Algebra-geometric techniques for C3 systems

Abstract

The purpose of this talk is to present a brief overview of work in mathematical systems theory which is directly applicable to the study and analysis of large complicated systems such as the so-called C3 systems. The material presented in this talk is based on joint work with Professor P. Krishnaprasad and various graduate students at Case Institute of Technology. I refer the reader to [1] for a full technical development of the systems theoretic concepts of this presentation. I acknowledge that this written material has been freely adopted from [l] and that Professor Krishnaprasad bears no responsibility for conceptual errors induced by this adaptation. In any physical system it is customary to develop a model of the system about some equilibrium point or some fixed trajectory. For simplicity we assume that we are modeling about an equilibrium point. Almost any useful such model can take into account small variations in system parameters. And, for routine control problems the ability to handle such small variations is usually adequate. However, few control problems are in practice routine and one often finds that the possibility of large excursions of parameters makes a totally automatic control system unacceptable to practitioners. Such problems in avionics has motivated us to consider continuous families of models that explicitly include critical parameters. In particular, the problem of developing reliable avionic systems leads one in a natural way to study families of systems. Control engineers have always implicitly been aware of families of systems. Our main contribution has been in the development of techniques to directly study such systems. Whereas for a fixed linear system the well-established techniques of linear algebra and differential equations generally suffice for a detailed study, we find that the more esoteric (but not more difficult) fields of differential geometry and topology and algebraic geometry play a key role in any such study of families of linear systems. For more complicated systems, such as C3 systems, obviously one cannot describe a mathematical model that completely mimics the system--nor would one want to. Instead, as is always done, one models portions of the system, or makes aggregated models that can be studied with known techniques. This talk will focus on such possible models for simplified C3 systems.