Region Of Asymptotic Stability Research Articles

Stabilization of Steady Rotations of a Spherical Robot on a Vibrating Base Using Feedback

This paper treats the problem of a spherical robot with an axisymmetric pendulum driverolling without slipping on a vibrating plane. The main purpose of the paper isto investigate the stabilization of the upper vertical rotations of the pendulumusing feedback (additional control action). For the chosen type of feedback,regions of asymptotic stability of the upper vertical rotations of the pendulum are constructedand possible bifurcations are analyzed. Special attention is also given to the question ofthe stability of periodic solutions arising as the vertical rotations lose stability.

Global stability of road vehicle motion with driver control

The paper contributes to unveil how drivers—either human or not—may lose control of road vehicles after a disturbance. First, a simple vehicle-and-driver model is considered: Its motion is characterized by the existence of limit cycles whose amplitude depend on vehicle forward velocity (both oversteering and understeering vehicles may exibit this property). Such limit cycles are originated by a Hopf bifurcation occurring at a relatively high vehicle forward velocity. A mathematical proof of the existence of Hopf bifurcations is given. The existence of Hopf bifurcations and saddle limit cycles is confirmed by experimental tests performed by a dynamic driving simulator with a complex vehicle model and human in the loop. By a Zubov method, a Lyapunov function is derived to compute the region of asymptotic stability for the simple vehicle-and-driver model. A necessary and sufficient condition is derived for global asymptotic stability. Such a condition refers to the variation of the kinetic energy which must vanish at the end of the disturbed motion. This occurrence has been detected at the driving simulator too. Just a single stable equilibrium has been found inside the domain of attraction in all of the examined cases.

Large-signal modeling and stable region estimation of DC microgrid with virtual DC machine control

In order to enhance the anti-disturbance capability of a DC microgrid, the virtual DC machine control (VDMC) is usually utilized. However, the existing literature on VDMC mainly focuses on the small-signal modeling based on linearization technique, which cannot be used to describe the system dynamics under large disturbance. Furthermore, the power electronics-based loads have the features of constant power characteristic, which increases the nonlinear of the DC microgrid. In such case, it becomes difficult to obtain the system stability region accurately via the small-signal methods. To settle such problem, the simplified large-signal model of the DC microgrid system with VDMC is established via the TS fuzzy modelling method. Thereupon, an approach to estimate the asymptotic stability region of the system is put forward. The region of asymptotic stability can be used to estimate the maximum undertaking changes of both the DC bus voltage and load power. Finally, the accuracy of the proposed model and the effectiveness of the presented methods are verified by simulations and experiments.

Mixed-potential-function-based large-signal stability analysis of DC microgrid with constant power loads

With the development of power grid technology, a large number of constant power loads (CPL) are introduced into the power grid system. The negative damping characteristic of CPL poses a great challenge to the stability of power grid under the condition of large signal disturbance. This paper focuses on the large signal stability and dynamic response characteristics of DC microgrid with CPL. The mixed potential function (MPF) method is adopted to model the large signal characterstic of load converter controlled by the traditional PI control strategy and the new virtual DC motor (VDM) control strategy. The large signal stability criteria and the asymptotic stability region of the DC microgrid system under the two control strategies are derived. The dynamic response characteristics of the DC microgrid are compared under three large signal operation conditions of the load step, load linear change and the load fault. The results show that VDM control strategy can effectively improve dynamic response characteristics of the DC microgrid system compared with those of PI control, and enhance the stability and anti-interference ability of the DC microgrid system.

Local Convergence of Multi-Agent Systems Toward Rigid Lattices

Geometric pattern formation is an important emergent behavior in many applications involving large-scale multi-agent systems, such as sensor networks deployment and collective transportation. Attraction/ repulsion virtual forces are the most common control approach to achieve such behavior in a distributed and scalable manner. Nevertheless, for most existing solutions only numerical and/or experimental evidence of their convergence is available. Here, we revisit the problem of achieving pattern formation in spaces of any dimension, giving sufficient conditions to prove analytically that under the influence of appropriate virtual forces, a large-scale multi-agent swarming system locally converges towards a stable and robust rigid lattice configuration. Our theoretical results are complemented by exhaustive numerical simulations confirming their effectiveness and estimating the region of asymptotic stability of the rigid lattice configuration.

