Fondamenti di Automatica (Esercizi e Teoria)

Controlli automatici - uniroma2 - Galeani

Laplace trasformata di G1

EKGIR Is dtflt)F(s) inte inSt= e zy-Rea -1

Tabella trasformate notevoli

1) (tf(t) 2(t =- += 21)(t322( 3t 1)3 ++ 2t + +- ++= 2x(t)2(t5 3+4 2 1)t2 =2+ ++ =+- - += 274 5 5)2 5= + - + +-45- 34 +2= -+ - et)

D2) ete(t ef(t) t -+= = +↓Le-at] 1site=> = Sta2 -+ sin3)(S - 3)xeu(x xen(x)(() sem)ca()+ = +

D3) =f(t) reu(2t M[xn12t)cos)+) rul))M +cal2t) == pe Fas dalE 25=Ment In Se=4) si)=n)= + (2)+) (0s(t)(()(a(x+ xu(x) neu()= -=4 [ca(wt)D4) )f(t) cos(Wt-45 % xen(wt)(1 real-15% %< 25 == --2 (ca(wt) ren(wt)]ru(wt)2)[ca(wt) 25 + =+ == 2 penlat)] dio(at) ==> s s,a/ se 2k(ser) 25) wi]8)=25 ==> + ,+-

Er3)xeu(x xen(x)(() sem)D5) ca()+ = +=f(t) Mxen /3t M[xu(3t)(0+4) (13txu(4)](4)S= =+ 2 penlat)](a(4)2(xen(st)] dinlat)a sM Meeul)2[ca(3t)]= ==>+ -,M(n) 3 + e- !D6) (e (e 7)) 7) 7tetf(t) (eet tet t = =++= + = ↓Levat] sta=(t)=> ==3 32 -+ *: woteatcos(wt)D7) /se F (S)27tf(t)f(t) S-= = =(et((t))=> sta= 22e-at)2 dialat) Es(1) = -,ds I- astwit-a) I0=1 as Im)- casatr-(s-a)sesI= s)s-wasta'+ ~1.2+ (S a w2)-2us + +(25-2a)] Eastätzl2]IwY-(s-=() senstatw-(2s"-zas-2as+za. = =2(S 2 w2)-2us a+ +L I Ia((s-2us++waS 25" ~Ens za2w- + 2as2 as2as++ -=s -+ --- -- -- 22(52 a w2)2us + + (s--2asscas=is(-s+2as-a)(s-zas wi- (- a) +a+ w+a+ -w2)(S a-2as + +7 s at(2s 2a)2a)(s a( wi)2as 2as25 -+-- ++ -+-= w2)(S a-2as + + at( 2a)2a)(s sa 25( wi)2as 2as+25 +--- +++ -+-- w2)(S a-2as + + 2as- (2)]5wY ( -a(as + -++- + w2)(S a-2as + +2ws w2- +- -w2)(S a-2as + + talt) +xult)ezeu(H)t(ca(t)-f(t) e =+D8) = +e= staa-datasettlatL (tD9) <en(wt)1)8(t) zeu(wt)xen(wt)+= ++ = enlat)] E2 It(penlat)] ==> s ,a=it whD10) et tt ttet(t tet tf(t) ete+ = +++= ↓Le-at] 1=(t)=> = Sta:Se=+ + -ANTITRASFORMATA LAPLACEDI (tettamente) FRAN SEMPLICI=>o--NS) voF(S) gaobD(S) -grado H(S)F(S) GN(S) (propria) +0 >= ==->: - him F(S)46 =D(s) ↳ S - >grao 2(S+1) 2)(S -> H(S)* SEMPLICIFCS)+ FRATTGfa >- ==grado2)" -> 2(S + oppureaduzionipoli di D(S)= 0: (S)Dsoluzioni Wi disdi F(s)N(S) aut:=0Zei -: R(S)60 ja+ /impropria) APPLICABILENON02If(t) cj estdSF(S)= so-se Resiohi)(MetodoSEMPLICI#I son*I I+F(S) ...-== pi)D(S) 0poli① 41Favare -> P2:: = , ...Residui② CalcoloPOLO -)N(S-pi)lineRiSEMPLICE = S-> Pi D(S)"occone"Pi nua evolta èzcha semplificoD(S) primaeduzione di limiteitfacciopoivoltasoleuna * + aPOLO Fes) t= Em ... ⑧0 2(MULTIPLO s S pi)--~I"occone" piùPi voltaredi esponente)(m-una d (x))DCS) ·dieduzione ,((s** Fpvolte 4- -più : !+e) as(m esponen- S- P- >semplificaprima devivopei limiteitfacciopei semplicepeloal↓ Rim diuguale uné caso Mei4POLI jbI R apoli += =Ri saa complesse ,it dei semplici i ecome case jbCOMPLESSI MeR2 a =. -=CONIUGATI autitzasformata③ Calcolo I(poiche e pitSize EsPitPOLO Ri- .SEMPLICE P t- .: R CS Im .~= p tesponente .-t1 2 e- Re -Pit .=m .~RitaPOLO P-> t2 .-M -e+-m -MULTIPLO ... +40) zeu(wit +4)(wit zen(Wit +4)Cos 4- 4ot== ,POLI f(t= Mes Mese Meta(witMetatswit Metelo swit e twitta)Mett 4)+=compressi # _: -)(0 )S-(0jwS- -jwCONIUGAN + . i, ,A1) -1F(s) 9z 0-> == 2S + Ilim F(s)6 1=== 5-> diordineS maggior faceF(s)H(S) sewe= Non12G -2== - 1 =>>= Fratti semplici- - :2z 2t-solt)F(s) 1 e==> - -↳5 8 2+ tsolt)F(s) 1 -+ e==>1-S 2 -oppure -. 1X -

