1. . Modeling OBJECTIVE Revision on Laplace transform
Poles and zeros in relation to transfer function.
Examples of modelling.

2. Obtained a Laplace Transform for aunit step
If is a function of time for , the Laplace Transform is defined as
where
is a complex frequency
Example:
Obtained a Laplace Transform for aunit step
f(t) 1 t

3. Laplace Transform’s table for common functions
Unit Impulse, 1
Unit ramp,
Unit ramp,
Exponential,
Sine,
Cosain,
Damped sine,
Damped cosain,
Damped ramp,

4. Characteristic of Laplace Transform
(1) Linear If
and
are constant and
and
are Laplace Transform

5. (2) Differential Theorem
For higher order systems
where
Let
and d

6. (3) Integration Theorem
Let
where
is the initial value of the function.
(4) Initial value Theorem
Initial value means
and as the frequency is inversed of time, this implies that
, thus

7. (5) Final value Theorem In this respect , gives Exampleas
, gives
Example
Consider a second order
Using differential property and assume intial condition is zero
Rearrangge
Inverse Lapalce

8. Linear system
Exhibit properties of .
Superposition: .
Homogeneity:

9. Nonlinear system A x x We must find the equilibrium point
Using Taylor series
Neglecting higher order term

10. Example
Linearize
about
Thus

11. . . . . Signal Flow Graph Y(s) + R(s) + + E(s) G(s) - Consider
Variables are reperesented as nodes.
Transmittence with directed branch.
Source node: node that has only outgoing branches.
Sink node: node that has only incoming branches. . .
R(s)
G(s)
Y(s)
As signal flow graph
D(s)
D(s) 1 + 1
R(s) +
E(s)
F(s) +
Y(s)
R(s)
E(s)
G(s)
F(s)
Y(s) 1
Y(s) - -1
H(s)
B(s)
B(s)

12. Cascade connection 

13. Parallel connection
Two parallel branch

14. Mason rule  Total transmittence for every single loop where
 Total transmittence for every 2 non-touching loops
 Total transmittence for every 3 non-touching loops
 Total transmittence for every m non-touching loops
 Total transmittence for k paths from source to sink nodes.
where:
 Total transmittence for every single non-touching loop of ks’ paths
 Total transmittence for every 2 non-touching loop of ks’ paths
 Total transmittence for every 3 non-touching loop of ks’ paths.  Total transmittence for every n non-touching loop of ks’ paths.

15. + + - - Example: Determine the transfer function of Y R
the following block diagram. + + Y R P Q - - H

16. and .
etc.
Transfer function

17. Example:
Determine R + + Y B C A - - - D E .
and

18. Poles and Zeros Consider a transfer function Singular point of
approaching to infinity is when
. The roots are called poles
approaching to zero is when
Singular point of
The roots are called zeros . .

19. example
Satah-s
3.3166 

20. Rank: Difference between the numerator and denomonitor, n+j-i
In general transfer function can be written as
where
K dc gain
Type: Highest factored s of the denomonitor, n
Order: Highest order s of the denomonitor, n+j
Rank: Difference between the numerator and denomonitor, n+j-i

21. Example
Type 2,
Order 4
Rank 4-2=2.

22. Mechanical Displacement

23. applied force (N) y mass displacement (m)
f viscous friction (N.m.rad-1.s-1)
k spring constant (N.m-1)
Newton’s law
where m is a mass (kg),
a acceleration (m.s-2)
F force (N).
From Lenz’s law

24. Taking Laplae transform and assume zero initial condition
Rearrange
In block diagram

25. Mechanical rotation

26. Where
J is the mass inertia (kg.m2)
 is the angular speed (rad.s-1)
f is the viscous friction (N.m.rad-1.s-1)
T is the applied torque (N.m)
From Newton’s law
where  is the angular acceleration (rad.s-2).
Defined
and obtained the Lapace of the above equations

27. Rearrange
The block diagram

28. RLC Circuit
Input voltage
and output voltage of

29. Applying Kirchoff Voltage Law to the input loop
While the output voltage
Laplace Transform
Rearrange
The block diagram

30. Fluid system resistance of the pipe and tanks cross section and
tanks height
and
input, output and middle flow rate

33. + - + - 1 1 - 1 + -1

35. Servo Mechanisme
Mechanical torque

36. Input loop where is a constant Laplaced dan Rearrange in term of and
Electrical torque

37. where
is a consatant
Rearrange
where
and
are constants