Linear and Non-linear Dynamics in an Endo-observer's frame of reference

Abstract

As an attempt to overcome the unfortunate division between data and physical modelling, this thesis is devoted to the development of a framework which allows the derivation of a physical model that at the same time includes errors and uncertainties about this system which are as unspecified with respect to their statistical properties as possible. The work can be summarized as: 1. Grey box modelling: An appropriate semi-physical model of a ballistic missile with very limited knowledge about the aerodynamical properties of the airframe is being developed. It is shown which kind of dynamics the missile may exhibit and with which methods these dynamics can be analyzed. It is demonstrated that the missile may also show chaotic behaviour, the methods to measure this type of dynamics are presented. The lack of data points may, however, prevent the applicability of these methods which therefore suggests the theoretical framework of an: 2. Endo-observer for linear systems: It is proved that real-life observers require a framework such as a stochastic one for incorporating their uncertainties into a physical model. It is shown how minimally specified uncertainties can enter the dynamics of a free particle. The dynamics turn out to be similar to those given by the Schrödinger equation. 3. Endo-observer for the nonlinear state space: The concepts of physical dynamics with uncertainties which have been developed for the free particle are exhaustively re-developed for the case when the variance of the uncertainties remains within an arbitrary but fixed bound defined prior to the experiment. The dynamics of a nonlinear vector field with uncertainties are expressed in a form which is very much similar to the Schrödinger equation with a 'momentum' consisting of the 'mechanical momentum' from which a 'electromagnetical momentum' is subtracted which in this case here is the vector field of the state space. 4. endo-obsserver with bias: It is argued that an observer which has to select permanently among all available variable and parameter values has to favour the more probable ones rather than the less probable ones. The dynamics then lead to the well known nonlinear Schrödinger equation.

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