Systems Analysis and Control

Size: px
Start display at page:

Download "Systems Analysis and Control"

Transcription

1 Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 5: Calculating the Laplace Transform of a Signal

2 Introduction In this Lecture, you will learn: Laplace Transform of Simple Signals. Step, Exponentials, Ramps. Impulse Properties of the Laplace Transform. delay, scaling convolution etc. Transfer Functions State Space to Transfer Function M. Peet Lecture 5: Control Systems 2 / 3

3 Previously: The Laplace Transform Definition. The Laplace Transform of the signal u(t) is û(s) = e st u(t)dt The Laplace Transform is not the same as the Fourier transform. Assumes that signals begin at time t = and u(t) = for t <. Tends to flatten out peaks. Signal: u(t) = sin at Laplace Transform: 2 a û(s) = s 2 a Frequency(w) magnitude of u(iw) M. Peet Lecture 5: Control Systems 3 / 3

4 The Laplace Transform Example: Step Function Consider the input signal: { t < u(t) = t The Laplace transform is û(s) = = s e st dt [ e st ] = [ ] s = s Note that lim t e st =. Because we assume the real part of s is negative. M. Peet Lecture 5: Control Systems 4 / 3

5 Review: Integration by parts A very useful formula for finding the Laplace Transform is integration by parts. b a b u(s)v (s)ds = [u(s)v(s)] s=b s=a u (s)v(s)ds a M. Peet Lecture 5: Control Systems 5 / 3

6 The Laplace Transform Example: Ramp Function Consider the input signal: { t < u(t) = t t = t (t) Recall (t) is the step function. The Laplace transform is Ramp Function t û(s) = = te st dt [ s te st ] + s = [ ] s 2 [e st ] dt e st dt = s 2 [ ] = s 2 M. Peet Lecture 5: Control Systems 6 / 3

7 The Laplace Transform Example: Quadratic Function The Quadratic is similar to the Ramp. { t < u(t) = t 2 t = t 2 (t) Recall (t) is the step function. The Laplace transform is Quadratic Function t û(s) = = = s 3 t 2 e st dt [ s t2 e st ] + s te st dt M. Peet Lecture 5: Control Systems 7 / 3

8 The Laplace Transform Example: exponential The Sinusoid is created from the complex exponentials. Lets do exponentials first..9 Consider the input signal:.8 {.7 t <.6 u(t) = e at.5 t = e at (t) Exponential Function Recall (t) is the step function. The Laplace transform is û(s) = = e (a s)t dt [ a s e(a s)t = s a ] t M. Peet Lecture 5: Control Systems 8 / 3

9 The Laplace Transform Linearity Input u Λ Output u^=λu The Laplace Transform is itself a LINEAR SYSTEM. The Input is u The Output is û Lets call the system L û = Lu As for any linear system, L(αu + βu 2 ) = αlu + βlu 2 = αû + βû 2 We can compute the Laplace transform by using simple parts. Example: Sinusoids cos ωt = eıωt + e ıωt 2 M. Peet Lecture 5: Control Systems 9 / 3

10 The Laplace Transform Example: Sinusoids Recall that cos ωt = eıωt + e ıωt 2 But the Laplace transform of e at is L(e at ) = s a So the Laplace transforms of e ıωt and e ıωt are û (s) = L(e ıωt ) = s ıω and û 2 (s) = L(e ıωt ) = s + ıω Which shows us how to find L(cos(ωt)) ( e ıωt + e ıωt ) ( ) ( ) e ıωt e ıωt L(cos(ωt)) = L = L + L = ( 2 s ıω + ) = ( ) 2s = s + ıω 2 (s ıω)(s + ıω) s s 2 + ω 2 = û 2 + û2 2 M. Peet Lecture 5: Control Systems / 3

11 The Laplace Transform Example: Sinusoids Likewise, for the sin function, sin ωt = eıωt e ıωt 2i Which shows us how to find L(cos(ωt)) L(sin(ωt)) = û 2 û2 2i = ( 2i s ıω ) s + ıω = ( ) 2ıω ω = 2i (s ıω)(s + ıω) s 2 + ω 2 M. Peet Lecture 5: Control Systems / 3

