Au oma ic Semiquali a i e Analysis: Applica ion
o a Biome allu gical Sys em
R.
M.
Gasca,
J.
A.
O ega,
M.
Ta o
Depa amen o
de Lenguajes y Sis emas In o ma icos,
Facul ad
de
In o ma ica
y
Es adis ica
A da. Reina Me cedes
s/n,
Se illa
(Espana)
e-mail:{gasca,jao ega,m o o }@lsi.us.es
Abs ac .
The
aim o his wo k is
he
ep esen a ion
and
analysis o
semiquali a i e
models.
Thei
quali a i e
knowledge is
ep esen ed
by
means
o
quali a i e
ope a o s
and
en elope unc ions. A
semiquali a i e
model is
ans o med
in o
a amily o
quan i a i e
models.
In his
pape
he
analysis o a model is
p oposed
as a
cons ain
sa is ac-
ion p oblem.
Cons ain
sa is ac ion is
an
umb ella
e m
o a a ie y o
echniques o A i icial In elligence
and
ela ed
disciplines.
In
his
pape
a en ion
is ocused
on
in e als consis ency echniques.
The
semiquali a-
i e analysis is
au oma ically
made
by means o consis ency echniques.
The
p esen ed
me hod
is
applied
o
a
indus ial
biome allu gical
sys em
in
o de
o
show how inc ease
he
capaci y o
p oduc ion.
1
In oduc ion
In enginee ing and science, he models made up o he s udy o dynamical
sys ems a e no mally composed o quan i a i e and quali a i e knowledge. This
knowledge
is
composed by
bo h
o hem.
I
is
known as semiquali a i e know-
ledge. Real models con ain quan i a i e, quali a i e and semiquali a i e know-
ledge. All his knowledge mus be conside ed when hese models a e s udied.
The
echniques de eloped o analyze and simula e quan i a i e models a e
well known. A g ea a ie y o echniques has been s udied o he ep esen a ion
and
he
manipula ion o quali a i e knowledge, such as algeb a
o
signs, in e al
a i hme ic, uzzy se s, and o de o magni ude easoning.
In
o de o analyze indus ial models, i is necessa y some imes
o
sol e con-
lic s on
he
eques o accu acy and lexibili y.
The
models o dynamical sys ems
should p o ide di e en le els o nume ical abs ac ion o hei elemen s. These
le els may be a pu ely quali a i e desc ip ion
[8),
semiquali a i e
[2), [6),
nume-
ical based on in e als
[11),
quan i a i e and mixed o all le els
[7].
On
o he hand, he sys ems dynamics ob ains he di e en ial equa ions o
a sys em om i s s uc u e. This echnique could
ob ain
di e en quali a i e
beha io s o a gi en s uc u e.
The
analysis o hese beha io s cons i u es he
quali a i e analysis o dynamical sys ems. The
ma hema ical
quali a i e heo y
o dynamical sys ems in ol ed s udying quali a i ely he beha iou (e.g. asymp-
o ic beha iou ) o ime e ol ing sys ems.
322
In
o de
o
au oma e
he quali a i e analysis o dynamic sys ems se e al ap-
plica ions ha e been de eloped. They combine echniques
o
nume ical me hods
wi h symbolic compu a ion, and me hods p oceeding om
he
knowledge o
he
science and
he
ma hema ics. These applica ions begin wi h
he
op-le el speci i-
ca ions
o
physical model. They p epa e simula ion expe imen s, and accomplish
hem.
Also hey in e p e he nume ical esul s, and hey o mula e
he
esul s in
quali a i e e ms. Among hem,
we
can ci e
PLR
[9],
bi u ca ion
in e p e e
[1],
KAM
[12],
POINCARE
[10],
and
MAPS
[13].
In his pape , a
me hod
o
ca y
ou
he analysis o dynamical sys ems au oma ically
is
shown.
The
semiquali-
a i e analysis
is
p oposed as a se o in e al cons ain sa is ac ion p oblems.
They a e sol ed applying consis ency echniques
[5].
