Boundary Control Method Research Articles

Robust vibration control for a flexible cable gantry crane system subject to boundary disturbance

This paper addresses the robust vibration control problem for a gantry crane system which is governed by a partial differential equation. The control objectives are to transport the cargo from the initial position to the desired position and suppress the vibration of the cable and payload in the presence of boundary disturbance. To achieve the objectives, a sliding-mode controller is proposed by repressing the boundary disturbance. By using the operator semigroup theory, the well-posedness of the closed-loop system is guaranteed. The asymptotic stability of the closed-loop system is proven by using the extended LaSalle’s invariance principle. Both numerical simulations and experiments are provided to illustrate the effectiveness of the proposed boundary control method.

Determination of black holes by boundary measurements

For a wave equation with time-independent Lorentzian metric consider an initial-boundary value problem in [Formula: see text], where [Formula: see text] is the time variable and [Formula: see text] is a bounded domain in [Formula: see text]. Let [Formula: see text] be a subdomain of [Formula: see text]. We say that the boundary measurements are given on [Formula: see text] if we know the Dirichlet and Neumann data on [Formula: see text]. The inverse boundary value problem consists of recovery of the metric from the boundary measurements. In the author’s previous works a localized variant of the boundary control method was developed that allows the recovery of the metric locally in a neighborhood of any point of [Formula: see text] where the spatial part of the wave operator is elliptic. This allows the recovery of the metric in the exterior of the ergoregion. The goal of this survey paper is to recover the black holes. In some cases the ergoregion coincides with the black hole. In the case of two space dimensions we recover the black hole inside the ergoregion assuming that the ergosphere, i.e. the boundary of the ergoregion, is not characteristic at any point of the ergosphere.

On the electric impedance tomography problem for nonorientable surfaces with internal holes

Let ( M , g ) (M,g) be a compact (in general, nonorientable) surface with boundary ∂ M \partial M and let Γ 0 \Gamma _0 , …, Γ m − 1 \Gamma _{m-1} be connected components of ∂ M \partial M . Let u = u f ( x ) u=u^{f}(x) be a solution to the problem Δ g u = 0 \Delta _{g}u=0 in M M , u | Γ 0 = f u\big |_{\Gamma _0}=f , u | Γ j = 0 u\big |_{\Gamma _j}=0 , j = 1 j=1 , …, m ′ m’ , ∂ ν u | Γ j = 0 \partial _{\nu }u\big |_{\Gamma _j}=0 , j = m ′ + 1 j=m’+1 , …, m − 1 m-1 , where ν \nu is the outward normal. With this problem, one associates the DN map Λ : f ↦ ∂ ν u f | Γ 0 \Lambda \colon f\mapsto \partial _{\nu }u^{f}\big |_{\Gamma _0} . The purpose is to determine M M from Λ \Lambda . To this end, an algebraic version of the boundary control method is applied. The key instrument is the algebra A \mathfrak {A} of functions holomorphic on the appropriate orientable double cover of M M . It is proved that A \mathfrak {A} is determined by Λ \Lambda up to isometric isomorphism. The spectrum of the algebra A \mathfrak {A} provides a relevant copy M ′ M’ of M M . This copy is conformally equivalent to M M while its DN map coincides with Λ \Lambda .

Vibration Control of Marine Flexible Riser with Uncertain Model

When flexible risers in the ocean are subjected to various environmental loads in the ocean, vibration will inevitably occur. Due to the nonlinearity and uncertainty of the actual riser system, it is impossible to obtain an accurate mathematical model of the riser system. The existing control algorithms designed based on the precise model of the riser system cannot meet the actual control requirements. A flexible riser boundary control method based on fuzzy control algorithm is proposed to suppress the vibration of marine flexible riser systems with uncertain models. Firstly, the dynamic models of existing marine flexible risers were studied. In response to the uncertainty of the riser system model, the control experience was transformed into an automatic control strategy in the form of fuzzy language control rules to improve the adaptability of the controller. Then, combining boundary control technology with fuzzy control, a boundary controller is constructed to stabilize the riser in a small neighborhood of its original position and reduce fatigue damage to the riser. Finally, the effectiveness of the designed fuzzy boundary control algorithm was verified through MATLAB simulation results, and the vibration suppression effect of the algorithm on the vertical tube was better than that of traditional PD control.

