TIME RESPONSE I hi chapr you will lar h followig: How o fid im rpo from h rafr fucio How o u pol ad zro o drmi h rpo of a corol ym How o dcrib h rai rpo of fir- ad cod-ordr ym How o viw h ffc of oliarii o h ym im rpo How o fid im rpo from h a-pac rpraio
Thi lcur i dvod o h aalyi of ym rai rpo. Fir w mu dcrib om valuabl aalyi ad dig ool, pol, zro ad ym rpo. Th pol of a rafr fucio ar h valu of h Laplac raform variabl,, ha cau h rafr fucio o bcom ifii or ay roo of h domiaor of h rafr fucio ha ar commo roo of h umraor. Th zro of a rafr fuio ar h valu of h Laplac raform variabl,, ha cau h rafr fucio bcom zro, or ay roo of h umraor of h rafr fucio ha ar commo roo of h domiaor. POLES AND ZEROS OF A FIRST ORDER SYSTEM : AN EXAMPLE Giv h rafr fucio G i Figur a, a pol xi a -5 ad a zro xi a -. h valu ar plod o h complx a pla i Figurb uig a x for h pol ad aοfor h zro. Two how h propri of h pol ad zro, l u fid h ui p rpo of h ym. Muliplyig h rafr fucio of Figura by a p fucio yild A B / 5 3/ 5 whr ad 3 A B C 5 5 5 5 Thu, c 5 3 5 5 5 All of h ar ummarizd i h follwig figurc ad w draw h followig cocluio : 5
Cocluio :. A pol of h ipu fucio gra h form of h forcd rpo ha i, h pol a h origi grad a p fucio a h oupu.. A pol of h rafr fucio gra h form of h aural rpo ha i, h pol a -5 grad -5. 3. A pol o h ral axi gra a xpoial rpo of h form -α whr α i h pol locaio o h ral axi. Thu, h farhr o h lf a pol i o h gaiv ral axi, h far h xpoial rai rpo will dcay o zro agai, h pol a -5 grad -5. Th zro ad pol gra h ampliud for boh h forcd ad aural rpo L u look a a xamplha dmora h chiqu of uig pol o obai h form of h ym rpo
Exampl : A ym ha a rafr fucio G 7 6 8 Wri, by ipcio, h oupu, c, i gral rm if h ipu i a ui p. Awr : c A B - C -7 D -8 E - By ow, w lard ha pol drmi h aur of h im rpo: Pol of h ipu fucio drmi h form of h forcd rpo, ad pol of h rafr fucio drmi h h form of h aural rpo. Zro ad pol of ipu or rafr fucio coribu o h ampliud of h compo par of h oal rpo. Fially, pol o yh ral axi gra xpoial rpo. FIRST-ORDER SYSTEMS W ow dicu fir-ordr ym wihou zro o dfi a prformac pcificaio for uch a ym. A fir-ordr wihou zro ca b dcribd by h rafr fuio giv i h figura. If h ipu i a ui p, whr R /, h Laplac raform of h p rpo i C, whr a C R G. Takig h ivr Laplac a raform, h p rpo i giv by c c c f whr h ipu pol a h origi grad h forcd rpo c f, ad h ym pol a a, a how i Figurb. a
a c c f c * L u xami h igificac of paramr a, h oly paramr dd o dcrib h rai rpo. Wh /a, a a.37 ** or c.37.63 *** / a / a / a W ow u h quaio o dfi hr rai rpo pcificaio. Tim Coa : W call /a h im coa of h rpo. From quaio *, h im coa ca b dcribd a h im for -a o dcay o 37% of i iiial im. Alraly, from quaio **, h im coa i h im i ak for h p rpo o ri o 63% of i fial valu. Thu, w ca call h paramr a a xpoial frqucy. Ri Tim, T r : Ri im i dfid a h im for h wavform o go from. o.9 of i fial valu. Ri im i foud by olvig quaio * ad olvig for im a c.9 ad c.. Hc T.3.. r a a a Slig Tim, T : Slig im i dfid a h im for h rpo o rach ad ay wihi % of i fial valu. Lig c.98 i quaio * ad olvig for, w fid h lig im o b T a
SECOND-ORDER SYSTEMS : INTRODUCTION L u ow xd h cocp of pol, zro ad rai rpo o cod ordr ym. Compard o h impliciy of a fir ordrym, a cod ordr ym xhibi a wid rag of rpo ha mu b aalyzd ad dcribd. To bcom familiar wih h wid rag of rpo bfor formalizig our dicuio, w ak a look a umrical xampl of h cod ordr ym rpo how i h figur. All xampl ar drivd from Figura, h gral ca, which ha wo fii pol ad o zro. By aigig appropria valu o paramr a ad b, w ca how all poibl cod,ordr rai rpo. W ow xplai ach rpo ad how how w ca u h pol o drmi h aur of h rpo wihou goig hrough h procdur of a parial-fracio xpaio followd by h ivr Laplac raform.
