<s>
Industrial	B-General_Concept
process	I-General_Concept
data	I-General_Concept
validation	I-General_Concept
and	I-General_Concept
reconciliation	I-General_Concept
,	O
or	O
more	O
briefly	O
,	O
process	O
data	B-General_Concept
reconciliation	I-General_Concept
(	O
PDR	O
)	O
,	O
is	O
a	O
technology	O
that	O
uses	O
process	O
information	O
and	O
mathematical	O
methods	O
in	O
order	O
to	O
automatically	O
ensure	O
data	B-General_Concept
validation	I-General_Concept
and	I-General_Concept
reconciliation	I-General_Concept
by	O
correcting	O
measurements	O
in	O
industrial	O
processes	O
.	O
</s>
<s>
Mathematically	O
the	O
model	O
can	O
be	O
expressed	O
by	O
a	O
nonlinear	O
system	O
of	O
equations	O
in	O
the	O
variables	O
,	O
which	O
incorporates	O
all	O
the	O
above-mentioned	O
system	B-Application
constraints	I-Application
(	O
for	O
example	O
the	O
mass	O
or	O
heat	O
balances	O
around	O
a	O
unit	O
)	O
.	O
</s>
<s>
systematic	O
errors	O
(	O
or	O
gross	O
errors	O
)	O
due	O
to	O
sensor	O
calibration	B-General_Concept
or	O
faulty	O
data	O
transmission	O
.	O
</s>
<s>
For	O
ease	O
in	O
deriving	O
and	O
implementing	O
an	O
optimal	O
estimation	O
solution	O
,	O
and	O
based	O
on	O
arguments	O
that	O
errors	O
are	O
the	O
sum	O
of	O
many	O
factors	O
(	O
so	O
that	O
the	O
Central	O
limit	O
theorem	O
has	O
some	O
effect	O
)	O
,	O
data	B-General_Concept
reconciliation	I-General_Concept
assumes	O
these	O
errors	O
are	O
normally	O
distributed	O
.	O
</s>
<s>
This	O
use	O
of	O
average	O
values	O
,	O
like	O
a	O
moving	O
average	O
,	O
acts	O
as	O
a	O
low-pass	B-Algorithm
filter	I-Algorithm
,	O
so	O
high	O
frequency	O
noise	O
is	O
mostly	O
eliminated	O
.	O
</s>
<s>
The	O
result	O
is	O
that	O
,	O
in	O
practice	O
,	O
data	B-General_Concept
reconciliation	I-General_Concept
is	O
mainly	O
making	O
adjustments	O
to	O
correct	O
systematic	O
errors	O
like	O
biases	O
.	O
</s>
<s>
Data	B-General_Concept
reconciliation	I-General_Concept
is	O
a	O
serious	O
issue	O
for	O
enterprise-control	O
integration	O
.	O
</s>
<s>
In	O
the	O
late	O
1960s	O
and	O
1970s	O
unmeasured	O
variables	O
were	O
taken	O
into	O
account	O
in	O
the	O
data	B-General_Concept
reconciliation	I-General_Concept
process.	O
,	O
PDR	O
also	O
became	O
more	O
mature	O
by	O
considering	O
general	O
nonlinear	O
equation	O
systems	O
coming	O
from	O
thermodynamic	O
models.	O
,	O
</s>
<s>
Data	B-General_Concept
reconciliation	I-General_Concept
is	O
a	O
technique	O
that	O
targets	O
at	O
correcting	O
measurement	O
errors	O
that	O
are	O
due	O
to	O
measurement	O
noise	O
,	O
i.e.	O
</s>
<s>
Given	O
measurements	O
,	O
data	B-General_Concept
reconciliation	I-General_Concept
can	O
mathematically	O
be	O
expressed	O
as	O
an	O
optimization	O
problem	O
of	O
the	O
following	O
form	O
:	O
</s>
<s>
is	O
the	O
reconciled	O
value	O
of	O
the	O
-th	O
measurement	O
(	O
)	O
,	O
is	O
the	O
measured	O
value	O
of	O
the	O
-th	O
measurement	O
(	O
)	O
,	O
is	O
the	O
-th	O
unmeasured	O
variable	O
(	O
)	O
,	O
and	O
is	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
-th	O
measurement	O
(	O
)	O
,	O
</s>
<s>
In	O
other	O
words	O
,	O
one	O
wants	O
to	O
minimize	O
the	O
overall	O
correction	O
(	O
measured	O
in	O
the	O
least	O
squares	O
term	O
)	O
that	O
is	O
needed	O
in	O
order	O
to	O
satisfy	O
the	O
system	B-Application
constraints	I-Application
.	O
</s>
<s>
Additionally	O
,	O
each	O
least	O
squares	O
term	O
is	O
weighted	O
by	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
corresponding	O
measurement	O
.	O
</s>
<s>
The	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
related	O
to	O
