SISTEM KENDALI KLASIK

Transkripsi

1

2 SISTEM KENDALI KLASIK Pmodln Mmik Anlii Digrm Bod, Nyqui, Nichol Sp & Impul Rpon Gin / Ph Mrgin Roo Locu Diin Simuli

3 SISTEM KONTROL LOOP TERTUTUP

4 PLANT PEMBANGKIT DAYA UAP

5 SISTEM KENDALI GENERATOR

6 KOMPONEN DISAIN SISTEM KENDALI

7 MODEL MATEMATIKA Bgimn mmbu modl mmik?

8 MODEL MATEMATIKA Rncngn dri im kndli mmbuuhkn rumu modl mmik dri im. Mngp hru dngn modl mmik? Agr ki dp mrncng dn mngnlii im kndli. Milny: Bgimn hubungn nr inpu dn oupu. Bgimn mmprdiki/mnggmbrkn prilku dinmik dri im kndli rbu.

9 Du mod unuk mngmbngkn modl mmik dri im kndli:. Fungi Pindh (Trnfr Funcion dlm domin frkuni (mnggunkn Trnformi Lplc.. Prmn-prmn Rung Kdn (S Spc Equion dlm domin wku.

10 RANGKAIAN RLC Prmn Difrnil Bi dp mnggmbrkn prilku dinmik im fiik (hubungn inpu v oupu pri im mknik mnggunkn Hukum Nwon dn im klirikn mnggunkn Hukum Kirchoff. Conoh: Rngkin RLC, jik V( dlh Inpu; i( dlh Oupu V( R i( L C Mnggunkn KVL: v( v ( v ( v ( R L C di( v( vr( L i( d d C Mnggunkn prmn difrnil : Apkh dp mnjdi prmn ljbr drhn? Apkh mudh mnggmbrkn hubungn nr Inpu dn Oupu dri im? Dpkh dibu mnjdi un-un rpih?

11 Jik jwbnny dlh idk unuk kig prnyn di, mk ki mmbuuhkn rnformi Lplc. Trnformi Lplc mmbrikn: Rprni dri Inpu, Oupu dn Sim bgi unun rpih. Hubungn ljbr drhn nr Inpu, Oupu dn Sim. Krbn dri Trnformi Lplc : Bkrj dlm domin frkuni. Brlku hny pbil im dlh linir..

12 x( Lplc Trnform TRANSFORMASI LAPLACE X( Tim Domin Tim Domin Circui Circui L -Domin Circui j Complx Frquncy L Y( Typ of -Domin Circui Wih nd Wihou Iniil Condiion y( Invr Lplc Trnform

13 TRANSFORMASI LAPLACE Trnformi Lplc dlh mod oprionl yng dp digunkn unuk mnylikn prmn difrnil linir. Dp mngubh fungi umum (fungi inuoid, inuoid rdm, fungi kponnil mnjdi fungi-fungi ljbr vribl komplk. Opri difrnii dn ingri dp digni dngn opri ljbr pd bidng komplk. Solui prmn difrnil dp diprolh dngn mnggun-kn bl rnformi Lplc u urin pchn pril. Mod rnformi Lplc mmungkinkn pnggunn grfik unuk mrmlkn kinrj im np hru mnylikn prmn difrnil im. Diprolh cr rnk bik komponn rnin mupun komponn kdn unk (dy.

14 VARIABEL KOMPLEKS Vribl komplk: = + j dngn : dlh komponn ny j dlh komponn my j j Bidng o

15 FUNGSI KOMPLEKS Suu fungi komplk: G( = G x + jg y dngn : G x dn G y dlh brn-brn ny Im G y Bidng G( G q O G x R Br dri brn komplk: Gy Sudu : q n G x G( G x G y

16 TURUNAN FUNGSI ANALITIK Turunn fungi nliik G( dibrikn olh: d d G( lim G( G( lim G Hrg urunn idk rgnung pd pmilihn linn. Krn = + j, mk dp mndki nol dngn k-rhingg linn yng brbd

17 Unuk linn = (linn jjr dngn umbu ny d d G( G x lim G j y G x G j Unuk linn = j (linn jjr umbu my, mk d d G( G x lim j Jik du hrg urunn ini m Gx Syr Cuchy-Rimnn Gy G x j j j Gy Gy j G x Gy G y Gx G j x y G y

18 CONTOH SOAL Tinju G( briku, p nliik? G( Jwb: G( j G x jg y j dimn G x dn G y Dp dilih bhw, kculi =- (yiu =-,=, G( mmnuhi yr Cuchy-Rimnn: G x G y G y G x Dngn dmikin G(=/(+ dlh nliik di luruh bidng kculi pd =-.

