Dr. Ian R. Manchester

 Brianna Cobb
 8 months ago
 Views:
Transcription
1 Dr Ian R. Manchester
2 Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the splane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus 7 Root Locus 2 Assign 2 Due 8 Bode Plots 9 Bode Plots 2 Assign 3 Due 10 State Space Modeling 11 State Space Design Techniques 12 Advanced Control Topics 13 Review Assign 4 Due Slide 2
3 The concept of a linear time invariant (LTI) Response of LTI systems to basic inputs Impulse Response Step Response Frequency Response The Laplace Transform Transfer Functions of linear systems Assignment 1 Slide 3
4 M f(t) y(t) In the last lecture, we considered this mechanical system We derived the following differential equation: m y = " F m y = f (t) # Ky(t) # K d y (t) m y + K d y (t) + Ky(t) = f (t) Slide 4
5 u(t) Plant y(t) The inputoutput relationship of physical systems can usually be modelled as a differential equation: Can we solve it? If we know u(t) can we find y(t)? Slide 5
6 This is called a Linear Time Invariant (LTI) System u 1 u 2 Plant y u 3 Superposition y(t) = f (u 1 (t) + u 2 (t) +...) = f (u 1 (t)) + f (u 2 (t)) +... Homogeneity y(t) = f (au(t)) = af (u(t)) Slide 6
7 vid Slide 7
8 Slide 8
9 Response of a car suspension system to a pothole Flexing modes of an aircraft wing Or a highprecision industrial robot arm The response of blood glucose concentration, insulin production, etc after eating a meal Slide 9
10 An impulse is an infinitely short pulse at t =0 Any signal can be thought of as the summation (integral) of many impulses at different points in time By the principal of superposition, if we can find the response of the system to one impulse, we will be able to find the response to an arbitrary input f(t) t Slide 10
11 Suppose the input consisted of just three impulses at times t = 0, 1, 2 of size 7, 8, 9. What is the value of y(t) at time t = 5? y = f (u 1 + u 2 + u 3 ) = f (u 1 ) + f (u 2 ) + f (u 3 ) = 7h(5) + 8h(4) + 9h(3) Slide 11
12 The output is the sum (integral) of each impulse response of the system to each individual impulse of the input, delayed by the appropriate time y(t) = % $ 0 h(")u(t #")d" We call this convolution, and it is written as y(t) = h(t) "u(t) Slide 12
13 Assume we have a system described by the following differential equation y (t) + ky(t) = u(t) The impulse response for this system is Slide 13
14 If we want to find the response of the system to a sinusoidal input, sin(!t) Slide 14
15 Now what happens if we have multiple components in the system? u(t) h (t) y 1 (t) h 1 (t) y(t) Slide 15
16 This provides us with a rich basis function for describing functions Through Euler s formula, we find Fourier analysis tells us that this is sufficient for representing any signal Slide 16
17 We can also consider the response of a system to sinusoidal inputs Fourier theory tells us that all signals can be decomposed into a sums of sinusoids Our basis signals are complex exponentials: u(t) = e st = e (" + j# )t Slide 17
18 In general Where Slide 18
19 Step input f (t) =1 Laplace Tr lim R " # % 0 R & = lim ( R " #' e $st dt e $st $s ) + * R 0 e $sr $1 = lim R " # $s If s > 0 = 1 s Slide 19
20 What about that nasty integral in the Laplace operation? We normally use tables of Laplace transforms rather than solving the preceding equations directly This greatly simplifies the transformation process Slide 20
21 Slide 21
22 Convolution in time is equivalent to multiplication in the Laplace domain U(s) H(s) Y 1 (s) H 1 (s) Y(s) Slide 22
23 Starting with an impulse response, h(t), and an input, u(t), find y(t) u(t), h(t) convolution y(t) L L 1 U(s), H(s) Multiplication, algebraic manipulation Y(s) Slide 23
24 The transfer function H(s) of a system is defined as the ratio of the Laplace transforms output and input with zero initial conditions H(s) = Y(s) U(s) It is also Laplace transform of the impulse response Slide 24
25 Recall the system we examined earlier y (t) + ky(t) = u(t) To find the transfer function for this system, we perform the following steps sy(s) " y(0) + ky(s) = U(s) Y(s)(s + k) = U(s) H(s) = Y(s) U(s) = 1 s + k or Slide 25
