Structural properties and poles assignability of LTI singular systems under output feedback

This paper studies the structural properties of both finite poles and the infinite pole of linear time-invariant singular systems under output feedback. Three main problems are studied, namely, (1) the algebraic structures of both finite poles and the infinite pole; (2) the assignability of finite poles and the elimination of the infinite pole by output feedback; and (3) the controllability and observability of the system with minimal number of inputs and outputs. New generic solutions to these problems are presented in terms of some new concepts defined in this paper including the geometric multiplicity of the infinite pole, the finite and impulsive output feedback cycle indices of the system. Determination of these multiplicities and indices are discussed. An assignability equivalence is established between the variable finite poles and the poles of a controllable and observable non-singular system. The number of the independent infinite poles that can be reduced is given in terms of the system matrices. The minimal number of inputs and outputs that guarantee controllability and observability are shown to be the output feedback cycle indices.