﻿ 三维自适应终端滑模协同制导律
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Three dimensional guidance law for cooperative operation based on adaptive terminal sliding mode
SI Yujie, XIONG Hua, SONG Xun, ZONG Rui
Beijing Institute of Electronic System Engineering, Beijing 100854, China
Abstract: Aiming at the problem of multi-missile cooperative operation and the directed topological communication between multi-missiles, a dual-layer cooperative guidance law based on the terminal sliding mode method is designed. The guidance command of line-of-sight direction can guarantee multiple-missile attacking the target at the same time. The three-dimensional guidance law normal to line-of-sight can ensure that each missile attacks the target at the desired line-of-sight angles, so as to exert the maximum lethality of each missile. Moreover, the constraint of line-of-sight angles is equivalent to the trajectory planning for the ballistic problem of the missile, which can avoid the collision between the missiles before attacking the target to some extent. At the same time, new adaptive laws are designed for the designed sliding mode guidance laws, accelerating the convergence speed of the sliding mode surface and attenuating the chattering phenomenon caused by the symbolic function. Based on the Lyapunov stability theory, the correctness of the guidance law is proved. Finally, a mathematical simulation experiment is conducted to verify the effectiveness and the superiority of the proposed guidance law.
Keywords: cooperative operation    terminal sliding mode    adaptive law    three-dimensional guidance    mathematical simulation

1 模型建立

 ${\ddot R - R\dot \theta _{\rm{L}}^2 - R\dot \phi _{\rm{L}}^2{\rm{co}}{{\rm{s}}^2}{\theta _{\rm{L}}} = {a_{{\rm{LT}}}} - {a_{{\rm{LM}}}}}$ （1）
 ${R{{\ddot \theta }_{\rm{L}}} + 2\dot R{{\dot \theta }_{\rm{L}}} + R\dot \phi _{\rm{L}}^2{\rm{sin}}{\theta _{\rm{L}}}{\rm{cos}}{\theta _{\rm{L}}} = {a_{{\rm{ZT}}}} - {a_{{\rm{ZM}}}}}$ （2）
 $\begin{array}{l} - R{{\ddot \phi }_{\rm{L}}}{\rm{cos}}{\theta _{\rm{L}}} - 2\dot R{{\dot \phi }_{\rm{L}}}{\rm{cos}}{\theta _{\rm{L}}} + 2R{{\dot \phi }_{\rm{L}}}{{\dot \theta }_{\rm{L}}}{\rm{sin}}{\theta _{\rm{L}}} = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {a_{{\rm{YT}}}} - {a_{{\rm{YM}}}} \end{array}$ （3）
 图 1 三维空间几何图 Fig. 1 Geometry in a three-dimensional space

 ${\ddot R = R\dot \theta _{\rm{L}}^2 + R\dot \phi _{\rm{L}}^2{\rm{co}}{{\rm{s}}^2}{\theta _{\rm{L}}} - {a_{{\rm{LM}}}} + {a_{{\rm{LT}}}}}$ （4）
 ${\left[ {\begin{array}{*{20}{l}} {{{\ddot \theta }_{\rm{L}}}}\\ {{{\ddot \phi }_{\rm{L}}}} \end{array}} \right] = \mathit{\boldsymbol{M}} + \mathit{\boldsymbol{D}} + \mathit{\boldsymbol{Bu}}}$ （5）

2) 视线跟踪角误差θLi-θLfiϕLi-ϕLfi在有限时间内收敛到区域θLi-θLfiΩ1和|ϕLi-ϕLfi|≤Ω1，这里Ω1是充分小的正的常数。

 ${\varOmega _1} = {\rm{max}}\left\{ {{\eta _i},{\rm{min}}\left\{ {{{\left( {\frac{\varOmega }{{{\varphi _i}}}} \right)}^{\frac{1}{2}}},{{\left( {\frac{\varOmega }{{{\lambda _i}}}} \right)}^{\frac{1}{2}}}} \right\}} \right\}$ （48）

3) 视线角速率$\dot{\theta}_{\mathrm{L} i}$$\dot{\phi}_{\mathrm{L}i}在有限时间内收敛到区域\left|\dot{\theta}_{\mathrm{L}{i}}\right|<\varOmega_{2}$$\left|\dot{\phi}_{\mathrm{L}{i}}\right|<\varOmega_{2}$