12 The Laplace Transform Example: Impulse functions Definition 2. An Impulse is a significant strike which occurs in an infinitesimally short time. Mathematically, we model the impulse as a Dirac delta function, δ(t). Question: What is the Laplace transform of an Impulse? Answer: By definition of the Dirac Delta: ˆδ(s) = = e st t= = e st δ(t)dt magnitude The Impulse has a Uniform Frequency Response!!!! The opposite of the step function ((s) = Lδ(t)). time M. Peet Lecture 5: Control Systems 2 / 3

13 Laplace Transform Table Simple Signals Summary: We can find the Laplace Transform for many simple signals u(t) û(s) Step (t) s Power t m m! s m+ Exponential e at t m e at (m )! s+a (s+a) m Power Exponential a Sine sin(at) s 2 +a 2 Cosine cos(at) s s 2 +a 2 Impulse δ(t) Remember that all functions are only for t. We always have u(t) = for t < Question: How to use these functions to solve for more complex functions? M. Peet Lecture 5: Control Systems 3 / 3

14 The Laplace Transform Other Properties Properties: What properties of the Laplace transform can we exploit? We have already discussed Linearity. Other Examples: Delay: The signal is delayed u(t) u(t τ) Time-Scale: The signal is slowed down or speeded up u(t) u(at) Differentiation: The signal is a differential in time: u(t) u (t) How do these changes affect the the Laplace transform? û(s)????? M. Peet Lecture 5: Control Systems 4 / 3

15 The Laplace Transform Delay What is the effect of a delay on the Laplace Transform? Use a change of variables: t = t τ, dt = dt û delayed (s) = L(u(t τ)) = = e st u(t τ)dt e s(t +τ) u(t )dt = e sτ e st u(t )dt = e sτ û(s) M. Peet Lecture 5: Control Systems 5 / 3

16 The Laplace Transform Time-Scaling What is the effect of scaling time? Use a change of variables: t = at, dt = adt û scaled (s) = L(u(at)) = = a = aû e st u(at)dt ( s a) e st a u(t )dt M. Peet Lecture 5: Control Systems 6 / 3

17 Example: Watch The Order of Operations What is the Laplace Transform of u(t) = sin(a(t τ)) We know that û sin (s) = L(sin(t)) = s 2 +. The change is sin t sin(at) sin(a(t τ)) Time-Scaling: û s (s) = ( aû s a Time-Delay: û d (s) = e τs û (s) Thus we have û ds (s) = e τs û s (s) = e τs ( = e τs a s 2 + a 2 ) ( a s ) 2 a + ) M. Peet Lecture 5: Control Systems 7 / 3

18 The Laplace Transform Properties Differentiation A very common case is when a signal has been differentiated. e.g.: ẋ(t) = Ax(t) + Bu(t) In this case we use integration by parts: L (ẋ(t)) = e st ẋ(t)dt = [ e st ẋ(t) ] + s e st x(t)dt = ẋ() + s = ẋ() + sˆx(s) e st x(t)dt M. Peet Lecture 5: Control Systems 8 / 3

19 The Laplace Transform Properties Multiple Differentiation The differentiation property can be applied recursively: L (... x (t)) = ẍ() + sl(ẍ(t)) = ẍ() sẋ() + s 2 L(ẋ(t)) = ẍ() sẋ() s 2 x() + s 3ˆx(s) General Formula: L(x (n) (t)) = s nˆx(s) s n x() x n () M. Peet Lecture 5: Control Systems 9 / 3

20 The Laplace Transform Properties Integration Importantly, the Laplace transform can also be applied to integration: An immediate consequence of differentiation ( ) L f(t) = f() + sl(f(t)) If f(t) = t x(τ)dτ, then f(t) = x(t), and f() =, so ( ) ˆx(s) = L(x(t)) = L f(t) = f() + sl(f(t)) ( t ) = sl x(τ)dτ Hence ( t ) L x(τ)dτ = ˆx(s) s M. Peet Lecture 5: Control Systems 2 / 3

