Automatic Control Systems

Size: px
Start display at page:

Download "Automatic Control Systems"

Transcription

1 Automatic Cotrol Sytem Lecture-5 Time Domai Aalyi of Orer Sytem Emam Fathy Departmet of Electrical a Cotrol Egieerig emfmz@yahoo.com

2 Itrouctio Compare to the implicity of a firt-orer ytem, a eco-orer ytem exhibit a wie rage of repoe that mut be aalyze a ecribe. Varyig a firt-orer ytem' parameter (τ, K) imply chage the pee a offet of the repoe Wherea, chage i the parameter of a eco-orer ytem ca chage the form of the repoe.

3 Itrouctio A geeral eco-orer ytem i characterize by the followig trafer fuctio. C( ) R( ) u-ampe atural frequecy of the eco orer ytem, which i the frequecy of ocillatio of the ytem without ampig. ampig ratio of the eco orer ytem, which i a meaure of the egree of reitace to chage i the ytem output. 3

4 Example# Determie the u-ampe atural frequecy a ampig ratio of the followig eco orer ytem. C( ) R( ) 4 4 Compare the umerator a eomiator of the give trafer fuctio with the geeral orer trafer fuctio. C( ) R( ) 4 ra / ec

5 Itrouctio R C ) ( ) ( Two pole of the ytem are 5 Accorig the value of, a eco-orer ytem ca be et ito oe of the four categorie:

6 . Overampe - whe the ytem ha two real itict pole ( >). jω. Uerampe - whe the ytem ha two complex cojugate pole (0 < <) jω δ δ -c -b -a -c -b -a 3. Uampe - whe the ytem ha two imagiary pole ( = 0). jω 4.Critically ampe - whe the ytem ha two real but equal pole ( = ). jω δ δ -c -b -a -c -b -a 6

7 7

8 8

9 9

10 S-Plae Natural Uampe Frequecy. Ditace from the origi of - plae to pole i atural uampe frequecy i ra/ec. jω δ

11 S-Plae Let u raw a circle of raiu 3 i -plae. If a pole i locate aywhere o the circumferece of the circle the atural uampe frequecy woul be 3 ra/ec. jω δ -3 3

12 S-Plae Therefore the -plae i ivie ito Cotat Natural Uampe Frequecy (ω ) Circle. jω δ 4

13 S-Plae Dampig ratio. Coie of the agle betwee vector coectig origi a pole a ve real axi yiel ampig ratio. jω co δ 5

14 S-Plae Therefore, -plae i ivie ito ectio of cotat ampig ratio lie. 9

15 The atural frequecy of cloe loop pole of orer ytem i ra/ec a ampig ratio i 0.5. C( ) R( ) 4 4 Imagiary Axi Example- 3 Pole-Zero Map

16 Example- Determie the locatio of cloe loop pole o that the ampig ratio remai ame but the atural uampe frequecy i ouble. 5 Pole-Zero Map C( ) R( ) Imagiary Axi

17 S-Plae The vertical locatio of the pole i the frequecy of the ocillatio i the repoe. The horizotal locatio of the pole i the reciprocal of the time cotat of the expoetial ecay. So, the further the pole i to the left i the -plae, the fater the repoe ie out. j j

18 Time-Domai Specificatio 3

19 Time-Domai Specificatio For 0< < a ω > 0, the orer ytem repoe ue to a uit tep iput look like 4

20 Time-Domai Specificatio The elay (t ) time i the time require for the repoe to reach half the fial value the very firt time. T ζ ω 5

21 Time-Domai Specificatio The rie time i the time require for the repoe to rie from 0% to 90%, 5% to 95%, or 0% to 00% of it fial value. For uerampe eco orer ytem, the 0% to 00% rie time i ormally ue. For overampe ytem, the 0% to 90% rie time i commoly ue. t r co

22 Time-Domai Specificatio The peak time i the time require for the repoe to reach the firt peak of the overhoot. t p M p e

23 Time-Domai Specificatio The maximum overhoot i the maximum peak value of the repoe curve meaure from uity. If the fial teay-tate value of the repoe iffer from uity, the it i commo to ue the maximum percet overhoot. It i efie by The amout of the maximum (percet) overhoot irectly iicate the relative tability of the ytem. 8

24 Time-Domai Specificatio The ettlig time i the time require for the repoe curve to reach a tay withi a rage about the fial value of ize pecifie by abolute percetage of the fial value (uually %, % or 5%). t k k=4 for (%) k=3 for (5%) 9

25 Step Repoe of uerampe Sytem C ) ( The partial fractio expaio of above equatio i give a C ) ( C ) ( R C ) ( ) ( C ) ( Step Repoe 30

26 Step Repoe of uerampe Sytem C ( ) Above equatio ca be writte a C( ) Where, i the frequecy of traiet ocillatio a i calle ampe atural frequecy. The ivere Laplace traform of above equatio ca be obtaie eaily if C() i writte i the followig form: C( ) 3

27 Step Repoe of uerampe Sytem C ) ( C ) ( C ) ( t e t e t c t t i co ) ( 3

28 Step Repoe of uerampe Sytem t e t e t c t t i co ) ( t t e t c t i co ) ( Whe 0 t t c co ) ( 33

29 Step Repoe of uerampe Sytem c( t) e t cot if 0. a 3 ra / ec i t

30 Step Repoe of uerampe Sytem c( t) e t cot if 0. 5 a 3 ra / ec.4 i t

31 Step Repoe of uerampe Sytem c( t) e t cot if 0. 9 a 3 ra / ec.4 i t

32 Step Repoe of uerampe Sytem b=0 b=0. b=0.4 b=0.6 b=

33 Step Repoe of uerampe Sytem w=0.5 w= w=.5 w= w=

34 E of Lec 39

35 Example Coier the ytem how i followig figure, where ampig ratio i 0.6 a atural uampe frequecy i 5 ra/ec. Obtai the rie time t r, peak time t p, maximum overhoot M p, a ettlig time % a 5% criterio t whe the ytem i ubjecte to a uit-tep iput. 40

