exponential stability of uniform euler bernoulli beams with non collocated boundary controllers 2014...

7/27/2019 Exponential Stability of Uniform Euler Bernoulli Beams With Non Collocated Boundary Controllers 2014 Journal of

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J. Math. Anal. Appl. 409 (2014) 851867

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Journal of Mathematical Analysis andApplications

journal homepage: www.elsevier.com/locate/jmaa

Exponential stability of uniform EulerBernoulli beams withnon-collocated boundary controllers

Yun Lan Chen , Gen Qi XuDepartment of Mathematics, Tianjin University, Tianjin, 300072, China

a r t i c l e i n f o

Article history:

Received 7 June 2012Available online 27 July 2013Submitted by David Russell

Keywords:

EulerBernoulli beamNon-collocated controllersAsymptotic analysisRiesz basisExponential stability

a b s t r a c t

We study thestability of a robot systemcomposedof two EulerBernoulli beams with non-collocated controllers. By the detailed spectral analysis, we prove that the asymptotical

spectra of the system are distributed in the complex left-half plane and there is a sequenceof the generalized eigenfunctions that forms a Riesz basis in the energy space. Since thereexist at most finitely many spectral points of the system in the right half-plane, to obtainthe exponential stability, we show that onecan choosesuitablefeedbackgainssuch that all

eigenvalues of the system are located in the left half-plane. Hence the Riesz basis propertyensures that the systemis exponentially stable. Finally we give some simulation forspectraof the system.

2013 Elsevier Inc. All rights reserved.

1. Introduction

In recentyears there hasbeen an increasedinterestin systems composed of multiple robotssincethey exhibit cooperativebehaviors. Such systems are of interest for several reasons: on one hand, the tasks may be inherently too complex for a singlerobot to accomplish, or performance benefits canbe gained from using multiple robots; on theother hand, building andusingseveral simple robots can be easier, cheaper, more flexible and more fault-tolerant than having a single powerful robot foreach separate task. Therefore, numerous cooperation problems emerge in the engineering. Thus the research on single robotsystems was naturally extended to that on multiple-robot systems since ultimately a single robot, no matter how capable,is spatially limited. The main questions are how to cooperate to achieve the predictable aim.

The multiple-robot systems are different from other distributed systems because of their implicit real-worldenvironment, they have presumably more difficulty in modeling than traditional components of distributed systemenvironments (i.e., computers, databases, networks). In the present paper, we consider how two robot-arms shouldcooperate to be qualified in one stable movement. Our system shown as in Fig. 1 consists of two robot arms, their commontask is to grab stably the huge mass M. To complete this task, we make the exterior forces Fi, i = 1, 2 act on two root endssuch that the other ends can clamp the mass, and we observe the velocity of the mass end so that the system completes the

performance. To model this system, we regard the mass as a tip mass and suppose that the arms are the same and uniformwith length one, whose positions are denoted by wi(x, t) with the root ends x = 0 and the arm motions described by theEulerBernoulli beams. To complete the performance, we observe the rotation angles wix(0, t), rotation angle velocitieswixt(0, t) and the velocity of other ends wit(1, t). As feedback controls we assume the control forces of the form

Fi = Fi(wixt(0, t), wix(0, t), wit(1, t)), i = 1, 2.The performance requirement is to make the displacement continuous at the other end x = 1, that is,

w1(1, t) = w2(1, t), t > 0,

This research was supported by the Natural Science Foundation of China (Grant NSFC-61174080, 61104130). Corresponding author.E-mail addresses: [email protected] (Y.L. Chen), [email protected] (G.Q. Xu).