26 The output response of a system is the sum of two related responses The natural response describes dissipation, oscillation, or unstable growth from initial conditions The forced response describes how the system reacts to external inputs Together these elements determine the overall response of the system Slide 26
27 M f(t) y(t) Earlier, we considered this mechanical system Now we can attempt to find y(t) M y (t) + K d y (t) + Ky(t) = f (t) M(s 2 Y(s) " sy(0) " y (0)) + K d (sy(s) " y(0)) + KY(s) = F(s) Slide 27
28 The natural response of the system can be found by assuming no input force Taking inverse Laplace transform will give us the system response Slide 28
29 Suppose we wish to find the natural response of the system to an initial displacement of 0.5m Assume the mass of the system is 1kg and the damping constant k d is 4 Nsec/m and the spring constant k is 3 N/m Slide 29
30 By Partial Fraction expansion we find Slide 30
31 What if we change the spring constant to 20 N/m? Slide 31
32 Assuming zero initial conditions, the forced response will be This is the transfer function for this system Slide 32
33 Suppose we wish to find the response of the system to a unit step input force f(t) Assume the mass of the system is 1kg and the systems constant k d is 4 Nsec/m and k is 3 N/m Slide 33
34 We have found We wish to write Y(s) in terms of its partialfraction expansion Slide 34
35 So We now take inverse Laplace transforms of these simple transforms Slide 35
36 This represents the step response for this system This plot shows the response y(t) vs time Slide 36
37 %! % Example 1: Step response for! % Y(s)/F(s) = 1/(s^2 + 4s + 3)! %! sys = tf(1,[1,4,3]);! step(sys);! hold on;! pause;! %! % Solving for time domain response yields! % y(t) = 1/31/2*exp(t) + 1/6*exp(3*t)! %! t = 0:0.1:6;! y_t = 1/31/2*exp(t) + 1/6*exp(3*t);! plot(t, y_t, 'r');! pause;! Slide 37
38 The poles of a transfer function are the values of the LT variables, s, that cause the transfer function to become infinite The zeros of a transfer function are the values of the LT variables, s, that cause the transfer function to go to zero A qualitative understanding of the effect of poles and zeros can help us to quickly estimate performance Slide 38
39 Recall Im{s} x x x Re{s} Slide 39
40 What if we change the spring constant k to 20 N/m? Im{s} x x Re{s} x Slide 40
41 Notice that the step response has changed in response to the change in the modified parameter k The steady state response and transient behaviour are different Slide 41
42 %! % Example 1: Step response for! % Y(s)/F(s) = 1/(s^2 + 4s + 20)! %! sys = tf(1,[1,4,20]);! step(sys);! hold on;! pause;! Slide 42
43 A first order system without zeros can be described by: The resulting impulse response is When! > 0, pole is located at s < 0, exponential decays = stable! < 0, pole is at s > 0, exponential grows = unstable Step Response Slide 43
44 From the preceding we can conclude that in general, if the poles of the system are on the left hand side of the splane, the system is stable Poles in the right hand plane will introduce components that grow without bound stable Im{s} unstable Re{s} Slide 44
45 Consider the modelling of an automobile suspension system. Wheel can be modelled as a stiff spring with the suspension modelled as a spring and damper in parallel Interested in describing the motion of the mass in response to changing conditions of the road K s K w M K d y(t) u(t) Slide 45
46 The Linear Time Invariant abstraction allows us to completely understand system response by looking at certain basic responses (impulse, step, frequency) The Laplace transform (of signals) and transfer functions (of systems) are a very convenient representations for analysis Slide 46
47 Nise Section and Franklin & Powell Section 3.1 Murray and Åström (online) Chapter 8 (somewhat different presentation) Slide 47
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) splane Time response p =0 s p =0,p 2 =0 s 2 t p =
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 informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 8: Response Characteristics Overview In this Lecture, you will learn: Characteristics of the Response Stability Real Poles
