Gramians, Energy Functionals, and Balanced Truncation for Linear Dynamical Systems With Quadratic Outputs

• 成果类型: Article
• 署名作者: Benner, Peter; Goyal, Pawan; Duff, Igor Pontes
• 署名单位: Max Planck Society; Otto von Guericke University
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2021.3086319
• 发表日期: 2022
• 页码: 886-893
• 关键词: Balanced truncation (BT); controllability and observability; dynamical systems; energy functionals; Gramians; model order reduction

摘要

In this article, we investigate a balancing-based model order reduction method for dynamical systems with a linear dynamical equation and a quadratic output function. With this aim, we propose a new algebraic observability Gramian for the system based on the Hilbert space adjoint theory. We then show that the proposed Gramian satisfies a particular type of generalized Lyapunov equation and we investigate its connection to energy functionals. It allows one to find states that are hard to control and hard to observe via an appropriate balancing transformation. Truncation of such states yields reduced-order models. Finally, based on H-2 energy considerations, we derive error bounds, depending on the neglected singular values. We demonstrate the efficiency of the proposed methodology using a numerical example.