# Feedback Controller Stabilizing Vibrations of  a Flexible Cable Related to an Overhead Crane

Authors: Abdelhadi Elharfi
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

The problem of stabilizing vibrations of flexible cable related to an overhead crane is considered. The cable vibrations are described by a hyperbolic partial differential equation  (HPDE) with an update boundary condition. We provide in this paper a systematic way to derive a boundary feedback law which restores in a closed form the cable vibrations to  the desired zero equilibrium. Such a control law is explicitly constructed in terms of the solution of an appropriate kernel PDE. The pursued approach combines the “backstepping method” and “semigroup theory”.


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## Body

## 1. Introduction

In this paper, we are concerned with the problem of boundary feedback stabilization of a second-order HPDE describing vibrations of a flexible cable related to an overhead crane. As illustrated in Figure1, the rigid load with mass M is related to cart of the overhead crane by a flexible cable.Figure 1The cable displacementz(t,x), at time t and height x, is mathematically modeled by the following hyperbolic equation: (1.1)z̈(t,x)=[ɛ(x)zx(t,x)]x+b(x)zx(t,x)+a(x)z(t,x),in(0,∞)×(0,1),Mz̈(t,0)=ɛ(0)zx(t,0),in(0,∞),mz̈(t,1)=-ɛ(1)zx(t,1)+βz(t,1)-u(t),in(0,∞),z(0,x)=z0(x),ż(0,x)=z1(x),in(0,1),
coupled with the update boundary condition imposed at the level x=0,(1.2)zx(t,0)=ρz(t,0).
The parameter ɛ(x)=g(M+x) denotes the tension force of the cable at the height x, g being the gravitational acceleration, m the mass of the cart, and M the mass of the rigid load. It is assumed that the line density of the cable is homogeneous and equal to 1. The vibrations in system (1.1) are not only being diffused and bifurcated ((ɛzx)x+bzx) but also a destabilizing displacement (az) is generated. Here, β, ρ are two constants and u(t) is a control placed at the extremity x=1. The boundary condition (1.2) corresponds to situations where the displacement z is subject to a dispositive effect when the rigid load is arrived to the soil, that is, x=0. Such effect arises in (1.2) as an external force which depends on the displacements. System (1.1)-(1.2) serves also as a linearized model of strings. Hereafter, we assume that the parameters a, b and the initial data z0, z1 satisfy the regularity conditions(H)z0∈H2,z1∈H1,a∈C1[0,1],b∈C2[0,1],withb(0)=0,
where H1, H2 are the usual Sobolev spaces on (0,1), see Section 2.The control objective that we are interested in, is to construct a feedback controlleru which restores the displacements z(t) to the equilibrium z≡0 (as t→∞). From a practical point of view, Rao [1] treated the stabilization problem of suppressing vibrations of the distributed overhead crane model with one rigid load, when a=b=β=0. The exponential stability of the closed loop is proved by exploiting an energy functional. In the study by Rahn et al. in [2], a study has been conducted to develop control algorithms for flexible cable crane models. An appropriate coupling amplification controller which asymptotically stabilizes all modes of a linear gantry crane model is constructed. Sano and Otanaka [3] generalized the stabilization problem of a flexible cable with two rigid loads. The model is described by two HPDEs, but the model contained a defect by neglecting the mass of the cart, that is, m=0. The defect of [3] is surmounted by H. Sano in [4] by using the LaSallse's invariance principle. Kim and Hong [5] augmented the simple model with an axially moving system concept. The crane was modeled as an axially moving string system. The dynamics of the moving string is derived using Hamilton's principle for systems with changing mass. Simplified versions of the concerned model was the subject, with respect to the stability, of several works by deferent approaches, see, for example, [6–10].In comparison with the existing works, the model treated in this paper generalizes HPDEs describing vibrations of overhead crane cable, (a≠0,b≠0,β≠0). Moreover, the concerned model contains a perturbing actuation due to the update boundary condition (1.2) which has a nonneglected effect on the behavior analysis of the cable displacements. The proposed method provides a systematic way to construct a boundary feedback law which restores the cable displacements z(t), described by the HPDE (1.1)-(1.2), to the desired equilibrium z≡0, as t→∞. Further, the boundary feedback law is explicitly represented in terms of the solution of an adequate kernel PDE.The paper is organized as follows: in Section2, we derive an appropriate kernel PDE, and we convert system (1.1)-(1.2) into a well-known open-loop system. A control law for the new system is constructed, and the well-posedness of the resulting closed-loop system is shown in Section 3. In Section 4, we derive a feedback controller which asymptotically stabilizes the solution of the closed-loop system associated with (1.1)-(1.2).


