Fondamenti di automatica - parziale 2

SPECIFICHE DI1) Stabilità interna: T(s) deve avere tutti i poli con Re(s)<0 e non ci sono cancellazioni poli/zeri2) Specifiche Statiche (comportamento a regime) precisione a regime3) Specifiche Dinamiche (comportamento transitorio) prontezza, stabilità-di riferimentosegna canonicid[rx(t)]tk 1Rk(s)rx(t)r(t) == == 1kk! +S roct)*11k· roct) gradino0 unitazio= = >1 rct)k rampa unitacla· = = * ri(t)E2rzct)k pazabolica· rampa= = >* (2(t)>yy(t)((s)e(t)/k(t) + -6(s)x(s)>L(S) =7 -- 1SCS/K C =(S(S) > ((5)1 +Hp: stabile internamentesistemaein ect) 0 eimitatooppure=t > kB. [Cs) 1(CS) [(0)con= =se RKCS) 11SCS).ECS) == . 1jk +L(s)1 +21 - 11 [(s)5k11e(t) -= 2. 11 =5k ++L(s) 11 + +- Internastab.eapezLose2 1-1ect) -= 1se 5kKBL(S) ++ (=)ein eskkect) (= [ 1e· >0= +7 >- eimitato)(ezzozeein ect) e· =7 >-ileim,ke = sk1 12 -e(t) .= Sk 1kskBL(S) ++ S20 1 E2eir(t) -e(t) +=de ecze: Cin (Ale(t) =t >0 (3 0=sk+kBIC)1SECS)=Quindi

Una volta che hai formattato il testo, dovrebbe apparire così:

eineimA S= = 15 k5>0 -- - =K > e se1 11 1 1-1e(t) -- I .= = 1e)5K+1Se S(kkBLCS) +se -KBL(S)+ +↳ è cimItaTOnonL’errore di inseguimento a regime all’ingresso canonico rk(t) è limitato se e=k (ovvero il numero e di poli in s=0 di L(s)è uguale all’ordine dell’ingresso canonico) è nullo se e>k (ovvero il numero di poli in s=0 di L(s) è maggioredell’ordine di ingresso canonicoEsempio:t/k (k(t) y(t)ect)(6(s) -7 >(S) > =- -6(s) 10= 5)S(1 + dIGCS)nKB 1Co pollins 010 ===(CS) VK10(CS)sTabletà 10Kk > Inierna= =- s[1 5)E +kB 10K=1e = it[cs) = sIE!.#I①①2 105 recM ↓setro(t)in 7+Idk0 , ((s) (o(s)I X-- 7--tkdk(t) 1Dx(S)= == 1kk! +SdK /kX se(S(S) scs)> ↳[CS) [C0) 1kB. con((s) == se siabileAp: internamentesistema uscitaldisturbidiProblema Relezione del (ineine 1y(t)1 0=-> 0ae1yct))1) ein 0=t 0>- erdi(CS)di poee s=0innumeroilsolo sese e kdidk(t)dell'ordinemaggiore limitato1xct)) cerroreein

yc02) =->t s=0dILCS) ee disolo lese potse numero ine kd1dk(t)all'ordineuguale S 10kkB =Esempio: 1e(CS)k10 10k(s)6(5) 10 == == [CS)5) 1S)s(1s(1 =+ + S1 +-mes Sw(S)v(s)Gu(t) Gy(S)y(s)y(t)u(t) y(t) = ==di 30105/20232.30de+ ↓Tre,6(s)+ >Y+(s)7- r(t)()e(t):y(t) 0==((S)6(S) [(0)(CS) (CS),kB 1== = se r se>S'CS)>r >T(S) SCS)((S) S(S)T(s) 1 1T(S) =+ == ((S)11 (CS) ++ di cocsiScalder SiCsYdiSpecifiche controlloSpecifiche [dipeecisionel1) stabieEk (e(t))eim0,1,2,rxc ) ek+ = = ... =o+t - t#al dek(t)milit k 0,1,2..= =k!1y(t)ein Tk1 =t 0x +-20deCt) A= xy(t)-de(t)A8 &im 1y(t)1 0=t d3- +·h ==> stabilesistema>0 internamenteS·e ovverolCs) haso,xy(t)N 0almenoBo polo sinFüte-z un =integrantelcovvero un>t (e(t)1⑧ ein 0=8 t ->0)diCt) (r(t) duct)D= ==dict)xDo >t0 y(t)diCt) 6(S)? ⑤ein S6CS)11y(t) eimF= =3 > -> x01((S) ((s)0(S)0t 1x -+- + +eGCS)Y(s) 2s) 6(S) D HCS) E== += e(s)6(s)2 S+ 1 + gli stessna di:poll6(S)((S)6(S)1 +6(S)Quindi

