双极限齐次性(二)、推导二阶非线性多智能体固定时间一致性协议(第二部分)

2.3、证明系统是全局渐进稳定的

证明 $\dot{z}=\psi$ 是全局渐进稳定的

选择李雅普诺夫函数 $V=V_1+V_2+V_3$
$$\begin{array}{rl}{V}_1& =\frac{l_1}{1+{\alpha }_1}\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}|{\tilde{x}}_i-{\tilde{x}}_j{|}^{1+{\alpha }_1}+\frac{l_2}{1+{\alpha }_2}\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}|{\tilde{x}}_i-{\tilde{x}}_j{|}^{1+{\alpha }_2}\\ V_2& =c{\tilde{x}}^T\hat{L}\tilde{x}\\ V_3& =\sum _{i=1}^N{p}_i({\tilde{x}}_i+{\tilde{v}}_i{)}^2\end{array}$$
对其求导
$$\begin{array}{rl}{\dot{V}}_1& =l_1\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}\lceil {\tilde{x}}_i-{\tilde{x}}_j{\rfloor }^{{\alpha }_1}({\tilde{v}}_i-{\tilde{v}}_j)+l_2\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}\lceil {\tilde{x}}_i-{\tilde{x}}_j{\rfloor }^{{\alpha }_2}({\tilde{v}}_i-{\tilde{v}}_j)\\ {\dot{V}}_2& =2c{\tilde{x}}^T\hat{L}\tilde{v}=2c{\tilde{x}}^T\hat{L}(\tilde{x}+\tilde{v}-\tilde{x})=2c{\tilde{x}}^T\hat{L}(\tilde{x}+\tilde{v})-2c{\tilde{x}}^T\hat{L}x\le c(\tilde{x}+\tilde{v}{)}^T\hat{L}(\tilde{x}+\tilde{v})-c{\tilde{x}}^T\hat{L}\tilde{x}\\ {\dot{V}}_3& =2\sum _{i=1}^N{p}_i({\tilde{x}}_i+{\tilde{v}}_i)({\tilde{v}}_i+{\dot{\tilde{v}}}_i)=2\sum _{i=1}^N{p}_i({\tilde{x}}_i+{\tilde{v}}_i)({\tilde{v}}_i+u_i+f(x_i,v_i)-\sum _{j=1}^N{p}_{j}f(x_j,v_j))\end{array}$$
推导 $V_3$ 的 $v_i$ 部分
$$\begin{array}{rl}2\sum _{i=1}^N{p}_i({\tilde{x}}_i+{\tilde{v}}_i)v_i& =2\sum _{i=1}^N{p}_i({\tilde{x}}_i+v_i)({\tilde{x}}_i+v_i-{\tilde{x}}_i)\\ & \le 2p_{max}\sum _{i=1}^N({\tilde{x}}_i+{\tilde{v}}_i{)}^2+2p_{max}\sum _{i=1}^N|{\tilde{x}}_i+{\tilde{v}}_i||{\tilde{x}}_i|\\ & \le 2p_{max}(\tilde{x}+\tilde{v}{)}^T(\tilde{x}+\tilde{v})+2p_{max}(\tilde{x}+\tilde{v}{)}^{T}x\\ & \le 2p_{max}(\tilde{x}+\tilde{v}{)}^T(\tilde{x}+\tilde{v})+p_{max}{\tilde{x}}^T\tilde{x}+p_{max}(\tilde{x}+\tilde{v}{)}^T\tilde{x}\\ & \le 3p_{max}(\tilde{x}+\tilde{v}{)}^T(\tilde{x}+\tilde{v})+p_{max}{\tilde{x}}^T\tilde{x}\end{array}$$
推导 $V_3$ 的 $f(x_i,v_i)-{\sum }_{j=1}^N{p}_{j}f(x_j,v_j)$ 部分，由于 ${\sum }_{i=1}^N{p}_i(x_i+v_i)=0$， 可得 ${\sum }_{i=1}^N{p}_i(x_i+v_i)(f(\bar{x},\bar{v})-{\sum }_{j=1}^N{p}_{j}f(x_j,v_j))=0$ 。基于假设1可得