the	O
accuracy	O
of	O
the	O
measurement	O
.	O
</s>
<s>
For	O
example	O
,	O
at	O
a	O
95%	O
confidence	O
level	O
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
about	O
half	O
the	O
accuracy	O
.	O
</s>
<s>
Data	B-General_Concept
reconciliation	I-General_Concept
relies	O
strongly	O
on	O
the	O
concept	O
of	O
redundancy	B-Application
to	O
correct	O
the	O
measurements	O
as	O
little	O
as	O
possible	O
in	O
order	O
to	O
satisfy	O
the	O
process	O
constraints	O
.	O
</s>
<s>
Here	O
,	O
redundancy	B-Application
is	O
defined	O
differently	O
from	O
redundancy	B-General_Concept
in	I-General_Concept
information	I-General_Concept
theory	I-General_Concept
.	O
</s>
<s>
Instead	O
,	O
redundancy	B-Application
arises	O
from	O
combining	O
sensor	O
data	O
with	O
the	O
model	O
(	O
algebraic	O
constraints	O
)	O
,	O
sometimes	O
more	O
specifically	O
called	O
"	O
spatial	O
redundancy	B-Application
"	O
,	O
"	O
analytical	O
redundancy	B-Application
"	O
,	O
or	O
"	O
topological	O
redundancy	B-Application
"	O
.	O
</s>
<s>
Redundancy	B-Application
can	O
be	O
due	O
to	O
sensor	O
redundancy	B-Application
,	O
where	O
sensors	O
are	O
duplicated	O
in	O
order	O
to	O
have	O
more	O
than	O
one	O
measurement	O
of	O
the	O
same	O
quantity	O
.	O
</s>
<s>
Redundancy	B-Application
also	O
arises	O
when	O
a	O
single	O
variable	O
can	O
be	O
estimated	O
in	O
several	O
independent	O
ways	O
from	O
separate	O
sets	O
of	O
measurements	O
at	O
a	O
given	O
time	O
or	O
time	O
averaging	O
period	O
,	O
using	O
the	O
algebraic	O
constraints	O
.	O
</s>
<s>
Redundancy	B-Application
is	O
linked	O
to	O
the	O
concept	O
of	O
observability	O
.	O
</s>
<s>
Rigorous	O
definitions	O
of	O
observability	O
,	O
calculability	O
,	O
and	O
redundancy	B-Application
,	O
along	O
with	O
criteria	O
for	O
determining	O
it	O
,	O
were	O
established	O
by	O
Stanley	O
and	O
Mah	O
,	O
for	O
these	O
cases	O
with	O
set	O
constraints	O
such	O
as	O
algebraic	O
equations	O
and	O
inequalities	O
.	O
</s>
<s>
Topological	O
redundancy	B-Application
is	O
intimately	O
linked	O
with	O
the	O
degrees	O
of	O
freedom	O
(	O
)	O
of	O
a	O
mathematical	O
system	O
,	O
i.e.	O
</s>
<s>
At	O
least	O
2	O
measurements	O
are	O
needed	O
to	O
estimate	O
all	O
the	O
variables	O
,	O
and	O
3	O
would	O
be	O
needed	O
for	O
redundancy	B-Application
.	O
</s>
<s>
When	O
speaking	O
about	O
topological	O
redundancy	B-Application
we	O
have	O
to	O
distinguish	O
between	O
measured	O
and	O
unmeasured	O
variables	O
.	O
</s>
<s>
If	O
the	O
system	O
is	O
calculable	O
with	O
the	O
measurements	O
given	O
,	O
then	O
the	O
level	O
of	O
topological	O
redundancy	B-Application
is	O
defined	O
as	O
,	O
i.e.	O
</s>
<s>
Another	O
way	O
of	O
viewing	O
the	O
level	O
of	O
redundancy	B-Application
is	O
to	O
use	O
the	O
definition	O
of	O
,	O
which	O
is	O
the	O
difference	O
between	O
the	O
number	O
of	O
variables	O
(	O
measured	O
and	O
unmeasured	O
)	O
and	O
the	O
number	O
of	O
equations	O
.	O
</s>
<s>
the	O
redundancy	B-Application
is	O
the	O
difference	O
between	O
the	O
number	O
of	O
equations	O
and	O
the	O
number	O
of	O
unmeasured	O
variables	O
.	O
</s>
<s>
The	O
level	O
of	O
total	O
redundancy	B-Application
is	O
the	O
sum	O
of	O
sensor	O
redundancy	B-Application
and	O
topological	O
redundancy	B-Application
.	O
</s>
<s>
We	O
speak	O
of	O
positive	O
redundancy	B-Application
if	O
the	O
system	O
is	O
calculable	O
and	O
the	O
total	O
redundancy	B-Application
is	O
positive	O
.	O
</s>
<s>
One	O
can	O
see	O
that	O
the	O
level	O
of	O
topological	O
redundancy	B-Application
merely	O
depends	O
on	O
the	O
number	O
of	O
equations	O
(	O
the	O
