# Observer Design for a Flexible Structure
with Distributed and Point Sensors

Alexander Zuyev and Julia Kalosha  Institute of Applied Mathematics and
Mechanics, National Academy of Sciences of Ukraine, Sloviansk

###### Abstract

The paper is devoted to the observability study of a dynamic system, which
describes the vibrations of an elastic beam with an attached rigid body and
distributed control actions. The mathematical model is derived using
Hamilton’s principle in the form of the Euler–Bernoulli beam equation with
hinged boundary conditions and interface condition at the point of attachment
of the rigid body. It is assumed that the sensors distributed along the beam
provide output information about the deformation in neighborhoods of the
specified points of the beam. Based on the variational form of the equations
of motion, the spectral problem for defining the eigenfrequencies and
eigenfunctions of the beam oscillations is obtained. Some properties of the
eigenvalues ??and eigenfunctions of the spectral problem are investigated.
Finite-dimensional approximations of the dynamic equations are constructed in
the linear manifold spanned by the system of eigenfunctions. For these
Galerkin approximations, observability conditions for the control system with
incomplete information about the state are derived. An algorithm for observer
design with an arbitrary number of modal coordinates is proposed for the
differential equation on a finite-dimensional manifold. Based on a quadratic
Lyapunov function with respect to the coordinates of the finite-dimensional
state vector, the exponential convergence of the observer dynamics is proved.
The proposed method of constructing a dynamic observer makes it possible to
estimate the full system state by the output signals characterizing the motion
of particular point only. Numerical simulations illustrate the exponential
decay of the norm of solutions of the system of ordinary differential
equations that describes the observation error.

##  1 Introduction

Control theory for flexible structures is a thoroughly studied field from
theoretical point of view. Controllability and stabilization problems for
different models of flexible beams have been the subject of interest of many
authors for several decades; see, for instance, [1, 2, 3, 4] and references
therein.

In real engineering structures that consist of networks of interconnected
beams, there is often a need to provide a feedback control which takes into
account an output information about the state of the system. A general
challenge is that most of the real settings have limitations for recording and
processing output signals. Namely, only partial information about the motion
of certain reference points is usually available for control purposes. That is
why the questions of observability and observer design become important for
estimating the complete state of a dynamic model [5, 6, 7]. Controllability
and observability problems have been studied for strings, plates, beams and
multi-link networks in [11, 12, 13, 14, 15, 16, 17, 18].

An analytical approach for designing a Luenberger observer for a finite-
dimensional linear system is based on constructing an auxiliary matrix of gain
parameters such that the error dynamics is asymptotically stable. Then the
above gain parameters can be defined by checking the Routh–Hurwitz or the
Liénard–Chipart conditions for the corresponding characteristic polynomial.
For mathematical models of flexible structures with multiple degrees of
freedom, this approach requires the analysis of high order polynomial
inequalities with parameter dependent coefficients. The implementation of this
scheme may be a very difficult technical task. An alternative approach for
observer design is based on Lyapunov’s direct method. The idea is to take a
suitable quadratic form as a Lyapunov function candidate and define the gain
parameters in such a way that the observation error dynamics has stable
trivial equilibrium. Moreover, it is possible to prove that the trivial
solution is asymptotically stable with the help of the Barbashin–Krasovskii
theorem (or LaSalle’s invariance principle). This technique has been applied
to observer design and observed-based stabilization in [8, 9, 10] for
mathematical models of flexible manipulators.

In the present paper, we study the mathematical model that describes a
flexible beam attached to a rigid body (shaker). The beam is simply supported
at the ends by two hinges and is actuated by distributed piezoelectric
controllers and the shaker. In previous studies, the stabilization problem
with a state feedback was investigated for this infinite-dimensional system.
The control design that ensures strong asymptotic stability of the trivial
equilibrium was proposed in [19]. Now, we widen our study to the system with
output and question ourselves whether it is possible to fully reconstruct the
state vector having at our disposal only output signals on a subregion of the
domain. In this work, the observability problem is investigated for a finite-
dimensional approximation of the dynamics. An observer design that guarantees
the exponential decay of the observation error is proposed.

