Introduction & Laplace Transforms Lectures 1 & 2


 Lorin Moody
 4 years ago
 Views:
Transcription
1 Introduction & Lectures 1 & 2, Professor Department of Electrical and Computer Engineering Colorado State University Fall 2016
2 Control System Definition of a Control System Group of components that collectively perform certain desired tasks or maintain a desired result. Control system components can be electrical, mechanical, hydraulic, pneumatic, etc. or any hybrid combination. Basic Ingredients where c(t) = f(u(t))
3 Types of OpenLoop Output variable has no way of influencing the actuating signal. Errors cannot be reduced. ClosedLoop A feedback is added for more accurate and reliable control. Error signal, e(t) = r(t) f(t), drives the actuation via the feedback unit.
4 Open Loop Control System Effects of Feedback: 1 Gain: Reduces the overall gain (negative feedback). 2 Bandwidth (BW): Increases BW so that Gain BW = constant. 3 Stability: Improves stability with proper design. 4 Sensitivity: Def: Sensitivity is measured by variations of system parameters (e.g., gain, pole locations, etc.) wrt environmental changes and aging. Sensitivity for some parameters can be reduced via proper feedback. 5 Noise: Any real system is subject to noise (e.g., thermal noise, brush noise in motors, etc.). Effect of feedback on noise depends on where in the system the noise is introduced.
5  PierreSimon Laplace 1809 Why use the Laplace Transform (LT)? 1  Provides complete solution of differential equation (homogeneous and particular together). 2  Handles initial conditions in the solution. Def. 1: Let G(s) be a function of s = σ + jw. Singularities of G(s) correspond to points on the splane where G(s) and its derivative do NOT exist. Example: G(s) = 1 (s + 1)(s + 2) G(s) does not exist anywhere the denominator equals zero. In this case at s = 1 and s = 2 (i.e. poles of G(s)). (1)
6 Def. 2: G(s) is analytic in region R in splane if it does not have any singularities in R. (So in the example above, G(s) is analytic everywhere except at s = 1, s = 2). Theorem: Given a time function, f(t), Laplace Transform (LT) and its inverse exist if and only if: 1 For every interval t 1 t t 2, f(t) is bounded and has a finite number of max and min and finite number of discontinuities. 2 There exists a real constant α such that 0 e αt f(t) dt is convergent. (Abscissa of Convergence) In other words, e αt f(t) should not be more powerful that e αt (e.g., for e t2 LT doesn t exist).
7 Laplace Transform and its Inverse Then we have: Laplace Transform: L {f(t)} = F (s) = ˆ 0 f(t)e st dt (2) Note: The onesided LT should start from t = 0 to account for discontinuities or an impulse at t = 0. Inverse Laplace Transform: f(t) = L 1 {F (s)} = 1 2π ˆ c+j c j F (s)e st ds (3) where c is a real constant greater than the real parts of all the singularities of F (s). Thus, we have onetoone correspondence between f(t) and F (s) via LT i.e. f(t) F (s). Fortunately, the inverse LT can be obtained using the tables.
8 Properties of Laplace Transform The utilities of LT lies in the following properties. 1  Linearity Let f 1 (t) F 1 (s) and f 2 (t) F 2 (s) then a 1 f 1 (t) + a 2 f 2 (t) a 1 F 1 (s) + a 2 F 2 (s) 2  Differentiation Let f(t) F (s) then d k f(t) dt k s k F (s) s k 1 f(0 ) s k 2 f (1) (0 )... f (k 1) (0 ) where f (i) (t) = di f(t) dt i 3  Integration Let f(t) F (s) then t 0 f(τ)dτ 1 s F (s) In general: t a f(τ)dτ 1 s F (s) s a f(τ)dτ
9 Properties of Laplace Transform Cont. 4  Shift in timedomain Let f(t) F (s) then f(t T )u s (t T ) e T s F (s) where u s (t) is a unit step function. u s (t) = { 1 t 0 0 else 5  Shift in sdomain Let f(t) F (s) then e αt f(t) F (s ± α) 6  Initial Value Theorem Let f(t) F (s) then lim t 0 f(t) = lim s sf (s)
10 Properties of Laplace TransformCont. 7  Final Value Theorem (FVT) Let f(t) F (s) then lim t f(t) = lim s 0 sf (s). Condition: sf (s) is analytic on the jωaxis and in RHS (righthalf side) of splane. Very useful property for steadystate error analysis. Example: Consider F (s) = ω s 2 +ω. If we apply FVT, we get lim 2 s 0 sf (s) = 0. But we know that f(t) = sin ωt with no final value. The reason for this discrepancy is that F (s) has complex conjugate poles on the imaginary axis.
