Special Mathematics Laplace Transform
|
|
- Milo Bridges
- 5 years ago
- Views:
Transcription
1 Special Mathematics Laplace Transform March 28
2 ii
3 Nature laughs at the difficulties of integration. Pierre-Simon Laplace 4 Laplace Transform Motivation
4 Properties of the Laplace transform the Laplace transform of a function f(t) is defined by: L[f(t)](p) = f(s)e sp ds and it transforms a function f(t) into another depending on p usually denoted with L[f(t)](p) or F (p). sometimes the function f(t) is called the source function. it is a linear transform: L[af(t) + bg(t)](p) = al[f(t)](p) + bl[g(t)](p) the Laplace transform is one to one: L[f] = L[g] = f = g. change of scale property: L[f(at)](p) = ( p ) a L[f(t)] a exponential scaling: time delay: L[e at f(t)](p) = L[f(t)](p a) L[f(t a)](p) = e ap L[f(t)](p) transform of the integral: [ t ] L f(s)ds (p) = p L[f(t)](p) Convolution of two functions: is another function defined as the integral: (f * g)(t) = and it provides the nice property: sometimes called Borel s theorem. transform the derivative: t f(t s)g(s)ds L[(f * g)(t)](p) = L[f(t)](p) L[g(t)](p) L[f (n) (t)](p) = p n L[f(t)](p) p n f() p n 2 f ()... f (n ) () derivate the transform: (L[f(t)](p)) (n) = ( ) n L[t n f(t)](p) the Laplace transform can be used to compute improper integrals according to the formula: [ ] f(t) F (p)dp = L (p) t if we recognize the integrand as being a Laplace transform of some source function f(t). 2
5 A formula for the inverse L : f(t) = L [F (p)](t) = Res(F (p)e pt ) all poles of F (p) Example: Find the inverse transform of F (p) = (p + 3) 2 (p ) In this case the function F (p) has a pole of order 2 in p = 3 and one of order in p =. One has the formulae: Res(F (p)e pt e pt, ) = lim(p ) p (p + 3) 2 (p ) = e t ( + 3) 2 (in the above residue t behaves like a parameter) Res(F (p)e pt, 3) = lim ( (p + 3) 2 e pt ) p 3 (p + 3) 2 = te 3t e 3t (p ) thus: [ f(t) = L (p + 3) 2 (p ) (derivation with respect to p) ] (t) = et 6 te 3t 4 e 3t, t. 6 Initial value theorem: lim t t> f(t) = lim p L[f(t)](p) p Final value theorem: lim f(t) = lim p L[f(t)](p) t p 3
6 Function Time t-domain f(t) Laplace (frequency) domain F (p) unit impulse δ(t) delayed impulse δ(t a) e ap unit step u(t) p ramp function t u(t) p 2 n-th power t n u(t) n! p n+ α-th power t α u(t) Γ(α+) p α+ exponential e ωt u(t) p ω general exponential a ωt u(t) p ln ω sine sin(ωt) u(t) ω p 2 +ω 2 cosine cos(ωt) u(t) p p 2 +ω 2 hyperbolic sine sinh(ωt) u(t) ω p 2 ω 2 hyperbolic cosine cosh(ωt) u(t) p p 2 ω 2 exp. decaying sine wave e at sin(ωt) u(t) ω (p+a) 2 +ω 2 exp. decaying cosine wave e at cos(ωt) u(t) p+a (p+a) 2 +ω 2 exp. decaying sinh wave e at sinh(ωt) u(t) ω (p+a) 2 ω 2 exp. decaying cosh wave e at cosh(ωt) u(t) p+a (p+a) 2 ω 2 4
7 Solved problems Problem. Find the source functions for the following Laplace transforms: a) F (p) = p 2 3p + 2 b) G (p) = p 2 (p 2 + ). Solution: a) We decompose the Laplace transform in: F (p) = making use of the formula: p 2 3p + 2 = (p ) (p 2) = p 2 p L [ e at] (p) = p a obtained from the table, one gets the source function: f (t) = e 2t e t. b) We decompose again the function making of the theory of rational functions: Ap + B Cp + D p 2 (p 2 = + ) p 2 + p 2 +, The identity yields, after identification, the coefficients : which implies Using now the formulae: L [t n ] (p) = one gets the source function A =, B =, C =, D =, G (p) = p 2 p 2 +. n! p n+, L [sin ωt] (p) = ω p 2 + ω 2, g (t) = t sin t. Problem 2. Solve the Cauchy problem: x + 2x + 5x = x () = x () =, using the Laplace transform. 5
