连续、离散时间零化神经网络(ZNN)求解动态矩阵线性方程、动态矩阵逆、动态矩阵Moore-Penrose逆问题

柔情只为你懂 2022-12-26 00:51 137阅读 0赞

## 1. 连续、离散ZNN求解动态矩阵线性方程问题： ##

动态矩阵线性方程问题的定义：![A(t)x(t)=b(t)][A_t_x_t_b_t]

定义误差函数：![\\epsilon(t)=A(t)x(t)-b(t)][epsilon_t_A_t_x_t_-b_t]

零化神经网络ZNN的演化公式：![\\dot\\epsilon(t)=-\\gamma\\epsilon(t)][dot_epsilon_t_-_gamma_epsilon_t]

综合推导可得用于求解动态矩阵线性方程问题的连续时间ZNN模型：

![A(t)\\dot x(t)=-\\dot A(t)x(t)+\\dot b(t)-\\gamma (A(t)x(t)-b(t))][A_t_dot x_t_-_dot A_t_x_t_dot b_t_-_gamma _A_t_x_t_-b_t]

欧拉前向差分定义：

![\\dot x\_n=\\frac\{x\_\{n+1\}-x\_n\}\{g\}][dot x_n_frac_x_n_1_-x_n_g]

上式中的g表示采样间隔

即可推导得出用于求解动态矩阵线性方程问题的离散时间ZNN模型：

![x\_\{n+1\}=x\_n + A^\{\\dagger\}\_n(-(A\_n-A\_\{n-1\})x\_n+b\_n-b\_\{n-1\}-\\gamma g(A\_nx\_n - b\_n))][x_n_1_x_n _ A_dagger_n_-_A_n-A_n-1_x_n_b_n-b_n-1_-_gamma g_A_nx_n - b_n]

## 2. 连续、离散ZNN求解动态矩阵逆问题： ##

动态矩阵线性方程问题的定义：![A(t)x(t)=I][A_t_x_t_I]

定义误差函数：![\\epsilon(t)=A(t)x(t)-I][epsilon_t_A_t_x_t_-I]

零化神经网络ZNN的演化公式：![\\dot\\epsilon(t)=-\\gamma\\epsilon(t)][dot_epsilon_t_-_gamma_epsilon_t]

综合推导可得用于求解动态矩阵逆问题的连续时间ZNN模型：

![A(t)\\dot x(t)=-\\dot A(t)x(t)-\\gamma (A(t)x(t)-I)][A_t_dot x_t_-_dot A_t_x_t_-_gamma _A_t_x_t_-I]

欧拉前向差分定义：

![\\dot x\_n=\\frac\{x\_\{n+1\}-x\_n\}\{g\}][dot x_n_frac_x_n_1_-x_n_g]

上式中的g表示采样间隔

根据欧拉前向展开公式进一步推导：

![A\_n \\frac\{x\_\{n+1\}-x\_n\}\{g\}=-\\frac\{A\_n - A\_\{n-1\}\}\{g\}x\_n-\\gamma(A\_nx\_n-I)][A_n _frac_x_n_1_-x_n_g_-_frac_A_n - A_n-1_g_x_n-_gamma_A_nx_n-I]

![A\_n x\_\{n+1\}-A\_n x\_n=-(A\_n-A\_\{n-1\})x\_n-\\gamma g(A\_nx\_n-I)][A_n x_n_1_-A_n x_n_-_A_n-A_n-1_x_n-_gamma g_A_nx_n-I]

最终整理得到用于求解动态矩阵逆问题的离散时间ZNN模型：

![x\_\{n+1\}=x\_n-x\_n(-(A\_n-A\_\{n-1\})x\_n-\\gamma g(A\_nx\_n-I))][x_n_1_x_n-x_n_-_A_n-A_n-1_x_n-_gamma g_A_nx_n-I]

## 3. 连续、离散ZNN求解动态矩阵Moore-Penrose逆问题： ##

动态矩阵的Moore-Penrose逆定义：

![M^\\text\{T\}(t)=M^\{\\dagger\}(t)M(t)M^\{\\text\{T\}\}(t), m<n][M_text_T_t_M_dagger_t_M_t_M_text_T_t_ m_n]

![M^\\text\{T\}(t)=M^\{\\text\{T\}\}(t)M(t)M^\{\\dagger\}(t), m>n][M_text_T_t_M_text_T_t_M_t_M_dagger_t_ m_n]

