Including Quali a i e Knowledge in
Semiquali a i e Dynamical Sys ems
J.
A.
O ega, R.
M.
Gasca,
M.
Ta o
Uni e sidad de Se illa
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:{jao ega,gasca,m o o }@lsi.us.es
Abs ac .
A new
me hod
o
inco po a e quali a i e knowledge in semi-
quali a i e sys ems
is
p esen ed. In hese sys ems quali a i e knowledge
may be exp essed in hei
pa ame e s,
ini ial condi ions
and/o
ec o
ields.
The
ep esen a ion o quali a i e knowledge
is
made
by means o
in e als, con inuous quali a i e unc ions
and
en elope unc ions.
A dynamical sys em is de ined by di e en ial equa ions
wi h
quali a i e
knowledge.
This
de ini ion
is
ans o med
in o a amily o dynamical sys-
ems. In his
pape
he
semiquali a i e analysis is ca ied
ou
by means
o
cons ain
sa is ac ion p oblems, using in e al consis ency echniques.
1
In oduc ion
In enginee ing
and
science, knowledge
abou
dynamical sys ems
may
be ep e-
sen ed in se e al ways.
The
models cons uc ed o
s udying
hem
a e no mally
composed
o
quali a i e
as well as
quan i a i e
knowledge. Models which only in-
co po a e
quan i a i e
knowledge
(quan i a i e
models) ha e been well s udied.
The
echniques de eloped o analyse
and
simula e a e well known, oo.
On
he
o he
hand,
a g ea a ie y o echniques ha e been
s udied
o ep e-
sen a ion
and
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.
The
knowledge composed o
quan i a i e
and quali a i e knowledge
is
known
as semiquali a i e. Real models con ain
quan i a i e,
quali a i e
and
semiquali-
a i e
knowledge,
and
all
o
hem
need
o
be conside ed when
hey
a e s udied.
The e o e, some imes i
is
necessa y 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 pu ely
quali a-
i e desc ip ions
[7],
semiquali a i e
[1],
[5],
nume ical based on in e als
[14],
quan i a i e
and mixed
o
all le els
[6).
In
he
six ies,
he
me hodology o sys em dynamics was p oposed.
I
inco -
po a ed
quali a i e
knowledge
o
models by means
o
a iables
and
quan i a i e
unc ions
sui ably
chosen.
Bu ,
i
is
no
un il
he
eig hies when
he
in e es o
s udying
quali a i e
knowledge independen ly o i s
quan i a i e
ep esen a ion
330
eme ges. This in e es appea s a ound quali a i e simula ion
[7],
and quali a-
i e analysis
[11].
Ma hema ical concep s o quan i a i e analysis o dynamical
sys ems ha e been applied in o quali a i e simula ion and analysis (see
[8]).
Quali a i e me hods o s udying dynamical sys ems began
a
he end o he
las cen u y, by he F ench ma hema ician Hen i Poinca e.
The
subsequen e o-
lu ion o hese wo ks has o igina ed he quali a i e heo y
o
dynamical sys ems.
In
[10]
he
echniques o ca y ou he analysis o he quali a i e models we e
in oduced,
ha
is, he s udy o equilib ium egions, s abili y and bi u ca ion
poin s.
In his pape , he quali a i e knowledge
is
ep esen ed by means o eal in-
e als, con inuous quali a i e unc ions and en elope unc ions.
The
in e als
include all he eal alues whe e he quali a i e label o such magni ude
is
ound.
A con inuous quali a i e unc ion s ands o a amily o unc ions de ined by
means o landma ks. An en elope unc ion s ands o a amily o unc ions in-
cluded be ween a eal supe io unc ion and an in e io one.
I
is
also p esen ed A me hod
o
ans o m semiquali a i e models in o a
cons ain ne wo k
is
p esen ed, as well.
The
in e al cons ain sa is ac ion p o-
blems a e sol ed applying consis ency echniques
[3].
2
Semiquali a i e
models
A semiquali a i e model
is
ep esen ed by
«P(:i:,
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,
:i:
he
a ia ion o
he
s a e
a iables wi h he ime,
«Po
he cons ain s in
he
ini ial condi ions, and
«P
he cons ain s depending on
:i:,
x
and
p.
They a e exp essed by means o he
ope a o s de ined in he nex sec ion. They a e composed o a iables, cons an s,
a i hme ic ope a o s, unc ions, quali a i e unc ions
and/o
en elope unc ions.
The e o e he equa ions
(1)
s and o a amily o dynamical sys ems depending
on
p
and
xo.
The
in eg a ion o quali a i e knowledge
is
made by adding cons ain s o
he ne wo k. They a e cons ain s combined wi h 'and' and 'o ' ope a o s. This
ep esen a ion will help us
o
ob ain he beha io o he sys em i
we
apply an
app op ia e algo i hm o sol e he esul ing cons ain ne wo k (see
[3]).