36 Example t r t r Rie Time 3. 4 co ta ( ) 0.93 ra t r

37 Example t p Peak Time 3. 4 t p Settlig Time (4%) t 3 Settlig Time (%) t 4 4 t t

38 Example Maximum Overhoot M p e 00 M p e M p M p 9.5% 43

39 .4 Example Step Repoe. Mp Amplitue Rie Time Time (ec) 44

40 Example 7 Whe the ytem how i Figure(a) i ubjecte to a uit-tep iput, the ytem output repo a how i Figure(b). Determie the value of a a c from the repoe curve. a ( c ) 5

ELEC 372 LECTURE NOTES, WEEK 4 Dr. Amir G. Aghdam Concordia University

ELEC 372 LECTURE NOTES, WEEK 4 Dr. Amir G. Aghdam Concordia University ELEC 37 LECTURE NOTES, WEE 4 Dr Amir G Aghdam Cocordia Uiverity Part of thee ote are adapted from the material i the followig referece: Moder Cotrol Sytem by Richard C Dorf ad Robert H Bihop, Pretice Hall

More information

Time Response. First Order Systems. Time Constant, T c We call 1/a the time constant of the response. Chapter 4 Time Response

Time Response. First Order Systems. Time Constant, T c We call 1/a the time constant of the response. Chapter 4 Time Response Time Repoe Chapter 4 Time Repoe Itroductio The output repoe of a ytem i the um of two repoe: the forced repoe ad the atural repoe. Although may techique, uch a olvig a differetial equatio or takig the

More information

Course Outline. Problem Identification. Engineering as Design. Amme 3500 : System Dynamics and Control. System Response. Dr. Stefan B.

Course Outline. Problem Identification. Engineering as Design. Amme 3500 : System Dynamics and Control. System Response. Dr. Stefan B. Course Outlie Amme 35 : System Dyamics a Cotrol System Respose Week Date Cotet Assigmet Notes Mar Itrouctio 8 Mar Frequecy Domai Moellig 3 5 Mar Trasiet Performace a the s-plae 4 Mar Block Diagrams Assig

More information

Introduction to Control Systems

Introduction to Control Systems Itroductio to Cotrol Sytem CLASSIFICATION OF MATHEMATICAL MODELS Icreaig Eae of Aalyi Static Icreaig Realim Dyamic Determiitic Stochatic Lumped Parameter Ditributed Parameter Liear Noliear Cotat Coefficiet

More information

State space systems analysis

State space systems analysis State pace ytem aalyi Repreetatio of a ytem i tate-pace (tate-pace model of a ytem To itroduce the tate pace formalim let u tart with a eample i which the ytem i dicuio i a imple electrical circuit with

More information

The Performance of Feedback Control Systems

The Performance of Feedback Control Systems The Performace of Feedbac Cotrol Sytem Objective:. Secify the meaure of erformace time-domai the firt te i the deig roce Percet overhoot / Settlig time T / Time to rie / Steady-tate error e. ut igal uch

More information

System Control. Lesson #19a. BME 333 Biomedical Signals and Systems - J.Schesser

System Control. Lesson #19a. BME 333 Biomedical Signals and Systems - J.Schesser Sytem Cotrol Leo #9a 76 Sytem Cotrol Baic roblem Say you have a ytem which you ca ot alter but it repoe i ot optimal Example Motor cotrol for exokeleto Robotic cotrol roblem that ca occur Utable Traiet

More information

EECE 301 Signals & Systems Prof. Mark Fowler

EECE 301 Signals & Systems Prof. Mark Fowler EECE 30 Sigal & Sytem Prof. Mark Fowler Note Set #8 C-T Sytem: Laplace Traform Solvig Differetial Equatio Readig Aigmet: Sectio 6.4 of Kame ad Heck / Coure Flow Diagram The arrow here how coceptual flow

More information

ME 375 FINAL EXAM Friday, May 6, 2005

ME 375 FINAL EXAM Friday, May 6, 2005 ME 375 FINAL EXAM Friay, May 6, 005 Divisio: Kig 11:30 / Cuigham :30 (circle oe) Name: Istructios (1) This is a close book examiatio, but you are allowe three 8.5 11 crib sheets. () You have two hours

More information

Mechatronics II Laboratory Exercise 5 Second Order Response

Mechatronics II Laboratory Exercise 5 Second Order Response Mechatroics II Laboratory Exercise 5 Seco Orer Respose Theoretical Backgrou Seco orer ifferetial equatios approximate the yamic respose of may systems. The respose of a geeric seco orer system ca be see

More information

ECM Control Engineering Dr Mustafa M Aziz (2013) SYSTEM RESPONSE

ECM Control Engineering Dr Mustafa M Aziz (2013) SYSTEM RESPONSE ECM5 - Cotrol Egieerig Dr Mutafa M Aziz (3) SYSTEM RESPONSE. Itroductio. Repoe Aalyi of Firt-Order Sytem 3. Secod-Order Sytem 4. Siuoidal Repoe of the Sytem 5. Bode Diagram 6. Baic Fact About Egieerig

More information

CONTROL SYSTEMS. Chapter 7 : Bode Plot. 40dB/dec 1.0. db/dec so resultant slope will be 20 db/dec and this is due to the factor s

CONTROL SYSTEMS. Chapter 7 : Bode Plot. 40dB/dec 1.0. db/dec so resultant slope will be 20 db/dec and this is due to the factor s CONTROL SYSTEMS Chapter 7 : Bode Plot GATE Objective & Numerical Type Solutio Quetio 6 [Practice Book] [GATE EE 999 IIT-Bombay : 5 Mark] The aymptotic Bode plot of the miimum phae ope-loop trafer fuctio