More informationModeling and Control Overview
Modeling and Control Overview D R. T A R E K A. T U T U N J I A D V A N C E D C O N T R O L S Y S T E M S M E C H A T R O N I C S E N G I N E E R I N G D E P A R T M E N T P H I L A D E L P H I A U N I
More informationPoles, Zeros and System Response
Time Response After the engineer obtains a mathematical representation of a subsystem, the subsystem is analyzed for its transient and steady state responses to see if these characteristics yield the desired
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 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 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 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 informationCHAPTER 5: LAPLACE TRANSFORMS
CHAPTER 5: LAPLACE TRANSFORMS SAMANTHA RAMIREZ PREVIEW QUESTIONS What are some commonly recurring functions in dynamic systems and their Laplace transforms? How can Laplace transforms be used to solve
More informationAPPLICATIONS FOR ROBOTICS
Version: 1 CONTROL APPLICATIONS FOR ROBOTICS TEX d: Feb. 17, 214 PREVIEW We show that the transfer function and conditions of stability for linear systems can be studied using Laplace transforms. Table
More informationLecture 12. AO Control Theory
Lecture 12 AO Control Theory Claire Max with many thanks to Don Gavel and Don Wiberg UC Santa Cruz February 18, 2016 Page 1 What are control systems? Control is the process of making a system variable
More informationLinear Filters and Convolution. Ahmed Ashraf
Linear Filters and Convolution Ahmed Ashraf Linear Time(Shift) Invariant (LTI) Systems The Linear Filters that we are studying in the course belong to a class of systems known as Linear Time Invariant
More informationSolving a RLC Circuit using Convolution with DERIVE for Windows
Solving a RLC Circuit using Convolution with DERIVE for Windows Michel Beaudin École de technologie supérieure, rue NotreDame Ouest Montréal (Québec) Canada, H3C K3 mbeaudin@seg.etsmtl.ca  Introduction
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 informationFrequency Response of Linear Time Invariant Systems
ME 328, Spring 203, Prof. Rajamani, University of Minnesota Frequency Response of Linear Time Invariant Systems Complex Numbers: Recall that every complex number has a magnitude and a phase. Example: z
More informationModule 4 : Laplace and Z Transform Problem Set 4
Module 4 : Laplace and Z Transform Problem Set 4 Problem 1 The input x(t) and output y(t) of a causal LTI system are related to the block diagram representation shown in the figure. (a) Determine a differential
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 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 informationINTRODUCTION TO DIGITAL CONTROL
ECE4540/5540: Digital Control Systems INTRODUCTION TO DIGITAL CONTROL.: Introduction In ECE450/ECE550 Feedback Control Systems, welearnedhow to make an analog controller D(s) to control a lineartimeinvariant
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 Homework 12  Solutions. 1. The transfer function of the system is given to be H(s) = s j j
EE3054  Homework 2  Solutions. The transfer function of the system is given to be H(s) = s 2 +3s+3. Decomposing into partial fractions, H(s) = 0.5774j s +.5 0.866j + 0.5774j s +.5 + 0.866j. () (a) The
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 informationModeling and Analysis of Systems Lecture #3  Linear, TimeInvariant (LTI) Systems. Guillaume Drion Academic year
Modeling and Analysis of Systems Lecture #3  Linear, TimeInvariant (LTI) Systems Guillaume Drion Academic year 20152016 1 Outline Systems modeling: input/output approach and LTI systems. Convolution
More informationModeling and Analysis of Systems Lecture #8  Transfer Function. Guillaume Drion Academic year
Modeling and Analysis of Systems Lecture #8  Transfer Function Guillaume Drion Academic year 20152016 1 Inputoutput representation of LTI systems Can we mathematically describe a LTI system using the
More informationLecture 27 Frequency Response 2
