Laplace Transformation of Linear Time-Varying Systems

Transcription

1 Laplace Tranformaon of Lnear Tme-Varyng Syem Shervn Erfan Reearch Cenre for Inegraed Mcroelecronc Elecrcal and Compuer Engneerng Deparmen Unvery of Wndor Wndor, Onaro N9B 3P4, Canada Aug. 4, 9

2 Oulne of he Preenaon From LTV Elemen o LTV Syem Obervaon on LTV Syem Generalzed-Delay Syem Repreenaon Crcularly Symmerc Funcon Bvarae Tme and Bfrequency Characerzaon Two-Dmenonal Laplace Tranform DLT The Hankel Tranformaon The Melln Tranformaon An Illurave Example Concluon Reference 8/4/9 Aug. 4, 9

3 Lnear Tme Varyng Elemen A ngle-npu ngle-oupu SISO dynamc yem elemen of fne order characerzed by npu-oupu relaonhp ad o be lnear f he followng hold for each : y h x Where h he yem funcon defne he repone a me, denoe he lope of he y-x curve n a recangular coordnae yem. y h3 h x h 8/4/9 3 Aug. 4, 9

4 Lnear Tme Varyng Syem A SISO dynamc yem operaon hown ymbolcally by: y ο{ x } The yem operaor lnear f and only f he followng relaon hold: ο { α + β } αο{ } + βο{ } α + β x x x The yem npu can be any funcon ncludng an mpule or a dela funcon: y δ x ; h δ y y 8/4/9 4 Aug. 4, 9

5 Obervng an Impule Repone Funcon Obervaon The followng ymbolc deny hold: h δ h δ h δ Obervaon - The produc dfferen from zero a he pon. Obervaon 3 - The mpule repone of he yem ha a crcular ymmerc propery wh repec o argumen and. y δ y ; ; δ 8/4/9 5 Aug. 4, 9

6 Obervng an Impule Repone Funcon Obervaon 4 In he,-plane, due o he crcular ymmery and becaue dela funcon an even funcon, we can defne a bvarae repone funcon a: y ; h, x, δ Obervaon 5 - The ordnary oupu repone a he pon : or, equvalenly: + y h, x, δ d h x + y h, x, δ d h x Queon Can a yem funcon be equal o an npu funcon? I he yem oupu condered o be ll lnear? 8/4/9 6 Aug. 4, 9

7 Real-Varable Funcon Repreenaon n,-plane fp o θ p A yem funcon, whch ha he mple roaonal propery of crcular ymmery hown n h fgure. Fgure how converon of a one-dmenonal profle of a yem funcon of p o a wo-dmenonal funcon of a complex varable of and. The common vew of he ndependen gnal and yem funcon are he projecon over -ax pure real and -ax pure magnary, repecvely. 8/4/9 7 Aug. 4, 9

8 Crcularly Symmerc Funcon h θ h, θ h co θ, n θ hym [ h, + h, ] H, φ H ym [ H, + H, ] Roaonal propery of a crcularly ymmerc funcon L 8/4/9 8 Aug. 4, 9

9 Impule-Repone Syem Repreenaon The yem repone for a crcularly ymmerc yem funcon can equvalenly be wren a: y The more famlar mpule repone, ung fng propery of he dela funcon wll be: y δ δ, h, δ + h, δ d h The lm of negraon can be exended o nfny. 8/4/9 9 Aug. 4, 9

10 Laplace Tranform of he Impule Funcon The ordnary unlaeral Laplace ranform of obaned a: + L{ δ } δ e d e Th a funcon of he varable me. δ A econd ranformaon yeld: L D + { δ } e e d + 8/4/9 Aug. 4, 9

11 Two-Dmenonal Laplace Tranform DLT Defnon of DLT The ordnary unlaeral DLT defned a: H, h, e e d d + + Invere Tranformaon - The nvere DLT gven by: L H, h, { } D σ+ j σ+ j σ j σ j π j, H e e d d 8/4/9 Aug. 4, 9

12 Inegral Repreenaon Conder a SISO lnear dynamc yem of fne order characerzed by fundamenal equaon: + y h, x d Where h, he yem funcon defne he repone n he frquadran of a recangular coordnae yem. x, Lnear Tme-Varyng LTV Syem y, 8/4/9 Aug. 4, 9

