﻿ 过虚拟交班点的能量最优制导律
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1. 北京理工大学 宇航学院, 北京 100081;
2. 北京理工大学 无人机自主控制技术北京市重点实验室, 北京 100081;
3. 北方华安工业集团, 齐齐哈尔 161006

Energy-optimal guidance law with virtual hand-over point
LI Chendi1,2, WANG Jiang1,2, LI Bin1,2, HE Shaoming1,2, ZHANG Tong3
1. School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China;
2. Beijing Key Laboratory of UAV Autonomous Control, Beijing Institute of Technology, Beijing 100081, China;
3. Hua An Industry Group Company Limited, Qiqihar 161006, China
Abstract: Aiming at the problem of fixed target missile guidance with virtual hand-over point, a global energy optimal guidance law with/without terminal angle constraint is designed based on the optimization theory in Hilbert space. By linearizing the model, the proposed optimal guidance model is transformed into a linear quadratic opti-mal control problem. Then the concept of Zero-Effort-Miss (ZEM) is used to reduce the order of the system, deriving the analytical solution. The proposed guidance law can ensure the missile pass through the virtual hand-over point accurately and reached the desired terminal angle. The simulation results show that, compared with the classical guidance law, the proposed guidance law can significantly reduce the energy consumption of global control.
Keywords: optimal guidance     virtual hand-over point     terminal angle constraint     path following     proportional navigation guidance     trajectory shaping guidance

1 运动模型建立 1.1 非线性运动模型

 图 1 弹目相对运动关系 Fig. 1 Missile-target engagement relationship

 $\left\{ {\begin{array}{*{20}{l}} {{{\dot r}_i} = - V\cos {\sigma _i}}\\ {{{\dot q}_i} = - \frac{{V\sin {\sigma _i}}}{{{r_i}}}\;\;\;\;i \in \left\{ {1,2} \right\}}\\ {\dot \theta = \frac{a}{V}} \end{array}} \right.$ （1）

 ${\sigma _i} = \theta - {q_i}$ （2）
1.2 模型线性化

 $\left\{ {\begin{array}{*{20}{l}} {{{\dot y}_i} = {v_i}}\\ {{{\dot v}_i} = - {a_ \bot }\cos \left( {{q_i} - {q_{\rm{R}}}} \right)} \end{array}\quad i \in \left\{ {1,2} \right\}} \right.$ （3）

 ${a_ \bot } = a\cos {\sigma _i}$ （4）

 $\left\{ {\begin{array}{*{20}{l}} {{{\dot y}_i} = {v_i}}\\ {{{\dot v}_i} = - a} \end{array}\;\;\;\;i \in \left\{ {1,2} \right\}} \right.$ （5）

 $\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot x}} = \mathit{\boldsymbol{Ax}} + \mathit{\boldsymbol{B}}a}\\ {\mathit{\boldsymbol{y}} = \mathit{\boldsymbol{Cx}}} \end{array}} \right.$ （6）

 $\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{A}}_1}}&{\bf{0}}\\ {\bf{0}}&{{\mathit{\boldsymbol{A}}_2}} \end{array}} \right],{\mathit{\boldsymbol{A}}_i} = \left[ {\begin{array}{*{20}{l}} 0&1\\ 0&0 \end{array}} \right]}\\ {\mathit{\boldsymbol{B}} = {{\left[ {0, - 1,0, - 1} \right]}^{\rm{T}}}}\\ {\mathit{\boldsymbol{C}} = \left[ {1,0,1,0} \right]} \end{array}} \right.$ （7）
2 最优控制问题描述

 $J = \left\{ {\begin{array}{*{20}{l}} {\int_t^{{t_{{\rm{f}},1}}} {{a^2}\left( \tau \right){\rm{d}}\tau } + \int_{{t_{{\rm{f}},1}}}^{{t_{{\rm{f}},2}}} {{a^2}\left( \tau \right){\rm{d}}\tau } \quad t \le {t_{{\rm{f,1}}}},}\\ {\int_t^{{t_{{\rm{f}},2}}} {{a^2}\left( \tau \right){\rm{d}}\tau } \quad {t_{{\rm{f}},1}} < t \le {t_{{\rm{f}},2}}} \end{array}} \right.$ （8）

