Transfer Functions. Chapter 5. Transfer Functions. Derivation of a Transfer Function. Transfer Functions

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1 5/4/6 PM : Trnfer Function Chpter 5 Trnfer Function Defined G() = Y()/U() preent normlized model of proce, i.e., cn be ued with n input. Y() nd U() re both written in devition vrible form. The form of the trnfer function indicte the dnmic behvior of the proce. Trnfer Function Provide vluble inight into proce dnmic nd the dnmic of feedbck tem. Provide mjor portion of the terminolog of the proce control profeion. Derivtion of Trnfer Function dt M F T F T ( F F ) T T? T T T T T T? T T Dnmic model of CST therml mixer Appl devition vrible dtˆ M FT?? FT ( FF) T Eqution in term of devition vrible.

2 5/4/6 PM : Derivtion of Trnfer Function Pole on Complex Plne T G( ) T T F T F T M F F ( ) ( ) F M F F Appl Lplce trnform to ech term conidering tht onl inlet nd outlet temperture chnge. Determine the trnfer function for the effect of inlet temperture chnge on the outlet temperture. Note tht the repone i firt order. Pole of the Trnfer Function Indicte the Dnmic pone G ( )( Y A ) ( ) ( ( b c)( d) B C b c) ( d) Exponentil Dec t pt ( t) A e B e in( t ) Ce For, b, c, nd d poitive contnt, trnfer function indicte exponentil dec, ocilltor repone, nd exponentil growth, repectivel. Time

3 5/4/6 PM : Dmped Sinuoidl Time Untble Behvior If the output of proce grow without bound for bounded input, the proce i referred to untble. If the rel portion of n pole of trnfer function i poitive, the proce correponding to the trnfer function i untble. If n pole i locted in the right hlf plne, the proce i untble. Exponentill Growing Sinuoidl Behvior (Untble) Time Routh Stbilit Criterion... where. n n n n i A necer nd ufficient condition for ll the root of the polnomil to hve negtive rel prt i tht ll the element of the firt column of the Routh rr re poitive. Note tht the tbilit of tem cn be eed b ppling the Routh tbilit criterion to the denomintor of the tem trnfer function.

4 5/4/6 PM : Routh Arr for 3 rd Order Stem 3 3 Zero of Trnfer Function The zero of trnfer function re the vlue of tht render N()=. If n of the zero re poitive, n invere repone i indicted. If ll the zero re negtive, overhoot cn occur in certin itution Routh Stbilit Anli Exmple Deterine if thi tem i tble: Gp () 3 39 Routh Arr: ; ; 3; Therefore, thi tem i untble 9 Combining Trnfer Function Conider the CST therml mixer in which heter i ued to chnge the inlet temperture of trem nd temperture enor i ued to meure the outlet temperture. Aume tht heter behve firt order proce with known time contnt.

5 5/4/6 PM : Combining Trnfer Function Block Digrm Algebr T G T G( ) T T, pec T G T H F M F F Trnfer function for the ctutor Trnfer function for the proce Trnfer function for the enor Serie of trnfer function Summtion nd ubtrction Divider Combining Trnfer Function Block Digrm Algebr G o G G( ) G U() C() D() E() Y() G () G () + + G o F M F F H F() G 3 () H()

6 5/4/6 PM : Block Digrm Algebr We wnt to determine Y / U o we trt with Y E H (ummtion function) E ( ) / C ( ) G( G ) (erie of trnfer function) U C F (divider function) H / F G (trnfer function definition) Subtitute into t eqn: 3 Y() G () G () U() G () U() 3 rrnge: G Y / U G G G OA 3 Trnfer Function for Nonliner Proce Conider nonliner firt-order ODE d f(, u) f(, u) Y() u Eqution 5.7.3: G ( ) U() f(, u) u, u, Wht if the Proce Model i Nonliner Before trnforming to the devition vrible, linerize the nonliner eqution. Trnform to the devition vrible. Appl Lplce trnform to ech term in the eqution. Collect term nd form the deired trnfer function. Or inted, ue Eqution Advntge of Eqution Eqution w derived bed on linering the nonliner ODE, ppling devition vrible, ppling Lplce trnform nd olving for Y()/U(). Therefore, Eqution i much eier to ue thn deriving the trnfer function for both liner nd nonliner firt-order ODE.

7 5/4/6 PM : Appliction of Eqution dl Conider: Fin k L f( L, Fin) dl Initill,, L L From the ODE, F () k L f( L, Fin ) f( L, Fin) ; F L L in L, F () in in L, F () G ( ) bed on the initil condition /( L ) in Overview The trnfer function of proce how the chrcteritic of it dnmic behvior uming liner repreenttion of the proce.