TRACKING CONTROL OF A FLEXIBLE BEAM BY NONLINEAR BOUNDARY FEEDBACK

This paper is concerned with tracking control of a dynamic model consisting of a flexible beam rotated by a motor in a horizontal plane at the one end and a tip body rigidly attached at the free end. The well-posedness of the closed loop systems considering the dissipative nonlinear boundary feedback is discussed and the asymptotic stability about difference energy of the hybrid system is also investigated.


Introduction and System Formulation
Mechanical flexibility in motion control systems attracted more attention in recent years.
Motivated by [4], in which a hybrid system describing the overhead crane model was studied, we will consider in this paper a flexible beam rotated by a motor in a horizontal plane at one end and a top body rigidly attached at the free end.This model fits a large class of real applications such as links of robot system and space-shuttle arms in which high speed manipulation and long and slender geometrical dimensions are the major factors causing mechanical vibration.To achieve high speed and precision end point positioning of the flexible beam (which must be gua- ranteed in any condition variations such as payload) the boundary control is one of the major stra- tegies in production and space applications.
Let t be the length of the beam, p the uniform mass density per unit length, E1 the uniform flexural rigidity and rn be the mass of the tip body attached at the free end of the link, Ira the moment of inertia of the motor and J the moment of inertia associated with the tip body.Tak- ing the motor's torque as the control input and neglecting rotary inertia and shear deformation effects and actuator dynamics, the total transversal displacement y(x, t) at position x and time 1This research was supported by the National Natural Science Foundation of China.can be described by the following coupled differential equation: v,(x, t)+ E,(, t) O, 0 < : < e, ,(0, t) O, E(O, t)-.t(O, t)+ u(t) O, Elyx,(, t) Mytt(e, t) O,   EIy(e, t) + gytt(e, t) O. (1) Let the terminal state trajectory be XOd(t), where O'S(t)-O, i.e., the tracked state would be uniform motion or fixed in some direction for the flexible beam.Thus the difference displacement e(x,t) y(x,t)--XOd(t will satisfy the same equation (1).By this fact, we will, here and throughout this paper, use y(x,t) to represent the error displacement as well as total displacement.It is obvious that the feedback control should make the energy of the dynamic system (1) be decreasing with time.
Let us briefly outline the content of this paper.In section 2, we design a dissipative nonlinear feedback control with angular velocity of motor and show the well-posedness of the closed loop system.It is also shown that the energy in this case will tend to zero as time goes to infinity.Section 3 is devoted to the uniform decay estimate of the closed loop system with the angular acceleration feedback.
2. Well-posedness of the Problem and the Asymptotic Stabihty We design a nonlinear dissipative feedback control by u(t) cyx(O t) f(Yxt(O, t)) (where c > 0 is a positive constant) and study the following closed-loop system flYtt(x, t) A-ElYxxxz(x, t) O, 0 < x < , t > O, where the feedback function f such that f C C(R) is increasing with f(0) 0 and sf(s) > 0 for s -fi 0.
the energy of the system: Notice that the norm of state is just -21 [py2 + EIY2xx]dx + -y(O, t) + -ImYt(O t) + -My (, 1 2 0 and formally Lemma 1: Under the assumption (3), the operator A defined by ( 4) is maximal monotone on with the domain D(J) that is dense in .
Then a simple calculation yields This means that A is monotone.
Assume that U (I + .A)U o for some U 0 G .Then, (U0, U0) _< (U, U0) 0, which implies that U o=U=0.ThusD(A) isdensein Since the operator t is maximal monotone with the dense domain D(t) in the energy space , by applying a method developed in [2] to the evolution equation (2), we obtain the following existence result.
Proof: 0 E %(A) immediately follows from the definition.Thus, we only consider the second condition.Let {Vn} C_ , I I Vn I I <-C, be a bounded series and let {Un} satisfy (I + A)U n V n.
By the Sobolev embedding theorem, there is a subsequence of Un, still indexed by n for notational simplicity, and U 0 E H2(0, ) x L2(0, ) x [2, such that   UnUo in the topology of %.
0 Thus E(t) 0 by the arbitrariness of T and E(t) E(O).The proof is complete.By Theorem 1, Lemmas 2 and 3, and using LaSalle's invariance principle [1], we have immediately" Theorem 2: Let E(t) be the energy of the hybrid system (2).Then, tli_,mE t O.

The Uniform Decay
To get the uniform decay, we design, in this section, a boundary feedback control as u(t) ImYxtt(O t) ayx(O t) f(yxt(O, t)).
Following the same line as that of section 2, we also have Theorem 3: The operator A defined by ( 17) is maximal monotone with the dense domain in the space and hence generates an asymptotically stable nonlinear semigroup of contractions on In the sequel, we always assume that the initial data belongs to the domain of operator A and hence the solution of equation ( 14) has the regularity properties expressed by (ii) of Theorem 1.Let 6 > 0 and (x) ax-at-1, where a is a constant to be determined.Define + 'yx(, t)[M()yt( t) + J'()yct(, t)] + (2 + 6)'[ / ytydx + -y(., t)yt( t) + yx(, t)Yxt(g t)] o Lemma 4: Let fl(t) be defined by (lS).
C, C2, and C 3 such that (18) (20) Proof: By the defined form of fl(t) and Sobolev's embedding theorem, we can always find a constant C O such that I/() _< CoE() once the constants a and 6 are determined.To prove the second condition, we find the derivative/3'(t) directly.