TIME RESPONSE Introduction

Transcription

1 TIME RESPONSE Iroducio Time repoe of a corol yem i a udy o how he oupu variable chage whe a ypical e ipu igal i give o he yem. The commoly e ipu igal are hoe of ep fucio, impule fucio, ramp fucio ad iuoidal fucio. ep igal impule igal iuoidal igal ramp igal raie ae eadyae The ime repoe of a corol yem coi of wo par: he raie repoe ad he eady-ae repoe. Traie repoe i he maer i which he yem goe form iiial ae o he fial (deired) ae. Seady-ae repoe i he behaviour i which he yem oupu behave a he ime approache ifiiy.

2 Differeial Mahemaical Model The model derived i he previou chaper are i he form of differeial equaio ad he ime repoe of hee model are he repecive differeial equaio oluio. Laplace raform i uually ued o olve he differeial equaio. Geerally, he differeial equaio i wrie a: d y d y a... a d d dy d a y = b 0 m m d u du... b b 0u m d d (0) (0) wih he iiial codiio ( dy d y y 0),... ad > m. d d EXAMPLE I Derive he differeial equaio for he ralaioal yem how below: x i k x o m c

3 Laplace Traform Defiiio: L f() = F() = 0 e f()d where = σ jϖ which i a complex variable Laplace raform ad ivere Laplace raformaio i wrie a: or Liearly, Hece,

4 EXAMPLE II Produce he Laplace raform for f() = which i alo kow a ui ep. EXAMPLE III Produce he Laplace raform for f() = e -a.

5 EXAMPLE IV Produce he Laplace raform for f() = a. Le However, i i o alway eceary o derive he Laplace raform of f() each ime. Laplace raform able ca coveiely be ued o fid he raform of a give fucio f(). The able below how Laplace raform of ime fucio ha frequely appear i liear corol aalyi.

6 LAPLACE TRANSFORM PAIRS f ( ), > 0 F() Laplace defiiio, y() ui impule pada = 0 ui ep u() ui ramp polyomial, =,, 3,. expoe e -α ie wave coie wave damped ie wave e -α iβ damped coie wave e -α co β fir differeiaio ecod differeiaio h differeiaio

7 EXAMPLE V Fid he ime repoe; x o (), for a ui-ep ipu wih iiial ae x o (0) = 0. K = C= x i x o

8 Time Repoe Laplace Traform Applicaio The ue of Laplace raform i olvig differeial equaio i eay ad i doe wih he aid of Laplace raform able. The differeial equaio i fir raformed io Laplace form wih all variable iiial codiio ake io coideraio. The Laplace form equaio i a ordiary algebraic equaio which ca eaily bee olved. The ime repoe i obaied by mea of iverig he Laplace raformaio of he oupu variable. Parial-fracio expaio echique i ormally ued beforehad o ai i implifyig he oluio. The advaage of he parial-fracio expaio approach i ha he idividual erm are very imple fucio which ca eaily bee olved uig he ivere Laplace able. EXAMPLE I Fid he ime repoe, x o (), for ui-ep ipu, x i () =, wih iiial codiio x o (0) k c = 0 da x& o (0) = 0. Take raio of = ad = 3. m m x i k x o m c

9 Remaider Theorem Parial fracio ca be olved uig Remaider Theorem. (The Remaider Theorem ca alo be ued for olvig complex roo). Y() = N() D() A i = ( ) i i.e. A i = N() ( i D() ) = i EXAMPLE II Solve: d y d dy d 5 6y = e ; dega y(0) = da y &(0) =

10 EXAMPLE III Solve: 3 X o () = ( )( ) EXAMPLE IV Solve: Xo () = ( 5 5)

11 Time Repoe Fir Order Syem Claificaio of Corol Syem Corol yem i claified accordig o a cerai defiiio a which he performace of he corol yem ca be prediced. Coider he uiy feedback corol yem how below. The ope-loop rafer fucio i defie a kg(), ad i geeral ca be wrie a: or where > m. Syem claificaio i doe accordig o::. Order :- highe order of a he deomiaor. Rak :- ( m) 3. Cla/Type :- l highe order of a he umeraor u _ k G() y )... )...( )( ( )... )...( )( ( ) ( f e d d c b a a k kg l = = = = = l k k k l m k k k l l l l l m m m m B A K B B B B B A A A A A k kg )... ( )... ( ) ( umeraor deomiaor

