9.5 The Transfer Function
|
|
- Roxanne Madeline Watts
- 5 years ago
- Views:
Transcription
1 Lecture Notes on Control Systems/D. Ghose/ 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 + b n 1 dt + b d 2 u 2 dt + + b d m u 2 m dt m with zero initial conditions on all derivatives. Taking the Laplace transform on both sides, we get, a 0 Y (s)+a 1 sy (s)+a 2 s 2 Y (s)+ + a n s n Y (s) b 0 U(s)+b 1 su(s)+b 2 s 2 U(s)+ + b m s m U(s) From which, Y (s) U(s) N(s) D(s) b 0 + b 1 s + b 2 s b m s m a 0 + a 1 s + a 2 s a n s n mk1 b k s k nj1 a j s j Where, G(s) is called the transfer function. It is defined from differential equations for which the initial condition is zero. G(s) issaidtobeproperifm n. G(s) is said to be strictly proper if m<n. Note: We will understand these terms in a better way a little later. However, in most cases we will be dealing with strictly proper systems. Let us do some more algebraic manipulations. Let, Then, K b m a n bk b k b m, for k 0,...,m 1 ā j a j a n, for j 0,...,n 1 b ms m + b m 1 s m b 1 s + b 0 a n s n + a n 1 s n a 1 s + a 0
2 Lecture Notes on Control Systems/D. Ghose/ where, m n b ( ) m s m + b m 1 b m s m b 1 b m s + b 0 b ( m ) a n sn + a n 1 a n s n a 1 a n s + a 0 a n b ( m s m + b m 1 s m b 1 s + b ) 0 a n s n +ā n 1 s n 1 + +ā 1 s +ā 0 ( s m + K b m 1 s m b 1 s + b ) 0 s n +ā n 1 s n 1 + +ā 1 s +ā 0 K (s z 1)(s z 2 ) (s z m ) (s p 1 )(s p 2 ) (s p n ) 1. The numerator roots z 1,,z m are called system zeros. 2. The denominator roots p 1,,p n are called the system poles. Figure 9.9: A sketch of the poles and zeros along the real line. The denominator polynomial is called the characteristic polynomial. 4. The transfer function is the Laplace transform of the impulse response function 4. An Example. An example of why m>n(or, a system which is not proper) is a bad idea. Newton s law says f m v. Let us identify the input and the output.
3 Lecture Notes on Control Systems/D. Ghose/ Figure 9.10: Force applied on a mass Figure 9.11: Case 1 Case 1: Let us say that input is v and output is f. Also, let v be a step input. Looks like a bad idea!!! Case 2: Let the input be f and the output be v, and let f be the unit force. Figure 9.12: Case 2 Looks OK. Now, look at the transfer function. 4 To see this, put u(t) δ(t), then U(s) 1. So, Y (s) G(s) andy(t) L 1 [G(s)]
4 Lecture Notes on Control Systems/D. Ghose/2012 f(t) m v F (s) msv (s) (assuming zero initial condition) For Case 1: F (s) msv (s) F (s) V (s) ms (which is an improper transfer function) Note that V (s) e st 0 F (s) me st 0. Thus, a delayed step function is caused by a s delayed impulse signal. For Case 2: V (s) 1 V (s) F (s) 1 ms F (s) ms (which is strictly proper). So, F (s) e st 0 V (s) me st 0 1 me st 0. This may be interpreted as the integral of s s 2 s s a delayed step function, giving rise to a ramp function. 9.6 Initial and Final Value Theorems Initial Value Theorem Since [ ] df L sf (s) f(0) dt and since, s e st 0, we have, So, Therefore, Final Value Theorem [ ] df lim L s 0 dt Therefore, and hence, [ df lim L s dt ] df lim s 0 dt e st dt 0 lim [sf (s) f(0)] 0 s f(0) lim s sf (s) df lim s 0 0 dt e st dt 0 lim[sf (s) f(0)] f( ) f(0) s 0 df dt f( ) f(0) dt lim sf (s) f( ) lim f(t) s 0 t However, the final value theorem is meaningful only if the following conditions are met.
