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# MATHEMATICAL SCIENCES VIGRE SEMINAR CAAM 699

Rice University

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Empirical Interpolation Method EIM Saifon Chaturantabut Advisor Dr Sorensen CAAM699 Ice Unlversl Department of Computational and Applied Mathematics R ty September 26 2008 Outline 0 IntroductionMotivation 0 Motivation 0 Interpolation Method 9 Empirical Interpolation Method EIM 0 Algorithm 0 Remark Properties 9 Examples and Results 0 Ex ElM for a nonlinear function 0 Ex ElM for PDE with nonlinearity 0 Error and Computational Time l I 0 Consider nonaffine parameter dependence function 50 6 L Q7 where Q spatial domainand p e 9 parameter domain 0 Define fs E539 1M6 9 o 15 could be a lowdimensional space a 15 2 spanq1 qm with small m a variation of 15 in it e Q can be captured by a selected points 21 zm e 2 with small m a Goal Approximate s by g in the form 30 ZMXWWL with 1 3ZW 5ZIW i17m7m7 for all p e 9 Closedform for Bhtw m I l l I Coefficient Approximation Matrix Form 0 Given Basis 0X E q1X i i i qmX Interpolation Points 2 E 21HizmT Q quot 0 Let Coef cient 50 E 17 751710017 6 Rm Function atz 52 M E SZ1M i i i szmiT E R quot 1121 1221 Qmz1 02 07122 072zzz qu22 6 Emmy cm39zm qztzm qmtzm 3 Z 36sz W 02quotSZM ax w 0xoz sz M I l I EAS S 52 5mm JIM 2W Ill OUTPUT 3m 271 WUWZW 11 3m 0ltxgtltaltzgtgt4sltwgt u NTERPOLAT ON POMS 1 221 5 01 Wemb e EIM Empirical Interpolation Method ElMPatera 2004 EIM Algorithm Given a set of basis functions mm the set of m EIM interpolation points z 0 Setzi UMX o z 0 ForL 9 z1 argesssupxenlgixquot em 5111 q1zw1 2m Solve 4 from QL WZL UPLquot 5A2 Define rim am 70 mp Set zL erg ess Supxe rLX HZmT is constructed as follows Note KOHaL wi cwgt1 e W WW X em sum squnT 6 RH 1Lquot 21 4117 6 QL T 0 x m xx mm m Final Output Basis qh wqm EIM points 21 V V V 2m sa 0x cmx 33102 cwzm 5Xv M 1 mXv M 0X02 15zv EIM points and bases Ex Input KL 1 are eigenvectors of discrete Laplacian Dr Sorensen 0 Sem argcsssupXEQ Q x W 4m If 0121m111 0 ForL 2 m 0 Saws 4 Vrom 0L711L71pL71 L1L71 e Denna W em 7 aL wL 9 Se ZLargcssSupX Q rLX 9 Se Nome 14 my 0V 1 L e HM pmmsz1 zL71arethezeros m qL x and my mum m n nutd imam Whammy a va fllcurr l a WWW t Ewpdr v my a m Ddfllcurr l mm xal v mmun x m pdrlV wwl mama Mmnma ism mmmum o mmmwmx 1 a mmmmw l l Remarks on EIM o EIM constructs an approximation for nonaffine parameter dependence function by using a method of greedy selection Given linear independent set 1 m Then for L 1 m o rLzL 7 0 since 51 are linearly independent 0 3 atranstormation between Inpm b35152 w Emma m EIM basis 0 qm a a 7 qml where 5121 P1 P 91m r222 2 Pg m R r222 I P3 6 Rmxmy 1721 rmzm a spanqmqt span515t I l l l Remarks on EMcont o rLz0foralli1mL71 qjzi0for0 iltj L o QLzL qjz e RLXL is lower triangular with QLzLl 0 ifi ltj QLzL i1 ifij forij1L l0LZLrl S 1 ifi gt1 0 QLzL is invertible since detQLzL 1 0 The EIM steps are welldefined 0 The EIM points will not repeat c There exists a unique EIM approx 3m since Q quotz quot is invertible o The EIM is hierarchyzfor L 1mm 71 small Z17m72L g zhuszH qhwqu g Cihmym l If given span m1 g span 1 m we can extend EIM points ZLML 2LT1 U Zmwand bases m7 qLT1 thm1 I l l l Posteriori Error bound of EIM approx gm Suppose 5W1 is not in span51m5m and 5M 6 Wm i span5175m1c S i SMlM 6 9 Define 00 3 l5zm1 mzm1lv 5XW i MQm1x7 where zm1 and qm1x are the m 1 h EIM point and basis Then 50 7 mXv