 ${y_i}\left( {{t_{{\rm{f}},i}}} \right) = 0\;\;\;\;i \in \left\{ {1,2} \right\}$ （9）

 $\theta \left( {{t_{{\rm{f}},2}}} \right) = {\theta _{\rm{d}}}$ （10）

3 过定点的最优制导律设计

3.1 系统降阶

 ${Z_i} = \left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{C}}_i}{\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}_i}\left( {{t_{{{\rm{f}}},i}},t} \right)\mathit{\boldsymbol{x}}_i^{\rm{T}}}&{t \le {t_{{\rm{f}},i}}}\\ {{Z_i}\left( {{t_{{\rm{f}},i}}} \right)}&{t > {t_{{\rm{f}},i}}} \end{array}\;\;\;\;i \in \left\{ {1,2} \right\}} \right.$ （11）

 ${\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}_i}\left( {{t_{{{\rm{f}}},i}},t} \right) = \left[ {\begin{array}{*{20}{c}} 1&{{t_{{\rm{f}},i}} - t}\\ 0&1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&{{t_{{\rm{go}},i}}}\\ 0&1 \end{array}} \right]$ （12）

 ${Z_i} = \left\{ {\begin{array}{*{20}{l}} {{y_i} + {t_{{\rm{go}},i}}{v_i}}&{t \le {t_{{\rm{f}},i}}}\\ {{Z_i}\left( {{t_{{\rm{f}},i}}} \right)}&{t > {t_{{\rm{f}},i}}} \end{array}\;\;\;\;i \in \left\{ {1,2} \right\}} \right.$ （13）

 ${{\dot Z}_i} = \left\{ {\begin{array}{*{20}{l}} { - {t_{{\rm{go,}}i}}a}&{t \le {t_{{\rm{f}},i}}}\\ 0&{t > {t_{{\rm{f}},i}}} \end{array}\quad i \in \left\{ {1,2} \right\}} \right.$ （14）

3.2 最优制导律设计

 ${Z_i}\left( {{t_{{\rm{f}},i}}} \right) = 0\;\;\;\;i \in \left\{ {1,2} \right\}$ （15）

 ${Z_i}\left( {{t_{{\rm{f}},i}}} \right) = {Z_i}\left( t \right) = \int_t^{{t_{{\rm{f}},i}}} - \left( {{t_{{\rm{f,}}i}} - \tau } \right)a\left( \tau \right){\rm{d}}\tau \;\;\;t \le {t_{{\rm{f,}}i}}$ （16）

 ${Z_i}\left( t \right) = \int_t^{{t_{{\rm{f}},i}}} {\left( {{t_{{\rm{f,}}i}} - \tau } \right)a\left( \tau \right){\rm{d}}\tau } \;\;\;t \le {t_{{\rm{f,}}i}}$ （17）

 ${x_{\min }} = \sum\limits_{i = 1}^n {{b_i}{\alpha _i}}$

bi满足：

 $\left\{ {\begin{array}{*{20}{l}} {\left( {{\alpha _1}|{\alpha _1}} \right){b_1} + \left( {{\alpha _2}|{\alpha _1}} \right){b_2} + \cdots + \left( {{\alpha _n}|{\alpha _1}} \right){b_n} = {c_1}}\\ {\left( {{\alpha _1}|{\alpha _2}} \right){b_1} + \left( {{\alpha _2}|{\alpha _2}} \right){b_2} + \cdots + \left( {{\alpha _n}|{\alpha _2}} \right){b_n} = {c_2}}\\ {\;\; \vdots }\\ {\left( {{\alpha _1}|{\alpha _n}} \right){b_1} + \left( {{\alpha _2}|{\alpha _n}} \right){b_2} + \cdots + \left( {{\alpha _n}|{\alpha _n}} \right){b_n} = {c_n}} \end{array}} \right.$