Ovrdampd Rpo : For hi rpo, C 9 9 9 9 7.85.6 Thi fucio ha a pol a h origi ha com from h ui p ipu ad wo ral pol ha com from h ym. Th ipu pol a h origi gra h coa forcd rpo; ach of wo ym pol o h ral axi gra a xpoial aural frqucy. Hc h oupu could hav b wri a c K K -7.85 K 3 -.6. Thi rpo how i figur calld ovrdampd. W ha h pol ll u h form of h rpo wihou h diou calculaio of h ivr Laplac raform. 9 Udrdampd Rpo : For hi rpo, C 9 Thi fucio ha a pol a h origi ha com from h ui p ad wo complx pol ha com from h ym. Th pol ha gra h aural rpo ar a -±j 8. Th ral par of h pol mach h xpoial dcay frqucy o iuoid ampliud, whil h imagiary par of h pol mach h frqucy of h ioidal ocillaio. Th iuoidal frqucy i giv h am of dampd frqucy of ocillaio, ω d. Thi rpo how i figur calld udrdampd.
Udrdampd Rpo : For hi rpo, 9 C 9 Thi fucio ha a pol a h origi ad wo imagiary pol. Th pol a h origi gra h coa forcd rpo, ad h wo ym pol o h imagiary axi a ±j3 gra a Siuoidal aural rpo. Hc, h oupu ca b imad a ck K co3-φ Thi yp of rpo i calld udampd. Criically Dampd Rpo : For hi rpo, C 9 6 9 9 3 Thi fucio ha a pol a h origi ad wo mulipl ral pol. Th ipu pol a h origi gra h coa forcd rpo, ad wo pol a h ral axi a -3 gra a aural xpoial rpo. Hc h oupu ca b imad a c K K -3 K 3-3 W ow ummariz our obrvaio. I hi cio w dfid h followig aural rpo ad foud hir characriic.
. Ovrdampd rpo : Pol : Two ral a σ, -σ Naural rpo : Two xpoial wih im coa qual o h rciprocal of h σ σ pol locaio, or c K K. Udrdampd rpo : Pol : Two complx a -σ d ±jω d Naural rpo : Dampd ioidal wih a xpoial vlop, or σ d c A co ω φ d 3. Udampd rpo : Pol : Two imagiary a ±jω Naural rpo : Udampd ioid, or c A co ω φ. Criically dampd rpo : Pol : wo ral a σ Naural rpo : Two xpoial rm, or c K K σ σ Figur. Sp rpo for cod-ordr ym dampig ca
THE GENERAL SECOND-ORDER SYSTEM I hi cio w will dfi wo phyically maigful pcificaio for cod-ordr ym. Naural Frqucy, ω : Th aural frqucy of a cod ordr ym i h frqucy of ocillaio of h ym wihou dampig. For xampl, h frqucy of ocillaio of a ri RLC circui wih h riac hord would b aural frqucy. Dampig Raio, ζ : W dfi h dampig raio, ζ, o b Expoial dcay frqucy Naural priod cod ζ Naural frqucy rad/c π Expoial im coa L u ow rvi our dcripio of h cod-ordr ym o rflc h w dfiiio. A gral cod-ordr ym ca b raformd o how h quaii ζ adω. Coidr h gral ym G b a b Wihou dampig, h pol would b o h jω axi, ad h rpo would b a b udampd iuoid. For h pol o b purly imagiary, a. Hc G By dfiiio, h aural frqucy, ω, i h frqucy of ocillaio of hi ym. Sic h pol of h ym ar o jω axi a ±jω b, ω b ad b ω. Now, wha i h valu of a? b