12 EXAMPLE I Deermie he order, rak ad cla of a yem wih he followig ope-loop rafer fucio: a. G() = b. G() = 3 ( )( ) c. G() = ( )( 4) Fir-Order Syem I geeral, a fir-order yem i repreeed by: X i () K T X o () where K i he yem gai. Example of Fir-Order K q i T q o C C R x i x o

13 Whe yem gai K =, he ime repoe for fir-order yem wih ui-ep ipu ca be obaied a follow:

14 EXAMPLE I The rafer fucio which relae ipu volage, v, ad oupu orque, τ, of a DC moor i repreeed by a fir order rafer fucio. A ime repoe e wih a 6 vol ipu volage reulig i a eady-ae oupu orque of 0 N-cm ad i ook 0.4 ecod o reach.6 N-cm. Fid he rafer fucio of he moor.

15 Time Repoe Secod-Order Syem Secod-Order Syem Geerally, a ecod-order yem i repreeed by he rafer fucio how below: u Kω ξω ω y where K i he yem gai, ω i udamped aural frequecy ad ξ i dampig raio. The value of hee parameer deermie he repoe of ecod-order yem ad are alo he deig parameer. Example of ecod-order yem K m C x i x o

16 Whe gai K =, ime repoe of ecod-order yem for a ui-ep ipu ca be obaied a follow: y() = ξ ξω e i( ω d α) where, damped aural frequecy, ω d = ω ξ ad phae hif, α = a ξ ξ

17 Effec of Dampig Raio o he Time Repoe of Secod-Order Syem a. Over damped, ξ > y() b. Udamped, ξ = 0 y() c. Damped, 0 < ξ < y()

18 Amog commo behavioural idicaor o be looked for are: i. how fa he yem repoe oward a ipu ii. how doe he yem ocillae iii. how log doe i ake o reach he fial value Thee idicaor ca be ralaed io he followig parameer of a ecod-order yem, y() M p % r a. rie ime, r - he ime he oupu repoe ake o rie from 0% o 00% π a ξ ξ r = ( ω ξ ) = π a ω d ξ ξ

19 b. peak ime - he ime of idividual peak of he ime repoe. π =, =, 3 ω ξ ω ξ ω ξ 3 π = 5 π c. overhoo - howig he overhoo value, he differece bewee he fir peak ad he eady ae value i perceage. ξπ/ ξ p Perceage of overhoo, M = e 00% d. elig ime, - ime eeded for he oupu o reach ad ay wihi a accepable oupu limi (he accepable oupu limi i ormally bewee % o 5% of he fial value). accepable limi = e ξω ξ for limi of %, 4 ξω e. damped aural frequecy, ω d ω d = ω ξ

20 EXAMPLE I The rafer fucio of a ecod-order yem i y = u deermie he gai, udamped aural frequecy ad dampig raio of hi yem. EXAMPLE II A ecod-order yem ha a udamped aural frequecy of ω = rad/ ad dampig raio of ξ = 0.. Fid he damped aural frequecy, he fir ad ecod peak, ad perceage of overhoo.

21 EXAMPLE III A ecod-order yem i how i he Figure below. For a proporioal corol value of K p = 0, deermie he aural frequecy, perceage of overhoo ad elig ime for he yem if he ipu i a ui-ep. u. _ K p ( )(0. ) y 6

22 EXAMPLE IV A uiy feedback yem i how i he Figure below. Deermie he gai K ad he appropriae value of parameer p a uch he followig pecificaio ca be me: Fae repoe wih perceage of overhoo le ha 5% ad elig ime le ha 4 ecod. u K _ ( p) y

23 The Effecivee of a Feedback Syem Whe deigig a feedback yem, he effecivee of he deig i achievig i deired objecive ha o be meaured. The effecivee of he yem i meaured by lookig a i repoe a eady ae. The effecivee of a feedback yem ca be deermied by referrig o he eady ae error. For example, coider he feedback yem how below: u() _ e() G() y() H() The relaiohip bewee he error ad he ipu ca be wrie a follow: e() = u() Hy() GH or e() = u() Sice we oly iereed i he repoe a eady-ae, he complee oluio of he yem i o eceary. The oluio a eady-ae ca be acquired eaily uig he Fial Value Theorem. Fial Value Theorem The Fial Value Theorem i defied a: e = lim f() = lim [.e()] 0 e = lim 0 or.u() GH()

