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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. 复杂系统控制与智能协同技术重点实验室, 北京 100074;
3. 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;
4. 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

Distributed cooperative guidance for maneuvering targets with directed conmunication topologies
DONG Xiaofei1,2,3, REN Zhang1,3,4, CHI Qingxi2, LI Qingdong1,3
1. School of Automation Science and Electrical Engineering, Beihang University, Beijing 100083, China;
2. Science and Technology on Complex System Control and Intelligent Agent Cooperation Laboratory, Beijing 100074, China;
3. Science and Technology on Aircraft Control Laboratory, Beihang University, Beijing 100083, China;
4. Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing 100083, China
Abstract: This paper studies the distributed cooperative guidance for maneuvering targets with directed communication topologies. First, a guidance system model is established based on the geometric relationship between the aircraft and the target, in which the nonlinear problem is solved by feedback linearization. In this design, the unknown maneuver of the target is observed by the extended state observer, and the estimation of the unknown maneuver of the target is applied to the design of the guidance law. In this process, the influence of the target maneuvering on the time-to-go is eliminated by means of direct compensation. Then, the designed guidance law is brought into the guidance model, transforming the problem of cooperative guidance into the problem of consensus. Next, the necessary and sufficient conditions for the convergence of the designed cooperative guidance law are obtained by conducting the pole analysis. Finally, the designed cooperative guidance law and the method of the parameter selection are analyzed by adopting the simulation analysis.
Keywords: maneuvering target    distributed cooperative guidance    extended state observer    consensus    feedback linearization

1) 本文所提出的协同制导律能够应用于机动目标，而文献[1-19]中的方法仅能应用于固定目标。由于目标随机机动带来的问题使得用于固定目标的协同制导律设计会变得更加复杂，同时机动目标相对于固定目标来说适用范围更加广泛。

2) 文献[20-27]研究了针对机动目标的协同制导问题，但是文献[20-25]中需要目标加速度信息的测量值，这在工程中通常难以直接获得。本文的协同制导律设计中没有用到目标的加速度信息。通过分布式扩张状态观测器同时对追踪方程的状态以及由目标加速度带来的干扰进行估计，同时通过极点分析的方式给出了协同制导律收敛的充分必要条件。

3) 虽然文献[26-27]没有用到目标加速度信息，但在通讯拓扑方面，文献[26]中的方法是集中式的，文献[27]中需要的通讯拓扑是无向的，而本文中的通讯拓扑是无向的并且仅需要满足拓扑中有生成树的条件。另外文献[27]需要飞行器具备轴向速度调整能力，而本文则不需要。

1 预备知识及问题描述 1.1 代数图论相关知识

1.2 问题描述

 图 1 针对机动目标的协同制导过程的几何关系 Fig. 1 Geometry relationship against maneuvering target in cooperative guidance process

 ${\eta _i}(t) = {q_i}(t) - {\theta _i}(t)$ （1）

 $\left\{ {\begin{array}{*{20}{l}} {{{\dot R}_i}(t) = {V_T}{\rm{cos}}{\eta _{{\rm{T}}i}}(t) - {V_i}{\rm{cos}}{\eta _i}(t)}\\ {{{\dot q}_i}(t) = \frac{{{V_i}}}{{{R_i}(t)}}{\rm{sin}}{\eta _i}(t) - \frac{{{V_T}}}{{{R_i}(t)}}{\rm{sin}}{\eta _{{\rm{T}}i}}(t)}\\ {{{\dot \theta }_i}(t) = {n_i}(t)g/{V_i}} \end{array}} \right.$ （2）

ri(t)=Ri(t)/Vi，由式(1)和式(2)可得

 $\left\{ {\begin{array}{*{20}{l}} {{{\dot r}_i}(t) = \frac{{{V_{\rm{T}}}}}{{{V_i}}}{\rm{cos}}{\eta _{Ti}}(t) - {\rm{cos}}{\eta _i}(t)}\\ {{{\dot r}_i}(t) = - \frac{{{V_{\rm{T}}}}}{{{V_i}}}{\rm{sin}}{\eta _{{\rm{T}}i}}(t){{\dot \eta }_{{\rm{T}}i}}(t) + {\rm{sin}}{\eta _i}(t)}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {\frac{{{\rm{sin}}{\eta _i}(t)}}{{{r_i}(t)}} - \frac{{{V_{\rm{T}}}}}{{{V_i}{r_i}(t)}}{\rm{sin}}{\eta _{{\rm{T}}i}}(t) - {n_i}(t)g/{V_i}} \right)} \end{array}} \right.$ （3）

