CONTROL SYSTEMS MEC 2825

Dr. Mohammad Abdelrahman

Semester I 2014/2015

Control Systems MEC 3825 Three Credit Hrs Core Course Assessment: Midterm 30% - Assignments and Quizzes 20% - Projects 10% - Final Examination 40%. Venue: E2-1-4 Time: 11:30 -12:50 pm T-TH

Lecture: Dr. Mohammad Abdelrahman

Room: E1-2-16.5

Contacts: Email: m_abdelrahman@iium.edu.my

Office: 03-6196-4557

H/P: 017-240-8923

Consultation Hrs: 2-3 pm M/W

References

1. Norman S. Nise Control Systems Engineering, Wiley & Sons, Sixth Edition, 2011.

2. F. Golnaraghi and B. Kuo Automatic Control Systems, Wiley & Sons, Sixth Edition, 2010.

3. K. Ogata Modern Control Engineering, Pearson Education International, 2002.

Course Overview

Introduction; Modeling in Frequency Domain, Modeling in Time Domain, Time Response, Reduction of Multiple Systems, Stability, Steady State Errors, Root Locus Technique, Design via Root Locus.

Chapter One Introduction Learning Outcomes

Define a control system and describe some applications Describe historical developments leading to modern day

control theory Describe the basic features and configurations of control

systems Describe control systems analysis and design objectives Describe a control system's design process Describe the benefit from studying control systems

Case Study Learning Outcomes An antenna azimuth position control system

Advantages of Control Systems

We build control systems for four primary reasons:

1. Power amplification

2. Remote control

3. Convenience of input form

4. Compensation for disturbances

For example, a radar antenna, positioned by the low-power rotation of a knob at the input, requires a large amount of power for its output rotation. A control system can produce the needed power amplification,

or power gain.

System Configurations

Open-Loop Systems A generic open-loop system is shown in It starts with

a subsystem called an input transducer, which converts the form of the input to that used by the controller.

Closed-Loop (Feedback Control) Systems The disadvantages of open-loop systems, namely

sensitivity to disturbances and inability to correct for these disturbances, may be overcome in closed-loop systems

Analysis and Design Objectives

Analysis is the process by which a system's performance is determined. For example, we evaluate its transient response and steady-state error to determine if they meet the desired specifications.

Design is the process by which a system's performance is created or changed. For example, if a system's transient response and steady-state error are analyzed and found not to meet the specifications, then we change parameters or add additional components to meet the specifications.

Transient Response

Steady-State Response

Stability

Total response = Natural response + Forced response

Other Considerations

The Design Process

Kirchhoff's voltage law: The sum of voltages around a closed path equals zero.

Kirchhoffs current law: The sum of electric currents flowing from a node equals zero.

Newton's laws: The sum of forces on a body equals zero;3 the sum of moments on a body equals zero.

Summary

A control system has an input, a process, and an output. Control systems can be open loop or closed loop

Control systems analysis and design focuses on three primary objectives:

Producing the desired transient response

Reducing steady-state errors

Achieving stability

A system must be stable in order to produce the proper transient and steady-state response.

The design of a control system follows these steps: 1. Determine a physical system and specifications from

requirements. 2. Draw a functional block diagram. 3. Represent the physical system as a schematic. 4. Use the schematic to obtain a mathematical model, such as

a block diagram. 5. Reduce the block diagram. 6. Analyze and design the system to meet specified

requirements and specifications that include stability, transient response, and steady-state performance.

Example 1.1 If it takes 10 turns to move the wiper arm from A to C, draw a block diagram of the potentiometer showing the input variable, the output variable, and (inside the block) the gain, which is a constant and is the amount by which the input is multiplied to obtain the output.

Example 1.2 Draw a functional block diagram for a closed-loop system that stabilizes the roll.

Chapter Two Modeling in the Frequency Domain Learning Outcomes

Find the Laplace transform of time functions and the inverse Laplace transform

Find the transfer function from a differential equation and solve the differential equation using the transfer function

Find the transfer function for linear, time-invariant electrical networks

Find the transfer function for linear, time-invariant translational mechanical systems

Find the transfer function for linear, time-invariant rotational mechanical systems

Find the transfer functions for gear systems with no loss and for gear systems with loss

Find the transfer function for linear, time-invariant electromechanical systems

Produce analogous electrical and mechanical circuits Linearize a nonlinear system in order to find the transfer function

Case Study Learning Outcomes

Given the antenna azimuth position control system, you will be able to find the transfer function of each subsystem.

Given a model of a human leg or a nonlinear electrical circuit, you will be able to linearize the model and then find the transfer function.

Development of mathematical models from schematics of physical Systems

We will discuss two methods: (1) transfer functions in the frequency domain

and (2) state equations in the time domain. These topics are covered in this chapter and in Chapter 3, respectively.

Laplace Transform Review The Laplace transform is defined as

The inverse Laplace transform, which allows us to find f(t) given F(s), is

Example 2.1

Example 2.2.1

Partial-Fraction Expansion

G(s) Has Multiple-Order Poles

Example 2.2.2

Example 2.2.3

Show that the completed partial-fraction expansion for the following transfer function

is given by:

The Transfer Function General nth-order, linear, time-invariant differential equation:

Taking the Laplace transform of both sides,

If we assume that all initial conditions are zero,

Example 2.4

Find the transfer function represented by

Find the response, c(t) to an input, r{t) = u(t), a unit step, assuming zero initial conditions.

Matlab Solution

Electrical Network Transfer Functions

Find the transfer function relating the capacitor voltage, Vc(s), to the input voltage, V(s)

Repeat the previous example using mesh analysis and transform methodswithout writing a differential equation.

Repeat the previous example using nodal analysis and without writing a differential equation.

Repeat the previous example using voltage division and the transformed circuit.

Given the network of Figure (a), find the transfer function, I2(s)/V(s).

Find the transfer function, Vc(s)/V(s), for the circuit in Figure (b). Use nodal analysis.

For the network of shown figure, find the transfer function, Vc(s)/V(s), using nodal analysis and a transformed circuit with current sources.

Mesh Equations via Inspection