Equilibrium Point Of System Research Articles

Phase-space diffusion of charged particles induced by random fluctuations of relative gyrophases in the presence of finite amplitude circularly polarized electromagnetic waves

A stochastic model to describe the phase-space diffusion of charged particles induced by random fluctuations of relative gyrophases in the presence of parallel propagating, circularly polarized electromagnetic waves is discussed. The perturbation analysis around the equilibrium points of the noiseless system shows the coexistence of classical diffusion and trapping oscillation. Even if the equation of motions for pitch angle does not include noise terms, the pitch angle diffusion occurs due to the noise term in the equation of the relative gyrophase and the existence of the finite amplitude wave. The resultant theory is validated by using numerical results of test particle simulations; when distributions of the relative gyrophases and pitch angle cosines are close to Gaussian, the classical diffusion and trapping oscillation are observed. With increasing wave amplitude and/or the strength of noise, the pitch angle diffusion becomes subdiffusive.

Dynamical Analysis of a Fractional-Order Hantavirus Infection Model

Abstract This paper considers a Hantavirus infection model consisting of a system of fractional-order ordinary differential equations with logistic growth. The fractional-order model describes the spread of Hantavirus infection in a system consisting of a population of susceptible and infected mice. The existence, uniqueness, non-negativity and boundedness of the solutions are established. In addition, the local and global asymptotic stability of the equilibrium points of the fractional order system and the basic reproduction number are studied. The impact of basic reproduction number and carrying capacity on the stability of the fractional order system are also theoretically and numerically investigated.

Analysis of a dynamics duopoly game with two content providers

Abstract This paper explores the idea of Bertrand duopoly in a context of bounded rationality, where two content providers (CP) endeavor to maximize their own profit. Each CP varies his price and fixes the level of his credibility of content. We suppose that one CP adjusts his price strategy according to a kind of smoothed information which average his marginal profit in the previous period and the one in a delayed period, while another CP makes its price strategy as the best response, which is a function the opponent action in the current period. Through a detailed analysis, we calculate explicitly the equilibrium points of the dynamical system, and we show the stability of its Nash equilibrium under some conditions. Numerical results reveal that while varying the model parameters (speed of adjustment and side payment), complex dynamic behaviors would occur, such as chaos. In addition, an appropriate method of chaos controlling is applied to force the system to go back to its stabilization behavior.

Winner-take-all competition with heterogeneous dynamic agents

Winner-take-all competition exists widely in nature and has been applied in many engineering fields. This paper mainly investigates a group of heterogeneous dynamic agents, which produce the winner-take-all competition. For the heterogeneous system consisting of first- and second-order dynamic agents, we propose two different kinds of protocols with and without velocity measurements, respectively. Firstly, we employ the Lasalle’s invariant principle to solve the equilibrium points of the proposed system. Secondly, we prove that the proposed protocols can solve the winner-take-all problems for the heterogeneous systems. The results reveal that winner is independent from the dynamics of agents, but is determined by inputs. Finally, some examples are also gave to verify the validity of the proposed protocols.

Conformable Fractional Order Lotka–Volterra Predator–Prey Model: Discretization, Stability and Bifurcation

Abstract The purpose of this study is to discuss dynamic behaviors of conformable fractional-order Lotka–Volterra predator–prey system. First of all, the piecewise constant approximation is used to obtain the discretize version of the model then, we investigate stability, existence, and direction of Neimark–Sacker bifurcation of the positive equilibrium point of the discrete system. It is observed that the discrete system shows much richer dynamic behaviors than its fractional-order form such as Neimark–Sacker bifurcation and chaos. Finally, numerical simulations are used to demonstrate the accuracy of analytical results.

Global dynamic behavior of a predator–prey model under ratio-dependent state impulsive control

This paper studies the global dynamic behavior of a prey–predator model with square root functional response under ratio-dependent state impulsive control strategy. It is shown that the boundary equilibrium point of the controlled system is globally asymptotically stable. An order-k periodic orbit is obtained by employing the Brouwer’s fixed point theorem. Furthermore, the critical values are determined for the existence of orbitally asymptotically stable order-1 and order-2 periodic orbits in finite time. These critical values play an important role in determining different kinds of order-k periodic orbits and can also be used for designing the control parameters to obtain the desirable dynamic behavior of the controlled prey–predator system. Moreover, it is found that the local equilibrium point is also globally asymptotically stable under the control strategy. Numerical examples are provided to validate the effectiveness and feasibility of the theoretical results.

