Oulie Sabiliy Sab Sabiliy of Digial Syem Ieral Sabiliy Exeral Sabiliy Example Roo Locu v ime Repoe Fir Orer Seco Orer Sabiliy e Jury e Rouh Crierio Example Sabiliy A very impora propery of a yamic yem i i abiliy: Ieral abiliy Exeral abiliy Ieral abiliy i cocere wih he repoe of all ieral variable a.k.a ae. Exeral abiliy uie he ipu-oupu p behaviour of a yamic yem. Chaper ME 54 Chaper ME 54 Exeral Sabiliy Mo commo efiiio of appropriae repoe for exeral abiliy i ha for every boue ipu BI, he yamic yem houl have a boue oupu BO. If hi coiio i aifie, he yem i ai o be BIBO able. Sabiliy for Coiuou-ime Syem A yem i aympoically i able if i reur o i iiial equilibrium ae afer he applicaio of a impule. Sabiliy i a propery of he yem a i i NO epee o a pecific ipu or iiial coiio. A coiuou-ime ime yem CS i able if a oly if he real par of i pole are all ricly egaive. Pole mu be locae a lef-ha ie of he -plae! Chaper ME 54 4 Chaper ME 54 5
Mappig bewee - a -plae Sabiliy of Corol Syem Coier he rafer fucio of he cloe-loop corol yem how: Le Sice σ ω where ˆ ˆ e σ ω σ ω e e e σ e{ co 444 ω 4 i 44 ω Raiu Depic ui circle Chaper ME 54 6 Y c p B R A where Hece, A um{ } N c p A a N... a a N Nr Nc A a ri pi ci pi ci 44 i 4 4 i 444444444 real pole where N N r N c i calle he yem orer. complex cougae pole Chaper ME 54 7 c p Sabiliy y Co For a BIBO able yem, all pole [i.e. roo of characeriic polyomial A] mu lie iie i he ui circle. Syem wih omia pole reiig iie lefha emi-circular regio exhibi force ocillaio a half he amplig frequecy. Very ueirable feaure! Example pe Deermie he abiliy of he followig yem: 7 Y.658 6.6 R.98.96 Soluio: 7.64.98 Rearrage he rafer fucio i power of o - : 7 7 Y.658 6.6.64 R.98.96.98 7 7 Chaper ME 54 8 Chaper ME 54 9
Soluio Roo Co Soluio Malab Scrip Co he characeriic polyomial A become A.98.96.98 Malab fucio roo come hay a hi poi: roo[ -.98.96 -.98] Hece, he roo of A are fou o be p.55; p,.967 ±.499499 Sice p i ouie he ui circle, he yem i UNSABLE! *** Defie he icree-ime ime F he la argume - ell Malab ha he amplig ime i upecifie. f[.658e-7 6.6e-7.64e-7],[ -.98.96 -.98],-; *** Calculae impule repoe for amplig ep... [y,k] impule,; *** Plo he reul emk,y,'.'; xlabel'ime Iex'; ylabel'impule Repoe'; Chaper ME 54 Chaper ME 54 Soluio Repoe Co epoe Impule Re.45.4.5..5..5..55 For a ui impule ipu, i oupu grow wihou a bou! 4 6 8 ime Iex Chaper ME 54 x eig Sabiliy Liear ime- ivaria Syem x Ieal Sampler y y* y y y o e he abiliy, a ui impule i applie o he yem a i ime repoe i oberve. Le u aume ha he impule repoe i alo ample for ake of argume. Chaper ME 54
ime Repoe of a Fir Orer CS ime Repoe v. Roo Locu of CS Le he rafer fucio of a fir orer yem be a y y y he yem ha oe real roo: p -a. Sice X L{δ}, Y X a 5 4 herefore, he ime repoe become y y y L { Y } L { } e a Chaper ME 54 4 Chaper ME 54 5 ime Repoe v. Roo Locu of DS Sice y e -a, he ample repoe y* become * y y k e Hece, i Z-raform lea o ak Y Z e e { y k } a a Noe ha X. he correpoig icree-ime yem ha oly oe pole a p e -a. Roo Locu of DS Co Im -plae y* y* y* y* Re y* Force ocillaio 5 4 wih f / are oberve i hi y* regio! - y* 5 4 Chaper ME 54 6 Chaper ME 54 7
-plae Aalyi of Seco Orer CS C. Co. ^ co he rafer fucio of a eco-orer yem i expree a Y X Kω ζω ω Kω ζω ω ζω ω where ζ i he ampig raio; ω i he aural frequecy; ω i calle ampe aural frequecy: ω ˆ ω ζ -plae Aalyi Co.56.8. If he complex cougae pole are locae alog he lie, he ampig raio oe o chage. If he pole are locae o a paricular emi-circle, he aural frequecy i kep coa. Chaper ME 54 8 Chaper ME 54 9 -plae Aalyi of a Seco Orer DS he rafer fucio of a eco-orer icree-ime yem become Y X e b b ζω ζω co ω e he roo of he characeriic polyomial are ζω p, e [co ω ± i ω ] -plae Aalyi Co ζ ω Mappig i lighly more complicae! Number alog he raial irecio how ecreaig ampig raio. Number ao alog he perimeer of he ui circle iicae coa aural frequecy. Chaper ME 54 Chaper ME 54
Malab Scrip o Prouce gri Sabiliy Crieria w lipace,pi,'; ea lipace,.9,'; p ero,; p p; qr-; Plo coa ampig coour for i : w*-eai/qr-eai^ ; w*-eai/qr-eai^ - ; p:,i exp; p:,i exp; e cloe all; plo[p p]; hol o xlabel'real'; ylabel'imagiary'; axi'quare' Plo coa w* coour ea lipace,,'; w lipacepi/,pi,'; for i : wi*-ea *qr-ea.^; wi*-ea - *qr-ea.^; p:,i exp; p:,i exp; e plo[p p]; gri Compario! Moer compuer program uch a Malab irecly give he pole of a paricular rafer fucio. Help u eermie he abiliy of ha yem righ away! Someime, he corol egieer wih o e for abiliy for a eire cla of yem: Coroller parameer gai are uually embee io he coefficie of he characeriic polyomial. he rage of corol parameer eig regio ha yiel a able yem are ough. I uch cae, abiliy e are uilie: Jury e w raform alog wih he Rouh-Hurwi Crierio Chaper ME 54 Chaper ME 54 Jury e Develope by Jury a Blachar i 96 for icree-ime yem. A Jury array i forme employig he coefficie of characeriic polyomial A. Mo of he array eleme are calculae via a umerical compuaio i proceure. Similar o he Rouh crierio, cerai eleme of he array mu be all o-egaive egaive o ge a able yem. Uforuaely, he calculaio of array eleme ca be quie complicae a eiou! Chaper ME 54 4 Rouh Crierio for Dicree-ime Syem o apply he well-kow Rouh Crierio, a coiuou-ime equivale for he icree-ime yem mu be obaie hrough ui raformaio: Some egieer replace by w a call he reulig operaio a w raform which ae he fac ha he equaio above o o irecly ake u back o he origial -plae! Ayhow, Y R F B' F' A' where F i he cloe-loop rafer fucio. he coefficie of A i ue o form Rouh array. Chaper ME 54 5
Example Example Co Deermie he abiliy of he followig yem wih ec uig Rouh crierio: Soluio: Y.967 F.484484 R.948 ui biliear raformaio lea o F F'.484.967.948 hu, F'.999 he characeriic polyomial become A'. Rouh Array: :. All fir-colum coefficie are : bigger ha ero yem i margially able. : ε Chaper ME 54 6 Chaper ME 54 7