- Email: [email protected]
Stabilization of Bernoulli Beam with Massive End-point1
STABILIZATION OF BERNOULLI BEAM WITH MASSIVE END-POINTl W.L. Chan* and De-Xing Feng** ·Chinese University of Hong Kong, Department of Mathematics, Shalin, N. T., Hong Kong ··Institute of Systems Science, Academia Sinica, Beijing 100080, PRC
Abstract This paper deals with the stabilization problem of the bending vibration for a Bernoulli beam with massive end-point by nonlinear feedback on the boundary. First, the spectral structure of the system is considered and the exact spectrum distribution of the system is obtained for the case of uniform beam. Then certain nonlinear feedback control on the boundary is applied to stabilize the transverse vibration, and the asymptotic stability is proved for the corresponding closed loop system. Finally, simulation results show that the suggested feedback control scheme is rather satisfactory. Keywords Bernoulli beam, spectral structure, nonlinear boundary control, feedback stahilization, asymptotic stability.
Stat ement of t he Proble m
1
D(A) =
p(x)y(x,t) + (EJ(X)Y")" (x,t)
= 0,
°
< x < I, 0< t, (1.1)
(1.5)
where I is the length of the beam, p is the mass per unit length of the beam, El is the stiffness coefficient of the beam, and y(x, t) stands for the transverse displacement of the beam at location x at time t. Assume that the beam is clamped at x = and carries a load of mass M with a moment of inertia J around its center of mass at the end-point. So the boundary conditions are as follows:
where ii(t)
= y'(O, t) = 0,
(1.2)
to the end-point, respectively, and where and below the prime denotes derivative with respect to the space variable x and the dot stands for derivative with respect to the time variable t. On the functions p(x) and p(x) ~ El(x), we assume that there exist constants Po,Pl> Po,PI such that
(HI) p( .) E C(O,I), PI 2: p(x) 2: Po > O,'v'x E [0,1); (H2) p(.) E C2[O,I), PI 2: p(x) 2: Po> 0, 'v'x E [0,1).
2
In order to formulate an abstract version of the problem, we define a product IIilbert space 1I = JR2 X L~(O , I) endowed with the scalar product
- b(w")'(l) ~(W")(l)
[
I
= [1/M, 0,Or,
~
=
Bas ic P roperties of A For .,0,
J, E D(A), by integrating by parts, we have
l'
+ JT/1'n + p(x)'1'I(x)'1'2(x)dx E H , i = 1, 2. Then we define in H a linear
(.pI,oh)H = M{d.2
A.p =
bl
In recent years, boundary feedback stabilization of elastic systems has been considered by many aut hors (e.g., see (5),[6)). (6) investigated boundary feedback of a nonuniform Bernoulli beam with one end-point clamped and another end-point free, and obtained the asymptotic stabilization result. The purpose of the present paper is to show the asymptotic stability of the closed loop system by using certain nonlinear boundary feedback in our model case. Obviously, the spectral properties of A is important for the system (1.5). Thus in §2, we begin with the basic properties of A. On the basis of these results, a nonlinear feedback on the boundary is presented and the solution of the corresponding closed loop system is given by using the nonlinear semigroup method in §3. In §4, the asymptotic stability of the closed loop system is proved by using the Lasalle's Invariance Principle. Finally, simulation results are given in §5.
My(l,t) - (Ely")"(I,t) = I1I(t), (1.3) JY'(I, t) + Ely"(I, t) = 112(t), (1.4) where I1I(t) and 112(t) are control force and moment applied
for .p; = [(; ,'1;,'1';)' operator A by
= [y(l,t),y'(I,t),Y(',tW,
[O,l/J,O)' .
°
y(O, t)
{.,o I (pcp")" E L;(O, I),CP(O) = '1"(0) = O},
Then the bending model described by (1.1)~(1.4) can be written as the following abstract evolution equation on H:
Consider the following bending model of a non uniform Bernoulli beam (e.g., see (1), (4)):
Consequently, it is not difficult to prove that A is a self-adjoint and positive operator in JI .
1 ,
T heo rem 2.1 With the hypotheses (H1) and (H2), the opemtor A ill H is self-adjoint and positive definite . Moreover, A has compact resolvent.
