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1. INTRODUCTION :
Automatic control is the research area and theoretical base for mechanization
and automation, employing methods from mathematics and engineering. A
central concept is that of the system which is to be controlled, such as a radder,
propeller or ballistic missile. The systems studied within automatic control are
mostly the linear systems. Automatic control is also a methodology or
philosophyof analyzing and designing a system that can self-regulate a plant (such
as a machine or a process) operating condition or parameters by the controller
with minimal human intervention. A regulator such as a thermostat is an example
of a device studied in automatic control.
An automatic control system is a preset closed-loop control system that
requires no operatoraction. This assumes the process remains in the
normal range for the control system. Anautomatic control system has two
process variables associated with it: a controlled variable anda manipulated
variable.A controlled variable is the process variable that is maintained at a
specified value or within aspecified range..A manipulated variable is the process
variable that is acted on by the control system to maintainthe controlled variable
at the specified value or within the specified range.
Functions of Automatic Control-In any automatic control system, the four basic
functions that occur are:MeasurementComparisonComputationCorrection
Elements of Automatic Control-The three functional elements needed to perform
the functions of an automatic control systemare:A measurement elementan error
detection elementA final control element.
Components :
Sensor(s), which measure some physical state such as temperature or liquid
level.
2. Responder (s), which may be simple electrical or mechanical systems or
complex special purpose digital controllers or general purpose computers.
Actuator (s), which effect a response to the sensor(s) under the command
of the responder, for example, by controlling a gas flow to a burner in a
heating system or electricity to a motor in a refrigerator or pump.
Feedback Control System :
A feedback or closed loop control system is one where the input has control over
the output(controlled variable). In this system the controlled variable is measured
and fed back to the controller through a path (or loop). Some or all of the system
outputs are measured and used by the controller. The controller then compares a
desired plant value with the actual measured output value and acts to reduce the
difference between the two to zero value.
Types of feedback:
When feedback acts in response to an event/phenomenon, it can influence the
input signal in one of two ways:
An in-phase feedback signal when feedback acts in response to an
event/phenomenon, it can influence the input signal in, where a positive-going
wave on the input leads to a positive-going change on the output, will amplify the
input signal, leading to more modification. This is known as positive feedback.
A feedback signal which is inverted, where a positive-going change on the input
leads to a negative-going change on the output, will dampen the effect of the
input signal, leading to less modification. This is known as negative .
3. I/PI Comparator Control Unit Signal Final Control Process O/P
Unit Processing Unit Plant
/P
Detecting
Unit
Measuring
Unit
Signal Conditioner Transmitting
Unit
This is the basic block diagram of feedback control system. The
different types of system functions and their order will be discussed
below.
A transfer function (also known as the system function or network function) is a
mathematical representation, in terms of spatial or temporal frequency, of the
relation between the input and output of a linear time-invariant system. With
optical imaging devices, for example, it is the Fourier transform of the point
spread function (hence a function of spatial frequency) i.e. the intensity
distribution caused by a point object in the field of view. The term is often used
exclusively to refer to linear, time-invariant systems (LTI) Most real systems have
non-linear input/output characteristics, but many systems, when operated within
nominal parameters (not "over-driven") have behavior that is close enough to
4. linear that LTI system theory is an acceptable representation of the input/output
behavior.
In its simplest form for continuous-time input signal x(t) and output y(t), the
transfer function is the linear mapping of the Lap lace transform of the input, X(s),
to the output Y(s):
Type 0 System:
G(s) is said to be type 0 if the number of poles of G(s) at the origin is equal to
zero.
Given a transfer function G(s),
G(s) is type 0 when n = 0.
When we compute the steady-state error, G(s) must correspond to the transfer
function of the forward loop.
5. Type 1 System:
G(s) is said to be type 1 if the number of poles of G(s) at the origin is equal to one.
Given a transfer function G(s),
G(s) is type 1 when n = 1.
When we compute the stedy-state error,G(s) must correspond to the transfer
function of the forward loop.
Type 2 System:
G(s) is said to be type 2 if the number of poles of G(s) at the origin is equal to two.
Given a transfer function G(s),
6. G(s) is type 1 when n = 2.
When we compute the stedy-state error,G(s) must correspond to the transfer
function of the forward loop.
Time response of a first order control system subjected to unit step
input function :
Output for the system-- C(s)=R(s)*[1/(sT+1)] , R(s)=(1/s)
Therefore, C(s)=(1/s)-(1/(s+(1/T)))
Taking inverse laplace transform, we get:
c(t)=1-e^(-t/T)
The error is given as:
e(t)= e^(-t/T)
The steady state error:
ess= Lim e^(-t/T)=0
t∞
7. Time response of a second order control system subjected to unit step
input function :
Output of the system—C(s)=R(s)*[(w^2) / (s^2+2*z*w*s+w^2)]
Where, z=zeta—damping ratio
w=natural frequency of oscillation
After taking the inverse laplace, we get:
c(t)=1- [{(e^(-zwt))/{(1-z^2)^(1/2)}} * x] ,
where,x=sin[{w*((1-z^2)*t))^(1/2)} + tan^(-1){((1-z^2)^(1/2)) / z}]
The error is given as:
e(t)= [{(e^(-zwt))/{(1-z^2)^(1/2)}} * x] ,
where, x=sin[{w*((1-z^2)*t))^(1/2)} + tan^(-1){((1-z^2)^(1/2)) / z}]
The steady state error is given as:
ess= Lim [{(e^(-zwt))/{(1-z^2)^(1/2)}} * x]
t∞
The steady-state error depends on the type of inputs and the type of
the forward-loop. It can be computed easily using error constants :
Static position error coefficient Kp :
8. Kp= lim G(s)
s0
For a type 0 system,
Kp=lim *K(Tas+1)(Tbs+1)…+/*(T1s+1)(T2s+1)…+ = K
s0
For a type 1 or higher system,
Kp=lim *K(Tas+1)(Tbs+1)…+/*(s^N) (T1s+1)(T2s+1)…+ = ∞ , for N>=1
s0
Static velocity error coefficient Kv :
Kv= limsG(s)
s0
For a type 0 system,
Kv=lim *s*K(Tas+1)(Tbs+1)…+/*(T1s+1)(T2s+1)…+ = 0
s0
For a type 1 system,
Kv=lim *s*K(Tas+1)(Tbs+1)…+/*s* (T1s+1)(T2s+1)…+ = K
s0
For a type 2 or higher system,
Kv=lim [s*K(Tas+1)(Tbs+1)…+/*(s^N) (T1s+1)(T2s+1)…+ =∞ , for N>=2
s0
9. Static acceleration error coefficient Ka :
Ka= lim (s^2)G(s)
s0
For a type 0 system,
Ka=lim *(s^2)*K(Tas+1)(Tbs+1)…+/*(T1s+1)(T2s+1)…+ = 0
s0
For a type 1 system,
Ka=lim *(s^2)*K(Tas+1)(Tbs+1)…+/*s* (T1s+1)(T2s+1)…+ = 0
s0
For a type 2 or higher system,
Ka=lim *(s^2)*K(Tas+1)(Tbs+1)…+/*(s^2) (T1s+1)(T2s+1)…+ =K
s0
MATLAB :
MATLAB, which stands for MATrix LABoratory, is a technical computing
environmentfor high-performance numeric computation and visualization.
SIMULINK is a part of MATLAB that can be used to simulate dynamic systems. To
facilitate model definition, SIMULINK adds a new class of windows called block
diagramwindows. In these windows, models are created and edited primarily by
mousedrivencommands. Part of mastering SIMULINK is to become familiar
withmanipulating model components within these windows.
