Search Results (10)

We explore nonlocal symmetries in a class of Hamiltonian dynamical systems governed by second-order differential equations. Specifically, we establish an algorithm for deriving nonlocal symmetries by utilizing the Jacobi metric and the Eisenhart–Duval lift to geometrize the dynamical systems. The geometrized systems often exhibit additional local symmetries compared to the original systems, some of which correspond to nonlocal symmetries for the original formulation. This novel approach allows us to determine nonlocal symmetries in a systematic way. Within this geometric framework, we demonstrate that the second-order differential equation $\ddot{q}-F\left(q\right)=0$ admits an infinite number of nonlocal symmetries generated by the infinite-dimensional conformal algebra of a two-dimensional Riemannian manifold. Applications to higher-dimensional systems are also discussed. Full article

In the age of fast digital transformation, Docker containers have become one of the central technologies for flexible and scalable application deployment. However, this has opened a new dimension of challenges in security, which are skyrocketing with increased technology adoption. This paper discerns these challenges through a manifold approach: first, comprehensive static analysis by Trivy, and second, real-time dynamic analysis by Falco in order to uncover vulnerabilities in Docker environments pre-deployment and during runtime. One can also find similar challenges in security within the Internet of Things (IoT) sector, due to the huge number of devices connected to WiFi networks, from simple data breaches such as brute force attacks and unauthorized access to large-scale cyber attacks against critical infrastructure, which represent only a portion of the problems. In connection with this, this paper is calling for the execution of robust AWS cloud security solutions: IoT Core, CloudWatch, and GuardDuty. IoT Core provides a secure channel of communication for IoT devices, and CloudWatch offers detailed monitoring and logging. Additional security is provided by GuardDuty’s automatized threat detection system, which continuously seeks out potential threats across network traffic. Armed with these technologies, we try to build a more resilient and privacy-oriented IoT while ensuring the security of our digital existence. The result is, therefore, an all-inclusive work on security in both Docker and IoT domains, which might be considered one of the most important efforts so far to strengthen the digital infrastructure against fast-evolving cyber threats, combining state-of-the-art methods of static and dynamic analyses for Docker security with advanced, cloud-based protection for IoT devices. Full article

This paper proposes a decentralized adaptive event-triggered fault-tolerant cooperative control (ET-FTCC) scheme for multiple unmanned aerial vehicles (UAVs) and unmanned ground vehicles (UGVs) with actuator faults and external disturbances under denial-of-service (DoS) attacks. The multiple UAVs and UGVs have a larger search radius, which is important in both the civilian and military domains. The different dynamics between UAVs and UGVs result in unbalanced interactions in the communication topologies, which increases the complexity of cooperative control. DoS attacks are conducted in both sensor and control channels. The dynamic models of UAVs and UGVs are introduced firstly, and the unified heterogeneous multiagent system model with actuator faults is established. The composite observer is designed to obtain the information of state and lumped disturbance, which is used to design the controller. In order to save the limited communication network resources, the event-triggered mechanism is introduced. The transformed error is presented by using the prescribed performance function (PPF). Then, the sliding-mode manifold is presented by combining the event-triggered control scheme to achieve the tracking purpose with actuator faults, external disturbances, and DoS attacks. Based on the Lyapunov function approach, the tracking errors are bounded within the prescribed boundary. Finally, the effectiveness of the proposed method is verified by qualitative analysis and quantitative analysis of the simulation results. This study can enhance the security and reliability of heterogeneous multiagent systems, providing technical support for the safe operation of unmanned systems. This paper mainly solves the FTCC problem of second-order nonlinear heterogeneous multiagent systems, and further research is needed for the FTCC problem of higher-order nonlinear heterogeneous multi-agent systems. In addition, the system may encounter multiple cyber attacks. As one of the future research works, we can extend the results of this paper to high-order nonlinear systems under multiple cyber attacks, which contain DoS attacks and deception attacks, and achieve fault-tolerant cooperative control of heterogeneous multiagent systems. Full article

The present paper investigates a fixed-time tracking control with fractional-order dynamics for a quadrotor subjected to external disturbances. After giving the formulation problem of a quadrotor system with six subsystems like a second-order system, a fractional-order sliding manifold is then designed to achieve a fixed-time convergence of the state variables. In order to cope with the upper bound of the disturbances, a switching fixed-time controller is added to the equivalent control law. Based on the switching law, fixed-time stability is ensured. All analysis and stability are proved using the Lyapunov approach. Finally, the higher performance of the proposed controller fixed-time fractional-order sliding mode control (FTFOSMC) is successfully compared to the two existing techniques through numerical simulations. Full article

The present tutorial paper constitutes the second of a series of tutorials on manifold calculus with applications in system theory and control. The aim of the present tutorial, in particular, is to explain and illustrate some key concepts in manifold calculus such as covariant derivation and manifold curvature. Such key concepts are then applied to the formulation, to the control, and to the analysis of non-linear dynamical systems whose state-space are smooth (Riemannian) manifolds. The main flow of exposition is enriched by a number of examples whose aim is to clarify the notation used and the main theoretical findings through practical calculations. Full article

