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

Course Assessment Lecture 3 hours per week Number of units 3 Final Examination 50 marks Class Test 1 10 marks Class Test 2 10 marks Mini Project 15 marks Assignment/Quiz 15 marks

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.

Lecturer Dr. Md. Mijanur Rahman mijanur@unimap.edu.my 018 9418701 016 6781633

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

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

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

Examples

Examples (contd…) Human Control

System Control

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

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

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

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

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

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

Closed-Loop Speed Control of Rotating Disk

GPS Control

Satellite Control

Satellite Control (Contd…)

Servo Control

Introduction to Scilab Xcos

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.

Example of Diff. Equation

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

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

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

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

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

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

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

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

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

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

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

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

Example 2 (contd…)

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

Example 3 Non zero initial condition

Example 3 (contd…)

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.

Solution (a)

Solution (b)