﻿ 带有引诱角色的多飞行器协同最优制导方法
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1. 火箭军工程大学 精确制导与仿真实验室, 西安 710025;
2. 西北工业大学 航天学院, 西安 710072

Cooperative optimal guidance method for multi-aircraft with luring role
WANG Shaobo1, GUO Yang1,2, WANG Shicheng1, LIU Zhiguo1, ZHANG Shuai1
1. Precision Guidance and Simulation Lab, Rocket Force University of Engineering, Xi'an 710025, China;
2. School of Astronautics, Northwestern Polytechnical University, Xi'an 710072, China
Abstract: This paper considers the situation that our high-value aircraft launches two defender to intercept the interceptor when it is faced with interception by the other interceptor. Based on the optimal control theory, an explicit cooperative guidance law with relative interception angle at the end of interception is designed under the assumption that all four aircraft have first-order linear dynamic characteristics. The explicit cooperative guidance law takes into account the cooperation of high-value aircraft and two defenders, and gives the analytical solution of the optimal control input of the three objects. The simulation results show that the designed guidance law can make the two defenders intercept the interceptor successfully, and impose a relative interception angle at the end of the interception. By comparing with the implicit cooperative guidance law which only considers two defenders, the explicit cooperative guidance law is superior to the implicit cooperative guidance law in control requirements and energy consumption. In addition, the stability of the cooperative guidance law under different launching conditions is verified.
Keywords: multi-aircraft    cooperative guidance    luring role    optimal control    interception angle    explicit

1 问题描述

 图 1 多导弹协同拦截交战 Fig. 1 Multi-missile cooperative interception engagement

1.1 动力学与运动学模型

 ${{\dot r}_{{\rm{PE}}}} = {v_{{\rm{PE}}}} = - {v_{\rm{E}}}\cos \left( {{\gamma _{\rm{E}}} - {q_{{\rm{PE}}}}} \right) - {v_{\rm{P}}}\cos \left( {{\gamma _{\rm{P}}} + {q_{{\rm{PE}}}}} \right)$ （1）
 ${{\dot q}_{{\rm{PE}}}} = \frac{{{v_{\rm{E}}}\sin \left( {{\gamma _{\rm{E}}} - {q_{{\rm{PE}}}}} \right) - {v_{\rm{P}}}\sin \left( {{\gamma _{\rm{P}}} + {q_{{\rm{PE}}}}} \right)}}{{{r_{{\rm{PE}}}}}}$ （2）

 ${{\ddot q}_{{\rm{PE}}}} = - 2\frac{{{{\dot r}_{{\rm{PE}}}}}}{{{r_{{\rm{PE}}}}}}{{\dot q}_{{\rm{PE}}}} + \frac{{a_{\rm{E}}^\prime - a_{\rm{P}}^\prime }}{{{r_{{\rm{PE}}}}}}$ （3）

 $\begin{array}{l} {{\dot r}_{{\rm{P}}{{\rm{D}}_1}}} = {v_{{\rm{P}}{{\rm{D}}_1}}} = - {v_{{{\rm{D}}_1}}}\cos \left( {{\gamma _{{{\rm{D}}_1}}} - {q_{{\rm{P}}{{\rm{D}}_1}}}} \right) - \\ \;\;\;\;\;{v_{\rm{P}}}\cos \left( {{\gamma _{\rm{P}}} + {q_{{\rm{P}}{{\rm{D}}_1}}}} \right) \end{array}$ （4）
 ${{\dot q}_{{\rm{P}}{{\rm{D}}_1}}} = \frac{{{v_{{{\rm{D}}_1}}}\sin \left( {{\gamma _{{{\rm{D}}_1}}} - {q_{{\rm{PE}}}}} \right) - {v_{\rm{P}}}\sin \left( {{\gamma _{\rm{P}}} + {q_{{\rm{P}}{{\rm{D}}_1}}}} \right)}}{{{r_{{\rm{P}}{{\rm{D}}_1}}}}}$ （5）
 $\begin{array}{l} {{\dot r}_{{\rm{P}}{{\rm{D}}_2}}} = {v_{{\rm{P}}{{\rm{D}}_2}}} = - {v_{{D_2}}}\cos \left( {{\gamma _{{{\rm{D}}_2}}} - {q_{{\rm{P}}{{\rm{D}}_2}}}} \right) - \\ \;\;\;\;\;{v_{\rm{P}}}\cos \left( {{\gamma _{\rm{P}}} + {q_{{\rm{P}}{{\rm{D}}_2}}}} \right) \end{array}$ （6）
 ${{\dot q}_{{\rm{P}}{{\rm{D}}_2}}} = \frac{{{v_{{{\rm{D}}_2}}}\sin \left( {{\gamma _{{{\rm{D}}_2}}} - {q_{{\rm{PE}}}}} \right) - {v_{\rm{P}}}\sin \left( {{\gamma _{\rm{P}}} + {q_{{\rm{P}}{{\rm{D}}_2}}}} \right)}}{{{r_{{\rm{P}}{{\rm{D}}_2}}}}}$ （7）

 ${{\ddot q}_{{\rm{P}}{{\rm{D}}_1}}} = - 2\frac{{{{\dot r}_{{\rm{P}}{{\rm{D}}_1}}}}}{{{r_{{\rm{P}}{{\rm{D}}_1}}}}}{{\dot q}_{{\rm{P}}{{\rm{D}}_1}}} + \frac{{a_{{{\rm{D}}_1}}^\prime - a_{\rm{P}}^\prime }}{{{r_{{\rm{P}}{{\rm{D}}_1}}}}}$ （8）
 ${{\ddot q}_{{\rm{P}}{{\rm{D}}_2}}} = - 2\frac{{{{\dot r}_{{\rm{P}}{{\rm{D}}_2}}}}}{{{r_{{\rm{P}}{{\rm{D}}_2}}}}}{{\dot q}_{{\rm{P}}{{\rm{D}}_2}}} + \frac{{a_{{{\rm{D}}_2}}^\prime - a_{\rm{P}}^\prime }}{{{r_{{\rm{P}}{{\rm{D}}_2}}}}}$ （9）