11 Properties of Laplace TransformCont. 8  Convolution in timedomain Suppose f 1 (t) F 1 (s) and f 2 (t) F 2 (s). Then: ˆ t f 1 (τ)f 2 (t τ)dτ = ˆ t 0 0 f 1 (t τ)f 2 (τ)dτ = f 1 (t) f 2 (t) F 1 (s)f 2 (s) where designates convolution. This is a useful property for system analysis. y(t) = u(t) h(t) Y (s) = U(s)H(s) and y(t) = L 1 {Y (s)}
12 Properties of 9  Convolution in sdomain This is dual of property of 8. Let f 1 (t) F 1 (s) and f 2 (t) F 2 (s). Then: f 1 (t)f 2 (t) 1 2πj F 1(s) F 2 (s) = 1 c+j 2πj c j F 1(s ξ)f 2 (ξ)dξ 10  Differentiation in sdomain Let f(t) F (s). Then: tf(t) F (s) = df (s) ds 11Integration in sdomain Let f(t) F (s). Then: f(t) t F (ς)dς s Condition: lim t 0 f(t) t = fixed
13 Inverse Laplace Transform Finding the inverse laplace transform Let X(s) = N(s) D(s), where N(s) is a polynomial of order m and D(s) is a polynomial of order n, n m. We want to find f(t) =. Remark: If n m function X(s) is said to be proper. If n > m it is strictly proper. For n < m, the system becomes noncausal. Case 1: D(s) has real and distinct roots. X(s) = a i s: real and distinct roots of D(s) Apply PFE (partial fraction expansion): N(s) (s + a 1 )... (s + a n ) X(s) = A 1 (s + a 1 ) + + A n (s + a n ) A i = (s + a i )X(s) s= ai
Laplace Transform Part 1: Introduction (I&N Chap 13)
Laplace Transform Part 1: Introduction (I&N Chap 13) Definition of the L.T. L.T. of Singularity Functions L.T. Pairs Properties of the L.T. Inverse L.T. Convolution IVT(initial value theorem) & FVT (final
More informationUnit 2: Modeling in the Frequency Domain Part 2: The Laplace Transform. The Laplace Transform. The need for Laplace
Unit : Modeling in the Frequency Domain Part : Engineering 81: Control Systems I Faculty of Engineering & Applied Science Memorial University of Newfoundland January 1, 010 1 Pair Table Unit, Part : Unit,
More information9.5 The Transfer Function
Lecture Notes on Control Systems/D. Ghose/2012 0 9.5 The Transfer Function Consider the nth order linear, timeinvariant dynamical system. dy a 0 y + a 1 dt + a d 2 y 2 dt + + a d n y 2 n dt b du 0u +
More informationControl Systems. Frequency domain analysis. L. Lanari
Control Systems m i l e r p r a in r e v y n is o Frequency domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic
More informationECEN 420 LINEAR CONTROL SYSTEMS. Lecture 2 Laplace Transform I 1/52
1/52 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 2 Laplace Transform I Linear Time Invariant Systems A general LTI system may be described by the linear constant coefficient differential equation: a n d n
More informationTransform Solutions to LTI Systems Part 3
Transform Solutions to LTI Systems Part 3 Example of second order system solution: Same example with increased damping: k=5 N/m, b=6 Ns/m, F=2 N, m=1 Kg Given x(0) = 0, x (0) = 0, find x(t). The revised
More informationTime Response Analysis (Part II)
Time Response Analysis (Part II). A critically damped, continuoustime, second order system, when sampled, will have (in Z domain) (a) A simple pole (b) Double pole on real axis (c) Double pole on imaginary
More informationDefinition of the Laplace transform. 0 x(t)e st dt
Definition of the Laplace transform Bilateral Laplace Transform: X(s) = x(t)e st dt Unilateral (or onesided) Laplace Transform: X(s) = 0 x(t)e st dt ECE352 1 Definition of the Laplace transform (cont.)
