🤖 AI Summary
This work addresses the rational realization problem for first-order differential-algebraic input-output equations, focusing on existence criteria and constructive methods under observability and real-coefficient constraints. Methodologically, it establishes an equivalence theorem between the existence of a rational realization and that of an observable rational realization for first-order systems; develops a decidable criterion for real rational realizability via differential-algebraic geometry and field extension theory; and devises a fully algorithmic, symbolically computed construction procedure that systematically generates both observable and real-coefficient rational realizations. Furthermore, several key results are extended to higher-order differential-algebraic equations. The contributions provide a rigorous yet practical algebraic framework for modeling, identification, and realization theory of linear and nonlinear dynamical systems, bridging theoretical algebra with applied system theory.
📝 Abstract
Given a single (differential-algebraic) input-output equation, we present a method for finding different representations of the associated system in the form of rational realizations; these are dynamical systems with rational right-hand sides. It has been shown that in the case where the input-output equation is of order one, rational realizations can be computed, if they exist. In this work, we focus first on the existence and actual computation of the so-called observable rational realizations, and secondly on rational realizations with real coefficients. The study of observable realizations allows to find every rational realization of a given first order input-output equation, and the necessary field extensions in this process. We show that for first order input-output equations the existence of a rational realization is equivalent to the existence of an observable rational realization. Moreover, we give a criterion to decide the existence of real rational realizations. The computation of observable and real realizations of first order input-output equations is fully algorithmic. We also present partial results for the case of higher order input-output equations.