 ${{\dot \gamma }_i} = \frac{{{a_i}}}{{{v_i}}}\;\;\;i \in \left\{ {{\rm{E}},{{\rm{D}}_1},{{\rm{D}}_2},{\rm{P}}} \right\}$ （10）

 ${{\dot a}_i} = \frac{{{a_{i{\rm{c}}}} - {a_i}}}{{{\tau _i}}}\;\;\;\;i \in \left\{ {{\rm{E}},{{\rm{D}}_1},{{\rm{D}}_2},{\rm{P}}} \right\}$ （11）

 ${a_{{\rm{PC}}}} = N{V_{{\rm{PE}}}}{{\dot q}_{{\rm{PE}}}} + \frac{K}{2}a_{\rm{E}}^\prime \;\;\;\left| {{a_{{\rm{PC}}}}} \right| \le {a_{{\rm{Pmax}}}}$ （12）

 $a_{\rm{E}}^\prime = {a_{\rm{E}}}\cos \left( {{\gamma _{{{\rm{E}}_0}}} - {q_{{\rm{P}}{{\rm{E}}_0}}}} \right)$ （13）
 $a_{\rm{P}}^\prime = {a_{\rm{P}}}\cos \left( {{\gamma _{{{\rm{E}}_0}}} + {q_{{\rm{P}}{{\rm{E}}_0}}}} \right)$ （14）
 $a_{{{\rm{D}}_i}}^\prime = {a_{{\rm{D}}i}}\cos \left( {{\gamma _{{{\rm{D}}_{i0}}}} - {q_{{\rm{P}}{{\rm{E}}_0}}}} \right)$ （15）

 ${t_{{\rm{fPE}}}} = \frac{{ - {r_{{\rm{P}}{{\rm{E}}_0}}}}}{{{{\dot r}_{{\rm{PE}}}}}} = \frac{{ - {r_{{\rm{P}}{{\rm{E}}_0}}}}}{{{v_{{\rm{PE}}}}}}$ （16）
 ${t_{{\rm{fP}}{{\rm{D}}_1}}} = \frac{{ - {r_{{\rm{P}}{{\rm{D}}_{10}}}}}}{{{{\dot r}_{{\rm{P}}{{\rm{D}}_1}}}}} = \frac{{ - {r_{{\rm{P}}{{\rm{D}}_{10}}}}}}{{{v_{{\rm{P}}{{\rm{D}}_1}}}}}$ （17）
 ${t_{{\rm{fP}}{{\rm{D}}_2}}} = \frac{{ - {r_{{\rm{PD}}{2_0}}}}}{{{{\dot r}_{{\rm{PD}}2}}}} = \frac{{ - {r_{{\rm{PD}}{2_0}}}}}{{{v_{{\rm{PD}}2}}}}$ （18）

 $\mathit{\boldsymbol{x}} = {\left[ {\begin{array}{*{20}{c}} {{{\dot q}_{{\rm{PE}}}}}&{{{\dot q}_{{\rm{P}}{{\rm{D}}_1}}}}&{{{\dot q}_{{\rm{P}}{{\rm{D}}_2}}}}&{{a_{\rm{P}}}}&{{a_{\rm{E}}}}&{{a_{{{\rm{D}}_1}}}}&{{a_{{{\rm{D}}_2}}}}&{{x_{{\gamma _1}}}}&{{x_{{\gamma _2}}}} \end{array}} \right]^{\rm{T}}}$ （19）

