随机和区间非齐次线性哈密顿系统的比较研究及其应用

随着计算机技术的飞速发展,更高效、更稳定和长时间模拟能力更强的数值算法需求迫切.哈密顿系统辛算法与传统算法相比在稳定性和长期模拟方面具有显著优越性.但动力系统中不可避免地存在大量不同程度的不确定性,动力学分析中需要考虑这些不确定性的影响以确保合理有效性. 然而,目前考虑参数不确定性的哈密顿系统响应分析的研究基础还比较薄弱. 为此,本文考虑随机和区间参数不确定性,对两种不确定性非齐次线性哈密顿系统分析计算结果进行了比较研究,从而突破了传统哈密顿系统的局限性, 并应用于结构动力响应评估中. 首先,针对确定性非齐次线性哈密顿系统, 提出了考虑确定性扰动的参数摄动法;在此基础上, 分别提出了随机、区间非齐次线性哈密顿系统的参数摄动法,得到了它们响应界限的数学表达; 随后,用数学理论推导得到了区间响应范围包含随机响应范围的相容性结论; 最后,两个数值算例在较小时间步长下验证了所提方法在结构动力响应中的可行性和有效性,体现了随机、区间哈密顿系统响应结果之间的包络关系,并在较大时间步长下与传统方法相比较凸显了哈密顿系统辛算法的数值计算优势、与蒙特卡洛模拟方法相比较验证了所提方法的精度.      

区间参数结构的动力响应优化

讨论区间参数结构的动态响应问题的区间优化方法．利用摄动理论和函数区间扩张,将区间优化问题转化为近似的确定性优化问题．由于区间设汁变量的中值和不确定性半径均可取作优化参数,昕以可得到比确定性优化更多的优化信息．将该方法应j用于桁架结构,算例表明该方法是有效的．     

DYNAMIC OPTIMIZATION FOR UNCERTAIN STRUCTURES USING INTERVAL METHOD

An interval optimization method for the dynamic response of structures with interval parameters is presented. The matrices of structures with interval parameters are given. Combining the interval extension with the perturbation, the method for interval dynamic response analysis is derived. The interval optimization problem is transformed into a corresponding deterministic one. Because the mean values and the uncertainties of the interval parameters can be elected design variables, more information of the optimization results can be obtained by the present method than that obtained by the deterministic one. The present method is implemented for a truss structure. The numerical results show that the method is effective.     

Robust attitude control for rapid multi-target tracking in spacecraft formation flying

A robust attitude tracking control scheme for spacecraft formation flying is presented. The leader spacecraft with a rapid mobile antenna and a camera is modeled. While the camera is tracking the ground target, the antenna is tracking the follower spacecraft. By an angular velocity constraint and an angular constraint, two methods are proposed to compute the reference attitude profiles of the camera and antenna, respectively. To simplify the control design problem, this paper first derives the desired inverse system （DIS）, which can convert the attitude tracking problem of 3D space into the regulator problem. Based on DIS and sliding mode control （SMC）, a robust attitude tracking controller is developed in the presence of mass parameter uncertainties and external disturbance. By Lyapunov stability theory, the closed loop system stability can be achieved. The numerical simulations show that the proposed robust control scheme exhibits significant advantages for the multi-target attitude tracking of a two-spacecraft formation.     

FREQUENCY DOMAIN CRITERIA FOR ROBUST D-STABILITY OF MIMO SYSTEMS BASED ON LMI METHOD

The problem of checking robust D-stability of multi-in and multi-out (MIMO) systems was studied. Three system models were introduced, i.e. multilinear polynomial matrix, polytopic polynomial matrix and feedback system model. Furthermore, the convex property of each model with respect to the parametric uncertainties was established respectively. Based on this, sufficient conditions for D-stability were expressed in terms of linear matrix inequalities (LMIs) involving only the convex vertices. Therefore, the robust D-stability was tested by solving an LMI optimal problem.     

