Some guidelines for an easier application of Popov's stability test

This paper compares the graphical and root-locus versions of Popov's stability test. The main purpose of the paper is to make possible the application of Popov's Method for a variety of non-linear elements.The various sectors of confinement are analyzed in detail for different non-linearities, and results are summarized in a table. The paper represents by itself a guide, permitting easier practical application of Popov's criterion, under different conditions.     

Some problems of second method of Lyapunov. In discrete systems

The geometric properties of the solution set of Lyapunov equation of linear time-invariant discrete system are discussed. Furthermore, the stabitility of piecewise linear discrete systems is studied and some sufficient conditions are obtained for the asymptotical stability of piecewise linear discrete systems in which each sub-system is stable. The results are applied to second order piecewise linear systems.Supported by the National Natural Science Foundation of China.     

Stability and Oscillations in a Time-Delayed Vehicle System with Driver Control

In this paper, linear stability and chaotic motion of a time-delayednonlinear vehicle system are studied. The stability is determined bycomputing the spectrum associated with a system of linear retardedfunctional differential equations, which reveals that a loss ofstability occurs following a Hopf bifurcation. Beyond the critical valuefor linear stability, the system exhibits limit cycle motions.Subharmonic, quasi-periodic and chaotic motions are observed for asystem excited by a periodic disturbance.     

The asymptotic stability analysis in stochastic logistic model with Poisson growth coefficient

The asymptotic stability of a discrete logistic model with random growth coefficient is studied in this paper. Firstly, the discrete logistic model with random growth coefficient is built and reduced into its deterministic equivalent system by orthogonal polynomial approximation. Then, the linear stability theory and the Jury criterion of nonlinear deterministic discrete systems are applied to the equivalent one. At last, by mathematical analysis, we find that the parameter interval for asymptotic stability of nontrivial equilibrium in stochastic logistic system gets smaller as the random intensity or statistical parameters of random variable is increased and the random parameter's influence on asymptotic stability in stochastic logistic system becomes prominent.     

Estimation of the Robust Stability Bound for a Discrete System

New robust stability bounds are established for a linear large-scale discrete system. The three approaches used in the problem are compared.     

The asymptotic stability of the linear discrete large-scale systems

In this paper,we directly use the linear norm Liapunov function to investigate the stability of the linear discrete large-scale systems and obtain some criteria for the asymptotic stability of such a system.     

The small disturbance problems associated with the algebraic riccati equation of continuous linear time-invariant systems

The stability problem of the disturbed algebraic Riccati equation of continuous linear time-invariant systems is discussed in this paper. Through matrix norm analysis the estimation (expressed in terms of the disturbance range of the system parameters) of the disturbance range in the solution of the disturbed algebraic Riccati equation is established. Apparently this method is quite convenient for the practical computational purposes.     

On the absolute instability in a boundary-layer flow with compliant coatings

The question of absolute instabilities occuring in a boundary-layer flow with compliant coatings is reassessed. Compliant coatings of the Kramer's type are considered. Performing a local, linear absolute/convective stability analysis, a family of spring-backed elastic plates with damping is shown to be absolutely unstable for sufficiently thin plates. The absolute instability arises from the coalescence between an upstream propagating evanescent mode and the Tollmien–Schlichting wave. To reinforce the local, linear stability results the global stability behaviour of the system is investigated, integrating numerically the full nonparallel and nonlinear two-dimensional Navier–Stokes system coupled to the dynamical model. Injecting Gaussian-type, spatially localized flow disturbances as initial conditions, the spatio-temporal evolution of wave packets is computed. The absolute stability behaviour is retrieved in the global system, for a compliant panel of finite length. It is demonstrated numerically that the global stability behaviour of the wall, triggered by finite-end-effects, may be independent of the disturbance propagation in the flow.     

