[control] multi agent system summary. 5. system consolidation.

【 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 .

5. System merge

5.1 First order one-dimensional system

$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_1\\ {\dot{p}}_2\\ {\dot{p}}_3\end{array}\right]& =0_{N\times N}\cdot X+I_N\cdot U\end{array}\text{\hspace{1em}()}$$

$$\begin{array}{rl}\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\end{array}\right]& =-\alpha L\cdot X\end{array}\text{\hspace{1em}()}$$

$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_1\\ {\dot{p}}_2\\ {\dot{p}}_3\end{array}\right]& =0_{N\times N}\cdot X+I_N\cdot U\\ & =0_{N\times N}\cdot X+I_N\cdot (-\alpha L\cdot X)\\ & =-\alpha L\cdot X\end{array}\text{\hspace{1em}()}$$

5.2 First order two-dimensional system

5.2.1 Mode one

$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_1^x\\ {\dot{p}}_1^y\\ {\dot{p}}_2^x\\ {\dot{p}}_2^y\\ {\dot{p}}_3^x\\ {\dot{p}}_3^y\end{array}\right]& =0_{2N\times 2N}\cdot X+I_{2N}\cdot U\end{array}\text{\hspace{1em}()}$$

$$\begin{array}{rl}\left[\begin{array}{c}{u}_1^x\\ u_1^y\\ u_2^x\\ u_2^y\\ u_3^x\\ u_3^y\end{array}\right]& =-\alpha L\otimes I_2\cdot X\end{array}\text{\hspace{1em}()}$$

$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_1^x\\ {\dot{p}}_1^y\\ {\dot{p}}_2^x\\ {\dot{p}}_2^y\\ {\dot{p}}_3^x\\ {\dot{p}}_3^y\end{array}\right]& =0_{2N\times 2N}\cdot X+I_{2N}\cdot U\\ & =0_{2N\times 2N}\cdot X+I_{2N}\cdot (-\alpha L\otimes I_2\cdot X)\\ & =-\alpha L\otimes I_2\cdot X\end{array}\text{\hspace{1em}()}$$

5.2.2 Mode two

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

$$\begin{array}{rl}\left[\begin{array}{c}{u}_1^x\\ u_2^x\\ u_3^x\\ u_1^y\\ u_2^y\\ u_3^y\end{array}\right]& =I_2\otimes -\alpha L\cdot X\end{array}\text{\hspace{1em}()}$$

$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_1^x\\ {\dot{p}}_2^x\\ {\dot{p}}_3^x\\ {\dot{p}}_1^y\\ {\dot{p}}_2^y\\ {\dot{p}}_3^y\end{array}\right]& =0_{2N\times 2N}\cdot X+I_{2N}\cdot U\\ & =0_{2N\times 2N}\cdot X+I_{2N}\cdot (I_2\otimes -\alpha L\cdot X)\\ & =I_2\otimes -\alpha L\cdot X\end{array}\text{\hspace{1em}()}$$

5.3 Second order one-dimensional system

5.3.1 Mode one

$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_1\\ {\dot{v}}_1\\ {\dot{p}}_2\\ {\dot{v}}_2\\ {\dot{p}}_3\\ {\dot{v}}_3\end{array}\right]& =I_N\otimes \left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\cdot X+I_N\otimes \left[\begin{array}{c}0\\ 1\end{array}\right]\cdot U\end{array}\text{\hspace{1em}()}$$

$$\begin{array}{rl}\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\end{array}\right]& =(-L)\otimes \left[\begin{array}{cc}\alpha & \beta \end{array}\right]\cdot X\end{array}\text{\hspace{1em}()}$$

