TMHMA MAJHMATIKWN JewrÐa Elègqou: Ask seic Grammikˆ Sust mata 'Askhsh 1: DÐnetai pðnakac: A = (a) Na brejoôn oi idiotimèc kai ta ant

Bản ghi

1 TMHMA MAJHMATIKWN JewrÐa Elègqou: Ask seic Grammikˆ Sust mata 'Askhsh : DÐnetai pðnakac: A = (a) Na brejoôn oi idiotimèc kai ta antðstoiqa idiodianôsmata tou A. (b) An o A eðna apl c dom c na brejeð o pðnakac T pou ton diagwniopoieð - diaforetikˆ na brejeð h kanonik morf Jordan tou A. (g) Na brejeð o pðnakac e At. 'Askhsh 2: DeÐxte oti e (A+B)t = e At e Bt an AB = BA. 'Askhsh 3: DeÐxte ìti ϕ (t) = (/t 2 /t) T kai ϕ 2 (t) = (2/t 3 /t 2 ) T eðnai lôseic thc exðswshc ẋ(t) = A(t)x(t), grammikˆ anexˆrthtec sto R, ìpou: ( 4 A(t) = t 2 ) t 2 Ne brejeð epðshc o pðnakac metaforˆc katˆstashc Φ(t, τ) kai h lôsh pou ikanopoieð thn arqik sunj kh x() = (, ) T. 'Askhsh 4: DeÐxte ìti giˆ to sôsthma ẋ(t) = A(t)x(t) ìpou ( ) A (t) A A(t) = 2 (t) A 22 (t) tìte Φ(t, t ) = ( Φ (t, t ) Φ 2 (t, t ) Φ 22 (t, t ) ) ìpou Φ ii (t) ikanopoieð thn exðswsh t Φ ii(t, t ) = A ii (t)φ ii (t, t ) kai ìpou: t Φ 2(t, t ) = A (t)φ 2 (t, t ) + A 2 (t)φ 22 (t, t ), Φ 2 (t, t ) = UpologÐste epomènwc ton pðnaka metaforˆc Φ(t, ) tou sust matoc ẋ(t) = A(t)x(t) ìpou ( ) e 2t A(t) = 'Askhsh 5: H diaforik exðswsh: d 3 y(t) dt d2 y(t) dt 2 + dy(t) dt + 2y(t) = u(t) dðnei th sqèsh eisìdou u(t) kai exìdou y(t) enìc sust matoc. Na gðnei perigraf tou sust matoc se morf q rou katˆstashc.