5 Lecture Notes on Control Systems/D. Ghose/ The Laplace transform of f(t) and df exists. dt 2. lim t f(t) exists.. All poles of F (s) are on the left half plane except for one which may be at the origin. 4. No poles on the imaginary axis. 9.7 Partial Fraction Expansion The reason we introduced the Laplace transform is to devise an easy way to find the system response. Figure 9.1: Input-output representation One way to do this would be by using the convolution integral, y(t) t 0 g(t τ)u(τ)dτ But if the Laplace transforms are known for g and u, then Y (s) G(s)U(s) and then y(t) is obtained by finding the inverse Laplace transform. To achieve this, we need to break up Y (s) into pieces for which the inverse Laplace transforms are available, and then use the Laplace transform tables to find the inverse Laplace transform of the complete Y (s). Let, N(s) D(s) b ms m + b m 1 s m b 1 s + b 0 a n s n + a n 1 s n a 1 s + a 0 mk1 (s z k ) K nj1 (s p j ) Note that G(s) is not necessarily the transfer function of the system. It could be the output Y (s) or any other function in the s-domain having a numerator polynomial and a denominator polynomial of appropriate order.
6 Lecture Notes on Control Systems/D. Ghose/ Case 1: Distinct Poles: p i p j,i j Then, the partial fraction expansion of G(s) is n k i s p i where, k i are called the residues. How do we find k i? To find k i : Multiply G(s) by(s p i ) and let s p i. i1 Example. Let n k j k i s pi lim (s p i ) s pi lim (s p i ) j1 (s p j ) It is easy to find the poles of G(s), s s 2 +s +2 s (s +1)(s +2) Since the poles are distinct (p 1 1,p 2 2), we may expand G(s) into partial fractions as, To find the residues, So, k 1 lim (s +1) lim s 1 s 1 k 2 lim (s +2) lim s 2 s 2 k 1 s +1 + k 2 s +2 s s +2 s s +1 1 s s Verify that the above is indeed the same as the original G(s). Case 2: A pole of multiple order or repeated pole. N(s) (s p 1 )(s p 2 ) (s p i 1 )(s p i ) l (s p i+1 ) (s p n )
7 Lecture Notes on Control Systems/D. Ghose/ In the above, the pole p i has order l>1. All other poles have order 1. Then the partial fraction expansion is given by, where, k 1 s p 1 + k 2 s p k i+1 s p i A 1 s p i + k n k i 1 s p i 1 Simple poles Simple poles s p n A 2 (s p i ) + + A l Repeated poles 2 (s p i ) l k j (s p j )G(s) spj For simple poles and for the repeated pole, A l (s p i ) l G(s) spi [ d { A l 1 (s pi ) l G(s) }] ds sp i A l 2 1 [ d 2 { (s pi ) l G(s) } ] 2! ds 2. A 2 A 1 [ 1 d (l 2) (l 2)! 1 (l 1)! sp i { (s pi ) l G(s) } ] ds (l 2) [ d (l 1) { (s pi ) l G(s) } ds (l 1) sp i ] sp i Example. Consider a system which has a response given by the differential equation, ẍ +2ẋ + x 0, x(0) a, ẋ(0) b Taking Laplace transform on both sides, s 2 X(s) sx(0) ẋ(0) + 2(sX(s) x(0)) + X(s) 0 s 2 X(s) sa b +2(sX(s) a)+x(s) 0 From which, X(s) as +(2a + b) s 2 +2s +1 The partial fraction expansion is then, as +(2a + b) (s +1) 2 X(s) A 1 s +1 + A 2 (s +1) 2
8 Lecture Notes on Control Systems/D. Ghose/ where, A 2 (s +1) 2 X(s) as +(2a + b) s 1 s 1 a +2a + b a + b [ ] d A 1 ds {(s +1)2 X(s) a s 1 So, X(s) a s +1 + a + b (s +1) 2 Verify that the above is indeed the same as the original X(s). Case : Complex poles TRANSFER FUNCTION HAVING DISTINCT POLES + TRANSFER FUNCTION HAVING REPEATED POLES + TRANSFER FUNCTION HAVING COMPLEX POLES The complex roots are expressed as, When b>0, k 1 s + k 2 (s + a) 2 + b 2 (s + a) 2 + b 2 (s + a) 2 (jb) 2 (s + a + jb)(s + a jb) Since the complex roots are distinct, one can use the method of distinct roots as given earlier to obtain the residues. But in that case the residues will also be complex. On further manipulations we can get back the real numbers. Finally we can use the following inverse Laplace transforms, [ ] L 1 b e at sin bt (s + a) 2 + b 2 [ ] L 1 s + a e at cos bt (s + a) 2 + b 2 Example.