聚变等离子体理想动力系统的非线性热不稳定性

本文讨论氖氖聚变等离子体理想动力系统的非线性热不稳定性问题，这个系统与外界有物质与能量交换。采用几何上零维近似的演化方程。对自治系统，讨论了在各种不同粒子约束定律和能量约束定律下的非线性热不稳定性问题，给出稳定性判据。研究了自治系统不稳定解的非线性演化。     

Stability of a compressible boundary layer relative to a localized disturbance

The problem of stability of an incompressible boundary layer relative to a localized disturbance is considered in a linear approximation. It is shown that the stability analysis reduces to the study of a discrete spectrum of eigenvalues of the corresponding boundary value problem. By means of numerical integration, analysis of the character of the emerging instability is carried out for an unstable mode for the Mach number M = 4.5.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, pp. 107–114, May–June, 1973.The author is grateful to V. V. Struminskii for his interest in this work; he is grateful to L. B. Aizin for useful discussion of the subjects broached, and to A. A. Maslov for the great help in carrying out the numerical calculations.     

Higher order stability effects in a natural convection boundary layer over a vertical heated wall

 The stability of a laminar boundary layer flow under natural convection on a vertical isothermally heated wall is studied analytically. The analysis is performed by using two different two-dimensional linear models: (1) The non-parallel flow model in which the steady mean flow as well as the disturbance amplitude functions can change in the streamwise direction; (2) The parallel flow model in which the effects of the mean flow and disturbance changes in the streamwise direction are neglected. The linear non-parallel stability analysis is based on the so-called parabolised stability equations (PSEs) which have been successfully applied to the stability analysis of forced convection boundary layers. In this study the PSE equations are applied to natural convection boundary layers in order to show the difference between parallel and non-parallel stability analysis. A second part of this study deals with the effects of variable properties, which are always present in natural convection flows. They are analysed by an extended version of the Orr–Sommerfeld equation (EOSE). Received on 31 May 2000     

Bifurcation analysis of a linear Hamiltonian system with two kinds of impulsive control

In this paper, the dynamical behavior of a linear Hamiltonian system under two kinds of impulsive control is discussed by means of both theoretical and numerical ways. The existence and stability of the periodic solution are investigated. Moreover, the conditions of existence for a Neimark?CSacker bifurcation are derived by using a discrete map. Numerical results for phase portraits, periodic solutions, and bifurcation diagrams are in good agreement with the theoretical analysis.     

The two‐grid stabilization of equal‐order finite elements for the stokes equations

This work presents a two‐grid stabilized method of equal‐order finite elements for the Stokes problems. This method only offsets the discrete pressure space by the residual of pressure on two grids to circumvent the discrete Babu?ka–Brezzi condition. The method can be done locally in a two‐grid approach without stabilization parameter by projecting the pressure onto a finite element space based on coarse mesh. Also, it leads to a linear system with minimal additional cost in implement. Optimal error estimates are obtained. Finally, some numerical simulations are presented to show stability and accuracy properties of the method. Copyright © 2010 John Wiley & Sons, Ltd.     

Modeling the Dynamics of Ensemble-Averaged Linear Disturbances in Homogeneous Shear Flow

In order to expand the predictive capability of single-point turbulence closure models to account for the early-stage transition regime, a methodology for the formulation and calibration of model equations for the ensemble-averaged disturbance kinetic energy and energy dissipation rate is presented. The calibration is based on homogeneous shear flow where disturbances can be described by rapid distortion theory (RDT). The relationship between RDT and linear stability theory is exploited in order to obtain a closed set of modeled equations. The linear disturbance equations are solved directly so that the numerical simulation yields a database from which the closure coefficients in the ensemble-averaged disturbance equations can be determined. This revised version was published online in July 2006 with corrections to the Cover Date.     

