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 noncatastrophic), 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.