2017
DOI: 10.1016/j.automatica.2017.04.017
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Boundary cooperative control by flexible Timoshenko arms

Abstract: This paper discusses a cooperative control problem by two one-link flexible Timoshenko arms. The goal is to control a grasping force to collect an object with the two flexible arms, and to simultaneously suppress the vibrations of the arms. To solve this problem, we propose a boundary controller that is based on a dynamic model represented by a hybrid PDE-ODE model; the exponential stability of the closed-loop system is then proven by the frequency domain method. Finally, several numerical simulations are carr… Show more

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Cited by 70 publications
(43 citation statements)
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“…In contrast, the boundary perturbations appear in the boundary conditions of the PDEs. This second type of perturbation naturally appears in numerous boundary control problems such as heat equations [8], transport equations [19], diffusion or diffusive equations [2], and vibration of structures [8] with numerous practical applications, e.g., in robotics [12,13] and aerospace engineering [22]. While many results have been reported regarding the ISS property with respect to distributed disturbances [1,9,10,24,25,26,27,31], the establishment of ISS properties with respect to boundary disturbances remains challenging [2,18,19,20].…”
Section: Introductionmentioning
confidence: 99%
“…In contrast, the boundary perturbations appear in the boundary conditions of the PDEs. This second type of perturbation naturally appears in numerous boundary control problems such as heat equations [8], transport equations [19], diffusion or diffusive equations [2], and vibration of structures [8] with numerous practical applications, e.g., in robotics [12,13] and aerospace engineering [22]. While many results have been reported regarding the ISS property with respect to distributed disturbances [1,9,10,24,25,26,27,31], the establishment of ISS properties with respect to boundary disturbances remains challenging [2,18,19,20].…”
Section: Introductionmentioning
confidence: 99%
“…As P −k−1 = P k , the study for k ∈ Z reduces to k ∈ N. The discriminant of P k is given by disc(P k ) =k 2 π 2 (k 2 β 2 π 2 − 4α). Based on Assumption 3.1, disc(P k ) = 0, thus disc(P k ) > 0 for k ≥ k0 and disc(P k ) < 0 for 0 ≤ k ≤ k0 − 1, providing the eigenvalues λ k,ǫ given by (9). Finally the associated eigenvectors are characterized by x1(ξ) = a(e ikπξ − e −ikπξ ) = 2ai sin(kπξ), and x2 = λ k,ǫ x1, providing (10).…”
Section: A Characterization Of the Spectral Properties Of A0mentioning
confidence: 99%
“…We need to check that γ 2 k,ǫ is a summable sequence. To do so, considering k ≥ k0, Re λ k,ǫ = λ k,ǫ , which gives along with (9) and (11) γ k,ǫ = √ 2 kπ k2 β 2 π 2 − 4α 1 +k 2 απ 2 |λ k,ǫ | 2 .…”
Section: Proof Of the Eiss Propertymentioning
confidence: 99%
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“…However, these studies have been treated only the vibration suppression problem. On the other hand, several studies have investigated the grasping by flexible arms based on the hybrid PDE-ODE model [20][21][22][23]. These studies discussed the cooperative control by two one-link flexible arms and realized stable grasping of an object.…”
Section: Introductionmentioning
confidence: 99%