ENGG 1203 Tutorial. Difference Equations. Find the Pole(s) Finding Equations and Poles

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1 ENGG 03 Tutoial Systms ad Cotol 9 Apil Laig Obctivs Z tasfom Complx pols Fdbac cotol systms Ac: MIT OCW 60, 6003 Diffc Equatios Cosid th systm pstd by th followig diffc quatio y[ ] x[ ] (5y[ ] 3y[ ]) wh x[] ad y[] pst th th sampls of th iput ad output sigals, spctivly Pol(s) of this systm: 3 ad 05 Dos th uit-sampl spos of th systm covg o divg as? Divg Fid th Pol(s) Lt Dtmi th pol(s) of H3 ad th pol(s) of Fidig Equatios ad Pols Fo 09 3

2 Covsio btw Bloc Diagams () Th systm that is pstd by th followig diffc quatio y[] x[] y[ ] y[ ] ca also b pstd by th lft bloc diagam It is possibl to choos cofficits fo th ight bloc diagam so that th systms pstd by th lft ad ight bloc diagams a quivalt 5 Covsio btw Bloc Diagams () Fo th lft diagam, Fo th ight diagam, Y X Y X A B p0r pr R R Th two systms a quivalt A B R R p0r pr Equatig domiatos ad umatos, p 0, p, A /3 ad B /3 6 Complx Numbs () Complx Numbs () Complx umb x (Ral) yi (Imagiay) Complx cougat of x yi is dfid to b x yi Pola xpssios Agumt φ ad Modulus/magitud/absolut valu Th agumt o phas of is th agl of th adius OP with th positiv al axis 7 8

3 Qustio: Complx Pols () Cosid a systm of th fom H ( ) wh i class w hav divd that th pols at ± (a) Wh 0, th two pols a at {0,} Stch i th diagam blow th tactois of th two pols as icass fom 0 W hav daw o fo you At what valu of that th two pols mt? Imagiay Qustio: Complx Pols () (b) Wh > /, th tm ( ) isid th squa oot is gativ W wit, istad, that th pols a at ± Not that th two pols a ow complx, ad a cougats of ach oth Daw th tactois of th pols as icas fom ¼ What is th magitud of th pols? Th two pols mt wh 0, o / Ral 9 0 Qustio: Complx Pols (3) As icass fom ¼, th al pat of th pols stays at ½, whil th imagiay pat icass i magitud Imagiay Qustio: Complx Pols () (c) Now lt us cosid, spcifically, that ½ Expss th pols of th systm i pola fom Lt b th o with th positiv imagiay valu, ad its cougat Wh ½, th oots a at ½ ± ½ Th magitud is Th magitud of th pols is Ral ( ) Thfo,, whil th agl is ± ta - ±

4 Qustio: Complx Pols (5) 3 (d) Show that w ca comput th patial factio of th tasf fuctio ad aiv at ) ( H W ca combi th patial factios so that ( ) ( ) ( )( ) W ow (- - )(- - ) - ½ - bcaus ad a th oots of th quadatic quatio I th umato, Qustio: Complx Pols (6) Thfo, 0 ad that () Show that w ca xpss th output y[] to a impuls x[] δ[] as Th cosi fuctio xplais th oscillatio, whil lads to a dcay i th magitud With th patial factio, w ow that Qustio: Complx Pols (7) ( ) cos 5 ( ) ( ) y ] [ Qustio: Complx Pols (8) ( ) ( ) ) ( cos ) ( cos 6 ( ) θ θ θ si cos Oscillat but covg to o

5 Qustio: Complx Pols (9) pols y[] wh x[]δ[] 0 << / al pols lss tha ; covgs /8 ± / << complx pols with magitud lss tha ; oscillats & covgs / ± cos( ) complx pols with magitud qual ; oscillats oly 3 ± cos( ) 3 6 > complx pols with magitud gat tha ; oscillats & divgs 7 ± 3 cos( 036) 7 7 Fdbac () Lt H pst a systm with iput X ad output Y Assum that th systm fuctio fo H ca b witt as a atio of polyomials i R with costat, al-valud cofficits I this poblm, w ivstigat wh th systm H is quivalt to th followig fdbac systm wh F is also a atio of polyomials i R with costat, al-valud cofficits 8 Fdbac () Fdbac (3) Exampl : Systms ad a quivalt wh Lt E pst th output of th add Th Exampl : Systms ad a quivalt wh E Which xpssios fo F guaats quivalc of Systms ad? 9 0

