高斯白噪声激励下分数阶Duffing-Van der Pol系统的稳态响应

应用随机平均法研究了高斯白噪声激励下含有分数阶阻尼项的Duffing-Van der Pol系统的稳态响应.首先应用基于广义谐和函数的随机平均法得到系统关于幅值的平均伊藤微分方程并建立相应的平稳FPK方程,求解该平稳FPK方程的近似理论解得到系统幅值的稳态概率密度.分析幅值、位移和速度的稳态概率密度探究分数阶阻尼项以及其它参数对系统稳态响应的影响.发现降低分数阶的阶数可以增强系统的响应而增大分数阶的系数可以减弱系统响应.最后对原系统进行Monte Carlo数值模拟验证近似理论解的有效性.     

多自由度拟不可积哈密顿系统的随机分数阶最优控制

推广了适用于分数阶系统控制的随机分数阶最优控制策略,提出了高斯白噪声激励下多自由度拟不可积哈密顿系统以响应最小化为目标的随机分数阶最优控制策略.首先,应用拟不可积哈密顿系统随机平均法,将受控系统简化为关于能量的部分平均伊藤方程.然后,将控制性能指标中关于控制力的部分表示为分数阶形式,结合随机动态规划原理,建立并求解部分平均系统的无界遍历控制的随机动态规划方程,获得了随机分数阶最优控制律.最后,采用一个算例验证了随机分数阶控制策略的控制效果和控制效率.研究表明,随机分数阶最优控制策略对传统的整数阶随机动力学系统同样适用,能比传统的整数阶控制策略取得更好的控制效果.另外,随着激励强度增加,整数阶控制策略的控制效率显著降低;而分数阶控制策略的控制效率虽比整数阶控制策略的控制效率略低,但随着激励强度的增加,分数阶控制策略的控制效率缓慢上升并趋于平稳,可以有效地缓解控制效率与控制效果之间的矛盾.     

含有分数阶有色关联噪声的分数阶系统的卡尔曼滤波器设计

提出基于Tustin生成函数的分数阶卡尔曼滤波器设计方法,以解决含有相互关联的分数阶有色过程噪声和分数阶有色测量噪声的连续时间线性分数阶系统的状态估计问题.通过Tustin生成函数方法,对连续时间线性分数阶系统进行离散化,将分数阶系统的微分方程转化为差分方程.利用增广向量法,将分数阶状态方程和分数阶有色噪声作为新的增广状态向量,从而将分数阶有色噪声转化为高斯白噪声.然后,提出一种基于Tustin生成函数的分数阶卡尔曼滤波算法,有效地实现对含有相互关联的分数阶有色过程噪声和分数阶有色测量噪声的连续时间线性分数阶系统的状态估计.与基于Grddotunwald-Letnikov差分的离散化方法相比,所提出的基于Tustin生成函数的卡尔曼滤波算法得到的状态估计精度更高,状态估计效果更好.最后,通过仿真结果验证所提出算法的有效性.     

色噪声激励下一类驰振能量采集器的稳态响应研究

使用具有指数相关性的色噪声近似外界环境干扰,研究色噪声激励下驰振能量采集器(GEH)的稳态响应.首先,介绍了系统的理论机电耦合运动方程,并对其进行无量纲化处理.其次,通过广义谐波变换对方程进行等效解耦,然后使用能量包线随机平均法得到系统稳态响应的解析解.最后,使用四阶龙格库塔算法进行数值模拟,在验证解析解有效性的同时分析了风速、色噪声参数对系统稳态响应的影响.研究结果表明,较大的噪声强度有利于提升系统的能量采集性能,且在风速较低时效果更明显；噪声相关时间对系统稳态响应的影响与噪声强度相反.当系统的振动模式由风激励主导时,系统的响应表现为带有附加噪声的周期性振动；随着风速增加,增强的风激励削弱了色噪声激励对系统稳态响应的影响.此外,本文还研究了系统结构参数和电气参数对平均输出功率的影响,研究结果为优化系统设计提供了有效的理论指导.     

