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1. 北京航空航天大学 自动化科学与电气工程学院, 北京 100083;
2. 北京航空航天大学 飞行器控制一体化技术国防科技重点实验室, 北京 100083;
3. 北京航空航天大学 大数据科学与脑机智能高精尖创新中心, 北京 100083

Time-varying formation control and disturbance rejection for UAV-UGV heterogeneous swarm system
ZHOU Siquan1,2, DONG Xiwang1,2,3, LI Qingdong1,2, REN Zhang1,2,3
1. School of Automation Science and Electrical Engineering, Beihang University, Beijing 100083, China;
2. Science and Technology on Aircraft Control Laboratory, Beihang University, Beijing 100083, China;
3. Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing 100083, China
Abstract: This paper studies the time-varying formation control and disturbance rejection problems for heterogeneous UAV-UGV swarm system. The UAV-UGV swarm system is designed to achieve the desired time-varying output formation under unknown external disturbances. First, the kinetic and dynamic mathematical models of each UAV/UGV are established. In addition, the cooperative control models for heterogeneous swarm system are constructed by using the algebraic graph theory. Then, a distributed time-varying output formation controller is designed based on the internal model theory. The distributed formation center estimator and disturbance rejection compensator are designed and added to the formation controller. Third, an algorithm is presented to design the formation control parameters, where the feasibility conditions for achieving the time-varying output formation are given. The stability of the closed-loop UAV-UGV swarm system is proved. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed formation control approach.
Keywords: UAV-UGV    heterogeneous system    distributed control    time-varying output formation    disturbance rejection

1 基础理论与系统建模 1.1 数学基础与图论

1.2 线性系统理论

 ${{\mathit{\boldsymbol{x}}_i} = {{[x_i^X,v_i^X,x_i^Y,v_i^Y]}^{\rm{T}}}}$

 $\left\{ {\begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\dot x}}}_i}(t) = {\mathit{\boldsymbol{A}}_i}{\mathit{\boldsymbol{x}}_i}(t) + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{u}}_i}(t)}\\ {{\mathit{\boldsymbol{y}}_i}(t) = {\mathit{\boldsymbol{C}}_i}{\mathit{\boldsymbol{x}}_i}(t)} \end{array}} \right.$ （1）

 ${\mathit{\boldsymbol{A}}_c} = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\hat A}}}&{\mathit{\boldsymbol{\hat B}}}\\ {{\mathit{\boldsymbol{Q}}_2}\mathit{\boldsymbol{\hat C}}}&{{\mathit{\boldsymbol{Q}}_1} + {\mathit{\boldsymbol{Q}}_2}\mathit{\boldsymbol{\hat D}}} \end{array}} \right]$ （2）

 $\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{XA}} = \mathit{\boldsymbol{\hat AX}} + \mathit{\boldsymbol{\hat BZ}} + \mathit{\boldsymbol{\hat E}}}\\ {\mathit{\boldsymbol{ZA}} = {\mathit{\boldsymbol{Q}}_1}\mathit{\boldsymbol{Z}} + {\mathit{\boldsymbol{Q}}_2}(\mathit{\boldsymbol{\hat CX}} + \mathit{\boldsymbol{\hat DZ}} + \mathit{\boldsymbol{\hat F}})} \end{array}} \right.$ （3）

 $\mathit{\boldsymbol{\hat CX}} + \mathit{\boldsymbol{\hat DZ}} + \mathit{\boldsymbol{\hat F}} = {{\bf 0}}$ （4）
1.3 无人机-无人车系统建模[29]

 图 1 麦克纳姆轮无人车示意图 Fig. 1 Diagram of Mecanum wheel UGV

 $\left[ {\begin{array}{*{20}{l}} {{v_x}}\\ {{v_y}}\\ \omega \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\frac{R}{4}{\rm{tan}}\alpha }&{ - \frac{R}{4}{\rm{tan}}\alpha }&{ - \frac{R}{4}{\rm{tan}}\alpha }&{\frac{R}{4}{\rm{tan}}\alpha }\\ {\frac{R}{4}}&{\frac{R}{4}}&{\frac{R}{4}}&{\frac{R}{4}}\\ { - \frac{{R{\rm{tan}}\alpha }}{{4({l_x} + {l_y})}}}&{\frac{{R{\rm{tan}}\alpha }}{{4({l_x} + {l_y})}}}&{ - \frac{{R{\rm{tan}}\alpha }}{{4({l_x} + {l_y})}}}&{\frac{{R{\rm{tan}}\alpha }}{{4({l_x} + {l_y})}}} \end{array}} \right]\left[ {\begin{array}{*{20}{l}} {{\omega _{m1}}}\\ {{\omega _{m2}}}\\ {{\omega _{m3}}}\\ {{\omega _{m4}}} \end{array}} \right]$ （5）