0022-247X/$ see front matter 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jmaa.2013.07.048
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852 Y.L. Chen, G.Q. Xu / J. Math. Anal. Appl. 409 (2014) 851867

Fig. 1. Two robot arms.

here we neglect the size of mass. Thus, the motion of the system is governed by the partial differential equations:

2wi(x, t)

t2+

4wi(x, t)

x4= 0, i = 1, 2, x (0, 1), t > 0,

M2w(1, t)

t2=

2

i=1 3wi(1, t)

x3, t > 0,

w1(0, t) = w2(0, t) = 0, t > 0, 2wi(0, t)

x2= Fi(wixt(0, t), wix(0, t), wit(1, t)), i = 1, 2, t > 0,

w1(1, t) = w2(1, t) = w(1, t),w1(1, t)

x= w2(1, t)

x,

2i=1

2wi(1, t)

x2= 0, t > 0,

wi(x, 0) = wi0, wit(x, 0) = wi1, i = 1, 2,where x stands for the position and t the time and M is the tip mass at x = 1.

First, we consider the steady state of the system, wi, namelywi satisfy differential equation

d4wi(x)dx4

= 0, x (0, 1), i = 1, 2,w1(0) = w2(0) = 0,d2wi(0)

dx2= Fi(0, wix(0), 0), i = 1, 2,

w1(1) = w2(1), dw1(1)dx

= dw2(1)dx

,

2i=1

d2wi(1)dx2

= 0,

2

i=1d3

wi(1)

dx3

=0.

From this we can determine the control forcesFi = F(0, wix, 0), i = 1, 2, that are constant forces.Next we determine the forces Fi, i = 1, 2. Let Fi(z, u,v), i = 1, 2 be twice continuously differentiable functions.

Linearizing Fi at fixed point Wi = (0,wix(0), 0) yieldsFi(wixt(0, t), wix(0, t), wit(1, t)) = Fi(0,wix(0), 0) + Fi

z

Wi wixt(0, t)+ Fi

u

Wi (wix(0, t) wix(0)) + FivWi wit(1, t) + o(wi wi).

Let yi(x, t) = wi(x, t)

wi(x) and denote by

i = F

izWi , i = Fiu Wi , i = Fiv Wi , i = 1, 2.


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Y.L. Chen, G.Q. Xu / J. Math. Anal. Appl. 409 (2014) 851867 853

Then the behavior of the error system is determined by the following linearized equations

2yi(x, t)

t2+

4yi(x, t)

x4= 0, i = 1, 2,x (0, 1),

Md2y(1, t)

dt2=

2i=1

3yi(1, t)

x3,

y1(0, t) = y2(0, t) = 0, 2yi(0, t)

x2= ui(t), i = 1, 2,

y1(1, t) = y2(1, t) = y(1, t),y1(1, t)

x= y2(1, t)

x,

2i=1

2yi(1, t)

x2= 0,

yi(x, 0) = wi0 wi, yit(x, 0) = wi1 wi, i = 1, 2,

(1.1)

where

ui(t) = iy

i(0, t)

x+ i

2yi(0, t)

xt+ i

yi(1, t)

t, i = 1, 2.

For simplification, we can take Fi = Fi + ui(t). In control input ui(t), the terms yi(0,t)x and 2yi(0,t)xt are collocated to theforceFi, but the term yi(1,t)t is from the tip mass end. In the performance, we ensure the position term y1(1, t) = y2(1, t)mainly based on the term

yi(1,t)

t, so j = 0 is necessary. Therefore the system (1.1) can be regarded as a linear system

with the non-collocated feedback control law (namely, the actuators and the sensor are located at different positions) andi, i, i can be regarded as the feedback gain constants.

Finally, under guaranteeing the performance, we seek for the conditions on the coefficients i, i and i that ensure thesystem (1.1) is exponentially stable. From the description above we see that guaranteeing the performance of the systemrequests the coefficients j = 0,j = 1, 2. Under these restrictions, the stability of the original nonlinear system is equivalentto that of(1.1) by determining the parameters i, i and i. Therefore, in the present paper, we mainly pay our attention todetermine the parameters i, i and i.

