A Computational Rational Function, Its Difference-Differential Calculus, and Practical Applications

Abstract

Noncommutative rational functions, which extend classical rational functions to settings where variables do not commute, play a crucial role in various domains including system theory, control, optimization, and formal language theory. This paper provides a comprehensive exploration of these functions, focusing on their construction, properties, and the mathematical tools needed for their analysis. We begin by defining noncommutative rational functions and discussing their algebraic structure, highlighting the absence of a canonical form akin to coprime fraction representations in the commutative case. The realization theory for noncommutative rational functions is then developed, extending classical concepts such as state space models, controllability, observability, and minimality to the noncommutative setting. This theory provides a structured framework for representing noncommutative rational functions in a minimal state space form, which is essential for applications in system theory and control. To further analyze these functions, we introduce a difference-differential calculus specifically designed for noncommutative rational functions. This calculus generalizes traditional differentiation and finite difference operators to the noncommutative context, allowing for the detailed study of how these functions change with respect to noncommutative variables. The calculus also enables the development of higher-order operators and finite difference formulas, which are useful in various analytical and computational applications.