非线性动力学系统全局分析之外的胞映射方法新发展

在20世纪80年代由徐皆苏教授创建的胞映射方法一直受非线性科学界同仁的欢迎.近几年胞映射方法有了许多新的应用和算法.本文介绍了一些控制应用和算法的文献.另外,还介绍和讨论胞映射方法应用与多目标优化问题的研究和方法,多目标优化控制设计和非线性代数方程找零解.文中指出胞映射方法在井行计算的帮助下,现在可以解决中等高维空间中的各类问题,新的应用还会不断出现.     

A two scaled numerical method for global analysis of high dimensional nonlinear systems

This paper first analyzes the features of two classes of numerical methods for global analysis of nonlinear dynamical systems, which regard state space respectively as continuous and discrete ones. On basis of this understanding it then points out that the previously proposed method of point mapping under cell reference (PMUCR), has laid a frame work for the development of a two scaled numerical method suitable for the global analysis of high dimensional nonlinear systems, which may take the advantages of both classes of single scaled methods but will release the difficulties induced by the disadvantages of them. The basic ideas and main steps of implementation of the two scaled method, namely extended PMUCR, are elaborated. Finally, two examples are presented to demonstrate the capabilities of the proposed method.     

A multi-objective optimal PID control for a nonlinear system with time delay

It is generally difficult to design feedback controls of nonlinear systems with time delay to meet time domain specifications such as rise time, overshoot, and tracking error. Furthermore, these time domain specifications tend to be conflicting to each other to make the control design even more challenging. This paper presents a cell mapping method for multi-objective optimal feedback control design in time domain for a nonlinear Duffing system with time delay. We first review the multi-objective optimization problem and its formulation for control design. We then introduce the cell mapping method and a hybrid algorithm for global optimal solutions. Numerical simulations of the PID control are presented to show the features of the multi-objective optimal design.     

A subspace expanding technique for global zero finding of multi-degree-of-freedom nonlinear systems

A subspace expanding technique(SET) is proposed to efficiently discover and find all zeros of nonlinear functions in multi-degree-of-freedom(MDOF) engineering systems by discretizing the space into smaller subdomains, which are called cells. The covering set of the cells is identified by parallel calculations with the root bracketing method. The covering set can be found first in a low-dimensional subspace, and then gradually extended to higher dimensional spaces with the introduction of more equations and variables into the calculations. The results show that the proposed SET is highlyefficient for finding zeros in high-dimensional spaces. The subdivision technique of the cell mapping method is further used to refine the covering set, and the obtained numerical results of zeros are accurate. Three examples are further carried out to verify the applicability of the proposed method, and very good results are achieved. It is believed that the proposed method will significantly enhance the ability to study the stability, bifurcation,and optimization problems in complex MDOF nonlinear dynamic systems.     

Basin boundaries and robustness of nonlinear dynamic systems

Summary We present a method of computing a series of sets that approximates the basin of attraction for a given solution of a nonlinear dynamic system. The construction technique uses the general cell-mapping method in conjunction with an adaptive subdivision of the initial cell space. Our method provides a series of sets that contains the basin of attraction. Thus, we can approximate the robustness of a given solution from below. Received 22 April 1999; accepted for publication 3 June 1999     

The global bifurcation structure of a kind of digit mapping

The global structure of the mapping T_n:x→[x~2]_n,is studied.The symmetricunconnected substructures of T_2 is coincident with[1]by computer,but for n=3 thesymmetry of these substructures vanishes.As n is increasing,the global bifurcationstructure of T_n is shown.Finally,similar results for the mapping T_n,μ:x→[μx~2 ]_n are alsoproved.     

Stochastic Optimal Control of Nonlinear Systems via Short-Time Gaussian Approximation and Cell Mapping

A novel strategy to obtain global solutions of stochasticoptimal control problems with fixed state terminal conditions and controlbounds is proposed in this paper. The solution is global in the sense that theoptimal control solutions for all the initial conditions in a region of thestate space are obtained. The method makes use of Bellman's principle ofoptimality, the cumulant neglect closure method and the short-time Gaussianapproximation. A Markov chain with a control dependent transition probabilitymatrix is built using the generalized cell mapping method. This allows toevaluate the transient and steady state response of the controlled system. Themethod is applied to several linear and nonlinear systems leading to excellentcontrol performances.     