 $\left\{ {\begin{array}{*{20}{l}} {\mathop {{\rm{lim}}}\limits_{t \to \infty } |{r_i}(t) - {r_j}(t)| = 0}\\ {\mathop {{\rm{lim}}}\limits_{t \to \infty } |{{\dot r}_i}(t) - {{\dot r}_j}(t)| = 0} \end{array}} \right.$ （4）

2 分布式协同制导律设计

2.1 分布式扩张状态观测器设计

 $\left\{ \begin{array}{l} {d_i}(t) = - \frac{{{V_{\rm{T}}}}}{{{V_i}}}{\rm{sin}}{\eta _{{\rm{T}}i}}(t){{\dot \eta }_{{\rm{T}}i}}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{{V_{\rm{T}}}{\rm{sin}}{\eta _i}(t){\rm{sin}}{\eta _{{\rm{T}}i}}(t)}}{{{V_i}{r_i}(t)}}\\ {u_i}(t) = \frac{{{V_i}{\rm{sin}}{\eta _i}{{(t)}^2}}}{{{r_i}(t)}} - \frac{{{\rm{sin}}{\eta _i}(t){n_i}(t)g}}{{{V_i}}} \end{array} \right.$ （5）

 $\left\{ {\begin{array}{*{20}{l}} {{{\dot x}_{1i}}(t) = {x_{2i}}(t)}\\ {{{\dot x}_{2i}}(t) = {u_i}(t) + {d_i}(t)} \end{array}} \right.$ （6）

 $\left\{ {\begin{array}{*{20}{l}} {\mathop {{\rm{lim}}}\limits_{t \to \infty } |{x_{1i}} - {x_{1j}}| = 0}\\ {\mathop {{\rm{lim}}}\limits_{t \to \infty } |{x_{2i}} - {x_{2j}}| = 0} \end{array}} \right.$ （7）

 $\left\{ {\begin{array}{*{20}{l}} {{{\dot x}_k}(t) = {x_{k + 1}}(t),k = 1,2, \cdots ,n - 1}\\ {{{\dot x}_n}(t) = u(t) + f(\mathit{\boldsymbol{x}}(t),t)}\\ {y(t) = {x_1}(t)} \end{array}} \right.$ （8）

 $\left\{ {\begin{array}{*{20}{l}} {{{\dot {\hat x}}_k}(t) = {{\hat x}_{k + 1}}(t) + {\beta _{0k}}({x_1}(t) - {{\hat x}_1}(t))}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k = 1,2, \cdots ,n - 1}\\ {{{\dot {\hat x}}_n}(t) = u(t) + {{\hat x}_{n + 1}}(t) + {\beta _{0n}}({x_1}(t) - {{\hat x}_1}(t))}\\ {y(t) = {x_1}(t)} \end{array}} \right.$ （9）

 $\left\{ {\begin{array}{*{20}{l}} {{{\dot {\hat x}}_{1i}}(t) = {{\hat x}_{2i}}(t) + {\beta _1}{\rho _i}(t)}\\ {{{\dot {\hat x}}_{2i}}(t) = {u_i}(t) + {{\hat x}_{3i}}(t) + {\beta _2}{\rho _i}(t)}\\ {{{\dot {\hat x}}_{3i}}(t) = {\beta _3}{\rho _i}(t)} \end{array}} \right.$ （10）

 $\left\{ {\begin{array}{*{20}{l}} {{{\dot {\tilde x}}_{1i}}(t) = {{\dot {\tilde x}}_{2i}}(t) - {\beta _1}\sum\limits_{j = 1}^{N - 1} {({{\dot {\tilde x}}_{1i}}(} t) - {{\dot {\tilde x}}_{1j}}(t))}\\ {{{\dot {\tilde x}}_{2i}}(t) = {{\dot {\tilde x}}_{3i}}(t) - {\beta _2}\sum\limits_{j = 1}^{N - 1} {({{\dot {\tilde x}}_{1i}}(} t) - {{\dot {\tilde x}}_{1j}}(t))}\\ {{{\dot {\tilde x}}_{3i}}(t) = - {\beta _3}\mathop \sum \limits^{N - 1} ({{\dot {\tilde x}}_{1i}}(t) - {{\dot {\tilde x}}_{1j}}(t))} \end{array}} \right.$ （11）
2.2 分布式协同制导律设计