.p = [cp(l) , '/(1), '1'r E D(A) ,
~(W")"(-)
This research was supported by the National Natural Science Foundation oC China
859
Theorem 2_5 With the uniform beam case, .for the eigenvalues {~n I n ~ I} of A, we have the following asymptotic expression:
Proof From the above, we only need to prove the compactness of the resolvent of A. In the sequel, Hm(o, I) denotes the usual mth order Sobolev space (e.g., see [10)). By the Sobolev embedding theorem, HI(O, I) is compactly embedded in C[O, I]. From the definite positivity of A, (-00,0) c p(A), the resolvent set of A. For ~ < and = [~, l1.Jr EH, denote cp = R(~; A)i, then cp E D(A) satisfies
°
~cp
i
- Acp = j.
with On
-+
0 as n
-+
00.
Proof Analyzing the above characteristic equation carefully, the detail of the proof is omitted here. •.
(2.1)
From (2.1) we have
3
(2.2)
We now turn to the control system (1.5). Assume that we are given measurements fI(l, t) and fI'(I, t), i.e. velocity and angular velocity of the transverse displacement at the end-point of the beam . Thus to stabilize the transverse vibration of the beam , we can apply the following form of nonlinear boundary feedback at the end-point of the beam:
Notice that, by the extension theorem (see [10)) and the Fourier transformation, there exist constants Cl and C2 such that ClllvIlJl2(O,I) ~ C2I1vIlH2(O,I),
Vv E H 2(0,1).
(2.3)
~ C 3 I1iIlH,
(2.4)
It follows from (2.2) that
IIc,oIlJl2(o,l)
+ Ic,o(O)1 + Ic,o'(O)1
Nonlinear Boundary Feedback
UI(t) E -h8gl(fI(l,t», { U2(t) E -j-8g (fI (I, t», 2 '
with a constant C3. Thus it is easy to prove that for any bounded sequence {in} C H, R(~; A)in contains a convergent subsequence, which implies the compactness of R(~; A). •
(3.1)
where gl, 92 : 1R -+ 1R are two lower semi continuous, proper convex functions, and 8gj denotes the subdifferential of gj for j = 1,2. In the sequel, we make the following hypothesis: (113) 89j(0) = {O}, and 0 ~ 8gj(r),Vr 0 for j = 1,2.
Then by Theorem 2.1, A has the pure point spectrum: utA) = Pk I k ~ I}, with 0 < ~I ~ ~2 ~ .... We now show that each ~n is simple. To this end, we need a lemma.
t-
From [2], we have that
Lemma 2.2 Let the assumptions (HI) and (H2) hold. If a function", defined on [0, I] satisfies (pc,o"J"(x) = ~p(x)c,o(x),
x E (0,1),
(2.5)
c,o'(l) = c,o"(I) = 0, c,o(l) > 0, (pc,o")'(l) < 0, where
~
D(AI/2)
(A I/ 2tj>,A I/ 2tp)H =
(2.6)
Proof The main idea of the proof is to integrate (2.5) repeatedly, the detail is omitted here. •
cp
(cp ,tp)v =
=
~p(x)c,o(x),
+ ~M
J
tj>,tp E D(A I /2).
10' p(x)c,o"(x)t/I"(x)dx,
Vtj>,tp E V.
We define the lower semicontinuous convex function 9
JR2 (2.7)
pc,o"(I) - M
(pc,o")'(l)
10' 1 (X)","(x) t/I"(x )dx,
(Atj>,Ij,)H =
x E (0,1),
",(0) = ",(0) = 0,
1
Vtj>,tp E D(A I/ 2).
Let V' denote the dual of V . We can view the operator A as the bounded operator from V into V', given by
be an eigenelement of A in H, then
Proof Since tj> is an eigenelement of A, there must be a positive number ~ such that (pc,o")"(x~
10' p(x)c,o"(x)t/I"(x)dx ,
Then we can define the Hilbert space V = D(A 1/2) with the inner product
> 0, then c,o(O) > O.
Theorem 2.3 Let c,o'(I) t- 0.