Lecture 27 Frequency Response 2 Fundamentals of Digital Signal Processing Spring, 2012 WeiTa Chu 2012/6/12 1 Application of Ideal Filters Suppose we can generate a square wave with a fundamental period
More informationFrequency Response and Continuoustime Fourier Series
Frequency Response and Continuoustime Fourier Series Recall course objectives Main Course Objective: Fundamentals of systems/signals interaction (we d like to understand how systems transform or affect
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 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 3793 Matlab Project 3 Solution
ECE 3793 Matlab Project 3 Solution Spring 27 Dr. Havlicek. (a) In text problem 9.22(d), we are given X(s) = s + 2 s 2 + 7s + 2 4 < Re {s} < 3. The following Matlab statements determine the partial fraction
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 informationLecture 9 Timedomain properties of convolution systems
EE 12 spring 2122 Handout #18 Lecture 9 Timedomain properties of convolution systems impulse response step response fading memory DC gain peak gain stability 9 1 Impulse response if u = δ we have y(t)
More informationEssence of the Root Locus Technique
Essence of the Root Locus Technique In this chapter we study a method for finding locations of system poles. The method is presented for a very general setup, namely for the case when the closedloop
More informationThe Continuoustime Fourier
The Continuoustime Fourier Transform Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn Outline Representation of Aperiodic signals:
More informationDEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2010
[E2.5] IMPERIAL COLLEGE LONDON DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2010 EEE/ISE PART II MEng. BEng and ACGI SIGNALS AND LINEAR SYSTEMS Time allowed: 2:00 hours There are FOUR
More informationAutonomous Mobile Robot Design
Autonomous Mobile Robot Design Topic: Guidance and Control Introduction and PID Loops Dr. Kostas Alexis (CSE) Autonomous Robot Challenges How do I control where to go? Autonomous Mobile Robot Design Topic:
More informationRecitation 11: Time delays
Recitation : Time delays Emilio Frazzoli Laboratory for Information and Decision Systems Massachusetts Institute of Technology November, 00. Introduction and motivation. Delays are incurred when the controller
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 informationDynamic System Response. Dynamic System Response K. Craig 1
Dynamic System Response Dynamic System Response K. Craig 1 Dynamic System Response LTI Behavior vs. NonLTI Behavior Solution of Linear, ConstantCoefficient, Ordinary Differential Equations Classical
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 informationCourse Outline. Higher Order Poles: Example. Higher Order Poles. Amme 3500 : System Dynamics & Control. State Space Design. 1 G(s) = s(s + 2)(s +10)
Amme 35 : System Dynamics Control State Space Design Course Outline Week Date Content Assignment Notes 1 1 Mar Introduction 2 8 Mar Frequency Domain Modelling 3 15 Mar Transient Performance and the splane
More informationIntroduction ODEs and Linear Systems
BENG 221 Mathematical Methods in Bioengineering ODEs and Linear Systems Gert Cauwenberghs Department of Bioengineering UC San Diego 1.1 Course Objectives 1. Acquire methods for quantitative analysis and
More informationAnalog Signals and Systems and their properties
Analog Signals and Systems and their properties Main Course Objective: Recall course objectives Understand the fundamentals of systems/signals interaction (know how systems can transform or filter signals)
More informationBIBO STABILITY AND ASYMPTOTIC STABILITY
BIBO STABILITY AND ASYMPTOTIC STABILITY FRANCESCO NORI Abstract. In this report with discuss the concepts of boundedinput boundedoutput stability (BIBO) and of Lyapunov stability. Examples are given to
More informationLecture 9 Infinite Impulse Response Filters
Lecture 9 Infinite Impulse Response Filters Outline 9 Infinite Impulse Response Filters 9 FirstOrder LowPass Filter 93 IIR Filter Design 5 93 CT Butterworth filter design 5 93 Bilinear transform 7 9
More informationControl Systems I. Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback. Readings: Emilio Frazzoli