i xw 9 ll539v m39vll S 00 Note in general the assumption that s M e me1 does not hold and the error bound above may not be exact or applicable I l l I Ex EIM for a nonlinear function sX M 1 7 Xcos37mx 1e 1x wherex 6 711 and M e 17r Plot of Approximate Functions dim 10 with Exact Functions in black solid line x 1 15 yk 11713 13855 1 A 3141e A 05 1 i m o s 05 1 x 1 1 05 0 05 1 Ex EIM for a nonlinear functioncont sm 1 7xws37mx12 1xw mu mg 17 r 7 mm a i m aquot mmHula m mm Mm mum 15 m In m l I Ex Application of the EIM on PDEs with nonlinearities Ex Unsteady 1D Burgers39Equation 2 2 gm 0 7 vaixmx r 6 smx r o smx r y07 t m t 0 t2 0 yX70 JoX7 X 6 0 E 071 Discretized system Galerkin 0 Fullorder system dim N FE basis gmHi1 M v WW 7 MW 0 MX 1 EL WOWf 0 Reducedorder system dim k lt N reduced basis 425 1 Hif liq17 N371 0 W06 1 ELM941 Eg 425 f 1 can be constructed from POD I l l I Ex Application of the EIM on PDEs with nonlinearities cont Problem ol reduced system lrom direct POD Suppose dgtX IJXUK FE basis IJX E zJ1XHt llX7 where Reduced basis X E 4151X7 A A t 7 kX7 Projection E U17 t t t UK 6 RIVth Note r wholt0 f WXTSWXvtdX N M NVt UImamltxgt7sltwltxgtuwltrgtgtdx KgtltN Ngtlt1 Computational Complexity still depends on N WANT N t C N t quota kmltltNquot nde endentofN y Wm M p mgtlt1 C and FIG0 can be constructed from the EIM I l x l I Ex Application of the EIM on PDEs with nonlinearitiescont How EIM works 0 Approximation from EIM 5WXUki7f 2 3X7 t 0XOZquots27 f EM Basis 0X E 511X7 t t t qmX EM Interpolation Points 2 E 1 t t t 2m T o Approximation for the Nonlinear term Wt U w xTswxukvmdx v Q kgtltN Ngtlt1 u II XTQXdxgt Qzquot sz t 90 W mxi I l i Z V kx m precompuwd Plots of Numerical Solutions from EIM w h POD ba 5 smnMsmmwuamm1am1nn swmsmmmmm n 10quot 10quot w m g a a 20 30 60 50 so say wspwammm s wmmmamm m Figure EIMPOD ACCURACY vs COMPLEXITY Relame Errorsum uy mrymWuy mHamemnsnmlswms mm POD M cm W W Wm WM MODE Wequot m Em a w m amp n2 m m m E 39amp gm m 1H m s m s m mmawonmswn NumbemrPonass used Figure LEFT Error Eavg 27 RIGHT CPU time sec HJh wfJHX Proper Orthogonal Decomposition POD Saifon Chaturantabut Advisor Dr Sorensen C Department of Computational and Applied Mathematics Rice University September 5 2008 Outline 0 IntroductionMotivation 0 Low Rank Approximation and POD 0 Model Reduction for ODEs and PDEs 9 Proper Orthogonal DecompositionPOD 0 Introduction to POD 0 POD in Euclidean Space 0 POD in General Hilbert Space 9 Numerical Results 0 Solutions from Full and Reduced Systems of PDE 1D Burgers Equation Q Problem POD for PDEs with nonlinearities 0 Nonlinear Approximation 0 Error and Computational Time l l l I Problem in finite dimensional space 0 Given y1myn E R quot Letg spany1y2myn C R quot r rank 0 y1myn e R quot possible almost linearly dependent NOT a good basis for 9 0 Goal Find orthonormal basis vectors 4251 t t t qbk that best approximates 9 for given k lt r 0 Solution Use Low Rank Approximation 0 Form a matrix of known data Y Y1 Yn ER W T li ilrrr m r rrlrlylii hm Low Rank Approximation Singular Value Decomposition SVD Let Y 6 RM r rankY and k lt r Problem Low Rank Approximation m ntllyi Yllrz rankt7 k J Solution 57 Ukzkva with min error H Y7 M2 ELM 02 where Y UXVT is SVD of Y X diaga1ma e R X with 012022Ht20rgt0 Optimal orthonormal basis of rank k POD basis W 210 in ML UI39 L1 J l l source I 39 wnlfmm 39 39 quot r nm 50 Low Rank Approximation 4 J7 W aw source I 39 wnlfmm 39 