Second-order Systems Research Articles

The Nonlinear Dynamics and Chaos Control of Pricing Games in Group Robot Systems

System stability control in resource allocation is a critical issue in group robot systems. Against this backdrop, this study investigates the nonlinear dynamics and chaotic phenomena that arise during pricing games among finitely rational group robots and proposes control strategies to mitigate chaotic behaviors. A system model and a business model for group robots are developed based on market mechanism mapping, and the dynamics of resource allocation are formulated as a second-order discrete nonlinear system using game theory. Numerical simulations reveal that small perturbations in system parameters, such as pricing adjustment speed, product demand coefficients, and resource substitution coefficients, can induce chaotic behaviors. To address these chaotic phenomena, a control method combining state feedback and parameter adjustment is proposed. This approach dynamically tunes the state feedback intensity of the system via a control parameter M, thereby delaying bifurcations and suppressing chaotic behaviors. It ensures that the distribution of system eigenvalues satisfies stability conditions, allowing control over unstable periodic orbits and period-doubling bifurcations. Simulation results demonstrate that the proposed control method effectively delays period-doubling bifurcations and stabilizes unstable periodic orbits in chaotic attractors. The stability of the system’s Nash equilibrium is significantly improved, and the parameter range for equilibrium pricing is expanded. These findings provide essential theoretical foundations and practical guidance for the design and application of group robot systems.

Consensus of second-order multi-agent systems based on PIDD-like control protocol with time delay

This paper concentrates on investigating the consensus for general second-order linear multiagent systems subjected to time-varying delay. Initially, a novel PIDD-like consensus control protocol with time-varying delay is designed, which includes information on position and its integration, as well as velocity and its differentiation. Subsequently, based upon model transformation, the consensus problem is transformed into the asymptotic stability of the error system. Through constructing of a novel augmented vector Lyapunov-Krasovskii functional, including delay-product terms and multiple integral terms, and then by adopting auxiliary-function-based integral inequalities and the delay-dependent-matrix-based reciprocally convex inequality to address the derivative of the constructed functional, the less conservative consensus conditions for multi-agent systems are obtained. Additionally, numerical simulations are given to demonstrate that the presented consensus conditions not only ensure the consensus of position and velocity states but also exhibit better dynamic performance.

A Simple Model of Turbine Control Under Stochastic Fluctuations of Internal Parameters

This article considers a model of a wind power generation system. It is assumed that the wind torque is transmitted to the generator via a gear. At the same time, the gear itself can have backlash with stochastic parameters. This kind of nonlinearity simulates an inevitable aging and wear of the mechanical parts of wind power generation systems over time. The purpose of the study was to identify a control system that would allow for establishing and maintaining the stability of the desired characteristics. The control system is formalized in the form of a second-order linear system. Numerical experiments demonstrated that the suggested control system is robust to stochastic perturbations resulting from both external and internal factors.

Time-varying formation tracking for multi-agent systems based on prescribed-time control

In this paper, a time-varying formation tracking method based on prescribed-time control is presented for second-order multi-agent systems. The control objective is that the formation system forms and maintains the desired time-varying configuration within an arbitrarily specified time. Addressing the problem that the existing prescribed-time control theorem may lead to divergence of system caused by infinite control input, a novel prescribed-time control theorem based on a novel time-varying scale function is proposed. Different from the existing prescribed-time control results, the proposed theorem introduces convergence time, lead time and prescribed time to avoid the occurrence of the infinite control input. Meanwhile, it can estimate the actual convergence time and has faster convergence rate. Based on proposed theorem, a distributed prescribed-time observer is designed to estimate leader’s states for each follower (suppose that only a few followers can obtain the states of leader) and a prescribed-time formation tracking controller is designed with the estimations of the observer to track leader. Both the observer and controller achieve the prescribed-time stability. Finally, a simulation example is proposed to verify the efficiency of the main results.

Observer-based event-triggered formation tracking control for second-order multi-agent systems in constrained region

Observer-based event-triggered formation tracking control for second-order multi-agent systems in constrained region

Parameterizing history-dependent dynamics of skeletal muscle with a new nonlinear stiffness and damping formulation