Fixed Final Time Optimal Control via Simple Cell Mapping

A cell mapping strategy based on Bellman'sPrinciple of optimality (BP) is proposed to solve the fixed final time optimalcontrol of strongly nonlinear systems with state and control bounds. Thefixed final time problem is transformed into a fixed final time-fixed finalstate optimal control problem by reversing the time. This transformationallows to solve the problem in the framework of the BP. Backward searchingalgorithms within the cell mapping context are used to obtain the solution ofthe new problem. This approach reduces considerably the computational effortrequired for the original problem when it is solved by a forward searchingapproach. The vibration control problem of damped and undamped variablestiffness oscillators with bounded stiffness tuning range is studied todemonstrate the effectiveness of the approach. A numerical study on theconvergence of the method is also presented.     

Solution of Fixed Final State Optimal Control Problems via Simple Cell Mapping

A strategy is proposed to solve the fixed final state optimalcontrol problem using the simple cell mapping method. A non-uniform timestep simple cell mapping is developed to create a general database fromwhich solutions of various optimal control problems can be obtained. Atwo-stage backward search algorithm is proposed to eliminate degeneratedpaths often associated with the simple cell mapping. The proposed methodcan accurately delineate the switching curves and eliminate false limitcycles in the solution. The method is applied to two optimal controlproblems with bang-bang control. The well-known minimum time controlproblem of moving a point mass from any initial condition to the originof the phase plane is studied first. This example has exact solutionsavailable which provide a yardstick to examine the accuracy of themethod. The cell size dependence of the solution accuracy is studiednumerically. The second example is a variable stiffness feedback controlproblem with tuning range saturation. The strategy proposed is able toprovide the switching curves in the phase plane. This result has notbeen obtained before.     

The global analysis of higher order nonlinear dynamical systems and the application of cell-to-cell mapping method

In this paper, the general characteristics and the topological consideration of the global behaviors of higher order nonlinear dynamical systems and the characteristics of the application of cell-to-cell mapping method in this analysis are expounded. Specifically, the global analysis of a system of two weakly coupled van der Pol oscillators using cell-to-cell mapping method is presented.The analysis shows that for this system, there exist two stable limit cycles in 4-dimensional state space, and the whole 4-dimensional state space is divided into two almost equal parts which are, respectively, the two asymototically stable domains of attraction of the two periodic motions of the two stable limit cycles. The validities of these conclusions about the global behaviors are also verified by direct long term numerical integration. Thus, it can be seen that the cell-to-cell mapping method for global analysis of fourth order nonlinear dynamical systems is quite effective.     

胞映射搜索最优控制路径的新策略

在胞映射求解最优控制问题的现有方法的基础上，提出了一种利用胞映射搜索最优控制路径的新策略。该策略基于负步长逆向数值积分技巧，采用了一种新的搜索过程，使得求解问题的效率大为提高。同现有的方法相比，新策略节省了几倍，甚至十几倍的时间。而且计算结果可靠，有很强的工程应用价值，为求解高维的最优控制问题可能会提供一定的基础。     

含非线性摩擦系数的单自由度颤振系统全局动力学行为数值研究

摩擦系数模型取更具普适性的Stribeck非线性模型,基于事件驱动理论,利用C与Matlab联合仿真的方法开发了干摩擦颤振问题的快速求解程序。给出改进的胞映射算法,对含非线性摩擦的单自由度摩擦颤振系统的演化过程及其全局性态进行数值分析和研究,得到系统在任意的初始状态下的响应特性、系统收敛域的数值计算分析结果。分析结果表...     

