Necessary and sufficient criteria for absolute stability of the direct control system

ＮＥＣＥＳＳＡＲＹＡＮＤＳＵＦＦＩＣＩＥＮＴＣＲＩＴＥＲＩＡＦＯＲＡＢＳＯＬＵＴＥＳＴＡＢＩＬＩＴＹＯＦＴＨＥＤＩＲＥＣＴＣＯＮＴＲＯＬＳＹＳＴＥＭＺｈａｎｇＪｉ－ｙｅ（张继业）;ＳｈｕＺｈｏｎｇ－ｚｈｏｕ（舒仲周）（ＤｅｐａｒｔｍｅｎｔｏｆＥｎｇｉ...     

On the criterion for the absolute stability of the control system

In this paper, firstly we give the criterion for the absolute stability of the second canonical form for the control system, including the equation of the longitudinal motions of a plane as a particular example. The corresponding result in [8], [9] is a particular example given in this paper. Secondly, we give the criteria for the absolute stability of the first canonical form in the usual case and in the critical case. Finally, we give some criteria for the absolute stability of the general form for the direct control system.All the results in this paper merely depend upon the relations between the parameters of the system itself to give an explicit algebraic discriminant.     

The criterions for absolute stability of direct control systems of lurie type

In this paper, we give necessary and sufficient conditions for absolute stability of several classes of direct control systems, and discuss the absolute stability of the first canonical form of control system. The corresponding results in references [3,5,6] and [7] are improved.This paper is completed under the direct guidance and encouragement of professor Liao Xiaoxin and professor Deng Zhong-qi. The writer especially acknowledges their help.     

On the Proper Form of the Amplitude Modulation Equations for Resonant Systems

The complex amplitude modulation equations of a discrete dynamicalsystem are derived under general conditions of simultaneous internal andexternal resonances. Alternative forms of the real amplitude and phaseequations are critically discussed. First, the most popular polar formis considered. Its properties, known in literature for a multitude ofspecific problems, are here proven for the general case. Moreover, thedrawbacks encountered in the stability analysis of incomplete motions(i.e. motions containing some zero amplitudes) are discussed as aconsequence of the fact the equations are not in standard normal form.Second, a so-called Cartesian rotating form is introduced, which makesit possible to evaluate periodic solutions and analyze their stability,even if they are incomplete. Although the rotating form calls for theenlargement of the space and is not amenable to analysis of transientmotions, it systematically justifies the change of variables sometimesused in literature to avoid the problems of the polar form. Third, amixed polar-Cartesian form is presented. Starting from the hypothesisthat there exists a suitable number of amplitudes which do not vanish inany motion, it is proved that the mixed form leads to standard formequations with the same dimension as the polar form. However, if suchprincipal amplitudes do not exist, more than one standard form equationshould be sought. Finally, some illustrative examples of the theory arepresented.     

Hopf Bifurcation of the Generalized Lorenz Canonical Form

Hopf bifurcation of a unified chaotic system – the generalized Lorenz canonical form (GLCF) – is investigated. Based on rigorous mathematical analysis and symbolic computations, some conditions for stability and direction of the periodic obits from the Hopf bifurcation are derived.     

Several criterions of absolute stability for the second canonical form of control system

In this paper several explicit criterions of absolute stability for the real second canonical form of control system are giyen, which generalize and include the known result in ref [2]. By appiyung then criterions to the well known equations of the longitudinal motion of aireraft aresult that generalizes and includes the known results in refs. [3,4,2] is obtained.     

Explicit criteria of absolute stability for the real second canonical form of control system

In this paper,some explicit criteria of absolute stability for the trivial solution of the real second canonical form of nonlinear control system are given,which include and improve the criteria in paper[1].By applying these criteria to the well-known equation of the longitudinal motion of aircraft,some results are obtained,which include and improve the corresponding results in papers[1,2,3,4].     

