juan a. ortega, jesus torres, rafael m. gasca, departamento de lenguajes y sistemas informáticos...
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Juan A. Ortega, Jesus Torres, Rafael M. Gasca, Departamento de Lenguajes y Sistemas Informáticos
University of Seville (Spain)
A new methodology for analysis of semiqualitative dynamic models
with constraints
A new methodology for analysis of semiqualitative dynamic models
with constraints
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Model that evolves in the time Qualitative and quantitative knowledge Constraints
+
Objectives
Semiqualitative model with constraints
Semiqualitative model with constraints
Study its
temporal evolution Obtain its
behaviour patterns
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Two interconnected tank system
Objectives
p
r1 r2
• Evolve in the time
t0
x2
x1
t1 t2 t3 • • • tf
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Two interconnected tanks system
Objectives
• Qualitative and quantitative knowledge - p is a moderadately positive influent
- x1,x2 contain a slightly positive quantity
of liquid at the initial time
p
r1 r2
x2
x1
0.4 x2
0.6 x2
g1r1 = g1 ( x1 – x2 )
h1
5
8
y0
x0 +0
0
r2 = h1 ( x2 )
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Two interconnected tank system
Objectives
p
r1 r2
x2
x1
• Constraints
- Height of the tanks is moderately positive
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Two interconnected tank system
Objectives
p
r1 r2
x2
x1
• Evolve in the time• Qualitative and quantita- tive knowledge• Constraints
Semiqualitative model withconstraints
Semiqualitative model withconstraints
Study its
temporal evolution
Obtain its
behavior patterns
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Two interconnected tanks system
– Study its temporal evolution
Objectives
p
r1 r2
x2
x1
• If always the system reaches a stable equilibrium
• If it is reached an equilibrium where x1 < x2
• If sometime the height of a tank is overflowed
• If sometime x1 < x2
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Two interconnected tanks system
– Obtain its behaviour patterns
Objectives
p
r1 r2
x2
x1
• Depending on the influent p:– “a tank is overflowed”
– “a tank is no overflowed and always x1>x2“
– “a tank is no overflowed and sometime x1<x2”
if p > 0.4 then a tank is overflowed
if p > 0.1 & p < 0.4 then
a tank is no overflowed & always x1>x2 if p < 0.1 then
a tank is no overflowed & sometime x1<x2
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Outline
Semiqualitative methodology Semiqualitative models Qualitative knowledge Generation of trajectories database Query/classification language Theoretical study of the conclusions Application to a logistic growth model with a
delay Conclusions and further work
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Semiqualitative methodology
DynamicSystem
LabelledDatabase
Classification QueriesLearning
Transformation techniquesStochastic techniques
Quantitative Models M
F
SemiqualitativeModel S
Trajectory Database
Quantitative simulation
T
Modelling
Answers
System Behaviour
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A formalism to incorporate qualitative knowledge– qualitative operators and labels
– envelope functions
– qualitative continuous functions This methodology allows us to study all the states of a
dynamic system: stationary and transient states. Main idea: “A semiqualitative model is transformed into
a family of quantitative models. Every quantitative model has a different quantitative behaviour, however, they may have similar quantitative behaviours”
Semiqualitative methodology
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Semiqualitative models
(x,x,y,q,t), x(t0) = x0 , 0 (q,x0 )
variables, parameters, ... numbers and intervals arithmetic operators and functions qualitative knowledge
qualitative operators and labels envelope functions qualitative continuous functions
• x: state variables x: derivative of x q: parameters y: auxiliary variables : constraints
dxdt
•
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Qualitative operators– Every operator is defined by means of a real interval Iop.
– This interval is given by the experts– Unary qualitative operators U(e)
• Every qualitative variable has its own unary operators defined
Ux = {VNx , MNx , LNx , A0x , LPx , MPx , VPx }– Binary qualitative operators B(e1,e2)
• They are applied between two qualitative magnitudes
B = {=, , , «, , ~<, , ~>, , »}
Qualitative knowledge
Qualitative operators
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A envelope function represents the family of functions included between a upper function g and a lower one g into a domain I.