2 'Askhsh 6: 'Ena grammikì polumetablhtì sôsthma anaparðstatai apì to parakˆtw zeôgoc diaforik n exis sewn: d 2 y (t) dt 2 + dy (t) + 2y (t) 2y 2 (t) = u (t) dt d 2 y 2 (t) dt 2 y (t) + y 2 (t) = u 2 (t) (a) Na gðnei perigraf tou sust matoc se morf q rou katˆstashc. metaforˆc metaxô twn dianusmˆtwn eisìdou-exìdou. (b) Na brejeð h sunˆrthsh 'Askhsh 7: DeÐxte ìti an y(t) h bhmatik apìkrish grammikoô qronikˆ anexˆrthtou sust matoc, tìte ẏ(t) h kroustik tou apìkrish (h apìkrish jhshc). Jewr ste ìti kai stic dôo peript seic h arqik katˆstash tou sust matoc eðnai mhdenik. 'Askhsh 8: LÔste thn akìloujh exðswsh qrhsimopoi ntac to je rhma sunèlixhc tou metasqhmatismoô Laplace: t y(t) = t + sin(t τ)y(τ)dτ ìpou y(t) =, t <. Upìdeixh: L(sin t) = s 2 +. 'Askhsh 9: H bhmatik apìkrish causal grammikoô qronikˆ anexˆrthtou sust matoc eðnai y(t) = t 2 e t, t. Na brejeð h apìkrish toô sust matoc se sunˆrthsh eisìdou u(t) = e t, t. Jewr ste ìti kai stic dôo peript seic h arqik katˆstash tou sust matoc eðnai mhdenik. 'Askhsh : GrammikopoieÐste to sôsthma: ẍ + (3 + ẋ 2 )ẋ + ( + x + x 2 )u =, gôrw apì to shmeðo isorropðac x = ẋ = kai thn sunˆrthsh eisìdou u(t) =. Eustˆjeia 'Askhsh : 'Estw asumptwtikˆ eustajèc sôsthma ẋ(t) = Ax(t) (dhl. o A pðnakac Hurwitz). DikaiologeÐste an oi parakˆtw protˆseic eðna alhjeðc h yeudeðc (me apìdeixh h antiparˆdeigma): (a) O pðnakac A T A eðnai Hurwitz. (b) O pðnakac e A eðnai Hurwitz. (g) O pðnakac A + A T eðnai Hurwitz. (d) O pðnakac A k eðnai Hurwitz gia k =, 3, 5,.... (e) O pðnakac A eðnai Hurwitz. (st) O pðnakac A αi eðnai Hurwitz gia α R, α >. 'Askhsh 2: DÐnetai to sôsthma ẋ(t) = Ax(t) + Bu(t) ìpou: A = 3 2, B = JewroÔme thn sunˆrthsh eisìdou u(t) = Gx(t) ìpou G = [g g 2 g 3 ] me g, g 2, g 3 R. KajorÐste touc periorismoôc epi twn stoiqeðwn tou G ste to sôsthma kleistoô brìgqou na eðnai asumptwtikˆ eustajèc. Sust mata Anˆdrashc 'Askhsh 3: DÐdetai sôsthma me sunˆrthsh metaforˆc: G(s) = s(s + )(s + 2) 2

3 (a) Na brejeð h antðstoiqh sunˆrthsh suqnot twn thc G(iω) kai oi antðstoiqec sunart seic mètrou kai fˆshc. (b) DeÐxte ìti asumptwtikˆ G(iω) ω 3 (dhl. lim ω (ω 3 G(iω) ) = ) kai ìti lim ω arg(g(iω)) = 3π/2. (g) Na brejeð to perij rio enðsqushc tou sust matoc. (To perij rio enðsqushc orðzetai wc G(iω o ) ìpou ω o h suqnìthta sthn opoða arg(g(iω o )) = π). (d) Na brejeð to perij rio fˆshc tou sust matoc. (To perij rio fˆshc orðzetai wc π arg(g(iω c )) ìpou ω c h suqnìthta sthn opoða G(iω c ) = ). 'Askhsh 4: 'Ena sôsthma anˆdrashc perigrˆfetai apo dôo sust mata se sôndesh pou orðzontai wc ex c: (i) SÔsthma me sunˆrthsh metaforˆc me eðsodo E(s) kai èxodo U(s). (ii) SÔsthma me sunˆrthsh metaforˆc G (s) = k(s + β) s + G 2 (s) = s(s + 2)(s + 3) me eðsodo U(s) kai èxodo Y (s). (a) Sqediˆste to diˆgramma bajmðdwn tou sust matoc kleistoô brìgqou me eðsodo R(s) kai èxodo Y (s), ìpou E(s) = R(s) Y (s) kai upologðste thn antðstoiqh sunˆrthsh metaforˆc. (b) Na brejoôn oi perioqèc t n tim n twn paramètrwn k kai β gia tic opoðec to kleistì sôsthma eðnai eustajèc. (g) An β = 3, na brejoôn oi timèc tou k gia tic opoðec to kleistì sôsthma eðnai eustajèc. (d) An to kleistì sôsthma eðnai eustajèc kai r(t) eðnai monadiaðo b ma (dhl. r(t) =, t, kai r(t) =, t < ) deðxte ìti lim t y(t) = gia kˆje dunat tim t n k kai β. (e) An to kleistì sôsthma eðnai eustajèc kai: r(t) = t t = t < na brejeð to ìrio: lim t e(t) (wc sunˆrthsh t n k kai β). Upìdeixh: Gia to (b) qrhsimopoieðste to krit rio Routh. Gia to (d) kai (e) qrhsimopoieðste to je rhma telik c tim c tou metasqhmatismoô Laplace. 'Askhsh 5: Hlekrokinht rac èqei sunˆrthsh metaforˆc Θ(s) V a (s) = k v s( + st ) ìpou k v, T >. H eðsodoc V a diamorf netai wc V a (t) = k(θ r (t) θ(t) k T θ(t)) ìpou k kai kt eðnai jetikèc parˆmetroi pou epilègontai apì ton sqediast tou sust matoc. (a) Na brejeð h sunˆrthsh metaforˆc tou sust matoc kleistoô brìgqou Θ(s)/Θ r (s) (b) DeÐxte oti to qarakthristikì polu numo tou sust matoc eðnai thc morf c q(s) = s 2 + 2ζω n s + ω 2 n kai ekfrˆste tic metablhtèc ω n kai ζ wc sunˆrthsh twn paramètrwn k v, T, k kai k T. (g) An k v = kai T = exetˆste an oi kˆtwji stìqoi sqedðashc mporoôn na epiteuqjoôn (tautìqrona): (i) ζ, (ii) ω n =, kai (iii) e ss, ìpou e ss = lim t (θ r (t) θ(t)) kai ìpou θ r (t) = t, t, θ r (t) =, t <. (d) Na brejeð h sunˆrthsh exìdou θ(t) an k v = T = k = k T = kai θ r (t) monadiaðo b ma. 'Askhsh 6: SÔsthma anˆdrashc apoteleðtai apì: 3

