【控制】多智能体系统总结。1. 系统模型。2.控制目标。3.模型转换。
【控制】多智能体系统总结。4.控制协议。
【控制】多智能体系统总结。5.系统合并。

4. 控制协议

4.1 一阶一维系统

$$u_i=\alpha \sum _{j\in N_i}{a}_{ij}(p_j-p_i)\text{\hspace{1em}()}$$

$$\begin{array}{rlrlr}{u}_1& =\alpha a_{11}(p_1-p_1)& +\alpha a_{12}(p_2-p_1)& +\alpha a_{13}(p_3-p_1)& +\cdots \\ u_2& =\alpha a_{21}(p_1-p_2)& +\alpha a_{22}(p_2-p_2)& +\alpha a_{23}(p_3-p_2)& +\cdots \\ u_3& =\alpha a_{31}(p_1-p_3)& +\alpha a_{32}(p_2-p_3)& +\alpha a_{33}(p_3-p_3)& +\cdots \end{array}\text{\hspace{1em}()}$$

$$\begin{array}{rl}\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\end{array}\right]& =\left[\begin{array}{ccc}-\alpha d_1& \alpha a_{12}& \alpha a_{13}\\ \alpha a_{21}& -\alpha d_2& \alpha a_{23}\\ \alpha a_{31}& \alpha a_{32}& -\alpha d_3\end{array}\right]\left[\begin{array}{c}{p}_1\\ p_2\\ p_3\end{array}\right]\\ & =-\alpha L\cdot X\end{array}\text{\hspace{1em}()}$$

4.2 一阶二维系统

$$\begin{array}{rl}{u}_i^x& =\alpha \sum _{j\in N_i}{a}_{ij}(p_j^x-p_i^x)\\ u_i^y& =\alpha \sum _{j\in N_i}{a}_{ij}(p_j^y-p_i^y)\end{array}\text{\hspace{1em}()}$$

$$\begin{array}{rlrlr}{u}_1^x& =\alpha a_{11}(p_1^x-p_1^x)& +\alpha a_{12}(p_2^x-p_1^x)& +\alpha a_{13}(p_3^x-p_1^x)& +\cdots \\ u_1^y& =\alpha a_{11}(p_1^y-p_1^y)& +\alpha a_{12}(p_2^y-p_1^y)& +\alpha a_{13}(p_3^y-p_1^y)& +\cdots \\ u_2^x& =\alpha a_{21}(p_1^x-p_2^x)& +\alpha a_{22}(p_2^x-p_2^x)& +\alpha a_{23}(p_3^x-p_2^x)& +\cdots \\ u_2^y& =\alpha a_{21}(p_1^y-p_2^y)& +\alpha a_{22}(p_2^y-p_2^y)& +\alpha a_{23}(p_3^y-p_2^y)& +\cdots \\ u_3^x& =\alpha a_{31}(p_1^x-p_3^x)& +\alpha a_{32}(p_2^x-p_3^x)& +\alpha a_{33}(p_3^x-p_3^x)& +\cdots \\ u_3^y& =\alpha a_{31}(p_1^y-p_3^y)& +\alpha a_{32}(p_2^y-p_3^y)& +\alpha a_{33}(p_3^y-p_3^y)& +\cdots \end{array}\text{\hspace{1em}()}$$

4.2.1 方式一

$$\begin{array}{rl}\left[\begin{array}{c}{u}_1^x\\ u_1^y\\ u_2^x\\ u_2^y\\ u_3^x\\ u_3^y\end{array}\right]& =\left[\begin{array}{cccccc}-\alpha d_1& 0& \alpha a_{12}& 0& \alpha a_{13}& 0\\ 0& -\alpha d_1& 0& \alpha a_{12}& 0& \alpha a_{13}\\ \alpha a_{21}& 0& -\alpha d_2& 0& \alpha a_{23}& 0\\ 0& \alpha a_{21}& 0& -\alpha d_2& 0& \alpha a_{23}\\ \alpha a_{31}& 0& \alpha a_{32}& 0& -\alpha d_3& 0\\ 0& \alpha a_{31}& 0& \alpha a_{32}& 0& -\alpha d_3\end{array}\right]\left[\begin{array}{c}{p}_1^x\\ p_1^y\\ p_2^x\\ p_2^y\\ p_3^x\\ p_3^y\end{array}\right]\\ & =-\alpha L\otimes I_2\cdot X\end{array}\text{\hspace{1em}()}$$

