Control Systems. Lecture 9 Frequency Response. Frequency Response
|
|
- Amy Ramsey
- 5 years ago
- Views:
Transcription
1 Conrol Syem Lecure 9 Frequency eone Frequency eone We now know how o analyze and deign yem via -domain mehod which yield dynamical informaion The reone are decribed by he exonenial mode The mode are deermined by he ole of he reone Lalace Tranform We nex will look a decribing yem erformance via frequency reone mehod Thi guide u in ecifying he yem ole and zero oiion
2 Sinuoidal Seady-Sae eone co A U A u Conider a able ranfer funcion wih a inuoidal inu: Where he naural yem mode lie Thee are in he oen lef half lane e< A he inu mode and -" Only he reone due o he ole on he imaginary axi remain afer a ufficienly long ime Thi i he inuoidal eady-ae reone The Lalace Tranform of he reone ha ole A z z z K U Y n m 3 Sinuoidal Seady-Sae eone Inu Tranform eone Tranform eone Signal Sinuoidal Seady Sae eone co in in co co A A A u in co A A U N N k k k k k U Y * reone naural reone forced * N N e k e k e k e k ke y SS e k ke y * forced reone naural reone 4
3 3 Sinuoidal Seady-Sae eone Calculaing he SSS reone o eidue calculaion Signal calculaion [ ] [ ] in co in co lim lim lim e A e A A A U Y k % & ' * % & ' * co A u K k e k e e k e k k L y K K " # $ % & ' co * co A y 5 Sinuoidal Seady-Sae eone eone o i Ouu frequency inu frequency Ouu amliude inu amliude Ouu hae inu hae The Frequency eone of he ranfer funcion i given by i evaluaion a a funcion of a comlex variable a We eak of he amliude reone and of he hae reone They canno indeendenly be varied» Bode relaion of analyic funcion heory co co A y A u 6
4 Frequency eone Find he eady ae ouu for v Aco _ V L Comue he eady ae ouu A v SS co an L / L V Comue he -domain ranfer funcion T Volage divider T L Comue he frequency reone & L # T, T an $! L % " - [ ] 7 Bode Diagram Log-log lo of magt, log-linear lo of argt veru " Bode Diagram Magniude Phae deg Magniude Phae deg Frequency rad/ec 8 4
5 Frequency eone u Aco Aco y Sable Tranfer Funcion Afer a ranien, he ouu ele o a inuoid wih an amliude magnified by and hae hifed by. Since all ignal can be rereened by inuoid Fourier erie and ranform, he quaniie and are exremely imoran. Bode develoed mehod for quickly finding and for a given and for uing hem in conrol deign. 9 Bode Diagram z z zm K n r r r [ ] z z z K z z z m i m n e K r r rn The magniude and hae of when i given by: r r K r K z z r r r z m n z z z, Nonlinear in he magniude m Linear in he hae n 5
6 Why do we exre Bode Diagram in decibel? log z z z m K r r rn By roerie of he logarihm we can wrie: z z z logr logr logr logr logr logr log log K m m K z z z r r r r r r r r The magniude and hae of when i given by: m z z z K m r Linear in he magniude n? Linear in he hae n n Bode Diagram Why do we ue a logarihmic cale? Le have a look a our examle: T T L Exreing he magniude in : T L L & L # $! % " T Aymoic behavior: log log, * L ' &, L # log$ * '! % " : T % : T log& # log ' / L $ / L log log L LINEA FUNCTION in log!!! We lo a a funcion of log. 6
7 Bode Diagram Why do we ue a logarihmic cale? Le have a look a our examle: T T L Exreing he hae: L L & T log $ % Aymoic behavior: "" """ 9 L #! an " & $ % L #! " 45 L LINEA FUNCTION in log!!! We lo a a funcion of log. 3 Bode Diagram Log-log lo of magt, log-linear lo of argt veru " Bode Diagram Magniude Phae deg Magniude Phae deg /dec Frequency rad/ec 4 7
8 Bode Diagram Decade: Any frequency range whoe end oin have a : raio A cuoff frequency occur when he gain i reduced from i maximum aband value by a facor / : % log& T # log T log log T 3 MAX MAX MAX ' $ Bandwih: frequency range anned by he gain aband Le have a look a our examle: T * L ' % & # T "! / L T / Thi i a low-a filer!!! Paband gain, Cuoff frequency/l The Bandwih i /L! 5 eneral Tranfer Funcion Bode Diagram & n, # o $ * ' ζ! $ % n n!" m K τ The magniude hae i he um of he magniude hae of each one of he erm. We learn how o lo each erm, we learn how o lo he whole magniude and hae Bode Plo. q Clae of erm: Ko m τ n &, # $ * ' ζ! $ % n n!" q 6 8
9 eneral Tranfer Funcion: DC gain Magniude Magniude and Phae: Ko log K # "! ± π Phaedeg o if K if K o o > < # Frequencyrad/ec # Frequencyrad/ec 7 eneral Tranfer Funcion: Pole/zero a origin Magniude and Phae: m, m m log π m Magniude/ # # #3 m dec Phaedeg # #4 #6 #8 # #4 # Frequency/rad/ec # # Frequencyrad/ec 8 9
