Section 2: The Z-Transform

Transcription

1 Scion : h -rnsform Digil Conrol Scion : h -rnsform In linr discr-im conrol sysm linr diffrnc quion chrcriss h dynmics of h sysm. In ordr o drmin h sysm s rspons o givn inpu, such diffrnc quion mus b solvd. Wih h -rnsform mhod, h soluions o linr diffrnc quions bcom lgbric in nur. Jus s h Lplc rnsformion rnsforms linr im-invrin diffrnil quions ino lgbric quions in s h -rnsformion rnsforms linr im-invrin diffrnc quions ino lgbric quions in. h min obciv of his scion is o prsn dfiniions of h -rnsform, bsic horms ssocid wih h -rnsform, nd mhods for finding h invrs -rnsform.. h Smpling Procss In gnrl, discr-im signls ris in sysm whn smpling oprion is crrid ou on coninuous-im signl. h smpling procss cn b liknd o h oprion of swich, s Figur., nmly whn h swich is opn no informion bou h coninuous-im signl is cpurd, bu whn h swich is closd h signl psss hrough nd hnc informion is ghrd. h swich closs priodiclly wih priod. h oprion of h swich is no insnnous, hnc if h inpu is h coninuous signl f h oupu from h swich, dnod fk, consiss of shdd pulss, s illusrd in Figur.b. his oupu puls rin fk my b pproximd by n impuls rin whr ch impuls funcion hs n r qul o h vlu of h inpu signl h smpling insn n which is h im of h impuls. f f * f f f f * Coninuous signl Smpld signl 3 3 b c Figur.: h smpling procss: h smplr s swich; b Smpld oupu s pulss; c pproximion o fk in rms of impulss. Alrnivly h smpling procss my b sn s h modulion of n impuls rin by h inpu signl f, s Figur.. h xprssion for h moduld rin, fk, is hn f k f δ f δ f δ... Eqn.. f k δ k k king h Lplc rnsform of quion. nd rclling h L [δ], nd b L [f- ] Fs -s, yilds L [ f k ] F * s 3//6 3 C.J. Downing &. O Mhony Dp. Elcronic Eng., CI,

2 Scion : h -rnsform Digil Conrol Figur.. Modulion of h inpu signl by n impuls rin o obin smpld signl k s s f f f... ks f k Eqn.. As n xmpl considr h funcion f - nd rcll h Eqn..3 L [ f ] F s s Howvr, if w smpl h funcion f o g h smpld or moduld signl fk, on is hn inrsd in h Lplc rnsform of fk. Equion. is usd o obin h Lplc rnsform of h smpld signl fk i.. L [ f k] F* s k k k ks k s Eqn..4 which is n infini sris h cnno b sily summd. In gnrl h Lplc rnsform of h smplr oupu is in h form of n infini sris. Consqunly h smplr oupu cnno b dscribd in rms of convnionl rnsfr funcion rling oupu o inpu rnsforms nd h sndrd chniqus for conrol sysm nlysis nd dsign r no pplicbl. his difficuly cn b ovrcom, nd h nlysis simplifid, by dfining h -rnsform.. h -rnsform h -rnsform dscribs h bhviour of signl h smpling insns. h -rnsform of n rbirry signl my b obind by pplying h following procdur Drmin h vlus of f h smpling insns o obin fk; fk is now in h form of n impuls moduld rin. k h Lplc rnsform of h succssion of impulss o obin F*s. F*s now conins rms of h form s. 3 Mk h subsiuion ino F*s o giv F. s Eqn..5 3//6 4 C.J. Downing &. O Mhony Dp. Elcronic Eng., CI,

3 Scion : h -rnsform Digil Conrol No lso h F Fs nd F F*s. Fs is h Lplc rnsform of h signl f nd s such is coninuous-im dscripion of h signl f i.. i conins informion s o wh is hppning bwn smpling insns s wll s h smpling insns. h -rnsform conins informion bou h signl h smpling insns only, nd hnc h wo dscripions cnno b quivln. Furhrmor F cnno b qul o F*s s his would imply h h Lplc vribl s is qul o h discr-im vribl. From quion.5 i is clr h his is no so, in fc h rlionship is nd h quivlnc is s s ln F F * ln From quion. w hd h h Lplc rnsform of h moduld rin f* is L [ ] * s s f k F s f f f... Hnc sps on nd wo of -rnsform procdur r compl. No h F*s my b r-wrin s - F * s f f f... s s nd mking h subsiuion s yilds F f f f... f f f... f k k Hnc h -rnsform of n rbirry funcion f is dfind by Rcll k [ f ] f k Eqn..6 k k s L [ f ] F s f d.3 Exmpls -rnsform of h Uni Sp Funcion h uni sp funcion is dfind s u < h uni sp funcion is smpld vry sconds, yilding discr-im or smpld uni sp signl dnod s uk nd dfind by uk ; k,,,3,... 3//6 5 C.J. Downing &. O Mhony Dp. Elcronic Eng., CI,

