Algebraic Structure of Linear Dynamical Systems. III. Realization Theory Over a Commutative Ring

Area of Science:

• Control Theory
• Abstract Algebra
• Systems Theory

Background:

• Linear dynamical systems are fundamental in control theory and signal processing.
• Realization theory, focused on constructing system models from input-output data, was previously limited to fields.
• Extending these concepts to more general algebraic structures is crucial for broader applications.

Purpose of the Study:

• To generalize the realization theory of linear dynamical systems.
• To investigate the applicability of existing realization criteria in a wider algebraic context.
• To establish the conditions under which finite realizations exist over commutative rings.

Main Methods:

• The study extends established realization theory techniques.
• It applies these methods to linear dynamical systems defined over commutative rings.
• The core methodology involves analyzing the structure of systems over Noetherian integral domains.

Main Results:

• The theory of linear dynamical systems realization is successfully extended to a broad class of commutative rings.
• A key finding is that the existence criterion for a finite realization is invariant.
• This criterion holds without modification when transitioning from fields to Noetherian integral domains.

Conclusions:

• The generalization of realization theory to commutative rings is achieved.
• The invariance of the finite realization existence criterion simplifies theoretical analysis.
• This work expands the scope of realization theory, enabling its application in more abstract algebraic settings.