Euler-Lagrange Research Articles

A Lyapunov-based control design for centralised networked control systems under false-data-injection attacks

ABSTRACT A central processing unit receives data from all agents and transmits control commands in a Networked Control System (NCS) which is centralised. Centralised NCSs have numerous applications in industrial settings due to their efficiency, simplicity and cost-effective design. However, centralised NCSs are vulnerable to false data injection (FDI) attacks. Despite the fact that researchers have developed detection and mitigation defense mechanisms during past several years, most of these methods have focused on systems with linear dynamics. Furthermore, the existing literature only assumes the injection of FDI attacks on measurement signals. In this paper, we assume that an adversary has injected the FDI attack into both state measurements and control signals with nonlinear dynamics while considering communication noises and disturbances. We propose a secure nonlinear control design that mitigates FDI attacks in real time by combining learning and model-based approaches. We used Lyapunov stability analysis to design the controller, estimator and updating laws of the neural network (NN). In addition, we selected a network of two robots with Euler–Lagrange dynamics to illustrate the robustness of the proposed controller and estimator.

БІРӨЛШЕМДІ КОНСЕРВАТИВТІ ЖҮЙЕНІ ЗЕРТТЕУ ӘДІСІ

The article examines the movements of systems with equal degrees of freedom in a potential, stationary external field using the laws of theoretical mechanics. In order to study the system, the following generalized form of the Lagrange function was used [1,2]: If the force F acting on a particle depends only on the x coordinate, the Lagrange function is transformed as follows [1,2]: где, , , . We have written down the Lagrange equation for plane mathematical, physical, and cyclonic pendulums as follows [3,4]: +mgl где, , +mg l The moment of inertia compared to the axis of rotation J, the distance to the center of mass with the axis of rotation l, for the case under consideration , J , . For any conservative system, the law of conservation is respected: The motion of a particle in one-dimensional space is investigated using the Lagrange function and the laws of conservation of energy.

TauFlowNet: Revealing latent propagation mechanism of tau aggregates using deep neural transport equations

Mounting evidence shows that Alzheimer's disease (AD) is characterized by the propagation of tau aggregates throughout the brain in a prion-like manner. Since current pathology imaging technologies only provide a spatial mapping of tau accumulation, computational modeling becomes indispensable in analyzing the spatiotemporal propagation patterns of widespread tau aggregates from the longitudinal data. However, current state-of-the-art works focus on the longitudinal change of focal patterns, lacking a system-level understanding of the tau propagation mechanism that can explain and forecast the cascade of tau accumulation. To address this limitation, we conceptualize that the intercellular spreading of tau pathology forms a dynamic system where each node (brain region) is ubiquitously wired with other nodes while interacting with the build-up of pathological burdens. In this context, we formulate the biological process of tau spreading in a principled potential energy transport model (constrained by brain network topology), which allows us to develop an explainable neural network for uncovering the spatiotemporal dynamics of tau propagation from the longitudinal tau-PET scans. Specifically, we first translate the transport equation into a GNN (graph neural network) backbone, where the spreading flows are essentially driven by the potential energy of tau accumulation at each node. Conventional GNNs employ a l2-norm graph smoothness prior, resulting in nearly equal potential energies across nodes, leading to vanishing flows. Following this clue, we introduce the total variation (TV) into the graph transport model, where the nature of system's Euler-Lagrange equations is to maximize the spreading flow while minimizing the overall potential energy. On top of this min-max optimization scenario, we design a generative adversarial network (GAN-like) to characterize the TV-based spreading flow of tau aggregates, coined TauFlowNet. We evaluate our TauFlowNet on ADNI and OASIS datasets in terms of the prediction accuracy of future tau accumulation and explore the propagation mechanism of tau aggregates as the disease progresses. Compared to the current counterpart methods, our physics-informed deep model yields more accurate and interpretable results, demonstrating great potential in discovering novel neurobiological mechanisms through the lens of machine learning.

Soft-landing dynamics of a type of four-legged space lander

Analyzing the landing dynamics response is essential for enhancing the success rate of surface explorations on celestial bodies like Mars, the moon, and asteroids. This paper develops a comprehensive dynamics model for a space lander equipped with four sets of buffer legs. Each set comprises a primary strut, two secondary struts, and a footpad. The compression force of the buffer struts is determined through interpolation of real experimental data, while the contact force between the footpad and the ground is calculated using Archimedes law for granular media. The Lagrange equation of the second kind is employed to construct the dynamical model for the lander with minimal degrees of freedom. The proposed dynamics model is validated through experiments involving touchdowns of a single legged lander and a four legged lander. Moreover, key physical quantities such as the buffer force, energy absorption and body acceleration are computed for both 1-2-1 and 2-2 landing configurations. The results indicate that the buffer strut is capable of absorbing approximately half of the impact energy, leading to a 50-fold reduction in the shock acceleration of the main body in both landing configurations. However, it is observed that the maximum compression of the buffer strut is greater in the 1-2-1 landing configuration than in the 2-2 landing configuration. Furthermore, these simulations can be completed within seconds, enabling extensive simulations for the design and optimization of soft landing processes.