有界不确定参数结构位移范围的区间摄动法

当结构参数具有误差或有界不确定性时,区间数学可以在不同知识不确定变量的概率分布的情况下定量地考察不确定参数对结构响应的影响。为计算出不确定结构参数对结构位移影响范围的上下界,文中提出了的两种区骚动国方法。     

Adaptive interval type-2 fuzzy sliding mode control for unknown chaotic system

In this paper, a novel adaptive interval type-2 fuzzy sliding mode control (AIT2FSMC) methodology is proposed based on the integration of sliding mode control and adaptive interval type-2 fuzzy control for chaotic system. The AIT2FSMC system is comprised of a fuzzy control design and a hitting control design. In the fuzzy control design, an interval type-2 fuzzy controller is designed to mimic a feedback linearization (FL) control law. In the hitting control design, a hitting controller is designed to compensate the approximation error between the FL control law and the interval type-2 fuzzy controller. The parameters of the interval type-2 fuzzy controller, as well as the uncertainty bound of the approximation error, are tuned adaptively. The adaptive laws are derived in the sense of Lyapunov stability theorem, thus the stability of the system can be guaranteed. The proposed control system compared to adaptive fuzzy sliding mode control (AFSMC). Simulation results show that the proposed control systems can achieve favorable performance and robust with respect to system uncertainties and external disturbances.     

含概率与区间混合不确定性的系统可靠性分析方法

系统可靠性问题中通常存在大量的不确定参数,传统方法一般是基于概率模型对系统进行可靠性分析,但是实际工程中由于数据缺乏或试验条件的限制往往难以得到参数的精确概率分布.本文将结构体系一部分样本信息充足的不确定变量用随机变量进行描述,而另一部分样本缺乏的用区间表示,并提出了一种新的含概率与区间混合不确定性的系统可靠性分析方法.首先,基于一个高效求解方法获得单失效模式下结构的最小可靠度指标;再针对多失效模式下含概率与区间混合不确定性问题建立了系统可靠性分析模型;考虑各失效模式之间的相关性,通过线性相关度计算方法求得相关系数矩阵;最后提出了串联体系和并联体系可靠度求解方法.3个数值算例表明,该方法可以实现含概率与区间混合的多个非线性失效模式下系统可靠度的计算.通过对比传统的概率可靠性分析方法,本文方法只需要少量的不确定信息便可确保系统更加安全,更适合复杂结构系统可靠性的分析和设计.     

Sensitivity‐guided decision‐making for wind farm micro‐siting

This paper presents a quantitative risk assessment for design and development of a renewable energy system to support decision‐making among design alternatives. Throughout the decision‐making phases, resources are allocated among exploration and exploitation tasks to manage the uncertainties in design parameters and to adapt designs to new information for enhanced performance. The resource allocation problem is formulated as a sequential decision feedback loop for a quantitative analysis of exploration and exploitation trade‐offs. We support decision‐making by tracking the evolution of uncertainties, the sensitivity of design alternatives to the uncertainties, and the performance, reliability, and robustness of each design. This is achieved by analyzing the uncertainties in the wind resource, the turbine performance and operation, and the models that define the power curve and wake deficiency. Comparison of the performance, reliability, and robustness of aligned and staggered turbine layouts before and after wind assessment experiments aids in improving micro‐siting decisions. The results demonstrate that design decisions can be supported by efficiently allocating resources towards improved estimates of achievable design objectives and by quantitatively assessing the risk in meeting those objectives. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite element analysis of post-buckling dynamics in plates—Part I: An asymptotic approach

Various static and dynamic aspects of post-buckled thin plates, including the transition of buckled patterns, post-buckling dynamics, secondary bifurcation, and dynamic snapping (mode jumping phenomenon), are investigated systematically using asymptotical and non-stationary finite element methods. In part I, the secondary dynamic instability and the local post-secondary buckling behavior of thin rectangular plates under generalized (mechanical and thermal) loading is investigated using an asymptotic numerical method which combines Koiter’s nonlinear instability theory with the finite element technique. A dynamic multi-mode reduction method—similar to its static single-mode counterpart: Liapunov–Schmidt reduction—is developed in this perturbation approach. Post-secondary buckling equilibrium branches are obtained by solving the reduced low-dimensional parametric equations and their stability properties are determined directly by checking the eigenvalues of the resulting Jacobian matrix. Typical post-secondary buckling forms—transcritical, supercritical and subcritical bifurcations are observed according to different combinations of boundary conditions and load types. Geometric imperfection analysis shows that not only the secondary bifurcation load but also changes in the fundamental post-secondary buckling behavior are affected. The post-buckling dynamics and the global analysis of mode jumping of the plates are addressed in part II.     