胞映射方法的研究和进展

介绍了胞映射方法的研究和进展. 归纳了目前胞映射方法的几种主要研究方法, 主要包括简单胞映射、广义胞映射、图胞映射、图胞映射的符号分析方法、图胞映射的面向集合方法、邻接胞映射、庞加莱型的简单胞映射、插值胞映射以及胞参照点映射方法, 分析了各类方法的基本特点和特色, 简述了这几种胞映射方法的最新国内外进展, 综述了胞映射方法在控制及相关领域的应用研究及进展, 给出了胞映射方法研究的一些展望, 提出了胞映射方法研究可能率先突破的几个研究方向.     

Application of Cell Mapping methods to a discontinuous dynamic system

The Cell Mapping method is a robust tool for investigating nonlinear dynamic systems. It is capable of finding the attractors and corresponding basins of attraction of a system under investigation. To investigate the applicability of the Cell Mapping method to discontinuous systems, a forced zero-stiffness impact oscillator is chosen as an application. The numerical integration algorithm, the basic element in the Cell Mapping method, is adjusted to overcome the discontinuity. Four types of Cell Mapping techniques are applied: Simple Cell Mapping, Generalized Cell Mapping, Interpolated Cell Mapping, and Mixed Cell Mapping. The last type is a new modification to existing types. Each type of Cell Mapping is briefly explained. The results are compared to the exact solutions. The Interpolated Cell Mapping and Mixed Cell Mapping methods are found to produce the most accurate results for this case.     

Localization of energy in nonlinear systems with two degrees of freedom

Energy pumping in a two-degrees-of-freedom system with linear and essentially nonlinear oscillators is studied. The kinetic energy envelopes of the linear and nonlinear subsystems are chosen to be the main characteristics of the process under consideration. A criterion that the energy of the linear oscillator excited at time zero is completely pumped over into the nonlinear oscillator is established together with an additional condition whereby the energy does not return to the linear subsystem. Optimal energy pumping mode is established using global optimization. The effect of the parameters of the system on the main characteristics is assessed __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 115–125, May 2007.     

Optimal control of quadratic functionals for affine nonlinear systems

In this paper we analyze the optimal control problem for a class of affine nonlinear systems under the assumption that the associated Lie algebra is nilpotent. The Lie brackets generated by the vector fields which define the nonlinear system represent a remarkable mathematical instrument for the control of affine systems. We determine the optimal control which corresponds to the nilpotent operator of the first order. In particular, we obtain the control that minimizes the energy of the given nonlinear system. Applications of this control to bilinear systems with first order nilpotent operator are considered.     

结构随机动力非线性反应的整体灵敏度分析

以随机参数或参数子集对目标函数总方差的相对贡献作为该参数或参数子集的整体随机灵敏度度量,定义了一类整体随机灵敏度指标。结合结构非线性随机反应分析的密度演化方法,给出了计算整体随机灵敏度指标的基本算法。该灵敏度指标物理意义明确、计算实施方便。以滞回结构非线性反应分析为例,考察了以位移反应和滞回耗能为目标函数的不同随机参数(子集)的灵敏度指标。分析表明,不同随机参数(子集)对不同目标函数的灵敏度差异甚大,并且随机参数可能对结构反应的总方差产生负贡献。     

Global analysis of Ivlev's type predator-prey dynamic systems

Consider a class of Ivlev's type predator-prey dynamic systems with prey and predator both having linear density restricts. By using the qualitative methods of ODE, the global stability of positive equilibrium and existence and uniqueness of non-small amplitude stable limit cycle are obtained. Especially under certain conditions, it shows that existence and uniqueness of non-small amplitude stable limit cycle is equivalent to the local un-stability of positive equilibrium and the local stability of positive equilibrium implies its global stability. That is to say, the global dynamic of the system is entirely determined by the local stability of the positive equilibrium.     

基于填充函数与DCD的全局优化算法及其反演

岩土工程中以监测位移为已知信息的反演问题可通过带未知变量约束空间的优化模型去求解.该模型中的优化函数常具有非线性、非凸性等特点,使得反演结果容易陷入局部最优的困境.为了应对在运用优化算法反演此类问题时存在的困境,并提高其算法效率,依据填充函数优化思想与DCD(Dynamic Canonical Descent)思想在反...