More information

ECE 422 Power System Operations & Planning 6 Small Signal Stability. Spring 2015 Instructor: Kai Sun

ECE 422 Power System Operations & Planning 6 Small Signal Stability. Spring 2015 Instructor: Kai Sun ECE 4 Power Sytem Operatio & Plaig 6 Small Sigal Stability Sprig 15 Itructor: Kai Su 1 Referece Saadat Chapter 11.4 EPRI Tutorial Chapter 8 Power Ocillatio Kudur Chapter 1 Power Ocillatio The power ytem

More information

CONTROL ENGINEERING LABORATORY

CONTROL ENGINEERING LABORATORY Uiverity of Techology Departmet of Electrical Egieerig Cotrol Egieerig Lab. CONTROL ENGINEERING LABORATORY By Dr. Abdul. Rh. Humed M.Sc. Quay Salim Tawfeeq M.Sc. Nihad Mohammed Amee M.Sc. Waleed H. Habeeb

More information

2.004 Dynamics and Control II Spring 2008

2.004 Dynamics and Control II Spring 2008 MIT OpeCourseWare http://ocw.mit.edu 2.004 Dyamics ad Cotrol II Sprig 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts Istitute of Techology

More information

Class 07 Time domain analysis Part II 2 nd order systems

Class 07 Time domain analysis Part II 2 nd order systems Class 07 Time domai aalysis Part II d order systems Time domai aalysis d order systems iput S output Secod order systems of the type α G(s) as + bs + c Time domai aalysis d order systems iput S α as +

More information

MAHALAKSHMI ENGINEERING COLLEGE TIRUCHIRAPALLI

MAHALAKSHMI ENGINEERING COLLEGE TIRUCHIRAPALLI MAHALASHMI ENGINEERING COLLEGE TIRUCHIRAPALLI-63. QUESTION BAN DEPARTMENT: ECE SUBJECT CODE: EC55 SEMESTER - III SUBJECT NAME: CONTROL SYSTEMS UNIT- SYSTEMS AND THEIR REPRESENTATION. Write Mao gai formula.

More information

EE Control Systems

EE Control Systems Copyright FL Lewis 7 All rights reserved Updated: Moday, November 1, 7 EE 4314 - Cotrol Systems Bode Plot Performace Specificatios The Bode Plot was developed by Hedrik Wade Bode i 1938 while he worked

More information

Last time: Completed solution to the optimum linear filter in real-time operation

Last time: Completed solution to the optimum linear filter in real-time operation 6.3 tochatic Etimatio ad Cotrol, Fall 4 ecture at time: Completed olutio to the oimum liear filter i real-time operatio emi-free cofiguratio: t D( p) F( p) i( p) dte dp e π F( ) F( ) ( ) F( p) ( p) 4444443

More information

Mechatronics. Time Response & Frequency Response 2 nd -Order Dynamic System 2-Pole, Low-Pass, Active Filter

Mechatronics. Time Response & Frequency Response 2 nd -Order Dynamic System 2-Pole, Low-Pass, Active Filter Time Respose & Frequecy Respose d -Order Dyamic System -Pole, Low-Pass, Active Filter R 4 R 7 C 5 e i R 1 C R 3 - + R 6 - + e out Assigmet: Perform a Complete Dyamic System Ivestigatio of the Two-Pole,

More information

ECEN620: Network Theory Broadband Circuit Design Fall 2014

ECEN620: Network Theory Broadband Circuit Design Fall 2014 ECE60: etwork Theory Broadbad Circuit Deig Fall 04 Lecture 3: PLL Aalyi Sam Palermo Aalog & Mixed-Sigal Ceter Texa A&M Uiverity Ageda & Readig PLL Overview & Applicatio PLL Liear Model Phae & Frequecy

More information

Lecture 30: Frequency Response of Second-Order Systems

Lecture 30: Frequency Response of Second-Order Systems Lecture 3: Frequecy Repoe of Secod-Order Sytem UHTXHQF\ 5HVSRQVH RI 6HFRQGUGHU 6\VWHPV A geeral ecod-order ytem ha a trafer fuctio of the form b + b + b H (. (9.4 a + a + a It ca be table, utable, caual

More information

Chapter (a) ζ. ω. 5 2 (a) Type 0 (b) Type 0 (c) Type 1 (d) Type 2 (e) Type 3 (f) Type 3. (g) type 2 (h) type (a) K G s.

Chapter (a) ζ. ω. 5 2 (a) Type 0 (b) Type 0 (c) Type 1 (d) Type 2 (e) Type 3 (f) Type 3. (g) type 2 (h) type (a) K G s. Chapter 5 5 1 (a) ζ. ω 0 707 rad / ec (b) 0 ζ 0. 707 ω rad / ec (c) ζ 0. 5 1 ω 5 rad / ec (d) 0. 5 ζ 0. 707 ω 0. 5 rad / ec 5 (a) Type 0 (b) Type 0 (c) Type 1 (d) Type (e) Type 3 (f) Type 3 (g) type (h)

More information

Dr. Seeler Department of Mechanical Engineering Fall 2009 Lafayette College ME 479: Control Systems and Mechatronics Design and Analysis

Dr. Seeler Department of Mechanical Engineering Fall 2009 Lafayette College ME 479: Control Systems and Mechatronics Design and Analysis Dr. Seeler Departmet of Mechaical Egieerig Fall 009 Lafayette College ME 479: Cotrol Systems ad Mechatroics Desig ad Aalysis Lab 0: Review of the First ad Secod Order Step Resposes The followig remarks

More information

a 1 = 1 a a a a n n s f() s = Σ log a 1 + a a n log n sup log a n+1 + a n+2 + a n+3 log n sup () s = an /n s s = + t i

a 1 = 1 a a a a n n s f() s = Σ log a 1 + a a n log n sup log a n+1 + a n+2 + a n+3 log n sup () s = an /n s s = + t i 0 Dirichlet Serie & Logarithmic Power Serie. Defiitio & Theorem Defiitio.. (Ordiary Dirichlet Serie) Whe,a,,3, are complex umber, we call the followig Ordiary Dirichlet Serie. f() a a a a 3 3 a 4 4 Note

