[control] multi agent system summary. 1. system model. 2. control objectives. 3. model transformation.

【 control 】 Summary of multi-agent system .1. System model .2. Control objectives .3. Model transformation .
【 control 】 Summary of multi-agent system .4. Control agreement .
【 control 】 Summary of multi-agent system .5. System merge .

1. System model

1.1 First order one-dimensional system

$$\{\begin{array}{rl}{\dot{p}}_i& =u_i\end{array}\text{\hspace{1em}()}$$

1.2 First order two-dimensional system

$$\{\begin{array}{rl}{\dot{p}}_i& =v_i\\ {\dot{v}}_i& =u_i\end{array}\text{\hspace{1em}()}$$

1.3 Second order one-dimensional system

$$\{\begin{array}{rl}{\dot{p}}_i^x& =u_i^x\\ {\dot{p}}_i^y& =u_i^y\end{array}\text{\hspace{1em}()}$$

1.4 Second order two-dimensional system

$$\{\begin{array}{rl}{\dot{p}}_i^x& =v_i^x\\ {\dot{p}}_i^y& =v_i^y\\ {\dot{v}}_i^x& =u_i^x\\ {\dot{v}}_i^y& =u_i^y\end{array}\text{\hspace{1em}()}$$

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2. Control objectives

The control target is the final state of all agents

2.1 First order one-dimensional system

$$\begin{array}{rl}\underset{t\to \infty }{\lim }\Vert p_j-p_i\Vert & =0\end{array}\text{\hspace{1em}()}$$

2.2 First order two-dimensional system

$$\begin{array}{rl}\underset{t\to \infty }{\lim }\Vert p_j^x-p_i^x\Vert & =0\\ \underset{t\to \infty }{\lim }\Vert p_j^y-p_i^y\Vert & =0\end{array}\text{\hspace{1em}()}$$

2.3 Second order one-dimensional system

$$\begin{array}{rl}\underset{t\to \infty }{\lim }\Vert p_j-p_i\Vert & =0\\ \underset{t\to \infty }{\lim }\Vert v_j-v_i\Vert & =0\end{array}\text{\hspace{1em}()}$$

2.4 Second order two-dimensional system

$$\begin{array}{rl}\underset{t\to \infty }{\lim }\Vert p_j^x-p_i^x\Vert & =0\\ \underset{t\to \infty }{\lim }\Vert p_j^y-p_i^y\Vert & =0\\ \underset{t\to \infty }{\lim }\Vert v_j^x-v_i^x\Vert & =0\\ \underset{t\to \infty }{\lim }\Vert v_j^y-v_i^y\Vert & =0\end{array}\text{\hspace{1em}()}$$

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3. Model transformation

3.1 First order one-dimensional system

The system model of a single agent is
$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_i\end{array}\right]& =\left[\begin{array}{c}0\end{array}\right]\left[\begin{array}{c}{p}_i\end{array}\right]+\left[\begin{array}{c}1\end{array}\right]\left[\begin{array}{c}{u}_i\end{array}\right]\\ & =0\cdot X+1\cdot U\end{array}\text{\hspace{1em}()}$$

The system model of multiple agents is
$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_1\\ {\dot{p}}_2\\ {\dot{p}}_3\end{array}\right]& =\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}{p}_1\\ p_2\\ p_3\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\end{array}\right]\\ & =0_{N\times N}\cdot X+I_N\cdot U\end{array}\text{\hspace{1em}()}$$

3.2 First order two-dimensional system

The system model of a single agent is
$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_i^x\\ {\dot{p}}_i^y\end{array}\right]& =\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}{p}_i^x\\ p_i^y\end{array}\right]+\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}{u}_i^x\\ u_i^y\end{array}\right]\\ & =0_{2\times 2}\cdot X+I_2\cdot U\end{array}\text{\hspace{1em}()}$$

3.2.1 Mode one

The system model of multiple agents is
$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_1^x\\ {\dot{p}}_1^y\\ {\dot{p}}_2^x\\ {\dot{p}}_2^y\\ {\dot{p}}_3^x\\ {\dot{p}}_3^y\end{array}\right]& =\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{p}_1^x\\ p_1^y\\ p_2^x\\ p_2^y\\ p_3^x\\ p_3^y\end{array}\right]+\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{u}_1^x\\ u_1^y\\ u_2^x\\ u_2^y\\ u_3^x\\ u_3^y\end{array}\right]\\ & =0_{2N\times 2N}\cdot X+I_{2N}\cdot U\end{array}\text{\hspace{1em}()}$$

3.2.2 Mode two

The system model of multiple agents is
$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_1^x\\ {\dot{p}}_2^x\\ {\dot{p}}_3^x\\ {\dot{p}}_1^y\\ {\dot{p}}_2^y\\ {\dot{p}}_3^y\end{array}\right]& =\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{p}_1^x\\ p_2^x\\ p_3^x\\ p_1^y\\ p_2^y\\ p_3^y\end{array}\right]+\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{u}_1^x\\ u_2^x\\ u_3^x\\ u_1^y\\ u_2^y\\ u_3^y\end{array}\right]\\ & =0_{2N\times 2N}\cdot X+I_{2N}\cdot U\end{array}\text{\hspace{1em}()}$$

