Coprime factorizations and stabilizability of infinite-dimensional linear systems

It has been known that a matrix-valued transfer function is dynamically stabilizable iff it has a doubly coprime factorization. We extend this to operator-valued functions and also to controllers with internal loop. We then present several other equivalent conditions, such as having a stabilizable and detectable realization. Our results lead to the extension of the classical results on dynamic stabilization and dynamic partial stabilization to general proper operator-valued functions. Part of the results are new even for scalar-valued functions.

I. INTRODUCTION

A proper rational matrix-valued function has the following properties (among others): (i)

has a stabilizing dynamic controller.

(ii)

has a doubly coprime factorization.

(iii)

has a right coprime factorization.

(iv)

has a stabilizable and detectable realization.

(v)

The LQR Riccati equation and its dual equation for some realization of have nonnegative solutions.