4.2.2 方式二

$$\begin{array}{rl}\left[\begin{array}{c}{u}_1^x\\ u_2^x\\ u_3^x\\ u_1^y\\ u_2^y\\ u_3^y\end{array}\right]& =\left[\begin{array}{cccccc}-\alpha d_1& \alpha a_{12}& \alpha a_{13}& 0& 0& 0\\ \alpha a_{21}& -\alpha d_2& \alpha a_{23}& 0& 0& 0\\ \alpha a_{31}& \alpha a_{32}& -\alpha d_3& 0& 0& 0\\ 0& 0& 0& -\alpha d_1& \alpha a_{12}& \alpha a_{13}\\ 0& 0& 0& \alpha a_{21}& -\alpha d_2& \alpha a_{23}\\ 0& 0& 0& \alpha a_{31}& \alpha a_{32}& -\alpha d_3\end{array}\right]\left[\begin{array}{c}{p}_1^x\\ p_2^x\\ p_3^x\\ p_1^y\\ p_2^y\\ p_3^y\end{array}\right]\\ & =I_2\otimes -\alpha L\cdot X\end{array}\text{\hspace{1em}()}$$

4.3 二阶一维系统

$$u_i=\alpha \sum _{j\in N_i}{a}_{ij}(p_j-p_i)+\beta \sum _{j\in N_i}{a}_{ij}(v_j-v_i)\text{\hspace{1em}()}$$

$$\begin{array}{rlrlr}{u}_1& =\alpha a_{11}(p_1-p_1)+\beta a_{11}(v_1-v_1)& +\alpha a_{12}(p_2-p_1)+\beta a_{12}(v_2-v_1)& +\alpha a_{13}(p_3-p_1)+\beta a_{13}(v_3-v_1)& +\cdots \\ u_2& =\alpha a_{21}(p_1-p_2)+\beta a_{21}(v_1-v_2)& +\alpha a_{22}(p_2-p_2)+\beta a_{22}(v_2-v_2)& +\alpha a_{23}(p_3-p_2)+\beta a_{23}(v_3-v_2)& +\cdots \\ u_3& =\alpha a_{31}(p_1-p_3)+\beta a_{31}(v_1-v_3)& +\alpha a_{32}(p_2-p_3)+\beta a_{32}(v_2-v_3)& +\alpha a_{33}(p_3-p_3)+\beta a_{33}(v_3-v_3)& +\cdots \end{array}\text{\hspace{1em}()}$$

4.3.1 方式一

$$\begin{array}{rl}\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\end{array}\right]& =\left[\begin{array}{cccccc}-\alpha d_1& -\beta d_1& \alpha a_{12}& \beta a_{12}& \alpha a_{13}& \beta a_{13}\\ \alpha a_{21}& \beta a_{21}& -\alpha d_2& -\beta d_2& \alpha a_{23}& \beta a_{23}\\ \alpha a_{31}& \beta a_{31}& \alpha a_{32}& \beta a_{32}& -\alpha d_3& -\beta d_3\end{array}\right]\left[\begin{array}{c}{p}_1\\ v_1\\ p_2\\ v_2\\ p_3\\ v_3\end{array}\right]\\ & =(-L)\otimes \left[\begin{array}{cc}\alpha & \beta \end{array}\right]\cdot X\end{array}\text{\hspace{1em}()}$$

4.3.2 方式二

$$\begin{array}{rl}\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\end{array}\right]& =\left[\begin{array}{cccccc}-\alpha d_1& \alpha a_{12}& \alpha a_{13}& -\beta d_1& \beta a_{12}& \beta a_{13}\\ \alpha a_{21}& -\alpha d_2& \alpha a_{23}& \beta a_{21}& -\beta d_2& \beta a_{23}\\ \alpha a_{31}& \alpha a_{32}& -\alpha d_3& \beta a_{31}& \beta a_{32}& -\beta d_3\end{array}\right]\left[\begin{array}{c}{p}_1\\ p_2\\ p_3\\ v_1\\ v_2\\ v_3\end{array}\right]\\ & =\left[\begin{array}{cc}\alpha & \beta \end{array}\right]\otimes (-L)\cdot X\end{array}\text{\hspace{1em}()}$$