The rest of this paper is organized as follows. The mechanical system under
consideration is described in section 2, and its equations of motion are
derived from the variational principle in section 3. Some properties of
eigenvalues of the associated spectral problem are discussed in sections 4. A
finite-dimensional approximation of the dynamical equations based on
Galerkin’s method is proposed in section 5. The main results (observability
conditions together with observer design scheme) are presented in section 6.
Finally, we illustrate the convergence of the proposed observer with some
numerical simulations in section 7.

##  2 Modeling of a flexible beam with a shaker

Consider a flexible beam supported at both ends and attached to the shaker.
Let $l$ be the length of the beam, and let $w(x,t)$ denote the deflection of
the centerline of the beam at a point $x\in[0,l]$ and time $t$ (here $x=0$
stands for the upper end and $x=l$ for the lower one, correspondingly). We
assume that the shaker is attached at the point $x=l_{0}$, and $m$ is the mass
of the moving part of the shaker. Then the kinetic energy of the system
considered is

$K=\frac{1}{2}\int_{0}^{l}\rho(x){\dot{w}}^{2}(x,t)\,dx+\frac{m}{2}\dot{w}^{2}(l_{0},t),$
(1)

where $\rho(x)$ is the mass per unit length of the beam. We use dots to denote
derivatives with respect to $t$, and primes to denote derivatives with respect
to $x$. According to the Euler–Bernoulli beam model, the potential energy can
be written as follows:

$U=\frac{1}{2}\int_{0}^{l}E(x)I(x)\left(w^{\prime\prime}(x,t)\right)^{2}\,dx+\frac{\varkappa}{2}w^{2}(l_{0},t).$
(2)

Here $E(x)$ and $I(x)$ are the Young modulus and the moment of inertia of the
cross section area, respectively, and $\varkappa$ is the stiffness coefficient
of the shaker. In the sequel, we assume that damping may be neglected.

##  3 Variational form of the dynamical equations

Suppose that $w(x,t)$ defines the motion of the system considered for
$t\in[0,\tau]$, $\tau>0$. We have the following geometric boundary conditions:

$w(0,t)=w(l,t)=0.$

By using Hamilton’s principle, we get

$\delta\int_{0}^{\tau}(K-U)\,dt+\int_{0}^{\tau}\left\\{F\delta
w(l_{0},t)+\sum_{j=1}^{k}M_{j}\int_{0}^{l}\psi_{j}(x)\delta
w^{\prime\prime}(x,t)dx\right\\}dt=0,$ (3)

for each admissible variation $\delta w(x,t)$ of class
$C^{2}\left([0,l]\times[0,\tau]\right)$ satisfying the boundary conditions

$\delta w|_{t=0}=\delta w|_{t=\tau}=\delta w|_{x=0}=\delta w|_{x=l}=0.$

Here $F$ is the control force implemented by the shaker. In formula (3), we
assume also that there are $k$ piezoelectric actuators attached to the beam.
We describe the action of the $j$-th actuator by its density of the torque
$M_{j}$ and the shape function $\psi_{j}(x)$. These characteristics may be
computed by using results of the paper [20]. We assume that

${\rm supp}\,\psi_{j}\cap\\{0,l_{0},l\\}=\emptyset\quad\text{for
each}\;j=1,2,\dots,k.$

By performing the integration by parts with respect to $t$ in (3), we obtain

$\int_{0}^{l}\left(\rho\ddot{w}\delta
w+\left(EIw^{\prime\prime}-\sum_{j=1}^{k}M_{j}\psi_{j}\right)\delta
w^{\prime\prime}\right)dx+\left.\left(m\ddot{w}+\varkappa w-F\right)\delta
w\right|_{x=l_{0}}=0.$ (4)

Then another integration by parts with respect to $x$ yields the following
partial differential equation:

