Spectral Analysis and Variable Structural Control of an Elastic Beam

http://dx.doi.org/10.4236/jamp.2013.15015

Spectral Analysis and Variable Structural

Control of an Elastic Beam

Xuezhang Hou

Mathematics Department, Towson University, Baltimore, USA Email: [email protected]

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 conditions 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 structural control is proposed and a significant 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 conditions. 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 generation of the system operator correspond- 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]:

 

_{ }

 

 

 





2 2 2

1

2 2 2

3 2

1

2 2

, ,

,

cos

u x t u x t

p x

t x x

u x t p x

x t x

L x g





   

 

  

 _ _

  _  _

  



 _ _

 _   _

 _ 

(1)

 

_{ }

 

 

 





2 2 2

2

2 2 2

3 2

2

2 2

, ,

,

cos cos sin

v x t v x t

p x

t x x

v x t p x

x t x

L x x





    

 

 _  

 

 

  _  _

  



 _ _

 _   _



 

 _     _

(2)

with the following boundary conditions:

 

0,

 

0, 0,



1

 

, 0,



1

  

, , u t u t  p u l t  p u l t mg

(3)

 

0, (0, ) 0,



2

 

, 0,



2

  

, 0,

v t v t  p v l t  p v l t 

(4)

and initial conditions

 

   

 

   

 

 

 

 

 

 

 

0 1

0 1

0 1

1 0

, 0 , , 0 ( );

, 0 , , 0 ,

0 , 0 ; 0

0 ; 0 , 0

u x u x u x u x

v x v x v x v x

0

1

,

,

     

     

 

 

  

  

  

 

where u x t v x t

   

, , , are vertical and horizontal bend- ing vibration displacements of the forearm of the robot respectively; p x1

 

1

, and are vertical and hori- zontal local bending rigidity of the forearm of the robot respectively; and

 

2 p x

 , 1, 2, and 2 are positive

 





0_i  p x_i _i   . i 1, 2 ; , ,

, ,

   are th e re-

spective base stand azimuth, the angle of eleva- tion-depression of the forearm and the rear-arm of the robot;   1 2 3

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; m is the

mass of the tip body; g is the acceleration of gravity; here ,  , m, g and I1, I2, I3 are all positive

constants; the symbols and u u express  u t and u x respectively.



2. Evolution Equation of the Beam System

We start this section with defining following operators

 

2

 

2

1

2 2 ,

i i

f x

A f p x

x x

    

 _ _

 _  _

 



 





 





 

 

 



 



  







4

2

0, , , , are absolutely continuous function on 0, ,

0, , 0 0 0, , 0. , 0 1, 2

i i

i i i

D A f H l f f f p f l

p f L l f f p f l t p f l t i



  

 

 

         

It should be noted that f in satisfying (1) can be written as

 

1 D A

f  u u, where the function suits (1) and (3), and the function suits (3) and following differential equation

u u

 

2 2

1

2 2 0.

u p x

x x

 

 



 

 _  _



(5)

By solving Equation (5) and (3) we find

 

₀

_{ }

1

1 d d .

x y l

z

u x mg z y

p z

 

_{ }

 (6)

Obviously, f in must suit Equation (1) if suits (1) and (3), as well as suits (5) and (3)

 

1 D A

u u

Lemma 2.1 The operators A1 and A2 are positive

self-adjoint operators in L2

 

0,l , moreover, A11 

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 D A

 

1 to find

   





 

   





 

     

1

1 ₀ 1

1 1 0 1

1 0

, d

d

d 0

l

l

l

.

A f f p x f x f x x

p x f x f x x

p x f x f x x













  



 



 

 







Since 01 p x1

 

1 , we have

2 1

1 , 1 0

A f f   f  , (7) and hence, A1 is a symmetric operator.

In order to show that A1 is self-adjoint, it suffices to

show that there is a constant c0 such that

 

1 , 1

A f c f f D A (see [11]). In fact, we can see from (7) that

2 1

1 1 , 1

A f f  A f f   f

Applying the boundary conditions of in , we can get the inequality [12].

f D(A1)

 

2 4

 

2

0 d ₁₂ 0 d

l l l

,

f x x f x x





and hence

 

2 2

1

1 1 4 0

12

d l

A f f f x x c f

l

   

 _{ } 

 



where

1 1 4

12

0

c l

 

  . It follows that

1 ,

A f c f (8)

and so A1 is a positively defined self-adjoint operator.

It is easy to see from (8) that A11 

exists. Now set

1

A f g, and f A g11 

 , then (8) gives us

1 1

1 ,

A g g

c

 _

this means that mapping A11:H4

 

0,l H4

 

0,l

 _

is bounded, and

1 1

1 .

A c

 _

Thus, 1 1

A is a compact operator by Sobolev embe- ding theorem [13].

