<s>
The	O
step	B-Algorithm
response	I-Algorithm
of	O
a	O
system	O
in	O
a	O
given	O
initial	O
state	O
consists	O
of	O
the	O
time	O
evolution	O
of	O
its	O
outputs	O
when	O
its	O
control	O
inputs	O
are	O
Heaviside	O
step	O
functions	O
.	O
</s>
<s>
In	O
electronic	O
engineering	O
and	O
control	O
theory	O
,	O
step	B-Algorithm
response	I-Algorithm
is	O
the	O
time	O
behaviour	O
of	O
the	O
outputs	O
of	O
a	O
general	O
system	O
when	O
its	O
inputs	O
change	O
from	O
zero	O
to	O
one	O
in	O
a	O
very	O
short	O
time	O
.	O
</s>
<s>
Formally	O
,	O
knowing	O
the	O
step	B-Algorithm
response	I-Algorithm
of	O
a	O
dynamical	O
system	O
gives	O
information	O
on	O
the	O
stability	O
of	O
such	O
a	O
system	O
,	O
and	O
on	O
its	O
ability	O
to	O
reach	O
one	O
stationary	O
state	O
when	O
starting	O
from	O
another	O
.	O
</s>
<s>
This	O
section	O
provides	O
a	O
formal	O
mathematical	O
definition	O
of	O
step	B-Algorithm
response	I-Algorithm
in	O
terms	O
of	O
the	O
abstract	O
mathematical	O
concept	O
of	O
a	O
dynamical	O
system	O
:	O
all	O
notations	O
and	O
assumptions	O
required	O
for	O
the	O
following	O
description	O
are	O
listed	O
here	O
.	O
</s>
<s>
For	O
a	O
general	O
dynamical	O
system	O
,	O
the	O
step	B-Algorithm
response	I-Algorithm
is	O
defined	O
as	O
follows	O
:	O
</s>
<s>
Conversely	O
,	O
for	O
an	O
LTI	O
system	O
,	O
the	O
derivative	O
of	O
the	O
step	B-Algorithm
response	I-Algorithm
yields	O
the	O
impulse	O
response	O
:	O
</s>
<s>
The	O
step	B-Algorithm
response	I-Algorithm
can	O
be	O
described	O
by	O
the	O
following	O
quantities	O
related	O
to	O
its	O
time	O
behavior	O
,	O
</s>
<s>
Below	O
the	O
step	B-Algorithm
response	I-Algorithm
of	O
a	O
simple	O
two-pole	O
amplifier	O
is	O
presented	O
,	O
and	O
some	O
of	O
these	O
terms	O
are	O
illustrated	O
.	O
</s>
<s>
In	O
LTI	O
systems	O
,	O
the	O
function	O
that	O
has	O
the	O
steepest	O
slew	O
rate	O
that	O
does	O
n't	O
create	O
overshoot	B-Algorithm
or	O
ringing	B-Algorithm
is	O
the	O
Gaussian	O
function	O
.	O
</s>
<s>
This	O
section	O
describes	O
the	O
step	B-Algorithm
response	I-Algorithm
of	O
a	O
simple	O
negative	O
feedback	O
amplifier	O
shown	O
in	O
Figure	O
1	O
.	O
</s>
<s>
This	O
feedback	O
amplifier	O
is	O
analyzed	O
to	O
determine	O
how	O
its	O
step	B-Algorithm
response	I-Algorithm
depends	O
upon	O
the	O
time	O
constants	O
governing	O
the	O
response	O
of	O
the	O
main	O
amplifier	O
,	O
and	O
upon	O
the	O
amount	O
of	O
feedback	O
used	O
.	O
</s>
<s>
Its	O
step	B-Algorithm
response	I-Algorithm
is	O
of	O
the	O
same	O
form	O
:	O
an	O
exponential	O
decay	O
toward	O
the	O
new	O
equilibrium	O
value	O
.	O
</s>
<s>
As	O
the	O
feedback	O
factor	O
β	O
is	O
increased	O
,	O
the	O
step	B-Algorithm
response	I-Algorithm
will	O
get	O
faster	O
,	O
until	O
the	O
original	O
assumption	O
of	O
one	O
dominant	O
pole	O
is	O
no	O
longer	O
accurate	O
.	O
</s>
<s>
In	O
the	O
case	O
that	O
the	O
open-loop	O
gain	O
has	O
two	O
poles	O
(	O
two	O
time	O
constants	O
,	O
τ1	O
,	O
τ2	O
)	O
,	O
the	O
step	B-Algorithm
response	I-Algorithm
is	O
a	O
bit	O
more	O
complicated	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
unit	O
step	B-Algorithm
response	I-Algorithm
