发表机构
School for the Engineering of Matter, Transport and Energy, Arizona State University（亚利桑那州立大学物质、运输与能源工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种针对PDE传递函数的有理逼近方法，通过Krylov子空间降阶和可控性矩阵投影，实现指数稳定并给出有限频率$H_\infty$误差界，数值示例验证了其性能。
AI 中文摘要
偏微分方程（PDEs）的传递函数是无理函数，且难以获得。与有理传递函数不同，针对无理传递函数的控制与分析仅有少数方法可用。因此，为了对PDE进行鲁棒分析与控制，我们需要使用传递函数的有理逼近，最好具有可证明的$H_\infty$误差界。本文中，我们将Krylov子空间模型降阶扩展到由偏积分方程（PIEs）——PDE的一种等价表示——建模的无限维系统。首先，我们提出一种PIE系统的状态变换，使得我们能够计算可控性矩阵。然后，为了获得有理逼近，我们将PIE状态投影到由截断的可控性矩阵张成的向量空间上。接下来，我们证明PIE系统的指数稳定性和适定性的充分条件保证了降阶系统的指数稳定性，并提供构造性的有限频率$H_\infty$误差界。最后，我们提供四个说明性数值示例来展示所提出方法的性能。
英文摘要
Transfer functions of Partial Differential Equations (PDEs) are irrational and difficult to obtain. Unlike for rational transfer functions, only a few methods are available for the control and analysis of irrational transfer functions. Thus, for robust analysis and control of PDEs, we need to use a rational approximation of the transfer function, preferably with provable $H_\infty$ error bounds. In this paper, we extend Krylov subspace model reduction to infinite-dimensional systems modeled by Partial Integral Equations (PIEs) -- an equivalent representation of PDEs. First, we propose a state transformation of the PIE system that allows us to compute a controllability matrix. Then, to obtain a rational approximation, we project the PIE states onto the vector space spanned by the truncated controllability matrix. Next, we show that sufficient conditions for exponential stability and well-posedness of the PIE system guarantee the exponential stability of the reduced-order system and provide constructive finite-frequency $H_\infty$ error bounds. Finally, we provide four illustrative numerical examples to demonstrate the performance of the proposed method.
