受宽带随机激励的最优反馈控制

文章提出了基于平稳响应的随机最优反馈控制策略，研究了受宽带随机激励的Hamilton系统的最优反馈控制。首先应用拟Hamilton系统随机平均法建立平均后的Ito随机微分方程；接着求解相应的Fokker—Plank—Kol—mogorov（FPK）方程得到系统精确平稳解；然后将控制力的均方值和Hamilton能量均值结合作为系统控制指标，得出最优反馈控制规律；最后用两个例子详细说明了这一控制方法的实施过程的有效性，数字模拟结果与理论结果完全吻合。     

Feedback minimization of the first-passage failure of a hysteretic system under random excitations

The stochastic optimal bounded control of a hysteretic system for minimizing its first-passage failure is presented. The hysteretic system subjected to random excitation is firstly replaced by an equivalent nonlinear non-hysteretic system. The controlled non-hysteretic system is reduced to a one-dimensional controlled diffusion process by using the stochastic averaging of the energy envelope method. The dynamical programming equations and their associated boundary and final-time conditions for the problems of maximization of reliability and mean first-passage time are formulated. The optimal control law is derived from the dynamical programming equations and the control constraints. The dynamical programming equations for the maximum reliability problem and the mean first-passage time problem are finalized and solved numerically. Finally, numerical results are worked out to illustrate the application and effectiveness of the proposed method.     

Reliability of quasi integrable generalized Hamiltonian systems

The reliability of quasi-integrable generalized Hamiltonian systems is studied. An m-dimensional integrable generalized Hamiltonian system has M Casimir functions C₁,…,C_M and n (n=(m−M)/2) independent first integrals H_M+1,…,H_M+n in involution. When an integrable generalized Hamiltonian system is subjected to light dampings and weakly stochastic excitations, it becomes a quasi-integrable generalized Hamiltonian system. The averaged Itô equations for slowly processes C₁,…,C_M,H_M+1,…,H_M+n can be obtained by using stochastic averaging method, from which a backward Kolmogorov equation governing the conditional reliability function and a Pontryagin equation governing the conditional mean of the first passage time are established. The conditional reliability function and the conditional mean of first passage time are obtained by solving these equations together with suitable initial condition and boundary conditions. Finally, an example of a 5-dimensional quasi-integrable generalized Hamiltonian system is worked out in detail and the solutions are confirmed by using a Monte Carlo simulation of the original system.     

Time delay induced effects on control of linear systems under random excitation

Recursive formulas in terms of statistics of the response of linear systems with time delay under normal white noise input are developed. Two alternative methods are presented, in order to capture the time delay effects. The first is given in an approximate solution obtained by expanding the control force in a Taylor series. The second, available for the stationary solution (if it exists) gets the variance of the controlled system, with time delay in an analytical form. The efficacy loss in terms of statistics of the response is discussed in detail.     

Nonlinear random vibration of the slender deep-water pier under seismic excitation

The present study investigates the nonlinear random vibration of the deep-water pier exposed to horizontal seismic excitation. First of all, the stochastic dynamic model of the pier is formulated. During the process, the pier is simplified as a cantilever beam fixed on the rock foundation, the seismic excitation is treated as Gaussian white noise, the hydrodynamic pressure is described with the radiation wave theory, and the equation for the nonlinear kinematic of the pier is deduced by the means of Kane’s method. Then, with the application of the stochastic averaging (SA) technique, the Fokker–Plank–Kolmogorov (FPK) equation governing the transient probability density function (PDF) of the amplitude envelope and the backward Kolmogorov (BK) equation for the conditional reliability function (CRF) are derived, respectively. The closed-form stationary PDF can be yielded directly from the reduced FPK equation, while the CRF and the conditional PDF of first-passage time are given after solving the BK equation numerically. Numerical discussions are performed to illustrate the trend of excitation intensity, mass ratio, immersion ratio, and inner and outer hydrodynamic effect on the stationary response and first-passage failure are examined, respectively. It has been shown that increases in the excitation intensity, mass ratio, and immersion ratio can amplify the response and reduce the reliability of the deep-water pier system. The hydrodynamic effect also leads to an amplification of the system response and a reduction in the reliability of the system. Similarly, the presence of inner water can also exacerbate these effects, and this phenomenon becomes more pronounced with increasing immersed ratios. Additionally, the analytical solution is validated by the result obtained by pertained Monte Carlo simulations (MCS). It is noted that this work will be helpful for the optimal seismic design of deep-water piers.     

