Asymptotic analysis of a 2D overhead crane with input delays in the boundary control
Abstract.
The paper investigates the asymptotic behavior of a 2D overhead crane with input delays in the boundary control. A linear boundary control is proposed. The main feature of such a control lies in the facts that it solely depends on the velocity but under the presence of timedelays. We endup with a closedloop system where no displacement term is involved. It is shown that the problem is wellposed in the sense of semigroups theory. LaSalle’s invariance principle is invoked in order to establish the asymptotic convergence for the solutions of the system to a stationary position which depends on the initial data. Using a resolvent method it is proved that the convergence is indeed polynomial.
Key words and phrases:
Overhead crane; boundary velocity control; timedelay; asymptotic behavior2010 Mathematics Subject Classification:
34B05, 34D05, 70J25, 93D15Contents
1. Introduction
Overhead cranes are extensively utilized in a variety of industrial and construction sites. Usually, it consists of a hoisting mechanism such as a hoisting cable and a hook and a support mechanism like a girder (trolley) [2]. The aim of using such cranes is to horizontally transport pointtopoint a suspended mass/load. It is wellknown that cables possess the inherent flexibility characteristics and can only develop tension [2]. Such natural features inevitably cause deflection in transversal direction of the cable. Furthermore, the suspended load is always subject to swings due to several reasons. Thereby, the behavior of the overhead crane system with flexible cable can generate complex system dynamics (see [2] for more details).
We shall consider in the present work an overhead crane system which consists of a motorized platform of mass moving along an horizontal rail. A flexible cable of length , holding a load mass , is attached to the platform (see Fig. 1). Furthermore, it is assumed that:
(i) The cable is completely flexible and nonstretching.
(ii) The length of the cable is constant.
(iii) Transversal and angular displacements are small.
(iv) Friction is neglected.
(v) The masses and are point masses.
(vi) The angle of the cable with respect to the vertical axis is small everywhere.
Under the above assumptions, the overhead crane is modeled by a hybrid PDEODE system (see [7] and [27]). For sake of completeness, we shall provide some details about the derivation of such a model (the reader is referred to [7] and [27] for more details).
Let be the the tension of the cable, be the angle between and the axis, and consider a portion of the cable of length . Newton’s law leads to
We can write due to the assumption of smallness of transversal and angular displacements. On the other hand, since the tension of the cable is essentially due to the action on its lower part, we have , which is the modulus of tension of the cable and will be denoted by . This, together with the above equation imply that
(1.1) 
We turn now to the equation of the platform part of the system (see Fig. 2). Taking into account the external controlling force , we have
which can be rewritten
(1.2) 
as and .
Using similar arguments for the the load mass (see Fig. 3), we have
(1.3) 
Combining (1.1)(1.3), we have the system
(1.4) 
where is supposed to satisfy the following conditions
(1.5) 
For simplicity and without loss of generality, we shall set the length .
As mentioned above, the objective is to seek a delayed control depending solely on the velocity so that the solutions of the closedloop system asymptotically converge to an equilibrium point in a suitable functional space.
The boundary stabilization of the system (1.4) has been the object of a considerable mathematical research. There are two categories of research articles: in the first category, at least one of the dynamical terms in the boundary conditions is neglected. In other words, either or does not appear in the system or even both terms are not present. For instance, it has been shown in [27] that the feedback law
exponentially stabilizes the system (1.4) with under appropriate assumptions on the function . Another stabilization result for the system (1.4) with has also been established in [15] via the action of the following feedback:
where is an additional control to be applied on the load mass. In [7], the asymptotic stabilization has been proved as long as a dynamical control is acting on the boundary . We also mention that a stabilization result has been obtained in [12] by proposing the feedback law
with and is a function in . Of course, such a result has been established under some conditions on the feedback gains as well as the function . Similar findings have been obtained in [8] for other types of controls containing a displacement term. We conclude this discussion about the first category of articles available in the literature by pointing out that it has been noticed in [11] that in all references cited above, either the boundary conditions in (1.4) or the stabilizing feedback law involves the displacement term . This is mainly due to the fact that most of the authors defined the energynorm of the system by
This observation has motivated the authors in [11] to consider a displacement term in the equation and propose a general class of feedback law containing only the velocity. In fact, the closedloop system in [11] has the following form
(1.6) 
in which and are two nonlinear functions. The multiplier method has been successfully used in [11] to get precise decay rate (polynomial or exponential) estimates of the energy of the system (1.6) according to the type of assumptions on the functions and . Recently, the backstepping approach has been successfully applied to a variant of the system (1.4) leading to an exponentially stabilizing boundary feedback controller [8]. In the same spirit, the following feedback law
has been suggested in [29] in the case where and and the Riesz basis property has been shown.