39 quot r nm l I Model Reduction for ODEs o Fullorder system dim N 0 Eva Kym 90 New 0 Reducedorder system dim k lt N Let y uky for UK 6 RNXquot with orthonormal columns UIUK 6 RR Uk rVU KUkYf 90 UM1i 757370 UZKUk W UEQU UNUkyt w w 70 Kim gm Ulemo How to construct Uk Use POD I l l I Model Reduction for PDEs Ex Unsteady 1D Burgers39Equation 8 62 8 yX t2 ajXJ 7 vaTZJXJ a 7 o x e o1r2 0 HQ t y1 7 t 0 t2 0 yX70 yoX7 X 6 07117 Discretized system Galerkin 9 Fullorder system dim N FE basis gmHi1 Mh ym thym 7 tht o s mx 0 251 solmitt 0 Reducedorder system dim k lt N orthonormal basis 425 1 370 m 7 W0 o s mx 0 2L1 asmxmr How to construct 15L1 Use POD Proper Orthogonal DecompositionPOD a POD is a method forfinding a POD basis l POD basis 2 lowdimensional representation of 0 a largescale dynamical 2 systems eg signal 20 05 i 40 05 i analysis turbulent fluid flow POD basis PODbeSisw a large data set eg image Z 0 E SVD in Euclidean space 4 4 0 05 l 0 05 l o Extracts basis functions containing characteristics from the system of interest 0 Generally gives a good approximation with substantially lower dimension w pusswam m l l Definition of POD 0 Let X be Hilbert Spacele Complete Inner Product Space with inner product lt gt and norm ll ll lt gt 0 Given y1 t t t yn e X Define y E snapshoti Vi Let 9 E spany1y2myn C X 0 Given k g n POD generates a set of orthonormal basis of dimension k which minimizes the error from approximating the snapshots POD basis E Optimal solution of n Z 7 jAsz SI abi 57 l j1 where 9X Ef1ltJj igt ix an approximation of y using ML 0 Solve by SVD I l Derivation for POD in Euclidean Space Eg X Rm minmgi EL My 7 Ei1yT i iii 2 Ewmm sit T u 1ifi H i 1761170 ing l 7717m Lagrange Function X k L 17w k7117 puykk E 17w7 k ZAijqbiTqui ij ij1 KKT Necessary Conditions n T i Zj1JjJj i Alva 3 L 0 gtAijom j lt15 415139 57 l I Optimal Condition Symmetric mbym eigenvalue problem WT i Ai i J i i whereAiY y1 yn for1mk i i Error for POD basis n k r ZiiJk ZJqT i iii 2 AI 1 i1 ik1 o Ai i YYTasI EF1JjJjT i A 2 00741502 0 Since 425qu 61 and spanY spanq51mq5 from SVD then y Ef1yIT qb for 1mn and n k n r r Z in 7 ZOOWIWHS Z Z JoW502 Z i 1 i1 1 ik1 ik1 I l l l SOLUTION for POD basis in R quot 0 Recall Optimality Condition and the POD Error YYTqbiq5i1mk 211 lyi 7 EL JjTa5i ll ELM A 0 Optimal solution POD basis 71 uni Lagrange multiplier A a 39 Wk can be obtained by the SVD of Y e Rm or EVD of YYT e Rm ie Y UXVT YYTuia2ui1mr where X diag a1ma e R with 012 02 2 2 a gt 0 u1tttu e Rm and V V1mv e Rn have orthonormal columns a This is equivalent to SVD solution for Low Rank Approximation I l l I SOLUTION for POD basis in a Hilbert Space X Two approaches 9 0 Define linear symmetric operator 9w Zf1ltwygty w e X Find eigenfunction u e X 9 0 Define linear symmetric operatorg gaygt16 RM Find Rm eigenvector V E cf YTYw 7214 I o Sol r oizvam of UK Wrzgi 0 NOTE w e R but um e X I l l I Remark Practical way for computing the POD basis for discretized PDEs 0 Let gmHi1 C X E H Q be finite eementFE basis 0 FE snapshotssoutions yj fe X at time QM71 N w nm Wwvi i1 5 Whygt1 Z Vik WWWgt1 WMVE RM kz1 where M ltg0 g0jgt e RNXN o POD basis n 1 45 2mm Ij1 where 2w m with v1 v2 t t t vk corresponding to 01 2 02 2 2 UK gt 0 I l l I Numerical Results for PDEs Recall the 1D Burgers39Equation X e 0 11 Z 0 2 2 WK 0 7 dam 0 2 o mu Smgu ar va ues of the Snapshots yory1ror20 mo YoXgt x e 0 11 sdwnew SSzmrvlmvomnmc Saarm SSmlnvuamlnmsma M K