  This work introduces the empirical cross gramian for
multiple-input-multiple-output systems. The cross gramian is a tool for
reducing the state space of control systems, which conjoins controllability and
observability information into a single matrix and does not require balancing.
Its empirical gramian variant extends the application of the cross gramian to
nonlinear systems. Furthermore, for parametrized systems, the empirical
gramians can also be utilized for sensitivity analysis or parameter
identification and thus for parameter reduction. This work also introduces the
empirical joint gramian, which is derived from the empirical cross gramian. The
joint gramian not only allows a reduction of the parameter space, but also the
combined state and parameter space reduction, which is tested on a linear and a
nonlinear control system. Controllability- and observability-based combined
reduction methods are also presented, which are benchmarked against the joint
gramian.