A discrete mutualism model: Analysis and exploration of a financial application

We perform a stability analysis on a discrete analogue of a known, continuous model of mutualism. We illustrate how the introduction of delays affects the asymptotic stability of the system's positive nontrivial equilibrium point. In the second part of the paper we explore the insights that the model can provide when it is used in relation to interacting financial markets. We also note the limitations of such an approach.

Switched Systems With Multiple Equilibria Under Disturbances: Boundedness and Practical Stability

This paper addresses robustness to external disturbances of switched discrete and continuous systems with multiple equilibria. First, we prove that if each subsystem of the switched system is input-to-state stable (ISS), then under switching signals that satisfy an average dwell-time bound, the solutions are ultimately bounded within a compact set. The size of this set varies monotonically with the supremum norm of the disturbance signal. These results generalize existing ones in the common equilibrium case to accommodate multiple equilibria. Then, we relax the (global) ISS conditions to consider equilibria that are locally exponentially stable (LES), and we establish practical stability for such switched systems under disturbances. Our motivation for studying this class of switched systems arises from certain motion planning problems in robotics, where primitive movements, each corresponding to an equilibrium point of a dynamical system, must be composed to obtain more complex motions. As a concrete example, we consider the problem of realizing safe adaptive locomotion of a three-dimensional biped under persistent external force by switching among motion primitives characterized by LES limit cycles. The results of this paper, however, are relevant to a much broader class of applications, in which composition of different modes of behavior is required.

Research on Nonlinear Control Method of Underactuated Gantry Crane Based on Machine Vision Positioning

The movement of the gantry crane is controlled by an symmetry underactuated system, and has poor robustness in precise positioning. A new active control method based on the machine vision positioning is proposed in this paper, and the trajectories are planned after the detection of starting and ending points. A new type of energy storage function is given in this paper, and a coupling control law is derived to minimize the load swing in the process of precise positioning. The equilibrium point of the closed-loop system is checked though Lyapunov and LaSalle’s theorems, and the calculation results are verified through experimental investigations. The results show that the equilibrium points are asymptotically stable, and the proposed control method is of better robustness. The findings provide a new kind of control method with higher efficiency, and can help with the precise control of gantry cranes.

Nonlinear Self-Excited Vibration Mechanism for Under-Actuated Oil-Well Drilling System with Stick-Slip Vibration

Exploration of oil or gas wells is usually carried out by using drilling system. When the kilometers of drilling system is externally disturbed, the drilling system is prone to undesired stick-slip vibration of the drill-string. The stick-slip vibration of the drill-string belongs to the nonlinear self-excited vibration. In order to study the vibration mechanism, the model for the nonlinear torque on the bit by using Karnopp friction model and the two degrees of freedom model for drilling system were established. The equivalent damping torque formula and the energy variation formula of the drilling system were proposed. The mechanism of the nonlinear self-excited vibration of the drilling system was explained. The simulation results indicated that the stick-slip vibration of the drill-string was caused by the negative value of the equivalent damping force of the drilling system. The equivalent damping torque made the positive work on the system, so that the system absorbed the vibration energy from the outside, which destroyed the stability of the system equilibrium point. The feedback effect in the speed of the drill-bit regulates the system energy to maintain constant vibration without attenuation. In each vibration cycle, the energy input by the system is equal to the energy dissipated by the system. The energy absorbed by the drilling system is mainly converted into the potential energy.

Deep-Learning Tracking for Autonomous Flying Systems Under Adversarial Inputs

We propose a game-theory-based deep-learning tracking control scheme to enable a holonomic flying system to perform an autonomous trajectory tracking task, when considering saturating actuators, adversarial inputs, and nonquadratic cost functionals. The problem is formulated as a two-player zero-sum game, whose online solution is computed by learning the saddle point strategies in real time. Three approximators, namely a critic and two actors, are tuned online using data generated in real time along the system trajectories. The adaptive control character of the algorithm requires a persistence of excitation condition to be a priori validated, which is relaxed by using. a deep-learning approach, based on experience replay with multiple layers. A robustifying control term is added to eliminate the effect of residual errors, leading to asymptotic stability of the equilibrium point of the closed-loop system. A simulation of a target tracking application, where the measurements available to the aerial system are perturbed by persistent adversaries, is performed to validate the effectiveness of the proposed approach.

