Energy Functions and Algebraic Gramians for Bilinear Systems

In linear system theory, Gramian matrices help quantify the input-tostate and state-to-output interactions in a state space model. In this paper two different existing generalizations of this idea are examined for the case of a stable bilinear system. One method is based on using the L2norm to measure signal sizes and leads to the notion of the controllability and observability functions, which are referred to collectively as energy functions. The other method is motivated by introducing algebraic generalizations of the linear system Lyapunov equations, the solutions of which are called the algebraic Gramians. While these generalizations are distinct, in this paper some new relationships between the two approaches are presented.