Transcript ortega

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)
Objectives



Model that evolves in the time
Qualitative and quantitative knowledge
Constraints
Semiqualitative model
with constraints
Study its
temporal evolution
Obtain its
behaviour patterns
+
Objectives

Two interconnected tank system
p
x1
x2
r1
r2
• Evolve in the time
t0
t1
t2
t3
• • •
tf
Objectives
Two interconnected tanks system

p
• Qualitative and quantitative knowledge
x1
- p is a moderadately positive influent
- x1,x2 contain a slightly positive quantity
of liquid at the initial time
x2
r1
r1 = g 1 ( x 1 – x 2 )
r2
g1
0.6
r2 = h 1 ( x 2 )
x2
0.4
x2
h1
8
y0
0
0
5
x0
+
Objectives

Two interconnected tank system
p
x1
x2
r1
r2
• Constraints
- Height of the tanks is moderately positive
Objectives

Two interconnected tank system
p
x1
x2
r1
• Evolve in the time
• Qualitative and quantitative knowledge
• Constraints
r2
Semiqualitative
model with
constraints
Study its
Obtain its
temporal evolution
behavior patterns
Objectives

Two interconnected tanks system
p
x1
x2
r1
r2
– Study its temporal evolution
•
•
•
•
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
Objectives

Two interconnected tanks system
p
x1
x2
r1
r2
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
– Obtain its behaviour patterns
• 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”
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
Semiqualitative methodology
Dynamic
System
Modelling
Semiqualitative
Model
S
Transformation techniques
Stochastic techniques
Quantitative
Models M
F
Quantitative simulation
System Behaviour
Learning
Trajectory
Database
Classification
Labelled
Database
T
Queries
Answers
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”
Semiqualitative models
•
(x,x,y,q,t),
x(t0) = x0 ,
0 (q,x0 )

x: state variables
dx
•

 x: derivative of x
dt
 q: parameters
 y: auxiliary variables
 : constraints

variables, parameters, ...
 numbers and intervals
 arithmetic operators and functions
 qualitative knowledge
 qualitative operators and labels
 envelope functions
 qualitative continuous functions
Qualitative knowledge
Qualitative 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 = {=,  ,  , «, , ~<, , ~>, , »}
Qualitative knowledge
Envelope functions

A envelope function represents the family of functions
included between a upper function g and a lower one g
into a domain I.
y=g(x), <g(x), g(x), I>
g
y
g
I
x
x I • g(x)  g(x)
Qualitative knowledge
Qualitative continuous functions

A qualitative continuous function represents a constraint involving 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 }
y2
h
y1
–
x0
x1
x2
0
x3
x4
+
y0
h {(–, +),–,(x0,0), –,(x1,y0),+,(x2,0),+,(0,y1),+,(x3,y2), –,(x4,0),–,(+,–)}
Transformation techniques

Semiqualitative model S
•
(x,x,y,q,t),
x(t 0) = x 0 ,  0(q,x 0 )
Transformation
rules
•x=f(x,y,p,t), x(t ) = x , pI , x I
0
0
p
0
0

Family of quantitative models F
Generation of trajectories database

Database generation T
T:={ }
for i=1 to N
M := Choose Model (F)
r := Quantitative Simulation (M)
T := T  r

r1
•
•
•
rn
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
T
Query/classification language

Abstract
Syntax
Queries
Query/classification language

Abstract
Syntax
Classification
Query/classification language
p
x1
x2
r1
r2
If always the system reaches a stable equilibrium
 rT EQ
true
If it is reached an equilibrium where x1 < x2
 rT EQ  (always (t ~ tF  x1<x2))
false
If sometime x1 < x2
 rT sometime x1< x2
true
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
Exponential
Asymptotic behaviour
growth
Logistic
growth
t
Decay and
extinction
t
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

x• = (n h1(y) – m) x,
y = delay(x),
x >0,
h1  {(–, –),+,(x0,0),+,(0,1),+,(x1,y0), –,(1,0),–,(+,–)}
y0
0 x0 [LPx,MPx], 1[MP, VP], LPhx1(m), LPx (n)
–
x0
0
–
x1
1
+
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
Application to a logistic growth model
with a delay

Queries
If an equilibrium is always reached
 rT EQ
True, therefore there are no limit cycles
If there is a logistic growth equilibrium
 rT EQ  always (t ~ tF  x 0)
True (1st behaviour pattern)
If the decay equilibrium is reached without oscillations
 rT 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
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

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All time-series were classified with a label

The obtained conclusions are in accordance when a mathematical reasoning is carried out
Application to a logistic growth model
with a delay
X/t

X
2. 5
X
2. 5
Recovered equilibrium
2
2
1. 5
Extinction
1. 5
1
1
0. 5
0. 5
t
10
20
t
10
20
30
40
50
X
6
Retarded catastrophe
4
2
t
10
20
30
40
50
30
40
50
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