Transportation Control of Double-Pendulum Cranes With a Nonlinear Quasi-PID Scheme: Design and Experiments

In real-world applications, industrial cranes commonly suffer from effects caused by the so-called double-pendulum phenomenon in many situations. However, at present, the double-pendulum phenomenon is usually directly roughly neglected when designing control methods. For double-pendulum cranes, most currently available approaches are open loop control; the existing feedback methods are mostly developed based on linearized dynamic models (around the equilibrium point) or designed without adding integral terms in the control laws, which may cause positioning errors in the presence of unmodeled dynamics. To address these problems, this paper proposes a new quasi-proportional integral derivative control method to effectively control underactuated double-pendulum crane systems. Then, we provide rigorous theoretical analysis for the equilibrium point of the closed-loop system based on the original nonlinear dynamic equations. To our knowledge, this paper gives the first plant-parameter-free controller that incorporates both integral action and actuating constraints without any linearizing operations during controller design or closed-loop analysis, which theoretically ensures that the controller can work well in the presence of unmodeled dynamics (e.g., insufficient friction compensation), actuating constraints, and large swing angles (i.e., not satisfying linearization conditions). Finally, hardware experimental results are provided to examine the effectiveness of the suggested control method.

Synchronized Vibration Transition of Three Exciters in Non-resonant Vibration System

The synchronized vibration transition has been proposed in the non-resonant vibration system with three exciters. Based on former man, the movement equations of self-synchronous vibration system with three rotors are replaced by differential equation of phase difference angle first, and the necessary conditions of synchronous movement for system are analyzed, stability and bifurcation of equilibrium points of vibration system are discussed. Firstly, dynamics model are established, differential equation of phase difference angle is deduced based on the dynamics equation of the vibration system. Then, the necessary conditions of synchronous movement are established, stability and bifurcation of equilibrium points of vibration system are discussed using Lyapunov theories. Finally, the effects of system parameters on synchronization stability about self-synchronous vibration system are investigated with numerical simulations.

An Increased Nonlinear Coupling Motion Controller for Underactuated Multi-TORA Systems: Theoretical Design and Hardware Experimentation

In mechanical engineering, multi-translational oscillator with rotational actuator (multi-TORA) systems have been introduced to study the self-synchronized phenomenon as well as to investigate the vibration damping problem associated with many vibrational mechatronic systems, e.g., hand-held drills. Due to the lack of available actuators, multi-TORA systems are typically underactuated. Multi-TORA systems consist of a series of single TORA subsystems connected to and coupled with each other by elastic springs. For practical multi-TORA systems, the plant parameters are usually unknown or difficult to measure. Moreover, they exhibit strong nonlinear coupling behaviors. These factors bring much difficulty for both controller design and analysis. The control problem for underactuated multi-TORA systems with parametric uncertainties is challenging and still open. To address the above issues, this paper proposes a nonlinear increased motion control scheme for multi-TORA systems with parametric uncertainties. Specifically, a novel energy function is constructed and some extra coupling terms are introduced into the proposed controller for improving the transient performance. Then, based on Lyapunov techniques, we rigorously prove the asymptotic stability for the equilibrium point of the closed-loop system. As far as we know, this paper gives the first smooth control law to yield global asymptotic control results for underactuated multi-TORA systems suffering from unknown/uncertain plant parameters. To verify the effectiveness of the proposed controller, a series of hardware experiments are carried out on a self-built double-TORA hardware testbed, which indicate that the controller achieves effective control results in various conditions.

Nonlinear viscous cosmological models in f(R,T) gravity

This article presents nonlinear viscous cosmological model of the universe that arises in the modified gravity theory in which the Lagrangian of the gravitational action contains a general function [Formula: see text], where [Formula: see text] and [Formula: see text] denote the curvature scalar and the trace of the energy–momentum tensor, respectively, in the flat Friedmann-Robertson-Walker background metric with isotropic fluid. We obtain cosmological solutions in [Formula: see text] theory of gravity, specially for the choice [Formula: see text] with nonlinear viscosity described by nonlinear Israel Stewart theory. The physical and geometrical properties of the models in [Formula: see text] gravity, where [Formula: see text] and [Formula: see text] are coupling parameters, with nonlinear Israel Stewart theory are studied in detail. The analysis of the variation of bulk viscous pressure, energy density, scale factor, Hubble parameter and deceleration parameter with cosmic evolution is done in the nonlinear Israel Stewart theory with [Formula: see text] gravity. The stability analysis of the equilibrium points of the dynamical system associated with the exponential evolution of the universe in [Formula: see text] gravity theory with nonlinear dissipative effects is also studied.

