随机系统的概率密度函数形状调节

杨恒占; 钱富才; 高韵; 谢国

doi:10.7498/aps.63.240508

摘要

针对受高斯白噪声激励的非线性随机系统, 提出了使状态响应的概率密度函数形状跟踪期望形状的调节方法. 首先, 确立了非线性随机系统的多项式反馈机制, 同时对系统中的非线性部分进行多项式展开; 然后, 以Fokker-Planck-Kolmogorov方程为工具, 导出了与控制增益相关的各阶矩递推方程, 并根据跟踪问题的要求, 构造了矩逼近优化问题, 用梯度搜索法求解该优化问题, 获得了调节函数; 再依据特征函数与概率密度函数构成Fourier对的关系, 对状态响应的概率密度函数进行重构; 最后, 通过两个例子仿真, 验证了本文方法的有效性.

关键词:

Abstract

For nonlinear stochastic systems which are excited by Gaussian white noise, an innovational regulation method is proposed to control the shape of the probability density function of state response to track a desired shape. Firstly, a polynomial feedback scheme is established, and the nonlinear part is replaced by polynomials expansion. Then the recursive equations of the moments which are related to control gain are derived under Fokker-Planck-Kolmogorov theory framework. Meanwhile, regarding the tracking requirement, an optimization problem about the moment approximation is constructed, and the gain of regulation function is obtained by solving this optimization problem using the gradient method. Furthermore, the probability density function of state response is reconstructed from the relationship of the Fourier transform pairs between the characteristic function and probability density function. Finally, two examples are given to demonstrate the effectiveness of the method developed in this paper.

Keywords:

作者及机构信息

1.
西安理工大学自动化与信息工程学院, 西安 710048;

2.
西安工业大学, 新型网络与检测控制国家地方联合工程实验室, 西安 710021

基金项目: 国家自然科学基金(批准号: 61273127, 61304204)和高等学校博士点专项科研基金(批准号: 20116118110008)资助的课题.

Authors and contacts

1.
School of Automation and Information Engineering, Xi'an University of Technology, Xi'an 710048, China;

2.
The National-Local Joint Engineering Laboratory for New Network and Detection Control, Xi'an Technological University, Xi'an 710021, China

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 61273127, 61304204), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20116118110008).

参考文献

[1]	Li D, Qian F C, Fu P L 2002 IEEE Trans. Autom. Control 47 2010
[2]	Li D, Qian F C, Fu P L 2008 Automatica 44 119
[3]	Li D, Qian F C, Gao J J 2009 IEEE Trans. Autom. Control 54 2225
[4]	Sain M K 1966 IEEE Trans. Autom. Control 11 118
[5]	Sain M K, Liberty S R 1971 IEEE Trans. Autom. Control 16 431
[6]	Huang J W, Feng J C, L S X 2014 Acta Phys．Sin. 63 050502 (in Chinese) [黄锦旺, 冯久超, 吕善翔 2014 物理学报 63 050502]
[7]	Li C, Xu W, Wang L, Li D X 2013 Chin. Phys. B 22 110205
[8]	Zhu Z W, Zhang Q X, Xu J 2014 Chin. Phys. B 23 088201
[9]	Yue X L, Xu W, Zhang Y, Wang L 2014 Acta Phys．Sin. 63 060502 (in Chinese) [岳晓乐, 徐伟, 张莹, 王亮 2014 物理学报 63 060502]
[10]	Forbes M G, Guay M, Forbes J F 2004 J. Precess Contr. 14 399
[11]	Karny M 1996 Automatica 32 1719
[12]	Liu F, Ouyang Z C 2014 Chin. Phys. B 23 070512
[13]	Yeontaek C, Sang G J 2011 Chin. Phys. B 20 050501
[14]	Hu H B, Du P, Huang S H, Wang Y 2013 Chin. Phys. B 22 074703
[15]	Guo L, Wang H 2010 Stochastic Distribution Control System Design: A Convex Optimization Approach (London: Springer)
[16]	Guo L, Wang H 2005 IEEE Trans. Syst. Man Cybern. B: Cybern. 35 65
[17]	Guo L, Yin L 2009 IET Control Theory Appl. 3 575
[18]	Fuller A T 1969 Int. J. Control 9 603
[19]	Zhuang B Z, Chen N L, Gao Z 1986 The Random Vibration Theory of the Nonlinear and Application (Hangzhou: Zhejiang University Press) (in Chinese) [庄表中, 陈乃立, 高瞻 1986 非线性随机振动理论及应用 (浙江大学出版社)]
[20]	Xie W X, Xu W, Lei Y M, Cai L 2005 Acta Phys．Sin. 54 1105 (in Chinese) [谢文贤, 徐伟, 雷佑铭, 蔡力 2005 物理学报 54 1105]
[21]	Langley R 1985 J. Sound Vibr. 101 41
[22]	Paola M D, Ricciardi G, Vasta M 1995 Probab. Eng. Mech. 10 1
[23]	Zhu C X, Zhu W Q 2011 Automatica 47 539