21 The Laplace Transform Properties Convolution Recall the solution of a state-space system: y(t) = t Ce A(t s) Bu(s)ds + Du(t) This is a special kind of integral called the Convolution Integral. Definition 3. The Convolution Integral of u(t) with v(t) is defined as (u v)(t) := t u(τ)v(t τ)dτ Thus roughly speaking y(t) = u(t) ( Ce At B ) M. Peet Lecture 5: Control Systems 2 / 3

22 The Laplace Transform Properties Convolution (f g)(t) := t f(τ)g(t τ)dτ M. Peet Lecture 5: Control Systems 22 / 3

23 The Laplace Transform Properties Convolution The convolution of two signals has a remarkable property: L(u v) = û(s)ˆv(s) Proof. Since u(t) = v(t) = for t <, we have that u(τ)v(t τ) = for τ < or τ > t. Thus: t u(τ)v(t τ)dτ = u(τ)v(t τ)dτ Now using the delay property: L(u v) = = = t e st u(τ)v(t τ)dτdt = u(τ) e st v(t τ)dtdτ u(τ)e sτ ˆv(s)dτ = û(s)ˆv(s) e st u(τ)v(t τ)dτdt M. Peet Lecture 5: Control Systems 23 / 3

24 Properties of the Laplace Transform Summary Summary: Simple functions and properties can be combined to calculate the Laplace Transform. u(t) û(s) Delay u(t τ) e τs û(s) Time-Scaling u(at) ( ) s aû a Differentiation u (t) sû(s) u() t Integration u(τ)dτ s û(s) d Frequency Differentiation tu(t) Frequency Shift e dsû(s) at u(t) û(s a) Convolution u(τ)v(t τ)dτ û(s)ˆv(s) t M. Peet Lecture 5: Control Systems 24 / 3

25 The Inverse Laplace Transform An Important Case Consider the simple case of a cosine with an exponential: u(t) = e σt sin ωt Approach: L(sin ωt) = ω s 2 +ω 2 Exponential means frequency shift: e σt u(t) û(s σ). Hence the Laplace transform is ω û(s) = (s σ) 2 + ω 2 = ω s 2 2σs + (σ 2 + ω 2 ) Problem: Can we go the other direction? Calculate u(t) if û(s) = s 2 + bs + c Solution: Use a combination sinusoid and frequency shift First recall that ω L(sin ωt) = s 2 + ω 2 M. Peet Lecture 5: Control Systems 25 / 3

26 An Important Case, continued If û(s) = s 2 +bs+c, the roots of the denominator are s,2 = b/2 ± 2 b2 4c Hence û(s) = (s + b/2 + 2 b2 4c)(s + b/2 2 b2 4c) = ((s + b/2) + 2 b2 4c)((s + b/2) 2 b2 4c) = ((s + b/2) 2 4 (b2 4c)) Now assume 4c > b 2. Then û(s) = ĥ(s + b/2), where ĥ(s) = s (4c b2 ). M. Peet Lecture 5: Control Systems 26 / 3

27 Numerical Example This is a simple sine function: where ω = c b2 4 ĥ(s) = s (4c b2 ). h(t) = sin (ωt) ω Recall the frequency-shift property: L û(s) = L ĥ(s + b 2 ) = e b 2 t h(t). We conclude that ) u(t) = e b 2 t sin ( 4c b24 t 4c b2 4 Numerical Example: Let û(s) = s 2 +2s+2. Taking the roots, we find û(s) = (s + ) 2 +. This is a sine function with a frequency shift of. Therefore u(t) = e t sin t M. Peet Lecture 5: Control Systems 27 / 3

28 The Laplace Transform of a State-Space System Transfer Functions Recall the State-Space System ẋ = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) x() = Applying the Laplace Transform to the first signal: sˆx(s) x() = Aˆx(s) + Bû(s) where we used Linearity Differentiation Property Let ẋ() = and we have (si A)ˆx(s) = Bû(s) or ˆx(s) = (si A) Bû(s) M. Peet Lecture 5: Control Systems 28 / 3

