Spring 2006 Process Dynamics, Operations, and Control Lesson 2: Mathematics Review
|
|
- Patrick Shields
- 6 years ago
- Views:
Transcription
1 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
2 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
3 Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw disurbanc y rspons.5 K 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
4 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
5 Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw disurbanc rspons 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
6 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( = ( 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
7 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 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 ( 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
8 Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw y = + = + = ( ( ( 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 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
9 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 ( y = x( 3 d x( = U( y( = (.5-6 rvisd 5 Jan 9
10 Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw y = = = U( = U( 3 = U( 5 = U( 5 U( d U( d [ ] [ ] ( 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 rspons 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
11 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
Boyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors
Boc/DiPrima 9 h d, Ch.: Linar Equaions; Mhod of Ingraing Facors Elmnar Diffrnial Equaions and Boundar Valu Problms, 9 h diion, b William E. Boc and Richard C. DiPrima, 009 b John Wil & Sons, Inc. A linar
More informationElementary Differential Equations and Boundary Value Problems
Elmnar Diffrnial Equaions and Boundar Valu Problms Boc. & DiPrima 9 h Ediion Chapr : Firs Ordr Diffrnial Equaions 00600 คณ ตศาสตร ว ศวกรรม สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา /55 ผศ.ดร.อร ญญา ผศ.ดร.สมศ
More informationTransfer function and the Laplace transformation
Lab No PH-35 Porland Sa Univriy A. La Roa Tranfr funcion and h Laplac ranformaion. INTRODUTION. THE LAPLAE TRANSFORMATION L 3. TRANSFER FUNTIONS 4. ELETRIAL SYSTEMS Analyi of h hr baic paiv lmn R, and
More informationLet s look again at the first order linear differential equation we are attempting to solve, in its standard form:
Th Ingraing Facor Mhod In h prvious xampls of simpl firs ordr ODEs, w found h soluions by algbraically spara h dpndn variabl- and h indpndn variabl- rms, and wri h wo sids of a givn quaion as drivaivs,
More information2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35
MATH 5 PS # Summr 00.. Diffrnial Equaions and Soluions PS.# Show ha ()C #, 4, 7, 0, 4, 5 ( / ) is a gnral soluion of h diffrnial quaion. Us a compur or calculaor o skch h soluions for h givn valus of h
More informationAn Indian Journal FULL PAPER. Trade Science Inc. A stage-structured model of a single-species with density-dependent and birth pulses ABSTRACT
[Typ x] [Typ x] [Typ x] ISSN : 974-7435 Volum 1 Issu 24 BioTchnology 214 An Indian Journal FULL PAPE BTAIJ, 1(24), 214 [15197-1521] A sag-srucurd modl of a singl-spcis wih dnsiy-dpndn and birh pulss LI
More informationChapter 12 Introduction To The Laplace Transform
Chapr Inroducion To Th aplac Tranorm Diniion o h aplac Tranorm - Th Sp & Impul uncion aplac Tranorm o pciic uncion 5 Opraional Tranorm Applying h aplac Tranorm 7 Invr Tranorm o Raional uncion 8 Pol and
More informationMidterm exam 2, April 7, 2009 (solutions)
Univrsiy of Pnnsylvania Dparmn of Mahmaics Mah 26 Honors Calculus II Spring Smsr 29 Prof Grassi, TA Ashr Aul Midrm xam 2, April 7, 29 (soluions) 1 Wri a basis for h spac of pairs (u, v) of smooh funcions
More informationCharging of capacitor through inductor and resistor
cur 4&: R circui harging of capacior hrough inducor and rsisor us considr a capacior of capacianc is conncd o a D sourc of.m.f. E hrough a rsisr of rsisanc R, an inducor of inducanc and a y K in sris.
More informationVoltage v(z) ~ E(z)D. We can actually get to this wave behavior by using circuit theory, w/o going into details of the EM fields!
Considr a pair of wirs idal wirs ngh >, say, infinily long olag along a cabl can vary! D olag v( E(D W can acually g o his wav bhavior by using circui hory, w/o going ino dails of h EM filds! Thr
More informationOn the Speed of Heat Wave. Mihály Makai
On h Spd of Ha Wa Mihály Maai maai@ra.bm.hu Conns Formulaion of h problm: infini spd? Local hrmal qulibrium (LTE hypohsis Balanc quaion Phnomnological balanc Spd of ha wa Applicaion in plasma ranspor 1.
