Sampling. Controller Implementation of Control. Algorithm. Aliasing. Aliasing Example. Example Prefiltering. Prefilters
|
|
- Linette Flynn
- 5 years ago
- Views:
Transcription
1 Conrollr Implmnaion of Conrol Algorims Sampling Compr A/D Algorim D/A Procss y Aliasing Compr Implmnaion of Conrol Algorims Sampld conrol ory Discrizaion of coninos im dsign Exampl: PID algorim discrizaion conrol mods ning AD-convrr acs as samplr A/D DA-convrr acs as a old dvic Normally, zro-ordr-old is sd picwis consan conrol signals Aliasing 5 Tim ω N = ω s / = Nyqis frqncy, (ω s = sampling frq.) Frqncis abov Nyqis frqncy ar foldd and appar as low-frqncy signals. T fndamnal alias frqncy for a frqncy f > f N is givn by f = (f + f N ) mod ( f s ) f N Aliasing Exampl Fd war aing in a sip boilr Prssr Sam Fd Valv war Tmprar Pmp Condnsd war 38 min min To boilr Tmprar Abov: f =.9, f s =, f N =.5, f =. Prssr 3. min 4 Tim Prfilrs Ani-aliasing filr Analog low-pass filr a liminas all frqncis abov Nyqis frqncy (a) Exampl Prfilring (b) Analog filr -6 ordr Bssl or Brwor Difficlis wi canging (sampling inrval) Digial Filr Fixd, fas sampling wi fixd analog filr Conrol algorim a a slowr ra ogr wi digial LP-filr Easy o cang sampling inrval 3 3 (c) (d) 3 3 Tim Tim ω d =.9, ω N =.5, ω alias =. 6 ordr Bssl wi ω B =.5 T filr may av o b incldd in dsign. 5 6
2 Dsign of Digial Conrollrs Sampld Conrol Tory Digial conrollrs can b dsignd in wo diffrn ways: Discr im dsign sampld (digial) conrol ory sif opraors (z-ransforms) (k) =k y(k)+k (k ) a dsign paramr Coninos im dsign + discrizaion Laplac ransform U(s) =G c (s)e(s) approxima coninos dsign fas, fixd sampling 7 Compr Clock A-D { } { ( k )} y( () y( ) k ) Algorim D-A Procss T basic ida: Look a sampling insancs only! 8 Sampld Conrol Tory Disk Driv Exampl k ( ) ( ) Hold k D-A Procss Compr y( ) A-D y( ) Samplr y k y k Conrol of arm of a disk driv Coninos im conrollr G(s) = k Js U(s) = b a U c(s) s + b s + a Y(s) Discr im conrollr (coninos im dsign + discrizaion) Sysm ory analogos o coninos im linar sysms ( k )= ( b a c( k ) y( k )+x( k )) x( k + ) =x( k )+ ((a b)y( k ) ax( k )) Br prformanc can b acivd Problms wi inrsampl bavior 9 Disk Driv Exampl Incrasd sampling priod y: = adin(in) :=*(b/a*c-y+x) do() x:=x+*((a-b)*y-a*x) Clock Algorim a) =.5/ω b) =.8/ω (a) Op (b) Op Sampling priod =./ω Op.5 5 Inp Tim (ω ) Inp Tim (ω ) Inp.5 5 Tim (ω )
