This paper extends interpolatory model reduction to systems with (up to) quadratic-bilinear dynamics and quadratic-bilinear outputs. These systems are referred to as QB-QB systems and arise in a number of applications, including fluid dynamics, optimal control, and uncertainty quantification. In the interpolatory approach, the reduced order models (ROMs) are based on a Petrov-Galerkin projection, and the projection matrices are constructed so that transfer function components of the ROM interpolate the corresponding transfer function components of the original system. To extend the approach to systems with QB outputs, this paper derives system transfer functions and sufficient conditions on the projection matrices that guarantee the aforementioned interpolation properties. Alternatively, if the system has linear dynamics and quadratic outputs, one can introduce auxiliary state variables to transform it into a system with QB dynamics and linear outputs to which known interpolatory model reduction can be applied. This transformation approach is compared with the proposed extension that directly treats quadratic outputs. The comparison shows that transformation hides the problem structure. Numerical examples illustrate that keeping the original QB-QB structure leads to ROMs with better approximation properties.

本文将插值模型简化​​扩展到具有（最多）二次双线性动力学和二次双线性输出的系统。这些系统被称为 QB-QB 系统，出现在许多应用中，包括流体动力学、最优控制和不确定性量化。在插值方法中，降阶模型 (ROM) 基于 Petrov-Galerkin 投影，并且构建投影矩阵，以便 ROM 的传递函数分量对原始系统的相应传递函数分量进行插值。为了将该方法扩展到具有 QB 输出的系统，本文推导了系统传递函数和保证上述插值属性的投影矩阵的充分条件。或者，如果系统具有线性动力学和二次输出，则可以引入辅助状态变量以将其变换为具有QB动力学和线性输出的系统，可以对其应用已知的插值模型简化​​。这种变换方法与直接处理二次输出的建议扩展进行了比较。比较表明，变换隐藏了问题结构。数值例子表明，保持原始的 QB-QB 结构可以使 ROM 具有更好的近似特性。