四元数Lyapunov方程的酉结构解及最佳逼近

针对四元数Lyapunov方程AX+XA*=C,在A、C为正规矩阵条件下,利用四元数矩阵的Frobenius范数酉乘积不变性和矩阵的右特征值分解,得到了该方程存在酉结构解的充分必要条件及其通解表达式。同时对预先给定的酉矩阵M,应用四元数矩阵的迹不等式和矩阵的分块方法,在该方程的酉结构解集中得到与M的最佳逼近解。最后给出求解步骤,通过数值算例和四元数矩阵的复表示运算,检验了所得理论结果的正确性及所给方法的可行性。

一类非线性振荡电路中的Lyapunov指数分析

通过Duffing方程研究了一类非线性振荡电路中的复杂动力学行为,分析了带有激振力的Duffing方程在参数改变时对系统动力学行为的影响。当系统的分岔参数有微小的改变时,系统呈现出非常丰富多样的动力学行为。分岔图显示有周期泡现象产生。利用Poincar啨映射图分析了系统混沌吸引子的特性,通过仿真系统的分岔图准确的刻画出系统的周期运动和混沌运动,通过计算Duffing方程时间序列的Lyapunov指数谱和维数谱分析了系统混沌特性,揭示了此类系统通向混沌的过程与系统的动力学行为的复杂性,验证了该系统的分岔图与Lyapunov指数谱图和维数谱图的一致性。此项研究得到了一些具有理论和工程价值的结论,为其他系统的研究提供了可靠的理论依据和有效的数值方法。

基于盒子维数和最大Lyapunov维数的瞬态信号特征提取

认知无线电是一种通过无线频谱感知来有效提高频谱利用率的无线通信技术。它除了面临传统无线通信的各种安全问题外,还会面临一些新的问题,比如模仿主用户攻击(PUE)。提出基于分形维数的瞬态信号特征提取来预防认知无线网络中的模仿主用户攻击,并通过实验提取无线发射机的分形维数,为下一步信号识别提供可靠的特征参数,达到预防PUE攻击的目的。

Liu系统的Lyapunov维数估计

基于Leonov提出的Lyapunov维数理论,通过构造合适的Lyapunov函数,给出了Liu系统不变集的Lyapunov维数估计式.最后并给出了Liu系统混沌吸引子的Lyapunov维数估计.

基于分形维数的转子—机匣系统故障诊断研究

基于相空间重构,在优选重构参数的基础上,将分形维数应用于航空发动机转子—机匣系统的故障诊断。建立以关联维数为特征量的故障诊断方法,实际诊断航空发动机转子—机匣系统故障;提出以Lyapunov维数为特征量的故障诊断新方法,并实际应用于航空发动机转子—机匣系统的故障诊断,弥补关联维数等分形维数计算中无标度区难以精确确定、限制分形维数在故障诊断中的实际应用的不足。

Φ~6-DVP振子在单势阱参数下的混沌特性

利用Melnikov方法,分析了含有5次方恢复系数项的Φ6-Duffing-van der Pol振子系统在单势阱参数条件下产生Smale意义下的混沌的必要条件.通过Poincar啨截面图、分岔图、Lyapunov指数谱和Lyapunov维数等理论和数值方法,阐明了系统运动在单势阱参数下随周期激励信号变化的动态特性、复杂性和系统的非线性特征.

Robust admissibility analysis of uncertain switched singular systems:a switched Lyapunov function approach

The robust admissibility analysis of a class of uncertain discrete-time switched linear singular（SLS） systems for arbitrary switching laws is addressed. The parameter uncertainty is assumed to be norm-bounded. First, by using the switched Lyapunov function approach, some new sufficient conditions ensuring the nominal discrete-time SLS system to be regular, casual and asymptotically stable for arbitrary switching laws are derived in terms of linear matrix inequalities. Then, the robust admissibility condition for the uncertain discrete-time SLS systems is presented. The obtained results can be viewed as an extension of previous works on the switched Lyapunov function approach from the regular switched linear systems to the switched linear singular cases. Numerical examples show the reduced conservatism and effectiveness of the proposed conditions.

