Lecture 25 Outline: LTI Systems: Causality, Stability, Feedback

Transcription

1 Lecure 5 Oulie: LTI Sye: Caualiy, Sabiliy, Feebac oucee: Reaig: 6: Lalace Trafor , , 73; 7: 7: Feebac. -4.5, 8-7. W 8 oe, ue oay. Free -ay eeio W 9 will be oe oay, ue e Friay (o lae W) Fial ea i Jue 7, 8:3a-:3a. ore o fial ea, ecio e wee, a era O o Friay OceaOe Robo Tour will be afer cla Friay (:3-:) Luch rovie aferwar. Ca arrage earae our for hoe w/coflic Caualiy a Sabiliy i LTI Sye LTI Sye ecribe by iffereial equaio Eale: orer Lowa Sye Feebac i LTI Sye

2 Review of La Lecure Lalace for Circui alyi Tur DE io algebraic equaio Eale: Lalace alyi of Orer LFP Iverio of Raioal Lalace Trafor ai iea: Cover cole equaio io a u of er where he ivere of each er i ow Se : Erac ricly roer ar of (): Iver D(): Se : Parial Fracio Eaio o ) ( ) ( Y D, B / a B u e u e!

3 Caualiy a Sabiliy i LTI Sye Caual LTI ye: iule reoe h()() LTI ye i caual if h()=, <, o i h() righ-ie For () raioal, a caual ye ha i ROC o he righ of he righ-o ole Se reoe i h()*u() ()/ Sable LTI Sye LTI ye i boue-iu boue-ouu (BIBO) able if all boue iu reul i boue ouu ye i able iff i iule reoe i aboluely iegrable; ilie (jw) ei, equivalely jwroc caual ye wih () raioal i able if & oly if all ole of () lie i he lef-half of he -lae Equivalely, all ole have Re()< ROC efie ilicily for caual able LTI ye

4 LTI Sye Decribe by Differeial Equaio (DE) Fiie-orer coa-coefficie liear DE ye Pole are roo of (), zero are roo of B() If ye i caual: for ()=(), iiial coiio are zero: y( - )=y () ( - )= =y (-) ( - ) ROC of () i righ-half lae o he righ of he righ-o ole Ca olve DE wih o-zero iiial coiio uig he uilaeral Lalace rafor: o covere i hi cla a we focu o caual able ye Era crei reaig: Lalace , eale b y a b Y a a b B ) ( ) ( e L u u

5 Seco Orer Lowa Sye w z R LC C L () R L C y() aural frequecy w a aig coefficie z w z b y y zw w y w zw w Y w Y w zw w w We fir facor (): w z z Three regie:, Uerae: <z<, iic, cole cojugae Criically Dale: z=, equal a real Overae: <z<, iic a real Pole i lef-half of -lae h() caual (jw ) ei

6 Frequecy Reoe: w jw jw zw jw w Uerae: <z<, iic, cole cojugae Criically Dale: z=, equal a real Overae: <z<, iic a real

7 Feebac i LTI Sye S e g G h oivaio for Feebac Ca ae a uable ye able Ca ae rafer fucio cloer o eire (ieal) oe Ca ae ye le eiive o iurbace Ca have egaive effec: ae a able ye uable Trafer Fucio T() of Feebac Sye: Y()=G()E(); E()=()±Y()() y Equivale Sye G G Y G T ( ) G y

8 Two Eale Sabilizig a uable ye: orer ye a>, ole i righ-half of -lae () S e() a y() r() K Sye i able if K>a, igle ole i lef half of -lae Feebac ca ae a able ye uable T K K K e T Uable if K K >

9 ai Poi caual ye wih () raioal i able if & oly if all ole of () lie i he lef-half of he -lae (all ole have Re()<) ROC efie ilicily for caual able LTI ye Sye ecribe by iffereial equaio eaily characerize uig Lalace aalyi If ye i caual: for ()=(), iiial coiio are zero ROC of () i righ-half lae o he righ of he righ-o ole Seco orer ye characerize by 3 regie: uerale, criically ae, overae Feebac very ueful i LTI ye eig Ca ae a uable ye able Ca ae he ye rafer fucio cloer o ieal oe Ca iigae he effec of iurbace Ca alo have ueire effec: ae a able ye uable