Journal of Applied Mathematics and Physics, 2013, 1, 98-104
Published Online November 2013 (http://www.scirp.org/journal/jamp)
http://dx.doi.org/10.4236/jamp.2013.15015
Open Access JAMP
Spectral Analysis and Variable Structural
Contro l of an Elas t i c B e am
Xuezhang Hou
Mathematics Department, Towson University, Baltimore, USA
Email: xhou@towson.edu
Received September 6, 2013; revised October 8, 2013; accepted October 19, 2013
Copyright © 2013 Xuezhang Hou. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
An elastic beam system formulated by partial differential equations with initial and boundary cond itions is investigated
in this paper. An evolution equation corresponding with the beam system is established in an appropriate Hilbert space.
The spectral analysis and semigroup generation of the system operator of the beam system are discussed. Finally, a
variable structur al control is proposed and a sign ificant result that the solution of the system is exponentially stable un-
der a variable structural control with some appropriate conditions is obtained.
Keywords: Spectral Analysis; Semigroups of Linear Operators; Elastic Beam System; Variable Structural Control
1. Introduction
A great attention has been paid to the dynamics and con-
trol of flexible robot (see [1-5]) in the past thirty years
since the high-speed performance and low energy con-
sumption are highly demanded. In this paper, as a con-
tinuation of our work [6-9], we shall investigate an elas-
tic robot system formulated by partial differential equa-
tions with initial-boundary value conditio n s. By means of
functional analysis and semigroups of linear operators,
the beam system is described as an evolution equation in
an appropriate Hilbert space. Spectral properties and
semigroup generatio n of the system operator corr espond-
ing to the evolution equation are studied. Several signifi-
cant results are obtained.
Let us consider a robot system composed of two
link-arm and three joints, an electrical machinery is in-
stalled on each joint, the beam connecting with based
stand is rigid and forearm is elastic. By means of the
space kinetic and Hamilton's variation principle, we can
obtain the following second-order hyperbolic system that
describes the motion of the elastic beam system [10]:
  
 

22
2
1
22 2
3
2
1
22
,,
,
cos
uxt uxt
px
tx x
uxt
px
xtx
Lxg
 




 









(1)

 

22
2
2
22 2
3
2
2
22
,,
,
cos cossin
vxt vxt
px
tx x
vxt
px
xtx
Lx x
 




 







 
(2)
with the following boundary conditio ns:
  
11
0,0, 0,, 0,,,ututpu ltpultmg
 
 
(3)
  
22
0,(0, )0,,0,,0,vtvtpv ltpvlt
 
 
(4)
and initial condition s
 

 
01
01
01
10
,0,,0();
,0, ,0,
0,0;0
0;0 ,0
ux uxux ux
vxvxvxv x
0
1
,
,
 





 
where
,, ,uxt vxt are vertical and horizontal bend-
ing vibration displacements of the forearm of the robot
respectively;
1
px
1
, and are vertical and hori-
zontal local bending rigidity of the forearm of the robot
respectively; and

2
px
, 1
, 2
, and 2
are positive
constants such that
X. Z. HOU 99
 
0.1
ii i
pxi ,2;,,
 
,,

are the re-
spective base stand azimuth, the angle of eleva-
tion-depression of the forearm and the rear-arm of the
robot; 123

L
are the control moments of the forces on
the electrical machineries installed on the three joints;
and are the length of the forearm and the rear-arm
respectively;
l
is the damping coefficient of the struc-
ture;
is the line density of the forearm; is the
m
mass of the tip body;
g
is the acceleration of gravity;
here
,
, , m
g
and 1
I
, 2
I
, 3
I
are all positive
constants; the symbols and express
u
uut
and
ux respectively.
2. Evolution Equation of the Beam System
We start this section with defining following operators
 
2
2
1
22
,
ii
f
x
Afp x
xx











 





4
2
0,,,,are absolutely continuous function on0,,
0, ,000,,0.,01,2
ii
iii
DAfHlf ffpfl
pfLl ffpfltpflti
 

 
 

It should be noted that
f
in satisfying (1)
can be written as

1
DA
f
uu
, where the function suits
(1) and (3), and the function suits (3) and following
differential equation
u
u

22
1
22
0.
u
px
xx





(5)
By solving Equation (5) and (3) we find
 
01
1dd .
xy
l
z
ux mgzy
pz

(6)
Obviously,
f
in must suit Equation (1) if
suits (1) and (3), as well as suits (5) and (3)