The second category of research papers takes into consideration the dynamics of both the load mass and platform mass. Within this context, it has been proved in [13] that the system (1.4) can be strongly (but nonuniformly) stabilized by means of the control
where is a suitable function. This motivated several authors to propose controls of higher orders to reach the uniform exponential stability. Indeed, the uniform stabilization holds if
It turned out that the same result result can be achieved by the control
where and are positive constants satisfying Motivated by the work of [11], a feedback control depending only on the velocity has been proposed in [14] for the system (1.4) and an asymptotic convergence result has been established (see also [1]).
All the papers mentioned above do not take into consideration timedelay. In turn, it is wellknown that delays are inevitable in practice as they naturally arises in most systems due to the time factor needed for the communication among the controllers, the sensors and the actuators of systems or in some cases due to the dependence of the state variables on past states. Furthermore, it has been noticed that the presence of a delay in a system could be a source of poor performance and instability [17][19] (see also [28][4], [5] and [6]).
The present work places primary emphasis on the analysis of the system (1.4) under the action of the following input delay
(1.7) 
where , and is the timedelay.
It is worth mentioning that the absence of the displacement term in the closedloop system prevents the applicability of classical Poincaré inequalities. To overcome this difficulty, an appropriate energynorm is suggested.
The main contribution of the present work is threefold:

Show that despite the presence of the delay term in the proposed feedback control law, the closedloop system possesses the asymptotic convergence property of its solutions to an equilibrium state which depends on the initial conditions.

Provide the rate of convergence of solutions of the closedloop system to the equilibrium state, in contrast to the work [14] where such a result has not been achieved.
The paper is organized as follows. The next section is devoted to the proof of existence and uniqueness of the solutions to the closedloop system. Section 3 deals with the asymptotic behavior of solutions via the use of LaSalle’s principle. Section 4 is devoted to the polynomial convergence of solutions. Finally, the paper closes with conclusions and discussions.
2. Wellposedness of the system
With the feedback law in (1.7), we obtain the closedloop system
(2.1) 
where obeys the condition (1.5), and .
Our immediate task is to seek an appropriate energy associated to (2.1). To proceed, let
(2.2) 
where is a positive constant. Using (2.1) and integrating by parts, a formal computation yields
(2.3) 
Applying Young’s inequality, the latter becomes
(2.4) 
for any positive constant . Subsequently, we introduce the following additional energy functional
(2.5) 
where
(2.6) 
and and are constants to be determined. Following the same arguments as for , we get
(2.7) 
Thereafter, we define the total energy of the system (2.1) as follows
(2.8) 
This, together with (2.4) and (2.7), imply that
(2.9) 
In order to make the energy decreasing, we shall assume that
(2.10) 
and then choose such that
(2.11) 
whereas the other constants are
(2.12) 
In light of (2.9) and (2.10)(2.12), we deduce that
(2.13) 
and hence the energy is decreasing.
Remark 1.
Here and elsewhere throughout the paper, we shall use the following definitions and notations for the Hilbert space and the Sobolev space , more precisely
equipped with its usual norm
and
endowed with the standard norm
Let us return now to our closedloop system (2.1). Using the wellknown change of variables [16]
(2.14) 
the system (2.1) becomes
(2.15) 
Let and consider the state variable Then, our state space is defined by
equipped with the following real inner product (the complex case is similar)
(2.16) 
in which satisfies the condition (2.11), while and is a positive constant to be determined. Note that is positive due to (2.10).