不确定广义时滞系统的参数依赖指数镇定

研究了具有凸多面体不确定广义时滞系统的时滞相关型指数镇定问题。首先，利用Lyapunov稳定性定理和线性矩阵不等式工具，给出了标称的广义时滞系统正则、无脉冲和指数稳定的时滞相关型充分条件。在此基础上，采用参数依赖Lyapunov函数方法，设计了使闭环系统正则、无脉冲和指数稳定的时滞相关型状态反馈控制器，并给出了相应控制器的显式表示。

Stability analysis for affine fuzzy system based on fuzzy Lyapunov functions

An analysis method based on the fuzzy Lyapunov functions is presented to analyze the stability of the continuous affine fuzzy systems. First, a method is introduced to deal with the consequent part of the fuzzy local model. Thus, the stability analysis method of the homogeneous fuzzy system can be used for reference. Stability conditions are derived in terms of linear matrix inequalities based on the fuzzy Lyapunov functions and the modified common Lyapunov functions, respectively. The results demonstrate that the stability result based on the fuzzy Lyapunov functions is less conservative than that based on the modified common Lyapunov functions via numerical examples. Compared with the method which does not expand the consequent part, the proposed method is simpler but its feasible region is reduced. Finally, in order to expand the application of the fuzzy Lyapunov functions, the piecewise fuzzy Lyapunov function is proposed, which can be used to analyze the stability for triangular or trapezoidal membership functions and obtain the stability conditions. A numerical example validates the effectiveness of the proposed approach.

Stability of second order Hopfield neural networks with time delays

Dynamical behaviors of a class of second order Hopfield neural networks with time delays is investigated.The existence of a unique equilibrium point is proved by using Brouwer's fixed point theorem and the counter proof method,and some sufficient conditions for the global asymptotic stability of the equilibrium point are obtained through the combination of a suitable Lyapunov function and an algebraic inequality technique.

Oil–water two-phase flow pattern analysis with ERT based measurement and multivariate maximum Lyapunov exponent

Oil–water two-phase flow patterns in a horizontal pipe are analyzed with a 16-electrode electrical resistance tomography(ERT) system. The measurement data of the ERT are treated as a multivariate time-series, thus the information extracted from each electrode represents the local phase distribution and fraction change at that location. The multivariate maximum Lyapunov exponent(MMLE) is extracted from the 16-dimension time-series to demonstrate the change of flow pattern versus the superficial velocity ratio of oil to water. The correlation dimension of the multivariate time-series is further introduced to jointly characterize and finally separate the flow patterns with MMLE. The change of flow patterns with superficial oil velocity at different water superficial velocities is studied with MMLE and correlation dimension, respectively, and the flow pattern transition can also be characterized with these two features. The proposed MMLE and correlation dimension map could effectively separate the flow patterns, thus is an effective tool for flow pattern identification and transition analysis.

Using Lyapunov function to design optimal controller for AQM routers

It was shown that active queue management schemes implemented in the routers of communication networks sup-porting transmission control protocol (TCP) flows can be modelled as a feedback control system. In this paper based on Lyapunov function we developed an optimal controller to improve active queue management (AQM) router’s stability and response time, which are often in conflict with each other in system performance. Ns-2 simulations showed that optimal controller outperforms PI controller significantly.

Optimization of support vector machine power load forecasting model based on data mining and Lyapunov exponents

According to the chaotic and non-linear characters of power load data,the time series matrix is established with the theory of phase-space reconstruction,and then Lyapunov exponents with chaotic time series are computed to determine the time delay and the embedding dimension.Due to different features of the data,data mining algorithm is conducted to classify the data into different groups.Redundant information is eliminated by the advantage of data mining technology,and the historical loads that have highly similar features with the forecasting day are searched by the system.As a result,the training data can be decreased and the computing speed can also be improved when constructing support vector machine(SVM) model.Then,SVM algorithm is used to predict power load with parameters that get in pretreatment.In order to prove the effectiveness of the new model,the calculation with data mining SVM algorithm is compared with that of single SVM and back propagation network.It can be seen that the new DSVM algorithm effectively improves the forecast accuracy by 0.75%,1.10% and 1.73% compared with SVM for two random dimensions of 11-dimension,14-dimension and BP network,respectively.This indicates that the DSVM gains perfect improvement effect in the short-term power load forecasting.

Application of Lyapunov exponent algorithm in balise signal chaotic oscillator detection

To improve the detection accuracy of the balise uplink signal transmitted in a strong noise environment,we use chaotic oscillator to detect the balise uplink signal based on the characteristics of the chaotic system that is sensitive to initial conditions and immune to noise.Combining with the principle of Duffing oscillator system used in weak signal detection and uplink signal feature,the methods and steps of using Duffing oscillator to detect the balise signal are presented.Furthermore,the Lyapunov exponent algorithm is used to calculate the critical threshold of the Duffing oscillator detection system.Thus,the output states of the system can be quantitatively judged to achieve demodulation of the balise signal.The simulation results show that the chaotic oscillator detection method for balise signal based on Lyapunov exponent algorithm not only improves the accuracy and efficiency of threshold setting,but also ensures the reliability of balise signal detection.

