边界控制中的输入延迟欧拉-伯努利光束稳定化策略
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"该资源是一篇研究论文，发表在《系统与控制信函》(Systems & Control Letters)期刊2012年的第61期，页码为1069–1078。文章主要关注的是具有输入延迟的欧拉-伯努利光束在边界控制中的稳定化问题。作者是Yingfeng Shang和Genqi Xu，来自中国天津大学数学系。"
文章详细内容：
在现代工程和物理系统中，欧拉-伯努利光束模型常用于描述细长结构如桥梁、管道等的动态行为。这篇论文探讨了在边界控制中，当存在输入延迟时，如何实现欧拉-伯努利光束的稳定化。边界控制是指在结构的端点处施加控制力，以调整系统的动态特性。
论文指出，如果边界控制器的输出是控制器输入u(t)的线性组合，即αu(t) + βu(t - τ)，其中τ表示输入延迟，那么系统的行为会受到α和β值的影响。已知在没有延迟（τ = 0）的情况下，如果β=0，系统可以通过集中式反馈控制法实现指数稳定性。然而，当α=0且τ>0时，相同的控制策略会导致系统不稳定。
文章的核心贡献在于设计了一种动态控制器，即使对于任何|α| ≠ |β|和τ>0的情况，也能确保系统实现指数稳定性。这意味着无论α和β的绝对值是否相等，只要它们不相同，结合适当的延迟，都可以通过这种新的控制策略使系统稳定下来。
此外，论文还讨论了当|α|=|β|时系统的稳定性问题。这种情况下的分析可能更为复杂，因为这将涉及到反馈控制和输入延迟相互作用的微妙平衡。作者通过深入的理论分析和可能的数值模拟，为理解和解决这类问题提供了新的见解。
关键词包括：欧拉-伯努利光束、输入延迟、反馈控制、指数稳定性，这些都是文章研究的关键领域。最后，版权声明表明该工作由Elsevier B.V.所有，所有权利保留，这强调了文章的原创性和学术价值。
Systems & Control Letters 61 (2012) 1069–1078
Contents lists available at SciVerse ScienceDirect
Systems & Control Letters
journal homepage: www.elsevier.com/locate/sysconle
Stabilization of an Euler–Bernoulli beam with input delay in the boundary
control
✩
Ying Feng Shang, Gen Qi Xu
∗
Department of Mathematics, Tianjin University, Tianjin 300072, China
a r t i c l e i n f o
Article history:
Received 12 April 2011
Received in revised form
7 July 2012
Accepted 24 July 2012
Keywords:
Euler–Bernoulli beam
Input delay
Feedback control
Exponential stability
a b s t r a c t
In this paper, we consider a cantilever Euler–Bernoulli beam with controller delay. Suppose that the
output of the boundary controller is of the form αu(t) + βu(t − τ ), where u(t) is the controller input. It
is well known that a system with a collocated feedback control law is exponentially stable if β = 0, and
it is unstable with the same control law when α = 0. In the present paper, we shall design a dynamic
controller that makes the system stabilize exponentially for any |α| = |β| and τ > 0. Furthermore, we
discuss the stability of the system for |α| = |β|.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
It is well known that a cantilever Euler–Bernoulli beam with
input delay in the boundary control is described by (see [1])
w
tt
(x, t) + w
xxxx
(x, t) = 0, 0 < x < 1, t > 0,
w(0, t) = w
x
(0, t) = w
xx
(1, t) = 0,
w
xxx
(1, t) = αu(t) + βu(t − τ ),
w(x, 0) = w
0
(x), w
t
(x, 0) = w
1
(x),
u(θ) = f (θ ), θ ∈ (−τ , 0),
(1.1)
where α, β ∈ R with |α| + |β| = 0 are the controller constants,
τ > 0 denotes the delay time, and u(θ) = f (θ ), θ ∈ [−τ , 0] is a
bounded measurable function. If β = 0 and α > 0, which means
that the controller has no delay, the system can be exponentially
stabilized by u(t) = w
t
(1, t); see [2,3]. However, such a feedback
law no longer stabilizes the system if the controller has full delay,
i.e., β = 0 and α = 0; for a similar result, we refer to [4,5]. When
α > 0 and β ∈ R, the relationship between α and β concerning the
stability of this system is discussed in [1]. The result in [1] shows
that a system with the same feedback control law is exponentially
stable when α > |β| and unstable when |β| > α > 0. When
α = |β|, the system is asymptotically stable for τ ∈ ∪
k
{τ >
0, f
k
(τ ) = 0}, where f
k
(x) is a real function defined by [1, Formula
(15), Theorem VI.2].
✩
This research was supported by the National Science Natural Foundation in
China (NSFC–61174080).
∗
Corresponding author. Tel.: +86 2281831255.
E-mail addresses: xugq_2001@yahoo.com, gqxu@tju.edu.cn (G.Q. Xu).
In the past five years, a similar issue has been studied exten-
sively for wave equations. In 2006, Xu et al. [6] discussed the sta-
bility of one-dimensional wave equations with partial time delay
in the boundary feedback control. Using a detailed spectral analy-
sis, they proved that the wave equation is exponentially stable if
α > β > 0, but the system is unstable for 0 < α < β. Nicaise
and Pignotti [7] extended this result to multi-dimensional wave
equations with a delay term in the boundary or internal feedbacks.
Nicaise and Valein [8] obtained similar results for one-dimensional
wave networks with delay in the nodal feedbacks. In 2009, Ben-
hassi et al. [9] characterized the stabilization of a class of abstract
second-order evolution equations with delay for α > β > 0. The
results mentioned above show that α > β > 0 is a criterion of sta-
bility of a system with delay damping. However, we see from [1]
that, for practical control systems, the coefficients α and β are de-
termined by the controller; this represents an inherent property of
the controller. Usually one cannot determine whether α > β holds
or not. Therefore, how to design an anti-delay control law for any
α, β ∈ R with |α|+|β| = 0 is an important task. We note that this
issue has not been discussed until now. In the present paper, we
propose a dynamic feedback law to stabilize system (1.1) for any
α, β ∈ R with |α| = |β|.
The idea of the dynamic control strategy in this paper comes
from the classical Smith predictor. Suppose that the full state of the
system is measurable, and that y(t) is the observation value of the
endpoint velocity of (1.1), which has no time delay. It is well known
that, if the output feedback law u(t) = y(t) is adopted, system (1.1)
is unstable for α = 0 and β = 0. Since the state of the system is
measurable, the classical Smith predictor can give an estimate of
the future state of the system. It should be pointed out that the
0167-6911/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
doi:10.1016/j.sysconle.2012.07.012
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