3 Dad-ba conrol =.4/ω Br prformanc? ( k )= c ( k )+ c ( k ) s y( k ) s y( k ) r ( k ) Sampling of Sysms Look a sysm from poin of viw of compr Posiion Clock Vlociy Inp Tim (ω ) Howvr, long sampling priods also av problms opn loop bwn sampls disrbanc and rfrnc cangs a occr bwn sampls 3 will rmain ndcd nil nx sampl {( k )} () y() { y( k )} D-A Sysm A-D Zro-ordr-old sampling of a sysm L inps b picwis consan Look a sampling poins only Us linariy and calcla sp rsponss wn solving sysm qaion 4 Sampling a coninos-im sysm Sysm dscripion T Gnral Cas dx d = Ax()+B() y() = Cx()+D() Solv sysm qaion x() = A( k) x( k )+ = A( k) x( k )+ = A( k) x( k )+ k k k A( s ) B(s ) ds = Φ(, k )x( k )+Γ(, k )( k ) A( s ) ds B( k ) ( cons.) As ds B( k ) (variabl cang) wr x( k+ ) = Φ( k+, k )x( k )+Γ( k+, k )( k ) y( k ) = Cx( k )+D( k ) Φ( k+, k ) = A(k+ k) k+ k Γ( k+, k ) = As ds B 5 6 Priodic sampling Assm priodic sampling, i.. k = k, n x(k + ) = Φx(k)+ Γ(k) y(k) = Cx(k)+D(k) wr Φ = A Γ = As ds B NOTE: Tim-invarian linar sysm! I is also possibl o sampl a sysm wi im dlay Hnc Exampl: Sampling of dobl ingraor dx = d y = x + x Φ = A = Γ = s ds = Svral ways o calcla Φ and Γ. Malab 7 8
4 Sabiliy rgion In coninos im sabiliy rgion is complx lf alf plan, i. sysm is sabl if all pols ar in lf alf plan. In discr im sabiliy rgion is ni circl. Conrol Dsign A larg variy of conrol dsign mods ar availabl in digial conrol ory,.g.: sa-fdback conrol pol-placmn LQ conrol obsrvr-basd sa fdback conrol LQG conrol op fdback conrol... Cors in Compr-Conrolld Sysms. 9 Sampling Inrval Discrizaion of Coninos Tim Dsign Nmbr of sampls pr ris im, T r, of closd loop sysm N r = T r 4 Basic idas: Rs dsign () A-D H(z) G(s) { ( k) } { y( k) } Algorim D-A y() Wi long sampling inrvals i may ak long bfor disrbancs ar dcd Clock G(s) is dsignd basd on analog cniqs Wan o g: A/D + Algorim + D/A G(s) Mods: Approxima s, i.., G(s) H(z) Or mods Approximaion Mods Sabiliy of Approximaions Forward Diffrnc (Elr s mod) dx() d Backward Diffrnc dx() d Tsin: x( + ) x() s = z x() x( ) s = z z s = z z + 3 How is coninos-im sabiliy rgion (lf alf plan) mappd? Forward diffrncs Backward diffrncs Tsin 4
5 Sampling Inrval An Exampl: PID Conrol T fasr br. Rl-of-mb: ω c.5.5 wr ω c is cross-ovr frqncy of coninos-im sysm ( frqncy wr gain is ) Sbsanially largr n wi discr-im dsign. Mor robs agains variaions in sampling inrval. T olds conrollr yp T mos widly sd Plp & Papr 86% Sl 93% Oil rfinris 93% Mc o larn!! 5 6 T Txbook Algorim Proporional Trm () = (() + T I (τ )dτ + d() TD ) d max U(s) = (E(s) + st I E(s) + T D se(s)) min Proporionalband = P + I + D max > = + < < min < 8 7 Propris of P-Conrol Errors wi P-conrol.5.5 S poin and masrd variabl Conrol variabl 4 c=5 c=5 c= c= c= c= Conrol signal: Error: Error rmovd if:. qals infiniy. = = + = 5 5 Solion: Aomaic way o obain saionary rror incrasd mans fasr spd, incrasd nois snsiiviy, wors 9 sabiliy 3