8 Solution: Using the Laplace transform we have: which implies: thus: L [x] (p) = F (p), L [x ] (p) = pf (p) x () = pf (p), L [x ] (p) = p 2 F (p) p x () x () = p 2 F (p) p, F (p) = p 2 F (p) p + 2 [pf (p) ] + 5F (p) =, = and finally one gets: p + 2 p 2 + 2p + 5 = p + (p + ) (p + ) p + (p + ) (p + ) x (t) = e t cos 2t + 2 e t sin 2t ( = e t cos 2t + ) 2 sin 2t. Problem 3. Solve the initial value problem x (t) + x (t) 2x(t) = t where x () = x () = and x () =. Solution: The above equation is a linear nonhomogeneous equation. apply the Laplace transform to both terms: We L [x] (p) = F (p), L [x ] (p) = p 2 F (p) p x () x () = p 2 F (p) L [x ] (p) = p 3 F (p) p 2 x () p x () x () = p 3 F (p) +, L [t] (p) = p 2, Using the linearity of the Laplace transform the above identities provide an algebraic equation with the unknown F (p): Now: p 3 F (p) + +p 2 F (p) 2F (p) = p 2 F (p) = p 2 p 2 (p 3 + p 2 2) = ( p) ( + p) p 2 (p ) (p 2 + 2p + 2) = p + p 2 (p 2 + 2p + 2) = 2 p (p + ) 2 + 6
9 and finally we get the particular solution of this initial value problem using the table with the common values of the Laplace transform: x (t) = L [F (p)](t) = 2 t + 2 e t sin t. Problem 4. Solve the following system of linear equations: x = x + 2y + t y = 2x + y + t, with the initial data x () = 2, y () = 4. Solution: We apply the Laplace transform to this system: L [x] (p) = X (p), L [y] (p) = Y (p) L [x ] (p) = px (p) x () = px (p) 2 L [y ] (p) = py (p) y () = py (p) 4, L [t] (p) = p 2, which leads to the algebraic linear system: px (p) 2 X (p) 2Y (p) = p 2 2X (p) + py (p) 4 Y (p) = p 2 thus: X (p) + Y (p) = ( 6 + 2p ) p 3 2 X (p) Y (p) = 2 p +, implies: X (p) = 3 p 3 + p 2 (p 3) p + = 3 p 3 9 p 3 p p 3 p + = 28 9 p 3 9 p 3 p 2 p + and leads to the source function: x (t) = 28 9 e3t 9 3 t e t. In order to find the source function y (t) we can use the first equation of the given system of differential equations. 7
10 The derivative of x (t) is: and so: x (t) = 28 3 e3t 3 + e t y (t) = x x t = e3t 9 3 t + e t. Eventually the solution of the given linear system is: x (t) = 28 9 e3t 9 t e t 3 y (t) = 28 9 e3t 9 t + e t 3 Problem 5. Solve the equation x + x = cos t, with x () =, x () = 2. Solution: We try to apply the Laplace transform to this equation. We can not find the Laplace tranform of for the moment, thus we pursue using cos t the expression L [ cos t ] (p) in the right side. For the left side: L [x] (p) = X (p), L [x ] (p) = p 2 X (p) p x () x () = p 2 X (p) 2, provides the equation: with the solution: or equivalently: We ll make use of the formula: to get: In conclusion: p 2 X (p) 2 + X (p) = L. [ ] (p), cos t X (p) = 2 p [ ] p 2 + L (p) cos t [ ] X (p) = 2L (sin t) + L (sin t) L (p) cos t L[f(t)](p) L[g(t)](p) = L[(f * g)(t)](p) x (t) = 2 sin t + sin t * t x (t) = 2 sin t + = 2 sin t + t sin (t τ) cos t cos τ dτ sin t cos τ sin τ cos t dτ cos τ = 2 sin t + t sin t + cos t ln (cos t). 8
11 t Problem 6. Solve the equation x (t) = 2 sin 4t + sin 4 (t u) x (u) du. Solution: The given equation can be written in the equivalent form: t x (t) x (u) sin 4 (t u) du = 2 sin 4t. The Laplace transform of the right part is: L [2 sin 4t] = In the left side Borel s theorem implies: L t One gets the equation: with the solution: X (p) = 8 p 2 + 6, x (u) sin 4 (t u) du (p) = L [x (t) * sin 4t] (p) X (p) X (p) = X (p) 4 p = 8 p 2 + 6, 8 p = 8 p 2 + ( 2 3 ) 2 = p p 2 + ( 2 3 ) 2 which provides the solution of the given integral equation: x (t) = 8 ( 2 3 sin 2 ) 3t = 4 3 ( sin 2 ) 3t. 3 Problem 7. Solve the Cauchy problem: x + tx x = x () = x () =, using the Laplace transform. Solution: We apply the Laplace transform: L [x] (p) = X (p), L [tx ] (p) = [px (p)] + x () = = X (p) px (p) L [x ] (p) = p 2 X (p) px () x () = p 2 X (p), 9