Non-collocated control has been widely used in engineering practice due to its more feasiblity (e.g. see, [1,4,18,11,14,5,20,22,27,29]); however, there is few theoretical study including stability analysis on such systems from the view ofmathematical control. The first difficulty for the non-collocated control comes from the non-minimum-phase of the open-loop form, a small increment of the feedback gain will lead to an unstable closed-loop system. The second difficulty arisesfrom the non-dissipativity for closed-loop form, which gives rise to difficulty in applying the traditional Lyapunov methodsor the energy multiplier methods to analyze the stability. Comparing with the huge works on the stabilization of collocatedPDEs in existed literature, the study for non-collocated PDEs is fairly scarce. To analyze the stability of the system with non-collocated feedback, some authors used the finite-dimensional approximations of the observers for infinite-dimensionalsystems (for analytic systems), we refer the reader to the literature [2,9,10,12,13]. There were some authors using non-collocated feedback controllers to stabilize the system (e.g. see [17,8,7]). To remove the negative effect of non-collocatedterm, however, these authors must use more complex feedback control signal. The method used by them, however, does notfit our model. Recently our paper [3] adopt direct feedback manner to stabilize exponentially a wave system with variablecoefficients, in which the key point is to choose the suitable feedback gains. In the present paper, we shall use the idea in [3]

to determine the parameters i, i and i, i = 1, 2.In what follows, we describe the method used in this paper. Consider the energy functional of the system (1.1), which is

defined as

E(t) = 12

2j=1

10

[|yjxx(x, t)|2 + |y2jt(x, t)|2]dx +1

2My2t(1, t) +

2j=1

1

2j|yjx(0, t)|2.

Formally differentiating with respect to t yields

dE(t)

dt=

2j=1

10

[yjxxyjxxt +yjtyjtt]dx + Myt(1, t)ytt(1, t) +2

j=1jyjx(0, t)yjxt(0, t)

=2

j=1[jy2jxt(0, t) jyjt(1, t)yjxt(0, t)].


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Clearly, the energy of the system is not dissipative for all j, j, which means a non-minimum-phase system. Our aim is to

choose suitable j, j such that the energy of the system decays exponentially, at least decays, i.e.,dE(t)

dt< 0. In the present

paper, the main task is to prove that this can be done.To analyze the stability of(1.1), we employ the asymptotic analysis technique to obtain the asymptotic frequency of the

system (the asymptotic spectrum of thesystem operator). Furthermore we prove theRiesz basis property of theeigenvectorsof this system. Note that the Riesz basis property implies the spectrum determined growth condition. Hence we can assertthe stability of the system can be determined via spectrum of the system operator. This approach has been extensively used

in the system analysis, for instance, [21,6,19,24,23] for single beam or string system and [25] for the serially connectedTimoshenko beams. Our model is different from the literature mentioned above, however, there may exist finitely manyeigenvalues of the system in the right half-plane, although its asymptotic spectra are in the left half-plane. So we shallchoose suitable parameters i and i such that all spectra are located in the left half-plane.

The rest is organized as follows. In Section 2, we discuss the well-posedness of the system (1.1). In Section 3, we carryout a complete asymptotic analysis for the spectrum of the system operator. In Section 4, we prove the Riesz basis propertyof the eigenvectors and generalized eigenvectors of the system operator. In Section 5, we discuss the exponential stabilityof the system under some conditions and give some simulation for the spectrum of the system. Section 6 concludes.

2. Well-posedness of the system

In this section, we shall study the well-posedness of system (1.1). We begin with formulating the system into anappropriate Hilbert state space.