 $\left\{ {\begin{array}{*{20}{l}} {x = {v_x}{\rm{cos}}\theta - {v_y}{\rm{sin}}\theta }\\ {\dot y = {v_x}{\rm{cos}}\theta + {v_y}{\rm{cos}}\theta }\\ {\dot \theta = \omega } \end{array}} \right.$ （6）

 $\left\{ \begin{array}{l} \begin{array}{*{20}{l}} {\ddot x = ( - {\rm{sin}}\phi {\rm{sin}}\psi - {\rm{cos}}\phi {\rm{sin}}\theta {\rm{cos}}\psi )\frac{{{u_1}}}{m}}\\ {\ddot y = ( - {\rm{cos}}\phi {\rm{sin}}\theta {\rm{sin}}\psi + {\rm{sin}}\phi {\rm{cos}}\psi )\frac{{{u_1}}}{m}} \end{array}\\ \begin{array}{*{20}{l}} {\ddot z = - {\rm{cos}}\phi {\rm{cos}}\theta \frac{{{u_1}}}{m} + g}\\ {\ddot \phi = \frac{{{u_2}L}}{{{I_{xx}}}} + \dot \theta \dot \psi \frac{{{I_{yy}} - {I_{zz}}}}{{{I_{xx}}}}}\\ {\ddot \theta = \frac{{{u_3}L}}{{{I_{yy}}}} + \dot \phi \dot \psi \frac{{{I_{zz}} - {I_{xx}}}}{{{I_{yy}}}}}\\ {\ddot \psi = \frac{{{u_4}}}{{{I_{zx}}}} + \dot \phi \frac{{{I_{xx}} - {I_{yy}}}}{{{I_{zz}}}}} \end{array} \end{array} \right.$ （7）

 $\left[ {\begin{array}{*{20}{l}} {{u_1}}\\ {{u_2}}\\ {{u_3}}\\ {{u_4}} \end{array}} \right] = \left[ {\begin{array}{*{20}{l}} b&b&b&b\\ 0&b&0&{ - b}\\ b&0&{ - b}&0\\ d&{ - d}&d&{ - d} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\omega _1^2}\\ {\omega _2^2}\\ {\omega _3^2}\\ {\omega _4^2} \end{array}} \right]$ （8）

2 问题描述

 图 2 无人机/无人车控制结构 Fig. 2 Control structure of UAV/UGV

 $\left\{ {\begin{array}{*{20}{l}} {{{\dot x}_i} = {v_{ix}}}\\ {{{\dot v}_{ix}} = ( - {\phi _i}{\rm{sin}}{\psi _i} - {\theta _i}{\rm{cos}}{\psi _i})g}\\ {{{\dot y}_i} = {v_{yi}}}\\ {{{\dot v}_{iy}} = ( - {\phi _i}{\rm{cos}}{\psi _i} - {\theta _i}{\rm{sin}}{\psi _i})g} \end{array}} \right.$ （9）

 $\left\{ {\begin{array}{*{20}{l}} {{u_{ix}} = ( - {\phi _i}{\rm{sin}}{\psi _i} - {\theta _i}{\rm{cos}}{\psi _i})g}\\ {{u_{iy}} = ( - {\phi _i}{\rm{cos}}{\psi _i} - {\theta _i}{\rm{sin}}{\psi _i})g} \end{array}} \right.$ （10）

 $\left\{ {\begin{array}{*{20}{l}} {\dot x(t) = v(t)}\\ {u(t) = v(t)} \end{array}} \right.$ （11）

 $\left\{ {\begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\dot x}}}_i}(t) = {\mathit{\boldsymbol{A}}_i}{\mathit{\boldsymbol{x}}_i}(t) + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{u}}_i}(t)}\\ {{\mathit{\boldsymbol{y}}_i}(t) = {\mathit{\boldsymbol{C}}_i}{\mathit{\boldsymbol{x}}_i}(t)} \end{array}} \right.$ （12）

 ${{\mathit{\boldsymbol{u}}_i} = {{[u_i^X,u_i^Y]}^{\rm{T}}}}$
 ${{\mathit{\boldsymbol{A}}_i} = {\mathit{\boldsymbol{I}}_2} \otimes \left[ {\begin{array}{*{20}{l}} 0&1\\ 0&0 \end{array}} \right]}$
 ${{\mathit{\boldsymbol{B}}_i} = {\mathit{\boldsymbol{I}}_2} \otimes \left[ {\begin{array}{*{20}{l}} 0\\ 1 \end{array}} \right]}$
 ${{\mathit{\boldsymbol{C}}_i} = {\mathit{\boldsymbol{I}}_2} \otimes \left[ {\begin{array}{*{20}{l}} 1&0 \end{array}} \right]}$
 ${{\mathit{\boldsymbol{x}}_i} = {{\left[ {\begin{array}{*{20}{l}} {x_i^X}&{,x_i^Y} \end{array}} \right]}^{\rm{T}}}}$

