Laplace transforms
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LAPLACE TRANSFORMS. Prepared by Mrs. Azduwin Binti Khasri. LAPLACE TRANSFORMS. A mathematical tool. Important analytical method for solving LINEAR ORDINARY DIFFERENTIAL EQUATIONS. Application to nonlinear ODEs? Must linearize first. Transform ODEs to algebraic equation.

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LAPLACE TRANSFORMS

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Laplace transforms

LAPLACE TRANSFORMS

Prepared by Mrs. AzduwinBintiKhasri


Laplace transforms1

LAPLACE TRANSFORMS

  • A mathematical tool.

  • Important analytical method for solving LINEAR ORDINARY DIFFERENTIAL EQUATIONS.

  • Application to nonlinear ODEs? Must linearize first.

  • Transform ODEs to algebraic equation.

  • Laplace transforms play a key role in important process

    control concepts and techniques.

  • Examples: Transfer functions ,Frequency response,Control

    system design, Stability analysis


Laplace transforms

  • DEFINITION

The Laplace transform of a function, f(t), is defined as

F(s)= symbol for the Laplace transform,

L =Laplace transform operator

f(t) =some function of time, t.

s= Complex Independent Variable

Note: The L operator transforms a time domain function f(t) into an s domain function, F(s).


Laplace transforms

INVERSE LAPLACE TRANSFORM, L-1

By definition, the inverse Laplace transform operator, L-1, converts an s-domain function back to the corresponding time domain function:

Important Properties:

Both L and L-1 are linear operators. Thus, satisfies the superposition principle;


Laplace transforms

where:

- x(t) and y(t) are arbitrary functions

- a and b are constants

Similarly,


Laplace transforms of common functions

LAPLACE TRANSFORMS OF COMMON FUNCTIONS

  • Constant Function

  • Let f(t) = a (a constant).

  • Then from the definition of the Laplace transform in (3-1),


Laplace transforms

  • Step Function

  • The unit step function is widely used in the analysis of process control problems. It is defined as:

Because the step function is a special case of a “constant”, it follows from (3-4) that


Laplace transforms

  • Derivatives

  • This is a very important transform because derivatives appear in the ODEs for dynamic models.

initial condition at t = 0

Similarly, for higher order derivatives:


Laplace transforms

where:

- n is an arbitrary positive integer

-

Special Case: All Initial Conditions are Zero

SupposeThen

In process control problems, we usually assume zero initial conditions. Reason: This corresponds to the nominal steady state when “deviation variables” are used.


Laplace transforms

4. Trigonometric function

Using integrating by parts or Euler Identity.


Laplace transforms

5. Exponential Functions

Consider where b > 0. Then,

6. Rectangular Pulse Function

It is defined by:


Laplace transforms

Time, t

The Laplace transform of the rectangular pulse is given by


Laplace transforms

7.Impulse Function (or Dirac Delta Function)

The impulse function is obtained by taking the limit of the

rectangular pulse as its width, tw, goes to zero but holding

the area under the pulse constant at one. (i.e., let )


Laplace transforms

SOLUTION OF ODEs BY LAPLACE TRANSFORMS

  • Procedure:

  • Take the L of both sides of the ODE.

  • Rearrange the resulting algebraic equation in the s domain to solve for the L of the output variable, e.g., Y(s).

  • Perform a partial fraction expansion.

  • Use the L-1 to find y(t) from the expression for Y(s).


Table 3 1 laplace transforms

Table 3.1. Laplace Transforms


Table 3 1 laplace transforms cont

Table 3.1. Laplace Transforms (cont.)


Table 3 1 laplace transforms cont1

Table 3.1. Laplace Transforms (cont.)


Laplace transforms

EXAMPLE 1

Solve the ODE,

First, take L of both sides of (3-26),

Rearrange,

Take L-1,

From Table 3.1,


Laplace transforms

PARTIAL FRACTION EXPANSIONS

Basic idea: Expand a complex expression for Y(s) into simpler terms, each of which appears in the Laplace Transform table. Then you can take the L-1 of both sides of the equation to obtain y(t).

Example:

Perform a partial fraction expansion (PFE)

where coefficients and have to be determined.


Partial fraction expansions method 1

PARTIAL FRACTION EXPANSIONSMETHOD 1

Multiply both sides by (s + 1)(s+4)


Laplace transforms

PARTIAL FRACTION EXPANSIONSMETHOD 2

Specify 2 value of s, and solve simultaneously.


Laplace transforms

PARTIAL FRACTION EXPANSIONSMETHOD 3-HEAVISIDE EXPANSION

To find : Multiply both sides by s + 1 and let s = -1

To find : Multiply both sides by s + 4 and let s = -4


General partial fraction expansions

GENERAL PARTIAL FRACTION EXPANSIONS

Consider a general expression,

Here D(s) is an n-th order polynomial with the roots all being real numbers which are distinct so there are no repeated roots.

The PFE is:

Note:D(s) is called the “characteristic polynomial”.


Laplace transforms

EXAMPLE 2

Solve the ODE;


General partial fraction expansions1

GENERAL PARTIAL FRACTION EXPANSIONS

  • Special Situations:

  • Two other types of situations commonly occur when D(s) has:

  • Complex roots: e.g.,

  • Repeated roots (e.g., )

  • For these situations, the PFE has a different form.


Complex factor

COMPLEX FACTOR

HEAVISIDE EXPANSION


Repeated factor

REPEATED FACTOR

HEAVISIDE EXPANSION


Laplace transforms

EXAMPLE 3

By using HEAVISIDE EXPANSION, solve the ODE;


Laplace transforms

IMPORTANT PROPERTIES OF LAPLACE TRANSFORMS

  • Final Value Theorem

It can be used to find the steady-state value of a closed loop system (providing that a steady-state value exists.)

Statement of FVT:

providing that the limit exists (is finite) for all where Re (s) denotes the real part of complex variable, s.


Laplace transforms

2. Initial Value Theorem

3. Transform of an Intergal


Laplace transforms

4. Time Delay

Time delays occur due to fluid flow, time required to do an analysis (e.g., gas chromatograph). The delayed signal can be represented as

Also,


The end thank you

THE ENDTHANK YOU


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