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

Himpe C, Ohlberger M

Abstract

This work introduces the empirical cross gramian for multiple-input-multiple-output systems. The cross gramian is a tool for model reduction of the state space of control systems, which conjoins controllability and observability information into a single matrix and does not require balancing. Its empirical variant extends the application of the cross gramian to nonlinear systems. For parametrized systems, the empirical gramians can also be utilized for sensitivity analysis and thus for parameter identification and reduction. This work also introduces the empirical joint gramian, which is derived from the 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 Bayesian inverse problem. A controllability and an observability based combined reduction method are presented which are benchmarked against the joint gramian.