Let Hk

(0, 1) be the usual Sobolev space. SetY := {y = (y1,y2) H2(0, 1) H2(0, 1)|y1(0) = y2(0) = 0,y1(1) = y2(1),y1(1) = y2(1)}

and Z = {z = (z1,z2) L2[0, 1] L2[0, 1]}. Let state spaceH = Y Z C, (2.1)

equipped the inner product, for any (y,z,p), (f,g, h) H, via

(y,z,p), (f,g, h)H =2

j=1

10

yj (x)f

j (x) +zj(x)gj(x)

dx +

2j=1

jy

j(0)f

j (0) + Mph,

by which the induced norm is

(y, v,p)2

=2

i=1 1

0[|yi (x)|

2

+ |vi(x)|2

]dx +2

i=1i|yi(0)|

2

+ M|p|2

. (2.2)

Obviously, (H, ) is a Hilbert space.Define the operator A inH by

A(y, v,p) =

v, y(4), 1M

2j=1

yj (1)

(2.3)

with domain

D(A) =

(y, v,p) Y [H4(0, 1)]2 Y C,yj (0) = jyj(0) + jvj (0) + jvj(1), j = 1, 2,

p=

v1

(1)=

v2

(1),

2

j=1 yj (1) = 0,

. (2.4)

With the operator A at hand, we can write system (1.1) into an evolutionary equation in HdY(t)

dt= AY(t), t > 0,

Y(0) = Y0(2.5)

where Y(t) = (y(, t), yt

(, t), yt

(1, t)) and Y0 = (U0, U1, U2) H is the initial data given.To discuss the well-posedness of system (1.1), we need the following definition of dissipative operator.Let X be a Banach space and X be its dual space, and denote by x,x the value ofx X at x X. For every x X, the

duality set ofx, F(x) X, is defined asF(x) = {x X|x,x = x2 = x2}.

Definition 2.1 ([15]). A linear operator A is dissipative if for every x D(A), there is a x F(x) such that Ax,x 0.


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Lemma 2.1. LetHandA be defined by (2.1), (2.3) and (2.4) respectively. ThenA is a closed and densely defined linear operator

in H. If j jj,j = 1, 2, then A ( 1M2

j=1 jj)I is a dissipative operator in H.

Proof. It is easy to check that A is a densely defined and closed linear operator. Here we only prove A ( 1M

2j=1 jj)I is

dissipative.For any F = (y, v,p) D(A), a directly calculation gives

AF, FH =2

i=11

0

vi (x)yi (x) y(4)i (x)vi(x) dx + 2i=1

yi (1)p +2

i=1ivi (0)yi(0)

=2

i=1

vi (0)yi (0) + ivi (0)yi(0)

+

2i=1

10

vi (x)yi (x)dx

10

yi (x)vi (x)dx

.

So we have

2AF, FH =2

i=1

vi (0)yi (0) + vi (0)yi (0)

+

2i=1

i

viy

i(0) +yi(0)vi (0)

.

Using the connective and boundary conditions, we get that

AF, FH = 2

i=1i|vi (0)|2 + i(vi(1)vi (0))

2

i=1i|vi (0)|2 +

2i=1

i

i|vi(1)|2 +

1

i|vi (0)|2

2

i=1

i +

i

i

|vi (0)|2 +

ii

MM|p|2

2

i=1

i +

i

i

|vi (0)|2 +

ii

MF, FH

where i > 0. So it holds that

A 1M

2i=1

iiI

F, FH

2

i=1

i + ii

|vi (0)|2 0ifii i, i = 1, 2. The desired result follows.

Lemma 2.2. LetH and A be defined as before. Then 0 (A) and A1 is compact on H, and hence the spectrum (A) of Aconsists of all isolated eigenvalues of finite algebraic multiplicities. In particular, all eigenvalues appear in conjugate pairs on the

complex plane.

Proof. For any F = (f,g, h) H, we consider the solvability of equationAY = F, Y = (y, v,p) D(A),

that is, y, v and p satisfy the equations

v(x) = f(x), (2.6)y(4)i (x) = gi(x), i = 1, 2, (2.7)1

M

2i=1

yi (1) = h, (2.8)

and the boundary conditions

y1(0) = y2(0) = 0, y1(1) = y2(1),yi (0) = iyi(0) + ivi (0) + ivi(1), i = 1, 2,p = v1(1) = v2(1), y1(1) = y2(1),

2i=1

yi (1) = 0.(2.9)