By similar manner, it can be shown that A2 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 _{H L}2

 

_{0,l L}2

 

_0,l

as a state space of Equations (1) and (2), on which inner product and norm are defined as follows:

 

u v, H 



u v1, 1

 

 u v2, 2



, u v, H,

here is the inner

roduct on





T

1, 2 ,

u u u v



v v1, 2



T,

 

 ,

 

_{ }

 

 

_

_

cos ,

cos cos sin

u t L x g

W t F t

v t L x x

                 _{    }          

 

 

 

1 1 2 2 0 , . 0 A

A D A D A D A

A

 

_{ }  

 

Then the Equations (1) and (2) with the initial-boun- dary conditions can be written as follows:

 

 

 

 

 

0 0,

 

0 10

W t AW t AW t F t

W W W W

           

 (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:

 1  1    2 2

, , , , ,

H H

u v   Au Av  u v u v ,

.

where u



u_{   1},u₂



T,v



v_{   1},v₂



T

Let



_{   }



_{   } T

1, 2 , 1 , 2 d d u u u u W u  W t.

 

 

 

0

_{ }

0

, = ,

I

D D A D A

F t A A

    _{ }  _{  }         

then the robot system (9) can be described first-order abstract evolution equation as follows:

 

_{ }

_{ }

 



   



T 0

d

d

0 0 ,

u t

u t t

t

u u W W

      _{ }      _   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 



T t





0

 

e t

T t M 

where   sup Re



  

 





0

Proof We shall prove Theorem 3.1 in two different cases,

Case 1.



1 2

2 _k k 1, 2,

   



. For the sake of sim- plicity, we denote the eigenpairs of  by



_n,e_n



,n1, 2,. For every real ,

 





sup Re

      

e

, we see from [11] that . For any u , since



constitutes a

 

    en





Riesz basis of , u n₁an n 







 

. A simple computation

shows that





1

1

1

n n

n n

I u a e

        



   and           1 =1 1 1 1 m

n m n

n _n

m

n m m n

n _n I u a e a e               _               

It should be noted that



 



m



  _n



m 1 be- cause   sup Re



  

 





and  . There- fore,













1 =1 1 1 m n n m m n

I u a e



   

 

 _  _{ }

    _



 _ u

We thus arrive at the following result:









1 1

, , 1, 2

m m I m        _  _{   }    _ ,

It follows from the theorem 5.3 of [14] that is the infinitesimal generator of a -Semigroup



0

C T t

 

on

, and

 T t

 

Met, where M 1.

Case 2. *

1 2

2 k

  for some positive integer k*. We see from [11] that for any u,

   





*  

* *

*

*

1 2 1 (2)

*

1 1 1

0

k k

j j

j j _{j j j}

n n

k k

k k k k k

k j j _{k j}

k k

u a  a  b _ b 

                    _{ }   

 

 



  Since

* * *, 1, 2, ,

j j

k k k j jk*

      and * * * _* * * * * * * * * 1 1 1 2 2 2

0 ₀ 0

0 . 2

j

j j _j

j j

j

j j

k

k k _k

k k k k k k k k I

A A A

Since *

1 2

2 k

  , we refer to 1) of Theorem 1 to find

 

* * * * * * * * * * * * 2 1 1 2 2 2 * 1 2 1 1 2 2 4 2

2 (2 ) 4

2 4 4 2

2 2 ,

k k k k

k k k k k k k k k                 _   _              _   _       2     and so * * _{* *} * * * * * 1 2 * 1 2 2 * 0 0 . j

j _{j j} j

j

j

k

k

k _{k k} k

k k k k

k    _{ }      _     _{ }         _ _     _{  }    _{ }          _ _ _      0   

Hence, the space spanned by

* *

*

1 * *

* 1 0 0 , , , , j k j k k k k k   _  _    _{   }  _{    }  _{ }  _{ }_      * 2 k j 

is an invariant

subspace of dimensions of , denoted by .

From theory of finite dimensional space, we assert that

* k

M



_k*



p



_k*



p

 

 

,

 M  M    

and therefore





*



* _{sup Re}



   

   M

k 

 



sup Re   





* . k

. Actually, we can arrange the vectors spanning M_k* as follows

* *

*

1 2 *

* * * 1 2 0 0 0 , , , , , , j k j k k k k k k   _                    _{ }     _{ }    Set * * * * 1 2 2 , 0 k k k k            _ _   

then M_k* has the form





*

* 0

there are in the diagonal

0 k k j s     _            M 

Apply the result of [14] to conclude that generates a 0 -semigroup satisfying



 

C T t1

 

 

1 1e

t

T t M  (11) On the other hand, since the family



k1, k1, , kjk, kjk



k k*

   



 _  _

 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

 T t2

  

, t0



M

 



 

M , we have













1 1 * 1 , 1

1, 2, , ;

j j

j j

k k

k

k k k

k

I

I j n k

                    k     _  

and





k , it follows from [16] that

1

k

I

 

 M M



k



 

 M   and









2 sup Re k

   

   M 

 





sup Re     . Thus, there is M2 such

that

 

2 2e

t

T t M  (12)

Since * is finite dimensional, it is a closed sub-

space of , and so _k* k

k

M

 M M , where  ex-

presses orthogonal sum in Hilbert space . Now, we define



 

def

 

T t2

 

T t _T t₁  (obviously,

   