A2) 1F(S) 92 2=- >= 3)2)(s(S ++ (528(3⑦ 2)(s 3) (3((x) 0 0 Pr+ +- - 2= == -+ - 3-==(s)+& FIS) = (S#Lim ·FCS)=RI (S-P1) him 2) 1-·= =+35-PI S 2- - ~(St ↓sis(im himFCS)R (S-P2) 2== = -2 S 3- - 2tsent 3t2③ ( xF(S) +- e+( e1)= e -- =A3) 1S + 182-> =(S 3)2)(s ++ (S8(3⑦ 2)(s 3) (((x) 0 0+ +- - E= = + -(s)+& FIS) = S 1 =+(32)Cim FCS)RI= (S-P1) him 1= = -· +3) =5-PI S 2- - S~ 1(S ( +im himFCS) 3R 2(S-2) :+ === 3) =2 sS 3- - e- 3t=2 e*③ (x +F(S) --2) 2 e= 2 2t - +. . =A4) sF(s) =Eng =G FIS) === = s si -2- 555 55 - 6SkS 1+H(s) F(s) - -mG =- - -= 52 35 6 2) (s 3)25 (S+++ ++2-F(s)=> 1 += 2) (s 3)(S ++neI 4(S !3 P⑦ 3) -2)(s 2=(1(s) 0 -0 ,+ -+- == x s==(s)+& FIS) = -2(S2)Lin FCS)RI= (S-P1) him 2:= =· =+3) -5-PI S 2- - ~(+ 3) =eimFCS)R S(S-42) 2:== M · = =2 3)S 3- - ( t2)-2 L - -③ 2 (- 3)t-F(S) -= 22C=-2 +5 =-+2 S 3+ +ze 3t2e= +a)A5) F(S) 82- 0- =StstaslineCin F(S)G = = 1= =-e St)F(S)H(S) 2 =- 1= =-5 s+hS -++ - -22I 52 45 4+ ++F(S) 1=> = H 2 (s① (s +2)2) (s0 0+D(S) + ->->0= ==4 (s Pr+S P22 z0 = -2 =-= =-> -> - 2m =025 2S = -=+ +im= in52)2② H(s) == m 1- 12--Pr (2R1 =e Se i(S H(S)im 2= =2 -., ((s H(s)]/se1 4) mRe =1 - ., (2 -1)1 2-③ -< 2) s-( (1]se1 1= - = "2 --IH(s) 2③ is= -Solta sett etF(S) 1 (2) t+=> = +2 etect-2=Solt) t+