12 in order to transform the equation into another differential equation: X (p) + 2 p2 p X (p) = p. This is a linear nonhomogeneous equation, written in the standard form as: X (p) + a (p) X (p) = b (p) with the general solution: X (p) = e a (p) dp k + b (p) e a (p) dp dp. In our case a (p) = 2 p2 p and b (p) = p lead to: X (p) = k p 2 e 2 p 2 + p 2. Making use of the intial data x () =, the transform X (p) has to be, for the initial value theorem states: lim p px (p) = lim x (t) =. t On has to impose the condition k =, which gives X (p) = p 2. Eventually, the solution of the Cauchy problem is x (t) = t.
13 Proposed problems Problem. Find the source function for the Laplace transform F (p) = p 3 5p 2 + 6p. Problem 2. Solve the Cauchy problem x x 6x = x () = x () =, using the Laplace transform technique. Problem 3. Solve the equation: x + 2x + 2x + x =, with the initial data x () = x () = x () =. Problem 4. Solve the linear system of differential equations: x + 4x + 4y =, x () = 3. y + 2x + 6y =, y () = 5 Problem 5. Solve the integral equation: t x (t) = x (u) cos (t u) du, cu x () =. Problem 6. Solve the Cauchy problem tx + x + x = x () = x () =, using the Laplace transform.
14 2
ENGIN 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 informationLaplace 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 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 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 informationf (t) K(t, u) d t. f (t) K 1 (t, u) d u. Integral Transform Inverse Fourier Transform
Integral Transforms Massoud Malek An integral transform maps an equation from its original domain into another domain, where it might be manipulated and solved much more easily than in the original domain.
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 n-th order linear, time-invariant 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 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 informationPractice Problems For Test 3
Practice Problems For Test 3 Power Series Preliminary Material. Find the interval of convergence of the following. Be sure to determine the convergence at the endpoints. (a) ( ) k (x ) k (x 3) k= k (b)
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 informationSeries FOURIER SERIES. Graham S McDonald. A self-contained Tutorial Module for learning the technique of Fourier series analysis
Series FOURIER SERIES Graham S McDonald A self-contained Tutorial Module for learning the technique of Fourier series analysis Table of contents Begin Tutorial c 24 g.s.mcdonald@salford.ac.uk 1. Theory
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 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 informationThe Laplace Transform
The Laplace Transform Generalizing the Fourier Transform The CTFT expresses a time-domain signal as a linear combination of complex sinusoids of the form e jωt. In the generalization of the CTFT to the
More informationPractice Problems For Test 3
Practice Problems For Test 3 Power Series Preliminary Material. Find the interval of convergence of the following. Be sure to determine the convergence at the endpoints. (a) ( ) k (x ) k (x 3) k= k (b)
More informationMATH 251 Examination II April 4, 2016 FORM A. Name: Student Number: Section:
MATH 251 Examination II April 4, 2016 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must
More information(f g)(t) = Example 4.5.1: Find f g the convolution of the functions f(t) = e t and g(t) = sin(t). Solution: The definition of convolution is,
.5. Convolutions and Solutions Solutions of initial value problems for linear nonhomogeneous differential equations can be decomposed in a nice way. The part of the solution coming from the initial data
More informationTHE UNIVERSITY OF WESTERN ONTARIO. Applied Mathematics 375a Instructor: Matt Davison. Final Examination December 14, :00 12:00 a.m.
