The aim in model order reduction is to approximate an input-output map described by a large-scale dynamical system with a low-dimensional and cheaper-to-evaluate reduced order model. While high fidelity can be achieved by a variety of methods, only a few of them allow for rigorous error control. In this paper, we propose a rigorous error bound for the reduction of linear systems with balancing-related reduction methods. More specifically, we consider the simulation over a finite time interval and provide an a posteriori adaption of the standard a priori bound for Balanced Truncation and Balanced Singular Perturbation Approximation in that setting, which improves the error estimation while still yielding a rigorous bound. Our result is based on an error splitting induced by a Fourier series approximation of the input and a subsequent refined error analysis. We make use of system-theoretic concepts, such as the notion of signal generator driven systems, steady-states and observability. Our bound is also applicable in the presence of nonzero initial conditions. Numerical evidence for the sharpness of the bound is given.

翻译：模型订单削减的目的是近似一个由大型动态系统描述的输入-输出图,该动态系统具有低维和低廉价到低评价的减少订单模型。虽然通过各种方法可以实现高度忠诚,但只有少数方法允许严格的错误控制。在本文中,我们提出一个严格的错误,用平衡相关的减少方法缩小线性系统。更具体地说,我们考虑在一定时间间隔内进行模拟,并为该环境中的平衡调整和平衡 Singurbization Apprximation 提供标准的事后调整,从而改进误差估计,同时仍然产生严格的约束。我们的结果是基于输入的四倍数近似和随后改进的错误分析引起的差错分。我们使用系统理论概念,例如信号发电机驱动系统的概念、稳定状态和可耐性。我们的约束也适用于非零度初始条件的存在。为约束的精确度提供了数字证据。