﻿ 基于事件触发的异构智能体分布式编队控制
 舰船科学技术  2022, Vol. 44 Issue (14): 83-88    DOI: 10.3404/j.issn.1672-7649.2022.14.019 PDF

Distributed formation control of heterogeneous agents based on event trigger
DENG Zhi-liang, SHI Wei-jia
School of Automation, Nanjing University of Information Science and Technology, Nanjing. 210044, China
Abstract: Aiming at the formation control problem of heterogeneous multi-agents, this paper proposes an event-triggered formation control algorithm for heterogeneous multi-agents. First, based on the mathematical model of UAVs and unmanned ships, the observer is designed in combination with the event trigger mechanism, and the sampling mechanism is introduced into the event trigger function, which can naturally eliminate the Zeno phenomenon. Then, based on the estimated information of the observer, a distributed formation control protocol for heterogeneous multi-agents is designed. This control algorithm reduces the number of communications between the UAV and the UAV, and reduces the communication resource consumption of the system while ensuring that the UAV and the UAV form a formation. In this paper, the Lyapunov function proves that the proposed algorithm satisfies the global asymptotic stability, and the effectiveness of the algorithm is verified by numerical simulation experiments.
Key words: heterogeneous agent     distributed formation     formation control     event trigger mechanism
0 引　言

1 预备知识及问题描述 1.1 图论

 图 1 无向图 ${G}_{1}$ Fig. 1 Undirected graph ${G}_{1}$
 $\begin{split} & V\left({G}_{1}\right)=\left\{{v}_{1},{v}_{2},{v}_{3},{v}_{4},{v}_{5}\right\};\\ &\varepsilon \left({G}_{1}\right)=\left\{{e}_{12},{e}_{13},{e}_{23},{e}_{34},{e}_{35},{e}_{45}\right\}。\end{split}$

1.2 系统建模

 $\left\{\begin{split} &{\dot{x}}_{i}={u}_{i}\cos{\varphi }_{i}-{v}_{i}\sin{\varphi }_{i}，\\ &{\dot{y}}_{i}={u}_{i}\sin{\varphi }_{i}+{v}_{i}\cos{\varphi }_{i}，\\ &{\dot{\varphi }}_{i}={r}_{i}，\\ &{\dot{u}}_{i}=\frac{{m}_{22}}{{m}_{11}}{v}_{i}{r}_{i}-\frac{{d}_{11}}{{m}_{11}}{u}_{i}+\frac{{\tau }_{ui}}{{m}_{11}}，\\ &{\dot{v}}_{i}=-\frac{{m}_{11}}{{m}_{22}}{u}_{i}{r}_{i}-\frac{{d}_{22}}{{m}_{22}}{v}_{i}，\\ &{\dot{r}}_{i}=\frac{{m}_{11}-{m}_{22}}{{m}_{33}}{u}_{i}{v}_{i}-\frac{{d}_{33}}{{m}_{33}}{r}_{i}+\frac{{\tau }_{ri}}{{m}_{33}}。\end{split}\right.$ (1)

 图 2 无人船模型示意图 Fig. 2 Schematic diagram of unmanned ship model

 图 3 四旋翼无人机结构示意图 Fig. 3 Schematic diagram of the four-rotor UAV structure

 $\begin{split}&{R}_{x}(\varphi )=\left[\begin{array}{ccc}1& 0& 0\\ 0& {C}_{\varphi }& -{S}_{\varphi }\\ 0& {S}_{\varphi }& {C}_{\varphi }\end{array}\right] ，\;\;{R}_{y}(\theta )=\left[\begin{array}{ccc}{C}_{\theta }& 0& {S}_{\theta }\\ 0& 1& 0\\ -{S}_{\theta }& 0& {C}_{\theta }\end{array}\right] ，\\ &{R}_{\textit{z}}(\psi )=\left[\begin{array}{ccc}{C}_{\psi }& -{S}_{\psi }& 0\\ {S}_{\psi }& {C}_{\psi }& 0\\ 0& 0& 1\end{array}\right] 。\\[-25pt]\end{split}$ (2)

