On the algebraic structure of quadratic and bilinear dynamical systems

A number of important control and estimation problems in the field of aerospace and avionics involve dynamical models that are quadratic and bilinear functions of system states and inputs. In this paper, formal definitions of free and forced quadratic/bilinear dynamical systems are given and the algebraic structure of this class of systems is explored with a view to setting up a systematic approach for deriving state and measurement models. Properties of quadratic and bilinear vectors are investigated, and relationships between these and linear vectors established. Systematic procedure for constructing quadratic state and bilinear state-input vectors is derived. The quadratic/bilinear vector modeling technique is applied to the formulation of a state-space model for a second-order approximation of a general non-linear system.