A Note on the Cross Gramian for Non-Symmetric Systems

The cross gramian matrix is a tool for model reduction and system identification, but it is only computable for square control systems. For symmetric control systems the cross gramian possesses a useful relation to the associated system’s Hankel singular values. Yet, many real-life models are neither square nor symmetric. In this work, concepts from decentralized control are used to generalize the cross gramian to be applicable for non-square and non-symmetric systems. To illustrate this new non-symmetric cross gramian, it is then applied in the context of model reduction.