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Stabilization of a rotating body beam without damping

June 1998
IEEE Transactions on Automatic Control 43(5):608 - 618

DOI:10.1109/9.668828

Source
IEEE Xplore

Authors:

Jean-Michel Coron

Sorbonne University

Brigitte D'Andrea Novel

Institut Mines-Télécom

This paper deals with the stabilization of a rotating body-beam system with torque control. The system we consider is the one studied by Baillieul and Levi (1987). Xu and Baillieul proved (1993) that, for any constant angular velocity smaller than a critical one, this system can be stabilized by means of a feedback torque control law if there is damping. We prove that this result also holds if there is no damping

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608 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 5, MAY 1998

Stabilization of a Rotating Body

Beam Without Damping

Jean-Michel Coron and Brigitte d’Andr

ea-Novel

Abstract—This paper deals with the stabilization of a rotating

body-beam system with torque control. The system we consider

is the one studied by Baillieul and Levi in [1]. In [12] it has been

proved by Xu and Baillieul that, for any constant angular velocity

smaller than a critical one, this system can be stabilized by means

of a feedback torque control law if there is damping. We prove

that this result also holds if there is no damping.

Index Terms—Distributed control system, elastic beam, hybrid

system, nonlinear control, stabilization.

I. INTRODUCTION

HE GOAL of this paper is to study the stabilization of

a system, already considered in [1], consisting of a disk

with a beam attached to its center and perpendicular to the

disk’s plane. The beam is conﬁned to another plane which is

perpendicular to the disk and rotates with the disk; see Fig. 1.

The dynamics of motion is (see [1] and [2])

(1)

(2)

where

is the length of the beam, is the mass per unit length

of the beam,

is the ﬂexural rigidity per unit length of the

beam,

is the angular velocity of the disk at time

is the disk’s moment of inertia, is the beam’s

displacement in the rotating plane at time

with respect to

the spatial variable

is the damping term, and is

the torque control variable applied to the disk at time

(see

Fig. 1).

If there is no damping

, and therefore (1) reads

(3)

Two types of damping are considered in [12].

1) Viscous damping:

with .

2) Structural damping:

with .

Manuscript received January 24, 1997. Recommended by Associate Editor,

C.-Z. Xu. This work was supported by DRET under Grant 951170 and by the

PRC-GDR “Automatique” of the CNRS and the MST.

J.-M. Coron is with the Universit´e de Paris-Sud, Analyze Num´erique

et EDP, 91405 Orsay Cedex, France (e-mail: Jean-Michel.Coron@math.u-

psud.fr).

B. d’Andr

ea-Novel is with the Centre de Robotique,

Ecole des Mines de

Paris, 75272 Paris Cedex 06, France.

Publisher Item Identiﬁer S 0018-9286(98)03589-2.

Fig. 1. The body-beam structure.

The asymptotic behavior of the solutions of (1) and (2) when

there is no control (i.e.,

) has been studied by Baillieul

and Levi in [1] and by Bloch and Titi in [4].

For both types of damping, Xu and Baillieul have con-

structed in [12] a feedback torque control law which globally

asymptotically stabilizes the equilibrium point

with

(4)

where

is an explicit critical angular velocity. This critical

angular velocity is given by

(5)

where

is the ﬁrst eigenvalue of the unbounded linear

operator

in with domain

(6)

They also prove that

is optimal: if , they prove that

there is no feedback law which asymptotically stabilizes

The asymptotically stabilizing feedback law constructed in

[12] is linear, and the stabilization is strong and exponential.

In [8] Laousy et al. have constructed a globally asymptotically

stabilizing feedback in the case where there is no damping but

when there is a control also on the free boundary of the beam

(

The goal of this paper is to investigate the stabilization

problem when there is no damping and no control on the free

boundary. We construct in this case a (nonlinear) feedback

torque control law which globally asymptotically stabilizes the

equilibrium point

, provided that (4) holds.

Our paper is organized as follows: in Section II we introduce

some notations and construct stabilizing feedback laws. In

0018–9286/98$10.00  1998 IEEE

CORON AND D’ANDR

EA-NOVEL: STABILIZATION OF A ROTATING BODY BEAM 609

Sections III and IV we prove the asymptotic stability for

and respectively.