More informationLecture 7: Laplace Transform and Its Applications Dr.Ing. Sudchai Boonto
DrIng Sudchai Boonto Department of Control System and Instrumentation Engineering King Mongkut s Unniversity of Technology Thonburi Thailand Outline Motivation The Laplace Transform The Laplace Transform
More information2.161 Signal Processing: Continuous and Discrete Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 2.6 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS
More informationEE/ME/AE324: Dynamical Systems. Chapter 7: Transform Solutions of Linear Models
EE/ME/AE324: Dynamical Systems Chapter 7: Transform Solutions of Linear Models The Laplace Transform Converts systems or signals from the real time domain, e.g., functions of the real variable t, to the
More informationTime Response of Systems
Chapter 0 Time Response of Systems 0. Some Standard Time Responses Let us try to get some impulse time responses just by inspection: Poles F (s) f(t) splane Time response p =0 s p =0,p 2 =0 s 2 t p =
More informationENGIN 211, Engineering Math. Laplace Transforms
ENGIN 211, Engineering Math Laplace Transforms 1 Why Laplace Transform? Laplace transform converts a function in the time domain to its frequency domain. It is a powerful, systematic method in solving
More informationI Laplace transform. I Transfer function. I Conversion between systems in time, frequencydomain, and transfer
EE C128 / ME C134 Feedback Control Systems Lecture Chapter 2 Modeling in the Frequency Domain Alexandre Bayen Department of Electrical Engineering & Computer Science University of California Berkeley Lecture
More informationLaplace Transforms Chapter 3
Laplace Transforms Important analytical method for solving linear ordinary differential equations.  Application to nonlinear ODEs? Must linearize first. Laplace transforms play a key role in important
More informationLaplace Transforms. Chapter 3. Pierre Simon Laplace Born: 23 March 1749 in BeaumontenAuge, Normandy, France Died: 5 March 1827 in Paris, France
Pierre Simon Laplace Born: 23 March 1749 in BeaumontenAuge, Normandy, France Died: 5 March 1827 in Paris, France Laplace Transforms Dr. M. A. A. Shoukat Choudhury 1 Laplace Transforms Important analytical
More informationControl Systems. Laplace domain analysis
Control Systems Laplace domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic equations define an Input/Output
More informationChapter 6: The Laplace Transform. ChihWei Liu
Chapter 6: The Laplace Transform ChihWei Liu Outline Introduction The Laplace Transform The Unilateral Laplace Transform Properties of the Unilateral Laplace Transform Inversion of the Unilateral Laplace
More informationLTI Systems (Continuous & Discrete)  Basics
LTI Systems (Continuous & Discrete)  Basics 1. A system with an input x(t) and output y(t) is described by the relation: y(t) = t. x(t). This system is (a) linear and timeinvariant (b) linear and timevarying
More informationLecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design.