= {cp I cp E H 2 (0, 1),
-+
JR :
g(O
= (1/M)91(€d + (1/J)g2({z),
V~
= [€I'~2r E JR 2.
From [12],
O.
If
Then 8g is a ma..ximal monotone graph in 1R 2 X JR 2. Thus the control problem (1.1)-(1.4) together with the non linear feedback (3.1) becomes
Theorem 2.4 Under the assumptions (HI) and (H2), eflch eigenvalue of A is simple.
(3.2)
°
=
°
Proof Let
We now define the bounded linear operator B : V
Obviously, the adjoint
Clearly, cp satisfies (2.7). But c,o'(I) = 0, hence by Theorem 2.3,
1 and
n°
-+
JR2 by
of B is given by
=
Therefore, (3.2) can be written as
For the uniform beam case, we may obtain the precise spectrum distribution of A. In fact, in this case, p(x) = p,p(x) = p are all constant, then by a direct calculation, each eigenvalue ~ of A must be a solution of the following algebraic equation:
~(tJ+ ~y(t) + ~o8g(B!) 30 on V', {
where 0
+ cosh z cos z) + ,8z3(coshzsinz + sinhzcosz) =
= M/pi,
,8
= Ye E V,
y(O)
= YI
(3.3)
E H.
Thus we can use the framework in [9] to study the stability of (3.3).
oz(coshzsinz - sinhzcosz) - 0,8z4(1- cosh zcos z) - (1
y(O)
0,
Theorem 3.1 With the hypotheses (HI), (H2), A+BO(8g)B is a maximal monotone grnph in V X V'.
= J/pI3, and z = 1(~p/p)1/4. 860
Lemma 4.2 Under the hypolheses (Hl)~(II4), the resol+ AA )-1 is compact from 11. 10 11. for all A > O. •
Proof It is enough to note that B is surjective from V onto
JR 2 . The desired maximal monotonicity of A + B"({}g)B is an immediate result of Theorem 2.1 of [9].
•
We now define the product Hilbert space 11. = V define a multi valued operator A : 11. -+ 2'H by
Let S(I) be a strongly continuous nonlinear semigroup on a closed subset C of a Hilbert space H . For x E C,
x Hand
,),(x) ~ {S(t)x
~ ] = [ A~+ ;"~{}9)B;P ],
A [ D(A) =
vent (I
{[~, ;pr
E
I t ~ O}
is called the orbit through x and
11.1 A~+ B"(og)B;P E H,;P E V}.
w(x)
~
{y
Eel
3tk
-+ 00
s.t. Y = lim S(tk)X} k-oo
Note that in the present case, the range of B"({}g)B is in H, so
is called the (possibly empty) w-limit set of x. When Set) is generated by -A, the structure of w-limit sets characterizes the asymptotic behavior of weak solutions of the evolution equation
D(A) = D(A) x V. From [9], we have Theorem 3.2 Under the hypolheses(Hl) ~ (H3), A is maximal monolone on 11.. •
ti(t) + Au(t) 3 O. Theorem 4.3 Let S(t) be the nonlinear semigroup generated by -A in 11. = V x H . Suppose Ihat [Yo, Yd' E 11. = V x H, then the orbit ')'(YO, Yt! o![Yo, Yd' is relatively compact in 11., arid the corresponding w-limit sel w(Yo, YI) is nonempty.
Then (3.3) can be rewritten in the form of the first order nonlinear evolution equation:
d [ y( I) ]
[ y( I) ]
~ y(~) ~A Y~I)
1
y(O) = Yo ,
30,
Proof By Lemma 4.2, (I + AA))-I is compact in 11. for A > O. Moreover, by the assumption (H3), 0 E n(A) is obvious, where n(A) stands for the range of A. Accordingly, the compactness of ')'(Yo, yd in 11. is a result of [7], and the last assertion immediately follows from the relative compactness of
(3.4)
y(O) = YI.
Since A is maximal monotone on 11., -A generates a nonlinear semigroup of contraction 5(1) on D(A) = 11.. Therefore by the nonlinear semigroup theory [3], for any Yo E V, YI EH, (3.4) has the unique weak solution
')'(Yo, 111).