II. N

OTATIONS AND STABILIZING FEEDBACK LAWS

Of course, by suitable scaling arguments, we may assume

that

. Let be the usual Sobolev

space

Let

The space with inner product

is a Hilbert space. For , let

We consider the unbounded linear operator in

with domain

It is well known that is an unbounded skew-adjoint operator

and therefore generates a unitary group

of bounded linear

operator on

. With this notation, our control system (2) and

(3) reads

(7)

(8)

with

(9)

By (9), we may consider

as the control.

Let us now give our stabilizing feedback law when

. Without loss of generality we may assume .

In order to explain how we have constructed our stabilizing

feedback law, let us ﬁrst consider (7) as a control system where

is the state and is the control. Then a natural candidate

for a control Lyapunov function for this system is

(10)

By the deﬁnition of

(11)

From (4), (5), (10), and (11), we get the existence of a constant

such that

(12)

Moreover, the time derivative of

along the trajectories of

(7) is given by

Therefore, in order to stabilize (7) where is the state and

is the control, it is natural to propose the feedback law

where is a function of class such that

(13)

Note that control system (7) and (8) is obtained by adding

an integrator to (7). Therefore, following Byrnes and Isidori

[5] or Tsinias [11], a natural candidate for a control Lyapunov

function for control system (7) and (8) is

(14)

Then the time derivative of

along the trajectories of (7) and

(8) is

(15)

and a natural candidate for a stabilizing feedback law is, with

(16)

With this feedback law one has, using (15)

(17)

Hence, by (13)

(18)

We require that

is of class , so that

is Lipschitz on any bounded set, and therefore the Cauchy

610 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 5, MAY 1998

problem associated to (7) and (8) has, for this feedback law

and for each initial datal in

, one and only one (maximal)

weak solution deﬁned on an open interval containing zero; see,

e.g., [9]. For technical reasons, we require also that

s.t. (19)

In Section III we will prove the following theorem.

Theorem 1: The feedback law

deﬁned by (16) globally

strongly asymptotically stabilizes the equilibrium point

of the control system (7) and (8).

Let us recall that by “globally strongly asymptotically

stabilizes the equilibrium point

,” one means that:

1) for every solution of (7)–(8) and (16)

(20)

2) for every

, there exists such that, for every

solution of (7)–(8) and (16),

Let us now turn to the case where . In order to

explain how we have constructed our stabilizing feedback law,

let us ﬁrst consider again (7) as a control system where

the state and

is the control. Then natural candidates for a

control Lyapunov function and a stabilizing feedback law are,

respectively,

and , where

satisﬁes on and on . One

can prove that (see Appendix A) such feedbacks always give

weak asymptotic stabilization, i.e., one gets instead of (20)

weakly in as (21)

But it is not clear that such feedbacks give strong asymp-

totic stabilization. It is possible to prove that one gets such

stabilization for the particular case where the feedback is

(22)

Let us recall that control system (7) and (8) is obtained by

adding an integrator to control system (7). Unfortunately,

deﬁned by (22) is not of class , and so one cannot use the

techniques given in [5] and [11]. The smoothness of this

also not sufﬁcient to apply the desingularization techniques

introduced in [10]. For these reasons, we use a different

control Lyapunov function and a different feedback law to

asymptotically stabilize control system (7). For the control

Lyapunov function, we take

where satisﬁes

(23)

so that, by (11)

(24)

Computing the time derivative

of along the trajectories

of (7) one gets

(25)

where, for simplicity, we write

for and where

(26)

Let us impose that

(27)

s.t. (28)

It is then natural to consider the feedback law for (7)

vanishing at zero and such that on

(29)

where

is such that

(30)

s.t. (31)

(32)

Note that, using (11), one gets that for every

(33)

which, with (27), (28), and (32), implies that

(34)

From (11), (26), and (27), one gets

(35)

From (34) and (35), we get that

is well deﬁned by (29) and

is of class

on . This regularity is sufﬁcient to apply

the desingularization technique of [10]: we note that (29) is

equivalent to

(36)

CORON AND D’ANDR

EA-NOVEL: STABILIZATION OF A ROTATING BODY BEAM 611

and therefore, following [10], one considers the following

control Lyapunov function for control system (7) and (8):

where, for simplicity, we write for and for .

