Tree-based solution representations for quadratic bilinear systems and their consequences in model order reduction

Abstract

We investigate quadratic bilinear systems by developing novel tree-based representations of their solutions. The proposed framework decomposes the solution into a sequence of coupled bilinear subsystems whose components admit explicit expansions indexed by full binary trees. These representations yield sufficient conditions for the existence of global solutions and lead to new output bounds in terms of reachability Gramians. Motivated by these estimates, we introduce time-limited and infinite-horizon reachability and observability Gramians, establish sufficient conditions for their existence, and characterize them through nonlinear matrix equations. The associated Gramians are employed to identify dominant state-spaces and to derive exact reduced-order models obtained by removing Gramian kernels. Building on these results, we develop a balanced truncation method for quadratic bilinear systems and prove an error bound for the reduced-order approximation. The proposed framework provides a unified connection between tree-based solution representations, nonlinear Gramian theory, and balanced truncation for quadratic bilinear systems, closing several theoretical gaps in the analysis of Gramian-based model reduction for this class of systems.