Robustness of exponential stability of stochastic differentialdelay equations

Regard the stochastic differential delay equation dx(t)=[(A+A¯(t))+(B+B¯(t-τ))x(t-τ)] dt+g(t, x(t), x(t-τ))dw(t) as the result of the effects of uncertainty, stochastic perturbation, and time lag to a linear ordinary differential equation x˙(t)=(A+B)x(t). Assume the linear system is exponentially stable. In this paper we characterize how much the uncertainty, stochastic perturbation, and time lag the linear system can bear such that the stochastic delay system remains exponentially stable. The result can also be extended to nonlinear systems     

Global exponential stability of competitive neural networks with different time scales

The dynamics of cortical cognitive maps developed by self-organization must include the aspects of long and short-term memory. The behavior of such a neural network is characterized by an equation of neural activity as a fast phenomenon and an equation of synaptic modification as a slow part of the neural system. We present a new method of analyzing the dynamics of a biological relevant system with different time scales based on the theory of flow invariance. We are able to show the conditions under which the solutions of such a system are bounded being less restrictive than with the K-monotone theory, singular perturbation theory, or those based on supervised synaptic learning. We prove the existence and the uniqueness of the equilibrium. A strict Lyapunov function for the flow of a competitive neural system with different time scales is given and based on it we are able to prove the global exponential stability of the equilibrium point.     

Stochastic perturbation methods for spike-timing-dependent plasticity

Online machine learning rules and many biological spike-timing-dependent plasticity (STDP) learning rules generate jump process Markov chains for the synaptic weights. We give a perturbation expansion for the dynamics that, unlike the usual approximation by a Fokker-Planck equation (FPE), is well justified. Our approach extends the related system size expansion by giving an expansion for the probability density as well as its moments. We apply the approach to two observed STDP learning rules and show that in regimes where the FPE breaks down, the new perturbation expansion agrees well with Monte Carlo simulations. The methods are also applicable to the dynamics of stochastic neural activity. Like previous ensemble analyses of STDP, we focus on equilibrium solutions, although the methods can in principle be applied to transients as well.     

Asymptotic tracking of a nonminimum phase nonlinear system withnonhyperbolic zero dynamics

A two-cart with an inverted-pendulum system is a nonlinear, nonminimum phase system with nonhyperbolic zero dynamics. Devasia introduced this system to study the asymptotic tracking problem for nonlinear systems with nonhyperbolic zero dynamics and pointed out that the nonhyperbolicity may be challenging to the application of the standard inversion-based tracking technique. We first show that nonhyperbolicity is not necessary for the applicability of the output regulation theory. In particular, the problem of asymptotic tracking of the two-cart with an inverted-pendulum system to a class of sinusoidal reference inputs is actually solvable by the standard output regulation theory. Moreover, an approximation method for calculating the center manifold equation associated with the output regulation problem for general nonlinear systems is given. This approach does not rely on the hyperbolicity condition and, hence, applies to a large class of nonlinear systems     

Almost sure convergence of extremum seeking algorithm using stochastic perturbation

This paper presents an extremum seeking (ES) algorithm where the perturbation signal is a martingale difference sequence (m.d.s.) with a vanishing variance. The measurement noise at the plant output is modeled by a superposition of deterministic component, and a non-stationary colored noise signal. The optimizing set point of the uncertain reference-to-output equilibrium map is estimated by a stochastic approximation (SA)-type algorithm. The algorithm has a vanishing gain sequence dependent on the set point estimates. By utilizing powerful tools of the martingale convergence theory it is proved that with probability one the set point estimates converge to the optimizing equilibrium point, in spite of the presence of a measurement noise. This result is derived without requiring boundedness or any prior condition on the set point estimates.     

厚壁圆筒在热、磁耦合场作用下的动态响应

运用力学和电磁场的知识对厚壁圆筒结构建立平衡方程,并通过 Laplace 和 Hankel 积分变换对物理方程进行变换,得到一个可解的方程形式.提出一种解析方法求解在热磁冲击作用下厚壁圆筒的动应力和磁场矢量扰动,得到柱体内动应力响应历程和分布规律及磁场矢量扰动的响应历程和分布规律.实例计算表明该方法是简单、有效的,并给出了一些有实际意义的结果.     