Then, by (24)

Moreover, if one computes the time derivative of along

the trajectories of (7) and (8), one gets, using in particular (25)

(37)

where

(38)

with (39), as shown at the bottom of the page. Hence it is

natural to deﬁne feedback law

(40)

and, for every

(41)

Note that by (34)–(36)

(42)

Moreover, by (26), (28), and (31)–(33), there exists

such that

s.t. (43)

Using (26), (35), (36), (38), (39), and (41)–(43), one easily

checks that

is Lipschitz on any bounded set of .

Therefore, the Cauchy problem associated to (7) and (8) has,

for feedback law

, one and only one (maximal) solution

deﬁned on an open interval containing zero. By (32), (37),

(38), (40), and (41), one has

(44)

In Section IV, we prove the following.

Theorem 2: The feedback law

deﬁned by (40) and (41)

globally strongly asymptotically stabilizes the equilibrium

point

for the control system (7) and (8).

III. P

ROOF OF THEOREM 1

Throughout this section,

is deﬁned by (16). The Proof of

Theorem 1 is divided in two parts.

1) First we prove that the trajectories of (7) and (8) are

precompact in

for .

2) Then we conclude by LaSalle’s theorem.

The main difﬁcult point is to prove 1). More precisely, one

needs to prove that the energy associated to the high-frequency

modes is uniformly small [see (64)]. In order to prove this

uniform smallness, a key point is that all these modes satisfy

the same equation as

[see (62)]. Finally, in Lemma 1 we

get some estimates on

for any solution of (62) which

will allow us to prove the uniform smallness.

A. Precompactness of the Trajectories

Let

Then system (7) and (8) is equivalent to

(45)

(46)

Let

be a trajectory of (45) and (46). By (12), (14), and

(18)

is bounded in (47)

In this subsection we prove that

is precompact in (48)

Let us point out that one cannot apply the classical method [6]

due to Dafermos and Slemrod, since the operator from

into

is not monotone.

From (13) and (17) we get

(49)

(50)

(39)

612 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 5, MAY 1998

By (47)

is bounded

which, with (13) and (19), implies the existence of

such

that for all

(51)

From (49) and (51) we get that

(52)

(53)

From (50) and (53) we get

(54)

Note that (47) and (54) give

(55)

The unbounded linear operator

in with

domain given by (6) is self-adjoint with compact resolvent.

Thus, it has a discrete spectrum

, with

, and its eigenvectors may be

taken to form an orthonormal basis in

; let us recall

that its eigenvalues are simple.

For

, let us deﬁne

(56)

where the

, are deﬁned by

(57)

(58)

Note that the convergence in (57) is in

and the

convergence in (58) is in

. Moreover,

and and therefore .

Let us assume for the moment that the following lemma

holds.

Lemma 1: There exist

and such that

for every

and for every

such that

(59)

one has

(60)

(61)

For

and , let [respectively, ]be

the

-orthogonal projection of [respectively, ]

on the

-closed linear subspace spanned by the .

Let

. Then belongs to and

satisﬁes

(62)

with

(63)

By (47), in order to prove (48), it sufﬁces to prove that

s.t.

(64)

Indeed, let

be given and let us denote by

the open ball of centered at and of radius . By (64) there

exists a positive integer

such that

(65)

But, by (47), the set

is a bounded subset

and, since it is also a subset of the ﬁnite dimensional

space spanned by the

, this set is precompact.

Therefore, there exists a ﬁnite number of functions

of such that

(66)

From (65) and (66) we get

Hence, can be covered by ﬁnitely many balls

of radius

for every and so is precompact. Since by

(47)

is also precompact, we get (48).