 $a = \left\{ {\begin{array}{*{20}{l}} {{\lambda _1}\left( {{t_{{\rm{f}},1}} - t} \right) + {\lambda _2}\left( {{t_{{\rm{f}},2}} - t} \right)}&{t \le {t_{{\rm{f}},1}}}\\ {{\lambda _2}\left( {{t_{{\rm{f}},2}} - t} \right)}&{{t_{{\rm{f}},1}} < t \le {t_{{\rm{f}},2}}} \end{array}} \right.$ （18）

ttf, 1为例，求解拉格朗日乘子的表达式。将式(18)代入式(17)，可得

 $\begin{array}{l} {Z_1}\left( t \right) = {\lambda _1}\int_t^{{t_{1,1}}} {\left( {{t_{{\rm{f}},1}} - \tau } \right)} \left( {{t_{{\rm{f}},1}} - \tau } \right){\rm{d}}\tau + \\ \;\;\;\;\;\;\;\;\;\;\;{\lambda _2}\int_t^{{t_{{\rm{f}},1}}} {\left( {{t_{{\rm{f}},i}} - \tau } \right)} \left( {{t_{{\rm{f}},2}} - \tau } \right){\rm{d}}\tau = \\ \;\;\;\;\;\;\;\;\;\;\;{\lambda _1}\int_t^{{t_{{\rm{f}},1}}} {{{\left( {{t_{{\rm{f}},1}} - \tau } \right)}^2}} {\rm{d}}\tau + \\ \;\;\;\;\;\;\;\;\;\;\;{\lambda _2}\int_t^{{t_{{\rm{f}},1}}} {\left[ {\left( {{t_{{\rm{f}},2}} - {t_{{\rm{f}},1}}} \right)\left( {{t_{{\rm{f}},1}} - \tau } \right) + } \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\left. {{{\left( {{t_{{\rm{f}},1}} - \tau } \right)}^2}} \right]{\rm{d}}\tau = {\lambda _1}\frac{1}{3}t_{{\rm{go}},1}^3 + {\lambda _2}\left[ {\frac{1}{2}\left( {{t_{{\rm{f}},2}} - } \right.} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\left. {\left. {{t_{{\rm{f}},1}}} \right)t_{{\rm{go}},1}^2 + \frac{1}{3}t_{{\rm{go}},1}^3} \right] = {\lambda _1}\frac{1}{3}t_{{\rm{go}},1}^3 + \\ \;\;\;\;\;\;\;\;\;\;\;{\lambda _2}\left[ {\frac{1}{2}\left( {{t_{{\rm{go}},2}} - {t_{{\rm{go}},1}}} \right)t_{{\rm{go}},1}^2 + \frac{1}{3}t_{{\rm{go}},1}^3} \right] = \\ \;\;\;\;\;\;\;\;\;\;\;{\lambda _1}\frac{1}{3}t_{{\rm{go}},1}^3 + {\lambda _2}\left( {\frac{1}{2}{t_{{\rm{go}},2}}t_{{\rm{go}},1}^2 - \frac{1}{6}t_{{\rm{go}},1}^3} \right) \end{array}$ （19）

 ${Z_2}\left( t \right) = {\lambda _2}\frac{1}{3}t_{{\rm{go}},2}^3$ （20）

 $\mathit{\boldsymbol{Z}} = \mathit{\boldsymbol{G\lambda }}$ （21）

 $\mathit{\boldsymbol{G}} = \left[ {\begin{array}{*{20}{c}} {\frac{1}{3}t_{{\rm{go}},1}^3}&{\frac{1}{2}{t_{{\rm{go}},2}}t_{{\rm{go}},1}^2 - \frac{1}{6}t_{{\rm{go}},1}^3}\\ {\frac{1}{2}{t_{{\rm{go}},2}}t_{{\rm{go}},1}^2 - \frac{1}{6}t_{{\rm{go}},1}^3}&{\frac{1}{3}t_{{\rm{go}},2}^3} \end{array}} \right]$ （22）

 $\mathit{\boldsymbol{\lambda }} = {\mathit{\boldsymbol{G}}^{ - 1}}\mathit{\boldsymbol{Z}}$ （23）

$t_{\mathrm{f}, 1} <t \leqslant t_{\mathrm{f}, 2}$时，Z1=0，ZEM向量降为一维，即Z=[Z2]，此时$\boldsymbol{G}=\left[\frac{1}{3} t_{\mathrm{go}, 2}^{3}\right]$

