A declarative theory for rational controllers

The author presents a computationally effective representation theory for a class of digital controllers referred to as rational controllers. The theory, which is expressed in terms of first-order predicate logic with some meta-extensions, characterizes the dynamic behavior of an element of the class using equations and inequalities that declare the structure of the controller in an algebraic variety V whose algebras satisfy the central factorization principle. The central factorization principle states that an element of an algebraic variety V is either primitive or can be expressed in finitely many different ways, in terms of operations of the algebra, on primitive elements. Important elements of V are the algebra of rational sets over the ring of real numbers, the algebra of rational trees, the algebra of rational functions on a module, the algebra of modular lattices, and direct products and limits of these algebras