More information

Last time: Ground rules for filtering and control system design

Last time: Ground rules for filtering and control system design 6.3 Stochatic Etimatio ad Cotrol, Fall 004 Lecture 7 Lat time: Groud rule for filterig ad cotrol ytem deig Gral ytem Sytem parameter are cotaied i w( t ad w ( t. Deired output i grated by takig the igal

More information

Lecture 11. Course Review. (The Big Picture) G. Hovland Input-Output Limitations (Skogestad Ch. 3) Discrete. Time Domain

Lecture 11. Course Review. (The Big Picture) G. Hovland Input-Output Limitations (Skogestad Ch. 3) Discrete. Time Domain MER4 Advaced Cotrol Lecture Coure Review (he ig Picture MER4 ADVANCED CONROL EMEER, 4 G. Hovlad 4 Mai heme of MER4 Frequecy Domai Aalyi (Nie Chapter Phae ad Gai Margi Iput-Output Limitatio (kogetad Ch.

More information

High-Speed Serial Interface Circuits and Systems. Lect. 4 Phase-Locked Loop (PLL) Type 1 (Chap. 8 in Razavi)

High-Speed Serial Interface Circuits and Systems. Lect. 4 Phase-Locked Loop (PLL) Type 1 (Chap. 8 in Razavi) High-Speed Serial Iterface Circuit ad Sytem Lect. 4 Phae-Locked Loop (PLL) Type 1 (Chap. 8 i Razavi) PLL Phae lockig loop A (egative-feedback) cotrol ytem that geerate a output igal whoe phae (ad frequecy)

More information

Chapter 7: The z-transform. Chih-Wei Liu

Chapter 7: The z-transform. Chih-Wei Liu Chapter 7: The -Trasform Chih-Wei Liu Outlie Itroductio The -Trasform Properties of the Regio of Covergece Properties of the -Trasform Iversio of the -Trasform The Trasfer Fuctio Causality ad Stability

More information

ELEC 372 LECTURE NOTES, WEEK 1 Dr. Amir G. Aghdam Concordia University

ELEC 372 LECTURE NOTES, WEEK 1 Dr. Amir G. Aghdam Concordia University EEC 37 ECTURE NOTES, WEEK Dr Amir G Aghdam Cocordia Uiverity Part of thee ote are adapted from the material i the followig referece: Moder Cotrol Sytem by Richard C Dorf ad Robert H Bihop, Pretice Hall

More information

Dynamic Response of Linear Systems

Dynamic Response of Linear Systems Dyamic Respose of Liear Systems Liear System Respose Superpositio Priciple Resposes to Specific Iputs Dyamic Respose of st Order Systems Characteristic Equatio - Free Respose Stable st Order System Respose

More information

Brief Review of Linear System Theory

Brief Review of Linear System Theory Brief Review of Liear Sytem heory he followig iformatio i typically covered i a coure o liear ytem theory. At ISU, EE 577 i oe uch coure ad i highly recommeded for power ytem egieerig tudet. We have developed

More information

MODERN CONTROL SYSTEMS

MODERN CONTROL SYSTEMS MODERN CONTROL SYSTEMS Lecture 1 Root Locu Emam Fathy Department of Electrical and Control Engineering email: emfmz@aat.edu http://www.aat.edu/cv.php?dip_unit=346&er=68525 1 Introduction What i root locu?

More information

Generalizing the DTFT. The z Transform. Complex Exponential Excitation. The Transfer Function. Systems Described by Difference Equations

Generalizing the DTFT. The z Transform. Complex Exponential Excitation. The Transfer Function. Systems Described by Difference Equations Geeraliig the DTFT The Trasform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 1 The forward DTFT is defied by X e jω = x e jω i which = Ω is discrete-time radia frequecy, a real variable.

More information

Performance-Based Plastic Design (PBPD) Procedure

Performance-Based Plastic Design (PBPD) Procedure Performace-Baed Platic Deig (PBPD) Procedure 3. Geeral A outlie of the tep-by-tep, Performace-Baed Platic Deig (PBPD) procedure follow, with detail to be dicued i ubequet ectio i thi chapter ad theoretical

More information

EE 205 Dr. A. Zidouri. Electric Circuits II. Frequency Selective Circuits (Filters) Bode Plots: Complex Poles and Zeros.

EE 205 Dr. A. Zidouri. Electric Circuits II. Frequency Selective Circuits (Filters) Bode Plots: Complex Poles and Zeros. EE 5 Dr. A. Zidouri Electric Circuits II Frequecy Selective Circuits (Filters) Bode Plots: Complex Poles ad Zeros Lecture #4 - - EE 5 Dr. A. Zidouri The material to be covered i this lecture is as follows:

More information

CDS 101: Lecture 8.2 Tools for PID & Loop Shaping

CDS 101: Lecture 8.2 Tools for PID & Loop Shaping CDS : Lecture 8. Tools for PID & Loop Shapig Richard M. Murray 7 November 4 Goals: Show how to use loop shapig to achieve a performace specificatio Itroduce ew tools for loop shapig desig: Ziegler-Nichols,

More information

Chapter 2 Feedback Control Theory Continued

Chapter 2 Feedback Control Theory Continued Chapter Feedback Cotrol Theor Cotiued. Itroductio I the previous chapter, the respose characteristic of simple first ad secod order trasfer fuctios were studied. It was show that first order trasfer fuctio,

More information

COMM 602: Digital Signal Processing

COMM 602: Digital Signal Processing COMM 60: Digital Sigal Processig Lecture 4 -Properties of LTIS Usig Z-Trasform -Iverse Z-Trasform Properties of LTIS Usig Z-Trasform Properties of LTIS Usig Z-Trasform -ve +ve Properties of LTIS Usig Z-Trasform

