Stability via Quotient Modules

We use quotient modules and quotient behaviors for algebraically treating the stability of autonomous and of IO behaviors. The prototypical example for such a quotient is the quotient field (e.g., \({\mathbb {Q}}\)) of a commutative integral domain (e.g., \({\mathbb {Z}}\)). The technique for rings and modules is presented and proven in Sections 9.1 and 9.2 with an emphasis on principal ideal domains. In Sect. 9.3, we apply this method to autonomous behaviors and in Sect. 9.4 to IO behaviors. In Sect. 9.5, we introduce the ring of proper and stable rational functions which will be essential in Chap. 10 for constructing stabilizing compensators with proper transfer matrices.