   

0

1 2 2 1

T t T t T t T t  ). We shall next prove an interesting result that T t

 



is exactly a 0-semigroup

on generated by . The semigroup properties of C



 

T t can be easily presented as follows:

1)

 

 

T

 

0 I I

*

1 2 _k k I

0 0

T T   _M  _M  _

2)













   

   

 

 

 

 

   





1 2

1 1 2 2

1 2 1 2

, 0

T t s T t s T t s

T t T s T t T s

T t T t T s T s

T t T s t s

    

_  _{ } _

_   _{ }  _

 

3) For every x, xx_k*xk, where x_k*M_k*,

k k

x M ,

 

 

 





 

 

 

 

 

 

 

 







 



* * * * * 1 2 0 0 1 2 0 0 1 2 1 2 0 1 2 0 0 lim lim lim lim lim lim lim k k t t k t t k k k t k k t t k k

T t x T t T t x x

T t x T t T t x

T t T t x

T t x T t x

T t x T t x

x x x

               _  _   _  _ _  _    _  _     

4) For any xD

 

 , we have xx_k*xk ,

and

* k

x M *

1 1e

t

T t M  ,





 

   



 



* *

* *

* * *

1

0

1 2

0

0

lim

lim

lim

k k

k k

k k

t

k

k k k

t

t

x x x x x

T t x x t

T t x T t x x x

t Tx x

t













   



 

 _ _

 

 _{  } 

 



 

  

   



Thus, T t

 

defined by the orthogonal sum of T t1

 

and 2 is exactly -Semigroup on generated by . Taking

 

t T

 0

C

max





1, 2



M  M M from (11) and (12), leads to the following result

 

e t



0



T t M  t 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 unmodelled dynamics.

In this section, let us consider the robot system (10) equipped with a feedback controller w u t t





 

,



:

 

_{ }

_{ }

_{ }

 

0

d

, d

0 .

u t

u t w u t u t

t

u u



  

 

 _





 

 

   ,

(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 SC u

 

 0 where  is an arbitrary given positive number, and use a continuous control w u t

 

,

to take place of w u



,t



in system (13), we have

 

_{ }

_{ }

 

0

d

, d

0

u t

u w u t u t

t

u u



  

 

 _



,

  

  

  



(14)

where u  u t, and the solution of (4.1) belongs to

the boundary layer S u

 

  .

Let S u

 

 Cu0. Applying to the first equa- tion of (13) leads to the following equivalent control

C

 

,t

eq

w u :

 

 

1



 



, ,

eq

w u t   C  C y u t . 

(15)

with assumption that the



C



1 exists. Substituting

 

,

eq

u u t into (13) yields

 

1

 

1

 

, .,

u_I C  C_u_I C  C_ u t (16) which is called the equivalent control equation.

Denote P

 

C 1C, 0 



IP



 and

  

 P

  

u t, .

0 t I



  , then (16) is equivalent to the

following equation:



0 0 ,

uu u t



(17) We turn now to prove the following result.

Theorem 4.1 If the following conditions are satisfied: 1)

 

C 1 exists, and P is a closed operator; 2) 0

 

u t,



 satisfies Lipschitz condition in u with the constant L;

3) the control w u t

 

, is bounded in any bounded re-gion, and the solution of (13) is unique and bounded in the boundary layer S u

 

 ; then for each solution

 

u t of (8) satisfying S u

 

0 0, u0D

 

0 ,

0 0

u u  , u0D

 





   

, we have

 

0

lim u t u t 0, uniformly on 0,T

 

 _

Proof Since P is bounded, it is clear that

 

, .

P u  P u uD 

Conditions that (ii) together with the above inequality imply that 0P is an infinitesimal generator

of an analytic semigroup , on in virtue of [14: Theorem 3.2.1], and so there are constants ,

 

T t t0 

1

N

1

 , N2, 2, satisfying

 

1

 

2

1e , 2e

t t

T t N  T t N  In the boundary layer, it is easy to see that

 

 

1



 



 

1

,

w u t    C  C u t  C  Cu (18) Substituting (18) into (14), we see that







  



  

0 , .

u I P y I P t Pu

u I P u t Pu

    

   

_ _

    _

  

and therefore, the solution u t

 

of (14) can be expressed as follows [14]

 

 





  



  

0 ₀

0

, d

d , t

t

u t T t u T t s L P u s s

T t s Pu s s

   

 





 _  _ 

 



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

 

u t



  

 



  

 

 

 

 

0

0 ₀ 0

0 0

d

d

0. t

t

T t s Pu s s

Pu t TPu T t s Pu s s

Pu t T t Pu T t Pu Pu t



   

   







 

 _  _ 

 _  _  

 (21)

 

 

0 ₀





  

, d .

t

u t T t u  



T t s IP  u s s Employing egration by parts, and estimating

Subtract (20) from (19), and employ condition (1) and the inequality

 

2

2e

t

T t N  to find

   

2

   

2e

T

u t u t N  LN 2

2 e ₀ d .

t T

IP 



u s u s s

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 Gequa-. 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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,