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Solving the differential equation (2.7) yields

yi(x) = yi(0) +xyi(0) +x2

2yi (0) +

x3

3!yi (0)

x0

(x r)33! gi(r)dr. (2.10)

For simplification, we denote

Gi(x) = x0

(x

r)3

3! gi(r)dr, i = 1, 2,

and

k1 = G1(x) G2(x)|x=1, k2 = G1x(x) G2x(x)|x=1,k3 = 1G1(x) G1(x)|x=0 1f1x(0) 1f1(1),k4 = 2G2(x) G2(x)|x=0 2f2x(0) 2f2(1),k5 = G1(x) + G2(x)|x=1, k6 = G1 (x) + G2 (x)|x=1 + Mh.

Set

X = [y1(0),y1(0),y1 (0),y2(0),y2(0),y2 (0)]T,K = [k1, . . . , k6]

T

.

(2.11)

Substituting (2.10) into the boundary conditions (2.9) leads to the following algebraic equations

CX = K (2.12)where the coefficient matrix is

C =

11

2

1

3! 1 1

2 1

3!1 1

1

21 1 1

2

1 1 0 0 0 00 0 0 2

1 0

0 1 1 0 1 1

0 0 1 0 0 1

.

A direct calculation shows det[C] = 12

1 + (1 + 12 ) = 0. So Eqs. (2.12) have a unique solution X and hence we candetermine uniquely functions yi(x) H4(0, 1), i = 1, 2. Clearly, (y, v,p) = (y,f,f1(1)) D(A) andA(y, v,p) = (f,g, h).The inverse operator theorem asserts that A1 is bounded on H, which indicates that 0 (A). In addition, the fact thatD(A) Y[H4(0, 1)]2 YC H shows thatA1 is compact due to Sobolevs Embedding theorem. The spectral theoryof linear compact operator shows that (A) consists of all isolated eigenvalues of finite algebraic multiplicities. Note thatA is a real linear operator on H, so its spectrum distributes symmetrically with respect to the real axis. This completes theproof.

Thanks to the above lemmas and LumerPhillips theorem (see, [15]), we have the following result.

Theorem 2.1. LetA and H be defined as before. Then Agenerates a C0 semigroup T(t) on H. Hence, the system (1.1) is well-posed, that is, for any Y0 H, the system (1.1) has a unique solution Y(t) = T(t)Y0.

3. Asymptotic eigenvalue problem

To investigate the properties of the semigroup T(t) generated by A, we need to find out some spectral properties ofA. We know from Lemma 2.2 that the spectrum ofA consists of all isolated eigenvalues, so we need only to discuss theeigenvalue ofA and its asymptotical distribution. In this section, we shall calculate the asymptotic values of eigenvaluesofA.

Let

C, we consider the existence of a nonzero solution of the equation

A(y, v,p) = (y, v,p), (y, v,p) D(A).


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This is equivalent to v = y,p = v1(1) = v2(1) = y(1) and yi satisfies the following boundary eigenvalue problem

2yi(x) +y(4)i (x) = 0,y1(0) = y2(0) = 0,yi (0) (i + i)yi(0) iyi(1) = 0,y1(1) = y2(1) = y(1), y1(1) = y2(1),

2

i=1 yi (1) = 0,2

i=1yi (1) M2y(1) = 0.

(3.1)

To solve the above equations, let = 2, we rewrite above equations into the equations about parameter as follows:

4yi(x) +y(4)i (x) = 0,y1(0) = y2(0) = 0,yi (0) (i + 2i)yi(0) 2iyi(1) = 0, i = 1, 2,y1(1) = y2(1) = y(1), y1(1) = y2(1),

2

i=1 yi (1) = 0,2

i=1yi (1)

1

2M4

2i=1

yi(1) = 0.

