Prove d’esame svolte di Fondamenti di automatica

Novembre Prova 25 2009

VR1 Vi=↑ ㄴintic' >'소 vir√K ~ VieV σㅓ ( )ticR ia うし」 =∠in tic3 3ino VicWㅜVa inRzV *=.is *= a Rn((Xn Xz) Lx2+ =ㄴcX X 2} 孨 XarC "e ~, R2X2Xe( )*RMaglia *u *I 0C t =: - - u2Recte Re Xta 0u =- 1--CD RexaR u×+=- -,Recxl ReX × ut= . -2(= )* Re × × utu。 - -2cX1-LX2-R2XzMaglia # o: =* R〆 コメューメー- =ε2Y2 RzX2 X1+= - ( )북* Rzxat Xe= -*Sy ε ( )D Re × × u+-= - ra.菠 圭 (メ )R ×= + ×- 2 、zaxz Raxe xe+= = -ke =Ra 王=く 王C 1=μμμ文μ二μ μμ [ 북 (천 )1 xa: ye w+-. -C )동 xex2 .y 2X2 X1= +-総 ”Y 2× ×e -= 2at YA{ T -- *-- TLT :℃ eaxi"ftotn" .;블 ±ㅇA = 어Cㅇ · 2ㅇ -클 ㅓλXI A =- λσ λσ · α+( )det λ λ 2 =+= ( ペ )λ +α= =3 λ2×= =λ^( )λ λ 0王* == C=λ λ 0王+ =8= Λ - 0 =± ㅇ2 ± 걸- _== -2 2Re oc= 0λ = =Re32 ., Rm = 들 ㅓㅇAOI = A- ㅇ ㅇ ㄹㅇ ㅇV 了 是: instabile~ autovaloremsistema instabileeIl一足入入入入入入 一 一 土 ETe-{" -cTnt-℃xn. Yox-"fthtato ."{ 意S 4 *れる=e るSXz 4Xz 2u+= -SXz 2Xz 24+= -Y 77× ××← t= 2 3 vCas-2u(s 2) sou 40-V 2us 2us-or5 +++ + -SXz s × 2202Xz ω←+ == = = _=ョ 程 SS 22S +2+ +( ] 20× as + =s u(2s 4)阴 -*X = =; ョ )(S 2+ s as +u(s 2) u(Ezu(s 2Us- 2)2) 2uSUS-2u- -s- -礼 -SXn == == _1 s(ss(s 2) s(s 2)s(s 2) S 2)2+ +++ +2) u(su( 2)S +- -X = = ^9 ( 23( ←, ) s s +2s +r(s 2) い+←ε℃ ㆀ- =+“ )( () s tass z+ jusu(s ous(s 2)2)+ + +--= =(s )2+s 8Us28Us" 16USus 2u + +- - -= =s2(s 2)+16us2 2u17Us --= =s2(s 2)+(16s2 2)v 17s --= sP ( )s a+osz _ ) 1652tts 17s- 22告 - - -( )G s == = ( )2 sa( )s stas 2+Poli 乡 C-: § Sそ 20t == -Pe PPz 2∞ 0 == = -3,.autovalori eranoalGli trovati precedente XzX2punto -2 00 ===n ,,Poli autovalori= formaèsistema in minimaIl一彡一 μ l )tta" a? -x1 X2+( )tu 0= = { (( ) ]*0 _×^ は+= - -_3メ×0 = - ☆当( )( ) ± *×ett 0 × ×* X= ==ー - 12 e23( ( )) × さ1t X* 0 ㅓ* = = =--- 4a 3× さX 1 = == _. -2±X =2 ㅓ洼 ** == ±ョSistema linearizzato : Xn 11Xz + -- - ㅇBA [ Dt ㅢ 0=== =Xa3ㅓ ㅓ-{ 0 c? ^ 州器 GX1 0X2+{ ”“災災約 器器!^ 0* wdy Gxn dXz+={ : 7 70×? ! :‰ 沿 ×c -: ! ,=! 点ε* by = OxaOxn +::: { retuY × メ←= 2ㅇㄹA = ㅓ -x 2 ㅇ-I A =- λㅢ 3+( ) ( )det λ+ 32λ= =-λ λ 2 6可←= =- -=λ ー 。 0← = ( ). 64 2 万=-= はー1 5 2-λ = ㄹ一ΣΣ士入入μμ士一μμ 方 instabileReλ 2 0 >s=」 Raλ 3 co= -王 nL equilibrio instabileè{ … - u々 Y × ×t= 9 2②ㅇA = 3ㅓ λ 2XI A =- 9 3x(x 3)det 2+ += =λ λ 23 0→ += = 2g 14= =.