## 2. Preliminaries

To simplify the reading, we denoteΔ∶={(x,y):0≤y≤x≤1}. Hi, i=0,1,2, are the usual Sobolev spaces on the interval (0,1). 〈·,·〉 will denote the inner product on the Hilbert space H0=L2. If (𝒜,D(𝒜)) is the generator of a C0-semigroup 𝒯 on a Hilbert space 𝒳, we denote by 𝒳1 the space D(𝒜) endowed with the graph norm ∥x∥∶=∥x∥𝒳+∥𝒜x∥𝒳.First, using the transformation(2.1)z̃(t,x)∶=e∫0x(b(s)/2ɛ(s))dsz(t,x),
with the compatible changes of parameters (2.2)ɛ̃(x)∶=ɛ(x),ã(x)∶=a(x)-b′(x)2-b2(x)4ɛ(x),β̃∶=β-b(1)2,ρ̃=ρ,ũ(t)∶=e∫01(b(s)/2ɛ(s))dsu(t),
one can eliminate the bifurcation term (bz) from (1.1). In fact, direct computations give(2.3)z̃̈-(ɛz̃x)x-ãz̃={z̈-(ɛzx)x-bzx-az}e∫0x(b(s)/2ɛ(s))ds,Mz̃̈(t,0)-ɛ(0)z̃x=0,mz̃̈(t,1)+ɛ(1)z̃x(t,1)=β̃z̃(t,1)-ũ(t).
Then, z satisfies (1.1) if and only if z̃ satisfies (1.1) with the parameters 0, ã, β̃, and ũ, instead of b, a, β, and u. Moreover, provided that b∈C2, the parameters ã∈C1. So, without loss of generality, we set in what follows b≡0.The following lemma is due to [11, Lemma 2.4]. It describes an integral transformation which will be used to convert (1.1)-(1.2) into a well-known one.Lemma 2.1.
Letk∈H2(Δ), and define the bounded operator Tk:Hi→Hi by
(2.4)(Tkφ)(x)∶=φ(x)+∫0xk(x,y)φ(y)dy.
Then, Tk has a linear bounded inverse Tk-1:Hi→Hi,  i=0,1,2.Next, assume thatz(t) satisfies (1.1)-(1.2) and set for t≥0,  x∈[0,1](2.5)w(t,x)∶=(Tkz(t))(x)=z(t,x)+∫0xk(x,y)z(t,y)dy.
By integrating by parts from 0 to x, we get for t>0,(2.6)ẅ(t,x)=z̈(t,x)+∫0xk(x,y)z̈(t,y)dy=z̈(t,x)+∫0xk(x,y)[[ɛ(y)zy(t,y)]y+a(y)z(t,y)]dy=z̈(t,x)+ɛ(x)k(x,x)zx(t,x)-ɛ(0)k(x,0)zx(t,0)-ɛ(x)ky(x,x)z(t,x)+ɛ(0)ky(x,0)z(t,0)+∫0x[[ɛ(y)ky(x,y)]y+a(y)k(x,y)]z(t,y)dy.