•
Theorem 4.4 Assume thal Ihe hypotheses (Hl)~(H4) Iwld. Then for any [Yodid' E 11., we have 11[y(t),Y(IWII'H
and moreover, if Yo E D( A), YI E V, then the above weak solution becomes the strong solution and [y(I), y(tW E D(A) x V for all t ~ o.
4
Proof We only need to prove that w(yo, yd = {O}. For this purpose, it needs only to prove this for [Yo, yd' E D(A). So we assume that [Yo, YIl' E D(A), and take any Zo E w(Yo, Yd. Denote Z(I) ~ [z(t),i(t)], = S(t)Zo. ( 4.4)
In this section, we consider the asymptotic stability of the closed loop system (3.4) by using the Lasalle's Invariance Principle (see [7]). We always assume (Hl)~(II3) to be true. Let [Y(I), Y(Il]' be the solution of (3.4). The energy associated with this solution is given by
=
Clearly, Z(n = S(I)Zo E w(Yo,Yd. Following [8], Set) is an isometry on w(yo, fit), so
t(t)
~1I[Y(t), y(tWII~ = ~ (IIY(t)lI~ + lIy(t)II~)
~
l'
[p(x)ly"(x,tW
E -y(l, I)ogl (y(l, I)) - y'(i, t){}g2(y'(l, I» .
z(l, t)
[p(x)y"(x, t)y"(x, t)
+M!i(/, t)y(l, t)
I ~ O.
z'(l, I)
= 0,
I ~ 0,
= 0;
ji'(I , t)
+ pz"(I,t) = o.
Thus to complete the proof, we only need to prove that whenever [y(t), y(t W satisfies the following equations:
+ p(x)y(x, t)!,i(x, t)]dx
+ Jy'(l, t)y'(l, t)
p(x)z(x,t) + (pz")"(x,l)
= 0,
z(O, I) = z'(O, I) = 0,
~
I
M z(l, I) - (pz")'(l, t) = 0,
= !i(/,t)[Afy(l,t) - (py")'(I,t)] E
= 0,
Mz(l,I) - (pz")'(I,I)
So by using the boundary conditions (1.3) (1.4) and feedback control law (3.1), we have
l'
,
which, by (3.2), implies that
(4.2)
Proof We first prove 2). let [Yo, yd' E D(A), then from
=
= (1/2)ll[z(t»)(I)n~.
E(I)
Consequently, by the assumption (H3) on gl , g2, we obtain
[3] , [yet) , y(tW = S(t)[Yo, YI]' is the strong solution of (3.4).
d~;t)
I ~ 0,
o E i(l,t)ogl{Z(l,t» + Z'(I , I){}g2(Z'(I,t» (4.1)
Lemma 4.1 1) E(I) is nonincreasing wilh respect to I ~ 0; 2) if the inilial dala [Yo, yd' E D(A) , then for all I ~ 0,
t(O
= 0,
Then from (4.2) it follows that
+ p(x)ly(x, tW]dx
+~[Mly(l,IW + JIY'(l,IW] .
(4.3)
-> 00,
where [Y(t),Y(tl]' = S(t)[Yo,yd' for t ~ O.
Asymptotic Stability
E(t) =
0 as t
-+
Jz'(l , I)+pz"(l,t) = 0,
+y'(l, t)[Jy'(i, t) + py"(l, t)] -W, t)ogl (!i(/, t» - y'(l, t)Qg2(y'(l, t»,
i(I,I)
= O,i'(I,I) = 0,
0 < x < I, I> 0,
0, I ~ 0,
(4.5)
,t~O,
I ~ 0,
[£(0»)(0)]' E w(Yo, yt!
which is 2). Since 8g 1 and {}g2 are monotone in JR x JR by the assumption (113), the assertion 1) is derived from 2). •
with [Yo, Yd' E D(A), then
We make the following further assumption on gl and g2:
[z(I»)(t)]' =: 0,
(H4) ogl and Og2 map every bounded set in JR into relatively compa.ct set in JR.
I ~ O.
