Balanced Model Reduction of Gradient Systems

Gradient systems are an important class of systems, e.g., a large class of linear and nonlinear electrical circuits are in fact gradient systems. The structure of the gradient system is important, and if model reduction is applied, it is thus of interest to preserve the gradient structure of the system, i.e., the reduced order model should (at least) be a gradient system again. In this paper we study balanced realizations for gradient systems. For linear gradient systems a balanced realization has a specific structure for the metric, implying that balancing based model order reduction preserves the gradient system structure of the system. For nonlinear gradient systems, such structure for the metric is not obtained, resulting in additional conditions to preserve the gradient structure of the system. Both balanced truncation and singular perturbation order reduction are studied. Finally, a relation with cross Gramians and balancing of gradient systems is studied.