K 0 m 20 an m 50 so 7 l I Problem POD for PDEs with nonlinearities If we apply the POD basis directly to construct a discretized system the original system of order N MW Mil1 Nip1 0 become a system of order k ltltN 70 MW 7 N371 0 where the nonlinearterm W U NhUkVt kgtltN Ngtlt1 Computational Complexity still depends on NI I l l l Nonlinear Approximation Recall the problem from bf Direct POD 170 5 Nhruvrr MN er WANT yt H C Nyt quota quot9 Independent of N kxnm nmx1 2 Approaches e Precomputing Technique Simple nonlinear 9 Empirical Interpolation Method EIM General nonlinear ACCURACY vs COMPLEXITY Relame Errorsum uy mrymWuy mHamemnsnmlswms mm POD M cm W W Wm WM MODE Wequot m Em a w m amp n2 m m m E 39amp gm m 1H m s m s m mmawonmswn NumbemrPonass used Figure LEFT Error Eavg 27 RIGHT CPU time sec HJh wfJHX QUESTION I i Empirical Interpolation Method EIM Patera 2004 o Approximates nonlinear parametrized functions 0 Let sX u be a parametrized function with spatial variable X 6 Q and parameter u e 9 a Function approximation from EIM aw ZN ammo nb1 where spanmmmi 2 w 2 s w M e 9 and MM is specified from the interpolation points 2m n quot1 in Q39 sz M quamo nb1 for1mnm I i EIM Numerical Example sX M 1 7 Xcos37mx 1e 1x wherex 6 711 and M e 17r Plot of Approximate Functions dim 10 with Exact Functions in black solid line x 1 15 11713 13855 1 A 3141e A 05 1 i m o s 05 1 x 1 1 05 0 05 1 I i Plots of Numerical Solutions from 3 Approaches 0 Direct POD 0 Precomputed POD Sninrmwmirn 9an Ami in Sninmmiswmiummmim i o EIMPOD ma Singuiarvaiues ofihe Snapshois m5 4mg n M n n 1 Sninihmi whxwillmdmlli mm m in 2U 3H m 5H EU I Figure SVD Figure Direct POD I i QOnIgtm 639 Don uwSQEoowi 99 Izusgiz cgiiw giiza si Conclusions 0 Unlque Contrlbutlon Clear RemWhitmmMlthWimpav description of the EIM Successful ErecamlmcledsnapshotsmingdmeremdimalbasislmmEMiPOD Implementation of EIM with POD O EIM is comparable to widely accepted methods such as precomputing technique Average Relative Elia D The results suggest that EIM with POD basis is a promising model reduction technique for more general nonlinear PDEs Future Work 9 Extend to higher dimensions o Extend to PDEs with more general noanean es cum me sec 0 Apply to practical problem eg optimal control problem m 20 so 1 so Numeral POD hm used I l I I Overview Nonlinear PDE gt7 r FEGalerkin FULL Discretized System ODE dim N gt7 r Solve ODE U SNAPSHOTS yXfz21 U U gt7 7 PODrGalerkin REDUCED Discretized System ODE Linear Term im N NonUnear Term dim N N A A Nonlinear Approx EM Precomputing REDUCED Discretized System ODE Linear Term dim k lt N NonLinear Term dim nquot lt N Circuit Synthesis An MNA Approach Timo Reis Model Reduction Seminar Equations 1 Example Outline 0 Motivation 9 Descriptor Systems 9 Circuit Equations 9 Circuit Synthesis 6 Example 0 Conclusion and Outlook Motivation Problem Formulation Given is a descriptor system EXU Axt But ya 0x0 Find an electrical circuit such that forthe voltages uVt ut of voltage and current sources and currents I39Vt 01 of voltage and current sources hold an mi fulfill the given descriptor system for some state vector Xt Motivation v quot Cil cui Equations Problem Formulation Given is a descriptor system EXU Axt But ya 0x0 Find an electrical circuit such that forthe voltages uVt ut of voltage and current sources and currents I39Vt 01 of voltage and current sources hold WU 7 W ut 7 01 and yt 7 mm fulfill the given descriptor system