Simulation of light propagation in medium with an ultrasonically induced refractive index gradient

Modulation of the refractive index in a medium by external stimuli enables fast and reversible control of light propagation. This technology for controlling light has led to new discoveries in a wide range of research fields from physics to life sciences and has played a major role in the development of photonics devices. In this article, we focus on ultrasound as an external stimulus and have devised a method to control the refractive index of a medium using ultrasound. Our research group has previously discovered that a giant refractive-index gradient (Δn on the order of 10−2) was induced when water was irradiated with high-frequency (100 MHz range), high-intensity (on the order of MPa) ultrasound. Here, we report ray-tracing simulations in a medium with a refractive-index gradient induced by ultrasonic radiation. A numerical model of the refractive-index gradient was developed based on the experimental data, and ray-tracing simulations were performed using the Euler–Lagrange equation. The ray-tracing simulation results were close numerically to the profiles of the laser beam observed in the experiment when the laser beam was incident on the refractive-index-gradient medium.

Wild solutions to scalar Euler-Lagrange equations

We study very weak solutions to scalar Euler-Lagrange equations associated with quadratic convex functionals. We investigate whether W 1 , 1 W^{1,1} solutions are necessarily W loc 1 , 2 W^{1,2}_{\operatorname {loc}} , which would make the theories by De Giorgi-Nash and Schauder applicable. We answer this question positively for a suitable class of functionals. This is an extension of Weyl’s classical lemma for the Laplace equation to a wider class of equations under stronger regularity assumptions. Conversely, using convex integration, we show that outside this class of functionals, there exist W 1 , 1 W^{1,1} solutions of locally infinite energy to scalar Euler-Lagrange equations. In addition, we prove an intermediate result which permits the regularity of a W 1 , 1 W^{1,1} solution to be improved to W loc 1 , 2 W^{1,2}_{\operatorname {loc}} under suitable assumptions on the functional and solution.

Restriction on molecular fluxionality by substitution: A case study for the 1,10-dicyanobullvalene.

We show herein that 1,10-dicyano substitution restricts the paragon fluxionality of bullvalene to just 14 isomers which isomerize along a single cycle. The restricted fluxionality of 1,10-dicyanobullvalene (DCB) is investigated by means of: (i) Bonding analyses of the isomer structures using the adaptive natural density partitioning (AdNDP). (ii) Quantum dynamical simulations of the isomerizations along the cyclic intrinsic reaction coordinate of the potential energy surface (PES). The PES possesses 14 equivalent potential wells supporting 14 isomers which are separated by 14 equivalent potential barriers supporting 14 transition states. Accordingly, at low temperatures, DCB appears as a hindered molecular rotor, without any delocalization of the wavefunction in the 14 potential wells, without any nuclear spin isomers, and with completely negligible tunneling. These results are compared and found to differ from those for molecular boron rotors. (iii) Born-Oppenheimer molecular dynamics (BOMD) simulations of thermally activated isomerizations. (iv) Calculations of the rate constants in the frame of transition state theory (TST) with reasonable agreement achieved with the BOMD results. (v) Simulations of the equilibration dynamics using rate equations for the isomerizations with TST rate coefficients. Accordingly, in the long-time limit, isomerizations of the 14 isomers, each with Cs symmetry, approach the "14 Cs → C7v" thermally averaged structure. This is a superposition of the 14 equally populated isomer structures with an overall C7v symmetry. By extrapolation, the results for DCB yield working hypotheses for so far un-explored properties e.g. for the equilibration dynamics of C10H10.

