Laplace’s Transform: Re-visit�

Presented by

Dr. Asim Halder

Dept. of Applied Electronics and Instrumentation Engineering

Haldia Institute of Technology

Basic Mathematics for Linear Control Systems

Pre-requisite

Pre-requisite for Linear Control Theory

• Laplace transform and linear algebra are the back-bone of Linear Control Theory.
• Hence, it is of our great interest to learn and understand these mathematical tools.
• But, before moving to the main topic, let’s see the advantages of the use of Laplace’s Transform.
• Some of the important advantages, for LTI systems, are-
• It includes the boundary conditions/ initial conditions.
• The differential equations of the system are transformed to the algebraic equation.
• The work is systematized.
• Standard transformation table reduce the rigorous mathematics.
• Discontinuous inputs can be treated.
• Steady-state and transient components of the solution are obtained

simultaneously.

Pre-requisite

Definition of the Laplace Transform

• The Laplace transform of time dependent function f(t) can define as-

(1.1)

• In (1.1) ‘L’ denotes the Laplace transform; F(s) represents the s-domain expression of f(t); where, ‘s’ is a complex quantity.
• It can also be seen that the integration limit is from ‘0’ to ‘infinity’, so it is immaterial what value f(t) has from ‘0’ to time.

Derivation of Laplace Transform of Standard Functions

• Unit Step Function u(t):

The Laplace transform of the unit step-function u(t) is-

Pre-requisite

Since u(t) has the value ‘1’ over the limit of integration, so-

if

• Decaying Exponential :

The exponent ‘α’ is a positive real number.

(1.2)

if

(1.3)

• Cosine Function :

Pre-requisite

Here,‘ω’ is a positive real number.

• Now expressing in exponential form

(1.4)

(1.5)

• Then

if

(1.6)

• Ramp Function r(t) = t u(t) :

Pre-requisite

(1.7)

• This expression is integrated by parts by using-

(1.8)

• Let u = t and . Then and .

• Thus

if

(1.9)

• Laplace Transform Theorems :

• There are various theorems which are useful and helpful in evaluating the transforms, these are-

• Theorem 1: Linearity

• If ‘a’ is a constant or is independent of ‘s’ and ‘t’, and if f(t) is transformable then-

(1.10)

Pre-requisite

• Theorem 2: Superposition

• If and are both transformable then the principle of superposition applies -

(1.11)

• Theorem 3: Complex differentiation

• If the Laplace transform of f(t) is F(s) then-

(1.12)

• Here, multiplication by time in the real domain results in differentiation w. r. t. ‘s’ in the s-domain.

• Theorem 4: Translation in time domain

• If the Laplace transform of f(t) is F(s) and ‘a’ is a positive real number, the Laplace transform of the translated function

is

(1.13)

• Translation in the positive ‘t’ direction in the real domain becomes multiplication by the exponential in the s-domain.

• Theorem 5: Translation in s-domain

• If the Laplace transform of f(t) is F(s) and ‘a’ is either real and complex then-

(1.14)

• Multiplication of in the real domain becomes translation in the s-domain.

Pre-requisite

• Theorem 6: Real Differentiation

• If the Laplace transform of f(t) is F(s) and the 1^st derivative of f(t) w. r. t. time is

transformable then-

(1.15)

• The term is the value of the right-hand limit of the function f(t) as the origin t = 0 is approached from the right side.
• For simplicity, the + sign following zero is omitted, although its presence is implied.
• The transform of the 2^nd derivative is-

(1.16)

where is the value of the limit of the derivative of f(t) as the

origin t = 0 is approached from the right side.

• The transform of the derivative is

Pre-requisite

(1.17)

One point to be noted here that the transform includes the initial conditions, where as in the classical method of solution, initial conditions are introduced separately to evaluate the coefficient of the solution of the differential equation. When all initial conditions are zero, the Laplace transform of the derivative of f(t) is simply .

• Theorem 7: Real Integration

• If the Laplace transform of f(t) is F(s), its integral is

transformable and the value of its transform is

(1.18)

The term is the constant of integration and is equal to the value of the integral as the origin is approached from the positive right side.

Pre-requisite

The transform of double integral is

(1.19)

The transform of the order integral is

(1.20)

• Theorem 8: Final Value

If f(t) and Df(t) are Laplace transformable, if the Laplace transform of f(t) is F(s), and if the limit of f(t) as exists then

(1.21)

The theorem states that behavior of f(t) in the neighborhood of is related to the behavior of sF(s) in the neighborhood of s = 0. If sF(s) has

poles (values of ‘s’ for which becomes infinite) on the imaginary axis (excluding the origin) or in the right-half ‘s’ plane, there is no finite final value of f(t) and the theorem can not be used.

If f(t) is sinusoidal the theorem is invalid, since has poles at

and does not exist.

Pre-requisite

However, for poles of sF(s) at the origin, s = 0, this theorem gives the final value of

This correctly describes the behavior of f(t) as .

• Theorem 9: Initial Value

If the function f(t) and its 1^st derivative are Laplace transformable, if the Laplace transform of f(t) is F(s), and if exists then

(1.22)

The theorem states that the behavior of f(t) in the neighborhood of t = 0 is related to the behavior of sF(s) in the neighborhood of . There are no limitations on the locations of the poles of sF(s).

• Theorem 10: Complex Integrations

If the Laplace transform f(t) is F(s) and if has a limit as

then

(1.23)

It states that the division by the variable in the real domain results in integration w. r. t. ‘s’ in s-domain.

Thank you