Chaper 7: Invere-Repone Syem Normal Syem Invere-Repone Syem
Baic Sar ou in he wrong direcion End up in he original eady-ae gain value Two or more yem wih differen magniude and cale in parallel Main yem Oppoing yem
Requiremen Oppoing yem mu have lower eady-ae gain han he main yem, i.e. > Oppoing yem mu have faer repone ime A lea one righ hand zero in final equaion
Mahemaical Explanaion / Baic equaion for wo yem in parallel g g g g Overall eady-ae value i obained from y Iniial lope i deermined from dy dy dy y y y, 0, 0 d d d Show dy d 0 y y y dy d y e 0 uing he iniial value heorem! /
Mahemaical Explanaion / Under he following condiion here will be an invere repone: A final equaion i derived from he baic equaion: Noaion: One righ-hand zero in equaion and where, g
Some Invere repone Syem Real-Life Example A drum boiler The reboiler ecion of a diillaion column An exohermic ubular caalyic reacor
A drum boiler
Dynamic Behavior of Syem wih Single, Righ-Half Plane Zero Equaion g g, where 0 Ha wo pole and one zero in Righ-half plane Uni ep repone for yem: y
Mah coninued Ulimae repone: lim y lim y lim 0 0 The final value heorem of Laplace ranform lim 0 The iniial lope i acquired wih dy d lim y lim The iniial value heorem of Laplace ranform lim
Figure of Invere Repone Syem Poiive final proce gain Negaive final proce gain
Three Differen Example wih one RHP Zero Effec of muliple non-poiive pole and a poiive zero a One RHP zero, wo pole b One RHP zero, hree pole c One RHP zero, one regular zero, hree pole
Three Differen Example - Coninued A B C g 3 5 g 3 5 4 g 3 5 4
Dynamic Behavior of Syem wih Muliple, Righ-Half Plane Zero Effec of muliple RHP zero The number of RHP zero explain he number of urn in yem repone curve Odd number of RHP zero oppoie ep repone direcion wih final eadyae gain Even number of RHP zero ame ep repone direcion wih final eadyae gain
Three Differen Example abou Muliple RHP Zero Effec of muliple RHP zero a Two RHP zero, hree pole b Three RHP zero, four pole c Four RHP zero, five pole
Three Differen Example abou Muliple RHP Zero - Coninued Figure and equaion for muliple RHP zero a b c g 3 5 4 g 3.5 5 4 3.5 g 3.5 6 5 4 3.5 7
Chaper 8: Time delay yem Mo yem in proce indury include Timedelay The ime-delay i he lag beween he conrol acion and he reul in he proce Example: a hea-exchanger
Example /5 L = Lengh of perfecly inulaed pipe, v = flow peed of incompreible fluid in he pipe, T 0 = original conan emperaure in he pipe, T i = New emperaure a ime = 0 in pipe inle a z=0, L Tranpor delay i calculaed from: v The pipe i divided in maller diviion wih lengh Δz
Example /5 From energy balance uniform croecional area in pipe: AC p z T vac p T T * z vac p T T * z z T * i he reference emperaure reference emperaure = he emperaure, where pecific enhalpy of he liquid i conidered a zero
Example 3/5 Dividing he eq. wih Δz and eing we ge: Thi eq. i a PDE Parial Differenial Equaion The deviaion variable: In erm of y, we ge: L z z z T v z T 0 ; 0,,, 0 z 0,, z z y v z y 0,, T z T z y
Example 4/5 Le ake a Laplace-ranform from our equaion y z, v 0 e y z, d z 0 y z, Reul: fir-order, linear Ordinary Differenial Equaion in yz,, oluion: y z, C e z v y z, v C i arbirary conan and may be evaluaed a y 0, C d dz 0
Example 5/5 If we e z=l, and ake in o conideraion, ha we ge: y L, y 0, e L v The Tranfer funcion: For any inpu u: In ime-domain: g e y L, e u y L, u Thi indicae ha yem oupu i exacly he ame a he yem inpu, wih he excepion ha he oupu i delayed by he ime uni α
The Pure Time-Delay Proce Repone of PTD proce i only a delayed verion of he inpu Tranfer funcion: g e By aking a Taylor erie for e we ranform he original finie-order denominaor polynomial in o infinie-order one: g 3 3! 3!... n n! n...
PTD Proce Repone o Variou Inpu:
N Fir-Order Syem in Serie Cae: N fir-order yem in erie, ranfer funcion: g Limi are e for yem: N N N lim N g N lim N N N I defined, ha lim N g N e The limi: PTD-yem e lim N N N i he ranfer funcion of
Dynamic Behavior of Syem wih Time Delay The Pure Time-Delay Syem defined a infinie erie i concepually equivalen o a yem wih infinie amoun of fir-order yem in erie high order yem funcion imilarly o baic PTD yem A ime-delay yem i idenified by erm he yem ranfer funcion e in
Iohermal CSTR Now, le conider he following proce, an iohermal CSTR:
Iohermal CSTR - Coninued The proce ha wo procee conneced in erie he pipe and he reacor The pipe funcion a PTD yem relaion beween oupu and inpu concenraion: c yem repone: y e u The reacor i fir-order yem, ranfer funcion g reacor p ip e Enire yem oupu/repone: g oal y c 0 pipe e
A. Very High Order Syem Approximaion Nh-order model characerized by N + parameer If approximaed wih order model plu a ime delay, 3 parameer ued Efficiency Approximaion i no accurae, bu ofen i i accurae enough All model are only idealizaion of real procee, impler model migh be beer
B. Raional Tranfer Funcion Approximaion Mehod ued o replace he rancendenal funcion e -α wih raional funcion I i ofen ueful o ue raional form Some analyi problem are made eaier Cerain proce analyi ool require raional form
Padé Approximaion A approximae way o preen low-order polynomial in a a erie of Higher he Padé order, beer he approximae Fir order approximaion: nd order approximaion: 3rd order approximaion: e e e 3 e 0 0 3 3 0 0 3 Malab demo
Padé Approximaed Uni Sep Repone of PTD Syem
High Order Syem Approximaion A way o approximae he ime delay by finie number of fir order procee: e N N No a effecive a Padé require higher order o achieve ame reul a Padé
High Order Syem Approximaed Uni Sep Repone of PTD Syem
Model Equaion for Syem Conaining Time Delay. Sae Space. Tranform Domain 3. Frequency Repone 4. Impule Repone
Sae-Space-model Ordinary SISO ae-pace model wihou delay: New inpu delay α, ae delay β, an oupu delay η and a diurbance delay δ, reformed equaion: x c y d bu Ax d dx T 0 0 0 0 x c x c y d d b u u b A x x A d dx T T
Tranform-Domain Model Ordinary ranfer model wihou delay One inpu and oupu: Muliple inpu and oupu: d g u g y d e g g * marice funcion ranfer are and where, G G d G u G y d d
Frequency-Repone Model Ordinary ranfer model wihou delay: y g u g d d he yem wih delay i preened a: g j * g j e j Uing he following ideniy we can remove he exponenial e j co jin
Impule-Repone Model Ordinary ranfer model wihou delay: he yem wih delay i preened a fir order yem: d g u g y d e g g, / 0 0,