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

Given a discrete-time linear systemC, a shortest basis forCis a set of linearly independent generators forCwith the least possible lengths. A basisBis 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 intervalJ, the generators inBwhose span is inJis a basis for the subsystemCJ). The dimensions of the minimal state spaces and minimal transition spaces ofCare simply the numbers of generators in a shortest basisBthat are active at any given state or symbol time, respectively. A minimal linear realization forCin controller canonical form follows directly from a shortest basis forC, and a minimal linear realization forCin observer canonical form follows directly from a shortest basis for the orthogonal systemC⊥. This approach seems conceptually simpler than that of classical minimal realization theory.