 ${C}_{b}^{e}=\left[\begin{array}{ccc}{C}_{\theta }{C}_{\psi }& {S}_{\varphi }{S}_{\theta }{C}_{\psi }-{C}_{\varphi }{S}_{\psi }& {C}_{\varphi }{S}_{\theta }{C}_{\psi }+{S}_{\varphi }{S}_{\psi }\\ {C}_{\theta }{S}_{\psi }& {S}_{\varphi }{S}_{\theta }{S}_{\psi }+{C}_{\varphi }{C}_{\psi }& {C}_{\varphi }{S}_{\theta }{S}_{\psi }-{S}_{\varphi }{C}_{\psi }\\ -{S}_{\theta }& {S}_{\varphi }{C}_{\theta }& {C}_{\varphi }{C}_{\theta }\end{array}\right]。$ (3)

 \left\{\begin{aligned}&{U}_{1}={F}_{1}+{F}_{2}+{F}_{3}+{F}_{4}，\\ &{U}_{2}={F}_{2}-{F}_{4}，\\ &{U}_{3}={F}_{1}-{F}_{3}，\\ &{U}_{4}={F}_{1}-{F}_{4}+{F}_{2}-{F}_{3}。\end{aligned}\right. (4)

 \left\{\begin{aligned}&\ddot{\varphi }=\frac{L}{{I}_{x}}{U}_{2}，\ddot{\theta }=\frac{L}{{I}_{y}}{U}_{3}，\ddot{\psi }=\frac{1}{{I}_{\textit{z}}}{U}_{4}，\\ &\ddot{x}=\frac{{U}_{1}}{m}\left({C}_{\varphi }{C}_{\psi }{S}_{\theta }+{S}_{\varphi }{S}_{\psi }\right)，\\ &\ddot{y}=\frac{{U}_{1}}{m}\left({C}_{\varphi }{S}_{\psi }{S}_{\theta }-{S}_{\varphi }{C}_{\psi }\right)，\\ &\ddot{\textit{z}}=\frac{{U}_{1}}{m}{C}_{\varphi }{C}_{\theta }-g。\end{aligned}\right. (5)
1.3 问题描述

2 控制器设计 2.1 观测器设计

 ${\dot{\beta }}_{i}\left(t\right)={u}_{i}\left(t\right)。$ (6)

 $\underset{t\to \infty }{{\rm{lim}}}{\beta }_{i}\left(t\right)=\underset{t\to \infty }{{\rm{lim}}}{\beta }_{j}\left(t\right)=\frac{\displaystyle\sum _{i=1}^{n}{\beta }_{i}\left(0\right)}{n} 。$ (7)

 $\left\{\begin{array}{c}{\dot{\beta }}_{i}\left(t\right)=-\displaystyle\sum _{j=1}^{n}{a}_{ij}\left({\widehat{\beta }}_{i}(t)-{\widehat{\beta }}_{j}(t)\right)，\\ {\dot{\xi }}_{i}\left(t\right)=-\displaystyle\sum _{j=1}^{n}{a}_{ij}\left({\widehat{\xi }}_{i}(t)-{\widehat{\xi }}_{j}(t)\right)。\end{array}\right.$ (8)

 ${e}_{i}\left(t\right)={\widehat{\beta }}_{i}\left(t\right)-{\beta }_{i}\left(t\right) 。$ (9)

 ${d}_{i}{{e}_{i}}^{2}\left(kh\right)\geqslant \sigma {{\gamma }_{i}}^{2}\left(kh\right)。$ (10)

 $\dot{\beta }\left(t\right)=-L\widehat{\beta }\left(t\right) 。$ (11)

 $\frac{1}{2}-h{\lambda }_{n}-2\sigma > 0 。$ (12)

2.2 控制律设计

2.2.1 虚拟控制律设计

 $\left\{\begin{array}{c}{e}_{xi}={x}_{i}-{\beta }_{i}，\\ {e}_{yi}={y}_{i}-{\xi }_{i}。\end{array}\right.$ (13)