 ${{\mathit{\boldsymbol{u}}_i} = {{\left[ {\begin{array}{*{20}{l}} {u_i^X}&{,u_i^Y} \end{array}} \right]}^{\rm{T}}}}$
 ${{\mathit{\boldsymbol{A}}_i} = {{{\bf 0}}_{2 \times 2}}}$
 ${{\mathit{\boldsymbol{B}}_i} = {\mathit{\boldsymbol{I}}_2}}$
 ${{\mathit{\boldsymbol{C}}_i} = {\mathit{\boldsymbol{I}}_2}}$

 ${\mathit{\boldsymbol{\dot w}}_i}(t) = {\mathit{\boldsymbol{S}}_{wi}}{\mathit{\boldsymbol{w}}_i}(t)$ （13）

 $\left\{ {\begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\dot x}}}_i}(t) = {\mathit{\boldsymbol{A}}_i}{\mathit{\boldsymbol{x}}_i}(t) + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{u}}_i}(t) + {\mathit{\boldsymbol{W}}_i}{\mathit{\boldsymbol{w}}_i}(t)}\\ {{\mathit{\boldsymbol{y}}_i}(t) = {\mathit{\boldsymbol{C}}_i}{\mathit{\boldsymbol{x}}_i}(t)} \end{array}} \right.$ （14）

 $\mathop {{\rm{lim}}}\limits_{t \to \infty } ({\mathit{\boldsymbol{y}}_i}(t) - {\mathit{\boldsymbol{h}}_{{y_i}}}(t) - {\mathit{\boldsymbol{r}}_y}(t)) = {{\bf 0}}\;\:i = 1,2, \cdots ,N$ （15）

3 时变编队控制器设计与分析 3.1 时变编队控制器设计

 $\left\{ \begin{array}{l} {{\mathit{\boldsymbol{\dot p}}}_i}(t) = {\mathit{\boldsymbol{S}}_p}{\mathit{\boldsymbol{p}}_i}(t) - \alpha {\mathit{\boldsymbol{K}}_p}\sum\limits_{j = 1}^N {{\mathit{\boldsymbol{w}}_{ij}}} ({\mathit{\boldsymbol{\xi }}_i}(t) - {\mathit{\boldsymbol{\xi }}_j}(t))\\ {{\mathit{\boldsymbol{\dot q}}}_i}(t) = {{\mathit{\boldsymbol{\bar Q}}}_{1i}}{\mathit{\boldsymbol{q}}_i}(t) + {{\mathit{\boldsymbol{\bar Q}}}_{2i}}({\mathit{\boldsymbol{y}}_i}(t) - {\mathit{\boldsymbol{h}}_{yi}}(t) - {\mathit{\boldsymbol{\xi }}_i}(t))\\ {\mathit{\boldsymbol{u}}_i}(t) = {\mathit{\boldsymbol{K}}_{1i}}{\mathit{\boldsymbol{x}}_i}(t) + {\mathit{\boldsymbol{K}}_{2i}}({\mathit{\boldsymbol{p}}_i}(t) + {\mathit{\boldsymbol{\delta }}_i}(t)) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{K}}_{3i}}{\mathit{\boldsymbol{q}}_i}(t) + {\mathit{\boldsymbol{v}}_i}(t) \end{array} \right.$ （16）

3.2 时变编队控制器参数选取算法

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{X}}_i}{\mathit{\boldsymbol{S}}_p} = {\mathit{\boldsymbol{A}}_i}{\mathit{\boldsymbol{X}}_i} + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{U}}_i}}\\ {{\mathit{\boldsymbol{C}}_i}{\mathit{\boldsymbol{X}}_i} - \mathit{\boldsymbol{F}} = {{\bf 0}}} \end{array}} \right.$ （17）

Step 1  选取使得调节器方程成立的矩阵对解(Xi, Ui)。对于给定的时变输出编队构型hy(t)，检验是否存在编队补偿输入vi(t)使得如下编队可行性条件成立：

 $\mathop {{\rm{lim}}}\limits_{t \to \infty } ({\mathit{\boldsymbol{X}}_i}({\mathit{\boldsymbol{S}}_p}{\mathit{\boldsymbol{\delta }}_i} - {\mathit{\boldsymbol{\dot \delta }}_i}) + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{v}}_i}) = 0$ （18）

Step 2  选取K2i使得Ai+BiK1i是Hurwitz的，令K2i=Ui-K1iXi

Step 3  选取增益系数$\alpha>\frac{1}{2 \operatorname{Re}\left(\lambda_{2}\right)}, \lambda_{2}$表示拉普拉斯矩阵的最小非零特征值。令增益矩阵Kp=PFT，正定矩阵P满足以下代数Riccati方程：

 $\mathit{\boldsymbol{PS}}_p^{\rm{T}} + {\mathit{\boldsymbol{S}}_p}\mathit{\boldsymbol{P}} - \mathit{\boldsymbol{P}}{\mathit{\boldsymbol{F}}^{\rm{T}}}\mathit{\boldsymbol{FP}} + \mathit{\boldsymbol{Q}} = {{\bf 0}}$ （19）

