改进的残余模态分离用于偏微分方程有限维控制:在欧拉-伯努利梁中的应用

IF 2.5 3区 计算机科学 Q3 AUTOMATION & CONTROL SYSTEMS
Anton Selivanov , Emilia Fridman
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引用次数: 0

摘要

考虑具有粘性和开尔文-沃伊特阻尼的简支欧拉-伯努利梁。我们的目标是用一个压电驱动器来衰减未知分布扰动的影响。我们展示了如何设计一个基于有限数量的主导模式的状态反馈控制器,以保证L2增益不大于给定值。如果忽略剩余的(无穷多个)模式,则计算出的L2增益是错误的。这是因为在反馈设计中没有考虑到控制对截断模式的影响时发生的溢出现象。我们提出了一个简单的修改H∞成本,以防止溢出。关键思想是将控制视为截断模式中的扰动,并使用有界实引理找到相应的L2增益。这些L2增益被添加到主导模式的H∞成本中的控制权值中,从而防止溢出。对具有实际参数的铝梁进行了数值模拟,验证了该方法的有效性。所提出的方法适用于其他类型的偏微分方程,如热方程、波方程和Kuramoto-Sivashinsky方程,以及它们的半线性版本。所提出的方法给出了一个Lyapunov函数,该函数也可用于保证成本控制、区域稳定性分析和输入到状态稳定性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Improved residual mode separation for finite-dimensional control of PDEs: Application to the Euler–Bernoulli beam
We consider a simply-supported Euler–Bernoulli beam with viscous and Kelvin–Voigt damping. Our objective is to attenuate the effect of an unknown distributed disturbance using one piezoelectric actuator. We show how to design a state-feedback controller based on a finite number of dominating modes that guarantees that the L2 gain is not greater than a given value. If the remaining (infinitely many) modes are simply ignored, the calculated L2 gain is wrong. This happens because of the spillover phenomenon that occurs when the effect of the control on truncated modes is not accounted for in the feedback design. We propose a simple modification of the H cost that prevents spillover. The key idea is to treat the control as a disturbance in the truncated modes and find the corresponding L2 gains using the bounded real lemma. These L2 gains are added to the control weight in the H cost for the dominating modes, which prevents spillover. A numerical simulation of an aluminum beam with realistic parameters demonstrates the effectiveness of the proposed method. The presented approach is applicable to other types of PDEs, such as the heat, wave, and Kuramoto–Sivashinsky equations, as well as their semilinear versions. The proposed method gives a Lyapunov functional that can also be used for guaranteed cost control, regional stability analysis, and input-to-state stability.
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来源期刊
Systems & Control Letters
Systems & Control Letters 工程技术-运筹学与管理科学
CiteScore
4.60
自引率
3.80%
发文量
144
审稿时长
6 months
期刊介绍: Founded in 1981 by two of the pre-eminent control theorists, Roger Brockett and Jan Willems, Systems & Control Letters is one of the leading journals in the field of control theory. The aim of the journal is to allow dissemination of relatively concise but highly original contributions whose high initial quality enables a relatively rapid review process. All aspects of the fields of systems and control are covered, especially mathematically-oriented and theoretical papers that have a clear relevance to engineering, physical and biological sciences, and even economics. Application-oriented papers with sophisticated and rigorous mathematical elements are also welcome.
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