三稳态风致振动能量俘获系统分析

使用具有指数相关性的色噪声近似外界环境干扰,研究色噪声激励下驰振能量采集器(GEH)的稳态响应.首先,介绍了系统的理论机电耦合运动方程,并对其进行无量纲化处理.其次,通过广义谐波变换对方程进行等效解耦,然后使用能量包线随机平均法得到系统稳态响应的解析解.最后,使用四阶龙格库塔算法进行数值模拟,在验证解析解有效性的同时分析了风速、色噪声参数对系统稳态响应的影响.研究结果表明,较大的噪声强度有利于提升系统的能量采集性能,且在风速较低时效果更明显；噪声相关时间对系统稳态响应的影响与噪声强度相反.当系统的振动模式由风激励主导时,系统的响应表现为带有附加噪声的周期性振动；随着风速增加,增强的风激励削弱了色噪声激励对系统稳态响应的影响.此外,本文还研究了系统结构参数和电气参数对平均输出功率的影响,研究结果为优化系统设计提供了有效的理论指导.     

宽带噪声激励下带有分数阶控制器的强非线性系统的随机平均技术

运用随机平均法研究了宽带噪声激励下带有分数阶PI~λD~μ控制器的强非线性系统.首先,应用广义谐波平衡技术,将分数阶PI~λD~μ控制力分解为幅值依赖的等效拟线性阻尼力和拟线性回复力,得到了受控整数阶等效非线性系统.然后,运用基于广义谐和函数的随机平均法得到关于幅值的平均伊藤微分方程.最后,建立并求解相应的简化Fokker-Planck-Kolmogorov(FPK)方程,得到稳态概率密度函数.作为算例,考察了Duffing-van de Pol振子.数值结果表明随机平均法能够达到较高的精度,分数阶PI~λD~μ控制器能够对系统响应进行有效的控制.此外,宽带噪声参数ξ_i、ω_i及D_i改变时,本文提出的方法仍具有较好的适用性,分数阶控制器仍同样具有非常好的控制效果.     

一种考虑非高斯L´evy 量测噪声下的改进分数阶卡尔曼滤波

针对量测噪声模型为非高斯L´evy 噪声, 研究离散线性随机分数阶系统的卡尔曼滤波设计问题. 通过剔除极大值的方法得到近似高斯白噪声的L´evy 噪声, 基于最小二乘原理, 提出一种考虑非高斯L´evy 量测噪声下的改进分数阶卡尔曼滤波算法. 与传统的分数阶卡尔曼滤波相比, 改进的分数阶卡尔曼滤波对非高斯L´evy 噪声具有更好的滤波效果. 最后, 通过模拟仿真验证了所提出算法的正确性和有效性.

     

含有分数阶PID控制器的随机动力系统可靠性分析

随着粘弹性材料在工程结构中的广泛应用,刻画工程结构中粘弹性材料遗传特性和长记忆性的分数阶微积分成为研究的热点,特别是具有分数阶微积分特点的PID控制器更是从理论上和应用上受到关注.本文研究高斯白噪声激励下含有分数阶PID控制器的随机结构动力系统的可靠性问题.利用慢变过程的特征以及广义积分的性质,对分数阶PID控制器在数学上进行了近似处理,之后应用能量包络随机平均法确定了可靠性函数满足的后向Kolmogorov方程以及首次穿越时间统计矩满足的广义Pontryagin方程.结果表明：在分数阶控制器中,较小的分数阶α和较大的分数阶β均可以得到较为理想的可靠性结果,并且这些均与蒙特卡洛仿真结果一致,验证了方法的有效性和正确性.     

相关乘性和加性高斯白噪声激励下周期势系统的随机共振

研究了外加周期信号作用下,相关高斯乘性和加性白噪声激励下周期势系统的随机共振.利用线性响应理论,计算了系统输出信号的功率谱密度、振幅、相位差.研究结果表明:当加性噪声强度和关联系数不变的情况下,通过调整乘性噪声强度可以出现随机共振;关联系数的正负以及大小对随机共振的影响较小.当乘性噪声强度较小时,输出信号的振幅和相位差曲线有一个单峰出现,即出现随机共振现象,能量从噪声向信号进行转化.随着噪声强度的增大,随机共振现象消失,噪声由增大系统的有序程度渐渐变为增大系统的无序程度.     

