6 December 2013 H. T. Hoang - www4.hcmut.edu.vn/~hthoang/ 1
|
|
|
- Clemence Cox
- 6 years ago
- Views:
Transcription
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 Homepage: www4.hcmu.edu.vn/~hhoang/ 6 December 213 H. T. Hoang - www4.hcmu.edu.vn/~hhoang/ 1
2 Chaper 5 ANALYSIS OF CONTROL SYSTEM PERFORMANCE 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 2
3 Conen Performance crieria Seady ae error Tranien repone The opimal performance index Relaionhip beween frequency domain performance and ime domain performance. 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 3
4 Performance crieria 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 4
5 Performance crieria: Seady ae error R +_ E G Y r y fb e Y fb H e e Error: i he difference beween he e-poin inpu and he feedback ignal. e r y E R Y fb Seady-ae error: i he error when ime approaching infiniy. fb e lim e e lim E 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 5
6 Performance crieria Percen of Overhoo POT Overhoo: refer o an oupu exceeding i eady-aeae value. y overhoo y max y y max y y y y No overhoo Percenage of Overhoo POT i an index o quanify he overhoo of a yem, POT i calculaed a: POT ymax y 1% y y 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 6
7 Performance crieria Seling ime and rie ime Seling ime : i he ime required for he repone of a yem o reach and ay wihin a range abou he eadyae value of ize pecified by abolue percenage of he eady-ae value uually 2% or 5% Rie ime r : i he ime required for he repone of a yem o rie from 1% o 9% of i eady-aeae value. y y 1+y y y 1y.9y.1y r 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 7
8 Seady-ae error 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 8
9 Seady-ae error R E Y +_ G Y fb H Error expreion: E 1 R G H Seady-ae error: e R lim E lim 1 G H Remark: Seady-ae error no only depend on he rucure and parameer of he yem bu alo depend on he inpu ignal. 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 9
10 Seady-ae error o ep inpu Sep inpu: R 1 / Seady-ae error: e 1 1 K p wih K limg H y fb p poiion conan y fb 1 1 GH doe no have any deal inegral facor GH ha a lea 1 ideal inegral facor 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 1
11 Seady-ae error o ramp inpu Ramp inpu: 2 R 1/ e 1 wih Kv lim G H K v velociy conan y fb y fb y fb r r r e e e = GH doe no have deal inegral facor GH ha 1 ideal inegral facor GH ha a lea 2 ideal inegral facor 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 11
12 Seady-ae error o parabolic inpu Parabolic inpu: R 1/ 1 e 2 wih Ka lim G H K a 3 acceleraion conan y fb y fb y fb r e r e r e = GH ha le han 2 ideal inegral facor GH ha 2 ideal inegral facor GH ha more han 2 ideal inegral facor 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 12
13 Tranien repone 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 13
14 Fir-order yem R K T 1 Y K Tranfer funcion: G T 1 Fir order yem ha 1 real pole: Tranien repone: Y p R G /T y K1 e 1 T 1 K. T December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 14
15 Fir-order yem con Im y 1/T Re 1+.K K 1.K.63K T Pole zero plo of a fir order yem Tranien repone of he fir order y K 1 e /T 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 15
16 Fir-order yem Remark Fir order yem ha only one real pole a 1/T, i ranien repone doen have overhoo. Time conan T: i he ime required for he ep repone of he yem o reach 63% i eady-ae value. The furher he pole 1/T of he yem i from he imaginary axi, he maller he ime conan and he faer he ime repone of he yem. Seling ime of he fir order yem i: 1 T ln where =.22 2% crierion or =.55 5% crierion 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 16
17 Fir-order yem The relaionhip beween he pole and he ime repone The furher he pole of he yem i from he imaginary axi, he maller he ime conan and he faer he ime repone of he yem. Im y Re K Pole zero plo Tranien repone of a fir order yem of he fir order 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 17
18 Second-order order ocillaing yem R T 2 2 K 2T 1 Y The ranfer funcion of he econd-order ocillaing yem: G K 2T 1 K 2n 2 n T The yem ha wo complex conjugae pole: p 2 1 Kn Tranien repone: Y R G e 2 n 2 1,2 n jn 1 n 2 y K 1 in n n, 1 T n n co 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 18
19 Second-order order ocillaing yem con co = n n Im j n 1 2 Re y 1+.K K 1.K j n 1 2 Pole zero plo of a econd order ocillaing yem Tranien repone of a econd order ocillaing yem 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 19
20 Second-order order ocillaing yem Remark A econd order ocillaion yem ha wo conjugaed complex pole, i ranien repone i a ocillaion ignal. If =, ranien repone i a able ocillaion ignal = a he frequency n n i called naural ocillaion frequency. If <<1, ranien repone i a decaying ocillaion ignal i called damping conan, =.6 he larger he value, he cloer he pole are o he real axi he faer he repone decay. =.2 =.4 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 2