Moreover, (2.7)[ɛwx]x=[ɛ(x)zx(t,x)]x+ɛ(x)k(x,x)zx(t,x)+[ɛ(x)k(x,x)]xz(t,x)+ɛ(x)kx(x,x)z(t,x)+∫0x[ɛ(x)kx(x,y)]xz(t,y)dy.
Taking into account of (1.2), we obtain (2.8)ẅ-[ɛwx]x=[a(x)-2ɛ(x)ddx(k(x,x))-ɛ′(x)k(x,x)]z(t,x)+∫0x[a(y)k(x,y)+([ɛ(y)ky(x,y)]y-[ɛ(x)kx(x,y)]x)]z(t,y)dy+[ky(x,0)-ρk(x,0)]ɛ(0)z(t,0).
Then, ẅ-[ɛ(x)wx]x=0,  in(0,∞)×(0,1), if and only if the kernel k verifies the PDE(2.9)[ɛ(x)kx(x,y)]x-[ɛ(y)ky(x,y)]y=a(y)k(x,y),0≤y≤x≤1,ky(x,0)=ρk(x,0),0≤x≤1,k(x,x)=12ɛ(x)∫0xa(s)ɛ(s)ds,0≤x≤1.
We note that the third (boundary) equation of (2.9) is obtained by solving the first order differential equation(2.10)2ɛ(x)ddx(k(x,x))+ɛ′(x)k(x,x)=a(x),
with the initial condition k(0,0)=0. Due to [12], for a given C2-function ɛ, the kernel PDE (2.9) has a unique solution k∈H2(Δ), see also [13] for ɛ=const. Further, the function k can be approximated numerically via scheme of successive approximations.Now, letk be the solution of (2.9). In view of (2.8), the new state w satisfies(2.11)ẅ(t,x)=[ɛ(x)wx(t,x)]x,in(0,∞)×(0,1),Mẅ(t,0)=ɛ(0)wx(t,0),in(0,∞),mẅ(t,1)=-ɛ(1)wx(t,1)-U(t),in(0,∞),w(0,x)=:w0(x),ẇ(0,x)=:w1(x),in(0,1),
with w0=Tkz0,  w1=Tkz1, and(2.12)U(t)=:u(t)-c0z(t,1)-c1zx(t,1)-〈p,z(t)〉,
where(2.13)c0∶=β+ɛ(1)k(1,1)-mɛ(1)ky(1,1),c1∶=mɛ(1)k(1,1),p(y)∶=(ɛ(1)+mg)kx(1,y)+mɛ(1)kxx(1,y),y∈(0,1).
We note here that the expression of U(t) is obtained using (1.1) and the first equation of (2.9). We summarize these results in the following.Lemma 2.2.
Letk be the solution of (2.9), and consider c0, c1, p with representation (2.13). Then, the isomorphism Tk converts (1.1)-(1.2) into (2.11)-(2.12).