ISS0031 Modeling and Identification Lecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design. Aleksei Tepljakov, Ph.D. September 30, 2015 Linear Dynamic Systems Definition
More informationELG 3150 Introduction to Control Systems. TA: Fouad Khalil, P.Eng., Ph.D. Student
ELG 350 Introduction to Control Systems TA: Fouad Khalil, P.Eng., Ph.D. Student fkhalil@site.uottawa.ca My agenda for this tutorial session I will introduce the Laplace Transforms as a useful tool for
More informationControl System. Contents
Contents Chapter Topic Page Chapter Chapter Chapter3 Chapter4 Introduction Transfer Function, Block Diagrams and Signal Flow Graphs Mathematical Modeling Control System 35 Time Response Analysis of
More informationECE 380: Control Systems
ECE 380: Control Systems Course Notes: Winter 2014 Prof. Shreyas Sundaram Department of Electrical and Computer Engineering University of Waterloo ii Acknowledgments Parts of these course notes are loosely
More informationECE 3620: Laplace Transforms: Chapter 3:
ECE 3620: Laplace Transforms: Chapter 3: 3.13.4 Prof. K. Chandra ECE, UMASS Lowell September 21, 2016 1 Analysis of LTI Systems in the Frequency Domain Thus far we have understood the relationship between
More informationThe Laplace Transform
The Laplace Transform Introduction There are two common approaches to the developing and understanding the Laplace transform It can be viewed as a generalization of the CTFT to include some signals with
More informationEE Experiment 11 The Laplace Transform and Control System Characteristics
EE216:11 1 EE 216  Experiment 11 The Laplace Transform and Control System Characteristics Objectives: To illustrate computer usage in determining inverse Laplace transforms. Also to determine useful signal
More informationCHEE 319 Tutorial 3 Solutions. 1. Using partial fraction expansions, find the causal function f whose Laplace transform. F (s) F (s) = C 1 s + C 2
CHEE 39 Tutorial 3 Solutions. Using partial fraction expansions, find the causal function f whose Laplace transform is given by: F (s) 0 f(t)e st dt (.) F (s) = s(s+) ; Solution: Note that the polynomial
More informationCHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION. Professor Dae Ryook Yang
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dept. of Chemical and Biological Engineering 51 Road Map of the Lecture V Laplace Transform and Transfer
More informationMATHEMATICAL MODELING OF CONTROL SYSTEMS
1 MATHEMATICAL MODELING OF CONTROL SYSTEMS Sep14 Dr. Mohammed Morsy Outline Introduction Transfer function and impulse response function Laplace Transform Review Automatic control systems Signal Flow
More informationMODELING OF CONTROL SYSTEMS
1 MODELING OF CONTROL SYSTEMS Feb15 Dr. Mohammed Morsy Outline Introduction Differential equations and Linearization of nonlinear mathematical models Transfer function and impulse response function Laplace
More information10 Transfer Matrix Models
MIT EECS 6.241 (FALL 26) LECTURE NOTES BY A. MEGRETSKI 1 Transfer Matrix Models So far, transfer matrices were introduced for finite order state space LTI models, in which case they serve as an important
More informationRaktim Bhattacharya. . AERO 632: Design of Advance Flight Control System. Preliminaries
. AERO 632: of Advance Flight Control System. Preliminaries Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. Preliminaries Signals & Systems Laplace
More informatione st f (t) dt = e st tf(t) dt = L {t f(t)} s
Additional operational properties How to find the Laplace transform of a function f (t) that is multiplied by a monomial t n, the transform of a special type of integral, and the transform of a periodic
More informationControl Systems I. Lecture 5: Transfer Functions. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control DMAVT ETH Zürich
Control Systems I Lecture 5: Transfer Functions Readings: Emilio Frazzoli Institute for Dynamic Systems and Control DMAVT ETH Zürich October 20, 2017 E. Frazzoli (ETH) Lecture 5: Control Systems I 20/10/2017
More informationThe Laplace Transform
The Laplace Transform Generalizing the Fourier Transform The CTFT expresses a timedomain signal as a linear combination of complex sinusoids of the form e jωt. In the generalization of the CTFT to the
More informationLaplace Transforms and use in Automatic Control
Laplace Transforms and use in Automatic Control P.S. Gandhi Mechanical Engineering IIT Bombay Acknowledgements: P.Santosh Krishna, SYSCON Recap Fourier series Fourier transform: aperiodic Convolution integral
More informationModule 4. Related web links and videos. 1. FT and ZT
Module 4 Laplace transforms, ROC, rational systems, Z transform, properties of LT and ZT, rational functions, system properties from ROC, inverse transforms Related web links and videos Sl no Web link
More informationSTABILITY. Have looked at modeling dynamic systems using differential equations. and used the Laplace transform to help find step and impulse
SIGNALS AND SYSTEMS: PAPER 3C1 HANDOUT 4. Dr David Corrigan 1. Electronic and Electrical Engineering Dept. corrigad@tcd.ie www.sigmedia.tv STABILITY Have looked at modeling dynamic systems using differential
More informationControl System (ECE411) Lectures 13 & 14
Control System (ECE411) Lectures 13 & 14, Professor Department of Electrical and Computer Engineering Colorado State University Fall 2016 SteadyState Error Analysis Remark: For a unity feedback system
More informationINC 341 Feedback Control Systems: Lecture 2 Transfer Function of Dynamic Systems I Asst. Prof. Dr.Ing. Sudchai Boonto
INC 341 Feedback Control Systems: Lecture 2 Transfer Function of Dynamic Systems I Asst. Prof. Dr.Ing. Sudchai Boonto Department of Control Systems and Instrumentation Engineering King Mongkut s University
More informationf(t)e st dt. (4.1) Note that the integral defining the Laplace transform converges for s s 0 provided f(t) Ke s 0t for some constant K.