THE UNIVERSITY OF WESTERN ONTARIO London Ontario Applied Mathematics 375a Instructor: Matt Davison Final Examination December 4, 22 9: 2: a.m. 3 HOURS Name: Stu. #: Notes: ) There are 8 question worth
More informationMATH 251 Examination II April 3, 2017 FORM A. Name: Student Number: Section:
MATH 251 Examination II April 3, 2017 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must
More informationMA 201, Mathematics III, July-November 2018, Laplace Transform (Contd.)
MA 201, Mathematics III, July-November 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 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 information(an improper integral)
Chapter 7 Laplace Transforms 7.1 Introduction: A Mixing Problem 7.2 Definition of the Laplace Transform Def 7.1. Let f(t) be a function on [, ). The Laplace transform of f is the function F (s) defined
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 informationFourier transforms, Generalised functions and Greens functions
Fourier transforms, Generalised functions and Greens functions T. Johnson 2015-01-23 Electromagnetic Processes In Dispersive Media, Lecture 2 - T. Johnson 1 Motivation A big part of this course concerns
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 informationMA 266 Review Topics - Exam # 2 (updated)
MA 66 Reiew Topics - Exam # updated Spring First Order Differential Equations Separable, st Order Linear, Homogeneous, Exact Second Order Linear Homogeneous with Equations Constant Coefficients The differential
More informationThe Laplace Transform
C H A P T E R 6 The Laplace Transform Many practical engineering problems involve mechanical or electrical systems acted on by discontinuous or impulsive forcing terms. For such problems the methods described
More informationChapter 6 The Laplace Transform
Ordinary Differential Equations (Math 2302) 2017-2016 Chapter 6 The Laplace Transform Many practical engineering problems involve mechanical or electrical systems acted on by discontinuous or impulsive
More informationLinear Differential Equations. Problems
Chapter 1 Linear Differential Equations. Problems 1.1 Introduction 1.1.1 Show that the function ϕ : R R, given by the expression ϕ(t) = 2e 3t for all t R, is a solution of the Initial Value Problem x =
More informationMathematical Methods and its Applications (Solution of assignment-12) Solution 1 From the definition of Fourier transforms, we have.
For 2 weeks course only Mathematical Methods and its Applications (Solution of assignment-2 Solution From the definition of Fourier transforms, we have F e at2 e at2 e it dt e at2 +(it/a dt ( setting (
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 5: Calculating the Laplace Transform of a Signal Introduction In this Lecture, you will learn: Laplace Transform of Simple
More informationMATH 251 Examination II November 5, 2018 FORM A. Name: Student Number: Section:
MATH 251 Examination II November 5, 2018 FORM A Name: Student Number: Section: This exam has 14 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work
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 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 informationMA 201, Mathematics III, July-November 2016, Laplace Transform
MA 21, Mathematics III, July-November 216, Laplace Transform Lecture 18 Lecture 18 MA 21, PDE (216) 1 / 21 Laplace Transform Let F : [, ) R. If F(t) satisfies the following conditions: F(t) is piecewise
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 informationMATH 251 Final Examination December 16, 2015 FORM A. Name: Student Number: Section:
MATH 5 Final Examination December 6, 5 FORM A Name: Student Number: Section: This exam has 7 questions for a total of 5 points. In order to obtain full credit for partial credit problems, all work must
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 time-invariant (b) linear and time-varying
More informationLINEAR RESPONSE THEORY
MIT Department of Chemistry 5.74, Spring 5: Introductory Quantum Mechanics II Instructor: Professor Andrei Tokmakoff p. 8 LINEAR RESPONSE THEORY We have statistically described the time-dependent behavior
More informationMath 216 Second Midterm 19 March, 2018
Math 26 Second Midterm 9 March, 28 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that
More informationECE 3620: Laplace Transforms: Chapter 3:
ECE 3620: Laplace Transforms: Chapter 3: 3.1-3.4 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 informationContinuous-Time Frequency Response (II) Lecture 28: EECS 20 N April 2, Laurent El Ghaoui
EECS 20 N April 2, 2001 Lecture 28: Continuous-Time Frequency Response (II) Laurent El Ghaoui 1 annoucements homework due on Wednesday 4/4 at 11 AM midterm: Friday, 4/6 includes all chapters until chapter
More informationDifferential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Undetermined Coefficients Page 1