More information

Lecture 13. Graphical representation of the frequency response. Luca Ferrarini - Basic Automatic Control 1

Lecture 13. Graphical representation of the frequency response. Luca Ferrarini - Basic Automatic Control 1 Lecture 3 Graphical represetatio of the frequecy respose Luca Ferrarii - Basic Automatic Cotrol Graphical represetatio of the frequecy respose Polar plot G Bode plot ( j), G Im 3 Re of the magitude G (

More information

Adaptive control design for a Mimo chemical reactor

Adaptive control design for a Mimo chemical reactor Automatio, Cotrol ad Itelliget Sytem 013; 1(3): 64-70 Publihed olie July 10, 013 (http://www.ciecepublihiggroup.com/j/aci) doi: 10.11648/j.aci.0130103.15 Adaptive cotrol deig for a Mimo chemical reactor

More information

COMPLEX NUMBERS AND DE MOIVRE'S THEOREM SYNOPSIS. Ay umber of the form x+iy where x, y R ad i = - is called a complex umber.. I the complex umber x+iy, x is called the real part ad y is called the imagiary

More information

STABILITY OF THE ACTIVE VIBRATION CONTROL OF CANTILEVER BEAMS

STABILITY OF THE ACTIVE VIBRATION CONTROL OF CANTILEVER BEAMS Iteratioal Coferece o Vibratio Problem September 9-,, Liboa, Portugal STBILITY OF THE CTIVE VIBRTIO COTROL OF CTILEVER BEMS J. Tůma, P. Šuráe, M. Mahdal VSB Techical Uierity of Otraa Czech Republic Outlie.

More information

EE 508 Lecture 6. Dead Networks Scaling, Normalization and Transformations

EE 508 Lecture 6. Dead Networks Scaling, Normalization and Transformations EE 508 Lecture 6 Dead Network Scalig, Normalizatio ad Traformatio Filter Cocept ad Termiology 2-d order polyomial characterizatio Biquadratic Factorizatio Op Amp Modelig Stability ad Itability Roll-off

More information

Course Outline. Designing Control Systems. Proportional Controller. Amme 3500 : System Dynamics and Control. Root Locus. Dr. Stefan B.

Course Outline. Designing Control Systems. Proportional Controller. Amme 3500 : System Dynamics and Control. Root Locus. Dr. Stefan B. Amme 3500 : System Dyamics ad Cotrol Root Locus Course Outlie Week Date Cotet Assigmet Notes Mar Itroductio 8 Mar Frequecy Domai Modellig 3 5 Mar Trasiet Performace ad the s-plae 4 Mar Block Diagrams Assig

More information

CALCULUS BASIC SUMMER REVIEW

CALCULUS BASIC SUMMER REVIEW CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=

More information

u t u 0 ( 7) Intuitively, the maximum principles can be explained by the following observation. Recall

u t u 0 ( 7) Intuitively, the maximum principles can be explained by the following observation. Recall Oct. Heat Equatio M aximum priciple I thi lecture we will dicu the maximum priciple ad uiquee of olutio for the heat equatio.. Maximum priciple. The heat equatio alo ejoy maximum priciple a the Laplace

More information

THE CONCEPT OF THE ROOT LOCUS. H(s) THE CONCEPT OF THE ROOT LOCUS

THE CONCEPT OF THE ROOT LOCUS. H(s) THE CONCEPT OF THE ROOT LOCUS So far i the tudie of cotrol yte the role of the characteritic equatio polyoial i deteriig the behavior of the yte ha bee highlighted. The root of that polyoial are the pole of the cotrol yte, ad their

More information

Written exam Digital Signal Processing for BMT (8E070). Tuesday November 1, 2011, 09:00 12:00.

Written exam Digital Signal Processing for BMT (8E070). Tuesday November 1, 2011, 09:00 12:00. Techische Uiversiteit Eidhove Fac. Biomedical Egieerig Writte exam Digital Sigal Processig for BMT (8E070). Tuesday November, 0, 09:00 :00. (oe page) ( problems) Problem. s Cosider a aalog filter with

More information

Frequency Response Methods

Frequency Response Methods Frequecy Respose Methods The frequecy respose Nyquist diagram polar plots Bode diagram magitude ad phase Frequecy domai specificatios Frequecy Respose Methods I precedig chapters the respose ad performace

More information

The z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j

The z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.

More information

Rational Function. To Find the Domain. ( x) ( ) q( x) ( ) ( ) ( ) , 0. where p x and are polynomial functions. The degree of q x

Rational Function. To Find the Domain. ( x) ( ) q( x) ( ) ( ) ( ) , 0. where p x and are polynomial functions. The degree of q x Graphig Ratioal Fuctios R Ratioal Fuctio p a + + a+ a 0 q q b + + b + b0 q, 0 where p a are polyomial fuctios p a + + a+ a0 q b + + b + b0 The egree of p The egree of q is is If > the f is a improper ratioal

More information

CHAPTER NINE. Frequency Response Methods

CHAPTER NINE. Frequency Response Methods CHAPTER NINE 9. Itroductio It as poited earlier that i practice the performace of a feedback cotrol system is more preferably measured by its time - domai respose characteristics. This is i cotrast to

More information

ECE 522 Power Systems Analysis II 3.2 Small Signal Stability

ECE 522 Power Systems Analysis II 3.2 Small Signal Stability ECE 5 Power Sytem Aalyi II 3. Small Sigal Stability Sprig 18 Itructor: Kai Su 1 Cotet 3..1 Small igal tability overview ad aalyi method 3.. Small igal tability ehacemet Referece: EPRI Dyamic Tutorial Chapter

More information

Professor: Mihnea UDREA DIGITAL SIGNAL PROCESSING. Grading: Web: MOODLE. 1. Introduction. General information