Step 4  令$\overline{\boldsymbol{A}_{i}}=\boldsymbol{A}_{i}+\boldsymbol{B}_{i} \boldsymbol{K}_{1 i}, \quad\left(\boldsymbol{Q}_{1 i}, \boldsymbol{Q}_{2 i}\right)$包含Swi的最小p-copy内模，设计K3i=[K3i1, K3i2]使得$\left[\begin{array}{ccc} \overline{\boldsymbol{A}_{i}}+\boldsymbol{B}_{i} \boldsymbol{K}_{3 i 1} & \boldsymbol{B}_{i} \boldsymbol{K}_{3 i 2} \\ \boldsymbol{Q}_{2 i} \boldsymbol{C}_{i} & \boldsymbol{Q}_{1 i} \end{array}\right]$是Hurwitz的，并选取$\bar{Q}_{1 i}=\left[\begin{array}{cc} \overline{\boldsymbol{A}_{i}}+\boldsymbol{B}_{i} \boldsymbol{K}_{3 i 1} & \boldsymbol{B}_{i} \boldsymbol{K}_{3 i 2} \\ \boldsymbol{Q}_{2 i} \boldsymbol{C}_{i} & Q_{1 i} \end{array}\right], \overline{\boldsymbol{Q}}_{2 i}=\left[\begin{array}{c} {\bf 0} \\ \boldsymbol{Q}_{2 i} \end{array}\right]$

3.3 时变编队控制器稳定性分析

 $\mathit{\boldsymbol{\dot p}} = ({\mathit{\boldsymbol{I}}_N} \otimes {\mathit{\boldsymbol{S}}_p})\mathit{\boldsymbol{p}} - \alpha (\mathit{\boldsymbol{L}} \otimes {\mathit{\boldsymbol{K}}_p}\mathit{\boldsymbol{F}})\mathit{\boldsymbol{p}}$ （20）

λi表示拉普拉斯矩阵的特征值，在假设1的条件下，有λ1=0，0 < Re(λ2)≤…≤Re(λN)。存在非奇异矩阵$\boldsymbol{T}=\left[\tilde{\boldsymbol{t}}_{1}, \boldsymbol{\boldsymbol { T }}\right], \tilde{\boldsymbol{t}}_{1}={\bf 1}_{N}, \boldsymbol{\mathcal { T }}=\left[\tilde{\boldsymbol{t}}_{2}, \tilde{\boldsymbol{t}}_{3}, \cdots, \tilde{\boldsymbol{t}}_{N}\right]$, 使得T-1LT=J=diag{0, J}, JR(N-1)×(N-1)表示对应特征值λi(i=2, 3, …, N)的约当标准型。记T-1=[t1, T], T=[t2, t3, …, tN], tiRN(i=1, 2, …, N), 令ϑ=(T-1Iq)p=[ϑ1T, ϑT], ϑ=[ϑ1T, ϑ2T, …, ϑNT]T, 则编队中心估计器可写成

 $\left\{ {\begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\dot \vartheta}} }_1} = {\mathit{\boldsymbol{S}}_p}{\mathit{\boldsymbol{\vartheta _1}}}}\\ {\mathit{\boldsymbol{\dot {\bar \vartheta}}} = (({\mathit{\boldsymbol{I}}_{N - 1}} \otimes {\mathit{\boldsymbol{S}}_p}) - \alpha (\mathit{\boldsymbol{\bar J}} \otimes {\mathit{\boldsymbol{K}}_p}\mathit{\boldsymbol{F}})){\mathit{\boldsymbol{\bar \vartheta}}} } \end{array}} \right.$ （21）

$\boldsymbol{p}_{\alpha}=\left(\boldsymbol{T} \otimes \boldsymbol{I}_{q}\right)\left[\begin{array}{c} \boldsymbol{\vartheta}_{1} \\ \boldsymbol{0} \end{array}\right], \boldsymbol{p}_{\alpha}^{-}=\left(\boldsymbol{T} \otimes \boldsymbol{I}_{q}\right)\left[\begin{array}{l} \boldsymbol{0} \\ \overline{\boldsymbol{\vartheta}} \end{array}\right]$, 则有p=pα+pα-pαpα-是线性无关的。进而有$\boldsymbol{p}_{\alpha}=\left(\boldsymbol{T} \otimes \boldsymbol{I}_{q}\right)\left[\begin{array}{c} \boldsymbol{\vartheta}_{1} \\ \boldsymbol{0} \end{array}\right]={\bf 1}_{N} \otimes \boldsymbol{\vartheta}_{1}, \boldsymbol{p}_{\alpha}^-=\boldsymbol{p}-\boldsymbol{p}_{\alpha}=\boldsymbol{p}-{\bf 1}_{N} \otimes \boldsymbol{\vartheta}_{1}$, 由于pαpα-是线性无关的，且T是非奇异矩阵，可知$\boldsymbol{p}_{\alpha}^- \rightarrow {\bf 0} \Leftrightarrow \bar{\vartheta} \rightarrow {\bf 0}$。因此，编队中心估计器的一致性问题可以转换为ϑ的稳定性问题。故只需选取合适的增益系数α和增益矩阵Kp使得矩阵(IN-1Sp)-α(JKpF)是Hurwitz的，即矩阵组Sp-αλiKpF(i=2, 3, …, N)是Hurwitz的，则有ϑ0