More informationCSE 245: Computer Aided Circuit Simulation and Verification
CSE 45: Compur Aidd Circui Simulaion and Vrificaion Fall 4, Sp 8 Lcur : Dynamic Linar Sysm Oulin Tim Domain Analysis Sa Equaions RLC Nwork Analysis by Taylor Expansion Impuls Rspons in im domain Frquncy
More informationfiziks Institute for NET/JRF, GATE, IIT JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES MATEMATICAL PHYSICS SOLUTIONS are
MTEMTICL PHYSICS SOLUTIONS GTE- Q. Considr an ani-symmric nsor P ij wih indics i and j running from o 5. Th numbr of indpndn componns of h nsor is 9 6 ns: Soluion: Th numbr of indpndn componns of h nsor
More informationLecture 1: Numerical Integration The Trapezoidal and Simpson s Rule
Lcur : Numrical ngraion Th Trapzoidal and Simpson s Rul A problm Th probabiliy of a normally disribud (man µ and sandard dviaion σ ) vn occurring bwn h valus a and b is B A P( a x b) d () π whr a µ b -
More informationLecture 4: Laplace Transforms
Lur 4: Lapla Transforms Lapla and rlad ransformaions an b usd o solv diffrnial quaion and o rdu priodi nois in signals and imags. Basially, hy onvr h drivaiv opraions ino mulipliaion, diffrnial quaions
More informationGeneral Article Application of differential equation in L-R and C-R circuit analysis by classical method. Abstract
Applicaion of Diffrnial... Gnral Aricl Applicaion of diffrnial uaion in - and C- circui analysis by classical mhod. ajndra Prasad gmi curr, Dparmn of Mahmaics, P.N. Campus, Pokhara Email: rajndraprasadrgmi@yahoo.com
More informationApplied Statistics and Probability for Engineers, 6 th edition October 17, 2016
Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 CHATER Scion - -. a d. 679.. b. d. 88 c d d d. 987 d. 98 f d.. Thn, = ln. =. g d.. Thn, = ln.9 =.. -7. a., by symmry. b.. d...6. 7.. c...
More informationwhereby we can express the phase by any one of the formulas cos ( 3 whereby we can express the phase by any one of the formulas
Third In-Class Exam Soluions Mah 6, Profssor David Lvrmor Tusday, 5 April 07 [0] Th vrical displacmn of an unforcd mass on a spring is givn by h 5 3 cos 3 sin a [] Is his sysm undampd, undr dampd, criically
More information4.1 The Uniform Distribution Def n: A c.r.v. X has a continuous uniform distribution on [a, b] when its pdf is = 1 a x b
4. Th Uniform Disribuion Df n: A c.r.v. has a coninuous uniform disribuion on [a, b] whn is pdf is f x a x b b a Also, b + a b a µ E and V Ex4. Suppos, h lvl of unblivabiliy a any poin in a Transformrs
More informationS.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15]
S.Y. B.Sc. (IT) : Sm. III Applid Mahmaics Tim : ½ Hrs.] Prlim Qusion Papr Soluion [Marks : 75 Q. Amp h following (an THREE) 3 6 Q.(a) Rduc h mari o normal form and find is rank whr A 3 3 5 3 3 3 6 Ans.:
More informationOn the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument
Inrnaional Rsarch Journal of Applid Basic Scincs 03 Aailabl onlin a wwwirjabscom ISSN 5-838X / Vol 4 (): 47-433 Scinc Eplorr Publicaions On h Driais of Bssl Modifid Bssl Funcions wih Rspc o h Ordr h Argumn
More informationDEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018
DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Aoc. Prof. Dr. Burak Kllci Spring 08 OUTLINE Th Laplac Tranform Rgion of convrgnc for Laplac ranform Invr Laplac ranform Gomric valuaion