A6) 1)(s(S 4)F(C) + + 0&2= -> =2)(5 + amastR((s) I6 = === = -+=H(S) 1F(S) G= - -- Kjbec a += npate25 GanteimmaginanazealeF(S) 1 + dimecanteriore= Pians camplero/Gauss= ~Y sim"->- e>X↓ =Tj I 1- -= ,5 Fzs①D(S) 020 = ++ + =-= 32(52) jjkjjk 2== -Ijz ==S jk)2e 1)j2-> = jk)-jk)(( ( 1)jj2k (+( 1)) 2=== ----- =das complemjE complem numeinP partestence reale= la- concomugati pote immaginance cane oppectojk seguoP2 = - * jb/c ac -- =55S 5jk)(s 5x5(S jx2) 22 2 ++=> += =- +-+ =② HIs) += = i)(S-p > e(SeHCS)Ra hinp .= , = jS -, ristic-Sr(s-sc)- : jE(jV) +6j(5 k)+22 See+- -j2k .!it i s= -= = 2)2)E) i)(i(r) !( 2 ij + -- + ==.= Ri j5R2 +== (S-P2) . St(s+H(S) jk)R2 him.= =Pr jS-- ) 2(s+5)- +ein . -= jjv)(-insiSt- --6 j(5 k)- +R /aI2 .== e2 - -m+ -j2k -- .!it i s= -= ==(r) 2)j)ij +-== . 2)z)(i)fj (j))-i i== ++ = += s5j&③ -H(S) += +(s jk) jz)(S- + s5j& +-1F(S) +==> + jz)(s (Sjk) +da - /)+d")[2]f(t) 2< += ↓ ↓polarefarma farealgebrica j4jb M0 ejaC= -+ ( )= = M2Mi M1modulo 62 62Im = =,- = ,1m-i 8...... =- N i4ME b 0- 34= 4= - 0 3, == -Y ,Re ,und =·!/ 3X [e)+ !I=solt) +Brer) tanti im Szow stir)(1Be=Solt) + +ale-t = asrte' + est jtM -=Solt) Ree e -++ -j(w1t 4) j(W1t 4)+ - ++Ae+4)cos(Wat E)y)= neu(Wytcas(Wst2 +4 ++ ==> ~ /son24)j(Wet ne+ j(W1t 4) -+--2(WIt 4) e i 13sen =+ asen- 2πi2j -41StessaM1 M2 M InexI== -4 -41-= j41 jEtsitMe'I - -=Solt) Me 22 ++ 41)j(kt j(X2t 41)+ +-=So(t) M Me e++ [e"(kt +4) x)]-j(xt=So(t) +M 1.++ e[e"(kt +4) x))-(2 +=So(t) + ZM -++ e 3)es(k +=solt) ++ -- +Ya)(s(kt=Solt) +Ya)ca(rt2 M+ -=>=Solt) ces(v2t 3)2 2 6 -0. -+ , ,So(t) 5 (rt-0 3)2 ,co.+- , ,generaleIn : -iwalir)y(s) =fa elo-swasteloss west MetMesYYIt) +.