Lagrangian statistics of concentrated emulsions

The dynamics of stabilised concentrated emulsions presents a rich phenomenology including chaotic emulsification, non-Newtonian rheology and ageing dynamics at rest. Macroscopic rheology results from the complex droplet microdynamics and, in turn, droplet dynamics is influenced by macroscopic flows via the competing action of hydrodynamic and interfacial stresses, giving rise to a complex tangle of elastoplastic effects, diffusion, breakups and coalescence events. This tight multiscale coupling, together with the daunting challenge of experimentally investigating droplets under flow, has hindered the understanding of concentrated emulsions dynamics. We present results from three-dimensional numerical simulations of emulsions that resolve the shape and dynamics of individual droplets, along with the macroscopic flows. We investigate droplet dispersion statistics, measuring probability density functions (p.d.f.s) of droplet displacements and velocities, changing the concentration, in the stirred and ageing regimes. We provide the first measurements, in concentrated emulsions, of the relative droplet–droplet separations p.d.f. and of the droplet acceleration p.d.f., which becomes strongly non-Gaussian as the volume fraction is increased above the jamming point. Cooperative effects, arising when droplets are in contact, are argued to be responsible of the anomalous superdiffusive behaviour of the mean square displacement and of the pair separation at long times, in both the stirred and in the ageing regimes. This superdiffusive behaviour is reflected in a non-Gaussian pair separation p.d.f., whose analytical form is investigated, in the ageing regime, by means of theoretical arguments. This work paves the way to developing a connection between Lagrangian dynamics and rheology in concentrated emulsions.

Trajectory Tracking Algorithm Study of Coal Mine Water Detector Drilling Bar Installation

Mechanical water detection is recognized as the most reliable and safe production technology for coal mines, mainly for the detection of water hazards in pre-mining operations. Intelligent water detectors are currently the main research direction in mechanical water detection, and the automatic installation of drilling bars is the key to achieving intelligent water detection. Improving the connection accuracy in the process of installing drilling bars is an important research topic for the improvement of control links. To improve the connection accuracy of the drilling bars at the time of supplying material, we used the modified Denavit–Hartenberg method to analyze the motion gestures of the supplied material device and the Lagrange equation to establish a dynamic analysis model. We aimed at better control precision by improving the sliding mode control algorithm and at increasing the convergence rate of tracking errors with a sliding controller based on an exponential approximation law and using saturated functions instead of the symbol functions in the reaching law to weaken the vibration in the control process. We then used particle swarm optimization (PSO) to find the optimum combination parameters of the sliding mode controllers and test the performance of the sliding mode controllers before and after PSO with MATLAB/Simulink. The results showed that the optimized controller has a strong resistance to parameter fluctuations, and the system responds quickly, achieves a good performance, and improves the convergence rate of tracking errors.

Bogomol’nyi-like equations in gravity theories

Using the Bogomol’nyi–Prasad–Sommerfield Lagrangian method, we show that gravity theory coupled to matter in various dimensions may possess Bogomol’nyi-like equations, which are first-order differential equations, satisfying the Einstein equations and the Euler–Lagrange equations of classical fields (U(1) gauge and scalar fields). In particular we consider static and spherically symmetric solutions by taking proper ansatzes and then we find an effective Lagrangian density that can reproduce the Einstein equations and the Euler–Lagrange equations of the classical fields. We consider the BPS Lagrangian density to be linear function of first-order derivative of all the fields. From these two Lagrangian desities we are able to obtain the Bogomol’nyi-like equations whose some of solutions are well-known such as Schwarzschild, Reissner–Nordström, Tangherlini black holes, and the recent black holes with scalar hair in three dimensions (Phys. Rev. D 107, 124047). Using these Bogomol’nyi-like equations, we are also able to find new solutions for scalar hair black holes in three and four dimensional spacetime. Furthermore we show that the Bogomol’nyi–Prasad–Sommerfield Lagrangian method can provide a simple alternative proof of black holes uniqueness theorems in any dimension.

Validation of the added mass effect in the narrow gaps of nuclear reactor pressure vessels

Nuclear reactor pressure vessels (RPVs) must be robust to external environmental excitations such as seismic loads. Precise numerical models that predict the dynamic responses to external loads are crucial for evaluating the safety of RPVs. Notably, in RPVs submerged in reactor coolant, it is necessary to examine the added mass effect of the fluid via fluid-structure interaction (FSI) analysis, employing either an analytical approach or a numerical analysis. The present study focuses on modeling and analyzing the added mass effect in the narrow gaps associated with the dense structural layout in the reactor. In this context, this paper validates analytical and numerical models that accurately describe the added mass effect for two cases: (i) simple concentric cylinders with a narrow gap filled with fluid and (ii) a complex small-scale RPV with narrow gaps also filled with fluid. Initially, the analytical model, employing Lagrange's equation of motion and velocity potential, and the numerical model, utilizing FE modeling in acoustic FSI through ANSYS, were validated through experimental modal testing for the added mass effect in the narrow gap between simple concentric cylinders. The analytical model for the concentric cylinders indicates that the added mass diverges to infinity as the gap approaches zero, which should be examined carefully. Through experiments for the concentric cylinders, the analytical and FEM-based added mass effects were validated with a natural frequency error of 3 %. Subsequently, based on the results for the concentric cylinders, a complex numerical model of the small-scale reactor with complex boundary conditions was developed and cross-validated through corresponding experimental modal testing of the small-scale RPV. The added mass model for the narrow gaps in the RPV was validated by substantial agreement of eight representative modes, as demonstrated by a 5 % natural frequency error.

