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Stochastic averaging of quasi integrable and resonant Hamiltonian systems excited by fractional Gaussian noise with Hurst index 1/2

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Acta Mechanica Solida Sinica 2017年第1期30卷 11-19页

作者： Q.F.Lü M.L.Deng W.Q.Zhu Department of Mechanics State Key Laboratory of Fluid Power and Mechatronic SystemsKey Laboratory of Soft Machines and Smart Devices of Zhejiang ProvinceZhejiang University

A stochastic averaging method of quasi integrable and resonant Hamiltonian systems under excitation of fractional Gaussian noise （fGn） with the Hurst index 1/2 〈 H 〈 1 is proposed. First, the definition and the basic property of fGn and related fractional Brownian motion （iBm） are briefly introduced. Then, the averaged fractional stochastic differential equations （SDEs） for the first integrals and combinations of angle variables of the associated Hamiltonian systems are derived. The stationary probability density and statistics of the original systems are then obtained approximately by simulating the averaged SDEs numerically. An example is worked out to illustrate the proposed stochastic averaging method. It is shown that the results obtained by using the proposed stochastic averaging method and those from digital simulation of original system agree well.

关键词： Quasi integrable and resonant Hamiltonian system Fractional Brownian motion Fractional Gaussian noise Stochastic averaging method Internal resonant

Stochastic averaging of quasi partially integrable Hamiltonian systems under fractional Gaussian noise

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Journal of Zhejiang University-Science A(Applied Physics & Engineering) 2017年第9期18卷 704-717页

作者： Qiang-feng LU Mao-lin DENG Wei-qiu ZHU Department of Mechanics Zhefiang University Hangzhou 310077 China State Key Laboratory of Fluid Power and Mechatronic Systems Zhejiang University Hangzhou 310027 China Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province Zhejiang University Hangzhou 310027 China

A stochastic averaging method for predicting the response of quasi partially integrable and non-resonant Hamiltoniansystems to fractional Gaussian noise （fGla） with the Hurst index 1/2〈H〈l is proposed. The averaged stochastic differential equa-tions （SDEs） for the first integrals of the associated Hamiltonian system are derived. The dimension of averaged SDEs is less thanthat of the original system. The stationary probability density and statistics of the original system are obtained approximately fromsolving the averaged SDEs numerically. Two systems are worked out to illustrate the proposed stochastic averaging method. It isshown that the results obtained by using the proposed stochastic averaging method and those from digital simulation of originalsystem agree well, and the computational time for the former results is less than that for the latter ones.

关键词： Fractional Brownian motion (fBm) Fractional Gaussian noise (fGn) Quasi partially integrable Hamiltonian system Stochastic averaging method Stationary response

Transient stochastic response of quasi integerable Hamiltonian systems

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Acta Mechanica Sinica 2013年第4期29卷 602-611页

作者： Zhong-Hua Liu Jian-Hua Geng Wei-Qiu Zhu Department of Civil Engineering Xiamen University Department of Mechanics State Key Laboratory of Fluid Power Transmission and Control Zhejiang University

The approximate transient response of quasi in- tegrable Hamiltonian systems under Gaussian white noise excitations is investigated. First, the averaged It6 equa- tions for independent motion integrals and the associated Fokker-Planck-Kolmogorov （FPK） equation governing the transient probability density of independent motion integrals of the system are derived by applying the stochastic averag- ing method for quasi integrable Hamiltonian systems. Then, approximate solution of the transient probability density of independent motion integrals is obtained by applying the Galerkin method to solve the FPK equation. The approxi- mate transient solution is expressed as a series in terms of properly selected base functions with time-dependent coeffi- cients. The transient probability densities of displacements and velocities can be derived from that of independent mo- tion integrals. Three examples are given to illustrate the ap- plication of the proposed procedure. It is shown that the re- suits for the three examples obtained by using the proposed procedure agree well with those from Monte Carlo simula- tion of the original systems.

