STOCHASTIC OPTIMAL CONTROL FOR THE RESPONSE OF QUASI NON-INTEGRABLE HAMILTONIAN SYSTEMS

A strategy is proposed based on the stochastic averaging method for quasi non- integrable Hamiltonian systems and the stochastic dynamical programming principle.The pro- posed strategy can be used to design nonlinear stochastic optimal control to minimize the response of quasi non-integrable Hamiltonian systems subject to Gaussian white noise excitation.By using the stochastic averaging method for quasi non-integrable Hamiltonian systems the equations of motion of a controlled quasi non-integrable Hamiltonian system is reduced to a one-dimensional av- eraged It stochastic differential equation.By using the stochastic dynamical programming princi- ple the dynamical programming equation for minimizing the response of the system is formulated. The optimal control law is derived from the dynamical programming equation and the bounded control constraints.The response of optimally controlled systems is predicted through solving the FPK equation associated with It stochastic differential equation.An example is worked out in detail to illustrate the application of the control strategy proposed.

FIRST-PASSAGE TIME OF QUASI-NON-INTEGRABLE-HAMILTONIAN SYSTEM

Studies on first-passage failure are extended to the multi-degree-of-freedom quasi-non-integrable-Hamiltonian systems under parametric excitations of Gaussian white noises in this paper. By the stochastic averaging method of energy envelope, the system's energy can be modeled as a one-dimensional approximate diffusion process by which the classical Pontryagin equation with suitable boundary conditions is applicable to analyzing the statistical moments of the first-passage time of an arbitrary order. An example is studied in detail and some numerical results are given to illustrate the above procedure.

STOCHASTIC HOPF BIFURCATION IN QUASIINTEGRABLE-HAMILTONIAN SYSTEMS

A new procedure is developed to study the stochastic Hopf bifurcation in quasi- integrable-Hamiltonian systems under the Gaussian white noise excitation.Firstly,the singular bound- aries of the first-class and their asymptotic stable conditions in probability are given for the averaged Ito differential equations about all the sub-system's energy levels with respect to the stochastic aver- aging method.Secondly,the stochastic Hopf bifurcation for the coupled sub-systems are discussed by defining a suitable bounded torus region in the space of the energy levels and employing the theory of the torus region when the singular boundaries turn into the unstable ones.Lastly,a quasi-integrable- Hamiltonian system with two degrees of freedom is studied in detail to illustrate the above procedure. Moreover,simulations by the Monte-Carlo method are performed for the illustrative example to verify the proposed procedure.It is shown that the attenuation motions and the stochastic Hopf bifurcation of two oscillators and the stochastic Hopf bifurcation of a single oscillator may occur in the system for some system's parameters.Therefore,one can see that the numerical results are consistent with the theoretical predictions.

A Class of Quasi-exact Solutions of Rabi Hamiltonian

A class of quasi-exact solutions of the Rabi Hamiltonian,which describes a two-level atom interacting witha single-mode radiation field via a dipole interaction without the rotating-wave approximation,are obtained by using awavefunction ansatz.Exact solutions for part of the spectrum are obtained when the atom-field coupling strength and thefield frequency satisfy certain relations.As an example,the lowest exact energy level and the corresponding atom-fieldentanglement at the quasi-exactly solvable point are calculated and compared to results from the Jaynes-Cummings andcounter-rotating cases of the Rabi Hamiltonian.

Quasi-Exactly Solvable Time-Dependent Hamiltonians

A generalized method which helps to find a time-dependent Schr&#214dinger equation for any static potential is established. We illustrate this method with two examples. Indeed, we use this method to find the time-dependent Hamiltonian of quasi-exactly solvable Lamé equation and to construct the matrix 2 × 2 time-dependent polynomial Hamiltonian.

Establishment of infinite dimensional Hamiltonian system of multilayer quasi-geostrophic flow & study on its linear stability

A multilayer flow is a stratified fluid composed of a finite number of layers with densities homogeneous within one layer but different from each other. It is an intermediate system between the single-layer barotropic model and the continuously stratified baroclinic model. Since this system can simulate the baroclinic effect simply, it is widely used to study the large-scale dynamic process in atmosphere and ocean. The present paper is concerned with the linear stability of the multilayer quasi-geostrophic flow, and the associated linear stability criteria are established. Firstly, the nonlinear model is turned into the form of a Hamiltonian system, and a basic flow is defined. But it cannot be an extreme point of the Hamiltonian function since the system is an infinite-dimensional one. Therefore, it is necessary to reconstruct a new Hamiltonian function so that the basic flow becomes an extreme point of it. Secondly, the linearized equations of disturbances in the multilayer quasi-geostrophic flow are derived by introducing infinitesimal disturbances superposed on the basic flows. Finally, the properties of the linearized system are discussed, and the linear stability criteria in the sense of Liapunov are derived under two different conditions with respect to certain norms.