6 Ingral Trm Aomaic Rs b +st i = + = ( + Ti ) ()d (PI) c + Saionary rror prsn d incrass incrass y incrass rror is no saionary U = E+ U + st i ( )U = + st i U = st i U + st i + st i + s i U = ( + )E st i 3 3 Propris of PI-Conrol S poin and masrd variabl.5 Ti= Ti= Ti=5.5 Ti= 5 5 Conrol variabl Ti= Ti= Prdicion A PI-conrollr conains no prdicion T sam conrol signal is obaind for bo s cass: Ti=5 Ti= 5 5 I P I P id id rmovs saionary rror smallr T I implis wors sabiliy, fasr sady-sa rror rmoval Drivaiv Par Propris of PD-Conrol Rglrfl () + T d d() d S poin and masrd variabl Td=. Td=.5 () ( + T d).5 Td= 5 5 id 6 4 Conrol variabl Td=. Td=.5 Td= P: () =() PD: ( ) d() () = ()+T d ( + T d ) d T d = Prdicion orizon T D oo small, no inflnc T D oo larg, dcrasd prformanc In indsrial pracic D-rm is ofn rnd off. 36
7 Alrnaiv forms So far w av dscribd dirc (posiion) vrsion of PID conrollr on paralll form Or forms: sris form U = ( + st )( + st D )E I = ( + T D T + st + st D )E I I Diffrn paramr vals incrmnal (vlociy) form U = s ΔU ΔU = (s + T I + s T D + st D /N )E Ingraion xrnal o algorim (.g. sp moor) or inrnal Algorim Modificaions Limiaions of drivaiv gain Modificaions ar ndd o mak conrollr pracically sfl Limiaions of drivaiv gain Drivaiv wiging Spoin wiging Handl conrol signal limiaions W do no wan o apply drivaion o ig frqncy masrmn nois, rfor following modificaion is sd: st D st D + st D /N N = maximm drivaiv gain, ofn 39 4 Drivaiv wiging Spoin wiging An advanag o also s wiging on spoin. T spoin is ofn consan for long priods of im Spoin ofn cangd in sps D-par bcoms vry larg. Drivaiv par applid on par of spoin or only on masrmn signal. rplacd by β = (y sp y) = (β y sp y) st D D(s) = + st D /N (γ Y sp(s) Y(s)) Ofn, γ = in procss conrol, γ = in srvo conrol A way of inrodcing fdforward from rfrnc signal (posiion a closd loop zro) Improvd s-poin rsponss. 4 4
8 .5 Spoin wiging.5 S poin and masrd variabl ba= ba=.5 ba= Conrol variabl Conrol Signal Limiaions All acaors sara. Problms for conrollrs wi ingraion. Wn conrol signal saras ingral par will conin o grow ingraor (rs) windp. Wn conrol signal saras ingral par will ingra p o a vry larg val. Tis may cas larg ovrsoos..5.5 Op y and yrf ba= ba=.5 ba= Conrol variabl Ani-Rs Windp Tracking Svral solions xis: conrollrs on vlociy form (Δ is s o if saras) limi spoin variaions (saraion nvr racd) condiional ingraion (ingraion is swicd off wn conrol is far from sady-sa) racking (back-calclaion) wn conrol signal saras, ingral is rcompd so a is nw val givs a conrol signal a saraion limi o avoid rsing ingral d o,.g., masrmn nois, rcompaion is don dynamically, i.., rog a LP-filr wi a im consan T (T r ) Tracking Tracking y Tds = r y T i s v Acaor s T.5 y = r y T ds Acaor modl Acaor.5 3 T i s T + s
