Minimal realizations of linear systems: The "shortest basis" approach

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 C_J). 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^\perp. This approach seems conceptually simpler than that of classical minimal realization theory.

💡 Research Summary

The paper introduces a novel framework for constructing minimal realizations of controllable discrete‑time linear systems by means of a “shortest basis.” A shortest basis B for a system C is a set of linearly independent generators whose support lengths are as small as possible. The authors prove that B is shortest if and only if it satisfies two equivalent properties: (i) the predictable‑span property (PSP), which comprises predictable delay, predictable degree, and non‑catastrophic behavior, and (ii) the subsystem‑basis property (SBP), which requires that for any finite interval J the generators whose support lies entirely in J form a basis of the restricted subsystem C_J.

From PSP (or SBP) the dimensions of the minimal state space and the minimal transition space follow directly: at any time t the number of generators active (i.e., whose support includes t) equals the minimal state dimension, and at any symbol instant k the number of active generators equals the minimal transition dimension. Consequently, once a shortest basis is known, the minimal dimensions are read off without any matrix rank calculations.

Using B, a controller‑canonical realization is built by assigning each active generator to a state variable that shifts forward with each time step; the output is the linear combination of the last symbols of the active generators. This yields a minimal realization whose state‑space dimension matches the active‑generator count. Dually, by constructing a shortest basis for the orthogonal system C^⊥, an observer‑canonical realization is obtained, providing a minimal realization that emphasizes output‑observability.

The approach bypasses the traditional machinery of Kalman‑Bucy decomposition, Ho‑Kálmán controllability/observability tests, and similarity transformations. It works uniformly for time‑varying or non‑canonical forms, because the shortest‑basis computation reduces to finding a minimal‑length independent generator set—a problem that can be solved by greedy or graph‑theoretic algorithms. The authors argue that this yields a conceptually simpler, more transparent path to minimal realizations, and they suggest extensions to continuous‑time, nonlinear, and real‑time estimation contexts.