Neural-network-based output-feedback control with stochastic communication protocols

This paper is concerned with the neural-network-based (NN-based) output-feedback control issue for a class of nonlinear systems. For the purpose of effectively mitigating the phenomena of data congestion/collision, the stochastic communication protocols are favorably utilized to orchestrate the data transmissions, and the resultant closed-loop plant is represented by a so-called protocol-induced Markovian jump system with uncertain transition probability matrices. Taking such an uncertainty probability into account, a novel iterative adaptive dynamic programming (ADP) algorithm is developed to obtain the desired suboptimal solution with the help of auxiliary quasi-HJB equation, and the algorithm convergence is also investigated via the intensive use of the mathematical analysis. In this ADP framework, an NN-based observer with a novel adaptive tuning law is first adopted to reconstruct the system states. Then, based on the reconfigurable system, an actor–critic NN scheme with online learning is developed to realize the considered control strategy. Furthermore, in light of the Lyapunov theory, some sufficient conditions are derived to guarantee the stability of the zero equilibrium point of the closed-loop system as well as the boundedness of the estimation errors for critic and actor NN weights. Finally, a simulation example is employed to demonstrate the effectiveness of the developed suboptimal control scheme.

Derivative Feedback Control for a Class of Uncertain Linear Systems Subject to Actuator Saturation

This manuscript considers a class of linear systems with time-invariant uncertainties, where only the derivative of the state vector is considered for feedback. In this scenario, the proposed strategy uses auxiliary dynamics, whose state is accessible for feedback, to control the original plant. It is proposed a design procedure by means of linear matrix inequalities, adding an auxiliary dynamics and subject to actuator saturation. If the conditions are feasible, they assure that the equilibrium point of the closed-loop system is locally asymptotically stable, for all initial conditions in an ellipsoidal region, which is within a given region defined for the plant and the new dynamics. Although the proposed design includes an auxiliary dynamics, it ensures the stability and decay rate proprieties for the original plant. Simulations examples illustrate the effectiveness of the proposed approach.

A Moving Target Defense Control Framework for Cyber-Physical Systems

This paper considers the problem of efficiently and securely controlling cyber-physical systems that are operating in uncertain, and adversarial environments. To mitigate sensor, actuator attacks, and performance loss due to such attacks, we formulate a secure control algorithm that consists of a proactive and a reactive defense mechanism. The proactive mechanism, which is based on the principles of moving target defense, utilizes a stochastic switching structure to dynamically and continuously alter the parameters of the system, while hindering the attacker's ability to conduct successful reconnaissance to the system. The unpredictability of the current actuator and sensor is optimized using an information entropy measure, which is induced by probabilistic switching. The reactive mechanism on the other side, detects potentially attacked components, namely sensors and actuators, by leveraging online data to compute an integral Bellman error. A rigorous mathematical framework is presented to guarantee the stability of the equilibrium point of the closed-loop system, and provide a quantified bound on the performance loss when utilizing both reactive and proactive mechanisms. Simulation results show the efficacy of the proposed approaches on a benchmark aircraft model.

Dynamic Feedback Antiswing Control of Shipboard Cranes Without Velocity Measurement: Theory and Hardware Experiments

As a class of typical representatives for conveyance, shipboard crane systems are usually fixed on ship decks to transport cargoes on the sea, which is greatly different from land-fixed cranes. Due to some disturbances induced by sea waves, payload positions are always difficult to control precisely. In addition, unless equipped with velocity sensors, it would be difficult to obtain velocity signals for feedback control, by noting that the traditional way of numerical differentiation operations (to recover velocities from positions/angles) may induce extra noises in practice. In this paper, we propose an observer-based dynamic feedback control method to deal with the foregoing issues. Specifically, both boom/rope positioning and payload swing elimination can be achieved simultaneously by utilizing only measurable displacement/angle feedback signals. Moreover, as far as we know, this paper gives the first input-saturated control method including nonlinear coupling terms to realize the objective of effective positioning and swing suppression without the requirement of velocity feedback, which is developed on the basis of the complicated nonlinear shipboard crane dynamics with no linearization operations during controller design or stability analysis. Meanwhile, the corresponding velocity signals are accurately recovered online by means of the suggested observer. In addition, the amplitudes of the control inputs can be guaranteed within the allowable ranges to avoid falling into saturation. The asymptotic stability for the equilibrium point of the crane system in closed loop with the controller and the observer is proven by rigorous analysis with Lyapunov techniques and LaSalle's invariance theorem. After a series of hardware experiments, we validate the effectiveness and robustness of the presented controller.

Extreme multistability in a new hyperchaotic meminductive circuit and its circuit implementation

In this paper, a new hyperchaotic meminductive circuit was designed. Based on the normalized mathematical model, the origin point as the unique equilibrium point of system was calculated. Especially, using the two-dimension complexity algorithms, the dynamical behaviors of system were analyzed. In the analysis, some peculiar physical phenomenon for instance coexisting evolution and infinitely many coexisting attractors were found. Finally, the practical circuit was physically implemented and the parameters of all the components are displayed. The results of numerical simulation show that the new circuit system is extremely multistable. It provides a theoretical guidance for the research of the chaotic related field.