9 Discrizaion Discrizaion I-par: I() = T I (τ )dτ P-par: P (k) =(β y sp (k) y(k)) Forward diffrnc di d = T I I( k+ ) I( k ) = T I ( k ) I(k+) := I(k) + (*/Ti)*(k) T I-par can b prcalclad in UpdaSas 49 Backward diffrnc T I-par canno b prcalclad, i(k) = f((k)) Ors 5 Discrizaion Discrizaion D-par (assm γ = ): st D = D + st D /N ( Y(s)) T D dd N d + D = T dy D d Forward diffrnc (nsabl for small T D ) Backward diffrnc T D D( k ) D( k ) + D( k ) N y( k ) y( k ) = T D T D D( k )= T D + N D( k ) T D N T D + N (y( k) y( k )) 5 Tracking: v := P + I + D; := sa(v,max,min); I := I + (*/Ti)* + (/Tr)*( - v); 5 Bmplss Transfrs Bmplss Mod Cangs Avoid bmps in conrol signal wn canging opraing mod (manal - ao - manal) canging paramrs canging bwn diffrn conrollrs Canging opraing mod y Iss: Mak sr a conrollr sas av corrc vals, i.., sam vals bfor and afr cang Incrmnal Form: + y sp y MCU Inc PID M A s 53 54
10 Bmplss Mod Cangs Bmplss paramr cangs Dirc Posiion form: + T m y sp y T s PD T i s T + M A + A cang in a paramr wn in saionariy sold no rsl in a bmp in conrol signal. For xampl: v := P + I + D; I := I +(*/Ti)*; or v := P + (/Ti)*I + D; I := I + *; T lar rsls in a bmp in if or Ti ar cangd Bmplss paramr cangs Canging Conrollrs Mor involvd siaion wn spoin wiging is sd. T qaniy P + I sold b invarian o paramr cangs. Conrollr Conrollr Swic Procss I nw = I old + old (β old y sp y) nw (β nw y sp y) 57 Similar o canging bwn manal and ao L conrollrs rn in paralll L conrollr a is no aciv rack on a is aciv. Alrnaivly, xc only aciv conrollr and iniializ nw conrollr o is corrc val wn swicing (savs CPUim) 58 PID cod PID-conrollr wi ani-rs windp and manal and ao mods (γ = ). y = yin.g(); = yrf - y; D = ad * D - bd * (y - yold); v = *(ba*yrf - y) + I + D; if (mod == ao) { = sa(v,max,min)} ls = sa(man,max,min); O.p(); I = I + (*/Ti)* + (/Tr)*( - v); if (incrmn) { inc = } ls {if (dcrmn){ inc = -} ls inc = ;} man = man + (/Tm) * inc + (/Tr) * ( - man) yold = y ad and bd ar prcalclad paramrs givn by backward diffrnc approximaion of D-rm. 59
DYNAMICS and CONTROL
DYNAMICS an CONTROL Mol IV(I) IV(II) Conrol Sysms Dsign Conrol sysm aramrs Prsn by Pro Albros Profssor of Sysms Enginring an Conrol - UPV Mols: Examls of sysms an signals Mols of sysms an signals Conroll
More informationSpring 2006 Process Dynamics, Operations, and Control Lesson 2: Mathematics Review
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
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 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 informationProblem 2. Describe the following signals in terms of elementary functions (δ, u,r, ) and compute. x(t+2) x(2-t) RT_1[x] -3-2 = 1 2 = 1
EEE 03, HW NAME: SOLUTIONS Problm. Considr h signal whos graph is shown blow. Skch h following signals:, -, RT [], whr R dnos h rflcion opraion and T 0 dnos shif dlay opraion by 0. - RT_[] - -3 - Problm.
