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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) 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 , pI , 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 rT EQ true If it is reached an equilibrium where x1 < x2 rT EQ (always (t ~ tF x1<x2)) false If sometime x1 < x2 rT 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 rT EQ True, therefore there are no limit cycles 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) 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 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