 $\begin{array}{*{20}{c}} {{u_i}(t) = - {k_1}{x_{1i}}(t) - {k_2}\sum\limits_{j = 1}^{n - 1} {{a_{ij}}} (({x_{1i}}(t) - {x_{1j}}(t)) + }\\ {({{\hat x}_{2i}}(t) - {{\hat x}_{2j}}(t))) - {{\hat x}_3}(t)} \end{array}$ （12）

 $\left\{ {\begin{array}{*{20}{l}} {{x_{1i}} = {x_{2i}}}\\ {{{\dot x}_{2i}} = - {k_1}{x_{1i}}(t) - {k_2}\sum\limits_{j = 1}^{n - 1} {{a_{ij}}} (({x_{1i}} - {x_{1j}}) + }\\ {\;\:{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({x_{2i}} - {x_{2j}}) - ({{\tilde x}_{2i}} - {{\tilde x}_{2j}})) + {{\tilde x}_{3i}}} \end{array}} \right.$ （13）

 ${{\xi _{1i}}(t) = {x_{1i}}(t),{\xi _{2i}}(t) = {x_{2i}}(t),{\xi _{3i}}(t) = {{\tilde x}_{1i}}(t)}$
 ${{\xi _{4i}}(t) = {{\tilde x}_{2i}}(t),{\xi _{5i}}(t) = {{\tilde x}_{3i}}(t)}$
 ${{\mathit{\boldsymbol{\xi }}_i}(t) = {{[{\xi _{1i}}(t),{\xi _{2i}}(t),{\xi _{3i}}(t),{\xi _{4i}}(t),{\xi _{5i}}(t)]}^{\rm{T}}}}$
 ${\mathit{\boldsymbol{\xi }}(t) = {{[{\mathit{\boldsymbol{\xi }}_1}(t),{\mathit{\boldsymbol{\xi }}_2}(t), \cdots ,{\mathit{\boldsymbol{\xi }}_n}(t)]}^{\rm{T}}}}$

 $\begin{array}{l} \mathit{\boldsymbol{\dot \xi }}(t) = {\mathit{\boldsymbol{I}}_N} \otimes \left[ {\begin{array}{*{20}{c}} 0&1&0&0&0\\ { - {k_1}}&0&0&0&1\\ 0&0&0&1&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{array}} \right]\mathit{\boldsymbol{\xi }}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{L}} \otimes \left[ {\begin{array}{*{20}{c}} 0&0&0&0&0\\ { - {k_2}}&{ - {k_2}}&0&{{k_2}}&0\\ 0&0&{ - {\beta _1}}&0&0\\ 0&0&{ - {\beta _2}}&0&0\\ 0&0&{ - {\beta _3}}&0&0 \end{array}} \right]\mathit{\boldsymbol{\xi }}(t) \end{array}$ （14）

λi(i=1, 2, …, N)为拉普拉斯矩阵L的特征值，其中特征值λ1=0对应的特征向量记为u1=1N，记0 < Re(λ2)≤…≤Re(λN)。

U-1LU=J，其中：U=[u1, u2, …, uN]，$\boldsymbol{U}^{-1}=\left[\begin{array}{l} \boldsymbol{\boldsymbol { u }}_{1}, \boldsymbol{\boldsymbol { u }}_{2}, \cdots, \tilde{\boldsymbol{u}}_{N} \end{array}\right]^{\mathrm{H}}$JL的约旦标准型，uiCN$\tilde{\boldsymbol{u}}_{i} \in {\bf C}^{N}(i=1, 2, \cdots, N)$

c1R5, c2R5, c3R5, c4R5, c5R5是线性无关的向量，同时记pj=uicq(j=5(i-1)+q; i=1, 2, …, N; q=1, 2, 3, 4, 5)，那么一致子空间可以定义为C(U), 其由p1=u1c1=1Nc1p2=u1c2=1Nc2，…, p5=u1c5=1Nc5扩张而成。非一致子空间可以相应定义为C(U), 其由p6, p7, …, p5N扩张而成。由于pj(j=1, 2, …, 5N)是线性无关的，那么可以得到引理4。