29 The Laplace Transform of a State-Space System Transfer Functions Applying the Laplace Transform to the output equation, y(t) = Cx(t) + Du(t) we get ŷ(s) = C ˆx(s) + Dû(s) Solving for ŷ(s), we get ŷ(s) = ( C(sI A) B + D ) û(s) Thus we have a Transfer Function for the system which can be used to construct a solution for any input using the frequency-domain representation. Theorem 4. The Transfer Function for a state-space system is Ĝ(s) = C(sI A) B + D M. Peet Lecture 5: Control Systems 29 / 3

30 Summary What have we learned today? Laplace Transform of Simple Signals. Step, Exponentials, Ramps. Impulse Properties of the Laplace Transform. delay, scaling convolution etc. Transfer Functions State Space to Transfer Function Next Lecture: More on Transfer Functions M. Peet Lecture 5: Control Systems 3 / 3

ENGIN 211, Engineering Math. Laplace Transforms

ENGIN 211, Engineering Math. Laplace Transforms ENGIN 211, Engineering Math Laplace Transforms 1 Why Laplace Transform? Laplace transform converts a function in the time domain to its frequency domain. It is a powerful, systematic method in solving

More information

Computing inverse Laplace Transforms.

Computing 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 information

2.161 Signal Processing: Continuous and Discrete Fall 2008

2.161 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 information

Chapter 6: The Laplace Transform 6.3 Step Functions and

Chapter 6: The Laplace Transform 6.3 Step Functions and Chapter 6: The Laplace Transform 6.3 Step Functions and Dirac δ 2 April 2018 Step Function Definition: Suppose c is a fixed real number. The unit step function u c is defined as follows: u c (t) = { 0

More information

Lecture 7: Laplace Transform and Its Applications Dr.-Ing. Sudchai Boonto

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

More information

LTI Systems (Continuous & Discrete) - Basics

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

More information

Definition of the Laplace transform. 0 x(t)e st dt

Definition 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 information

ECEN 420 LINEAR CONTROL SYSTEMS. Lecture 2 Laplace Transform I 1/52

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

More information

e st f (t) dt = e st tf(t) dt = L {t f(t)} s

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

More information

The Laplace transform

The 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 information

GATE EE Topic wise Questions SIGNALS & SYSTEMS

GATE 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 information

+ + LAPLACE TRANSFORM. Differentiation & Integration of Transforms; Convolution; Partial Fraction Formulas; Systems of DEs; Periodic Functions.

+ + LAPLACE TRANSFORM. Differentiation & Integration of Transforms; Convolution; Partial Fraction Formulas; Systems of DEs; Periodic Functions. COLOR LAYER red LAPLACE TRANSFORM Differentiation & Integration of Transforms; Convolution; Partial Fraction Formulas; Systems of DEs; Periodic Functions. + Differentiation of Transforms. F (s) e st f(t)

More information

Control Systems. Frequency domain analysis. L. Lanari

Control 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 information

MA 201, Mathematics III, July-November 2018, Laplace Transform (Contd.)

MA 201, Mathematics III, July-November 2018, Laplace Transform (Contd.) MA 201, Mathematics III, July-November 2018, Laplace Transform (Contd.) Lecture 19 Lecture 19 MA 201, PDE (2018) 1 / 24 Application of Laplace transform in solving ODEs ODEs with constant coefficients

More information

Control 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 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 information

(f g)(t) = Example 4.5.1: Find f g the convolution of the functions f(t) = e t and g(t) = sin(t). Solution: The definition of convolution is,

(f g)(t) = Example 4.5.1: Find f g the convolution of the functions f(t) = e t and g(t) = sin(t). Solution: The definition of convolution is, .5. Convolutions and Solutions Solutions of initial value problems for linear nonhomogeneous differential equations can be decomposed in a nice way. The part of the solution coming from the initial data

More information

Linear dynamical systems with inputs & outputs

Linear dynamical systems with inputs & outputs EE263 Autumn 215 S. Boyd and S. Lall Linear dynamical systems with inputs & outputs inputs & outputs: interpretations transfer function impulse and step responses examples 1 Inputs & outputs recall continuous-time

More information

The formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d

The formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d Solving ODEs using Fourier Transforms The formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d 2 dx + q f (x) R(x)

