复杂网络采样数据同步的主要结果

### 复杂网络采样数据同步的主要结果 #### 1. 误差系统重写 为了实现复杂动态网络的采样数据同步，设计了一组采样数据控制器 \(L_i (i = 1, 2, \cdots, N)\)，使得误差系统 (5.7) 渐近稳定，即系统 (5.1) 实现采样数据同步。 定义如下变量： - \(e(t) = [e_1^T(t), e_2^T(t), \cdots, e_N^T(t)]^T\) - \(g(e(t)) = [g^T(e_1(t)), g^T(e_2(t)), \cdots, g^T(e_N(t))]^T\) - \(\overline{\Gamma} = G \otimes \Gamma\) - \(L = diag[L_1, L_2, \cdots, L_N]\) - \(\overline{U} = \frac{(I_N \otimes U)^T (I_N \otimes V)}{2} + \frac{(I_N \otimes V)^T (I_N \otimes U)}{2}\) - \(\overline{V} = -\frac{(I_N \otimes U)^T + (I_N \otimes V)^T}{2}\) 则误差动态方程 (5.7) 可重写为： \(\dot{e}(t) = g(e(t)) + c\overline{\Gamma} e(t - \tau(t)) + Le(t_p)\) (5.8) #### 2. 定理 5.1 在假设 5.1 下，若存在矩阵 \(P > 0\)，\(Q_1 > 0\)，\(Q_2 > 0\)，\(Q_3 > 0\)，\(W_1 > 0\)，\(W_2 > 0\)，\(Z_1 > 0\)，\(Z_2 > 0\)，\(N_1 > 0\)，\(N_2 > 0\)，\(N_3 > 0\)，矩阵 \(Y\)，\(X_1\)，\(X_2\)，\(F_1\)，\(F_2\) 以及标量 \(\sigma > 0\)，使得以下矩阵不等式成立，则误差动态系统 (5.8) 渐近稳定： - \(\Pi_1 = \Pi + d\begin{bmatrix}0_{nN\times nN} & I_{nN\times nN} & 0_{nN\times 6nN}\end{bmatrix}^T N_1\begin{bmatrix}0_{nN\times nN} & I_{nN\times nN} & 0_{nN\times 6nN}\end{bmatrix} + 2d\begin{bmatrix}0_{nN\times 2nN} & I_{nN\times nN} & 0_{nN\times 5nN}\end{bmatrix}^T N_2\begin{bmatrix}0_{nN\times 2nN} & I_{nN\times nN} & 0_{nN\times 5nN}\end{bmatrix} < 0\) (5.9) - \(\Pi_2 =\begin{pmatrix}\Pi & Y \\ * & -\frac{1}{d}N_1\end{pmatrix} < 0\) (5.10) - \(\begin{pmatrix}W_2 & X_1 \\ * & W_2\end{pmatrix} \geq 0\) (5.11) - \(\begin{pmatrix}Z_2 & X_2 \\ * & Z_2\end{pmatrix} \geq 0\) (5.12) 其中： - \(\Pi = \Psi + \begin{bmatrix}Y_{8nN\times nN} & 0_{8nN\times nN} & -Y_{8nN\times nN} & 0_{8nN\times 5nN}\end{bmatrix} + \begin{bmatrix}Y_{8nN\times nN} & 0_{8nN\times nN} & -Y_{8nN\times nN} & 0_{8nN\times 5nN}\end{bmatrix}^T\) - \(\Psi =\begin{pmatrix}\Psi_{11} & \Psi_{12} & \Psi_{13} & \Psi_{14} & \Psi_{15} & \Psi_{16} & 0 & \Psi_{18} \\ * & \Psi_{22} & \Psi_{23} & 0 & 0 & \Psi_{26} & 0 & \Psi_{28} \\ * & * & \Psi_{33} & \Psi_{34} & 0 & 0 & 0 & 0 \\ * & * & * & \Psi_{44} & 0 & 0 & 0 & 0 \\ * & * & * & * & \Psi_{55} & \Psi_{56} & \Psi_{57} & 0 \\ * & * & * & * & * & \Psi_{66} & \Psi_{67} & 0 \\ * & * & * & * & * & * & \Psi_{77} & 0 \\ * & * & * & * & * & * & * & \Psi_{88}\end{pmatrix}\) 