$$\begin{array}{rl}& 2\sum _{i=1}^N{p}_i({\tilde{x}}_i+{\tilde{v}}_i)(f(x_i,v_i)-\sum _{j=1}^N{p}_{j}f(x_j,v_j))\\ & =2\sum _{i=1}^N{p}_i({\tilde{x}}_i+{\tilde{v}}_i)(f(x_i,v_i)-f(\bar{x},\bar{v}))\\ & =2\rho p_{max}\sum _{i=1}^N|{\tilde{x}}_i-{\tilde{v}}_i|(|{\tilde{x}}_i|+|{\tilde{v}}_i|)\\ & =2\rho p_{max}\sum _{i=1}^N|{\tilde{x}}_i-{\tilde{v}}_i|(|{\tilde{x}}_i|+|{\tilde{v}}_i+{\tilde{x}}_i-{\tilde{x}}_i|)\\ & =2\rho p_{max}{\tilde{x}}^T\tilde{x}+4\rho p_{max}(\tilde{x}+\tilde{v}{)}^T(\tilde{x}+\tilde{v})\end{array}$$
推导 $V_3$ 的 $u_{3i}$ 部分
$$\begin{array}{rl}2\sum _{i=1}^N{p}_i({\tilde{x}}_i+{\tilde{v}}_i)u_{3i}& =-2k\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}({\tilde{x}}_i+{\tilde{v}}_i)({\tilde{x}}_i+{\tilde{v}}_i-{\tilde{x}}_j-{\tilde{v}}_j)\\ & =-k\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}({\tilde{x}}_i+{\tilde{v}}_i-{\tilde{x}}_j-{\tilde{v}}_j{)}^2\\ & =-2k(\tilde{x}+\tilde{v}{)}^T\hat{L}(\tilde{x}+\tilde{v})\end{array}$$
推导 $V_3$ 的 $u_{1i}+u_{2i}$ 部分
$$\begin{array}{rl}& 2\sum _{i=1}^N{p}_i({\tilde{x}}_i+{\tilde{v}}_i)(u_{1i}+u_{2i})\\ & =-2k\sum _{i=1}^N{p}_i{a}_{ij}({\tilde{x}}_i+{\tilde{v}}_i)(l_1\lceil x_i-x_j{\rfloor }^{{\alpha }_1}+l_1\lceil v_i-v_j{\rfloor }^{{\alpha }_1^{\prime }}+l_2\lceil x_i-x_j{\rfloor }^{{\alpha }_2}+l_2\lceil v_i-v_j{\rfloor }^{{\alpha }_2^{\prime }})\\ & =-2k\sum _{i=1}^N{p}_i{a}_{ij}{\tilde{x}}_i(l_1\lceil x_i-x_j{\rfloor }^{{\alpha }_1}+l_1\lceil v_i-v_j{\rfloor }^{{\alpha }_1^{\prime }}+l_2\lceil x_i-x_j{\rfloor }^{{\alpha }_2}+l_2\lceil v_i-v_j{\rfloor }^{{\alpha }_2^{\prime }})\\ & -2k\sum _{i=1}^N{p}_i{a}_{ij}{\tilde{v}}_i(l_1\lceil x_i-x_j{\rfloor }^{{\alpha }_1}+l_1\lceil v_i-v_j{\rfloor }^{{\alpha }_1^{\prime }}+l_2\lceil x_i-x_j{\rfloor }^{{\alpha }_2}+l_2\lceil v_i-v_j{\rfloor }^{{\alpha }_2^{\prime }})\\ & =-l_1\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}|{\tilde{x}}_i-{\tilde{x}}_j{|}^{1+{\alpha }_1}-l_2\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}|{\tilde{x}}_i-{\tilde{x}}_j{|}^{1+{\alpha }_2}\\ & -l_1\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}|{\tilde{v}}_i-{\tilde{v}}_j{|}^{1+{\alpha }_1^{\prime }}-l_2\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}|{\tilde{v}}_i-{\tilde{v}}_j{|}^{1+{\alpha }_2^{\prime }}\\ & -l_1\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}\lceil {\tilde{x}}_i-{\tilde{x}}_j{\rfloor }^{{\alpha }_1}({\tilde{v}}_i-{\tilde{v}}_j)-l_2\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}\lceil {\tilde{x}}_i-{\tilde{x}}_j{\rfloor }^{{\alpha }_2}({\tilde{v}}_i-{\tilde{v}}_j)\\ & -l_1\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}\lceil {\tilde{v}}_i-{\tilde{v}}_j{\rfloor }^{{\alpha }_1^{\prime }}({\tilde{x}}_i-{\tilde{x}}_j)-l_2\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}\lceil {\tilde{v}}_i-{\tilde{v}}_j{\rfloor }^{{\alpha }_2^{\prime }}({\tilde{x}}_i-{\tilde{x}}_j)\end{array}$$
由引理4(也是一个缩放不等式)以及参数关系 $1+{\alpha }_1<1+{\alpha }_1^{\prime }<2<1+{\alpha }_2^{\prime }<1+{\alpha }_2$ 可以得到
$$\begin{array}{rl}& -l_1\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}\lceil {\tilde{v}}_i-{\tilde{v}}_j{\rfloor }^{{\alpha }_1^{\prime }}({\tilde{x}}_i-{\tilde{x}}_j)\\ & \le l_1\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}|{\tilde{v}}_i-{\tilde{v}}_j{|}^{{\alpha }_1^{\prime }}|{\tilde{x}}_i-{\tilde{x}}_j|\\ & \le l_1\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}\left(\frac{{\alpha }_1^{\prime }}{1+{\alpha }_1^{\prime }}|{\tilde{v}}_i-{\tilde{v}}_j{|}^{1+{\alpha }_1^{\prime }}+\frac{1}{1+{\alpha }_1^{\prime }}|{\tilde{x}}_i+{\tilde{x}}_j{|}^{1+{\alpha }_1^{\prime }}\right)\\ & \le l_1\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}\left(\frac{{\alpha }_1^{\prime }}{1+{\alpha }_1^{\prime }}|{\tilde{v}}_i-{\tilde{v}}_j{|}^{1+{\alpha }_1^{\prime }}+\frac{1}{1+{\alpha }_1^{\prime }}|{\tilde{x}}_i+{\tilde{x}}_j{|}^{1+{\alpha }_1}+\frac{1}{1+{\alpha }_1^{\prime }}|{\tilde{x}}_i+{\tilde{x}}_j{|}^2\right)\end{array}$$
和上面一样，同样可得
$$\begin{array}{rl}& -l_1\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}\lceil {\tilde{v}}_i-{\tilde{v}}_j{\rfloor }^{{\alpha }_2^{\prime }}({\tilde{x}}_i-{\tilde{x}}_j)\\ & \le l_1\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}\left(\frac{{\alpha }_2^{\prime }}{1+{\alpha }_2^{\prime }}|{\tilde{v}}_i-{\tilde{v}}_j{|}^{1+{\alpha }_2^{\prime }}+\frac{1}{1+{\alpha }_2^{\prime }}|{\tilde{x}}_i+{\tilde{x}}_j{|}^{1+{\alpha }_2}+\frac{1}{1+{\alpha }_2^{\prime }}|{\tilde{x}}_i+{\tilde{x}}_j{|}^2\right)\end{array}$$
带入 $V_3$ 的 $u_{1i}+u_{2i}$ 部分
