# IPACS Electronic library

## Linear Algebra techniques in stability problems of systems over rings

Andres Saez-Schwedt, Tomas Sanchez-Giralda
A linear system over a commutative ring $R$ is a pair $(A,B)$, where $A$ is an $n\times n$ matrix and $B$ is an $n\times m$ matrix with coefficients in $R$. Systems over rings are a generalization of linear control systems, which are used in the study of evolution processes which can be modelled as differential or difference equations, with states, inputs and outputs.

We will study two different aspects of systems theory:
(i) the problems of pole assignment (PA), coefficient assignment (CA) and feedback cyclization (FC), which basically consist of replacing $A$ by a matrix of the form $A+BK$ such that the characteristic polynomial of $A+BK$ has some desired properties, and
(ii) the `feedback' classification of systems: the pair $(A,B)$ is equivalent to $(PAP^{-1}+PBK,PBQ)$, for matrices $P,Q,K$ of appropriate sizes and $P,Q$ invertible.

After giving an outline of the main linear algebraic techniques used, we will present an almost-canonical form for systems with coefficients in rings for which the above mentioned FC problem is solvable.