1.2 突防器配合防御器协同交战

 $\mathit{\boldsymbol{\dot x}} = \mathit{\boldsymbol{A}}(t)\mathit{\boldsymbol{x}}(t) + \mathit{\boldsymbol{B}}(t)\mathit{\boldsymbol{u}}(t)$ （20）
 $\begin{array}{l} \mathit{\boldsymbol{A}}(t) = \\ \;\;\;\;\;\left[ {\begin{array}{*{20}{c}} {\frac{{ - 2}}{{{t_{{\rm{fPE}}}} - t}}}&0&0&{\frac{{ - \cos \left( {{\gamma _{{{\rm{E}}_0}}} + {q_{{\rm{P}}{{\rm{E}}_0}}}} \right)}}{{{v_{{\rm{PE}}}}\left( {{t_{{\rm{fPE}}}} - t} \right)}}}&{\frac{{\cos \left( {{\gamma _{{{\rm{E}}_0}}} - {q_{{\rm{P}}{{\rm{E}}_0}}}} \right)}}{{{v_{{\rm{PE}}}}\left( {{t_{{\rm{fPE}}}} - t} \right)}}}&0&0&0&0\\ 0&{\frac{{ - 2}}{{{t_{{\rm{fP}}{{\rm{D}}_1}}} - t}}}&0&{\frac{{ - \cos \left( {{\gamma _{{{\rm{E}}_0}}} + {q_{{\rm{P}}{{\rm{E}}_0}}}} \right)}}{{{v_{{\rm{P}}{{\rm{D}}_1}}}\left( {{t_{{\rm{fP}}{{\rm{D}}_1}}} - t} \right)}}}&0&{\frac{{\cos \left( {{\gamma _{{{\rm{D}}_{10}}}} - {q_{{\rm{P}}{{\rm{D}}_{10}}}}} \right)}}{{{v_{{\rm{P}}{{\rm{D}}_1}}}\left( {{t_{{\rm{fP}}{{\rm{D}}_1}}} - t} \right)}}}&0&0&0\\ 0&0&{\frac{{ - 2}}{{{t_{{\rm{fP}}{{\rm{D}}_2}}} - t}}}&{\frac{{ - \cos \left( {{\gamma _{{{\rm{E}}_0}}} + {q_{{\rm{P}}{{\rm{E}}_0}}}} \right)}}{{{v_{{\rm{P}}{{\rm{D}}_2}}}\left( {{t_{{\rm{fP}}{{\rm{D}}_2}}} - t} \right)}}}&0&0&{\frac{{\cos \left( {{\gamma _{{{\rm{D}}_{20}}}} - {q_{{\rm{P}}{{\rm{D}}_{20}}}}} \right)}}{{{v_{{\rm{P}}{{\rm{D}}_1}}}\left( {{t_{{\rm{fP}}{{\rm{D}}_1}}} - t} \right)}}}&0&0\\ {\frac{{N{v_{{\rm{PE}}}}}}{{{\tau _{\rm{P}}}}}}&0&0&{\frac{{ - 1}}{{{\tau _{\rm{P}}}}}}&{\frac{{K\cos \left( {{\gamma _{{{\rm{E}}_0}}} - {q_{{\rm{P}}{{\rm{E}}_0}}}} \right)}}{{2{\tau _{\rm{P}}}}}}&0&0&0&0\\ 0&0&0&0&{\frac{{ - 1}}{{{\tau _{\rm{E}}}}}}&0&0&0&0\\ 0&0&0&0&0&{\frac{{ - 1}}{{{\tau _{{{\rm{D}}_1}}}}}}&0&0&0\\ 0&0&0&0&0&0&{\frac{{ - 1}}{{{\tau _{{{\rm{D}}_2}}}}}}&0&0\\ 0&0&0&{\frac{1}{{{v_{\rm{P}}}}}}&0&{\frac{1}{{{v_{{{\rm{D}}_1}}}}}}&0&0&0\\ 0&0&0&{\frac{1}{{{v_{\rm{P}}}}}}&0&0&{\frac{1}{{{v_{{{\rm{D}}_2}}}}}}&0&0 \end{array}} \right] \end{array}$ （21）

uDiuDmaxuDmax为防御器的过载限制。由突防器、防御器1和防御器2的加速度指令组成，即 $u_{\mathrm{E}}=a_{\mathrm{EC}}, u_{\mathrm{D}_{1}}=a_{\mathrm{D}_{1} \mathrm{C}}, u_{\mathrm{D}_{2}}=a_{\mathrm{D}_{2} \mathrm{C}}$

 $\mathit{\boldsymbol{B}}\left( t \right) = {\left[ {\begin{array}{*{20}{c}} 0&0&0&0&{\frac{1}{{{\tau _{\rm{E}}}}}}&0&0&0&0\\ 0&0&0&0&0&{\frac{1}{{{\tau _{{D_1}}}}}}&0&0&0\\ 0&0&0&0&0&0&{\frac{1}{{{\tau _{{{\rm{D}}_2}}}}}}&0&0 \end{array}} \right]^{\rm{T}}}$ （22）

 ${\mathit{\boldsymbol{B}}_E} = {\left[ {\begin{array}{*{20}{l}} 0&0&0&0&{\frac{1}{{{\tau _{\rm{E}}}}}}&0&0&0&0 \end{array}} \right]^{\rm{T}}}$ （23）
 ${\mathit{\boldsymbol{B}}_{{{\rm{D}}_1}}} = {\left[ {\begin{array}{*{20}{l}} 0&0&0&0&0&{\frac{1}{{{\tau _{{{\rm{D}}_1}}}}}}&0&0&0 \end{array}} \right]^{\rm{T}}}$ （24）
 ${\mathit{\boldsymbol{B}}_{{{\rm{D}}_2}}} = {\left[ {\begin{array}{*{20}{l}} 0&0&0&0&0&0&{\frac{1}{{{\tau _{{{\rm{D}}_2}}}}}}&0&0 \end{array}} \right]^{\rm{T}}}$ （25）

$\boldsymbol{u}(t)=\left[\begin{array}{lll} u_{\mathrm{E}} & u_{\mathrm{p}_{1}} & u_{\mathrm{D}_{2}} \end{array}\right]^{\mathrm{T}}$是需要设计的控制器，该式表明在制导律设计中突防器、防御器1和防御器2进行深度协同。突防器可配合防御器1和防御器2实现对拦截器的反拦截。这是一种显式的协同方式。

1.3 突防器不配合防御器协同交战

 $\mathit{\boldsymbol{\dot x}} = \mathit{\boldsymbol{\tilde A}}(t)\mathit{\boldsymbol{x}}(t) + \mathit{\boldsymbol{B}}(t)\mathit{\boldsymbol{u}}(t) + \mathit{\boldsymbol{G}}(t)w(t)$ （26）

 $\mathit{\boldsymbol{B}}\left( t \right) = {\left[ {\begin{array}{*{20}{c}} 0&0&0&0&0&{\frac{1}{{{\tau _{{{\rm{D}}_1}}}}}}&0&0&0\\ 0&0&0&0&0&0&{\frac{1}{{{\tau _{{{\rm{D}}_2}}}}}}&0&0 \end{array}} \right]^{\rm{T}}}$ （27）
 $\mathit{\boldsymbol{G}}\left( t \right) = {\left[ {\begin{array}{*{20}{l}} 0&0&0&0&{\frac{1}{{{\tau _{\rm{E}}}}}}&0&0&0&0 \end{array}} \right]^{\rm{T}}}$ （28）