位移反馈分数阶PID控制对单自由度线性振子的影响

研究了基于位移反馈分数阶PID控制的单自由度线性振子的自由振动,通过平均法得到了系统的一阶近似解析解.发现分数阶PID控制器的比例环节以等效线性刚度的形式影响系统的动力学特性,积分环节以等效线性负阻尼和等效线性刚度的形式影响系统的动力学特性,微分环节以等效线性阻尼和等效线性刚度的形式影响系统的动力学特性.对近似解析解和数值解进行了比较,二者吻合良好,验证了近似解析解的正确性.从近似解析解和分数阶系统的特征方程两个角度对系统的稳定性进行了分析.最后利用系统的时间响应性能指标分析了位移反馈分数阶PID控制器的系数和分数阶阶次变化时,对单自由度线性振子控制性能的影响.     

Mode-superposition correction method for deterministic and stochastic analysis of structural systems

The role played by the modal analysis in the framework of structural dynamics is fundamental from both deterministic and stochastic point of view. However the accuracy obtained by means of the classical modal analysis is not always satisfactory. Therefore it is clear the importance of methods able to correct the modal response in such a way to obtain the required accuracy. Many methods have been proposed in the last years but they are meaningful only when the forcing function is expressed by an analytical function. Moreover in stochastic analysis they fail for white noise excitation. In the paper a method able to give a very accurate response for both deterministic and stochastic input is presented. This method is based upon the use of Ritz vectors together with the classical modal analysis. Numerical applications for both deterministic and stochastic inputs show the great accuracy of the proposed method.     

An efficient dynamical systems method for solving singularly perturbed integral equations with noise

In this paper we apply the dynamical systems method (DSM) proposed by A. G. Ramm, and the variational regularization method (VRM), to obtain numerical solution to some singularly perturbed ill-posed problems contaminated by noise. The results obtained by these methods are compared to the exact solution for the model problems. It is found that the dynamical systems method is preferable because it is easier to apply, highly stable, robust, and it always converges to the solution even for large size models.     

多自由度粘弹性非线性随机系统的瞬态响应

研究了高斯白噪声激励下多自由度粘弹性非线性系统的瞬态响应.首先,通过将粘弹性项对系统的作用近似地简化为对原系统阻尼部分以及刚度部分的修正,得到近似的不具粘弹性项的等效非线性随机系统.然后,应用基于广义谐和函数的随机平均法,导出关于幅值瞬态概率密度的平均Fokker-Planck-Kolmogorov方程.该方程的解可通过多重级数式表示,基函数为幅值相关正交函数,系数为时间函数.应用Galerkin方法,关于时间的系数可由一阶线性微分方程组解得,从而得出幅值响应的瞬态概率密度、状态空间概率密度及幅值统计矩的半解析表达式.最后,以耦合的二自由度Duffing-van der Pol振子系统为例,通过与原系统数值模拟结果的比较分析验证了所提出的半解析方法的有效性,并讨论了粘弹性对系统响应的影响.     

On disturbance rejection in magnetic levitation

Magnetic levitation systems belong to an important and challenging class of control engineering problems with nonlinear uncertain dynamics, multiple disturbances and large sensor noise. To obtain a simple and practical solution that does not depends on the exact model information, a time-varying active disturbance rejection solution is proposed and validated in both numerical and experimental results. The proposed method is confirmed with rigorous analysis of transient performance and noise attenuation. Moreover, the proposed solution is tested in a magnetic levitation ball system with disturbances and measurement noise. The results show that the proposed solution is effective and practical.     

The spectral density of the nonlinear damping model: single DOFcase

An analytical method of computing the frequency response of single degree of freedom (DOF) oscillators with nonlinear damping is described. The author proposes an energy-type nonlinear damping model and the corresponding stationary probability density with white noise input can be obtained explicitly. A theorem is presented which gives an interesting result, in terms of the Krylov-Bogoliubov approximation, concerning the modeling and identification of nonlinear internal damping in flexible structures. This analysis also serves as a contribution to random vibration theory by providing a method of computing the first- and second-order statistics (steady-state probability density, correlation function, and spectral density) of nonlinearity damped oscillators with white noise input     

Identification of ARX systems with non-stationary inputs — asymptotic analysis with application to adaptive input design

A key problem in optimal input design is that the solution depends on system parameters to be identified. In this contribution we provide formal results for convergence and asymptotic optimality of an adaptive input design method based on the certainty equivalence principle, i.e. for each time step an optimal input design problem is solved exactly using the present parameter estimate and one sample of this input is applied to the system. The results apply to stable ARX systems with the input restricted to be generated by white noise filtered through a finite impulse response filter, or a binary signal obtained from the latter by a static nonlinearity.     