2
Semiquali a i e
models
A dynamical sys em can be conside ed as he cons ain s
<P(x,
x,p), x( o)
=
xo,
<Po(p,xo)
(
1)
being
x
he
s a e
a iables o
he
sys em,
p
he
pa ame e s,
x
he
a ia ion
o
he
s a e
a iables wi h he ime,
<Po
he cons ain s among pa ame e s and
ini ial condi ions, and
<P
he
cons ain s on
x,
x
and
p.
The
dynamical sys em
ep esen ed in
(1)
can symbolically be ans o med in o a se o con ain s wi h
a iables, pa ame e s and in e als. In his pape ,
we
only
s udy
sys ems
ha
can be ans o med as
x
=
l(x,p),
x( o) =
Xo,
<Po(p,
xo)
(2)
The
ec o ield
may be composed
o
quan i a i e
and quali a i e a iables,
cons an s, a i hme ic ope a o s, unc ions and en elope unc ions, exp essed as i
is indica ed in ou p e ious pape
[4],
whe e quali a i e a iables and en elope
unc ions a e ans o med
o
in e al exp essions.
I
we
ake in o accoun he
s ablished concep s in
ha
pape , he dynamical sys em (2)
is
ans o med in
x
=
(x, ,p),
x(
0) =
xo,
<Po
(p,
,
xo)
(3)
whe e
E
II
a e new pa ame e s, p E II,
xo
E II, and
does
no
con ain en elope
unc ions, being
II
he
se o closed in e als o
JR.
These unc ions ep esen a
dynamical sys ems amily depending on p, x0
and
.
I
is
deno ed as semiquali-
a i e model and i is ep esen ed u he on
x
=
(x,p),
x( o) =
xo,
<Po(p,xo)
(4)
whe e
p
and ha e been joined in an unique
pa ame e s
ec o
p.
323
3
Semiquali a i e
analysis
Quali a i e analysis o a dynamical sys em in ends
o
analyze he
phase po ai
o
phase space
o
he
sys em. The
phase space
o he dynamical sys em is cons i-
u ed
by
he
a iables o
s a e
x,
and he
ex ended phase space
by a iables and
pa ame e s
x,
p.
The
phase
po ai
is o med by he p ojec ion o he ajec o-
ies o he dynamical sys em in he ex ended phase space.
The
phase
po ai
is
in e p e ed as a co espondence be ween he di e en ial equa ions and he ec-
o ield. In his pape semiquali a i e sys ems
ha
hey a e s able s uc u ally
a e s udied. In hem, li le pe u ba ions keep hei quali a i e beha io s.
The
i s s ep o semiquali a i e analysis o a dynamical sys em (
4)
is
he
de e mina ion o he
equilib ium egions.
They a e de ined by he cons ain s
Equilib ium(x,p)::: {
(x,p)
=
0,
(5)
The
s udy
o solu ions o
(5)
le us know he s uc u e o he phase
po ai .
Each s able equilib ium egion is an
a ac o
egion.
The
s abili y
o each equilib ium egion
is
ela ed
o
he eal
pa
o he
eigen alues o he Jacobian o he sys em.
I
has been
demons a ed
in he bibli-
og aphy
ha
in he s able ixed poin s he eal
pa
o he eigen alues
is
nega i e.
In o de
o
apply he s abili y c i e ia, i
is
necessa y
o
cons uc he ollowing
de e minan s. They a e o med wi h he coe icien s o he cha ac e is ic polyno-
mial
Pn
o
he
Jacobian
ma ix
A
o he dynamical sys em.
The
Jacobian
ma ix
o
(4)
is
A=
Dx (x,p),
and
Pn
is
de ined as
Pn(, )
=
de (A-
, I)=
aoAn
+
a1, n-
1
+
...
+
an-1,
+an
(6)
In o de o de e mine
he
s abili y condi ions, he ma ices a e de ined
(
a1
a3 as ...
a2i-1)
. _ d
ao
a2
a4 ...
a2;-
2
b . . _ 1
g,
-e
emg z - , ... ,
n
... ...
...
...
.
..
0 0 0 0
a;
(7)
The
elemen s
ak
o
g;
a e he coe icien s o
Pn
o
k
>
n,
and
0 o
k
::S
n.
Bo h
ak
and
g;
a e symbolic exp essions dependen on
x,
p.