4.4 二阶二维系统

$$\begin{array}{rl}{u}_i^x& =\alpha \sum _{j\in N_i}{a}_{ij}(p_j^x-p_i^x)+\beta \sum _{j\in N_i}{a}_{ij}(v_j^x-v_i^x)\\ u_i^y& =\alpha \sum _{j\in N_i}{a}_{ij}(p_j^y-p_i^y)+\beta \sum _{j\in N_i}{a}_{ij}(v_j^y-v_i^y)\end{array}\text{\hspace{1em}()}$$

$$\begin{array}{rlrlr}{u}_1& =\alpha a_{11}(p_1^x-p_1^x)+\beta a_{11}(v_1^x-v_1^x)& +\alpha a_{12}(p_2^x-p_1^x)+\beta a_{12}(v_2^x-v_1^x)& +\alpha a_{13}(p_3^x-p_1^x)+\beta a_{13}(v_3^x-v_1^x)& +\cdots \\ u_2& =\alpha a_{21}(p_1-p_2)+\beta a_{21}(v_1-v_2)& +\alpha a_{22}(p_2-p_2)+\beta a_{22}(v_2-v_2)& +\alpha a_{23}(p_3-p_2)+\beta a_{23}(v_3-v_2)& +\cdots \\ u_3& =\alpha a_{31}(p_1-p_3)+\beta a_{31}(v_1-v_3)& +\alpha a_{32}(p_2-p_3)+\beta a_{32}(v_2-v_3)& +\alpha a_{33}(p_3-p_3)+\beta a_{33}(v_3-v_3)& +\cdots \end{array}\text{\hspace{1em}()}$$

4.4.1 方式一

$$\begin{array}{rl}\left[\begin{array}{c}{u}_1^x\\ u_1^y\\ u_2^x\\ u_2^y\\ u_3^x\\ u_3^y\end{array}\right]& =\left[\begin{array}{cccccccccccc}-\alpha d_1& 0& -\beta d_1& 0& \alpha a_{12}& 0& \beta a_{12}& 0& \alpha a_{13}& 0& \beta a_{13}& 0\\ 0& -\alpha d_1& 0& -\beta d_1& 0& \alpha a_{12}& 0& \beta a_{12}& 0& \alpha a_{13}& 0& \beta a_{13}\\ \alpha a_{21}& 0& \beta a_{21}& 0& -\alpha d_2& 0& -\beta d_2& 0& \alpha a_{23}& 0& \beta a_{23}& 0\\ 0& \alpha a_{21}& 0& \beta a_{21}& 0& -\alpha d_2& 0& -\beta d_2& 0& \alpha a_{23}& 0& \beta a_{23}\\ \alpha a_{31}& 0& \beta a_{31}& 0& \alpha a_{32}& 0& \beta a_{32}& 0& -\alpha d_3& 0& -\beta d_3& 0\\ 0& \alpha a_{31}& 0& \beta a_{31}& 0& \alpha a_{32}& 0& \beta a_{32}& 0& -\alpha d_3& 0& -\beta d_3\end{array}\right]\left[\begin{array}{c}{p}_1^x\\ p_1^y\\ v_1^x\\ v_1^y\\ p_2^x\\ p_2^y\\ v_2^x\\ v_2^y\\ p_3^x\\ p_3^y\\ v_3^x\\ v_3^y\end{array}\right]\\ & =-L\otimes \left[\begin{array}{cc}\alpha & \beta \end{array}\right]\otimes I_2\cdot X\end{array}\text{\hspace{1em}()}$$