4 SÔsthma me sunˆrthsh metaforˆc G (s) = s+ me eðsodo U(s) kai èxodo Y (s). Antistajmist me sunˆrthsh metaforˆc G 2 = s me eðsodo E(s) kai èxodo U(s). Na brejeð: (a) h sunˆrthsh metaforˆc tou sust matoc kleistoô brìgqou me eðsodo R(s) kai èxodo Y (s) pou prokôptei an orðsoume E(s) = R(s) Y (s) kai, (b) h sunˆrthsh exìdou y(t) tou sust matoc an h eðsodoc r(t) eðnai monadiaðo b ma. 'Askhsh 7: 'Estw sôsthma me sunˆrthsh metaforˆc: G(s) = p(s) s n q(s) ìpou n N kai p(s), q(s) polu numa wc proc s me deg(p(s)) n + deg(p(s)) kai ìpou p() kai q(). H eðsodoc tou sust matoc eðnai E(s) kai h èxodoc Y (s). Me qr sh monadiaðac arnhtik c anˆdrashc E(s) = R(s) Y (s) (ìpou R(s) o metasqhmatismìc Laplace exwterik c sunˆrthshc eisìdou r(t)) kataskeuˆzoume to antðstoiqo sôsthma kleistoô brìgqou me sunˆrthsh metaforˆc H sunˆrthsh euaisjhsðac orðzetai wc: T (s) = Y (s) R(s) = G(s) + G(s) S(s) = E(s) R(s) = + G(s) 'Estw ìti to sôsthma kleistoô brìgqou eðnai asumptwtikˆ eustajèc kai èstw r(t) = t m u(t) ìpou u(t) h monadiaða bhmatik sunˆrthsh ( R(s) = m! s m+ ) kai ìpou m N. DeÐxte ìti: lim e(t) = t m!q() lim e(t) = t p() lim e(t) = t an n > m an n = m an n < m Poiì eðnai to sumpèrasma gia tic asumptwtikèc idiìthtec tracking tou sust matoc? Elegximìthta/Parathrhsimìthta 'Askhsh 8: 'Estw sôsthma Σ(A, B, C): ẋ(t) = 2 2 x(t) + u(t), y(t) = ( ) x(t) (a) EÐnai to sôsthma pl rwc elègximo? Na brejeð o elègximoc upìqwroc. (b) EÐnai to sôsthma pl rwc parathr simo? Na brejeð o mh-parathr simoc upìqwroc. (g) Ean to sôsthma den eðnai pl rwc parathr simo na brejeð metasqhmatismìc isodunamðac T : z = T x ste ( ) ( ) ( ) ( ) ż (t) Â z(t) ˆB ż(t) = = + u(t), y(t) = ( Ĉ ż 2 (t) Â 2 Â 22 z(t) ) ( ) z ˆB 2 z 2 4

