Relationships between internal and external stability for infinite-dimensional systems with applications to a servo problem

Abstract

A study is made of the relationships among various stability motions for a class of infinite-dimensional systems, which contains a class of systems not covered by existing methods, e.g. those having infinitely many unstable poles. It is proved that: internal L/sup 2/-stability and exponential stability are equivalent; and internal stability implies H/sup infinity /-stability. Several necessary and sufficient conditions for internal stability are derived. In particular, under certain conditions, a canonical realization is internally stable if it is externally stable. These results are applied to the servo problem involving this class of systems. It is shown that: (i) an internal model is necessary for tracking; (ii) an internal model along with closed-loop stability implies tracking. A typical example, called a repetitive control system, is discussed to illustrate the results.