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Time
Domain
AnalysisofControl
SystemsThird
ClassChapter
3Review2Definition
of
time
response;Typical
test
signals;
Performance
specifications
of
steady-stateresponse
–
steady-state
error;Impact
of
disturbance
to
steady-state
error;
Impact
of
parameter
variation
to
steady-staterror.Outline3Transient
ResponsePerformance
CriteriaTransient
Response
of
1st-Order
SystemsTransient
Response
of
2nd-Order
SystemsApproximation
of
High-Order
SystemsTransient
Response4
Transient
response
is
the
response
of
a
systea
certain
input
which
dies
out
as
time
becomeslarge;
For
linear
control
systems,
the
characterizathe
transient
response
is
oftendone
by
use
ofunit-step
function
as
input;
The
response
of
a
control
system
when
the
inpuis
a
unit-step
function
is
called
the
unit-steresponse.MaximumOvershootPerformance
CriteriaCommonly
used
performance
criteria
characterizing
transient
responset1.0y(t)tstr
tp0.01.0+2%1.0-2%Maximum
OvershootRise
Time5Peak
TimeSettling
TimePerformance
CriteriaMaximum
OvershootRise
Time6Peak
TimeSettling
TimeThe
difference
between
maximum
value
andthe
steady-state
value
of
output.Rise
time
is
defined
as
the
time
required
for
the
step
respto
rise
from
zero
to
the
final
value
of
output.
(In
some
othsituation,
rise
time
is
defined
as
the
time
required
for
thresponse
to
rise
from
10
to
90
percent
of
the
final
value.
)Peak
time
is
defined
as
the
time
required
for
the
step
respto
rise
from
zero
to
the
maximum
value.Settling
time
is
defined
as
the
time
required
for
the
step
response
to
decrease
and
stay
within
a
specified
percenta
of
its
final
value.
A
frequently
used
figure
is
2%Performance
CriteriaMaximum
Overshoot
is
used
to
measure
the
relative
stability
of
a
consystem.
A
system
with
a
large
overshoot
is
usually
undesirable.Rise
time
,
peak
time ,
settling
Time
are
used
to
measure
the
quicknesthat
a
system
responds
to
the
unit-step
input.The
methods
commonly
used
to
study
the
performance
of
a
system
in
transient:Computer
simulation
–
numerical
method;
Referring
to
transient
performance
of
the
first-order
and
second
ordesystem,
estimate
those
of
higher
order
systems7Transient
Response
of
the
1st-Order
Systems8Transient
Response
of
the
2nd-Order
SystemsNatural
undamped
frequencyTime
constant,
reciprocal
of
the
natural
undamped
frequencyDamping
ratio9Transient
Response
of
the
2nd-Order
SystemsRoots
of
the
characteristic
equation:SetReal
part
of
the
rootsImaginary
part
of
the
roots10Transient
Response
of
the
2nd-Order
SystemsRelationship
between
the
characteristic-equation
roots
andand11Transient
Response
of
the
2nd-Order
SystemsWhenthe
unit-step
response
is:Undamped12Transient
Response
of
the
2nd-Order
SystemsWhenthe
unit-step
response
is:13UnderdampedTransient
Response
of
the
2nd-Order
SystemsWhenthe
unit-step
response
is:Critically
damped14Transient
Response
of
the
2nd-Order
SystemsWhenthe
unit-step
response
is:Overdamped15Step
Response
of
Systems
with
Different
Damping
Ratios16Transient
Response
of
the
2nd-Order
SystemsTwo
useful
experimental
formulas
to
estimate
the
maximum
overshoot
andsettling
time
of
the
step
response
of
2nd-order
under
damped
systems17ExampleQ:
please
find
the
maximum
overshoot
and
settling
time
of
the
following
systemwhereA:18s=tf("s");z=0.3;g=1/(s^2+2*s*z+1);step(g,20)grid
onRelationship
Between
the
Damping
Ratio
and
the
Maximum
Overshootζ↑
σ%↓19Relationship
Between
the
Damping
Ratio
and
the
Settling
TimeActualEstimated20Relationship
Between
the
Natural
Undamped
Frequency
and
Other
Criteria21ExampleQ:
please
design
a
second-order
system
with
a
damping
ratio
equal
to
0.707and
a
settling
time
equal
to
0.5s.