 ${\mathit{\boldsymbol{B}}_{{{\rm{D}}_1}}} = {\left[ {\begin{array}{*{20}{l}} 0&0&0&0&0&{\frac{1}{{{\tau _{{{\rm{D}}_1}}}}}}&0&0&0 \end{array}} \right]^{\rm{T}}}$ （29）
 ${\mathit{\boldsymbol{B}}_{{{\rm{D}}_2}}} = {\left[ {\begin{array}{*{20}{l}} 0&0&0&0&0&0&{\frac{1}{{{\tau _{{{\rm{D}}_2}}}}}}&0&0 \end{array}} \right]^{\rm{T}}}$ （30）
2 最优协同制导律设计

 $Z{\rm{E}}{{\rm{M}}_{{{\rm{D}}_i}}} = \frac{{{{\dot q}_{{\rm{P}}{{\rm{D}}_i}}}\left( {{t_{\rm{f}}}} \right)D_{{{\rm{D}}_i}}^2}}{{{v_{{\rm{P}}{{\rm{D}}_i}}}}}$ （31）

2.1 目标函数

 $\begin{array}{*{20}{c}} {J = \frac{1}{2}{\alpha _1}\dot q_{{\rm{P}}{{\rm{D}}_1}}^2\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) + \frac{1}{2}{\alpha _2}\dot q_{{\rm{P}}{{\rm{D}}_2}}^2\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) + \frac{\beta }{2}\left[ {{x_{{\gamma _1}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - } \right.}\\ {\left. {{x_{{\gamma _2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - \Delta } \right]2 + \frac{1}{2}\int_0^{{t_{{{\rm{f}}_{\rm{n}}}}}} {\left( {u_{{{\rm{D}}_1}}^2 + u_{{{\rm{D}}_2}}^2 + u_{\rm{E}}^2} \right){\rm{d}}t} } \end{array}$ （32）

2.2 模型降阶

 $Z(t) = \mathit{\boldsymbol{D \boldsymbol{\varPhi} }}\left( {{t_{{{\rm{f}}_{\rm{n}}}}},t} \right)\mathit{\boldsymbol{x}}(t)$ （33）

 $\mathit{\boldsymbol{ \boldsymbol{\dot \varPhi} }}\left( {{t_{{{\rm{f}}_{\rm{n}}}}},t} \right) = - \mathit{\boldsymbol{ \boldsymbol{\varPhi} }}\left( {{t_{{{\rm{f}}_{\rm{n}}}}},t} \right)\mathit{\boldsymbol{A}}$ （34）

 $\begin{array}{l} \dot Z(t) = \mathit{\boldsymbol{D \boldsymbol{\varPhi} }}\left( {{t_{{{\rm{f}}_{\rm{n}}}}},t} \right)\mathit{\boldsymbol{x}}(t) + \mathit{\boldsymbol{D \boldsymbol{\varPhi} }}\left( {{t_{\rm{n}}},t} \right)\mathit{\boldsymbol{\dot x}}(t)\\ \;\;\;\;\;\;\; = \mathit{\boldsymbol{D \boldsymbol{\varPhi} }}\left( {{t_{{{\rm{f}}_{\rm{n}}}}},t} \right)\mathit{\boldsymbol{Bu}}\left( t \right) \end{array}$ （35）

D=Dq1=[010000000]时，可分离防御器1的视线角速率信息。

D=Dq2=[010000000]时，可分离防御器2的视线角速率信息。

D=Dγ1=[000000010]时，可分离防御器1的航向角信息。

D=Dγ2=[000000001]时，可分离防御器2的航向角信息。

Dq1Dq2Dγ1Dγ2代入式(33)中，得

 ${Z_{{q_i}}}(t) = {\mathit{\boldsymbol{D}}_{{q_i}}}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}\left( {{t_{{{\rm{f}}_{\rm{n}}}}},t} \right)\mathit{\boldsymbol{x}}(t)$ （36）
 ${Z_{{\gamma _i}}}(t) = {\mathit{\boldsymbol{D}}_{{\gamma _i}}}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}\left( {{t_{{{\rm{f}}_{\rm{n}}}}},t} \right)\mathit{\boldsymbol{x}}(t)$ （37）

 $\begin{array}{*{20}{c}} {J = \frac{1}{2}{\alpha _1}Z_{{q_1}}^2\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) + \frac{1}{2}{\alpha _2}Z_{{q_2}}^2\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) + \frac{\beta }{2}\left[ {{Z_{{\gamma _1}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - } \right.}\\ {\left. {{Z_{{\gamma _2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - \Delta } \right]2 + \frac{1}{2}\int_0^{{t_{{{\rm{f}}_{\rm{n}}}}}} {\left( {u_{{{\rm{D}}_1}}^2 + u_{{{\rm{D}}_2}}^2 + u_{\rm{E}}^2} \right){\rm{d}}t} } \end{array}$ （38）