关键词： Transient response ~ Stochastic averagingmethod ~ Galerkin method ~ Quasi integrable Hamiltoniansystem

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拟哈密顿系统非线性随机最优控制

力学进展 2013年第1期43卷 39-55页

作者：朱位秋应祖光浙江大学航空航天学院力学系杭州310027

主要介绍近十几年来拟哈密顿系统非线性随机最优控制理论方法及其应用的研究成果,包括基于拟哈密顿系统随机平均法与随机动态规划原理的非线性随机最优控制基本策略,即响应极小化控制、随机稳定化、首次穿越损坏最小化控制、以概率密度... 详细信息

主要介绍近十几年来拟哈密顿系统非线性随机最优控制理论方法及其应用的研究成果,包括基于拟哈密顿系统随机平均法与随机动态规划原理的非线性随机最优控制基本策略,即响应极小化控制、随机稳定化、首次穿越损坏最小化控制、以概率密度为目标的控制,为将它们应用于工程实际而作的部分可观测系统最优控制、有界控制、时滞控制、半主动控制、极小极大控制的进一步研究,以及综合考虑这些实际问题的非线性随机最优控制的综合策略,非线性随机最优控制在滞迟系统、分数维系统等中的若干应用,介绍与这些研究有关的背景,并指出今后有待进一步研究的问题.

关键词：拟哈密顿系统非线性随机动力学非线性随机最优控制随机平均随机动态规划

Asymptotic stability with probability one of MDOF nonlinear oscillators with fractional derivative damping

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Science China(Physics,Mechanics & Astronomy) 2013年第11期56卷 2200-2207页

作者： CHEN LinCong LI HaiFeng LI ZhongShen ZHU WeiQiu College of Civil Engineering Huaqiao University College of Mechanical Engineering and Automation Huaqiao University Department of Mechanics State Key Laboratory of Fluid Power Transmission and ControlZhejiang University

In this paper,the asymptotic stability with probability one of multi-degree-of-freedom(MDOF)nonlinear oscillators with fractional derivative damping parametrically excited by Gaussian white noises is investigated.A stochastic averaging method and the Khasminskii’s procedure are employed to evaluate the largest Lyapunov exponent,whose sign determines the stability of the *** an example,two coupled nonlinear oscillators with fractional derivative damping is worked out to demonstrate the proposed procedure and to examine the effect of fractional order on the stochastic stability of *** particular,the case of factional order more than 1 is studied for the first time.

关键词： fractional derivative damping nonlinear oscillator stochastic stability Lyapunov exponent

Stochastic response of fractional-order van der Pol oscillator

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Theoretical & Applied Mechanics Letters 2014年第1期4卷 68-72页

作者： Lincong Chen Weiqiu Zhu College of Civil Engineering Huaqiao University Department of Mechanics Zhejiang University

We studied the response of fractional-order van de Pol oscillator to Gaussian white noise excitation in this letter. An equivalent integral-order nonlinear stochastic system is obtained to replace the given system based on the principle of minimum mean-square error. Through stochastic averaging, an averaged Ito equation is deduced. We obtained the Fokker–Planck–Kolmogorov equation connected to the averaged Ito equation and solved it to yield the approximate stationary response of the system. The analytical solution is confirmed by using Monte Carlo simulation.

关键词： fractional-order van de Pol oscillator Gaussian white noise stationary response equivalent nonlinear system method stochastic averaging

谐和与宽带噪声综合激励下多自由度非线性不确定系统的最优极小极大控制

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谐和与宽带噪声综合激励下多自由度非线性不确定系统的最优极小极...