Find
the
poles
of
the
system
and
calculate
inatural
undamped
frequency.A:
becauseandThenAccording
towe
can
getTherefores-z
wn450jw22s1s2High-order
Systems’
Time
ResponseLaplace
transform
of
the
step
response
of
a
high-order
systemTime
response
of
the
system
with
a
step-function
as
input2324High-order
Systems’
Time
Responses-
a-
5a
The
time
response
of
a
high-order
system
is
the
linear
combination
ofthose
of
1st
and
2nd
order
systems;
Poles
of
a
high-order
system
which
is
far
away
from
the
imaginary
axis
has
less
impact
on
the
system’s
transient
response
than
those
close
to
thimaginary
axis.
The
farther
the
distance
is,
the
smaller
the
impact
is.
Zeros
of
a
high-order
system
also
has
impact
on
the
system’s
timeresponse.
They
mainly
affect
the
magnitude
and
phase
of
a
dynamic
mode.
The
performance
of
a
high-order
system
is
mainly
determined
by
itscontrolling
polesControlling
PolejwControlling
PoleImpact
of
Additional
Poles
and
ZerosOriginal
system:25Impact
of
Additional
ZerosWith
additional
zero:26Impact
of
Additional
PolesWith
additional
pole:2728Conclusion
on
the
Impact
of
Additional
Poles
and
Zerosy(t)y1
(t)1
×1s2
+
0.8s
+
1
sY
(s)=s2×s
+
0.8s+
1s
+
1
1Y1(s)
=y(t)y2
(t)s
0.8s
1
s1
1×+2
+Y
(s)
=1
12×(s2
+
0.8s+
1)(s+
1)
sY
(s)=Decrease
Damping
RationIncrease
Damping
RationApproximation
of
High-order
Systems
by
Low-Order
SystemsExample:29To
keep
the
final
value
the
sameApproximation
of
High-order
Systems
by
Low-Order
SystemsExample:30Approximation
of
High-order
Systems
by
Low-Order
Systems31Approximation
CriterionThe
criterion
of
finding
the
low-order
,
given ,
is
thatthe
following
relation
should
be
satisfied
as
close
as
possible:This
condition
implies
that
the
amplitude
characteristics
of
the
two
systin
the
frequency
domain
( )
are
similar.32Wrap-up33Transient
ResponsePerformance
CriteriaTransient
Response
of
1st-Order
SystemsTransient
Response
of
2nd-Order
SystemsApproximation
of
High-Order
SystemsAssignment34Page
729,(4)10,11,12Review
questionsQ1:
The
maximum
overshoot
of
a
unit-step
response
of
the
second
order
prototypesystem
will
never
exceed
100
percent
when
the
damping
ratio
and
the
naturalfrequency
are
all
positive.
(T)
(F)35Review
questionsQ2:
For
the
second-order
prototype
system,
when
the
undamped
natural
frequencyincreases,
the
maximum
overshoot
of
the
output
stays
the
same.(T)
(F)36Review
questionsQ3:
The
maximum
overshoot
of
the
following
system
will
never
exceed
100%
when,
and
T
are
all
positive.(T)
(F)37Reference
answers:Q3:
False.Let ,
and
T
varies
from
0
to
5,
and
then
the
relationship
between
T
andthe
overshoot
can
be
shown
in
the
following
graph:Reference
answers:Let ,
and
T
varies
from
0
to
500,
and
then
the
relationship
between
T
andthe
overshoot
can
be
shown
in
the
following
graph:40Review
questionsQ4:
Increasing
the
undamped
natural
frequency
will
generally
reduce
the
rise
timeof
the
step
response(T)
(F)Reference
answers:Q4:
True.Let ,
and
varies
from
0.5
to
1.5,
and
then
the
relationship
between
andthe
rising
time
can
be
shown
in
the
following
graph:Reference
answers:Q4:
True.Let ,
and
varies
from
0.5
to
1.5,
and
then
the
relationship
between
andthe
rising
time
can
be
shown
in
the
foll
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