Direct control method for improving stability and reliability of nonlinear stochastic dynamical systems

Optimal control for improving the stability and reliability of nonlinear stochastic dynamical systems is of great significance for enhancing system performances. However, it has not been adequately investigated because the evaluation indicators for stability (e.g. maximal Lyapunov exponent) and for reliability (e.g. mean first-passage time) cannot be explicitly expressed as the functions of system states. Here, a unified procedure is established to derive optimal control strategies for improving system stability and reliability, in which a physical intuition-inspired separation technique is adopted to split feedback control forces into conservative components and dissipative components, the stochastic averaging is then utilized to express the evaluation indicators of performances of controlled system, the optimal control strategies are finally derived by minimizing the performance indexes constituted by the sigmoid function of maximal Lyapunov exponent (for stability-based control)/the reciprocal of mean first-passage time (for reliability-based control), and the mean value of quadratic form of control force. The unified procedure converts the original functional extreme problem of optimal control into an extremum value problem of multivariable function which can be solved by optimization algorithms. A numerical example is worked out to illustrate the efficacy of the optimal control strategies for enhancing system performance.     

Time-delay stochastic optimal control and stabilization of quasi-integrable Hamiltonian systems

Innovative procedures for the time-delay stochastic optimal control and stabilization of quasi-integrable Hamiltonian systems subject to Gaussian white noise excitations are proposed. First, the problem of time-delay stochastic optimal control of quasi-integrable Hamiltonian systems is formulated and converted into the problem of stochastic optimal control without time delay. Then the converted control problem is solved by applying the stochastic averaging method for quasi-integrable Hamiltonian systems and the stochastic dynamical programming principle. The time-delay feedback stabilization of quasi-integrable Hamiltonian systems is formulated as an ergodic control problem with an un-determined cost function which is determined later by minimizing the largest Lyapunov exponent of the controlled system. As an example, a two-degree-of-freedom quasi-integrable Hamiltonian system with time-delay feedback control forces is investigated in detail to illustrate the procedures and their effectiveness.     

Reliability analysis for multi-state systems subject to distinct random shocks

This paper analyzes the competing and dependent failure processes for multi-state systems suffering from four typical random shocks. Reliability analysis for discrete degradation is conducted by explicitly modeling the state transition characteristics. Semi-Markov model is employed to explore how the system vulnerability and potential transition gap affect the state residence time. The failure dependence is specified as that random shocks can not only lead to different abrupt failures but also cause sudden changes on the state transition probabilities, making it easier for the system to stay at the degraded states. Reliability functions for all the exposed failure processes are presented based on the corresponding mechanisms. Interactions between different failure processes are also taken into account to evaluate the actual reliability levels in the context of degradation and distinct random shocks. An illustrative example of a multi-state air conditioning system is studied to demonstrate how the proposed method can be applied to the engineering practice.     

随机激振下直线振动筛的振幅分析

振幅是直线振动筛重要工作参数之一,直接影响到振动筛分效率及工作可靠性.考虑激振力幅值大小及方向的随机性,在建立随机模型基础上,研究随机激振参数(偏心块质量矩、激振力方向角、激振频率)对振动筛振幅的影响规律.引入相对均差系数及变差系数,定量描述振动筛振幅的变异规律.结果表明:偏心块质量矩随机对振幅的均值影响很小,但是引起的振幅动态偏离度不容忽视;激振力方向角随机引起振幅的均值变异较大,动态偏离度亦大,是引起振幅变异最敏感的激振参数;当偏心块质量矩、激振力方向角、激振频率同时随机且变差系数达到15%后,振幅变异梯度迅速增大,振幅的均值变异及动态偏离度都很大.振动筛结构在制造与安装过程中随机因素引起的振幅变异必须加以分析与有效控制.     

A solution method for linear and geometrically nonlinear MDOF systems with random properties subject to random excitation

A method for computing the lower-order moments of response of randomly excited multi-degree-of-freedom (MDOF) systems with random structural properties is proposed. The method is grounded in the techniques of stochastic calculus, utilizing a Markov diffusion process to model the structural system with random structural properties. The resulting state-space formulation is a system of ordinary stochastic differential equations with random coefficients and deterministic initial conditions which are subsequently transformed into ordinary stochastic differential equations with deterministic coefficients and random initial conditions. This transformation facilitates the derivation of differential equations which govern the evolution of the unconditional statistical moments of response. Primary consideration is given to linear systems and systems with odd polynomial nonlinearities, for in these cases there is a significant reduction in the number of equations to be solved. The method is illustrated for a five-story shear-frame structure with nonlinear interstory restoring forces and random damping and stiffness properties. The results of the proposed method are compared to those estimated by extensive Monte-Carlo simulation.     

随机地震激励作用下自复位结构的平稳响应

自复位结构是一种震后不需修复或仅需少量修复即可继续投入使用的新型抗震结构。目前有关自复位结构地震响应的研究多是在确定性激励下进行,鲜有涉及随机激励环境。假定地震动加速度过程为金井清过滤白噪声模型,研究了随机地震激励下单自由度自复位体系的平稳响应。应用广义谐波平衡技术分解旗帜形的恢复力,建立原系统的等效非线性随机系统。通过Stratonovich-Khasminskii极限定理作随机平均,得到关于幅值的近似一维扩散过程。建立并求解对应的FPK方程,得到关于幅值的稳态概率密度函数并进行参数分析。数值结果表明,能量耗散系数与屈服位移的减小能降低系统响应,同时,随着这两个参数的变化,系统会出现随机P分岔现象。通过蒙特卡罗数值模拟法验证了解析解的有效性。     

Response spectral density of linear systems with external and parametric non-Gaussian, delta-correlated excitation

An exact, closed form, analytical expressions for a response spectral density of a certain type of systems, subjected to non-Gaussian, stationary, delta-correlated noise are derived. A new, extended mean square stability conditions are derived for such systems.     

