  When solving partial differential equations numerically, usually a high order
spatial discretisation is needed. Model order reduction (MOR) techniques are
often used to reduce the order of spatially-discretised systems and hence
reduce computational complexity. A particular MOR technique to obtain a reduced
order model (ROM) is balanced truncation (BT), a method which has been
extensively studied for deterministic linear systems. As so-called type I BT it
has already been extended to bilinear equations, an important subclass of
nonlinear systems. We provide an alternative generalisation of the linear
setting to bilinear systems which is called type II BT. The Gramians that we
propose in this context contain information about the control. It turns out
that the new approach delivers energy bounds which are not just valid in a
small neighbourhood of zero. Furthermore, we provide an $H_\infty$-error bound
which so far is not known when applying type I BT to bilinear systems.

Optimization and Control