基于受控耗散Hamiltonian系统模型的光伏准Z源T型三电平并网逆变器控制策略

基于端口受控耗散哈密顿系统(PCHD)模型的无源控制策略无需对受控对象进行线性化处理,它从能量成型与注入阻尼方面讨论受控对象的稳定性,简化了控制系统的结构。首先,利用状态空间法推导出准Z源T型三电平逆变器数学模型;其次,结合无源PCHD模型的控制规律选取能量函数与注入阻尼,并进行控制系统的设计。该控制策略不但使系统具有优良的动、静特性,且逆变器输出电流谐波低、经济性高;最后,仿真和硬件实验验证了基于PCHD模型的无源控制策略的正确性和有效性。

Energy diffusion controlled reaction rate in dissipative Hamiltonian systems

In this paper the energy diffusion controlled reaction rate in dissipative Hamiltonian systems is investigated by using the stochastic averaging method for quasi Hamiltonian systems. The boundary value problem of mean first- passage time （MFPT） of averaged system is formulated and the energy diffusion controlled reaction rate is obtained as the inverse of MFPT. The energy diffusion controlled reaction rate in the classical Kramers bistable potential and in a two-dimensional bistable potential with a heat bath are obtained by using the proposed approach respectively. The obtained results are then compared with those from Monte Carlo simulation of original systems and from the classical Kraraers theory. It is shown that the reaction rate obtained by using the proposed approach agrees well with that from Monte Carlo simulation and is more accurate than the classical Kramers rate.

基于拟哈密顿理论的随机电力系统暂态稳定性分析

蒙特卡洛法分析随机稳定性所需的计算量大,且难以同时考虑随机过程中发生的大量不确定性。基于拟哈密顿系统随机平均方法,提出随机电力系统暂态稳定性分析法。首先,根据扩展等面积法,将受扰多机系统动态映射为两群系统,建立其随机微分方程模型。忽略相关的非哈密顿因素,按哈密顿能量函数确定其安全区域。然后,考虑随机扰动、阻尼等的影响,对等效拟哈密顿系统进行随机平均,求出支配暂态能量转移的平均扩散方程,并基于扩散理论,根据系统条件可靠性分析切除时间、阻尼系数及激励强度等对随机电力系统暂态稳定性的影响。通过4机2区随机系统验证了该方法的有效性。

基于拟可积Hamilton系统的铁路桥梁动力可靠度计算研究

铁路运输提速增载,迫切需要对铁路桥梁动力可靠度进行研究。本文基于振型空间,首先推导出结构体系的广义动能和势能,继而基于拟Hamilton系统理论,推导出铁路混凝土桥梁的拟可积Hamilton系统方程。只考虑横向和扭转位移,得到铁路混凝土桥梁的条件可靠性函数所满足的BK方程及其定量边界、初值条件,可用中心差分法进行求解。以实际桥梁为算例,用上述方程求解其在列车荷载下的动力可靠度,得出或验证了与实际情况相符的若干重要结论,结果表明:动力可靠度和概率密度峰值,随桥梁初始能量的增大而减小,随桥梁边界能量的增大而增大;不同跨度桥梁分析结果与实际情况相符,说明基于拟可积Hamilton系统理论计算铁路桥梁的动力可靠度是可行的。

提高送端多直流落点系统暂态稳定性的非线性控制策略

为改善送端多直流落点系统暂态稳定性,提出了一种高压直流输电(HVDC)与发电机励磁协调的非线性控制策略。通过建立送端多直流落点系统非线性微分代数方程模型,将结构保持模型下的能量函数与广义拟Hamilton理论结合,同时兼顾发电机励磁与HVDC控制的协同作用,设计出一种HVDC与发电机励磁的综合控制规律。该方法从能量的角度出发,充分考虑电力系统的强非线性特性,而且得到的控制器结构简单,易于工程实现。仿真结果表明,该控制规律能够有效提高系统阻尼,起到功率支援的作用,改善系统的暂态稳定性。

碰振准哈密顿系统局部亚谐轨道的Melnikov方法

将Melnikov方法应用于碰振准哈密顿系统的局部亚谐轨道,推导出了局部亚谐轨道的Melnikov函数。该函数可以用于确定存在局部亚谐轨道及极限环分岔的条件。给出的局部亚谐轨道Melnikov函数和同宿轨道Melnikov函数共同揭示了准哈密顿系统的诸多动力学特征;然后,通过数值方法模拟受迫碰振振子运动,验证了局部亚谐Melnikov函数的正确性。

拟可积哈密顿系统中噪声诱发的混沌运动

研究拟可积哈密顿系统在谐和与噪声激励联合作用下的混沌运动。通过对噪声性质的假 定，并利用动力系统理论，给出了高维梅尔尼科夫方法应用于随机拟可积哈密顿系统的推广形 式．根据这种推广的方法，研究了谐和与高斯白噪声激励联合作用下两自由度拟可积哈密顿系 统的同宿分岔，得出了系统发生混沌运动的参数阈值，并由此讨论了噪声对系统的混沌运动的 影响．蒙特－卡罗方法模拟、李雅普诺夫指数等数值结果表明。

拟哈密顿系统非线性随机最优控制

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.