A family of approximate inertial manifolds for a Van der Pol/FitzHugh–Nagumo perturbation problem

We investigate the stability of a family of approximate inertial manifolds (AIMs) obtained from an ODE containing a perturbation parameter. For two choices of the parameter, the dynamics associated with the equations are already well known: in one case, we have a Van der Pol equation, while in the other setting we obtain a FitzHugh–Nagumo equation. Recently, it has been shown that (a modified version of) each equation admits a sequence of AIMs which converges to the inertial manifold. We show that our model admits a family of AIMs depending on the perturbation parameter. We then investigate the stability of the family of AIMs as the perturbation parameter approaches two different vanishing coefficient limits. These results are intended to shed insight into the continuity properties of inertial manifolds.     

Stabilization of stochastic nonlinear systems driven by noise ofunknown covariance

This paper poses and solves a new problem of stochastic (nonlinear) disturbance attenuation where the task is to make the system solution bounded by a monotone function of the supremum of the covariance of the noise. This is a natural stochastic counterpart of the problem of input-to-state stabilization in the sense of Sontag (1989). Our development starts with a set of new global stochastic Lyapunov theorems. For an exemplary class of stochastic strict-feedback systems with vanishing nonlinearities, where the equilibrium is preserved in the presence of noise, we develop an adaptive stabilization scheme (based on tuning functions) that requires no a priori knowledge of a bound on the covariance. Next, we introduce a control Lyapunov function formula for stochastic disturbance attenuation. Finally, we address optimality and solve a differential game problem with the control and the noise covariance as opposing players; for strict-feedback systems the resulting Isaacs equation has a closed-form solution     

Dynamical analysis of continuous higher-order hopfield networks for combinatorial optimization

In this letter, the ability of higher-order Hopfield networks to solve combinatorial optimization problems is assessed by means of a rigorous analysis of their properties. The stability of the continuous network is almost completely clarified: (1) hyperbolic interior equilibria, which are unfeasible, are unstable; (2) the state cannot escape from the unitary hypercube; and (3) a Lyapunov function exists. Numerical methods used to implement the continuous equation on a computer should be designed with the aim of preserving these favorable properties. The case of nonhyperbolic fixed points, which occur when the Hessian of the target function is the null matrix, requires further study. We prove that these nonhyperbolic interior fixed points are unstable in networks with three neurons and order two. The conjecture that interior equilibria are unstable in the general case is left open.     

一种平衡策略下网络多输入多输出系统的预编码方案

针对在多小区多输入多输出（MIMO）系统预处理中,最大化目标用户信干噪比与最小化干扰泄露之间存在的相互矛盾,提出一种基于平衡策略的预编码算法,构建预编码算法数学模型,引入干扰代价函数,并借助拉格朗日方程对所提算法的预编码向量进行求解,同时对算法的实现步骤进行了说明。最后在仿真中,对效用函数的收敛速度与迭代次数的关系进行了仿真,并基于系统最大和速率对该算法与现有算法的性能进行了比较,仿真结果验证了该算法在收敛速度与系统吞吐量方面的优越性能。     

Further results on robustness of (possibly discontinuous) sample and hold feedback

We demonstrate that sample and hold state feedback control (possibly discontinuous with respect to the state) is robust when the closed loop system possesses an appropriate Lyapunov function. We first show that if a Lyapunov decrease over sampling periods exists for the nominal system, this decrease can be maintained with some degradation relative to a sufficiently small additive perturbation. We then proceed to catalog several applications of this robustness, e.g., robustness to measurement noise, computational delays, or fast actuator dynamics.     

