基于随机平均的非线性随机最优控制

首先建立非线性随机系统的最优控制问题,并介绍通过随机平均法导出平均系统、再由随机动态规划原理确定控制律的平均系统的非线性随机最优控制方法。然后,对于非线性随机系统的动态规划方程,提出应用随机平均法简化该方程、从而得到最优平均控制律的方法,并证明该最优平均控制律等价于平均系统的最优控制律。最后用一个例子说明方法及等价性,并指出在一定条件下,最优平均控制律将是动态规划方程的精确解。     

随机激励下一类包装振动系统的随机平均

对一类在拟周期和随机联合激励作用下的杜芬系统使用标准随机平均法,求取平均方程的扩散和漂移系数,并推导出系统在非共振与共振情形下的FPK方程.     

随机系统均方稳定Lyapunov型方程

研究随机系统的均方稳定性．得到了一个Lyapunov型方程，并证明随机系统是均方稳定的充分必要条件是该Lyapunov型方程有正定解.     

随机最优控制下矩形薄板受面内随机参数激励的首次穿越研究

根据建立了四边简支受控矩形薄板受面内高斯白噪声激励的振动模型,并用Galerkin变分法将其化简为二自由度常微分非线性动力学方程。又利用拟不可积Hamilton系统平均理论将方程等价为一个一维的Ito随机过程,随后结合随机动态规划方法,得到了使系统可靠性最大的随机最优控制策略。最后建立了受控系统的条件可靠性函数所满足的Backward Kolmogorov(BK)方程,根据初始条件和边界条件得出数值结果。数值结果表明,随机最优控制对系统的可靠性提升有明显作用     

随机激励下高维包装振动系统的可靠性分析

研究了两自由度包装振动系统在受高斯白噪声外激励下的可靠性.利用由随机平均法导出的关于系统的独立运动积分的随机平均方程,通过系统本身的特性以及对漂移与扩散系数作出的一些假定,给出了系统的可靠性函数以及首通损坏的条件转移概率密度函数所满足的后向柯尔莫哥洛夫方程和福克-普朗克方程,并进一步详细地讨论了其边界与可解性条件.数值结果表明,关于系统的可靠性函数以及首通损坏的条件转移概率密度函数方面的结果是比较合理的,这也表明文中的分析方法是正确的.     

电磁激振器的非线性随机参变振动

本文研究激振设备之一——电磁激振器在白噪声电流激励下的随机振动特性。文章推导了电磁激振器在直流和白噪声随机电流作用下的非线性随机参变方程,指出系统的响应构成各态历经的马尔科夫过程,运用随机参变振动理论中的马尔科夫方法、矩方程法和解决非线性问题的准矩截断法求出了晌应的统计特性以及对于输入电流、系统参数的限制条件。     

随机质量板的振动响应及其统计分析

以Rayleigh-Ritz能量法和Lagrange动力学方程为基础,提出了一种分析随机质量板的振动响应方法,并对其统计特性进行了分析.考虑到工程中各种因素导致板的不确定性,在分析板的振动响应时,首先将板当作是各向同性的均匀质量板.应用Rayleigh-Ritz能量法分析板的振动能量,将由各种不确定性导致的板的振动响应作为由板上的附加质量块引起的因素来考虑,通过附加质量块在板上的分布来建立其动能,然后应用Lagrange动力学方程建立随机质量板的运动方程,求解该运动方程来分析板的振动响应.该方法可以根据分析对象的边界条件和分析频率需要,选取合适的基函数,提高在高频时板的振动响应的分析精度.应用Monte Carlo数值仿真验证了该方法.结果表明,由该方法得到的随机质量板的特征频率空间与由随机矩阵理论预测的结果是一致的;验证了该方法的正确性.     