3.3 Second order one-dimensional system

The system model of a single agent is
$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_i\\ {\dot{v}}_i\end{array}\right]& =\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{p}_i\\ v_i\end{array}\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]\left[\begin{array}{c}{u}_i\end{array}\right]\end{array}\text{\hspace{1em}()}$$

3.3.1 Mode one

The system model of multiple agents is
$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_1\\ {\dot{v}}_1\\ {\dot{p}}_2\\ {\dot{v}}_2\\ {\dot{p}}_3\\ {\dot{v}}_3\end{array}\right]& =\left[\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{p}_1\\ v_1\\ p_2\\ v_2\\ p_3\\ v_3\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 1& 0& 0\\ 0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\end{array}\right]\\ & =I_N\otimes \left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\cdot X+I_N\otimes \left[\begin{array}{c}0\\ 1\end{array}\right]\cdot U\end{array}\text{\hspace{1em}()}$$

3.3.2 Mode two

The system model of multiple agents is
$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_1\\ {\dot{p}}_2\\ {\dot{p}}_3\\ {\dot{v}}_1\\ {\dot{v}}_2\\ {\dot{v}}_3\end{array}\right]& =\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{p}_1\\ p_2\\ p_3\\ v_1\\ v_2\\ v_3\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\end{array}\right]\\ & =\left[\begin{array}{cc}{0}_{N\times N}& I_N\\ 0_{N\times N}& 0_{N\times N}\end{array}\right]\cdot X+\left[\begin{array}{c}{0}_{N\times N}\\ I_N\end{array}\right]\cdot U\\ & =\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\otimes I_N\cdot X+\left[\begin{array}{c}0\\ 1\end{array}\right]\otimes I_N\cdot U\end{array}\text{\hspace{1em}()}$$

3.4 Second order two-dimensional system

The system model of a single agent is

$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_i^x\\ {\dot{p}}_i^y\\ {\dot{v}}_i^x\\ {\dot{v}}_i^y\end{array}\right]& =\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{p}_i^x\\ p_i^y\\ v_i^x\\ v_i^y\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& 0\\ 1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}{u}_i^x\\ u_i^y\end{array}\right]\\ & =a\otimes I_2\cdot X_i+b\otimes I_2\cdot U_i\end{array}\text{\hspace{1em}()}$$

3.4.1 Mode one

The system model of multiple agents is
$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_1^x\\ {\dot{p}}_1^y\\ {\dot{v}}_1^x\\ {\dot{v}}_1^y\\ {\dot{p}}_2^x\\ {\dot{p}}_2^y\\ {\dot{v}}_2^x\\ {\dot{v}}_2^y\\ {\dot{p}}_3^x\\ {\dot{p}}_3^y\\ {\dot{v}}_3^x\\ {\dot{v}}_3^y\end{array}\right]& =\left[\begin{array}{cccccccccccc}0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{p}_1^x\\ p_1^y\\ v_1^x\\ v_1^y\\ p_2^x\\ p_2^y\\ v_2^x\\ v_2^y\\ p_3^x\\ p_3^y\\ v_3^x\\ v_3^y\end{array}\right]+\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{u}_1^x\\ u_1^y\\ u_2^x\\ u_2^y\\ u_3^x\\ u_3^y\end{array}\right]\\ & =I_N\otimes \left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\cdot X+I_N\otimes \left[\begin{array}{cc}0& 0\\ 0& 0\\ 1& 0\\ 0& 1\end{array}\right]\cdot U\end{array}\text{\hspace{1em}()}$$

3.4.2 Mode two

The system model of multiple agents is
$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_1^x\\ {\dot{p}}_2^x\\ {\dot{p}}_3^x\\ {\dot{p}}_1^y\\ {\dot{p}}_2^y\\ {\dot{p}}_3^y\\ {\dot{v}}_1^x\\ {\dot{v}}_2^x\\ {\dot{v}}_3^x\\ {\dot{v}}_1^y\\ {\dot{v}}_2^y\\ {\dot{v}}_3^y\end{array}\right]& =\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{p}_1^x\\ p_2^x\\ p_3^x\\ p_1^y\\ p_2^y\\ p_3^y\\ v_1^x\\ v_2^x\\ v_3^x\\ v_1^y\\ v_2^y\\ v_3^y\end{array}\right]+\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{u}_1^x\\ u_2^x\\ u_3^x\\ u_1^y\\ u_2^y\\ u_3^y\end{array}\right]\\ & =\left[\begin{array}{cccc}{0}_{N\times N}& 0_{N\times N}& I_N& 0_{N\times N}\\ 0_{N\times N}& 0_{N\times N}& 0_{N\times N}& I_N\\ 0_{N\times N}& 0_{N\times N}& 0_{N\times N}& 0_{N\times N}\\ 0_{N\times N}& 0_{N\times N}& 0_{N\times N}& 0_{N\times N}\end{array}\right]\cdot X+\left[\begin{array}{cc}{0}_{N\times N}& 0_{N\times N}\\ 0_{N\times N}& 0_{N\times N}\\ I_N& 0_{N\times N}\\ 0_{N\times N}& I_N\end{array}\right]\cdot U\\ & =\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\otimes I_N\cdot X+\left[\begin{array}{cc}0& 0\\ 0& 0\\ 1& 0\\ 0& 1\end{array}\right]\otimes I_N\cdot U\end{array}\text{\hspace{1em}()}$$

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