$\rho\ddot{w}(x,t)+\left(EIw^{\prime\prime}(x,t)\right)^{\prime\prime}=\sum_{j=1}^{k}\psi_{j}^{\prime\prime}(x)M_{j},\quad
x\in(0,l)\setminus\\{l_{0}\\},$ (5)

with boundary and interface conditions

$\begin{gathered}w|_{x=0}=w|_{x=l}=0,\;w^{\prime\prime}|_{x=0}=w^{\prime\prime}|_{x=l}=0,\\\
\left.\left(m\ddot{w}+\varkappa
w\right)\right|_{x=l_{0}}=\left.(EIw^{\prime\prime})^{\prime}\right|_{x=l_{0}-0}-\left.(EIw^{\prime\prime})^{\prime}\right|_{x=l_{0}+0}+F,\end{gathered}$
(6)

for $w(x,t)$ of class $C^{2}\left([0,l]\times[0,\tau]\right)$ such that the
derivatives $w^{\prime\prime\prime}(x,t)$ and
$w^{\prime\prime\prime\prime}(x,t)$ exist for all
$x\in[0,l]\setminus\\{l_{0}\\}$ and $t\in[0,\tau]$.

##  4 Spectral problem

To analyse the spectrum of oscillations of the mechanical system considered,
we first study the control system (5)–(6) with ${M_{1}=\ldots=M_{k}=0}$ and
$F=0$. For simplicity, we assume that the beam is homogeneous, i.e. $\rho$,
$E$, and $I$ are treated as constants in the sequel.

By substituting $w(x,t)=W(x)q(t)$ into the homogeneous part of (5)–(6), we get
$\ddot{q}(t)=-\lambda q(t)$ together with the following spectral problem:

$\begin{gathered}\frac{d^{4}}{dx^{4}}W(x)=\lambda\frac{\rho}{EI}W(x),\quad
x\in(0,l)\setminus\\{l_{0}\\},\\\
W(0)=W(l)=0,\;W^{\prime\prime}(0)=W^{\prime\prime}(l)=0,\;W\in C^{2}[0,l],\\\
W^{\prime\prime\prime}(l_{0}-0)-W^{\prime\prime\prime}(l_{0}+0)=\frac{\varkappa-\lambda
m}{EI}W(l_{0}).\end{gathered}$ (7)

The eigenfunctions of the above problem are

$W(x)=\left\\{\begin{array}[]{ll}C_{1}\sin(\mu x)+C_{2}\sinh(\mu x),&x\leq
l_{0},\\\
B_{1}\sin(\mu(x-l))+B_{2}\sinh(\mu(x-l)),&x>l_{0},\end{array}\right.$

where $\mu=\left(\frac{\lambda\rho}{EI}\right)^{1/4}$, and the constants
$C_{1},C_{2},B_{1},B_{2}$ satisfy the following linear algebraic system:

$\left(\begin{array}[]{cccc}\sin\mu l_{0}&-{\rm sh\,}\mu
l_{0}&-\sin\mu(l_{0}-l)&{\rm sh\,}\mu(l_{0}-l)\\\ \sin\mu l_{0}&{\rm sh\,}\mu
l_{0}&-\sin\mu(l_{0}-l)&-{\rm sh\,}\mu(l_{0}-l)\\\ \cos\mu l_{0}&{\rm ch\,}\mu
l_{0}&-\cos\mu(l_{0}-l)&-{\rm ch\,}\mu(l_{0}-l)\\\ \cos\mu
l_{0}+\frac{\varkappa-m\lambda}{EI\mu^{3}}\sin\mu
l_{0}&\frac{\varkappa-m\lambda}{EI\mu^{3}}{\rm sh\,}\mu l_{0}-{\rm ch\,}\mu
l_{0}&-\cos\mu(l_{0}-l)&{\rm
ch\,}\mu(l_{0}-l)\end{array}\right)\left(\begin{array}[]{c}C_{1}\\\ C_{2}\\\
B_{1}\\\ B_{2}\end{array}\right)=0.$ (8)

Thus, the eigenvalues $\lambda$ of the spectral problem (7) are obtained by
solving the equation

$\Delta(\lambda)=0,$ (9)

where $\Delta(\lambda)$ is the determinant of the $4\times 4$-matrix of (8). A
rigorous investigation of the characteristic equation (9) with regard to the
asymptotic behavior of its roots is carried out in [21].