3
Rep esen a ion
o
quali a i e
knowledge
We
shall ocus ou a en ion in dynamical sys ems whe e he e may be quali a-
i e knowledge in hei pa ame e s, ini ial condi ions
and/o
ec o ield. They
cons i u e he semiquali a i e di e en ial equa ions o he sys em.
Fi s ,
we
need o ake in o accoun
ha
he
ep esen a ion o he quali a i e
knowledge is ca ied ou
by
means o ope a o s. They ha e associa ed eal in e -
als. This ep esen a ion p o ides he ollowing ad an ages: easy in eg a ion o
331
quali a i e and quan i a i e knowledge
[2];
and
i
makes possible he de ini ion
o he ange o quali a i e a iables and pa ame e s o he sys em. This de ini-
ion
is
p o ided by expe s, and i allows o echniques de eloped on in e als
analysis and cons ain sa is ac ion p oblem
o
be used
[4],
[13]
and
[3].
3.1
Quali a i e
pa ame e s
and
ini ial
condi ions
The
quali a i e ep esen a ion o pa ame e s
and/o
ini ial condi ions o dyna-
mical sys ems
may
be ca ied ou by means o he quali a i e ope a o s U and
B.
They
ep esen , espec i ely, he se o una y and bina y quali a i e ope a o s.
Fo example, U
=
{ e y
nega i e, mode a ely nega i e, sligh ly nega i e, sligh ly
posi i e, mode a ely posi i e, e y posi i e } and
B
=
{much
less han, mode a-
ely less han, sligh ly less han,
much
g ea e han, ... }. Each ope a o op E U o
op E B has associa ed a eal in e al
op.
This eal in e al deno es a quan i y.
I
s ands o hose alues whe e he magni ude has he quali a i e label.
The
bina y
quali a i e ope a o s a e classi ied in wo classes acco ding
o
hei
ypes:
• Ope a o s ela ed
o
he di e ence. They can be exac ly equal o
=,
smalle
o
equal o
~,
and la ge
o
equal o
2::.
• Ope a o s ela ed
o
he quo ien . They can be
much
less han
«,
mo-
de a ely less han -
<,
sligh ly less han
~<,
app oxima ely equal o
:::i,
sligh ly
g ea e han
>~,
mode a ely g ea e han
>
-,
and
much
g ea e han
»
3.2
En elope
unc ions
These unc ions es ablish a possible ange o alues o i s image o each gi en
alue. They ep esen he amily o unc ions included be ween wo de ined unc-
ion, a supe io one
g :
IR
--+
IR
and ano he in e io one
g :
IR
--+
JR.
Le be
y =
g(x)
an en elope unc ion (see igu e
l.a).
I
is ep esen ed by
means o
(£
(X)
,
g(
X)
,
J) ,
' :/x
E
I:
£(x)
~
g(x)
(2)
whe e I
is
he
de ini ion domain in he eal line o g, and x is a a iable.
3.3
Quali a i e
con inuous
unc ions
Le be y
=
h(x)
a quali a i e con inuous unc ion.
I
ep esen s a unc ional
ela ionship wi h
x
as independen a iable, and y as dependen a iable.
y
=
h(x),
h
={PI,
s1,
P2,
...
sk-I,
Pk} wi h
Pi=
(di, ei)
(3)
whe e each
Pi
is a poin .
I
s ands o an
impo an
quali a i e landma k o h.
Each
Pi
is
ep esen ed by means o a pai (
di, ei) whe e
di
is
he associa ed qua-
li a i e landma k
o
he
a iable x and
ei
o
y.
Poin s a e sepa a ed by
he
sign
Si
o he de i a i e in he in e al be ween a poin and he ollowing.
The
sign
Si
is
+
i he unc ion
is
s ic ly mono onic inc easing in
ha
in e al, -i i
is
s ic ly mono onic dec easing, and 0 i i
is
cons an .
The
de ini ion o a unc ion
332
g y
a)
b)
X X
- -
Fig.
1. Quali a i e unc ions
is
always comple ed wi h he landma ks which deno e
he
cu
poin s wi h he
axes, and
he
poin s whe e
he
sign om he de i a i e changes (a
maximum
o a minimum o h). Quali a i e in e p e a ion o h (see igu e
l.b)
o each
P;
is:
{
x
=
d;::::}
y
=
e;
y-
h(x)
=
0
= {
s;
:
+::::}
e;
<
y
<
ei+l
di
<
x
<
di+l
::::}
si
= -
::::}
e;_!
y
>
ei+l
si-
0
=>
y-
e,
A special case o con inuous unc ion
is
ha
whe e
he
sign o all in e als is he
same,
ha
is, s
1 =
...