5 ìpou Σ(Â, ˆB, Ĉ) pl rwc parathr simo kai se kanonik morf parathrhsimìthtac: Â = a a a 2... a n, ˆB = b b 2 b 3... b n, Ĉ = ( ) 'Askhsh 9: 'Estw A R n n kai b R n. DeÐxte ìti an perissìtera apì èna grammikˆ anexˆrthta idiodianôsmata antistoiqoôn se mða idiotim tìte to sôsthma Σ(A, b) den eðnai pl rwc elègximo. ('Estw ˆv T kai ˆvT 2 dôo grammikˆ anexˆrthta aristerˆ idiodianôsmata pou antistoiqoôn sthn idiotim λ = λ 2 = λ. Parathr ste ìti an ˆv T b = α kai ˆv 2 T b = α 2, tìte (α ˆv α2 ˆv 2) T b = ). 'Askhsh 2: DeÐxte ìti Σ(A, B) pl rwc elègximo an kai mìno an Σ(A+αI, B) pl rwc elègximo, ìpou α R. 'Askhsh 2: JewroÔme to sôsthma: ẋ(t) = Ax(t) + Bu(t) ìpou ( A = a ) (, B = b kai ìpou a, b R. Na brejeð to qwrðo tou epipèdou (me epilog axìnwn tic timèc twn paramètrwn a kai b) ìpou to sôsthma eðnai pl rwc elègximo. 'Askhsh 22: DeÐxte ìti Σ(A, B) pl rwc elègximo an kai mìno an Σ(A, BB T ) pl rwc elègximo. ) Anˆdrash katastˆsewn/parathrhtèc 'Askhsh 23: DÐdetai to sôsthma ẋ = Ax + Bu, ìpou: A = 3 4, B = Na brejeð pðnakac F ste σ(a + BF ) = { ± i, 2 ± i}. 'Askhsh 24: Sqediˆste parathrht gia to sôsthma talˆntwshc ẋ(t) = v(t), v(t) = ω 2 x(t) ìtan h mètrhsh eðnai h metablht taqôthtac. Epilèxte kai tic dôo idiotimèc tou parathrht wc s = ω. 'Askhsh 25: 'Estw sôsthma talantwt qwrðc apìsbesh: ẋ = x 2, ẋ 2 = ω 2 x + u. Qrhsimopoi ntac wc mètrhsh thn metablht taqôthtac, y = x 2, sqediˆste antistajmist parathrht /anˆdrashc katastˆsewn ste na elègxete thn metablht x. Epilèxte idiotimèc anˆdrashc { ω ± iω } kai thn idiotim tou parathrht wc s = ω (pollaplìthtac dôo). 5

6 PÐnakac metasqhmatism n Laplace f(t) F (s) f(t) F (s) s δ(t) cos ωt s 2 +ω 2 /s sin ωt ω s 2 +ω 2 t /s 2 cosh at s s 2 a 2 t 2 2/s 3 sinh at a s 2 a 2 t n n! s n+ e at cos ωt s a (s a) 2 +ω 2 e at s a e at sin ωt ω (s a) 2 +ω 2 te at (s+a) 2 t n e at (n )! (s+a) n G. Qalikiˆc,