Optimized Second-Order Dynamical Systems and Their RLC Circuit Models with PWL Controlled Sources
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1 ADIOENGINEEING VO. NO. SEPTEMBE Optimized Second-Ode Dynmicl Systems nd Thei icit Models with PW ontolled Soces Jiří POSPÍŠI Zdeně KOKA Stnislv HANUS Jiří PETŽEA Jomí BZOBOHATÝ Dept. of dio Electonics Bno Univesity of Technology Pyňov 8 6 Bno zech eplic Dept. of Micoelectonics Bno Univesity of Technology Údolní 5 6 Bno zech eplic pospisil@feec.vt.cz zooh@feec.vt.cz Astct. omplementy ctive cicit models with voltge-contolled voltge soce (VVS nd centcontolled cent soce (S fo the second-ode tonomos dynmicl system eliztion e poposed. The min dvntge of these eqivlent cicits is the simple eltion etween the stte model pmetes nd thei coesponding cicit pmetes which leds lso to simple design fomls. Keywods Dynmicl systems second-ode systems stte models ctive eqivlent cicits PW contolled soces optimized design fomls.. Intodction Atonomos piecewise-line (PW systems of clss cn e descied y the genel stte mti fom [] [] & A h ( w T ; ( the nomlized elementy PW feedc fnction (Fig. T T T ( w ( w w h ( contins the egions D nd D (D -. The dynmicl ehvio of the system is detemined y two chcteistic polynomils elted to these individl egions []. All the systems of lss hving the sme chcteistic polynomils e qlittively eqivlent nd they e elted y line topologicl conjgcy []. Typicl systems of this clss e the h s model oth its cnonicl foms [] nd lso the ecently deived optimized stte model hving the minimm sm of eltive eigenvle sensitivity sqes with espect to chnge of the individl stte mti pmetes [7]. Jst this low-sensitivity model is vey sefl s pototype fo the pcticl chotic system eliztion in fom of electonic cicit. It povides the possiility to tilize loc-decomposed fom of the stte mti so tht the design pocede cn e stted fom the optimized second-ode system nd then etended y simple wy to the optimized highe-ode cse [7] [9]. The stte model cn e sed s mthemticl tool fo the nmeicl simltion of dynmicl system ehvio s well s pototype fo the electonic cicit eliztion sing ville cicit techniqe. Fom the complete stte eqtions eithe the genel integto-sed cicit locdigm (typicl fo oth cnonicl foms o the coesponding ctive cicit (typicl fo h s oscillto cn diectly e deived. In oth cses only single PW netwo element is sed tilizing vios types of ctive electonic locs opeting in oth voltge nd cent modes (op-mps cent conveyos tns-impednce mplifies etc.. Fo the optimized low-sensitivity model fist the coesponding integto-sed loc digm hs een deived fo oth the second- nd the thid-ode cses [9]. Intention of the ppe is to popose the coesponding ctive cicits whee nlie the h s model the cicit pmetes hve diect eltions to the model pmetes. h(w T D - D D w T Fig.. Simple memoyless PW feedc fnction.. Second-Ode Stte Models with Optimized Eigenvle Sensitivities The most feqently occing tonomos dynmicl systems hve thei comple conjgte eigenvles in oth egions of PW fnction (Fig. i.e. fo the inne egion (D it is (µ µ ± jµ nd fo the ote egions (D - D it is ( ± j. Then the ssocited chcteistic polynomils e defined s follows (D : P( s ( s µ ( ( s µ det ( s A (D - D : Q( s ( s ( s det ( s A (
2 J. POSPÍŠII Z. KOKA S. HANUS J. PETŽEA J. BZOBOHATÝ OPTIMIZED SEOND-ODE DYNAMIA SYSTEMS whee eltion etween stte mtices cn e epessed [] A w T A ( nd is the nity mti. The optimized low-sensitivity stte model ( hve een chosen in the simplified nd decomposed comple fom [7] in which the coesponding stte mtices e µ µ K A A (5 µ K µ nd the optimizing coefficient K is given s the el oot of the qdtic eqtion K K( M i. e. K M ± M ( M whee the iliy pmete M is ( µ '- ' ( µ "- " M > ( µ ". µ " " In the vectos [ ] T nd w [ w w ] T one of the pmetes cn e chosen e.g. w while the othes e les nd one el eigenvle in oth ote nd inne egions (i.e. ± j - el; µ µ ± jµ µ - el. Then the stte mti nd the vectos hve the fom ' " A ' ( w w whee ( µ ( nd pmetes nd w e given y (6. Sstitting