Let $\lambda_{1}$, $\lambda_{2}$, $\dots$ be eigenvalues of (7), and let
$W_{1}(x)$ ,$W_{2}(x)$, $\dots$ be the corresponding eigenfunctions. Let us
define

$\left<W_{i},W_{j}\right>_{H}:=\int_{0}^{l}\rho
W_{i}(x)W_{j}(x)\,dx+mW_{i}(l_{0})W_{j}(l_{0}).$ (10)

We have the following lemma.

Lemma 1. If $\lambda_{i}\neq\lambda_{j}$ then
$\left<W_{i},W_{j}\right>_{H}=0$.

Proof. The integration by parts of the expression
$\int_{0}^{l}W_{i}^{\prime\prime}(x)W_{j}^{\prime\prime}(x)dx$ and exploiting
the interface conditions
$W_{j}^{\prime\prime\prime}(l_{0}+0)-W_{j}^{\prime\prime\prime}(l_{0}-0)=\frac{\lambda
m-\varkappa}{EI}W_{j}(l_{0})$, $j=1,\dots,N$, gives

$\int_{0}^{l}W_{j}(x)\frac{d^{4}}{dx^{4}}W_{i}(x)dx+\frac{\lambda_{i}m-\varkappa}{EI}W_{i}(l_{0})W_{j}(l_{0})$

on the one hand, and

$\int_{0}^{l}W_{i}(x)\frac{d^{4}}{dx^{4}}W_{j}(x)dx+\frac{\lambda_{j}m-\varkappa}{EI}W_{i}(l_{0})W_{j}(l_{0})$

on the other hand. Calculating the difference between these two expressions,
we obtain

$\int_{0}^{l}\left(W_{j}(x)\frac{d^{4}}{dx^{4}}W_{i}(x)-W_{i}(x)\frac{d^{4}}{dx^{4}}W_{j}(x)\right)dx+\frac{m}{EI}(\lambda_{i}-\lambda_{j})W_{i}(l_{0})W_{j}(l_{0})=0.$

As both $W_{i}(x)$ and $W_{j}(x)$ satisfy (7),

$(\lambda_{i}-\lambda_{j})\left<W_{i},W_{j}\right>_{H}=0,$

which leads to $\left<W_{i},W_{j}\right>_{H}=0$ for
$\lambda_{i}\neq\lambda_{j}$. $\square$

Lemma 2. Each eigenvalue of the spectral problem (7) is a non-negative real
number.

Proof. Consider the differential operator ${\cal L}:D({\cal
L})\to\tilde{H}=L^{2}[0,l]\times{\mathbb{R}}$ acting as

$\zeta=\left(\begin{array}[]{c}u(x)\\\ p\\\ \end{array}\right)\mapsto{\cal
L}\zeta=\left(\begin{array}[]{c}\frac{EI}{\rho}u^{\prime\prime\prime\prime}(x)\\\
\frac{\varkappa}{m}p-\frac{EI}{m}\left(u^{\prime\prime\prime}(l_{0}-0)-u^{\prime\prime\prime}(l_{0}+0)\right)\\\
\end{array}\right)$

with the domain

$D({\cal L})=\left\\{\begin{aligned} &\zeta\in H^{2}[0,l]\cup
H^{4}(0,l_{0})\cup H^{4}(l_{0},l)\times{\mathbb{R}}:\\\
&u(0)=u(l)=0,\;u^{\prime\prime}(0)=u^{\prime\prime}(l)=0,\\\
&u^{\prime\prime}(l_{0}-0)=u^{\prime\prime}(l_{0}+0),\;p=u(l_{0})\end{aligned}\right\\}\subset\tilde{H}.$

