# Dr. Ian R. Manchester

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Dr Ian R. Manchester

2 Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the s-plane 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 input-output 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 high-precision 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) = 1/3-1/2*exp(-t) + 1/6*exp(-3*t)! %! t = 0:0.1:6;! y_t = 1/3-1/2*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 s-plane, 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) s-plane Time response p =0 s p =0,p 2 =0 s 2 t p =

### Dr 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 s-plane 4 27 Mar Block Diagrams Assign 1 Due 5 3 Apr Feedback System Characteristics

### Systems 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

### Modeling 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

### Poles, 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

### LTI 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

### Review 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

### Chapter 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

### Dynamic 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

### CHAPTER 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

### APPLICATIONS 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

### Lecture 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

### Linear 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

### Solving 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 Notre-Dame Ouest Montréal (Québec) Canada, H3C K3 mbeaudin@seg.etsmtl.ca - Introduction

### ECEN 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

### Frequency 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

### Module 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

### Laplace 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

### f(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

### INTRODUCTION 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 linear-time-invariant

### The 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

### EE 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

### MAE143A 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

### Modeling and Analysis of Systems Lecture #3 - Linear, Time-Invariant (LTI) Systems. Guillaume Drion Academic year

Modeling and Analysis of Systems Lecture #3 - Linear, Time-Invariant (LTI) Systems Guillaume Drion Academic year 2015-2016 1 Outline Systems modeling: input/output approach and LTI systems. Convolution

### Modeling 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 2015-2016 1 Input-output representation of LTI systems Can we mathematically describe a LTI system using the

### Lecture 27 Frequency Response 2

Lecture 27 Frequency Response 2 Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/6/12 1 Application of Ideal Filters Suppose we can generate a square wave with a fundamental period

### Frequency Response and Continuous-time Fourier Series

Frequency Response and Continuous-time Fourier Series Recall course objectives Main Course Objective: Fundamentals of systems/signals interaction (we d like to understand how systems transform or affect

### Unit 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,

### Control 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

### ECE 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

### I 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

### Lecture 9 Time-domain properties of convolution systems

EE 12 spring 21-22 Handout #18 Lecture 9 Time-domain 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)

### Essence 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 set-up, namely for the case when the closed-loop

### The Continuous-time Fourier

The Continuous-time Fourier Transform Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn Outline Representation of Aperiodic signals:

### DEPARTMENT 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

### Autonomous 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:

### Recitation 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

### Notes 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...................

### Dynamic System Response. Dynamic System Response K. Craig 1

Dynamic System Response Dynamic System Response K. Craig 1 Dynamic System Response LTI Behavior vs. Non-LTI Behavior Solution of Linear, Constant-Coefficient, Ordinary Differential Equations Classical

### Introduction & 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

### Course 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 s-plane

### Introduction 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

### Analog 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)

### BIBO STABILITY AND ASYMPTOTIC STABILITY

BIBO STABILITY AND ASYMPTOTIC STABILITY FRANCESCO NORI Abstract. In this report with discuss the concepts of bounded-input boundedoutput stability (BIBO) and of Lyapunov stability. Examples are given to

### Lecture 9 Infinite Impulse Response Filters

Lecture 9 Infinite Impulse Response Filters Outline 9 Infinite Impulse Response Filters 9 First-Order Low-Pass Filter 93 IIR Filter Design 5 93 CT Butterworth filter design 5 93 Bilinear transform 7 9

### Control 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 D-MAVT ETH Zürich October 13, 2017 E. Frazzoli (ETH)

### Control 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 Input-Output

### Lecture 2. Introduction to Systems (Lathi )

Lecture 2 Introduction to Systems (Lathi 1.6-1.8) Pier Luigi Dragotti Department of Electrical & Electronic Engineering Imperial College London URL: www.commsp.ee.ic.ac.uk/~pld/teaching/ E-mail: p.dragotti@imperial.ac.uk

### Control Systems I. Lecture 2: Modeling. Suggested Readings: Åström & Murray Ch. 2-3, Guzzella Ch Emilio Frazzoli

Control Systems I Lecture 2: Modeling Suggested Readings: Åström & Murray Ch. 2-3, Guzzella Ch. 2-3 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 29, 2017 E. Frazzoli