More informationEconomics 302 (Sec. 001) Intermediate Macroeconomic Theory and Policy (Spring 2011) 3/28/2012. UW Madison
Economics 302 (Sc. 001) Inrmdia Macroconomic Thory and Policy (Spring 2011) 3/28/2012 Insrucor: Prof. Mnzi Chinn Insrucor: Prof. Mnzi Chinn UW Madison 16 1 Consumpion Th Vry Forsighd dconsumr A vry forsighd
More informationWave Equation (2 Week)
Rfrnc Wav quaion ( Wk 6.5 Tim-armonic filds 7. Ovrviw 7. Plan Wavs in Losslss Mdia 7.3 Plan Wavs in Loss Mdia 7.5 Flow of lcromagnic Powr and h Poning Vcor 7.6 Normal Incidnc of Plan Wavs a Plan Boundaris
More informationChapter 3: Fourier Representation of Signals and LTI Systems. Chih-Wei Liu
Chapr 3: Fourir Rprsnaion of Signals and LTI Sysms Chih-Wi Liu Oulin Inroducion Complx Sinusoids and Frquncy Rspons Fourir Rprsnaions for Four Classs of Signals Discr-im Priodic Signals Fourir Sris Coninuous-im
More informationLaPlace Transform in Circuit Analysis
LaPlac Tranform in Circui Analyi Obciv: Calcula h Laplac ranform of common funcion uing h dfiniion and h Laplac ranform abl Laplac-ranform a circui, including componn wih non-zro iniial condiion. Analyz
More informationH is equal to the surface current J S
Chapr 6 Rflcion and Transmission of Wavs 6.1 Boundary Condiions A h boundary of wo diffrn mdium, lcromagnic fild hav o saisfy physical condiion, which is drmind by Maxwll s quaion. This is h boundary condiion
More informationLecture 2: Current in RC circuit D.K.Pandey
Lcur 2: urrn in circui harging of apacior hrough Rsisr L us considr a capacior of capacianc is conncd o a D sourc of.m.f. E hrough a rsisr of rsisanc R and a ky K in sris. Whn h ky K is swichd on, h charging
More informationChapter 5 The Laplace Transform. x(t) input y(t) output Dynamic System
EE 422G No: Chapr 5 Inrucor: Chung Chapr 5 Th Laplac Tranform 5- Inroducion () Sym analyi inpu oupu Dynamic Sym Linar Dynamic ym: A procor which proc h inpu ignal o produc h oupu dy ( n) ( n dy ( n) +
More informationBoyce/DiPrima 9 th ed, Ch 7.8: Repeated Eigenvalues
Boy/DiPrima 9 h d Ch 7.8: Rpad Eignvalus Elmnary Diffrnial Equaions and Boundary Valu Problms 9 h diion by William E. Boy and Rihard C. DiPrima 9 by John Wily & Sons In. W onsidr again a homognous sysm
More informationA THREE COMPARTMENT MATHEMATICAL MODEL OF LIVER
A THREE COPARTENT ATHEATICAL ODEL OF LIVER V. An N. Ch. Paabhi Ramacharyulu Faculy of ahmaics, R D collgs, Hanamonda, Warangal, India Dparmn of ahmaics, Naional Insiu of Tchnology, Warangal, India E-ail:
More informationEXERCISE - 01 CHECK YOUR GRASP
DIFFERENTIAL EQUATION EXERCISE - CHECK YOUR GRASP 7. m hn D() m m, D () m m. hn givn D () m m D D D + m m m m m m + m m m m + ( m ) (m ) (m ) (m + ) m,, Hnc numbr of valus of mn will b. n ( ) + c sinc
More informationMEM 355 Performance Enhancement of Dynamical Systems A First Control Problem - Cruise Control
MEM 355 Prformanc Enhancmn of Dynamical Sysms A Firs Conrol Problm - Cruis Conrol Harry G. Kwany Darmn of Mchanical Enginring & Mchanics Drxl Univrsiy Cruis Conrol ( ) mv = F mg sinθ cv v +.2v= u 9.8θ
More informationFIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS I: Introduction and Linear Systems
FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS I: Inroducion and Linar Sysms David Lvrmor Dparmn of Mahmaics Univrsiy of Maryland 9 Dcmbr 0 Bcaus h prsnaion of his marial in lcur will diffr from
More information7.4 QUANTUM MECHANICAL TREATMENT OF FLUCTUATIONS *