F=D=ein5 x01 ((s)G(S)- +((s,60) Ics),(0)6(s) (s) y, 11 == ==kB k6Cs) kakosk(s)=S) = = gse Sc59 +-(s)-kG6(S) (s)59 kos6(s)(S)1 =+ kck(s)6(s)95+kck6[(s)8(S)1 ++ gsc + 5]2(H(s) [H(s)]16- 1(t)F22 -lyct))ein eim +=+= t- ->+o-> F1ct)) Fl1ein =- t +x- perche(S) condizioneein KosDF == 5 deveaccada:0> essere- kck6[(s)GCs)g5+ integratore cinun+ugualeQuesto 0limite C30LaèRisultato: l’errore sull’uscita a regime prodotto da un disturbo di(t) costante è nullo se e solo se il sistema èstabile internamente e C(s) ha almeno un polo in s=0 (ovvero un integrante) S KG=0Sec E= Dg 0=KCkG1IF1 = +K6ECS)eim k5 9070 58- Kc6)Jcs)+Specifiche Dinamiche sYI e,c(s)u,o(s)M >-e Y,~T(s)s =2 Non superiori a un certo valore massimo datoTCS)2) is =x,,,11I 1 (0,1),TCS) 8 wnso5 =1 + SCS)28n 2 TS+1 -e 2= n -TS TSS iwn1 ③ - ⑧WnTS5 T D-i- -5 >S8i0 05 5 1(8)1 i accosTS -= w 521 - dominiosut(s)Specifiche dinamiche delnel TempofrequenzaNel dominio della y(t)Acosat,

yreg(t) = TCS) ytr(t))y(t + I = yr(t)2 0im =+->-> T(xw)AlT((w)/cos(wtyreg(t) +=T(xw)w 01(T(xw)/ /maxIT (3w)MrMr picco=ter di rison.il 0w = risonanzapulsaz.cur = 3dBbandacub a= IT(0)/|T(xwb)) = 2TCs) frequenzedinamicheSpecifiche insuMt -- bwb 1r -- iMr wn- ③swb IWD SI 7(8 IInir 00: suT(S)Specifiche dinamichegradinoS=5 1T5risposta al TSMrArfrequenza relazione 3WbTS=wbIcubrisposta inwbis5 1+S-G Mr .....- I3..... 0.8 - i↳070,918 0 i's0.703[0.85,1]relazione 1 5+ =Mr-(L(3w)dB1 ((s)T(S) =decdB1-20 ((s)1 +dlpuls.al aur= (T(s)((s) ((S)W =wn3 HL(w) I T(s)((s)T(S) L(S)40dBIdeC- - = --+(s) T(s))((s)(1I -=i 1TCS)L(S) == 28T(S)12SW -wD ! 1 =LCS) (s)5cn-- = -1mo SC1 128 +- 28wnE (n/28kB = 1[CS)Im = 11 + 28W⑪ >Re8 ((xwe)E my n= +1L(3wY)1 1= someM ---wo 1 I2un I0.S 0 >SI0.1 e<5 I0 (CS)frequenzaSpecifiche caratterisia: myparametriin wy,suwo 1 I0.8.CD 0.8]wbu (0.5wy +=Ssmpeo. iTra 0.6 >S'10m4 x⑧65 -100 MrdB 20e0g,14 Mr=0 Iazcos(1-1Myapp. = 2πr2 ider,dryt esr y>

1. Errore di inseguimento a regime nullo se r(t) = A (A diverso da 0): e(t) = 0
2. Errore di inseguimento a una rampa unitaria in ingresso (ovvero r(t) = t) non superiore a 32l: e(t) = 1 - 2^(1 - 2kC)/(s)
3. Errore a regime sull'uscita prodotto da un disturbo di(t) = B: e(t) = 1/(1 + kC)
4. Tempo di salita Ts: Ts = 3/(wb * sqrt(0.5))
5. Sovraelongazione: 0.2 = [(s) * (0.5)] / (k * sqrt(2))

Specifiche stabili (specifiche di precisione):

• Errore di inseguimento a regime nullo se r(t) = A (A diverso da 0): e(t) = 0
• Errore di inseguimento a una rampa unitaria in ingresso (ovvero r(t) = t) non superiore a 32l: e(t) = 1 - 2^(1 - 2kC)/(s)
• Errore a regime sull'uscita prodotto da un disturbo di(t) = B: e(t) = 1/(1 + kC)
• Tempo di salita Ts: Ts = 3/(wb * sqrt(0.5))
• Sovraelongazione: 0.2 = [(s) * (0.5)] / (k * sqrt(2))

SINTESI PER TENTATIVA:

e(t) = (1 - 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 2kC)/(s)) * (1 + 2^(1 - 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unoa equindiAnn vera soloZero sese ec 1=trct)3) t= y(t)f(t)e(t)x = - ein 1e(t) 0.1=0.1 2 1 =t o+- eatabellinadevo quazdaze> (8[(s),[(0)x(s6(s) 1(s) = ==-0.1 Eal1 mil8!es1kB| 0.12x .1