A Semiqualitative Methodology for Reasoning about Dynamic Systems

A new methodology is proposed in this paper in order to study semiqualitative models of dynamic systems . It is also described a formalism to incorporate qualitative information into these models . This qualitative information may be composed of. qualitative operators, envelope functions, qualitative labels and qualitative continuous functions . This methodology allows us to study all the states of a dynamic system : the stationary and the transient states. It also allows us to obtain behaviours patterns of semiqualitative dynamic systems . The main idea of the methodology follows : a semiqualitative model is transformed into a family of quantitative models . Every quantitative model has a different quantitative behaviour, however they may have similar qualitative behaviours . A semiqualitative model is transformated into a set of quantitative models. The simulation of every quantitative model generates a trajectory in the phase space . A database is obtained with these quantitative behaviours. It is proposed a language to carry out queries about the qualitative properties of this database of trajectories . This language is also intended to classify the different qualitative behaviours of our model . This classification helps us to describe the semiqualitative behaviour of a system by means of hierarchical rules obtained by means of machine learning. The completeness property is characterized by statistical means . A theoretical study about the reliability of the obtained conclusions is presented . The methodology is applied to a logistic growth model with a delay . Introduction In science and engineering, knowledge about dynamic systems may be quantitative, qualitative, and semiqualitative . When these models are studied all this knowledge should be taken into account . In the literature, different levels of numeric abstraction have been considered . They may be: purely qualitative (Kuipers 1994), semiqualitative (Kay 1996), (Ortega, Gasca and Toro 1998x) and (Berleant and Kuipers 1997), numeric interval (Vescovi, Farquhar and Iwasaki 1995) and (Corliss 1995), and quantitative . QR99 Loch Awe, Scotland {iaortega,gasca,mtorol Clsi.us.es Different approximations have been developed in the literature when qualitative knowledge is taken into account : distributions of probability, transformation of non-linear to piecewise linear relationships, MonteCarlo method, constraint logic programming (Hickey 1994), fuzzy sets (Bonarini and Bontempi 1994), causal relations (Bousson and Trave-Massuyes 1994), and combination of all levels of qualitative and quantitative abstraction (Kay 1996), (Ortega, Gasca and Toro 1998b) and (Gasca 1998) . In this paper, qualitative knowledge of dynamic systems may be qualitative operators, envelope functions, qualitative labels and qualitative continuous functions. The paper is organised as follows : firstly, the proposed methodology is explained, that is, the transformation techniques of a semiqualitative model into a family of quantitative models, and the stochastic methods applied to obtain a database of quantitative trajectories . Secondly, the kind of qualitative knowledge we are using is introduced and the concept of semiqualitative model is defined . Thirdly, the query/classification language on this database is described . The language allows us to classify this database, and in such case, a labeled database is obtained . Machine learning algorithms are applied in order to obtain the different qualitative behaviours of the system . This behaviour is expressed by means of a set of hierarchical qualitative rules . In forthcoming papers, these machine learning algorithms will be describe in detail . Finally, a theoretical study about the reliability of the obtained conclusions is presented . This methodology is applied to a logistic growth model with a delay. Proposed Methodology There is enough bibliography that studies stationary states, however, the study of transient states is also necessary. For example, it is very important in production industrial systems in order to carry out optimizations about their efficiency . Stationary and transient states of a semiqualitative dynamic system may be studied with the proposed methodology. It is shown in figure 1 .