ELM 322, Control Systems
Control Systems
Spring 2015
Dr. Erkan Zergeroğlu
202 Computer Engineering Department
Email: ezerger@bilmuh.gyte.edu.tr
CLASS TIME: See the Schedule
OFFICE HOURS: To be announced.
Introduction to Control Systems
● A control system is basically an interconnection of
components that provide a desired system response
● It is desirable whenever some quantity in a event, has to
behave in a desirable way, such as;
– Temparature in a room
– Speed of an automobile
– Concentration or pH value of a liquid
● On most cases feedback is used to regulate the system
response
Control System
● A control system consists of subsystems and processes
assembled for the purpose of obtaining a desired output
with a desired performance for a given (specified) input.
How does 'control' work ?
● Subsystems and processes assembles for the purpose of
shaping the outputs of the overall process
● One way is comparing the actual behivour of the system
with the desired behavior, then take corrective actions
based on the difference.
● Examples ?? (Videos)
Definitions
● System : is a set of interacting or interdependent
components forming an integrated whole. Every system is
delineated by its spatial and temporal boundaries,
surrounded and influenced by its environment, described
by its structure and purpose and expressed in its
functioning
● Control System : is a device, or set of devices, that
manages, commands, directs or regulates the behavior of
other devices or systems. Industrial control systems are
used in industrial production for controlling equipment or
machines.
● Process : The device , plant or system under control
Open and Closed loop
● There are two common classes of control systems,
– In open loop control systems output is generated
based on inputs (a).
– In closed loop control systems current output is taken
into consideration and corrections are made based on
feedback (b). A closed loop system is also called a
feedback control system
Open and Closed loop Systems
Classification of Control Systems
● Regulator Systems (Process Control):
– The controlled variable or output is required to be held
as close as posible to a usually constant desired value
or input despite any disturbaces
● Servomechanism (Tracking Control):
– The input varies amd the output is requierd to follow or
track it as closely as possible
Analysis and Design (Synthesis)
● Analysis :
– What is the performance of a given system in response
to the changes on the input or external disturbances
● Design :
– If the performance is unsatisfactory how can it be
improved ? (Best if it can be done without changing the
process, actuator and the power amplification blocks)
Performance
● Improve transient response
● Reduce steady state errors
● Reduce the effects of external disturbace
Example: Mass Damper Spring
Spring :
Damper :
Mass :
The overall system equation is
Result is a second order differential equation of variable x
Example : RLC Circuit
Kirchoff's Voltage Law :
Electric charge
After some mathematical manipulations
Result is a second order differential equation of variable q
Motivation
Both of the systems given in the examples are governed
by similar looking Differential Eqautions.
Control is the “shaping“ of the output of the Differential
Equations by changing the input to the system (mostly by
feedback)
Differential Equations
Definition :
A differential equation is an eqaution which involves
differentials or derivatives.
Like
also known as Newton's law of motion.
Definitions
– In this class, we will be concerned with ordinary
differential equations of the form
(1)
where and are functions of time ,
are constants
Definitions
Time Invariant
The differential equation of the form (1) is said to be time invariant if
the coefficients, denoted by and do not explicitly depend on
time.
Causality
A system is causal if the outputs depends only on the present and
past values of the input.
Linear
Given a system with two different inputs, say X1 and X2 with
corresponding outputs, say Y1 and Y2
For two constants, say C1 and C2 . The corresponing output of
C1*X1+C2*X2 should be equal to C1*Y1+C2*Y2
The “D“ Notation
For the differential equation
define and
which enables us to rewrite the original differential
equation as
Characteristic Equation
More generally
is called the characteristic equation.
● The roots of the characteristic equation is refered as the
“poles“ (or eigen values ) of the system and are very
important in determining the properties of the differential
equation (therefore the system itself !!!) as poles determine
the shape of (the output).
What's Next ?
● Given x(t) it would be adventageous to have a general
procedure for finding the y(t)
● It would also be adventageous to develop analysis
techniques for examing systems without knowing the
inputs
● Using Laplace Transformation Tools might be one way of
achieving these.
Laplace Transform
● What is it used for ?
Laplace Transform is used to convert differential equations to
algebraic equations
● Why ?
Because it is easier to work with algebraic equations compared to
differential equations
● Definition (one sided)
where F is the Laplace Transform of f and s is a complex number
Examples for Laplace Transform
1) Unit Step Function
2) A Ramp function
Note that we hardly use this method, instead we use Laplace
transfrom tables and some algebraic manipulations
Laplace Transform
● Laplace Transform of commonly used functions
Properties of Laplace Transform
Additional Property
Convolution
where, the convolution operator is defined as
That is convolution integral in time domain is avoided by
multiplication in frequency domain
● These properties can be used to manipulate a function into a
form in which transition between time to frequency domains
are eased
Solutions by Laplace Transformation
Method
– Take the Laplace Transformation of both sides of the
differential equation in order to work in frequency
domain
– Solve the algebraic equation for the unknown variable
in frequency domain
– Take the inverse Laplace Transform of the result to find
the time domain expression.
Example ( a warm up )
Find for
Solution : In frequency domain eqaution becomes
then
partial fraction expansion yields
use inverse Laplace trasform to find the result in time domain
Example 1
Find h(t) for
Solution : First ensure that the order of denomunator is
greater than that of the numerator (long division)
Example 1
Therefore
easy
need to work
on this part
Need to find these K's
Example 1
That is
inverse Laplace Transform yields
Example 2
Find h(t) for
repeated roots
Solution :
Example 2
Then
taking the inverse Laplace Transfrom yields
Example (from basic circuit th)
Given
find how the voltage across the
resistor behaves when the input
voltage is
Solution :
Do this as HW#1
(a 1F capacitor might be hard to find)
How about imaginary roots ?
Show the behaviour of the output for the system
when the input is
Solution : Apply Laplace Transformation to obtain
Note that
Example Cont.
no real roots
We can still apply PFE
A is easy to find, for B and C apply whatever you learned
at highschool
Example Cont.
Ok ! Don't know how ?
Try something different
Now we can use the Laplace Transform Tables
Example Cont.
Inverse Laplace Transform yields
Problem : How does this funciton really behave ?
How output Behaves ?
One way
Frequency domain
and another
Time Domain