Then the spectral problem (7) can be rewritten as

${\cal L}\zeta=\lambda\zeta.$

Let us rewrite the bilinear form (10) in terms of elements of $\tilde{H}$:

$\left<\zeta_{i},\zeta_{j}\right>_{\tilde{H}}=\left<\left(\begin{array}[]{c}u_{i}\\\
p_{i}\\\ \end{array}\right),\left(\begin{array}[]{c}u_{j}\\\ p_{j}\\\
\end{array}\right)\right>_{\tilde{H}}:=\int\limits_{0}^{l}\rho
u_{i}(x)u_{j}(x)dx+mp_{i}p_{j}.$

A direct computation leads to

$\left<{\cal
L}\zeta_{i},\zeta_{j}\right>_{\tilde{H}}=\int\limits_{0}^{l}EIu_{i}^{\prime\prime\prime\prime}(x)u_{j}(x)dx+\varkappa
p_{i}p_{j}-EI\left(u_{i}^{\prime\prime\prime}(l_{0}-0)-u_{i}^{\prime\prime\prime}(l_{0}+0)\right)p_{j}$

and

$\left<\zeta_{i},{\cal
L}\zeta_{j}\right>_{\tilde{H}}=\int\limits_{0}^{l}EIu_{i}(x)u_{j}^{\prime\prime\prime\prime}(x)dx+\varkappa
p_{i}p_{j}-EI\left(u_{j}^{\prime\prime\prime}(l_{0}-0)-u_{j}^{\prime\prime\prime}(l_{0}+0)\right)p_{i}.$

The integration by parts gives $\left<{\cal
L}\zeta_{i},\zeta_{j}\right>_{\tilde{H}}=\left<\zeta_{i},{\cal
L}\zeta_{j}\right>_{\tilde{H}}$. Besides, it is easily verifiable that
$\left<\zeta,{\cal
L}\zeta\right>_{\tilde{H}}=\int\limits_{0}^{l}EI(u^{\prime\prime}(x))^{2}dx+\varkappa
p^{2}\geq 0$ for all $\zeta\in D({\cal L})$. So, ${\cal L}$ is positive self-
adjoint operator with respect to the bilinear form
$\left<\cdot,\cdot\right>_{\tilde{H}}$; thus the eigenvalues of ${\cal L}$ are
real and non-negative. $\square$

##  5 Galerkin’s method

Let us consider a finite set of eigenvalues $\lambda_{1}$, $\lambda_{2}$, … ,
$\lambda_{N}$ and the corresponding eigenfunctions $W_{1}(x)$, $W_{2}(x)$, …,
$W_{N}(x)$ of the spectral problem (7). Then we substitute

$w(x,t)=\sum_{j=1}^{N}W_{j}(x)q_{j}(t)$

into the variational form of equations of motion (4), assuming that (4) holds
for each

$\delta w(\cdot,t)\in{\rm span}\left\\{W_{1},W_{2},\dots,W_{N}\right\\},\quad
t\geq 0.$

As a result, we get the Galerkin system with respect to
$q_{1}(t),q_{2}(t),\dots,q_{N}(t)$:

$\ddot{q}_{j}+\lambda_{j}q_{j}=\frac{W_{j}(l_{0})}{\|W_{j}\|^{2}_{H}}u_{0}+\sum_{i=1}^{k}\frac{\int_{0}^{l}\psi_{i}(x)W_{j}^{\prime\prime}(x)\,dx}{\|W_{j}\|^{2}_{H}}u_{i},\;\;j=1,2,\dots,N,$
(11)

where $u_{0}=F$, $u_{i}=M_{i}$, $i=1,\dots,k$.