Stability of nonlinear time-varying perturbed differential equations

The goal of this paper is twofold. The first part presents a converse Lyapunov theorem for the notion of uniform practical exponential stability of nonlinear differential equations in presence of small perturbation. This class of nonlinear differential equations can be viewed as parametric differential equations. The second part provides the classical perturbation method of seeking an approximate solution as a finite Taylor expansion of the exact solution. The practical asymptotic validity on the approximate is established on infinite-time interval. Finally, we give a numerical example to prove the validity of our methods.     

Dynamic stability of uncertain laminated beams subjected to subtangential loads

Because of the inherent complexity of fiber-reinforced laminated composites, it can be challenging to manufacture composite structures according to their exact design specifications, resulting in unwanted material and geometric uncertainties. Thus the understanding of the effect of uncertainties in laminated structures on their static and dynamic responses is highly important for a reliable design of such structures. In this research, we focus on the probabilistic stability analysis of laminated structures subject to subtangential loading, a combination of conservative and nonconservative tangential loads, using the dynamic criterion. In order to study the dynamic behavior by including uncertainties into the problem, three models were developed: exact Monte Carlo simulation, sensitivity-based Monte Carlo simulation, and probabilistic FEA. These methods were integrated into the existing finite element analysis. Also, perturbation and sensitivity analysis have been used to study nonconservative problems to study the stability analysis using the dynamic criterion.     

An interval perturbation method for exterior acoustic field prediction with uncertain-but-bounded parameters

This paper proposes two interval analysis methods, called the first-order interval parameter perturbation method (FIPPM) and the modified interval parameter perturbation method (MIPPM), for use in exterior acoustic field prediction when there are uncertainties in both the material properties and the external load. Interval variables are used to quantitatively describe the uncertain parameters in the face of limited information. The conventional first-order Taylor expansion and perturbation terms are employed in the FIPPM, while the MIPPM introduces modified Taylor series to approximate the non-linear interval matrix and vector. The high-order terms of the Neumann expansion are retained to calculate the interval matrix inverse. A numerical example is given by comparing the results with a Monte Carlo simulation to demonstrate the feasibility and effectiveness of the proposed methods at evaluating the sound pressure ranges in an exterior acoustic field.     

计算不确定结构系统静态响应的一种可靠方法

不确定性广泛存在于工程结构分析和设计过程之中，不能简单地予以忽略。目前，概率方法、模糊方法和区间方法是不确定性建模的三种主要方法。本文把具有不确定性的结构材料参数、几何参数和所受外力用区间数描述，通过求解线性区间方程组准确地计算了结构静态响应。计算结果易于扩张是区间计算的一个主要缺陷，本文提出了一种有效避免这一问题的方法。该方法把区间函数的计算和区间线性方程组的求解转化为相应的全局优化问题，来确定解中的每个区间元素的边界值，并采用一种智能性算法(实数编码遗传算法)来求解这些全局优化问题。本文首先采用数学和结构分析算例对该方法的正确性和有效性进行了验证，然后把该方法与有限元方法相结合计算不确定结构系统的响应范围，并和求解同类问题的方法进行了比较。     

On robust stability of a class of polynomial families

In this paper,we discuss the robust stability of a class of polynomial families more general than the interval polynomial family and diamond polynomial family.We prove that the Hurwitz stability of some special cases of this class of polynomial families can be determined by checking finite polynomials.We also give an example to illustrate that it is not always possible to determine the Hurwitz stability of all this class of polynomial families by checking finite polynomials.     