=
(4.6)
In Section 2, we have shown that utA) {An I n ~ I}. Let {~n I n ~ I} be the corresponding normalized eigenelement
Thus from [9] , we have
861
sequence, which forms an orthonormal basis of H. Define the linear operator Ao on H by
Ao=
[0-1] A 0
R e fer e n ces (1) Balakrishnan, A.V., Control of flexible flight structures, in « Matlll?mati'lue et Applications", Paris, Gauthier- Villars,
'
1988, pp. 23-34.
then the Co semigroup T(t) on H generated by -Ao is given by {]
T(t)
[
-
00
=
'1
L: n='
[(an cos .,!:f,;t
+ (f3n/.,!:f,;) sin .,!:f,;t) <{;n
(f3n cos .,!:f,;t - .,!:f,;a n
) sin.,!:f,;t <{;n
(2) Balakrishnan, A.V., Damping operators in continuum models of flexible structures: Explicit models for proportional damping in beam bending with end-bodies, Appl. Math. Optim., 21(1990), 3, 315-324.
1,
(3) Bnizis , H., Operateurs Maximaux Monotones et Semigroupes de Contraction dans les Espaces de Hi/bert, North Holland, 1973.
(4.7) where an = ({,
[i(t),i(tW = T(t)[zo,i,l',
(4) Chassiakos, A.G . & G.A. Bekey, Pointwise control of a flexible manipulator arm, Proc. of the IFAC Symposium on R.obotics and Automation, Barcelona, Spain, 1985, 181185.
t ~ 0,
so it follows that
f (an cos";>:;'t + (f3n/";>:;') i(t) = f (f3n cos";>:;'t - ";>:;'a n
i(t) =
sin ..;>:;.t) <{;n,
(4 .8)
n=1
~1. Delfour, A. M. !(rall & G. Payre, Modeling, stabilization and control of serially connected beams, SIAM J. Control & Optimization, 25(1987), 3, 526-546.
(5) Chen , G. ,
sin ..;>:;.t)
(4 .9)
n=1
(6) Conrod, F. & M. Pierre, Stabilization of Euler-Bernoulli beam by non linear boundary feedback, INRlA, RR No.1235, June 1990.
The series of (4.8) and (4.9) converge in V and in H, respectively, and the convergence is uniform with respect to t E IR. Therefore. it follows from (4.9) that
z'(I,t) =
f
(f3ncos";>:;'t-an";>:;'sin";>:;'t)
t
~ O.
(7) Dafermos, C.M. & M .Slemrod, Asymptotic behavior of nonlinear contraction semigroups, J. Funct. Anal., 13(1973), 97- 106.
n=1
(4.10) This means that f3ncp'(I) and -an.,!:f,;'P'n(l) are the Fourier coefficients of the uniformly almost periodic function O. Hence f3ncp~(I) = an
(8) Gibson, J .S., A note of stabilization of infinite dimensional linear oscillators by compact linear feedback, SIAM J. Control8: Optimization, 18(1980),3,311-316.
n ~ l.
Dut by Theorem 2.3,
(9) Lasiecka, I., Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary, JoufIlal of Differential Equa.tions, 79 ( 1989), 340-38l.
Particularly, if g,(O and g2(0 are linear functions, then the assumption (H3) is satisfied. Thus we have
(10) Lions, J .L. & E.Magenes, Problemes aux Limites Nonhomogenes et Applications, Vo!. 1 et 2, Dunod, Paris, 1968.
Theorem 4.5 For the lineal" boundary feedback contml system
p(x)y(x,t)
+ (py")"(x,t) =
y(O, t) = y'(O, t) = 0,
0,
(11) Wang, W.Z., An eigenvalue problem of flexible arms, Mathematics in Theory and Practice, 1991, 3.(in Chinese)
O
(12) Rockafellar, R.T., Convex Analysis, Princeton University Press, Princeton , New York, 1972.
t > 0,
MW,t) - (pii")'(I,t) = -/(\y(l,t), t ~ 0,
(4.11)
[13J Showalter, R.E ., Hi/bert Space Methods for Partial Differential Equations, Pitman, London-San FranciscoMclbourul, 1973.