System (11) can be equivalently written as

$\dot{z}=Az+Bu,\quad
z=(q_{1},p_{1},\dots,q_{N},p_{N})^{T},\;u=(u_{0},u_{1},\dots,u_{k})^{T},$ (12)

where $A={\rm diag}(A_{1},A_{2},\dots,A_{N})$,

$A_{j}=\left(\begin{array}[]{cc}0&1\\\
-\lambda_{j}&0\end{array}\right),\;B=\left(\begin{array}[]{cccc}0&0&\ldots&0\\\
b_{10}&b_{11}&\ldots&b_{1k}\\\ \vdots&\vdots&\ddots&\vdots\\\ 0&0&\ldots&0\\\
b_{N0}&b_{N1}&\ldots&b_{Nk}\end{array}\right),$
$b_{j0}=\frac{W_{j}(l_{0})}{\|W_{j}\|^{2}_{H}},\;b_{ji}=\frac{\int_{0}^{l}\psi_{i}(x)W_{j}^{\prime\prime}(x)\,dx}{\|W_{j}\|^{2}_{H}},\;i=1,2,\dots,k.$
(13)

##  6 Luenberger observer design

Assume that the state $w(x,t)$ of the flexible beam can measured by $r$ strain
gauges located at points $x=l_{1},\dots,l_{r}$ and, moreover, the shaker
displacement $w(l_{0},t)$ is measured as function of time. The current,
generated by the $s$-th gauge, is proportional to
$\dot{w}^{\prime\prime}(l_{s},t)$. By assuming that the measuring device
implements a current integrator, we treat the signals

$y_{0}(t)=w(l_{0},t),\;y_{s}(t)=w^{\prime\prime}(l_{s},t),\quad s=1,\dots,r,$
(14)

as the output for the mathematical model (5)–(6). Then, as the representation

$w(x,t)=\sum_{j=1}^{N}W_{j}(x)q_{j}(t)$

is used to derive Galerkin’s system (12), we express the output (14) in terms
of functions $q_{1}(t),\dots,q_{N}(t)$ as follows:

$y_{0}(t)\approx\sum_{j=1}^{N}W_{j}(l_{0})q_{j}(t),\;y_{s}(t)\approx\sum_{j=1}^{N}W_{j}^{\prime\prime}(l_{s})q_{j}(t),\quad
s=1,\dots,r.$

Thus, we assume that the output signal $y(t)=Cz(t)\in{\mathbb{R}}^{r+1}$ is
available for the finite-dimensional control system (12), where

$y=Cz,\,C=\left(\begin{array}[]{lllllll}c_{01}&0&\ldots&c_{0N}&0\\\
c_{11}&0&\ldots&c_{1N}&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
c_{r1}&0&\ldots&c_{rN}&0\end{array}\right),\;c_{0j}=W_{j}(l_{0}),\,c_{sj}=W_{j}^{\prime\prime}(l_{s}),\,s=\overline{1,r}.$
(15)

Lemma 3. Let the eigenvalues $\lambda_{1}$, $\lambda_{2}$, …, $\lambda_{N}$ be
distinct and, for each $j=\overline{1,N}$, there is an $s\in\\{0,...,r\\}$
such that $c_{sj}\neq 0$. Then system (12) with output (15) is observable.

Proof. Let us prove that ${\rm rank}\,H=2N$, where the matrix

$H=\left(\begin{array}[]{l}C\\\ CA\\\ \vdots\\\ CA^{2N-1}\end{array}\right)$

by a suitable permutation of its rows and columns can be transformed to the
following block form:

$\left(\begin{array}[]{cc}H_{0}&0\\\ \vdots&\vdots\\\ H_{r}&0\\\ \hline\cr
0&H_{0}\\\ \vdots&\vdots\\\ 0&H_{r}\\\ \end{array}\right),\quad
H_{s}=\left(\begin{array}[]{ccc}c_{s1}&\ldots&c_{sN}\\\
-\lambda_{1}c_{s1}&\ldots&-\lambda_{N}c_{sN}\\\ \vdots&\ddots&\vdots\\\
-\lambda_{1}^{N-1}c_{s1}&\ldots&-\lambda_{N}^{N-1}c_{sN}\\\
\end{array}\right),\quad s=0,\dots,r.$