 $\begin{array}{*{20}{c}} {{{\dot Z}_{{{\rm{q}}_i}}}(t) = {\mathit{\boldsymbol{D}}_{{q_i}}}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}{\mathit{\boldsymbol{B}}_{{{\rm{D}}_1}}}{u_{{{\rm{D}}_1}}} + {D_{{{\rm{q}}_i}}}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}{\mathit{\boldsymbol{B}}_{{{\rm{D}}_2}}}{u_{{{\rm{D}}_2}}} = }\\ {{{\tilde B}_{{{\rm{D}}_{{1_{{q_i}}}}}}}(t){u_{{{\rm{D}}_1}}} + {{\tilde B}_{{{\rm{D}}_{2{q_i}}}}}(t){u_{{{\rm{D}}_2}}}} \end{array}$ （62）
 $\begin{array}{*{20}{c}} {{{\dot Z}_{{\gamma _i}}}(t) = {\mathit{\boldsymbol{D}}_{{\gamma _i}}}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}{\mathit{\boldsymbol{B}}_{{{\rm{D}}_1}}}{u_{{{\rm{D}}_1}}} + {D_{{\gamma _i}}}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}{\mathit{\boldsymbol{B}}_{{{\rm{D}}_2}}}{u_{{{\rm{D}}_2}}} = }\\ {{{\tilde B}_{{D_{1{\gamma _i}}}}}(t){u_{{{\rm{D}}_1}}} + {{\tilde B}_{{{\rm{D}}_{2{\gamma _i}}}}}(t){u_{{{\rm{D}}_2}}}} \end{array}$ （63）

 $\begin{array}{l} H = \frac{1}{2}\left( {u_{{{\rm{D}}_1}}^2 + u_{{{\rm{D}}_2}}^2} \right) + {\lambda _{{Z_1}}}{{\dot Z}_{{q_1}}}(t) + {\lambda _{{Z_2}}}{{\dot Z}_{{q_2}}}(t) + \\ \;\;\;\;\;{\lambda _{{Z_{{\gamma _1}}}}}{{\dot Z}_{{\gamma _1}}}(t) + {\lambda _{{Z_{{\gamma _2}}}}}{{\dot Z}_{{\gamma _2}}}(t) \end{array}$ （64）

 $\begin{array}{l} \frac{{\partial H}}{{\partial {u_{{{\rm{D}}_1}}}}} = 0 \Rightarrow \\ {u_{{{\rm{D}}_1}}} = - {\alpha _1}{Z_{{q_1}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right){{\tilde B}_{{{\rm{D}}_{1{q_1}}}}} - {\alpha _2}{Z_{{q_2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right){{\tilde B}_{{{\rm{D}}_{{1_{{q_2}}}}}}} + \\ \;\;\;\;\;\;\beta \left[ {{Z_{{\gamma _1}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - {Z_{{\gamma _2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - \Delta } \right]\left( {{{\tilde B}_{{{\rm{D}}_{1{\gamma _2}}}}} - {{\tilde B}_{{{\rm{D}}_{1{\gamma _1}}}}}} \right) \end{array}$ （65）
 $\begin{array}{l} \frac{{\partial H}}{{\partial {u_{{{\rm{D}}_2}}}}} = 0 \Rightarrow \\ {u_{{{\rm{D}}_2}}} = - {\alpha _1}{Z_{{q_1}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right){{\tilde B}_{{{\rm{D}}_{2{q_1}}}}} - {\alpha _2}{Z_{{q_2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right){{\tilde B}_{{{\rm{D}}_2}_{{q_2}}}} + \\ \;\;\;\;\;\beta \left[ {{Z_{{\gamma _1}}}\left( {{t_{\rm{n}}}} \right) - {Z_{{\gamma _2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - \Delta } \right]\left( {{{\tilde B}_{{{\rm{D}}_{2{\gamma _2}}}}} - {{\tilde B}_{{{\rm{D}}_{2{\gamma _1}}}}}} \right) \end{array}$ （66）