1
DA
uu
Lemma 2.1 The operators 1
A
and 2
A
are positive
self-adjoint operators in
2
L0,l, moreover, 1
1
A
and
1
2
A
exist, and they are compact operators.
Proof Apply integration by parts with the definition of
A
and the boundary conditions included in
1
DA to
find
 


 


  
1
11
0
11
0
11
0
,d
d
d0
l
l
l.
A
ffpxf xfxx
pxf xfxx
pxfxfxx


 
 

Since , we have

111
0px

 
2
1
11
,0Af ff


,
(7)
and hence, 1
A
is a symmetric operator.
In order to show that 1
A
is self-adjoint, it suffices to
show that there is a constant such that
0c

11
,
A
fcffDA (see [11]).
In fact, we can see from (7) that
2
1
11 1
,
A
ff Afff



Applying the boundary conditions of in ,
we can get the inequality [12]. f)( 1
AD
 
4
22
00
dd
12
ll
l,
f
xx fxx


and hence

22
1
11
40
12 d
l
A
fffx xcf
l





where 11
4
12 0cl

. It follows that
1,
A
fcf (8)
and so 1
A
is a positively defined self-adjoint operator.
It is easy to see from (8) that 1
1
A
exists. Now set
1
A
fg
, and 1
1
f
Ag
, then (8) gives us
1
11,
A
gg
c
this means that mapping

14 4
1:0, 0,
A
HlHl
is
bounded, and
1
11.Ac
Thus, 1
1
A
is a compact operator by Sobolev embe-
ding theorem [13].
By similar manner, it can be shown that 2
A
is a posi-
tively defined self-adjoint operator, and 1
2
A
exists as a
compact operator, and the proof is complete.
We now choose Hilbert space
22
0, 0,
H
LlLl
as a state space of Equations (1) and (2), on which inner
product and norm are defined as follows:
 
1122
,, ,,,
H
uvu vuvuvH ,
here is the inner
roduct on

T
12
,,uuu

T
12
,,vvv

,
p
20,Ll. Let
Open Access JAMP
X. Z. HOU
100
 
  
cos
,cos cossin
ut Lxg
Wt Ft
vt Lx x
 

 

 

 
 
 
112
2
0,.
0
A
A
DA DA DA
A




Then the Equations (1) and (2) with the initial-boun-
dary conditions can be written as fo llows:
 
 
010
0,0
WtAWt AWtFt
WWWW


 
(9)
For the sake of establishing an evolution equation of
the system (1) and (2), we introduce a Hilbert space
H
H, on which inner product is defined as fol-
lows:
  
11 22
,, ,,,
HH
u vAuAvuvu v
 ,
.
where




TT
12 12
,, ,uuu vvv

Let
 

 
T
12 12
,, ,dduuuu WuWt
.
 
0
0,= ,
IDDADAFt
AA






then the robot system (9) can be described first-order
abstract evolution equation as follows:
  
 

T
0
d
d
00,
ut ut t
t
uuWW




0.
(10)
3. Spectral Analysis and Semigroup
Generation
We have discussed the spectral properties and semigroup
generation of the system operator in the system (10),
and obtained the following significant results:
Theorem 3.1 The operator is an infinitesimal
generator of a -semigroup on , and there
are constants such that
0
C
M
Tt
0

et
TtM
where

sup Re0

 
Proof We shall prove Theorem 3.1 in two different
cases,
Case 1.
1
2
21,2,
kk


. For the sake of sim-
plicity, we denote the eigenpairs of by

,,1,2,
nn
en
. For every real
,


sup Re

 

e
, we see from [11] that
. For any u, since
constitutes a


n
e
Riesz basis of ,
1nn
n
ua
. A simple computation
shows that

1
1
1
nn
nn
I
ua e


and





1
=1
1
1
1
m
nn
m
nn
m
nn
mm
nn
Iu
ae
ae






It should be noted that


1
m
m
n
 
 be-
cause

sup Re

 and
 . There-
fore,
  
1
=1
11
m
nn
mm
n
I
uae
 




u

We thus arrive at the following result:
 
11,,1,2
m
m
Im



 

,
It follows from the theorem 5.3 of [14] that is the
infinitesimal generator of a -Semigroup
0
C
Tt on
, and
et
TtM
, where 1
M
.
Case 2. *
1
2
2k
for some positive integer . We
*
k
see from [11] that for any u
,
 