CORON AND D’ANDR

EA-NOVEL: STABILIZATION OF A ROTATING BODY BEAM 613

From (62), we get

(67)

Taking the scalar product of (62) with

in the Hilbert

space

and using (67) we get, with

(68)

Clearly, for every

Hence, using (11), using (61) with

and using (60) with

we get that for every

and all

(69)

with

From (68) we get

which, with (47), (52), (55), (63), and (69) gives that for every

, there exists such that for every

(70)

Therefore, for every

and every

(71)

Indeed, (71) is true if

and, if ,it

sufﬁces to apply (70) with

replaced by such that

and

Moreover, since is compact, there exists

such that

(72)

which, with (71), proves (64).

Finally, let us prove Lemma 1. Let us ﬁrst point out that

we may assume that

(73)

Indeed, let us write

(74)

where

and are

deﬁned by

(75)

(76)

Taking the scalar product of (75) with

in , we get, with

(77)

Let us denote by

, various positive constants

which are independent of

and , but may depend on

. From (76) and (77) and Gronwall’s lemma, one gets the

existence of

such that

(78)

which implies that, for some

(79)

Let

and let . We require that the

are also independent of . From (78) and the

Cauchy–Schwarz inequality, we get the existence of

such

that for every

(80)

Let us deﬁne

and by

requiring

(81)

(82)

614 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 5, MAY 1998

(83)

(84)

The convergences in (81) and (82) [respectively, (83) and (84)]

are in

[respectively, ]. Then

(85)

Let us ﬁrst study (61). Let us recall that (see, e.g., [7, pp.

74–75])

(86)

Hence by (80) and (85), there exists

such that for

every

(87)

Similarly, using (78) and (86), one gets the existence of

such that for every

(88)

Moreover, by (74) and (76)

(89)

and by (95), which is proved below

(90)

Taking

, we get from (79) and (87)–(90)

(91)

which, with (89), shows that in order to prove (61), we may

assume, without loss of generality, (73).

Let us now turn to (60). It follows from (79) and (87)–(90)

that

(92)

Using (89) and (92) with

small enough, one gets that in

order to prove (60) we may assume without loss of generality

that (73) holds.

From now on, we assume that (73) holds. Then, by (59) we

can write, with

(93)

(94)

The convergences in (93) and (94) are in

and respectively. We have

(95)

CORON AND D’ANDR

EA-NOVEL: STABILIZATION OF A ROTATING BODY BEAM 615

which, in particular, implies (90) for . Moreover, we

have

By (86) we have, for large enough and if we let

Therefore, by a classical result on lacunary Fourier series (see,

e.g., [7, Lemmas 1.4.4 and 1.4.5]), for

large enough there

exists

such that

(96)

which, with (95), gives (60) and (61) for

large enough.

B. LaSalle’s Theorem

By (13), (48), and LaSalle’s theorem, in order to ﬁnish

the proof of Theorem 1, it sufﬁces to check that if

is such that

(97)

in (98)

then

(99)

But (99) follows directly from (60), (97), and (98).

Remark 1: Let us point out that the main difﬁculty in the

proof of Theorem 1 is the strong convergence of

to zero

(as ). Indeed the convergence of to zero

can be directly deduced from the fact that

is in

and its derivative is bounded (see (54), (8), (16), (47), and [8,

Lemma 1]). Moreover, the weak convergence of

to zero

can be proved by proceeding as in Appendix A.

IV. P

ROOF OF THEOREM 2

Throughout this section,

is deﬁned by (41) and .

We proceed as in the previous section, i.e.,

1) ﬁrst we prove that the trajectories of (7) and (8) are

precompact in

for ;

2) then conclude by LaSalle’s theorem.

A. Precompactness of the Trajectories

Let

be a solution of (7) and (8). By (44)

is bounded in on (100)

In order to prove that

is precompact in (101)

we are going to check that, again

s.t. (102)

From (7) we get

(103)

which implies that

(104)

Let

and be deﬁned by

(105)

(106)

By (36), (105), and (106), we have

(107)

For

, let

(108)

where the

, are deﬁned by (57) and (58) with .

It is clear that for any

there exists such

that for any

(109)

It follows easily from (109) and from the Proof of Lemma 1

that the following generalization of this lemma holds.