Intermediate asymptotics and renormalization group theory

The principles of the renormalization group (RG) are presented pedagogically from the point of view of intermediate asymptotics (IA), which is familiar to hydrodynamicists and applied mathematicians. To demonstrate the equivalence of RG and IA approaches, a typical statistical mechanical problem, conventionally studied by the renormalized perturbation approach, is reconsidered from the IA point of view, and renormalized perturbation theory is applied to a partial differential equation conventionally studied by IA. This example is important because it is an explicit demonstration that the RG can be applied to partial differential equations without adding a noise source. We suggest that the ideas explained in this article may be applicable to the Navier-Stokes equation.Presented at the Third Nobeyama Workshop on Supercomputers and Experiments in Fluid Mechanics, September 1989.     

On the analysis of a recurrent neural network for solving nonlinearmonotone variational inequality problems

We investigate the qualitative properties of a recurrent neural network (RNN) for solving the general monotone variational inequality problems (VIPs), defined over a nonempty closed convex subset, which are assumed to have a nonempty solution set but need not be symmetric. The equilibrium equation of the RNN system simply coincides with the nonlinear projection equation of the VIP to be solved. We prove that the RNN system has a global and bounded solution trajectory starting at any given initial point in the above closed convex subset which is positive invariant for the RNN system. For general monotone VIPs, we show by an example that the trajectory of the RNN system can converge to a limit cycle rather than an equilibrium in the case that the monotone VIPs are not symmetric. Contrary to this, for the strictly monotone VIPs, it is shown that every solution trajectory of the RNN system starting from the above closed convex subset converges to the unique equilibrium which is also locally asymptotically stable in the sense of Lyapunov, no matter whether the VIPs are symmetric or nonsymmetric. For the uniformly monotone VIPs, the aforementioned solution trajectory of the RNN system converges to the unique equilibrium exponentially.     

An Unconditionally Stable ADI Method for the Linear Hyperbolic Equation in Three Space Dimensions

An unconditionally stable alternating direction implicit (ADI) method of O(k 2 +h 2 ) of Lees type for solving the three space dimensional linear hyperbolic equation u tt +2 f u t + g 2 u = u xx + u yy + u zz + f ( x , y , z , t ), 0<x, y, z<1, t>0 subject to appropriate initial and Dirichlet boundary conditions is proposed, where f >0 and g S 0 are real numbers. For this method, we use a single computational cell. The resulting system of algebraic equations is solved by three step split method. The new method is demonstrated by a suitable numerical example.     

Small-amplitude limit cycles of Liénard systems

This paper is concerned with the study of second order differential equations of Liénard type: (A) $$\ddot x + f(x)\dot x + g(x) = 0$$ where f and g are polynomials. The equation (A) can also be written as a system of the form (B) $$\dot x = y - F(x),\dot y = - g(x),$$ , where \(F(x) = \mathop \smallint \limits_0^x f(\xi )d\xi \) . The results described here are mainly concerned with small amplitude limit cycles; that is, limit cycles which may be bifurcated from the origin on perturbation of the coefficients of F and g. The problem is to estimate the maximum number of limit cycles which various classes of systems of the form (B) can have; this is a special case of the second part of Hilbert’s sixteenth problem. Most of the calculations have been carried out on a computer using the REDUCE symbolic manipulation package.     

不可压缩超弹性球体中微孔运动的分岔和混沌

针对一类径向横观各向同性不可压缩neo Hookean材料组成的球体,研究了周期扰动载荷作用下球体中心处微孔的动力学行为.根据平衡微分方程和初边值条件等导出描述微孔径向对称运动的强非线性的非自治常微分方程,通过对微分方程解的定性分析,讨论了微孔的定性行为：(1) 在常值载荷作用下,讨论了材料参数和结构参数对系统平衡点的影响,特别分析了微孔的二次转向分岔;通过对系统势阱的分析,讨论了微孔在常值载荷作用下的周期和振幅跳跃现象.(2) 在周期扰动载荷作用下,利用时程曲线,Poincaré截面和最大Lyapunov指数分别讨论了二次转向分岔情形下微孔的拟周期和混沌运动,给出了系统产生混沌的条件,并进一步分析了周期扰动载荷对微孔混沌运动的影响.     

