Dynamic stabilization of an Euler–Bernoulli beam under boundary control and non-collocated observation

Abstract

We study the dynamic stabilization of an Euler–Bernoulli beam system using boundary force control at the free end and bending strain observation at the clamped end. We construct an infinite-dimensional observer to track the state exponentially. A proportional output feedback control based on the estimated state is designed. The closed-loop system is shown to be non-dissipative but admits a set of generalized eigenfunctions, which forms a Riesz basis for the state space. As consequences, both the spectrum-determined growth condition and exponential stability are concluded.

Introduction

Most approaches to controller design for systems described by partial differential equations (PDEs) use static collocated feedback. This means that the actuators and sensors are located in the same areas and are such that the control operator is the adjoint of the observation operator. Lyapunov or energy multiplier methods are often used for the stability analysis of the resulting closed-loop systems. It has been known for a long time that the performance of these closed-loop systems may not be good, see [4]. Several articles considered non-collocated control for specific systems described by PDEs using simulation and experiments [3], [15], [18], [21] but mathematical investigations of such controllers are quite few. The first difficulty behind is that the open-loop form of a non-collocated systems is usually not minimum-phase. This results in the closed-loop system unstable under a small increment of large feedback controller gains. Secondly, the closed loop of a non-collocate system is usually non-dissipative. The well-posedness of non-dissipative systems is a big challenge and furthermore, the traditional Lyapunov function or the energy multiplier methods are not easy to be used for the analysis of the stability of the non-dissipative systems.

The first effort on stabilizing compensators for infinite-dimensional systems was made in [8] where a generalization of the Luenberger compensator was given for the system in which both input and output operators are bounded. The earlier results about compensators for the distributed parameter systems can be found in [1]. For finite-dimensional compensators with unbounded input and bounded/unbounded output operators systems, we refer to [6], [14]. The abstract formulation for an observer-based stabilizing controller of regular well-posed linear infinite-dimensional systems was established in [23]. In past several years, some efforts are particularly made for flexible arms. The passivity property of a non-collocated single-link flexible manipulator with a parameterized output was studied in [15]. It was showed that for a non-collocated truncated passive transfer function, a PD controller is sufficient to stabilize the overall system [13] defined two transmission zero condition numbers that quantify zero sensitivity precisely and lead to a clear geometrical interpretation of the circumstances under which a non-collocated structure will posses ill-conditioned zeros. Recently, the estimated state feedbacks are designed through backstepping observers in [20] to stabilize a class of one-dimensional parabolic PDEs. The abstract observers design for a class of well-posed regular infinite-dimensional systems can be found in [7] but the stabilization was not addressed.

The objective of this paper is to generalize our recent work on the stabilization of wave equation under boundary control with non-collocated observation to the Euler–Bernoulli beam equation [9]. But there are some differences for these two kinds of systems. First, recent study shows that the stabilization of Euler–Bernoulli beam equations by backstepping methods is much hard than that for wave equations. It could be done only for some special boundary conditions through the control of linearized Schrödinger equations. Secondly, in some cases, Lyapunov methods are not effective in proving the stability for non-collocated beam equation but are effective for most of wave equations.

The problem we are concerned with is the following Euler–Bernoulli beam equation with boundary control and non-collocated observation:

$$\{\begin{array}{c}{w}_{tt}(x,t)+w_{xxxx}(x,t)=0,0<x<1,t>0\text{,}\\ w(0,t)=w_x(0,t)=w_{xx}(1,t)=0,t\ge 0\text{,}\\ w_{xxx}(1,t)=u\left(t\right),y\left(t\right)=w_{xx}(0,t),t\ge 0\text{,}\end{array}$$

where $u$ is the boundary shear control (or input) at $x=1$ and the observation (or output) $y$ is the bending strain at $x=0$. This problem comes from a dynamic model of vibrating beam for which one end is clamped to a control motor shaft and rotated by the motor at an angular velocity $\dot{\theta }\left(t\right)$ in the horizontal plane ([16], pp. 176):