Aumig h udrdampd ym, h complx pol hav a ral par, σ, qual o -a/. Th magiud of hi valu i h xpoial dcay frqucy. Hc, Expoial dcay frqucy σ a/ ζ Naural frqucyrad/cod ω ω from which a ζω. Our gral cod ordr rafr fucio fially look lik hi: G ω ζω ω I h followig xampl w fid umrical valu of ζ adω by machig h rafr fucio o fial quaio w foud abov 36 Exampl : Giv h rafr fucio, fidζadω. G. 36 Soluio : No haω 36 adω 6. Alo, ζω ad ζ.35 W hav dfid ζ ad ω, l u rla h quaii o h pol locaio. Solvig for h pol of h rafr fucio i quaio ω G ζω ω yild ζω ± ω ζ,
ζω ω ζ, ± From h quaio abov, w ha h variou ca of cod-ordr rpo ar a fucio of ζ ; hy ar ummarizd i h abl blow. Figur. Scod-ordr rpo a a fucio of dampig raio
I h followig xampl w will fid h umrical valu of ζ ad drmi h aur of rai rpo Exampl : For ach of h ym how i h figur, fid h valu of ζ ad rpor h kid of rpo xpcd. Soluio : Fir mach h form of h ym o h form how i quaio b ω G ad G a b ζω ω Sic a ζω adω b ζa/ b. Uig h valu of a ad b from ach of h ym of h figur, w fidζ.55 for yma, which i ovrdampd, ic ζ>; ζ for ymb, which i hu criically dampd; adζ.89 for ymc, which i hu udrdampd, ic ζ<.
Udrdampd Scod Ordr Sym I addiio oζadω, mor pcificaio idigou o h udrdampd ca will b dfid. Th pcificaio ar dfid a follow: Figur. Scod-ordr udrdampd rpo pcificaio. Ri im, T r : h im rquird for h wavform o go from. of h fial valu o.9 of h fial valu.. Pak im, T p : Th im rquird o rach h fir, or maximum, pak. 3. Prc ovrhoo, %OS : h amou ha h wavform ovrhoo h ady-a, or fial, valu a h pak im, xprd a a prcag of h ady-a valu.. Slig im, T : Th im rquird for h rai dampd acillaio o rach ad ay wihi ±% of h ady-a valu.
Evaluaio of T p, %OS ad T : To fid h valu of h paramr, w u h quaio of π ζπ / ζ T p ω ζ %OS T ζω W ca u h graphic i figur.6 o obai h valu of ri im, T r. Figur.6 Normalizd ri im v. dampig raio for a cod-ordr udrdampd rpo L u look a xampl
Exampl : Giv h rafr fucio G, fid T p, %OS, T ad T r. 5 Soluio : ω ad ζ.7. Uig h rlad quaio, T p.75 cod, %OS.838, T.533. Uig h rlad graphic, ormalid ri im i approximaly.3 cod. Dividig byω yild T r.3 cod. W ow hav xprio ha rla pak im, prc ovrhoo ad lig im o h aural frqucy ad h dampig raio. Now l u rla h quaii o h locaio of pol ha gra h characriic. Th pol plo for a gral udrdampd cod-ordr ym i i figur blow. W from h Pyhagora horm ha h radial diac from h origi o h pol i h aural frqucy, ω, ad h coθ ζ. Now comparig h quaio w hav valuad formrly wih h pol locaio, w valua pak im ad lig im i rm of pol locaio : T p ω π π ω T ζ d ζω σ d whr ω d i h imagiary par of h pol ad i calld h dampd frqucy of ocillaio, adσ d i h magiud of h ral par of pol ad i xpoial dampig frqucy
