Normalized coprime factorizations and the graph metric for linear time-varying systems

It is shown that a finite-dimensional linear time-varying continuous-time system admits normalized coprime factorizations if and only it admits a stabilizable and detectable realization, or equivalently, if and only if it is internally stabilizable via dynamic output feedback. In the process, a simple proof that stabilizability and detectability are sufficient to ensure the existence of stabilizing solutions to standard continuous-time control and filter Riccati equations is given. State-space formulas for normalized coprime factorizations are given. These are used to define the graph metric for the set of internally stabilizable finite-dimensional linear time-varying plants. As an application a robustness estimate is derived for feedback stability of linear time-varying plants using nonlinear time-varying controllers