We can apply wo s abili y c i e ia. Fi s is he
Rou h-Hou wi z c i e ion.
Fo his c i e ion he p edica e
S able_Pol
is
de ined as
S able_Pol(Pn(, ))
=:
{
g1
>
0,
... ,gn > 0
(8)
Second
is
he
Linea d-Chipa d c i e ion.
I
de ines he p edica e
S able_Pol
as
S able_Pol(Pn(, ))
:=
{ a
1 >
O,
....
,an>
O,
gn-1 > O,gn-3 >
0,
...
The e o e he cons ain s
ha
de ine he s able equilib ium egions a e
S able(x )
= {
Equilib ium(x,p),
A=
Dx (x,p),
,p
-
Pn
=
Pc(A),
S able_Pol(Pn(, ))
(9)
(10)
324
whe e
Dx
s ands
o he Jacobian, and Pc
s ands
o
he
se o cons ain o cha-
ac e is ic polynomial.
I
cons ain s (10) a e sa is ied by an equilib ium egion,
i
is
s able. O he wise i is
no
s able.
The
s udy
o he bi u ca ions poin s o a sys em in ends
o
di ide he
pa am-
e e s space in egions.
The
sys em has he same numbe
and
ype o
a ac o s
in hese egions.
The
on ie s o hese egions a e o med by bi u ca ion poin s.
An
a
ac o
appea s, disappea s o changes o ype, when
we
c oss a de e mined
on ie .
The
mos elemen al classi ica ion o bi u ca ion poin s dis inguishes
hem
in o s a ics and dynamics. The s a ics bi u ca ion poin s a e
he
simples . They
appea in hose poin s whe e he numbe o
a ac o s
poin s a ies.
The
de-
e minan
o
he
Jacobian
ma ix
is
annuled in hem,
ha
is, he cha ac e is ic
polynomial has a null oo .
The
dynamic
bi u ca ion poin s in ol e
limi
cycles o s ange a ac o s. We
s udy he
Hop
bi u ca ion, whe e an
a ac o
poin
is
con e ed in o a limi
cycle o ice e sa. In hese bi u ca ion poin s he cha ac e is ic polynomial o
he Jacobian
ma ix
has a pai o oo s wi h eal
pa
equal
o
ze o.
{
E.quilib imn(x,p), {
l~quilib ium(x,p),
A=
Dx (x,p),
A=
Dx (x,p),
S a_Bi (x,p)
:=
Pn
= Pc(A),
Din_Bi (x,p)
:=
Pn
= Pc(A),
(11)
Pn
=
,
Qn-1,
Pn
=
(,
2 + w2) Qn-2,
S able_Pol(Qn-1) S able_Pol(Qn-2)
I
is in e es ing
o
no ice
ha
all he p edica es de ined be ween
(5)
and (11)
a e o muled as in e al cons ain sa is ac ion p oblems. They a e sol ed by
adequa e consis ency echniques
[5].
4 A
biome allu gical
sys em
4.1
Desc ip ion
and
de e mina ion
o
he
model
Fo a long ime, i has been obse ed
na u al
ans o ma ions o he sulphu
and i on compounds. They a e o igina ed om
he
dissolu ion o mine als. P es-
ence o i on-oxiding bac e ia in mining a eas and hei acid d ainages has been
epo ed epea edly.
Thiobacillus Fe ooxidans
is
conside ed
o
be
he
mos
impo an
o ganism
o
he
bac e ial leaching o mine als. In indi ec leaching
he
bac e ia gene a e
e ic i on by oxidizing soluble e ous i on.
The
global eac ion
is
(11)
This
me hod
o p oduc ion o acidi ied e ic solu ions is used because e ic i on
in
u n
oxidizes o he me als in mine al, ans o ming
hem
in
he
soluble o m,
and because
i
a oids ecological con amina ion p oblem o indus ial ex ac ion
o me als om he ocks.