确定性周期与随机激励联合作用下非线性系统非平稳响应的统计线性化方法

提出一种用于求解确定性周期与非平稳随机激励联合作用下,单自由度非线性系统非平稳响应的统计线性化方法。将系统响应分解为确定性周期和零均值随机分量之和,则原非线性运动方程可等效地化为一组耦合的、分别以确定性和随机动力响应为未知量的非线性微分方程。利用统计线性化方法将非平稳随机激励作用下的非线性随机动力方程化为等效线性方程,得到关于线性随机响应二阶矩的李雅普诺夫方程。联立李雅普诺夫方程与谐波激励作用下的确定性微分方程,并利用数值算法对其进行求解。以蒙特卡洛模拟验证了此方法的适用性和精度。     

随机结构重特征值分析的递推随机有限元法

利用递推随机有限元方法研究了具有随机参数结构的重特征值问题。采用随机收敛的非正交多项式展式表示未知的随机重特征值和随机特征向量，建立了和摄动法类似的一系列确定的递推方程，通过求解这些速推方程，得到了重特征值的统计值。算例表明，同基于二阶泰勒展开的摄动随机有限元法相比，递推随机有限元法的结果能在较宽的随机涨落范围内更好地逼近蒙特卡洛模拟结果。     

随机结构动力分析的扩阶系统方法

本文发展了一类用于随机结构系统动力分析的扩阶系统方法。基于随机空间的正交展开理论,文中推导给出了扩阶系统动力方程的一般公式。这组公式可以适用于同时具有随机质量、随机阻尼、随机刚度参数的多自由度动力系统。通过引入虚拟结构的概念,文中建议了形成扩阶系统动力矩阵的有限元方法。通过对比算例分析,证实了扩阶系统方法的可行性与可靠性。最后,讨论了一个很重要现象：随机系统均值反应与相应均值参数系统反应之间的差异并指出了此差异的意义。     

二自由度碰撞振动系统的随机响应

用拟不可积哈密顿系统随机平均法研究了二自由度磁撞振动系统的随机响应，先将二自由度随机激励的碰撞振动系统表示成随机激励的耗散的哈密顿系统形式，然后用拟不可积哈密顿系统的随机平均法得到了以系统哈密顿函数为基本变量的一维It^↑o随机微分方程，最后用数值方法求解与该方程相应的稳态FPK方程，得到系统响应的平稳概率密度。两个算例的结果与数字模拟结果的比较表明了随机平均法在二自由度磁撞振动系统的随机响应分析中的有效性。     

迟滞的磁流变阻尼器的随机最优控制力

用Bouc-Wen迟滞模型描述磁流变阻尼器的动力学特性,分离阻尼器控制力的半主动部分和被动部分,被动部分结合到受控系统.先将该系统变换成等价的非迟滞的非线性随机控制系统,再运用随机平均法导出关于能量的It随机微分方程.根据随机动态规划原理,建立控制总能量的动态规划方程,并由此确定非clip的最优控制力.最后通过数值结果表明该控制力的有效性.     

邻接高耸结构的随机最优控制

文基于随机动态规划原理与随机平均法，提出耦合相邻高耸结构的随机最优控制方法。先建立任意层数并在任意层高处控制联接的耦合结构的缩聚模型，再运用随机平均法导出关于模态能量的It6随机微分方程，应用随机动态规划原理建立动态规划方程，由此可确定最优控制律。将结构的响应控制化为模态能量控制，缩减控制系统的维数。用高斯随机过程模拟地震激励，可计及其功率谱特性。数值结果表明该耦合结构控制方法的有效性。     

Stochastic stability of SDOF linear viscoelastic system under wideband noise excitation

The stochastic moment stability and almost-sure stability of a single-degree-of-freedom (SDOF) viscoelastic system subject to parametric fluctuation is investigated by using the method of higher-order stochastic averaging. The stochastic parametric excitation is modeled as a wideband noise, which is taken as Gaussian white noise and real noise. The viscoelastic material is assumed to follow ordinary Maxwell linear constitutive relation. For small damping and weak stochastic fluctuation, analytical expressions are derived for the moment Lyapunov exponent and the Lyapunov exponent, which indicate moment stability and almost-sure stability respectively. The effects of various system and loading parameters on the stochastic stability are discussed. Both analytical and simulation results show that higher-order stochastic averaging improves the accuracy compared with the first-order stochastic averaging. However, results of the third-order averaging are almost overridden by those of second-order averaging and the third-order averaging involves far more calculation. It is advisable to consider a balance between accuracy achievement and calculation endeavor when using higher-order stochastic averaging.     