=
Bk-l
=
s.
I
is
a s ic ly mono ononic unc ion. This
unc ion can be exp essed in a sho way by h
=
M
s { P
1,
P
2,
.•.
,
Pk}
4
F om
quan i a i e
and
quali a i e
knowledge
o
cons ain
ne wo ks
The
quali a i e knowledge
is
added by means
o
a se o cons ain s. They a e
combined wi h 'and' and 'o ' ope a o s.
The
cons ain s ob ained om quali a-
i e knowledge a e:
• Quali a i e pa ame e s and ini ial condi ions
Fo each ope a o ,
i
is ob ained a cons ain acco ding
o
i s ype. Le
be a new a iable gene a ed.
Iu,
h a e he in e als associa ed
o
he
una y
and
bina y ope a o s. In e als
Iu
a e s ablished in acco dance wi h
[12],
and
in e als h wi h
[9].
*
Le u be an una y ope a o u E
U.
Le e be an a i hme ic exp ession.
The
esul ing cons ain s a e
u(e):={e- =O,
Eiu
*
Le b be a bina y ope a o b
E
B.
Le
op
he
bina y ope a o , and le
e1,
e2
be
wo a i hme ic exp essions.
I
b is an ope a o ela ed
o
he di e ence (
=,
~,
2:)
hen
he
esul ing cons ain s a e
333
and i b
is
an ope a o ela ed
o
he quo ien ,
• En elope unc ions
Fo each en elope unc ion y = g (
x)
he
cons ain (
4)
is ob ained
g(x) =
O: !_(x)
+
(1-
o:)g(x)
o:E[0,1]
(4)
This cons ain s ands o a amily o unc ions included be ween g
and
g.
I
is
in e es ing
o
no ice
ha
i
o:
= 0
:::::?
g(x)
= g(x) and i
o:
= 1
=?-g(x)
= g(x)
and any o he alue o
o:
in
[0,1]
s ands o any included alue be ween
g(x)
-and
~x).
-
• Quali a i e unc ions
Fo each quali a i e unc ion y = h(x) de ined as
(3)
is ca ied ou he ollo-
wmg:
* Fo each
landma k
d;
o
e;
appea ed in he de ini ion o h, i is added
o
he
se o a iables o he model a new a iable wi h a domain (
-oo,
+oo
).
*
The
ollowing linea cons ain s due o
he
de ini ion o h a e added
o
he
cons ain ne wo k
{
d1
<
d2
<
...
< dk,
e1
o
e2
o ... o
ek-1
o ek,
y _ h(x) = O =
((x
= d1, y = e1); ... ;
(x
= dk, y = ek);
(d1
< x < d2,e1
01
yo1
e2); ... ;
(dk-1 <X< dk,
e1
Ok-1 y Ok-1 ek))
(5)
whe e
(5)
is a se o linea cons ain s combined wi h and and o ope a o s,
espec i ely deno ed by comma
(,
) and semicolon
(;
) .
The
ope a o o
is
de ined
as
{
x > y i
s;
-1
=
s;
= +
X
0;
y = X < y i
Si
-1
=
Si
= -
X=
y
i
Si-1
=
Si
=
0;
Si-1
=F
Si
The
se o cons ain s ob ained by he inclusion o quali a i e knowledge o
he
model
is
o muled as an in e al cons ain sa is ac ion p oblem.
As
we
ha e
indica ed, a cons ain -based easoning me hod by means o in e al consis ency
echniques
is
applied (see
[3]).
The
esul s ob ained a e a se
o eal
in e als by
he
a iables and cons ain s o de among hem.
5
An
example
Le be wo in e connec ed anks (see igu e 2).
The
semiquali a i e model
is
Va iables: V =
{x1,x2, 1, 2,s,p}
Landma ks:
L = {a,
b}
Func ions: h1
:=
M+
{(0, 0),
a,
b}
91
=< 2x, x,
[0,
oo]
>
Cons ain s:
1
=
h1(s),
2
= 91(x2), s =
x1-
x2
~
-
~
-" "
d"
( )
d
-
P-
1,
d
-
1
- 2,
pos1
1 e
me
mm
p

334
Fig.
2. The in e connec ed anks sys em
The cons ain ne wo k ob ained applying he p oposed echniques is
V =
{xl,x2, l, 2,s,p,a,b}
0
<a,
0 <
b,
( (s <
0,
1 < 0); (s =
0,
1
= 0);
(0
< s
<a,
0 <
1
<b);
(s=a, 1=b);
(s>a, l>b)),
2
= a(x2) +
(1-
a)2x2, a E
[0,
1],
u-
~-
E
(3
5]
s =
x1-
x2,
d
-
p-
1, d -
1-
2, p ,
(6)
The concep s in oduced in
(10]
ha e been applied in o de o ca y ou he
analysis o his model. The cons ain ne wo k
ha
s ands o he equilib ium
egions o he model a e eplacing he exp essions
dxl/
d ,
dx2/
d
by ze o in (6).
