Rez Teme Lab

8/15/2019 Rez Teme Lab

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Constantin Daniel AlexandruGrupa 332 AA

Probleme pentru colocviul de laborator

--- CALCUL NUMERIC ÎN AUTOMATICĂ ---

Laboratorul 2 - Rezolvarea ecuaţiilor matriceale liniare Slvester !i Liapunov

Problema " : Se consideră date matricile mxm RS ∈ suerior !idia"onală nxn

RT ∈ dia"onală #imxn

RC ∈ $ Se cere un ro"ram MATLA% e&icient entru re'ol(area ecua)iei matriciale S*l(estercontinue S+ , +T C:

&or i m:-.:.&or / .:n

+0i1/20C0i1/2-S0i1i,.23+0i,.1/2240S0i1i2,T0/1/22

end end

Problema 2 : Se consideră date matricile mxm RS ∈ suerior !idia"onală1 nxn

RT ∈ in&eriortriun"5iulară #i mxn

RC ∈ $ Se cere un ro"ram MATLA% e&icient entru re'ol(area ecua)iei

matriciale S*l(ester discrete S+T - + C:

S.67S8679or i m:-.:. 9or /n:-.:.

9or /,.:n S.S.,S0i1i23+0i123T01/27 end 9or /:n S8S8,S0i1i,.23+0i,.123T01/27 end +0i1/20C0i1/2-S.-S822400S0i1i23T0/1/2-.2 endend

Problema 3 : Se consideră date matricile mxm R A∈ suerior !idia"onală #i mxm

RC ∈ $

Se cere un ro"ram MATLA% e&icient entru re'ol(area ecua)iei matriciale Liauno( continueC XA X A

T =+

&or i .:m &or / .:m +0i1/20C0i1/2-A0i-.1i23+0i-.1/2-+0i1/-.23A0/-.1/2240A0i1i2,A0/1/22 endend



Problema # : Se consideră date matricile mxm R A∈ suerior !idia"onală #i mxm

RC ∈ $Se cere un ro"ram MATLA% e&icient entru re'ol(area ecua)iei matriciale Liauno( discrete C X XA A

T =−

&or /.:m &or i/:m +0i1/2 0C0i1/2 ; 0A0i-.1i23+0i-.1/-.23A0/-.1/2,A0i1i23+0i1/-.23A0/-.1/2,A0I-.1I23+0I-

.1/23A0/1/224 40A0i1i23A0/1/2-. 2 endend

Laboratorul 3 - Calculul $uncţiilor de matrice% &xponenţiala matriceal'

Problema " : Al"oritmul <arlett =n (ersiunile >e linii? #i >e coloane?$

&unction @9<arlletLinii0T1&2nlen"t50T27

9'eros0n27&or in:-.:. 90i1i2&e(al0&1T0i1i227 &or /i,.:n s67 &or i,.:/-. ss,T0i123901/2-90i123T01/27 end 90i1/20T0i1/23090/1/2-90i1i22,s240T0/1/2-T0i1i227 end

end

&unction @9<arlletColoane0T1&2nlen"t50T27

9'eros0n27&or /.:n 90/1/2&e(al0&1T0/1/227 &or i/-.:-.:. s67 &or i,.:/-. ss,T0i123901/2-90i123T01/27 end 90i1/20T0i1/23090/1/2-90i1i22,s240T0/1/2-T0i1i227 end

endProblema 2 : Imlementarea al"oritmului de calcul al eBonen)ialei matriceale !a'at earoBima)ia Ta*lor$

&unction @9Ta*lor0A1t2Sc5ema 0.482D8normanorm0A27m675ile t3normaF. tt487

mm,.7endnlen"t50A27

+e*e0n279e*e0n27 G7

&or .: +.43t3A3+7

99,+7end

Problema 3 : Al"oritmul <arlett entru calculul &unc)iilor de matrice in&erior triun"5iulare cuelementele dia"onale distincte$&unction @9<arlettIn&0T1&2nlen"t50T27





M 'eros0l1ni-.27&or i.:l M0i1:2 roots00i1:227endRad 'eros0.1ni,l27 67

&or i.:l &or /.:ni-. aar 67 &or i.: i& 0Rad0.1i2 M0i1/22 aar aar ,.7 end end i&0aar 62 ,.7 Rad0.12 M0i1/27

end end

end

n , .7 . 'eros0.1n27 . ol*0Rad0.1:227A 'eros0n-.27

&or i.:n-. A0.1i2 -.0.1i,.27 A0i1i,.2 .7end% 'eros0n-.1.27%0.1.2 67C 'eros0l1n-.27&or i.:l &or /.:n-. C0i1/2 N0i1/27 end

end

Problema 2 : Se (or scrie ro"rame MATLA% rorii entru imlementarea al"oritmilor de calcul olinomial necesare maniulării matricelor de trans&er$