This paper introduces a novel mathematical modeling framework for skeletal muscle (SM), capturing its history-dependent characteristics during submaximal contractions and exploring the influence of muscle mechanical parameters and related diseases. The proposed model utilizes a history-dependent nonlinear second-order system, encompassing various contractions: isometric, isotonic, and cyclic. Unlike previous approaches, it incorporates nonlinear stiffness-damping formulation (NSDF) to accurately represent SM’s passive properties, independent of neural excitation. Allowing SM mass deflection for realistic bilateral movement, the model incorporates force–length and force–velocity relationships, yielding accurate SM dynamics predictions with less than 10% normalized force error deviation compared to neural excitation-based model. In addition, it exposes the time-varying natural frequency (TV-NF) of SM using the NSDF and damper coefficients, contributing to our understanding of SM diseases. The model shows a 28.07% rise in initial passive stiffness (IPS) and a 13.18% increase in TV-NF during passive stretching for both healthy and Osgood–Schlatter disease scenarios. During cyclic contraction, IPS increases by 9.19%, TV-NF by 4.49%, and collaborative stiffness by 7.52%. Despite slight disparities in force and muscle length, the healthy muscle generates more power due to the diseased muscle’s higher stiffness limiting length change. This research enhances biomechanical understanding in developing robotics and prostheses by considering SM’s history-dependent dynamics.

A double closed loop digital hydraulic cylinder position system based on switching active disturbance rejection control

A switching active disturbance rejection control (SADRC) strategy was proposed to solve the composite disturbance challenge arising from gap, LuGre friction, hydraulic spring force, and external load disturbance in the double closed-loop digital hydraulic cylinder position control system. Firstly, leveraging the established mathematical model of the double closed-loop digital hydraulic cylinder, the high-order state equation was derived. Subsequently, the high-order double closed-loop digital hydraulic cylinder control system was transformed into a second-order integral series control system using ADRC strategy. Given the pronounced nonlinear characteristics of double closed-loop digital hydraulic cylinders, combined with the characteristics of switching control, a SADRC strategy was proposed, and the stability of the closed-loop system was proved based on Lyapunov theory and switching rules. Finally, the effectiveness of the proposed control method was verified through simulation and experiment. Results indicate that the system's response speed employing the SADRC strategy outperforms the PID control strategy by 32.56%, with a 2.0% reduction in error. The average error in the response speed of the digital hydraulic cylinder and the MATLAB simulation value of the tracking error are 9.0% and 1.50% respectively. Notably, the simulation and experimental results exhibit a consistent overall trend, affirming the feasibility and effectiveness of the control strategy.

Robust multivariable PID controller design for MIMO second order systems: A linear matrix inequality approach

This paper presents new linear-quadratic-regulator-based multivariable PID controller design conditions for MIMO uncertain second-order systems. The gain matrices of the multivariable PID controller are computed by solving an LMI problem which minimizes a cost function consisting of the expectation of the sum of quadratic forms relating the system states and control effort. The linear-quadratic-regulator-based design is much more intuitive to tuning than the eigenvalue assignment within a desired set or region for placement, in which a balance between the state trajectories and control energy is pursued through the tuning parameters. The proposed design methodology was tested on an experimental platform consisting of a mobile inverted pendulum robot, an underactuated and challenging control problem to ensure stability and performance. The obtained results support the theoretical findings.

Fixed-time consensus of second-order multi-agent systems based on event-triggered mechanism under DoS attacks

<p>This paper investigates fixed-time consensus (FXTC) for second-order nonlinear multi-agent systems under denial of service (DoS) attacks using event-triggered control. First, consensus in second-order nonlinear multi-agent systems with directed topologies is studied under a static event-triggered mechanism. Building upon this, dynamic auxiliary variables are introduced, and a dynamic event-triggered mechanism is designed. Consensus control protocols are proposed for both leader-follower and leaderless scenarios. Using Lyapunov stability theory and algebraic graph theory, the fixed-time consensus of multi-agent systems with directed topologies under DoS attacks is analyzed. Furthermore, Zeno behavior is excluded. Finally, numerical examples are presented to validate the theoretical results.</p>

Rendezvous analysis and control design for multiple mobile agents with preserved network connectivity on hyperboloid

This paper studies the rendezvous problem for a second-order multi-agent system on the hyperboloid under a proximity-based communication topology. Using the Riemannian geometrical structure on the hyperboloid, we design a nonlinear feedback control law in which each agent has a limited sensing range. By analyzing the switching behavior of the communication topology, we show that the switching time of the resulting closed-loop system is finite, and the connectivity of the communication topology is preserved if the initial network is connected. In addition, we also provide an analysis of the asymptotic behavior of the second-order closed-loop system on the hyperboloid and prove that the proposed feedback control law enables all agents to converge to the same position.

