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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.
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关键词:
- 非线性随机系统 /
- 概率密度函数 /
- Fokker-Planck-Kolmogorov方程 /
- 矩
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:
- nonlinear stochastic systems /
- probability density function /
- Fokker-Planck-Kolmogorov equation /
- moment
[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 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 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 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
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[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 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 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 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
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