Phase plane method is an important graphical methods to deal with problems related to a second-order autonomous system.

Transcription

1 NCTU Dpam of Elcical ad Compu Egiig Sio Cous <Dyamic Sysm alysis ad Simulaio By Pof. Yo-Pig Ch 8. Phas Pla Mhod. Phas Pla B. Naual Mods of TI Sysms C. Phas Poais Phas pla mhod is a impoa gaphical mhods o dal wih poblms lad o a scod-od auoomous sysm.. Phas Pla Fis, l s ioduc h basic cocps of h phas pla by a scod-od auoomous sysm which is mahmaically pssd as blow: & ( ) f ( ( ) ( ) ), ( ), & ( ) f( ( ), ( ) ), ( ) (8-) (8-) wihou ay ipus. I is kow ha h sysm sas () ad () a uiquly dmid by iiial alus ( ) ad ( ) codiios ( ) ad ( ). Claly, diff iiial will sul i diff sas () ad () fo. If w adop h alus of h wo sa aiabls as wo as, h a coodia is fomd i Figu 8-, calld h phas pla. W ca daw all h pois ( (), ()) fo o obai a ajcoy wih spc o spcifid iiial codiios ( ) ad ( ). Figu 8- shows a ampl of h ajcoy ( (), ()) fo <<5, wh a aow is usd o show h dicio of h ajcoy fom o 5 sc. ( (), ()) ( (3), (3)) 3 ( (4), (4)) 4 ( (), ()) 5 ( (5), (5)) ( (), ()) Figu 8-8-

2 NCTU Dpam of Elcical ad Compu Egiig Sio Cous <Dyamic Sysm alysis ad Simulaio By Pof. Yo-Pig Ch Sic ach ajcoy is uiquly dmid by a s of iiial codiios, ha mas o iscio iss amog all h ajcois, cp a a sigula poi (, ) saisfyig & ( ) ad ( ) ( ) & fo all, i.., f (8-3),, ( ) f (8-4),, I fac, (, ) is a quilibium poi ad o cssay o b uiqu. s of ajcois obaid fom diff iiial codiios is calld a phas poai, as show i Figu 8-, wh P is a quilibium poi o a sigula poi. P Figu 8- B. Naual Mods of TI Sysms Th aual mods flcs h iisic popis of sysms which a o cid by ay al ipus. I oh wods, h aual mods a h sysm bhaio causd oly by is iiial codiios. s mploy h simpls d od dyamic sysm as a ampl o plai how o dmi h aual mods of a TI sysm. Cosid h followig d od TI sysm wihou ay ipu: o i a mai fom ( ) a ( ) a ( ) &, () (8-5) ( ) a ( ) a ( ) &, () (8-6) &, (8-7) 8-

3 wh a a, a a ( ) ( ) NCTU Dpam of Elcical ad Compu Egiig Sio Cous <Dyamic Sysm alysis ad Simulaio By Pof. Yo-Pig Ch ad. Claly, h oigi (,) i h phas pla is a quilibium poi sic h uh of lads o &. Fom (-8), h sa co ca b sold as (8-8) wih dfid as I (8-9) a fom simila o. I is kow ha ay squa mai ca b asfomd io a mai i diagoal fom o oda fom, pssd as (8-) wh is iibl. I is asy o show ha Th a h ypical foms of, dod as,,, (8-) c, o (8-) c wh h disgoal ms a h igalus of ad ca b sold fom is chaacisic quaio a a I (8-3) a a Tha mas hy a h soluios of ( a a ) a a a a (8-4) ad boh of hm may b al o compl i cojuga ad may b disic o pad. ccodigly, h a h cass of h igalus: Cas- disic al igalus, Cas- pad al igalus, Cas-3 compl igalus i cojuga, α jω ad α jω. c I is kow ha fo a igalu i h iss a las o igco i such ha (8-5) i i i c 8-3

4 NCTU Dpam of Elcical ad Compu Egiig Sio Cous <Dyamic Sysm alysis ad Simulaio By Pof. Yo-Pig Ch 8-4 Fo Cas-, basd o (8-5) w ha (8-6) which suls i [ ] [ ] (8-7) No ha is h fis ypical fom i (8-) ad is iibl ad calld h igcomai sic i coais all h igcos of. Claly, (8-7) ca b aagd as, a fom gi i (8-). Bsids, w ha (8-8) Hc, h sa co i (8-8) is obaid as I I (8-9) o i h followig fom ( ) α α (8-) ( ) α α (8-) wh α, α, α ad α dpd o () ad (). Th fucios ad a calld h aual mods of his sysm. Fo Cas- of pad igalu, w may obai o o wo igcos cospodigly. s assum oly o igco is foud, h w ha (8-)