of	O
the	O
system	O
is	O
:	O
</s>
<s>
The	O
phenomenon	O
of	O
oscillation	O
about	O
the	O
final	O
value	O
is	O
called	O
ringing	B-Algorithm
.	O
</s>
<s>
The	O
overshoot	B-Algorithm
is	O
the	O
maximum	O
swing	O
above	O
final	O
value	O
,	O
and	O
clearly	O
increases	O
with	O
μ	O
.	O
</s>
<s>
Likewise	O
,	O
the	O
undershoot	B-Algorithm
is	O
the	O
minimum	O
swing	O
below	O
final	O
value	O
,	O
again	O
increasing	O
with	O
μ	O
.	O
</s>
<s>
The	O
settling	B-Algorithm
time	I-Algorithm
is	O
the	O
time	O
for	O
departures	O
from	O
final	O
value	O
to	O
sink	O
below	O
some	O
specified	O
level	O
,	O
say	O
10%	O
of	O
final	O
value	O
.	O
</s>
<s>
The	O
dependence	O
of	O
settling	B-Algorithm
time	I-Algorithm
upon	O
μ	O
is	O
not	O
obvious	O
,	O
and	O
the	O
approximation	O
of	O
a	O
two-pole	O
system	O
probably	O
is	O
not	O
accurate	O
enough	O
to	O
make	O
any	O
real-world	O
conclusions	O
about	O
feedback	O
dependence	O
of	O
settling	B-Algorithm
time	I-Algorithm
.	O
</s>
<s>
However	O
,	O
the	O
asymptotes	O
[	O
1−exp( −ρt	O
)	O
]	O
and	O
[	O
1+exp( −ρt	O
)	O
]	O
clearly	O
impact	O
settling	B-Algorithm
time	I-Algorithm
,	O
and	O
they	O
are	O
controlled	O
by	O
the	O
time	O
constants	O
of	O
the	O
open-loop	O
amplifier	O
,	O
particularly	O
the	O
shorter	O
of	O
the	O
two	O
time	O
constants	O
.	O
</s>
<s>
That	O
suggests	O
that	O
a	O
specification	O
on	O
settling	B-Algorithm
time	I-Algorithm
must	O
be	O
met	O
by	O
appropriate	O
design	O
of	O
the	O
open-loop	O
amplifier	O
.	O
</s>
<s>
The	O
open-loop	O
amplifier	O
decides	O
settling	B-Algorithm
time	I-Algorithm
.	O
</s>
<s>
As	O
an	O
aside	O
,	O
it	O
may	O
be	O
noted	O
that	O
real-world	O
departures	O
from	O
this	O
linear	O
two-pole	O
model	O
occur	O
due	O
to	O
two	O
major	O
complications	O
:	O
first	O
,	O
real	O
amplifiers	O
have	O
more	O
than	O
two	O
poles	O
,	O
as	O
well	O
as	O
zeros	O
;	O
and	O
second	O
,	O
real	O
amplifiers	O
are	O
nonlinear	O
,	O
so	O
their	O
step	B-Algorithm
response	I-Algorithm
changes	O
with	O
signal	O
amplitude	O
.	O
</s>
<s>
How	O
overshoot	B-Algorithm
may	O
be	O
controlled	O
by	O
appropriate	O
parameter	O
choices	O
is	O
discussed	O
next	O
.	O
</s>
<s>
Using	O
the	O
equations	O
above	O
,	O
the	O
amount	O
of	O
overshoot	B-Algorithm
can	O
be	O
found	O
by	O
differentiating	O
the	O
step	B-Algorithm
response	I-Algorithm
and	O
finding	O
its	O
maximum	O
value	O
.	O
</s>
<s>
The	O
result	O
for	O
maximum	O
step	B-Algorithm
response	I-Algorithm
Smax	O
is	O
:	O
</s>
<s>
The	O
final	O
value	O
of	O
the	O
step	B-Algorithm
response	I-Algorithm
is	O
1	O
,	O
so	O
the	O
exponential	O
is	O
the	O
actual	O
overshoot	B-Algorithm
itself	O
.	O
</s>
<s>
It	O
is	O
clear	O
the	O
overshoot	B-Algorithm
is	O
zero	O
if	O
μ	O
=	O
0	O
,	O
which	O
is	O
the	O
condition	O
:	O
</s>
<s>
To	O
be	O
more	O
adventurous	O
than	O
a	O
design	O
allowing	O
for	O
no	O
overshoot	B-Algorithm
we	O
can	O
introduce	O
a	O
factor	O
α	O
in	O
the	O
above	O
relation	O
:	O
</s>
<s>
and	O
let	O
α	O
be	O
set	O
by	O
the	O
amount	O
of	O
overshoot	B-Algorithm
that	O
is	O
acceptable	O
.	O
</s>
<s>
Comparing	O
the	O
top	O
panel	O
( α	O
=	O
4	O
)	O
with	O
the	O
lower	O
panel	O
(α=	O
