Model Structure Evolution Continuous Fraction Descriptions

The paper deals with a form of complexity inherent in the new paradigm of Structure Evolving Systems (SES), [N.Karcanias (2008)]. This new family emerges in many challenging applications where there is variability of the system structure, its components and environment, in a way that implies evolution of system structure and properties. Of special interest are problems of integrated system design where there is variability and evolution of the models used from early to late stages of design. We are interested in working out a representation of the evolution of this complexity and then having a representation of the chains of such models, study the evolution of structural properties within the derived chains. The central question driving our study is: “How do system structural properties evolve, when model complexity increases from the low to high complexity?”. Given that progressing from simple to more complex requires deep knowledge of the particular system, we reverse the process and we consider chains of models generated by model reduction; in this process we use the additional requirement that model simplification may be expressed in a structural sense that may allow study of evolution of system properties. We consider linear systems and we focus on the SISO case and use the McMillan degree as an indicator of complexity. The need for parameterizing the chain of models motivates the use of methods for approximation based on Continuous Fractions. Different forms of Continuous Fractions are considered and their role in parameterizing Chains of Models are considered, as well as their link to control related representations and Control problems. Central to deriving such chain representations are different forms of the Euclidian division for polynomials. The paper provides the basis for studying structural properties evolution in the context of Structure Evolving Systems.