SISTEM KENDALI OTOMATIS Transformasi Laplace

SISTEM KENDALI OTOMATIS Trnformi Lplc

Opn Loop/Clod Loop Sytm Input/ Dird output Controllr Control ignl Actutor Actuting ignl Plnt Plnt output Input/ Dird output + - Error ignl Controllr Control ignl Actutor Actuting ignl Plnt Plnt output Snor

Itilh-itilh dlm SKO Plnt : Sutu prltn tu objk fiik yng ditur/dikndlikn Pro : Opri yng dikndlikn Sitm : Gbungn komponn yng bkrjm untuk mncpi tu tujun Gnggun : Sutu inyl (intrnl/ktrnl) yng mmpunyi pngruh mrugikn output itm

Itilh-itilh dlm SKO Input (Dird Output) : Output yng diinginkn Error : Sliih ntr input dn output yng trjdi pd t itu Sinyl kontrol : Sinyl dri kontrollr

Modl Mtmtik Rncngn dri itm kndli mmbutuhkn rumu modl mtmtik dri itm. Mngp hru dngn modl mtmtik? Agr kit dpt mrncng dn mngnlii itm kndli. Milny: Bgimn hubungn ntr input dn output. Bgimn mmprdiki/mnggmbrkn prilku dinmik dri itm kndli trbut.

Trnformi Lplc Mngubh fungi dri itm fii (domin wktu) k fungi vribl komplk (domin ) Mnydrhnkn prmn mtmti yng mngndung opri turunn/diffrnil tu intgrl mnjdi prmn yng brii prklin tu pmbgin bi Dpt mngubh fungi umum (fungi inuoid, inuoid trdm, fungi kponnil) mnjdi fungi-fungi ljbr vribl komplk Mtod ini mmungkinkn untuk mrml kinrj itm mnggunkn grfi tnp hru mnylikn prmn diffrnil Komponn trnin dn tdy tt diprolh cr rntk

Pnylin Mnggunkn Trnformi Lplc Scr drhn produr dr pmchn mnggunkn mtod trnformi Lplc dlh: Prmn difrnil yng brd dlm kwn wktu (t), ditrnformikn k kwn vribl komplk() dngn trnformi Lplc. Untuk mmprmudh pro trnformi dpt digunkn tbl trnformi lplc. Prmn yng diprolh dlm kwn trbut dlh prmn ljbr dri vribl yng mrupkn oprtor Lplc. Pnylin yng diprolh kmudin ditrnformi-blikkn k dlm kwn wktu. Hil trnformi blik ini mnghilkn pnylin prmn dlm kwn wktu.

x(t) Lplc Trnform X() Tim Domin Tim Domin Circuit Circuit L L -Domin Circuit j Complx Frquncy Typ of -Domin Circuit Y() With nd Without Initil Condition y(t) Invr Lplc Trnform

Dfinii Trnformi Lplc t L[ f ( t)] F( ) f ( t) dt dngn: f(t) = fungi wktu t, dngn f(t)= untuk t< = vribl komplk

Ltihn Hitung Trnformi Lplc Unit Stp u(t) t

Hitung Trnformi Lplc Unit Rmp f(t) t f ( t) At untuk t

Hitung Trnformi Lplc dri f t = t

Hitung Trnformi Lplc dri fungi inu

( (b f(t) n t t in(t) F()=L[f(t)] (t) u(t) t co(t) h(t) ch(t) / n!/ /( ) /( /( /( /( / (n) bt in(t) /[( b) ] co(t) ( b) /[( b) bt bt bt t ) /(b ) t ) /(b ) ) ) /( )( ) ) b) /( )( b) ] b b

F () L[f (t)] () L[f (t)] F SIFAT LINIERITAS L[c c c.l[f.f ().f (t) c (t)] c c.f.f () c,c (t)].l[f (t)] Contn t

SIFAT TRANSLASI ) Jik F()=L[f(t)] L[ t f (t)] F( ) L[ t f (t)] [ t f (t)] t dt f (t) ()t dt F( ) Contoh L[Co(t)] 4 t L[ Co(t)] ( ) 4 5

Trnli [tim] f(t) g(t) b) Jik g(t) = f(t-) for t> = for t< L[g(t)] L[g(t)] f (t F() )] t dt f (u) (u) du Contoh 3 3! 6 L[t ] 4 4 3 g(t) (t ), t g(t), t L[g(t)] f (u) 6 4 u du t

Prubhn kl wktu L[f (.t)] F( ) L[f (.t)] f (.t)] t dt f (u) u du F( ) Contoh L[Sin (t)] L[Sin (3t)] 3 3 3 9

TEOREMA DIFERENSIASI Trnformi Lplc dri turunn fungi f(t) dibrikn bgi df ( t) L dt df ( t) dt Intgri bgin dmi bgin mmbrikn df ( t) L dt df(t) L dt t dt t f ( t) f ( t f() ) L f(t) Trnformi Lplc ngt brgun krn mngubh prmn difrnil mnjdi prmn ljbr drhn. t dt