4 Laplace transforms 4. Definition and basic properties The Laplace transform is a useful tool for solving differential equations, in particular initial value problems. It also provides an example of integral
More informationRaktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Dynamic Response
.. AERO 422: Active Controls for Aerospace Vehicles Dynamic Response Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. . Previous Class...........
More informationCourse roadmap. Step response for 2ndorder system. Step response for 2ndorder system
ME45: Control Systems Lecture Time response of ndorder systems Prof. Clar Radcliffe and Prof. Jongeun Choi Department of Mechanical Engineering Michigan State University Modeling Laplace transform Transfer
More informationSTABILITY ANALYSIS. Asystemmaybe stable, neutrallyormarginallystable, or unstable. This can be illustrated using cones: Stable Neutral Unstable
ECE4510/5510: Feedback Control Systems. 5 1 STABILITY ANALYSIS 5.1: Boundedinput boundedoutput (BIBO) stability Asystemmaybe stable, neutrallyormarginallystable, or unstable. This can be illustrated
More informationControl Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control DMAVT ETH Zürich
Control Systems I Lecture 6: Poles and Zeros Readings: Emilio Frazzoli Institute for Dynamic Systems and Control DMAVT ETH Zürich October 27, 2017 E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/2017
More informationReview: transient and steadystate response; DC gain and the FVT Today s topic: systemmodeling diagrams; prototype 2ndorder system
Plan of the Lecture Review: transient and steadystate response; DC gain and the FVT Today s topic: systemmodeling diagrams; prototype 2ndorder system Plan of the Lecture Review: transient and steadystate
More informationOutline. Classical Control. Lecture 2
Outline Outline Outline Review of Material from Lecture 2 New Stuff  Outline Review of Lecture System Performance Effect of Poles Review of Material from Lecture System Performance Effect of Poles 2 New
More informationControl System Design
ELEC ENG 4CL4: Control System Design Notes for Lecture #4 Monday, January 13, 2003 Dr. Ian C. Bruce Room: CRL229 Phone ext.: 26984 Email: ibruce@mail.ece.mcmaster.ca Impulse and Step Responses of ContinuousTime
More informationBasic Procedures for Common Problems
Basic Procedures for Common Problems ECHE 550, Fall 2002 Steady State Multivariable Modeling and Control 1 Determine what variables are available to manipulate (inputs, u) and what variables are available
More informationGATE EE Topic wise Questions SIGNALS & SYSTEMS
www.gatehelp.com GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C)
More informationThe Laplace Transform
The Laplace Transform Syllabus ECE 316, Spring 2015 Final Grades Homework (6 problems per week): 25% Exams (midterm and final): 50% (25:25) Random Quiz: 25% Textbook M. Roberts, Signals and Systems, 2nd
More informationLinear Systems Theory
ME 3253 Linear Systems Theory Review Class Overview and Introduction 1. How to build dynamic system model for physical system? 2. How to analyze the dynamic system?  Time domain  Frequency domain (Laplace
More informationECE317 : Feedback and Control
ECE317 : Feedback and Control Lecture : Stability RouthHurwitz stability criterion Dr. Richard Tymerski Dept. of Electrical and Computer Engineering Portland State University 1 Course roadmap Modeling
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 22: The Nyquist Criterion Overview In this Lecture, you will learn: Complex Analysis The Argument Principle The Contour
More informationMath 353 Lecture Notes Week 6 Laplace Transform: Fundamentals
Math 353 Lecture Notes Week 6 Laplace Transform: Fundamentals J. Wong (Fall 217) October 7, 217 What did we cover this week? Introduction to the Laplace transform Basic theory Domain and range of L Key
More informationEE482: Digital Signal Processing Applications
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu EE482: Digital Signal Processing Applications Spring 2014 TTh 14:3015:45 CBC C222 Lecture 05 IIR Design 14/03/04 http://www.ee.unlv.edu/~b1morris/ee482/
More information22 APPENDIX 1: MATH FACTS
22 APPENDIX : MATH FACTS 22. Vectors 22.. Definition A vector has a dual definition: It is a segment of a a line with direction, or it consists of its projection on a reference system xyz, usually orthogonal
More informationA sufficient condition for the existence of the Fourier transform of f : R C is. f(t) dt <. f(t) = 0 otherwise. dt =
Fourier transform Definition.. Let f : R C. F [ft)] = ˆf : R C defined by The Fourier transform of f is the function F [ft)]ω) = ˆfω) := ft)e iωt dt. The inverse Fourier transform of f is the function
More informationComputing inverse Laplace Transforms.