A wavelet regularization method for diffuse radar-target imaging and speckle-noise reduction

We consider the problem of forming radar images under a diffuse-target statistical model for the reflections off a target surface. The desired image is the scattering functionS(f, ), which describes the second-order statistics of target reflectivity in delay—Doppler coordinates. Our estimation approach is obtained by application of the maximum-likelihood principle and a regularization procedure based on a wavelet representation for the logarithm ofS(f, ). This approach offers the ability to capture significant components of lnS(f, ) at different resolution levels and guarantees nonnegativity of the scattering function estimates. We show that the radar imaging problem can be set up as a problem of inference on the wavelet coefficients of an image corrupted by additive noise. A simple hypothesis-testing technique for solving the problem at a prespecified significance level is studied. The significance level of the test is selected according to the desired noise/resolution trade-off. The regularization technique is applicable to a broad class of speckle-noise reduction problems.     

An experimental study of noise and asynchrony in elementary cellular automata with sampling compensation

This article focuses on the set of 32 legal elementary cellular automata (ECA). We perform an exhaustive study of the systems’ response under: (i) α-asynchronous dynamics, from full asynchronism to perfect synchrony, (ii) κ-scaling, which extends α-asynchrony to compensate for less cell activity, and (iii) ?-noise scheme, a perturbation that affects the local transition function and causes a cell to probabilistically miscalculate the new state when it is updated. We propose a new classification in three classes under asynchronous conditions: α-invariant, α-robust, and α-dependent. We classify the 32 legal ECA according to the degree of behavioural modification, and we show that our classifying scheme provides results coherent with the density-based classification. We also show that κ-scaling provides results comparable to synchronous systems, both quantitatively and qualitatively. Subsequently, we analyse the effects of including different levels of noise in synchronous systems. We identify different responses to noise, including systems that are robust to asynchrony and susceptible to noise. To conclude, we investigate the behavioural changes caused by simultaneous asynchrony and noise in models tolerant to both perturbations. We describe a number of effects caused by the interplay of noise and asynchrony, thus further reinforcing that both aspects are pertinent for future studies.     

Recursive pointwise design for nonlinear systems

This paper presents a recursive pointwise design (RPD) method for a class of nonlinear systems represented by x(t)=f(x(t))+g(x(t))u(t). A main feature of the RPD method is to recursively design a stable controller by using pointwise information of a system. The design philosophy is that f(x(t)) and g(x(t)) can be approximated as constant vectors in very small local state spaces. Based on the design philosophy, we numerically determine constant control inputs in very small local state spaces by solving linear matrix inequalities (LMIs) derived in this paper. The designed controller switches to another constant control input when the states move to another local state space. Although the design philosophy is simple and natural, the controller does not always guarantee the stability of the original nonlinear system x(t)=f(x(t))+g(x(t))u(t). Therefore, this paper gives ideas of compensating the approximation caused by the design philosophy. After addressing outline of the pointwise design, we provide design conditions that exactly guarantee the stability of the original system. The controller design conditions require to give the maximum and minimum values of elements in the functions f(x(t)) and g(x(t)) in each local state - space. Therefore, we also present design conditions for unknown cases of the maximum and minimum values. Furthermore, we construct an effective design procedure using the pointwise design. A feature of the design procedure is to subdivide only infeasible regions and to solve LMIs again only for the subdivided infeasible regions. The recursive procedure saves effort to design a controller. A design example demonstrates the utility of the RPD method.     

基于6DOF模型及动网格的动静压轴承刚度阻尼数值计算

针对传统雷诺方程求解三维油膜流场特性的局限性,提出基于6DOF模型及动网格的动静压轴承刚度阻尼计算方法.以具有典型结构形式的液体动静压轴承为例,通过加载6DOF自定义程序,采用非线性迭代方法计算外载荷作用下轴心轨迹的瞬态变化过程,得到轴颈在外载荷作用下的静平衡位置;通过嵌入UDF宏程序以动网格更新方法实现对轴颈在静平衡位置的扰动,求解Navier-Stokes方程得到轴颈扰动前后位置变化后的瞬态油膜力,利用差分法求得动静压轴承油膜刚度和阻尼,并分析了不同转速下轴承刚度和阻尼的变化规律.