Application of the Laplace Transform to LTI Differential ... · Application of the Laplace...

Signals and Systems Lecture: 8

Dr. Ayman Elshenawy Elsefy Page | 1

Application of the Laplace Transform to LTI Differential systems

1. Definition of the two-sided Laplace transform

In the previous lectures, we have seen that If a complex exponential signal

is applied to a LTI system the output will be the same signal with

the same frequency ( is a system function or eignfunction) multiplied a magnitude value

H(s) (system value or eign value) as in the following:

( ) ( ) ( ) ( )

Where

( ) ∫ ( )

If then

( ) ∫ ( )

( )

( )( )

If

( ) ∫ ( )

( )

( )

* ( )+ ( )

( ) ∫ ( ) ( )

( ) ( ) ∫ , ( ) -



( ) ∫ ( )

* ( ) +

* ( )+ ( ) ( ) ( )

The Laplace transform of h(t) can be interpreted as the Fourier transform of the original

function h(t) multiplied by a real exponential signal which may be decaying or growing

depending on the value of .

( ) * ( )+ ( )

Equation 3 and 4 indicates the relation between Laplace transform and Fourier transform.

Example(1): if ( ) ( ) find the Fourier transform and Laplace transform.

Fourier transform:

( ) ∫ ( )

∫

∫ ( )

[ ( ) ]

[ ( ) ( ) ]

, -

Laplace transform:

* ( )+ * ( ) + ( ) ∫ ( ) ( )

∫ ( )

∫ ( )

[ ( ) ]

[ ( ) ( ) ]

, -

The condition can be stated as Real{s}>-a as s= and real{s}=

By substitute s=

( )

* + ( )



Example(2): if ( ) ( ) find the Fourier transform and Laplace transform.

Fourier transform:

( ) ∫ ( )

∫

∫ ( )

[ ( ) ]

[ ( ) ( ) ]

, -

Laplace transform:

* ( )+ * ( ) + ( ) ∫ ( ) ( )

∫ ( )

∫ ( )

[ ( ) ]

[ ( ) ( ) ]

, -

* +

The condition can be stated as Real{s}<-a as s= and real{s}=

By substitute s=

( )

* + ( )

What is the difference between equation (5) and (6)?

Although ( ) ( ) and ( )

( ) are two different signals

- * ( )+ * ( )+

- The set of values of s for the first signal ( ) is different than the set of values for

( ).

- Equation (5) and (6) are rational algebraic expression.

- For equations (5) and (6) the algebraic expression is not enough without the range of s

for each of them.



Region of convergence (ROC)

In general, the range of values of s for which the integral of Laplace transform converges is

referred to as Region of convergence (ROC). Thus ROC consists of all values of s for

which the Fourier transform for * ( ) + converges.

- Property 1: ROC of X(S) consists of strips parallel to the jw-axis in the s-plane

- For rational X(s) the ROC doesn’t contain any pole.

- If x(t) is right sided and if the line * + InROC, then all values real{s} > will

be in the ROC.

- If x(t) is left sided and if the line * + in ROC, then all values real{s} < will

be in the ROC.

- If x(t) is tow sided signal then the ROC is considered as a strip

Example 3:

Find the Laplace transform of the following signal:

( ) ( ) ( )

( )

* +

( )

* +



( )

* +

Example 3:

Find the Laplace transform of the following signal:

( ) ( ) ( ) ( )

Substitute by

( ) ( ) (

) ( )

( ) ( )

( ) ( )

( ) ( )

( )

* +

( ) ( )

( ) * +

( ) ( )

( ) * +

( )

(

( )

( )) * +

Example 4: The following differential equations represent a LTI system

( )

( ) ( )

Find the Laplace transform

( ) ( ) ( )

, - ( ) ( )

( )

( )

Example 5: The following differential equations represent a LTI system

( )

( )

( )

( )

( )

( )

, - ( ) , - ( )



( )

( )

From example 1, 2,3,4,5 we can conclude that:

-The Laplace transform ( ) is always relational polynomial in the complex variable s

( ) ( )

( )

when:

a. The signal x(t) is represented as a linear combination of real or complex exponential

signal.

b. A system model of constant coefficients linear differential equations

2. Zeros and Poles of rational Laplace transform

Complex or real exponential signals have rational Laplace transforms. That is, they are

ratios of a numerator polynomial and a denominator polynomial.

( ) ( )

( )

The zeros of the numerator N(s) are called the zeros of the Laplace transform. If is a

zero, then ( ) . The zeros of the denominator ( ) are called the poles of the Laplace

transform. If is a pole then ( ) . The representation of X(s) through it’s poles and

zeros in the s-plane is referred to as the pole-zero plot of X(s).

For differential LTI systems, the zeros of the characteristic polynomial are equal to the

poles of the Laplace transform of the impulse response.

3. Convolution Property



4. Stability Analysis of LTI system using Laplace Transform

So far, we have seen that bounded-input bounded-output (BIBO) stability of a continuous-

time LTI system is equivalent to its impulse response being absolutely integrable, in which

case its Fourier transform converges. Also, the stability of an LTI differential system is

equivalent to having all the zeros of its characteristic polynomial having a negative real part.

For the Laplace transform, the first stability condition translates into the following:

An LTI system is stable if and only if the ROC of its transfer function contains the

axis.



The first term is constant and corresponds to an impulse. The two other terms are partial

fractions that can correspond to right-sided or left-sided signals, depending on the chosen

ROCs. The stability condition for a causal LTI system with a proper, rational transfer function

is stated as follows. Note that a large class of causal differential LTI systems has rational

transfer functions.



5. Application of the Laplace Transform to LTI Differential Systems

5.1 The Transfer Function of LTI System



5.2 Poles and Zeros of the Transfer Function of LTI System



6. Block Diagram realization of LTI differential System

Block diagrams are useful to analyze LTI differential systems composed of subsystems.

They are also used to represent a realization of an LTI differential system. As a combination of

three basic elements: the integrator, the gain, and the summing junction. The reason is that the

space of all transfer functions forms an algebra with the usual arithmetic operations (+,-,×,/).

That is, the sum, difference, multiplication, or division of two transfer functions yields another

transfer function.

This feedback system represents a car cruise control system where P(s) is the car’s

dynamics, F(s) is a scaling factor to get, for example, km/h units, and K(s) is the controller,

here a pure integrator that integrates the error between the desired speed and the actual car



speed to obtain a throttle command u(t) for the engine. Thus, we need to find the transfer

function relating the input ( ) to the scaled measured speed of the car v(t) and compute its

poles. Let us denote the Laplace transforms of the signals with “hats.” We have



It can be seen on this plot that there is too much overshoot in the speed response.

7. Realization of Transfer Function

It is possible to obtain a realization of the transfer function of an LTI differential system as

a combination of three basic elements shown in Figure: the integrator, the gain, and the

summing junction.

7.1 First Order System

7.2 Second Order System

Consider the transfer function ( )

. It can be realized with a feedback

interconnection of the three basic elements in a number of ways.



Parallel form: expand the transfer function as a sum of two first-order transfer

functions. It can be obtained by expanding the transfer function into partial first-order

fractions with real coefficients or complex coefficients for complex poles.

Cascading form: break up the transfer function as a cascade (multiplication) of two

first-order transfer functions.

Direct Form: is another way to realize the second-order transfer function, it may be

called controllable canonical form. To develop this form, consider the system equation

Application of the Laplace Transform to LTI Differential ... · Application of the Laplace...

Documents

Transcript of Application of the Laplace Transform to LTI Differential ... · Application of the Laplace...