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 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 information2. Transfer function. Kanazawa University Microelectronics Research Lab. Akio Kitagawa
. ransfr funion Kanazawa Univrsiy Mirolronis Rsarh Lab. Akio Kiagawa . Wavforms in mix-signal iruis Configuraion of mix-signal sysm x Digial o Analog Analog o Digial Anialiasing Digial moohing Filr Prossor
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 informationBoyce/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 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 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 informationFeedback Control and Synchronization of Chaos for the Coupled Dynamos Dynamical System *
ISSN 746-7659 England UK Jornal of Informaion and Comping Scinc Vol. No. 6 pp. 9- Fdbac Conrol and Snchroniaion of Chaos for h Copld Dnamos Dnamical Ssm * Xdi Wang Liin Tian Shmin Jiang Liqin Y Nonlinar
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 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 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 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 informationt=0 t>0: + vr - i dvc Continuation
hapr Ga Dlay and rcus onnuaon s rcu Equaon >: S S Ths dffrnal quaon, oghr wh h nal condon, fully spcfs bhaor of crcu afr swch closs Our n challng: larn how o sol such quaons TUE/EE 57 nwrk analys 4/5 NdM
More informationDigital Image Processing
5 Imag Rsoraion and Rconsrcion Digial Imag Procssing Jn-i Chang Dparmn o Compr cinc Naional Chiao ng Unirsi 5. A Modl o h Imag Dgradaion/Rsoraion Procss 5. Nois Modls 5.3 Rsoraion in h Prsnc o Nois Onl
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 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 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 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 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 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 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 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 informationA L A BA M A L A W R E V IE W
A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N
More informationLecture 1: Contents of the course. Advanced Digital Control. IT tools CCSDEMO
Goals of he course Lecure : Advanced Digial Conrol To beer undersand discree-ime sysems To beer undersand compuer-conrolled sysems u k u( ) u( ) Hold u k D-A Process Compuer y( ) A-D y ( ) Sampler y k
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 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 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 informationdy 1. If fx ( ) is continuous at x = 3, then 13. If y x ) for x 0, then f (g(x)) = g (f (x)) when x = a. ½ b. ½ c. 1 b. 4x a. 3 b. 3 c.
AP CALCULUS BC SUMMER ASSIGNMENT DO NOT SHOW YOUR WORK ON THIS! Complt ts problms during t last two wks of August. SHOW ALL WORK. Know ow to do ALL of ts problms, so do tm wll. Itms markd wit a * dnot
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 informationNAME: SOLUTIONS EEE 203 HW 1
NAME: SOLUIONS EEE W Problm. Cosir sigal os grap is so blo. Sc folloig sigals: -, -, R, r R os rflcio opraio a os sif la opraio b. - - R - Problm. Dscrib folloig sigals i rms of lmar fcios,,r, a comp a.
More informationState Observer Design
Sa Obsrvr Dsgn A. Khak Sdgh Conrol Sysms Group Faculy of Elcrcal and Compur Engnrng K. N. Toos Unvrsy of Tchnology Fbruary 2009 1 Problm Formulaon A ky assumpon n gnvalu assgnmn and sablzng sysms usng
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 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 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 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 informationThe pn junction: 2 Current vs Voltage (IV) characteristics
Th pn junction: Currnt vs Voltag (V) charactristics Considr a pn junction in quilibrium with no applid xtrnal voltag: o th V E F E F V p-typ Dpltion rgion n-typ Elctron movmnt across th junction: 1. n
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 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 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 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 informationANALYSIS IN THE FREQUENCY DOMAIN
ANALYSIS IN THE FREQUENCY DOMAIN SPECTRAL DENSITY Dfinition Th spctral dnsit of a S.S.P. t also calld th spctrum of t is dfind as: + { γ }. jτ γ τ F τ τ In othr words, of th covarianc function. is dfind
More informationINTRODUCTION TO AUTOMATIC CONTROLS INDEX LAPLACE TRANSFORMS
adjoint...6 block diagram...4 clod loop ytm... 5, 0 E()...6 (t)...6 rror tady tat tracking...6 tracking...6...6 gloary... 0 impul function...3 input...5 invr Laplac tranform, INTRODUCTION TO AUTOMATIC
More informationTHE LAPLACE TRANSFORM
THE LAPLACE TRANSFORM LEARNING GOALS Diniion Th ranorm map a ncion o im ino a ncion o a complx variabl Two imporan inglariy ncion Th ni p and h ni impl Tranorm pair Baic abl wih commonly d ranorm Propri
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 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 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 informationPartial Fraction Expansion
Paial Facion Expanion Whn ying o find h inv Laplac anfom o inv z anfom i i hlpfl o b abl o bak a complicad aio of wo polynomial ino fom ha a on h Laplac Tanfom o z anfom abl. W will illa h ing Laplac anfom.