Professor: Mihnea UDREA DIGITAL SIGNAL PROCESSING. Grading: Web:   MOODLE. 1. Introduction. General information Geeral iformatio DIGITL SIGL PROCESSIG Profeor: ihea UDRE B29 mihea@comm.pub.ro Gradig: Laboratory: 5% Proect: 5% Tet: 2% ial exam : 5% Coure quiz: ±% Web: www.electroica.pub.ro OODLE 2 alog igal proceig

More information

Analysis of Dynamic Systems

Analysis of Dynamic Systems ME 43 Syem Dyamic & Corol Chaper 8: Time Domai Aalyi of Dyamic Syem Syem Chaper 8 Time-Domai Aalyi of Dyamic Syem 8. INTRODUCTION Pole a Zero of a Trafer Fucio A. Bazoue Pole: The pole of a rafer fucio

More information

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense, 3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [

More information

Digital Signal Processing, Fall 2010

Digital Signal Processing, Fall 2010 Digital Sigal Proeig, Fall 2 Leture 3: Samplig ad reotrutio, traform aalyi of LTI ytem tem Zheg-ua Ta Departmet of Eletroi Sytem Aalborg Uiverity, Demar t@e.aau.d Coure at a glae MM Direte-time igal ad

More information

REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION

REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION I liear regreio, we coider the frequecy ditributio of oe variable (Y) at each of everal level of a ecod variable (X). Y i kow a the depedet variable.

More information

Rational Functions. Rational Function. Example. The degree of q x. If n d then f x is an improper rational function. x x. We have three forms

Rational Functions. Rational Function. Example. The degree of q x. If n d then f x is an improper rational function. x x. We have three forms Ratioal Fuctios We have three forms R Ratioal Fuctio a a1 a p p 0 b b1 b0 q q p a a1 a0 q b b1 b0 The egree of p The egree of q is is If the f is a improper ratioal fuctio Compare forms Epae a a a b b1

More information

Chapter #5 EEE Control Systems

Chapter #5 EEE Control Systems Sprig EEE Chpter #5 EEE Cotrol Sytem Deig Bed o Root Locu Chpter / Sprig EEE Deig Bed Root Locu Led Cotrol (equivlet to PD cotrol) Ued whe the tedy tte propertie of the ytem re ok but there i poor performce,

More information

Chapter 1. Complex Numbers. Dr. Pulak Sahoo

Chapter 1. Complex Numbers. Dr. Pulak Sahoo Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistat Professor Departmet of Mathematics Uiversity Of Kalyai West Begal, Idia E-mail : sahoopulak1@gmail.com 1 Module-2: Stereographic Projectio 1 Euler

More information

LECTURE 10. Single particle acceleration: Phase stability. Linear Accelerator Dynamics:

LECTURE 10. Single particle acceleration: Phase stability. Linear Accelerator Dynamics: ETURE 0 Sigle particle acceleratio: Phae tability iear Accelerator Dyamic: ogituial equatio of motio: Small amplitue motio ogituial emittace a aiabatic ampig arge amplitue motio We ow begi to coier the

More information

Chapter 4 : Laplace Transform

Chapter 4 : Laplace Transform 4. Itroductio Laplace trasform is a alterative to solve the differetial equatio by the complex frequecy domai ( s = σ + jω), istead of the usual time domai. The DE ca be easily trasformed ito a algebraic

More information

MEM 255 Introduction to Control Systems: Analyzing Dynamic Response

MEM 255 Introduction to Control Systems: Analyzing Dynamic Response MEM 55 Itroductio to Cotrol Systems: Aalyzig Dyamic Respose Harry G. Kwaty Departmet of Mechaical Egieerig & Mechaics Drexel Uiversity Outlie Time domai ad frequecy domai A secod order system Via partial

More information

11/19/ Chapter 10 Overview. Chapter 10: Two-Sample Inference. + The Big Picture : Inference for Mean Difference Dependent Samples

11/19/ Chapter 10 Overview. Chapter 10: Two-Sample Inference. + The Big Picture : Inference for Mean Difference Dependent Samples /9/0 + + Chapter 0 Overview Dicoverig Statitic Eitio Daiel T. Laroe Chapter 0: Two-Sample Iferece 0. Iferece for Mea Differece Depeet Sample 0. Iferece for Two Iepeet Mea 0.3 Iferece for Two Iepeet Proportio

More information

Introduction to Signals and Systems, Part V: Lecture Summary

Introduction to Signals and Systems, Part V: Lecture Summary EEL33: Discrete-Time Sigals ad Systems Itroductio to Sigals ad Systems, Part V: Lecture Summary Itroductio to Sigals ad Systems, Part V: Lecture Summary So far we have oly looked at examples of o-recursive

More information

Unit 4: Polynomial and Rational Functions

Unit 4: Polynomial and Rational Functions 48 Uit 4: Polyomial ad Ratioal Fuctios Polyomial Fuctios A polyomial fuctio y px ( ) is a fuctio of the form p( x) ax + a x + a x +... + ax + ax+ a 1 1 1 0 where a, a 1,..., a, a1, a0are real costats ad

More information

EE 508 Lecture 6. Scaling, Normalization and Transformation

EE 508 Lecture 6. Scaling, Normalization and Transformation EE 508 Lecture 6 Scalig, Normalizatio ad Traformatio Review from Lat Time Dead Network X IN T X OUT T X OUT N T = D D The dead etwork of ay liear circuit i obtaied by ettig ALL idepedet ource to zero.

More information

MAS160: Signals, Systems & Information for Media Technology. Problem Set 5. DUE: November 3, (a) Plot of u[n] (b) Plot of x[n]=(0.

