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 1 , Baozhu GUO 2 , Kunyi YANG 2 (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: w tt (x, t)+ w xxxx (x, t)+ σ(x)u(x, t)=0, w(0,t)= w x (0,t)=0, w xx (1,t)= w xxx (1,t)=0, y(t)= w t (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, 1 2 ), b > a, 0, otherwise. (2) We choose the energy state space as H = H 2 E (0, 1) × L 2 (0, 1), H 2 E (0, 1) = {f | f ∈ H 2 (0, 1), f (0) = f 0 (0) = 0}. H is equipped with the obvious inner product in- duced norm |(f,g)| 2 H = w 1 0 [|f 00 (x)| 2 + |g(x)| 2 ]dx for any (f,g) ∈ H. The input and output spaces are U = L 2 (0, 1) and Y = C 1 , 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, f 00 (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.
Stability analysis for an Euler-Bernoulli beam under local internal control and boundary observation
May 17, 2023
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