325
I
he equa ion o Michaelis-Men ion
is
applied
o
he
eac ion ( 11), hen
oxida ion
a e
V
is
calcula ed as ollows
[5]
V =
Vmax
km
+ [5]
(12)
whe e
Vmax
is
maximum
a e
ha
i can be eached by inc easing in
he
subs a e
concen a ion,
[5]
is
subs a e concen a ion, and
km
is
Michaelis cons an . This
cons an
s ands
o he concen a ion which he eac ion
a e
is
hal
o
he
max-
imum
a e. This equa ion has wo p oblems: he concen a ion bac e ian
is
no
cons an and i canno be applied
o
he bac e ian g ow h because i
is
exponen-
cial. Due
o
he
complexi y o he ac o s
ha
ake
pa
in he bac e ia oxida ion
o Fe(II) in
Ro a ing
Biological Con ac o s ( RBC), as shown in igu e (1).
I
has
no been possible
o
de e mine a gene al
ma hema ical
model o his p ocess.
Howe e , i has been p o ed
ha
he bioxida ion eac ion con inues a kine ic o
i s o de wi h espec o he subs a e concen a ion. In he expe imen a ion
he e a e wo in e connec ed RBC. In
hem
i
is
in oduced a
low
Q wi h an
e ous i on concen a ion.
Disk
Di isio
~--~---------.---.
D i e sha
Ou luen
+-C:
:J
+-
In luen
L-~--~--~--~--~
La e al
Raised
G ound
Fig.
1.
A
Ro a ing
Biological Con ac o s ( RBC)
The
equa ions o he model o his dynamical sys em a e
(13)
4.2
Expe imen al
da a
Acco ding o he expe imen al esul s
[3]
i has been de e mined he quasi-
equilib ium poin s o he sys em. They ha e been ob ained s udying di e en
in luen
lows
p1 and alues o i on concen a ion in such
lows
P2.

326
Acco ding o he
da a
supplied
by
he expe s
P5
is simila
o
p9 and hei
o de o absolu e magni ude
is
mode a ely posi i e, P7 is e y posi i e, and p3
is
sligh ly g ea e
ha
P7. The e o e, i
i
is
associa ed
he
co esponding in e als
o
he p e iously exp essed quali a i e ope a o s,
i
is ob ained
P1
= 0.61ljh,p2 = 3.96gjl,
P3
= [5.6,5.8),p4 = 0.741,
P5
= [0.4,0.5),
P6
= 0.015,
P7
= [5.3, 5.6), Ps = 0.781,
pg
= [0.4, 0.5), Plo = 0.01
Using hese
da a
and applying he exposed echniques,
we
ca y
ou
he semi-
quali a i e analysis o hese dynamical sys ems.
4.3
Semiquali a i e
analysis
The
semiquali a i e analysis o his sys em
is
ca ied ou
o
s udy
how
o
inc ease
he capaci y o p oduc ion, when sys ems pa ame e s a e a ied.
The
equilib ium
egions o he sys em a e de e mined sol ing he ne wo k o con ain s
!
(P1P2-
P3P4x,"'+ps- P1x1)
P6
= 0,
(Pl
X1-
P7
Ps
x2
".[:P•
-
P1X2)
PlD
= 0,
Equilib ium(x,p)
::::=
0.4:::; p5
:::;
0.5, 0.4:::; p9
:::;
0.5, 5.6
:=:;
P3
:=:;
5.8,
5.3
:S
P7
:S
5.6,
P1
= 0.61,
P2
= 3.96,
P4
= 0.74,
P6
= 0.015, Ps = 0.78,
P10
= 0.01
I
i is applied in e al a i hme ic he esul s ob ained a e oo wide. Ne e heless
i
is
applied in e al consis ency echniques de eloped in
[5]
and
we
will ob ain
a na owing equilib ium egion
Equilib ium(x,p)
= {[0.397,0.49], x [0.0198,0.034]}
This solu ion includes all expe imen al esul s ob ained om di e en expe ience
da a.
The
Jacobian
ma ix
o his model
is
The
cha ac e is ic polynomial o A
is
Pn(.. )
=
ao..
2 +
a1..
+
a2
=
..
2 +
(-au-
a22).. +
aua22-
a12a21
and acco ding
o
he Liena d-Chipa d c i e ion,
S able_Pol(Pn(.. ))
::::=
{(-au-
a22) >
0,
aua22-
a12a21 > 0
Subs i u ing
Pi
o hei alues and simpli ying,
he
cons ain s
ha
de ine he
s abili y a e
327
These cons ain s a e sa is ied wi h he ob ained equilib ium egion
and
he e o e
i
is
conclude
ha
he
egion
is
s able.