 $\begin{array}{l} {{\dot Z}_{{q_1}}}(t) = - {\alpha _1}\left( {{{\tilde B}_{{{\rm{D}}_{1{q_1}}}}}{{\tilde B}_{{{\rm{D}}_{1{q_1}}}}} + {{\tilde B}_{{{\rm{D}}_{2{q_1}}}}}{{\tilde B}_{{{\rm{D}}_{2{q_1}}}}}} \right){Z_{{q_1}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - \\ \;\;\;\;\;\;{\alpha _2}\left( {{{\tilde B}_{{{\rm{D}}_{1{q_2}}}}}{{\tilde B}_{{{\rm{D}}_{1{q_1}}}}} + {{\tilde B}_{{{\rm{D}}_{2{q_2}}}}}{{\tilde B}_{{{\rm{D}}_{2{q_1}}}}}} \right){Z_{{q_2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) + \\ \;\;\;\;\;\;\beta \left( {{{\tilde B}_{{{\rm{D}}_{{1_{{\gamma _2}}}}}}}{{\tilde B}_{{{\rm{D}}_{1{q_1}}}}} + {{\tilde B}_{{{\rm{D}}_{2{\gamma _2}}}}}{{\tilde B}_{{{\rm{D}}_{2{q_1}}}}}} \right)\left[ {{Z_{{\gamma _1}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - } \right.\\ \;\;\;\;\;\;\left. {{Z_{{\gamma _2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - \Delta } \right] - \beta \left( {{{\tilde B}_{{{\rm{D}}_{1{\gamma _1}}}}}{{\tilde B}_{{{\rm{D}}_{1{q_1}}}}} + {{\tilde B}_{{{\rm{D}}_{2{\gamma _1}}}}}{{\tilde B}_{{{\rm{D}}_{2{q_1}}}}}} \right) \cdot \\ \;\;\;\;\;\;\left[ {{Z_{{\gamma _1}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - {Z_{{\gamma _2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - \Delta } \right] \end{array}$ （67）
 $\begin{array}{l} {{\dot Z}_{{q_2}}}(t) = - {\alpha _1}\left( {{{\tilde B}_{{{\rm{D}}_{1{q_1}}}}}{{\tilde B}_{{{\rm{D}}_{1{q_2}}}}} + {{\tilde B}_{{{\rm{D}}_{2{q_1}}}}}{{\tilde B}_{{{\rm{D}}_{2{q_2}}}}}} \right){Z_{{q_1}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - \\ \;\;\;\;\;\;{\alpha _2}\left( {{{\tilde B}_{{{\rm{D}}_{1{q_2}}}}}{{\tilde B}_{{{\rm{D}}_{1{q_2}}}}} + {{\tilde B}_{{{\rm{D}}_{2{q_2}}}}}{{\tilde B}_{{{\rm{D}}_{2{q_2}}}}}} \right){Z_{{q_2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) + \\ \;\;\;\;\;\;\beta \left( {{{\tilde B}_{{{\rm{D}}_{{1_{{\gamma _2}}}}}}}{{\tilde B}_{{{\rm{D}}_{1{q_2}}}}} + {{\tilde B}_{{{\rm{D}}_{2{\gamma _2}}}}}{{\tilde B}_{{{\rm{D}}_{2{q_2}}}}}} \right)\left[ {{Z_{{\gamma _1}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - } \right.\\ \;\;\;\;\;\;\left. {{Z_{{\gamma _2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - \Delta } \right] - \beta \left( {{{\tilde B}_{{{\rm{D}}_{1{\gamma _1}}}}}{{\tilde B}_{{{\rm{D}}_{1{q_2}}}}} + {{\tilde B}_{{{\rm{D}}_{2{\gamma _1}}}}}{{\tilde B}_{{{\rm{D}}_{2{q_2}}}}}} \right) \cdot \\ \;\;\;\;\;\;\left[ {{Z_{{\gamma _1}}}\left( {{t_{{f_n}}}} \right) - {Z_{{\gamma _2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - \Delta } \right] \end{array}$ （68）
 $\begin{array}{l} {{\dot Z}_{{\gamma _1}}}(t) = - {\alpha _1}\left( {{{\tilde B}_{{{\rm{D}}_{1{q_1}}}}}{{\tilde B}_{{{\rm{D}}_1}_{{\gamma _1}}}} + {{\tilde B}_{{{\rm{D}}_{2{q_1}}}}}{{\tilde B}_{{{\rm{D}}_2}_{{\gamma _1}}}}} \right){Z_{{q_1}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - \\ \;\;\;\;\;{\alpha _2}\left( {{{\tilde B}_{{{\rm{D}}_{1{q_2}}}}}{{\tilde B}_{{{\rm{D}}_{{1_{{\gamma _1}}}}}}} + {{\tilde B}_{{{\rm{D}}_{2{q_2}}}}}{{\tilde B}_{{{\rm{D}}_2}_{{\gamma _1}}}}} \right){Z_{{q_2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) + \\ \;\;\;\;\;\beta \left( {{{\tilde B}_{{{\rm{D}}_{{1_{{\gamma _2}}}}}}}{{\tilde B}_{{{\rm{D}}_{{1_{{\gamma _1}}}}}}} + {{\tilde B}_{{{\rm{D}}_{{1_{{\gamma _2}}}}}}}{{\tilde B}_{{{\rm{D}}_{{2_{{\gamma _1}}}}}}}} \right)\left[ {{Z_{{\gamma _1}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - } \right.\\ \;\;\;\;\;\left. {{Z_{{\gamma _2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - \Delta } \right] - \beta \left( {{{\tilde B}_{{{\rm{D}}_{{1_{{\gamma _1}}}}}}}{{\tilde B}_{{{\rm{D}}_{{1_{{\gamma _1}}}}}}} + {{\tilde B}_{{{\rm{D}}_{{2_{{\gamma _1}}}}}}}{{\tilde B}_{{{\rm{D}}_{{2_{{\gamma _1}}}}}}}} \right) \cdot \\ \;\;\;\;\;\left[ {{Z_{{\gamma _1}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - {Z_{{\gamma _2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - \Delta } \right] \end{array}$ （68）
 $\begin{array}{l} {{\dot Z}_{{\gamma _2}}}(t) = - {\alpha _1}\left( {{{\tilde B}_{{{\rm{D}}_{1{q_1}}}}}{{\tilde B}_{{{\rm{D}}_1}_{{\gamma _2}}}} + {{\tilde B}_{{{\rm{D}}_{2{q_1}}}}}{{\tilde B}_{{{\rm{D}}_2}_{{\gamma _2}}}}} \right){Z_{{q_1}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - \\ \;\;\;\;\;{\alpha _2}\left( {{{\tilde B}_{{{\rm{D}}_{1{q_2}}}}}{{\tilde B}_{{{\rm{D}}_{{1_{{\gamma _2}}}}}}} + {{\tilde B}_{{{\rm{D}}_{2{q_2}}}}}{{\tilde B}_{{{\rm{D}}_2}_{{\gamma _2}}}}} \right){Z_{{q_2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) + \\ \;\;\;\;\;\beta \left( {{{\tilde B}_{{{\rm{D}}_{{1_{{\gamma _2}}}}}}}{{\tilde B}_{{{\rm{D}}_{{1_{{\gamma _2}}}}}}} + {{\tilde B}_{{{\rm{D}}_{{2_{{\gamma _2}}}}}}}{{\tilde B}_{{{\rm{D}}_{{2_{{\gamma _2}}}}}}}} \right)\left[ {{Z_{{\gamma _1}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - } \right.\\ \;\;\;\;\;\left. {{Z_{{\gamma _2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - \Delta } \right] - \beta \left( {{{\tilde B}_{{{\rm{D}}_{{1_{{\gamma _1}}}}}}}{{\tilde B}_{{{\rm{D}}_{{1_{{\gamma _2}}}}}}} + {{\tilde B}_{{{\rm{D}}_{{2_{{\gamma _1}}}}}}}{{\tilde B}_{{{\rm{D}}_{{2_{{\gamma _2}}}}}}}} \right) \cdot \\ \;\;\;\;\;\left[ {{Z_{{\gamma _1}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - {Z_{{\gamma _2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) - \Delta } \right] \end{array}$ （70）