10 eneral Tranfer Funcion: eal ole/zero Magniude and Phae: τ n n log τ n an τ Aymoic behavior: $$ $ << / τ $$ $ n τ >> / τ n log """ << / τ """ n 9 >> / τ 9 eneral Tranfer Funcion: eal ole/zero Magniude 5 #5 # #5 # #5 #3 #35 n 3 n dec #4 # 3 Frequencyrad/ec n, τ / / τ n 3 gn n
11 eneral Tranfer Funcion: eal ole/zero Phae4deg # # #3 #4 #5 #6 #7 #8 #9 n 45 # # 3 Frequency4rad/ec n, τ / / τ n 45 n 9 eneral Tranfer Funcion: Comlex ole/zero Magniude and Phae: Aymoic behavior: &, # $ * ' ζ! $ % n n!" / & # q log- $! -. % n " & ζ / # % / n " n q an $! q & #, $ ζ! * % n " * $$ $ << n $$ $ q >> n n q 4log """ << n """ q 8 >> n
12 Magniude # #4 #6 #8 # eneral Tranfer Funcion: Comlex ole/zero q ζ # # 3 Frequencyrad/ec MAX q 4 dec q,, ζ.5 n n q ζ ζ r n r q ζ. 3 ζ gn q 3 Phaedeg # #4 #6 #8 # # #4 #6 #8 eneral Tranfer Funcion: Comlex ole/zero q 9 # # # Frequencyrad/ec q,, ζ.5 n. / τ q 9 q 8 4
13 Frequency eone: Pole/Zero in he HP Same. The effec on i ooie han he able cae. An unable ole behave like a able zero An unable zero behave like a able ole Examle: Thi frequency reone canno be found exerimenally bu can be comued and ued for conrol deign. 5 Neural Sabiliy U Y - K oo locu condiion: K, 8 A oin of neural abiliy L condiion hold for K, 8 Sabiliy: A K < K > 8 If K lead o inabiliy If K lead o inabiliy 6 3
14 Sabiliy Margin The AIN MAIN M i he facor by which he gain can be raied before inabiliy reul. M < M < M i equal o / K K where 8. UNSTABLE SYSTEM a he frequency The PHASE MAIN PM i he value by which he hae can be raied before inabiliy reul. PM < UNSTABLE SYSTEM PM i he amoun by which he hae of -8 when K K exceed 7 Sabiliy Margin M PM 8 4
15 Frequency eone u Aco Aco y Sable Tranfer Funcion e BODE lo { } Im{ } e NYQUIST lo 9 e Nyqui Diagram { } Im{ } e How are he Bode and Nyqui lo relaed? They are wo way o rereen he ame informaion Im { } e { } 3 5
16 Nyqui Diagram Find he eady ae ouu for v Aco V _ L Comue he eady ae ouu A v SS co an L / L V Comue he -domain ranfer funcion T Volage divider T L Comue he frequency reone & L # T, T an $! L % " - [ ] 3 Phae deg Magniude Bode Diagram Bode Diagram Log-log lo of magt, log-linear lo of argt veru " Magniude Phae deg -/dec / L Frequency rad/ec [ ] rad / ec, πf, [ f ] Hz 3 6
17 7 Nyqui Diagram 33, : T T! " # $ % & L T L T an, 9, : T T L L L L L L T { } { } Im, e L L T L T T!!!!!! L T { } e T { }, Im T Nyqui Diagram 34 / L { } { } e Im v.
18 Nyqui Diagram eneral rocedure for keching Nyqui Diagram: Find Find Find * uch ha e{*}; Im{*} i he inerecion wih he imaginary axi. Find * uch ha Im{*}; e{*} i he inerecion wih he real axi. Connec he oin 35 Examle: Nyqui Diagram : :!!!!!! 3 e Im { } { }, e{ } 36 8
19 Nyqui Diagram Examle: 37 Nyqui Diagram from Bode Diagram dec 6 dec
20 Neural Sabiliy U Y - K oo locu condiion: K, 8 A oin of neural abiliy L condiion hold for K, 8 Sabiliy: A K < K > 8 If K lead o inabiliy If K lead o inabiliy 39 Sabiliy Margin The AIN MAIN M i he facor by which he gain can be raied before inabiliy reul. M < M < M i equal o / K K where 8. UNSTABLE SYSTEM a he frequency The PHASE MAIN PM i he value by which he hae can be raied before inabiliy reul. PM < UNSTABLE SYSTEM PM i he amoun by which he hae of -8 when K K exceed 4
21 Sabiliy Margin M PM 4 Sabiliy Margin /M PM 4
22 Nyqui Sabiliy Crierion Cae : No ole/zero wihin conour Cae : Pole/zero wihin conour Argumen Princile: A conour ma of a comlex funcion will encircle he origin Z-P ime, where Z i he number of zero and P i he number of ole of he funcion inide he conour. 43 Nyqui Sabiliy Crierion Le u conider hi conour and cloed-loo yem U Y K - The cloed-loo ole are he oluion roo of: The evaluaion of H will encircle he origin only if H ha a HP zero or ole K 44
23 Nyqui Sabiliy Crierion Le u aly he argumen rincile o he funcion H K. If he lo of K encircle he origin, he lo of K encircle - on he real axi. 45 By wriing Nyqui Sabiliy Crierion b a Kb K K a a we can conclude ha he ole of K are alo he ole of. Auming no ole of in he HP, an encirclemen of he oin - by K indicae a zero of K in he HP, and hu an unable ole of he cloed-loo yem. A clockwie conour of C encloing a zero of K will reul in K encircling he - oin in he clockwie direcion. A clockwie conour of C encloing a ole of K will reul in K encircling he - oin in he counerclockwie direcion. The ne number of clockwie encirclemen of he oin -, N, equal he number of zero cloed-loo ole in he HP, Z, minu he number of ole oen-loo ole in he HP, P: N Z P 46 3