This cons ain sa is ac ion p oblem has an unique solu ion, hence he e is an
equilib ium egion whe e
i
is
sa is ied
ha
x1
> x2
(7)
being
lx
2 a eal in e al.
I
i
is
applied in e al a i hme ic o ( 6), he in e als
ob ained o x1
,x
2 a e oo wide. The equilib ium egion
is
[O.,oo]
X [O.,oo]. In
o de o na ow his solu ion
i
is
applied he consis ency echniques
[3],
and
hen
i
is
ob ained o x2 he in e al (1.06487, 5.0]. The in e als o
x1,
s a e
posi i es, and using he cons ain s = x1 -x2, hen esul s
(7)
a e concluded.
The e o e he solu ion o
lx
2 is closed o he eal solu ion [1.5, 5.0].
The esul s
(7)
may be in e p e ed
as:
he sys em has an unique equilib ium
whe e he heigh o he i s
ank
is highe
ha
he second one.
The
heigh o
second
ank
is in he eal in e al [1.5,
5.].
In a simila
way,
i
is
ob ained he ne wo k
ha
s ands o he s abili y o such
egion. This ne wo k is also sa is ied, hence
i
is an s able equilib ium egion. The
cons ain ne wo k
ha
de ine he bi u ca ions poin s a e no sa is ied. The e o e
i
is
concluded
ha
he e a e no bi u ca ions.
6
Conclusions
This pape p o ides a me hod o including quali a i e knowledge in semiqua-
li a i e dynamical sys ems. Quali a i e knowledge is ep esen ed by means o
in e als, con inuous quali a i e unc ions and en elope unc ions. This know-
ledge helps us o make analysis o
ha
kind o sys ems.
335
We
ha e applied he me hod p oposed o se e al examples. The ob ained
esul s ha e been sa is ac o y. The echnique p esen ed
is
app op ia e o p e-
dic i e p oblems in indus ial p ocesses whe e he e
is
quali a i e in o ma ion o
hei componen s. A he momen ,
we
a e applying he me hod o
s udy
a eal
biome all u gic sys em.
Re e ences
1.
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)
2.
Gasca
R.M.,
Ta o
M.,
O ega
J.A., P opagaci6n
de
es icciones in eg ando
conocimien o
cuali a i o y
cuan i a i o
Bole {n de
AEPIA
num
6 (1996) 23-30
3.
Gasca R.M.
Razonamien o
y
Simulaci6n
en
Sis emas
que in eg an
conocimien o
cuali a i o y
cuan i a i o
Ph.D.
Thesis, Uni e sidad de Se illa (1998)
4.
Hy iinen E.
Cons ain
easoning based on
in e al
a i hme ic:
he ole ance p opa-
ga ion A i icial In elligence no.
58
(1992) 71-112
5.
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 (1993) 606-613
6.
Kay H.
Re ining
imp ecise models
and
hei
beha io s
Ph.D.
Tesis, Uni e si y o
Texas (1996)
7.
Kuipe s
B.J.,
Quali a i e
simula ion,
A i ical In eligence 29, (1986) 289-338
8.
Lee
W.W.,
Kuipe s
B.J.,
A quali a i e
me hod
o
cons uc
phase
po ai s, P oc.
11 h
Na ional Con . on A i icial In elligence (1993) 614-619
9.
Ma o ounio is M.L., S ephanopoulos G. Fo mal
o de -o -magni ude
easoning
P o-
cess Enginee ing
Compu e
Chemical Enginee ing no.
12
(1988) 867-880
10.
O ega
J.A.,
Gasca R.M., Ta o M.,
Au oma izaci6n
del analisis y
simulaci6n
de
modelos cuali a i os
median e
es icciones P oc. VII Con . Asoc.
Espanola
pa a
In eligencia A i icial, (1997) 85-94
11. 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
(1990) 313-364
12. T a e-Massuyes
1.,
and
Pie a
N., The o de
o
magni ude
models as quali a i e
algeb as P oc. o
11 h
In . Join Con .
on
A i icial In elligence,
De oi
MI (1989)
1261-1266
13. VanHen en yck P., McAlles e D.,
Kapu
D.,
Sol ing
polynomial
sys ems
using
a
b anch
and
p une
algo i hm. Tech.
epo
CS-95-01 Uni e si y o P o idence (1995)
14. Vesco i M.,
Fa quha
A., Iwasaki Y.,
Nume ical
in e al
simula ion:
combined
qua-
li 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
In . Join Con .
on
AI (1994) 1806-1812