b_lecture6 time-domain specifications automatic control system
DESCRIPTION
Automatic control SystemTRANSCRIPT
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Time-Domain Analysis of Control Systems
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Introduction
Because time is used as an independent variable in most control systems, it is usually of interest to evaluate the output response with respect to time, or simply, the time response.
In the analysis problem, a reference input signal is applied to a system, and the performance of the system is evaluated by studying the system response in the time domain.
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The inputs to many practical control systems are not
exactly known ahead of time.
In many cases, the actual inputs of a control system
may vary in random fashion with respect to time.
(For instance, in a radar-tracking system for anti-aircraft
missiles, the position and speed of the target to be tracked
may vary in an unpredictable manner).
It is difficult to design a control system so that it will
perform satisfactorily to all possible forms of input
signals.
Introduction
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For the purpose of analysis and design, it is necessary to
assume some basic types of test inputs so that the
performance of a system can be evaluated.
By selecting these basic test signals properly, not only is
the mathematical treatment of the problem systematized,
but the response due to these inputs allows the prediction
of the systems performance to other more complex inputs.
In the design problem, performance criteria may be
specified with respect to those test signals so that the
system may be designed to meet the criteria.
Introduction
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Typical Test Signals For The Time Response of Control Systems
Step-Function Input:
0 0
0 )(
t
tRtr
)(tr
t
R
0)()( tuRtror s
The step function is very useful as a signal since its initial
instantaneous jump in amplitude reveals a great deal
about a systems quickness in responding to inputs with abrupt changes.
)(tus is the unit-step function
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Ramp-Function Input:
0 0
0 )(1)(
t
ttRttRtr
0 t
)(tr
Slope=R
The ramp function has ability to test how the system would
respond to a signal that changes linearly with time.
Typical Test Signals For The Time Response of Control Systems
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Parabolic-Function Input:
0 0
0 2)(1
2)(
2
2
t
ttR
ttR
tr
0 t
)(tr
The parabolic function represents a signal that is one order
faster than the ramp function.
Typical Test Signals For The Time Response of Control Systems
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1). Unit step response
s
ssRs1
ssLth
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Typical Responses to Typical Test Signals of
Control Systems
2). Unit ramp response
2
1
sssRssCt
2
1 1
ssLtct
3). Impulse response
sssRssK 1 sLtk 1
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Relationship between these responses
t
0
t
t20
1 1( ) ( ) ( ) , h(t) ( )
1 1( ) ( ) ( ) , c (t) ( )t
H s S K s k ds s
C s S H s h dss
dt
tdcssCsH
dt
tdhssHsK
tt
)(h(t) , )()(
)(k(t) , )()(
Typical Responses to Typical Test Signals of
Control Systems
{ ( ) , ( ) , ( ) }k t impulse response h t step response c t ramp responset
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The Unit-Step Response
and Time-Domain Specification
Basic and macroscopically requirements to
design a control system:
The system must be stable (stability)First requirement.
The control should be accurate (accuracy).
The response should be quick-acting (rapidity).
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For linear control systems,the characterization of the
transient response is often done by use of the response of
a linear control system when the input is a unit-step
function.
The Unit-Step Response
and Time-Domain Specification
Many control systems are dominated by a second order
pair of poles. So look at time response (to a unit-step
input) of
2
2 22n
n n
C ss
R s s s
The response of a system could be : )()()( tCtCsC st
Transient portion and steady-state portion )(tCt )(tCs
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The Unit-Step Response
and Time-Domain Specification
The typical uint-step response of a second-order system
Percent overshot %100% max
ss
ss
y
yy
)(ty
t
maxy
ssy
0
overshoot
pt Peak time
Peak overshoot is important, both because it is a measure
(to a degree) of stability, and for practical reasons, overshoot
should be minimized (think of an elevator!).
For under-damped systems
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Settling Time: The settling time is defined as the time
required for the step response to decrease and stay
within a specified percentage of its final value.
A frequently used figure is 5 percent.
)(ty
t0
00.1
95.0
st
05.1
The Unit-Step Response
and Time-Domain Specification
The typical uint-step response of a second-order system
Setting time is a measure of rapidity (quick-acting ) of the system.
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)(ty
t0
00.1
50.0
dt
Delay Time: The delay time is defined as the time
required for the step response to reach 50 percent
of its final value.
The Unit-Step Response
and Time-Domain Specification
The typical uint-step response of a second-order system
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Rise Time: For under-damped systems with an overshoot, the
rise time is defined as the time required for the step response
to rise from 0 to 100% of its final value. Rise time is a measure of rapidity (quick-acting ) of the system.
)(ty
t0
00.1
rt
The Unit-Step Response
and Time-Domain Specification
The typical uint-step response of a second-order system
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Rise Time: If the system is over-damped, then the peak time
is not defined, and the 10-90% rise time is normally used. Rise time is a measure of rapidity (quick-acting ) of the system.
)(ty
t0
00.1
rt
10.0
90.0
The Unit-Step Response
and Time-Domain Specification
The typical uint-step response of a second-order system
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)(ty
t0
sse
Steady-State Error: The steady-state error of a
system response is defined as the discrepancy
between the output and the reference input when
the steady state(t) is reached.
The Unit-Step Response
and Time-Domain Specification
1sse c
The typical uint-step response of a second-order system
Steady-State Error is a measure of accuracy of the system.
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The Unit-Step Response
and Time-Domain Specification
=2% or 5%
The typical uint-step response of a second-order system
Each of the above parameters may be important in the
design of the control system.