24 Nyqui Sabiliy Crierion U Y - When i hi ranfer funcion Sable? NYQUIST: The cloed loo i aymoically able if he number of counerclockwie encirclemen N negaive of he oin - by he Nyqui curve of i equal o he number of ole of wih oiive real ar unable ole P. Corollary: If he oen-loo yem i able P, hen he cloed-loo yem i alo able rovided make no encirclemen of he oin - N Nyqui Sabiliy Crierion
25 Secificaion in he Frequency Domain. The croover frequency c, which deermine bandwih BW, rie ime r and eling ime.. The hae margin PM, which deermine he daming coefficien ζ and he overhoo M. 3. The low-frequency gain, which deermine he eady-ae error characeriic. 49 Secificaion in he Frequency Domain The hae and he magniude are NOT indeenden! Bode ain-phae relaionhi: W u o π W u ln M ln dm du u ln / o coh u / du 5 5
26 Secificaion in he Frequency Domain The croover frequency: c BW c 5 Secificaion in he Frequency Domain The Phae Margin: PM v. M 5 6
27 Secificaion in he Frequency Domain The Phae Margin: PM v. ζ PM ζ 53 Frequency eone Phae Lead Comenaor T D, α < αt in α in log MAX MAX MAX &, $ log* % T ', # log* ' αt! " I i a high-a filer and aroximae PD conrol. I i ued whenever ubanial imrovemen in daming i needed. I end o increae he eed of reone of a yem for a fixed low-frequency gain. 54 7
28 Frequency eone Phae Lead Comenaor. Deermine he oen-loo gain K o aify error or bandwidh requiremen: - To mee error requiremen, ick K o aify error conan K, K v, K a o ha e ecificaion i me. - To mee bandwidh requiremen, ick K o ha he oen-loo croover frequency i a facor of wo below he deired cloed-loo bandwidh.. Deermine he needed hae lead α baed on he PM ecificaion. inmax α inmax 3. Pick MAX o be a he croover frequency. 4. Deermine he zero and ole of he comenaor. z/t MAX α / / α T MAX α -/ 5. Draw he comenaed frequency reone and check PM. 6. Ierae on he deign. Add addiional comenaor if needed. 55 Frequency eone Phae Lag Comenaor T D α, α > αt I i a low-a filer and aroximae PI conrol. I i ued o increae he low frequency gain of he yem and imrove eady ae reone for fixed bandwidh. For a fixed low-frequency gain, i will decreae he eed of reone of he yem. 56 8
29 Frequency eone Phae Lag Comenaor. Deermine he oen-loo gain K ha will mee he PM requiremen wihou comenaion.. Draw he Bode lo of he uncomenaed yem wih croover frequency from e and evaluae he lowfrequency gain. 3. Deermine α o mee he low frequency gain error requiremen. 4. Chooe he corner frequency /T he zero of he comenaor o be one decade below he new croover frequency c. 5. The oher corner frequency he ole of he comenaor i hen / α T. 6. Ierae on he deign 57 9
Frequency Response. We now know how to analyze and design ccts via s- domain methods which yield dynamical information
Frequency Repone We now now how o analyze and deign cc via - domain mehod which yield dynamical informaion Zero-ae repone Zero-inpu repone Naural repone Forced repone The repone are decribed by he exponenial
More informationu(t) Figure 1. Open loop control system
Open loop conrol v cloed loop feedbac conrol The nex wo figure preen he rucure of open loop and feedbac conrol yem Figure how an open loop conrol yem whoe funcion i o caue he oupu y o follow he reference
More information6.302 Feedback Systems Recitation : Phase-locked Loops Prof. Joel L. Dawson
6.32 Feedback Syem Phae-locked loop are a foundaional building block for analog circui deign, paricularly for communicaion circui. They provide a good example yem for hi cla becaue hey are an excellen
More information6 December 2013 H. T. Hoang - www4.hcmut.edu.vn/~hthoang/ 1
Lecure Noe Fundamenal of Conrol Syem Inrucor: Aoc. Prof. Dr. Huynh Thai Hoang Deparmen of Auomaic Conrol Faculy of Elecrical & Elecronic Engineering Ho Chi Minh Ciy Univeriy of Technology Email: hhoang@hcmu.edu.vn
More informationSample Final Exam (finals03) Covering Chapters 1-9 of Fundamentals of Signals & Systems
