In this paper, me prove, in a constructive way, the equivalence between pie
cewise affine systems and a broad class of hybrid systems described by inte
racting linear dynamics, automata, and propositional logic. By focusing our
investigation on the former class, we show through counterexamples that ob
servability and controllability properties cannot be easily deduced from th
ose of the component linear subsystems. Instead, rye propose practical nume
rical tests based on mixed-integer linear programming.