 $a = \left\{ {\begin{array}{*{20}{l}} {{{\left( {{\mathit{\boldsymbol{G}}^{ - 1}}\mathit{\boldsymbol{Z}}} \right)}^{\rm{T}}}{{\left[ {{t_{{\rm{go}},1}},{t_{{\rm{go}},2}}} \right]}^{\rm{T}}}}&{t \le {t_{{\rm{f}}.{\rm{i}}}}}\\ {\frac{{3{Z_2}}}{{t_{{\rm{go}},2}^2}}}&{{t_{{\rm{i}},1}} < t \le {t_{{\rm{f}},2}}} \end{array}} \right.$ （24）
3.3 讨论与分析

 ${q_i} - {q_{\rm{R}}} = \frac{{{y_i}}}{{{r_i}}}$ （25）

 ${{\dot q}_i} = \frac{{{v_i}{r_i} - {y_i}{{\dot r}_i}}}{{r_i^2}} = \frac{{{v_i}{t_{{\rm{go}},i}} + {y_i}}}{{r_i^2/V}} = \frac{{{Z_i}}}{{Vt_{{\rm{go}},i}^2}}$ （26）

 ${Z_i} = Vt_{{\rm{go}},i}^2{{\dot q}_i}$ （27）

 $a = 3V{{\dot q}_2}\quad {t_{{\rm{f}},1}} < t \le {t_{{\rm{f}},2}}$ （28）

 $\begin{array}{l} a = \frac{{6\left( {2t_{{\rm{go}},2}^2 - {t_{{\rm{go}},1}}{t_{{\rm{go}},2}}} \right)}}{{t_{{\rm{go}},1}^2\left( {{t_{{\rm{go}},2}} - {t_{{\rm{go}},1}}} \right)\left( {4{t_{{\rm{go}},2}} - {t_{{\rm{go}},1}}} \right)}}{Z_1} - \\ \;\;\;\;\;\frac{{ - 6}}{{\left( {{t_{{\rm{go}},2}} - {t_{{\rm{go}},1}}} \right)\left( {4{t_{{\rm{go}},2}} - {t_{{\rm{go}},1}}} \right)}}{Z_2} \end{array}$ （29）

 $a = {N_1}V{{\dot q}_1} - {N_2}V{{\dot q}_2}$ （30）

 $\left\{ \begin{array}{l} {N_1} = \frac{{6{t_{{\rm{go}},2}}\left( {2{t_{{\rm{go}},2}} - {t_{{\rm{go}},1}}} \right)}}{{\left( {{t_{{\rm{go}},2}} - {t_{{\rm{go}},1}}} \right)\left( {4{t_{{\rm{go}},2}} - {t_{{\rm{go}},1}}} \right)}}\\ {N_2} = \frac{{6t_{{\rm{go}},2}^2}}{{\left( {{t_{{\rm{go}},2}} - {t_{{\rm{go}},1}}} \right)\left( {4{t_{{\rm{go}},2}} - {t_{{\rm{go}},1}}} \right)}} \end{array} \right.$ （31）

 ${{\dot q}_1} = {{\dot q}_0}{\left( {\frac{{{t_{{\rm{go}},1}}}}{{{t_{{\rm{f}},1}}}}} \right)^{{N_1} - 2}} + \frac{{{N_2}{{\dot q}_2}}}{{{N_1} - 2}}\left[ {1 - {{\left( {\frac{{{t_{{\rm{go}},1}}}}{{{t_{{\rm{f}},1}}}}} \right)}^{{N_1} - 2}}} \right]$ （32）

 $\mathop {\lim }\limits_{{t_{{\rm{go}},1}} \to {0^ + }} {{\dot q}_1} = \frac{{{N_2}}}{{{N_1} - 2}}{{\dot q}_2}$ （33）

 $\begin{array}{l} \mathop {\lim }\limits_{{t_{{\rm{go}},1}} \to {0^ + }} a = \frac{{{N_1}{N_2}}}{{{N_1} - 2}}V{{\dot q}_2} - {N_2}V{{\dot q}_2} = \frac{{2{N_2}}}{{{N_1} - 2}}V{{\dot q}_2} = \\ \;\;\;\;\;\;\;\;\;\;\;\frac{{6t_{{\rm{go}},2}^2}}{{2t_{{\rm{go}},2}^2 - t_{{\rm{go}},1}^2 + 2{t_{{\rm{go}},1}}{t_{{\rm{go}},2}}}}V{{\dot q}_2} \end{array}$ （34）