6 What s Cooig () Sous vid ("ud vacuum") cooig ivolvs cooig food at a vy pcis, fixd tmpatu T (low ough to p it moist, but high ough to ill ay pathogs) I this poblm, w modl th bhavio of th hat ad wat bath usd fo such cooig Lt I b th cut goig ito th hat, ad c b th popotioality costat such that Ic is th at of hat iput Th systm is thus dscibd by th followig diagam: What s Cooig () Diffc quatio of th systm: T [ ] ( ) T[ ] T[ ] ci[ Th systm fuctio: T c H I D)( Lt 05, 3, ad c Dtmi th pols of H M Ic MD M T M Ic D M Ic TD T D D D Pols at 05 ad 3 ( D ] ) What s Cooig (3) Psoal Savigs () Lt th systm stat at st (all sigals a o) Suppos I[0] 00 ad I[] 0 fo >0 What is th plot wh 05 ad 0? What is th plot wh ad 05? You ad you fid Wallac hav accouts i th bas Each moth, you ba dposits you itst fom last moth ito you accout, lavig you w balac qual to α tims you old balac Wallac s ba is simila but th costat is γ istad of α Each moth, you ma a additioal dposit (ito you accout) of x[] dollas plus β tims th balac i Wallac s accout fom last moth Each moth, Wallac withdaws (fom h accout) δ tims th balac i you accout fom last moth 3

7 Psoal Savigs () Psoal Savigs (3) W wish to dscib th balacs i ths ba accouts as a lia systm Lt y[] ad w[] pst last moth s balacs i you accout ad i Wallac s accout, spctivly Lt x[] pst th iput to th systm, ad lt w[] pst th output Dtmi a systm fuctio to dscib th latio btw th sigals X ad W (Th systm fuctio should ot dpd o Y) 5 6 Psoal Savigs () Sigals ad Fdbac Cotol () X : Dsid tmpatu T st Dtmi if Wallac s balac oscillats ad divgs α 0, β 05, γ 0, δ 05 Oscillats ov tim; Magitud covgs α, β, γ, δ 5 Oscillats ov tim; Magitud divgs α 05, β 0, γ, δ 0 Not oscillats ov tim; Magitud covgs α 5, β 0, γ, δ 0 Not oscillats ov tim; Magitud divgs 7 Y : Actual hot wat tmpatu T HW Th sso ads T HW with gai s, but has a uit dlay Th valv also tas a uit dlay to spod, ad has som psistc about its motio, chaactid by v Th popotioal cotoll, with sttig a, covts th tmpatu diffc btw T st ad th sso tmpatu to a sigal iput to th valv 8

8 Sigals ad Fdbac Cotol () Sigal ad Fdbac Cotol (3) Th diffc quatio: y[] a x[ ] v y[ ] s a y[ ] Assum th systm stats at st, ad th iput is a uit sampl sigal x[] (i x[] [,0,0,0, ]), suppos v 09 ad a 05 Wh s 0 y[0] 0 y[] 05 () 09 (0) 0 05 y[] 05 (0) 09 (05) 0 05 y[3] 05 (0) 09 ( 05) y[] 05 (0) 09 (005) Wh s 035 y[0] 0 y[] 05 () 09 (0) ( 035)(05) (0) 05 y[] 05 (0) 09 (05) ( 035)(05) (0) 05 y[3] 05 (0) 09 ( 05) ( 035)(05) (05) 095 y[] 05 (0) 09 (095) ( 035)(05) ( 05) 05 Wh s 5 y[0] 0 y[] 05 x[ ] 09 y[ ] 05 s y[ ] y[] 05 () 09 (0) (5)(05) (0) 05 y[] 05 (0) 09 (05) (5)(05) (0) 05 y[3] 05 (0) 09 ( 05) (5)(05) (05) 003 y[] 05 (0) 09 (003) (5)(05) ( 05) Qustio: Bloc Diagams fo Cotol () Qustio: Bloc Diagams fo Cotol () A gic bloc diagam fo fdbac cotol systms is show blow: I this poblm, w will substitut difft cott fo th vaious blocs ad comput th tasf fuctio Th possibl blocs iclud: Wh ths blocs a substitutd fo ith cotoll o systm, th flow is fom lft to ight; wh thy a substitutd fo sso, th blocs a flippd hoiotally ad th flow is fom ight to lft 3 3

9 Qustio: Bloc Diagams fo Cotol (3) (a) Lt th cotoll b D, th systm b A, ad th sso b B What is th ovall tasf fuctio ad th valu of th pol(s)? Fo this ad subsqut pats, lt cotoll C(), systm P(), ad sso G() Th ovall tasf fuctio is C( ) P( ) H ( ) C( ) P( ) G( ) Systm "A"is,"B"is, "C"is, ad "D"is Qustio: Bloc Diagams fo Cotol () Fo this pat, ()( ) H ) ()( )( ) ( ) ( Hc, w hav a fist-od systm with o pol at (b) Lt th cotoll b A, th systm b C, ad th sso b D What is th ovall tasf fuctio ad th valu of th pol(s)? H ( ) ( ) Hc, w hav a fist-od systm with o pol at 33 3 Qustio: Bloc Diagams fo Cotol (5) (c) Lt th cotoll b D, th systm b C, ad th sso b B What is th ovall tasf fuctio ad th valu of th pol(s)? H ( ) ( ) Hc, w hav a fist-od systm with o pol at Qustio: Bloc Diagams fo Cotol (6) (d) Lt th cotoll b B, th systm b C, ad th sso b B What is th ovall tasf fuctio ad th valu of th pol(s)? H ( ) Hc, w hav a scod-od systm with two pols at ± 35 36