Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension

The paper is concerned with the finite-time stabilization of a hybrid PDE-ODE system describing the motion of an overhead crane with a flexible cable. The dynamics of the flexible cable is described by the wave equation with a variable coefficient which is an affine function of the curvilinear abscissa along the cable. Using several changes of variables, a backstepping transformation, and a finite-time stable second-order ODE for the dynamics of a conveniently chosen variable, we prove that a global finite-time stabilization occurs for the full system constituted of the platform and the cable. The kernel equations and the finite-time stable ODE are numerically solved in order to compute the nonlinear feedback law, and numerical simulations validating our finite-time stabilization approach are presented.

and that the acceleration of the load mass could be neglected, they obtained the following system: The initial conditions are y(s, 0) = y 0 (s), y t (s, 0) = y 1 (s), X p (0) = X 0 p ,Ẋ p (0) = X 1 p . In the above system, s denotes the curvilinear abscissa (i.e. the arclength) along the cable, y = y(s, t) is the horizontal displacement at time t of the point on the cable of curvilinear abscissa s, X p is the abscissa of the platform, ρ the mass per unit length of the cable, and V the force applied to the platform. As usual, y tt = ∂ 2 y/∂t 2 , y ss = ∂ 2 y/∂s 2 , etc., andẌ p = d 2 X p /dt 2 .

Previously obtained control results for the crane
An asymptotic (but not exponential) stabilization of (1.1)-(1.4) was established in [1], while an exponential stabilization was subsequently derived in [2] by using the cascaded structure of the system and a backstepping approach (see [16] for developments concerning this approach). A similar result was obtained for system (1.8)-(1.11) with a constant tension (but with Dirichlet boundary conditions) in [23]. The dynamics of the load mass was taken into account in [21].
The backstepping approach is a powerful tool for the design of stabilizing controllers in the context of finite dimensional systems (see for example [29]), but the cascaded structure of flexible mechanical systems coupling ODE and PDE is also a useful property in regard to stabilization, as for the overhead crane with flexible cable. One can also refer to [11], where the authors proposed a class of nonlinear asymptotically stabilizing boundary feedback laws for a rotating body-beam without natural damping. Let us also mention that in [3] the authors considered the case of a variable length flexible cable.

Finite-time stabilization of 2 × 2 hyperbolic systems
Solutions of certain hyperbolic systems with transparent boundary conditions can reach the equilibrium state in finite time. Such a property, called finite-time stability in [6,24] or super-stability in [30], was first noticed in [18] for the usual wave equation. The extension of such a property to the wave equation on a network was investigated in [6,30]. It turns out that the finite-time stability also holds for one-dimensional first order quasilinear hyperbolic systems of diagonal form without source terms [17,24]. If source terms are incorporated in such systems, the finite-time stability is in general lost when using transparent boundary conditions, but a rapid stabilization still occurs [14]. In the linear case, however, the use of a boundary feedback law based on a backstepping transformation allows to recover the finite-time stability for hyperbolic systems with source terms [12].

Finite-time stability of an abstract evolution system
The finite-time stability of an ODE evolution systeṁ is defined as follows.
Definition 1.1 (Finite-time stability of an ODE evolution system [9]). Let Φ = Φ(x, t), x ∈ V, t ∈ R + , be the flow associated with an ODE evolution system in a finite-dimensional space V ⊆ R n , with x = 0 an equilibrium point, that is, Φ(0, t) = 0 for all t ∈ R + .
We say that the flow Φ(x, z, t) (x,z,t)∈H×[0,1]×R+ is globally finite-time stable (i) if there exists a nondecreasing function T : (0, +∞) → (0, +∞) called the settling-time function such that Φ(x, z, t) = 0 for all x ∈ H \ {0}, all z ∈ [0, 1] and all t ≥ T ( x H ) (finite-time convergence); (ii) and if the equilibrium point x = 0 is Lyapunov stable, that is, for each > 0, there exists some δ > 0 such that 1.6. Aim and structure of the present paper The aim of the paper is to design a boundary feedback law U (t) leading to the finite-time stability of the system with affine tension. For that purpose, we first use several changes of variables to transform the original system (1.8)-(1.11) into a 2×2 hyperbolic system with coupling terms. Next, following [12], we define a target system for which the application of transparent boundary conditions gives a finite-time stability, and we define a backstepping transformation leading to this target system. Finally, we show that the finite-time stabilization of a certain quantity φ defined in terms of X p (t) and z = z(x, t) (see below (4.2)) yields the finite-time stabilization of both the platform and the cable.
The paper is scheduled as follows. In Section 2, the original system is transformed into a 2×2 hyperbolic system with coupling terms. In Section 3, the 2×2 hyperbolic system is transformed into the target system by using some Volterra integral transformation borrowed from [12]. Methods for numerically solving the kernel equations are described. Section 4 is dedicated to the proof of our main result concerning the finite-time stabilization of the coupled PDE-ODE system. Our theoretical result is illustrated by a simulation of the finite-time stabilization of the overhead crane in Section 5. Finally, we give some words of conclusion in Section 6.

