On rational quadratic differential forms

In linear system theory, we often encounter the situation of investigating some quadratic functionals which represent Lyapunov functions, energy storage, performance measures, e.t.c. Such a quadratic functional is called a quadratic differential form (QDF) in the context of the behavioral approach. In the past works, a QDF is usually defined in terms of a polynomial matrix. The contribution of this paper is to present a new and more general formulation of QDF’s in terms of rational functions rather than polynomials. A QDF defined by rational functions is called a rational QDF. Unlike polynomial QDF’s, a rational QDF defines a set of values of a quadratic functional. It turns out that several basic features of polynomial QDF’s (nonnegativity, average nonnegativity, e.t.c.) can be generalized to the case of rational QDF’s.