Optimizing Variational Problems through Weighted Fractional Derivatives

In this article, we explore a variety of problems within the domain of calculus of variations, specifically in the context of fractional calculus. The fractional derivative we consider incorporates the notion of weighted fractional derivatives along with derivatives with respect to another function. Besides the fractional operators, the Lagrange function depends on extremal points. We examine the fundamental problem, providing the fractional Euler–Lagrange equation and the associated transversality conditions. Both the isoperimetric and Herglotz problems are also explored. Finally, we conclude with an analysis of the variational problem, incorporating fractional derivatives of any positive real order.

Study on lateral vibration characteristics of BHA with double-span drill collar

The severe vibration of BHA directly affect the drilling efficiency, borehole trajectory, and wellbore quality. Therefore, it is crucial for developing dynamic mechanics of drill string system by studying the mechanical behavior of BHA. In this paper, considering the continuity condition of double-span drill collar, the expression of lateral vibration of drill collars is firstly obtained by using the assumption mode method. Then, the lateral vibration equation of double-span drill collar is derived by Lagrange equation. Finally, a self-programming model is established based on dynamic mechanics of drill string system, the lateral vibration characteristics of double-span drill collar in four different types of BHA is further determined by the fourth-order Runge-Kutta algorithm. It is obviously found that the lateral vibration is enhanced with increasing the length of drill collar; the rotation speed of BHA, weight on bit (WOB), and the size of stabilizer have a significant influence on the lateral vibration of drill collar. In the practical application, the lengths of the first-span and the second-span drill collar are given as 10 and 19 m, respectively; and the stabilizer diameter approached to bit diameter is recommend in practical gas drilling engineering; under WOB of 10–40 kN, the rotation speed of the drilling string should be controlled nearby 60 r/min. The theoretical investigation is a guidance for structure design of BHA and optimization of drilling parameter.

Theory of elastic wave propagation in a fluid-saturated multi-porous medium with multi-permeability

This paper establishes the concept of elastic wave propagation in a multi-porous medium with different permeabilities by assuming there are n distinct pore fluid phases. The dynamic equation of motion of elastic wave propagation through this multi-porous medium is derived based on Lagrangian mechanics. In this regard, the generalized form of mass coefficients and then the energy loss due to the fluid phases in terms of dissipation coefficients are presented for low-frequency limits with the help of Darcy’s Law of multi-phases system. The elastic coefficients of the constitutive equation in terms of compliance matrix are identified using a series of Gedanken experiments. Some significant results regarding the compressional and rotational waves in a multi-porosity medium are derived. The validation of the theory has been shown by comparing it with the existing theory of single and double porosity. It is observed that there are ( n + 1 ) compressional waves corresponding to solid and fluid phases, whereas only one rotational wave is associated with the solid phase. The concept of multi-porosity theory can contribute to a deeper understanding of wave behaviour in a porous medium.

Dijkstra algorithm based minimum acceleration/snap quadrotor UAV trajectory planning

This paper focuses on path planning using the Dijkstra/A* algorithm and addresses the trajectory generation problem using the minimum acceleration/snap approach. Firstly, the mathematical model of the optimization problem is formulated, specifically utilizing the Euler-Lagrange equation to derive the necessary conditions for achieving minimum acceleration/snap trajectories, which are represented by either 3rd or 7th order polynomials. The coefficients of these polynomials are determined by imposing appropriate constraints on velocity, acceleration, and higher-order derivatives. Subsequently, a detailed comparison of the performances of the two optimization methods in complex environments is conducted, evaluating differences in position and velocity. Based on the deterministic analysis, conclusive results are obtained.

