1. Modern Control System EKT 308
General Introduction
Introduction to Control System
Brief Review
- Differential Equation
- Laplace Transform

2. Course Assessment Lecture 3 hours per week Number of units 3
Final Examination 50 marks
Class Test marks
Class Test marks
Mini Project marks
Assignment/Quiz 15 marks

3. Course Outcomes
CO1: : The ability to obtain the mathematical model for electrical and mechanical systems and solve state equations.
CO2: : The ability to perform time domain analysis with response to test inputs and to determine the stability of the system.
CO3: The ability to perform frequency domain analysis of linear system and to evaluate its stability using frequency domain methods.
CO4: The ability to design lag, lead , lead-lag compensators for linear control systems.

4. Lecturer Dr. Md. Mijanur Rahman mijanur@unimap.edu.my 018 9418701

5. Text Book References
Dorf, Richard C., Bishop, Robert H., “Modern Control Systems”, Pearson, Twelfth Edition, 2011
Nise , Norman S. , “Control Systems Engineering”, John Wiley and Sons , Fourth Edition, 2004.
Kuo B.C., "Automatic Control Systems", Prentice Hall, 8th Edition, 1995
Ogata, K, "Modern Control Engineering"Prentice Hall, 1999
Stanley M. Shinners, “Advanced Modern Control System Theory and Design”, John Wiley and Sons, 2nd Edition. 1998

6. What is a Control System ?
A device or a set of devices
Manages, commands, directs or regulates the behavior of other devices or systems.

7. What is a Control System ? (contd….)
Process (Plant) to be controlled
Process with a controller

8. Examples

9. Examples (contd…)
Human Control

10. System Control

11. Classification of Control Systems
Control systems are often classified as • Open-loop Control System • Closed-Loop Control Systems Also called Feedback or Automatic Control System

12. Open-Loop Control System
Day-to-day Examples
Microwave oven set to operate for fixed time
Washing machine set to operate on fixed timed sequence.
No Feedback

13. Open-Loop Speed Control of Rotating Disk
For example, ceiling or table fan control

14. What is Feedback? Feedback is a process whereby some
proportion of the output signal of a
system is passed (fed back) to the input.
This is often used to control the dynamic
behavior of the System

15. Closed-Loop Control System
Utilizes feedback signal (measure of the output)
Forms closed loop

16. Example of Closed-Loop Control System
Controller:
Driver
Actuator:
Steering
Mechanism
The driver uses the difference between the actual and the desired direction to generate a controlled adjustment of the steering wheel

17. Closed-Loop Speed Control of Rotating Disk

18. GPS Control

19. Satellite Control

20. Satellite Control (Contd…)

21. Servo Control

22. Introduction to Scilab
Xcos

23. Differential Equation
N-th order ordinary differential equation
Often required to describe physical system
Higher order equations are difficult to solve directly.
However, quite easy to solve through Laplace transform.

24. Example of Diff. Equation

25. Example of Diff. Equation (Contd…)
Newton’s second law:

28. Table 2.2 (continued) Summary of Governing Differential Equations for Ideal Elements

29. Laplace Transform
A transformation from time (t) domain to complex frequency (s) domain
Laplace Transform is given by

30. Laplace Transform (contd…)
Example: Consider the step function.
u(t)
u(t) = 1 for t >= 0
u(t) = 0 for t < 0 1 t -1

31. Inverse Laplace Transform
Transformation from s-domain back to t-domain
Inverse Laplace Transform is defined as:
Where,
is a constant

32. Laplace Transform Pairs
Laplace transform and its inverse are seldom calculated through equations.
Almost always they are calculated using look-up tables.

33. Laplace Transform’s table for common functions
Unit Impulse, 1
Unit step,
Unit ramp,
Exponential,
Sine,
Cosine,
Damped sine,
Damped cosain,
Damped ramp,

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

35. Characteristic of Laplace Transform (contd…)
(2) Differential Theorem
For higher order systems
where
Let
and

36. Characteristic of Laplace Transform (contd…)
(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

37. Characteristic of Laplace Transform (contd…)
(5) Final value Theorem
In this respect as
, gives
Example1
Consider a second order
Using differential property and assume intial condition is zero
Rearrangge
Inverse Lapalce

38. Example 2 Assume, 0 initial conditions.
Taking Laplace transform, we obtain

39. Example 2 (contd…)

40. Example 2 (contd…) Thus the solution of the differential equation
From table, inverse Laplace transform is
Thus the solution of the differential equation

41. Example 3
Non zero initial condition

42. Example 3 (contd…)

43. Example 4 (a) Show that is a solution to
the following differential equation
(b) Find solution to the above equation using
Laplace transform with the following initial
condition.

44. Solution
(a)

45. Solution
(b)