$$\{\begin{array}{c}{z}_{tt}(x,t)+z_{xxxx}(x,t)=-x\ddot{\theta }\left(t\right),0<x<1,t>0\text{,}\\ z(0,t)=z_x(0,t)=z_{xx}(1,t)=-z_{xxx}(1,t)=0,t\ge 0\text{,}\\ y_z\left(t\right)=z_{tt}(1,t),t\ge 0\text{,}\end{array}$$

where $y_z\left(t\right)$ is the acceleration point output of the system. It is motivated by the acceleration point-sensing problem for weakly coupled wave equation considered in [2]. Let $w(x,t)=z_{xx}(1-x,t),\ddot{\theta }\left(t\right)=u\left(t\right),y\left(t\right)=-u\left(t\right)-y_z\left(t\right)$; we get (1.1) from (1.2).

We consider the system (1.1) in the energy state space $\mathbb{H}=H_E^2(0,1)\times L^2(0,1),H_E^2(0,1)=\left\{f\right|f\in H^2(0,1),f\left(0\right)=f^{\prime }\left(0\right)=0\}$. $\mathbb{H}$ is equipped with the obvious inner product induced norm ${\Vert (f,g)\Vert }_{\mathbb{H}}^2={\int }_0^1[{\left|f^{″}\left(x\right)\right|}^2+{\left|g\left(x\right)\right|}^2]dx$ for any $(f,g)\in \mathbb{H}$. And the input (output) space is $U={\mathbb{C}}^1$. Define a linear operator $\mathbb{A}:D\left(\mathbb{A}\right)(\subset \mathbb{H})\to \mathbb{H}$ as following:

$$\mathbb{A}(f,g)=(g,-f^{\left(4\right)})\text{,}D\left(\mathbb{A}\right)=\{(f,g)\in \mathbb{H}|\mathbb{A}(f,g)\in \mathbb{H},f^{″}\left(1\right)=f^{‴}\left(1\right)=0\}\text{.}$$

Then system (1.1) can be written as

$$\sum (\mathbb{A},\mathbb{B},\mathbb{C}):\{\begin{array}{c}\frac{d}{dt}\left(\begin{array}{c}\hfill w\hfill \\ \hfill w_t\hfill \end{array}\right)=\mathbb{A}\left(\begin{array}{c}\hfill w\hfill \\ \hfill w_t\hfill \end{array}\right)+\mathbb{B}u\left(t\right)\text{,}\\ \mathbb{B}=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill -\delta (x-1)\hfill \end{array}\right)\text{,}\\ y\left(t\right)=\mathbb{C}\left(\begin{array}{c}\hfill w\hfill \\ \hfill w_t\hfill \end{array}\right)=w_{xx}(0,t)\text{,}\\ \mathbb{C}=(〈{\delta }^{″}\left(x\right),\cdot 〉,0)\text{,}\end{array}$$

where $\delta (\cdot )$ denotes the Dirac delta function. Obviously, both $\mathbb{B}$ and $\mathbb{C}$ are unbounded operators.

Theorem 1.1

For each $u\in L_{\mathrm{loc}}^2(0,\infty )$ and initial datum $(w(\cdot ,0),w_t(\cdot ,0))\in \mathbb{H}$ , there exists a unique solution $(w,w_t)\in C(0,\infty ;\mathbb{H})$ to Eq.(1.1), and for each $T>0$ , there exists a $C_T>0$ independent of $u$ and $(w(\cdot ,0),w_t(\cdot ,0))$ such that

$${\Vert (w(\cdot ,T),w_t(\cdot ,T))\Vert }_{\mathbb{H}}^2+{\int }_0^T{\left|y\left(\tau \right)\right|}^{2}d\tau \le C_T[{\Vert (w(\cdot ,0),w_t(\cdot ,0))\Vert }_{\mathbb{H}}^2+{\int }_0^T{\left|u\left(\tau \right)\right|}^{2}d\tau ]\text{.}$$

Proof

By the well-posed linear infinite-dimensional system theory [5], [12], it is equivalent to showing that $\mathbb{C}$ is admissible for $e^{\mathbb{A}t}$, ${\mathbb{B}}^{\ast }$ is admissible for $e^{{\mathbb{A}}^{\ast }t}$ and the input–output map is bounded (see Definition 2.1, 2.5 of [12]). The former two facts are almost trivial, we need only to consider the boundedness of the input–output map under the zero initial condition: $w(x,0)=w_t(x,0)\equiv 0$.

