1. © Dr. Elmer P. Dadios - DLSU Fellow & Professor
Transfer Functions ,
The Laplace Transform converts linear differential equations into algebraic expressions which are easier to manipulate.
The Laplace Transform converts functions with a real dependent variable (such as time) into functions with a complex dependent variable (such as frequency, often represented by s).
The modeling equation gathered from the free body diagram is in the time domain.
Some analysis are easier to perform in the frequency domain.
In order to convert to the frequency domain, apply the Laplace Transform to determine the transfer function of the system.
© Dr. Elmer P. Dadios - DLSU Fellow & Professor

2. © Dr. Elmer P. Dadios - DLSU Fellow & Professor
Transfer Functions ,
A transfer function (also known as the network function) is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a (linear time-invariant) system.
In its simplest form for continuous-time input signal x(t) and output y(t), the transfer function is the linear mapping of the Laplace transform of the input, X(s), to the output Y(s):
where H(s) is the transfer function of the LTI system.
© Dr. Elmer P. Dadios - DLSU Fellow & Professor

3. © Dr. Elmer P. Dadios - DLSU Fellow & Professor
Transfer Functions ,
The general procedure to find the transfer function of a linear differential equation from input to output is to take the Laplace Transforms of both sides assuming zero conditions, and to solve for the ratio of the output Laplace over the input Laplace.
© Dr. Elmer P. Dadios - DLSU Fellow & Professor

4. How To Find the Transfer Function,
How To Find the Transfer Function
In most cases the governing equation will be linear, consisting of a variable and its derivatives.
The Laplace Transform allows a linear equation to be converted into a polynomial.
The most useful property of the Laplace Transform for finding the transfer function is the differentiation theorem.
Several properties are shown below:
Time Domain
Frequency Domain
Linearity
f(t) + g(t)
Function
x(t)
1st Derivative
x'(t)
2nd Derivative
x"(t)
nth Derivative
xn(t)
© Dr. Elmer P. Dadios - DLSU Fellow & Professor

5. How To Find the Transfer Function,
How To Find the Transfer Function
Note:  While linearity allows Laplace Transforms to be added, the same does not hold true for multiplication. 
f(t)g(t) does not equal F(s)G(s). 
The solution to multiplication requires convolution:
Refer to a differential equations book.
© Dr. Elmer P. Dadios - DLSU Fellow & Professor

6. How To Find the Transfer Function,
How To Find the Transfer Function
In order to convert the time dependent governing equation to the frequency domain, perform the Laplace Transform to the input and output functions and their derivatives. 
These transformed functions must then be substituted back into the governing equation assuming zero initial conditions. 
Because the transfer function is defined as the output Laplace function over the input Laplace function, rearrange the equation to fit this form.
© Dr. Elmer P. Dadios - DLSU Fellow & Professor

7. How To Find the Transfer Function,
How To Find the Transfer Function
Example: Find the transfer function of the second order equation below
f(t) - kx - bx' - mx" = 0
Solution: The notation of the Laplace Transform operation is L{ }.
© Dr. Elmer P. Dadios - DLSU Fellow & Professor

8. How To Find the Transfer Function,
How To Find the Transfer Function
Solution:
When finding the transfer function, ‘zero’ initial conditions must be assumed, so x(0) = x'(0) = 0.
Taking the Laplace Transform of the governing equation results in:
F(s) - k[X(s)] - b[sX(s)] - m[s2X(s)] = 0
Collecting all the terms involving X(s) and factoring leads to:
[ms2 + bs + k] X(s) = F(s)
The transfer function is defined as the output Laplace Transform over the input Laplace Transform, and so the transfer function of this second order system is:
X(s)/F(s) = 1/[ms2 + bs + k]
© Dr. Elmer P. Dadios - DLSU Fellow & Professor

9. HOW TO INPUT THE TRANSFER FUNCTION INTO MATLAB,
HOW TO INPUT THE TRANSFER FUNCTION INTO MATLAB
In order to enter a transfer function into MATLAB, the variables must be given numerical value, because MATLAB cannot manipulate symbolic variables without the symbolic toolbox. 
Enter the numerator and denominator polynomial coefficients separately as vectors of coefficients of the individual polynomials in descending order.
The syntax for defining a transfer function in MATLAB is:
transferfunction = tf(num, den)
Where: num is defined as the vector of numerator coefficients, and den is defined as the vector of denominator coefficients.
© Dr. Elmer P. Dadios - DLSU Fellow & Professor

10. HOW TO INPUT THE TRANSFER FUNCTION INTO MATLAB,
HOW TO INPUT THE TRANSFER FUNCTION INTO MATLAB
Example:
Input the transfer function X(s)/F(s) = 1/[ms2 + bs + k] into MATLAB:
For illustration purposes, this example uses m = 2, b = 5, and k = 3.
>> m = 2; >> b = 5; >> k = 3; >> num = [ 1 ]; >> den = [ m b k ]; >> tutorial_tf = tf(num, den)
MATLAB will assign the transfer function under the name tutorial_tf, and output the following:
Transfer function:        s^2 + 5 s + 3
© Dr. Elmer P. Dadios - DLSU Fellow & Professor

11. STEP RESPONSE USING THE TRANSFER FUNCTION,
STEP RESPONSE USING THE TRANSFER FUNCTION
Once the transfer function is entered into MATLAB it is easy to calculate the response to a step input.
To calculate the response to a unit step input, use:
step(transferfunction)
where transferfunction is the name of the transfer function of the system.
For steps with magnitude other than one, calculate the step response using:
step(u * transferfunction)
where u is the magnitude of the step and  transferfunction is the name of the transfer function of the system.
© Dr. Elmer P. Dadios - DLSU Fellow & Professor

12. © Dr. Elmer P. Dadios - DLSU Fellow & Professor,
EXAMPLE
Find the unit step response and the step response when u = 4 of  tutorial_tf using MATLAB:
Solution:
To find the unit step response:
>> step(tutorial_tf)
The MATLAB output will be the following plot of the unit step response:
© Dr. Elmer P. Dadios - DLSU Fellow & Professor

13. © Dr. Elmer P. Dadios - DLSU Fellow & Professor,
Solution:
To find the step response when u = 4:
>> u = 4;
>> step(u * tutorial_tf)
The MATLAB output will be the following plot of the step response:
© Dr. Elmer P. Dadios - DLSU Fellow & Professor

14. © Dr. Elmer P. Dadios - DLSU Fellow & Professor,
Setwork: Use MATLAB to find
1) IMPULSE RESPONSE USING THE TRANSFER FUNCTION 
2) BODE PLOT USING THE TRANSFER FUNCTION
3) STATE SPACE FROM TRANSFER FUNCTION
© Dr. Elmer P. Dadios - DLSU Fellow & Professor