More informationA Backstepping Simple Adaptive Control Application to Flexible Space Structures
Chins Journal of Aronauics 5 (01) 446-45 Conns liss availabl a cincdirc Chins Journal of Aronauics journal hompag: www.lsvir.com/loca/cja A Backspping impl Adapiv Conrol Applicaion o Flxibl pac rucurs
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 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 informationBifurcation Theory. , a stationary point, depends on the value of α. At certain values
Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local
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 informationUNSTEADY FLOW OF A FLUID PARTICLE SUSPENSION BETWEEN TWO PARALLEL PLATES SUDDENLY SET IN MOTION WITH SAME SPEED
006-0 Asian Rsarch Publishing work (ARP). All righs rsrvd. USTEADY FLOW OF A FLUID PARTICLE SUSPESIO BETWEE TWO PARALLEL PLATES SUDDELY SET I MOTIO WITH SAME SPEED M. suniha, B. Shankr and G. Shanha 3
More informationLecture 2. Basic Digital Communication Principles
Lcr Basic Principls Signals Basd on class nos by Pro: Amir Asi Basic Digial Commnicaion Principls In his lcr w prsn a rviw abo basic principls in digial commnicaion, som o i yo migh hav sn bor Digial vs.
More informationLecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields
Lctur 37 (Schrödingr Equation) Physics 6-01 Spring 018 Douglas Filds Rducd Mass OK, so th Bohr modl of th atom givs nrgy lvls: E n 1 k m n 4 But, this has on problm it was dvlopd assuming th acclration
More informationMATH 308: Diff Eqs, BDP10 EXAMPLES [Belmonte, 2019] 1 Introduction 1.1 Basic Mathematical Models; Direction Fields
MATH 308: Diff Eqs, BDP0 EXAMPLES Blmon, 09 Mos problms ar from NSS9 Inroducion Basic Mahmaical Modls; Dircion Filds / Plo a dircion fild for dy/dx = 4x/y (a) Vrify ha h sraigh lins y = ±x ar soluions
More informationChapter 6 Folding. Folding
Chaptr 6 Folding Wintr 1 Mokhtar Abolaz Folding Th folding transformation is usd to systmatically dtrmin th control circuits in DSP architctur whr multipl algorithm oprations ar tim-multiplxd to a singl
More informationNumbering Systems Basic Building Blocks Scaling and Round-off Noise. Number Representation. Floating vs. Fixed point. DSP Design.
Numbring Systms Basic Building Blocks Scaling and Round-off Nois Numbr Rprsntation Viktor Öwall viktor.owall@it.lth.s Floating vs. Fixd point In floating point a valu is rprsntd by mantissa dtrmining th
More informationwhere: u: input y: output x: state vector A, B, C, D are const matrices
Sa pac modl: linar: y or in om : Sa q : f, u Oupu q : y h, u u Du F Gu y H Ju whr: u: inpu y: oupu : a vcor,,, D ar con maric Eampl " $ & ' " $ & 'u y " & * * * * [ ],, D H D I " $ " & $ ' " & $ ' " &
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 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 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 informationSECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.
SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain
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 informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
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 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 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 information3-2-1 ANN Architecture
ARTIFICIAL NEURAL NETWORKS (ANNs) Profssor Tom Fomby Dpartmnt of Economics Soutrn Mtodist Univrsity Marc 008 Artificial Nural Ntworks (raftr ANNs) can b usd for itr prdiction or classification problms.
More informationLinear Systems Analysis in the Time Domain
Liar Sysms Aalysis i h Tim Domai Firs Ordr Sysms di vl = L, vr = Ri, d di L + Ri = () d R x= i, x& = x+ ( ) L L X() s I() s = = = U() s E() s Ls+ R R L s + R u () = () =, i() = L i () = R R Firs Ordr Sysms
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 informationCommunication Technologies
Communication Tchnologis. Principls of Digital Transmission. Structur of Data Transmission.2 Spctrum of a Data Signal 2. Digital Modulation 2. Linar Modulation Mthods 2.2 Nonlinar Modulations (CPM-Signals)
More informationProblem Set #2 Due: Friday April 20, 2018 at 5 PM.
