Volterra核函数法在轴承滚珠磨损中的特征提取及应用

针对滚动轴承滚珠磨损故障特征难以提取的问题,提出一种基于多脉冲激励法下的Volterra级数核的求解算法.该方法是一种非线性系统模型的“交叉”诊断法,利用轴承系统输入输出的采样信号,建立Volterra非线性辨识系统模型,并运用多脉冲激励Volterra低阶核求解算法,将得到的低阶核通过时域和频域进行对比来判断轴承当前所处的运行状态.该文以无心车床主轴轴承为例进行实验验证,并与传统的小波分析法对比得出:多脉冲激励法能够方便准确地提取轴承的故障特征,该方法对此类故障的诊断具有一定的借鉴意义.     

A note on exact finite difference schemes for modified Lotka–Volterra differential equations

We construct a class of modified Lotka–Volterra ordinary differential equations (ODE’s) and show that a nonlinear change of the dependent variables transform them into a set of coupled, linear ODE’s. Using the latter equations, we calculate the corresponding exact finite difference schemes using a technique given by Mickens. Next, we show how to reconfigure these relations to obtain the exact finite difference representation of the original modified, nonlinear Lotka–Volterra ODE’s.     

Well posedness and asymptotic behavior of a coupled system of nonlinear viscoelastic equations

In this paper, we consider a nonlinear system of two coupled viscoelastic equations which describes the interaction between two different fields arising in viscoelasticity. We prove the well-posedness and, for a wider class of relaxation functions, establish a generalized stability result for this system.     

Volterra Integral Equation Models for Semiconductor Devices

Van Roosbroeck's bipolar drift diffusion equations cover the qualitative behaviour of many semiconductor devices. The complexity of the model equations however prevents efficient implementations needed in circuit simulations. Under close-to-thermal-equilibrium biasing conditions (zero space charge assumption, low injection limit) the van Roosbroeck system can be replaced by a system of coupled non-linear Volterra integral equations of the second kind. Involving only the macroscopic quantities current, applied voltage and serial resistance this Volterra system can be handled with comparably little effort. Volterra integral equations models are formulated for a large class of semiconductor devices with abrupt pn-junctions. The model equations are made explicit for diodes, transistors and thyristors. A survey on various results concerning Volterra models describing the switching behaviour of pn-diodes is given. The integral equation model allows to recover all relevant properties of the voltage–current characteristics.     

Asymptotic behaviour of nonlinear difference equations

In this paper, we investigate the growth/decay rate of solutions of a class of nonlinear Volterra difference equations. Our results can be applied for the case when the characteristic equation of an associated linear difference equation has complex dominant eigenvalue with higher than one multiplicity. Illustrative examples are given for describing the asymptotic behaviour of solutions in a class of linear difference equations and in several discrete nonlinear population models.     

Nonlinear parametric models from volterra kernels measurements

Methods for nonlinear system identification are often classified, based on the employed model form, into parametric (nonlinear differential or difference equations) and nonparametric (functional expansions). These methods exhibit distinct sets of advantages and disadvantages that have motivated comparative studies and point to potential benefits from combined use. Fundamental to these studies are the mathematical relations between nonlinear differential (or difference, in discrete time) equations (NDE) and Volterra functional expansions (VFE) of the class of nonlinear systems for which both model forms exist, in continuous or discrete time. Considerable work has been done in obtaining the VFE's of a broad class of NDE's, which can be used to make the transition from nonparametric models (obtained from experimental input-output data) to more compact parametric models. This paper presents a methodology by which this transition can be made in discrete time. Specifically, a method is proposed for obtaining a parametric NARMAX (Nonlinear Auto-Regressive Moving-Average with exogenous input) model from Volterra kernels estimated by use of input-output data.     

非线性算子方程迭代解的存在性定理及其应用

在Ｂａｎａｃｈ空间中,利用锥理论和单调迭代方法研究了一类非线性算子方程的解和最小最大耦合解的存在与迭代逼近定理,并应用到Ｂａｎａｃｈ空间中非线性Ｖｏｌｔｅｒｒａ型积分方程和常微分方程的初值问题．     

Abstract Volterra operators

We propose a new general definition of Volterra operators. Several types of evolutionary operators, including Volterra ones in the sense of A.N. Tikhonov, satisfy this definition. For equations with generalized Volterra operators we introduce the notions of local, global, and maximally extended solutions. For solutions to nonlinear equations we formulate the existence, uniqueness, and extendability conditions. The theorems proved in this paper imply both known and new results on the solvability of concrete equations. We adduce an example of the application of obtained results to the study of the Cauchy problem for functional differential equations.     

The extended F-expansion method and its application for a class of nonlinear evolution equations

By means of a simple transformation technique, we have shown that the higher-order nonlinear Schrödinger equation in nonlinear optical fibers, a new Hamiltonian amplitude equation, generalized Hirota–Satsuma coupled system and generalized ZK-BBM equation can be reduced to the elliptic-like equation. Then, the extended F-expansion method is used to obtain a series of solutions including the single and the combined nondegenerative Jacobi elliptic function solutions and their degenerative solutions to the above mentioned class of nonlinear evolution equations.     

Influence of feedback controls on an autonomous Lotka–Volterra competitive system with infinite delays

In this paper, we consider an autonomous Lotka–Volterra competitive system with infinite delays and feedback controls. The extinction and global stability of equilibriums are discussed using the Lyapunov functional method. If the Lotka–Volterra competitive system is globally stable, then we show that the feedback controls only change the position of the unique positive equilibrium and retain the stable property. If the Lotka–Volterra competitive system is extinct, by choosing the suitable values of feedback control variables, we can make extinct species become globally stable, or still keep the property of extinction. Some examples are presented to verify our main results.     