- . ⇌3 さ±-= = … 2x1 Re1 0<= - Re王 < o=↓' equilibrio é asintoticamente stabilee↑acs) = Im4. ΛZeri 1s 0: =- 1± Z == ュ,Poli S 0: = 1 …S Re0+ = O*N sP PPe ±10 ュニー= = -3,,us o Llow ) chePiù zeri inpoli polo o+(vw)ω < 'P2 3Z2 머러 ( ) )1k= ω σωσ ∞→= -ㅇ ℃°° 0S180 36~ 180 · c+ →++ =+ ° a°° 0a°90180 Nyquist270∞ 180 In+ 0 =++ - ( /-) faseZericontributo di + allagoo - sempregoo +goo goo :goo gootδ∞ - - -- ( -) fase+contributo alladi Poli sempre-:入一入Σ士入己μ 足 十 igooiir.....· < << %270 /goaP 0= yccaso N ASA 40 =A -: = =-- En N definitoBcaso A non: -=- -E Ncasoc A 2 = P-: = --- )( oく 1899α00 = -. ° eL'unico diagramma Ain -180ilcuiinpunto passa1)2 “ ”( )(50 1)25 (9 )4t (O414 4e _- 9414 500=- .. -- ..)((1 99 xは o0 =132 = =に =. 50.414 ( 414( 1) 414(0 8285)50.00 - 171050 82850 343+ .03435050 829 .- + .0.+.. -. . Z.' 119f 0+ .= =34323432.0 + .017 ±1± . AHMA2A< == =_ - .µ 8O □. ☆1 2 11 이<- - .M ㅓ 4190Mc - = -- .4142.414u 0< . 0)(nonPer insistema Nc'èil è cuiMO mai puntoas nessunnon =sistemaIl e AS 419per M <0 . ×5 。 to( )G s = ( ) ( )2 S }3 + +Poli 3Sa 2 s- =- -,心μ cheZerineNon complessi sorraelongazioneci coniugatipoli possonosono creareVERO X(nelha ..6 G(s)sistemaUn tutti iasintoticamente stabile negativastrettamente dominiopartedi continuo)realepoli concombinazioneQuindi la esponenzialirisposta enel dominio decrescentiall'impulso ditempodel, una, :it'h ( )( ) Re λt eC c ocon i= .: haPoiché componentitutte esponenzialmente sidecrescenti,Le sono :h(t)Lim 0=ヒーコ ∞ h(t) tendeDunque la risposta esponenzialmenteall'impulso tzeroa per 0.>-VERO Xcheafferma inferiorecriterio sfasamentoBodeIl pulsazionecorrispondenzastabilesistema è eil calcolato tagliodellaindi diloCo se a(in co)fase°; fase% il sistema instabileèinstabile superiore eè > oilstabile margine180 dimargineildi180 Parolealtre sese ,a see:(che )Anche fase anche:°°contributo negativiintegratori si puòdue zeripositivimargineildanno di alzare-180 poli-go-90 con .con o=FALSO ャa. (( 3) = ↑ β5( )2 s$ )F == ( ) 읔h ff s ++)( 了 は 0=::: saráPer negativoil Polo sempreop >FALSO X( 2CGe (×s)Ger ( )s = G anchehannoGe(s)Gn(s)Se Geg(s) sarádunquesemipianosia sia sinistrosemipiano avrá asintoticamentesinistronelnel polipolisolo ,,stabile sufficienteè necessariaLa condizione , nonma .VERO Progetto statico.6 1 :a) y(t) sca(t)2 0 pera = =es(s)y - ㅓ gr 0=Mc1 +'ㅓ=)= □elt (sE (1 s= )= =) i。 」 照品ca ((。 。 =' )」) =品 品, ?2季) + s .. ㅇ 0ga 3Scelgo 1gr = t さagaga grgrga = == t =Libero 1Mr scelgo Mr =µ 素( )Re s = = (-180 1)e0 ; 01s0) -+ .n0=( ) ( 5(ん G)Re §ss ) ”-= 了e (‰( )‰ ~ *)は +慈節 +dinamicoProgetto : い号器=(2 。 )·1M = · 20log10db 20dbM = =dbน = Cologna - Cologin db 20logw20logw0 20 2020o =: =- -20log(w) zologw20 o==LO 1wω ==-1 SZeri £ S1 0 co+: == =_ -MNOZ too=Λ花 흐Pols loせは ㅢ s0 = == -0S S ±1 S 1000+ == =- -100 eoo PaPe Pa P - 0too oo, , == - =- = 3 ,พัμ Mi60 -… ㅢ2 - พะ /rad=10munumuuta od20- - Polo in 100Zeroesi annullano9 0- -60- -lot 10 100ㅓ ∅-zarctan--arctan1801800-1-90° arctanYm 10 010= - .. . ㅠ51 プにFgE980 go- - ー ポー 55ー= ~ <9595℃ -9 .g F- .180= ^ =5Sposto minimoalw, : Mㅇ… ㅓ-…muummunt *o ab *20- -1 a- -06- - '9 10 100ㅓ ℃lorModifico il guadagnoM 그 2uMe = ·W= =JWa 180 carctant-carctan-1° °Ym 90180 2 0 01.= .- -- . ㅠ に- ー g621 - .980 90 215 22 2 .o= -1 .= 0- 500H 6℃ go1 6300. √70= 1 .一一μ 弓方 ‰ e( oes.o)9 -- -号 ‰t( )22 s = ”( )' (‰) +→_+o )인에+×( 응( ‰ 。i-2 s(( .=(R 솥)s )= =d2 - 。+ )(ん ) s (-st e = 素素Ra()( s)R Re (3 s )= =o756 2 φ <m. ^≈ 常Ta =63 g 94 m .9 g630上 ,100 too ท0.639.2 16& .Ta 599.= sea=3ที二μ

Settembre Prova 24 2009

ㅓ { @ ( aet -)= (8 kuz ]) tay-(e t )(□ f(9n)92(t) kz29g(t) kzz9e(t) ke92(t)+= - -9 s e de( Ua tt +( 92( 93tkez -= t- (kas ()) )' 232)( ) ( (t ( ) ))y tt t9a 93t t= . zkz2k13 1= =kaka 0s=3 : ^)(f au = . a . .{ ade aet tue( -=) - taye ay92(t) k292k2z92 93+= - - + 1 a+ 1Gs ( )t ka993 t 9 Ua1-= t3 ay 91 9392 ++=X9 =e 1X92 = 2×43 = 3 -{ ir * * Ue= t- n^ txe 1⇌ kaXeXe keyXa 43+= +- - x1* xz k23xzx1= + 4c+3 +-Y × +×*= t る_“ , 站マメコメY = -μμ μ :앞 로:A = ± …ㅇH …1 ·-A λ- = ±ㅓ ㅇ-ㅓ λα·(x 1)(x 1)1)(xdet 1+ ++ == -( )( パー) ー^+ ハーλ+β *+ == =λ~ λ+± 3( =λ) λ= *( )λ ^* 0=λ= ε々λ λ 0+ = 14 1 ℃= =.- . ご11 ±-⇌ =αReλ 00 ==e z, Rexz 1 oa= -e2m = …ㅓ ㅇOI A =- -± 0ㅓ ㅇ ㅓ☆ 23 연== - instabile- autovalore Xm< 0> =- instabilesistemaIl e“ 儘品× メ U= + t」 22メY X= -、 王 A (A"AT cT)cT cTM =. 哇A C 2ㅓ ㅇ0 == → ±∞- - ±1억 ㅓ~ π「A f= =f o ㅇ^· · -さ 「 ' )「 'cT (AAA c == ㅇ1ㅇ σㅇ±들Mo = ± ㅇ- · ·· osservabiledet Sistemanon rango max0 → non=>=sistema autovalori osservabiliIl possiede non入∆一μ士ΣΣ 工 至 13 . { (y 2)xxn -=; ッ( ) Utu 0== { ( )0 * 2 ×= - -* 2メ0 メ= ^ - 2( ) X*2 Xメ ×0×^ + = = =~ 々 2 1王 ㄱ7 XXX 20 0= == ー2=X 王= -1σ ②-* * ==, · 2 王linearizzatoSistema :Xz Xn 2-- - ωA = D=E O 0=== ㅓ±±{ *災dt 1: '啜 : o × ztou{s ay δ xe= { 感談潼… *{ SX ュメェ=