More information

Introduction to Modern Control MT 2016

Introduction to Modern Control MT 2016 CDT Autonomous and Intelligent Machines & Systems Introduction to Modern Control MT 2016 Alessandro Abate Lecture 2 First-order ordinary differential equations (ODE) Solution of a linear ODE Hints to nonlinear

More information

Chapter 6: The Laplace Transform. Chih-Wei Liu

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

More information

Ordinary differential equations

Ordinary differential equations Class 11 We will address the following topics Convolution of functions Consider the following question: Suppose that u(t) has Laplace transform U(s), v(t) has Laplace transform V(s), what is the inverse

More information

Laplace Transform. Chapter 4

Laplace Transform. Chapter 4 Chapter 4 Laplace Transform It s time to stop guessing solutions and find a systematic way of finding solutions to non homogeneous linear ODEs. We define the Laplace transform of a function f in the following

More information

Lecture 29. Convolution Integrals and Their Applications

Lecture 29. Convolution Integrals and Their Applications Math 245 - Mathematics of Physics and Engineering I Lecture 29. Convolution Integrals and Their Applications March 3, 212 Konstantin Zuev (USC) Math 245, Lecture 29 March 3, 212 1 / 13 Agenda Convolution

More information

( ) f (k) = FT (R(x)) = R(k)

( ) f (k) = FT (R(x)) = R(k) Solving ODEs using Fourier Transforms The formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d 2 dx + q f (x) = R(x)

More information

Time Response of Systems

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 information

MathQuest: Differential Equations

MathQuest: 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 information

Math Shifting theorems

Math Shifting theorems Math 37 - Shifting theorems Erik Kjær Pedersen November 29, 2005 Let us recall the Dirac delta function. It is a function δ(t) which is 0 everywhere but at t = 0 it is so large that b a (δ(t)dt = when

More information

2.161 Signal Processing: Continuous and Discrete Fall 2008

2.161 Signal Processing: Continuous and Discrete Fall 2008 MIT OpenCourseWare http://ocw.mit.edu.6 Signal Processing: Continuous and Discrete Fall 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS

More information

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.

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

More information

Generalized sources (Sect. 6.5). The Dirac delta generalized function. Definition Consider the sequence of functions for n 1, Remarks:

Generalized sources (Sect. 6.5). The Dirac delta generalized function. Definition Consider the sequence of functions for n 1, Remarks: Generalized sources (Sect. 6.5). The Dirac delta generalized function. Definition Consider the sequence of functions for n, d n, t < δ n (t) = n, t 3 d3 d n, t > n. d t The Dirac delta generalized function

More information

Control 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 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 information

Identification Methods for Structural Systems

Identification 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 information

9.5 The Transfer Function

9.5 The Transfer Function Lecture Notes on Control Systems/D. Ghose/2012 0 9.5 The Transfer Function Consider the n-th order linear, time-invariant dynamical system. dy a 0 y + a 1 dt + a d 2 y 2 dt + + a d n y 2 n dt b du 0u +

More information

Raktim Bhattacharya. . AERO 632: Design of Advance Flight Control System. Preliminaries

Raktim 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 information

BIBO STABILITY AND ASYMPTOTIC STABILITY

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

More information

Laplace Transform Part 1: Introduction (I&N Chap 13)

Laplace Transform Part 1: Introduction (I&N Chap 13) Laplace Transform Part 1: Introduction (I&N Chap 13) Definition of the L.T. L.T. of Singularity Functions L.T. Pairs Properties of the L.T. Inverse L.T. Convolution IVT(initial value theorem) & FVT (final

More information

Control Systems. Laplace domain analysis

Control 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 information

The Laplace Transform

The 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 information

Review of Frequency Domain Fourier Series: Continuous periodic frequency components

Review of Frequency Domain Fourier Series: Continuous periodic frequency components Today we will review: Review of Frequency Domain Fourier series why we use it trig form & exponential form how to get coefficients for each form Eigenfunctions what they are how they relate to LTI systems

