Selivanov, A. orcid.org/0000-0001-5075-7229 and Fridman, E. (2025) Improved residual mode separation for finite-dimensional control of PDEs: Application to the Euler–Bernoulli beam. Systems & Control Letters, 197. 106048. ISSN 0167-6911
Abstract
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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2025 The Authors. Except as otherwise noted, this author-accepted version of a journal article published in Systems & Control Letters is made available via the University of Sheffield Research Publications and Copyright Policy under the terms of the Creative Commons Attribution 4.0 International License (CC-BY 4.0), which permits unrestricted use, distribution and reproduction in any medium, provided the original work is properly cited. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ |
Keywords: | Distributed parameter systems; Euler–Bernoulli beam; H∞ control; modal decomposition |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > School of Electrical and Electronic Engineering |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 21 Feb 2025 11:54 |
Last Modified: | 21 Feb 2025 11:54 |
Status: | Published |
Publisher: | Elsevier BV |
Refereed: | Yes |
Identification Number: | 10.1016/j.sysconle.2025.106048 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:223648 |