Differential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Undetermined Coefficients Page 1 Questions Example (3.5.3) Find a general solution of the differential equation y 2y 3y = 3te
More informationCITY UNIVERSITY LONDON
No: CITY UNIVERSITY LONDON BEng (Hons)/MEng (Hons) Degree in Civil Engineering BEng (Hons)/MEng (Hons) Degree in Civil Engineering with Surveying BEng (Hons)/MEng (Hons) Degree in Civil Engineering with
More informationMATH 251 Final Examination December 16, 2014 FORM A. Name: Student Number: Section:
MATH 2 Final Examination December 6, 204 FORM A Name: Student Number: Section: This exam has 7 questions for a total of 0 points. In order to obtain full credit for partial credit problems, all work must
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. 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 informationLecture 7: Laplace Transform and Its Applications Dr.-Ing. Sudchai Boonto
Dr-Ing 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 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 informationENGI 9420 Lecture Notes 1 - ODEs Page 1.01
ENGI 940 Lecture Notes - ODEs Page.0. Ordinary Differential Equations An equation involving a function of one independent variable and the derivative(s) of that function is an ordinary differential equation
More informationMAT389 Fall 2016, Problem Set 11
MAT389 Fall 216, Problem Set 11 Improper integrals 11.1 In each of the following cases, establish the convergence of the given integral and calculate its value. i) x 2 x 2 + 1) 2 ii) x x 2 + 1)x 2 + 2x
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 informationMath 251 December 14, 2005 Answer Key to Final Exam. 1 18pt 2 16pt 3 12pt 4 14pt 5 12pt 6 14pt 7 14pt 8 16pt 9 20pt 10 14pt Total 150pt
Name Section Math 51 December 14, 5 Answer Key to Final Exam There are 1 questions on this exam. Many of them have multiple parts. The point value of each question is indicated either at the beginning
More information06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 1
IV. Continuous-Time Signals & LTI Systems [p. 3] Analog signal definition [p. 4] Periodic signal [p. 5] One-sided signal [p. 6] Finite length signal [p. 7] Impulse function [p. 9] Sampling property [p.11]
More informationSolutions to Assignment 7
MTHE 237 Fall 215 Solutions to Assignment 7 Problem 1 Show that the Laplace transform of cos(αt) satisfies L{cosαt = s s 2 +α 2 L(cos αt) e st cos(αt)dt A s α e st sin(αt)dt e stsin(αt) α { e stsin(αt)
More informationMATH 251 Final Examination May 4, 2015 FORM A. Name: Student Number: Section:
MATH 251 Final Examination May 4, 2015 FORM A Name: Student Number: Section: This exam has 16 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all work must
More informatione iωt dt and explained why δ(ω) = 0 for ω 0 but δ(0) =. A crucial property of the delta function, however, is that
Phys 53 Fourier Transforms In this handout, I will go through the derivations of some of the results I gave in class (Lecture 4, /). I won t reintroduce the concepts though, so if you haven t seen the
More information1 otherwise. Note that the area of the pulse is one. The Dirac delta function (a.k.a. the impulse) can be defined using the pulse as follows:
The Dirac delta function There is a function called the pulse: { if t > Π(t) = 2 otherwise. Note that the area of the pulse is one. The Dirac delta function (a.k.a. the impulse) can be defined using the
More informationPartial Differential Equations for Engineering Math 312, Fall 2012
Partial Differential Equations for Engineering Math 312, Fall 2012 Jens Lorenz July 17, 2012 Contents Department of Mathematics and Statistics, UNM, Albuquerque, NM 87131 1 Second Order ODEs with Constant
More informationMath 221 Topics since the second exam
Laplace Transforms. Math 1 Topics since the second exam There is a whole different set of techniques for solving n-th order linear equations, which are based on the Laplace transform of a function. For
More informationECE : Linear Circuit Analysis II
Purdue University School of Electrical and Computer Engineering ECE 20200 : Linear Circuit Analysis II Summer 2014 Instructor: Aung Kyi San Instructions: Midterm Examination I July 2, 2014 1. Wait for
More informatione iωt dt and explained why δ(ω) = 0 for ω 0 but δ(0) =. A crucial property of the delta function, however, is that
Phys 531 Fourier Transforms In this handout, I will go through the derivations of some of the results I gave in class (Lecture 14, 1/11). I won t reintroduce the concepts though, so you ll want to refer
More informationMath 3313: Differential Equations Laplace transforms
Math 3313: Differential Equations Laplace transforms Thomas W. Carr Department of Mathematics Southern Methodist University Dallas, TX Outline Introduction Inverse Laplace transform Solving ODEs with Laplace
More informationMAE143A Signals & Systems - Homework 1, Winter 2014 due by the end of class Thursday January 16, 2014.