Sample Final Exam Covering Chaper 9 (final04) Sample Final Exam (final03) Covering Chaper 9 of Fundamenal of Signal & Syem Problem (0 mar) Conider he caual opamp circui iniially a re depiced below. I LI
More informationAn Improved Anti-windup Control Using a PI Controller
05 Third nernaional Conference on Arificial nelligence, Modelling and Simulaion An mroved Ani-windu Conrol Uing a P Conroller Kyohei Saai Graduae School of Science and Technology Meiji univeriy Kanagawa,
More informationTo become more mathematically correct, Circuit equations are Algebraic Differential equations. from KVL, KCL from the constitutive relationship
Laplace Tranform (Lin & DeCarlo: Ch 3) ENSC30 Elecric Circui II The Laplace ranform i an inegral ranformaion. I ranform: f ( ) F( ) ime variable complex variable From Euler > Lagrange > Laplace. Hence,
More information, the. L and the L. x x. max. i n. It is easy to show that these two norms satisfy the following relation: x x n x = (17.3) max
ecure 8 7. Sabiliy Analyi For an n dimenional vecor R n, he and he vecor norm are defined a: = T = i n i (7.) I i eay o how ha hee wo norm aify he following relaion: n (7.) If a vecor i ime-dependen, hen
More informationCHAPTER 7: SECOND-ORDER CIRCUITS
EEE5: CI RCUI T THEORY CHAPTER 7: SECOND-ORDER CIRCUITS 7. Inroducion Thi chaper conider circui wih wo orage elemen. Known a econd-order circui becaue heir repone are decribed by differenial equaion ha
More informationEE Control Systems LECTURE 2
Copyrigh F.L. Lewi 999 All righ reerved EE 434 - Conrol Syem LECTURE REVIEW OF LAPLACE TRANSFORM LAPLACE TRANSFORM The Laplace ranform i very ueful in analyi and deign for yem ha are linear and ime-invarian
More informationChapter 7: Inverse-Response Systems
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
More informationDesign of Controller for Robot Position Control
eign of Conroller for Robo oiion Conrol Two imporan goal of conrol: 1. Reference inpu racking: The oupu mu follow he reference inpu rajecory a quickly a poible. Se-poin racking: Tracking when he reference
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signal & Syem Prof. Mark Fowler Noe Se #27 C-T Syem: Laplace Tranform Power Tool for yem analyi Reading Aignmen: Secion 6.1 6.3 of Kamen and Heck 1/18 Coure Flow Diagram The arrow here how concepual
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 30 Signal & Syem Prof. ark Fowler oe Se #34 C-T Tranfer Funcion and Frequency Repone /4 Finding he Tranfer Funcion from Differenial Eq. Recall: we found a DT yem Tranfer Funcion Hz y aking he ZT of
More informationAN ANALYTICAL METHOD OF SOLUTION FOR SYSTEMS OF BOOLEAN EQUATIONS
CHAPTER 5 AN ANALYTICAL METHOD OF SOLUTION FOR SYSTEMS OF BOOLEAN EQUATIONS 51 APPLICATIONS OF DE MORGAN S LAWS A we have een in Secion 44 of Chaer 4, any Boolean Equaion of ye (1), (2) or (3) could be
More informationCONTROL SYSTEMS. Chapter 10 : State Space Response
CONTROL SYSTEMS Chaper : Sae Space Repone GATE Objecive & Numerical Type Soluion Queion 5 [GATE EE 99 IIT-Bombay : Mark] Conider a econd order yem whoe ae pace repreenaion i of he form A Bu. If () (),
More informationCONTROL SYSTEMS. Chapter 3 Mathematical Modelling of Physical Systems-Laplace Transforms. Prof.Dr. Fatih Mehmet Botsalı
CONTROL SYSTEMS Chaper Mahemaical Modelling of Phyical Syem-Laplace Tranform Prof.Dr. Faih Mehme Boalı Definiion Tranform -- a mahemaical converion from one way of hinking o anoher o make a problem eaier
More informationEE202 Circuit Theory II
EE202 Circui Theory II 2017-2018, Spring Dr. Yılmaz KALKAN I. Inroducion & eview of Fir Order Circui (Chaper 7 of Nilon - 3 Hr. Inroducion, C and L Circui, Naural and Sep epone of Serie and Parallel L/C
More information13.1 Circuit Elements in the s Domain Circuit Analysis in the s Domain The Transfer Function and Natural Response 13.
Chaper 3 The Laplace Tranform in Circui Analyi 3. Circui Elemen in he Domain 3.-3 Circui Analyi in he Domain 3.4-5 The Tranfer Funcion and Naural Repone 3.6 The Tranfer Funcion and he Convoluion Inegral
More informationLet. x y. denote a bivariate time series with zero mean.
Linear Filer Le x y : T denoe a bivariae ime erie wih zero mean. Suppoe ha he ime erie {y : T} i conruced a follow: y a x The ime erie {y : T} i aid o be conruced from {x : T} by mean of a Linear Filer.