more	O
equations	O
the	O
higher	O
the	O
redundancy	B-Application
)	O
and	O
the	O
number	O
of	O
unmeasured	O
variables	O
(	O
the	O
more	O
unmeasured	O
variables	O
,	O
the	O
lower	O
the	O
redundancy	B-Application
)	O
and	O
not	O
on	O
the	O
number	O
of	O
measured	O
variables	O
.	O
</s>
<s>
Simple	O
counts	O
of	O
variables	O
,	O
equations	O
,	O
and	O
measurements	O
are	O
inadequate	O
for	O
many	O
systems	O
,	O
breaking	O
down	O
for	O
several	O
reasons	O
:	O
(	O
a	O
)	O
Portions	O
of	O
a	O
system	O
might	O
have	O
redundancy	B-Application
,	O
while	O
others	O
do	O
not	O
,	O
and	O
some	O
portions	O
might	O
not	O
even	O
be	O
possible	O
to	O
calculate	O
,	O
and	O
(	O
b	O
)	O
Nonlinearities	O
can	O
lead	O
to	O
different	O
conclusions	O
at	O
different	O
operating	O
points	O
.	O
</s>
<s>
In	O
1981	O
,	O
observability	O
and	O
redundancy	B-Application
criteria	O
were	O
proven	O
for	O
these	O
sorts	O
of	O
flow	O
networks	O
involving	O
only	O
mass	O
and	O
energy	O
balance	O
constraints	O
.	O
</s>
<s>
Redundancy	B-Application
classification	O
follows	O
,	O
by	O
testing	O
for	O
a	O
path	O
of	O
unmeasured	O
streams	O
,	O
since	O
that	O
would	O
lead	O
to	O
an	O
unmeasured	O
cycle	O
if	O
the	O
measurement	O
was	O
removed	O
.	O
</s>
<s>
Redundancy	B-Application
can	O
be	O
used	O
as	O
a	O
source	O
of	O
information	O
to	O
cross-check	O
and	O
correct	O
the	O
measurements	O
and	O
increase	O
their	O
accuracy	O
and	O
precision	O
:	O
on	O
the	O
one	O
hand	O
they	O
reconciled	O
Further	O
,	O
the	O
data	B-General_Concept
reconciliation	I-General_Concept
problem	O
presented	O
above	O
also	O
includes	O
unmeasured	O
variables	O
.	O
</s>
<s>
Based	O
on	O
information	O
redundancy	B-Application
,	O
estimates	O
for	O
these	O
unmeasured	O
variables	O
can	O
be	O
calculated	O
along	O
with	O
their	O
accuracies	O
.	O
</s>
<s>
In	O
industrial	O
processes	O
these	O
unmeasured	O
variables	O
that	O
data	B-General_Concept
reconciliation	I-General_Concept
provides	O
are	O
referred	O
to	O
as	O
soft	O
sensors	O
or	O
virtual	O
sensors	O
,	O
where	O
hardware	O
sensors	O
are	O
not	O
installed	O
.	O
</s>
<s>
When	O
adding	O
thermodynamic	O
constraints	O
such	O
as	O
energy	O
balances	O
to	O
the	O
model	O
,	O
its	O
scope	O
and	O
the	O
level	O
of	O
redundancy	B-Application
increases	O
.	O
</s>
<s>
Indeed	O
,	O
as	O
we	O
have	O
seen	O
above	O
,	O
the	O
level	O
of	O
redundancy	B-Application
is	O
defined	O
as	O
,	O
where	O
is	O
the	O
number	O
of	O
equations	O
.	O
</s>
<s>
Including	O
energy	O
balances	O
means	O
adding	O
equations	O
to	O
the	O
system	O
,	O
which	O
results	O
in	O
a	O
higher	O
level	O
of	O
redundancy	B-Application
(	O
provided	O
that	O
enough	O
measurements	O
are	O
available	O
,	O
or	O
equivalently	O
,	O
not	O
too	O
many	O
variables	O
are	O
unmeasured	O
)	O
.	O
</s>
<s>
thumb|350px|The	O
workflow	O
of	O
an	O
advanced	O
data	B-General_Concept
validation	I-General_Concept
and	I-General_Concept
reconciliation	I-General_Concept
process	O
.	O
</s>
<s>
It	O
is	O
important	O
to	O
note	O
that	O
the	O
remediation	O
of	O
gross	O
errors	O
reduces	O
the	O
quality	O
of	O
the	O
reconciliation	O
,	O
either	O
the	O
redundancy	B-Application
decreases	O
(	O
elimination	O
)	O
or	O
the	O
uncertainty	O
of	O
the	O
measured	O
data	O
increases	O
(	O
relaxation	O
)	O
.	O
</s>
<s>
Therefore	O
,	O
it	O
can	O
only	O
be	O
applied	O
when	O
the	O
initial	O
level	O
of	O
redundancy	B-Application
is	O
high	O
enough	O
to	O
ensure	O
that	O
the	O
data	B-General_Concept
reconciliation	I-General_Concept
can	O
still	O
be	O
done	O
(	O
see	O
Section	O
2	O
,	O
)	O
.	O
</s>
