Balanced truncation for quadratic-bilinear control systems

We discuss model order reduction (MOR) for large-scale quadratic-bilinear (QB) systems based on balanced truncation. The method for linear systems mainly involves the computation of the Gramians of the system, namely reachability and observability Gramians. These Gramians are extended to a general nonlinear setting in Scherpen (Systems Control Lett. 21, 143-153 1993). These formulations of Gramians are not only challenging to compute for large-scale systems but hard to utilize also in the MOR framework. This work proposes algebraic Gramians for QB systems based on the underlying Volterra series representation of QB systems and their Hilbert adjoint systems. We then show their relation to a certain type of generalized quadratic Lyapunov equation. Furthermore, we quantify the reachability and observability subspaces based on the proposed Gramians. Consequently, we propose a balancing algorithm, allowing us to find those states that are simultaneously hard to reach and hard to observe. Truncating such states yields reduced-order systems. We also study sufficient conditions for the existence of Gramians, and a local stability of reduced-order models obtained using the proposed balanced truncation scheme. Finally, we demonstrate the proposed balancing-type MOR for QB systems using various numerical examples.