Stability of Spectral Divisors

Abstract

In numerical computations of minimal factors of a given transfer function questions concerning the conditioning of the factors turn up naturally. According to the division theory developed in the previous chapters, all minimal factorizations may be obtained in an explicit way in terms of supporting projections of minimal systems. This fact allows one to reduce questions concerning the conditioning of minimal factorizations to questions concerning the stability of divisors of a system. In the present chapter we study the matter of stability of spectral divisors mainly. In this case the investigation can be carried out for finite- as well as for infinite-dimensional state spaces. The invariant subspace method employed in this chapter will also be used to prove that “spectral” solutions of an operator Riccati equation are stable. The case of minimal non-spectral factorizations will be considered in the next chapter.