 $\mathop {\lim }\limits_{{t_{{\rm{go}},1}} \to {0^ + }} \frac{{6t_{{\rm{go}},2}^2}}{{2t_{{\rm{go}},2}^2 - t_{{\rm{go}},1}^2 + 2{t_{{\rm{go}},1}}{t_{{\rm{go}},2}}}} = 3$ （35）

 $\mathop {\lim }\limits_{{t_{{\rm{go}},1}} \to {0^ + }} a = 3V{{\dot q}_2}$ （36）

4 考虑终端落角约束的过定点最优制导律设计

4.1 最优制导律设计

 $\theta \left( {{t_{{\rm{f}},2}}} \right) = {\theta _d}$ （37）

 $\left\{ {\begin{array}{*{20}{l}} {{Z_i}\left( t \right) = \int_t^{{t_{{\rm{f}},i}}} {\left( {{t_{{\rm{f}},i}} - \tau } \right)a\left( \tau \right){\rm{d}}\tau } }&{t \le {t_{{\rm{f}},i}}}\\ {{\theta _{\rm{d}}} - \theta \left( t \right) = \int_t^{{t_{{\rm{f}},2}}} {\frac{{a\left( \tau \right)}}{V}{\rm{d}}\tau } }&{t \le {t_{{\rm{f}},2}}} \end{array}} \right.$ （38）

 $a = {a_Z} + {a_\theta }$ （39）

 $\left\{ \begin{array}{l} {a_Z} = \left\{ {\begin{array}{*{20}{l}} {{\lambda _1}\left( {{t_{{\rm{f}},1}} - t} \right) + {\lambda _2}\left( {{t_{{\rm{f}},2}} - t} \right)}&{t \le {t_{{\rm{f}},1}}}\\ {{\lambda _2}\left( {{t_{{\rm{f}},2}} - t} \right)}&{{t_{{\rm{f}},1}} < t \le {t_{\rm{f}}}} \end{array}} \right.\\ {a_\theta } = {\lambda _3}/V\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;t \le {t_{{\rm{f}},2}} \end{array} \right.$ （40）

 $\left\{ \begin{array}{l} {Z_1}(t) = {\lambda _1}\int_t^{{t_{{\rm{f}},1}}} {\left( {{t_{{\rm{f}},1}} - \tau } \right)} \left( {{t_{{\rm{f}},1}} - \tau } \right){\rm{d}}\tau + \\ \;\;\;\;\;\;\;\;\;\;{\lambda _2}\int_t^{{t_{{\rm{f}},1}}} {\left( {{t_{{\rm{f}},1}} - \tau } \right)} \left( {{t_{{\rm{f}},2}} - \tau } \right){\rm{d}}\tau + \\ \;\;\;\;\;\;\;\;\;\frac{{{\lambda _3}}}{V}\int_t^{{t_{{\rm{f}},1}}} {\left( {{t_{{\rm{f}},1}} - \tau } \right){\rm{d}}\tau } \\ {Z_2}(t) = {\lambda _1}\int_t^{{t_{{\rm{f}},1}}} {\left( {{t_{{\rm{f}},2}} - \tau } \right)} \left( {{t_{{\rm{f}},1}} - \tau } \right){\rm{d}}\tau + \\ \;\;\;\;\;\;\;\;\;\;\;{\lambda _2}\int_t^{{t_{{\rm{f,2}}}}} {\left( {{t_{{\rm{f,2}}}} - \tau } \right)} \left( {{t_{{\rm{f,2}}}} - \tau } \right){\rm{d}}\tau + \\ \;\;\;\;\;\;\;\;\;\;\frac{{{\lambda _3}}}{V}\int_t^{{t_{{\rm{f}},2}}} {\left( {{t_{{\rm{f}},2}} - \tau } \right){\rm{d}}\tau } \\ {\theta _{\rm{d}}} - \theta (t) = \frac{{{\lambda _1}}}{V}\int_t^{{t_{{\rm{f}},1}}} {\left( {{t_{{\rm{f}},1}} - \tau } \right){\rm{d}}\tau } + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{{\lambda _2}}}{V}\int_t^{{t_{{\rm{f}},2}}} {\left( {{t_{{\rm{f}},2}} - \tau } \right){\rm{d}}\tau } + \frac{{{\lambda _3}}}{{{V^2}}}\int_t^{{t_{{\rm{f}},2}}} {\rm{d}} \tau \end{array} \right.$ （41）