13 Obervaon on LTV Syem Mahemacally peakng, and repreen me varable of an appled gnal and he correpondng yem. Varaon of an npu gnal and an auonomou yem are ndependen of each oher. For crcularly ymmercal yem, wh no lo of generaly, we can rewre he repone a: y, e ln h, x 8/4/9 3 Aug. 4, 9

14 Generalzed-Delay Syem Repreenaon If h a pecewe connuou funcon and bounded by a fne number, -order and hgher-order dervave ex. The yem repone can equvalenly be wren a: h' ξ dξ y h ξ e x The yem repone can be wren more compacly a a generalzed-delay operaor: y e g, x 8/4/9 4 Aug. 4, 9

15 Crcularly Symmerc Funcon From an operaonal pon of vew, a crcularly ymmerc yem funcon h, can be wren a: h, h + 8/4/9 5 Aug. 4, 9

16 Bvarae-Tme and Bfrequency Obervaon 6 - A general yemheorec approach for characerzaon of lnear me-varyng LTV yem baed on he applcaon of a wodmenonal Laplace ranform DLT feable. Obervaon 7 - Th echnque appear o have remaned largely unknown o he analog gnal proceng communy up o now. L { f } F + Vecor: [, ] [, ] f e d θ θ H, f g L L Two convolvng real-varable funcon repreenaon n he meplane n polar coordnae 8/4/9 6 Aug. 4, 9

17 Fundamenal Inpu-Oupu Repreenaon The clacal heory of varable yem baed on he oluon of lnear ordnary dfferenal equaon wh varyng coeffcen. The varyng coeffcen are funcon of an ndependen varable, convenenly called he me. The me aumed o be real for phycal yem. n a y m k b k x k 8/4/9 7 Aug. 4, 9

18 Bfrequency Inpu-Oupu Syem Repreenaon h x X - H y Y Black box repreenaon of crcularly ymmerc lnear yem 8/4/9 8 Aug. 4, 9

19 Crcularly Symmerc Syem Tranformaon Y + Regon over whch he mpule-repone h, of a nonancpave yem defned he haded area. h, x e + dd Ieraed Laplace Tranform 8/4/9 9 Aug. 4, 9

20 Tme-Doman Repreenaon of Nonancpave Syem Funcon Defne he nan a whch he npu appled o he yem a he orgn for me. The nonancpave condon mple: h x u o for < h x u y, for > Then, we may defne h. o be zero for negave value of argumen: y δ, h δ 8/4/9 Aug. 4, 9

21 RESEARCH CENTRE FOR INTEGRATED MICROSYSTEMS - UNIVERSITY OF WINDSOR Aug. 4, 9 8/4/9 Le u defne: The DLT : Smlarly, we defne : Addng ogeher, we oban: Frequency-Doman Repreenaon of Nonancpave Syem Funcon < > for for h h,, H d h e d e H < > for for h h,, H H + { }, H H H h L D + +

22 RESEARCH CENTRE FOR INTEGRATED MICROSYSTEMS - UNIVERSITY OF WINDSOR Aug. 4, 9 The DLT of General LTV Syem 8/4/9 Conder a SISO LTV yem, nally a re, decrbed by: An npu x. appled o he yem a me ξ - Takng a DLT, we oban: m k k k k n d x d b d y d a m k k k k n d x d b d y d a, ξ dd e e d x d b dd e e d y d a m k k k k n, Th may demand more nal condon ha he problem requre!

23 More on he DLT Obervaon 9 The DLT of a crcularly ymmerc funcon wh he propery lm h ha a Hankel ranform of order zero. Hankel ranform are negral ranformaon whoe kernel are Beel funcon. The blaeral DLT of a LTV reor h / n he frequency doman /ω. The blaeral DLT of a crcular un-ep ua-, whoe value equal o uny for < a, n he frequency doman aj a ω /ω. Obervaon A change of me-cale -ln wll ranform he DLT of a crcularly ymmerc funcon no a Melln ranform. 8/4/9 3 Aug. 4, 9

24 The Hankel Tranform The Hankel ranform compable wh LTV yem decrbed by a general Beel equaon gven a: d d + d d n ± a N y x The Hankel ranform par are ymmerc becaue deal wh ymmerc funcon. The DLT of a crcularly ymmerc funcon wh he propery lm h ha a Hankel ranform of order zero. Th propery que ueful n applcaon of Hankel ranform o LTV yem. 8/4/9 4 Aug. 4, 9