21 Second-order order ocillaing yem Overhoo Tranien repone of he econd order ocillaing yem ha overhoo. exp.1% The percenage of overhoo: POT 2 1 POT % The relaionhip beween POT and The larger he value, he cloer he pole are o he real axi he maller he POT. The maller he value, he cloer he pole are o he imaginary axi he larger he POT 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 21
22 Second-order order ocillaing yem Seling ime Seling ime: 5% crierion: 3 n 2% crierion: 4 n 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 22
23 Second-order order ocillaing yem Relaionhip beween pole locaion and ranien repone The 2 nd order yem ha have he pole locaed in he ame ray aring from he origin have he ame damping conan, hen he percenage of overhoo are he ame.the furher he pole from he origin, he horer he eling ime. Im y co = Re K Pole zero plo of a econd Tranien repone of a econd order ocillaing yem order ocillaing yem 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 23
24 Second-order order ocillaing yem Relaionhip beween pole locaion and ranien repone con The 2 nd order yem ha have he pole locaed in he ame diance from he origin have he ame naural ocillaion frequency. The cloer he pole o he imaginary i axi, he maller he damping conan, hen he higher he POT. Im y n Re K Pole zero plo of a econd order ocillaing yem Tranien repone of a econd order ocillaing yem 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 24
25 Second-order order ocillaing yem Relaionhip beween pole locaion and ranien repone con The 2nd order yem ha have he pole locaed in he ame diance from he imaginary axi have he ame n, hen he eling ime are he ame. The furher he pole from he real axi, he maller he damping conan, hen he higher he POT Im y n Re K Pole zero plo of a econd order ocillaing yem Tranien repone of a econd order ocillaing yem 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 25
26 Tranien repone of high order yem High-order yem are he yem ha have more han 2 pole If a high order yem have a pair of pole locaed cloer o he imaginary axi han he oher hen he high order yem can be approximaed o a econd order yem. The pair of pole neare o he imaginary axi are called he dominan pole. Im Re High order yem have more han 2 pole y Repone of highh order yem Repone of econd order yem wih he dominan pole A high order yem can be approximaed by a 2nd order yem 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 26
27 Performance indice 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 27
28 Inegral performance indice IAE crierion Inegral of he Abolue Magniude of he Error J IAE e d ISE crierion Inegral of he Square of he Error J ISE 2 ITAE crierion Inegral of Time muliplied by he Abolue Value of he Error J ITAE e d e d 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 28
29 Opimal yem A conrol yem i opimal when he eleced performance index i minimized Second order yem: J IAE min when. 77 J ISE min when. 5 J min when. 77 ITAE y =.3 =.5 =.77 =.9 Tranien repone of econd order yem 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 29
30 ITAE opimal conrol ITAE i uually ued in deign of conrol yem An n-order yem i opimal according o ITAE crierion if he denominaor of i ranfer funcion ha he form: 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 3
31 ITAE opimal conrol con Opimal repone according o ITAE crierion y 1 order yem 2 nd order yem 3 rd order yem 4 h order yem 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 31
32 Relaionhip beween frequency domain performance & ime domain performance 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 32
33 Relaionhip beween frequency Relaionhip beween frequency repone & repone & eady ae error eady ae error R Y R G + Y lim lim j H j G H G K p lim lim j H j G j H G K v lim lim 2 2 j H j G j H G K a 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 33
34 Relaionhip beween frequency repone & eady ae error R Y + G Seady ae error of he cloed-loop yem depend on he magniude repone of he open-loop yem a low frequencie bu no a high frequencie. The higher he magniude repone of he open-loop yem a low frequencie, he maller he eady-ae error of he cloed-loop yem. In paricular, if he magniude repone of he open-loop yem i infiniy a frequency approaching zero, hen he eady-ae error of he cloed-loop yem o ep inpu i zero. 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 34
35 Relaionhip beween frequency repone & ranien repone R Y + G In he frequency range < c, becaue G j G cl G j G j j 1 1 G j G j 1 hen: In he frequency range > c, becaue G j 1 hen: G j G j G cl j G j 1 G j 1 Bandwidh of he cloed-loop loop yem i approximae he gain croover frequency of he open-loop yem. 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 35
36 Relaionhip beween frequency repone & ranien repone Bode plo of a open-loop yem Bode plo of he correponding cloed-loop yem 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 36