J(y)'(I,t)+ py"(I,t) = -/(2y'(I,t), t ~ 0, y(x,O) = yo(x),y(O,t) = y" where 1", /(2 are two positive constants, under the assumptions (HI) ~(H3) , the origin is asymptotically stable. 0
0.20 ~- -~---- ,--- - ~ - - -- ,----;-- --,----; ----,---- ,
Remark Gibson(8) showed that a linear oscillator in an
O. l ~
--
010
-
I
infinite dimensional Hilbert space, with no uniform decay rate, cannot be given a uniform decay rate with compact linear feedback. Therefore , in the present case, the exponential stability is not expected (different from (5)) and in this sense, the conclusion obtained here cannot be improved.
I
-- -
-~ I I
- -, - - -
O.O~ .::
I
I
I I
, ,
- - - .. ----1----
I
I
t - - -.,- - I I
I I
-r- - - - , - - - - , . - - - T - - - - , - -
:(\:
I
:
- , . ' --~. . ~----r
1
I
I
I I
I I
-- I I I
-to--- -.;- -
--r-----,-- -- ,
:
:
:
:
:
i
,
I
i
- ,
-;J ---~- --- i----~- --+-- + --- ~ ----~ - --- ~
v · uoo " - 005
- - - ~ - - - - :- - - - { - - - - :- - -
-7----:----~ --- ~- ---:
-015 +1~.,..j., 1 ~,~,,+I~,,~,~,hi~,+;~,~,~,;I-r.-~'+;~'~'''''';I-r.-~'+;=T"',;
Simulation R esults
().~
0.0
1.0
1.~
2.0
25
3.0
.15
4.0
4.5
time in second
Here we present the results of numerical simulation of the linear boundary feedback control system (4.11). We use the modal analysis to approximate the closed loop system (4.11) and solve the obtained finite dimensional system by fourth order Runge-Kutta method. For the purpose of simulation, we assume that the beam is uniform and related parameters are as follows: El = 900N-m2 , M = .5/(g-m2 , J = l/(g2 m , 1= 3m, p = 0.9J\g/m. The gain constants are taken as 1\'\ 4, A'2 6, and the initial vibration amplitute of the beam at the end-point is assumed (l/20 .. )m. The simulation results are shown in Figures 1 and 2. We found that the magnitude of various modes of vibration decreases rapidly as the mode number increases. So in the simulation, we only take the first three modes. From the Figures 1 and 2 we see that the feedback control scheme suggested here is rather satisfactory.
=
-~ I I
-(\10
5
-~I - - - - I~ - - - ~I - - --~, - - -!I --- ~-I --- ~- -~--.:I I I
Figure 1: End-point displacement of the link
~.: 0.00 -O~5 '.:;- -050
=
.
I:: : 0\::~ : :: ; ::::~ : ::; :::~: :::~:::~:: ::~ __
.J ____
I.._~~
"_r-' ---,_>
1:
:
-r- - - - , . --
- __
I
I
I
- t 00
- ' :!5
,
!
!
•
:
:
:
:
•
1
:
-I"" - - -T - - -.,- - - -,... - - - , - - - - I I
I
1
I
I
I
4 • • _ .1.. ___ J ___ .1.. __ • .1. __ .-1 ____ 1.. __ • .J ____ , I '
·~-075
: I
I
I
I
I
I
t
I
I
I
I
I
-1-~ - ---~ -- -~ ---- ~ - -- ~ -- -~--- -~- -- ~ ---- : I
I
I
I
,
,
,
I
r
I
I
,
t
I
I
I
I
1
1
1
1
'- _ _ J .
_ _ . '- _ _ _ J ___ .1.. __ _ 1 __ • ...1. _ __ L ___ .J. __ _ ,
- - -,---
-~
- - - , - -- -,----, - - -,- - -
-1"'---.,---- 1 I
I
I
1
1
1
- 1 50 +.,..,...,.,..j~"";"~rh-=_..,.,..j~"";"~rh-=tn-.,..., 0,0 o.~ 1.0 1.:. 2.0 2 .~ 3.0 J.~ ".0 •.~
time in second Figure 2: End-point displacement velocity of the link
862