for some state vector Xt Arising questions 0 For which class of descriptor systems can we find a circuit that realizes its inputoutputbehaVIour 0 Can we formulate numerical methods forthe realization Motivation v quot Cil cui Equations History of the problem 0 Wilhelm Cauer 1920s 1940s 0 Consideration of the problem from a frequency domain point of View 0 SingleInputSingle OutputSystems 0 Discovered that impedance functions of circuit satisfy some special conditions positive realness 0 Otto Brune 1930s1940s and Sidney Darlington 1950s 1970s gave further realization techniques in frequency domain 0 BDO Anderson and S Vongpanidlerd 1960s 1990s gave methods for the realization of multipleinputmultipleoutput ODE models X AX Bu y CX Du Realization based on matrix operations Motivation Descriptor Systems Circuit Equations History of the problem 0 Wilhelm Cauer 1920s 1940s 0 Consideration of the problem from a frequency domain point of view 0 SingleInputSingle OutputSystems o Discovered that impedance functions of circuit satisfy some special conditions positive realness 0 Otto Brune 1930s1940s and Sidney Darlington 1950s 1970s gave further realization techniques in frequency domain 0 BBQ Anderson and S Vongpanidlerd 1960s 1990s gave methods for the realization of multipleinputmultipleoutput ODE models X AX Bu y CX Du Realization based on matrix operations We are interested in timedomain methods for differentialalgebraic models Descriptor Systems V 39cu Equations Outline a Descriptor Systems Descriptor Systems Let a descriptor system EXt Axt But yt Cxt be given with E A 6 W B E Rmquot C 6 RM such that the pencil E 7 A is regular 0 The transfer function G E Rsqquot of this system is given by Gs CsE 7 A WB Descriptor Systems Definition 0 The transfer function G E Rsqquot of this system is given by Gs CsE 7 A lB 0 The system is called minimal if the state space dimension n is minimal among all descriptor systems with transfer function Gs Descriptor Systems Let a descriptor system EXU Axt But ya 0x0 be given with E A 6 W B E Rmquot C 6 RM such that the pencil E 7 A is regular Definition 0 The transfer function G E Rsqquot of this system is given by Gs CsE 7 A lB 0 The system is called minimal if the state space dimension n is minimal among all descriptor systems with transfer function G For i 12 lettwo minimal descriptor systems E39ltt AXt Bu yt Cxt with EA 6 RM 1 6 RMquot C 6 RM be given which have the same transferfunction Then there exist unique W T E Rm such that E2 WEiT A2 WAiT B2 WBi C2 CiT Moreover both W T are invertible Descriptor Systems Given a system EXU AXt l But yt Cxt with EA 6 RM B 6 RMquot C 6 RD and transferfunction 63 6 Rspxp 0 A system is passive if for all sufficiently smooth u E L2R 1R holds y E L2RRp with few uTry7dr 2 0 0 A system is lossess passive if for all sufficiently smooth u E L2RRp holds y E L2RRp with few uTry7dr 0 Descriptor Systems Definition 0 O A system is passive if for all sufficiently smooth u E L2R 1R holds y E L2RRp with few uTry7dr 2 0 A system is lossess passive if for all sufficiently smooth u E L2RRp holds y E L2RRp with few uTryqdr 0 A transferfunction Gs E Rspxp is positive real if it has no poles in 3 and for all s E 3 holds Gs Gs 2 0 A transferfunction Gs E Rspxp is lossess positive real if it is positive real and for all p E R such that G has no pole is iw holds Giw GT7iw 0 Descriptor Systems Given a system EXU AXt But yt Cxt with EA 6 RM B 6 