Turunn Prtm [Drivtiv firt ordr] L[f' (t)] L[f '(t)] t df L[ dt f(t)dt ] L[f (t)].f() f ( t f(t) ) t f(t)dt F() f( ) f(t) L[f '(t)].f() f ( ) f ( ) t 4

Turunn ord tinggi (Drivtiv of highr ordr) L[f '(t)] L[f"(t)] L[ (n) f (t)] df L[ ] L[f (t)] dt L[f (t)].f() n F() n f ().F() f (.f ( n ) ) f '() () (n) f ()... f () (n) L[f (t)] Jik dicontinuity pd n F() n i ni f ( (i).f () ) f ( ) L[f '(t)].f() f () [f ( ) f ( )] 5

Contoh Turunn L[Sin ( t)] d[in( t)] Co( t) dt L[Co( t)] L[Sin( t)] L[Co( t)] d[sin ( t)] Co( t) dt Sin( ) ( ) d[co( t)] d[co( t)] Sin ( t) Sin ( t) dt dt Co( ) L[Sin ( t)] L[Co( t)] ( ) 6

INTEGRASI t g (t) f (u)du] F() L[f (t)] g(t) f (t) L[g(t)] L[g(t)] g( ) F() L[ t f (u)du] F()

Prklin dngn fktor t df() d ' F () d d [ t f (t)dt Libnitz rul df() d [ t f (t)dt] t [ tf(t)]dt L[tf(t)] L[tf(t)] F () ' Rumu umum L[t n f (t)] ( ) n d n d F() n

Pmbgin dngn fktor t f (t) g(t) f (t) tg(t) t G() L[f (t)] dl[g(t)] F(u)du d dg() d F(u)du F() LimG() f (t) L[ ] t F(u)du

f (t kt) f (t) FUNGSI PERIODIK t, k L[f (t)] L[f (t)] L[f (t)] L[f (t)] F() F() n F() F() nt T T T t t n t f (t)dt f (t)dt T T T f (t)dt nt [ T T t t (ut) T T f (t)dt f (t)dt] 3T T t f (u T)du u f (u)du L[f (t)] f (t)dt... T T T (ut) u F() f (u T)du... f (u)du... T f (t) t T dt

Fungi priodik Sinu & Coinu t) jsin ( t) Co( t j dt dt t)] jl[sin ( t)] L[Co( ] L[ )t ( j t t j t j T T )t j ( t j dt ] [ L ] [ j ] [ j j dt T T T j T )t ( j T )t ( j t j j ) j )( j ( j j ] L[

Sift Trnformi Lplc

Trnformi Lplc Invr Dikthui: F()=L[f(t)] Bgimn mncri f(t) dri F()? f(t) L [F()] ) Mtod Tbl F() f (t) t

n i p t i n n i p... p p A() B() F() n i p t i t p n t p t p i n... f(t) b) Ekpni frki dngn kr-kr brbd Hrg k (ridu pd pol =-p k ) dpt diprolh dngn: k p k k n n k k k k p k k ) p ( p... ) p ( p... ) p ( p ) p ( A() B() Smu uku urin mnjdi nol, kculi k. Jdi ridu k diprolh: p k k k ) p ( A() B()

Contoh Sol Crilh trnformi Lplc blik dri 3 F() Jwb: ( )( Trnformi Lplc blik dri: ) L F() p ( -pt )( ) ( ) ( 3 3 ( ) ( )( ) 3 ( ) ( )( ) )

) ( L ) ( L F() L untuk t F() L t t

Contoh Sol 3) )( )( ( 4 F() 3) ( 7 ) 4( 3 ) 6( F() 7 4 3 6 3 t t t (t) f

. Dfinii input dri itm kndli otomti yng pling tpt dlh. Mukn dri itm yng mmpngruhi pro b. Output yng diinginkn c. Prngkt yng digunkn untuk mmukkn dt kdlm itm d. Sliih ntr mukn dn klurn

A B C D. Dri gmbr dit, inyl kontrol ditunjukkn olh bgin:. A b. B c. C d. D

3. Mnkh brikut ini yng mrupkn kgunn dri trnformi lplc (pilih lbih dri tu):. Mngubh prmn dlm domin wktu k vribl komplk (S) b. Mnydrhnkn prmn mtmti yng brii turunn/difrnil mnjdi prmn yng brii prklin dn pmbgin bi c. Mngubh prmn dlm domin wktu k domin frkuni d. Mngubh fungi umum (inuoid, kponnil, dll) k dlm vrbl komplk

4. Fungi yng mmpunyi knggotn dngn nili untuk t< dn nili untuk t, dlh. Unit tp b. Unit rmp c. Ekponnil d. Unit tunggl

5. Trnformi lplc dri unit tp dlh. S b. S c. / d. /

Tug. Tntukn trnformi lplc dri. f t = t 3 t b. f t = t c. f t = t in(3t) d. f t = t. Tntukn invr trnformi lplc dri. G = 3 +5 +9+7 (+)(+) b. F = + ( ++) c. F = ++3 + 3

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