4.4.2 方式二

$$\begin{array}{rl}\left[\begin{array}{c}{u}_1^x\\ u_2^x\\ u_3^x\\ u_1^y\\ u_2^y\\ u_3^y\end{array}\right]& =\left[\begin{array}{cccccccccccc}-\alpha d_1& \alpha a_{12}& \alpha a_{13}& 0& 0& 0& -\beta d_1& \beta a_{12}& \beta a_{13}& 0& 0& 0\\ \alpha a_{21}& -\alpha d_2& \alpha a_{23}& 0& 0& 0& \beta a_{21}& -\beta d_2& \beta a_{23}& 0& 0& 0\\ \alpha a_{31}& \alpha a_{32}& -\alpha d_3& 0& 0& 0& \beta a_{31}& \beta a_{32}& -\beta d_3& 0& 0& 0\\ 0& 0& 0& -\alpha d_1& \alpha a_{12}& \alpha a_{13}& 0& 0& 0& -\beta d_1& \beta a_{12}& \beta a_{13}\\ 0& 0& 0& \alpha a_{21}& -\alpha d_2& \alpha a_{23}& 0& 0& 0& \beta a_{21}& -\beta d_2& \beta a_{23}\\ 0& 0& 0& \alpha a_{31}& \alpha a_{32}& -\alpha d_3& 0& 0& 0& \beta a_{31}& \beta a_{32}& -\beta d_3\end{array}\right]\left[\begin{array}{c}{p}_1^x\\ p_2^x\\ p_3^x\\ p_1^y\\ p_2^y\\ p_3^y\\ v_1^x\\ v_2^x\\ v_3^x\\ v_1^y\\ v_2^y\\ v_3^y\end{array}\right]\\ & =\left[\begin{array}{cc}\alpha & \beta \end{array}\right]\otimes I_2\otimes (-L)\cdot X\end{array}\text{\hspace{1em}()}$$

3.1 动态一致性控制协议

控制协议为
$$u_i=\alpha \sum _{j\in N_i}{a}_{ij}(p_j-p_i)+\beta \sum _{j\in N_i}{a}_{ij}(v_j-v_i)\text{\hspace{1em}()}$$

其中
$a_{ij}$：表示邻接矩阵
$\alpha$：表示控制参数
$\beta$：表示控制参数

通过分析控制协议：
$$\begin{array}{rlrlr}{u}_1& =\alpha a_{11}(p_1-p_1)+\beta a_{11}(v_1-v_1)& +\alpha a_{12}(p_2-p_1)+\beta a_{12}(v_2-v_1)& +\alpha a_{13}(p_3-p_1)+\beta a_{13}(v_3-v_1)& +\cdots \\ u_2& =\alpha a_{21}(p_1-p_2)+\beta a_{21}(v_1-v_2)& +\alpha a_{22}(p_2-p_2)+\beta a_{22}(v_2-v_2)& +\alpha a_{23}(p_3-p_2)+\beta a_{23}(v_3-v_2)& +\cdots \\ u_3& =\alpha a_{31}(p_1-p_3)+\beta a_{31}(v_1-v_3)& +\alpha a_{32}(p_2-p_3)+\beta a_{32}(v_2-v_3)& +\alpha a_{33}(p_3-p_3)+\beta a_{33}(v_3-v_3)& +\cdots \end{array}\text{\hspace{1em}()}$$

可以得到以下控制协议的矩阵形式：

$$\begin{array}{rl}\left[\begin{array}{c}{u}_1^x\\ u_1^y\\ u_2^x\\ u_2^y\\ u_3^x\\ u_3^y\end{array}\right]& =\left[\begin{array}{cccccccccccc}-\alpha d_1& 0& -\beta d_1& 0& \alpha a_{12}& 0& \beta a_{12}& 0& \alpha a_{13}& 0& \beta a_{13}& 0\\ 0& -\alpha d_1& 0& -\beta d_1& 0& \alpha a_{12}& 0& \beta a_{12}& 0& \alpha a_{13}& 0& \beta a_{13}\\ \alpha a_{21}& 0& \beta a_{21}& 0& -\alpha d_2& 0& -\beta d_2& 0& \alpha a_{23}& 0& \beta a_{23}& 0\\ 0& \alpha a_{21}& 0& \beta a_{21}& 0& -\alpha d_2& 0& -\beta d_2& 0& \alpha a_{23}& 0& \beta a_{23}\\ \alpha a_{31}& 0& \beta a_{31}& 0& \alpha a_{32}& 0& \beta a_{32}& 0& -\alpha d_3& 0& -\beta d_3& 0\\ 0& \alpha a_{31}& 0& \beta a_{31}& 0& \alpha a_{32}& 0& \beta a_{32}& 0& -\alpha d_3& 0& -\beta d_3\end{array}\right]\left[\begin{array}{c}{p}_1^x\\ p_1^y\\ v_1^x\\ v_1^y\\ p_2^x\\ p_2^y\\ v_2^x\\ v_2^y\\ p_3^x\\ p_3^y\\ v_3^x\\ v_3^y\end{array}\right]\\ & =\left[\begin{array}{ccc}-d_1& a_{12}& a_{13}\\ a_{21}& -d_2& a_{23}\\ a_{31}& a_{32}& -d_3\end{array}\right]\otimes \left[\begin{array}{cc}\alpha & \beta \end{array}\right]\otimes I_2\cdot X_i\\ & =-L\otimes \left[\begin{array}{cc}\alpha & \beta \end{array}\right]\otimes I_2\cdot X_i\end{array}$$