Lemma 2: For any

, there exist

and such that, for every , for every

, and for every

such that

one has

(110)

616 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 5, MAY 1998

(111)

Let

(112)

By (35) and (105), one has on

(113)

By (27), (100), (112), and (113)

(114)

Let

be as in Lemma 1 and let, for

(115)

From (112), (113), and (115), we get that

(116)

Moreover, from (103) and (107), we obtain, for every

and for every

(117)

which implies that

(118)

For

and , let

(119)

Let us denote by

, various constants which are

independent of

and . Using (100), (117),

(118), and (119), using (110) for

, and using (111) for

, we get the existence of such that

(120)

Note that (30) and (44) give

(121)

(122)

From (30), (31), (100), and (121) we get

(123)

(124)

From (35), (100), (106), (122), and (124), we get

(125)

By (115) and the Poincar

e inequality, there exists

such that

(126)

By (7), (115), and (119), there exists

such that if

(127)

then

(128)

By the Cauchy–Schwarz inequality and (108)

which, with (86), (110) for

and ,

CORON AND D’ANDR

EA-NOVEL: STABILIZATION OF A ROTATING BODY BEAM 617

(120), (126), (127), and (128), gives the existence of

such that for every and every

(129)

By (116) and (119)

(130)

Straightforward computations show that

(131)

which, with (35), (100), and (123), gives

(132)

From (114), (123), (125), (129), (130), and (132), we get that

for every

, there exist such that for every ,

for every

, and for every

which, as in Section III-A, implies (102).

B. LaSalle’s Theorem

By LaSalle’s theorem, (30), (36), (44), and (101), in order

to ﬁnish the proof of Theorem 2, it sufﬁces to check that if

is such that

(133)

on (134)

then

. But, from (26), (133), and (134), we get that

and do not depend on time and so there

exists

such that

(135)

By (27), (35), and (135)

(136)

From (133)–(136) and (110) of Lemma 2 applied with

and ,weget .

PPENDIX A

In this Appendix, we prove (21). By LaSalle’s theorem

(for

with the weak topology) it sufﬁces to check that if

satisﬁes

(137)

(138)

then

(139)

By (137) and (138)

(140)

Therefore, since

, there exists

such that

(141)

Clearly, in order to prove (139), it sufﬁces to check that

(142)

Since

there exists a sequence

of positive real numbers tending to as tends to

such that for some

weakly in (143)

Since

is weakly continuous, we have, using

(143)

weakly in as

(144)

with

. From (141) and (144) we get, if

(145)

Taking the time derivative of (145) with respect to time, we get

(146)

Using (96) with

we get, with (146), ,

which with (145) implies (142).

CKNOWLEDGMENT

The authors would like to thank C. Z. Xu for useful

discussions.

618 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 5, MAY 1998

REFERENCES

[1] J. Baillieul and M. Levi, “Rotational elastic dynamics,” in Physica, vol.

27D. North-Holland, Amsterdam: Elsevier Sci., 1987, pp. 43–62.

[2]

, “Constrained relative motions in rotational mechanics,” Arch.

Rational Mechanics Anal., vol. 115, pp. 101–135, 1991.

[3] J. M. Ball and M. Slemrod, “Feedback stabilization of distributed

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1979.

[4] A. M. Bloch and E. Titi, “On the dynamics of rotating elastic beams,”

in New Trends in Systems Theory, G. Conte, A. Perdon, and B. Wyman,

Eds. Boston, MA: Birkh¨auser, 1991, pp. 128–135.

[5] C. I. Byrnes and A. Isidori, “New results and counterexamples in non-

linear feedback stabilization,” Syst. Contr. Lett., vol. 12, pp. 437–442,

1989.

[6] C. M. Dafermos and M. Slemrod, “Asymptotic behavior of nonlinear

contraction semi-groups,” J. Funct. Anal., vol. 13, pp. 97–106, 1973.

[7] W. Krabs, “On moment theory and controllability of one dimensional

vibrating systems and heating processes,” Lecture Notes in Control

and Information Sciences, vol. 173. Berlin, Germany: Springer-Verlag,

1992.

[8] H. Laousy, C. Z. Xu, and G. Sallet, “Boundary feedback stabilization

of a rotating body-beam system,” IEEE Trans. Automat. Contr., vol. 41,

pp. 241–245, 1996.