MAE43A Signals & Systems - Homework, Winter 4 due by the end of class Thursday January 6, 4. Question Time shifting [Chaparro Question.5] Consider a finite-support signal and zero everywhere else. Part
More informationLaplace Theory Examples
Laplace Theory Examples Harmonic oscillator s-differentiation Rule First shifting rule Trigonometric formulas Exponentials Hyperbolic functions s-differentiation Rule First Shifting Rule I and II Damped
More informationIdentification Methods for Structural Systems
Prof. Dr. Eleni Chatzi System Stability 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 be defined from
More informationControl Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich
Control Systems I Lecture 6: Poles and Zeros Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 27, 2017 E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/2017
More informationINHOMOGENEOUS LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS
INHOMOGENEOUS LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS Definitions and a general fact If A is an n n matrix and f(t) is some given vector function, then the system of differential equations () x (t) Ax(t)
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 informationStrauss PDEs 2e: Section Exercise 4 Page 1 of 6
Strauss PDEs 2e: Section 5.3 - Exercise 4 Page of 6 Exercise 4 Consider the problem u t = ku xx for < x < l, with the boundary conditions u(, t) = U, u x (l, t) =, and the initial condition u(x, ) =, where
More informationEE 3054: Signals, Systems, and Transforms Summer It is observed of some continuous-time 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 continuous-time LTI system that the input signal = 3 u(t) produces
More informationSolution of ODEs using Laplace Transforms. Process Dynamics and Control
Solution of ODEs using Laplace Transforms Process Dynamics and Control 1 Linear ODEs For linear ODEs, we can solve without integrating by using Laplace transforms Integrate out time and transform to Laplace
More informationChapter 31. The Laplace Transform The Laplace Transform. The Laplace transform of the function f(t) is defined. e st f(t) dt, L[f(t)] =
Chapter 3 The Laplace Transform 3. The Laplace Transform The Laplace transform of the function f(t) is defined L[f(t)] = e st f(t) dt, for all values of s for which the integral exists. The Laplace transform
More informationFaculty of Engineering, Mathematics and Science School of Mathematics
Faculty of Engineering, Mathematics and Science School of Mathematics SF Engineers SF MSISS SF MEMS MAE: Engineering Mathematics IV Trinity Term 18 May,??? Sports Centre??? 9.3 11.3??? Prof. Sergey Frolov
More informationOrdinary Differential Equations
Ordinary Differential Equations for Engineers and Scientists Gregg Waterman Oregon Institute of Technology c 2017 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International
More informationMath 216 Second Midterm 16 November, 2017
Math 216 Second Midterm 16 November, 2017 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material
More informationREVIEW NOTES FOR MATH 266
REVIEW NOTES FOR MATH 266 MELVIN LEOK 1.1: Some Basic Mathematical Models; Direction Fields 1. You should be able to match direction fields to differential equations. (see, for example, Problems 15-20).