## 3. Stabilization of the Transformed System

We proceed in this section to construct an appropriate controlU(t) which stabilizes the new open-loop system (2.11), and we show the well-posedness of the resulting closed-loop system. To do so, it is logical to think about the energy of the system as Lyapunov function. So, let us introduce the following energy associated with  (2.11)(3.1)E(t)=12{∫01[ɛ(x)wx(t,x)2+ẇ(t,x)2]dx+Mẇ(t,0)2+mẇ(t,1)2+αw(t,1)2},
where α is a positive constant. The integral term corresponds to the inner energy of the cable. The coefficient ẇ2(t,0)+ẇ2(t,1) is proportional to the kinetic energy of the cart. However, the term w2(t,1) guarantees the position convergence, it can be replaced by (w-wd)2 in order to reach any desired position wd by the cart, see [7] and the reference therein for a more discussions on the functional energy E associated with hybrid systems.Differentiating (3.1) with respect to t, we get by using  (2.11)(3.2)Ė(t)=-ẇ(t,1)(U(t)-αw(t,1)).
To cause E(t) to decrease, a simple choice of the feedback law is(3.3)U(t)=αw(t,1)+γẇ(t,1),γ>0.
Substituting (3.3) in (3.4), we obtain(3.4)Ė(t)=-γẇ(t,1)2.
This means that under the boundary feedback law (3.3), the energy E decreases with time t.Now, let us consider the Hilbert space𝒳∶=H1×L2×ℝ×ℝ endowed with the inner product (3.5)〈(f,g,ξ,η),(f̃,g̃,ξ̃,η̃)〉∶=∫01[ɛfxf̃x+gg̃]dx+Mξξ̃+mηη̃+αf(1)f̃(1),
and introduce the operator
(3.6)𝒟(𝒜0)∶={(f,g,ξ,η)⊤∈H2×H1×ℝ×ℝ:g(0)=ξ,g(1)=η},𝒜0(fgξη)=(g(ɛfx)xɛ(0)fx(0)M-[ɛ(1)fx(1)+αf(1)+γη]m),for(fgξη)∈𝒟(𝒜0).
Setting now v(t)∶=(w(t),ẇ(t),ẇ(t,0),ẇ(t,1))⊤, for t≥0. Then, (2.11)-(3.3) can be represented on the state space 𝒳 by the abstract Cauchy problem(3.7)v̇(t)=𝒜0v(t),t≥0,v(0)=v0,
where v0∶=(w0,w1,w1(0),w1(1))⊤. In the following lemma, we will confirm the well-posedness of (2.11)-(3.3).Lemma 3.1.
𝒜0 generates a C0-semigroup of contraction (𝒯t0)t≥0 on 𝒳.Proof.
Obviously,𝒜0 is densely defined. Moreover, by integrating by parts, we get
(3.8)〈𝒜0v,v〉𝒳=-γη2,
for v=(f,g,ξ,η)∈𝒟(𝒜0). Therefore, 𝒜0 is dissipative. By the Lumer-Philips theorem [14, page 85], the proof will be accomplished if one can show that (I-𝒜0) is surjective. In fact, for a given v0=(f0,g0,ξ0,η0)∈𝒳, we have to solve the functional equation
(3.9)(I-𝒜0)v=v0,v=(f,g,ξ,η)∈𝒟(𝒜0),
which means that
(3.10)f∈H2,g∈H1,g=f-f0,ξ=g(0),η=g(1),(3.11)g-(ɛfx)x=g0,Mξ-ɛ(0)fx(0)=Mξ0,ɛ(1)fx(1)+αf(1)+(m+γ)η=mη0.
Substituting (3.10) in (3.11), we obtain the PDE
(3.12)(ɛfx)x-f=-(f0+g0),ɛ(0)fx(0)-Mf(0)=-M(ξ0+f0(0)),ɛ(1)fx(1)+(α+m+γ)f(1)=mη0+(m+γ)f0(1).
Following the method of [15, VIII.4], one can see that the system (3.12) has a unique solution ϕ∈H2. Of course, the vector v=(ϕ,ψ;r0;r1)⊤ is given by
(3.13)ϕsolutionof(3.12),ψ=ϕ-f0,r0∶=[Mξ0+ɛ(0)ϕx(0)]M,r1∶=[mη0+(m-α-β)f0(1)-ɛ(1)ϕx(1)](α+β+γ),