[9] A. Pazy, Semigroups of Linear Operators and Applications to Partial

Differential Equations. New York: Springer-Verlag, 1983.

[10] L. Praly, B. d’Andr

ea-Novel, and J.-M. Coron, “Lyapunov design of

stabilizing controllers for cascaded systems,” IEEE Trans. Automat.

Contr., vol. 36, pp. 1177–1181, 1991.

[11] J. Tsinias, “Sufﬁcient Lyapunov-like conditions for stabilization,” Math.

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Jean-Michel Coron was born in Paris, France, in

1956. He obtained the dipl

ome d’ing

enieur from the

Ecole Polytechnique in 1978 and from the Corps des

Mines in 1981. He received the th

ese d’

Etat in 1982.

He was a Researcher at the Ecole Nationale

Sup

erieure des Mines de Paris, then Associate Pro-

fessor at Ecole Polytechnique, Professor at the Uni-

versit

e Paris-Sud, Director of Research at CNRS,

and Director of the Centre de Math

ematiques et de

Leurs Applications (CNRS and ENS de Cachan). He

is currently a Professor at the Universit

e Paris-Sud.

His research interests include nonlinear partial differential equations, calculus

of variations, and nonlinear control theory.

Brigitte d’Andr

ea-Novel was born in Villerupt,

France, in 1961. She graduated from the

Ecole

Sup´erieure d’Informatique

Electronique Automa-

tique, France, in 1984. She received the Doctorate

degree from the

Ecole des Mines de Paris in 1987

and the Habilitation degree from the University of

Paris XI, Orsay, in 1995.

Presently she is a Professor of Systems Control

Theory at the

Ecole des Mines de Paris, and she is

also responsible for a research group on Control of

Mechanical Systems at the “Centre de Robotique”

Ecole des Mines de Paris. Her current research interests include nonlinear

control theory and its applications to underactuated mechanical systems,

modeling and control of mobile robots with application to automated

highways, and boundary control of mechanical systems with ﬂexibilities

(described by coupled ODE and PDE).

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Rapid Stabilization of a Linearized Bilinear

1-D

Schr\"odinger Equation

Preprint

Nov 2016

We consider the one dimensional Schr\"odinger equation with a bilinear control and prove the rapid stabilization of the linearized equation around the ground state. The feedback law ensuring the rapid stabilization is obtained using a transformation mapping the solution to the linearized equation on the solution to an exponentially stable target linear equation. A suitable condition is imposed on the transformation in order to cancel the non-local terms arising in the kernel system. This conditions also insures the uniqueness of the transformation. The continuity and invertibility of the transformation follows from exact controllability of the linearized system.

Approximate stabilization of a quantum particle in a 1D infinite square potential well

Preprint

Jan 2008

We consider a non relativistic charged particle in a 1-dimensional infinite square potential well. This quantum system is subjected to a control, which is a uniform (in space) time depending electric field. It is represented by a complex probability amplitude solution of a Schrodinger equation on a 1-dimensional bounded domain, with Dirichlet boundary conditions. We prove the almost global approximate stabilization of the eigenstates by explicit feedback laws.

Exponential stabilization of cascade ode-linearized kdv system by boundary Dirichlet actuation

Preprint

Full-text available

Jan 2018

Habib Ayadi

In this paper, we solve the problem of exponential stabilization for a class of cascade ODE-PDE system governed by a linear ordinary differential equation and a 1-d linearized Korteweg-de Vries equation (KdV) posed on a bounded interval. The control for the entire system acts on the left boundary with Dirichlet condition of the KdV equation whereas the KdV acts in the linear ODE by a Dirichlet connection. We use the socalled backstepping design in infinite dimension to convert the system under consideration into an exponentially stable cascade ODE-PDE system.Then, we use the invertibility of such design to achieve the exponential stability for the ODE-PDE cascade system under consideration by using Lyapunov analysis.