Disturbance growth in two-fluid channel flow: The role of capillarity

This work examines the role of capillarity in the non-modal linear stability properties of three-dimensional disturbances in sheared two-layer flow of immiscible fluids of similar density. Capillarity reduces the transient growth of energy that occurs due to the non-normality of the linear stability problem according to a scaling of peak energy with We^1/2 over a wide range of Weber number, viscosity ratio and wavenumber. More importantly, the participation of capillary modes in non-modal growth leads to oscillatory energy growth and to larger disturbance growth rates, features that are confirmed by computing the numerical range and numerical abscissa of the non-normal disturbance evolution operator. Examination of energy components and disturbance structure reveals that early rapid growth and subsequent oscillations are due to the coupling of streamwise vortices – the two-fluid analog of lift-up – to the displaced interface.     

Stability of nonlinear comparison equations for discrete large-scale systems

On the stability analysis of large-scale systems by Lyapunov functions, it is necessary to determine the stability of vector comparison equations. For discrete systems, only the stability of linear autonomous comparison equations was studied in the past. In this paper, various criteria of stability for discrete nonlinear autonomous comparison equations are completely established. Among them, a criterion for asymptotic stability is not only sufficient, but also necessary, from which a criterion on the function class C ₁ is derived. Both of them can be used to determine the unexponential stability, even in the large, for discrete nonlinear (autonomous or nonautonomous) systems. All the criteria are of simple algebraic forms and can be readily used.Projects Supported by the National Natural Science Foundation of China, 1880359.     

提高多体系统离散时间传递矩阵法计算精度的研究

深入探讨多体系统离散时间传递矩阵法对平面、空间刚体-光滑铰多体系统运动响应的研究。提出提高该方法计算精度和计算稳定性的方法，导出相应的多端刚体传递矩阵。设计了角坐标的迭代循环，该变量不必采用近似形式，因而提高了计算精度和计算稳定性。其它方法有：1)增加泰勒展开式的高阶项；2)合理选择将速度、加速度表示为位移的线性函数的方法(本文简称线性化方法)；3)合理确定迭代循环的初值等。对4种多体系统进行了计算机仿真研究，表明本文提出的方法有效。     

Linear parameter varying feedforward control synthesis using parameter-dependent Lyapunov function

This paper presents the dynamic feedforward control synthesis for linear parameter varying (LPV) systems. It is assumed that all system matrices are dependent on varying parameters, which are measurable with sensor or observable. The parameters have bounded variation rates. Parameter-dependent Lyapunov function is used for the feedforward control synthesis such that the robust stability is assured for all varying parameters at the time of the operation. The method is formulated in terms of linear matrix inequalities for LPV feedforward controller that guarantees the stability of the transfer matrix having \(L_{2}\) -gain. This compensator is designed by adding on the feedback controller in two degrees of freedom control configuration. This controller can be used for the disturbance attenuation or decreasing the tracking error. The numerical examples and simulations are given to provide the applicability of the proposed solution.     

Development of a family of explicit algorithms for structural dynamics with unconditional stability

A new family of explicit integration algorithms is developed based on discrete control theory for solving the dynamic equations of motion. The proposed algorithms are explicit for both displacement and velocity and require no factorisation of the damping matrix and the stiffness matrix. Therefore, for a system with nonlinear damping and stiffness, the proposed algorithms are more efficient than the common explicit algorithms that provide only explicit displacement. Accuracy and stability properties of the proposed algorithms are analysed theoretically and verified numerically. Certain subfamilies are found to be unconditionally stable for any system state (linear elastic, stiffness softening or stiffness hardening) that may occur in earthquake engineering of a practical structure. With dual explicit expression and excellent stability property, the proposed family of algorithms can potentially solve complicated nonlinear dynamic problems.     

Mixed testing strategies in the quality check problem for stabilization algorithms

The maximin testing problem for the stabilization quality of linear systems with discrete initial and parametric disturbances is considered. A testing procedure is implemented by reducing the original dynamic game to a geometric game. The transition to a mixed expansion of the geometric game is proposed in the absence of saddle points. An optimal mixed disturbance strategy is shown to be applicable.