19 Turunn dg(/d pd =- dlh d d G( G x G j y Gy Gx j j Prhikn bhw urunn fungi nliik dp diprolh hny dngn mndifrniikn G( rhdp d d Tiik-iik pd bidng yng mnybbkn fungi G( nliik dibu iik ordinr, dngkn iik-iik pd bidng yng mnybbkn fungi G( idk nliik dibu iik ingulr. Tiik-iik ingulr yng mnybbkn fungi G( u urunnurunnny mndki k rhingg dibu pol

20 KUTUB-KUTUB DAN NOL-NOL Zro dri G( Pol dri G( Prmn krkrik roo numror roo dnominor dnominor dri G(= Im R pol zro Pol pol-zro

21 CONTOH SOAL Tnukn jumlh pol dn zro dri fungi G( briku: K( 3 G( ( ( Jwb: Mmpunyi pol di =- dn =- Mmpunyi buh zro di =-3. Jik iik-iik di k rhingg di mukkn: lim G( lim K Fungi G( mmpunyi buh zro di k rhingg. Jdi G( mmpunyi jumlh pol dn zro yng m, yiu 3 buh pol dn 3 buh zro (u zro rhingg dn du zro k rhingg.

22 PEMETAAN KONFORMAL Pmn Konforml dlh uu pmn yng mnjg ukurn mupun pngrin udu. Pmn Konforml digunkn dlm mmbh digrm mp kdudukn kr (roo locu dn kriri kbiln Nyqui. Hubungn fungionl: z=f( dp diinrprikn bgi pmn iik-iik pd bidng k iik-iik pd bidng z / bidng F(. Unuk ip iik P pd bidng rdp uu iik P pngnny pd bidng F(. P dlh byngn dri P. Unuk mmbukikn bhw pmn yng dinykn dngn uu fungi nliik z=f( dlh konforml, inju uu kurv hlu =(, yng mllui uu iik ordinr.

23 Jik ki uli z o =F( o, mk: Dngn dmikin, - o o k. z z z z o o F( F( o o o F( F( o ( o dlh udu nr umbu ny poiif dn vkor dri Jik mndki o pnjng kurv hlu (, mk - o dlh udu q nr umbu ny poiif dn gri inggung kurv rbu pd o. Dngn cr m, jik z mndki z o, mk z - z o mndki udu yng mrupkn udu nr umbu ny poiif dn gri inggung dri F( pd z. Dngn dmikin diprolh - q = F ( o o

24 Dngn kurv hlu yng lin = (, yng mllui iik o, ki dp mlkukn nlii rup hingg diprolh Olh krn iu - q = F ( o u - q = - q - = q - q Jdi ukurn dn pngrin udu pd pmn p dijg. Pmn yng dinykn dngn uu fungi nliik z=f( dlh konforml di ip iik yng mnybbkn F( rgulr dn F (.

25 DEFINISI TRANSFORMASI LAPLACE Trnformi Lplc dri f( didfiniikn bgi dngn: L[f(] F( f( f( = fungi wku, dngn f(= unuk < = vribl komplk d

26 Conoh fungi Dirc L[f (] F( f ( d f( ( L[ (] ( d ( f( L[f (] ( d 6

27 CONTOH Trnformi Lplc dri fungi ngg briku: f( = unuk < f( = A unuk > A Jwb: L{f(} A d A A

28 Trnformi Lplc dri fungi Rmp f( f( unuk L[r(] d d 8

29 Trnformi Lplc dri fungi kponnil briku: f( = unuk < - = A - unuk > A Jwb: L{A } A d A ( d ( A ( ( A

30 Trnformi Lplc dri fungi inuoid briku: f( = unuk < = A in unuk > Jwb: L{A in} A in d j = co + j in - jw = co - j in L{f(} A j A ( in j j j A j ( j d j j j j A

31 f( Sp funcion, u( b - - in( co( n ( b F( / /(+ /(+ / ( + / ( + n!/ n+ ( ( b

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

33 SIFAT LINIERITAS F ( L[f (] ( L[f (] F L[c c c.l[f (].F (.f ( c c c.f.f ( c,c (].L[f (] Con n

34 SIFAT TRANSLASI Jik F(=L[f(] L[ f (] F( L[ f (] [ f (] d f ( ( d F( Conoh L[Co(] 4 L[ Co(] ( 4 5

35 Trnli [im] f( g( b Jik g( = f(- for > = for < L[g(] L[g(] F( f ( ] d f (u (u du Conoh 3 3! 6 L[ ] g( (, g(, L[g(] f (u 6 4 u du 35

36 Prubhn kl wku L[f (.] F( L[f (.] f (.] d f (u u du F( Conoh L[Sin(] L[Sin(3]

37 TEOREMA DIFERENSIASI Trnformi Lplc dri urunn fungi f( dibrikn bgi df ( L d df ( d Ingri bgin dmi bgin mmbrikn df ( L d df( L d d f ( f ( f( L f( Trnformi Lplc ng brgun krn mngubh prmn difrnil mnjdi prmn ljbr drhn. d