Exampl : Giv h pol plo i figur, fidζ, ω, T p, %OS ad T. Soluio : Th dampig raio i giv by ζcoθco[arca7/3].39. Th aural frqucy, ω, i h radial diac from h origi o h pol, or ω 7 3 7.66. Th pak im i T p π π ω 7.9 Th prc ovrhoo i d cod ζπ / ζ OS 6 Th approxima lig im i T σ d 3.333 cod
Exampl : Giv h ym how i figur, fid J ad D o yild % ovrhoo ad a lig im of cod for a p ipu of orqu T. Soluio : Fir, h rafr fucio for h ym i / J G From h rafr fucio, ω ad Bu, from h problm am, or. Hc ζω. ω J K T ζω ζω D J K J K J D J ζω ζπ / ζ Alo,. W kow h formulaio for ovrhoo, which i ζ OS If w drawζfrom hi quaio, w fid ha a % ovrhoo impliζ.56. Thrfor D J ζ ω J K J K.5 From h problm am, K 5 N-m/rad. Combiig hi valu wih quaio D J ζω ad.5 which w hav foud abov, w g J K D. N-m-/rad ad J.6kg-m
SYSTEM RESPONSE WITH ADDITIONAL POLES AND ZEROS By ow, w aalyzd ym wih o or wo pol ad wih o zro. If a ym ha mor ha wo pol or zro w ca o u h formulaio w drivd for calculaio of prformac pcificaio. Bu, udr crai codiio, a ym wih mor ha wo pol or zro ca b approximad a a cod ordr ym ha ha ju wo domia pol. If h ral pol i fiv im farhr o h lf ha wo complxcojuga pol pair, w ca ay hi complx-cojuga pol pair i domia pol pair. A way o look a h ffc of a addiioal zroi a follow. L C b h rpo of a ym, T, wih uiy i h umraor. If w add a zro o h rafr fucio, yildig at, h Laplac raform of h rpo will b ac C ac Thu, a rpo of a ym wih a zro coi of wo par : Th drivaiv of h origial rpo ad a cald vrio of h origial rpo. If a i gaiv, a irig phomo occur. Th oupu may b oppoi ig of ipu a how i figur. Alog fir.6 cod, h oupu ha gaiv ig alhough h ipu ha poiiv ig p ipu. A ym ha xhibi hi phomo i kow a a omiimum pha ym. If a airpla wa a omiimum pha ym, i would iiially vr lf wh commadd o r righ.
EFFECT OF NONLINEARITIES UPON TIME RESPONSE I hi cio w will ir oliarii, uch a auraio, dad zo ad backlah io a ym o how h ffc of h oliarii upo h liar rpo. Th rpo wr obaid uig Simulik. L u aum h moor ad load from a Aa Poiio Corol Sym. Th ipu of h ym i applid volag o moor, ad h oupu of h ym i load agular vlociy. L u add a auraio±5 V o ym. A how i figur, h ffc of auraio i o limi h obaid vlociy. L u xami h ffc of dadzo oliariy.
Add a dadzo - o o ym ad compar h ym rpo wih ad wihou h dadzo oliariy. No ha h ipu i iuoidal. No ha, h rpo bgi wh h ipu volag hrhold. o h moor xcd a L u xami h ffc of backlah oliariy.
Add a backlah o ym ad compar h ym rpo wih ad wihou h backlah oliariy. No ha h ipu i iuoidal. A h moor rvr dircio, h oupu haf rmai aioary whil h moor bgi o rvr. Th rulig rpo i qui diffr from h liar rpo wihou backlah.