*
**
*
*
121 (2)
*
111
0
kk
jj
jj jj
j
n
n
kk
kk kk
k
kjj
kj
kk
uaa bb
 










 
Since
***
,1,2,,
jj
kkk
jj
*
k

and
*
**
*
**
*
*
**
**
1
11
2
22
00
0
0
.
2
j
jj
j
jj
j
jj
k
kk
k
kk
k
k
kk
kk
I
AA A
A




 

 



 

 

 

 













Open Access JAMP
X. Z. HOU 101
Since *
1
2
2k
, we refer to 1) of Theorem 1 to find

****
**
*
**
*
**
2
11
2
22
*
1
2
11
22
42
2(2)4
2442
22,
kkkk
kk
k
k
kk
k
kk
 




 


 

 



 
2
and so
*
**
**
****
1
2
*
1
22
*
0
0.
j
jj
jj
jj
k
k
kk
kk
kkkkk




 



 


 


 



 





0
Hence, the space spanned by
** *
1***
1
0
0
,, ,,
jkjk
kk k
k








 


 


 
*
2k
j
is an invariant
subspace of dimensions of , denoted by .
From theo ry of finit e dim e ns io na l space, w e as se rt that
*
k
M

 
** ,
pp
kk
 
 MM
and therefore

*
*sup Re

 Mk

sup Re


*
.
k
. Actually, we can arrange
the vectors spanning as follows
*
k
M
**
*
12 *
** *
12
0
00
,, ,,,,
jk
jk
kk
k
kk
 


 
 
 
 

Set
**
*
*
12
2,
0
kk
k
k







then *
k
M has the form

*
*
0
there areinthe diagonal
0
k
k
js





M
Apply the result of [14] to conclude that generates
a 0-semigroup satisfying

C

1
Tt
11
et
Tt M
(11)
On the other hand, since the family
11 *
,,,,
jj
kk
kkk k
kk

consists of the eigenvectors
of , the subspace spanned by them is an invari-
ant subspace of , and this family is just a Riesz basis
of [5]. Thus, form case 1, it is aware of the fast that
generates a 0-Semigroup in .
For
M
M
C
 
2,0Tt tM
M, we have



1
1*
1,
11, 2,,;
jj
jj
kk
k
kk k
k
I
I
jnk
 

 


 
k

and k
, it follows from [16] that

1
k
I
MM
k

M and
2sup Rek

M
sup Re


. Thus, there is 2
M
such
that
22
et
TtM
(12)
Since * is finite dimensional, it is a closed sub-
space of , and so *k
k
k
M
MM, where
ex-
presses orthogonal sum in Hilbert space . Now, we
define
2
T t
def
Tt1
Tt (obviously,
0
12 21
TtTtTtt T
). We shall next prove an
interesting result that
Tt
is exactly a 0-semigroup
on generated by . The semigroup properties of
C
Tt can be easily presented as follows:
1)
0T II
*
12 k
kI00TT

MM
2)
 
  
 
12
112 2
121 2
,0
Tt sTt sTt s
TtTs TtTs
TtTtTs Ts
TtTs ts
 

 
 
 



3) For every x
, *k
k
x
xx
, where ,
**
kk
xM
kk
x
M,
 
 
 
 




*
*
*
*
*
12
00
12
00
12
12
0
12
00
lim lim
lim lim
lim
lim lim
k
k
tt
k
tt
k
k
k
t
k
k
tt
k
k
TtxTtTtxx
TtxTtTtx
TtTtx
TtxTtx
TtxTtx
xxx


















4) For any
xD, we have *k
k
x
xx,
and
*
k
xM
*
11
et
Tt M
,
and so *
kkk
x
M, and
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X. Z. HOU
102

 

**
**
***
1
0
12
0
0
lim
lim
lim
kk
kk
kk
t
k
kkk
t
t
xxx xx
Ttxx
t
Ttx Ttxxx
t
Tx x
t









 
Thus, defined by the orthogonal sum of

Tt
1
Tt
and 2 is exactly -Semigroup on generated
by . Taking

tT
0
C
max
12
,
M
MM from (11) and (12),
leads to the following result
 
e0
t
Tt Mt

The proof of Theorem 11 is complete.
4. Stabilization with Variable Structural
Control
The variable structural system is a system whose struc-
ture is intentionally changed with a discontinuous control
and it drives the phase trajectory to a hyperplane or
manifold. This method is well-known for its robustness
to disturbance and parameter variations [15-18]. Conven-
tionally, the variable structure control is based on the
state-space approach in which a Lyapunov function need
to be constructed so that the derivative of the Lyapunov
function negative definite. As the method provides ro-
bustness characteristics, there exists a major problem,
that is, the chattering phenomenon, usually encountered
in the practical implementation. This phenomenon is
highly undesirable because it may excite the high-fre-
quency u n model l ed dynamics.
In this section, let us consider the robot system (10)
equipped with a feedbac k contr o l le r :