Let $F\left(t\right)=\frac{1}{2}{\int }_0^1[w_t^2(x,t)+w_{xx}^2(x,t)]dx$. Then for any $T>0$, we have $\dot{F}\left(t\right)=-w_t(1,t)u\left(t\right)$ and hence

$$F\left(t\right)\le \frac{\delta }{2}{\int }_0^T{w}_t^2(1,t)dt+\frac{1}{2\delta }{\int }_0^T{u}^2\left(t\right)dt,\forall t\in [0,T]\text{ and }\delta >0\text{.}$$

Next, let ${\rho }_1\left(t\right)={\int }_0^{1}x{w}_x(x,t)w_t(x,t)dx$. Then $\left|{\rho }_1\left(t\right)\right|\le F\left(t\right)$ and

$${\dot{\rho }}_1\left(t\right)=\frac{1}{2}{w}_t^2(1,t)-w_x(1,t)u\left(t\right)-\frac{1}{2}{\int }_0^1[w_t^2(x,t)+3w_{xx}^2(x,t)]dx\text{.}$$

Integrate over $[0,T]$ with respect to $t$ and make use of (1.5) to give

$$\frac{1}{2}{\int }_0^T{w}_t^2(1,t)dt\le F\left(T\right)+\frac{1}{2}{\int }_0^T{u}^2\left(t\right)dt+4{\int }_0^{T}F\left(t\right)dt\le (\frac{\delta }{2}+2\delta T){\int }_0^T{w}_t^2(1,t)dt+(\frac{1}{2\delta }+\frac{1}{2}+\frac{2T}{\delta }){\int }_0^T{u}^2\left(t\right)dt$$

and hence

$${\int }_0^T{w}_t^2(1,t)dt\le \frac{2}{1-(1+4T)\delta }(\frac{1}{2\delta }+\frac{1}{2}+\frac{2T}{\delta }){\int }_0^T{u}^2\left(t\right)dt$$

as $0<\delta <\frac{1}{1+4T}$. This together with (1.5) gives

$$F\left(t\right)\le C_{\delta ,T}{\int }_0^T{u}^2\left(t\right)dt,\forall t\in [0,T]\text{ and }0<\delta <\frac{1}{1+4T}\text{,}$$

where $C_{\delta ,T}$ is a constant depending on $\delta ,T$ only.

Now, let ${\rho }_2\left(t\right)={\int }_0^1(x-1)w_x(x,t)w_t(x,t)dx$. Then $\left|{\rho }_2\left(t\right)\right|\le F\left(t\right)$ and

$${\dot{\rho }}_2\left(t\right)=\frac{1}{2}{w}_{xx}^2(0,t)-\frac{1}{2}{\int }_0^1[w_t^2(x,t)+3w_{xx}^2(x,t)]dx\text{.}$$

Integrate over $[0,T]$ with respect to $t$ and make use of (1.8) to produce

$$\frac{1}{2}{\int }_0^T{y}^2\left(t\right)dt\le F\left(t\right)+3{\int }_0^{T}F\left(t\right)dt\le (C_{\delta ,T}+3T{C}_{\delta ,T}){\int }_0^T{u}^2\left(t\right)dt\text{,}$$

where $T>0$ and $0<\delta <\frac{1}{1+4T}$. The proof is complete.  □

The significance of Theorem 1.1 is that it not only gives the well-posedness of the open-loop system (1.1) but also shows that for any $L^2$ control, the output $y$ makes sense and is also in $L^2$. This fact is a key point to the design of the observer because for the observer, $y$ becomes input and the $L^2$ property of $y$ plays an important role in solvability of the observer. The remaining part of this paper is organized as follows. In Section 2, we construct an observer for the system (1.1) and show that this observer is exponentially convergent. Section 3 is devoted to the output feedback design via the estimated state. In Section 4, we analyze the asymptotic behavior of the eigenvalues. The Riesz basis and exponential stability are presented in Section 5. Some concluding remarks are given in last Section 6.