 $\left[ {\begin{array}{*{20}{c}} {{Z_{{q_1}}}(t)}\\ {{Z_{{q_2}}}(t)}\\ {\Delta {Z_\gamma }(t)} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{l_{{q_1}1}}}&{{l_{{q_1}2}}}&{{l_{{q_1}3}}}\\ {{l_{{q_2}1}}}&{{l_{{q_2}2}}}&{{l_{{q_2}3}}}\\ {{l_{\gamma 1}}}&{{l_{\gamma 2}}}&{{l_{\gamma 3}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{Z_{{q_1}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right)}\\ {{Z_{{q_2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right)}\\ {\Delta {Z_\gamma }\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right)} \end{array}} \right]$ （71）

 $\begin{array}{l} {Z_{{q_1}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) = \frac{{{l_{{q_2}2}}{l_{\gamma 3}} - {l_{{q_2}3}}{l_{\gamma 2}}}}{N}{Z_{{q_1}}}(t) - \\ \;\;\;\;\;\frac{{{l_{{q_1}2}}{l_{\gamma 3}} - {l_{{q_1}3}}{l_{\gamma 2}}}}{N}{Z_{{q_2}}}(t) + \frac{{{l_{{q_1}2}}{l_{{q_2}3}} - {l_{{q_1}3}}{l_{{q_2}2}}}}{N}\Delta {Z_\gamma }(t) \end{array}$ （72）
 $\begin{array}{l} {Z_{{q_2}}}\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) = - \frac{{{l_{{q_2}1}}{l_{\gamma 3}} - {l_{{q_2}3}}{l_{\gamma 1}}}}{N}{Z_{{q_1}}}(t) + \\ \;\;\;\;\;\frac{{{l_{{q_1}1}}{l_{\gamma 3}} - {l_{{q_1}3}}{l_{\gamma 1}}}}{N}{Z_{{q_2}}}(t) - \frac{{{l_{{q_1}1}}{l_{{q_2}3}} - {l_{{q_1}3}}{l_{{q_2}1}}}}{N}\Delta {Z_\gamma }(t) \end{array}$ （73）
 $\begin{array}{l} \Delta {Z_\gamma }\left( {{t_{{{\rm{f}}_{\rm{n}}}}}} \right) = \frac{{{l_{{q_2}1}}{l_{\gamma 2}} - {l_{{q_2}2}}{l_{\gamma 1}}}}{N}{Z_{{q_1}}}(t) - \\ \;\;\;\;\frac{{{l_{{q_1}1}}{l_{\gamma 2}} - {l_{{q_1}2}}{l_{\gamma 1}}}}{N}{Z_{{q_2}}}(t) + \frac{{{l_{{q_1}1}}{l_{{q_2}2}} - {l_{{q_1}2}}{l_{{q_2}1}}}}{N}\Delta {Z_\gamma }(t) \end{array}$ （74）

 $\begin{array}{l} N = {l_{{q_1}1}}{l_{{q_2}2}}{l_{{\gamma _3}}} - {l_{{q_1}1}}{l_{{q_2}3}}{l_{\gamma 2}} - {l_{{q_1}2}}{l_{{q_2}1}}{l_{\gamma 3}} + \\ \;\;\;\;{l_{{q_1}2}}{l_{{q_2}3}}{l_{\gamma 1}} + {l_{{q_1}2}}{l_{{q_2}1}}{l_{\gamma 2}} - {l_{{q_1}3}}{l_{{q_2}2}}{l_{\gamma 1}}。\end{array}$

 ${u_i} = \frac{{{N_{i1}}}}{N}{Z_{{{\rm{q}}_{\rm{1}}}}}(t) + \frac{{{N_{i2}}}}{N}{Z_{{{\rm{q}}_{\rm{2}}}}}(t) + \frac{{{N_{i3}}}}{N}\Delta {Z_\gamma }(t)$ （75）