RMquot C 6 RD and transferfunction Gs E Rspxp Definition 0 A system is passive if for all sufficiently smooth u E L2R 1R holds y E L2RRp with few uTry7dr 2 0 0 A system is lossess passive if for all sufficiently smooth u E L2RRp holds y E L2RRp with few uTryqdr 0 A transferfunction Gs E Rspxp is positive real if it has no poles in 3 and for all s E 3 holds Gs Gs 2 0 A transferfunction Gs E Rspxp is lossess positive real if it is positive real and for all p E R such that G has no pole is iw holds Giw GT7iw 0 0 O Parseval s identity implies A descriptor system s is lossless passive if and only if its transfer function is lossless positive real Descriptor Systems Auxiliary result Positive real lemma for lossless descriptor systems A minimal system EXU AXt But yt Cxt is lossless passive if and only if there exists X 6 RM such that ATgtltgtltTA 0 X78 7 CT 0 and ETX 2 0 Moreover in the case of solvability X is invertible and unique Descriptor Systems Definition A transfer function G 6 Ms is reciprocal if there exists ppm 6 N with pi p2 p such that for is diagp17lp2 and all s E there G has no pole holds as eGRs The matrix is is called external signature of the system A descriptor system is called reciprocal if its transfer function is reciprocal Descriptor Systems Definition A transfer function G 6 Ms is reciprocal if there exists ppm 6 N with pi p2 p such that for is diagp17lp2 and all s E there G has no pole holds as eGRs The matrix is is called external signature of the system A descriptor system is called reciprocal if its transfer function is reciprocal The transfer function of a reciprocal system has the form 7 6115 C3125 Glsl 76125 ems where GM GI1 e mst 622 3272 e Rsp2 p2 Conclusion 9 Circuit Equations Circuit Equations Equations for an electrical circuit 9 Rm R 39 Mt ran C ut L ma Urt Trim01mm 7 39 UTt ARIRU ALILa ATiTt Anina Avivt AiE2 0 0000 0 where et R C L T7 ARYACY ALVAy Av Any4n Aet uRt A650 WU resistances uct capacitances inductances ideal transformers irchhoff s current law Alec um Aiea um Aileama t Allit 5 Kirchhoff s voltage law vector of node potentials resistance capacitance and inductance matrix symm pos def transformer gain matrix Element incidence matrix of branches of resistances capacitances inductances voltagecurrent source and initialterminal por of ideal transformers 0 0 0 fv 0 0 0 jvfi 4 jv A 0 0 IV 0 V4 liuHubF 7V4 angaavg a 552 89 0000 oooo o1spee sgsmeue epou peygpow ueql suogwnba uncum 0 M h 4 o o 0 mm 0 0 0 jvg 0W 0 IV 7v jmga yvg o1spee sgsmeue epou peygpow ueql suogwnba uncum on Motivation Descriptor systems Circuit Equations Circuit Synthesis Example Conclusi Observations for the circuit descriptor system ACCAg 0 0 0 ARR V AL AriJrAnTr Av 7A o 0 L 0 0 AT 0 0 0 0 0 E A T LT T 527 0 0 o o o o ATyiTTAn o o o o o o o A 0 0 0 o 7 0 E and 7A AT are symmetric and positive semidefinite 0 if the circuit does not contain resistances then A A7 0 o 307 Descriptor gystelns Circuit Equations 39 39 39 Example Observations for the circuit descriptor system ACCAg 0 0 0 ARR V AL ATWATITT Av 7A 0 AT 0 0 0 0 0 E 0 LOOVA TLTT 507 0 0 0 0 0 0 ATI7TT An 0 0 0 0 0 0 0 T 0 0 0 0 7 0 E and 7A AT are symmetric and positive semidefinite 0 if the circuit does not contain resistances then A A7 0 o B CT Conclusion The transfer function Gs CsE 7 A iB of the circuit satisfies for s 6 Ci 63 Gs BTsE 7A iB BT E 7ATiB ZBTsE 7 A i2 ResE 7 A ATsE 7A B Descriptor gystelns Circuit Equations 39 39 