 $\left[\begin{array}{c}{\dot{e}}_{x}\\ {\dot{e}}_{y}\end{array}\right]=\left[\begin{array}{cc}\rm{cos}\varphi & -\rm{sin}\varphi \\ \rm{sin}\varphi & \rm{cos}\varphi \end{array}\right]\left[\begin{array}{c}u\\ v\end{array}\right]-\left[\begin{array}{c}\dot{\beta }\\ \dot{\xi }\end{array}\right] 。$ (14)

 $\left[\begin{array}{c}{\alpha }_{u}\\ {\alpha }_{v}\end{array}\right]=\left[\begin{array}{cc}\rm{cos}\varphi & \rm{sin}\varphi \\ -\rm{sin}\varphi & \rm{cos}\varphi \end{array}\right]\left[\begin{array}{c}\dot{\beta }-k{e}_{x}\\ \dot{\xi }-k{e}_{y}\end{array}\right] 。$ (15)

 $V=\frac{1}{2}{e}_{xi}^{2}+\frac{1}{2}{e}_{yi}^{2}。$ (16)

 $\left[\begin{array}{c}{\dot{e}}_{x}\\ {\dot{e}}_{y}\end{array}\right]=\left[\begin{array}{cc}\rm{cos}\varphi & -\rm{sin}\varphi \\ \rm{sin}\varphi & \rm{cos}\varphi \end{array}\right]\left[\begin{array}{c}u\\ v\end{array}\right]-\left[\begin{array}{c}\dot{\beta }\\ \dot{\xi }\end{array}\right] 。$ (17)

 $\dot{V}=-k{e}_{xi}^{2}-k{e}_{yi}^{2}\leqslant 0$ (18)

 $\left\{\begin{array}{c}{u}_{i}{{\rm{cos}}}{\phi }_{i}-{v}_{i}{{\rm{sin}}}{\phi }_{i}-{\dot{\beta }}_{i}=-ke_{xi}，\\ {u}_{i}{{\rm{sin}}}{\phi }_{i}+{v}_{i}{{\rm{cos}}}{\phi }_{i}-{\dot{\xi }}_{i}=-ke_{yi}。\end{array}\right.$ (19)

 $\left[\begin{array}{cc}\rm{cos}\varphi & -\rm{sin}\varphi \\ \rm{sin}\varphi & \rm{cos}\varphi \end{array}\right]\left[\begin{array}{c}u\\ v\end{array}\right]=\left[\begin{array}{c}\dot{\beta }-kex\\ \dot{\xi }-k{e}_{y}\end{array}\right]。$ (20)

2.2.2 纵向推进力控制律设计

 ${e}_{ui}={u}_{i}-{\alpha }_{ui}。$ (21)

 ${s}_{1}={\mu }_{1}{e}_{ui} 。$ (22)

 $\dot{{S}}_{1} ={\mu }_{1}{\dot{e}}_{ui} \\ ={\mu }_{1}\left({\dot{u}}_{i}-{\dot{\alpha }}_{ui}\right) \\ ={\mu }_{1}\left(\frac{{m}_{22}}{{m}_{11}}{v}_{i}{r}_{i}-\frac{{d}_{11}}{{m}_{11}}{u}_{i}+\frac{{\tau }_{ui}}{{m}_{11}}-{\dot{\alpha }}_{ui}\right) 。$ (23)

${\dot{s}}_{1}=0$ ，可取趋近律为：

 ${\dot{s}}_{1}=-{\varepsilon }_{1}{{\rm{sgn}}}({s}_{1})。$ (24)