Suma a 8 olinoame de "rade di&erite$$unction)c*+suma,ab.@1msi'e0a27@1nsi'e0!27c'eros0.1maB0m1n227i& mFn &or i.:m-n c0i2a0i27 end &or im-n,.:m c0i2!0i-m,n2,a0i27 endelse &or i.:n-m c0i2!0i27 end &or in-m,.:n c0i2a0i-n,m2,!0i27 endend

Îmăr)irea a 8 olinoame cu rorietetea: "radul rimului este mai mare dacHt "ardul celui de-aldoilea



$unction)c*+impartzire,ab.@1msi'e0a27@1nsi'e0!27(arm-n,.7c'eros0.1(ar27c0.2a0.24!0.27

i& 0mod0m182.2 &or i8:(ar s67 &or /.:i-. ss,c0/23!0(ar-/,.27 end c0i20a0i2-s24!0.27 endelse &or is:(ar-. s67

&or /.:i-. ss,c0/23!0(ar-27 end c0i20a0i2-s24!0.27 end c0(ar2a0m24!0n27end

Calcului coe&icien)ilor unui olinom cunoscHnd rădăcinile$unction)b*+coe$icientzi,a.@1msi'e0a27&or i.:m,. !0i267end !0.2.7&or i.:m &or /i,.:-.:8 !0/2!0/2- a0i23!0/-.27 endend

Problema 3 : 9iind date trei sisteme Si1 i.:1 rin metodele lor de trans&er1 să se scrie al"oritmii deconstruc)ie ai matricelor de trans&er ale sistemelor re'ultate rin interconectarea sistemelor datecon&orm sc5emelor din &i"urile:



(atricea de trans$er pentru ,a./

0a2: ( ) ( ).8..8.88...

Y U H H Y U H H Y −+−⋅⋅=

( ) ( )88.88..88.8

Y U H H Y U H H Y −+−⋅⋅=

Se re'ol(a sistemul si se scot eBresiile lui Y 1 si Y 2$ Je eBemlu entru Y 2 se o!tine:

...88.88...88.888..

88...888.888..888...8.

8

. H H H H H H H H H H H H

U H H H H U H H U H H H H U H H Y

−+++++−⋅⋅=

$unction )00"002*+mmatricetrans$er1a,0"[email protected]'e0K.27@m81n8si'e0K827@m1nsi'e0K27Kd.'eros0m8,m1n8,n27Kd.0.:m81.:n82K87Kd.0m8,.:m8,m1n8,.:n8,n2K7Ks.'eros0m8,m1n8,n27Ks.0.:m81n,.:n8,n2K87Ks.0m8,.:m8,m1.:n2K7KK.in(0e*e0m8,m1n8,n2-Ks.23Kd.7Kd.'eros0m.,m1n.,n27Kd.0.:m.1.:n.2K.7Kd.0m.,.:m.,m1n.,.:n.,n2K7Ks.'eros0m.,m1n.,n27Ks.0.:m.1n,.:n.,n2K.7Ks.0m.,.:m.,m1.:n2K7



KK8in(0e*e0m.,m1n8,n2-Ks.23Kd.7

(atricea de trans$er pentru ,b./&unction @KK.1KK8m*matricetrans&[email protected]'e0K.27

@m81n8si'e0K827@m1nsi'e0K27Kd.'eros0m8,m1n8,n27Kd.0.:m1.:n2K7Kd.0m,.:m8,m1n,.:n8,n2K87Ks.'eros0m8,m1n8,n27Ks.0.:m1n8,.:n8,n2K7Ks.0m,.:m8,m1.:n82K87KK.in(0e*e0m8,m1n8,n2-Ks.23Kd.7Kd.'eros0m.,m1n.,n27Kd.0.:m1.:n2K7

Kd.0m,.:m.,m1n,.:n.,n2K.7Ks.'eros0m.,m1n.,n27Ks.0.:m1n.,.:n.,n2K7Ks.0m,.:m.,m1.:n.2K.7KK8in(0e*e0m.,m1n8,n2-Ks.23Kd.7

Problema # : Se dă sistemul S 0A1%1C1J2 ale cărui matrice au structura:

Să se scrie o rocedură de rere'entare a lui S su! &orma unei coneBiuni a2 aralel #i !2 serie a douăsisteme S.1 S8 con(ena!il de&inite$a2 la coneBiunea serie:



++

+⋅=

20 ..888

...