More information

Basic Procedures for Common Problems

Basic 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 information

Fourier Series and Transforms. Revision Lecture

Fourier Series and Transforms. Revision Lecture E. (5-6) : / 3 Periodic signals can be written as a sum of sine and cosine waves: u(t) u(t) = a + n= (a ncosπnft+b n sinπnft) T = + T/3 T/ T +.65sin(πFt) -.6sin(πFt) +.6sin(πFt) + -.3cos(πFt) + T/ Fundamental

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 8: Response Characteristics Overview In this Lecture, you will learn: Characteristics of the Response Stability Real

More information

Systems Analysis and Control

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

More information

Laplace Transforms Chapter 3

Laplace 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 information

Unit 2: Modeling in the Frequency Domain Part 2: The Laplace Transform. The Laplace Transform. The need for Laplace

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,

More information

Analysis III for D-BAUG, Fall 2017 Lecture 10

Analysis III for D-BAUG, Fall 2017 Lecture 10 Analysis III for D-BAUG, Fall 27 Lecture Lecturer: Alex Sisto (sisto@math.ethz.ch Convolution (Faltung We have already seen that the Laplace transform is not multiplicative, that is, L {f(tg(t} L {f(t}

More information

Math 308 Exam II Practice Problems

Math 308 Exam II Practice Problems Math 38 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture and all suggested homework problems..

More information

Outline. Classical Control. Lecture 2

Outline. Classical Control. Lecture 2 Outline Outline Outline Review of Material from Lecture 2 New Stuff - Outline Review of Lecture System Performance Effect of Poles Review of Material from Lecture System Performance Effect of Poles 2 New

More information

Ordinary Differential Equations. Session 7

Ordinary 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 information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 2: Drawing Bode Plots, Part 2 Overview In this Lecture, you will learn: Simple Plots Real Zeros Real Poles Complex

More information

Math 307 Lecture 19. Laplace Transforms of Discontinuous Functions. W.R. Casper. Department of Mathematics University of Washington.

Math 307 Lecture 19. Laplace Transforms of Discontinuous Functions. W.R. Casper. Department of Mathematics University of Washington. Math 307 Lecture 19 Laplace Transforms of Discontinuous Functions W.R. Casper Department of Mathematics University of Washington November 26, 2014 Today! Last time: Step Functions This time: Laplace Transforms

More information

Control System Design

Control System Design ELEC ENG 4CL4: Control System Design Notes for Lecture #4 Monday, January 13, 2003 Dr. Ian C. Bruce Room: CRL-229 Phone ext.: 26984 Email: ibruce@mail.ece.mcmaster.ca Impulse and Step Responses of Continuous-Time

More information

Laplace Transforms and use in Automatic Control

Laplace 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 information

Part IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL. Glenn Vinnicombe HANDOUT 2

Part IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL. Glenn Vinnicombe HANDOUT 2 Part IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL Glenn Vinnicombe HANDOUT 2 Impulse responses, step responses and transfer functions. transfer function ū(s) u(t) Laplace transform pair

More information

(an improper integral)

(an improper integral) Chapter 7 Laplace Transforms 7.1 Introduction: A Mixing Problem 7.2 Definition of the Laplace Transform Def 7.1. Let f(t) be a function on [, ). The Laplace transform of f is the function F (s) defined

More information

LINEAR SYSTEMS. J. Elder PSYC 6256 Principles of Neural Coding

LINEAR SYSTEMS. J. Elder PSYC 6256 Principles of Neural Coding LINEAR SYSTEMS Linear Systems 2 Neural coding and cognitive neuroscience in general concerns input-output relationships. Inputs Light intensity Pre-synaptic action potentials Number of items in display

More information

2 Classification of Continuous-Time Systems

2 Classification of Continuous-Time Systems Continuous-Time Signals and Systems 1 Preliminaries Notation for a continuous-time signal: x(t) Notation: If x is the input to a system T and y the corresponding output, then we use one of the following

More information

Review of Linear Time-Invariant Network Analysis

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

More information

06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 1

06/12/ 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 information

The Laplace Transform

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

More information

MA Ordinary Differential Equations and Control Semester 1 Lecture Notes on ODEs and Laplace Transform (Parts I and II)

