Lyapunov approach to the boundary stabilisation of a beam equation with boundary disturbance

Guo, Bao‐Zhu; Kang, Wen

doi:10.1080/00207179.2013.861931

Cited by 44 publications

(27 citation statements)

References 24 publications

(19 reference statements)

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“…It is seen from Equations (25)-(28) that the proposed control needs feedback from the beam deflection velocity and slope at the controlled point; this is motivated by showing that, in order to reject the disturbances of the Euler-Bernoulli beam [34,35], the knowledge of both velocity and slope is required. This brings another advantage to the implementation of the proposed control considering the abundance of the control strategies for beam-type structures [1, 2, 4-6, 28-31] that entail the additional feedback from the shear force.…”

Section: Remarkmentioning

confidence: 99%

“…They studied in another work [31] adaptive boundary control of the inhomogeneous Timoshenko beam equation with constraints. Guo et al investigated in several papers [32][33][34][35] robust boundary stabilization of Euler-Bernoulli beam by active disturbance rejection control and sliding mode control approaches.…”

Section: Introductionmentioning

confidence: 99%

“…The control strategy consists of applying a transverse force and a twisting torque at the free end of the cantilever beam. A common signal used for controlling beams is the beam deflection velocity [34]. However, the result of Jin and Guo [35] shows that in order to reject very general disturbance, the additional angular signal of beam slope seems necessary.…”

Section: Introductionmentioning

confidence: 99%

“…Inequalities(33) and(34) result from the fact that the right-hand sides of two last inequalities do not depend on x.782 A. TAVASOLI AND V. ENJILELA APPENDIX E. PROOF OF THEOREM 3 Differentiating Equation (40) with respect to time, utilizing Inequality (D.2) in Appendix D, and inserting the control and adaptation laws given by Equations (36)-(39), we can obtain…”

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confidence: 99%

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Adaptive robust boundary control of coupled bending‐torsional vibration of beams with only one axis of symmetry

Tavasoli

Enjilela

2016

Adaptive Control & Signal

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In control design for vibration of beams in literature, the beam section is considered to have two axes of symmetry so that the bending and torsional vibrations are uncoupled; thus, the bending vibration is controlled independently without twisting the beam. However, if the cross section of a beam has only one axis of symmetry, the bending and torsional vibrations become coupled and the beam will undergo twisting in addition to bending. This paper addresses Lyapunov-based boundary control of coupled bending-torsional vibration of beams with only one axis of symmetry. The control strategy is based on applying a transverse force and a torque at the free end of the beam. The control design is directly based on the system partial differential equations (PDEs) so that spillover instabilities that are a result of model truncation are avoided. Three cases are investigated. Firstly, it is shown that when exogenous disturbances do not affect the beam, a linear boundary control law can exponentially stabilize the coupled bending-torsional vibration. Secondly, a nonlinear robust boundary control is established that exponentially stabilizes the beam in the presence of boundary and spatially distributed disturbances. Thirdly, to rule out the need for prior knowledge of disturbances upper-bound, the proposed robust control is redesigned to achieve an adaptive robust control that stabilizes the beam in the presence of disturbances with unknown upper-bound. The efficacy of the proposed controls is illustrated by simulation results.A. TAVASOLI AND V. ENJILELA Figure 1. Channel section as a common example of beams with only one axis of symmetry.The need for high speed and low energy consumption has increased the application of lightweight flexible beams in various mechanical systems, such as marine risers [1,2], rotors [3], wind turbine towers [4], flexible robots [5], and space mechanisms [6]. Because beams are made as slender as possible to avoid the penalties attached to excessive weight and cost, they lack sufficient rigidity and damping properties to effectively operate under applied loads [7]. Therefore, active control techniques [8] have been employed to suppress vibration of beam structures. Flexible beams are continuous mechanical systems [9, 10] with infinite degrees of freedom and are modeled through PDEs. This makes both practical and theoretical challenges in controlling beams. Because the control synthesis techniques for systems represented by PDEs have not been developed, the conventional approaches for controlling beams mainly rely on discretizing the system PDEs into some ordinary differential equations (ODEs) [11]. Although the mathematical complexities in dealing with PDEs are avoided by discretizing the model, a control system designed based on model truncation can incur spillover instabilities [12,13] because of excitation of residual modes. In addition, a finite-state approximation to a PDE may wrongly render fundamental system theoretic properties like controllability and observability into functions of the ...

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Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Adaptive robust boundary control of coupled bending‐torsional vibration of beams with only one axis of symmetry

Tavasoli

Enjilela

2016

Adaptive Control & Signal

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“…The control design and stability analysis of flexible mechanical systems based on PDEs has been extensively studied [10][11][12][13][14][15]. The asymptotic behavior of a partial state of a coupled PDE and ordinary differential equation (ODE) is investigated in [16].…”

Section: Introductionmentioning

confidence: 99%

Dynamic modeling and vibration control of a flexible aerial refueling hose

Liu

2016

Aerospace Science and Technology

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Dynamic modeling and adaptive robust boundary control of a flexible robotic arm with 2‐dimensional rigid body rotation

Tavasoli

Mohammadpour

2018

Adaptive Control & Signal

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This paper investigates the control of a single-link flexible robot manipulator with a tip payload appointed to rotate about 2 perpendicular axes in space. The control objective is to regulate the rigid body rotation of the manipulator with guaranteeing the stability of its vibration in the presence of exogenous disturbances. To achieve this, a Lyapunov-based control design procedure is used and accomplished in some steps. First, the partial differential equation (PDE) dynamic model governing the rigid-flexible hybrid motion of the arm is derived by applying Hamilton's principle. Next, based on the developed PDE model, an adaptive robust boundary control is established using the Lyapunov redesign approach. To this end, an adaptation mechanism is proposed so that the robust boundary control gains are dynamically updated online and there is no need for prior knowledge of disturbance upper bounds. The actuators and sensors are fully implemented at the arm boundary without using distributed actuators or sensors. Furthermore, in order to avoid control errors resulting from the spillover, control design is directly based on infinite-dimensional PDE model without resorting to model truncation. Simulation results illustrate the efficacy of the considered method. KEYWORDS adaptive upper bound, boundary control, distributed parameter systems, flexible arm, robust control Int J Adapt Control Signal Process. 2018;32:891-907.wileyonlinelibrary.com/journal/acs

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Lyapunov approach to the boundary stabilisation of a beam equation with boundary disturbance

Cited by 44 publications

References 24 publications

Adaptive robust boundary control of coupled bending‐torsional vibration of beams with only one axis of symmetry

Adaptive robust boundary control of coupled bending‐torsional vibration of beams with only one axis of symmetry

Dynamic modeling and vibration control of a flexible aerial refueling hose

Dynamic modeling and adaptive robust boundary control of a flexible robotic arm with 2‐dimensional rigid body rotation

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