More information18.03SC Unit 3 Practice Exam and Solutions
Sudy Guide on Sep, Dela, Convoluion, Laplace You can hink of he ep funcion u() a any nice mooh funcion which i for < a and for > a, where a i a poiive number which i much maller han any ime cale we care
More informationChapter 9 - The Laplace Transform
Chaper 9 - The Laplace Tranform Selece Soluion. Skech he pole-zero plo an region of convergence (if i exi) for hee ignal. ω [] () 8 (a) x e u = 8 ROC σ ( ) 3 (b) x e co π u ω [] ( ) () (c) x e u e u ROC
More informationd 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3
and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or
More information376 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD. D(s) = we get the compensated system with :
376 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Therefore by applying the lead compenator with ome gain adjutment : D() =.12 4.5 +1 9 +1 we get the compenated ytem with : PM =65, ω c = 22 rad/ec, o
More informationChapter 6. Laplace Transforms
Chaper 6. Laplace Tranform Kreyzig by YHLee;45; 6- An ODE i reduced o an algebraic problem by operaional calculu. The equaion i olved by algebraic manipulaion. The reul i ranformed back for he oluion of
More information* l/d > 5 and H/D = 1 Mass term of Morison dominates
1 3.6 LOADS Source and conequence of non-lineariie Sinuoidal wave and ylinder rucure: η =H/ in z λ H d x * l/d > 5 and H/D = 1 Ma erm of Morion dominae * linear wave eory: D fz;x= =.5 π c m D H/ gk co
More informationMath 333 Problem Set #2 Solution 14 February 2003
Mah 333 Problem Se #2 Soluion 14 February 2003 A1. Solve he iniial value problem dy dx = x2 + e 3x ; 2y 4 y(0) = 1. Soluion: This is separable; we wrie 2y 4 dy = x 2 + e x dx and inegrae o ge The iniial
More informationChapter 6. Laplace Transforms
6- Chaper 6. Laplace Tranform 6.4 Shor Impule. Dirac Dela Funcion. Parial Fracion 6.5 Convoluion. Inegral Equaion 6.6 Differeniaion and Inegraion of Tranform 6.7 Syem of ODE 6.4 Shor Impule. Dirac Dela
More informationLecture 4. Chapter 11 Nise. Controller Design via Frequency Response. G. Hovland 2004
METR4200 Advanced Control Lecture 4 Chapter Nie Controller Deign via Frequency Repone G. Hovland 2004 Deign Goal Tranient repone via imple gain adjutment Cacade compenator to improve teady-tate error Cacade
More informationELECTRONIC FILTERS. Celso José Faria de Araújo, M.Sc.
ELECTRONIC FILTERS Celo Joé Faria de Araújo, M.Sc. A Ideal Electronic Filter allow ditortionle tranmiion of a certain band of frequencie and ure all the remaining frequencie of the ectrum of the inut ignal.
More informationModule 2 F c i k c s la l w a s o s f dif di fusi s o i n
Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms
More informationLecture 5 Buckling Buckling of a structure means failure due to excessive displacements (loss of structural stiffness), and/or
AOE 204 Inroducion o Aeropace Engineering Lecure 5 Buckling Buckling of a rucure mean failure due o exceive diplacemen (lo of rucural iffne), and/or lo of abiliy of an equilibrium configuraion of he rucure
More informationSingle Phase Line Frequency Uncontrolled Rectifiers
Single Phae Line Frequency Unconrolle Recifier Kevin Gaughan 24-Nov-03 Single Phae Unconrolle Recifier 1 Topic Baic operaion an Waveform (nucive Loa) Power Facor Calculaion Supply curren Harmonic an Th
More informationCHAPTER. Forced Equations and Systems { } ( ) ( ) 8.1 The Laplace Transform and Its Inverse. Transforms from the Definition.
CHAPTER 8 Forced Equaion and Syem 8 The aplace Tranform and I Invere Tranform from he Definiion 5 5 = b b {} 5 = 5e d = lim5 e = ( ) b {} = e d = lim e + e d b = (inegraion by par) = = = = b b ( ) ( )
More informationFigure 1 Siemens PSSE Web Site
Stability Analyi of Dynamic Sytem. In the lat few lecture we have een how mall ignal Lalace domain model may be contructed of the dynamic erformance of ower ytem. The tability of uch ytem i a matter of
More informationBuckling of a structure means failure due to excessive displacements (loss of structural stiffness), and/or
Buckling Buckling of a rucure mean failure due o exceive diplacemen (lo of rucural iffne), and/or lo of abiliy of an equilibrium configuraion of he rucure The rule of humb i ha buckling i conidered a mode
More information6.8 Laplace Transform: General Formulas
48 HAP. 6 Laplace Tranform 6.8 Laplace Tranform: General Formula Formula Name, ommen Sec. F() l{ f ()} e f () d f () l {F()} Definiion of Tranform Invere Tranform 6. l{af () bg()} al{f ()} bl{g()} Lineariy
More informationControl Systems. Root locus.
Control Sytem Root locu chibum@eoultech.ac.kr Outline Concet of Root Locu Contructing root locu Control Sytem Root Locu Stability and tranient reone i cloely related with the location of ole in -lane How
More information( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is
UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires
More information1 birth rate γ (number of births per time interval) 2 death rate δ proportional to size of population
Scienific Comuing I Module : Poulaion Modelling Coninuous Models Michael Bader Par I ODE Models Lehrsuhl Informaik V Winer 7/ Discree vs. Coniuous Models d d = F,,...) ) =? discree model: coninuous model:
More informationControl Systems. Root locus.