上式表示的是右Moore-Penrose逆，下式表示的是左伪逆，在这里仅讨论右伪逆的推导方式。

构造误差函数：![\\epsilon(t) = X(t)M(t)M^\{\\text\{T\}\}(t)-M^\{\\text\{T\}\}(t)][epsilon_t_ _ X_t_M_t_M_text_T_t_-M_text_T_t]

零化神经网络ZNN的演化公式：![\\dot\\epsilon(t)=-\\gamma\\epsilon(t)][dot_epsilon_t_-_gamma_epsilon_t]

综合推导可得用于求解动态矩阵Moore-Penrose逆问题的连续时间ZNN模型：

![\\small \\dot X(t)M(t)M^T(t)=-X(t)\\dot M(t)M^T(t)-X(t)M(t)\\dot M^T(t)+\\dot M^T(t) -\\gamma (X(t)M(t)M^T(t)-M^T(t))][small _dot X_t_M_t_M_T_t_-X_t_dot M_t_M_T_t_-X_t_M_t_dot M_T_t_dot M_T_t_ -_gamma _X_t_M_t_M_T_t_-M_T_t]

进一步推导可得：

![\\small \\dot X(t)=(-X(t)\\dot M(t)M^T(t)-X(t)M(t)\\dot M^T(t)+\\dot M^T(t) -\\gamma (X(t)M(t)M^T(t)-M^T(t)))(M(t)M^T(t))^\{-1\}][small _dot X_t_-X_t_dot M_t_M_T_t_-X_t_M_t_dot M_T_t_dot M_T_t_ -_gamma _X_t_M_t_M_T_t_-M_T_t_M_t_M_T_t_-1]

使用欧拉前线公式离散化：

![\\small x\_\{n+1\}=x\_n(-x\_n(M\_n-M\_\{n-1\})M^T\_n-x\_nM\_n(M^T\_n-M^T\_\{n-1\})+M\_n^T-M\_\{n-1\}^T-\\gamma g(x\_nM\_nM^T\_n-M\_n^T))(M\_nM\_n^T\})^\{-1\}][small x_n_1_x_n_-x_n_M_n-M_n-1_M_T_n-x_nM_n_M_T_n-M_T_n-1_M_n_T-M_n-1_T-_gamma g_x_nM_nM_T_n-M_n_T_M_nM_n_T_-1]