Andri Tokmakoff, MIT Dparmn of Chmisry, 5/19/5 7-11 7.4 QUANTUM MECANICAL TREATMENT OF FLUCTUATIONS * Inroducion and Prviw Now h origin of frquncy flucuaions is inracions of our molcul (or mor approprialy
More informationUNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Answr Ky Nam: Da: UNIT # EXPONENTIAL AND LOGARITHMIC FUNCTIONS Par I Qusions. Th prssion is quivaln o () () 6 6 6. Th ponnial funcion y 6 could rwrin as y () y y 6 () y y (). Th prssion a is quivaln o
More informationPhys463.nb Conductivity. Another equivalent definition of the Fermi velocity is
39 Anohr quival dfiniion of h Fri vlociy is pf vf (6.4) If h rgy is a quadraic funcion of k H k L, hs wo dfiniions ar idical. If is NOT a quadraic funcion of k (which could happ as will b discussd in h
More informationInstructors Solution for Assignment 3 Chapter 3: Time Domain Analysis of LTIC Systems
Inrucor Soluion for Aignmn Chapr : Tim Domain Anali of LTIC Sm Problm i a 8 x x wih x u,, an Zro-inpu rpon of h m: Th characriic quaion of h LTIC m i i 8, which ha roo a ± j Th zro-inpu rpon i givn b zi
More informationAR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 )
AR() Procss Th firs-ordr auorgrssiv procss, AR() is whr is WN(0, σ ) Condiional Man and Varianc of AR() Condiional man: Condiional varianc: ) ( ) ( Ω Ω E E ) var( ) ) ( var( ) var( σ Ω Ω Ω Ω E Auocovarianc
More informationChap.3 Laplace Transform
Chap. aplac Tranorm Tranorm: An opraion ha ranorm a uncion ino anohr uncion i Dirniaion ranorm: ii x: d dx x x Ingraion ranorm: x: x dx x c Now, conidr a dind ingral k, d,ha ranorm ino a uncion o variabl
More informationReview Lecture 5. The source-free R-C/R-L circuit Step response of an RC/RL circuit. The time constant = RC The final capacitor voltage v( )
Rviw Lcur 5 Firs-ordr circui Th sourc-fr R-C/R-L circui Sp rspons of an RC/RL circui v( ) v( ) [ v( 0) v( )] 0 Th i consan = RC Th final capacior volag v() Th iniial capacior volag v( 0 ) Volag/currn-division
More informationA Condition for Stability in an SIR Age Structured Disease Model with Decreasing Survival Rate
A Condiion for abiliy in an I Ag rucurd Disas Modl wih Dcrasing urvival a A.K. upriana, Edy owono Dparmn of Mahmaics, Univrsias Padjadjaran, km Bandung-umng 45363, Indonsia fax: 6--7794696, mail: asupria@yahoo.com.au;
More informationCHAPTER. Linear Systems of Differential Equations. 6.1 Theory of Linear DE Systems. ! Nullcline Sketching. Equilibrium (unstable) at (0, 0)
CHATER 6 inar Sysms of Diffrnial Equaions 6 Thory of inar DE Sysms! ullclin Skching = y = y y υ -nullclin Equilibrium (unsabl) a (, ) h nullclin y = υ nullclin = h-nullclin (S figur) = + y y = y Equilibrium
More information3(8 ) (8 x x ) 3x x (8 )
Scion - CHATER -. a d.. b. d.86 c d 8 d d.9997 f g 6. d. d. Thn, = ln. =. =.. d Thn, = ln.9 =.7 8 -. a d.6 6 6 6 6 8 8 8 b 9 d 6 6 6 8 c d.8 6 6 6 6 8 8 7 7 d 6 d.6 6 6 6 6 6 6 8 u u u u du.9 6 6 6 6 6
More informationControl System Engineering (EE301T) Assignment: 2
Conrol Sysm Enginring (EE0T) Assignmn: PART-A (Tim Domain Analysis: Transin Rspons Analysis). Oain h rspons of a uniy fdack sysm whos opn-loop ransfr funcion is (s) s ( s 4) for a uni sp inpu and also
More informationLecture 1: Growth and decay of current in RL circuit. Growth of current in LR Circuit. D.K.Pandey