9 Lecture Notes on Control Systems/D. Ghose/ s +s 2 +6s +4 s + s 2 +2s 2 +2s +4s +4 s 2 (s +1)+2s(s +1)+4(s +1) (s 2 +2s +4)(s +1) (s +1)[(s +1) 2 +( ) 2 ) So, the poles are, p 1 1, p 2 1 j,p 1+j Since all the poles are distinct, by partial fraction expansion, k 1 s +1 + k 2 s +1+j + k s +1 j and, the residues are computed as, k 1 (s +1)G(s) s 1 (s +1) s 1 Similarly, k 2 (s +1+j )G(s) (s +1)(s +1 j ) 1 j + ( j )( j2 ) 2 j j 6 k (s +1 j )G(s) s 1 j s 1 j s 1+j
10 Lecture Notes on Control Systems/D. Ghose/ (s +1)(s +1+j ) 1+j + (j )(j2 ) 2+j 6 1 j 6 Substituting these values, (1/) + j( /6) + s +1+j s 1+j 2/ + (1/) j( /6) s +1 s +1 j 2/ s +1 +( 2/) s +1 (s +1) 2 + +(1/ ) (s +1) 2 + Applying the inverse transform we get, g(t) 2 e t 2 e t cos t + 1 e t sin t An alternative way to solve the same problem is by equating the coefficients. This avoids the complication of using imaginary numbers. Let, So, (s +1)[(s +1) 2 +] (s +1)[(s +1) 2 +] k 1 s +1 + k 2s + k (s +1) 2 + k 1[(s +1) 2 +]+(s +1)(k 2 s + k ) (s +1)[(s +1) 2 +] Since the denominators are the same, the numerators must also be the same. Thus, (k 1 + k 2 )s 2 +(2k 1 + k 2 + k )s +4k 1 + k Comparing the coefficients of the powers of s, Solving, k 1 + k 2 0 2k 1 + k 2 + k 1 4k 1 + k k 1 2 k 2 2 k 1
11 Lecture Notes on Control Systems/D. Ghose/ The rest follows in the same way as before. Question. How would you handle repeated complex roots?
Time 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 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 information9.2 The Input-Output Description of a System
Lecture Notes on Control Systems/D. Ghose/212 22 9.2 The Input-Output Description of a System The input-output description of a system can be obtained by first defining a delta function, the representation
More informationIntroduction & Laplace Transforms Lectures 1 & 2
Introduction & Lectures 1 & 2, Professor Department of Electrical and Computer Engineering Colorado State University Fall 2016 Control System Definition of a Control System Group of components that collectively
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 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 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 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 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 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 informationMODELING OF CONTROL SYSTEMS
1 MODELING OF CONTROL SYSTEMS Feb-15 Dr. Mohammed Morsy Outline Introduction Differential equations and Linearization of nonlinear mathematical models Transfer function and impulse response function Laplace
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 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 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 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 informationLaplace Transforms. Chapter 3. Pierre Simon Laplace Born: 23 March 1749 in Beaumont-en-Auge, Normandy, France Died: 5 March 1827 in Paris, France
Pierre Simon Laplace Born: 23 March 1749 in Beaumont-en-Auge, Normandy, France Died: 5 March 1827 in Paris, France Laplace Transforms Dr. M. A. A. Shoukat Choudhury 1 Laplace Transforms Important analytical
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 5-1 Road Map of the Lecture V Laplace Transform and Transfer
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 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 informationMATHEMATICAL MODELING OF CONTROL SYSTEMS
1 MATHEMATICAL MODELING OF CONTROL SYSTEMS Sep-14 Dr. Mohammed Morsy Outline Introduction Transfer function and impulse response function Laplace Transform Review Automatic control systems Signal Flow
More informationSome special cases
Lecture Notes on Control Systems/D. Ghose/2012 87 11.3.1 Some special cases Routh table is easy to form in most cases, but there could be some cases when we need to do some extra work. Case 1: The first
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 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 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 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 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 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 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 informationTime Response Analysis (Part II)
Time Response Analysis (Part II). A critically damped, continuous-time, 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 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 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 informationI Laplace transform. I Transfer function. I Conversion between systems in time-, frequency-domain, 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 informationNotes for ECE-320. Winter by R. Throne
Notes for ECE-3 Winter 4-5 by R. Throne Contents Table of Laplace Transforms 5 Laplace Transform Review 6. Poles and Zeros.................................... 6. Proper and Strictly Proper Transfer Functions...................