施引文献

[1]	Li D, Qian F C, Fu P L 2002 IEEE Trans. Autom. Control 47 2010
[2]	Li D, Qian F C, Fu P L 2008 Automatica 44 119
[3]	Li D, Qian F C, Gao J J 2009 IEEE Trans. Autom. Control 54 2225
[4]	Sain M K 1966 IEEE Trans. Autom. Control 11 118
[5]	Sain M K, Liberty S R 1971 IEEE Trans. Autom. Control 16 431
[6]	Huang J W, Feng J C, L S X 2014 Acta Phys．Sin. 63 050502 (in Chinese) [黄锦旺, 冯久超, 吕善翔 2014 物理学报 63 050502]
[7]	Li C, Xu W, Wang L, Li D X 2013 Chin. Phys. B 22 110205
[8]	Zhu Z W, Zhang Q X, Xu J 2014 Chin. Phys. B 23 088201
[9]	Yue X L, Xu W, Zhang Y, Wang L 2014 Acta Phys．Sin. 63 060502 (in Chinese) [岳晓乐, 徐伟, 张莹, 王亮 2014 物理学报 63 060502]
[10]	Forbes M G, Guay M, Forbes J F 2004 J. Precess Contr. 14 399
[11]	Karny M 1996 Automatica 32 1719
[12]	Liu F, Ouyang Z C 2014 Chin. Phys. B 23 070512
[13]	Yeontaek C, Sang G J 2011 Chin. Phys. B 20 050501
[14]	Hu H B, Du P, Huang S H, Wang Y 2013 Chin. Phys. B 22 074703
[15]	Guo L, Wang H 2010 Stochastic Distribution Control System Design: A Convex Optimization Approach (London: Springer)
[16]	Guo L, Wang H 2005 IEEE Trans. Syst. Man Cybern. B: Cybern. 35 65
[17]	Guo L, Yin L 2009 IET Control Theory Appl. 3 575
[18]	Fuller A T 1969 Int. J. Control 9 603
[19]	Zhuang B Z, Chen N L, Gao Z 1986 The Random Vibration Theory of the Nonlinear and Application (Hangzhou: Zhejiang University Press) (in Chinese) [庄表中, 陈乃立, 高瞻 1986 非线性随机振动理论及应用 (浙江大学出版社)]
[20]	Xie W X, Xu W, Lei Y M, Cai L 2005 Acta Phys．Sin. 54 1105 (in Chinese) [谢文贤, 徐伟, 雷佑铭, 蔡力 2005 物理学报 54 1105]
[21]	Langley R 1985 J. Sound Vibr. 101 41
[22]	Paola M D, Ricciardi G, Vasta M 1995 Probab. Eng. Mech. 10 1
[23]	Zhu C X, Zhu W Q 2011 Automatica 47 539