Control Systems I Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback Readings: Emilio Frazzoli Institute for Dynamic Systems and Control DMAVT ETH Zürich October 13, 2017 E. Frazzoli (ETH)
More informationControl Systems Design
ELEC4410 Control Systems Design Lecture 13: Stability Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 13: Stability p.1/20 Outline InputOutput
More informationLecture 2. Introduction to Systems (Lathi )
Lecture 2 Introduction to Systems (Lathi 1.61.8) Pier Luigi Dragotti Department of Electrical & Electronic Engineering Imperial College London URL: www.commsp.ee.ic.ac.uk/~pld/teaching/ Email: p.dragotti@imperial.ac.uk
More informationControl Systems I. Lecture 2: Modeling. Suggested Readings: Åström & Murray Ch. 23, Guzzella Ch Emilio Frazzoli
Control Systems I Lecture 2: Modeling Suggested Readings: Åström & Murray Ch. 23, Guzzella Ch. 23 Emilio Frazzoli Institute for Dynamic Systems and Control DMAVT ETH Zürich September 29, 2017 E. Frazzoli
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 information1 4 r q. (9.172) G(r, q) =
FOURIER ANALYSIS: LECTURE 5 9 Green s functions 9. Response to an impulse We have spent some time so far in applying Fourier methods to solution of di erential equations such as the damped oscillator.
More informationDynamic circuits: Frequency domain analysis
Electronic Circuits 1 Dynamic circuits: Contents Free oscillation and natural frequency Transfer functions Frequency response Bode plots 1 System behaviour: overview 2 System behaviour : review solution
More informationGeneralizing the DTFT!
The Transform Generaliing the DTFT! The forward DTFT is defined by X e jω ( ) = x n e jωn in which n= Ω is discretetime radian frequency, a real variable. The quantity e jωn is then a complex sinusoid
More informationCh 6.4: Differential Equations with Discontinuous Forcing Functions
Ch 6.4: Differential Equations with Discontinuous Forcing Functions! In this section focus on examples of nonhomogeneous initial value problems in which the forcing function is discontinuous. Example 1:
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 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 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 informationBasic concepts in DT systems. Alexandra Branzan Albu ELEC 310Spring 2009Lecture 4 1
Basic concepts in DT systems Alexandra Branzan Albu ELEC 310Spring 2009Lecture 4 1 Readings and homework For DT systems: Textbook: sections 1.5, 1.6 Suggested homework: pp. 5758: 1.15 1.16 1.18 1.19
More informationAN INTRODUCTION TO THE CONTROL THEORY
OpenLoop controller An OpenLoop (OL) controller is characterized by no direct connection between the output of the system and its input; therefore external disturbance, nonlinear dynamics and parameter
More informationLearn2Control Laboratory
Learn2Control Laboratory Version 3.2 Summer Term 2014 1 This Script is for use in the scope of the Process Control lab. It is in no way claimed to be in any scientific way complete or unique. Errors should
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 informationIntroduction to Feedback Control
Introduction to Feedback Control Control System Design Why Control? OpenLoop vs ClosedLoop (Feedback) Why Use Feedback Control? ClosedLoop Control System Structure Elements of a Feedback Control System
More informationSchool of Mechanical Engineering Purdue University. ME375 Feedback Control  1
Introduction to Feedback Control Control System Design Why Control? OpenLoop vs ClosedLoop (Feedback) Why Use Feedback Control? ClosedLoop Control System Structure Elements of a Feedback Control System
More informationSecond Order and Higher Order Systems
Second Order and Higher Order Systems 1. Second Order System In this section, we shall obtain the response of a typical secondorder control system to a step input. In terms of damping ratio and natural
More informationCDS 101/110: Lecture 6.2 Transfer Functions
CDS 11/11: Lecture 6.2 Transfer Functions November 2, 216 Goals: Continued study of Transfer functions Review Laplace Transform Block Diagram Algebra Bode Plot Intro Reading: Åström and Murray, Feedback