各元素具体表达式如下： |元素|表达式| | ---- | ---- | |\(\Psi_{11}\)|\(Q_1 + Q_2 - W_1 + Z_1 - \frac{1}{d}Z_2 - \frac{\pi^2}{4}N_3 - \sigma\overline{U}\)| |\(\Psi_{12}\)|\(-F_1 + P\)| |\(\Psi_{13}\)|\(\frac{1}{d}Z_2 - \frac{1}{d}X_2 + \frac{\pi^2}{4}N_3 + F_1L\)| |\(\Psi_{14}\)|\(\frac{1}{d}X_2\)| |\(\Psi_{15}\)|\(W_1\)| |\(\Psi_{16}\)|\(cF_1\overline{\Gamma}\)| |\(\Psi_{18}\)|\(-\sigma\overline{V} + F_1\)| |\(\Psi_{22}\)|\(\tau_1^2W_1 + (\tau_2 - \tau_1)^2W_2 + dZ_2 + d^2N_3 - F_2 - F_2^T\)| |\(\Psi_{23}\)|\(F_2L\)| |\(\Psi_{26}\)|\(cF_2\overline{\Gamma}\)| |\(\Psi_{28}\)|\(F_2\)| |\(\Psi_{33}\)|\(-\frac{1}{d}Z_2 + \frac{1}{d}(X_2 + X_2^T) - \frac{1}{d}Z_2 - dN_2 - \frac{\pi^2}{4}N_3\)| |\(\Psi_{34}\)|\(-\frac{1}{d}X_2 + \frac{1}{d}Z_2\)| |\(\Psi_{44}\)|\(-Z_1 - \frac{1}{d}Z_2\)| |\(\Psi_{55}\)|\(-Q_2 + Q_3 - W_1 - W_2\)| |\(\Psi_{56}\)|\(W_2 - X_1\)| |\(\Psi_{57}\)|\(X_1\)| |\(\Psi_{66}\)|\(-(1 - h)Q_1 - W_2 + X_1 + X_1^T - W_2\)| |\(\Psi_{67}\)|\(-X_1 + W_2\)| |\(\Psi_{77}\)|\(-Q_3 - W_2\)| |\(\Psi_{88}\)|\(-\sigma I\)| #### 3. 定理 5.1 证明 考虑误差动态系统 (5.8) 的如下不连续 Lyapunov - Krasovskii 泛函： \(V(e(t)) = \sum_{q = 1}^{7}[V_q(e(t))], t \in [t_p, t_{p + 1})\) (5.13) 其中各部分定义如下： - \(V_1(e(t)) = e^T(t)Pe(t)\) (5.14) - \(V_2(e(t)) = \int_{t - \tau(t)}^{t} e^T(s)Q_1e(s)ds + \int_{t - \tau_1}^{t} e^T(s)Q_2e(s)ds + \int_{t - \tau_2}^{t - \tau_1} e^T(s)Q_3e(s)ds\) (5.15) - \(V_3(e(t)) = \tau_1\int_{-\tau_1}^{0}\int_{t + \theta}^{t} \dot{e}^T(s)W_1\dot{e}(s)dsd\theta + (\tau_2 - \tau_1)\int_{-\tau_2}^{-\tau_1}\int_{t + \theta}^{t} \dot{e}^T(s)W_2\dot{e}(s)dsd\theta\) (5.16) - \(V_4(e(t)) = \int_{t - d}^{t} e^T(s)Z_1e(s)ds + \int_{-d}^{0}\int_{t + \theta}^{t} \dot{e}^T(s)Z_2\dot{e}(s)dsd\theta\) (5.17) - \(V_5(e(t)) = (d - t + t_p)\int_{t_p}^{t} \dot{e}^T(s)N_1\dot{e}(s)ds\) (5.18) - \(V_6(e(t)) = (t_{p + 1} - t)(t - t_p)e^T(t_p)N_2e(t_p)\) (5.19) - \(V_7(e(t)) = d^2\int_{t_p}^{t} \dot{e}^T(s)N_3\dot{e}(s)ds - \frac{\pi^2}{4}\int_{t_p}^{t} [e(s) - e(t_p)]^T N_3[e(s) - e(t_p)]ds\) (5.20) 计算 \(V(e(t))\) 的时间导数： - \(\dot{V}_1(e(t)) = 2e^T(t)P\dot{e}(t)\) (5.21) - \(\dot{V}_2(e(t)) \leq e^T(t)Q_1e(t) - (1 - h)e^T(t - \tau(t))Q_1e(t - \tau(t)) + e^T(t)Q_2e(t) - e^T(t - \tau_1)Q_2e(t - \tau_1) + e^T(t - \tau_1)Q_3e(t - \tau_1) - e