[1]	王路, 徐江荣. 两相湍流统一色噪声法概率密度函数模型. 物理学报, 2015, 64(5): 054704. doi: 10.7498/aps.64.054704
[2]	黄锦旺, 李广明, 冯久超, 晋建秀. 一种无线传感器网络中的混沌信号重构算法. 物理学报, 2014, 63(14): 140502. doi: 10.7498/aps.63.140502
[3]	套格图桑, 白玉梅. 非线性发展方程的Riemann theta 函数等几种新解. 物理学报, 2013, 62(10): 100201. doi: 10.7498/aps.62.100201
[4]	戈阳祯, 米建春. 圆柱热尾流中温度的概率密度函数. 物理学报, 2013, 62(2): 024702. doi: 10.7498/aps.62.024702
[5]	杨会会, 宁丽娟. 非线性漂移的Fokker-Planck方程的近似非定态解. 物理学报, 2013, 62(18): 180501. doi: 10.7498/aps.62.180501
[6]	成海英, 何文平, 何涛, 吴琼. 基于概率密度分布型变化的突变检测新途径. 物理学报, 2012, 61(3): 039201. doi: 10.7498/aps.61.039201
[7]	金伟, 万振茂, 刘要稳. 自旋转移矩效应激发的非线性磁化动力学. 物理学报, 2011, 60(1): 017502. doi: 10.7498/aps.60.017502
[8]	牛玉俊, 徐伟, 戎海武, 王亮, 冯进钤. 随机脉冲微分方程的p阶矩稳定性和参激白噪声作用下Lorenz系统的脉冲同步. 物理学报, 2009, 58(5): 2983-2988. doi: 10.7498/aps.58.2983
[9]	赵燕, 徐伟, 邹少存. 非高斯噪声激励下FHN神经元系统的定态概率密度与平均首次穿越时间. 物理学报, 2009, 58(3): 1396-1402. doi: 10.7498/aps.58.1396
[10]	吕大昭. 非线性发展方程的丰富的Jacobi椭圆函数解. 物理学报, 2005, 54(10): 4501-4505. doi: 10.7498/aps.54.4501
[11]	戎海武, 王向东, 徐伟, 孟光, 方同. 窄带随机噪声作用下Duffing振子的双峰稳态概率密度. 物理学报, 2005, 54(6): 2557-2561. doi: 10.7498/aps.54.2557
[12]	刘式适, 付遵涛, 王彰贵, 刘式达. Lam函数和非线性演化方程的扰动方法. 物理学报, 2003, 52(8): 1837-1841. doi: 10.7498/aps.52.1837
[13]	赵超樱, 谭维翰, 郭奇志. 由非简并光学参量放大系统获得压缩态光所满足的Fokker-Planck方程及其解. 物理学报, 2003, 52(11): 2694-2699. doi: 10.7498/aps.52.2694
[14]	朱学光, 匡光力, 赵燕平, 李有宜, 谢纪康. Fokker-Planck方程在快波加热中的应用. 物理学报, 1998, 47(7): 1137-1142. doi: 10.7498/aps.47.1137
[15]	卢志恒, 林建恒, 胡岗. 随机共振问题Fokker-Planck方程的数值研究. 物理学报, 1993, 42(10): 1556-1566. doi: 10.7498/aps.42.1556
[16]	屈支林, 胡岗. 非线性非势系统的Fokker-Planck方程的非定态解. 物理学报, 1992, 41(9): 1396-1405. doi: 10.7498/aps.41.1396
[17]	林仁明, 黄思先, 张林. 受驱动光学系统多光子量子统计理论(Ⅰ)——Fokker-Planck方程和良腔情况. 物理学报, 1988, 37(4): 573-581. doi: 10.7498/aps.37.573
[18]	郑伟谋. 双阱势Fokker-Planck方程准确解模型. 物理学报, 1986, 35(2): 247-253. doi: 10.7498/aps.35.247
[19]	吴君汝, A. LARRAZA, I. RUDNICK. 水表面波矩型谐振器非线性共振曲线的测量. 物理学报, 1985, 34(6): 796-800. doi: 10.7498/aps.34.796
[20]	胡岗. 非线性漂移的Fokker-Planck方程的非定态解. 物理学报, 1985, 34(5): 573-580. doi: 10.7498/aps.34.573

计量

文章访问数: 6499
PDF下载量: 594
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

搜索

留言板