which is a 𝒟(𝒜0)-solution of the functional equation (3.9). This leads us to conclude that I-𝒜0 is surjective.Since𝒜0 generates a C0-semigroup of contraction, then (0,∞)⊆ϱ(A0), and the resolvent ℛ(μ,𝒜0) is well-defined for all μ>0.Lemma 3.2.
ℛ(μ,𝒜0) is compact for all μ>0. In particular, 𝒯0 is relatively compact.Proof.
In view of [14, page 117], it remains to show that the injection j:𝒳1→X is compact. To do so, we introduce the auxiliary Hilbert space 𝒱∶=H2×H1×ℝ×ℝ with the inner product
(3.14)〈v,ṽ〉𝒱∶=〈f,f̃〉H2+〈g,g̃〉L2+ξξ̃+ηη̃+f(1)f̃(1),
for v=(f,g,ξ,η)∈𝒱 and ṽ=(f̃,g̃,ξ̃,η̃)∈𝒱. Obviously, 𝒳1⊆𝒱⊆𝒳. Moreover, from the Sobolev's embedding theorem, the embedding from H1 in L2 and the one of H2 in H1 are compact. It follows that the injection j1:𝒱↔𝒳 is compact. On the other hand, direct computations show that
(3.15)∥v∥𝒱≤C∥v∥𝒳1,
for some constant C. Therefore, the injection j2:𝒳1→𝒱 is continuous. Consequently, j∶=j1∘j2 is compact from 𝒳1 into 𝒳. Thus, 𝒜0 has a compact resolvent, and by [14, Corollary V.2.15], we conclude that the semigroup 𝒯0 is relatively compact.Taking into account of [14, page 318], and the fact that 𝒯0 is relatively compact, the following decomposition holds(3.16)𝒳=𝒳s⊕𝒳r,
where (3.17)𝒳s={v∈𝒳:∥𝒯t0v∥𝒳→0as(t→∞)},𝒳r=lin¯{v∈𝒳:∃σ∈ℝ,𝒜v=iσv}.
We point out here that, due to (H), the initial data v0 belongs to 𝒟(𝒜0). Since 𝒜0 generates a C0 semigroup 𝒯0, then, by [14, page 145], the evolution equation (3.7) has a unique solution v∈C1([0,∞),𝒳)∩C([0,∞),𝒟(𝒜0)), given by v(t)=𝒯t0v0. Which implies that the closed-loop system (2.11)-(3.3) has a unique solution w satisfying (3.18)w∈C1([0,∞),H1)∩C([0,∞),H2),ẇ∈C1([0,∞),L2)∩C([0,∞),H1),(w(t),ẇ(t),ẇ(t,0),ẇ(t,1))⊤=v(t)=𝒯t0v0.
Equation (3.18) explains that the solution of (2.11)-(3.3) is represented by the semigroup 𝒯0. For that reason, we will adopt in this work the concept of stability associated with semigroup theory as defined in [14]. One says that (2.11)-(3.3) is asymptotically stable, if  𝒯0 is strongly stable, that is, for all v∈𝒳,  ∥𝒯t0v∥𝒳→0 as (t→0). In the following theorem, we show how the controller (3.3) affects on either the stability and the energy of (2.11).Theorem 3.3.
(i) 𝒯0  is strongly stable. In particular, (2.11)-(3.3) is asymptotically stable.
(ii)  The energy E(t) decreases with time t: E(t)↘0 as t→∞.Proof.
In view of (3.16), it suffices to show that 𝒳r={0}. In fact, let v=(f,g,g(0),g(1))⊤∈𝒟(𝒜0) satisfying
(3.19)𝒜0v=iσv,
for some σ∈ℝ. Equation (3.19) is equivalent to
(3.20)f∈H2,g=iσf,(3.21)(ɛfx)x+σ2f=0,ɛ(0)fx(0)+Mσ2f(0)=0,ɛ(1)fx(1)+(α+iσγ-mσ2)f(1)=0.
By using the method of [15, VIII.4] and taking into consideration of the regularity condition f∈H2, one can see that f≡0 is the unique solution of (3.21). From (3.20), we conclude that g≡0. Therefore, 𝒳r={0} and 𝒳=𝒳s.
On the other hand, in view of the third equation of (3.18), we have
(3.22)E(t)=12∥𝒯t0v0∥𝒳2→0,(t→∞).
This proves the statement (ii).