1) Re(a1)>0

2) Re(a1)·Re(a1a2-a3)-(lma2)2>0

3) [Re(a1)Re(a2a3)-(Re(a3))2]·[Re(a1)Re(a1a2-a3)-(lm(a2))2]- [Re(a1)lm(a1a3)+(Re(a3)lm(a2))]2>0

 $\begin{array}{*{20}{l}} {{k_2} {\rm{Re}} ({\lambda _i}) > 0}\\ {{k_2} {\rm{Re}} ({\lambda _i})({k_1}{k_2} {\rm{Re}} ({\lambda _i}) + {k_2} {\rm{Re}} {{({\lambda _i})}^2} + {\rm{Im}} {{({\lambda _i})}^2})) - }\\ {k_2^2 {\rm{Im}} {{({\lambda _i})}^2} > 0} \end{array}$ （15）

 $\begin{array}{*{20}{l}} {{\beta _1} {\rm{Re}} ({\lambda _i}) > 0}\\ {{\beta _1} {\rm{Re}} ({\lambda _i})({\beta _1}{\beta _2}( {\rm{Re}} {{({\lambda _i})}^2} + {\rm{Im}} {{({\lambda _i})}^2}) - {\beta _3} {\rm{Re}} ({\lambda _i})) - }\\ {\beta _2^2 {\rm{Im}} {{({\lambda _i})}^2} > 0}\\ {[{\beta _1}{\beta _2}{\beta _3} {\rm{Re}} ({\lambda _i})( {\rm{Re}} {{({\lambda _i})}^2} + {\rm{Im}} {{({\lambda _i})}^2}) - \beta _3^2 {\rm{Re}} {{({\lambda _i})}^2}] \cdot }\\ {[{\beta _1} {\rm{Re}} ({\lambda _i})({\beta _1}{\beta _2}( {\rm{Re}} {{({\lambda _i})}^2} + {\rm{Im}} {{({\lambda _i})}^2}) - {\beta _3} {\rm{Re}} ({\lambda _i})) - }\\ {\beta _2^2 {\rm{Im}} {{({\lambda _i})}^2}] - {{[{\beta _2}{\beta _3} {\rm{Re}} ({\lambda _i}) {\rm{Im}} ({\lambda _i})]}^2} > 0} \end{array}$ （16）

$\tilde{\boldsymbol{U}}=\left[\begin{array}{l} \tilde{\boldsymbol{u}}_{2}, \tilde{\boldsymbol{u}}_{3}, \cdots, \tilde{\boldsymbol{u}}_{N} \end{array}\right]^{\mathrm{H}}, \quad \zeta(t)=\left(\tilde{\boldsymbol{\mu}}_{1}^{\mathrm{H}} \otimes \boldsymbol{I}_{5}\right) \xi(t)$并且$\boldsymbol{ς}(t)=\left(\widetilde{\boldsymbol{U}} \otimes \boldsymbol{I}_{5}\right) \boldsymbol{\xi}(t)$，那么系统(14)可以写为

 $\begin{array}{l} \mathit{\boldsymbol{\dot \zeta }}(t) = \left[ {\begin{array}{*{20}{c}} 0&1&0&0&0\\ { - {k_1}}&0&0&0&1\\ 0&0&0&1&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{array}} \right]\mathit{\boldsymbol{\zeta }}(t)\\ \mathit{\boldsymbol{ \boldsymbol{\dot \varsigma} }}(t) = \left\{ {{\mathit{\boldsymbol{I}}_{n - 1}} \otimes \left[ {\begin{array}{*{20}{l}} 0&1&0&0&0\\ { - {k_1}}&0&0&0&1\\ 0&0&0&1&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{array}} \right] + } \right.\\ \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{J}} \otimes \left[ {\begin{array}{*{20}{c}} 0&0&0&0&0\\ { - {k_2}}&{ - {k_2}}&0&{{k_2}}&0\\ 0&0&{ - {\beta _1}}&0&0\\ 0&0&{ - {\beta _2}}&0&0\\ 0&0&{ - {\beta _3}}&0&0 \end{array}} \right]} \right\}\mathit{\boldsymbol{\zeta }}(t) \end{array}$ （17）