,wut t
  

0
d,
d
0.
ut ut wutut
t
uu
 


 ,
(13)
where is a bounded linear operator acting on
into . We shall first introduce the equivalent control
theorem, and then apply the equivalent control theorem
to the robot system to obtain a significant result that the
solution of the system is exponentially stable under the
variable structural control.
Let now consider the
-neighborhood of sliding
mode where

0SCu
is an arbitrary given
positive number, and use a continuous control
,wut
to take place of in system (13), we have
,wu
t
  

0
d,
d
0
ut u wutut
t
uu

,


 
(14)
where uu 

t, and the solution of (4.1) belongs to
the boundary layer
Su
.
Let
0Su Cu
. Applying to the first equa-
tion of (13) leads to the following equivalent control
C
,t
eq
wu
:


1
,,
eq
wutCC yut
 
.

(15)
with assumption that the
exists. Substituting
1
C
,
eq
uut
into (13) yields
 
11
,.,uI CCuI CCut


 


 
(16)
which is called the equivalent cont rol eq uatio n.
Denote

1
P
CC
, and

0IP
,.Put
0tI
, then (16) is equivalent to the
following equation:
00
,uu u
t

 (17)
We turn now to prove the following result.
Theorem 4.1 If the following conditions are satisfied:
1)

1
C
exists, and is a closed operator; P
2)
0,ut
satisfies Lipschitz condition in u
with
the constant ;
L
3) the control
,wut
is bounded in any bounded re-
gion, and the solution of (13) is unique and bounded in
the boundary layer
Su
; then for each solution
ut
of (8) satisfying
00Su
, ,

00
uD
00
uu
,
0
uD
 
, we have

0
lim 0ut ut,uniformly on0,T
Proof Since is bounded, it is clear that P

,.PuPuu D

 
Conditions that (ii) together with the above inequality
imply that 0P
  is an infinitesimal generator
of an analytic semigroup , on in virtue
of [14: Theorem 3.2.1], and so there are constants ,

Tt
0t
1
N
1
, ,
2
N2
, satisfying

12
12
e, e
tt
Tt NTt N

In the boundary layer, it is easy to see that


11
,wutC CutC Cu

 

 (18)
Substituting (18) into (14), we see that

0,.
uIPyIP tPu
uIP utPu
 





and therefore, the solution
ut
of (14) can be
expressed as follows [14]
 

00
0
,d
d,
t
t
utTtuTtsLPuss
Tt sPus s




(19)
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X. Z. HOU
Open Access JAMP
103
the solution of (17) can be written as follows
(20)
int the
back term of the right side of (19) in view of [14], we
obtain

ut

 
 
0
00
0
00
d
d
0.
t
t
Tt sPuss
Pu tTPuT ts Puss
PutT tPuT t PuPut
 

 



(21)

00,d.
t
utTtuTt s IPuss
 


Employingegration by parts, and estimatingSubtract (20) from (19), and employ condition (1) and
the inequality
2
2et
Tt N
to find
 
2
2
e
T
ut utNLN
 

2
20
ed.
t
T
I Pususs
 
 
The consequence of Theorem 4.1 is now derived from
the well-known Gronwall inequality. The proof is com-
plete.
system is described
by partial differential equations with initial and boundary
ated. First, an abstract evolution equa-
. Krall and G. Payre, “Mod-
eling Stabilization and Control of Serially Connected
Beam,” SIAM Optimization, Vol
25, No. 3, 198
It can be seen from Theorem 4.1 that the robot system
(13), and therefore the robot system (10) are exponen-
tially stable under the variable structural control with
some appropriate conditions.
5. Conclusion
In the present paper, an elastic beam
conditions investig
tion is established in an appropriate Hilbert space. Then
the spectral analysis and semigroup generation of the
system operator of the beam system are studied and ap-
plied to prove an equivalent control theorem. Finally, a
significant result that the solution of the beam system is
exponentially stable under the variable structural control
with some appropriate conditions is proved by means of
the equivalent control theorem.
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