5 NCTU Dpam of Elcical ad Compu Egiig Sio Cous <Dyamic Sysm alysis ad Simulaio By Pof. Yo-Pig Ch 8-5 I is kow ha w ca ca a w co which saisfis (8-3) ad mos impoaly ad a idpd. s a sul, ww aai [ ] [ ] (8-4) No ha is h scod ypical fom i (8-) ad is iibl sic ad a idpd. Claly, (8-4) ca b aagd as, sam as (8-). Bsids, w ha (8-5) Hc, h sa co i (8-8) is obaid as 3 I I (8-6) o i h followig fom ( ) β γ, (8-7) ( ) β γ, (8-8) wh γ, γ, β ad β dpd o () ad (). Th fucios ad a calld h aual mods of his sysm. Fo Cas-3 of compl igalus i cojuga, w will skip h diaio ad jus show h suld sa aiabls which a ( ) si B cos ω ω α α (8-9) ( ) si B cos ω ω α α (8-3)

6 NCTU Dpam of Elcical ad Compu Egiig Sio Cous <Dyamic Sysm alysis ad Simulaio By Pof. Yo-Pig Ch wh,, B ad B dpd o () ad (). Th fucios α siω α cosω a calld h aual mods of his sysm. No ha if α, i.., h igalus a pu imagiay, h h aual mods a cosω ad siω. I is cla ha all h aual mods dpd o h igalus of h sysm mai. If a igalu is locad o h lf half compl pla, h h aual mod cospodig his igalu will cog o as. O h oh had, if a igalu is locad o h igh half compl pla, h h lad aual mod will icas o as. s fo h igalu o h imagiay ais, is aual mod will oscilla. ad C. Phas Poais Now, l s us som ampls o daw phas poais, a s of ajcois, ad discuss hi lad popis. cually, diff aual mods will sul i diff ajcois ad diff phas poais as wll. Cosid h scod-od TI sysm i caoical fom, which is dscibd as blow: ( ) ( ) &, () (8-3) ( ) a ( ) a ( ) &, () (8-3) Th sysm mai is a a ad h chaacisic quaio is I a a (8-33) a a whos oos a a ± a 4a, (8-34) If a a, h ad a al ad disic. Wihou loss of galiy, 4 assum, h h a h cass lisd as blow: I. < <, i.., a a, a < ad a <. 4 II., i.., a a, a ad a <

7 III., i.., a a ad a. 4 NCTU Dpam of Elcical ad Compu Egiig Sio Cous <Dyamic Sysm alysis ad Simulaio By Pof. Yo-Pig Ch Thi phas poais a dpicd i Figu 8-3(a)(b)(c), spcily. Fom (8-), h fis sa aiabl ( ) is ( ) α α (8-35) ad fom (8-3), h scod sa aiabl is ( ) ( ) α α & (8-36) Fo som iiial codiios () ad (), h coffici α i (8-35) may b qual o, h w ha ( ) α (8-37) which implis ( ) α ( ) (8-38) ( ) ( ) (8-39) i.., h lad ajcoy is h saigh li show i Figu 8-3(a)(b)(c). Similaly, if iiial codiios sul i α, h ( ) α (8-4) which implis ( ) α ( ) (8-4) ( ) ( ) (8-4) i.., h lad ajcoy is h saigh li show i Figu 8-3(a)(b)(c). I Cas-I wih < <, is phas poai is dpicd i Figu 8-3(a). s, du o h fac ha h phas poai mus saisfy ad, all h ajcois cp i ( ) α (8-43) ( ) α ( ) (8-44) i.., ( ) ( ). I oh wods, all h ajcois cp i h phas poai will appoach as. Claly, i is a sabl sysm sic h sysm sa will cog o h quilibium poi (,) as. 8-7

8 NCTU Dpam of Elcical ad Compu Egiig Sio Cous <Dyamic Sysm alysis ad Simulaio By Pof. Yo-Pig Ch (a) (b) (c) Figu 8-3 I Cas-II ad Cas III, hi phas poais a dpicd i Figu 8-3(b) ad Figu 8-3(c). s, du o h fac ha ajcois cp i h phas poai mus saisfy ( ) α ad, all h (8-45) ( ) α ( ) (8-46) i.., ( ) ( ). I oh wods, all h ajcois cp i h phas poai will appoach as. Claly, i is a usabl sysm sic h sysm sa will mo o (, ) o (, ) as, cp i Cas-II. How, Cas-II is sill ad as usabl bcaus ay mly small disubac will di h sysm o la ad mo o (, ) o (, ). (a) Figu 8-4 (b) If a a, h fom (8-34) w ha 4 lisd as blow: I. <, i.., a a ad a <. 4 a. Th a wo cass 8-8

