�
�
�
Recitation
11:
Time
delays
Emilio
Frazzoli
Laboratory
for
Information
and
Decision
Systems
Massachusetts
Institute
of
Technology
November
22,
2010.
1
Introduction
and
motivation
1.
Delays
are
incurred
when
the
controller
is
implemented
on
a
computer,
which
needs
some
time
to
compute
the
appropriate
control
input,
given
a
certain
error.
More
in
general,
the
evaluation
of
sensory
information
aimed
at
deciding
the
best
course
of
action,
will
require
a
finite
computation
time.
2.
In
some
systems,
delays
may
also
be
part
of
the
physical
plant. For
example,
on
an
airplane,
the
effect
of
lift
variation
at
the
main
wing
are
felt
on
the
tail
when
the
vortices
shed
by
the
wing
reach
the
tail
plane.
A
rather
extreme
example
is
remote
teleoperation:
communication
with
a
deepspace
spacecraft
or
planetary
rover
may
require
several
minutes.
2
The
frequency
response
of
a
time
delay
A
time
delay
is
an
operator
that
transforms
an
input
signal
t
�→
u
(
t
)
into
a
delayed
output
signal
t
�→
y
(
t
),
with
y
(
t
) =
u
(
t
−
T
),
where
T
≥
0
is
the
amount
of
the
delay.
Clearly,
this
is
a
linear
operator:
you
can
check
easily
that
the
delayed
version
of
a
linear
combination
of
signals
is
equal
to
the
linear
combination
of
the
delayed
signals.
Let
us
indicate
this
operator
with
Delay
T
.
In
order
to
compute
the
transfer
function
of
this
linear
operator,
let
us
compute
the
Laplace
transform
of
the
output
as
a
function
of
the
Laplace
transform
of
the
input.
We
get
∞
∞
∞
Y
(
s
) =
L
[
y
(
t
)]
=
y
(
t
)
e
−
st
dt
=
u
(
t
−
T
)
e
−
st
dt
=
u
(
τ
)
e
−
sτ
−
sT
dτ
=
e
−
sT
U
(
s
)
,
0
0
0
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�
�
from
which
we
conclude
that
Delay
T
(
s
) =
e
−
sT
.
What
is
the
frequency
response
of
the
time
delay?
We
need
to
study
the
function
ω
�→
Delay
T
(
jω
) =
e
−
jωT
.
Recall
that
the
polar
representation
of
a
complex
number
z
∈
C
is
z
=

z

e
−
j
∠
(
z
)
,
from
which
we
deduce
immediately
that

e
jωT

= 1
,
∠
e
−
jωT
=
−
ωT.
This
is
not
unexpected,
since
the
timedelayed
version
of
a
sinusoid
of
unit
amplitude
and
zero
phase
u
(
t
) =
sin
(
ωt
)
is
another
sinusoid,
y
=
sin
(
ω
(
t
−
T
)),
with
unit
magnitude
and
a
phase
delayed
by
ωT
.
(The
peak
of
the
input
is
reached
at
times
t
p
=
π/
(2
ω
) + 2
πn
,
where
the
phase
of
the
input
is
equal
to
π/
2;
at
those
times,
the
phase
of
the
output
is
π/
2
−
ωT
.)
See
Figure
1
.
1
0.5
0
0.5
1
0
5
10
15
20
y(t)
t
Figure
1:
Timedelayed
sinusoid.
The
blue
line
represents
the
signal
y
(
t
)
=
sin(2
π/
10
t
)
(for
t
≥
0);
the
red
line
is
the
same
signal
delayed
by
2.5
time
units,
i.e.,
Delay
2
.
5
[
y
].
Since
ω
= 2
π/
10
rad/s,
and
T
= 2
.
5
s
,
ωT
=
π/
2:
the
phase
of
the
timedelayed
signal
is
lagging
that
of
the
original
signal
by
exactly
π/
2
radians,
or
90
◦
degrees.
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 Fall '04
 EricFeron
 Feedback Control Systems, time delay, DelayT

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