$$\begin{array}{rl}& 2\sum _{i=1}^N{p}_i({\tilde{x}}_i+{\tilde{v}}_i)(u_{1i}+u_{2i})\\ & \le -l_1\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}\lceil {\tilde{x}}_i-{\tilde{x}}_j{\rfloor }^{{\alpha }_1}({\tilde{v}}_i-{\tilde{v}}_j)-l_2\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}\lceil {\tilde{x}}_i-{\tilde{x}}_j{\rfloor }^{{\alpha }_2}({\tilde{v}}_i-{\tilde{v}}_j)\\ & +\frac{l_1}{1+{\alpha }_1^{\prime }}\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}|{\tilde{x}}_i-{\tilde{x}}_j{|}^2+\frac{l_2}{1+{\alpha }_2^{\prime }}\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}|{\tilde{x}}_i-{\tilde{x}}_j{|}^2\\ & \le -l_1\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}\lceil {\tilde{x}}_i-{\tilde{x}}_j{\rfloor }^{{\alpha }_1}({\tilde{v}}_i-{\tilde{v}}_j)-l_2\sum _{i=1}^N\sum _{j=1}^N{p}_i{a}_{ij}\lceil {\tilde{x}}_i-{\tilde{x}}_j{\rfloor }^{{\alpha }_2}({\tilde{v}}_i-{\tilde{v}}_j)\\ & +\frac{2l_1}{1+{\alpha }_1^{\prime }}{\tilde{x}}^T\hat{L}\tilde{x}+\frac{2l_2}{1+{\alpha }_2^{\prime }}{\tilde{x}}^T\hat{L}\tilde{x}\end{array}$$
结合 ${\dot{V}}_1$ ${\dot{V}}_2$ ${\dot{V}}_3$ 可得
$$\begin{array}{rl}\dot{V}& \le (-c+\frac{2l_1}{1+{\alpha }_1^{\prime }}+\frac{2l_2}{1+{\alpha }_2^{\prime }}){\tilde{x}}^T\hat{L}\tilde{x}+(2\rho +1)p_{max}{\tilde{x}}^T\tilde{x}\\ & +(-2k+c)(\tilde{x}+\tilde{v}{)}^T\hat{L}(\tilde{x}+\tilde{v})+(3+4\rho )p_{max}(\tilde{x}+\tilde{v}{)}^T(\tilde{x}+\tilde{v})\\ & \le -((2\rho +1)p_{max}+l_3)/K{\tilde{x}}^T\hat{L}\tilde{x}+(2\rho +1)p_{max}{\tilde{x}}^T\tilde{x}\\ & -((4\rho +3)p_{max}+l_4)/K(\tilde{x}+\tilde{v}{)}^T\hat{L}(\tilde{x}+\tilde{v})+(3+4\rho )p_{max}(\tilde{x}+\tilde{v}{)}^T(\tilde{x}+\tilde{v})\\ & \le -l_3{\tilde{x}}^T\tilde{x}-l_4(\tilde{x}+\tilde{v}{)}^T(\tilde{x}+\tilde{v})\\ & \le 0\end{array}$$
这里用到了引理5，证明到最后一步就用到了参数定义时的不等式，我们发现这个 证明也是利用了巧妙的参数。到这里证明了 $z=\psi$ 是全局渐进稳定的，也就证明了系统是全局固定时间稳定的。

只能说，精彩！！！

三、仿真

仿真不再做了，从这个控制律的形式也能看出来，效果一定很好，这里难的是证明他的固定时间收敛性，更多的信息可以参考原文

四、参考文献

[1] H. Hong, W. Yu, J. Fu, and X. Yu, “A Novel Class of Distributed Fixed-Time Consensus Protocols for Second-Order Nonlinear and Disturbed Multi-Agent Systems,” IEEE Trans. Netw. Sci. Eng., vol. 6, no. 4, pp. 760–772, Oct. 2019, doi: 10.1109/TNSE.2018.2873060.