 ${\mathit{\boldsymbol{\dot \theta }}_i} = ({\mathit{\boldsymbol{S}}_p} - \alpha {\lambda _i}{\mathit{\boldsymbol{K}}_p}\mathit{\boldsymbol{F}}){\mathit{\boldsymbol{\theta }}_i}\;\:i = 2,3, \cdots ,N$ （22）

 ${\mathit{\boldsymbol{\dot V}}_i} = \mathit{\boldsymbol{\theta }}_i^{\rm{H}}({\mathit{\boldsymbol{P}}^{ - 1}}{\mathit{\boldsymbol{S}}_r} + \mathit{\boldsymbol{S}}_r^{\rm{T}}\mathit{\boldsymbol{P}} - 2\alpha {\rm{Re}} ({\lambda _i}){\mathit{\boldsymbol{F}}^{\rm{T}}}\mathit{\boldsymbol{F}}){\mathit{\boldsymbol{\theta }}_i}$ （23）

θi=P-1θi, 由算法Step 3中$\alpha>\frac{1}{2 \operatorname{Re}\left(\lambda_{2}\right)}$以及相应代数Riccati方程可得

 ${\mathit{\boldsymbol{\dot V}}_i} = \mathit{\boldsymbol{\bar \theta }}_i^{\rm{H}}({\mathit{\boldsymbol{P}}^{ - 1}}{\mathit{\boldsymbol{S}}_r} + \mathit{\boldsymbol{S}}_r^{\rm{T}}\mathit{\boldsymbol{P}} - 2\alpha {\rm{Re}} ({\lambda _i}){\mathit{\boldsymbol{F}}^{\rm{T}}}\mathit{\boldsymbol{F}}){\bar \theta _i}$ （24）

 $\begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\dot {\tilde x}}}}_i} = {{\mathit{\boldsymbol{\bar A}}}_i}{{\mathit{\boldsymbol{\tilde x}}}_i} + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{K}}_{3i}}{\mathit{\boldsymbol{q}}_i} + {\mathit{\boldsymbol{W}}_i}{\mathit{\boldsymbol{w}}_i} + {\mathit{\boldsymbol{X}}_i}({\mathit{\boldsymbol{S}}_p}{\mathit{\boldsymbol{\delta }}_i} - {{\mathit{\boldsymbol{\dot \delta }}}_i}) + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{v}}_i} + \alpha {\mathit{\boldsymbol{X}}_i}{\mathit{\boldsymbol{K}}_{pi}}\sum\limits_{j = 1}^N {{\mathit{\boldsymbol{w}}_{ij}}} ({\mathit{\boldsymbol{\xi }}_i}(t) - {\mathit{\boldsymbol{\xi }}_j}(t))} \end{array}$ （25）

 ${\mathit{\boldsymbol{\tilde n}}_i} = \alpha {\mathit{\boldsymbol{X}}_i}{\mathit{\boldsymbol{K}}_{pi}}\sum\limits_{j = 1}^N {{\mathit{\boldsymbol{w}}_{ij}}} ({\mathit{\boldsymbol{\xi }}_i}(t)) - {\mathit{\boldsymbol{\xi }}_j}(t))) \to {{\bf 0}}$ （26）

 ${\mathit{\boldsymbol{e}}_i} = {\mathit{\boldsymbol{y}}_i} - {\mathit{\boldsymbol{\xi }}_i} - {\mathit{\boldsymbol{h}}_{yi}} = {\mathit{\boldsymbol{C}}_i}({\mathit{\boldsymbol{x}}_i} - {\mathit{\boldsymbol{X}}_i}({\mathit{\boldsymbol{p}}_i} + {\mathit{\boldsymbol{\delta }}_i})) = {\mathit{\boldsymbol{C}}_i}{\mathit{\boldsymbol{\tilde x}}_i}$ （27）

