Necessary Conditions For Stability Research Articles

Input-output pairing criterion applied in the genetic algorithm for unstable linear systems

Abstract In this paper, a new approach towards input-output pairing for an unstable system has been proposed. First, it is demonstrated that the previous method of input-output pairing for unstable plants cannot find appropriate pairs as it only checks necessary conditions for stability and integrity. Then, a new approach using relative error matrix and genetic algorithm for finding appropriate pairs in unstable systems is proposed. As it is shown, this approach takes into consideration both static and dynamic information of plant in measuring interaction. Finally, the accuracy of proposed method is demonstrated by an example and closed loop simulation.

On stability criteria for kinetic magnetohydrodynamics

The existence of a potential energy functional in the zero-Larmor-radius collisionless plasma theory of Kruskal & Oberman (Phys. Fluids, vol. 1, 1958 p. 275), Rosenbluth & Rostoker (Phys. Fluids, vol. 2, 1959, p. 23) allows us to derive easily sufficient conditions for linear stability. However, this kinetic magnetohydrodynamics (KMHD) theory does not have a self-adjointness property, making it difficult to derive necessary conditions. In particular, the standard methods to prove that an instability follows if some trial perturbation makes the incremental potential energy negative, which rely on the self-adjointness of the force operator or on the existence of a complete basis of normal modes, are not applicable to KMHD. This paper investigates KMHD linear stability criteria based on the time evolution of initial-value solutions, without recourse to the classic bounds or comparison theorems of Kruskal–Oberman and Rosenbluth–Rostoker for the KMHD potential energy. The adopted approach does not solve the kinetic equations by integration along characteristics and does not require that the particle orbits be periodic or nearly periodic. Most importantly, the investigation of a necessary condition for stability does not require the self-adjointness of the force operator or the existence of a complete basis of normal modes. It is thereby shown that stability in isothermal ideal-MHD is a sufficient condition for stability in KMHD and that, with a proviso on the long-time behaviour of oscillations about stable equilibria, stability in the double-adiabatic fluid theory, including the variation of the parallel fluid displacement, would be a necessary condition for stability in KMHD.

Stability of chains of oscillators with negative stiffness normal, shear and rotational springs

We consider chains of particles with fixed ends in planar movement such that each particle has three degrees of freedom: two translational and one rotational. The particles are connected by one normal (longitudinal), one shear (transverse) and one rotational spring with a possibility that stiffness of some springs can be negative. We showed that the necessary condition of stability is that such a chain is allowed no more than three negative stiffness springs. The allowable negative stiffness springs can be arranged in two ways: (i) the chain contains one normal, one shear and one rotational negative stiffness springs or; (ii) the chain contains one normal and two rotational negative stiffness springs. The absolute values of the negative stiffnesses should not exceed certain threshold values that depend upon the stiffnesses and the number of the other (positive) springs. The positions of normal and shear negative stiffness springs do not affect the critical values, however the positions of the negative rotational stiffness springs are found to be important. The modal frequencies reduce when one of the negative stiffnesses tends to the critical value; the smallest frequency tends to zero. Damped chains also exhibit similar decrease of damping frequencies, but the lowest frequency tends to zero while the chain is still stable. At this point the damping bifurcates and produces two brunches: one increases with the increase in the value of negative stiffness, the other decreases. No giant damping is observed.

Stability analysis of amplitude death in delay-coupled high-dimensional map networks and their design procedure

The present paper studies amplitude death in high-dimensional maps coupled by time-delay connections. A linear stability analysis provides several sufficient conditions for an amplitude death state to be unstable, i.e., an odd number property and its extended properties. Furthermore, necessary conditions for stability are provided. These conditions, which reduce trial-and-error tasks for design, and the convex direction, which is a popular concept in the field of robust control, allow us to propose a design procedure for system parameters, such as coupling strength, connection delay, and input–output matrices, for a given network topology. These analytical results are confirmed numerically using delayed logistic maps, generalized Henon maps, and piecewise linear maps.

On the Stability of Static Poisson Networks Under Random Access

We investigate the stable packet arrival rate region of a discrete-time slotted random access network, where the sources are distributed as a Poisson point process. Each of the sources in the network has a destination at a given distance and a buffer of infinite capacity. The network is assumed to be random but static, i.e., the sources and the destinations are placed randomly and remain static during all the time slots. We employ tools from queueing theory as well as point process theory to study the stability of this system using the concept of dominance. The problem is an instance of the interacting queues problem, further complicated by the Poisson spatial distribution. We obtain sufficient conditions and necessary conditions for stability. Numerical results show that the gap between the sufficient conditions and the necessary conditions is small when the access probability, the density of transmitters, or the SINR threshold is small. The results also reveal that a slight change of the arrival rate may greatly affect the fraction of unstable queues in the network.