More informationChapter 6: The Laplace Transform. Chih-Wei Liu
Chapter 6: The Laplace Transform Chih-Wei Liu Outline Introduction The Laplace Transform The Unilateral Laplace Transform Properties of the Unilateral Laplace Transform Inversion of the Unilateral Laplace
More informationCourse roadmap. ME451: Control Systems. Example of Laplace transform. Lecture 2 Laplace transform. Laplace transform
ME45: Control Systems Lecture 2 Prof. Jongeun Choi Department of Mechanical Engineering Michigan State University Modeling Transfer function Models for systems electrical mechanical electromechanical Block
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 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 informationControl Systems. System response. L. Lanari
Control Systems m i l e r p r a in r e v y n is o System response L. Lanari Outline What we are going to see: how to compute in the s-domain the forced response (zero-state response) using the transfer
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 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 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 informationEE 380. Linear Control Systems. Lecture 10
EE 380 Linear Control Systems Lecture 10 Professor Jeffrey Schiano Department of Electrical Engineering Lecture 10. 1 Lecture 10 Topics Stability Definitions Methods for Determining Stability Lecture 10.
More informationNumerical Methods. Equations and Partial Fractions. Jaesung Lee
Numerical Methods Equations and Partial Fractions Jaesung Lee Solving linear equations Solving linear equations Introduction Many problems in engineering reduce to the solution of an equation or a set
More informationLecture 5 Rational functions and partial fraction expansion
EE 102 spring 2001-2002 Handout #10 Lecture 5 Rational functions and partial fraction expansion (review of) polynomials rational functions pole-zero plots partial fraction expansion repeated poles nonproper
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 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 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 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 informationLecture 4 Stabilization
Lecture 4 Stabilization This lecture follows Chapter 5 of Doyle-Francis-Tannenbaum, with proofs and Section 5.3 omitted 17013 IOC-UPC, Lecture 4, November 2nd 2005 p. 1/23 Stable plants (I) We assume that
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 informationThe z-transform Part 2
http://faculty.kfupm.edu.sa/ee/muqaibel/ The z-transform Part 2 Dr. Ali Hussein Muqaibel The material to be covered in this lecture is as follows: Properties of the z-transform Linearity Initial and final
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 informationName: Solutions Final Exam
Instructions. Answer each of the questions on your own paper. Put your name on each page of your paper. Be sure to show your work so that partial credit can be adequately assessed. Credit will not be given
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 informationECE 3084 QUIZ 2 SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY APRIL 2, Name:
ECE 3084 QUIZ 2 SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY APRIL 2, 205 Name:. The quiz is closed book, except for one 2-sided sheet of handwritten notes. 2. Turn off
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 informationMAE143A Signals & Systems - Homework 5, Winter 2013 due by the end of class Tuesday February 12, 2013.
MAE43A Signals & Systems - Homework 5, Winter 23 due by the end of class Tuesday February 2, 23. If left under my door, then straight to the recycling bin with it. This week s homework will be a refresher
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 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 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 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 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 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 information20.3. Further Laplace Transforms. Introduction. Prerequisites. Learning Outcomes
Further Laplace Transforms 2.3 Introduction In this Section we introduce the second shift theorem which simplifies the determination of Laplace and inverse Laplace transforms in some complicated cases.
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 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 informationSpecial Mathematics Laplace Transform
Special Mathematics Laplace Transform March 28 ii Nature laughs at the difficulties of integration. Pierre-Simon Laplace 4 Laplace Transform Motivation Properties of the Laplace transform the Laplace transform
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 informationECEN 420 LINEAR CONTROL SYSTEMS. Lecture 6 Mathematical Representation of Physical Systems II 1/67
1/67 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 6 Mathematical Representation of Physical Systems II State Variable Models for Dynamic Systems u 1 u 2 u ṙ. Internal Variables x 1, x 2 x n y 1 y 2. y m Figure
More informationMathQuest: Differential Equations
MathQuest: Differential Equations Laplace Tranforms 1. True or False The Laplace transform method is the only way to solve some types of differential equations. (a) True, and I am very confident (b) True,
More informationINTRODUCTION TO TRANSFER FUNCTIONS
INTRODUCTION TO TRANSFER FUNCTIONS The transfer function is the ratio of the output Laplace Transform to the input Laplace Transform assuming zero initial conditions. Many important characteristics of
More informationChap 4. State-Space Solutions and
Chap 4. State-Space Solutions and Realizations Outlines 1. Introduction 2. Solution of LTI State Equation 3. Equivalent State Equations 4. Realizations 5. Solution of Linear Time-Varying (LTV) Equations
More informationLecture Discrete dynamic systems
Chapter 3 Low-level io Lecture 3.4 Discrete dynamic systems Lecture 3.4 Discrete dynamic systems Suppose that we wish to implement an embedded computer system that behaves analogously to a continuous linear
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 informationRaktim Bhattacharya. . AERO 632: Design of Advance Flight Control System. Norms for Signals and Systems
. AERO 632: Design of Advance Flight Control System Norms for Signals and. Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. Norms for Signals ...