The present paper is concerned with an instance of automatic control for autonomous vehicles based on the theory of virtual attractive-repulsive potentials (VARP). The first part of this paper presents a review of the VARP control theory as developed specifically by B. Nguyen, Y.-L. Chuang, D. Tung, C. Hsieh, Z. Jin, L. Shi, D. Marthaler, A. Bertozzi and R. Murray, in the paper ‘Virtual attractive-repulsive potentials for cooperative control of second order dynamic vehicles on the Caltech MVWT’, which appeared in the Proceedings of the 2005 American Control Conference, (Portland, OR, USA) held in June 2005 (pp. 1084–1089). The aim of the first part of the present paper is to recall the mathematical and logical steps that lead to controlling an autonomous robot by a VARP-based control theory. The concepts recalled in the first part of the present paper, with special reference to the physical interpretation of the terms in the developed control field, serve as the starting point to develop a more convoluted control theory for (second-order) dynamical systems whose state spaces are (possibly high-dimensional) curved manifolds. The second part of this paper is, in fact, devoted to extending the classical VARP control theory to regulate dynamical systems whose state spaces possess the mathematical structure of smooth manifolds through manifold calculus. Manifold-type state spaces present a high degree of symmetry, due to mutual non-linear constraints between single physical variables. A comprehensive set of numerical experiments complements the review of the VARP theory and the theoretical developments towards its extension to smooth manifolds. Full article

The aim of the present tutorial paper is to recall notions from manifold calculus and to illustrate how these tools prove useful in describing system-theoretic properties. Special emphasis is put on embedded manifold calculus (which is coordinate-free and relies on the embedding of a manifold into a larger ambient space). In addition, we also consider the control of non-linear systems whose states belong to curved manifolds. As a case study, synchronization of non-linear systems by feedback control on smooth manifolds (including Lie groups) is surveyed. Special emphasis is also put on numerical methods to simulate non-linear control systems on curved manifolds. The present tutorial is meant to cover a portion of the mentioned topics, such as first-order systems, but it does not cover topics such as covariant derivation and second-order dynamical systems, which will be covered in a subsequent tutorial paper. Full article

In this paper, we put forth an evolve-then-correct reduced order modeling approach that combines intrusive and nonintrusive models to take hidden physical processes into account. Specifically, we split the underlying dynamics into known and unknown components. In the known part, we first utilize an intrusive Galerkin method projected on a set of basis functions obtained by proper orthogonal decomposition. We then present two variants of correction formula based on the assumption that the observed data are a manifestation of all relevant processes. The first method uses a standard least-squares regression with a quadratic approximation and requires solving a rank-deficient linear system, while the second approach employs a recurrent neural network emulator to account for the correction term. We further enhance our approach by using an orthonormality conforming basis interpolation approach on a Grassmannian manifold to address off-design conditions. The proposed framework is illustrated here with the application of two-dimensional co-rotating vortex simulations under modeling uncertainty. The results demonstrate highly accurate predictions underlining the effectiveness of the evolve-then-correct approach toward real-time simulations, where the full process model is not known a priori. Full article

The nonlinear orbital dynamics of a class of the perturbed restricted three-body problem is studied. The two primaries considered here refer to the binary system HD 191408. The third particle moves under the gravity of the binary system, whose triaxial rate and radiation factor are also considered. Based on the dynamic governing equation of the third particle in the binary HD 191408 system, the motion state manifold is given. By plotting bifurcation diagrams of the system, the effects of various perturbation factors on the dynamic behavior of the third particle are discussed in detail. In addition, the relationship between the geometric configuration and the Jacobian constant is discussed by analyzing the zero-velocity surface and zero-velocity curve of the system. Then, using the Poincaré–Lindsted method and numerical simulation, the second- and third-order periodic orbits of the third particle around the collinear libration point in two- and three-dimensional spaces are analytically and numerically presented. This paper complements the results by Singh et al. [Singh et al., AMC, 2018]. It contains not only higher-order analytical periodic solutions in the vicinity of the collinear equilibrium points but also conducts extensive numerical research on the bifurcation of the binary system. Full article

This paper presents the dynamic behaviors of a second-order asymmetric stochastic delay system with a Duffing oscillator as well as through the detection of weak signals, which are analyzed theoretically and numerically. The dynamic behaviors of the asymmetric system are analyzed based on the stochastic center manifold, together with Hopf bifurcation. Numerical analysis revealed that the time delay could enhance the noise immunity of the asymmetric system so as to enhance the asymmetric system’s ability to detect weak signals. The frequency of the weak signal under noise excitation was detected through the ‘act-and-wait’ method. The small amplitude was detected through the transition from the chaotic to the periodic state. Theoretical analysis and numerical simulation indicate that the application of the asymmetric Duffing oscillator with delay to detect weak signal is feasible. Full article