 $\left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{Z}}\\ {{\theta _{\rm{d}}} - \theta } \end{array}} \right] = \mathit{\boldsymbol{G\lambda }},\mathit{\boldsymbol{G}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{G}}_1}}&{{\mathit{\boldsymbol{G}}_{12}}}\\ {{\mathit{\boldsymbol{G}}_{21}}}&{{G_2}} \end{array}} \right]$ （42）

 ${\mathit{\boldsymbol{G}}_1} = \left[ {\begin{array}{*{20}{c}} {\frac{1}{3}t_{{\rm{go}},1}^3}&{\frac{1}{2}{t_{{\rm{go}},2}}t_{{\rm{go}},1}^2 - \frac{1}{6}t_{{\rm{go,1}}}^3}\\ {\frac{1}{2}{t_{{\rm{go}},2}}t_{{\rm{go}},1}^2 - \frac{1}{6}t_{{\rm{go}},1}^3}&{\frac{1}{3}t_{{\rm{go}},2}^3} \end{array}} \right]$ （43）

 $\left\{ {\begin{array}{*{20}{l}} {\int_t^{{t_{{\rm{f}},i}}} {\left( {{t_{{\rm{f}},i}} - \tau } \right)} {\rm{d}}\tau = - \left. {\frac{1}{2}{{\left( {{t_{{\rm{f}},i}} - \tau } \right)}^2}} \right|_t^{{t_{{\rm{f}},i}}} = \frac{{t_{{\rm{go}},i}^2}}{2}}\\ {\int_t^{{t_{{\rm{f}},2}}} {\rm{d}} \tau = {t_{{\rm{go}},2}}} \end{array}} \right.$ （44）

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{G}}_{12}} = \mathit{\boldsymbol{G}}_{21}^{\rm{T}} = {{\left[ {\frac{{t_{{\rm{go}},1}^2}}{{2V}},\frac{{t_{{\rm{go}},2}^2}}{{2V}}} \right]}^{\rm{T}}}}\\ {{G_2} = \frac{{{t_{{\rm{go}}.2}}}}{{{V^2}}}} \end{array}} \right.$ （45）

 $\mathit{\boldsymbol{\lambda }} = {\mathit{\boldsymbol{G}}^{ - 1}}\left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{Z}}\\ {{\theta _{\rm{d}}} - \theta } \end{array}} \right]$ （46）

 $\begin{array}{l} a = {\mathit{\boldsymbol{\lambda }}^{\rm{T}}}{\left[ {{t_{{\rm{go}},1}},{t_{{\rm{go}},2}},\frac{1}{V}} \right]^{\rm{T}}} = \\ \;\;\;\;\;{\left( {{\mathit{\boldsymbol{G}}^{ - 1}}\left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{Z}}\\ {{\theta _{\rm{d}}} - \theta } \end{array}} \right]} \right)^{\rm{T}}}{\left[ {{t_{{\rm{go}},1}},{t_{{\rm{go,}}2}},\frac{1}{V}} \right]^{\rm{T}}} \end{array}$ （47）

tf, 1 < ttf, 2时，Z1=0，ZEM向量降为一维，即Z=[Z2]，此时

 $\mathit{\boldsymbol{G}} = \left[ {\begin{array}{*{20}{c}} {\frac{{t_{{\rm{go}},2}^3}}{3}}&{\frac{{t_{{\rm{go}},2}^2}}{{2V}}}\\ {\frac{{t_{{\rm{go}},2}^2}}{{2V}}}&{\frac{{{t_{{\rm{go}},2}}}}{{{V^2}}}} \end{array}} \right]$ （48）