37 Relaionhip beween frequency repone & ranien repone R Y + G The higher he gain croover frequency of open-loop yem, he wider he bandwidh of cloed-loop yem he faer he repone of cloe-loop yem, he horer he eling ime. 4 qd 4 c The higher he phae margin of he open-loop yem, he maller he POT of cloed-loop yem. In mo of he cae, if he phae margin of he open-loop p yem i larger han 6 hen he POT of he cloed-loop yem i maller han 1%. 6 December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 37 c
38 Ex: relaionhip beween gain croover frequency & eling ime R Y + G G December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 38
39 Ex: relaionhip beween gain croover frequency and eling ime R Y + G G December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 39
40 Example of relaionhip beween phae margin and POT R Y + G G December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 4
41 Ex of relaionhip beween phae margin and POT con R Y + G G December 213 H. T. Hoàng - www4.hcmu.edu.vn/~hhoang/ 41
u(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
Sample 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
6.302 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
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
Chapter 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
To 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,
EECE 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
Design 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
EE 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
Control Systems. Lecture 9 Frequency Response. Frequency Response
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
Chapter 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
, 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
13.1 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
Module 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
EECE 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
CHAPTER 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
Chapter 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
CONTROL 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 () (),
d 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
CHAPTER 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
Introduction 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
CHE302 LECTURE VI DYNAMIC BEHAVIORS OF REPRESENTATIVE PROCESSES. Professor Dae Ryook Yang
CHE30 LECTURE VI DYNAMIC BEHAVIORS OF REPRESENTATIVE PROCESSES Professor Dae Ryook Yang Fall 00 Dep. of Chemical and Biological Engineering orea Universiy CHE30 Process Dynamics and Conrol orea Universiy
Instrumentation & Process Control
Chemical Engineering (GTE & PSU) Poal Correpondence GTE & Public Secor Inrumenaion & Proce Conrol To Buy Poal Correpondence Package call a -999657855 Poal Coure ( GTE & PSU) 5 ENGINEERS INSTITUTE OF INDI.
( ) ( ) 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
Problem Set If all directed edges in a network have distinct capacities, then there is a unique maximum flow.
CSE 202: Deign and Analyi of Algorihm Winer 2013 Problem Se 3 Inrucor: Kamalika Chaudhuri Due on: Tue. Feb 26, 2013 Inrucion For your proof, you may ue any lower bound, algorihm or daa rucure from he ex
EE202 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
Interpolation and Pulse Shaping
EE345S Real-Time Digial Signal Proceing Lab Spring 2006 Inerpolaion and Pule Shaping Prof. Brian L. Evan Dep. of Elecrical and Compuer Engineering The Univeriy of Texa a Auin Lecure 7 Dicree-o-Coninuou
Single 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
CONTRIBUTION TO THE THEORETICAL ANALYSIS OF THE ZIEGLER NICHOLS METHOD
Journal of ELECTRICAL ENGINEERING, VOL 54, NO 7-8, 3, 88 94 CONTRIBUTION TO THE THEORETICAL ANALYSIS OF THE ZIEGLER NICHOLS METHOD Mikuláš Huba Kaarína Žáková The high pracical ignificance of he I T d
LAPLACE 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
18.03SC 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
Homework 6 AERE331 Spring 2019 Due 4/24(W) Name Sec. 1 / 2
Homework 6 AERE33 Spring 9 Due 4/4(W) Name Sec / PROBLEM (5p In PROBLEM 4 of HW4 we used he frequency domain o design a yaw/rudder feedback conrol sysem for a plan wih ransfer funcion 46 Gp () s The conroller
An 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,
Discussion 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
Application of Combined Fourier Series Transform (Sampling Theorem)
Applicaion o Combined Fourier Serie ranorm Sampling heorem x X[] m m Sampling Frequency Deparmen o Elecrical and Compuer Engineering Deparmen o Elecrical and Compuer Engineering X[] x We need Fourier Serie
L1, L2, N1 N2. + Vout. C out. Figure 2.1.1: Flyback converter
page 11 Flyback converer The Flyback converer belongs o he primary swiched converer family, which means here is isolaion beween in and oupu. Flyback converers are used in nearly all mains supplied elecronic
1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.