cur : Growh and dcay of currn in circui Growh of currn in Circui us considr an inducor of slf inducanc is conncd o a DC sourc of.m.f. E hrough a rsisr of rsisanc and a ky K in sris. Whn h ky K is swichd
More informationCPSC 211 Data Structures & Implementations (c) Texas A&M University [ 259] B-Trees
CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 259] B-Trs Th AVL r and rd-black r allowd som variaion in h lnghs of h diffrn roo-o-laf pahs. An alrnaiv ida is o mak sur ha all roo-o-laf pahs
More informationSOLUTIONS. 1. Consider two continuous random variables X and Y with joint p.d.f. f ( x, y ) = = = 15. Stepanov Dalpiaz
STAT UIUC Pracic Problms #7 SOLUTIONS Spanov Dalpiaz Th following ar a numbr of pracic problms ha ma b hlpful for compling h homwor, and will lil b vr usful for suding for ams.. Considr wo coninuous random
More informationInstitute of Actuaries of India
Insiu of Acuaris of India ubjc CT3 Probabiliy and Mahmaical aisics Novmbr Examinaions INDICATIVE OLUTION Pag of IAI CT3 Novmbr ol. a sampl man = 35 sampl sandard dviaion = 36.6 b for = uppr bound = 35+*36.6
More informationI) Title: Rational Expectations and Adaptive Learning. II) Contents: Introduction to Adaptive Learning
I) Til: Raional Expcaions and Adapiv Larning II) Conns: Inroducion o Adapiv Larning III) Documnaion: - Basdvan, Olivir. (2003). Larning procss and raional xpcaions: an analysis using a small macroconomic
More informationMicroscopic Flow Characteristics Time Headway - Distribution
CE57: Traffic Flow Thory Spring 20 Wk 2 Modling Hadway Disribuion Microscopic Flow Characrisics Tim Hadway - Disribuion Tim Hadway Dfiniion Tim Hadway vrsus Gap Ahmd Abdl-Rahim Civil Enginring Dparmn,
More informationEE 434 Lecture 22. Bipolar Device Models
EE 434 Lcur 22 Bipolar Dvic Modls Quiz 14 Th collcor currn of a BJT was masurd o b 20mA and h bas currn masurd o b 0.1mA. Wha is h fficincy of injcion of lcrons coming from h mir o h collcor? 1 And h numbr
More informationREPETITION before the exam PART 2, Transform Methods. Laplace transforms: τ dτ. L1. Derive the formulas : L2. Find the Laplace transform F(s) if.
Tranform Mhod and Calculu of Svral Variabl H7, p Lcurr: Armin Halilovic KTH, Campu Haning E-mail: armin@dkh, wwwdkh/armin REPETITION bfor h am PART, Tranform Mhod Laplac ranform: L Driv h formula : a L[
More informationa dt a dt a dt dt If 1, then the poles in the transfer function are complex conjugates. Let s look at f t H t f s / s. So, for a 2 nd order system:
Undrdamd Sysms Undrdamd Sysms nd Ordr Sysms Ouu modld wih a nd ordr ODE: d y dy a a1 a0 y b f If a 0 0, hn: whr: a d y a1 dy b d y dy y f y f a a a 0 0 0 is h naural riod of oscillaion. is h daming facor.
More informationDE Dr. M. Sakalli
DE-0 Dr. M. Sakalli DE 55 M. Sakalli a n n 0 a Lh.: an Linar g Equaions Hr if g 0 homognous non-homognous ohrwis driving b a forc. You know h quaions blow alrad. A linar firs ordr ODE has h gnral form
More informationLogistic equation of Human population growth (generalization to the case of reactive environment).
Logisic quaion of Human populaion growh gnralizaion o h cas of raciv nvironmn. Srg V. Ershkov Insiu for Tim aur Exploraions M.V. Lomonosov's Moscow Sa Univrsi Lninski gor - Moscow 999 ussia -mail: srgj-rshkov@andx.ru
More informationVoltage v(z) ~ E(z)D. We can actually get to this wave behavior by using circuit theory, w/o going into details of the EM fields!