Control Sytem Root locu chibum@eoultech.ac.kr Outline Concet of Root Locu Contructing root locu Control Sytem Root Locu Stability and tranient reone i cloely related with the location of ole in -lane How
More informationA Theoretical Model of a Voltage Controlled Oscillator
A Theoreical Model of a Volage Conrolled Ocillaor Yenming Chen Advior: Dr. Rober Scholz Communicaion Science Iniue Univeriy of Souhern California UWB Workhop, April 11-1, 6 Inroducion Moivaion The volage
More informationA Design of an Improved Anti-Windup Control Using a PI Controller Based on a Pole Placement Method
KYOHE SAKA e al: A DESGN OF AN MPROVED ANT-WNDUP CONTROL USNG A P CONTROLLER A Deign of an mrove Ani-Winu Conrol Uing a P Conroller Bae on a Pole Placemen Meho Kyohei Saai Grauae School of Science an Technology
More informationHomework 12 Solution - AME30315, Spring 2013
Homework 2 Solution - AME335, Spring 23 Problem :[2 pt] The Aerotech AGS 5 i a linear motor driven XY poitioning ytem (ee attached product heet). A friend of mine, through careful experimentation, identified
More informationIntroduction to Congestion Games
Algorihmic Game Theory, Summer 2017 Inroducion o Congeion Game Lecure 1 (5 page) Inrucor: Thoma Keelheim In hi lecure, we ge o know congeion game, which will be our running example for many concep in game
More informationRoot Locus Diagram. Root loci: The portion of root locus when k assume positive values: that is 0
Objective Root Locu Diagram Upon completion of thi chapter you will be able to: Plot the Root Locu for a given Tranfer Function by varying gain of the ytem, Analye the tability of the ytem from the root
More informationCHAPTER HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG. 7.1 SECOND-ORDER SYSTEM Transfer Function
CHAPTER 7 HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG 7. SECOND-ORDER SYSTEM Tranfer Funcion Thi ecion inroduce a baic yem called a econd-order yem or a quadraic lag. Second-order yem are
More information5.1 - Logarithms and Their Properties
Chaper 5 Logarihmic Funcions 5.1 - Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We
More informationDiscussion Session 2 Constant Acceleration/Relative Motion Week 03
PHYS 100 Dicuion Seion Conan Acceleraion/Relaive Moion Week 03 The Plan Today you will work wih your group explore he idea of reference frame (i.e. relaive moion) and moion wih conan acceleraion. You ll
More informationModule 4: Time Response of discrete time systems Lecture Note 2
Module 4: Time Response of discree ime sysems Lecure Noe 2 1 Prooype second order sysem The sudy of a second order sysem is imporan because many higher order sysem can be approimaed by a second order model
More informationAdditional Methods for Solving DSGE Models
Addiional Mehod for Solving DSGE Model Karel Meren, Cornell Univeriy Reference King, R. G., Ploer, C. I. & Rebelo, S. T. (1988), Producion, growh and buine cycle: I. he baic neoclaical model, Journal of
More information20. Applications of the Genetic-Drift Model
0. Applicaions of he Geneic-Drif Model 1) Deermining he probabiliy of forming any paricular combinaion of genoypes in he nex generaion: Example: If he parenal allele frequencies are p 0 = 0.35 and q 0
More informationPhysics 240: Worksheet 16 Name
Phyic 4: Workhee 16 Nae Non-unifor circular oion Each of hee proble involve non-unifor circular oion wih a conan α. (1) Obain each of he equaion of oion for non-unifor circular oion under a conan acceleraion,
More informationExponential Sawtooth
ECPE 36 HOMEWORK 3: PROPERTIES OF THE FOURIER TRANSFORM SOLUTION. Exponenial Sawooh: The eaie way o do hi problem i o look a he Fourier ranform of a ingle exponenial funcion, () = exp( )u(). From he able
More informations-domain Circuit Analysis
Domain ircui Analyi Operae direcly in he domain wih capacior, inducor and reior Key feaure lineariy i preerved c decribed by ODE and heir I Order equal number of plu number of Elemenbyelemen and ource
More informationAli Karimpour Associate Professor Ferdowsi University of Mashhad
LINEAR CONTROL SYSTEMS Ali Karimour Aoiate Profeor Ferdowi Univerity of Mahhad Leture 0 Leture 0 Frequeny domain hart Toi to be overed inlude: Relative tability meaure for minimum hae ytem. ain margin.
More informationLaplace Transform. Inverse Laplace Transform. e st f(t)dt. (2)
Laplace Tranform Maoud Malek The Laplace ranform i an inegral ranform named in honor of mahemaician and aronomer Pierre-Simon Laplace, who ued he ranform in hi work on probabiliy heory. I i a powerful
More informationLAPLACE TRANSFORM AND TRANSFER FUNCTION
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions
More informationCHAPTER 2 Signals And Spectra
CHAPER Signals And Specra Properies of Signals and Noise In communicaion sysems he received waveform is usually caegorized ino he desired par conaining he informaion, and he undesired par. he desired par
More informationME 375 FINAL EXAM Wednesday, May 6, 2009
ME 375 FINAL EXAM Wedneday, May 6, 9 Diviion Meckl :3 / Adam :3 (circle one) Name_ Intruction () Thi i a cloed book examination, but you are allowed three ingle-ided 8.5 crib heet. A calculator i NOT allowed.
More informationVoltage/current relationship Stored Energy. RL / RC circuits Steady State / Transient response Natural / Step response
Review Capaciors/Inducors Volage/curren relaionship Sored Energy s Order Circuis RL / RC circuis Seady Sae / Transien response Naural / Sep response EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu Lecure
More information6.003 Homework #9 Solutions
6.003 Homework #9 Soluions Problems. Fourier varieies a. Deermine he Fourier series coefficiens of he following signal, which is periodic in 0. x () 0 3 0 a 0 5 a k a k 0 πk j3 e 0 e j πk 0 jπk πk e 0
More informationA study on the influence of design parameters on the power output characteristic of piezoelectric vibration energy harvester
A udy on he influence of deign arameer on he ower ouu characeriic of iezoelecric vibraion energy harveer N. Aboulfooh, J. Twiefel Iniue of Dynamic and Vibraion eearch Leibniz Univeriä Hannover Aelr. 67
More information- The whole joint distribution is independent of the date at which it is measured and depends only on the lag.