MAS160: Signals, Systems & Information for Media Technology. Problem Set 5. DUE: November 3, (a) Plot of u[n] (b) Plot of x[n]=(0. MAS6: Sigals, Systems & Iformatio for Media Techology Problem Set 5 DUE: November 3, 3 Istructors: V. Michael Bove, Jr. ad Rosalid Picard T.A. Jim McBride Problem : Uit-step ad ruig average (DSP First

More information

Curve Sketching Handout #5 Topic Interpretation Rational Functions

Curve Sketching Handout #5 Topic Interpretation Rational Functions Curve Sketchig Hadout #5 Topic Iterpretatio Ratioal Fuctios A ratioal fuctio is a fuctio f that is a quotiet of two polyomials. I other words, p ( ) ( ) f is a ratioal fuctio if p ( ) ad q ( ) are polyomials

More information

732 Appendix E: Previous EEE480 Exams. Rules: One sheet permitted, calculators permitted. GWC 352,

732 Appendix E: Previous EEE480 Exams. Rules: One sheet permitted, calculators permitted. GWC 352, 732 Aedix E: Previous EEE0 Exams EEE0 Exam 2, Srig 2008 A.A. Rodriguez Rules: Oe 8. sheet ermitted, calculators ermitted. GWC 32, 9-372 Problem Aalysis of a Feedback System Cosider the feedback system

More information

Exponential Moving Average Pieter P

Exponential Moving Average Pieter P Expoetial Movig Average Pieter P Differece equatio The Differece equatio of a expoetial movig average lter is very simple: y[] x[] + (1 )y[ 1] I this equatio, y[] is the curret output, y[ 1] is the previous

More information

GRADE 12 JUNE 2016 MATHEMATICS P2

GRADE 12 JUNE 2016 MATHEMATICS P2 NATIONAL SENIOR CERTIFICATE GRADE 1 JUNE 016 MATHEMATICS P MARKS: 150 TIME: 3 hours *MATHE* This questio paper cosists of 11 pages, icludig 1 iformatio sheet, ad a SPECIAL ANSWER BOOK. MATHEMATICS P (EC/JUNE

More information

Heat Equation: Maximum Principles

Heat Equation: Maximum Principles Heat Equatio: Maximum Priciple Nov. 9, 0 I thi lecture we will dicu the maximum priciple ad uiquee of olutio for the heat equatio.. Maximum priciple. The heat equatio alo ejoy maximum priciple a the Laplace

More information

Chapter 13: Complex Numbers

Chapter 13: Complex Numbers Sectios 13.1 & 13.2 Comple umbers ad comple plae Comple cojugate Modulus of a comple umber 1. Comple umbers Comple umbers are of the form z = + iy,, y R, i 2 = 1. I the above defiitio, is the real part

More information

ECE 422/522 Power System Operations & Planning/Power Systems Analysis II : 6 - Small Signal Stability

ECE 422/522 Power System Operations & Planning/Power Systems Analysis II : 6 - Small Signal Stability ECE 4/5 Power System Operatios & Plaig/Power Systems Aalysis II : 6 - Small Sigal Stability Sprig 014 Istructor: Kai Su 1 Refereces Kudur s Chapter 1 Saadat s Chapter 11.4 EPRI Tutorial s Chapter 8 Power

More information

CAMI Education linked to CAPS: Mathematics. Grade The main topics in the FET Mathematics Curriculum NUMBER

CAMI Education linked to CAPS: Mathematics. Grade The main topics in the FET Mathematics Curriculum NUMBER - 1 - CAMI Eucatio like to CAPS: Grae 1 The mai topics i the FET Curriculum NUMBER TOPIC 1 Fuctios Number patters, sequeces a series 3 Fiace, growth a ecay 4 Algebra 5 Differetial Calculus 6 Probability

More information

ANDRONOV-HOPF S BIFURCATION IN A DYNAMIC MODEL OF CELL POPULATION

ANDRONOV-HOPF S BIFURCATION IN A DYNAMIC MODEL OF CELL POPULATION Mathematical Moelig ANDRONOV-HOPF S BIFURCATION IN A DYNAMIC MODEL OF CELL POPULATION JuG Nekhozhia VA Sobolev Samara Natioal Reearch Uiverity Samara Ruia Abtract The mathematical moel of the ifferetial

More information

MCT242: Electronic Instrumentation Lecture 2: Instrumentation Definitions

MCT242: Electronic Instrumentation Lecture 2: Instrumentation Definitions Faculty of Egieerig MCT242: Electroic Istrumetatio Lecture 2: Istrumetatio Defiitios Overview Measuremet Error Accuracy Precisio ad Mea Resolutio Mea Variace ad Stadard deviatio Fiesse Sesitivity Rage

More information

Chapter 5. Root Locus Techniques

Chapter 5. Root Locus Techniques Chapter 5 Rt Lcu Techique Itrducti Sytem perfrmace ad tability dt determied dby cled-lp l ple Typical cled-lp feedback ctrl ytem G Ope-lp TF KG H Zer -, - Ple 0, -, - K Lcati f ple eaily fud Variati f

More information

Finite-length Discrete Transforms. Chapter 5, Sections

Finite-length Discrete Transforms. Chapter 5, Sections Fiite-legth Discrete Trasforms Chapter 5, Sectios 5.2-50 5.0 Dr. Iyad djafar Outlie The Discrete Fourier Trasform (DFT) Matrix Represetatio of DFT Fiite-legth Sequeces Circular Covolutio DFT Symmetry Properties

More information

Chapter #2 EEE Subsea Control and Communication Systems

Chapter #2 EEE Subsea Control and Communication Systems EEE 87 Chpter # EEE 87 Sube Cotrol d Commuictio Sytem Trfer fuctio Pole loctio d -ple Time domi chrcteritic Extr pole d zero Chpter /8 EEE 87 Trfer fuctio Lplce Trform Ued oly o LTI ytem Differetil expreio

More information

A New Accurate Analytical Expression for Rise Time Intended for Mechatronics Systems Performance Evaluation and Validation

