<s>
The	O
quantized	O
state	O
systems	O
(	O
QSS	B-Algorithm
)	O
methods	O
are	O
a	O
family	O
of	O
numerical	O
integration	O
solvers	O
based	O
on	O
the	O
idea	O
of	O
state	O
quantization	B-Algorithm
,	O
dual	O
to	O
the	O
traditional	O
idea	O
of	O
time	O
discretization	B-Algorithm
.	O
</s>
<s>
Unlike	O
traditional	O
numerical	B-Algorithm
solution	I-Algorithm
methods	I-Algorithm
,	O
which	O
approach	O
the	O
problem	O
by	O
discretizing	B-Algorithm
time	O
and	O
solving	O
for	O
the	O
next	O
(	O
real-valued	O
)	O
state	O
at	O
each	O
successive	O
time	O
step	O
,	O
QSS	B-Algorithm
methods	O
keep	O
time	O
as	O
a	O
continuous	O
entity	O
and	O
instead	O
quantize	B-Algorithm
the	O
system	O
's	O
state	O
,	O
instead	O
solving	O
for	O
the	O
time	O
at	O
which	O
the	O
state	O
deviates	O
from	O
its	O
quantized	O
value	O
by	O
a	O
quantum	O
.	O
</s>
<s>
They	O
inherently	O
allow	O
for	O
modeling	O
discontinuities	O
in	O
the	O
system	O
due	O
to	O
their	O
discrete-event	B-General_Concept
nature	O
and	O
asynchronous	O
nature	O
.	O
</s>
<s>
Finally	O
,	O
QSS	B-Algorithm
methods	O
satisfy	O
remarkable	O
global	O
stability	O
and	O
error	O
bounds	O
,	O
described	O
below	O
,	O
which	O
are	O
not	O
satisfied	O
by	O
classical	O
solution	O
techniques	O
.	O
</s>
<s>
By	O
their	O
nature	O
,	O
QSS	B-Algorithm
methods	O
are	O
therefore	O
neatly	O
modeled	O
by	O
the	O
DEVS	B-Application
formalism	O
,	O
a	O
discrete-event	B-General_Concept
model	O
of	O
computation	O
,	O
in	O
contrast	O
with	O
traditional	O
methods	O
,	O
which	O
form	O
discrete-time	O
models	O
of	O
the	O
continuous-time	O
system	O
.	O
</s>
<s>
They	O
have	O
therefore	O
been	O
implemented	O
in	O
[PowerDEVS],	O
a	O
simulation	O
engine	O
for	O
such	O
discrete-event	B-General_Concept
systems	O
.	O
</s>
<s>
Avoiding	O
this	O
condition	O
would	O
require	O
finding	O
a	O
reliable	O
technique	O
for	O
dynamically	O
lowering	O
the	O
quantum	O
in	O
a	O
manner	O
analogous	O
to	O
adaptive	B-Algorithm
stepsize	I-Algorithm
methods	O
in	O
traditional	O
discrete	O
time	O
simulation	O
algorithms	O
.	O
</s>
<s>
Notice	O
that	O
this	O
quantization	B-Algorithm
function	O
is	O
hysteretic	O
because	O
it	O
has	O
memory	O
:	O
not	O
only	O
is	O
its	O
output	O
a	O
function	O
of	O
the	O
current	O
state	O
,	O
but	O
it	O
also	O
depends	O
on	O
its	O
old	O
value	O
,	O
.	O
</s>
<s>
The	O
second-order	O
QSS	B-Algorithm
method	O
,	O
QSS2	O
,	O
follows	O
the	O
same	O
principle	O
as	O
QSS1	O
,	O
except	O
that	O
it	O
defines	O
as	O
a	O
piecewise	O
linear	O
approximation	O
of	O
the	O
trajectory	O
that	O
updates	O
its	O
trajectory	O
as	O
soon	O
as	O
the	O
two	O
differ	O
from	O
each	O
other	O
by	O
one	O
quantum	O
.	O
</s>
<s>
It	O
is	O
important	O
to	O
note	O
that	O
,	O
while	O
in	O
principle	O
a	O
QSS	B-Algorithm
method	O
of	O
arbitrary	O
order	O
can	O
be	O
used	O
to	O
model	O
a	O
continuous-time	O
system	O
,	O
it	O
is	O
seldom	O
desirable	O
to	O
use	O
methods	O
of	O
order	O
higher	O
than	O
four	O
,	O
as	O
the	O
Abel	O
–	O
Ruffini	O
theorem	O
implies	O
that	O
the	O
time	O
of	O
the	O
next	O
quantization	B-Algorithm
,	O
,	O
cannot	O
(	O
in	O
general	O
)	O
be	O
explicitly	B-Algorithm
solved	I-Algorithm
for	O
algebraically	O
when	O
the	O
polynomial	O
approximation	O
is	O
of	O
degree	O
greater	O
than	O
four	O
,	O
and	O
hence	O
must	O
be	O
approximated	O
iteratively	O
using	O
a	O
root-finding	O
algorithm	O
.	O
</s>
<s>
The	O
QSS	B-Algorithm
Methods	O
can	O
be	O
implemented	O
as	O
a	O
discrete	O
event	O
system	O
and	O
simulated	O
in	O
any	O
DEVS	B-Application
simulator	O
.	O
</s>
<s>
QSS	B-Algorithm
methods	O
constitute	O
the	O
main	O
numerical	O
solver	O
for	O
PowerDEVS[BK011]	O
software	O
.	O
</s>