(3.2)

Here we use the equalityy(1) = y1(1) = y2(1) = 12 (y1(1)+y2(1)). Obviously, the differential equation has general solution

yi(x) =4

j=1ai,je

jx

where all j are the distinct root of equation 4 = 1. Substituting these expressions into the boundary conditions in (3.2)

leads to

a11 + a12 + a13 + a14 = 0,a21 + a22 + a23 + a24 = 0,c11()a11 + c12()a12 + c13()a13 + c14()a14 = 0,c21()a21 + c22()a22 + c23()a23 + c24()a24 = 0,e1(a11 a21) + e2(a12 a22) + e3(a13 a23) + e4(a14 a24) = 0,1e

1(a11 a21) + 2e2(a12 a22) + 3e3(a13 a23) + 4e4(a14 a24) = 0,21e

1(a11 + a21) + 22e2(a12 + a22) + 23e3(a13 + a23) + 24e4(a14 + a24) = 0,31

M

2

e1(a11 + a21) +

32

M

2

e2(a12 + a22) +

33

M

2

e3(a13 + a23)

+

34 M

2

e4(a14 + a24) = 0

(3.3)

where

cij() = (j)2 ij

ij iej.

Denote by dj() = 3j M2 and

D() = det

1 1 1 1 0 0 0 00 0 0 0 1 1 1 1

c11() c12() c13() c14() 0 0 0 00 0 0 0 c21() c22() c23() c24()

e1 e2 e3 e4 e1 e2 e3 e41e

1 2e2 3e

3 4e4 1e1 2e2 3e3 4e4

21e1 22 e

2 23e3 24e

4 21e1 22e

2 23e3 24e

4

d1()e1 d2()e

2 d3()e3 d4()e

4 d1()e1 d2()e

2 d3()e3 d4()e

4

. (3.4)

Then we have the following result.


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Lemma 3.1. LetA be defined as before and D() be defined as (3.4), then = 2 C is an eigenvalue of A if and only ifD() = 0, i.e.,

(A) = { = 2|D() = 0, C}.

We are now in a position to determine the zeros ofD(). Due to the symmetry of the spectrum ofA with respect to thereal axis, we only need to discuss the case that arg (0, ) (complex up-half-plane), or equivalently arg (0,

2).

First, we consider the cases that arg (

0,

4 )and arg

(3

4 , ).

When arg (0, 4

), we have arg (0, 8

). We order j as follows

1 = e34 i, 2 = e

54 i, 3 = e

4 i, 4 = e

74 i

so that

(1) (2) < 0 < (3) (4), arg

0,

8

.

When arg ( 34

, ), we have arg ( 38

, 2

). Taking

1 = e34 i, 2 = e

4 i, 3 = e

54 i, 4 = e

74 i,

we have

(1)

(2) < 0

2C + C2|| + 12 C +

i||

C4|| + C24

2i=1 i

.

Since

Y2

= R(,A0)F

1

|| F

,

we find out

Y1 = R(,A)F (R(,A0)F + Y)

C5

|| +1

||

F, (A) (A0) R.

Therefore lim R(,A)F = 0.Note that G() is an entire function of finite exponential type, G() is uniformly bounded along the line = > 0.

The Phragmn-Linderlf theorem (cf. [28]) asserts that

|G()| M, C.

The Liouville theorem says that G() 0. Observe that G() = (F, R(,A)U0)H for any F H, so it must beR(,A)U0 = 0 which implies U0 = 0. Therefore Sp(A) = H.


14/17


In what follows, we shall study the Riesz basis generation of the (generalized) eigenvectors ofA, we need the followingresult that comes from [26].

Lemma 4.2. LetHbe a separable Hilbert space, and A be the generator of a C0 semigroup T(t) onH. Suppose that the followingconditions are satisfied:

(1) (A) = 1(A) 2(A), where 2(A) = {k}k=1 consists of isolated eigenvalues of A with finite multiplicity;(2) sup

k1ma(k) 0.

Then the following assertions are true.

(i) There exist two T(t)-invariant closed subspaces H1,H2 with the property that (A|H1 ) = 1(A) , (A|H2 ) =2(A), E(k,A)H2 forms a subspace Riesz basis forH2 and

H = H1

H2.