非线性流滞阻尼器耗能结构随机地震响应和首超时间分析

对非线性流滞阻尼器耗能结构在Kanai-Tajimi谱地震激励下的随机响应及其随机失效时间和动力可靠性进行了系统研究。首先建立了结构的非线性运动方程;然后,基于随机平均法,将结构响应幅值近似为一维markov扩散过程,获得了扩散过程漂移系数和扩散系数的解析表达式;其次,利用扩散过程与FPK方程的对应关系,获得了幅值平稳概率密度函数和幅值任意阶矩的解析表达式;再次,利用幅值与结构位移和速度的相互转化关系,获得了结构位移与速度的平稳联合概率密度函数和位移、速度方差以及位移期望穿越率的解析表达式;最后,利用扩散过程的后向Kolmogrov方程,基于首超失效模型,建立了结构动力可靠性函数方程和结构随机失效时间统计矩方程,并利用一维扩散过程的边界分类性质,将统计矩方程的奇异定性边界条件转化为等价的定量边界条件,进而获得了失效时间任意阶统计矩的解析解,并利用此矩,对结构动力可靠性和失效时间概率分布函数进行了近似分析,给出了算例,从而建立了结构非线性随机地震响应及其随机失效时间和动力可靠性的分析方法。     

Minimax optimal control of uncertain quasi-integrable Hamiltonian systems with time-delayed bounded feedback

A minimax optimal control strategy for uncertain quasi-integrable Hamiltonian systems with time-delayed bounded feedback control is proposed. First, a quasi-integrable Hamiltonian system with time-delayed bounded control forces and uncertain excitation and system parameters is converted into a set of Itô stochastic differential equations without time delay. Then, the partially averaged Itô stochastic differential equations for the energy processes are derived by using the stochastic averaging method for quasi-integrable Hamiltonian systems. For these equations together with an appropriate performance index, a worst-case optimal control strategy is derived via solving a stochastic differential game problem. The worst-case disturbances and the optimal bounded controls are obtained by solving a Hamilton–Jacobi–Isaacs (HJI) equation. Finally, two examples are worked out in detail to illustrate the application and effectiveness of the proposed method.     

线性奇异系统的H_2控制:状态反馈情形

研究一类线性奇异系统的H2控制问题,在较为一般的条件下,利用线性矩阵不等式方法给出一个使闭环系统无脉冲模,内稳定,并满足H2性能指标的状态反馈控制器存在的充要条件以及控制器的一族解.数值仿真证实了所提方法的有效性和实用性.     

Numerical and experimental studies of linear systems subjected to non-Gaussian random excitations

This paper presents a numerical simulation scheme for generating symmetric non-Gaussian random processes governed by prescribed kurtosis and spectral density. The generated process is represented as a continuous stationary random signal with occasional spikes superimposed on a Gaussian random background. The generated time history data records are used to simulate random excitations acting on linear single-degree-of-freedom systems. The results of the numerical simulation are compared with those measured experimentally. For a wide-band random excitation with kurtosis close to 3, the response kurtosis is found to be very sensitive to small changes in the excitation kurtosis. This is manifested by the appearance of significant spikes in the time history records when the excitation records do not display any significant spikes. The influence of the system damping is also examined for narrow-band and wide-band random excitations, and some differences are reported in the results.     

Transient and stationary response statistics of van der Pol oscillators subjected to broad band random excitation

The joint probability density function of the state space vector of a white noise excited van der Pol oscillator satisfies a Fokker-Planck-Kolmogorov (FPK) equation. The paper describes a numerical procedure for solving the transient FPK equation based on the path integral solution (PIS) technique. It is shown that by combining the PIS with a cubicB-spline interpolation method, numerical solution algorithms can be implemented giving solutions of the FPK equation that can be made accurate down to very low probability levels. The method is illustrated by application to two specific examples of a van der Pol oscillator.     

LMI的广义系统输出反馈H2控制

利用广义系统正常化方法,在较为一般的条件下,把存在输出反馈控制器使得闭环系统无脉冲模,内稳定且满足H2性能指标的充分条件表示成线性矩阵不等式（LMI）的形式.给出的控制器含有可调参数,便于工程应用.     

随机激励作用下硬涂层薄板振动有限元分析及减振预估

硬涂层减振是一项新兴的阻尼减振技术,硬涂层复合结构在随机激励载荷作用下的振动特性建模与分析方法还未开展研究.将有限元建模方法和虚拟激励法(pseudo excitation method,PEM)相结合,实现了随机激励作用下硬涂层悬臂薄板结构振动建模及减振性能预估.首先,基于等效单层法创建了双面涂敷硬涂层的薄板结构振动...