Linear Control Systems
Instructor:
Email:
Office hours:
M. A. Choudhry
drahmad@uettaxila.edu.pk
M-T: 1- 2 p.m.
Textbook:
Feedback Control Systems, 4th Ed.
C.L. Phillips and R.D. Harbor ,
Prentice-Hall
Matlab, Simulink, Control toolbox
and Mathematica
Supp
System Responses (Time Domain)
Reference: Phillips and Habor
First order systems:
Transient response
Steady state response
Step response
Ramp response
Impulse response
Second order systems
Transient response
Steady state response
Step response
Ramp resp
Frequency Response of Systems
(Reference: Phillips and Habor
Inthissectionwestudythesteadystateresponseof
systemstosinusoidalinputs.Thefrequencyresponse
hasmeaningfarbeyondthecalculationoftheresponse
tosinusoids. (t ) A cos(1t )
r
Suppose
As
R(s) 2
2
s 1
Root-Locus Analysis and Design
The Root Locus Procedure Indicates The
Characteristics Of A Control Systems
NaturalResponse.
An Other Approach For Analysis And
Design Of Control System Is Frequency
ResponseMethod.
The Frequency Response Procedure Gives
UsI
REVIEW SLIDES
Reference: Textbook by Phillips and Habor
Mathematical Modeling
Models of Electrical Systems
R-L-C series circuit, impulse voltage source:
di (t ) 1
Ri (t ) L
(t )dt 0
i
dt
C
Model of an RLC parallel circuit:
v (t )
dv (t ) 1
C
R
dt
L
t
v
Introduction to control systems
Reference: Phillips and Habor
The first applications of feedback control appeared in
Greece 300-1 BC ( a float regulator mechanism).
The first feedback system invented in modern Europe was
a temperature regulator in Holla
Steady State Accuracy (Reference: Phillips and Habor
Consider the following feedback system:
The output is given by:
Gc ( s )G p ( s )
C ( s)
R(s)
1 Gc ( s )G p ( s )
We can express Gc ( s )G p ( s ) as :
F (s)
Gc ( s )G p ( s) N
s Q( s)
where neither F
Mathematical Models of Physical
Systems
Reference: Phillips and Habor
Why mathematical models of physical systems
needed?
Design of engineering systems by trying and error
versus design by using mathematical models.
Physical laws such as Newtons second
Transfer Function
Reference: Phillips and Harbor
Suppose we have a constant-coefficient
linear differential equation with input f(t)
and output x(t).
After Laplace transform we have
X(s)=G(s)F(s)
We call G(s) the transfer function.
An Example
Linear dif
Laplace Transform
The Laplace transform of a function f(t) is defined
as:
st
F ( s ) [ f (t )] f (t )e dt
0
The inverse Laplace transform is defined as:
1
f (t ) [ F ( s )]
2 j
1
j
F (s)e
st
ds
j
where j 1 and the value of is determined
by the singu
Matlab Basics
Matlab
Control Toolbox
Simulink Toolbox
Matlab
Statements and variables
Matrices
Graphics
Scripts and M-files
Example:
>A=[1 2;4 6];
>A=[3 4;5 6]
A=
3 4
5 6
Operators: + - * / ^
As calculator:
>12.4/6.9
ans =
1.7971
The matrix A and the vari
Frequency Response of Systems
(Reference: Phillips and Habor
In this section we study the steady-state response of systems
to sinusoidal inputs. The frequency response has meaning far
beyond the calculation of the response to sinusoids.
r (t ) A cos(1t )