Exponential Stabilization of a Flexible Structure: A Delayed Boundary Force Control Versus a Delayed Boundary Moment Control

Article

Full-text available

Feb 2024

The main concern of this paper is to study the boundary stabilization problem of the disk-beam system. To do so, we assume that the boundary control is either of a force type control or a moment type control and is subject to the presence of a constant time-delay. First, we show that in both cases, the system is well-posed in an appropriate functional space. Next, the exponential stability property is established. Finally, the obtained outcomes are ascertained through numerical simulations.

Boundary stabilization of one-dimensional cross-diffusion systems in a moving domain: Linearized system

Preprint

Full-text available

Jul 2023

We study the boundary stabilization of one-dimensional cross-diffusion systems in a moving domain. We show first exponential stabilization and then finite-time stabilization in arbitrary small-time of the linearized system around uniform equilibria, provided the system has an entropic structure with a symmetric mobility matrix. One example of such systems are the equations describing a Physical Vapor Deposition (PVD) process. This stabilization is achieved with respect to both the volume fractions and the thickness of the domain. The feedback control is derived using the backstepping technique, adapted to the context of a time-dependent domain. In particular, the norm of the backward backstepping transform is carefully estimated with respect to time.

Rapid Stabilization of Timoshenko Beam by PDE Backstepping

Preprint

Full-text available

Jul 2022

In this paper, we present a rapid boundary stabilization of a Timoshenko beam with anti-damping and anti-stiffness at the uncontrolled boundary, by using PDE backstepping. We introduce a transformation to map the Timoshenko beam states into a (2+2) x (2+2) hyperbolic PIDE-ODE system. Then backstepping is applied to obtain a control law guaranteeing closed-loop stability of the origin in the H^1 sense. Arbitrarily rapid stabilization can be achieved by adjusting control parameters. Finally, a numerical simulation shows that the proposed controller can rapidly stabilize the Timoshenko beam. This result extends a previous work which considered a slender Timoshenko beam with Kelvin-Voigt damping, allowing destabilizing boundary conditions at the uncontrolled boundary and attaining an arbitrarily rapid convergence rate.

Rapid Stabilization of Timoshenko Beam by PDE Backstepping

Article

Jan 2023

In this paper, we present rapid boundary stabilization of a Timoshenko beam with anti-damping and anti-stiffness at the uncontrolled boundary, by using infinite-dimensional backstepping. We introduce a Riemann transformation to map the Timoshenko beam states into a set of coordinates that verify a 1-D hyperbolic PIDE-ODE system. Then backstepping is applied to obtain a control law guaranteeing closed-loop stability of the origin in the

L^{2}

sense. Arbitrarily rapid stabilization can be achieved by adjusting control parameters, and has not been achieved in previous results. Finally, a numerical simulation shows the effectiveness of the proposed controller. This result extends a previous work which considered a slender Timoshenko beam with Kelvin-Voigt damping, by allowing destabilizing boundary conditions at the uncontrolled boundary and attaining an arbitrarily rapid convergence rate.

Boundary stabilization of one-dimensional cross-diffusion systems in a moving domain: Linearized system

Article

Mar 2023
J DIFFER EQUATIONS

Backstepping-Based Exponential Stabilization of Timoshenko Beam with Prescribed Decay Rate

Article

Nov 2022

In this paper, we present a rapid boundary stabilization of a Timoshenko beam with anti-damping and anti-stiffness at the uncontrolled boundary, by using PDE backstepping. We introduce a transformation to map the Timoshenko beam states into a (2+2) × (2+2) hyperbolic PIDE-ODE system. Then backstepping is applied to obtain a control law guaranteeing closed-loop stability of the origin in the H¹ sense. Arbitrarily rapid stabilization can be achieved by adjusting control parameters. Finally, a numerical simulation shows that the proposed controller can rapidly stabilize the Timoshenko beam. This result extends a previous work which considered a slender Timoshenko beam with Kelvin-Voigt damping, allowing destabilizing boundary conditions at the uncontrolled boundary and attaining an arbitrarily rapid convergence rate.