= ((s(Watte) 4)j (W1t=Met +-e e+Met e4) E)cos(wit sen(Wet=2 2M+ Y ++=A8) esFIS) 82= 0:-13+lin 12 == - =5,0 as 13+ MEs E~ 13A(s) F(s) + =2 1=- =: -13 52 s1345+ ++eF(s) 1 +==> ↓ j5S 4①D(S) 0134) =--0 +-> =+ I= P 2 j3- -21j =21j3 I->- >-> j3>1 2 +=-2 (552 j3))(s(45 j3))13 12 2+ = - -+ - - - +-pa)HCs)& *= p)(s-cas( - -Pe)em SimR # (S) == inS- 2 j3- - e-Lim 53557531)5- 22 +- - 413=S = =2 48532j E j2 7= = -- -- = ,j1 85R1 ,R2 j)2 7= 3e- += = ,se se iiI=> ="Solt) 285[e- 83f(-2- 33t)eist27 , =3+ +, ↳eI2t=Golt) :13t 89172e-7 2- ,3+ 2. ., >2/35- 85)2= ,6 e-2t cos(3t 85)Solt) -114+ ,- ,A9) F(S) 182= - =13)s +e↓ PrE = 0Si s+S(S①D(S) s0 13) ->+ j5- + 0= = =1320 -> - j- 2I->sol I+ctippolimanio ->del .: 21j3>->coefficientitutti i nerotonameregene Pz 2 j3--= j3Pz 2 += - (552 j3))(s(45 j3))13 12 2+ = - -+ - - - +Apa)FIS)② * *= p)p)24 ++= ( j3) 53)s 2 -- +- Eslin(S-P1) in 0R1 08F(S) -=- = .13)S-P1 s + i( -P2) -SerieeimR2 #(S) == "S Pz-> ? -a e se-j 1-=j3)2 ti2)(2 53)( 12) E( -- ++-+- -=Ej3+2 j3)i )- -- 1=E i12 e+ =j6)j3)(( 2 - -- Rationalitations-Isti. re: Ej6)( j6)9( 9 - -+- unitariacirconferentej37 87 4S 4min 2 ki+- =,0 6 /4 Re= , 25]10zu :I# Ij37 87,ER3 0i 6e+= ,1 este③ ⑱F(S) +=d *. inst* [1 .) Ess-E s0 6 e 008 6 e0 + +,- ,, j37 87( j3)t2 j37 87( j3)t2-- -, +-,080 0 6e26 20=> ++ ,, , 2tj3t]j35 87-2t6/e 53t 37 87, -- - ,008 +=0 2 e+ ,, 8)))L e) (3t: 35+ ,08 0 6=0 +2+ ,, ((3t 37 87)2 87) +((3t- 3 ,-2 06 +080 ,.+= , 2 87)((3t- 31-1 2e +080 ,+= ,,A10) (F(S) = 81 3+ =3)+ 2(S P1E 2) Pc 2+ =02) (s -=(S =3)① 0D(S) -= ->0 +-> += 3)(S P3 30+ -= =E aFIS)⑧ = (S Pz)-Grz"(s-Pe)" E= F(S)Raz 1= = =)+(51)/s(s-anE (i, +))!( 5 s)lMar == "! - -....- 11 1- [ - )71 1= - = -.. -2 ...(S/2)(S-P3) sessSimR2 limFCS) 1·= = =5--5 3P3->③ FIS) = " " s" 2 stte2t- e- + +A22) Sys+F(S) 482= =s )2) 3) (P3& ⑯(5) P18(5 P2 20 2 m=0 = =+ -+ += - = -> Pa 3= = n- m =:FC)& = :(S-P2)" (2) 15+:Ma himF(S) 2= =, 32: (s - (ass-))F(S)])son" os)"Ra =2 ..., -sess-scos]) Fis"Il tisill- ....(S 3)4+ ....- =-2"FCS)(S-P2)timR2 (55)"Sihim "I2 =, 1Pase 39S-3-3: (s-1) " lata).F(s)])s sRa = "...-(ss-wisco) FiIl". Lisall..... "(S .2)4+ ...I 10--③ 1-2F(S) (53) "Ess= 152) + +<52) ::al 2 10 e3424 est2e e tt ++ +=122) F(S) 482 =-Si= 3((s +a)+ 2(S 2)