Large Signal Stability Criteria Combined with a 3D Region of Asymptotic Stability Method for Islanded AC/DC Hybrid Microgrids

Large disturbances frequently happen in isolated AC/DC Hybrid Microgrids. Unfortunately, constant power loads (CPLs) with negative impedance characteristics are equivalent to positive feedback, resulting in an increase in large disturbances. The system can easily become unstable. Consequently, large signal stability criteria are proposed in this paper. Combined with a three-dimensional region of asymptotic stability (3D RAS) method for islanded AC/DC Hybrid Microgrids, important parameters to increase stability margins were determined. Firstly, mixed potential theory was used to derive a large-signal stability criterion. The criteria gave constraints on filtering parameters, CPL power, power of the battery to charge and discharge, AC resistive loads, and DC bus voltage. Then, Lyapunov functions were constructed, and the Lasalle invariance principle was adopted to achieve 3D RAS. When large disturbances emerged, and simultaneously voltage and current varied in 3D RAS, the system always obtained stability and reached new steady-state equilibrium points. Finally, according to comparisons, bigger capacitances of the DC bus capacitor and the AC capacitor, larger battery discharging power and smaller charging power could significantly increase stability margins of islanded Microgrids. Simulations and experimental data have shown that the large signal stability criteria and the 3D RAS work.

AN APPROACH TO THE STABILITY OF THE GAUSE-LOTKA-VOLTERRA COMPETITION ECOLOGICAL MODEL USING THE ROUTH-HURWITZ CRITERION

In the qualitative study of systems of ordinary differential equations, we can analyze the dynamics of the solutions by observing whether small variations or modifications in the initial conditions produce small changes in the future, this intuitive idea was formalized and studied by Lyapunov and reflects the concept of stability. For both linear and nonlinear systems of differential equations, we can analyze stability using criteria to obtain Hurwitz type polynomials. In this paper, we study the stability of a prey-predator model where three species compete considering the Gause effect generated by the Gause-Lotka-Volterra system, using an analytical stability technique to obtain Hurwitz-type polynomials called Routh-Hurwitz criterion. This criterion provides the necessary and/or sufficient conditions to analyze the dynamics of the system by studying the location of the roots of the characteristic polynomial associated with the system. This article describes and presents an analytical method to analyze the stability of the Gause-Lotka-Volterra model, based on the study of Hurwitz type polynomials. This technique allows establishing the asymptotic stability of the system by analyzing the location of the eigenvalues of the matrix associated with said system. The research results allow establishing a region of asymptotic stability of the Gause-Lotka-Volterra model that allows analyzing the interaction dynamics between the species.

Study on the local and global performance of homogeneous platoon with a bidirectional ring interconnection

This paper investigates the local and global performance of the platoon where multiple identical vehicles move in a closed roadway and cooperate based on nearest-neighbour interactions. The information from the front neighbour, namely position, speed, and acceleration, is weighted non-equally component-wise to that from the back neighbour, which is quantified by the asymmetry level on each component. The local performance refers to the locally propagating attenuation for the bidirectional control, which captures whether the disturbance is amplified when propagating from a vehicle to its immediate neighbours. The global performance refers to the system transfer scalability, which characterises the scalability of system transfer matrix. Based on forward and backward wave transfer functions, several analytical necessary conditions of the locally propagating attenuation are obtained. Furthermore, two of these necessary conditions happen to be the same with the conditions for scalably asymptotic stability. Under mild assumptions, the locally propagating attenuation is sufficient for scalably asymptotic stability, the latter of which in turn implies the system transfer scalability. Numerical calculations further show that it is favourable for both performance measures to select the asymmetry levels on speed and acceleration with small magnitudes and located around the centre line of the band-like region of scalably asymptotic stability.

On Ziegler Effect with Arbitrary Large Viscous Friction

We study a mechanical system with two degrees of freedom simulating the motion of rotor blades on an elastic bushing of a medium-sized helicopter. For small values of some problem parameters, the destabilizing effect due to small linear viscous friction forces has been studied earlier. Here we study the problem with arbitrary large friction forces for arbitrary values of the problem parameters. In the plane of parameters, the regions of asymptotic stability and instability are calculated. As a result, necessary and sufficient conditions for the existence of a destabilizing effect under the action of potential, follower forces and arbitrary friction forces have been obtained. It is shown that, if some critical friction coefficient $k_{*}$ tends to infinity, then there exists a Ziegler area with arbitrarily large dissipative forces.

Теорема об области асимптотической устойчивости и ее приложения

The article considers a theorem on the region of asymptotic stability of solutions of ordinary differential equations, which is generalized to the case of a system of equations, where the right-hand side explicitly depends on time. It is shown how it can be used to answer the question of whether the equation under consideration has periodic solutions other than the trivial one, and how, using the constructed Lyapunov functions, one can find these periodic solutions.