Saionary Processes Sricly saionary - The whole join disribuion is indeenden of he dae a which i is measured and deends only on he lag. - E y ) is a finie consan. ( - V y ) is a finie consan. ( ( y, y s
More informationChapter 7 Response of First-order RL and RC Circuits
Chaper 7 Response of Firs-order RL and RC Circuis 7.- The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial
More informationAnalysis of Boundedness for Unknown Functions by a Delay Integral Inequality on Time Scales
Inernaional Conference on Image, Viion and Comuing (ICIVC ) IPCSIT vol. 5 () () IACSIT Pre, Singaore DOI:.7763/IPCSIT..V5.45 Anali of Boundedne for Unknown Funcion b a Dela Inegral Ineuali on Time Scale
More informationIntroduction to SLE Lecture Notes
Inroducion o SLE Lecure Noe May 13, 16 - The goal of hi ecion i o find a ufficien condiion of λ for he hull K o be generaed by a imple cure. I urn ou if λ 1 < 4 hen K i generaed by a imple curve. We will
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Civil and Environmental Engineering
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Deparmen of Civil and Environmenal Engineering 1.731 Waer Reource Syem Lecure 17 River Bain Planning Screening Model Nov. 7 2006 River Bain Planning River bain planning
More informationV L. DT s D T s t. Figure 1: Buck-boost converter: inductor current i(t) in the continuous conduction mode.
ECE 445 Analysis and Design of Power Elecronic Circuis Problem Se 7 Soluions Problem PS7.1 Erickson, Problem 5.1 Soluion (a) Firs, recall he operaion of he buck-boos converer in he coninuous conducion
More information6.003 Homework #9 Solutions
6.00 Homework #9 Soluions Problems. Fourier varieies a. Deermine he Fourier series coefficiens of he following signal, which is periodic in 0. x () 0 0 a 0 5 a k sin πk 5 sin πk 5 πk for k 0 a k 0 πk j
More informationFinal Spring 2007
.615 Final Spring 7 Overview The purpose of he final exam is o calculae he MHD β limi in a high-bea oroidal okamak agains he dangerous n = 1 exernal ballooning-kink mode. Effecively, his corresponds o
More informationECE 3510 Root Locus Design Examples. PI To eliminate steady-state error (for constant inputs) & perfect rejection of constant disturbances
ECE 350 Root Locu Deign Example Recall the imple crude ervo from lab G( ) 0 6.64 53.78 σ = = 3 23.473 PI To eliminate teady-tate error (for contant input) & perfect reection of contant diturbance Note:
More informationCHAPTER 12 DIRECT CURRENT CIRCUITS
CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As
More informationAlgorithmic Discrete Mathematics 6. Exercise Sheet
Algorihmic Dicree Mahemaic. Exercie Shee Deparmen of Mahemaic SS 0 PD Dr. Ulf Lorenz 7. and 8. Juni 0 Dipl.-Mah. David Meffer Verion of June, 0 Groupwork Exercie G (Heap-Sor) Ue Heap-Sor wih a min-heap
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signals & Sysems Prof. Mar Fowler Noe Se #1 C-T Signals: Circuis wih Periodic Sources 1/1 Solving Circuis wih Periodic Sources FS maes i easy o find he response of an RLC circui o a periodic source!
More informationProblem Set #3: AK models
Universiy of Warwick EC9A2 Advanced Macroeconomic Analysis Problem Se #3: AK models Jorge F. Chavez December 3, 2012 Problem 1 Consider a compeiive economy, in which he level of echnology, which is exernal
More informationRoot Locus Contents. Root locus, sketching algorithm. Root locus, examples. Root locus, proofs. Root locus, control examples
Root Locu Content Root locu, ketching algorithm Root locu, example Root locu, proof Root locu, control example Root locu, influence of zero and pole Root locu, lead lag controller deign 9 Spring ME45 -
More informationNetwork Flow. Data Structures and Algorithms Andrei Bulatov
Nework Flow Daa Srucure and Algorihm Andrei Bulao Algorihm Nework Flow 24-2 Flow Nework Think of a graph a yem of pipe We ue hi yem o pump waer from he ource o ink Eery pipe/edge ha limied capaciy Flow
More informationCS4445/9544 Analysis of Algorithms II Solution for Assignment 1
Conider he following flow nework CS444/944 Analyi of Algorihm II Soluion for Aignmen (0 mark) In he following nework a minimum cu ha capaciy 0 Eiher prove ha hi aemen i rue, or how ha i i fale Uing he
More informationY 0.4Y 0.45Y Y to a proper ARMA specification.