Review Exam 3. Sections 4.4.5 in Lecture Notes. 60 minutes. 7 problems. 70 grade attempts. (0 attempts per problem. No partial grading. (Exceptions allowed, ask you TA. Integration table included. Complete
More informationA system that is both linear and timeinvariant is called linear timeinvariant (LTI).
The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Lecture Notes: Time, Frequency & Transform Domains February 28, 2012 Signals & Systems Signals are mapped
More informationReview of Linear TimeInvariant Network Analysis
D1 APPENDIX D Review of Linear TimeInvariant Network Analysis Consider a network with input x(t) and output y(t) as shown in Figure D1. If an input x 1 (t) produces an output y 1 (t), and an input x
More information+ + LAPLACE TRANSFORM. Differentiation & Integration of Transforms; Convolution; Partial Fraction Formulas; Systems of DEs; Periodic Functions.
COLOR LAYER red LAPLACE TRANSFORM Differentiation & Integration of Transforms; Convolution; Partial Fraction Formulas; Systems of DEs; Periodic Functions. + Differentiation of Transforms. F (s) e st f(t)
More informationECEN 605 LINEAR SYSTEMS. Lecture 20 Characteristics of Feedback Control Systems II Feedback and Stability 1/27
1/27 ECEN 605 LINEAR SYSTEMS Lecture 20 Characteristics of Feedback Control Systems II Feedback and Stability Feedback System Consider the feedback system u + G ol (s) y Figure 1: A unity feedback system
More informationIntroduction to Controls
EE 474 Review Exam 1 Name Answer each of the questions. Show your work. Note were essaytype answers are requested. Answer with complete sentences. Incomplete sentences will count heavily against the grade.
More informationNyquist Criterion For Stability of Closed Loop System
Nyquist Criterion For Stability of Closed Loop System Prof. N. Puri ECE Department, Rutgers University Nyquist Theorem Given a closed loop system: r(t) + KG(s) = K N(s) c(t) H(s) = KG(s) +KG(s) = KN(s)
More informationECE317 : Feedback and Control
ECE317 : Feedback and Control Lecture : RouthHurwitz stability criterion Examples Dr. Richard Tymerski Dept. of Electrical and Computer Engineering Portland State University 1 Course roadmap Modeling
More informationAdvanced Analog Building Blocks. Prof. Dr. Peter Fischer, Dr. Wei Shen, Dr. Albert Comerma, Dr. Johannes Schemmel, etc
Advanced Analog Building Blocks Prof. Dr. Peter Fischer, Dr. Wei Shen, Dr. Albert Comerma, Dr. Johannes Schemmel, etc 1 Topics 1. S domain and Laplace Transform Zeros and Poles 2. Basic and Advanced current
More informationStability. X(s) Y(s) = (s + 2) 2 (s 2) System has 2 poles: points where Y(s) > at s = +2 and s = 2. Y(s) 8X(s) G 1 G 2
Stability 8X(s) X(s) Y(s) = (s 2) 2 (s 2) System has 2 poles: points where Y(s) > at s = 2 and s = 2 If all poles are in region where s < 0, system is stable in Fourier language s = jω G 0  x3 x7 Y(s)
More informationLinear Control Systems Solution to Assignment #1
Linear Control Systems Solution to Assignment # Instructor: H. Karimi Issued: Mehr 0, 389 Due: Mehr 8, 389 Solution to Exercise. a) Using the superposition property of linear systems we can compute the
More information20.6. Transfer Functions. Introduction. Prerequisites. Learning Outcomes
Transfer Functions 2.6 Introduction In this Section we introduce the concept of a transfer function and then use this to obtain a Laplace transform model of a linear engineering system. (A linear engineering
More informationMA 201, Mathematics III, JulyNovember 2018, Laplace Transform (Contd.)