其中
$d_1=a_{12}+a_{13}+\cdots$,
$d_2=a_{21}+a_{23}+\cdots$,
$d_3=a_{31}+a_{32}+\cdots$,
$\cdots$.

3.2 静态一致性控制协议

控制协议为
$$u_i=\alpha \sum _{j\in N_i}{a}_{ij}(p_j-p_i)-\beta v_i\text{\hspace{1em}()}$$

其中
$a_{ij}$：表示邻接矩阵
$\alpha$：表示控制参数
$\beta$：表示控制参数

通过分析控制协议：
$$\begin{array}{rl}{u}_1& =\alpha a_{11}(p_1-p_1)+\alpha a_{12}(p_2-p_1)+\alpha a_{13}(p_3-p_1)+\cdots +\beta v_1\\ u_2& =\alpha a_{21}(p_1-p_2)+\alpha a_{22}(p_2-p_2)+\alpha a_{23}(p_3-p_2)+\cdots +\beta v_2\\ u_3& =\alpha a_{31}(p_1-p_3)+\alpha a_{32}(p_2-p_3)+\alpha a_{33}(p_3-p_3)+\cdots +\beta v_3\end{array}\text{\hspace{1em}()}$$

可以得到以下控制协议的矩阵形式：

$$\begin{array}{rl}\left[\begin{array}{c}{u}_1^x\\ u_1^y\\ u_2^x\\ u_2^y\\ u_3^x\\ u_3^y\end{array}\right]& =\left[\begin{array}{cccccccccccc}-\alpha d_1& 0& -\beta & 0& \alpha a_{12}& 0& 0& 0& \alpha a_{13}& 0& 0& 0\\ 0& -\alpha d_1& 0& -\beta & 0& \alpha a_{12}& 0& 0& 0& \alpha a_{13}& 0& 0\\ \alpha a_{21}& 0& 0& 0& -\alpha d_2& 0& -\beta & 0& \alpha a_{23}& 0& 0& 0\\ 0& \alpha a_{21}& 0& 0& 0& -\alpha d_2& 0& -\beta & 0& \alpha a_{23}& 0& 0\\ \alpha a_{31}& 0& 0& 0& \alpha a_{32}& 0& 0& 0& -\alpha d_3& 0& -\beta & 0\\ 0& \alpha a_{31}& 0& 0& 0& \alpha a_{32}& 0& 0& 0& -\alpha d_3& 0& -\beta \end{array}\right]\left[\begin{array}{c}{p}_1^x\\ p_1^y\\ v_1^x\\ v_1^y\\ p_2^x\\ p_2^y\\ v_2^x\\ v_2^y\\ p_3^x\\ p_3^y\\ v_3^x\\ v_3^y\end{array}\right]\\ & =(\left[\begin{array}{ccc}-d_1& a_{12}& a_{13}\\ a_{21}& -d_2& a_{23}\\ a_{31}& a_{32}& -d_3\end{array}\right]\otimes \left[\begin{array}{cc}\alpha & 0\end{array}\right]\otimes I_2+\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right]\otimes \left[\begin{array}{cc}0& \beta \end{array}\right]\otimes I_2)\cdot X_i\\ & =(-L\otimes \left[\begin{array}{cc}\alpha & 0\end{array}\right]\otimes I_2+-I_N\otimes \left[\begin{array}{cc}0& \beta \end{array}\right]\otimes I_2)\cdot X_i\end{array}$$

其中
$d_1=a_{12}+a_{13}+\cdots$,
$d_2=a_{21}+a_{23}+\cdots$,
$d_3=a_{31}+a_{32}+\cdots$,
$\cdots$,
下标 ${}_N$ 表示智能体的个数。

最后

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