More informationTopic 3: Fourier Series (FS)
ELEC264: Signals And Systems Topic 3: Fourier Series (FS) o o o o Introduction to frequency analysis of signals CT FS Fourier series of CT periodic signals Signal Symmetry and CT Fourier Series Properties
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 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 informationVibrations Qualifying Exam Study Material
Vibrations Qualifying Exam Study Material The candidate is expected to have a thorough understanding of engineering vibrations topics. These topics are listed below for clarification. Not all instructors
More informationSection 6.5 Impulse Functions
Section 6.5 Impulse Functions Key terms/ideas: Unit impulse function (technically a generalized function or distribution ) Dirac delta function Laplace transform of the Dirac delta function IVPs with forcing
More informationMathematics portion of the Doctor of Engineering Qualifying Examination
Mathematics portion of the Doctor of Engineering Qualifying Examination. The exam will be made up by faculty members of the Department of Mathematics and Computer Science. Dr. Kathy Zhong ( zhongk@udmercy.edu
More informationExam. 135 minutes + 15 minutes reading time
Exam January 23, 27 Control Systems I (559L) Prof. Emilio Frazzoli Exam Exam Duration: 35 minutes + 5 minutes reading time Number of Problems: 45 Number of Points: 53 Permitted aids: Important: 4 pages
More informationThe Convolution Sum for DiscreteTime LTI Systems
The Convolution Sum for DiscreteTime LTI Systems Andrew W. H. House 01 June 004 1 The Basics of the Convolution Sum Consider a DT LTI system, L. x(n) L y(n) DT convolution is based on an earlier result
More informationCHAPTER 1 Basic Concepts of Control System. CHAPTER 6 Hydraulic Control System
CHAPTER 1 Basic Concepts of Control System 1. What is open loop control systems and closed loop control systems? Compare open loop control system with closed loop control system. Write down major advantages
More informationRobotics. Dynamics. Marc Toussaint U Stuttgart
Robotics Dynamics 1D point mass, damping & oscillation, PID, dynamics of mechanical systems, EulerLagrange equation, NewtonEuler recursion, general robot dynamics, joint space control, reference trajectory
More informationTransient Response of a SecondOrder System
Transient Response of a SecondOrder System ECEN 830 Spring 01 1. Introduction In connection with this experiment, you are selecting the gains in your feedback loop to obtain a wellbehaved closedloop
More informationApplications of SecondOrder Differential Equations
Applications of SecondOrder Differential Equations ymy/013 Building Intuition Even though there are an infinite number of differential equations, they all share common characteristics that allow intuition
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 informationMath 256: Applied Differential Equations: Final Review
Math 256: Applied Differential Equations: Final Review Chapter 1: Introduction, Sec 1.1, 1.2, 1.3 (a) Differential Equation, Mathematical Model (b) Direction (Slope) Field, Equilibrium Solution (c) Rate
More informationRichiami di Controlli Automatici
Richiami di Controlli Automatici Gianmaria De Tommasi 1 1 Università degli Studi di Napoli Federico II detommas@unina.it Ottobre 2012 Corsi AnsaldoBreda G. De Tommasi (UNINA) Richiami di Controlli Automatici
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 informationReview of Fourier Transform
Review of Fourier Transform Fourier series works for periodic signals only. What s about aperiodic signals? This is very large & important class of signals Aperiodic signal can be considered as periodic
More informationThe Relation Between the 3D Bode Diagram and the Root Locus. Insights into the connection between these classical methods. By Panagiotis Tsiotras
F E A T U R E The Relation Between the D Bode Diagram and the Root Locus Insights into the connection between these classical methods Bode diagrams and root locus plots have been the cornerstone of control
More informationEE3CL4: Introduction to Linear Control Systems
