Spectral Analysis and Variable Structural Control of an Elastic Beam

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 under a variable structural control with some appropriate conditions is obtained.


Introduction
A great attention has been paid to the dynamics and control of flexible robot (see [1][2][3][4][5]) in the past thirty years since the high-speed performance and low energy consumption are highly demanded.In this paper, as a continuation of our work [6][7][8][9], we shall investigate an elastic robot system formulated by partial differential equations 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 corresponding to the evolution equation are studied.Several significant results are obtained.
Let us consider a robot system composed of two link-arm and three joints, an electrical machinery is installed 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]: , , , cos , , , cos cos sin with the following boundary conditions: and initial conditions , 0 , , 0 , 0 , 0 ; 0 0 ; 0 , 0 where    

Evolution Equation of the Beam System
We start this section with defining following operators , , , are absolutely continuous function on 0, , 0, , 0 0 0, , 0. , 0 1, 2 It should be noted that f in satisfying (1) can be written as , where the function suits (1) and (3), and the function suits (3) and following differential equation By solving Equation ( 5) and (3) we find A  exist, and they are compact operators.
Proof Apply integration by parts with the definition of A and the boundary conditions included in 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 0 [11]).In fact, we can see from (7) that Applying the boundary conditions of in , we can get the inequality [12].
and so 1 A is a positively defined self-adjoint operator.It is easy to see from (8) that , then (8) gives us A  is a compact operator by Sobolev embeding theorem [13].
By similar manner, it can be shown that 2 A is a positively defined self-adjoint operator, and A  exists as a compact operator, and the proof is complete.
We now choose Hilbert space as a state space of Equations ( 1) and ( 2), on which inner product and norm are defined as follows: here is the inner roduct on Then the Equations ( 1) and ( 2) with the initial-boundary conditions can be written as follows: 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 follows: . where then the robot system (9) can be described first-order abstract evolution equation as follows: (10)

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 For the sake of plicity, we denote the eigenpairs of by 

 
, , 1,2, , we see from [11] that .For any u We thus arrive at the following result: It follows from the theorem 5.3 of [14] that is the infinitesimal generator of a -Semigroup , where for some positive integer .We * k see from [11] that for any u    , Hence, the space spanned by is an invariant subspace of dimensions of , denoted by .
From theory of finite dimensional space, we assert that Apply the result of [14] to conclude that generates a 0 -semigroup satisfying On the other hand, since the family , , , , consists of the eigenvectors of , the subspace spanned by them is an invariant 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 Since * is finite dimensional, it is a closed subspace of , and so M , where  expresses orthogonal sum in Hilbert space .Now, we define

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: , 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
Thus, defined by the orthogonal sum of is exactly -Semigroup on generated by .Taking 11) and ( 12), leads to the following result The proof of Theorem 11 is complete.

Stabilization with Variable Structural Control
The variable structural system is a system whose structure 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][16][17][18].Conventionally, 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 robustness 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-frequency unmodelled dynamics.In this section, let us consider the robot system (10) equipped with a feedback controller : 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.
 is an arbitrary given positive number, and use a continuous control   , w u t   to take place of in system (13), we have where t, and the solution of (4.1) belongs to the boundary layer . Applying to the first equation of (13) leads to the following equivalent control with assumption that the  exists.Substituting , then ( 16) is equivalent to the following equation: We turn now to prove the following result.Theorem 4.1 If the following conditions are satisfied: is bounded in any bounded region, and the solution of ( 13) is unique and bounded in the boundary layer 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 , In the boundary layer, it is easy to see that Substituting ( 18) into ( 14), we see that and therefore, the solution   u t  of ( 14) can be expressed as follows [14]  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  The consequence of Theorem 4.1 is now derived from the well-known Gronwall inequality.The proof is complete.
system is described by partial differential equations with initial and boundary ated.First, an abstract evolution equa-.Krall and G. Payre, "Modeling 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 exponentially stable under the variable structural control with some appropriate conditions.

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 applied 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.


Let now consider the  -neighborhood of sliding mode where