计算具有区间参数结构特征值范围的一种新方法

基于区间效学的包含单调性和区间函效所表述的实际物理意义，把广义区间特征值问题转化为两个以非确定参效为优化变量，以关心的特征值为目标函效的全局优化问题，并采用遗传算法对优化问题求解，计算得到结构特征值的区间范围。通过效值算例对本文方法的有效性进行了验证，并和区间摄动法的计算结果进行了比较。     

Adaptive control of rotary inverted pendulum system with time-varying uncertainties

In this paper, an adaptive controller is proposed to balance a rotary inverted pendulum with time-varying uncertainties. The goal of the control is to bring the pendulum close to the upright position regardless of the various uncertainties and disturbances. Its underactuated dynamics is first decoupled by Olfati’s transformation into a cascade form, and then an adaptive controller is designed to deal with the uncertainties in the new space. Based on the Lyapunov-like theory, the closed loop stability and boundedness of all internal signals can be proved. The simulation results show that the proposed scheme is capable of giving good performance, as desired.     

弹性关节空间机械臂级联智能滑模控制

谐波减速器和力矩传感器等柔性元件因其独特性能而广泛应用在空间机器人关节系统中，以获取高减速比．但同时这些柔性元件的存在为空间机械臂系统引入了关节柔性，使得对其稳定控制变得更为复杂．基于此，文中讨论了基于自适应回归小波神经网络(Self-Recurrent Wavelet Neural Networks, SRWNN)的弹性关节空间机械臂系统动力学建模及级联智能滑模控制．首先，利用级联系统理论及第二类拉格朗日方法推导出了由外环刚性臂子系统和内环关节电机转子子系统组成的系统级联动力学模型；其次，为两个子系统分别设计了内、外环自适应滑模回归小波神经网络控制．外环控制算法以期望轨迹为控制量，而其控制信号作为抑制弹性关节振动的内环控制算法的控制量，整个控制系统由内、外环控制系统叠加而成；而后，基于Lyapunov稳定性理论证明了整个控制系统的稳定性并设计了自适应回归小波神经网络的各权值参数在线学习算法．所提的控制算法有效地消除了模型不确定的影响，避免了复杂的求导计算和角加速度可测的要求，同时，控制系统设计过程中未涉及惯常奇异摄动双时标分解操作，在理论上适合任意大小的关节柔性刚度．最后，系统对比仿真试验证明了所提的级联智能控制算法优于惯常基于奇异摄动法和基于柔性铰补偿奇异摄动法的控制方案．     

Dynamic operation modes of AFM: Non-linear behavior and theoretical analysis of the stability of the AFM oscillator

The first part of this paper gives a theoretical study of the mechanics of contact of an AFM tip on viscous materials. Analytical expressions are derived showing the non-linear behaviors specifically related to the use of dynamic operation modes of AFM on viscous materials. A detailed analysis of the dissipated energy as a function of the tip indentation is presented. The second part is dedicated to a theoretical analysis investigating the domain of stability of the oscillator and the influence of the machine. The theoretical approach includes the electronic feedback loop used with the frequency modulation mode. Because the interaction between the tip and the sample produces a dynamical non-linear behavior, an unstable branch occurs that can change the stability of the oscillator. In particular, a sudden jump of the oscillating tip can be produced. In spite of the complexity of the problem, the analytical approach ends with two simple equations. The two equations provide an unambiguous way of discriminating between the contributions from the machine and the tip sample interaction.     

漂浮基柔性关节空间机器人自适应滑模运动控制及振动主动抑制

讨论了关节柔性且系统参数不确定的漂浮基空间机器人系统的动力学建模过程、运动轨迹跟踪控制算法设计及系统柔性振动的主动抑制问题。利用系统动量、动量矩守恒关系和拉格朗日法对系统动力学进行分析,并建立系统动力学方程。基于奇异摄动法将系统分解为表示系统刚性运动部分的慢变子系统和表示系统柔性运动部分的快变子系统。针对慢变子系统提出了一种自适应滑模控制算法。该控制算法是由基于滑模面的等效控制项、自适应控制项和PID反馈控制项组成。因此,它集合了滑模控制、自适应算法和PID技术的优点,且弥补了三种算法各自的缺点。该控制算法能够有效地补偿系统的转动误差和不确定参数,提高控制系统的精度。针对快变子系统,提出基于速度差值的反馈控制算法来抑制柔性关节引起的系统柔性振动,保证系统的稳定性。最后,通过仿真实验证明了提出的混合控制算法的有效性。