Takagi-Sugeno Multimodeling-Based Large Signal Stability Analysis of DC Microgrid Clusters

DC microgrid (DCMG) clusters represent interconnections of multiple DCMGs to enable flexible power flow, and hence advantages of high resilience, economic dispatch, loss minimization, and optimal load response with microgrid-based distributed generations can fully be taken. However, small-scale DCMGs are known to be weak in nature due to low inertia and high grid impedance. Meanwhile, dynamic analyses of DCMG clusters have been plagued by their high order dynamic nonlinear system models since numerous state variables are involved. This article proposes large signal stability analysis of DCMG clusters based on Takagi-Sugeno multimodeling approach. With the proposed method, the large signal Lyapunov stability of the DCMG cluster is reduced to the computation of a series of linear matrix inequalities, which will significantly simplify the analysis. The influences of circuit parameters, power flows, and topological change on large signal stability of the DCMG cluster are revealed, and asymptotic stability regions of the network are estimated as well. In addition to simulation verification, a laboratory prototype of a ring topology DCMG cluster with three 48 V DCMGs interconnected is also built to validate the analysis by experimental results.

Chaotic Particle Swarm Optimisation for Enlarging the Domain of Attraction of Polynomial Nonlinear Systems

A novel technique for estimating the asymptotic stability region of nonlinear autonomous polynomial systems is established. The key idea consists of examining the optimal Lyapunov function (LF) level set that is fully included in a region satisfying the negative definiteness of its time derivative. The minor bound of the biggest achievable region, denoted as Largest Estimation Domain of Attraction (LEDA), can be calculated through a Generalised Eigenvalue Problem (GEVP) as a quasi-convex Linear Inequality Matrix (LMI) optimising approach. An iterative procedure is developed to attain the optimal volume or attraction region. Furthermore, a Chaotic Particular Swarm Optimisation (CPSO) efficient technique is suggested to compute the LF coefficients. The implementation of the established scheme was performed using the Matlab software environment. The synthesised methodology is evaluated throughout several benchmark examples and assessed with other results of peer technique in the literature.

Implicit-explicit-compact methods for advection diffusion reaction equations

We provide a comparison of the dispersion properties, specifically the time-amplification factor, the scaled group velocity, the error in the phase speed and scaled numerical diffusion coefficient of four spatiotemporal discretization schemes utilized for solving a linear, one-dimensional (1D) as well as a linear/nonlinear, two-dimensional (2D) advection diffusion reaction (ADR) equation: (a) Explicit (RK2) temporal integration combined with the Optimal Upwind Compact Scheme (or OUCS3, J. Comp. Phys., 192, pg. 677–694 (2003)) and the central difference scheme (CD2), (b) Fully implicit mid-point rule for time integration coupled with the OUCS3 and the Lele’s compact scheme (J. Comp. Phys., 103, pg. 16–42 (1992)), (c) Implicit (mid-point rule)-explicit (RK2) or IMEX time integration blended with OUCS3 and Lele’s compact scheme (where the IMEX time integration follows the same ideology as introduced by Ascher et al., SIAM J. Numer. Anal., 32(3), pg. 797–823 (1995)), and (d) IMEX (mid-point/RK2) time integration melded with the New Combined Compact Difference scheme (or NCCD scheme, J. Comp. Phys., 228, pg. 6150–6168 (2009)). Analysis reveals the superior resolution features of the IMEX-OUCS3-Lele scheme and the IMEX-NCCD scheme including an enhanced region of asymptotic stability (a region with numerical amplification factor less than unity), a diminished region of spurious propagation characteristics (or a region of negative scaled group velocity) and a smaller region of nonzero phase speed error. In particular, the IMEX-NCCD scheme captures the correct propagation feature (or positive scaled group velocity) in the largest possible region in the wavenumber-Courant-Friedrichs-Lewy (CFL) number, parameter space, in comparison with the other three numerical methods. The in silico experiments investigating the role of q−waves in the numerical solution of the linear, 1D ADR equation divulge excellent Dispersion Relation Preservation (DRP) properties of the IMEX-NCCD scheme including minimal dissipation via implicit filtering and negligible unphysical oscillations (or Gibbs’ phenomenon) on coarser grids. The numerical solution of the 2D viscous Burgers’ Equation underline the supremacy of the IMEX method with regard to handling the ‘stiff’ derivative terms in contrast with a fully explicit time integration method and lower computational time versus with a fully implicit scheme. The DRP resolution of the IMEX-NCCD scheme is further benchmarked by solving the classical two-dimensional (2D), Patlak-Keller-Segel (PKS) nonlinear parabolic model. Numerical results reveal that the spiky structure of the solution is oscillation free and, when compared with the Explicit-OUCS3-CD2 method, the solution is better resolved by the IMEX-NCCD method.