Constrained relative motions in rotational mechanics

Article

Full-text available

Jun 1991

The dynamical effects of imposing constraints on the relative motions of component parts in a rotating mechanical system or structure are explored. It is noted that various simplifying assumptions in modeling the dynamics of elastic beams imply strain constraints, i.e., that the structure being modeled is rigid in certain directions. In a number of cases, such assumptions predict features in both the equilibrium and dynamic behavior which are qualitatively different from what is seen if the assumptions are relaxed. It is argued that many pitfalls may be avoided by adopting so-called geometrically exact models, and examples from the recent literature are cited to demonstrate the consequences of not doing this. These remarks are brought into focus by a detailed discussion of the nonlinear, nonlocal model of a shear-free, inextensible beam attached to a rotating rigid body. Here it is shown that linearization of the equations of motion about certain relative equilibrium configurations leads to a partial differential equation. Such spatially localized models are not obtained in general, however, and this leaves open questions regarding the way in which the geometry of a complex structure influences computational requirements and the possibility of exploiting parallelism in performing simulations. A general treatment of linearization about implicit solutions to equilibrium equations is presented and it is shown that this approach avoids unintended imposition of constraints on relative motions in the models. Finally, the example of a rotating kinematic chain shows how constraining the relative motions in a rotating mechanical system may destabilize uniformly rotating states.

Boundary Feedback Stabilization of a Rotating Body-Beam System

Article

Full-text available

Mar 1996

This paper deals with boundary feedback stabilization of a flexible beam clamped to a rigid body and free at the other end. The system is governed by the beam equation nonlinearly coupled with the dynamical equation of the rigid body. The authors propose a stabilizing boundary feedback law which suppresses the beam vibrations so that the whole structure rotates about a fixed axis with any given small constant angular velocity. The stabilizing feedback law is composed of control torque applied on the rigid body and either boundary control moment or boundary control force (or both of them) at the free end of the beam. It is shown that in any case the beam vibrations are forced to decay exponentially to zero

On The Dynamics of Rotating Elastic Beams

Article

Jan 1991

In this paper we show that the rotating elastic beam model proposed by Baillieul and Levi has an inertial manifold which is linear. We emphasize that the spectrum of the linear dissipative part of the model equations does not satisfy the gap condition.

Asymptotic behavior of nonlinear contraction semigroups

Article

May 1973

The asymptotic behavior of weak solutions of the equation {u(t) + Au(t) ϵ ƒ(t) (A maximal monotone in Hilbert space) is determined via a characterization of ω-limit sets of the contraction semigroup generated by −A.

Feedback stabilization of distributed semilinear control systems

Article

Mar 1979

This paper considers feedback stabilization for the semilinear control system [(u)\dot](t) = Au(t) + u(t)B(u(t)).\dot u(t) = Au(t) + \upsilon (t)B(u(t)). HereA is the infinitesimal generator of a linearC 0 semigroup of contractions on a Hilbert spaceH andB : H H is a nonlinear operator. A sufficient condition for feedback stabilization is given and applications to hyperbolic boundary value problems are presented.

Sufficient Lyapunov-Like Conditions for Stabilization

Article

Dec 1989

J. Tsinias

In this paper we study the stabilizability problem for nonlinear control systems. We provide sufficient Lyapunov-like conditions for the possibility of stabilizing a control system at an equilibrium point of its state space. The stabilizing feedback laws are assumed to be smooth except possibly at the equilibrium point of the system.

Rotational elastic dynamics

Article

Jul 1987
PHYSICA D

The combined dynamical effects of elasticity and a rotating reference frame are explored for structures in a zero gravity environment. A simple yet general approach to modeling is presented, and this approach is applied to analyze in detail the dynamics of a specific prototypical structure. Energy dissipation is included and its effects are studied in detail in a model problem. Bifurcations and stability are analyzed as well.

New Results and Examples in Nonlinear Feedback Stabilization

Article

Jun 1989
SYST CONTROL LETT

Using the general methodology of nonlinear zero dynamics we derive globally stabilizing state feedback laws for broad classes of nonlinear systems. While it is tempting to reinterpret this design philosophy in terms of high gain feedback, we present an example of a globally stabilizable nonlinear feedback system which cannot be stabilized, even locally, by any output feedback law - in particular, by a high gain law.

On Moment Theory and Controllability of One-Dimensional Vibrating Systems and Heating Processes

Article