 $\left\{ {\begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\dot {\tilde x}}}}_i} = {{\mathit{\boldsymbol{\bar A}}}_i}{{\mathit{\boldsymbol{\tilde x}}}_i} + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{K}}_{3i}}{\mathit{\boldsymbol{q}}_i} + {\mathit{\boldsymbol{W}}_i}{\mathit{\boldsymbol{w}}_i} + {{\mathit{\boldsymbol{\tilde h}}}_i} + {{\mathit{\boldsymbol{\tilde n}}}_i}}\\ {{{\mathit{\boldsymbol{\dot q}}}_i}(t) = {{\mathit{\boldsymbol{\bar Q}}}_{1i}}{\mathit{\boldsymbol{q}}_i}(t) + {{\mathit{\boldsymbol{\bar Q}}}_{2i}}({\mathit{\boldsymbol{y}}_i}(t) - {\mathit{\boldsymbol{h}}_{yi}}(t) - {\mathit{\boldsymbol{\xi }}_i}(t))}\\ {{\mathit{\boldsymbol{e}}_i} = {\mathit{\boldsymbol{C}}_i}{{\mathit{\boldsymbol{\tilde x}}}_i}} \end{array}} \right.$ （28）

$\boldsymbol{\varphi}_{i}=\left[\begin{array}{ll} \tilde{\boldsymbol{x}}_{i}^{\mathrm{T}}, & \boldsymbol{q}_{i}^{\mathrm{T}} \end{array}\right]^{\mathrm{T}}$, 则有

 $\left\{ {\begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\dot \varphi }}}_i} = {\mathit{\boldsymbol{A}}_{{\mathit{\boldsymbol{\varphi }}_i}}}{\mathit{\boldsymbol{\varphi }}_i} + {\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{\varphi }}_i}}}{\mathit{\boldsymbol{w}}_i} + {{\mathit{\boldsymbol{\tilde \eta }}}_i}}\\ {{\mathit{\boldsymbol{e}}_i} = {\mathit{\boldsymbol{C}}_{{\mathit{\boldsymbol{\varphi }}_i}}}{\mathit{\boldsymbol{\varphi }}_i}} \end{array}} \right.$ （29）

Q1iQ2i代入Aφi，并对其进行相似变换可得

 ${\mathit{\boldsymbol{A}}_{{\mathit{\boldsymbol{\varphi }}_i}}} \backsim \left[ {\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\bar A}}}_i} + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{K}}_{3i1}}}&{{\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{K}}_{3i2}}}&{{\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{K}}_{3i1}}}\\ {{\mathit{\boldsymbol{Q}}_{2i}}{\mathit{\boldsymbol{C}}_i}}&{{\mathit{\boldsymbol{Q}}_{1i}}}&{{\bf 0}}\\ {{\bf 0}}&{{\bf 0}}&{{{\mathit{\boldsymbol{\bar A}}}_i}} \end{array}} \right]$ （30）

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{X}}_{wi}}{\mathit{\boldsymbol{S}}_{wi}} = {{\mathit{\boldsymbol{\bar A}}}_i}{\mathit{\boldsymbol{X}}_{wi}} + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{K}}_{3i}}{\mathit{\boldsymbol{Z}}_{wi}} + {\mathit{\boldsymbol{W}}_i}}\\ {{\mathit{\boldsymbol{Z}}_{wi}}{\mathit{\boldsymbol{S}}_{wi}} = {{\mathit{\boldsymbol{\bar Q}}}_{2i}}{\mathit{\boldsymbol{C}}_i}{\mathit{\boldsymbol{X}}_{wi}} + {{\mathit{\boldsymbol{\bar Q}}}_{1i}}{\mathit{\boldsymbol{Z}}_{wi}}}\\ {{\mathit{\boldsymbol{C}}_i}{\mathit{\boldsymbol{X}}_{wi}} = {{\bf 0}}} \end{array}} \right.$ （31）

$\overline{\boldsymbol{X}}_{\boldsymbol{\varphi}_{i}}=\left[\boldsymbol{X}_{w i}^{\mathrm{T}}, \boldsymbol{Z}_{w i}^{\mathrm{T}}\right]^{\mathrm{T}}$, 则有

 $\left\{ {\begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\bar X}}}_{{\mathit{\boldsymbol{\varphi }}_i}}}{\mathit{\boldsymbol{S}}_{wi}} = {\mathit{\boldsymbol{A}}_{{\mathit{\boldsymbol{\varphi }}_i}}}{{\mathit{\boldsymbol{\bar X}}}_{{\mathit{\boldsymbol{\varphi }}_i}}} + {\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{\varphi }}_i}}}}\\ {{\mathit{\boldsymbol{C}}_{{\mathit{\boldsymbol{\varphi }}_i}}}{{\mathit{\boldsymbol{\bar X}}}_{{\mathit{\boldsymbol{\varphi }}_i}}} = {{\bf 0}}} \end{array}} \right.$ （32）

$\overline{\boldsymbol{\varphi}}_{i}=\boldsymbol{\varphi}_{i}-\overline{\boldsymbol{X}}_{\boldsymbol{\varphi}_{i}} \boldsymbol{w}_{i}$, 则有

