Minimal Realizations of Linear Systems: The “Shortest Basis” Approach

Given a discrete-time linear system
C
, a shortest basis for
C
is a set of linearly independent generators for
C
with the least possible lengths. A basis
B
is a shortest basis if and only if it has the predictable span property (i.e., has the predictable delay and degree properties, and is non-catastrophic), or alternatively if and only if it has the subsystem basis property (for any interval
J
, the generators in
B
whose span is in
J
is a basis for the subsystem
CJ
). The dimensions of the minimal state spaces and minimal transition spaces of
C
are simply the numbers of generators in a shortest basis
B
that are active at any given state or symbol time, respectively. A minimal linear realization for
C
in controller canonical form follows directly from a shortest basis for
C
, and a minimal linear realization for
C
in observer canonical form follows directly from a shortest basis for the orthogonal system
C
^⊥
. This approach seems conceptually simpler than that of classical minimal realization theory.