1 EE102B Spring 2018 Signal Procssing and Linar Systms II Goldsmith Problm St #2 Du: Friday April 20, 2018 at 5 PM. 1. Non-idal sampling and rcovry of idal sampls by discrt-tim filtring 30 pts) Considr
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 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 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 informationChapter 13 GMM for Linear Factor Models in Discount Factor form. GMM on the pricing errors gives a crosssectional
Chaptr 13 GMM for Linar Factor Modls in Discount Factor form GMM on th pricing rrors givs a crosssctional rgrssion h cas of xcss rturns Hors rac sting for charactristic sting for pricd factors: lambdas
More informationEE 350 Signals and Systems Spring 2005 Sample Exam #2 - Solutions
EE 35 Signals an Sysms Spring 5 Sampl Exam # - Soluions. For h following signal x( cos( sin(3 - cos(5 - T, /T x( j j 3 j 3 j j 5 j 5 j a -, a a -, a a - ½, a 3 /j-j -j/, a -3 -/jj j/, a 5 -½, a -5 -½,
More informationPFC Predictive Functional Control
PFC Prdiciv Funcional Conrol Prof. Car d Prada D. of Sm Enginring and Auomaic Conrol Univri of Valladolid, Sain rada@auom.uva. Oulin A iml a oibl Moivaion PFC main ida An inroducor xaml Moivaion Prdiciv
More informationFourier Series: main points
BIOEN 3 Lcur 6 Fourir rasforms Novmbr 9, Fourir Sris: mai pois Ifii sum of sis, cosis, or boh + a a cos( + b si( All frqucis ar igr mulipls of a fudamal frqucy, o F.S. ca rprs ay priodic fucio ha w ca
More informationPart 3 System Identification
2.6 Sy Idnificaion, Eiaion, and Larning Lcur o o. 5 Apri 2, 26 Par 3 Sy Idnificaion Prpci of Sy Idnificaion Tory u Tru Proc S y Exprin Dign Daa S Z { u, y } Conincy Mod S arg inv θ θ ˆ M θ ~ θ? Ky Quion:
More informationSUMMER 17 EXAMINATION
(ISO/IEC - 7-5 Crtifid) SUMMER 7 EXAMINATION Modl wr jct Cod: Important Instructions to aminrs: ) Th answrs should b amind by ky words and not as word-to-word as givn in th modl answr schm. ) Th modl answr
More informationPredictive Control for Time-Delayed Switching Control Systems
Priciv Conrol for im-dlay Swiching Conrol Sysms obby L. Shils Eric J. arh Michal Golfarb Dparmn of Mchanical Enginring Vanrbil Univrsiy Nashvill, N 37235 Vrsion: Spmbr 13, 2005 o appar in h SME Jornal
More informationPropositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018
Propositional Logic Combinatorial Problm Solving (CPS) Albrt Olivras Enric Rodríguz-Carbonll May 17, 2018 Ovrviw of th sssion Dfinition of Propositional Logic Gnral Concpts in Logic Rduction to SAT CNFs
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 informationECE602 Exam 1 April 5, You must show ALL of your work for full credit.
ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b
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 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 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 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 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 informationPredictive time optimal algorithm for a third-order dynamical system with delay
Jornal of Phsics: Confrnc Sris PAPER OPEN ACCESS Prdiciv im opimal algorihm for a hird-ordr dnamical ssm wih dla To ci his aricl: G A Pikina 7 J Phs: Conf Sr 89 78 iw h aricl onlin for pdas and nhancmns
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 informationEXST Regression Techniques Page 1
EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy
More information