![1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.](/thumbs/71/65799522.jpg "1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.")
. Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.
Study of simple inductive-capacitive series circuits using MATLAB software package
ecen Advance in ircui, Syem and Auomaic onrol Sudy of imple inducive-capaciive erie circui uing MAAB ofware package NIUESU IU, PĂSUESU DAGOŞ Faculy of Mechanical and Elecrical Engineering Univeriy of Peroani
Course Notes for EE227C (Spring 2018): Convex Optimization and Approximation
Course Noes for EE7C Spring 018: Convex Opimizaion and Approximaion Insrucor: Moriz Hard Email: hard+ee7c@berkeley.edu Graduae Insrucor: Max Simchowiz Email: msimchow+ee7c@berkeley.edu Ocober 15, 018 3
5.1 - 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
Lecture 8. PID control. Industrial process control ( today) PID control. Insights about PID actions
Lecture 8. PID control. The role of P, I, and D action 2. PID tuning Indutrial proce control (92... today) Feedback control i ued to improve the proce performance: tatic performance: for contant reference,
Motion In One Dimension. Graphing Constant Speed
Moion In One Dimenion PLATO AND ARISTOTLE GALILEO GALILEI LEANING TOWER OF PISA Graphing Conan Speed Diance v. Time for Toy Car (0-5 ec.) be-fi line (from TI calculaor) d = 207.7 12.6 Diance (cm) 1000
Performance Comparison of LCMV-based Space-time 2D Array and Ambiguity Problem
Inernaional journal of cience Commerce and umaniie Volume No 2 No 3 April 204 Performance Comparion of LCMV-baed pace-ime 2D Arra and Ambigui Problem 2 o uan Chang and Jin hinghia Deparmen of Communicaion
e 2t u(t) e 2t u(t) =?
EE : Signals, Sysems, and Transforms Fall 7. Skech he convoluion of he following wo signals. Tes No noes, closed book. f() Show your work. Simplify your answers. g(). Using he convoluion inegral, find
s-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
UT 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,
EE 315 Notes. Gürdal Arslan CLASS 1. (Sections ) What is a signal?
EE 35 Noes Gürdal Arslan CLASS (Secions.-.2) Wha is a signal? In his class, a signal is some funcion of ime and i represens how some physical quaniy changes over some window of ime. Examples: velociy of
The Brock-Mirman Stochastic Growth Model
c December 3, 208, Chrisopher D. Carroll BrockMirman The Brock-Mirman Sochasic Growh Model Brock and Mirman (972) provided he firs opimizing growh model wih unpredicable (sochasic) shocks. The social planner
Digital 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
Chapter 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
8.1. a) For step response, M input is u ( t) Taking inverse Laplace transform. as α 0. Ideal response, K c. = Kc Mτ D + For ramp response, 8-1
8. a For ep repone, inpu i u, U Y a U α α Y a α α Taking invere Laplae ranform a α e e / α / α A α 0 a δ 0 e / α a δ deal repone, α d Y i Gi U i δ Hene a α 0 a i For ramp repone, inpu i u, U Soluion anual
Dynamic Effects of Feedback Control!
Dynamic Effecs of Feedback Conrol! Rober Sengel! Roboics and Inelligen Sysems MAE 345, Princeon Universiy, 2017 Inner, Middle, and Ouer Feedback Conrol Loops Sep Response of Linear, Time- Invarian (LTI)
CONTROL 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
RPN TUNING STRATEGY FOR MODEL PREDICTIVE CONTROL
RPN TUNING TRATEGY FOR MODEL PREDICTIVE CONTROL J. O. Trierweiler #, L. A. Farina, and R. G. Duraiki Laboraory of Proce Conrol and Inegraion (LACIP) Deparmen of Chemical Engineering, Federal Univeriy of
Bifurcation and Chaotic Aspects in Peak Current Controlled Buck-Boost Converters
Dan Negoiecu, Dan acu, Viorel Popecu, Corina Ivan Bifurcaion and Chaoic Apec in Peak Curren Conrolled BuckBoo Converer DAN NEGOITESCU, DAN ASCU, VIOE POPESCU, COINA IVAN Deparmen of Applied Elecronic Poliehnica
LAB 5: Computer Simulation of RLC Circuit Response using PSpice
--3LabManualLab5.doc LAB 5: ompuer imulaion of RL ircui Response using Ppice PURPOE To use a compuer simulaion program (Ppice) o invesigae he response of an RL series circui o: (a) a sinusoidal exciaion.
