P.S. GandhiP.S. GandhiMechanical Engineering Mechanical Engineering IIT BombayIIT Bombay

ODEsODEs, Response and , Response and Fourier AnalysisFourier Analysis

Acknowledgements: Ms Mehzabin Amin

Recap

Linearity, Time invariance (LTI)Solution of ODE : MA 203

Linear, Constant-CoeffODEs

Consider the homogeneous equation

Solution?

( ) thy t Aeλ=

λ is a solution to the characteristic eqn

Remember MA203?☺

( ) ( 1)1 0( ) ( ) ... ( ) 0n n

ny t a y t a y t−−+ + + =

With initial conditions 1)1(1

0)1( )0(;........;)0(:)0( −− === n

on yyyyyoy

0.... 01

1 =+++ −− aa n

nn λλ

Linear, Constant-CoeffODEs contd..

If λi’s are distinct:

Ai’s are determined from the set of initial conditions

Exercise: what will be the solution if λi’s are not distinct?

1( ) itnH i iy t Aeλ==∑

Stability of Homogeneous ODE

Under what conditions can you guarantee thatyH(t) remains bounded?

Homogenous solution is stable if and only if, for every i:

0)Re( <iλ

First Order Example: Particular solution

What will be the solution?

Homogenous Solution:Solution to u(t) = 0. Also termed as the freeresponse

Particular Solution:Response to u(t). Initial conditions assumed zero. Also termed as the forced response

yoytbutayty ==+ )0();()()(

( )

0

( ) ( )t

at a t ry t e yo e bu dτ τ− − −= + ∫

Step Response of First Order System

with u(t) being the unit step input

Assuming a>0, sketch y(t) as a function of t

yoytbutayty ==+ )0();()()(

Looking at physical system with mathematical eye

Heat transfer system

Liquid level dynamics in tank system

Step Response of Second Order System

Consider the ODE:

with u(t) being the unit step input

Often useful to view this system as a mass-spring-damper system

0)0()0();()()()( 01

Sketch step response for cases:1. 0< ζ<12. ζ=13. ζ>1

===++ yytbutyatyaty

)()()(2)( 2 tbutywtywty nn =++ ζ

Spring mass system

Motor system withPD control L=0

Looking at physical system with mathematical eye

Need for More Tools

Solving Solving ODEsODEs for getting good values of for getting good values of kpkpkdkd, , kiki becomes cumbersome especially for becomes cumbersome especially for higher order systems (Time domain) higher order systems (Time domain) We need other perspectives toWe need other perspectives toa. analyze systems anda. analyze systems andb. get better insights into their behaviorb. get better insights into their behavior

System and Signals

• A system can be viewed as a map from input signals to output signals• Signals are considered to be functions of an independent variable (say time)• We will assume that a signal is an element of a vector space

Example: set of bounded signals

• Since signal is an element of vector space, it can be represented as a weighted sum of basis functions

Fourier series

• We are specifically interested in sinusoids being used as basis functions •We will use e jωt as basis functions in our series expansions• A periodic signal may be expressed as a sum of complex exponentials (sinusoids)

Periodic Signals

• A signal x(t) is said to be periodic if there exists T>0, such that x(t+T)=x(t) for all t• T is called the period• Fundamental period T0 is the smallest period which satisfies the above equation• Fundamental frequency ω0 = 2Π/T0• Fourier said that a signal with fundamental frequency ω0 may be expressed as a sum ofsinusoids whose frequencies are integral multiples of the fundamental frequency

• Fourier series expansion of periodic signal

• Given a signal x(t) with fundamental frequency ω0 , How to determine the Fourier series coefficients?

Can you prove this?

Fourier series contd…

Fourier series coefficients contd..

•

• ak may be complex. If x(t) is real valued, akwill also be real• ak e jω0t is called the component of Fourier series expansion

Example: Fourier series Expansion

• Consider the signal U(t) given below

U(t)=u0+u1+u2+… where,uk= ak e jω0t for k=0,1,2, and so onCan you find ak for this signal?

Example: Fourier series expansion

u0+u1 u0+u1+u3 u0+u1+u3+u5

•Can you see the summation converging towards the actual signal as the no of terms is increased?• what is u2,u4 etc? Evaluate

Fourier series coefficients

What if the period increases?

• As the period is made very large ie. it approaches infinity, the signal becomes an aperiodic signal as given below

Aperiodic Signals

For the above period x(t) let x1(t) be its periodic counterpart

Aperiodic Signals

• x1(t) is the periodic counterpart of x(t), we let T inf or ω0 0 in the Fourier series expansion of x1(t)

x(t) within one period is same as x1(t), it is 0 elsewhere

Aperiodic Signals contd..

Recap

Response of first and second order systems: Response of first and second order systems: looking at physical system with a looking at physical system with a ““mathematical mathematical eyeeye””Introduction to use of Fourier series to gain Introduction to use of Fourier series to gain insight into nature of signalsinsight into nature of signalsExtension of Fourier series representation to Extension of Fourier series representation to Fourier transform formula for Fourier transform formula for aperiodicaperiodic signals. signals. Concept: Concept: –– Take limit as TTake limit as T infinity or infinity or ωω00 0 of Fourier series 0 of Fourier series

formula. formula. –– Discrete Discrete ωω points kpoints kωω00 will become will become ωω as as ωω00 00

Fourier Transform

Fourier Transform of x(t)

• Transformation takes the time domain signal to a frequency domain•|X(jw)| is a measure of the significance of the term ejwt in the decomposition of x(t)• X(jw) is, in general, complex-valued

Impulse signal

• Above signal is called an impulse signal for T1 0• Can you find its Fourier transform?

Properties of Fourier Transform

• Linearity x1(t) X1(jω)x2(t) X2(jω)ax1(t)+bx2(t) aX1(jω)+bX2(jω)

• Integral relation x(t) X(jω)

• Deivative relation

• Delay Relation

Impulse Response

Response of a system to an impulse is called an impulse response of the system

Properties of a system revisited…

Linearity

Time-invariance

Signal in terms of impulse

• Any signal u(t) can be written as weighted sum of impulse functions

Response of system to any signal

h(t)*u(t)

Convolution Integral

• Linear, time-invariant (LTI) systems can be completely characterized by their impulse response

• Consider an LTI system whose impulse reponse is h(t), the response of the system to arbitrary input u(t) is given by:

y(t)=h(t)*u(t)

Frequency Response

LTIe jωt ?

Frequency response of the system = fourier transform of the impulse response

H(jω) e jωt

Complex Exponentials and LTI systems

LTIe jωt ? H(jω) e jωt

What will be the response of the system to a sum of complex exponentials?