More informationThe Laplace transform
The Laplace transform Samy Tindel Purdue University Differential equations - MA 266 Taken from Elementary differential equations by Boyce and DiPrima Samy T. Laplace transform Differential equations 1
More informationEE102 Homework 2, 3, and 4 Solutions
EE12 Prof. S. Boyd EE12 Homework 2, 3, and 4 Solutions 7. Some convolution systems. Consider a convolution system, y(t) = + u(t τ)h(τ) dτ, where h is a function called the kernel or impulse response of
More informationSecond Order Differential Equations Lecture 6
Second Order Differential Equations Lecture 6 Dibyajyoti Deb 6.1. Outline of Lecture Repeated Roots; Reduction of Order Nonhomogeneous Equations; Method of Undetermined Coefficients Variation of Parameters
More informationMA 242 Review Exponential and Log Functions Notes for today s class can be found at
MA 242 Review Exponential and Log Functions Notes for today s class can be found at www.xecu.net/jacobs/index242.htm Example: If y = x n If y = x 2 then then dy dx = nxn 1 dy dx = 2x1 = 2x Power Function
More informationMath 266 Midterm Exam 2
Math 266 Midterm Exam 2 March 2st 26 Name: Ground Rules. Calculator is NOT allowed. 2. Show your work for every problem unless otherwise stated (partial credits are available). 3. You may use one 4-by-6
More informationDifferential Equations
Differential Equations Math 341 Fall 21 MWF 2:3-3:25pm Fowler 37 c 21 Ron Buckmire http://faculty.oxy.edu/ron/math/341/1/ Worksheet 29: Wednesday December 1 TITLE Laplace Transforms and Introduction to
More informationFOURIER AND LAPLACE TRANSFORMS
FOURIER AND LAPLACE TRANSFORMS BO BERNDTSSON. FOURIER SERIES The basic idea of Fourier analysis is to write general functions as sums (or superpositions) of trigonometric functions, sometimes called harmonic
More informationVariation of Parameters
Variation of Parameters James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University April 13, 218 Outline Variation of Parameters Example One We eventually
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) s-plane Time response p =0 s p =0,p 2 =0 s 2 t p =
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 informationLinear Algebra and ODEs review
Linear Algebra and ODEs review Ania A Baetica September 9, 015 1 Linear Algebra 11 Eigenvalues and eigenvectors Consider the square matrix A R n n (v, λ are an (eigenvector, eigenvalue pair of matrix A
More informationMS&E 321 Spring Stochastic Systems June 1, 2013 Prof. Peter W. Glynn Page 1 of 7
MS&E 321 Spring 12-13 Stochastic Systems June 1, 213 Prof. Peter W. Glynn Page 1 of 7 Section 9: Renewal Theory Contents 9.1 Renewal Equations..................................... 1 9.2 Solving the Renewal
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 informationWork sheet / Things to know. Chapter 3
MATH 251 Work sheet / Things to know 1. Second order linear differential equation Standard form: Chapter 3 What makes it homogeneous? We will, for the most part, work with equations with constant coefficients
More informationMATH 251 Final Examination May 3, 2017 FORM A. Name: Student Number: Section:
MATH 5 Final Examination May 3, 07 FORM A Name: Student Number: Section: This exam has 6 questions for a total of 50 points. In order to obtain full credit for partial credit problems, all work must be
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 one-sided) Laplace Transform: X(s) = 0 x(t)e st dt ECE352 1 Definition of the Laplace transform (cont.)
More informationFinal Exam December 20, 2011
Final Exam December 20, 2011 Math 420 - Ordinary Differential Equations No credit will be given for answers without mathematical or logical justification. Simplify answers as much as possible. Leave solutions
More informationEC Signals and Systems
UNIT I CLASSIFICATION OF SIGNALS AND SYSTEMS Continuous time signals (CT signals), discrete time signals (DT signals) Step, Ramp, Pulse, Impulse, Exponential 1. Define Unit Impulse Signal [M/J 1], [M/J
More informationExercises. T 2T. e ita φ(t)dt.
Exercises. Set #. Construct an example of a sequence of probability measures P n on R which converge weakly to a probability measure P but so that the first moments m,n = xdp n do not converge to m = xdp.
More informationJim Lambers ENERGY 281 Spring Quarter Lecture 3 Notes
Jim Lambers ENERGY 8 Spring Quarter 7-8 Lecture 3 Notes These notes are based on Rosalind Archer s PE8 lecture notes, with some revisions by Jim Lambers. Introduction The Fourier transform is an integral
More information20. The pole diagram and the Laplace transform
95 0. The pole diagram and the Laplace transform When working with the Laplace transform, it is best to think of the variable s in F (s) as ranging over the complex numbers. In the first section below
More informationChapter 6: The Laplace Transform 6.3 Step Functions and
Chapter 6: The Laplace Transform 6.3 Step Functions and Dirac δ 2 April 2018 Step Function Definition: Suppose c is a fixed real number. The unit step function u c is defined as follows: u c (t) = { 0
More informationLecture 1 January 5, 2016
MATH 262/CME 372: Applied Fourier Analysis and Winter 26 Elements of Modern Signal Processing Lecture January 5, 26 Prof. Emmanuel Candes Scribe: Carlos A. Sing-Long; Edited by E. Candes & E. Bates Outline
More information