4 Scion : h -rnsform Digil Conrol u u k 3 n b Figur.3: Uni sp signl; coninuous-im rprsnion nd b discr-im or smpld rprsnion Figur.3 dpics h rlionship bwn h coninuous-im funcion u nd h smpld signl uk. o obin h -rnsform of h uni sp funcion quion.6 is pplid o obin - [ u ] [ uk ]... Rcll h sum of n infini gomric sris is givn by h formul S n r whr is h firs rm in h infini gomric sris nd r is h rio bwn ny wo rms of h sris providd r is lss hn on. In his cs nd r /, hrfor h -rnsform of h uni sp funcion is - [ u ] Eqn..7 nd h sris convrgs providd >. Rcll L [ u ] s... -rnsform of h Exponnil Funcion h xponnil funcion is dfind s f Assuming h his funcion is smpld vry sconds, h vlu of h funcion h smpling insns is simply fk -k. h -rnsform of f is hrfor givn by [ f ] F... hrfor h -rnsform of h xponnil funcion is - [ f ] Eqn..8 Rcll L [ ] s No h us s h coninuous-im pol locion is dfind by s -, h locion of h discrim pol is dfind by -. Sinc boh nd r consns - is lso consn nd h locion of h discr-im pol is fixd obviously nough!!. Howvr h pol posiion in h //6 6 C.J. Downing &. O Mhony Dp. Elcronic Eng., CI,

5 Scion : h -rnsform Digil Conrol pln dpnds on h posiion of h pol in h s-pln AND on h smpling inrvl,. Hnc by vrying h smpling r i is possibl o vry h posiion of discr-im pol. -rnsform of h Uni Rmp Funcion h uni rmp funcion is dfind by f wih f k k k,,,... i.. h vlus of f h smpling insns r,,, 3, By pplying h dfiniion of h - rnsform, quion.6, w g h 3 F... 3 o obin h summion in closd form, considr F 3... Eqn Muliplying hrough by d nd ingring yilds F d... K 3 whr K is consn of ingrion. h rms in form sndrd infini gomric sris nd my sily b summd o yild F d K Diffrniing h bov wih rspc o yilds F Hnc h -rnsform of h uni rmp funcion is givn by [ f ] Eq.. whil L [ ] s Scond-Ordr Exmpl Find h -rnsform of h funcion whos Lplc rnsform is F s s s h simpls mhod of obining h -rnsform of h bov funcion is o spli h scond-ordr rnsfr funcion ino firs-ordr rnsfr funcions whos -rnsforms w know vi pril frcion xpnsion i.. b s s s s whr, b-. Hnc From quions.7 nd.8 w hv h F s s s 3//6 7 C.J. Downing &. O Mhony Dp. Elcronic Eng., CI,

6 Scion : h -rnsform Digil Conrol s ; s Hnc F No h h Lplc rnsform of f nd h -rnsform of f hv h sm numbr of roos, hnc h -rnsform prsrvs h ordr of h quion. -rnsform of Sinusoid h sinusoidl funcion sin f my lrnly b xprssd s f sin wih ] [ Hnc f ] [ cos sin h -rnsform of h sinusoid cos sin ] [sin Eq.. whil L ] [sin s -rnsform of h Squnc [ n ] h -rnsform of h squnc n is givn by... ] [ 3 3 n n n n his infini sum is 3//6 8 C.J. Downing &. O Mhony Dp. Elcronic Eng., CI,

7 Scion : h -rnsform Digil Conrol S / hrfor h -rnsform of h squnc [ n ] Eq.. In his scion svrl xmpls hv bn prsnd which illusr how o obin h -rnsform of im funcion f by dirc pplicion of h on-sidd -rnsform, quion.6. I should b nod h lrniv mhods of obining h -rnsform r lso possibl. A bl of commonly ncounrd -rnsforms is vry usful whn solving problms in h fild of discr-im conrol sysms; bl. prsns such bl. bl.: bl of commonly ncounrd -rnsforms 3//6 9 C.J. Downing &. O Mhony Dp. Elcronic Eng., CI,

8 Scion : h -rnsform Digil Conrol.4 horms & Propris Muliplicion by Consn If F is h -rnsform of f, hn [ f ] [ f ] F Eq.. whr is consn. o prov his, no h by dfiniion k [ f ] f k f k F k k k Linriy Propry h -rnsform posssss n imporn propry: linriy. his mns h if i is possibl o obin h -rnsform of wo funcions, x nd g, nd α nd β r sclrs, hn h funcion f formd by h linr combinion - f α x β g hs h -rnsform [ F ] α X β G Eq..3 Exrcis: Prov his. horm No. : Muliplicion by Exponnil Funcion Assuming h h funcion f hs h -rnsform F, hn h -rnsform of h funcion f Proof: is [ f ] F Eq..4 k k f f k k k f k F In words, o g h -rnsform of h funcion - f, on obins h -rnsform of f i.. F nd rplcs h vribl wih. As n xmpl, considr h uni sp funcion, u, whos - rnsform is [ u ] hrfor [ u ] U k 3//6 C.J. Downing &. O Mhony Dp. Elcronic Eng., CI,