Necessary Conditions for Stabilization of Solutions to the Dirichlet Problem for Divergence Parabolic Equations

We establish necessary conditions for the stabilization to zero of solutions to the Dirichlet problem for a parabolic equation with elliptic operator of divergence form and coefficients depending on x and t.

Wave propagation and strain localization in a fully saturated softening porous medium under the non‐isothermal conditions

The thermo-hydro-mechanical (THM) coupling effects on the dynamic wave propagation and strain localization in a fully saturated softening porous medium are analyzed. The characteristic polynomial corresponding to the governing equations of the THM system is derived, and the stability analysis is conducted to determine the necessary conditions for stability in both non-isothermal and adiabatic cases. The result from the dispersion analysis based on the Abel–Ruffini theorem reveals that the roots of the characteristic polynomial for the THM problem cannot be expressed algebraically. Meanwhile, the dispersion analysis on the adiabatic case leads to a new analytical expression of the internal length scale. Our limit analysis on the phase velocity for the non-isothermal case indicates that the internal length scale for the non-isothermal THM system may vanish at the short wavelength limit. This result leads to the conclusion that the rate-dependence introduced by multiphysical coupling may not regularize the THM governing equations when softening occurs. Numerical experiments are used to verify the results from the stability and dispersion analyses. Copyright © 2016 John Wiley & Sons, Ltd.

On a Stochastic Lotka-Volterra Competitive System with Distributed Delay and General Lévy Jumps

This paper considers a stochastic competitive system with distributed delay and general Lévy jumps. Almost sufficient and necessary conditions for stability in time average and extinction of each population are established under some assumptions. And two facts are revealed: both stability in time average and extinction have closer relationships with the general Lévy jumps, firstly; and secondly, the distributed delay has no effect on the stability in time average and extinction of the stochastic system. Some simulation figures, which are obtained by the split-step θ-method to discretize the stochastic model, are introduced to support the analytical findings.

Stability analysis of discrete-time switched linear systems with unstable subsystems

The problem of stability for discrete-time switched linear systems (DSLSs) with unstable subsystems is investigated. Unlike most existing results that require each switching mode of the system to be asymptotically stable, this paper considers the case that each subsystem may be unstable. First, under certain hypotheses, a necessary condition of stability for DSLSs is obtained. Second, using the average dwell time (ADT) strategy, some sufficient conditions of exponential stability for switched linear systems are derived under two assumptions. Finally, two examples are presented to show the effectiveness of the proposed approaches.

Stability analysis of recurrent type-2 TSK fuzzy systems with nonlinear consequent part

A necessary condition for stability of a class of recurrent type-2 TSK fuzzy systems is presented. In this system, the antecedent part is indeed represented by interval Gaussian type-2 fuzzy set, and the consequent part is an ordinary nonlinear function of the system's inputs. In the proposed method, at first a type-2 fuzzy model is established, and then an LMI-based stability analysis of the model is fully discussed. Two first-order systems (one stable and one unstable) and two second-order systems (one stable and one unstable) are then considered as proper case studies. The simulation results easily approve the effectiveness of the proposed method.

Perfectly matched layers in negative index metamaterials and plasmas

This work deals with the stability of Perfectly Matched Layers (PMLs). The first part is a survey of previous results about the classical PMLs in non-dispersive media (construction and necessary condition of stability). The second part concerns some extensions of these results. We give a new necessary criterion of stability valid for a large class of dispersive models and for more general PMLs than the classical ones. This criterion is applied to two dispersive models: negative index metamaterials and uniaxial anisotropic plasmas. In both cases, classical PMLs are unstable but the criterion allows us to design new stable PMLs. Numerical simulations illustrate our purpose.

Comments on “Robust Stability and Stabilization of Fractional-Order Interval Systems With the Fractional Order <inline-formula> <tex-math notation="TeX">$\alpha$</tex-math></inline-formula>: The <inline-formula> <tex-math notation="TeX">$0<\alpha<1$</tex-math></inline-formula> Case”

This note considers the work entitled “Robust Stability and Stabilization of Fractional-Order Interval Systems with the Fractional Order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">$\alpha$</tex-math></inline-formula> : The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">$0&lt;\alpha&lt;1$</tex-math></inline-formula> Case” by Lu and Chen, where the authors meant to provide sufficient and necessary conditions for stability of fractional-order linear-time-invariant interval systems. As this note proves, conditions are only sufficient, since necessity relies unsuitably on the certain case. An example is provided.

Soliton stability criterion for generalized nonlinear Schrödinger equations.

