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

This paper studies the relationships among various stability notions for a class of infinite-dimensional systems, which contains a class of systems not covered by the existing method, e.g., those having infinitely many unstable poles. It is proved thai i) internal L2-stability and exponential stability are equivalent; ii) internal stability implies H??-stability. Several necessary conditions and sufficient conditions for internal stability are derived. In particular, it is proved that, 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.