 $\begin{split} &{\mu }_{1}\left(\frac{{m}_{22}}{{m}_{11}}{v}_{i}{r}_{i}-\frac{{d}_{11}}{{m}_{11}}{u}_{i}+\frac{{\tau }_{ui}}{{m}_{11}}-{\dot{\alpha }}_{ui}\right)=-{\varepsilon }_{1}{\rm{sgn}}\left({s}_{1}\right)，\\ &\frac{{m}_{22}}{{m}_{11}}{v}_{i}{r}_{i}-\frac{{d}_{11}}{{m}_{11}}{u}_{i}+\frac{{\tau }_{ui}}{{m}_{11}}-{\dot{\alpha }}_{ui}=-\frac{{\varepsilon }_{1}}{{\mu }_{1}}{\rm{sgn}}\left({s}_{1}\right)，\\ &{m}_{22}{v}_{i}{r}_{i}-{d}_{11}{u}_{i}+{\tau }_{ui}-{m}_{11}{\dot{\alpha }}_{ui}=-\frac{{m}_{11}{\varepsilon }_{1}}{{\mu }_{1}}{\rm{sgn}}\left({s}_{1}\right) 。\\[-34pt] \end{split}$ (25)

 ${\tau }_{ui}={m}_{11}{\dot{\alpha }}_{u}-\frac{{m}_{11}{\varepsilon }_{1}}{{\mu }_{1}}{\rm{sgn}}({s}_{1})-{m}_{22}{v}_{i}{r}_{i}+{d}_{11}{u}_{i}。$ (26)

 ${V}_{1}=\frac{1}{2}{s}_{1}^{2} 。$ (27)

 $\begin{split} {\dot{V}}_{1}=&{s}_{1}\cdot {\dot{s}}_{1} ={s}_{1}\cdot {\mu }_{1} \Biggr(\frac{{m}_{22}}{{m}_{11}}{v}_{i}{r}_{i}-\frac{{d}_{11}}{{m}_{11}}{u}_{i}+\\ &\dfrac{{m}_{11}{\dot{\alpha }}_{ui}-\dfrac{{m}_{11}{\varepsilon }_{1}}{{\mu }_{1}}{\rm{sgn}}\left({s}_{1}\right)-{m}_{22}{v}_{i}{r}_{i}+{d}_{11}{u}_{i}}{{m}_{11}}\Biggr)= \\ &-{s}_{1}{\varepsilon }_{1}{\rm{sgn}}\left({s}_{1}\right) -{\varepsilon }_{1}\left|{s}_{1}\right|\leqslant 0。\end{split}$ (28)
2.2.3 转向力矩控制律设计

 ${e}_{vi}={v}_{i}-{\alpha }_{vi} 。$ (29)

 ${s}_{2}={\dot{e}}_{vi}+{\mu }_{2}{e}_{vi} 。$ (30)

 ${\dot{S}}_{2}={\ddot{e}}_{vi}+{\mu }_{2}{\dot{e}}_{vi} =\left({\ddot{v}}_{i}-{\ddot{e}}_{vi}\right)+{\mu }_{2}\left({\dot{v}}_{i}-{\dot{\alpha }}_{vi}\right) 。$ (31)

 $f=-\left[{\ddot{\beta }}_{i}-k{\dot{e}}_{xi}\right]{{\rm{sin}}}{\phi }_{i}+\left[{\ddot{\xi }}_{i}-k{\dot{e}}_{yi}\right]{{\rm{cos}}}{\phi }_{i}。$ (32)

 ${\ddot{\alpha }}_{vi}=-{\dot{r}}_{i}{\alpha }_{ui}-{r}_{i}{\dot{\alpha }}_{ui}+\dot{f}。$ (33)

 ${\dot{s}}_{2}={\ddot{v}}_{i}+{\dot{r}}_{i}{\alpha }_{ui}+{r}_{i}{\dot{\alpha }}_{ui}-\dot{f}+{\mu }_{2}({\dot{v}}_{i}-{\dot{\alpha }}_{vi}) 。$ (34)

${\dot{s}}_{2}=0$ ，可取趋近律：

 ${\dot{s}}_{2}=-{\varepsilon }_{2}{\rm{sgn}}({s}_{2}) 。$ (35)