Some Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
Lecture 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
6.003 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
Physics 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,
Linear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and
Challenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k
Challenge Problems DIS 03 and 0 March 6, 05 Choose one of he following problems, and work on i in your group. Your goal is o convince me ha your answer is correc. Even if your answer isn compleely correc,
Module 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
EE 435. Lecture 31. Absolute and Relative Accuracy DAC Design. The String DAC
EE 435 Lecure 3 Absolue and Relaive Accuracy DAC Design The Sring DAC . Review from las lecure. DFT Simulaion from Malab Quanizaion Noise DACs and ADCs generally quanize boh ampliude and ime If convering
Curvature. Institute of Lifelong Learning, University of Delhi pg. 1
Dicipline Coure-I Semeer-I Paper: Calculu-I Leon: Leon Developer: Chaianya Kumar College/Deparmen: Deparmen of Mahemaic, Delhi College of r and Commerce, Univeriy of Delhi Iniue of Lifelong Learning, Univeriy
376 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
Module 4: Time Response of discrete time systems Lecture Note 1
Digital Control Module 4 Lecture Module 4: ime Repone of dicrete time ytem Lecture Note ime Repone of dicrete time ytem Abolute tability i a baic requirement of all control ytem. Apart from that, good
6.003 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
(1) (2) Differentiation of (1) and then substitution of (3) leads to. Therefore, we will simply consider the second-order linear system given by (4)
Phase Plane Analysis of Linear Sysems Adaped from Applied Nonlinear Conrol by Sloine and Li The general form of a linear second-order sysem is a c b d From and b bc d a Differeniaion of and hen subsiuion
ANALYSIS OF SOME SAFETY ASSESSMENT STANDARD ON GROUNDING SYSTEMS
ANAYSIS OF SOME SAFETY ASSESSMENT STANDARD ON GROUNDING SYSTEMS Shang iqun, Zhang Yan, Cheng Gang School of Elecrical and Conrol Engineering, Xi an Univeriy of Science & Technology, 710054, Xi an, China,
where the coordinate X (t) describes the system motion. X has its origin at the system static equilibrium position (SEP).
Appendix A: Conservaion of Mechanical Energy = Conservaion of Linear Momenum Consider he moion of a nd order mechanical sysem comprised of he fundamenal mechanical elemens: ineria or mass (M), siffness
TIME RESPONSE Introduction
TIME RESPONSE Iroducio Time repoe of a corol yem i a udy o how he oupu variable chage whe a ypical e ipu igal i give o he yem. The commoly e ipu igal are hoe of ep fucio, impule fucio, ramp fucio ad iuoidal
Chapter 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
EECE 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!
15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
dt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.
Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies
Randomized Perfect Bipartite Matching
Inenive Algorihm Lecure 24 Randomized Perfec Biparie Maching Lecurer: Daniel A. Spielman April 9, 208 24. Inroducion We explain a randomized algorihm by Ahih Goel, Michael Kapralov and Sanjeev Khanna for
Chapter 5 Digital PID control algorithm. Hesheng Wang Department of Automation,SJTU 2016,03
Chaper 5 Digial PID conrol algorihm Hesheng Wang Deparmen of Auomaion,SJTU 216,3 Ouline Absrac Quasi-coninuous PID conrol algorihm Improvemen of sandard PID algorihm Choosing parameer of PID regulaor Brief
A 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
Linear Time-invariant systems, Convolution, and Cross-correlation
Linear Time-invarian sysems, Convoluion, and Cross-correlaion (1) Linear Time-invarian (LTI) sysem A sysem akes in an inpu funcion and reurns an oupu funcion. x() T y() Inpu Sysem Oupu y() = T[x()] An
Control Systems Engineering ( Chapter 7. Steady-State Errors ) Prof. Kwang-Chun Ho Tel: Fax:
Control Sytem Engineering ( Chapter 7. Steady-State Error Prof. Kwang-Chun Ho kwangho@hanung.ac.kr Tel: 0-760-453 Fax:0-760-4435 Introduction In thi leon, you will learn the following : How to find the
CHAPTER 6: FIRST-ORDER CIRCUITS
EEE5: CI CUI T THEOY CHAPTE 6: FIST-ODE CICUITS 6. Inroducion This chaper considers L and C circuis. Applying he Kirshoff s law o C and L circuis produces differenial equaions. The differenial equaions
CHAPTER. 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 ( ) ( )
t )? How would you have tried to solve this problem in Chapter 3?