Considr a pair of wirs idal wirs ngh >, say, infinily long olag along a cabl can vary! D olag v( E(D W can acually g o his wav bhavior by using circui hory, w/o going ino dails of h EM filds! Thr
More informationPoisson process Markov process
E2200 Quuing hory and lraffic 2nd lcur oion proc Markov proc Vikoria Fodor KTH Laboraory for Communicaion nwork, School of Elcrical Enginring 1 Cour oulin Sochaic proc bhind quuing hory L2-L3 oion proc
More informationFinal Exam : Solutions
Comp : Algorihm and Daa Srucur Final Exam : Soluion. Rcuriv Algorihm. (a) To bgin ind h mdian o {x, x,... x n }. Sinc vry numbr xcp on in h inrval [0, n] appar xacly onc in h li, w hav ha h mdian mu b
More informationCHAPTER CHAPTER14. Expectations: The Basic Tools. Prepared by: Fernando Quijano and Yvonn Quijano
Expcaions: Th Basic Prpard by: Frnando Quijano and Yvonn Quijano CHAPTER CHAPTER14 2006 Prnic Hall Businss Publishing Macroconomics, 4/ Olivir Blanchard 14-1 Today s Lcur Chapr 14:Expcaions: Th Basic Th
More informationA Propagating Wave Packet Group Velocity Dispersion
Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to
More informationImpulsive Differential Equations. by using the Euler Method
Applid Mahmaical Scincs Vol. 4 1 no. 65 19 - Impulsiv Diffrnial Equaions by using h Eulr Mhod Nor Shamsidah B Amir Hamzah 1 Musafa bin Mama J. Kaviumar L Siaw Chong 4 and Noor ani B Ahmad 5 1 5 Dparmn
More informationCopyright 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Chapr Rviw 0 6. ( a a ln a. This will qual a if an onl if ln a, or a. + k an (ln + c. Thrfor, a an valu of, whr h wo curvs inrsc, h wo angn lins will b prpnicular. 6. (a Sinc h lin passs hrough h origin
More informationOn General Solutions of First-Order Nonlinear Matrix and Scalar Ordinary Differential Equations
saartvlos mcnirbata rovnuli akadmiis moamb 3 #2 29 BULLTN OF TH ORN NTONL DMY OF SNS vol 3 no 2 29 Mahmaics On nral Soluions of Firs-Ordr Nonlinar Mari and Scalar Ordinary Diffrnial uaions uram L Kharaishvili
More information1. Inverse Matrix 4[(3 7) (02)] 1[(0 7) (3 2)] Recall that the inverse of A is equal to:
Rfrncs Brnank, B. and I. Mihov (1998). Masuring monary policy, Quarrly Journal of Economics CXIII, 315-34. Blanchard, O. R. Proi (00). An mpirical characrizaion of h dynamic ffcs of changs in govrnmn spnding
More informationWhy Laplace transforms?
MAE4 Linar ircui Why Lalac ranform? Firordr R cc v v v KVL S R inananou for ach Subiu lmn rlaion v S Ordinary diffrnial quaion in rm of caacior volag Lalac ranform Solv Invr LT V u, v Ri, i A R V A _ v
More informationDecline Curves. Exponential decline (constant fractional decline) Harmonic decline, and Hyperbolic decline.
Dlin Curvs Dlin Curvs ha lo flow ra vs. im ar h mos ommon ools for forasing roduion and monioring wll rforman in h fild. Ths urvs uikly show by grahi mans whih wlls or filds ar roduing as xd or undr roduing.
More informationC From Faraday's Law, the induced voltage is, C The effect of electromagnetic induction in the coil itself is called selfinduction.
Inducors and Inducanc C For inducors, v() is proporional o h ra of chang of i(). Inducanc (con d) C Th proporionaliy consan is h inducanc, L, wih unis of Hnris. 1 Hnry = 1 Wb / A or 1 V sc / A. C L dpnds
More informationInverse Fourier Transform. Properties of Continuous time Fourier Transform. Review. Linearity. Reading Assignment Oppenheim Sec pp.289.
Convrgnc of ourir Trnsform Rding Assignmn Oppnhim Sc 42 pp289 Propris of Coninuous im ourir Trnsform Rviw Rviw or coninuous-im priodic signl x, j x j d Invrs ourir Trnsform 2 j j x d ourir Trnsform Linriy
More informationLagrangian for RLC circuits using analogy with the classical mechanics concepts
Lagrangian for RLC circuis using analogy wih h classical mchanics concps Albrus Hariwangsa Panuluh and Asan Damanik Dparmn of Physics Educaion, Sanaa Dharma Univrsiy Kampus III USD Paingan, Maguwoharjo,
More informationRevisiting what you have learned in Advanced Mathematical Analysis
Fourir sris Rvisiing wh you hv lrnd in Advncd Mhmicl Anlysis L f x b priodic funcion of priod nd is ingrbl ovr priod. f x cn b rprsnd by rigonomric sris, f x n cos nx bn sin nx n cos x b sin x cosx b whr
More informationChapter 6 Differential Equations and Mathematical Modeling
6 Scion 6. hapr 6 Diffrnial Equaions and Mahmaical Modling Scion 6. Slop Filds and Eulr s Mhod (pp. ) Eploraion Sing h Slops. Sinc rprsns a lin wih a slop of, w should d pc o s inrvals wih no chang in.