综合推导可得用于求解动态矩阵Moore-Penrose逆问题的离散时间ZNN模型：

[A_t_x_t_b_t]: https://private.codecogs.com/gif.latex?A%28t%29x%28t%29%3Db%28t%29
[epsilon_t_A_t_x_t_-b_t]: https://private.codecogs.com/gif.latex?%5Cepsilon%28t%29%3DA%28t%29x%28t%29-b%28t%29
[dot_epsilon_t_-_gamma_epsilon_t]: https://private.codecogs.com/gif.latex?%5Cdot%5Cepsilon%28t%29%3D-%5Cgamma%5Cepsilon%28t%29
[A_t_dot x_t_-_dot A_t_x_t_dot b_t_-_gamma _A_t_x_t_-b_t]: https://private.codecogs.com/gif.latex?A%28t%29%5Cdot%20x%28t%29%3D-%5Cdot%20A%28t%29x%28t%29&plus;%5Cdot%20b%28t%29-%5Cgamma%20%28A%28t%29x%28t%29-b%28t%29%29
[dot x_n_frac_x_n_1_-x_n_g]: https://private.codecogs.com/gif.latex?%5Cdot%20x_n%3D%5Cfrac%7Bx_%7Bn&plus;1%7D-x_n%7D%7Bg%7D
[x_n_1_x_n _ A_dagger_n_-_A_n-A_n-1_x_n_b_n-b_n-1_-_gamma g_A_nx_n - b_n]: https://private.codecogs.com/gif.latex?x_%7Bn&plus;1%7D%3Dx_n%20&plus;%20A%5E%7B%5Cdagger%7D_n%28-%28A_n-A_%7Bn-1%7D%29x_n&plus;b_n-b_%7Bn-1%7D-%5Cgamma%20g%28A_nx_n%20-%20b_n%29%29
[A_t_x_t_I]: https://private.codecogs.com/gif.latex?A%28t%29x%28t%29%3DI
[epsilon_t_A_t_x_t_-I]: https://private.codecogs.com/gif.latex?%5Cepsilon%28t%29%3DA%28t%29x%28t%29-I
[A_t_dot x_t_-_dot A_t_x_t_-_gamma _A_t_x_t_-I]: https://private.codecogs.com/gif.latex?A%28t%29%5Cdot%20x%28t%29%3D-%5Cdot%20A%28t%29x%28t%29-%5Cgamma%20%28A%28t%29x%28t%29-I%29
[A_n _frac_x_n_1_-x_n_g_-_frac_A_n - A_n-1_g_x_n-_gamma_A_nx_n-I]: https://private.codecogs.com/gif.latex?A_n%20%5Cfrac%7Bx_%7Bn&plus;1%7D-x_n%7D%7Bg%7D%3D-%5Cfrac%7BA_n%20-%20A_%7Bn-1%7D%7D%7Bg%7Dx_n-%5Cgamma%28A_nx_n-I%29
[A_n x_n_1_-A_n x_n_-_A_n-A_n-1_x_n-_gamma g_A_nx_n-I]: https://private.codecogs.com/gif.latex?A_n%20x_%7Bn&plus;1%7D-A_n%20x_n%3D-%28A_n-A_%7Bn-1%7D%29x_n-%5Cgamma%20g%28A_nx_n-I%29
[x_n_1_x_n-x_n_-_A_n-A_n-1_x_n-_gamma g_A_nx_n-I]: https://private.codecogs.com/gif.latex?x_%7Bn&plus;1%7D%3Dx_n-x_n%28-%28A_n-A_%7Bn-1%7D%29x_n-%5Cgamma%20g%28A_nx_n-I%29%29
[M_text_T_t_M_dagger_t_M_t_M_text_T_t_ m_n]: https://private.codecogs.com/gif.latex?M%5E%5Ctext%7BT%7D%28t%29%3DM%5E%7B%5Cdagger%7D%28t%29M%28t%29M%5E%7B%5Ctext%7BT%7D%7D%28t%29%2C%20m%3Cn
[M_text_T_t_M_text_T_t_M_t_M_dagger_t_ m_n]: https://private.codecogs.com/gif.latex?M%5E%5Ctext%7BT%7D%28t%29%3DM%5E%7B%5Ctext%7BT%7D%7D%28t%29M%28t%29M%5E%7B%5Cdagger%7D%28t%29%2C%20m%3En
[epsilon_t_ _ X_t_M_t_M_text_T_t_-M_text_T_t]: https://private.codecogs.com/gif.latex?%5Cepsilon%28t%29%20%3D%20X%28t%29M%28t%29M%5E%7B%5Ctext%7BT%7D%7D%28t%29-M%5E%7B%5Ctext%7BT%7D%7D%28t%29
[small _dot X_t_M_t_M_T_t_-X_t_dot M_t_M_T_t_-X_t_M_t_dot M_T_t_dot M_T_t_ -_gamma _X_t_M_t_M_T_t_-M_T_t]: https://private.codecogs.com/gif.latex?%5Cdpi%7B100%7D%20%5Csmall%20%5Cdot%20X%28t%29M%28t%29M%5ET%28t%29%3D-X%28t%29%5Cdot%20M%28t%29M%5ET%28t%29-X%28t%29M%28t%29%5Cdot%20M%5ET%28t%29&plus;%5Cdot%20M%5ET%28t%29%20-%5Cgamma%20%28X%28t%29M%28t%29M%5ET%28t%29-M%5ET%28t%29%29
[small _dot X_t_-X_t_dot M_t_M_T_t_-X_t_M_t_dot M_T_t_dot M_T_t_ -_gamma _X_t_M_t_M_T_t_-M_T_t_M_t_M_T_t_-1]: https://private.codecogs.com/gif.latex?%5Cdpi%7B100%7D%20%5Csmall%20%5Cdot%20X%28t%29%3D%28-X%28t%29%5Cdot%20M%28t%29M%5ET%28t%29-X%28t%29M%28t%29%5Cdot%20M%5ET%28t%29&plus;%5Cdot%20M%5ET%28t%29%20-%5Cgamma%20%28X%28t%29M%28t%29M%5ET%28t%29-M%5ET%28t%29%29%29%28M%28t%29M%5ET%28t%29%29%5E%7B-1%7D
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