Delay-range-dependent stability criteria for delayed discrete-time Lur’e system with sector-bounded nonlinearities

This paper is devoted to the absolute and robust stability for uncertain discrete-time Lur’e systems with interval time-varying delays and sector-bounded nonlinearities. Both the cases with time-invariant and time-varying nonlinearities are considered. By dividing the variation interval of the time delays into some subintervals, some new delay-range-dependent robust stability criteria are derived in the form of linear matrix inequalities (LMIs) via a modified Lyapunov-Krasovskii functional (LKF) approach. The criteria are less conservative than some existing results. Finally, some numerical examples are presented to show the effectiveness of the proposed approach.     

van der Pol型时滞系统的两参数余维一Hopf分岔及其稳定性

研究具有三次非线性时滞项的ｖａｎ ｄｅｒ Ｐｏｌ型时滞系统随两参数（时滞量和增益系数）余维一Ｈｏｐｆ分岔,说明了线性化特性方程随两参数变化时的根的分布和Ｈｏｐｆ分岔存在性;通过构造中心流形并且使用范式方法确定出Ｈｏｐｆ分岔的方向以及周期解的稳定性;分析了时滞量对所论系统发生Ｈｏｐｆ分岔的影响。     

Nonparallel stability of the flow in an axially rotating pipe

The linear stability of the developing flow in an axially rotating pipe is analyzed using parabolized stability equations (PSE). The results are compared with those obtained from a near-parallel stability approximation that only takes into account the axial variation of the basic flow. Though the PSE results obviously coincide with the near-parallel ones far downstream, when the flow has reached a Hagen-Poiseuille axial velocity profile with superimposed solid-body rotation, they differ significantly in the developing region. Therefore, the onset of instability strongly depends on the axial evolution of the perturbations. The PSE results are also compared with experimental data from Imao et al. [Exp. Fluids 12 (1992) 277], showing a good agreement in the frequencies and wavelengths of the unstable disturbances, that take the form of spiral waves. Finally, a simple method for detecting one of the conditions to characterize the onset of absolute instability using PSE is given.     

On the criteria of the absolute convective stability for compressible fluids

The conditions under which natural convection is absent from compressible fluids are investigated. It is shown that in the parameter “Rayleigh number-given temperature difference” plane there is a domain in which convection occurs for neither Rayleigh numbers. It is proposed to refer to this domain as the absolute convective stability region and to name the criterion determining the boundary of this region the absolute convective stability criterion. The necessary, sufficient, and necessary and sufficient conditions of the absolute convective stability for a viscous compressible fluid are derived. It is shown that in the particular case in which the thermal properties of the fluid and the adiabatic gradient are constant, these conditions coincide with the Schwarzschild criterion.     

Convective and absolute instabilities in counter-rotating spiral Poiseuille flow

We present results of an experimental study on the stability of Taylor–Couette flow in case of counter-rotating cylinders and an imposed axial through flow. We are able to confirm results form recent numerical investigations done by Pinter et al. [24] by measuring the absolute and convective stability boundaries of both propagating Taylor vortices (PTV) and spiral vortices (SPI). Thus our work shows that these theoretical concepts from hydrodynamic stability in open flows apply to experimental counter-rotating Taylor–Couette systems with an imposed axial through flow. PACS 47.20.-k, 05.45.-a, 47.15.fe     

Sub-harmonic resonant solutions of a harmonically excited dry friction oscillator

Special sub-harmonic solutions of a harmonically forced dry-friction oscillator are analysed. Although the typical non-sticking solutions are stable and symmetric, a continuum of possible asymmetric, marginally stable solutions exist at excitation frequencies Ω = 1/2n. We determine the explicit form of the one-parameter family of these solutions, and give the conditions under which our formulae are valid. The stability of the solutions is examined in the third-order approximation. Finally, our analytical results are checked by numerical simulations.     