Dynamic Event-Triggered Secure Cooperative Control of Second-Order Nonlinear Multi-Agent Systems Under DoS Attacks

Dynamic Event-Triggered Secure Cooperative Control of Second-Order Nonlinear Multi-Agent Systems Under DoS Attacks

Luenberger-like interval observer design in the physical framework of uncertain vector second-order dynamic systems

Luenberger-like interval observer design in the physical framework of uncertain vector second-order dynamic systems

Adaptive decentralized control for second-order large-scale nonlinear systems via fully actuated system approach

Adaptive decentralized control for second-order large-scale nonlinear systems via fully actuated system approach

Existence and multiplicity results for non-autonomous second-order Hamiltonian systems

Abstract In this article, using the least action principle and minimax methods in critical point theory, some existence and multiplicity results for periodic solutions of second-order Hamiltonian systems are obtained.

Non-homogeneous BVPs for second-order symmetric Hamiltonian systems

Abstract By making use of Bolle’s method, we show that the following problem has infinitely many solutions: x ¨ + V ′ ( x ) = 0 , x ( 0 ) cos α − x ˙ ( 0 ) sin α = x 0 , x ( 1 ) cos β − x ˙ ( 1 ) sin β = x 1 , \begin{array}{rcl}\ddot{x}+{V}^{^{\prime} }\left(x)& =& 0,\\ x\left(0)\cos \alpha -\dot{x}\left(0)\sin \alpha & =& {x}_{0},\\ x\left(1)\cos \beta -\dot{x}\left(1)\sin \beta & =& {x}_{1},\end{array} where α , β ∈ ( 0 , π ) x 0 , x 1 ∈ R n \alpha ,\beta \in \left(0,\pi ){x}_{0},{x}_{1}\in {{\rm{R}}}^{n} are given and V ∈ C 2 ( R n , R ) V\in {C}^{2}\left({{\rm{R}}}^{n},{\rm{R}}) is even and is super-quadratic at infinity.

Existence and controllability results for second-order non-autonomous neutral evolution system with hemivariational inequalities

The major goal of this research is to examine the approximate controllability results for second-order neutral non-autonomous evolution systems with hemivariational inequalities in Hilbert spaces. Initially, we derive the existence of a mild solution for a given system with the help of cosine functions, fixed point approach, and generalized Clarke's subdifferential. Further, the approximate controllability results were also established using certain necessary conditions. Finally, we show a theoretical application to the validate the main findings.

HE CAUCHY PROBLEM FOR PARABOLIC SYSTEM WITH VARIABLE COEFFICIENTS IN ANISOTROPIC ZYGMUND SPACES

The Cauchy problem for a second-order parabolic system with coefficients and the right hand side which belong to the Zygmund anisotropic space is considered. A smoothness scale of the Cauchy problem solutions in anisotropic Zygmund spaces is obtained. A priori estimates of solutions for uniformly elliptic systems in isotropic Zygmund spaces are derived.

Preview-Based Optimal Control for Trajectory Tracking of Fully-Actuated Marine Vessels

In this paper, the problem of preview optimal control for second-order nonlinear systems for marine vessels is discussed on a fully actuated dynamic model. First, starting from a kinematic and dynamic model of a three-degrees-of-freedom (DOF) marine vessel, we derive a fully actuated second-order dynamic model that involves only the ship’s position and yaw angle. Subsequently, through the higher-order systems methodology, the nonlinear terms in the system were eliminated, transforming the system into a one-order parameterized linear system. Next, we designed an internal model compensator for the reference signal and constructed a new augmented error system based on this compensator. Then, using optimal control theory, we designed the optimal preview controller for the parameterized linear system and the corresponding feedback parameter matrices, which led to the preview controller for the original second-order nonlinear system. Finally, a numerical simulation indicates that the controller designed in this paper is highly effective.

Event-triggered affine formation of second-order multi-agent systems via complex-valued Laplacian.

This paper investigates event-triggered affine formation control of second-order multi-agent systems with directed communication graph. An approach based on complex-valued Laplacian is used as a means of avoiding the use of global information. Two event-triggered strategies are proposed, both of which are capable of achieving global convergence and forming the desired formation without Zeno-behavior, while also optimizing the utilization of resources. Moreover, the dynamic event-triggered strategy conserves more communication resources by comparison. Finally, a numerical simulation example is provided to demonstrate the effectiveness of the obtained conclusions.

Line geometry of pairs of second-order Hamiltonian operators and quasilinear systems

We demonstrate that a pair consisting of a second-order homogeneous Hamiltonian structure in N components and its associated system of conservation laws is in bijective correspondence with an alternating three-form on a N + 2 -dimensional vector space. Additionally, we show that the three-form offers N + 2 linear equations in the Plücker coordinates that define the associated line congruence. We utilize these results to characterize systems of conservation laws with second-order structure for N ≤ 4 . We finally comment on how to extend this result for N = 6 .