More information3.4 Repeated Roots; Reduction of Order
3.4 Rpd Roos; Rducion of Ordr Rcll our nd ordr linr homognous ODE b c 0 whr, b nd c r consns. Assuming n xponnil soluion lds o chrcrisic quion: r r br c 0 Qudric formul or fcoring ilds wo soluions, r &
More informationContinous system: differential equations
/6/008 Coious sysm: diffrial quaios Drmiisic modls drivaivs isad of (+)-( r( compar ( + ) R( + r ( (0) ( R ( 0 ) ( Dcid wha hav a ffc o h sysm Drmi whhr h paramrs ar posiiv or gaiv, i.. giv growh or rducio
More information5. An object moving along an x-coordinate axis with its scale measured in meters has a velocity of 6t
AP CALCULUS FINAL UNIT WORKSHEETS ACCELERATION, VELOCTIY AND POSITION In problms -, drmin h posiion funcion, (), from h givn informaion.. v (), () = 5. v ()5, () = b g. a (), v() =, () = -. a (), v() =
More informationA MATHEMATICAL MODEL FOR NATURAL COOLING OF A CUP OF TEA
MTHEMTICL MODEL FOR NTURL COOLING OF CUP OF TE 1 Mrs.D.Kalpana, 2 Mr.S.Dhvarajan 1 Snior Lcurr, Dparmn of Chmisry, PSB Polychnic Collg, Chnnai, India. 2 ssisan Profssor, Dparmn of Mahmaics, Dr.M.G.R Educaional
More informationCircuit Transients time
Circui Tranin A Solp 3/29/0, 9/29/04. Inroducion Tranin: A ranin i a raniion from on a o anohr. If h volag and currn in a circui do no chang wih im, w call ha a "ady a". In fac, a long a h volag and currn
More informationThe transition:transversion rate ratio vs. the T-ratio.
PhyloMah Lcur 8 by Dan Vandrpool March, 00 opics of Discussion ransiion:ransvrsion ra raio Kappa vs. ransiion:ransvrsion raio raio alculaing h xpcd numbr of subsiuions using marix algbra Why h nral im
More informationDiscussion 06 Solutions
STAT Discussion Soluions Spring 8. Th wigh of fish in La Paradis follows a normal disribuion wih man of 8. lbs and sandard dviaion of. lbs. a) Wha proporion of fish ar bwn 9 lbs and lbs? æ 9-8. - 8. P
More information46. Let y = ln r. Then dy = dr, and so. = [ sin (ln r) cos (ln r)
98 Scion 7.. L w. Thn dw d, so d dw w dw. sin d (sin w)( wdw) w sin w dw L u w dv sin w dw du dw v cos w w sin w dw w cos w + cos w dw w cos w+ sin w+ sin d wsin wdw w cos w+ sin w+ cos + sin +. L w +
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More informationNikesh Bajaj. Fourier Analysis and Synthesis Tool. Guess.? Question??? History. Fourier Series. Fourier. Nikesh Bajaj
Guss.? ourir Analysis an Synhsis Tool Qusion??? niksh.473@lpu.co.in Digial Signal Procssing School of Elcronics an Communicaion Lovly Profssional Univrsiy Wha o you man by Transform? Wha is /Transform?
More informationPhysics 160 Lecture 3. R. Johnson April 6, 2015
Physics 6 Lcur 3 R. Johnson April 6, 5 RC Circui (Low-Pass Filr This is h sam RC circui w lookd a arlir h im doma, bu hr w ar rsd h frquncy rspons. So w pu a s wav sad of a sp funcion. whr R C RC Complx
More informationCircuits and Systems I
Circuis and Sysms I LECTURE #3 Th Spcrum, Priodic Signals, and h Tim-Varying Spcrum lions@pfl Prof. Dr. Volan Cvhr LIONS/Laboraory for Informaion and Infrnc Sysms Licns Info for SPFirs Slids This wor rlasd
More informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
More informationMath 3301 Homework Set 6 Solutions 10 Points. = +. The guess for the particular P ( ) ( ) ( ) ( ) ( ) ( ) ( ) cos 2 t : 4D= 2
Mah 0 Homwork S 6 Soluions 0 oins. ( ps) I ll lav i o you o vrify ha y os sin = +. Th guss for h pariular soluion and is drivaivs is blow. Noi ha w ndd o add s ono h las wo rms sin hos ar xaly h omplimnary
More information( ) C R. υ in RC 1. cos. ,sin. ω ω υ + +
Oscillaors. Thory of Oscillaions. Th lad circui, h lag circui and h lad-lag circui. Th Win Bridg oscillaor. Ohr usful oscillaors. Th 555 Timr. Basic Dscripion. Th S flip flop. Monosabl opraion of h 555
More informationDouble Slits in Space and Time
Doubl Slis in Sac an Tim Gorg Jons As has bn ror rcnly in h mia, a am l by Grhar Paulus has monsra an inrsing chniqu for ionizing argon aoms by using ulra-shor lasr ulss. Each lasr uls is ffcivly on an
More informationLecture 26: Leapers and Creepers
Lcur 6: Lapr and Crpr Scrib: Grain Jon (and Marin Z. Bazan) Dparmn of Economic, MIT May, 005 Inroducion Thi lcur conidr h analyi of h non-parabl CTRW in which h diribuion of p iz and im bwn p ar dpndn.