into ( we cn esily deive tht the stte mti ssocited with inne egion hs the lowe loc-tingl fom µ ' µ " K - A µ " K µ ( ' w µ so tht sch model hs vey low eigenvle sensitivities oth in ote nd inne egions of the PW feedc fnction. The complete stte eqtions of the optimized thidode PW tonomos system cn e then ewitten into otined s [9] & ' [ h ( w ] ( µ ' ' µ " K µ' [ h ( w ] µ ' ' " µ " K w µ ' '. (6c Then the complete stte eqtions of the optimized secondode PW tonomos system cn e witten s & h( w y y µ h w (7 [ ] ( y ( w y y& y h (8 whee the pmetes nd w e given y the fomls (6c. The coesponding integto-sed cicit loc digm sitle lso s pototype fo pcticl eliztion is shown in [7]. All the sensitivity fnctions e otined in comple fom so tht lso the sensitivities epessed septely fo the eigenvle el nd imginy pts cn esily e deived. Then the minimm sms of eltive eigenvle sensitivity sqes with espect to the chnge of the individl stte mti pmetes cn e epessed fo oth the el nd imginy pts genelly s S ( ( λ ij S λ ij (9 whee in the ote egions (D - D λ λ nd in the inne egion (D λ µ λ µ [8].. Thid-Ode Stte Models Utilizing the eslts fo the second-ode systems the thid-ode model with ppe loc-tingl stte mti contining comple decomposed second-ode smti cn e deived []. Sppose one pi of the comple conjgte eigenv [ h ( w ] ( & ' ( & µ [ h ( w ] h ( w ( whee the sic individl pmetes e septed. Howeve pmetes nd w e given y moe comple fomls (6c t the finl eqied effect i.e. the minimm eigenvle sensitivities hs een chieved y this.. Active icit Models with PW ontolled Soces To otin genel eslts in cicit model synthesis the second-ode system descied y the genel stte mti eqtion ( is consideed i.e. w A (5 y w w which evidently incldes lso the optimized stte model intodced in hpte. In the net pt two cicit models contining voltge- nd cent-contolled soces nd possessing simple design fomls e shown.. icit Model Utilizing VVS onside the tonomos cicit intodced in Fig. contining voltge-contolled voltge soce
3 ADIOENGINEEING VO. NO. SEPTEMBE 5 (VVS with PW tnsfe chcteistic fnction f( i hving thee segments (Fig. epessed s A i A A h. (6 ( ( ( i hoosing the cpcito voltge nd the indcto cent i s the stte viles oth Kichhoff s eqtions of this cicit cn e witten in the sic fom d i (7 E y i E t τ. (9c Then the coesponding nomlized cpcitnce indctnce nd ll esistnces e g g. ( di i i ( (7 Denoting sgn( the stte eqtions (8 cn e ewitten into the nomlized foms nd then ewitten to the complete (non-nomlized stte eqtion fom i.e. & [( A g g] [ A g ] y ( d A G ( G G A G i ( A A g h( y (8 A A h( i y& ( A [( A ] y ( di A A ( ( A A h( y. i. (8 A A omping them with the genel mti fom (5 the h( i following eqtions cn e otined di A d t A A d i d t i Fig.. Second-ode tonomos cicit with PW voltge soce. A -E A E i A D- D D i Fig.. Tnsfe PW chcteistic of VVS. Utilizing the efeence vles of voltge E (Fig. esistnce nd cpcitnce the nomlized stte viles inclding the time scling cn e given s [( A g g] ( A g ( A A g (cd w [( A ] ( A A w (cd ( A nd then tilized s independent fomls fo designing the individl cicit pmetes. Fo the cse when nd e chosen s fee pmetes the eslts e smmized in the following design fomls whee oth the genel nd optimized stte models e consideed. g w g µ K µ ( w ( w ( µ ( µ K µ K µ µ K µ A
4 6 J. POSPÍŠII Z. KOKA S. HANUS J. PETŽEA J. BZOBOHATÝ OPTIMIZED SEOND-ODE DYNAMIA SYSTEMS A ( ( µ µ K whee the pmete is genelly given s w. ( w. icit Model Utilizing S onside the tonomos cicit (Fig. contining cent-contolled cent soce (S with the PW tnsfe chcteistic fnction i f(i G hving thee segments (Fig. 5 epessed s ( i G ( B B h( i G i ( B di B d t B B d d t i i G Fig.. Atonomos nd ode cicit with PW cent soce. B -I i B I i B D- D D i G Fig. 5. Tnsfe PW chcteistic of S. hoosing the indcto cent i nd the cpcito voltge s the stte viles oth Kichhoff s eqtions fo this cicit cn e witten in the sic fom di i ( i i (5 d i i (5 nd then ewitten into the complete (non-nomlized stte eqtion fom i.e. d d ( i B B G B B