(ii) If supk1 E(k,A) < , thenD(A) H1 H2 H. (4.6)

(iii) H has the decomposition

H = H1 H2 (topological direct sum),if and only if

supn1

nk=1

E(k,A)

< . (4.7)Applying Theorem 4.1 and Lemma 4.2 to our problem, we have the following result.

Theorem 4.2. LetH and A be defined as (2.1) and (2.3), respectively. Then there is a sequence of eigenvectors and generalizedeigenvectors of A that forms a Riesz basis forH. In particular, the system associated with A satisfies the spectrum determined

growth condition.

Proof. Set 1(A) = {}, 2(A) = (A). Lemma 2.2 shows that A generates a C0 semigroup and the condition (1) inLemma 4.2 holds. Theorem 3.1 shows that, for sufficient large n, j,n are separable and ma(j,n) = 1, which means that theconditions (2) and (3) in Lemma 4.2 are fulfilled. Therefore, according to Lemma 4.2, there is a sequence of eigenvectors andgeneralized eigenvectors ofA that forms a Riesz basis for H2 due to ma(j,n) = 1 for sufficient large n. Theorem 4.1 showsthat the eigenvectors and generalized eigenvectors are complete in H, that is, H2 = H. Therefore the sequence is also aRiesz basis for H.

Note that when n is sufficient large, all j,n are simple eigenvalues ofA. There are probably finitely many eigenvaluesof multiplicity two. The Riesz basis property of the eigenvectors and generalized eigenvectors ofA ensures that the systemassociated with A satisfies the spectrum determined growth condition. The desired result follows.

5. Stability analysis of the cooperation system

5.1. Stability analysis

In this section, we shall discuss the stability of the system (1.1). It is well known that if(1, 2) = 0, then the system isexponentially stable. However, the performance of the nonlinear system requires j = 0. So the requirement is inherited tothe system (1.1). So we only need to consider the cases of (1, 2) > 0.

We know from Theorem 4.2 that the growth rate of the system is determined via the maximal real part of spectrum ofA. We denote by

s(,) = sup (A)

, (5.1)

where = (1, 2), = (1, 2). Then we have the following result.

Theorem 5.1. LetH and A be defined as (2.1) and (2.3), respectively. If we can choose the gain coefficients and such thats(, ) < 0, then the closed loop system (2.1) is exponentially stable.


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Fig. 2. The spectrum graph.

For sufficient small > 0, denote by the negative real

= 1 + 212

+ .

According to the asymptotic expression of the eigenvalues of A in Theorem 3.1, we see that there are at most finitelymany number of eigenvalues ofA in the half plane > . Since the asymptotic values of eigenvalues are not obviouslydependent on , we can assert that only the finite eigenvalues ofA depend upon the value of strongly. Therefore, we haveto check whether there exist > 0 such that s(, ) < 0. Obviously, if = 0, From the proof ofLemma 2.1 we can see thatA is a dissipative operator in H. In this case, we have s(, 0) < 0.

To obtain the stability of(1.1) for > 0, we need the following result, which comes from [16, Theorem 2.1].

Lemma 5.1. Suppose that B Rn is an open and connected set, h(, r ) is continuous in (, r ) C B and analytic in C, and zeros of h(, r ) in the right half plane { C| 0} are uniformly bounded. If for any r B1 B, whereB1 is a bounded, closed and connected set, h(,

r ) has no zero on the imaginary axis, then the sum of the orders of the zero of

h(,r ) in the open right half plane (

> 0) is a fixed number for B1, that is, it is independent of the parameter

r

B1.