 $\left\{ {\begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\dot {\tilde \varphi }}}}_i} = {\mathit{\boldsymbol{A}}_{{\mathit{\boldsymbol{\varphi }}_i}}}{{\mathit{\boldsymbol{\tilde \varphi }}}_i} + {{\mathit{\boldsymbol{\tilde \eta }}}_i}}\\ {{\mathit{\boldsymbol{e}}_i} = {\mathit{\boldsymbol{C}}_{{\mathit{\boldsymbol{\varphi }}_i}}}{{\mathit{\boldsymbol{\tilde \varphi }}}_i}} \end{array}} \right.$ （33）

4 仿真分析 4.1 仿真条件设置 4.1.1 无人机-无人车异构系统设置

 图 3 异构系统通信拓扑 Fig. 3 Communication topology of heterogeneous system

 ${\mathit{\boldsymbol{S}}_r} = {\mathit{\boldsymbol{I}}_2} \otimes \left[ {\begin{array}{*{20}{l}} 0&1\\ 0&0 \end{array}} \right],\mathit{\boldsymbol{F}} = {\mathit{\boldsymbol{I}}_2} \otimes \left[ {\begin{array}{*{20}{l}} 1&0 \end{array}} \right]。$

4.1.2 期望时变编队构型设计

 ${\mathit{\boldsymbol{h}}_{yi}}(t) = \left[ {\begin{array}{*{20}{c}} {r{\rm{cos}}(\omega t + (i - 1)\pi /2)}\\ {r{\rm{sin}}(\omega t + (i - 1)\pi /2)} \end{array}} \right]\;\:i = 1,2,3,4$

 ${{\mathit{\boldsymbol{\delta }}_{Xi}} = \left[ {\begin{array}{*{20}{c}} {r{\rm{cos}}(\omega t + (i - 1)\pi /2)}\\ { - r{\rm{cos}}(\omega t + (i - 1)\pi /2)} \end{array}} \right]\;\:i = 1,2,3,4}$
 ${{\mathit{\boldsymbol{\delta }}_{Yi}} = \left[ {\begin{array}{*{20}{c}} {r{\rm{sin}}(\omega t + (i - 1)\pi /2)}\\ {r{\rm{cos}}(\omega t + (i - 1)\pi /2)} \end{array}} \right]\;\:i = 1,2,3,4}$

4.1.3 时变输出编队控制器参数设计

 ${{\mathit{\boldsymbol{X}}_1} = {\mathit{\boldsymbol{I}}_2} \otimes [1\;\:0],{\mathit{\boldsymbol{U}}_1} = {\mathit{\boldsymbol{I}}_2} \otimes [0\;\:1]}$
 ${{\mathit{\boldsymbol{X}}_2} = {\mathit{\boldsymbol{I}}_4},{\mathit{\boldsymbol{U}}_2} = {{{\bf 0}}_{2 \times 4}}}$
 ${{\mathit{\boldsymbol{X}}_3} = {\mathit{\boldsymbol{I}}_2} \otimes [1\;\:0],{\mathit{\boldsymbol{U}}_3} = {\mathit{\boldsymbol{I}}_2} \otimes [0\;\:1]}$
 ${{\mathit{\boldsymbol{X}}_4} = {\mathit{\boldsymbol{I}}_4},{\mathit{\boldsymbol{U}}_4} = {{{\bf 0}}_{2 \times 4}}}$

 ${{v_{X1}} = 0,{v_{Y1}} = 0}$
 ${{v_{X2}} = - r{\omega ^2}{\rm{cos}}(\omega t + \pi /2)}$
 ${{v_{Y2}} = - r{\omega ^2}{\rm{sin}}(\omega t + \pi /2)}$
 ${{v_{X3}} = 0,{v_{Y3}} = 0}$
 ${{v_{X4}} = - r{\omega ^2}{\rm{cos}}(\omega t + 3\pi /2)}$
 ${{v_{Y4}} = - r{\omega ^2}{\rm{sin}}(\omega t + 3\pi /2)}$

 ${{\mathit{\boldsymbol{K}}_{11}} = - {\mathit{\boldsymbol{I}}_2},{\mathit{\boldsymbol{K}}_{21}} = {\mathit{\boldsymbol{I}}_2} \otimes [1\;\:1]}$
 ${{\mathit{\boldsymbol{K}}_{12}} = {\mathit{\boldsymbol{I}}_2} \otimes [ - 2 - 2],{\mathit{\boldsymbol{K}}_{22}} = {\mathit{\boldsymbol{I}}_2} \otimes [2\;\:2]}$
 ${{\mathit{\boldsymbol{K}}_{13}} = - {\mathit{\boldsymbol{I}}_2},{\mathit{\boldsymbol{K}}_{23}} = {\mathit{\boldsymbol{I}}_2} \otimes [1\;\:1]}$
 ${{\mathit{\boldsymbol{K}}_{14}} = {\mathit{\boldsymbol{I}}_2} \otimes [ - 2 - 2],{\mathit{\boldsymbol{K}}_{24}} = {\mathit{\boldsymbol{I}}_2} \otimes [2\;\:2]}$

