A minimal realization for control systems

This thesis study is based on the observability of the control systems on connected Lie groups, the minimal realization of affine control systems and almost observability concepts for linear control systems. Minimal realization is an applicable problem to the areas of engineering, physics and chemistry. Working on these terms by using mathematical point of view and differential geometric methods are very rare works in the literature. In the almost observability topic, a new approach to observability in control theory has been introduced by omitting the finite time in the time parameter. After the introduction of the thesis in the second part, the systems used to create minimal realization and almost observability concepts and the basic elements that form these systems are introduced. In third section, which is the first original part of the thesis, in order to create minimal realization of affine control systems on connected Lie groups, topological and algebraic properties of the set of indistinguishable points from the neutral element are investigated and a minimal realization for this system is established using the obtained Lie group structure and canonical projection. In the original part of the fourth chapter, for linear control systems on connected Lie groups, a new type of observability, the concept of almost observability, is introduced. For this, definition of almost indistinguishability has been given and it has been shown to be an equivalence relation. In particular, by taking a Lie group homomorphism of the output function of the system, we investigated the topological and algebraic structure of the image of almost indistinguishable points from the neutral element, and revealed the relation between local observability and local almost observability. Finally, almost minimal realization definition has been established by using the concept of almost observability rather than observability in definition of minimal realization.