A New Accurate Analytical Expression for Rise Time Intended for Mechatronics Systems Performance Evaluation and Validation Iteratioal Joural of Automatio, Cotrol ad Itelliget Systems Vol., No., 05, pp. 5-60 http://www.aisciece.org/joural/ijacis A New Accurate Aalytical Expressio for Rise Time Iteded for Mechatroics Systems

More information

Fig. 1: Streamline coordinates

Fig. 1: Streamline coordinates 1 Equatio of Motio i Streamlie Coordiate Ai A. Soi, MIT 2.25 Advaced Fluid Mechaic Euler equatio expree the relatiohip betwee the velocity ad the preure field i ivicid flow. Writte i term of treamlie coordiate,

More information

Module 4: Time Response of discrete time systems Lecture Note 1

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

More information

Appendix: The Laplace Transform

Appendix: The Laplace Transform Appedix: The Laplace Trasform The Laplace trasform is a powerful method that ca be used to solve differetial equatio, ad other mathematical problems. Its stregth lies i the fact that it allows the trasformatio

More information

Answer: 1(A); 2(C); 3(A); 4(D); 5(B); 6(A); 7(C); 8(C); 9(A); 10(A); 11(A); 12(C); 13(C)

Answer: 1(A); 2(C); 3(A); 4(D); 5(B); 6(A); 7(C); 8(C); 9(A); 10(A); 11(A); 12(C); 13(C) Aswer: (A); (C); 3(A); 4(D); 5(B); 6(A); 7(C); 8(C); 9(A); 0(A); (A); (C); 3(C). A two loop positio cotrol system is show below R(s) Y(s) + + s(s +) - - s The gai of the Tacho-geerator iflueces maily the

More information

1. Linear second-order circuits

1. Linear second-order circuits ear eco-orer crcut Sere R crcut Dyamc crcut cotag two capactor or two uctor or oe uctor a oe capactor are calle the eco orer crcut At frt we coer a pecal cla of the eco-orer crcut, amely a ere coecto of

More information

1it is said to be overdamped. When 1, the roots of

1it is said to be overdamped. When 1, the roots of Homework 3 AERE573 Fall 8 Due /8(M) SOLUTIO PROBLEM (4pt) Coider a D order uderdamped tem trafer fuctio H( ) ratio The deomiator i the tem characteritic polomial P( ) (a)(5pt) Ue the quadratic formula,

More information

( ) 2 + k The vertex is ( h, k) ( )( x q) The x-intercepts are x = p and x = q.

( ) 2 + k The vertex is ( h, k) ( )( x q) The x-intercepts are x = p and x = q. A Referece Sheet Number Sets Quadratic Fuctios Forms Form Equatio Stadard Form Vertex Form Itercept Form y ax + bx + c The x-coordiate of the vertex is x b a y a x h The axis of symmetry is x b a + k The

More information

Math 312 Lecture Notes One Dimensional Maps

Math 312 Lecture Notes One Dimensional Maps Math 312 Lecture Notes Oe Dimesioal Maps Warre Weckesser Departmet of Mathematics Colgate Uiversity 21-23 February 25 A Example We begi with the simplest model of populatio growth. Suppose, for example,

More information

Statistics and Chemical Measurements: Quantifying Uncertainty. Normal or Gaussian Distribution The Bell Curve

Statistics and Chemical Measurements: Quantifying Uncertainty. Normal or Gaussian Distribution The Bell Curve Statitic ad Chemical Meauremet: Quatifyig Ucertaity The bottom lie: Do we trut our reult? Should we (or ayoe ele)? Why? What i Quality Aurace? What i Quality Cotrol? Normal or Gauia Ditributio The Bell

More information

The structure of Fourier series

The structure of Fourier series The structure of Fourier series Valery P Dmitriyev Lomoosov Uiversity, Russia Date: February 3, 2011) Fourier series is costructe basig o the iea to moel the elemetary oscillatio 1, +1) by the expoetial

More information

MAT 271 Project: Partial Fractions for certain rational functions

MAT 271 Project: Partial Fractions for certain rational functions MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,

More information

Dynamic System Response

Dynamic System Response Solutio of Liear, Costat-Coefficiet, Ordiary Differetial Equatios Classical Operator Method Laplace Trasform Method Laplace Trasform Properties 1 st -Order Dyamic System Time ad Frequecy Respose d -Order

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science. BACKGROUND EXAM September 30, 2004.

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science. BACKGROUND EXAM September 30, 2004. MASSACHUSETTS INSTITUTE OF TECHNOLOGY Departmet of Electrical Egieerig ad Computer Sciece 6.34 Discrete Time Sigal Processig Fall 24 BACKGROUND EXAM September 3, 24. Full Name: Note: This exam is closed

More information

mx bx kx F t. dt IR I LI V t, Q LQ RQ V t,

mx bx kx F t. dt IR I LI V t, Q LQ RQ V t, Lecture 5 omplex Variables II (Applicatios i Physics) (See hapter i Boas) To see why complex variables are so useful cosider first the (liear) mechaics of a sigle particle described by Newto s equatio

More information

Module 2: z-transform and Discrete Systems

Module 2: z-transform and Discrete Systems Module : -Trasform ad Discrete Systems Prof. Eliathamy Amikairajah ajah Dr. Tharmarajah Thiruvara School of Electrical Egieerig & Telecommuicatios The Uiversity of New South Wales Australia The -Trasform

More information

CHAPTER 1 BASIC CONCEPTS OF INSTRUMENTATION AND MEASUREMENT

CHAPTER 1 BASIC CONCEPTS OF INSTRUMENTATION AND MEASUREMENT CHAPTER 1 BASIC CONCEPTS OF INSTRUMENTATION AND MEASUREMENT 1.1 Classificatio of istrumets Aalog istrumet The measure parameter value is isplay by the moveable poiter. The poiter will move cotiuously with

More information