M

8

M

.

u DC B x A

u B x A

x

x

re'ultă:

= 8.8

. 6

AC B

A

A si

= .8

.

D B

B

B $ F

A8.%8C8 si 8%8J.

**8C8+8,J8u8C8B8,J80C.+.,J.u2 [ ]8.8

C C D +

8

.

x

[email protected]

!2 la coneBiunea aralel se o!ser(ă ca 68.

= A 1 88 BG = 1 ..

C H = iar [ ]8.

D D D = $

Laboratorul - Calculul r'spunsului n timp al sistemelor liniare

Problema " : Scrie)i al"oritmul de calcul al răsunsului unui sistem liniar discret S descris rintr-orela)ie intrare - ie#ire de &orma:

reci'Hnd cu "ri/ă datele ini)iale necesare$

$unction +calcul1rasp",abu.nlen"t50a27dlen"t50!27mlen"t50u27

&or .:m sum.67 &or i.:n i& -iF6 sum.sum.,a0i23*0-i27 end end &or i.:d i& -i,.F6 sum.sum.,!0i23u0-i,.27 end

end *02sum.7end

Problema 2 : Scrie)i al"oritmul de calcul al rasunsului *0521 . : & 1 al unui sistem liniar discret daca in locul starii initiale B062 B6 se da starea &inala B0t& 2 $

$unction calcul1rasp2,A4CD5$x.&or &,.:-.:.

P0:12C3B,J3U0:127 BSL<<0A1B-%3U0:122



ends-au &olosit entru re'ol(area eBerci)iului anterior următoarele &unc)ii:

$unction )A6p*+GPP,A.nlen"t50A27 !67

U'eros0n27 .:.:n7&or .:n-. (.:.:n-,.7 &or i:n (0i-,.2a!s0A0i1227

end dmaB0(27 &or i:n i& da!s0A0i122 02i7

!rea7 end end &or /:n sA01/27 A01/2A0021/27 A0021/2s7 end &or i,.:n7 U0i12A0i124A0127 A0i1267 end &or i,.:n &or /,.:n A0i1/2A0i1/2-U0i123A01/27 end endend

$unction )x*+SL1GPP,Ab.nlen"t50!27@U1M1<<0A27&or .:n-. s!027 !02!00227 !0022s7

&or i,.:n !0i2!0i2-M0i123!027 end Bssutr0!1U27end

$unction )x*+s1sup1tr,ba.nlen"t50!27&or in:-.:. s!0i27 i& iQn &or i,.:n ss-a0i123B027 end end B0i2s4a0i1i27end

Problema 3 : <ro"ramele MATLA% entru imlementarea al"oritmilor de calcul al răsunsului ermanent al sistemelor liniare continue la intrări armonice$

sistemele sunt SIMO %n3.

$unction 7+rasp1perm1contin1armonice,A4CD8amma"8amma29:[email protected]%@.76C@"amma.13"amma8mlen"t50A27M'eros083m183m27M0.:m1.:m2-A7M0.:m1m,.:83m23e*e0m1m27



M0m,.:83m1.:m2-3e*e0m1m27M0m,.:83m1m,.:83m2-A7R@%3"amma.7%3"amma87(SL<<0M1R2(.(0.:m2(8(0m,.:83m2

.C3(.,J3"amma.78C3(8,J3"amma87@. 38C3,J3CPinitial"0A11%151t&27

s-a &olosit &unc)ia descrisă la eBercitiul anterior1 SL<<1 si &unc)ia seci&ică sistemelor SIMO:

$unction )7*+initial18,ACx:t$.&&iB0t&4527AdeBm053A27

%'eros0len"t50A21.27J'eros0len"t50C21.27Pdinitial0Ad1%1C1J1B27

Problema # : <ro"ramul MATLA% entru imlementarea al"oritmului de calcul al caracteristicilor de &rec(en)ă a unui sistem SISO continuu$ Se re(ede rere'entarea "ra&ică a 5odo"ra&ului0dia"rama N*uist21 a modulului #i &a'ei 0dia"rame %ode2

&unction @RE1IMcaract&rec(0a1!1c1d11n2&or .:n 02.6D0-.27end@M1K5ess0a2 !M3!cc3in(0M2@m1msi'e0a2&or .:n a6i3023e*e0m2-a @N1Rsc5ur0a62 @N1Rrs&8cs&0N1R2

!6N3! in(0a623!6 Tc3,d 02 RE@RE real0T2 IM@IM ima"0T2 A@A srt0real0T2D8,ima"0T2D82 9@9 atan0ima"0T24real0T22endss0a1!1c1d27@Re1Imn*uist012@MA1<KASE %OJE012

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Rez Teme Lab