Dynamics of a vibrating screen with two motor-vibrators

A method for calculating a vibrating screen with two motor vibrators is proposed. The first part of the article is devoted to the issue of synchronizing the rotation of the rotors of both drives. Their opposite rotation creates a directed force passing through the center of gravity of the screen and perpendicular to the plane of the vibrator axes. The interaction of forces in this plane determines the phase mismatch of the rotation angles of the vibrators θ. An expression is obtained for determining the angle θ depending on the frequency of their rotation vibrators ω. It is shown that at ω > 100 rad/s, motor-vibrators operate in self-synchronizing mode. In the second part of the article a dynamic calculation of the screen was carried out. Linear vibrations in the vertical plane of symmetry of the screen and rotational vibrations around a perpendicular to this plane are investigated. The Lagrange equation of the second kind is compiled without considering inelastic resistances. They are considered at the last stage of the calculation in the form of the imaginary part of the complex modulus of elastic bonds. The amplitude-frequency characteristic of the screen was constructed considering the rotation mismatch angle of the vibrators θ. Stabilization of the oscillation amplitude at high frequencies is shown.

Analysis of Vibration Characteristics of Horizontal Axis Wind Turbine Tower under Random Wind Action

As the main supporting structure of wind turbines, the service quality of towers directly affects the operational safety of wind turbines. Researching tower vibration characteristics, evaluating its service quality and early warning of damage is an important means to improve the operational safety of wind turbines. Firstly, the external load conditions of the wind turbine tower were determined through wind load calculation, and the load and dynamic theory of the variable pitch torque wind turbine tower were summarized. Secondly, based on the Euler-Lagrange energy equation, the Lagrange equation of the simplified dynamic model of the wind turbine tower is established, and the vibration characteristics of the tower under different wind speeds are analyzed. The research results can provide a theoretical basis for the research on the characteristics of the wind turbine tower.

Optimal trajectory planning combining model-based and data-driven hybrid approaches

Trajectory planning aims at computing an optimal trajectory through the minimization of a cost function. This paper considers four different scenarios: (i) the first concerns a given trajectory on which a cost function is minimized by a acting on the velocity along it; (ii) the second considers trajectories expressed parametrically, from which an optimal path and the velocity along it are computed; (iii), the case in which only the departure and arrival points of the trajectory are known, and the optimal path must be determined; and finally, (iv) the case involving uncertainty in the environment in which the trajectory operates. When the considered cost functions are expressed analytically, the application of Euler–Lagrange equations constitutes an appealing option. However, in many applications, complex cost functions are learned by using black-box machine learning techniques, for instance deep neural networks. In such cases, a neural approach of the trajectory planning becomes an appealing alternative. Different numerical experiments will serve to illustrate the potential of the proposed methodologies on some selected use cases.

A variational approach to deriving Hamiltonian functions under non-holonomic constraints — the case of the Gaussian thermostats

The Hamiltonian function (H), if it exists, provides a powerful platform for studying the dynamics of constrained many-particle systems. For systems under holonomic constraints, H can be derived from the principle of least action (PLA) formulated in terms of a suitable set of generalized coordinates. In molecular dynamics (MD) simulations, where the system temperature often prescribes a non-holonomic constraint, the challenge is to find the system’s Hamiltonian. The Gaussian Isokinetic (GIK) thermostat, the first deterministic physics-preserving thermostat in MD, was derived from the principle of least constraint (PLC) in the early 1980s: its H, interestingly, was discovered almost 15 years later. A mathematically rigorous connection of the GIK Hamiltonian with PLA, however, has been missing until now. In this article, we adopt a variational approach to establish the equivalence of PLC with PLA for a class of non-holonomically constrained many-particle systems, and present the general formulation of the Euler–Lagrange equations of motion through time rescaling. Making use of the particular forms of the Lagrange multiplier obtained from the principle of least constraint, we derive H for the GIK and the Gaussian Isoenergetic thermostats.

Finite-time control of uncertain Euler–Lagrange systems using ELM-based velocity observer

This paper is concerned with the finite-time trajectory tracking control of a class of Euler–Lagrange (EL) systems under unknown velocity information, dynamic uncertainties and disturbances. An extreme learning machine (ELM) algorithm is employed to approximate the uncertainties, while an adaptive algorithm is proposed to tune output weights of the ELM, as well as to eliminate the negative effects of residual errors and disturbances. Then an adaptive finite-time ELM-based velocity observer is developed to estimate the unavailable velocity states. Further, based on the estimations and the approximations of model uncertainties, an adaptive finite-time observer-based nonsingular terminal sliding mode (TSM) control strategy is constructed to guarantee the finite-time bounded tracking of the uncertain EL system by using the Lyapunov stability theorem. Simulation results on a robotic manipulator platform demonstrate the efficiency of the developed finite-time observation and control methods.