MA Ordinary Differential Equations and Control Semester 1 Lecture Notes on ODEs and Laplace Transform (Parts I and II) MA222 - Ordinary Differential Equations and Control Semester Lecture Notes on ODEs and Laplace Transform (Parts I and II Dr J D Evans October 22 Special thanks (in alphabetical order are due to Kenneth

More information

EE102 Homework 2, 3, and 4 Solutions

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

More information

Transfer Functions. Chapter Introduction. 6.2 The Transfer Function

Transfer Functions. Chapter Introduction. 6.2 The Transfer Function Chapter 6 Transfer Functions As a matter of idle curiosity, I once counted to find out what the order of the set of equations in an amplifier I had just designed would have been, if I had worked with the

More information

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 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)

More information

2.161 Signal Processing: Continuous and Discrete Fall 2008

2.161 Signal Processing: Continuous and Discrete Fall 2008 MIT OpenCourseWare http://ocw.mit.edu 2.161 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 information

Math Laplace transform

Math Laplace transform 1 Math 371 - Laplace transform Erik Kjær Pedersen November 22, 2005 The functions we have treated with the power series method are called analytical functions y = a k t k The differential equations we

More information

The Laplace Transform and the IVP (Sect. 6.2).

The Laplace Transform and the IVP (Sect. 6.2). The Laplace Transform and the IVP (Sect..2). Solving differential equations using L ]. Homogeneous IVP. First, second, higher order equations. Non-homogeneous IVP. Recall: Partial fraction decompositions.

More information

The Laplace Transform

The 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 information

Systems and Control Theory Lecture Notes. Laura Giarré

Systems and Control Theory Lecture Notes. Laura Giarré Systems and Control Theory Lecture Notes Laura Giarré L. Giarré 2017-2018 Lesson 7: Response of LTI systems in the transform domain Laplace Transform Transform-domain response (CT) Transfer function Zeta

More information

9.2 The Input-Output Description of a System

9.2 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 information

Special Mathematics Laplace Transform

Special 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 information

EE/ME/AE324: Dynamical Systems. Chapter 7: Transform Solutions of Linear Models

EE/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 information

Professor Fearing EE C128 / ME C134 Problem Set 7 Solution Fall 2010 Jansen Sheng and Wenjie Chen, UC Berkeley

Professor 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 information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 21: Stability Margins and Closing the Loop Overview In this Lecture, you will learn: Closing the Loop Effect on Bode Plot Effect

More information

20.6. Transfer Functions. Introduction. Prerequisites. Learning Outcomes

20.6. 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 information

Applied Differential Equation. October 22, 2012

Applied Differential Equation. October 22, 2012 Applied Differential Equation October 22, 22 Contents 3 Second Order Linear Equations 2 3. Second Order linear homogeneous equations with constant coefficients.......... 4 3.2 Solutions of Linear Homogeneous

More information

CH.6 Laplace Transform

CH.6 Laplace Transform CH.6 Laplace Transform Where does the Laplace transform come from? How to solve this mistery that where the Laplace transform come from? The starting point is thinking about power series. The power series

More information

Continuous-Time Frequency Response (II) Lecture 28: EECS 20 N April 2, Laurent El Ghaoui

Continuous-Time Frequency Response (II) Lecture 28: EECS 20 N April 2, Laurent El Ghaoui EECS 20 N April 2, 2001 Lecture 28: Continuous-Time Frequency Response (II) Laurent El Ghaoui 1 annoucements homework due on Wednesday 4/4 at 11 AM midterm: Friday, 4/6 includes all chapters until chapter

More information

EE 3054: Signals, Systems, and Transforms Summer It is observed of some continuous-time LTI system that the input signal.