分数高斯噪声激励下拟可积哈密顿系统的随机平均法

第十五届全国非线性振动暨第十二届全国非线性动力学和运动稳定性学术会议

作者：胡荣春应祖光朱位秋浙江大学航空航天学院力学系

本文介绍了谐和与宽带噪声激励下多自由度非线性不确定系统、基于随机平均法、随机动态规划原理和随机微分对策理论的一种随机极小极大最优控制策略。首先,考虑系统频率与谐和激励频率存在共振关系的情形下,运用广义谐和函数平均法,将... 详细信息

本文介绍了谐和与宽带噪声激励下多自由度非线性不确定系统、基于随机平均法、随机动态规划原理和随机微分对策理论的一种随机极小极大最优控制策略。首先,考虑系统频率与谐和激励频率存在共振关系的情形下,运用广义谐和函数平均法,将系统状态从速度和位移的快变量形式转化为能量与相位差的慢变量形式,得到部分平均的Ito随机微分方程;其次,给定控制性能指标,对于不确定多自由度非线性系统的随机最优控制,根据随机微分对策理论,将其转化为一个极小极大控制问题;再根据随机动态规划原理,建立关于能量与相位差的动态规划方程,即Hamilton-Jacobi-Isaacs(HJI)方程,由此得到最坏情形下的扰动参数与极小极大最优控制;然后,将最坏扰动参数与最优控制代入部分平均的Ito随机微分方程并完成平均,求解与完全平均的Ito随机微分方程相应的Fokker-Planck-Kolmogorov(FPK)方程,可得受控系统的响应量并计算控制效果;最后,将上述不确定拟哈密顿系统的随机最优控制策略应用于一个潜艇传动轴的纵向振动控制系统,并通过Monte Carlo数值模拟结果说明该随机极小极大控制策略的控制效果。

关键词：多自由度非线性系统不确定性极小极大最优控制随机动态规划原理随机平均法。

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分数高斯噪声激励下拟可积哈密顿系统的随机平均法

第十届动力学与控制学术会议

作者：吕强锋胡荣春朱位秋浙江大学航空航天学院应用力学研究所

近几十年来,基于高斯白噪声与扩散过程模型,非线性随机动力学取得了长足的发展。然而随着研究工作的深入,新的问题不断出现,其中最引人注目的要数反常扩散现象和动态系统响应的非Markov性,这些现象和性质在湍流、多孔介质渗流和软物质... 详细信息

近几十年来,基于高斯白噪声与扩散过程模型,非线性随机动力学取得了长足的发展。然而随着研究工作的深入,新的问题不断出现,其中最引人注目的要数反常扩散现象和动态系统响应的非Markov性,这些现象和性质在湍流、多孔介质渗流和软物质等研究中表现尤其突出。在理论研究方面,分数阶布朗运动非常适合用来模型化反常扩散过程,相应的激励和噪声则由分数阶高斯噪声来描述。目前,只得到了分数高斯噪声激励下奥恩斯坦-乌伦贝克(Ornstein-Uhlenbeck)系统的精确解。所以,有必要发展渐近或近似的分析方法,比如随机平均法,来预测高阶或高维系统在分数高斯噪声激励下的响应。针对上述情形,本文提出了拟可积哈密顿系统在分数高斯噪声激励下的随机平均法。首先,简要介绍了分数高斯噪声和分数布朗运动的定义、基本特点以及分数布朗运动的积分形式,并推导了分数高斯噪声的相关函数和谱密度。分数布朗运动的积分形式有对称积分、前向积分、后向积分,本文所采用的是后向积分。由于受分数高斯噪声激励的系统响应不是Markov过程,不能应用常规的伊藤微积分公式,本文根据后向积分推导了新的伊藤微分规则。然后,应用新的伊藤微分规则,根据分数噪声激励下的动态系统的平均原理推导了能量首次积分的平均随机微分方程。数值求解平均随机微分方程可以近似得到原方程的平稳概率密度和响应的统计特性,由于平均随机微分方程的维数要比原方程的少,数值计算时效率高。最后用一个算例来说明该方法的有效性,算例结果显示,用所提出的方法得出的结果与原方程的数值模拟结果吻合的很好。

关键词：拟可积哈密顿系统分数布朗运动分数高斯噪声随机平均法

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随机激励下粘弹性系统的稳定化控制

随机激励下粘弹性系统的稳定化控制