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

arXiv: https://arxiv.org/abs/1302.0634 · HTML: https://mlsanity.com/paper/1302.0634/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Feb 2013.

## Built on by (most cited first, 1 of 1)

- [Nonlinear Model Order Reduction of Dynamical Systems in Process Engineering: Review and Comparison](https://mlsanity.com/paper/2506.12819/) (2025)

## Abstract (arXiv)

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.