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

More information

Math 2C03 - Differential Equations. Slides shown in class - Winter Laplace Transforms. March 4, 5, 9, 11, 12, 16,

Math 2C03 - Differential Equations. Slides shown in class - Winter Laplace Transforms. March 4, 5, 9, 11, 12, 16, Math 2C03 - Differential Equations Slides shown in class - Winter 2015 Laplace Transforms March 4, 5, 9, 11, 12, 16, 18... 2015 Laplace Transform used to solve linear ODEs and systems of linear ODEs with

More information

ANALOG AND DIGITAL SIGNAL PROCESSING CHAPTER 3 : LINEAR SYSTEM RESPONSE (GENERAL CASE)

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

More information

ECE 3620: Laplace Transforms: Chapter 3:

ECE 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 information

L2 gains and system approximation quality 1

L2 gains and system approximation quality 1 Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.242, Fall 24: MODEL REDUCTION L2 gains and system approximation quality 1 This lecture discusses the utility

More information

Solutions to Assignment 7

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)

More information

ECEN 605 LINEAR SYSTEMS. Lecture 7 Solution of State Equations 1/77

ECEN 605 LINEAR SYSTEMS. Lecture 7 Solution of State Equations 1/77 1/77 ECEN 605 LINEAR SYSTEMS Lecture 7 Solution of State Equations Solution of State Space Equations Recall from the previous Lecture note, for a system: ẋ(t) = A x(t) + B u(t) y(t) = C x(t) + D u(t),

More information

Systems Analysis and Control

Systems 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 information

ECE 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, 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 information

Math 216 Second Midterm 17 November, 2016

Math 216 Second Midterm 17 November, 2016 Math 216 Second Midterm 17 November, 2016 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material

More information

THE UNIVERSITY OF WESTERN ONTARIO. Applied Mathematics 375a Instructor: Matt Davison. Final Examination December 14, :00 12:00 a.m.

THE UNIVERSITY OF WESTERN ONTARIO. Applied Mathematics 375a Instructor: Matt Davison. Final Examination December 14, :00 12:00 a.m. THE UNIVERSITY OF WESTERN ONTARIO London Ontario Applied Mathematics 375a Instructor: Matt Davison Final Examination December 4, 22 9: 2: a.m. 3 HOURS Name: Stu. #: Notes: ) There are 8 question worth

More information

Grammians. Matthew M. Peet. Lecture 20: Grammians. Illinois Institute of Technology

Grammians. Matthew M. Peet. Lecture 20: Grammians. Illinois Institute of Technology Grammians Matthew M. Peet Illinois Institute of Technology Lecture 2: Grammians Lyapunov Equations Proposition 1. Suppose A is Hurwitz and Q is a square matrix. Then X = e AT s Qe As ds is the unique solution

More information

Name: Solutions Final Exam

Name: 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 information

Linear systems, small signals, and integrators

Linear systems, small signals, and integrators Linear systems, small signals, and integrators CNS WS05 Class Giacomo Indiveri Institute of Neuroinformatics University ETH Zurich Zurich, December 2005 Outline 1 Linear Systems Crash Course Linear Time-Invariant

More information

221B Lecture Notes on Resonances in Classical Mechanics

221B Lecture Notes on Resonances in Classical Mechanics 1B Lecture Notes on Resonances in Classical Mechanics 1 Harmonic Oscillators Harmonic oscillators appear in many different contexts in classical mechanics. Examples include: spring, pendulum (with a small

More information

Math Fall Linear Filters

Math Fall Linear Filters Math 658-6 Fall 212 Linear Filters 1. Convolutions and filters. A filter is a black box that takes an input signal, processes it, and then returns an output signal that in some way modifies the input.

More information

Lecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design.

Lecture 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 information

Notes for ECE-320. Winter by R. Throne

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...................

More information

EECE 3620: Linear Time-Invariant Systems: Chapter 2

EECE 3620: Linear Time-Invariant Systems: Chapter 2 EECE 3620: Linear Time-Invariant Systems: Chapter 2 Prof. K. Chandra ECE, UMASS Lowell September 7, 2016 1 Continuous Time Systems In the context of this course, a system can represent a simple or complex

More information

EC Signals and Systems

EC Signals and Systems UNIT I CLASSIFICATION OF SIGNALS AND SYSTEMS Continuous time signals (CT signals), discrete time signals (DT signals) Step, Ramp, Pulse, Impulse, Exponential 1. Define Unit Impulse Signal [M/J 1], [M/J

More information

Lecture 12. AO Control Theory

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

More information

Control Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control D-MAVT ETH Zürich

Control Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control D-MAVT ETH Zürich Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 2, 2018 J. Tani, E. Frazzoli (ETH) Lecture 7:

More information