Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw. conx and dircion Imagin a sysm ha varis in im; w migh plo is oupu vs. im. A plo migh imply an quaion, and h quaion is usually an ODE (ordinary diffrnial quaion. Thrfor, w will rviw h mah of h firs-ordr ODE whil mphasizing how i can rprsn a namic sysm. W xamin how h sysm is affcd by is iniial condiion and by disurbancs, whr h disurbancs may b non-smooh, mulipl, or dlayd.. firs-ordr, linar, variabl-cofficin ODE Th dpndn variabl y( dpnds on is firs drivaiv and forcing funcion x(. Whn h indpndn variabl is, y is y. a ( + y( = Kx( y( = y (.- d In wriing (.- w hav arrangd a cofficin of + for y. Thrfor a( mus hav dimnsions of indpndn variabl, and K has dimnsions of y/x. W solv (.- by dfining h ingraing facor p( d p ( = xp (.- a( Noic ha p( is dimnsionlss, as is h quoin undr h ingral. Th soluion p( y( K p(x( y ( = + d (.-3 p( p( a( compriss conribuions from h iniial condiion y( and h forcing funcion Kx(. Ths ar known as h homognous (as if h righ-hand sid wr zro and paricular (dpnds on h righ-hand sid soluions. In h languag of namic sysms, w can hink of y( as h rspons of h sysm o inpu disurbancs Kx( and y(.. firs-ordr ODE, spcial cas for procss conrol applicaions Th indpndn variabl will rprsn im. For many procss conrol applicaions, a( in (.- will b a posiiv consan; w call i h im consan. + y( = Kx( y( = y (.- d Th ingraing facor (.- is rvisd 5 Jan
Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw d p ( = xp = (.- and h soluion (.-3 bcoms ( K y( = y + x(d (.-3 Th iniial condiion affcs h sysm rspons from h bginning, bu is ffc dcays o zro according o h magniud of h im consan - largr im consans rprsn slowr dcay. If no furhr disurbd by som x(, h firs ordr sysm rachs quilibrium a zro. Howvr, mos pracical sysms ar disurbd. K is a propry of h sysm, calld h gain. By is magniud and sign, h gain influncs how srongly y rsponds o x. Th form of h rspons dpnds on h naur of h disurbanc. Exampl: suppos x is a uni sp funcion a im. Bfor w procd formally, l us hink inuiivly. From (.-3 w xpc h rspons y o dcay oward zro from IC y. A im, h sysm will rspond o bing hi wih a sp disurbanc. Afr a long im, hr will b no mmory of h iniial condiion, and h sysm will rspond only o h disurbanc inpu. Bcaus his is consan afr h sp, w guss ha h rspons will also bcom consan. Now h mah: from (.-3 y( = y = y ( K + ( ( + KU( U( d (.-4 Figur.- shows h soluion. Noic ha h paricular soluion maks no conribuion bfor im. Th iniial condiion dcays, and wih no disurbanc would coninu o zro. A, howvr, h sysm rsponds o h sp disurbanc, approaching consan valu K as im bcoms larg. This immdia rspons, followd by asympoic approach o h nw sa sa, is characrisic of firs-ordr sysms. Bcaus h rspons dos no rack h sp inpu faihfully, h rspons is said o lag bhind h inpu; h firs-ordr sysm is somims calld a firs-ordr lag. rvisd 5 Jan
Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw disurbanc.5.5 3 4 5 6 y rspons.5 K.5 3 4 5 6 im Figur.- firs-ordr rspons o iniial condiion and sp disurbanc.3 picwis ingraion of non-smooh disurbancs Th soluion (.-3 is applid ovr succding im inrvals, ach fauring an iniial condiion (from h prcding inrval and disurbanc inpu. ( K y( + ( K ( = y( + c. < < < < y x(d x(d (.3- Exampl: suppos rvisd 5 Jan 3
Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw d x = + y = x ( y( = < < < < < (.3- In his problm, variabls, x, and y should b prsumd o hav appropria, if unsad, unis; in hs unis, boh gain and im consan ar of magniud. From (.3-, y( = ( ( + ( < < < < < (.3-3 Wih a zro iniial condiion and no disurbanc, h sysm rmains a quilibrium unil h ramp disurbanc bgins a =. Thn h oupu immdialy riss in rspons, lagging bhind h linar ramp. A =, h disurbanc cass, and h oupu dcays back oward quilibrium. rvisd 5 Jan 4
Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw disurbanc.5.5.5 rspons.8.7.6.5.4.3.. 3 4 5 6 3 4 5 6 im.4 mulipl disurbancs and suprimposiion Sysms can hav mor han on inpu. Considr a firs-ordr sysm wih wo disurbanc funcions. + y( = Kx( + K x ( y( = y (.4- d Applying (.-3 and disribuing h ingral across h disurbancs, w find ha h ffcs of h disurbancs on y ar addiiv. ( y K K ( = y + x(d + x (d (.4- This addiiv bhavior is a happy characrisic of linar sysms. Thus anohr way o viw problm (.4- is o dcompos i ino componn problms. Tha is, dfin y = y + y + y (.4-3 H rvisd 5 Jan 5
Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw and wri (.4- in hr quaions. W pu h iniial condiion wih no disurbancs, and ach disurbanc wih a zro iniial condiion. H + yh ( = d + y( = Kx( d + y( = K x ( d y ( = y y ( = H y ( = (.4-4 Equaions and iniial condiions (.4-4 can b summd o rcovr h original problm spcificaion (.4-. Th soluions ar y H ( = y K y( = K y( = ( x x (d (d (.4-5 and of cours hs soluions can b addd o rcovr original soluion (.4-. Thus w can viw h problm of mulipl disurbancs as a sysm rsponding o h sum of h disurbancs, or as h sum of rsponss from svral idnical sysms, ach rsponding o a singl disurbanc. Exampl: considr 3 + y = U( U( 3 y( = (.4-6 4 d 4 4 W firs plac h quaion in sandard form, in which h cofficin of y is +. + y = 3U( 4U( 3 y( = (.4-7 d Equaion (.4-7 shows us ha h im consan is, and ha h sysm rsponds o h firs disurbanc wih a gain of 3, and o h scond wih a gain of -4. Th soluion is y ( ( 3 ( 4U( 3 ( = + 3U( (.4-8 rvisd 5 Jan 6
Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw In Figur.4-, h individual soluion componns ar plod as solid racs; hir sum, which is h sysm rspons, is a dashd rac. Noic how h firs-ordr lag rsponds o ach nw disurbanc as i occurs. disurbancs.5 rspons 4 3 - - -3-4 -5 4 6 4 6 im 8 8 Figur.4- firs-ordr rspons o mulipl disurbancs Wriing h sp funcions xplicily in soluion (.4-8 mphasizs ha paricular disurbancs do no influnc h soluion unil h im of hir occurrnc. For xampl, if hy wr omid, som dcpivly corrc bu inappropria rarrangmn would lad o rrors. y = + 3 = + 3 3 = + 3 ( ( ( ( ( 4 4 + 4 + 4 ( 3 ( 3 ( 3 (do no do his! (.4-9 This noaion a las implis ha wo of h xponnial funcions hav dlayd onss. Howvr, furhr corrc-bu-inappropria rarrangmn maks hings vn wors. rvisd 5 Jan 7
Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw y = + = + = + 3 3 ( + 4 + 4 3 ( 3 + 4 ( 3 3 (do no do his! (.4- Th incorrc soluions ar plod wih (.4-8 in Figur.4-. Equaion (.4-9 has bcom disconinuous - h rspons aks non-physical laps a h ons of ach nw disurbanc. Equaion (.4- has los all dpndnc on h disurbancs and dcays from a non-physical iniial condiion. Evn wih h misaks, boh incorrc soluions lad o h corrc long-rm condiion. 5 rspons 5-5 4 6 im 8 soluion q.4.9 q.4. Figur.4- comparison of corrc and incorrc soluions.5 dlayd rspons o disurbancs Considr a sysm ha racs o a disurbanc, bu only afr som inrvning im inrval θ has passd. Tha is + y( = Kx( θ y( = y (.5- d Equaion (.5- shows h dpndnc of y, a any im, on h valu of x a arlir im - θ. Th soluion is wrin dircly from (.-3. ( K y( = y + x( θd (.5- W mus ingra h disurbanc considring h im dlay. Tak as an xampl a disurbanc x( occurring a im. Th plo shows h rvisd 5 Jan 8
Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw disurbanc, as wll as h disurbanc as h sysm xprincs i, which bgins a im + θ. W could xprss his disurbanc-as-xprincd as som nw funcion x (, occurring a im + θ. disurbanc as i occurs x( disurbanc as xprincd by sysm x( - θ = x ( + θ x( - θ - θ Alrnaivly, w could dfin a nw im variabl = θ (.5-3 and wri h inpu as x(. Th ingral in (.5- bcoms, hn, x( θd = x(d = θ +θ x( d (.5-4 Thrfor, soluion (.5- bcoms ( K θ y( = y + x( d (.5-5 θ Exampl: considr a sp disurbanc a im = ha affcs h sysm 3 im unis lar. Using (.5-5 + y = x( 3 d x( = U( y( = (.5-6 rvisd 5 Jan 9
Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw y = = = 3 3 3 3 3 U( = U( 3 = U( 5 = U( 5 U( d U( d + 3 3 [ ] 3 + 3 + 3 [ ] ( 5 [ ] U( d (.5-7 Figur.5- shows ha a ypical firs-ordr lag sp rspons occurs 3 im unis afr bing disurbd a =. disurbancs.5 4 6 8.8 rspons.6.4. 4 6 8 im Figur.5- sp rspons of firs ordr sysm wih dad im Th im dlay in rsponding o a disurbanc is ofn calld dad im. Dad im is diffrn from lag. Lag occurs bcaus of h combinaion of y and is drivaiv on h lf-hand sid of h quaion. Dad im rvisd 5 Jan
Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw occurs bcaus of a im dlay in procssing a disurbanc on h righhand sid..6 conclusion Plas bcom comforabl wih handling ODEs. Viw hm as sysms; idnify hir inpus and oupus, hir gains and im paramrs. rvisd 5 Jan