Estimating the Predictability of the Quasi-Biweekly Oscillation Using the Nonlinear Local Lyapunov Exponent Approach

The quasi-biweekly oscillation (QBWO) is a major intraseasonal variability (ISV) in the tropics. Based on bandpass-filtered outgoing longwave radiation (OLR) and wind field data, the predictability limits of the QBWO in boreal summer and boreal winter are investigated using the nonlinear local Lyapunov exponent (NLLE) approach The analysis shows that the evolution of the mean error growth of the QBWO in boreal summer and the evolution of the mean error growth in boreal winter are comparable Both curves exhibit rapid growth in the initial stage followed by a slowly fluctuating, ascending trend before saturation is reached. As a result, the potential predictability limits for the boreal summer QBWO are very close to those for the boreal winter QBWO, with a lead time of approximately three weeks. Given the current limitations in the simulation and prediction of ISV, including the QBWO, the results of this study provide a useful reference for assessing the predictability of the QBWO using model simulations.

Forecasting available parking space with largest Lyapunov exponents method

The techniques to forecast available parking space(APS) are indispensable components for parking guidance systems(PGS). According to the data collected in Newcastle upon Tyne, England, the changing characteristics of APS were studied. Thereafter, aiming to build up a multi-step APS forecasting model that provides richer information than a conventional one-step model, the largest Lyapunov exponents(largest LEs) method was introduced into PGS. By experimental tests conducted using the same dataset, its prediction performance was compared with traditional wavelet neural network(WNN) method in both one-step and multi-step processes. Based on the results, a new multi-step forecasting model called WNN-LE method was proposed, where WNN, which enjoys a more accurate performance along with a better learning ability in short-term forecasting, was applied in the early forecast steps while the Lyapunov exponent prediction method in the latter steps precisely reflect the chaotic feature in latter forecast period. The MSE of APS forecasting for one hour time period can be reduced from 83.1 to 27.1(in a parking building with 492 berths) by using largest LEs method instead of WNN and further reduced to 19.0 by conducted the new method.

Improved algorithm for calculating Lyapunov exponent and distinguishing chaos from noise

An improved algorithm for calculating the largest Lyapunov exponents (λ1) is presented based on Kantz algorithm. The presented algorithm can select a neighborhood in a certain extent according to the variety of the curves for calculating the largest Lyapunov exponent. And it can determine the linear zone based on the curves where branch is generated, thus, the largest Lyapunov exponent is obtained. The numerical experiments for the Hénon map prove that the proposed method is a direct method to identify whether a linear envelope to the curves exists in distinguishing chaos from noise, and it is superior to the Kantz algorithm.

Nonsmooth Control-Lyapunov Function and Stabilization of Nonlinear Switching Systems

In this paper we consider general nonlinear switching systems. Under an additional assumption, we prove that there exists a state space depending switching rule which stabilizes the system in a very general sense.

Local Lyapunov Exponents and characteristics of fixed/periodic points embedded within a chaotic attractor

A chaotic dynamical system is characterized by a positive averaged exponential separation of two neighboring tra- jectories over a chaotic attractor. Knowledge of the Largest Lyapunov Exponent λ1 of a dynamical system over a bounded attractor is necessary and sufficient for determining whether it is chaotic (λ1>0) or not (λ1≤0). We intended in this work to elaborate the connection between Local Lyapunov Exponents and the Largest Lyapunov Exponent where an alternative method to calculate λ1 has emerged. Finally, we investigated some characteristics of the fixed points and periodic orbits embedded within a chaotic attractor which led to the conclusion of the existence of chaotic attractors that may not embed in any fixed point or periodic orbit within it.

Quantifying local predictability of the Lorenz system using the nonlinear local Lyapunov exponent

The nonlinear local Lyapunov exponent（NLLE） can be used as a quantification of the local predictability limit of chaotic systems. In this study, the phase-spatial structure of the local predictability limit over the Lorenz-63 system is investigated. It is found that the inner and outer rims of each regime of the attractor have a high probability of a longer than average local predictability limit, while the center part is the opposite. However, the distribution of the local predictability limit is nonuniformly organized, with adjacent points sometimes showing quite distinct error growth.The source of local predictability is linked to the local dynamics, which is related to the region in the phase space and the duration on the current regime.