 ${{\mathit{\boldsymbol{\xi }}_\mathit{\boldsymbol{C}}}(t) = (\mathit{\boldsymbol{U}} \otimes {\mathit{\boldsymbol{I}}_5}){{[{\mathit{\boldsymbol{\zeta }}^{\rm{H}}}(t),0]}^{\rm{H}}}}$
 ${{\mathit{\boldsymbol{\xi }}_{\mathit{\boldsymbol{\bar C}}}}(t) = (\mathit{\boldsymbol{U}} \otimes {\mathit{\boldsymbol{I}}_5}){{[0,{\mathit{\boldsymbol{ \boldsymbol{\varsigma} }}^{\rm{H}}}(t)]}^{\rm{H}}}}$

c1, c2, c3, c4, c5为线性无关的向量，所以存在α1(t), α2(t), α3(t), α4(t), α5(t)以及α5k+j(t)(k=1, 2, …, N-1;j=1, 2, 3, 4, 5)使得

 $\begin{array}{l} \mathit{\boldsymbol{\zeta }}(t) = {\alpha _1}(t){\mathit{\boldsymbol{c}}_1} + {\alpha _2}(t){\mathit{\boldsymbol{c}}_2} + {\alpha _3}(t){\mathit{\boldsymbol{c}}_3} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\alpha _4}(t){\mathit{\boldsymbol{c}}_4} + {\alpha _5}(t){\mathit{\boldsymbol{c}}_5} \end{array}$
 $\begin{array}{*{20}{l}} {\mathit{\boldsymbol{ \boldsymbol{\varsigma} }}(t) = [{\alpha _6}(t)\mathit{\boldsymbol{c}}_1^{\rm{H}} + {\alpha _7}(t)\mathit{\boldsymbol{c}}_2^{\rm{H}} + {\alpha _8}(t)\mathit{\boldsymbol{c}}_3^{\rm{H}} + {\alpha _9}(t)\mathit{\boldsymbol{c}}_4^{\rm{H}} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\alpha _{10}}(t)\mathit{\boldsymbol{c}}_5^{\rm{H}}, \cdots ,{\alpha _{5N - 4}}(t)\mathit{\boldsymbol{c}}_1^{\rm{H}} + {\alpha _{5N - 3}}(t)\mathit{\boldsymbol{c}}_2^{\rm{H}} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\alpha _{5N - 2}}(t)\mathit{\boldsymbol{c}}_3^{\rm{H}} + {\alpha _{5N - 1}}(t)\mathit{\boldsymbol{c}}_4^{\rm{H}} + {\alpha _{5N}}(t)\mathit{\boldsymbol{c}}_5^{\rm{H}}{]^{\rm{H}}}} \end{array}$

 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{\xi }}_\mathit{\boldsymbol{C}}}(t) = (\mathit{\boldsymbol{U}} \otimes {\mathit{\boldsymbol{I}}_5})({\mathit{\boldsymbol{e}}_1} \otimes \mathit{\boldsymbol{\zeta }}(t)) = {{\mathit{\boldsymbol{\bar u}}}_1} \otimes \mathit{\boldsymbol{\zeta }}(t) = }\\ {{\alpha _1}(t){\mathit{\boldsymbol{p}}_1} + {\alpha _2}(t){\mathit{\boldsymbol{p}}_2} + \cdots + {\alpha _5}(t){\mathit{\boldsymbol{p}}_5} \in \mathit{\boldsymbol{C}}(\mathit{\boldsymbol{U}})} \end{array}$ （18）

 $\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{\xi }}_{\mathit{\boldsymbol{\bar C}}}}(t) = \sum\limits_{i = 2}^N {({\alpha _{5i - 4}}(t)(} {{\mathit{\boldsymbol{\bar u}}}_i} \otimes {\mathit{\boldsymbol{c}}_1}) + {\alpha _{5i - 3}}(t)({{\mathit{\boldsymbol{\bar u}}}_i} \otimes {\mathit{\boldsymbol{c}}_2}) + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cdots + {\alpha _{5i}}(t)({{\mathit{\boldsymbol{\bar u}}}_i} \otimes {\mathit{\boldsymbol{c}}_5})) = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{j = 6}^N {{\alpha _j}} (t){\mathit{\boldsymbol{p}}_j} \in \mathit{\boldsymbol{\bar C}}(\mathit{\boldsymbol{U}})} \end{array}$ （19）