Exercie 9) Ue Laplace ranform o wrie down he oluion o 2 x x = F in x = x x = v. wha phenomena do oluion o hi DE illurae (even hough we're forcing wih in co )? How would you have ried o olve hi problem
The Residual Graph. 11 Augmenting Path Algorithms. Augmenting Path Algorithm. Augmenting Path Algorithm
Augmening Pah Algorihm Greedy-algorihm: ar wih f (e) = everywhere find an - pah wih f (e) < c(e) on every edge augmen flow along he pah repea a long a poible The Reidual Graph From he graph G = (V, E,
Linear Motion, Speed & Velocity
Add Iporan Linear Moion, Speed & Velociy Page: 136 Linear Moion, Speed & Velociy NGSS Sandard: N/A MA Curriculu Fraework (2006): 1.1, 1.2 AP Phyic 1 Learning Objecive: 3.A.1.1, 3.A.1.3 Knowledge/Underanding
Additional 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
Let. 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.
!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)
"#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5
THE SINE INTEGRAL. x dt t
THE SINE INTEGRAL As one learns in elemenary calculus, he limi of sin(/ as vanishes is uniy. Furhermore he funcion is even and has an infinie number of zeros locaed a ±n for n1,,3 Is plo looks like his-
CHAPTER 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
1/8 1/31/2011 ( ) ( ) Amplifiers lecture. out. Jim Stiles. Dept. of o EECS
1/31/2011 Amplifiers lecure 1/8 Amplifiers An ideal amplifier is a wo-por circui ha akes an pu signal v and reproduces i exacly a is oupu, only wih a larger magniude! ( ) i ( ) v + ( ) A I v + ou ( ) (
EECE 301 Signals & Systems Prof. Mark Fowler
EECE 3 Signals & Sysems Prof. Mark Fowler Noe Se #2 Wha are Coninuous-Time Signals??? Reading Assignmen: Secion. of Kamen and Heck /22 Course Flow Diagram The arrows here show concepual flow beween ideas.
Lecture 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,
Rectilinear Kinematics
Recilinear Kinemaic Coninuou Moion Sir Iaac Newon Leonard Euler Oeriew Kinemaic Coninuou Moion Erraic Moion Michael Schumacher. 7-ime Formula 1 World Champion Kinemaic The objecie of kinemaic i o characerize
EECS 2602 Winter Laboratory 3 Fourier series, Fourier transform and Bode Plots in MATLAB
EECS 6 Winer 7 Laboraory 3 Fourier series, Fourier ransform and Bode Plos in MATLAB Inroducion: The objecives of his lab are o use MATLAB:. To plo periodic signals wih Fourier series represenaion. To obain
ECE 510 Lecture 4 Reliability Plotting T&T Scott Johnson Glenn Shirley
ECE 5 Lecure 4 Reliabiliy Ploing T&T 6.-6 Sco Johnson Glenn Shirley Funcional Forms 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley Reliabiliy Funcional Forms Daa Model (funcional form) Choose funcional form for
Sub Module 2.6. Measurement of transient temperature
Mechanical Measuremens Prof. S.P.Venkaeshan Sub Module 2.6 Measuremen of ransien emperaure Many processes of engineering relevance involve variaions wih respec o ime. The sysem properies like emperaure,
4. Electric field lines with respect to equipotential surfaces are
Pre-es Quasi-saic elecromagneism. The field produced by primary charge Q and by an uncharged conducing plane disanced from Q by disance d is equal o he field produced wihou conducing plane by wo following
Announcements: Warm-up Exercise:
Fri Apr 13 7.1 Sysems of differenial equaions - o model muli-componen sysems via comparmenal analysis hp//en.wikipedia.org/wiki/muli-comparmen_model Announcemens Warm-up Exercise Here's a relaively simple