非线性动力学系统全局分析之外的胞映射方法新发展

Jian-Qiao SUN; Fu-Rui XIONG

doi:10.6052/1000-0992-16-027

非线性动力学系统全局分析之外的胞映射方法新发展

doi: 10.6052/1000-0992-16-027

孙建桥^1, ,,
熊夫睿²

1.
加州大学默塞德分校工学院, 美国加利福尼亚州 95343
2.
中国核动力研究设计院, 成都 610041

详细信息

中图分类号: O322
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出版历程
- 收稿日期: 2016-08-23
- 网络出版日期: 2016-11-02
- 刊出日期: 2017-02-24

Cell mapping methods-beyond global analysis of nonlinear dynamic systems

Jian-Qiao SUN^{1
, ,},
Fu-Rui XIONG²

1.
School of Engineering, University of California, Merced, CA 95343, USA
2.
Nuclear Power Institute of China, Chengdu 610041, China

More Information

Corresponding author: Jian-Qiao SUN

摘要

摘要: 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并行计算的帮助下，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.
- 胞映射方法 /
- 全局分析 /
- 最优控制 /
- 多目标优化 /
- 非线性代数方程的零解
Abstract: The cell mapping methods created by Hsu in 1980s have been popular choices for the researchers in nonlinear science communities. There have been new applications and new algorithm developments of the cell mapping methods. This paper presents a discussion of the literature of some control applications and recent algorithm developments of the cell mapping methods. In particular, we present studies of multi-objective optimization problems with the cell mapping methods, multi-objective optimal control designs, and zeros finding of nonlinear algebraic equations. The problems solved with the cell mapping methods are now in moderately high dimensional space with the help of parallel computing.
- cell mapping methods /
- global analysis /
- optimal control /
- multi-objective optimization /
- finding zeros

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Figure 1. Two-dimensional projections of a six-dimensional strange attractor of the Lorenz system. Blue dots are the centers of the cells in the invariant set. Red dots showing the fine structure of the attractor are generated with interpolation

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Figure 2. The 5-dimensional Pareto front of the objective functions, projected on to 2-dimensional sub-spaces of the objective space. The color code indicates the level of M_{p, x1}, IAE_x2, max |x₂|, and t_{p, x1} in subplots from upper left in counterclockwise order. The conflicting nature among these objectives can be observed clearly. For example, the overshoot and peak time are conflicting for tracking control, which can be seen in the upper left plot of x₁

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Figure 3. Experimental square wave tracking response of the rotary flexible joint under a Pareto optimal sliding mode control. x₁ is the base angle and x₂ is the angle of the flexible link

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Figure 4. Stability boundary of the Mathieu system in the feedback gain space (k_p, k_d)

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