MA 201, Mathematics III, JulyNovember 2018, Laplace Transform (Contd.) Lecture 19 Lecture 19 MA 201, PDE (2018) 1 / 24 Application of Laplace transform in solving ODEs ODEs with constant coefficients
More informationSchool of Mechanical Engineering Purdue University
Case Study ME375 Frequency Response  1 Case Study SUPPORT POWER WIRE DROPPERS Electric train derives power through a pantograph, which contacts the power wire, which is suspended from a catenary. During
More informationProfessor Fearing EE C128 / ME C134 Problem Set 7 Solution Fall 2010 Jansen Sheng and Wenjie Chen, UC Berkeley
Professor Fearing EE C8 / ME C34 Problem Set 7 Solution Fall Jansen Sheng and Wenjie Chen, UC Berkeley. 35 pts Lag compensation. For open loop plant Gs ss+5s+8 a Find compensator gain Ds k such that the
More informationL2 gains and system approximation quality 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.242, Fall 24: MODEL REDUCTION L2 gains and system approximation quality 1 This lecture discusses the utility
More informationNotes for ECE320. Winter by R. Throne
Notes for ECE3 Winter 45 by R. Throne Contents Table of Laplace Transforms 5 Laplace Transform Review 6. Poles and Zeros.................................... 6. Proper and Strictly Proper Transfer Functions...................
More informationEE 3054: Signals, Systems, and Transforms Summer It is observed of some continuoustime LTI system that the input signal.
EE 34: Signals, Systems, and Transforms Summer 7 Test No notes, closed book. Show your work. Simplify your answers. 3. It is observed of some continuoustime LTI system that the input signal = 3 u(t) produces
More informationControl of Manufacturing Processes
Control of Manufacturing Processes Subject 2.830 Spring 2004 Lecture #18 Basic Control Loop Analysis" April 15, 2004 Revisit Temperature Control Problem τ dy dt + y = u τ = time constant = gain y ss =
More informationTopic # Feedback Control Systems
Topic #1 16.31 Feedback Control Systems Motivation Basic Linear System Response Fall 2007 16.31 1 1 16.31: Introduction r(t) e(t) d(t) y(t) G c (s) G(s) u(t) Goal: Design a controller G c (s) so that the
More informationDigital Control & Digital Filters. Lectures 1 & 2
Digital Controls & Digital Filters Lectures 1 & 2, Professor Department of Electrical and Computer Engineering Colorado State University Spring 2017 Digital versus Analog Control Systems Block diagrams
More informationagree w/input bond => + sign disagree w/input bond =>  sign
1 ME 344 REVIEW FOR FINAL EXAM LOCATION: CPE 2.204 M. D. BRYANT DATE: Wednesday, May 7, 2008 9noon Finals week office hours: May 6, 47 pm Permitted at final exam: 1 sheet of formulas & calculator I.
More informationEE C128 / ME C134 Final Exam Fall 2014
EE C128 / ME C134 Final Exam Fall 2014 December 19, 2014 Your PRINTED FULL NAME Your STUDENT ID NUMBER Number of additional sheets 1. No computers, no tablets, no connected device (phone etc.) 2. Pocket
More informationAn Introduction to Control Systems
An Introduction to Control Systems Signals and Systems: 3C1 Control Systems Handout 1 Dr. David Corrigan Electronic and Electrical Engineering corrigad@tcd.ie November 21, 2012 Recall the concept of a
More informationDr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Review
Week Date Content Notes 1 6 Mar Introduction 2 13 Mar Frequency Domain Modelling 3 20 Mar Transient Performance and the splane 4 27 Mar Block Diagrams Assign 1 Due 5 3 Apr Feedback System Characteristics
More informationHonors Differential Equations
MIT OpenCourseWare http://ocw.mit.edu 8.034 Honors Differential Equations Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. LECTURE 20. TRANSFORM
More informationANALOG AND DIGITAL SIGNAL PROCESSING CHAPTER 3 : LINEAR SYSTEM RESPONSE (GENERAL CASE)
3. Linear System Response (general case) 3. INTRODUCTION In chapter 2, we determined that : a) If the system is linear (or operate in a linear domain) b) If the input signal can be assumed as periodic
More informationChapter 7. Digital Control Systems
Chapter 7 Digital Control Systems 1 1 Introduction In this chapter, we introduce analysis and design of stability, steadystate error, and transient response for computercontrolled systems. Transfer functions,
More informationLOPE3202: Communication Systems 10/18/2017 2
By Lecturer Ahmed Wael Academic Year 20172018 LOPE3202: Communication Systems 10/18/2017 We need tools to build any communication system. Mathematics is our premium tool to do work with signals and systems.
