Natural Factorization of Linear Control Systems through Parallel Gathering of Simple Systems

Abstract

Linear systems over vector spaces and feedback morphisms form an additive category taking into account the parallel gathering of linear systems. This additive category has a minimal exact structure and thus a notion of simple systems as those systems have no subsystems apart from zero and themselves. The so-called single-input systems are proven to be exactly the simple systems in the category of reachable systems over vector spaces. The category is also proven to be semisimple in objects because every reachable linear system is decomposed in a finite parallel gathering of simple systems. Hence, decomposition result is fulfilled for linear systems and feedback morphisms, but category of reachable linear systems is not abelian semisimple because it is not balanced and hence fails to be abelian. Finally, it is conjectured that the category of linear systems and feedback actions is in fact semiabelian; some threads to find the result and consequences are also given.