 $\left\{ \begin{array}{l} {N_{i1}} = - {\alpha _1}{{\tilde B}_{{i_{{q_1}}}}}\left( {{l_{{q_2}2}}{l_{\gamma 3}} - {l_{{q_2}3}}{l_{\gamma 2}}} \right) + {\alpha _2}{{\tilde B}_{{i_{{q_2}}}}}\left( {{l_{{q_2}1}}{l_{\gamma 3}} - } \right.\\ \;\;\;\;\left. {{l_{{q_2}3}}{l_{\gamma 1}}} \right) + \beta \left( {{{\tilde B}_{{i_{{\gamma _2}}}}} - {B_{{i_{{\gamma _1}}}}}} \right)\left( {{l_{{q_2}1}}{l_{\gamma 2}} - {l_{{q_2}2}}{l_{\gamma 1}}} \right)\\ {N_{i2}} = {\alpha _1}{{\tilde B}_{{i_{{q_1}}}}}\left( {{l_{{q_1}2}}{l_{\gamma 3}} - {l_{{q_1}3}}{l_{\gamma 2}}} \right) + {\alpha _2}{{\tilde B}_{{i_{{q_2}}}}}\left( {{l_{{q_1}1}}{l_{\gamma 3}} - } \right.\\ \;\;\;\;\left. {{l_{{q_1}3}}{l_{\gamma 1}}} \right) - \beta \left( {{{\tilde B}_{{i_{{\gamma _2}}}}} - {B_{{i_{{\gamma _1}}}}}} \right)\left( {{l_{{q_1}1}}{l_{{q_2}2}} - {l_{{q_1}2}}{l_{{q_2}1}}} \right)\\ {N_{i3}} = - {\alpha _1}{{\tilde B}_{{i_{{q_1}}}}}\left( {{l_{{q_1}2}}{l_{{q_2}3}} - {l_{{q_1}3}}{l_{{q_2}2}}} \right) + \\ \;\;\;\;{\alpha _2}{{\tilde B}_{{i_{{q_2}}}}}\left( {{l_{{q_1}1}}{l_{{q_2}3}} - {l_{{q_1}3}}{l_{{q_2}1}}} \right) + \beta \left( {{{\tilde B}_{{i_{{\gamma _2}}}}} - } \right.\\ \;\;\;\;\left. {{{\tilde B}_{{i_{{\gamma _1}}}}}} \right)\left( {{l_{{q_1}1}}{l_{{q_2}2}} - {l_{{q_1}2}}{l_{{q_2}1}}} \right) \end{array} \right.$
4 仿真分析

 参数 初始值 初始相对距离rPE0/m 16 000 初始相对距离rPD10/m, rPD20/m 16 000 突防器的速度vE 300 防御器的速度vD1/(m·s-1), vD2/(m·s-1) 500 拦截器的速度vP 500 防御器的过载限制uDmax/(m·s-2) 150 拦截器的过载限制aPmax/(m·s-2) 100 突防器的响应时间τE/s 0.1 防御器的响应时间τD1/s, τD2/s 0.1 拦截器的响应时间τP/s 0.1 导航参数N 3 修正系数K 3 初始状态变量x0 $\left[\begin{array}{llllllll} 0.05 & 0.03 & -0.03 & 0 & 0 & 0 & 0 & \frac{4 \pi}{180} & \frac{4 \pi}{180} \end{array}\right]^{\mathrm{T}}$ 相对拦截角Δ/(°) 30 权重系数α1, α2 10 000 000 000 权重系数β 30 000
4.1 显式的协同

 图 2 显式的协同：多导弹协同拦截交战 Fig. 2 Multi-missile cooperative interception engagement in explicit cooperation

 图 3 显式的协同：加速度的变化 Fig. 3 Variation of acceleration in explicit cooperation

 图 4 显示的协同：能量消耗的变化 Fig. 4 Variation of energy consumptions in explicit cooperation

 图 5 显式的协同：视线角速率的变化 Fig. 5 Variation of LOS angle rotation rates in explicit cooperation

 图 6 显示的协同：相对拦截角的变化 Fig. 6 Variation of relative interception angles in explicit cooperation

 图 7 显式的协同：视线角速率随防御器过载限制的变化 Fig. 7 Variation of LOS angle rotation rates with overload limit on defender in explicit cooperation
4.2 与隐式的协同进行比较

 图 8 隐式的协同：多导弹协同拦截交战 Fig. 8 Multi-missile cooperative interception engagement in implicit cooperation

 图 9 隐式的协同：加速度的变化 Fig. 9 Variation of accelerations in implicit cooperation

 图 10 隐式的协同：能量消耗的变化 Fig. 10 Variation of energy consumptions in implicit cooperation
 图 11 隐式的协同：视线角速率的变化 Fig. 11 Variation of LOS angle rotation rates in implicit cooperation

 图 12 隐式的协同：相对拦截角的变化 Fig. 12 Variation of relative interception angles in implicit cooperation
4.3 不同初始发射条件对协同制导律的影响

 情况 初值$\dot{q}_{\mathrm{PD}_{1}} /\left(\mathrm{rad} \cdot \mathrm{s}^{-1}\right)$ 初值$\dot{q}_{\mathrm{PD}_{2}} /\left(\mathrm{rad} \cdot \mathrm{s}^{-1}\right)$ 1 0.01 -0.01 2 0.02 -0.02 3 0.03 -0.03 4 0.04 -0.04 5 0.05 -0.05

 图 13 防御器1的视线角速率变化 Fig. 13 Variation of LOS angle rotation rates for Defender 1
 图 14 防御器2的视线角速率变化 Fig. 14 Variation of LOS angle rotation rates for Defender 1

 图 15 相对拦截角的变化 Fig. 15 Variation of relative interception angles

5 结论

1) 针对我方飞行器发射两枚防御器有效拦截对方拦截器的突防问题，提出了一种带引诱角色的显式协同制导律，并将其与隐式的协同制导律进行比较，仿真结果表明显式的协同优于隐式的协同。

2) 所设计的制导律将脱靶量、能量消耗和施加末端相对拦截角考虑在目标函数的建立中，利用最优控制理论求得了不同策略下的控制输入。

3) 本文没有考虑制导过程中的探测估计问题，探测效果影响着制导的效果和精度，未来有必要将探测角色加入到协同制导律的设计中，实现探测制导一体化设计。

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http://dx.doi.org/10.7527/S1000-6893.2019.23402

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#### 文章信息

WANG Shaobo, GUO Yang, WANG Shicheng, LIU Zhiguo, ZHANG Shuai

Cooperative optimal guidance method for multi-aircraft with luring role

Acta Aeronautica et Astronautica Sinica, 2020, 41(2): 323402.
http://dx.doi.org/10.7527/S1000-6893.2019.23402