Derivation of the 2 × 2 hyperbolic system
The second order hyperbolic equation (1.8) with boundary conditions (1.9)-(1.10) and initial conditions is rewritten as a 2 × 2 quasilinear hyperbolic system by performing two transformations.

First transformation
Following [19], we set z(x, t) := y(s, t), where so that x(0) = 0 and x(1) = 1. With this transformation, the PDE (1.8) can be rewritten as 3) The PDE (2.2) has to be supplemented with the boundary conditions 4) and the initial conditions

Second transformation
2) is subsequently rewritten as a system of two first-order PDEs. Noticing that (∂ t − λ∂ x )(∂ t + λ∂ x )z = −λλ z x = (∂ t + λ∂ x )(∂ t − λ∂ x )z, we infer that the Riemann invariants S := z t + λz x and D := z t − λz x satisfy the system we obtain the system with boundary conditions W (t) denoting a feedback law that can be expressed in terms of the original control U (t) =Ẍ p (t) and the angle θ(t) = y s (1, t) as System (2.9)-(2.11) is in the form required to apply the backstepping transformation from [12].
Definition 3.1 (Backstepping transform [12]). The backstepping transform between the original state variables w(x, t) and the target state variables γ(x, t) is defined through a Volterra integral equation with the direct kernels K(x, ξ) decomposed as The inverse transformation is given by with the inverse kernels By a direct calculation, it is proven in [12] that the direct kernels K are the solution of the following Goursat system of two 2 × 2 first-order hyperbolic PDEs Fig. 2) with the boundary conditions It is also shown in [12] that the indirect kernels L satisfy a similar Goursat system of 4 × 4 first-order hyperbolic PDEs defined on T .
Note that it is also possible to compute the inverse kernels L from the direct kernels K [7]: If the coefficients i (x) and c i (x) in the Goursat systems are constant (independent of x or ξ), explicit solutions for the kernels have been obtained in [32]. In the general case, numerical methods have been designed in [7]. These methods approximate the left-hand side of the equations of the Goursat system (for instance (3.7)) by a directional derivative and interpolate between values at points of a triangular half of an equidistant square grid as shown in Figure 2.
and settling-time function

Finite-time stabilization of the coupled PDE-ODE system
In the case of the coupled (PDE-ODE) crane model (1.8)-(1.11), one does not control the dynamics of the system cable-platform through the boundary condition v(1, t) = W (t), but rather through the acceleration X p (t) = U (t) of the platform.
In that case, the transparent boundary condition β(1, t) = 0 in (3.4) is not necessarily satisfied.
Our goal is to design a feedback law U (t) in such a way that both β(1, t) and an auxiliary variable φ(t) vanish after a finite time depending on the initial data. Next, by construction of the auxiliary variable φ(t), both the target variable γ(x, t) (cable) and X p (t) (platform) will be stabilized in a finite time, after which the entire system will be at the equilibrium position. This is the subject of the next section.

Finite-time stability of the complete system
A finite-time stabilizer U (t) for the crane and the cable with affine tension is constructed as follows. Our goal is to have the condition β(1, t) = 0 satisfied for t large enough. We write where a(x) and b(x) are two functions to be defined. Then we havė where we used an integration by parts in the last line. Replacing in (4.3)Ẋ p (t) by its expression in (4.1) results inφ We define the functions a and b as with a 0 , b 0 some constants still to choose. Then and hence (4.4) becomesφ for α(0, t) = β(0, t), (q = 1). Now we let We obtainφ Later on, we shall define β(1, t) fromφ(t) using (4.9). It is thus important to prove that µ = 0. This is done in the following proposition.
Then the Goursat system satisfied by the kernels L reads (see [12]) with the boundary conditions The existence and uniqueness of the solution (L αα , L αβ , L βα , L ββ ) ∈ [C 0 (T )] 4 was already established in Theorem A.1 of [12]. However, the claims in Proposition 4.1 are still to be justified. We introduce the following 2 × 2 system: with the boundary conditions Proof of Lemma 4.2. We first express (f, g) as a fixed-point of a map from [C 0 (T , R + )] 2 into itself. Next, reducing the domain for (x, ξ) to a "trapezoid", we prove that the above map is a contraction. The full domain T is covered by iterating the above construction.
Using the stability of the origin in R 2 for (4.26) and the formulas (4.36), (4.40), we infer the stability of the origin in H for system (3.3), (4.26) and (4.28), which is thus finite-time stable (according to Def. 1.3).
The Theorem 4.4 proves the finite-time stability of the hybrid PDE-ODE system in the coordinates (X p (t), α(x, t), β(x, t)). We now return to the original coordinates (X p (t), z(x, t) = y(s, t)).
Introduce the space We are in a position to state the main result in this paper.
Replacing z t and z x by their expressions in terms of S, D and using (2.6)-(2.7), we obtain For the boundary conditions (2.4), we have that z(1, t) = X p (t) (by construction of z), and that For the initial conditions (2.5), we have that z(·, 0) = z 0 (by construction of z) and that Let us investigate the dynamics of X p . Set U (t) :=Ẍ p (t) for t ∈ R + .

