  We investigate model order reduction (MOR) for linear dynamical systems,
where a quadratic output is defined as a quantity of interest. The system can
be transformed into a linear dynamical system with many linear outputs. MOR is
feasible by the method of balanced truncation, but suffers from the large
number of outputs in approximate methods. To ameliorate this shortcoming we
derive an equivalent quadratic-bilinear system with a single linear output and
analyze the properties of this system. We examine MOR for this system via the
technique of balanced truncation, which requires a stabilization of the system.
Therein, the solution of two quadratic Lyapunov equations is traced back to the
solution of just two linear Lyapunov equations. We present numerical results
for several test examples comparing the two MOR approaches.