 $\begin{array}{l} a = {\mathit{\boldsymbol{\lambda }}^{\rm{T}}}{\left[ {{t_{{\rm{go}},2}},\frac{1}{V}} \right]^{\rm{T}}} = \\ \;\;\;\;\;\;{\left( {{\mathit{\boldsymbol{G}}^{ - 1}}\left[ {\begin{array}{*{20}{c}} {{Z_2}}\\ {{\theta _{\rm{d}}} - \theta } \end{array}} \right]} \right)^{\rm{T}}}{\left[ {{t_{{\rm{go}},2}},\frac{1}{V}} \right]^{\rm{T}}} \end{array}$ （49）
4.2 讨论与分析

 $a = {\lambda _2}{t_{{\rm{go,}}2}} + \frac{{{\lambda _3}}}{V} = \frac{{6{Z_2}}}{{t_{{\rm{go,}}2}^2}} - \frac{{2V}}{{{t_{{\rm{go,}}2}}}}\left( {{\theta _{\rm{d}}} - \theta } \right)$ （50）

 ${\theta _{\rm{d}}} - \theta = \frac{1}{V}\int_t^{{t_{\rm{f}}}} {a{\rm{d}}\tau }$ （51）

 $\begin{array}{l} {\theta _{\rm{d}}} - \theta = \frac{1}{V}\int_t^{{t_{\rm{f}}}} {a{\rm{d}}\tau } = \frac{1}{V}\int_t^{{t_{\rm{f}}}} - {{\dot v}_2}{\rm{d}}\tau = \\ \;\;\;\;\;\;\;\;\frac{1}{V}\left( {{v_2} - {v_{2{\rm{f}}}}} \right) = \frac{1}{V}\left( {{{\dot y}_2} - {{\dot y}_{2{\rm{f}}}}} \right) \end{array}$ （52）

 ${Z_2} = {y_2} + {t_{{\rm{go,}}2}}{v_2} = {y_2} + {{\dot y}_2}{t_{{\rm{go}},2}}$ （53）

 $\begin{array}{l} a = \frac{6}{{t_{{\rm{go}},2}^2}}\left( {{y_2} + {{\dot y}_2}{t_{{\rm{go}},2}}} \right) - \frac{{2V}}{{{t_{{\rm{go}},2}}}} \cdot \frac{1}{V}\left( {{{\dot y}_2} - {{\dot y}_{2{\rm{f}}}}} \right) = \\ \;\;\;\;\frac{{6{y_2} + 4{{\dot y}_2}{t_{{\rm{go}},2}} + 2{{\dot y}_{{\rm{2f}}}}{t_{{\rm{go}},2}}}}{{t_{{\rm{go}},2}^2}} \end{array}$ （54）

5 仿真验证

 参数 数值 导弹初始坐标/m (0，0) 启控时间/s 40 虚拟交班点/m (22000，5000) 目标点/m (25000，0) 速度/(m·s-1) 200 初始弹道倾角/(°) 40
5.1 无落角约束的最优制导律仿真

 图 2 无落角约束情况下的仿真结果对比 Fig. 2 Comparison of simulation results without angle constraint

5.2 考虑终端落角约束的最优制导律仿真

 图 3 考虑终端落角约束情况下的仿真结果对比 Fig. 3 Comparison of simulation results with angle constraint

6 结论

1) 本文在希尔伯特空间基于最优化理论，提出了一种针对固定目标，过虚拟交班点的全局最优制导律。在无落角约束下，能准确经过虚拟交班点，到达固定目标点，且全局所需控制能量比最优比例导引制导律减少了49%。

2) 在最优制导律的基础上考虑终端落角约束，推导出含终端落角约束的全局最优制导律，该制导律可以实现期望落角，且全局所需控制能量比PNG+TSG下减少了22%。

3) 该制导律形式简单，计算制导指令所需的信息较少，具有良好的工程应用价值。

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http://dx.doi.org/10.7527/S1000-6893.2019.23249

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#### 文章信息

LI Chendi, WANG Jiang, LI Bin, HE Shaoming, ZHANG Tong

Energy-optimal guidance law with virtual hand-over point

Acta Aeronautica et Astronautica Sinica, 2019, 40(12): 323249.
http://dx.doi.org/10.7527/S1000-6893.2019.23249