The Process of Model Building and Simulation of Ill-defined Systems: Application to Wastewater Treatment

In recent years, there has been a growing awareness of the ill-de nedness of environmental processes. To provide a frame of reference for discussions regarding ill-de ned systems, a taxonomy and terminology of the modelling and simulation of systems will be presented. Due to the complexity of ill-de ned systems, it is not only necessary to describe the nature of models, but also to describe some procedures according to which the modelling will proceed. This will enable the modeller to obtain the model which best ts his goals (optimal model). For meaningful description of models, di erent model formalisms will be presented. Furthermore, modelling procedures will be described at a generic level and di erent model formalisms will be presented. Throughout this presentation, Waste-Water Treatment Plants and processes occurring within these plants will serve as illustrations of the de nitions given. Introduction In recent years, mathematical models have gained importance in environmental studies. Environmental processes, such as those occurring in Waste-Water Treatment Plants (WWTP's), are often referred to as examples of ill-de ned systems. Compared to the modelling of well-de ned (e.g., electrical, mechanical) systems, ill-de ned systems modelling is more complex. In particular, the di culty in choosing the \right" model is very apparent. In the sequel a rigorous approach to modelling of ill-de ned systems is presented. Illustrations are given for the case of WWTP's. In order to develop a framework for the modelling of ill-de ned systems, some de nitions concerning modelling and simulation enterprise are given. Thereafter, a modelling procedure which may guide the modeller to nd the \right" model, is presented. This modelling procedure consists of interactions between information sources and activities. These information sources and activities will be discussed. Models, the subset of the modelling enterprise, may be described in di erent formalisms. A common formalism classi cation will be presented. Modelling and Simulation Concepts One of the most important de nitions in modelling and simulation is the de nition of a system. A system is de ned as a potential source of behaviour. It is observable when its behaviour can be transformed into data (information). Knowledge about given systems can be acquired through experiments. An experiment is de ned as the process of causing (by known stimuli) and observing the behaviour of a system. In other words, given the inputs, the system outputs will be observed. In order to perform experiments on a system, its experimental frame has to be de ned. The concept of experimental frame refers to a limited set of circumstances under which a system is to be observed or subjected to experimentation. As such, the experimental frame re ects the objectives of the experimenter who performs experiments on a system. A way to organise collected knowledge about a system, given its experimental frame, is by means of modelling and models. In a very broad sense, a model is anything which is capable of generating behaviour resembling the behaviour of a system (given its experimental frame). In this paper only parametric models will be discussed. A parametric model, is a model consisting of parameters, where parameters are de ned as constants or an experiment. All systems may roughly be divided into two subclasses: well-de ned systems and ill-de ned systems. However, there is a fundamental problem in classifying systems. All information of a system can only be given by means of a model. Therefore, in order to classify, one has to de ne the properties of a model describing the system. Klir [1] solves this problem by de ning epistemological levels at which the system may be observed and Zeigler [5] postulates a Base Model, a hypothetical model capable of describing