24 I ca be clearly bee ee ha he eady-ae error of a corol yem deped o he ype of ipu, u() ad ope-loop rafer fucio, GH(). For ui-ep, ui-ramp ad ui-parabolic ipu, he eady ae error ca be wrie a he followig:. Sep ipu. Ramp ipu 3. Parabolic ipu u() =, u() = / u() =, u() = / u() =, u() = / 3 Where k p i kow a diplaceme error coa k v k a i kow a velociy error coa i kow a acceleraio error coa The ope-loop rafer fucio GH() deermie he cla or ype of a yem. The followig Table how he eady ae error for differe clae of yem. Cla 0 Cla Cla Sep ipu k p 0 0 Ramp ipu kv 0 Parabolic ipu ka

25 EXAMPLE I Calculae he diplaceme error coa ad eady-ae error for a yem wih he followig ope-loop rafer fucio: 0 GH ( ) = 0 EXAMPLE II Fid ou he error coa for he yem how below. R() _ 4 K C()

26 Corol Acio For a corol egieer, hi/her ulimae objecive i o deig a coroller which i able o fulfil he deig pecificaio of he yem. The pecificaio iclude he eady-ae error, overhoo, rie ime ad elig ime. The coroller ipu ormally i he error bewee he ipu ad he oupu, e() a how below. u() _ e() coroller G c () m() proce G() y() Oe of he corol raegie uually ued i he PID (proporioal-iegral-derivaive) coroller where i rafer fucio i give by he followig equaio: m() G c () = = (K p K d e() K i ) a. Proporioal Acio, P For proporioal corol, P, oly he proporioal gai, K p i ued o improve he yem performace. Here, he error igal ielf i ued a he bai of corol. For a cla 0, he eady-ae error cao be elimiaed. Proporioal corol fucio i give by: m() = K p.e() he coroller rafer fucio i m() G () = = e() c K p

27 b. Iegral corol Acio, I The oupu of iegral corol i proporioal o he iegraio of he coroller ipu (i.e. iegraio of error): m() = K i 0 e() d he rafer fucio of hi coroller i, G m() () = e() c = K i where K i i kow a he iegral gai. The iegral acio will remove he eadyae error of cla 0 yem. c. Derivaive Corol Acio, D The derivaive coroller i ued o eablih error which move oward zero. I ca alo predic he error ad akig acio before he error occur. or i rafer fucio, G m() = K d de d m() () = e() c = K d The iegral ad derivaive coroller are ormally o ued aloe. I i uually combied ogeher wih he proporioal coroller o produce a beer corol acio. d. Proporioal ad Iegral Corol Acio (PI) The proporioal ad iegral coroller fucio i give by: m() G c () = = K p ( e() T i )

28 where, T i = K p /K i e. Proporioal ad Derivaive Corol Acio, PD I hi applicaio boh error ad i derivaive igal are ued a he bai of corollig. Derivaive coroller acio repoe o he rae of error chage, hece, i provide roger igal for faer error chage. The derivaive coroller predic he large error ad doe he correcio before he error occur. A eady ae he rae of error chage i zero, hu, he derivaive coroller ha o effec o he eady-ae error. The proporioal ad derivaive corol acio i give by, where, T d = K d /K p () G c = m() e() = K p ( T d ) f. Proporioal, Iegral ad Derivaive Corol Acio, PID Oe of he very commo corol raegie i he combiaio of proporioal, iegral ad derivaive coroller which i kow a PID (proporioal-iegral-derivaive). The coroller rafer fucio G c () i give by he followig equaio: m() G c () = = (K p K d e() K i ) m() or G c () = = K p ( Td ) e() T i I deigig he coroller, he corol egieer ha o chooe he appropriae value of K p, K i ad K d o fulfil he effecive pecificaio ad he feedback yem behaviour. Tuig of hee parameer ca be doe uig Ziegler-Nichol mehod; however, hi i beyod he cope of hi coure.

29 EXAMPLE I A fir order corol yem wih uiy feedback i a how i he Figure below. Skech he ime repoe of hi yem for ui-ep ipu whe T = for he followig corol acio: a. G c () = b. G c () = 9 c. G c () = / u() _ e() pegawal G c () m() proe T y()