### INC 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

### 1 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.

### Dynamic 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

### Generalizing 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 discrete-time radian frequency, a real variable. The quantity e jωn is then a complex sinusoid

### Ch 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:

### ANALOG 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

### EE 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

### Solution 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

### Basic concepts in DT systems. Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 1

Basic concepts in DT systems Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 1 Readings and homework For DT systems: Textbook: sections 1.5, 1.6 Suggested homework: pp. 57-58: 1.15 1.16 1.18 1.19

### AN INTRODUCTION TO THE CONTROL THEORY

Open-Loop controller An Open-Loop (OL) controller is characterized by no direct connection between the output of the system and its input; therefore external disturbance, non-linear dynamics and parameter

### Learn2Control 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

### Lecture 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

### Introduction to Feedback Control

Introduction to Feedback Control Control System Design Why Control? Open-Loop vs Closed-Loop (Feedback) Why Use Feedback Control? Closed-Loop Control System Structure Elements of a Feedback Control System

### School of Mechanical Engineering Purdue University. ME375 Feedback Control - 1

Introduction to Feedback Control Control System Design Why Control? Open-Loop vs Closed-Loop (Feedback) Why Use Feedback Control? Closed-Loop Control System Structure Elements of a Feedback Control System

### Second 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 second-order control system to a step input. In terms of damping ratio and natural

### CDS 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

### Topic 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

### An 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

### EE102 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

### Vibrations 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

### Section 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

### Mathematics 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

### Exam. 135 minutes + 15 minutes reading time

Exam January 23, 27 Control Systems I (5-59-L) 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

### The Convolution Sum for Discrete-Time LTI Systems

The Convolution Sum for Discrete-Time 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

### CHAPTER 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

### Robotics. Dynamics. Marc Toussaint U Stuttgart

Robotics Dynamics 1D point mass, damping & oscillation, PID, dynamics of mechanical systems, Euler-Lagrange equation, Newton-Euler recursion, general robot dynamics, joint space control, reference trajectory

### Transient Response of a Second-Order System

Transient Response of a Second-Order System ECEN 830 Spring 01 1. Introduction In connection with this experiment, you are selecting the gains in your feedback loop to obtain a well-behaved closed-loop

### Applications of Second-Order Differential Equations

Applications of Second-Order Differential Equations ymy/013 Building Intuition Even though there are an infinite number of differential equations, they all share common characteristics that allow intuition

### e 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

### Math 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

### Richiami 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

### Solutions 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)

### Review 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

### The Relation Between the 3-D 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

### EE3CL4: 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

### 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

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

### One-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

### The 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

### Linear 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

### Aspects of Continuous- and Discrete-Time Signals and Systems

Aspects of Continuous- and Discrete-Time 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

### Practice 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)

### Frequency-Domain C/S of LTI Systems

Frequency-Domain C/S of LTI Systems x(n) LTI y(n) LTI: Linear Time-Invariant system h(n), the impulse response of an LTI systems describes the time domain c/s. H(ω), the frequency response describes the

### HIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS IV: Laplace Transform Method. David Levermore Department of Mathematics University of Maryland

HIGHER-ORDER 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

c J. Fessler, June 9, 3, 6:3 (student version) 7. Ch. 7: Z-transform Definition Properties linearity / superposition time shift convolution: y[n] =h[n] x[n] Y (z) =H(z) X(z) Inverse z-transform by coefficient

### EQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FREE VIBRATION

1 EQUIVALENT SINGLE-DEGREE-OF-FREEDOM 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

### Solution 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

### Lecture 1 From Continuous-Time to Discrete-Time

Lecture From Continuous-Time to Discrete-Time Outline. Continuous and Discrete-Time Signals and Systems................. What is a signal?................................2 What is a system?.............................

### Department 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 11201-2983 Department of Electrical and Telecommunications Engineering Technology TEL (718) 260-5300 - FAX:

### I. 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

### Signals and Systems. Problem Set: The z-transform and DT Fourier Transform

Signals and Systems Problem Set: The z-transform and DT Fourier Transform Updated: October 9, 7 Problem Set Problem - Transfer functions in MATLAB A discrete-time, causal LTI system is described by the

### Fourier 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