1 / 17 EE3CL4: Introduction to Linear Control Systems Section 7: McMaster University Winter 2018 2 / 17 Outline 1 4 / 17 Cascade compensation Throughout this lecture we consider the case of H(s) = 1. We
More informationFigure XP3.1 (a) Mass in equilibrium, (b) Freebody diagram, (c) Kinematic constraint relation Example Problem 3.1 Figure XP3.1 illustrates a mass m
LECTURE 7. MORE VIBRATIONS ` Figure XP3.1 (a) Mass in equilibrium, (b) Freebody diagram, (c) Kinematic constraint relation Example Problem 3.1 Figure XP3.1 illustrates a mass m that is in equilibrium and
More informationOneSided Laplace Transform and Differential Equations
OneSided Laplace Transform and Differential Equations As in the dcretetime case, the onesided transform allows us to take initial conditions into account. Preliminaries The onesided Laplace transform
More informationThe Continuous Time Fourier Transform
COMM 401: Signals & Systems Theory Lecture 8 The Continuous Time Fourier Transform Fourier Transform Continuous time CT signals Discrete time DT signals Aperiodic signals nonperiodic periodic signals Aperiodic
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 informationAspects of Continuous and DiscreteTime Signals and Systems
Aspects of Continuous and DiscreteTime Signals and Systems C.S. Ramalingam Department of Electrical Engineering IIT Madras C.S. Ramalingam (EE Dept., IIT Madras) Networks and Systems 1 / 45 Scaling 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 informationFrequencyDomain C/S of LTI Systems
FrequencyDomain C/S of LTI Systems x(n) LTI y(n) LTI: Linear TimeInvariant system h(n), the impulse response of an LTI systems describes the time domain c/s. H(ω), the frequency response describes the
More informationHIGHERORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS IV: Laplace Transform Method. David Levermore Department of Mathematics University of Maryland
HIGHERORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS IV: Laplace Transform Method David Levermore Department of Mathematics University of Maryland 6 April Because the presentation of this material in lecture
More informationCh. 7: Ztransform Reading
c J. Fessler, June 9, 3, 6:3 (student version) 7. Ch. 7: Ztransform Definition Properties linearity / superposition time shift convolution: y[n] =h[n] x[n] Y (z) =H(z) X(z) Inverse ztransform by coefficient
More informationEQUIVALENT SINGLEDEGREEOFFREEDOM SYSTEM AND FREE VIBRATION
1 EQUIVALENT SINGLEDEGREEOFFREEDOM SYSTEM AND FREE VIBRATION The course on Mechanical Vibration is an important part of the Mechanical Engineering undergraduate curriculum. It is necessary for the development
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 informationLecture 1 From ContinuousTime to DiscreteTime
Lecture From ContinuousTime to DiscreteTime Outline. Continuous and DiscreteTime Signals and Systems................. What is a signal?................................2 What is a system?.............................
More informationDepartment of Electrical and Telecommunications Engineering Technology TEL (718) FAX: (718) Courses Description:
NEW YORK CITY COLLEGE OF TECHNOLOGY The City University of New York 300 Jay Street Brooklyn, NY 112012983 Department of Electrical and Telecommunications Engineering Technology TEL (718) 2605300  FAX:
More informationI. Impulse Response and Convolution
I. Impulse Response and Convolution 1. Impulse response. Imagine a mass m at rest on a frictionless track, then given a sharp kick at time t =. We model the kick as a constant force F applied to the mass
More informationSignals and Systems. Problem Set: The ztransform and DT Fourier Transform
Signals and Systems Problem Set: The ztransform and DT Fourier Transform Updated: October 9, 7 Problem Set Problem  Transfer functions in MATLAB A discretetime, causal LTI system is described by the
More informationFourier series. XE31EO2  Pavel Máša. Electrical Circuits 2 Lecture1. XE31EO2  Pavel Máša  Fourier Series
Fourier series Electrical Circuits Lecture  Fourier Series Filtr RLC defibrillator MOTIVATION WHAT WE CAN'T EXPLAIN YET Source voltage rectangular waveform Resistor voltage sinusoidal waveform  Fourier
More information