## 4. Closed-Loop Stability of the Cable Displacements

We will derive in this section a controlleru which restores, in a closed form, the cable displacements z(t) of the concerned system (1.1)-(1.2) to the equilibrium z≡0. In fact, by substituting (2.12) in (3.3), we reach, by using (2.5), the following expression of the control u(t)(4.1)u(t)=(α+c0)z(t,1)+c1zx(t,1)+γż(t,1)+〈q,z(t)〉+〈γk0,ż(t)〉,
where k0(y)∶=k(1,y),  q∶=αk0+p.The system (1.1)-(1.2), (4.1) is well posed, since it can be obtained from the well-posed system (2.11)-(3.3) via the isomorphism (Tk)-1. Which means that the closed-loop system (1.1)-(1.2), (4.1) has a unique solution z satisfying, in view of (3.18), the following regularity conditions:(4.2)z∈C1([0,∞),H1)∩C([0,∞),H2),ż∈C1([0,∞),L2)∩C([0,∞),H1).
Consider now the operator (4.3)𝒟(𝒜)∶=𝒟(𝒜0),𝒜(fgξη)=(g(ɛfx)x+afɛ(0)fx(0)M-[(ɛ(1)+c1)fx(1)+(α+c0-β)f(1)+γη+〈q,f〉+〈γk0,g〉]m),
for (f,g,ξ,η)⊤∈𝒟(𝒜).By direct computations one can prove that the functionζ defined by(4.4)zsolutionof(1.1)-(1.2),(4.1),ζ(t)∶=(z(t),ż(t),ż(t,0),ż(t,1))⊤,t≥0,
is the unique classical solution of the evolution equation(4.5)𝒵̇(t)=𝒜𝒵(t),t≥0,𝒵(0)=ζ0,
where ζ0=(z0,z1,z1(0),z1(1))⊤. This means that the operator 𝒜 generates a C0-semigroup 𝒯t given by 𝒯tζ0∶=ζ(t). Moreover, the fact that (Tk)-1 is bounded, then there exists a constant C>0 such that(4.6)∥z(t)∥H1≤C∥w(t)∥H1,∥ż(t)∥L2≤C∥ẇ(t)∥L2,|ż(t,0)|≤C|ẇ(t,0)|,|ż(t,1)|≤C|ẇ(t,1)|,
for t≥0. Thus, ∥ζ(t)∥𝒳≤C∥v(t)∥𝒳, and so ∥𝒯tζ0∥𝒳≤C∥𝒯t0v0∥𝒳. By Theorem 3.3, we deduce the strong stability of 𝒯t. Therefore, the closed-loop system (1.1)-(1.2), (4.1) is asymptotically stable. This proves the main result of this paper which can be reformulated in the following theorem.Theorem 4.1.
The semigroup𝒯t is strongly stable. In particular, the closed-loop system (1.1)-(1.2), (4.1) is asymptotically stable.Remark 4.2.
One can express the controller (4.1) using the solution  w of (2.11)-(3.3). In fact, let us denote by  l  the kernel of the inverse transformation
(4.7)z(t,x)=w(t,x)+∫0xl(x,y)w(t,y),
where w is the solution of (2.11)-(3.3). Substituting (4.7) in (1.1)-(1.2), we find the PDE governing the kernel l(4.8)[ɛ(x)lx(x,y)]x-[ɛ(y)ly(x,y)]y=-a(x)l(x,y),0≤y≤x≤1,ly(x,0)=ρl(x,0),0≤x≤1,l(x,x)=-12ɛ(x)∫0xa(s)ɛ(s)ds,0≤x≤1.
The PDE (4.8) is in the same class of (2.9). Hence, the PDE (4.8) has a unique solution l∈H2(Δ). Let now l be the solution of (4.8). Substituting (4.7) in (4.1), one obtains an expression of the controller (4.1) in terms of w and l.


## 5. Conclusion

The proposed approach represents a blinding of the so-called “backsteeping method” and “semigroup theory” to construct a controller which asymptotically stabilizes the solution of the HPDE (1.1)-(1.2). Various properties of parabolic PDEs and hyperbolic PDEs can be treated by using similar techniques. The idea of the study is to convert a complicated (parabolic or hyperbolic) PDE into a well-known one by using the famous integral transformation (2.4) with a kernel required to satisfy an adequate PDE. We also note that the isomorphism (2.4) transforms such PDEs without effects on their topological properties. Therefore, one can deal with others topological properties of complicated systems such as: regularity, controllability, and observability, and so forth.



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