ABSOLUTE AND CONVECTIVE BENDING INSTABILITIES IN FLUID-CONVEYING PIPES

The effect of internal plug flow on the lateral stability of fluid conveying pipes is investigated by determining the absolute or convective nature of the instability from the analytically derived linear dispersion relation. The fluid–structure interaction is modelled by following the work of Gregory & Paı̈doussis. The formulation of the fluid-conveying pipe problem is shown to be related to previous studies of a flat plate in the presence of uniform flow by Brazier-Smith & Scott and Crigthon & Oswell. The different domains of stability, convective instability, and absolute instability are explicitly derived in control parameter space. The effects of flow velocity, fluid–structure mass ratio, stiffness of the elastic foundation, bending rigidity and axial tension are considered. Absolute instability in flexural pipes prevails over a wide range of parameters. Convective instability is mostly found in tensioned pipes, which are modelled by a generalized linear Klein–Gordon equation. The impulse response is given in closed form or as an integral approximation and its behaviour confirms the results found directly from the dispersion equation.     

Stability and Hopf bifurcation of a delayed ratio-dependent predator-prey system

Since the ratio-dependent theory reflects the fact that predators must share and compete for food, it is suitable for describing the relationship between predators and their preys and has recently become a very important theory put forward by biologists. In order to investigate the dynamical relationship between predators and their preys, a so-called Michaelis-Menten ratio-dependent predator-prey model is studied in this paper with gestation time delays of predators and preys taken into consideration. The stability of the positive equilibrium is investigated by the Nyquist criteria, and the existence of the local Hopf bifurcation is analyzed by employing the theory of Hopf bifurcation. By means of the center manifold and the normal form theories, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. The above theoretical results are validated by numerical simulations with the help of dynamical software WinPP. The results show that if both the gestation delays are small enough, their sizes will keep stable in the long run, but if the gestation delays of predators are big enough, their sizes will periodically fluc-tuate in the long term. In order to reveal the effects of time delays on the ratio-dependent predator-prey model, a ratiodependent predator-prey model without time delays is considered. By Hurwitz criteria, the local stability of positive equilibrium of this model is investigated. The conditions under which the positive equilibrium is locally asymptotically stable are obtained. By comparing the results with those of the model with time delays, it shows that the dynamical behaviors of ratio-dependent predator-prey model with time delays are more complicated. Under the same conditions, namely, with the same parameters, the stability of positive equilibrium of ratio-dependent predator-prey model would change due to the introduction of gestation time delays for predators and preys. Moreover, with the variation of time delays, the positive equilibrium of the ratio-dependent predator-prey model subjects to Hopf bifurcation.     

局部彼得洛夫-伽辽金法分析各向异性板屈曲

基于Kirchhoff板理论和对挠度函数采用移动最小二乘近似函数进行插值，进一步研究无网格局部Petrov-Galerkin(MLPG)方法在各向异性板稳定问题中的应用．分析中，本质边界条件采用罚因子法施加，离散的特征值方程由板稳定控制方程的局部积分对称弱形式中得到．通过数值算例并与其他方法的结果进行比较，表明MLPG法求解各向异性薄板稳定问题具有收敛性好、精度高等一系列优点．     

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.     

van der Pol-Duffing时滞系统的稳定性和Hopf分岔

研究了具有三次项的ｖａｎ ｄｅｒ Ｐｏｌ－Ｄｕｆｆｉｎｇ非线性时滞系统的稳定性和Ｈｏｐｆ分岔,分析了当线性化特征方程随两参数（时滞量和增益系数）变化时特征根的分布;证明了Ｈｏｐｆ分岔的存在性,通过构造中心流形并且使用范式方法给出的Ｈｏｐｆ分岔的方向以及周期解的稳定性,讨论时滞量对该系统的Ｈｏｐｆ分岔的影响。     

A form function method for form finding of tension structure

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Stability of strain control systems of nano- and microdisplacement piezotransducers

The transfer functions of multilayer nano- and microdisplacement piezotransducers are obtained under the conditions of longitudinal and transverse piezo-optic effects. The absolute stability conditions are derived for the strain control systems of multilayer nano- and microdisplacement piezotransducers. Some compensating devices ensuring the stability of strain control systems of multilayer piezotransducers are chosen.