 $\mathop {{\rm{lim}}}\limits_{t \to \infty } \mathit{\boldsymbol{ \boldsymbol{\varsigma} }}(t) = 0$ （20）

J的结构可知，方程(20)等价于以下N-1个子系统的稳定性：

 ${\mathit{\boldsymbol{ \boldsymbol{\dot \varsigma} }}_i}(t) = \left[ {\begin{array}{*{20}{c}} 0&1&0&0&0\\ { - {k_1} - {k_2}{\lambda _i}}&{ - {k_2}{\lambda _i}}&0&{{k_2}{\lambda _i}}&1\\ 0&0&{ - {\beta _1}{\lambda _i}}&1&0\\ 0&0&{ - {\beta _2}{\lambda _i}}&0&1\\ 0&0&{ - {\beta _3}{\lambda _i}}&0&0 \end{array}} \right]{\mathit{\boldsymbol{ \boldsymbol{\varsigma} }}_i}(t)$ （21）

 $\begin{array}{*{20}{l}} {{f_i}(s) = ({s^2} + {k_2}{\lambda _i}s + {k_1} + {k_2}{\lambda _i}) \cdot }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({s^3} + {\lambda _i}{\beta _1}{s^2} + {\lambda _i}{\beta _2}s + {\lambda _i}{\beta _3})} \end{array}$ （22）

 $\begin{array}{l} {n_i}(t) = \frac{{{V_i}{\rm{sin}}({\eta _i}(t))}}{{g{r_i}(t)}} - \frac{{{V_i}}}{{g{\rm{sin}}({\eta _i}(t))}}[ - {k_1}{x_{1i}}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {k_2}\sum\limits_{j = 1}^{n - 1} {{a_{ij}}} (({x_{1i}}(t) - {x_{1j}}(t)) + ({{\hat x}_{2i}}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\hat x}_{2j}}(t))) - {{\hat x}_3}(t)] \end{array}$ （23）
3 数值仿真

 图 2 多飞行器通信拓扑关系 Fig. 2 Communication topology for multi-aircrafts

 $\mathit{\boldsymbol{L}} = \left[ {\begin{array}{*{20}{c}} 0&0&0&0\\ { - 1}&1&0&0\\ { - 1}&0&2&{ - 1}\\ 0&{ - 1}&0&1 \end{array}} \right]$

 飞行器 速度/(m·s-1) 距离/m 初始弹道偏角/(°) 1 260 10 440 0 2 268 10 050 0 3 265 10 198 0 4 261 10 770 0

 图 3 多飞行器飞行轨迹 Fig. 3 Trajectories of multi-aircrafts
 图 4 Ri/Vi的变化曲线 Fig. 4 Curve of Ri/Vi
 图 5 飞行器过载指令 Fig. 5 Acceleration commands of multiaircarft
 图 6 Δr的变化曲线 Fig. 6 Curves of Δr

4 结论

1) 基于反馈线性化方法及一致性理论，提出了针对机动目标的协同制导律，并通过极点分析给出了制导律参数的选取方法。

2) 所提出的协同制导律能够在有向拓扑、不需要目标加速度测量信息、不需要飞行器具备轴向机动的条件下实现对机动目标的协同攻击。

3) 在制导末端存在制导指令过大的问题，目前采用切换成比例导引的方式避免该问题，但是会导致命中时间在制导末端产生一定的误差，针对该问题仍需进一步研究。

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http://dx.doi.org/10.7527/S1000-6893.2019.23762

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#### 文章信息

DONG Xiaofei, REN Zhang, CHI Qingxi, LI Qingdong

Distributed cooperative guidance for maneuvering targets with directed conmunication topologies

Acta Aeronautica et Astronautica Sinica, 2020, 41(S1): 723762.
http://dx.doi.org/10.7527/S1000-6893.2019.23762