Potv ar ytm cotg of bytm wt ffrt fractoa orr EEE Tra. rct a Sytm vo. 58 o. 6 pp.. [5] Kaczork T. Potvty a racabty of fractoa ctrca crct cta Mcaca t tomatca vo. 5 o. pp. 5. [6] Kaczork T. Practca tabty of potv fractoa crt tm ar ytm B. Po. ca. Sc. Tc. vo. 56 o. pp [7] Kaczork T. acabty a cotroabty to zro tt for taar a potv fractoa crt tm ytm Jora Eropé Sytm tomaté JES vo. o. 6 8 pp [8] Kaczork T. acabty a obrvabty of fractoa potv coto tm ar ytm Proc. XV of. Sytm Mog a otro Spt. oz Poa. [9] Kaczork T. acabty a obrvabty of fractoa potv ctrca crct omptatoa Probm of Ectrca Egrg vo. o. pp [] Kaczork T. Sct Probm of Fractoa Sytm Tory Br Grmay: Sprgr Vrag. [] Kaczork T. Stabty of potv coto tm ytm wt ay B. Po. ca. Sc. Tc. vo. 57 o. pp [] Kaczork T. Staar a potv ctrca crct wt zro trafr matrc Poza Uvrty of Tcoogy camc Jora: Ectrca Egrg o. 85 pp [] Kaczork T. ogowk K. Fractoa ar Sytm a Ectrca rct St Sytm Dco a otro vo. Sprgr 5. [] Kaat T. ar ytm Prtc Ha Egwoo ff Nw York 98. [5] Kama. Matmatca crpto of ar ytm SM J. otro vo. o. pp [6] Kama. O t gra tory of cotro ytm Proc. Frt tr. ogr o tomatc otro oo UK: Bttrwort pp [7] obrock H. Stat pac a mtvarab tory Nw York US: J. Wy 97. (cv: rv: 8.. 6

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