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

Abstract

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 ...

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

“…3 Therefore, to achieve economical and realistic control systems and to overcome the hardware complexities of distributed control approaches, boundary control by stabilizing the flexible system only through its boundary without intruding into the domain sounds promising. 8,9 This has recently motivated different studies on the PDE-based boundary control of the flexible structures such as string, 10 rod, 11 beam, [12][13][14] plate, 15,16 marine riser, 17 satellite, 18 rotor, 19 wind turbine tower, 20 systems subject to moving mass, 21 moving belt, 22,23 and gantry crane. 24 The power of PDE boundary control has been also examined to overcome the challenges facing researchers in control of flexible arms.…”