提高送端多直流落点系统暂态稳定性的L_2增益控制

提出了一种改善送端多直流落点系统暂态稳定性的L2增益控制。首先将送端直流线路传送的有功功率简化为送端系统的正的有功负荷,构造送端多直流落点系统非线性微分代数方程模型;然后采用预置反馈的方法,得到包含多条直流线路的结构保持模型的拟哈密尔顿实现;最后基于拟哈密尔顿实现,针对系统外界干扰,通过非线性干扰抑制方法,得到高压直流与发电机励磁的控制规律,使得系统的干扰量到输出量的L2增益低于给定的水平。仿真结果表明,与传统控制器比较,所设计的控制器能够保证系统的渐进稳定性,有效抑制干扰,具有较强的鲁棒性。

一类拟 Hamilton 碰振系统的全局分岔及多解共存现象分析

研究了一类拟Hamilton碰振系统的全局动力学特性,参照同宿轨道的Melnikov函数形式,构造了周期轨道次谐Melnikov函数.并用一类拟Hamilton碰振系统详细介绍了其计算方法和运用,数值结果验证了构造的次谐Melnikov函数的有效性.另外用改进的胞映射方法对这类系统的全局分岔和多解共存现象进行了分析,发现随着外激励力的变动吸引子数量发生变化,各个吸引域形态复杂且相互缠绕.

随机非线性物价模型的最优控制

将政府对价格系统的宏观调控作为外部控制力,建立受控的随机非线性物价模型:利用拟Hamilton系统随机平均法和随机动态规划原理的非线性随机控制策略对系统实施最优控制,控制目标是实现系统的稳定性变大;并通过对比控制前后的Lyapunov指数值说明了控制的有效性.

基于拟不可积哈密顿理论的桥梁动力可靠度计算

为提高铁路桥梁动力可靠度计算的效率,考虑结构非线性,基于振型空间,导出了铁路桥梁动能和势能的表达式,进而根据拟Hamilton系统理论确定铁路混凝土桥梁的广义动量、广义速度、Hamilton函数及拟Hamilton系统方程.只考虑横向位移和扭转位移,导出了铁路混凝土桥梁的拟不可积Hamilton系统方程,得到条件可靠度函数应满足的后向Kolmogorov方程及其定量边界、初值条件,并用中心差分法求解该方程.以实际铁路桥梁为算例,用上述方程求解其在列车荷载作用下的动力可靠度.研究结果表明:非线性桥梁结构的动力可靠度和概率密度峰值随桥梁初始能量增大而减小,随桥梁临界能量增大而增大;不同跨度桥梁的分析结果与实际情况相符,说明基于拟不可积Hamilton系统理论计算铁路桥梁的非线性动力可靠度是可行的.

轨道随机激励对弹性约束轮对随机稳定性与随机分岔的影响

考虑轨道随机不平顺激励与结构自身随机参激建立了弹性约束轮对系统的伊藤随机微分方程组,运用拟不可积Hamilton系统理论和奇异边界性态求解了系统的随机局部稳定性和随机全局稳定性,通过分析稳态概率密度和联合概率密度得到了模型的随机Hopf分岔类型并讨论了分岔的条件.可知,分岔的发生不仅受到系统固有参数的影响同时也受随机因素的影响,即使满足一定的分岔条件分岔也并不一定会发生,系统分岔的发生是以概率特征来体现的,这和只能考虑系统固有参数下的确定性分岔有着明显差别;另外,不同随机强度下轮对系统有着不同的失稳临界速度,这和不能考虑随机因素作用下的确定性轮对系统只有一个确定的失稳临界速度有着本质区别.

拟不可积Hamilton系统响应的随机最优控制

提出了一种基于拟不可积Hamilton系统随机平均法和随机动态规划原理的控制策略 .它可以对受高斯白噪声激励的拟不可积Hamilton系统进行非线性随机最优控制 ,以使系统的响应最小化 .利用拟不可积Hamilton系统随机平均法可以将受控的拟不可积Hamilton系统降维成一维的It 随机微分方程 .利用随机动态规划原理可以为系统响应最小化问题建立动态规划方程 .在控制力为有界的条件下 ,从动态规划方程中可以确定出最优控制规律 .受控系统的响应是通过求解与It 随机微分方程相联系的FPK方程得到的 ,用一个例子阐述了这一随机最优控制策略的实施过程 .