To apply Lemma 5.1 to our case, we take

h(,r ) = D(,,)

where D(,,) = D() is defined as in (3.4) and = 2. Here we only consider the case that > 0 is fixed and variesin neighbor of the original point, B1 O(r) R2, that is, r = . Obviously, h(, r ) is analytic in and continuous inr . Theorem 3.1 ensures that zeros ofh(,

r ) in the right half-plane are uniformly bounded. By Lemma 5.1, there exists a

small neighbor B1 O(r) in which the zeros ofh(, r ) are constants in the right half-plane. Since r = = 0, there areno zeros ofh(,

r ) in the right half-plane. Therefore, there is no zero ofh(,

r ) in the right half-plane for all

r = B1.

Applying Lemma 5.1 to our model, using the continuity of s(,) with respect to , there exists () > 0 such that0 < < (), it holds that s(, ) < 0 since s(, 0) < 0. Therefore, we have the following result.

Theorem 5.2. LetA be defined by (2.3) and (2.4). Then for each > 0, there exists () > 0 such that s(,) < 0 provided

that 0 < < (). Hence the closed loop system (1.1) is exponentially stable.From Theorem 5.2 we know that one can choose smaller such that the system (1.1) is exponentially stable. However

we cannot obtain an estimate for () or the relation between and . Therefore, how to choose the gain parameters is stillan important question in practice. In the next subsection, we shall give some simulations to show the relationship between and that make the closed-loop system (1.1) stable exponentially.

5.2. Simulation

In this subsection, we give some simulations for the eigenvalues of the system (1.1) to show the relation between and .

(1) The first simulation is to check the stability of the system for (1, 2) < (1, 2).Set = (1, 1) and = (5, 5) or set = (0.1, 0.1) and = (2, 2). By Matlab scientific calculation, we obtain the

finitely many number spectral points of the system (1.1), whose distributions are shown in Figs. 2 and 3, respectively. InFigs. 2 and 3, the notation denotes all spectral points.


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Fig. 3. The spectrum graph.

Fig. 4. The relation between and the spectrum.

The simulation results (see Figs. 2 and 3) show that all eigenvalues are located on the left hand side of the imaginaryaxis. Hence the system (1.1) is exponentially stable with such feedback gains. These results also show that the feedbackcontrollers are feasible for (1, 2) reasonable large.

(2) The second simulation is to describe change ofs(,) with = (1, 2).In order to study the behavior of the rightmost eigenvalue of the system with the change of gain parameter , we keep

= (1, 2) fixed and change the value of .Set = (1, 1), we take of the form = (1, 1) where R. Let change from 0 to 20 and take step length h = 0.1,

calculate the value ofs(,). We obtain the calculation results whose graphics are shown in Fig. 4. But we cannot obtain

an analytic expression between and the maximal real part of the spectrum since the figure is so complex.The simulation result (see, Fig. 4) shows that all eigenvalues of the system (1.1) are in the left half-plane. In this case,

the maximal real part of spectrum ofA is still negative. This implies that the system is still exponentially stable. However,when changes in interval [6, 8], s(,) has a singular variety. The rightmost eigenvalue approximates to 0.2 when isabout 7.

6. Conclusion

We studied the stabilityproblem of a robot systemcomposed of twoEulerBernoullibeamswith non-collocated feedbackcontrollers. The advantage of this class of the non-collocated feedback controllers is simpler and more feasible in practice.With the help of the Riesz basis approach and the asymptotic analysis technique, we proved the exponential stability ofuniform EulerBernoulli beam equations under non-collocated boundary feedback controllers with suitable choice of the

feedback gains. The key point of this kind of controllers is the suitable choice of the feedback gains. Although we do notobtain an analytic expression of relation between (1, 2) and (1, 2), we have proved that such a relation exists.


17/17


The simulation result (see, Fig. 4) shows that for the variables and of the form = (1, 1) and = (1, 1) , R,all eigenvalues of the system (1.1) are in the left half-plane. Hence the corresponding closed loop systems are exponentiallystable. We observe that and need not have this form. Our further work will determine (1, 2) for certain fixed = (1, 2).

Acknowledgments

The authors would like to thank the referees for their useful and helpful comments and suggestions.

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