LAPLACE TRANSFORM SOLUTION OF STATE EQUATIONS I hi cio, w will u h Laplac raform o olv h a quaio for h a ad oupu vcor. Coidr h a ad oupu quaio Takig h Laplac raform of boh id of h a quaio yild X x AX BU x y Ax Bu Cx I ordr o pra X, rplac X wih IX, whr I i a * idiy marix, ad i h ordr of h ym. Combiig all of h X rm, w g I A X x BU Solvig for X by prmuliplyig boh id of h la quaio by I-A - yild Du X I A adj I d I x I A A [ x BU ] A BU Tkaig h Laplac raform of h oupu quaio yild Y CX DU
Eigvalu ad Trafr Fucio Pol : W aw ha h pol of h rafr fucio drmi h aur of h rai rpo of h ym. I hr a quival quaiy i h a-pac rpraio ha yild h am iformaio? Th roo of di-a i h igvalu of h ym marix A. L u how ha h igvalu ar qual o h pol of h ym rafr fucio. L h oupu, Y, ad h ipu, U, b calar quaii Y ad U, rpcivly. Furhr, coform h dfiiio of a rafr fucio, l x, h iiial a vcor, qual, h ull vcor. Subiuig h quaio X I A x I A BU adj I d I A [ x BU ] A io quaio Y CX DU ad olvig for h rafr fucio Y/U yild Y adj I A C B D U d I A Cadj I A B D d I A d I A Th roo of h domiaor of hi quaio ar h pol of h ym. Sic, h roo of X ad Y/U ar idical, h ym pol qual h igvalu. Hc, if a ym i rprd i a-pac, w ca fid h pol from di-a. Th followig xampl dmora olvig h a quaio uig h Laplac raform a wll a fidig h igvalu ad ym pol.
Exampl : Giv h ym rprd i a,pac by quaio [ ]x y x x x, 9 6 do h followig : a. Solv h prcdig a quaio ad obai h oupu for h giv ipu. b. Fid h igvalu ad h ym pol. Soluio : a Rmmbr h quaio 6 9 6 9 9 6 9 9 6 3 A I A I U, BU A I x A I X 3 3 3 9 37 3 3 X X X w g [ ] Y X X X X X Y 3-3 3.5 9-6.5 y raform : Laplac ivr h Takig a -. zro a cacld a - pol a whr.5 3 9 6.5 3 5 6 b Th roo of di-a giv u boh h pol of h ym ad h igvalu which ar -, -3 ad -.
TIME DOMAIN SOLUTION OF STATE EQUATIONS W ow look a aohr chiqu for olvig h a quaio. Rahr ha uig h Laplac raform, w olv h quaio dircly i h im domai uig a mhod cloly allid o claical oluio of diffrial quaio. W will fid ha h fial oluio coi of wo par ha ar diffr from h forcd ad aural rpo. Th oluio i h im domai i giv dircly by x A x Φ x A τ Bu τ dτ Φ τ Bu τ d τ whrφ A by dfiiio, ad which i calld h a-raiio marix. Noic ha h fir rmo h righ had id of h quaio i h rpo du o h iiial a vcor x. Noic alo ha i i h oly rm dpd o h iiial a vcor ad o h ipu. W call hi par of h rpo h zro-ipu rpo, ic i i h oal rpo if h ipu i zro. Th cod rm, calld h covoluio igral, i dpd oly o h ipu, u, ad h ipu marix, B, o h iiial a vcor. W call hi par of rpo h zro-a rpo, ic i i oal rpo if h iiial a vcor i zro.
L u xami h form h lm of Φ ak for liar, im ivaria ym. Th quaio X I A x I A BU, i h raform of Φx, zro ipu rpo. Thu, for h uforcd ym, ak h Laplac raform of x yild XI-A - x From which w ca ha I-A - i h Laplac raform of h a-raiio marix. W hav alrady ha h domiaor of I-A - i a polyomial i who roo ar h ym pol. Thi polyomial i foud from h quaio di- A. Sic, adj I A L [ I A ] L Φ d I A whr L - rpr h ivr Laplac raform opraor.
Exampl : For h a quaio ad iiial a vcor how i h quaio, whr u i a ui p, fid h a raiio marix ad h olv for x. x, 6 8 u x x Soluio : Fid h igvalu uig di-a. 68, from which - ad -. Sic ach rm of h a-raiio marix i h um of rpo grad by h pol or igvalu, w aum a a-raiio marix of h form Φ 8 7 6 5 3 K K K K K K K K WihΦ, w g 8 imulaou quaio o fid h valu of h coa K i. Afr h valu of all coa ar calculad, w g Φ Φ τ τ τ τ τ B Hc h fir rm of h a raiio marix will b Φ x ad h cod rm will b Φ d Bu 8 8 τ τ Th fial rul i foud a Φ Φ d Bu x x 7 7 8 7 7 8 τ τ τ