$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_1\\ {\dot{v}}_1\\ {\dot{p}}_2\\ {\dot{v}}_2\\ {\dot{p}}_3\\ {\dot{v}}_3\end{array}\right]& =I_N\otimes \left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\cdot X+I_N\otimes \left[\begin{array}{c}0\\ 1\end{array}\right]\cdot U\\ & =I_N\otimes \left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\cdot X+I_N\otimes \left[\begin{array}{c}0\\ 1\end{array}\right]\cdot ((-L)\otimes \left[\begin{array}{cc}\alpha & \beta \end{array}\right]\cdot X)\end{array}\text{\hspace{1em}()}$$

5.3.2 Mode two

$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_1\\ {\dot{p}}_2\\ {\dot{p}}_3\\ {\dot{v}}_1\\ {\dot{v}}_2\\ {\dot{v}}_3\end{array}\right]& =\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\otimes I_N\cdot X+\left[\begin{array}{c}0\\ 1\end{array}\right]\otimes I_N\cdot U\end{array}\text{\hspace{1em}()}$$

$$\begin{array}{rl}\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\end{array}\right]& =\left[\begin{array}{cc}\alpha & \beta \end{array}\right]\otimes (-L)\cdot X\end{array}\text{\hspace{1em}()}$$

$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_1\\ {\dot{p}}_2\\ {\dot{p}}_3\\ {\dot{v}}_1\\ {\dot{v}}_2\\ {\dot{v}}_3\end{array}\right]& =\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\otimes I_N\cdot X+\left[\begin{array}{c}0\\ 1\end{array}\right]\otimes I_N\cdot U\\ & =\left[\begin{array}{cc}{0}_{N\times N}& I_N\\ -\alpha L& -\beta L\end{array}\right]\cdot X\end{array}\text{\hspace{1em}()}$$

5.4 Second order two-dimensional system

5.4.1 Mode one

5.4.2 Mode two

$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_1^x\\ {\dot{p}}_2^x\\ {\dot{p}}_3^x\\ {\dot{p}}_1^y\\ {\dot{p}}_2^y\\ {\dot{p}}_3^y\\ {\dot{v}}_1^x\\ {\dot{v}}_2^x\\ {\dot{v}}_3^x\\ {\dot{v}}_1^y\\ {\dot{v}}_2^y\\ {\dot{v}}_3^y\end{array}\right]& =\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\otimes I_N\cdot X+\left[\begin{array}{cc}0& 0\\ 0& 0\\ 1& 0\\ 0& 1\end{array}\right]\otimes I_N\cdot U\end{array}\text{\hspace{1em}()}$$

$$\begin{array}{rl}\left[\begin{array}{c}{u}_1^x\\ u_2^x\\ u_3^x\\ u_1^y\\ u_2^y\\ u_3^y\end{array}\right]& =\left[\begin{array}{cc}\alpha & \beta \end{array}\right]\otimes I_2\otimes (-L)\cdot X\end{array}\text{\hspace{1em}()}$$

$$\begin{array}{rl}\left[\begin{array}{c}{\dot{p}}_1^x\\ {\dot{p}}_2^x\\ {\dot{p}}_3^x\\ {\dot{p}}_1^y\\ {\dot{p}}_2^y\\ {\dot{p}}_3^y\\ {\dot{v}}_1^x\\ {\dot{v}}_2^x\\ {\dot{v}}_3^x\\ {\dot{v}}_1^y\\ {\dot{v}}_2^y\\ {\dot{v}}_3^y\end{array}\right]& =\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\otimes I_N\cdot X+\left[\begin{array}{cc}0& 0\\ 0& 0\\ 1& 0\\ 0& 1\end{array}\right]\otimes I_N\cdot U\\ & =\left[\begin{array}{cccc}{0}_{N\times N}& 0_{N\times N}& I_N& 0_{N\times N}\\ 0_{N\times N}& 0_{N\times N}& 0_{N\times N}& I_N\\ -\alpha L& 0_{N\times N}& -\beta L& 0_{N\times N}\\ 0_{N\times N}& -\alpha L& 0_{N\times N}& -\beta L\end{array}\right]\end{array}\text{\hspace{1em}()}$$