More informationState Variable Analysis of Linear Dynamical Systems
Chapter 6 State Variable Analysis of Linear Dynamical Systems 6 Preliminaries In state variable approach, a system is represented completely by a set of differential equations that govern the evolution
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 informationThe Method of Laplace Transforms.
The Method of Laplace Transforms. James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University May 25, 217 Outline 1 The Laplace Transform 2 Inverting
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 informationSystems Engineering/Process Control L4
1 / 24 Systems Engineering/Process Control L4 Input-output models Laplace transform Transfer functions Block diagram algebra Reading: Systems Engineering and Process Control: 4.1 4.4 2 / 24 Laplace transform
More informationExam in Systems Engineering/Process Control
Department of AUTOMATIC CONTROL Exam in Systems Engineering/Process Control 27-6-2 Points and grading All answers must include a clear motivation. Answers may be given in English or Swedish. The total
More informationControl System. Contents
Contents Chapter Topic Page Chapter- Chapter- Chapter-3 Chapter-4 Introduction Transfer Function, Block Diagrams and Signal Flow Graphs Mathematical Modeling Control System 35 Time Response Analysis of
More informationOne-Sided Laplace Transform and Differential Equations
One-Sided Laplace Transform and Differential Equations As in the dcrete-time case, the one-sided transform allows us to take initial conditions into account. Preliminaries The one-sided Laplace transform
More informationIntroduction to Controls
EE 474 Review Exam 1 Name Answer each of the questions. Show your work. Note were essay-type answers are requested. Answer with complete sentences. Incomplete sentences will count heavily against the grade.
More informationControl Systems I. Lecture 5: Transfer Functions. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich
Control Systems I Lecture 5: Transfer Functions Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 20, 2017 E. Frazzoli (ETH) Lecture 5: Control Systems I 20/10/2017
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture : Different Types of Control Overview In this Lecture, you will learn: Limits of Proportional Feedback Performance
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 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 informationOrdinary Differential Equations. Session 7
Ordinary Differential Equations. Session 7 Dr. Marco A Roque Sol 11/16/2018 Laplace Transform Among the tools that are very useful for solving linear differential equations are integral transforms. An
More informationTRACKING AND DISTURBANCE REJECTION
TRACKING AND DISTURBANCE REJECTION Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 13 General objective: The output to track a reference
More informationSolution to Homework Assignment 1
ECE602 Fall 2008 Homework Solution September 2, 2008 Solution to Homework Assignment. Consider the two-input two-output system described by D (p)y (t)+d 2 (p)y 2 (t) N (p)u (t)+n 2 (p)u 2 (t) D 2 (p)y
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 informationReview of Linear Time-Invariant Network Analysis
D1 APPENDIX D Review of Linear Time-Invariant Network Analysis Consider a network with input x(t) and output y(t) as shown in Figure D-1. If an input x 1 (t) produces an output y 1 (t), and an input x
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 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 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 informationCourse Background. Loosely speaking, control is the process of getting something to do what you want it to do (or not do, as the case may be).
ECE4520/5520: Multivariable Control Systems I. 1 1 Course Background 1.1: From time to frequency domain Loosely speaking, control is the process of getting something to do what you want it to do (or not
More informationIntroduction. Performance and Robustness (Chapter 1) Advanced Control Systems Spring / 31
Introduction Classical Control Robust Control u(t) y(t) G u(t) G + y(t) G : nominal model G = G + : plant uncertainty Uncertainty sources : Structured : parametric uncertainty, multimodel uncertainty Unstructured
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. Linear systems?!? (Roughly) Systems which obey properties of superposition Input u(t) output
Linear Systems Linear systems?!? (Roughly) Systems which obey properties of superposition Input u(t) output Our interest is in dynamic systems Dynamic system means a system with memory of course including
More information