i h t G ( i G d ( G G B BG B B i h( i G. (6 Utilizing the efeence vles of voltge E (in Fig. 5 I E/ esistnce nd cpcitnce the nomlized stte viles inclding the time scling cn e defined s i t y τ. (7c E E Then the coesponding nomlized indctnce cpcitnce nd ll esistnces e g. (8 g Denoting sgn( the stte eqtions (6 cn e ewitten into the nomlized foms & y& B [( B ] ( B g ( B B h( g y ( [( B g g ] ( B B h( g y y (9 y. (9 omping them with the genel mti fom ( fo the second-ode system the following eqtions e otined [( B ] ( B ( B B g w (cd [( B g ] ( B g ( B B w (cd g nd then tilized s independent fomls fo the design of the individl cicit pmetes. Fo the cse when nd e chosen s fee pmetes the eslts e smmized in the following design fomls whee gin oth the genel nd optimized stte models e consideed. ( µ K µ g w ( w ( µ K µ µ K µ
5 ADIOENGINEEING VO. NO. SEPTEMBE 7 g µ K ( w µ B B ( ( µ µ K nd whee the pmete is genelly given s. ( w w Any othe detils ot the eliztion conditions of the individl cicit elements fo oth cicit models e pesented in []. The coesponding cicit models cn esily e developed fom cicits shown in Fig. nd Fig. nd then sed s the pototypes fo the pcticl eliztion of the optimized chotic oscillto. 5. onclsion This contition dels with the second-ode nonline dynmicl systems nd thei eliztions sing ctive cicits in which the VVS nd S with theesegment PW symmetic tnsfe chcteistics e consideed s the ctive elements. This is sitle especilly fo voltge- nd cent-mode eliztions. The dynmicl ehvio of sch system is detemined y two sets of comple conjgte stte mti eigenvles ssocited with the coesponding egions. The contition pesents the complete nd nomlized stte eqtions in which the simple eltion etween the model nd the cicit pmetes entils lso vey simple design fomls in the synthesis pocede eithe in genel o optimized (low eigenvle sensitivities foms. icits poposed epesent one possiility of the secondode system eliztion nd cn e esily etended lso fo the thid-ode system tilizing the loc decomposition of the stte mti [7]. Sch highe-ode eqivlent cicit cn model lso chotic ehvio of the system. Acnowledgements This esech is ptilly sppoted y the zech Gnt Agency gnt pojects no. ///A nd no. //69. It epesents the pts of the esech Pogms of the zech Ministy of Edction EZ: J/98: 6 nd EZ: J/98: 6. efeences [] POSPÍŠI J. KOKA Z. HOSKÁ J. Synthesis of optimized piecewise-line system sing simility tnsfomtion pt II: second ode systems. dioengineeing. vol. no. p. 8. [] POSPÍŠI J. BZOBOHATÝ J. Elementy cnonicl stte models of h s cicit fmily. IEEE Tns. ic. Syst.-I: Fndmentls. 996 vol. no. 8 p [] POSPÍŠI J. BZOBOHATÝ J. KOKA Z. HOSKÁ J. Simplest ODE eqivlents of h s eqtions. Inten. Jonl of Bifction & hos. vol. no. p.. [] WU. W. HUA. O. On line topologicl conjgcy of 'e systems. IEEE Tns. ic. Syst. - I: Fndmentls. 996 vol. no. p [5] KOKA Z. Using simility tnsfomtion fo nonline system synthesis. In Poc. dioeletoni. Bno p [6] POSPÍŠI J. BZOBOHATÝ J. KOKA Z. HOSKÁ J. DO- STÁ T. Dynmicl systems with low eigenvle sensitivities. In Poc. MI Innsc (Asti p [7] POSPÍŠI J. BZOBOHATÝ J. KOKA Z. HANUS S. DO- STÁ T. Optimized highe-ode dynmicl systems. In Poc. MI' Innsc (Asti vol. I p [8] HANUS S. eliztion of thid-ode chotic systems sing thei elementy cnonicl stte models. In Poc. dioeletoni 97 Btislv (Slovi 997 p. 5. [9] POSPÍŠI J. BZOBOHATÝ J. KOKA Z. HANUS S. MI- HÁEK V. Optimized stte model of piecewise-line dynmicl systems. dioengineeing. vol. no. p [] MIT J. Optimiztion of Dynmicl System Eigenvle Sensitivities Using Its Pmete Modifiction. (In zech. Diplom poject. Inst. of dio Electonics BUT Bno. Aot Athos... Jiří POSPÍŠI fo iogphy see Apil isse pge 9. Zdeně KOKA fo iogphy see Apil isse pge 9. Jomí BZOBOHATÝ fo iogphy see Apil isse pge 9. Stnislv HANUS fo iogphy see Deceme isse pge. Jiří PETŽEA fo iogphy see Jne isse pge 5.
Active RLC Circuit Model of the Second-order Dynamical System with Piecewise-linear Controlled Sources
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