The
cons ain s
ha
de ine he bi u ca ions a e
S B
. (
) = {Equilib ium(x,p), D.
B" (
) = {Equilib ium(x,p),
a_
z x, p _ 0 0
zn_
z x, p _ 0 0
a1
> ,
a2
=
a1
= ,
a2
>
When
i
is
applied cons ain sa is ac ion echniques
o
hese cons ain s, he e
a e no solu ions,
and
hence
he
sys em has no bi u ca ions.
5 Conclusions
This pape p oposes a
me hod
o
ca y
ou
au oma ically he semiquali a i e
analysis o dynamical sys ems by in e al consis ency echniques. Quali a i e
knowledge
is
ep esen ed by in e als, and hey a e quali a i e ope a o s and
en elope unc ions.
I
has been applied he p oposed app oach
o
sys ems appea ed in he bibli-
og aphy
and
he ob ained esul s a e qui e simila
o
hem.
In his pape ,
i
has
been s udied a eal biome allu gic sys em.
The
achie ed esul s ha e allowed
o
know how
o
inc ease he capaci y o p oduc ion.
In
he
u u e,
we
a e going
o
apply he p e ious echniques
o
o he
eal p ob-
lems.
We
also wan
o
ex end he analysis p ocess wi h he
s udy
o
o he
ypes
o
a ac o s,
dynamic bi u ca ions,
and
he inco po a ion o mul iple scales o
ime,
and
delays.
Re e ences
1. Abelson H., The bi u ca ion
in e p e e :
a s ep owa ds he
au oma ic
analysis
in
dynamical
sys ems.
In . J.
Compu e s
Ma h.
Applic. Vol 20,
n9,8
13-35, 1990.
2.
D o ak
D.Moni o ing
and
diagnosis
o
con inuous
dynamic
sys ems
using
semiquan-
i a i e
simula ion
Ph.D.
Disse a ion, Uni e si y o Texas. Tech.
Repo
AI92-170,
1992.
3.
Ga cia
M.J.,
La
Oxidaci6n
biol6gica
en
con inuo
del ion e oso
en
eac o es de
pelicula bac e ian sopo ada.
Ph.D.
Thesis. Uni e sidad de Se illa. 1992.
4.
Gasca R.M.,
O ega
J .A.,
Ta o
M.
Rep esen aci6n
y
simulaci6n
de
modelos
in e-
g ando
conocimien o
cuali a i o y
cuan i a i o.
In P oc. VII Con . Asoc. Espanola
pa a
In eligencia A i icial, 1997 ( o appea ).
5.
Gasca R.M.
Razonamien o
y
simulaci6n
en
sis emas
que
in eg an
conocimien o
cual-
i a i o
y
cuan i a i o
Ph.D.
Thesis Uni e sidad de Se illa, 1998.
6.
Kay
H., Kuipe s B.,
Nume ical
beha io en elopes
o
quali a i e models.
P oc.
11 h
Na ional
Con .
on
A i icial In elligence, 606-613, 1993.
7.
Kay H.
Re ining
Imp ecise models
and
hei
beha io s
Ph.D.
Tesis, Uni e si y o
Texas, 1996.
8.
Kuipe s
B.J.,
Quali a i e
simula ion,
A i ical In eligence 29,289-338, 1986.
9.
Sacks E.,
Au oma ic
quali a i e analysis
o
dynamic
sys ems
using
piecewise linea
app oxima ions,
A i icial In elligence 41, 313-364, 1990.
328
10. Sacks
E.,
Au oma ic
analysis
o
one-pa ame e
plana
o dina y
di e en ial equa-
ions
by
in elligen
nume ic
simula ion
A i icial In elligence ng48, 27-56, 1991.
11.
Vesco i M.,
Fa quha
A., Iwasaki
Y.,
Nume ical
in e al
simula ion:
combined qual-
i a i e
and
quan i a i e
simula ion
o bound beha io s
o
non-mono onic
sys ems
P oc.
14 h
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