Each $H_{s}$ is a Vandermonde-like matrix whose determinant can be calculated
as

$\det H_{s}=\prod\limits_{i=1}^{N}c_{si}\prod\limits_{1\leq j<n\leq
N}(\lambda_{n}-\lambda_{j}),\quad s=0,\dots,r.$

Under the assumptions of Lemma 3, the rectangular block
$\left(\begin{array}[]{cc}H_{0}\\\ \vdots\\\ H_{r}\\\ \end{array}\right)$ has
a nonzero minor of order $N$. So, ${\rm rank}H=2N$ and system (12), (15) is
observable by Kalman’s observability criterion. $\square$

The following result describes an explicit scheme for constructing the
Luenberger observer for system (12), (15).

Theorem 1. Let system (12), (15) be observable and $\lambda_{j}>0$ for all
$j=\overline{1,N}$. Then, for any $z(0)\in{\mathbb{R}}^{2N}$,
$\bar{z}(0)\in{\mathbb{R}}^{2N}$, and any $u\in L^{\infty}[0,\infty)$, the
corresponding solutions $z(t)$ and $\bar{z}(t)$ of system (12) and

$\dot{\bar{z}}(t)=(A-FC)\bar{z}(t)+Bu(t)+Fy(t)$ (16)

satisfy the property

$\|z(t)-\bar{z}(t)\|\to 0\quad\text{as}\;t\to+\infty.$ (17)

Here

$F=\left(\begin{array}[]{ccc}f_{10}&\ldots&f_{1r}\\\ 0&\ldots&0\\\
\vdots&\cdots&\vdots\\\ f_{N0}&\ldots&f_{Nr}\\\ 0&\ldots&0\\\
\end{array}\right),\quad
f_{js}=\gamma_{s}\frac{c_{sj}}{\lambda_{j}\|W_{j}\|^{2}},\quad
j=\overline{1,N},\;s=\overline{0,r},$

and $\gamma_{s}>0$ are arbitrary constants.

Proof. Consider the observation error
$e(t)=z(t)-\bar{z}(t)=(\Delta_{1},\delta_{1},\ldots,\Delta_{N},\delta_{N})^{T}$.
Then subtracting (16) from (12) yields the following dynamics:

$\dot{e}=(A-FC)e.$ (18)

Let us consider the following positive definite quadratic form:

$2W(e)=\sum\limits_{j=1}^{N}\|W_{j}\|^{2}(\delta_{j}^{2}+\lambda_{j}\Delta_{j}^{2}),$

and compute its time derivative along the trajectories of system (18):

$\dot{W}(e)=-\sum\limits_{s=1}^{r}\sum\limits_{j=1}^{N}\|W_{j}\|^{2}\lambda_{j}\Delta_{j}f_{js}\sum\limits_{i=1}^{N}C_{si}\Delta_{i}=-\sum\limits_{s=1}^{r}\gamma_{s}\left(\sum\limits_{j=1}^{N}C_{sj}\Delta_{j}\right)^{2}\leq
0.$

According to Lyapunov’s theorem, the trivial solution of (18) is stable.

Our goal is to check whether the system (18) admits nontrivial trajectories on
the set ${\cal K}=\\{e\in\mathbb{R}^{2N}:\dot{W}(e)=0\\}$. As far as
$\gamma_{s}>0$ for all $s=0,\dots,r$, the identity $\dot{W}(e(t))\equiv 0$
implies that $\sum\limits_{j=1}^{N}C_{sj}\Delta_{j}(t)\equiv 0$ or, in the
matrix form,

$Ce(t)\equiv 0,$ (19)

which in turn implies that the time derivatives $\frac{d^{n}}{dt^{n}}(Ce(t))$
vanish for all $n=0,1,2,...$. In view of (19), system (18) reduces to