含分数阶导数项的随机Duffing振子的稳态响应分析

本文对一个含有分数阶导数项阻尼的、Gaussian白噪声激励下的Duffing振子进行了稳态响应分析。首先,基于能量平衡理论,运用等效线性化方法,计算等效系统的线性阻尼及自然频率,建立统计意义下的等效线性化系统。然后,利用平均法建立随机Ito方程,得到随机响应的Markovian近似;给出描述振子振幅概率密度函数演化的Fokker-Planck方程,并得到它的稳态解。进一步,对于含有响应振幅的等效线性系统,借助由Laplace变换得到的转换函数,得到原系统的条件功率谱密度,结合振幅的稳态概率密度作为权重函数,给出原系统功率谱密度的估计,以及响应的统计量的估计。数值模拟的结果说明所提出的功率谱密度的近似解析表达式是可靠的,它甚至适用于Duffing振子具有强非线性回复力的情形,因为它可以较好的表现出功率谱密度共振频谱加宽及多峰现象的出现。     

Stochastic averaging in parametric regions near separatrices of integrability

Stochastic averaging in Hamiltonian framework, as a powerful tool for the dimensional reduction of strongly-nonlinear stochastic dynamical systems, achieves its enormous efficacy by distinguishing the integrability and resonance of associated conservative systems. Its implementation and mathematical expressions of drift and diffusion coefficients are completely different for different categories. For multi-degree-of-freedom systems, a slight changing of stiffness coefficients may induce an immediate conversion of the integrability, and correspondingly, notably different mathematical expressions, which means a sudden changing of dynamical behaviors. This work is devoted to this anti-intuitive phenomenon through a two-degree-of-freedom nonlinear stochastic system with adjustable parameters, by introducing the concept of the degree of integrability, applying stochastic averaging for quasi-integrable/quasi-nonintegrable systems to the parametric region near the separatrix of integrability, and comparing their accuracy of prediction. Numerical results illustrate that there exists a specific band around the separatrix of integrability, as system parameters fall in which, stochastic averaging for quasi-integrable systems achieves higher accuracy than that for quasi-nonintegrable systems even though the system itself is non-integrable. This work uncovers the existence of such specific band, and constitutes a necessary supplement to stochastic averaging in Hamiltonian framework.     

Stochastic response of nonlinear system in probability domain

A stochastic averaging procedure for obtaining the probability density function (PDF) of the response for a strongly nonlinear single-degree-of-freedom system, subjected to both multiplicative and additive random excitations is presented. The procedure uses random Van Der Pol transformation, Ito’s equation of limiting diffusion process and stochastic averaging technique as outlined by Zhu and others. However, the equations are rederived in generalized form and arranged in such a way that the procedure lends itself to a numerical computational scheme using FFT. The main objective of the modification is to consider highly irregular nonlinear functions which cannot be integrated in closed form and also to solve problems where analytical expressions for probability density function cannot be obtained. The procedure is applied to obtain the PDF of the response of Duffing oscillator subjected to additive and multiplicative random excitations represented by rational power spectral density functions (PSDFs). The results are verified by digital simulation. It is shown that the procedure provides results which compare very well with those obtained from simulation analysis not only for wide-band excitations but also for very narrow-band excitations, which are weak (when normalized with respect to mass of the system.) This paper is dedicated to Prof R N Iyengar of the Indian Institute of Science on the occasion of his formal retirement.     

Stochastic averaging of quasi-integrable Hamiltonian systems under bounded noise excitations

A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems to bounded noise excitations. Two cases of external resonance only and both internal and external resonances are considered separately. The reduced averaged Fokker–Planck–Kolmogorov equation is solved by using the combination of the finite difference method and the successive over relaxation method. An example of two coupled Duffing oscillators under bounded noise excitations is given to illustrate the application of the proposed procedure. It is shown that the analytical results agree well with those from digital simulation of original system.