39 Example Observations for the circuit descriptor system ACCAg 0 0 0 ARR V AL ATWATITT Av 7A 0 AT 0 0 0 0 0 E 0 LOOVA TLTT 507 0 0 0 0 0 0 ATI7TT An 0 0 0 0 0 0 0 T 0 0 0 0 7 0 E and 7A AT are symmetric and positive semidefinite 0 if the circuit does not contain resistances then A A7 0 o B CT Conclusion The transfer function Gs CsE 7 A iB of the circuit satisfies for s 6 Ci 63 Gs BTsE 7A iB BT E 7ATiB ZBTsE 7 A i2 ResE 7 A ATsE 7A B 2 0 Le the circuit system is positive real Motivatioi39i Desan ptor systems Circuit Equations Circuit Synthesis Example Conclusion Observations for the circuit descriptor system ACCAE 0 0 0 ARR V AL AritAnTT Av 7A 0 E 0 L 0 o W A A 0 0 0 V 5 CT 0 0 o o o 0 4774 o o 0 0 0 o o o o A 0 0 0 7 For 2 diagU m l u l w IHTL 26 diagl 7l7 IWL the MNA matrices satisfy LE ET A 7AT 23 079 37 ec Circuit Equations Synthesis Observations for the circuit descriptor system ACCAE 0 0 0 ARR V AL AritATzTT Ay 7A 0 AT 0 0 0 0 0 E 0 L 0 0 A L 5 CT o o o 0 4774 o o 0 0 0 o o o o A 0 0 0 o 7 For X diagl e7 lm lm lm is diaglm7 lm the MNA matrices satisfy LE ET A 7AT 23 079 87 90 Hence we have that the transferfunction Gs CsE 7 A iB satisfies Gs ZGTS ie circuit system is reciprocal Circuit Equations Synthesis Reformulation of the task Given is a passive and reciprocal descriptor system with transfer function Gs Find 0 incidence matrices AR AC A A7 A7 A AV 0 positive definite matrices R C L 0 a transformer gain matrix T7 such that 4 0 1T SACCAE ARR4A AL AVATJT AV T1 7 T Gs L i 0 J 7ATTTAI 0 0 0 o a i ii 0 J ui Nations Circuit Synthesis a Conclusion 9 Circuit Synthesis Circuit Synthesis First auxiliary result Let a minimal and passive descriptor system EXU Axt But yt Cxt that is reciprocal with external signature is given Then there exist W T E GMR such that diagUm 7 lpz be 7 EM 7 AM A12 7 T7 Bi 0 WETi 0 E22WAT7 74171 0 WBiCT 7 0 32 where EWAU 6 RM B 6 RM with further EM 2 0 E22 2 0 AM g 0 If the system is moreover lossless then we can find a transformation with AM 0 Descriptor systems 39 iii Equations Circuit Synthesis Example Next task Given is a system with 7 EM 0 7 AM A12 7 77 BM 0 E70 E227A7 A12 07sici0 B22 Introduce new states and perform blockdiagonal congruence transformations T 7 73961 79 such that 775739 TAT TTB CTT is in desired form 22 Ci uit Synthesis We demonstrate the technique for the lossless case 7 EM 0 7 0 A12 7 77 BM 0 50 E MeLAW OLBeCe o 322 Circuit Synthesis Example We demonstrate the technique for the lossless case 7 EM 0 7 0 A12 7 77 BM 0 E io Eni Aei oi B C 0 322 I 0 0 Transforman T1T1EMTM 0 0 v TZTZEZZTH 0 Circuit Synthesis Example leads to Desan ptor systems 39 iii Equations Circuit Synthesis Example leads to Introducing new states Motiv39atim39i Des rigitor Systems c cui Equations Circuit Synthesis Example Conclusion leads to ooooooo ooooooo Motivation Desan ptor systems Circuit Equations Circuit Synthesis Example leads to Transforming TiTA23T2 and eliminating some other blocks Conclusion Example Circuit Synthesls w m m E0 m Desan ptor systems Motivatiorr leads to 1J 0000000004 000000000 00000 Motivation Desan ptor systems Circuit Equations Circuit Synthesis Example leads to 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 A27 A2E 29 0 0 0 0 0 0 0 A35 A39 0 0 0 0 0 0 0 0 A45 A49 0 0 0 0 0 0 0 A57 A5E A59 7 A 0 0 7 0 0 0 0 0 0 0 0 4 0 0 4 0 0 0 0 0 0 4g8 4 4 4B 0 0 0 0 0 429 439 449 459 0 0 0 0 0 0 0 0 0 0 0 0 0 Transforming TiTA48 T2 I 0 and eliminating