Exponential Consensus of Multiagent Systems Based on Nonlinear Parabolic PDEs via Intermittent Boundary Control

The exponential consensus issue is investigated for leader-following multiagent systems (MASs) based on nonlinear parabolic partial differential equations (PDEs) on a directed graph via intermittent boundary control (IBC) methods. On the one hand, when the system’s state is accessible, an IBC protocol is designed based on the transfer information generated by the agent’s communication with its neighbors and the information coupling at the boundary <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x= l$</tex-math> </inline-formula> , and the sufficient conditions for exponential consensus of leader-following MASs are obtained. On the other hand, when the system’s state is not available completely, the agent’s state is estimated by the proposed observer, and the observer’s coupling information and the boundary coupling information of the agent are adopted to design an IBC protocol to solve the leader-following consensus issue. Finally, two numerical simulations are supplied to account for the obtained theoretical results.

Boundary Control Design of Three-Dimensional Vibrating Marine Riser System With Barrier Term

Abstract In this paper, a boundary control with barrier term is proposed for a typical three-dimensional flexible riser system by integrating backstepping technique with barrier Lyapunov functions. The proposed boundary control does not depend on the exact value of interference and can effectively restrain the vibration of the riser and constrain the angle between the riser and the platform. The proposed control has a small amount of calculation and strong robustness, and can effectively restrain the riser vibration when the environmental disturbance changes. In addition, based on the proposed boundary control method, the stability of the closed-loop system in space and time is proved by Lyapunov stability theorem when the initial conditions are satisfied. Numerical simulation shows the effectiveness and superiority of the controller.

Formula omitted] fuzzy intermittent boundary control for nonlinear parabolic distributed parameter systems

For nonlinear space-varying parabolic distributed parameter systems, this paper introduces an H∞ fuzzy intermittent boundary control, where the output measurements are only available at some specified boundary position (i.e., boundary measurements). Initially, a Takagi–Sugeno fuzzy parabolic partial differential equation is used to precisely describe the nonlinear space-varying parabolic distributed parameter system. Then, under boundary measurements, an H∞ fuzzy intermittent boundary control design based on the Takagi–Sugeno fuzzy parabolic partial differential equation model ensuring the exponential stability with an H∞ performance for closed-loop space-varying distributed parameter system is subsequently obtained via spatial linear matrix inequalities by employing inequality techniques and piecewise switching-time-dependent Lyapunov function. Furthermore, in order to solve the H∞ fuzzy intermittent boundary controller design of nonlinear space-varying parabolic distributed parameter systems under boundary measurements, we express spatial linear matrix inequalities as linear matrix inequalities and further present some linear matrix inequality based fuzzy intermittent boundary control design conditions respecting the property of membership functions. Finally, two simulation examples are offered to demonstrate the effectiveness of the proposed H∞ fuzzy intermittent boundary control method.

Boundary control for floating beam system in irregular waves under one end pinned to reduce hydroelastic response

The structural stability is of great importance in research and development of very large floating structure (VLFS). The vibration reduction on VLFS would not only prevent the structural fatigue but also improve the platform serviceability. In order to reduce the hydroelastic response on VLFS in the actual ocean environment for better land use, especially on the area close to boundaries, we investigate a boundary control method for a floating beam system with irregular waves in this paper. Through utilizing Lyapunov direct method, a boundary controller is proposed to reduce the hydroelastic response of the floating beam system restrained at its downstream end. The numerical simulation with experimental calibration is given to verify the effectiveness of the suggested control strategies on the floating beam system. Then, we further investigate the hydroelastic performance of floating beam system under boundary control with various wave amplitudes and bending stiffness. The results demonstrate that the boundary control method could be a potential solution to improve the serviceability of the system in the actual uncertain ocean environments.

A comparative study of two globally convergent numerical methods for acoustic tomography

The comparative study of two globally convergent numerical methods for acoustic tomography is carried out in two dimensions. These are the boundary control method and the quasi-reversibility method. The novelty is that in the latter a nonlinear inverse problem is reduced to a family of the linear integral equation of the first kind via the Lavrentiev approach and this reduction is used within the quasi-reversibility method. The analysis of its stability is carried out via Carleman estimates. The computational effectiveness of these methods is tested in the numerical experiments with the smooth and discontinuous coefficients to be recovered from the tomographic data.

Impact of Boundary Control Methods on Bound-Constrained Optimization Benchmarking

Benchmarking various metaheuristics and their new enhancements, strategies, and adaptation mechanisms has become standard in computational intelligence research. Recently, many challenges and issues regarding fair comparisons and recommendations toward good practices for benchmarking of metaheuristic algorithms, have been identified. This article is aimed at an important issues in metaheuristics design and benchmarking, which are boundary strategies or boundary control methods (BCMs). This work aims to investigate whether the choice of a BCM could significantly influence the performance of competitive algorithms. The experiments encompass the top three performing algorithms from IEEE CEC competitions 2017 and 2020 with six different BCMs. We provide extensive statistical analysis and rankings resulting in conclusions and recommendations for metaheuristics researchers and possibly also for the future direction of benchmark definitions. We conclude that the BCM should be considered another vital metaheuristics input variable for unambiguous reproducibility of results in benchmarking and for a better understanding of population dynamics, since the BCM setting could impact the optimization method performance.

Neural-network-based boundary control for a gantry crane system with unknown friction and output constraint

This paper presents a neural-network-based boundary control method for a gantry system with unknown friction and output constraint. Firstly, to tackle the unknown friction, a radial basis function neural network (RBFNN) is adopted to approximate it. Secondly, we employ a barrier Lyapunov function to handle the output constraint problem. Then, a neural-network-based boundary controller is proposed to deal with the aforementioned problems. Subsequently, based on the Lyapunov stability approach, the uniformly ultimately bounded stability of the state of the closed-loop system is guaranteed. Finally, the effectiveness of the developed control method is illustrated through both numerical simulations and physical experiments.

Boundary consensus control strategies for fractional-order multi-agent systems with reaction-diffusion terms

Multi-agent systems (MASs), as an essential research topic of distributed artificial intelligence, mainly refer to the coordination behavior among the agents to achieve a global goal. The issue dealing with the security threats of the oil and gas industry has always attracted the attention of numerous researchers. Meanwhile, MASs play a significant role in the security and defense of oil pipelines. This paper mainly studies the spatial boundary control for consensus of fractional-order MASs with reaction–diffusion term. First of all, two kinds of consensus boundary control methods are studied in fractional-order MASs, both based on Neumann boundary conditions. Furthermore, by employing fractional-order inequalities, consensus criteria of the MASs are presented in terms of LMIs. Finally, two numerical examples in the multi-agent simulation environment indicate the effectiveness of the proposed theoretical results.

Linearized boundary control method for an acoustic inverse boundary value problem

We develop a linearized boundary control method for the inverse boundary value problem of determining a potential in the acoustic wave equation from the Neumann-to-Dirichlet map. When the linearization is at the zero potential, we derive a reconstruction formula based on the boundary control method and prove that it is of Lipschitz-type stability. When the linearization is at a nonzero potential, we prove that the problem is of Hölder-type stability in two and higher dimensions. The proposed reconstruction formula is implemented and evaluated using several numerical experiments to validate its feasibility.

DEEP LEARNING AND SUCCESSIVE ITERATIONS FOR AN INVERSE PROBLEM OF THE VARIABLE DENSITY RECONSTRUCTION BY THE BOUNDARY CONTROL METHOD

For the first time, two approaches, the deep learning and successive iterations, are proposed for use and implemented for enhancing the images reconstructed by the boundary control method. The construction of successive iterations is based on the reduction of a nonlinear inverse problem for a time-domain wave equation to a linear integral equation of the first kind. The computational effectiveness of the numerical techniques is demonstrated in the numerical experiments in two and three dimensions.

Reconstruction and stability in Gelfand’s inverse interior spectral problem

Assume that $M$ is a compact Riemannian manifold of bounded geometry given by restrictions on its diameter, Ricci curvature and injectivity radius. Assume we are given, with some error, the first eigenvalues of the Laplacian $\Delta_g$ on $M$ as well as the corresponding eigenfunctions restricted on an open set in $M$. We then construct a stable approximation to the manifold $(M,g)$. Namely, we construct a metric space and a Riemannian manifold which differ, in a proper sense, just a little from $M$ when the above data are given with a small error. We give an explicit $\log\log$-type stability estimate on how the constructed manifold and the metric on it depend on the errors in the given data. Moreover a similar stability estimate is derived for the Gel'fand's inverse problem. The proof is based on methods from geometric convergence, a quantitative stability estimate for the unique continuation and a new version of the geometric Boundary Control method.

Lorentzian Calderón problem under curvature bounds

We introduce a method of solving inverse boundary value problems for wave equations on Lorentzian manifolds, and show that zeroth order coefficients can be recovered under certain curvature bounds. The set of Lorentzian metrics satisfying the curvature bounds has a non-empty interior in the sense of arbitrary, smooth perturbations of the metric, whereas all previous results on this problem impose conditions on the metric that force it to be real analytic with respect to a suitably defined time variable. The analogous problem on Riemannian manifolds is called the Calderón problem, and in this case the known results require the metric to be independent of one of the variables. Our approach is based on a new unique continuation result in the exterior region of double null cones. The approach shares features with the classical Boundary Control method, and can be viewed as a generalization of this method to cases where no real analyticity is assumed.

New notions and constructions of the boundary control method

&lt;p style='text-indent:20px;'&gt;The paper provides a list of mathematical objects, constructions and notions, which were revealed and introduced in the course of creation and development of the boundary control method. Among them, there are: amplitude integral, equidistant transforms of 3d vector fields, &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ s $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-points in acoustic scattering, formulas for Betti numbers via DN-map, eikonal algebras, wave spectrum of symmetric semi-bounded operator. We describe these objects and explain how they work for solving inverse problems.&lt;/p&gt;

A novel process of salt tolerance partial denitrification and anammox (ST-PDA) for treating saline wastewater

In the study, the salt-tolerant partial denitrification and Anammox (ST-PDA) process was established, meanwhile, the feasibility of salinity inhibition model as the boundary control method and the subsequent operation performance were studied. Study indicated that the performance of salt-tolerant PD sludge was the optimum under the 10 g·L-1 salinity, and AnAOB also maintained high activity at the salinity. Haldane and Aiba models verified that NO3–-N (substrate) and FNA (product) would have negative consequences for performance of ST-PDA system. However, the effect of FNA would be eliminated by self-alkalization in the denitrification process. A 90% nitrogen removal rate was achieved and the average effluent total nitrogen of 17.8 mg·L-1 was maintained in the system. The high throughput sequencing revealed that the species richness decreased with the salinity increased, while Thauera played a major role in nitrogen removal in saline environment. The study provides a novel insights for salt-containing industrial wastewater.

On an application of the Boundary control method to classical moment problems.

We establish relationships between the classical moments problems which are problems of a construction of a measure supported on a real line, on a half-line or on an interval from prescribed set of moments with the Boundary control approach to a dynamic inverse problem for a dynamical system with discrete time associated with Jacobi matrices. We show that the solution of corresponding truncated moment problems is equivalent to solving some generalized spectral problems.

Study on suppressing the vortex-induced vibration of flexible riser in frequency domain

Vortex-induced vibration (VIV) may cause severe fatigue damage on deep-sea flexible risers. In many researches on active control of VIV, numerical simulation is widely used because of its suitability for parametric studies and lower cost compared to experiments. However, the existing numerical simulations rarely consider the change of lift during the active control of VIV due to the complexity of the control method. Moreover, the calculation time of numerical simulation is relatively long in the time domain. To solve these problems, the active control proposed in this paper is carried out in the frequency domain. A boundary control method considering the change of lift force is proposed through an active control bending moment is applied to the top of riser. Compared with the experimental and numerical results of the flexible riser model under shear flow, the effectiveness of the proposed method is verified. In addition, the effects of different shear currents and different controlled bending moments on structural fatigue damage are studied. The results demonstrated that the reduction of fatigue damage is smaller when the control bending moment is small. As the control bending moment increases, the reduction of fatigue damage increases. However, when the control bending moment exceeds the critical value, the fatigue damage no longer decreases. From the total power perspective, the control energy and the proportion of energy in the system increase with the growth of the control moment. It is difficult to directly obtain the optimal control bending moment although there is an optimal control bending moment. Trial calculations are used to obtain the optimal control bending moment in this paper. The greater the shear currents, the greater the required control bending moment.