0.5	O
)	O
shows	O
lower	O
values	O
for	O
α	O
increase	O
the	O
rate	O
of	O
response	O
,	O
but	O
increase	O
overshoot	B-Algorithm
.	O
</s>
<s>
The	O
case	O
α	O
=	O
2	O
(	O
center	O
panel	O
)	O
is	O
the	O
maximally	O
flat	O
design	O
that	O
shows	O
no	O
peaking	O
in	O
the	O
Bode	B-Algorithm
gain	I-Algorithm
vs.	I-Algorithm
frequency	I-Algorithm
plot	I-Algorithm
.	O
</s>
<s>
That	O
design	O
has	O
the	O
rule	O
of	O
thumb	O
built-in	O
safety	O
margin	O
to	O
deal	O
with	O
non-ideal	O
realities	O
like	O
multiple	O
poles	O
(	O
or	O
zeros	O
)	O
,	O
nonlinearity	O
(	O
signal	O
amplitude	O
dependence	O
)	O
and	O
manufacturing	O
variations	O
,	O
any	O
of	O
which	O
can	O
lead	O
to	O
too	O
much	O
overshoot	B-Algorithm
.	O
</s>
<s>
The	O
amplitude	O
of	O
ringing	B-Algorithm
in	O
the	O
step	B-Algorithm
response	I-Algorithm
in	O
Figure	O
3	O
is	O
governed	O
by	O
the	O
damping	O
factor	O
exp( −ρt	O
)	O
.	O
</s>
<s>
That	O
is	O
,	O
if	O
we	O
specify	O
some	O
acceptable	O
step	B-Algorithm
response	I-Algorithm
deviation	O
from	O
final	O
value	O
,	O
say	O
Δ	O
,	O
that	O
is	O
:	O
</s>
<s>
this	O
condition	O
is	O
satisfied	O
regardless	O
of	O
the	O
value	O
of	O
β	O
AOL	O
provided	O
the	O
time	O
is	O
longer	O
than	O
the	O
settling	B-Algorithm
time	I-Algorithm
,	O
say	O
tS	O
,	O
given	O
by	O
:	O
</s>
<s>
where	O
the	O
τ1≫τ2	O
is	O
applicable	O
because	O
of	O
the	O
overshoot	B-Algorithm
control	O
condition	O
,	O
which	O
makes	O
τ1	O
=	O
αβAOL	O
τ2	O
.	O
</s>
<s>
Often	O
the	O
settling	B-Algorithm
time	I-Algorithm
condition	O
is	O
referred	O
to	O
by	O
saying	O
the	O
settling	O
period	O
is	O
inversely	O
proportional	O
to	O
the	O
unity	O
gain	O
bandwidth	O
,	O
because	O
1/	O
( 2πτ2	O
)	O
is	O
close	O
to	O
this	O
bandwidth	O
for	O
an	O
amplifier	O
with	O
typical	O
dominant	O
pole	O
compensation	O
.	O
</s>
<s>
As	O
an	O
example	O
of	O
this	O
formula	O
,	O
if	O
the	O
settling	B-Algorithm
time	I-Algorithm
condition	O
is	O
tS	O
=	O
8τ2	O
.	O
</s>
<s>
In	O
general	O
,	O
control	O
of	O
overshoot	B-Algorithm
sets	O
the	O
time	O
constant	O
ratio	O
,	O
and	O
settling	B-Algorithm
time	I-Algorithm
tS	O
setsτ2	O
.	O
</s>
<s>
This	O
method	O
uses	O
significant	O
points	O
of	O
the	O
step	B-Algorithm
response	I-Algorithm
.	O
</s>
<s>
Determine	O
the	O
time-spans	O
and	O
where	O
the	O
step	B-Algorithm
response	I-Algorithm
reaches	O
25%	O
and	O
75%	O
of	O
the	O
steady	O
state	O
output	O
value	O
.	O
</s>
<s>
The	O
procedure	O
outlined	O
in	O
the	O
Bode	B-Algorithm
plot	I-Algorithm
article	O
is	O
followed	O
.	O
</s>
<s>
If	O
α	O
is	O
increased	O
by	O
shortening	O
τ2	O
,	O
the	O
settling	B-Algorithm
time	I-Algorithm
tS	O
also	O
is	O
shortened	O
.	O
</s>
<s>
If	O
α	O
is	O
increased	O
by	O
lengthening	O
τ1	O
,	O
the	O
settling	B-Algorithm
time	I-Algorithm
tS	O
is	O
little	O
altered	O
.	O
</s>
<s>
As	O
an	O
aside	O
,	O
for	O
an	O
amplifier	O
with	O
more	O
than	O
two	O
poles	O
,	O
the	O
diagram	O
of	O
Figure	O
5	O
still	O
may	O
be	O
made	O
to	O
fit	O
the	O
Bode	B-Algorithm
plots	I-Algorithm
by	O
making	O
f2	O
a	O
fitting	O
parameter	O
,	O
referred	O
to	O
as	O
an	O
"	O
equivalent	O
second	O
pole	O
"	O
position	O
.	O
</s>