some other blocks Conclusion J 0000000000 J M m 00007000000 n 0000000000 E l mm0m000000 AA A T C 02305000000 5 AA A m s e m n y S m m m 00000 00000 S 7 9n n rs r 1 m oooooo ok f m q E m oooooooooo t C an r m ooooo00A Ao rm ram UUUUUUUUA 2 0 7A 7A 0 000000 0000 Desan ptor systems leads to E Motivatiorr Conclusion Example Circuit Synthesis w m m Em m Desan ptor systems Motivatioi39i leads to J 0000000000 0000000000 00000 5750000 EE 00000 00000 00000 00000 00000 0 0 0 0 0 0 0 00000 000000 Eliminating some unoontrolIableonobservable states Conclusion Example Circuit Synthesls w m m Em m Desan ptor systems Motivatiorr leads to 1J 00000000 00000000 0000 0000 0000 0000 00000 00000 Motivation Desan ptor systems Circuit Equations Circuit Synthesis Example Conclusion leadsto 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 E55 E56 0 0 0 5 c7 0 0 0 E576 E66 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 7 0 0 0 0 0 0 A26 A27 A2E 0 0 0 0 0 0 A37 A35 0 0 0 0 0 0 A46 A47 A45 7 A 0 0 7 0 0 0 0 0 0 0 4 0 gs 0 0 0 0 0 0 427 A377 7AM 0 0 0 0 0 428 ASTE 7AA 0 0 0 0 0 0 0 0 0 0 0 0 Introducing a new state Conclusion 1J 00000000 Example Circuit Synthesls w m m Em m Desan ptor systems leads to E Motivatiorr Motivatim39r v 39 39 quot 339 39i Equations Circuit Synthesis Example Columnechelonform A27 7 P29 R29 P29 AmiV7 ipaa 5 39 P39 e i em Conclusion 00070000000 0000000000 leads to E J 000000004 uuu mmm0000000 P AAA m 00000000 a 99 PPa 4 1K 7 0A0000000 r C ww e 5 PP 5 h a a m 0M0A47000000 w m pm 0000000000 m 0000000000000000000000 PP 0000004000 m m y 090 3 TAT 1 a 23 000000000AA4TA4 L RR 7 n 7 E u 99 IE pp A m 0000000 rAa0 7w V P 7 s m m 77 N cnlv AA TRB n In rm m IO 000000047940 F e 2 m h JP 5 C e e D n M m m m w m M Motiv39atim39i Des rigitor Systems c cui Equations Circuit Synthesis Example Conclusion ACCAJ2 0 0 0 0 AL Ti rAnTT Av 7A 0 0 L 0 0 AK 0 0 567 0 0 0 0 0 0 7TTTAZ 0 0 o 0 0 o o o o g o 0 0 o 7 with o o o o o 0 7 o 7 o o 0 P29 0 0 P29 0 o Ac0Ai 7 0 An0 Pia 0An0 aawv 0 0 o o o o o o 0 o o o 7 o 0 0 o o Example Conclusion 6 Example uit Equations 39 Example Consider As descriptor system 100000 010000 00 010000 7100000 10 001000 000100 00 57000100 A0071000 B017 000001 000010 00 000000 00 001 10 C021010 010010 Motivation1 Desan ptor systems Circuit Equations Circuii Synthesis Example Bringing into block form 0 5570 70 6436 70 1587 E 0 0 0 0 0 0 0 0 0 A 0 0 0 70 6256 71 0516 0 7625 71 0549 1 2702 0 2525 11931 701807 01607 70 6436 1 9137 70 9081 0 0 0 0 6258 1 0516 70 7625 0 0 0 70 5400 711 0 0 0 1564 2 0000 70 7709 Conclusion Motivation1 Desan ptor systems Circuit Equations Circuii Synthesis Example Conclusion Realization procedure leads to o o o o o o 71 o o o o o o 1 o o o 1 o o o 1 o o o o o o o o o o 1 o o o 1 o o o o 0 AL 71 o 0 Ac 0 0 AV 0 A 0 An 0 o 1 0 AT 0 o o o o o o o 1 o o o o 1 o o o o 71 o o o o o o o o o 1 o o o o 71 o o o o o o o o o 1 o o 2986 72 9851 721213 0 701981 y T 70 9950 70 0995 0 71071 2 9802 10198 701981 70 4925 721812 0 Motivatioi39i Desan ptor systems Circuit Equations Circuii Synthesis Example Conclusion A realizing circuit is therefore given by V 39cu Equations 1 m Conclusion Outline 6 Conclusion and Outlook Cil cu Equations 1 Example Conclusion Conclusion 0 Passive and Reciprocal Descriptor Systems 0 Transformation into MNA form 0 Interpretation as an electrical circuit Thanks a lot for your attention

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