9 Scion : h -rnsform Digil Conrol which is h sm s h obind prviously, s quion.8 i.. muliplicion by h uni sp funcion or uniy will no lr h rsul. horm No. : Muliplicion by Rmp Funcion If h funcion f, wih -rnsform F, is muliplid by h rmp funcion or h -rnsform of h combind funcion is givn by Proof: [ f ] df Eq..5 d df d f f f... d d f f 3 f f f 3f which is h -rnsform of f [compr wih quion.9]. Exmpl: Find h -rnsform of whr u is h uni sp funcion. d d [ u ] [ u ] du d [ u ] which gin is h sm s h obind prviously, quion.. horm No. 3: Iniil Vlu horm If f hs h -rnsform F nd if lim F xiss, hn h iniil vlu f of fis givn by f lim F Eq..6 Proof: f f F f... Clrly, king h limi s yilds 3//6 C.J. Downing &. O Mhony Dp. Elcronic Eng., CI,

10 Scion : h -rnsform Digil Conrol lim F f h bhviour of h signl in h nighbourhood of cn hus b drmind by h bhviour of F. h iniil vlu horm is convnin for chcking -rnsform clculions for possibl rrors. Sinc f is usully known, chck of h iniil vlu by lim F cn sily spo rrors in F, if ny xis. horm No. 4: Finl Vlu horm h finl vlu of f, h is, h vlu of f s is givn by [ F ] lim f lim Eq..7 horm No. 5: Rl rnslion Bckwrd Shif Considr Figur.4. f f f f f f f b Figur.4: Coninuous-im funcion dlyd by n smpls; Originl signl; b Dlyd vrsion h coninuous-im funcion hs vlus f, f, c. Whn dlyd by n smpls h rsuling funcion f-n hs vlus f n, f n, c. Now n n n Consqunly f f [ f ] f... [ f f f f n ]... n n n f f f... n n F Hnc n [ f n ] F Eq..8 horm No. 6: Rl rnslion Forwrd Shif h concp is illusrd by Figur.5. 3//6 C.J. Downing &. O Mhony Dp. Elcronic Eng., CI,

11 Scion : h -rnsform Digil Conrol f f f f f f f f3 b Figur.5: Coninuous-im funcion dvncd on smpl; Originl signl; b Advncd vrsion In h forwrd shif cs h signl is proc forwrd in im no rl physicl mning nd hnc h signl is shifd o h lf. In Figur.4 h coninuous-im signl is shifd o h lf by on smpling inrvl, o yild f. f hs h vlu f. f f 3 [ f ] f... f f f 3 [ f ] 3... f f f 3 [ f ] f f 3... And hrfor [ f ] f F [ f ] F f Eq..9 As spcil cs, if f, i.. ro iniil condiions pply, hn [ f ] F Eq.. nd muliplicion of h -rnsform of signl f by corrsponds o forwrd im shif of on smpling priod. Equion.9 cn sily b modifid o obin h following rlionship [ f ] [ f ] f F f f Eq.. Similrly, n n n n [ f n] F f f f... f n Eq...5 Conclusion In his scion w hv modlld squncs of smpls s sums of impuls funcions, h srngh of ch impuls corrsponding o h numricl vlu of h smpl i rprsnd.g. h squnc is modlld by 3,,,,,,, 3δ δ δ whr h impulss rprsn h smpls ims, nd rspcivly. o obin h corrsponding -rnsform, bl. my b uilisd. 3//6 3 C.J. Downing &. O Mhony Dp. Elcronic Eng., CI,

12 Scion : h -rnsform Digil Conrol Dscripion f Fs F Uni Impuls Uni Impuls Impuls of srngh w δ δ- -s - wδ- w -s w - bl.: Lplc nd -rnsform of Impuls Funcion hus ny smpld signl my b modlld using h -rnsform.g. for h squnc of smpls discussd bov on gs 3δ δ δ 3 3 Furhrmor h -rnsform my b usd o dscrib dlys.g. if h originl signl is dlyd by on smpling priod i is wrin s, 3,,,,,,, s 3δ δ δ 3 3 i.. h originl squnc muliplid by -. For his rson - is ofn rfrrd o s h dly opror or h bckwrd shif opror. 3 s.6 Exrciss Find h -rnsform of i cos ii 3 cos iii b n If show h G s s G b b s b b b 3 Wri down h -rnsform of h squnc u k {,,,,,, k,,,...6 rspcivly 4 Wri down h -rnsform of h dcying xponnil funcion f, smpld frquncy of H. 5 Find h -rnsform of h following smpld signls i rmp of slop, smpld vry scond ii dcying xponnil wih im consn of. sconds nd n iniil vlu of 5, smpld 5H. iii h smpld sp rspons of h sysm whos rnsfr funcion is s smpld 3H. 3//6 4 C.J. Downing &. O Mhony Dp. Elcronic Eng., CI,