 $\begin{split}&{\ddot{v}}_{i}+\left(\frac{{m}_{11}-{m}_{22}}{{m}_{33}}{u}_{i}{v}_{i}-\frac{{d}_{33}}{{m}_{33}}{r}_{i}+\frac{{\tau }_{ri}}{{m}_{33}}\right){\alpha }_{ui}+{r}_{i}{\alpha }_{ui}-\dot{f}+{\mu }_{2}{\dot{e}}_{vi}=\\ &-{\varepsilon }_{2}{\rm{sgn}}({s}_{2})\left(\frac{{m}_{11}-{m}_{22}}{{m}_{33}}{u}_{i}{v}_{i}-\frac{{d}_{33}}{{m}_{33}}{r}_{i}+\frac{{\tau }_{ri}}{{m}_{33}}\right){\alpha }_{ui}= -{\ddot{v}}_{i}-\\ &{r}_{i}{\alpha }_{ui}+\dot{f}-{\mu }_{2}{\dot{e}}_{vi}-{\varepsilon }_{2}{\rm{sgn}}({s}_{2})({m}_{11}-{m}_{22}){u}_{i}{v}_{i}-{d}_{33}{r}_{i}+{\tau }_{ri}=\\ &-\frac{{m}_{33}}{{\alpha }_{ui}}({\ddot{v}}_{i}+{r}_{i}{\alpha }_{ui}-\dot{f}+{\mu }_{2}{\dot{e}}_{vi}+{\varepsilon }_{2}{\rm{sgn}}({s}_{2}))。\\[-18pt] \end{split}$ (36)

 $\begin{split}{\tau }_{ri}=&-({m}_{11}-{m}_{22}){u}_{i}{v}_{i}+{d}_{33}{r}_{i}-\\ &\frac{{m}_{33}}{{\alpha }_{ui}}({\ddot{v}}_{i}+{r}_{i}{\dot{\alpha }}_{ui}-\dot{f}+{\mu }_{2}{\dot{e}}_{vi}+{\varepsilon }_{2}\rm{sgn}({s}_{2}))。\end{split}$ (37)

 ${V}_{2}=\frac{1}{2}{{s}_{2}}^{2}。$ (38)

 $\begin{split}&{\dot{V}}_{2}={\dot{s}}_{2}{s}_{2} ={s}_{2}\cdot {\ddot{v}}_{i}+ \left(\frac{{m}_{11}-{m}_{22}}{{m}_{33}}{u}_{i}{v}_{i}-\frac{{d}_{33}}{{m}_{33}}{r}_{i} +\right. \\ &\frac{ - \left({m}_{11} - {m}_{22}\right){u}_{i}{v}_{i} + {d}_{33}{r}_{i} }{{{m}_{33}}}-\\ &\left. \frac{ \dfrac{{m}_{33}}{{\alpha }_{ui}}\left({\ddot{v}}_{i}+{r}_{i}{\alpha }_{ui}-\dot{f}+{\mu }_{2}{\dot{e}}_{vi}+{\varepsilon }_{2}{{\rm{sgn}}}({s}_{2})\right)}{{m}_{33}}\right)，\\ &{\alpha }_{ui} +{r}_{i}{\dot{\alpha }}_{ui}-\dot{f}+{\mu }_{2}{\dot{e}}_{vi} ={s}_{2}\left({\ddot{v}}_{i}-{\ddot{v}}_{i}-{r}_{i}{\dot{\alpha }}_{ui}+\dot{f}-\right.\\ &\left.{\mu }_{2}{\dot{e}}_{vi}{-\varepsilon }_{2}{{\rm{sgn}}}({s}_{2})+{r}_{i}{\dot{\alpha }}_{ui}-\dot{f}+{\mu }_{2}{\dot{e}}_{vi}\right)=-{s}_{2}{\varepsilon }_{2}{{\rm{sgn}}}({s}_{2}) =\\ &-{\varepsilon }_{2}\left|{s}_{2}\right|\leqslant 0。\end{split}$ (39)

3 仿真及分析 3.1 仿真实验

 图 4 通信拓扑图 Fig. 4 Communication topology diagram

 图 5 各智能体状态轨迹观测值 Fig. 5 Observed value of each agent state trajectory

 图 6 事件触发间隔 Fig. 6 Event trigger interval

 图 7 位置误差 Fig. 7 Position error

 图 8 速度误差 Fig. 8 Speed error
3.2 仿真结果分析

4 结　语

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