Funnel control for fully actuated systems under a fragment of signal temporal logic specifications

Temporal logics have lately proven to be a valuable tool for various control applications by providing a rich specification language. Existing temporal logic-based control strategies discretize the underlying dynamical system in space and/or time. We will not use such an abstraction and consider continuous-time systems under a fragment of signal temporal logic specifications by using the associated robust semantics. In particular, this paper provides computationally-efficient funnel-based feedback control laws for a class of systems that are, in a sense, feedback equivalent to single integrator systems, but where the dynamics are partially unknown for the control design so that some degree of robustness is obtained. We first leverage the transient properties of a funnel-based feedback control strategy to maximize the robust semantics of some atomic temporal logic formulas. We then guarantee the satisfaction for specifications consisting of conjunctions of such atomic temporal logic formulas with overlapping time intervals by a suitable switched control system. The result is a framework that satisfies temporal logic specifications with a user-defined robustness when the specification is satisfiable. When the specification is not satisfiable, a least violating solution can be found. The theoretical findings are demonstrated in simulations of the nonlinear Lotka–Volterra equations for predator–prey models.     

Mathematical modeling and control of population systems: Applications in biological pest control

The aim of this paper is to apply methods from optimal control theory, and from the theory of dynamic systems to the mathematical modeling of biological pest control. The linear feedback control problem for nonlinear systems has been formulated in order to obtain the optimal pest control strategy only through the introduction of natural enemies. Asymptotic stability of the closed-loop nonlinear Kolmogorov system is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution of the Hamilton–Jacobi–Bellman equation, thus guaranteeing both stability and optimality. Numerical simulations for three possible scenarios of biological pest control based on the Lotka–Volterra models are provided to show the effectiveness of this method.     

Stability analysis of Volterra functional equations by realization theory

The method of realization of input–output operators in the form of abstract discrete-time control systems and the frequency-domain method are used for the stability/instability analysis of a class of nonlinear Volterra functional equations. To this end, we construct an associated time-invariant abstract discrete-time control system in some weighted function spaces.     

Numerical Solution Based on Hat Functions for Solving Nonlinear Stochastic Itô Volterra Integral Equations Driven by Fractional Brownian Motion

This paper presents a numerical method for solving nonlinear stochastic Itô Volterra integral equations driven by fractional Brownian motion with Hurst parameter \( H \in (0,1)\) via of hat functions. Using properties of the generalized hat basis functions and fractional Brownian motion, new stochastic operational matrix of integration is achieved and the nonlinear stochastic equation is transformed into nonlinear system of algebraic equations which by solving it, an approximation solution with high accuracy is obtained. In addition, error analysis of the method is investigated, and by some examples, efficiency and accuracy of the suggested method are shown.     

Extremes of Volterra series expansions with heavy-tailed innovations

In this paper, we look at the extremal behavior of Volterra series expansions generated by heavy-tailed innovations, via a point process formulation. Volterra series expansions are known to be the most general nonlinear representation for any stationary sequence. The so called complete convergence theorem on point processes we prove enable us to give in detail, the weak limiting behavior of various functionals of the underlying process including the asymptotic distribution of upper and lower order statistics. In particular, we investigate the limiting distribution of the sample maxima and the corresponding extremal index. The study of the extremal properties of finite order Volterra series expansions would be highly valuable in understanding the extremal behavior of nonlinear processes as well as understanding of order identification and adequacy of Volterra series when used as models in signal processing. In fact, such extremal properties may suggest a way of finding the order of a finite Volterra expansions which is consistent with the nonlinearities of the observed process.     

Identification of Parameters of Nonlinear Dynamical Systems Simulated by Volterra Polynomials

We study the identification methods for the nonlinear dynamical systems described by Volterra series. One of the main problems in the dynamical system simulation is the problem of the choice of the parameters allowing the realization of a desired behavior of the system. If the structure of the model is identified in advance, then the solution to this problem closely resembles the identification problem of the system parameters. We also investigate the parameter identification of continuous and discrete nonlinear dynamical systems. The identification methods in the continuous case are based on application of the generalized Borel Theorem in combination with integral transformations. To investigate discrete systems, we use a discrete analog of the generalized Borel Theorem in conjunction with discrete transformations. Using model examples, we illustrate the application of the developed methods for simulation of systems with specified characteristics.     

Global stability analysis of epidemiological models based on Volterra–Lyapunov stable matrices

In this paper, we study the global dynamics of a class of mathematical epidemiological models formulated by systems of differential equations. These models involve both human population and environmental component(s) and constitute high-dimensional nonlinear autonomous systems, for which the global asymptotic stability of the endemic equilibria has been a major challenge in analyzing the dynamics. By incorporating the theory of Volterra–Lyapunov stable matrices into the classical method of Lyapunov functions, we present an approach for global stability analysis and obtain new results on some three- and four-dimensional model systems. In addition, we conduct numerical simulation to verify the analytical results.     

Conditions for blow-up of solutions of some nonlinear Volterra integral equations

In this paper we give necessary and sufficient conditions for blow-up of solutions for a particular class of nonlinear Volterra equations. We also give some examples.     

A class of concave operators with applications

In this paper we provide some properties of a class of concave operators and apply these results in discussing three-point boundary value problems for differential equations and nonlinear Volterra integral equations.     

Existence,uniqueness and stability of solutions for a class of nonlinear integral equations under generalized Lipschitz condition

In this paper, we prove the existence, uniqueness and the stability of solutions for some nonlinear functional-integral equations by using generalized Lipschitz condition. We prove a fixed point theorem to obtain the mentioned aims in Banach space X:= C([a, b],R). As application we study some Volterra integral equations with linear, nonlinear and singular kernel.