Numerical calculation of the kernels
The direct kernels K are calculated by numerically solving system (3.7) [7], where the domain T is discretized using a uniform grid with ∆x = ∆ξ = 0.005 (see Fig. 2). Knowing the direct kernels K, the inverse kernels L are calculated numerically using (3.8). The obtained kernels are shown in Figures 3-4. The used controller gains correspond to the kernels L evaluated at the boundary (x = 1, ξ), that are shown in detail in Figure 5.

Time evolution of the controlled system
Once the direct and inverse kernels are known, the system (1.8)-(1.11) controlled by the law (4.27) can be simulated in the coordinates (X p (t), α(x, t), β(x, t)). Finally, the inverse state transformation will be applied in order to obtain the simulated time evolution in the original coordinates (X p (t), y(s)).
We consider zero initial conditions, except for X p (0) = 0.5 m, i.e., the system is initially at rest with the platform at a nonzero position.

Time evolution of φ(t)
The finite-time dynamics of the variable φ(t) (4.2) is simulated using the numerical method proposed in [27]. This method supposes that the system's vector field F is d-homogeneous [27,Def. 3.6], which corresponds to imposing that ν 1 = ν2 2−ν2 in (4.26). A nonlinear state transformation is provided with which an alternative continuous time system representationż =F (z) is obtained, that admits an implicit discretization scheme  preserving the finite-time stability property in discrete time. Performing the inverse state transformation then yields a discretization scheme preserving the finite-time stability in the original coordinates.
Rewrite the finite-time ODE dynamics (4.26) as d dt Let us check that the assumptions in [27], Theorem 4.1 are fulfilled. We have that the vector field F (x), with x := x 1 x 2 , is uniformly continuous on the unit sphere, d-homogeneous with homogeneity degree . Let G d = diag(r 1 , r 2 ) be the generator of the dilatation d(s) and take the symmetric matrix P := I 2 that satisfies P G d + G d P 0.

Time evolution of α(x, t) and β(x, t)
The two first order hyperbolic PDEs for β(x, t), (4.33), and α(x, t), (4.37), are solved numerically using a first order downwind-upwind scheme [28]. Uniform spatial (∆x = 0.05) and temporal (∆t = 0.01) discretization steps are chosen, respecting the CFL stability condition max(λ(x)) ∆t ∆x ≤ 1. At each time t, the following steps are executed. First the boundary condition β(1, t) is evaluated from (4.9) knowing the value ofφ(t), where µ = 2.379 is calculated from (4.10) by the trapezoidal rule on the spatial grid. Then, the first order downwind scheme β(x, t) = β(x, t − ∆t) + λ(x) ∆t ∆x (β(x + ∆x, t − ∆t) − β(x, t − ∆t)) translates the information for x from 1 to 0 on the spatial grid. Next, the boundary condition α(0, t) = β(0, t) is evaluated. Finally, the first order upwind scheme translates the information for x from 0 to 1 on the spatial grid. The simulated time evolution of β(x, t) and α(x, t) is shown in Figures 7 and 8. We observe that they are stabilized in a finite time whose value can be verified using the expression (cf.  Figure 6. Simulated time evolution of φ(t) andφ(t).

Time evolution of X p (t)
At each discrete time t, φ(t), α(x, t) and β(x, t) are known on a grid for x. The corresponding numerical value for X p (t) is computed from the definition for φ(t) (see (4.2)), numerically evaluating the integral by the trapezoidal rule on the spatial grid. The simulated time evolution of X p (t) is shown in Figure 9.

Time evolution of y(s, t)
Given the simulated time evolutions of (α(x, t), β(x, t)), one can perform the inverse transformations in order to express the movement of the cable in the original coordinates y(s, t):   For the numerical integration of z x (x, t) at a given time t, z(1, t) is set to X p (t) in order to satisfy the compatibility condition corresponding to (1.10). The spatial integration is then evaluated numerically using the trapezoidal rule for x from 1 to 0 on the spatial grid.
The obtained time evolution of y(s, t) = z(x, t) is shown in Figures 9-10. In conclusion, both the platform X p (t) and the cable y(s, t) have been stabilized at the origin in a finite time, namely T 1 ≈ 4.76 s.

Conclusions and future works
A finite-time controller for the motion of an overhead crane described by a coupled PDE-ODE system with varying tension along the cable is derived. Our feedback law incorporates existing results for the finite-time stabilization of 2 × 2 quasilinear hyperbolic systems, that involve kernels that are defined by a Goursat-type system of PDEs. Recent numerical methods are used to calculate the kernels and a simulation of the finite-time stabilization of the overhead crane is shown.
The computation of the proposed control law u(t) (4.27) needs the knowledge of the position y(s, t) of the cable on its entire length. An observer for this position would be needed if one wants to obtain a more realistic implementation using only the measurement of the position X p (t) of the platform and the angle θ(t) at the top of the cable.
Furthermore, the proposed crane model could be extended to the case with a variable length cable, and with an additional degree of freedom for the bench on which the platform moves (a perpendicular translation of the bench in the case of an overhead crane, or a rotation of the bench in the case of a tower crane).