An introduction to quadratic dierential forms

In modeling and control problems it is often necessary to study certain func-tionals of the system variables and their derivatives; when considering linear systems, such functionals are often quadratic. The parametrization of such functionals using two-variable polynomial matrices has been studied in detail in [WT1], resulting in the definition of bilinear-and quadratic di erential form (BDF and QDF respectively in the following) and in the development of a cal-culus for application in many areas. In this tutorial communication we review the main definitions and results regarding QDFs. We first examine bilinear di erential forms. These are functionals from C,(R, R,) 脳 C,(R, R,) to C,(R, R), defined as: