Feedback stabilisation of an Euler–Bernoulli beam with the boundary time-delay disturbance

In this paper, we consider the feedback stabilisation of an Euler–Bernoulli beam with the boundary time-delay disturbance. Due to unknown time-delay coefficient, the system might be exponentially increasing at the lack of control. We design the feedback control law based on Lyapunov function method. Different from usual use of Lyapunov function method, our approach is to combine the construction of Lyapunov functionals with the controller design, which will guarantee the system energy function decays exponentially. In this procedure, we deduce the inequality equations satisfied by the system parameters. We prove the well-posedness of the corresponding closed-loop system by using semigroup theory and the inequality equations are solvable. Moreover, the exponential decay rate of the system is estimated. In addition, some numerical simulations are also presented to support the obtained results.     

Stabilization of the Euler–Bernoulli equation via boundary connection with heat equation

In this paper, we are concerned with the stabilization of a coupled system of Euler–Bernoulli beam or plate with heat equation, where the heat equation (or vice versa the beam equation) is considered as the controller of the whole system. The dissipative damping is produced in the heat equation via the boundary connections only. The one-dimensional problem is thoroughly studied by Riesz basis approach: The closed-loop system is showed to be a Riesz spectral system and the spectrum-determined growth condition holds. As the consequences, the boundary connections with dissipation only in heat equation can stabilize exponentially the whole system, and the solution of the system has the Gevrey regularity. The exponential stability is proved for a two dimensional system with additional dissipation in the boundary of the plate part. The study gives rise to a different design in control of distributed parameter systems through weak connections with subsystems where the controls are imposed.     

Rapid stabilisation of an Euler–Bernoulli beam with the internal delay control

ABSTRACT

In this paper, we are concerned with rapid stabilisation of an Euler--Bernoulli beam with internal delayed control. Herein we introduce a new approach of the feedback control design from the system equivalence point of view. The design approach can be divided into several steps. First, we construct a target system of the desired stability. Second, we select a suitable integral transform that transforms the present system to the target system. In this procedure, one can get a corresponding feedback control. Third, we find a transform that transforms the target system to the present system, which provides the equivalence of both systems. Finally, we prove that the two transforms are bounded linear operators in an appropriate Hilbert space.     

Hierarchical Nash equilibrium seeking strategies of quadratic time-varying games with Euler–Lagrange players

This article investigates the distributed Nash equilibrium seeking problem of quadratic time-varying games with Euler–Lagrange (EL) players, where external disturbances and parametric uncertainties are involved. A gradient-based hierarchical algorithm consisting of a game layer and a control layer is proposed. Specifically, in the game layer, EL players communicate with neighbors through a graph to reach the consensus on potential aggregate values, which will be employed to calculate the gradient of each player's objective function, and then, a gradient-based sliding mode controller is developed to track time-varying gradient in the control layer. Thus, the convergence results are hierarchically obtained through the Lyapunov stability method. In addition, the hierarchical control strategy is extended to address the constrained problems through the utilization of a smooth penalty function. By appropriately choosing control parameters, the Nash equilibrium seeking errors can be arbitrarily small. The relation between the optimal solutions of the original problem and the dual one is further discussed. Finally, the proposed methods are numerically verified.     

Convergence of the Euler–Maruyama method for stochastic differential equations with Markovian switching

Stochastic differential equations with Markovian switching (SDEwMSs), one of the important classes of hybrid systems, have been used to model many physical systems that are subject to frequent unpredictable structural changes. The research in this area has been both theoretical and applied. Most of SDEwMSs do not have explicit solutions so it is important to have numerical solutions. It is surprising that there are not any numerical methods established for SDEwMSs yet, although the numerical methods for stochastic differential equations (SDEs) have been well studied. The main aim of this paper is to develop a numerical scheme for SDEwMSs and estimate the error between the numerical and exact solutions. This is the first paper in this direction and the emphasis lies on the error analysis.     

Stabilization of the Euler–Bernoulli plate with variable coefficients by nonlinear internal feedback

We consider the energy decay for solutions of the Euler–Bernoulli plate equation with variable coefficients where a nonlinear internal feedback acts in a suitable subregion of the domain. The Riemannian geometric method is used to deal with variable coefficient problems. When the feedback region has a structure similar to that for the wave equation with constant coefficients, we establish the stabilization of the system in the case of fixed boundary conditions. Several energy decay rates are established according to various growth restrictions on the nonlinear feedback near the origin and at infinity. We further show that, unlike for the case of constant coefficients, choices of such feedback regions depend not only on the type of boundary conditions but also on the curvature of a Riemannian metric, based on the coefficients of the system.     

Mean-square stability of the Euler–Maruyama method for stochastic differential delay equations with jumps

This paper deals with the mean-square (MS) stability of the Euler–Maruyama method for stochastic differential delay equations (SDDEs) with jumps. First, the definition of the MS-stability of numerical methods for SDDEs with jumps is established, and then the sufficient condition of the MS-stability of the Euler–Maruyama method for SDDEs with jumps is derived, finally a class scalar test equation is simulated and the numerical experiments verify the results obtained from theory.     

Cache-aware design of general-purpose Single-Producer–Single-Consumer queues

Data processing pipelines normally use lockless Single-Producer–Single-Consumer (SPSC) queues to efficiently decouple their processing threads and achieve high throughput, minimizing the cost of synchronization. SPSC queues have been widely studied, mostly for applications such as streaming data or network monitoring, where the main goal is maximizing throughput. There are now many applications, such as virtual-machine–virtual-machine communication, software-defined networking, and message-based kernels, where low latency is also important, and the tradeoffs between high-throughput and low-latency algorithms have not been studied equally well. Furthermore, at high or variable transaction rates, the effect of memory hierarchies and cache coherence subsystems may be dominant and yield surprising results. In this paper, we make two contributions. First, we provide a comprehensive study of the two main families of SPSC queues, namely, “Lamport” and “FastForward” queues, with a detailed analytical and experimental characterization of their behavior in terms of operating regimes, throughput, latency, and cache misses. Second, we propose two new queue variants, namely, improved FastForward and batched improved FastForward, which have better worst-case behavior than other variants in terms of cache misses, which is an important feature for a number of applications. Together, these two contributions provide practical guidelines to choose the best solution depending on the application requirements.     

Piece wise linear least–squares approximation of planar curves

Two algorithms for solving the piecewise linear least–squares approximation problem of plane curves are presented. The first is for the case when the L ₂ residual (error) norm in any segment is not to exceed a pre–assigned value. The second algorithm is for the case when the number of segments is given and a (balanced) L ₂ residual norm solution is required. The given curve is first digitized and either algorithm is then applied to the discrete points. For each segment, we obtain the upper triangular matrix R in the QR factorization of the (augmented) coefficient matrix of the resulting system of linear equations. The least–squares solutions are calculated in terms of the R (and Q) matrices. The algorithms then work in an iterative manner by updating the least–squares solutions for the segments via up dating the R matrices. The calculation requires as little computational effort as possible. Numerical results and comments are given. This, in a way, is a tutorial paper.     

A hybrid CBO–PSO algorithm for optimal design of truss structures with dynamic constraints

The vibration domain of structures can be reduced by imposing some constraints on their natural frequencies. For this purpose optimal design of structures under frequency constraints is required which involves highly non-linear and non-convex problems. In this paper an efficient hybrid algorithm is developed for solving such optimization problems. This algorithm utilizes the recently developed colliding bodies optimization (CBO) algorithm as the main engine and uses the positive properties of the particle swarm optimization (PSO) algorithm to increase the efficiency of the CBO. The distinct feature of the present hybrid algorithm is that it requires no parameter tuning. The CBO is known for being parameter independent, and avoiding the use of the traditional penalty method to handle the constraints upholds this property. Two mathematical constrained functions taken from the literature are studied to verify the performance of the algorithm. The algorithm is then applied to optimize truss structures with frequency limitations. The numerical results demonstrate the efficiency of the presented algorithm for this class of problems.     

Classical design of two-by-two systems with severe interaction†

Based on the concept of the interaction factor, a method for obtaining rapid graphical estimates of equivalent single-loop gain functions and interaction ratios is used in the design of 2 x 2 systems with severe interaction by classical techniques.     

Motion recognition using local auto-correlation of space–time gradients

In this paper, we propose a motion recognition scheme based on a novel method of motion feature extraction. The feature extraction method utilizes auto-correlations of space–time gradients of three-dimensional motion shape in a video sequence. The method effectively exploits the local relationships of the gradients corresponding to the space–time geometric characteristics of the motion. For recognizing motions, we apply the framework of bag-of-frame-features, which, in contrast to the standard bag-of-features framework, enables the motion characteristics to be captured sufficiently and the motions to be quickly recognized. In experiments on various datasets for motion recognition, the proposed method exhibits favorable performances as compared to the other methods, and faster computational time even than real time.     

Robot Head Motion Control with an Emphasis on Realism of Neck–Eye Coordination during Object Tracking

Important aspects of present-day humanoid robot research is to make such robots look realistic and human-like, both in appearance, as well as in motion and mannerism. In this paper, we focus our study on advanced control leading to realistic motion coordination for a humanoid’s robot neck and eyes while tracking an object. The motivating application for such controls is conversational robotics, in which a robot head “actor” should be able to detect and make eye contact with a human subject. Therefore, in such a scenario, the 3D position and orientation of an object of interest in space should be tracked by the redundant head–eye mechanism partly through its neck, and partly through its eyes. In this paper, we propose an optimization approach, combined with a real-time visual feedback to generate the realistic robot motion and robustify it. We also offer experimental results showing that the neck–eye motion obtained from the proposed algorithm is realistic comparing to the head–eye motion of humans.     

A stress–based approach to the optimal design of structures with unilateral behavior of material or supports

The paper presents a stress-based approach that copes with the optimal design of truss-like elastic structures in case of unilateral behavior of material or ground supports. The conventional volume-constrained minimization of compliance is coupled with a set of local stress constraints that are enforced, all over the domain or along prescribed boundaries, to control the arising of members with tension-only or compression-only strength. A Drucker–Prager failure criterion is formulated to provide a smooth approximation of the no-tension or no-compression conditions governing the stress field. A selection strategy is implemented to handle efficiently the multi-constrained formulation that is solved through mathematical programming. Benchmark examples are investigated to discuss the features of the achieved optimal designs, as compared with problems involving material and ground supports with equal behavior in tension and compression. Numerical simulations show that a limited set of constraints is needed in the first iterations to steer the solution of the energy-driven optimization towards designs accounting for the prescribed assumption of unilateral strength.     

Exponential stability of a network of serially connected Euler–Bernoulli beams

The aim is to prove the exponential stability of a system modelling the vibrations of a network of N Euler–Bernoulli beams serially connected. Using a result given by K. Ammari and M. Tucsnak, the problem is reduced to the estimate of a transfer function and the obtention of an observability inequality. The solution is then expressed in terms of Fourier series so that one of the sufficient conditions for both the estimate of the transfer function and the observability inequality is that the distance between two consecutive large eigenvalues of the spatial operator involved in this evolution problem is superior to a minimal fixed value. This property called spectral gap holds. It is proved using the exterior matrix method due to W. H. Paulsen. Two more asymptotic estimates involving the eigenfunctions are required. They are established using an adequate basis.     

Human Motion Tracking with a Kinematic Parameterization of Extremal Contours

This paper addresses the problem of human motion tracking from multiple image sequences. The human body is described by five articulated mechanical chains and human body-parts are described by volumetric primitives with curved surfaces. If such a surface is observed with a camera, an extremal contour appears in the image whenever the surface turns smoothly away from the viewer. We describe a method that recovers human motion through a kinematic parameterization of these extremal contours. The method exploits the fact that the observed image motion of these contours is a function of both the rigid displacement of the surface and of the relative position and orientation between the viewer and the curved surface. First, we describe a parameterization of an extremal-contour point velocity for the case of developable surfaces. Second, we use the zero-reference kinematic representation and we derive an explicit formula that links extremal contour velocities to the angular velocities associated with the kinematic model. Third, we show how the chamfer-distance may be used to measure the discrepancy between predicted extremal contours and observed image contours; moreover we show how the chamfer distance can be used as a differentiable multi-valued function and how the tracker based on this distance can be cast into a continuous non-linear optimization framework. Fourth, we describe implementation issues associated with a practical human-body tracker that may use an arbitrary number of cameras. One great methodological and practical advantage of our method is that it relies neither on model-to-image, nor on image-to-image point matches. In practice we model people with 5 kinematic chains, 19 volumetric primitives, and 54 degrees of freedom; We observe silhouettes in images gathered with several synchronized and calibrated cameras. The tracker has been successfully applied to several complex motions gathered at 30 frames/second.     

Convergence of linearized backward Euler–Galerkin finite element methods for the time-dependent Ginzburg–Landau equations with temporal gauge

We present error analysis of fully discrete Galerkin finite element methods for the time-dependent Ginzburg–Landau equations with the temporal gauge, where a linearized backward Euler scheme is used for the time discretization. We prove that the convergence rate is O(τ+h^r) if the finite element space of piecewise polynomials of degree r is used. Due to the degeneracy of the problem, the convergence rate is one order lower than the optimal convergence rate of finite element methods for parabolic equations. Numerical examples are provided to support our theoretical analysis.     

Distributed aggregative games for Euler–Lagrange systems with system parameter uncertainties

In this paper, an aggregative game of Euler–Lagrange (EL) systems is studied, where the parameters of the EL systems are not available. To seek the Nash equilibrium of the game, a novel distributed Nash equilibrium seeking algorithm is proposed, where the system parameters are not used in the feedback control. Moreover, a Lyapunov function is constructed such that EL players are proved to exponentially converge to the Nash equilibrium of the game. Finally, an example in the electricity market is provided to illustrate our result.     

Predictor-based control for an uncertain Euler–Lagrange system with input delay

Controlling a nonlinear system with actuator delay is a challenging problem because of the need to develop some form of prediction of the nonlinear dynamics. Developing a predictor-based controller for an uncertain system is especially challenging. In this paper, tracking controllers are developed for an Euler–Lagrange system with time-delayed actuation, parametric uncertainty, and additive bounded disturbances. The developed controllers represent the first input delayed controllers developed for uncertain nonlinear systems that use a predictor to compensate for the delay. The results are obtained through the development of a novel predictor-like method to address the time delay in the control input. Lyapunov–Krasovskii functionals are used within a Lyapunov-based stability analysis to prove semi-globally uniformly ultimately bounded tracking. Experimental results illustrate the performance and robustness of the developed control methods.