More informationMundell-Fleming I: Setup
Mundll-Flming I: Sup In ISLM, w had: E ( ) T I( i π G T C Y ) To his, w now add n xpors, which is a funcion of h xchang ra: ε E P* P ( T ) I( i π ) G T NX ( ) C Y Whr NX is assumd (Marshall Lrnr condiion)
More informationAN INTRODUCTION TO FOURIER ANALYSIS PROF. VEDAT TAVSANOĞLU
A IRODUCIO O FOURIER AALYSIS PROF. VEDA AVSAOĞLU 994 A IRODUCIO O FOURIER AALYSIS ABLE OF COES. HE FOURIER SERIES ---------------------------------------------------------------------3.. Priodic Funcions-----------------------------------------------------------------------3..
More information14.02 Principles of Macroeconomics Fall 2005 Quiz 3 Solutions
4.0 rincipl of Macroconomic Fall 005 Quiz 3 Soluion Shor Quion (30/00 poin la a whhr h following amn ar TRUE or FALSE wih a hor xplanaion (3 or 4 lin. Each quion coun 5/00 poin.. An incra in ax oday alway
More information10. If p and q are the lengths of the perpendiculars from the origin on the tangent and the normal to the curve
0. If p and q ar h lnghs of h prpndiculars from h origin on h angn and h normal o h curv + Mahmaics y = a, hn 4p + q = a a (C) a (D) 5a 6. Wha is h diffrnial quaion of h family of circls having hir cnrs
More informationFrequency Response. Lecture #12 Chapter 10. BME 310 Biomedical Computing - J.Schesser
Frquncy Rspns Lcur # Chapr BME 3 Bimdical Cmpuing - J.Schssr 99 Idal Filrs W wan sudy Hω funcins which prvid frquncy slciviy such as: Lw Pass High Pass Band Pass Hwvr, w will lk a idal filring, ha is,
More informationLinear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let
It is impossibl to dsign an IIR transfr function with an xact linar-phas It is always possibl to dsign an FIR transfr function with an xact linar-phas rspons W now dvlop th forms of th linarphas FIR transfr
More informationBasic Polyhedral theory
Basic Polyhdral thory Th st P = { A b} is calld a polyhdron. Lmma 1. Eithr th systm A = b, b 0, 0 has a solution or thr is a vctorπ such that π A 0, πb < 0 Thr cass, if solution in top row dos not ist
More informationUniversity of Kansas, Department of Economics Economics 911: Applied Macroeconomics. Problem Set 2: Multivariate Time Series Analysis
Univrsiy of Kansas, Dparmn of Economics Economics 9: Applid Macroconomics Problm S : Mulivaria Tim Sris Analysis Unlss sad ohrwis, assum ha shocks (.g. g and µ) ar whi nois in h following qusions.. Considr
More informationThe Science of Monetary Policy
Th Scinc of Monary Policy. Inroducion o Topics of Sminar. Rviw: IS-LM, AD-AS wih an applicaion o currn monary policy in Japan 3. Monary Policy Sragy: Inrs Ra Ruls and Inflaion Targing (Svnsson EER) 4.
More informationAsymptotic Solutions of Fifth Order Critically Damped Nonlinear Systems with Pair Wise Equal Eigenvalues and another is Distinct
Qus Journals Journal of Rsarch in Applid Mahmaics Volum ~ Issu (5 pp: -5 ISSN(Onlin : 94-74 ISSN (Prin:94-75 www.usjournals.org Rsarch Papr Asympoic Soluions of Fifh Ordr Criically Dampd Nonlinar Sysms
More informationChapter 1 Fundamental Concepts
Chaper 1 Fundamenal Conceps 1 Signals A signal is a paern of variaion of a physical quaniy, ofen as a funcion of ime (bu also space, disance, posiion, ec). These quaniies are usually he independen variables
More information