 ${{\mathit{\boldsymbol{Q}}_{11}} = {{{\bf 0}}_{2 \times 2}},{\mathit{\boldsymbol{Q}}_{21}} = {\mathit{\boldsymbol{I}}_2}}$
 ${{\mathit{\boldsymbol{Q}}_{12}} = {\mathit{\boldsymbol{I}}_2} \otimes \left[ {\begin{array}{*{20}{c}} 0&1\\ { - 1}&0 \end{array}} \right],{\mathit{\boldsymbol{Q}}_{22}} = {\mathit{\boldsymbol{I}}_2} \otimes \left[ {\begin{array}{*{20}{l}} 0\\ 1 \end{array}} \right]}$
 ${{\mathit{\boldsymbol{Q}}_{13}} = {{{\bf 0}}_{2 \times 2}},{\mathit{\boldsymbol{Q}}_{23}} = {\mathit{\boldsymbol{I}}_2}}$
 ${{\mathit{\boldsymbol{Q}}_{14}} = {\mathit{\boldsymbol{I}}_2} \otimes \left[ {\begin{array}{*{20}{c}} 0&1\\ { - 1}&0 \end{array}} \right],{\mathit{\boldsymbol{Q}}_{24}} = {\mathit{\boldsymbol{I}}_2} \otimes \left[ {\begin{array}{*{20}{l}} 0\\ 1 \end{array}} \right]}$

 ${{\mathit{\boldsymbol{K}}_{31}} = {\mathit{\boldsymbol{I}}_2} \otimes [\begin{array}{*{20}{c}} { - 1}&{ - 2} \end{array}],{\mathit{\boldsymbol{K}}_{311}} = - {\mathit{\boldsymbol{I}}_2},{\mathit{\boldsymbol{K}}_{312}} = - 2{\mathit{\boldsymbol{I}}_2}}$
 ${{\mathit{\boldsymbol{K}}_{32}} = {\mathit{\boldsymbol{I}}_2} \otimes [\begin{array}{*{20}{c}} { - 7}&{ - 3}&{ - 5}&{ - 5} \end{array}]}$
 ${{\mathit{\boldsymbol{K}}_{321}} = - {\mathit{\boldsymbol{I}}_2} \otimes [7\;\:3],{\mathit{\boldsymbol{K}}_{322}} = {\mathit{\boldsymbol{I}}_2} \otimes [5\;\: - 5]}$
 ${{\mathit{\boldsymbol{K}}_{33}} = {\mathit{\boldsymbol{I}}_2} \otimes [\begin{array}{*{20}{c}} { - 1}&{ - 2} \end{array}],{\mathit{\boldsymbol{K}}_{331}} = - {\mathit{\boldsymbol{I}}_2},{\mathit{\boldsymbol{K}}_{332}} = - 2{\mathit{\boldsymbol{I}}_2}}$
 ${{\mathit{\boldsymbol{K}}_{34}} = {\mathit{\boldsymbol{I}}_2} \otimes [\begin{array}{*{20}{c}} { - 7}&{ - 3}&5&{ - 5} \end{array}]}$
 ${{\mathit{\boldsymbol{K}}_{341}} = - {\mathit{\boldsymbol{I}}_2} \otimes [7\;\:3],{\mathit{\boldsymbol{K}}_{342}} = {\mathit{\boldsymbol{I}}_2} \otimes [5\;\: - 5]}$
4.2 仿真结果与分析

 图 4 异构时变输出编队控制轨迹 Fig. 4 Trajectory of heterogeneous time-varying output formation control

 图 5 XY平面输出编队与编队中心估计器轨迹 Fig. 5 Trajectory of output formation and FCE in XY plane
 图 6 XY平面输出编队与编队中心估计器不同时刻截图 Fig. 6 Screenshot of output formation and FCE in XY plane at different moments

 图 7 编队中心估计器输出误差 Fig. 7 Errors of FCE output

 图 8 输出编队X/Y轴误差 Fig. 8 Errors of output formation in X/Y axis
 图 9 输出编队误差的欧几里得范数 Fig. 9 Euclidean norm of output formation error

5 结论

1) 控制器中的编队中心估计项可对编队参考函数进行有效估计，并达到输出一致。

2) 控制器中扰动抑制补偿项可对未知外部扰动进行有效抑制。

3) 无人机-无人车异构系统在所设计的分布式控制器作用下可以构成期望的时变编队构型。

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http://dx.doi.org/10.7527/S1000-6893.2019.23767

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#### 文章信息

ZHOU Siquan, DONG Xiwang, LI Qingdong, REN Zhang

Time-varying formation control and disturbance rejection for UAV-UGV heterogeneous swarm system

Acta Aeronautica et Astronautica Sinica, 2020, 41(S1): 723767.
http://dx.doi.org/10.7527/S1000-6893.2019.23767