Sign-linear systems as cascades of automata and continuous variable systems

When the output of a linear system passes through a sign function before it can be measured, observability of the pair (A, C) is not enough for state observation. Thus a minimal dimensional sign-linear system is not necessarily observable. We prove that every sign-linear system can be reduced to a canonical (reachable and observable) system which takes the form of a cascade of a continuous system and a finite automaton