25 RESEARCH CENTRE FOR INTEGRATED MICROSYSTEMS - UNIVERSITY OF WINDSOR Aug. 4, 9 8/4/9 5 The Melln ranform compable wh LTV yem characerzed by a general Euler-Cauchy equaon gven a: The mpule repone of h nonancpave Euler-Cauchy LTV yem : where g. he mpule repone of a prooype LTI yem obaned by changng he me cale -ln The DLT n h cae become he followng Melln Tranform par: The Melln Tranform x d y d a n, u g h { } d h h M + { } d H j H M j j + σ σ π

26 An Illurave Example The DLT of mpule-repone obaned a: H, + + The nvere funcon ung able of DLT : Compare wh he mpule repone funcon for he nally relaxed crcu obaned by ung he nducor flux and capacor charge a ae varable: 8/4/9 6 Aug. 4, 9 h, e J o h e n

27 Concluon The DLT echnque are applcable o LTV yem. Th approach allow, n effec, wo-dmenonal ranform echnque o be ued for he me-varyng yem n he ame manner ha he convenonal frequency-doman echnque are ued n connecon wh fxed yem. The DLT mehod appled o an Euler-Cauchy yem and a Beel yem reul n a Melln ranform and Hankel ranform, repecvely. The DLT, Melln ranform, and Hankel ranform can be derved from he wo-dmenonal Fourer ranform. The work preened here open everal area for furher nvegaon n heory of varable yem. 8/4/9 7 Aug. 4, 9

28 For Furher Informaon. S. Efan, Exendng Laplace and Fourer ranform and he cae of varable yem: A peronal perpecve, Preenaon o IEEE Sgnal Proceng Long Iland Secon, NY, May 5, 7.. S. Erfan and N. Bayan, On Lnear Tme-Varyng Syem Characerzaon, Proc. IEEE EIT 9, Wndor, ON, Jun. 7-9, 9, pp S. Erfan and N. Bayan, Laplace, Hankel, and Melln Tranform of of Lnear Tme-Varyng Syem, Proc. IEEE 5h MWSCAS, Cancun, Mexco, Aug. -9, 9, pp V. A. Dkn and A. P. Prudnkuv, Operaonal Calculu n Two Varable and I Applcaon, Pergaman Pre, D. Voelker and G. Doech, De Zwerdmenonal Laplace Tranformaon, Brkhauer Verleg, Bael, A. H. Zemanan, Generalzed Inegral Tranformaon, Dover Publcaon, New York, NY, I. N. Sneddon, Fourer Tranformaon, Dover Publcaon, New York, NY, N. Bayan and S. Erfan, Frequency-doman realzaon of lnear me-varyng yem by wodmenonal Laplace ranformaon, Proc. NEWCAS, Monreal, Que., June -4, 8, pp A. D. Poularka, ed., The Tranform and Applcaon Handbook, nd Ed., CRC Pre, Boca Raon, FL,.. L. A. Zadeh, Tme-varyng nework, I, Proc. IRE, vol. 49, Ocober 96, pp /4/9 8 Aug. 4, 9

29 8/4/9 9 Aug. 4, 9

30 Shervn Erfan Shervn Erfan a profeor and former Elecrcal and Compuer Engneerng Deparmen Head a he Unvery of Wndor, Wndor, Onaro, Canada. H experence pan over 3 year wh AT&T Bell Lab, Lucen Technologe, Unvery of Puero Rco, Unvery of Mchgan-Dearborn, Seven Inue of Technology, and Iranan Naval Academy. H expere expand over many area from Syem Theory o Dgal Sgnal Proceng o Nework Secury Managemen. He ha been a conulan o he ndury for a number of year. Dr. Erfan ha publhed more han 7 echncal paper, hold hree paen, and he Senor Techncal Edor of he Journal of Nework and Syem Managemen and an aocae edor for Compuer & Elecrcal Engneerng: An Inernaonal Journal. He receved a combned B.Sc. and M.Sc. degree n Elecrcal Engneerng from he Unvery of Tehran n 97, and M.Sc. and Ph.D. degree, alo n Elecrcal Engneerng, from Souhern Mehod Unvery n 974 and 976, repecvely. He wa a Member of Techncal Saff a Bell Lab of Lucen Technologe n Holmdel, New Jerey, from 985 o. Aug. 4, 9