HG Jan 04 ECON 50 Exercises II - 0 Feb 04 (wih answers Exercise. Read secion 8 in lecure noes 3 (LN3 on he common facor problem in ARMA-processes. Consider he following process Y 0.4Y 0.45Y 0.5 ( where
More information6.003 Homework #8 Solutions
6.003 Homework #8 Soluions Problems. Fourier Series Deermine he Fourier series coefficiens a k for x () shown below. x ()= x ( + 0) 0 a 0 = 0 a k = e /0 sin(/0) for k 0 a k = π x()e k d = 0 0 π e 0 k d
More informationTP B.2 Rolling resistance, spin resistance, and "ball turn"
echnical proof TP B. olling reiance, pin reiance, and "ball urn" upporing: The Illuraed Principle of Pool and Billiard hp://billiard.coloae.edu by Daid G. Alciaore, PhD, PE ("Dr. Dae") echnical proof originally
More informationDigital Control System
Digital Control Sytem - A D D A Micro ADC DAC Proceor Correction Element Proce Clock Meaurement A: Analog D: Digital Continuou Controller and Digital Control Rt - c Plant yt Continuou Controller Digital
More informationGATE. Topicwise Solved Paper. Year By RK Kanodia & Ashish Murolia. For more GATE Resources, Mock Test and Study material
GATE ELECTRONICS & COMMUNICATION Toicwie Solved Paer Year 996 By RK Kanodia & Ahih Murolia F me GATE Reource, Mock Te and Sudy maerial Join he Communiy h://www.facebook.com/gaeec GATE Elecronic and Communicaion
More informationSwitching Characteristics of Power Devices
Swiching Characeriic of Power Device Device uilizaion can be grealy improved by underanding he device wiching charcaeriic. he main performance wiching characeriic of power device: he ave operaing area
More informationLinear Algebra Primer
Linear Algebra rimer And a video dicuion of linear algebra from EE263 i here (lecure 3 and 4): hp://ee.anford.edu/coure/ee263 lide from Sanford CS3 Ouline Vecor and marice Baic Mari Operaion Deerminan,
More information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More informationChapter 8 The Complete Response of RL and RC Circuits
Chaper 8 The Complee Response of RL and RC Circuis Seoul Naional Universiy Deparmen of Elecrical and Compuer Engineering Wha is Firs Order Circuis? Circuis ha conain only one inducor or only one capacior
More informationSignal and System (Chapter 3. Continuous-Time Systems)
Signal and Sysem (Chaper 3. Coninuous-Time Sysems) Prof. Kwang-Chun Ho kwangho@hansung.ac.kr Tel: 0-760-453 Fax:0-760-4435 1 Dep. Elecronics and Informaion Eng. 1 Nodes, Branches, Loops A nework wih b
More informationUT Austin, ECE Department VLSI Design 5. CMOS Gate Characteristics
La moule: CMOS Tranior heory Thi moule: DC epone Logic Level an Noie Margin Tranien epone Delay Eimaion Tranior ehavior 1) If he wih of a ranior increae, he curren will ) If he lengh of a ranior increae,
More informationFORECASTS GENERATING FOR ARCH-GARCH PROCESSES USING THE MATLAB PROCEDURES
FORECASS GENERAING FOR ARCH-GARCH PROCESSES USING HE MALAB PROCEDURES Dušan Marček, Insiue of Comuer Science, Faculy of Philosohy and Science, he Silesian Universiy Oava he Faculy of Managemen Science
More informationComputing with diode model
ECE 570 Session 5 C 752E Comuer Aided Engineering for negraed Circuis Comuing wih diode model Objecie: nroduce conces in numerical circui analsis Ouline: 1. Model of an examle circui wih a diode 2. Ouline
More informationCHAPTER 3 SIGNALS & SYSTEMS. z -transform in the z -plane will be (A) 1 (B) 1 (D) (C) . The unilateral Laplace transform of tf() (A) s (B) + + (D) (C)
CHAPER SIGNALS & SYSEMS YEAR ONE MARK n n MCQ. If xn [ ] (/) (/) un [ ], hen he region of convergence (ROC) of i z ranform in he z plane will be (A) < z < (B) < z < (C) < z < (D) < z MCQ. he unilaeral
More informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More informationMacroeconomics 1. Ali Shourideh. Final Exam
4780 - Macroeconomic 1 Ali Shourideh Final Exam Problem 1. A Model of On-he-Job Search Conider he following verion of he McCall earch model ha allow for on-he-job-earch. In paricular, uppoe ha ime i coninuou
More informationLecture 12. Aperture and Noise. Jaeha Kim Mixed-Signal IC and System Group (MICS) Seoul National University
Lecure. Aperure and Noie Analyi of locked omparaor Jaeha Kim Mixed-Signal I and Syem Group MIS Seoul Naional Univeriy jaeha@ieee.org locked omparaor a.k.a. regeneraive amplifier, ene-amplifier, flip-flop,
More informationME 391 Mechanical Engineering Analysis
Fall 04 ME 39 Mechanical Engineering Analsis Eam # Soluions Direcions: Open noes (including course web posings). No books, compuers, or phones. An calculaor is fair game. Problem Deermine he posiion of
More information6.003 Homework #13 Solutions
6.003 Homework #3 Soluions Problems. Transformaion Consider he following ransformaion from x() o y(): x() w () w () w 3 () + y() p() cos() where p() = δ( k). Deermine an expression for y() when x() = sin(/)/().
More informationControl Systems -- Final Exam (Spring 2006)
6.5 Conrol Syem -- Final Eam (Spring 6 There are 5 prolem (inluding onu prolem oal poin. (p Given wo marie: (6 Compue A A e e. (6 For he differenial equaion [ ] ; y u A wih ( u( wha i y( for >? (8 For
More information1 Motivation and Basic Definitions
CSCE : Deign and Analyi of Algorihm Noe on Max Flow Fall 20 (Baed on he preenaion in Chaper 26 of Inroducion o Algorihm, 3rd Ed. by Cormen, Leieron, Rive and Sein.) Moivaion and Baic Definiion Conider
More information