9 ., i.., a a ad a. 4 NCTU Dpam of Elcical ad Compu Egiig Sio Cous <Dyamic Sysm alysis ad Simulaio By Pof. Yo-Pig Ch Boh phas poais a dpicd i Figu 8-4(a)(b), spcily. I fac, hy a spcial cass of Cas-I ad Cas-II wh ad a mgd ogh. Fom h phas poais, i is asy o coclud ha Cas-I is sabl ad Cas- is usabl. (a) (b) (c) Figu 8-5 If a a, h ad a compl i cojuga. Th a wo cass lisd as blow: 4 < I., ±jω wh ω, ω o ω<. II., α±jω wh α o α<. Thi phas poais a dpicd i Figu-5(a)(b)(c) ad Figu-6(a)(b). Fo Cas-I, fom (8-9) wih α, h fis sa aiabl is ( ) cosω B siω C cos( ω θ ) (8-47) ad fom (8-3) h scod sa aiabl is ( ) ( ) Cω si( ω θ ) & (8-48) wh C is dmid by h iiial codiios () ad (). Claly, i is a oscillaoy sysm. Fom (8-47) ad (8-48), w ha ( ) (8-49) ω ( ) C whos phas poais a llipss o cicls, show i Figu 8-5(a)(b)(c) spcily fo ω, ω ad ω<. No ha h aow i hs figus is poiig fom lf o igh i h gio () sic & ( ) ( ), i.., ( ) is icasd as icass, ad fom igh o lf i h gio ()< sic & ( ) ( ), i.., ( ) < is dcasd as icass. Whil passig hough h ( ) ais, i.., ( ) ( ) h aow gos dow o up ically. &, 8-9

10 NCTU Dpam of Elcical ad Compu Egiig Sio Cous <Dyamic Sysm alysis ad Simulaio By Pof. Yo-Pig Ch (a) Figu 8-6 (b) Th abo dscipio of h aow dicio is also suiabl fo Cas-I o Cas-. Fo Cas-II, fom (3), h fis sa aiabl is α α ( ) ( cosω B siω) C cos( ω θ ) (8-5) ad fom (8-3) h scod sa aiabl is α α ( ) ( ) C ω si( ω θ ) Cα cos( ω θ ) α C cos( ω θ ) & (8-5) Claly, i is a dig oscillaoy sysm fo α as show i Figu 8-6(a) ad a auad oscillaoy sysm fo α< as show i Figu 8-6(b). Ulik a TI sysm, h phas poai of a olia sysm is of mo complicad sic i may ha mulipl quilibium pois o limi cycls. How, h local bhaio of a coiuous olia sysm aoud a quilibium poi ca b dmid by liaizaio which suls i a appoimd TI sysm a o h quilibium poi. If h appoimad TI sysm is sabl h h quilibium poi is sabl, ohwis i is usabl. Now, cosid a olia sysm ad dmi is sabiliy by h us of phas pla mhod. h sa quaios b & & ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (8-5) If ( ) ( ), h i is asy o sol h sa aiabls as ( ) cos( θ ) ad ( ) si( θ ) ( ) ( ). I oh wods, if h sysm is iiially locad o h cicl, h h sysm sa will kp i a cicula ajcoy. Whil h iiial codiio is a lil diad fom h cicl, fo ampl ( ) ( ) ε, 8-

11 NCTU Dpam of Elcical ad Compu Egiig Sio Cous <Dyamic Sysm alysis ad Simulaio By Pof. Yo-Pig Ch is bhaio is h god by & & ( ) ε( ) ( ) ( ) ( ) ε ( ) (8-53) ε whos sysm mai is. Cospodigly, is chaacisic quaio is ε ε (8-54) ε ± 4ε 4 ad h igalus a, ε ± j. Claly, fo ε h sysm is i h gio ousid h cicl ad bcoms usabl, i.., i mos away h cicl o h ifiiy. Fo ε< h sysm is i h gio isid h cicl ad bcoms sabl, i.., i mos away h cicl o h oigi. Figu 8-7 shows h phas poai icludig a limi cycl ( ) ( ). ( ) ( ) Figu 8-7 quaios N, l s cosid a aiabl sucu sysm, dscibd by h followig sa & & ( ) ( ) ( ) (. 5. 5sig( ( ) ( ) )) ( ) (8-55) I is obious ha h a wo sucus of h sysm, o fo ( ) ( ) ad h oh fo ( ) ( ). No ha h sysm is udfid fo ( ) ( ) < ha ( ) ( ) is god by. I cas, i.., i h fis ad hid quadas of h phas pla, h sysm 8-

12 & & ( ) ( ) ( ) 4 ( ) which pfoms as a llips. Whil ( ) ( ) NCTU Dpam of Elcical ad Compu Egiig Sio Cous <Dyamic Sysm alysis ad Simulaio By Pof. Yo-Pig Ch < quadas of h phas pla, h sysm is god by & & ( ) ( ) ( ) ( ) (8-56), i.., i h scod ad fouh (8-57) which is a cicl. Th suld phas poai is plod by h followig MTB simulaio: Ca m-fil: ss.m fucio dss(,) dzos(,); d()(); d()-(.5.5*sg(()*())*(); % ky i h followig isucios [,]od45(@ss,[:.:],[ ]) plo((:,),(:,)); labl( ); ylabl( ); gid I ca b cocludd ha h aiabl sucu sysm is sabl, hough boh of is sub-sysms a usabl. Poblms P.8- Cosid h followig sysm: ( ) ( ) &, ()5 ( ) ( ) ( ) &, () 4 8-

13 NCTU Dpam of Elcical ad Compu Egiig Sio Cous <Dyamic Sysm alysis ad Simulaio By Pof. Yo-Pig Ch Daw h ajcoy o h phas pla fom o i MTB. 8-3