Reduced order modelling of acoustical systems based on measurement data

在物理模型与现实世界系统存在较大偏差的情况下，基于测量的建模方法可能是一种很好的替代方法。传递函数形式的输入输出数据或同样的脉冲响应测量构成了这种方法的核心。然而，在实际应用中，为了获得较高的模型保真度，通常需要对许多输入输出对进行高维测量。因此，已建立的数据驱动方法可以迅速导致计算要求很高的模型。解决方案以状态空间模型的形式呈现，因为它们提供了对大量降阶建模方法的访问。数值线性代数领域的最新进展提出了随机低秩矩阵分解，可用于在高维数据上使用现有的数据驱动系统识别方法，从而使其首次在工程声学中得到实际应用。除了提高计算效率之外，状态空间的另一个可想象的优点是可以通过以空间位置为参数的参数化状态空间模型来减少测量工作量。本提案的主要目标是建立最先进的数据驱动系统识别方法，用于工程声学领域的声传输系统建模。建议的行动旨在增加和创造关于有目的地使用数据驱动的建模和插值方法的知识，特别是关于声学系统的特定动态特性，并促进与数学建模社区的跨学科科学交流。初步研究表明，将特征系统实现算法（ERA）改进为随机奇异值分解，可以很好地模拟室内声场。本文旨在用不同声学系统的测量数据来验证该方法，如空气和结构传输系统以及耦合振动声系统的辐射。此外，该方法将被扩展，以使声学系统中常见的输入输出延迟可以有效地建模。此外，还打算将模型参数纳入方法中，以便实现空间插值和减少测量工作量。

A measurement-based modelling approach can be a good alternative in cases where physical models exhibit large deviations from the real-world system. Input-output data in the form of transfer function or, equally, impulse response measurements form the core of such an approach.In practical applications, however, oftentimes high-dimensional measurements for many input-output pairings are needed in order to attain high model fidelity. Hence, established data-driven approaches can quickly lead to computationally demanding models. A solution ispresented in the form of state-space models because they provide access to a plethora of reduced order modelling methods. Recent advances inthe area of numerical linear algebra have brought forth randomized low-rank matrix factorizations that can be used to employ existing data-driven system identification methods on high-dimensional data, thus enabling their practical use in engineering acoustics for the first time. Besides the enhanced computational efficiency, another conceivable advantage of state-space is given by a possible reduction of the measurement effort which can be achieved through parametric state-space models with the spatial location as a parameter. The main goal of this proposal is the establishment of state-of-the-art data-driven system identification methods for the modelling of acoustical transmission systems in the area of engineering acoustics. The proposed actions are designed to increase and create knowledge about the purposeful employment of data-driven modelling and interpolation methods particularly regarding the specific dynamic properties of acoustical systems and to facilitate interdisciplinary scientific exchange with the community of mathematical modelling. Preliminary studies have shown that amodification of the Eigensystem Realization Algorithm (ERA) with a randomized Singular Value Decomposition is well suited for the modelling of an acoustic field in a room. It is intended to validate this method with measurement data from different acoustical systems such as air- and structure-borne transmission systems and the radiation of coupled vibro-acoustical systems. Furthermore, the method is to be augmented in way that the commonly present input-output delays of acoustical systems can be modelled efficiently. Additionally, it is intended to incorporate model parameters into the method such that spatial interpolation and reduced measurement effort can be achieved.