局部内部控制与边界观测的Euler-Bernoulli梁稳定性分析

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"文章研究了在局部内部分布式控制和边界点观测下的欧拉-伯努利梁系统。设计了一个无限维观察器来处理开环系统的状态。通过估计的状态反馈，得到了非耗散的闭环系统。详细的研究谱分析揭示了一组广义特征函数，它们构成了状态空间的Riesz基。由此，得出谱决定增长条件和指数稳定性的结论。关键词包括：欧拉-伯努利方程、观察器、Riesz基、可控性和可观测性、稳定性。" 在本文中，作者们深入探讨了欧拉-伯努利梁模型在受控和被观测情况下的稳定性分析。欧拉-伯努利梁方程是描述细长梁动态行为的经典数学模型，广泛应用于结构工程、航空航天和机械设计等领域。梁系统受到局部内部分布式控制，意味着控制力沿着梁的某一部分分布，而不仅仅是作用在端点。同时，系统还配备了边界点观测，意味着可以从梁的一端或两端获取关于系统状态的信息。文章的核心是设计了一个无限维观察器来估计开环系统的状态。观察器是控制理论中的一个重要工具，它能够根据系统的输出数据来估计系统的内部状态。在这个系统中，观察器的设计允许对未直接测量的状态进行估算，从而实现更全面的系统控制。通过将观察器与状态反馈相结合，作者们构建了一个非耗散的闭环控制系统。这种闭环系统在没有外部能量输入的情况下仍能保持其性能，这是非耗散系统的一个关键特性。谱分析是理解这种系统动态行为的关键，因为它揭示了系统的频域特性。作者们进行的详细谱分析发现了一组构成状态空间Riesz基的广义特征函数。Riesz基在函数分析和控制理论中具有重要意义，因为它确保了状态空间的完备性和系统表示的稳定性。谱决定增长条件和指数稳定性是控制理论中的核心概念。谱决定增长条件表明系统的动态特性（如增长速度）是由其特征值决定的。指数稳定性则意味着系统在小扰动后会以指数速率回归到平衡状态，这是稳定性的一个强形式。在梁系统中，这两点的结合意味着即使在控制和观测的复杂条件下，系统也能保证其动态性能的稳定。这篇文章对欧拉-伯努利梁系统进行了深入的分析，提出了一个有效的控制策略，并通过谱分析证明了系统的稳定性和可控性。这些研究结果对于理解和设计实际工程中的梁结构控制具有重要的理论和实践价值。

J Control Theory Appl 2008 6 (4) 341–350

DOI 10.1007/s11768-008-7217-5

Stability analysis for an Euler-Bernoulli beam under

local internal control and boundary observation

Junmin WANG

, Baozhu GUO

, Kunyi YANG

(1. Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China;

2. Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100080, China)

Abstract: An Euler-Bernoulli beam system under the local internal distributed control and boundary point observation

is studied. An inﬁnite-dimensional observer for the open-loop system is designed. The closed-loop system that is non-

dissipative is obtained by the estimated state feedback. By a detailed spectral analysis, it is shown that there is a set of

generalized eigenfunctions, which forms a Riesz basis for the state space. Consequently, both the spectrum-determined

growth condition and exponential stability are concluded.

Keywords: Euler-Bernoulli equation; Observer; Riesz basis; Controllability and observability; Stability

1 Introduction

Most output stabilization controls for systems described

by partial differential equations (PDEs) are designed in a

collocated way, that is, the actuators and sensors are in the

same positions and designed in a “collocated” fashion. This

is natural in the sense that the proportional output feedback

for a collocated system results in a dissipative closed-loop

system, and methods like the Lyapunov function methods

and the multiplier methods can be used to get the stability

of the system.

On the other hand, it has been found long ago in engi-

neering practice that the performance of the collocated con-

trol design is not always good enough [1]. Although the

non-collocated control has been widely used in engineer-

ing systems [2∼4], the theoretical studies on these systems

from the mathematical control point of view are quite few.

The ﬁrst difﬁculty is that the open-loops of non-collocated

systems are usually not minimum-phase. This leads to the

closed-loop system being unstable for large feedback con-

troller gains. Second, the closed-loops of non-collocated

systems are usually non-dissipative, which gives rise to

the difﬁculty in applying the traditional Lyapunov function

methods or the multiplier methods to the analysis of the

stability. Recently, the estimated state feedback is designed

through backstepping observers in [5] to stabilize a class of

one-dimensional parabolic PDEs. The abstract observer de-

sign for a class of well-posed regular inﬁnite-dimensional

systems can be found in [6] but the stabilization is not ad-

dressed.

Recently, some efforts have been made for non-

collocated system control. In [7] and [8], the stabilization

of wave and beam equations under boundary control with

non-collocated observation has been handled respectively.

The objective of this paper is to study the stabilization of

an Euler-Bernoulli beam system under local distributed in-

ternal control with point boundary observation. This results

in a typical non-collocated system of PDEs. Such a distrib-

uted control is feasible in engineering practice due to the

application of the smart materials. The pointwise measure-

ment is the common observation for distributed parameter

systems.

The system we are concerned with is the following Euler-

Bernoulli beam with local internal distributed control and

boundary point observation:











(x, t) + w

xxxx

(x, t) + σ(x)u(x, t) = 0,

w(0, t) = w

(0, t) = 0,

(1, t) = w

xxx

(1, t) = 0,

y(t) = w

(1, t),

(1)

where w(x, t) represents the transverse displacement of the

beam at position x ∈ [0, 1] and time t > 0, u(x, t) is the

locally distributed control (input), y(t) is the output (obser-

vation), and

σ(x) =







1, x ∈ (a, b) ∈ (0,

), b > a,

0, otherwise.

(2)

We choose the energy state space as H = H

(0, 1) ×

(0, 1), H

(0, 1) = {f| f ∈ H

(0, 1), f(0) = f

(0) =

0}. H is equipped with the obvious inner product in-

duced norm |(f, g)|

[|f

(x)|

+ |g(x)|

] dx for any

(f, g) ∈ H. The input and output spaces are U = L

(0, 1)

and Y = C

, respectively.

Deﬁne the operator A : D(A)(⊂ H) → H as follows:











A(f, g) = (g, −f

(4)

D(A) = {(f, g) ∈ H| A(f, g) ∈ H,

(1) = f

000

(1) = 0}.

(3)

Received 17 September 2007; revised 30 January 2008.

This work was supported by the National Natural Science Foundation of China and the Program for New Century Excellent Talents in University of

China.

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