Cross-Gramian-Based Combined State and Parameter Reduction for Large-Scale Control Systems

Abstract: 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.