The first result is stated below.
Proposition 1.
Proof.
It suffices to show the existence of two positive constants and such that
(2.17) 
where denotes the usual norm of , that is,
The righthand inequality is straightforward. Indeed, Young’s and Hölder’s inequalities yield
Moreover, by virtue of (1.5) and the wellknown trace continuity Theorem [3]
the above inequality leads to the desired result with depending on and .
With regard to the other inequality of (2.17), we proceed as follows:
(2.18) 
It follows from Young’s inequality that for any ,
(2.19)  
Combining (2.18) and (2.19), and choosing , we obtain
(2.20) 
A direct computation gives
(2.21) 
for any Inserting (2.21) into (2.20) and using (1.5) yields
(2.22) 
for any and . Finally, we choose such that
where Thus, (2.17) holds and the proof of Proposition 1 is achieved. ∎
We are now in a position to set our problem in the state space . Define a linear operator by
(2.23) 
The closedloop system (2.1) can now be formulated in terms of the operator by the evolution equation over
(2.24) 
in which and
The wellposedness result is stated below.
Theorem 1.
(i) The operator defined by (2.23) is densely defined in and generates on a semigroup of contractions . Moreover, , the spectrum of , consists of isolated eigenvalues of finite algebraic multiplicity only.
(ii) For any initial condition , the system (2.24) has a unique mild solution . In turn, if , then necessarily the solution is strong and belongs to .
Proof.
Let Then, in light of (2.16) and (2.23), a simple integration by parts gives
(2.25) 
and so the operator is dissipative due to the assumption (2.11).
Next, we claim that the operator is onto for sufficiently large. To ascertain the correctness of this claim, one has to show that given , there exists for which . Although this can be considered as a classical problem, one can easily verify that the latter is equivalent to solve the following system
(2.26) 
Solving the equation of in the above system, we obtain
(2.27) 
and hence
(2.28) 
This, together with (2.26) and (2.27), imply that one has only to seek satisfying
(2.29) 
Multiplying the first equation in (2.29) by , we get the weak formulation
(2.30) 
which in turn can be written in the form where is a bilinear form defined by
such that
and is a linear form given by
Applying LaxMilgram Theorem [10], one can deduce the existence of a unique solution of (2.29) as long as is large. This establishes that the range of is , for . Thus, according to semigroup theory [25], the operator is densely defined in and generates on a semigroup of contractions denoted by . As a direct consequence of the fact that, for , the range of is , it follow that exists and maps into . Finally, using Sobolev embedding [3], if follows that is compact and hence the spectrum of , consists of isolated eigenvalues of finite algebraic multiplicity only [23]. This completes the proof of the first assertion (i) in Theorem 1.
Concerning the proof of the second assertion, it suffices to use (i) and invoke semigroups theory [25]. ∎
3. Asymptotic behavior
We begin this section by recalling the following result.
Theorem 2.
[22] Let be the infinitesimal generator of a semigroup in a Hilbert space such that has compact resolvent. Then, is strongly stable if and only if it is uniformly bounded and , for any in the spectrum of .
It is clear from (2.23) that is an eigenvalue of whose eigenfunction is , where . Thus, Theorem 2 implies that the semigroup generated by is not stable. However, we are able to prove the main result of the paper which is stated next.
Theorem 3.
Proof.
The proof depends on an essential way on the application of LaSalle’s invariance principle [22]. Using a standard argument of density of in and the contraction of the semigroup , it suffices to prove Theorem 3 for smooth initial data . Let be the solution of (2.1). It follows from Theorem 1 that the trajectories set of solutions is a bounded for the graph norm and thus precompact by virtue of the compactness of the operator . Invoking LaSalle’s principle, we deduce that is non empty, compact, invariant under the semigroup and in addition as [22]. Clearly, in order to prove the convergence result, it suffices to show that reduces to . To this end, let and consider as the unique strong solution of (2.24). It is wellknown that is constant [22] and thus