2009/10/22 by G. David Forney, Forney, G. David
Computer Science · Engineering · #Control Systems and Identification #FOS: Computer and information sciences #FOS: Electrical engineering #FOS: Mathematics #Formal Methods in Verification #Information Theory (cs.IT) #Optimization and Control (math.OC) #Stability and Control of Uncertain Systems #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.0910.4336
openalex publication_date 2009/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a controllable 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.