  Model order reduction is a technique that is used to construct low-order
approximations of large-scale dynamical systems. In this paper, we investigate
a balancing based model order reduction method for dynamical systems with a
linear dynamical equation and a quadratic output function. To this aim, we
propose a new algebraic observability Gramian for the system based on Hilbert
space adjoint theory. We then show the proposed Gramians satisfy a particular
type of generalized Lyapunov equations and we investigate their connections to
energy functionals, namely, the controllability and observability. This allows
us to find the states that are hard to control and hard to observe via an
appropriate balancing transformation. Truncation of such states yields
reduced-order systems. Finally, based on $\mathcal H_2$ energy considerations,
we, furthermore, derive error bounds, depending on the neglected singular
values. The efficiency of the proposed method is demonstrated by means of two
semi-discretized partial differential equations and is compared with the
existing model reduction techniques in the literature.
