We discuss model order reduction (MOR) for large-scale quadratic-bilinear (QB) systems based on balanced truncation. The method for linear systems mainly involves the computation of the Gramians of the system, namely reachability and observability Gramians. These Gramians are extended to a general nonlinear setting in Scherpen (Systems Control Lett. 21, 143-153 1993). These formulations of Gramians are not only challenging to compute for large-scale systems but hard to utilize also in the MOR framework. This work proposes algebraic Gramians for QB systems based on the underlying Volterra series representation of QB systems and their Hilbert adjoint systems. We then show their relation to a certain type of generalized quadratic Lyapunov equation. Furthermore, we quantify the reachability and observability subspaces based on the proposed Gramians. Consequently, we propose a balancing algorithm, allowing us to find those states that are simultaneously hard to reach and hard to observe. Truncating such states yields reduced-order systems. We also study sufficient conditions for the existence of Gramians, and a local stability of reduced-order models obtained using the proposed balanced truncation scheme. Finally, we demonstrate the proposed balancing-type MOR for QB systems using various numerical examples.

我们讨论基于平衡截断的大规模二次双线性（QB）系统的模型降阶（MOR）。线性系统的方法主要涉及系统的格拉米安数的计算，即可达性和可观测性格拉米安数。这些 Gramians 被扩展到 Scherpen 中的一般非线性设置（Systems Control Lett. 21 , 143-153 1993）。这些格拉米公式不仅对大规模系统的计算具有挑战性，而且也很难在 MOR 框架中使用。这项工作基于 QB 系统及其希尔伯特伴随系统的底层 Volterra 级数表示，提出了 QB 系统的代数格拉米亚式。然后我们展示它们与某种类型的广义二次李亚普诺夫方程的关系。此外，我们根据提出的格拉米安量化了可达性和可观测性子空间。因此，我们提出了一种平衡算法，使我们能够找到那些同时难以达到和难以观察的状态。截断这些状态会产生降阶系统。我们还研究了格拉米亚存在的充分条件，以及使用所提出的平衡截断方案获得的降阶模型的局部稳定性。最后，我们使用各种数值示例演示了 QB 系统所提出的平衡型 MOR。