Time-delayed local feedback control for a chaotic finance system

In this paper, we consider the local control problem for a chaotic finance system via the time-delayed feedback. Using the Lyapunov–Krasovskii stability theorem, the quadratic system theory, some integral inequalities, and rigorous mathematical analysis, we obtain a local stabilization condition by means of linear matrix inequalities. Then we discuss the estimate of the region of asymptotic stability and give the corresponding optimization problem. Also, we address the local control problem under the nondelayed feedback. Finally, we present numerical simulations to show the effectiveness of the proposed results.

Dynamical Control for the Parametric Uncertain Cancer Systems

In this study, we consider a parametric uncertain Lotka-Volterra cancer model including three interacting cell populations of tumor cells, healthy host cells and immune effector cells. The biological parameter (i.e., cell growth rate) is described as a form of the triangular fuzzy number. By using grade mean value conversion, the imprecise fuzzy parameter is translated into the degree of optimism (λ-integral value λ ∈ [0,1]) interval. We derive the sufficient conditions for the existence of the region of asymptotic stability (RAS) in the fuzzy cancer model. The boundary crisis of transient chaos and properties of RAS are investigated under fuzzy environment. We present a dynamical perturbation control to avoid uncontrolled tumor cell growth and prevent healthy cell extinction.

Estimating the boundary of the asymptotic stability region of Lotka–Volterra system by using the trajectory reversing method

This paper proposes a topological approach for plotting the boundary of the region of asymptotic stability (RAS) of Lotka–Volterra predator-prey system. First, stability analysis was used to determine the specific saddle point that has eigenvalues with one positive and two negative real parts in a linearized Jacobian matrix. A set of initial states located around the saddle point on the specific eigenplane spanned by the two stable eigenvectors was then selected. Finally, the trajectory reversing method was used and the trajectories that had initial states on the eigenplane delineated the boundary of the asymptotic stability region. The trajectories of the initial states that started from the opposite sides of the RAS exhibited different dynamic behaviour. The numerical simulation are presented to demonstrate the effectiveness of the proposed approach.

Compensated θ-Milstein methods for stochastic differential equations with Poisson jumps

In this paper, we are concerned with numerical methods for solving stochastic differential equations with Poisson-driven jumps. We construct a class of compensated θ-Milstein methods and study their mean-square convergence and asymptotic mean-square stability. Sufficient and necessary conditions for the asymptotic mean-square stability of the compensated θ-Milstein methods when applied to a scalar linear test equation are derived. We compare the asymptotic mean-square stability region of the linear test equation with that of the compensated θ-Milstein methods with different θ values. Numerical results are given to verify our theoretical results.

The Control Parameter Determination Method for Bidirectional DC-DC Power Converters Interfaced Storage Systems Based on Large Signal Stability Analysis

In DC microgrid (DC-MG), the loads connected with converters under strict control areconsidered as CPLs (constant power loads). When the voltage of CPLs decreases, the currentincreases and the negative impedance characteristic of CPLs cause instability easily. Fortunately,appropriate control for energy storage units could improve the system stability. However, mosttraditional control methods for bidirectional DC-DC power converters (BDC) connected withbattery storage units do not quantitatively consider the stability influences of control parameters.This paper quantitatively analyzes the stability influence of the BDC current-mode controlparameters and the negative impact of CPLs and derives the control parameter determinationmethod for BDC interfaced storage systems. Large signal stability constraints are obtained in termsof mixed potential function. According to the constraints, the large signal stability is improved whenthe BDC control parameter kp increases, while the stability is degraded when the power of CPLsincreases. The control parameter determination method is very effective and convenient to apply,and the appropriate parameter kp for BDC is determined. The regions of asymptotic stability (RAS)identify that the proposed control parameter determination method could improve the systemstability effectively. The determination method is fully verified by the simulation and experimental results.

Effect of Asymmetric Control Constraints on Fixed-Wing UAV Trajectories

Reference trajectories, for the longitudinal states of a fixed-wing unmanned aerial vehicle subjected to asymmetric control constraints, are distinguished using the characteristics of regions of asymptotic stability calculated as ellipsoidal domains by solving linear matrix inequalities. These domains are used to determine the maximum allowable deviations from the reference, for which stable closed-loop tracking can be ensured. Constraints are imposed on a linear state feedback controller with the saturation modeled as a sector nonlinearity.

Study on split-step Rosenbrock type method for stiff stochastic differential systems

ABSTRACT In this paper, we present a split-step Rosenbrock type method for stiff stochastic differential systems. The method is proved to be mean-square (MS) convergent with strong order . For one- and two-dimensional Itô test equations with multiplicative noise, we analysis asymptotically MS stability and plot asymptotic MS stability regions. Numerical examples and simulations are given to illustrate the effectiveness of theoretical results.