38 Turunn Prm [Driviv fir ordr] L[f' (] L[f '(] df L[ d f(d ] L[f (].F( f ( f( f(d F( f( f( L[f'(].F( f( f ( 38

39 Turunn ord inggi L[f '(] L[f"(] L[ (n f (] df L[ ] L[f (] d L[f (].F( n F( n f (.F( f (.f ( n f '( ( (n f (... f ( (n L[f (] n F( Jik diconinuiy pd n i ni f ( (i.f ( f ( L[f '(].F( f ( [f ( f ( ] 39

40 4 Conoh Turunn ] L[Sin( ] L[Co( d ] d[sin( Co( ( Sin( ] L[Sin( ] L[Co( Co( d ] d[in( Sin( d ] d[co( d ] d[co( Sin( ( Co( ] L[Co( ] [Sin( L

41 APLIKASI RANGKAIAN RC R ( C v( Prmn rngkin dv ( RC v( d v( Trnformi Lplc: E( RCV( V( V([ RC] E( V( RC

42 INTEGRASI g ( f (udu] F( L[f (] g( f ( L[g(] L[g(] g( F( L[ f (udu] F(

43 PERKALIAN DENGAN FAKTOR T df( d ' F ( d d [ f (d Libniz rul df( d [ f (d] [ f (]d L[f (] L[f (] F ( ' Rumu umum L[ n f (] ( n d n d F( n

44 PEMBAGIAN DENGAN FAKTOR T f ( g( f ( g( G( L[f (] dl[g(] d F(udu dg( d F(udu F( LimG ( f ( L[ ] F(udu

45 FUNGSI PERIODIK f ( kt f (, k L[f (] L[f (] L[f (] L[f (] F( F( n F( F( nt T T T n f (d f (d T T T f (d nt [ T T (ut T T f (d f (d] 3T T f (u Tdu u f (udu L[f (] f (d... T T T (ut u F( f (u Tdu... f (udu... T f ( T d

46 FUNGSI PERIODIK SINUS & COSINUS jsin ( Co( j d d ] jl[sin( ] L[Co( ] L[ j ( j j T T ( j j d ] [ L ] [ j ] [ j j d T T T j T j ( T j ( j j j ( j ( j j ] L[

47 PERILAKU BATAS LIMIT : NILAI INISIAL L[f (] F( f ( Exponnil ordr Lim[ f (d] Lim[F(] f ( Lim[f (] f ( Lim[f (] lim[f(] { }...{ }

48 FUNGSI IMPULSIONAL ( ( E( ( RC V ( RC RC V ( RC Impul rpon v( RC CR CR RC V( (RC

49 FUNGSI TANGGA DENGAN KONDISI AWAL NOL ( u( E ( V( ( RC ( V ( RC v( [ cr ],63 v( CR [ CR ] RC

50 FUNGSI TANGGA DENGAN KONDISI AWAL Sp funcion dn iniil condiion v( E( RC[V( v(] V( V([ RC] RCv( V( ( RC RCv( RC v( cr [v( ] ( v( ( RC V v( v( CR [v( ] RC[v( ] V( RC

51 FUNGSI RAMP r( ( ( E RC ( ( V RC ( RC RC ( V CR RC RC v( RC CR v( CR d dv (RC (RC RC V(

52 ANALISIS HARMONIK ( in( E( V( ( ( RC A B C V( ( V( ( A B C

53 ( ( V ] co( in( RC [ ( v CR RC g( (RC Co( ] in( [in( Co( ( v CR Forcd Trnin

54 TRANSFORMASI LAPLACE INVERSE Dikhui: F(=L[f(] Bgimn mncri f( dri F(? f( L [F(] Mhod Anliik L [F(] f (..i i. i. F(.d Pd konour Bromwich b Mod Tbl F( f (

55 c Ekpni frki dngn kr-kr brbd F( B( A( p p... n p n n i i p i f( p p... n p n n i i p i Hrg k (ridu pd pol =-p k dp diprolh dngn: B( k n k ( pk ( pk... ( pk... ( pk A( p p p k p n k p k Smu uku urin mnjdi nol, kculi k. Jdi ridu k diprolh: B( A( k ( pk p k

56 CONTOH SOAL Crilh rnformi Lplc blik dri Jwb: F( Trnformi Lplc blik dri: -p L p F( ( ( ( 3 ( ( ( 3 3 ( ( ( 3 ( ( (

57 L L F( L L ( ( F( unuk

58 CONTOH SOAL 3 ( ( ( 4 F( 3 ( 7 4( 3 6( F( ( f

59 SIFAT-SIFAT TRANSFORMASI LAPLACE