$\dot{e}=Ae$ (20)

on the set ${\cal K}$. Computing the derivatives of $Ce(t)$ up to $(2N-1)$-st
along the trajectories of (20), one can see that

$\frac{d^{n}}{dt^{n}}Ce(t)=CA^{n}e(t),\quad n=0,\ldots,2N-1.$

So, the components of $e(t)$ satisfy the following algebraic system:

$CA^{n}e(t)=0,\quad n=0,\ldots,2N-1.$ (21)

As system (12), (15) is observable, system (21) admits only the trivial
solution $e(t)\equiv 0$ because of Kalman’s observability rank condition.
Thus, the trivial solution of (18) is asymptotically stable by the
Barbashin–Krasovskii theorem. $\square$

Remark. As the error dynamics (18) is linear in the considered case, the
obtained asymptotic stability result guarantees in fact the exponential
convergence in (17). Note that the assertion of Theorem 1 can be reformulated
in terms of partial asymptotic stability of the extended system (12), (15),
(16) with respect to the variables $z-\bar{z}$. This type of stability
conditions can be formally analyzed with extensions of the
Barbashin–Krasovskii–LaSalle results for partial stabilization of ordinary
differential equations (see, e.g., [22]) and dynamical systems in abstract
spaces [23].

##  7 Numerical simulations

The goal of this section is to illustrate the behavior of solutions of control
system (12), (15) and its dynamic observer (16) by numerical simulations for
the model with $N=6$ modes of oscillations. All computations are carried out
for the following realistic mechanical parameters (cf. [21]):

$\displaystyle l=1.875\,\text{m},\quad l_{0}=1.4\,\text{m},\quad E=7.1\cdot
10^{10}\,\text{Pa},\quad I=1.6875\cdot 10^{-10}\,\text{m}^{4},$
$\displaystyle\rho=2660\cdot 2.25\cdot 10^{-4}\,\text{kg}/\text{m},\quad
m=0.045\,\text{kg},\quad\varkappa=2630\,\text{N/m}.$

The model is driven by the periodic force $F=\sin 4t$ and the beam
displacement at $x=l_{0}$ is taken as the output, so that the piezo actuators
are switched off the data from piezo sensors is neglected. For numerical
simulations, we choose the observer gain parameter $\gamma_{1}=3$ and the
initial conditions $\bar{z}(0)=0$, $q_{i}(0)=0.1$, $p_{i}(0)=0.1$,
$i=1,\dots,N$. The overall weighted error
$\|e(t)\|^{2}=\frac{\varkappa}{2}\sum\limits_{j=1}^{N}(\Delta_{j}(t))^{2}+\frac{m}{2}\sum\limits_{j=1}^{N}(\delta_{j}(t))^{2}$
is presented in Fig. 1, while high frequency components of the observation
error are depicted in Figs. 2 and 3.

Fig. 1. The time plot of $\|e(t)\|^{2}$.

Fig. 2. The time plot of
$\frac{\varkappa}{2}(\Delta_{N}(t))^{2}+\frac{m}{2}(\delta_{N}(t))^{2}$.

Fig. 3. The time plot of
$\frac{\varkappa}{2}(q_{N}(t)-\bar{q}_{N}(t))^{2}+\frac{m}{2}(p_{N}(t)-\bar{p}_{N}(t))^{2}$.

The above time plots confirm the convergence of the proposed observer for the
considered multidimensional model.

##  8 Conclusions and future work

In this paper, we have proposed an explicit analytic approach for constructing
the Luenberger observer for a class of finite-dimensional models of flexible
structures. This observer design allows estimating the full system state for
an arbitrary dimension of the output vector. We do not solve here the output
stabilization problem leaving this task for future investigations. In the
previous work [19], the stabilization problem was solved for the beam-shaker
model by a feedback law depending on the infinite-dimensional state vector.
So, another direction of further study is related to the observer-based
stabilization of this class of distributed parameter flexible structures.

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