基于熵测度的跳频信号谱图分析

为了有效分析跳频信号并估计其参数,依据跳频信号谱图和Wigner-Ville分布(WVD)的特征分析,提出了一种自适应跳频信号时频分析方法。该方法依据熵测度进行谱图窗函数宽度选择,获取优化的跳频信号谱图表示。理论分析和仿真结果表明,基于熵测度的谱图分析方法能够在大于0dB的高斯白噪声环境下,给出跳频周期参数的准确估计;与平滑伪Wigner分布及其自适应方法相比,该方法能够在较低信噪比下有效降低参数估计的方差,提高参数估计的准确性;同时对于长观测信号,具有更快的运算速度。     

The design of optimal reduced order stochastic observers fordiscrete-time linear systems

The minimum variance state estimation of linear discrete-time systems with random white noise input and partially noisy measurements is investigated. An observer of minimal-order that attains the minimum-variance estimation error is found. The structure of this observer is shown to depend strongly on the geometry of the system. This geometry dictates the length of the delays that are applied on the measurements in order to obtain the optimal estimate. The transmission properties of the observer are investigated for systems that are left invertible and free of measurement noise. An explicit expression is found for the transfer function matrix of the observer, from which a simple solution to the linear discrete-time singular optimal filtering problem is obtained     

一种高阶无迹卡尔曼滤波方法

现有的研究中,高阶无迹变换（Unscented transform,UT）还不存在具体的解析解,因此,无法利用高阶无迹变换获得具备更高精度的高阶无迹卡尔曼滤波器（Unscented Kalman filter,UKF）.为了解决这一问题,本文在五阶容积变换（Cubature transform,CT）的基础上,通过引入一个自由参数κ,得到高阶无迹变换的解析解,从而获得了高阶无迹卡尔曼滤波器（Unscented Kalman filter,UKF）.同时验证了现有的五阶容积变换和五阶无迹变换分别是本文所提出的高阶无迹变换在κ=2和κ=6-n时的两个特例.进而分析和讨论了高阶无迹卡尔曼滤波器在系统不同维数条件下κ值的最优选取,并讨论了其稳定性.纯方位跟踪模型和弹道目标再入模型仿真验证了本文方法的正确性,且与现有方法相比具有更高的精度.     

Blind adaptive carrier phase recovery for QAM systems

This paper provides a study of adaptive phase recovery in quadrature amplitude modulation (QAM) based communication systems. Here, we modify the traditional fourth-power phase recovery algorithm (FP-PRA) to propose three improved algorithms, and analyze their performances in the presence of additive white Gaussian noise, phase noise, frequency-offset, and inter-symbol interference. We demonstrate that it is possible to obtain the optimal values of step-sizes (or loop-gains) in closed-form in the presence of phase noise and/or frequency-offset. In particular, we discuss two methods to improve FP-PRA. The first method involves utilizing the idea of partitioning the QAM constellation into QPSK-like and not-QPSK-like annular regions. The phase synchronizer is allowed to update only when a derotated QAM symbol lies in QPSK-like region; otherwise, the update process is stopped. The second method exploits an evolving idea of QAM-to-QPSK transformation, and uses transformed symbols to estimate phase mismatch. We provide a new interpretation of this transformation method and relate it to the quadrant-wise centroid of the rotated constellation. Furthermore, we discuss the feasibility of this method for both square and cross-QAM, and, identify and verify numerically the existence of false-locks in the case of cross-QAM. To the best of our knowledge, the ideas of constellation partitioning and constellation transformation have not appeared earlier in the context of adaptive phase recovery. We discuss adaptive blind estimation of optimal step-sizes in the presence of phase noise and frequency-offset. Finally, we discuss the modification of the proposed stochastic gradient methods to transform them into batch processing algorithms so as to make them more suitable for higher data rate systems. Numerical experiments indicate that the proposed algorithms can outperform the traditional FP-PRA algorithm for a number of practical QAM sizes under different mismatch conditions, and that our analytical findings are in close agreement with the simulation results.