Sin t

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A new methodology for analysis of
semiqualitative dynamic models with constraints
Juan A. Ortega, Jesus Torres, Rafael M. Gasca,
Departamento de Lenguajes y Sistemas
Informáticos University of Seville (Spain)
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Objectives

• Model that evolves in the time
• Qualitative and quantitative knowledge
• Constraints

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Objectives

• Two interconnected tank system

• Evolve in the time

t0
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Objectives

• Two interconnected tanks system

• Qualitative and quantitative knowledge

- p is a moderadately positive influent - x1,x2
contain a slightly positive quantity of liquid
at the initial time
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Objectives

• Two interconnected tank system

• Constraints

- Height of the tanks is moderately positive
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Objectives

• Two interconnected tank system

• Evolve in the time
• Qualitative and quantita-
• tive knowledge
• Constraints

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Objectives

• Two interconnected tanks system
• Study its temporal evolution

• If always the system reaches a stable equilibrium
• If it is reached an equilibrium where x1 lt x2
• If sometime the height of a tank is overflowed
• If sometime x1 lt x2

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Objectives

• Two interconnected tanks system
• Obtain its behaviour patterns

• Depending on the influent p
• a tank is overflowed
• a tank is no overflowed and always x1gtx2
• a tank is no overflowed and sometime x1ltx2

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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
Semiqualitative Model
Modelling
S
Dynamic System
Transformation techniques Stochastic techniques
Quantitative simulation
T
System Behaviour
Trajectory Database
Classification
Learning
Queries
Answers
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Semiqualitative methodology

• 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

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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

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Qualitative knowledgeQualitative operators

• 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 , ? , ? , , ??, lt, ?, gt, ??,

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Qualitative knowledgeEnvelope functions

• A envelope function represents the family of
  functions included between a upper function g and
  a lower one g into a domain I.

yg(x), ltg(x), g(x), Igt ?x ?I g(x) ? g(x)
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Qualitative knowledgeQualitative continuous
functions

• A qualitative continuous function represents a
  constraint in-volving the values of y and x
  according to the properties of h

yh(x) h ? P1, s1, P2, ..., sk-1, Pk with Pi
( di, ei ), si ? , -, 0
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Transformation techniques

• Semiqualitative model S
• Family of quantitative models F

Transformation rules
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Generation of trajectories database

• Database generation T
• T
• for i1 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
• vChoose Value (Domain (p))
• substitute p by v in M
• for every function h ? F
• HChoose H (h)
• substitute h by H in M

r1
T
rn

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Query/classification language

• Abstract
• Syntax

Queries
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Query/classification language

• Abstract
• Syntax

Classification
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Query/classification language
If always the system reaches a stable equilibrium
? r?T? EQ If it is reached an equilibrium where
x1 lt x2 ? r?T? EQ ? (always (t tF ?
x1ltx2)) If sometime x1 lt x2 ? r?T? sometime
x1lt x2
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Application to a logistic growth model with a
delay

• 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

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Application to a logistic growth model with a
delay

• Let S be a semiqualitative model of these systems
  where a delay has been added. Its differential
  equations are

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Application to a logistic growth model with a
delay

• 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

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Application to a logistic growth model with a
delay

• Queries

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Application to a logistic growth model with a
delay
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Application to a logistic growth model with a
delay

• X/t

Recovered equilibrium
Extinction
Retarded catastrophe
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Conclusions and further work

• 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