Qualitative knowledge
Envelope functions
x
I
y gg
y=g(x), <g(x), g(x), I> x I • g(x) g(x)
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Qualitative knowledge
Qualitative continuous functions A qualitative continuous function represents a constraint in-
volving the values of y and x according to the properties of h
y=h(x) h {P1, s1, P2, ..., sk-1, Pk} with Pi =( di, ei ), si { +, -, 0 }
h {(–, +),–,(x0,0), –,(x1,y0),+,(x2,0),+,(0,y1),+,(x3,y2), –,(x4,0),–,(+,–)}
x0 x1 x2 x3 x4
y2
y1
y0
– +
h
0
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Semiqualitative model S
Family of quantitative models F
Transformation techniques
(x,x,y,q,t), x(t ) = x , (q,x )0000
Transformationrules
x=f(x,y,p,t), x(t0) = x0, pIp, x0I0•
•
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Database generation TT:={ }for i=1 to N
M := Choose Model (F) r := Quantitative Simulation (M) T := T r
Choose Model (F)for every interval parameter and qualitative variable p F
v:=Choose Value (Domain (p)) substitute p by v in M
for every function h F H:=Choose H (h) substitute h by H in M
Generation of trajectories database
r1
rn
T•••
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Abstract Syntax
Query/classification language
QueriesQueries
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Abstract Syntax
Query/classification language
ClassificationClassification
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If always the system reaches a stable equilibrium rT EQ
If it is reached an equilibrium where x1 < x2
rT EQ (always (t ~ tF x1<x2))
If sometime x1 < x2
rT sometime x1< x2
If always the system reaches a stable equilibrium rT EQ
If it is reached an equilibrium where x1 < x2
rT EQ (always (t ~ tF x1<x2))
If sometime x1 < x2
rT sometime x1< x2
p
r1 r2
x2
x1
true
false
true
Query/classification language
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It is very common to find growth processes in which an initial phase of exponential growth is followed by another phase of approaching to a saturation value asymptotically
They abound in natural, social and socio-technical systems:– evolution of bacteria,– mineral extraction– economic development– world population growth
t
Logisticgrowth
Decay andextinction
Application to a logistic growth model with a delay
t
Exponentialgrowth Asymptotic behaviour
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Let S be a semiqualitative model of these systems where a delay has been added. Its differential equations are
x = (n h1(y) – m) x,y = delay(x),x >0,h1 {(–, –),+,(x0,0),+,(0,1),+,(x1,y0), –,(1,0),–,(+,–)}
•
x0 [LPx,MPx], [MP, VP], LPx (m), LPx (n)0
y0
h11
x0x1 1
–
– +0
Application to a logistic growth model with a delay
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We would like– to know if an equilibrium is always reached
– to know if there is logistic growth equilibrium
– to know if all the trajectories reach the decay equilibrium without oscillations
– to classify the database in accordance with the behaviours of the system
Applying the proposed methodology is obtained a time-series database
Application to a logistic growth model with a delay
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Queries
If an equilibrium is always reached rT EQ
If an equilibrium is always reached rT EQ
True, therefore there are no limit cycles
If there is a logistic growth equilibrium rT EQ always (t ~ tF x 0)
If there is a logistic growth equilibrium rT EQ always (t ~ tF x 0)
True (1st behaviour pattern)
If the decay equilibrium is reached without oscillations rT EQ always (t ~ tF x 0 ) (length([ x 0],{x}) 0)
If the decay equilibrium is reached without oscillations rT EQ always (t ~ tF x 0 ) (length([ x 0],{x}) 0) •
False, there are two ways to reach this equilibrium, with and without oscillations (2nd y 3rd behaviour patterns )
Application to a logistic growth model with a delay
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All time-series were classified with a label The obtained conclusions are in accordance when a mathema-
tical reasoning is carried out
Behaviour patterns
[r, EQ length([x 0],{x})>0 always (t ~ tF x 0)] recoved equil.
[r, EQ length([x 0],{x})>0 always (t ~ tF x 0)] ret. catast.
[r, EQ length([x 0],{x}) 0 always (t ~ tF x 0)] extinction
[r, EQ length([x 0],{x})>0 always (t ~ tF x 0)] recoved equil.
[r, EQ length([x 0],{x})>0 always (t ~ tF x 0)] ret. catast.
[r, EQ length([x 0],{x}) 0 always (t ~ tF x 0)] extinction
••
•
Application to a logistic growth model with a delay
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X/t
10 20 30 40 50t
0.5
1
1.5
2
2.5
X
Recovered equilibrium
10 20 30 40 50t
0.5
1
1.5
2
2.5
X
Extinction
10 20 30 40 50t
2
4
6
X
Retarded catastrophe
Application to a logistic growth model with a delay
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A new methodology has been presented in order to automates the analysis of dynamic systems with qualitative and quantitative knowledge
The methodology applied a transformation process, stochastic techniques and quantitative simulation.
Quantitative simulations are stored into a database and a query/classification language has been defined
In the future– the language will be enrich with operators for comparing trajectories, and for comparing
regions of the same trajectory.
– Clustering algorithms will be applied in other to obtain automatically the behaviours of the systems
– Dynamic systems with explicit constraints and with multiple scales of time are also one of our future points of interest
Conclusions and further work