Backstepping Boundary Controllers and Observers for the Slender Timoshenko Beam: Part I--Design

Krstić, Miroslav; Siranosian, Antranik Antonio; Smyshlyaev, Andrey

doi:10.1109/acc.2006.1656581

Cited by 65 publications

(60 citation statements)

References 13 publications

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“…The fourth author is with the Engineering Sciences and Applications Division, Los Alamos National Laboratory. end [5]. This simplifies the stability analysis and the selection of design parameters for achieving desired performance.…”

Section: Introductionmentioning

confidence: 98%

“…This simplifies the stability analysis and the selection of design parameters for achieving desired performance. Our observer design in [5] is a dual of the damping feedback ideas combined with backstepping, adapted to the observer error system.…”

Section: Introductionmentioning

confidence: 99%

“…Our design, first introduced in [5], is a novel combination of the classical "damping boundary feedback" [3], [6], [7], [8], [11], [12], [13] ideas with backstepping. For physical realism, we assume some small amount of KV damping as is the case with all physical systems, but note that this is not a requirement for the implementation of our theory or the proof of stability as shown in a companion paper for the undamped shear beam [4].…”

Section: Introductionmentioning

confidence: 99%

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Backstepping Boundary Controllers and Observers for the Slender Timoshenko Beam: Part II---Stability and Simulations

Krstić

Siranosian

Smyshlyaev

et al. 2006

Proceedings of the 45th IEEE Conference on Decision and Control

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We present the stability analysis and numerical results for our controller and observer designs for the slender Timoshenko beam with Kelvin-Voigt (internal material) damping. Our designs, introduced in a companion ACC '06 paper, use a combination of classical "damping boundary feedback" ideas with backstepping. The main advantage of our techniques over the previous boundary damping results is that they apply to the readily implementable "anti-collocated" architecture, which uses actuation only at the beam's base and sensing only at the beam's tip. I. INTRODUCTIONThe Timoshenko beam model [3], [6], [7], [8], [11], [12], [13] consists of two coupled second-order-in-space/secondorder-in-time wave equations. This model is the most involved of the four beam models as it includes the effects of lateral displacement, bending moment, shear deformation and rotary inertia [2]. We consider a version of the Timoshenko beam model with Kelvin-Voigt (KV) damping (internal material damping) and design output feedbacks which employ measurements only at the tip of the beam and apply actuation only at the base. Our design, first introduced in [5], is a novel combination of the classical "damping boundary feedback" [3], [6], [7], [8], [11], [12], [13] ideas with backstepping. For physical realism, we assume some small amount of KV damping as is the case with all physical systems, but note that this is not a requirement for the implementation of our theory or the proof of stability as shown in a companion paper for the undamped shear beam [4].While damping feedback ideas have traditionally required actuation to be applied at the free end, making them appropriate primarily for passive control implementations, combining them with backstepping allows actuation to be moved to the beam base. This architecture makes active control more readily implementable to several applications. Besides producing an increase in the rate of exponential stability of the unforced beam model, our design readily extends to situations where the beam is forced by persistent disturbances at the tip as well as situations in which tip effects, such as nonlinear multi-equilibrium forces, produce unstable beam dynamics.Our backstepping boundary control is such that, after a change of variable, the beam model is converted into a wave equation (target system) for a high-tension string with a damper on one end and a high-stiffness spring on the other

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Backstepping Boundary Controllers and Observers for the Slender Timoshenko Beam: Part II---Stability and Simulations

Krstić

Siranosian

Smyshlyaev

et al. 2006

Proceedings of the 45th IEEE Conference on Decision and Control

Self Cite

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“…If we then substitute (16), (18), and (14) into (11) and arrange the remaining terms as coefficients of the system state variable v, the resulting equation is:…”

Section: A Solution Of the Kernel Functionmentioning

confidence: 99%

“…Boundary control has been an attractive technique both practically and mathematically as it constrains all physical control actuators to a single location to minimize intrusiveness in the system, and also because all control terms are isolated within the boundary conditions of the PDE. Boundary control has been utilized in previous studies for the control of EulerBernoulli beams [3], [15], [21], [23], other flexible beam models [12], [16], [17], as well as other classes of PDEs [5], [6], [9], [24].…”

Section: Introductionmentioning

confidence: 99%

Sliding mode boundary control for an Euler-Bernoulli beam with boundary disturbances and parameter variations

Karagiannis

Radisavlejevic-Gajic

2014

2014 American Control Conference

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This work addresses the problem of stability and control of an Euler-Bernoulli beam with sliding-pinned boundary conditions. The control comes in the form of an applied displacement and bending moment on the pinned end. The proposed controller accounts for bounded exogenous disturbances on the pinned end boundary conditions, representing a number of possible practical issues such as model parameter uncertainties or unknown external forces. This controller takes advantage of a technique for stabilizing an Euler-Bernoulli beam through a Schrödinger equation representation and a previously developed integral transformation which provides the opportunity to select arbitrary damping for the system. The contribution of this work is to introduce unknown boundary disturbances into the system boundary terms and the development of a Sliding Mode Controller which effectively drives the system states to an exponentially stable infinite-dimensional sliding surface which then eliminates the system vibration with arbitrary damping. The resulting control efforts developed are continuous functions and therefore do not create a chattering effect on control actuators.

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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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Backstepping Boundary Controllers and Observers for the Slender Timoshenko Beam: Part I--Design

Cited by 65 publications

References 13 publications

Backstepping Boundary Controllers and Observers for the Slender Timoshenko Beam: Part II---Stability and Simulations

Backstepping Boundary Controllers and Observers for the Slender Timoshenko Beam: Part II---Stability and Simulations

Sliding mode boundary control for an Euler-Bernoulli beam with boundary disturbances and parameter variations

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

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