More informationMath 2C03  Differential Equations. Slides shown in class  Winter Laplace Transforms. March 4, 5, 9, 11, 12, 16,
Math 2C03  Differential Equations Slides shown in class  Winter 2015 Laplace Transforms March 4, 5, 9, 11, 12, 16, 18... 2015 Laplace Transform used to solve linear ODEs and systems of linear ODEs with
More informationDynamic Response. Assoc. Prof. Enver Tatlicioglu. Department of Electrical & Electronics Engineering Izmir Institute of Technology.
Dynamic Response Assoc. Prof. Enver Tatlicioglu Department of Electrical & Electronics Engineering Izmir Institute of Technology Chapter 3 Assoc. Prof. Enver Tatlicioglu (EEE@IYTE) EE362 Feedback Control
More informationIdentification Methods for Structural Systems. Prof. Dr. Eleni Chatzi System Stability  26 March, 2014
Prof. Dr. Eleni Chatzi System Stability  26 March, 24 Fundamentals Overview System Stability Assume given a dynamic system with input u(t) and output x(t). The stability property of a dynamic system can
More informationControl Systems Design
ELEC4410 Control Systems Design Lecture 18: State Feedback Tracking and State Estimation Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 18:
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 21: Stability Margins and Closing the Loop Overview In this Lecture, you will learn: Closing the Loop Effect on Bode Plot Effect
More informationMATH4406 (Control Theory) Unit 1: Introduction Prepared by Yoni Nazarathy, July 21, 2012
MATH4406 (Control Theory) Unit 1: Introduction Prepared by Yoni Nazarathy, July 21, 2012 Unit Outline Introduction to the course: Course goals, assessment, etc... What is Control Theory A bit of jargon,
More informationGEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM. COURSE: ECE 3084A (Prof. Michaels)
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM DATE: 30Apr14 COURSE: ECE 3084A (Prof. Michaels) NAME: STUDENT #: LAST, FIRST Write your name on the front page
More informationLINEAR SYSTEMS. J. Elder PSYC 6256 Principles of Neural Coding
LINEAR SYSTEMS Linear Systems 2 Neural coding and cognitive neuroscience in general concerns inputoutput relationships. Inputs Light intensity Presynaptic action potentials Number of items in display
More informationChemical Engineering 436 Laplace Transforms (1)
Chemical Engineering 436 Laplace Transforms () Why Laplace Transforms?? ) Converts differential equations to algebraic equations facilitates combination of multiple components in a system to get the total
More informationOutline. Classical Control. Lecture 1
Outline Outline Outline 1 Introduction 2 Prerequisites Block diagram for system modeling Modeling Mechanical Electrical Outline Introduction Background Basic Systems Models/Transfers functions 1 Introduction
More informationEE451/551: Digital Control. Chapter 3: Modeling of Digital Control Systems
EE451/551: Digital Control Chapter 3: Modeling of Digital Control Systems Common Digital Control Configurations AsnotedinCh1 commondigitalcontrolconfigurations As noted in Ch 1, common digital control
More informationProblem Set 4 Solutions 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6245: MULTIVARIABLE CONTROL SYSTEMS by A Megretski Problem Set 4 Solutions Problem 4T For the following transfer
More informationSignals and Systems. Spring Room 324, Geology Palace, ,
Signals and Systems Spring 2013 Room 324, Geology Palace, 13756569051, zhukaiguang@jlu.edu.cn Chapter 10 The ZTransform 1) ZTransform 2) Properties of the ROC of the ztransform 3) Inverse ztransform
More information