A new system structure theory. Application to system properness

Abstract

A new definition is proposed for the concept of structure at infinity. To each input component is attached a rational integer, which, for a single input single output system defined by a single differential equation, is the difference between the the order in the output and the order in the input of the differential equation defining the system. The m-tuple of these rational integers is the new structure at infinity of the system, m stands for the number of input components. Associated to the structure at infinity is also defined a p-tuple of rational integers representing a new notion of essential structure. The old structure at infinity is shown to be recoverable through the new one. The new definition is intrinsic in the sense that it does not depend neither on an algorithm which serves to compute the structure at infinity and the essential structure, nor on a particular set of differential equations used to define the system. Moreover, the new notions are not tied to affine control systems, they fit well for any differential algebraic system, in state form, or in input output form, or even with general latent variable. An algorithm for computing the structure at infinity and the essential structure for arbitrary differential algebraic systems is also proposed. In this paper the new theory is applied to the longstanding problem of a satisfactory definition of system properness. A system is said to be proper if its structure at infinity consists of natural integers. This may look abstract but results are presented that make it sound like a good candidate for system properness definition. In forthcoming publications, the new system structure theory is applied to the basic noninteracting control problems.