A stability criterion for solitons of the driven nonlinear Schrödinger equation (NLSE) has been conjectured. The criterion states that p'(v)<0 is a sufficient condition for instability, while p'(v)>0 is a necessary condition for stability; here, v is the soliton velocity and p=P/N, where P and N are the soliton momentum and norm, respectively. To date, the curve p(v) was calculated approximately by a collective coordinate theory, and the criterion was confirmed by simulations. The goal of this paper is to calculate p(v) exactly for several classes and cases of the generalized NLSE: a soliton moving in a real potential, in particular a time-dependent ramp potential, and a time-dependent confining quadratic potential, where the nonlinearity in the NLSE also has a time-dependent coefficient. Moreover, we investigate a logarithmic and a cubic NLSE with a time-independent quadratic potential well. In the latter case, there is a bisoliton solution that consists of two solitons with asymmetric shapes, forming a bound state in which the shapes and the separation distance oscillate. Finally, we consider a cubic NLSE with parametric driving. In all cases, the p(v) curve is calculated either analytically or numerically, and the stability criterion is confirmed.

Моделирование процессов управления, устойчивость и стабилизация систем с программными связями

Results of researches on modeling of dynamics of difficult systems contain ing different physical elements, solving dynamics problems and control of a rigid body and robotics systems are stated. Necessary conditions for stability of de cisions of dynamics equations are defined, and the algorithm of the constraints perturbations equations construction, guaranteeing constraint stabilization is of fered at the numerical solution. Some results of the applied problems solution illustrate efficiency of the described methods

Характеристические показатели периодических решений гамильтоновых систем и необходимые условия устойчивости

Характеристические показатели периодических решений гамильтоновых систем и необходимые условия устойчивости

Second-order explicit difference schemes for the space fractional advection diffusion equation

In this paper, two kinds of explicit second order difference schemes are developed to solve the space fractional advection diffusion equation. The discretizations of fractional derivatives are based on the weighted and shifted Grünwald difference operators developed in [Meerschaert and Tadjeran, J.Comput.Appl.Math. 172 (2004) 65–77; Tian et al., arXiv:1201.5949; Li and Deng, arXiv:1310.7671]. The stability of the presented difference schemes are discussed by means of von Neumann analysis. The analysis shows that the presented numerical schemes are both conditionally stable. The necessary conditions of stability is discussed. Finally, the results of numerical experiments are given to illustrate the performance of the presented numerical methods.

Stability analysis of cascade networks via fluid models

In this work, the fluid approach methodology is first applied to find a sufficient stability condition for a two-station cascade network: customers that are awaiting service at the first queue can move to the second station, whenever it is free, to be served there immediately, but the opposite is not allowed. Each station is fed by a renewal input with general i.i.d. inter-arrival times and general i.i.d. service times (possibly different in the two stations). Then we extend the stability analysis to cascade networks with an arbitrary number N of stations and find a sufficient stability condition. In such networks, an awaiting customer from the queue j≤N−1 jumps to station j+1 if free, to be served there. Finally, we obtain a necessary condition for stability of the N-station cascade network, which matches the sufficient condition when N=2.

Twisted Skyrmion String

We study nonlinear sigma model, especially Skyrme model without twist and Skyrme model with twist: twisted Skyrmion string. Twist term, mkz, is indicated in vortex solution. Necessary condition for stability of vortex solution has consequence that energy of vortex is minimum and scale-free (vortex solution is neutrally stable to changes in scale). We find numerically that the value of vortex minimum energy per unit length for twisted Skyrmion string is 20.37 x 1060 eV/m.

Stability analysis of reset positive systems with discrete-time triggering conditions

This paper discusses the problem of stability analysis of reset positive systems based on discrete-time triggering conditions. A reset control system is positive, if for any nonnegative initial condition and input, its states and outputs are nonnegative whenever the reset actions occur. First, the relationship between reset positive systems (RPSs) with discrete-time triggering conditions and discrete-time switched positive systems (SPSs) is established, by which the stability problem of RPSs is transformed to that of SPSs. Then, some sufficient and necessary conditions of stability and tracking problem for RPSs are given.

A necessary condition for stability of kinematically indeterminate pin-jointed structures with symmetry

Stability conditions are the key to transform kinematically indeterminate structures into prestressed structures or deployable structures. From the viewpoint of symmetry, a necessary condition is presented for the stability of symmetric pin-jointed structures with kinematic indeterminacy. The condition is derived from the positive definiteness of the quadratic form of the tangent stiffness matrix. Numerical examples verify that the proposed necessary stability condition is in accord with the conventional theory of structural rigidity, and is considered to be more comprehensible. It is robust and easy to implement. Results show that a symmetric prestressed structure is guaranteed to possess integral prestress modes, if the necessary condition is satisfied. Further, a pin-jointed structure with fully symmetric mechanism modes is proved to be unstable, if it does not satisfy the condition.