  We discuss balanced truncation model order reduction for large-scale
quadratic-bilinear (QB) systems. Balanced truncation 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 (1993), where it is shown that Gramians for nonlinear
systems are the solutions of state-dependent nonlinear Hamilton-Jacobi
equations. Therefore, they are not only difficult to compute for large-scale
systems but also hard to utilize in the model reduction framework. In this
paper, we propose algebraic Gramians for QB systems based on the underlying
Volterra series representation of QB systems and their Hilbert adjoint systems.
We then show their relations with a certain type of generalized quadratic
Lyapunov equation. Furthermore, we present how these algebraic Gramians and
energy functionals relate to each other. Moreover, we characterize the
reachability and observability of QB systems based on the proposed algebraic
Gramians. This allows us to find those states that are hard to control and hard
to observe via an appropriate transformation based on the Gramians. Truncating
such states yields reduced-order systems. Additionally, we present a truncated
version of the Gramians for QB systems and discuss their advantages in the
model reduction framework. We also investigate the Lyapunov stability of the
reduced-order systems. We finally illustrate the efficiency of the proposed
balancing-type model reduction for QB systems by means of various
semi-discretized nonlinear partial differential equations and show its
competitiveness with the existing moment-matching methods for QB systems.
