Algebraically Defined Gramians for Nonlinear Systems

In this paper, controllability and observability Gramian definitions central to the model reduction method for bilinear state space systems due to Al-Baiyat, et al. are extended to the nonlinear affine control case. While in general these algebraically defined Gramians lack a direct physical interpretation, their numerical tractability is superior to the established general nonlinear Gramian generalizations known as the energy functions. The latter can only be determined at present by either solving a pair of Hamilton-Jacobi-Bellman equations or by Monte-Carlo simulation. Connections between the algebraic Gramians developed here and existing nonlinear Gramian generalizations are presented. A connection to approximation theory for nonlinear operators is also described. Cascade lemma for set-stable systems