  The paper presents realization theory of discrete-time linear switched
systems. A discrete-time linear switched system is a hybrid system, such that
the continuous sub-system associated with each discrete state is linear. In
this paper we present necessary and sufficient conditions for an input-output
map to admit a discrete-time linear switched state-space realization. The
conditions are formulated as finite rank conditions of a generalized
Hankel-matrix. In addition, we present a characterization of minimality of
discrete-time linear switched systems in terms of reachability and
observable.Further, we prove that minimal realizations are unique up to
isomorphism. We also discuss procedures for converting a linear switched system
to a minimal one and we present an algorithm for constructing a state-space
representation from input-output data.The paper uses the theory rational formal
power series in non-commutative variables. The latter theory was successfully
applied to bilinear and state-affine systems in the past.