 ${\varOmega _2} = {\varphi _i}{\varOmega _1} + {\lambda _i}\varOmega _1^r + \varOmega$ （49）

 ${V_3} = \frac{1}{2}{\mathit{\boldsymbol{\sigma }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }} + \frac{1}{2}{({\mathit{\boldsymbol{\varepsilon }}_i} - {\mathit{\boldsymbol{y}}_i})^{\rm{T}}}({\mathit{\boldsymbol{\varepsilon }}_i} - {\mathit{\boldsymbol{y}}_i})$ （50）

V3进行求导可得：

 $\begin{array}{l} \begin{array}{*{20}{l}} {{{\dot V}_3} = {\mathit{\boldsymbol{\sigma }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }} - {{({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i})}^{\rm{T}}}{{\mathit{\boldsymbol{\dot y}}}_i} = {\mathit{\boldsymbol{\sigma }}^{\rm{T}}}[{\mathit{\boldsymbol{D}}_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} {k_i} {\rm{sig}} {{(\mathit{\boldsymbol{\sigma }})}^{{a_i}}} - {\mathit{\boldsymbol{Q}}_i}({\alpha _i}{\mathit{\boldsymbol{y}}_i} + {\mathit{\boldsymbol{M}}_i}(\mathit{\boldsymbol{\sigma }}))] - }\\ {{\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} {\alpha _i}{{({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i})}^{\rm{T}}}\left( {\frac{1}{{{\alpha _i}}}\left\| \mathit{\boldsymbol{\sigma }} \right\| - {c_i}{\mathit{\boldsymbol{y}}_i}} \right) \le } \end{array}\\ \begin{array}{*{20}{l}} {{\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} - {k_i}{\mathit{\boldsymbol{\sigma }}^{\rm{T}}} {\rm{sig}} {{(\mathit{\boldsymbol{\sigma }})}^a} + \sum\limits_{j = 1}^2 | {s_{ji}}|{\varepsilon _{ji}} - {\mathit{\boldsymbol{\sigma }}^{\rm{T}}}{\mathit{\boldsymbol{Q}}_i}(\alpha {\mathit{\boldsymbol{y}}_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} {\mathit{\boldsymbol{M}}_i}(\mathit{\boldsymbol{\sigma }})) - {\alpha _i}{{({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i})}^{\rm{T}}}(\frac{1}{{{\alpha _i}}}\left\| \mathit{\boldsymbol{\sigma }} \right\| - {c_i}{\mathit{\boldsymbol{y}}_i}) \le } \end{array}\\ {\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} \begin{array}{*{20}{l}} {{\kern 1pt} - {k_i}{\mathit{\boldsymbol{\sigma }}^{\rm{T}}} {\rm{sig}} {{(\mathit{\boldsymbol{\sigma }})}^{{a_i}}} - {\mathit{\boldsymbol{\sigma }}^{\rm{T}}}{\mathit{\boldsymbol{Q}}_i}{\mathit{\boldsymbol{M}}_i}(\mathit{\boldsymbol{\sigma }}) + {\alpha _i}{{({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i})}^{\rm{T}}} \cdot }\\ \begin{array}{l} {c_i}{\mathit{\boldsymbol{y}}_i} \le - {k_i}{\mathit{\boldsymbol{\sigma }}^{\rm{T}}} {\rm{sig}} {(\mathit{\boldsymbol{\sigma }})^{{a_i}}} - {\rm{min}}({\gamma _{0i}}({{\rm{e}}^{{\gamma _{1i}}|{s_{1i}}{|^{{p_i}}}}} - \\ \begin{array}{*{20}{l}} {{\gamma _{2i}}),{\gamma _{00i}}({{\rm{e}}^{{\gamma _{1i}}|{s_{2i}}{|^{{p_i}}}}} - {\gamma _{2i}}))\left\| \mathit{\boldsymbol{\sigma }} \right\| + {\alpha _i}({\mathit{\boldsymbol{\varepsilon }}_i} - }\\ {{\alpha _i}{\mathit{\boldsymbol{y}}_i}{)^{\rm{T}}}{c_i}{\mathit{\boldsymbol{y}}_i} \le - {2^{(1 + {a_i})/2}}{k_i}{{\left( {\frac{1}{2}{\mathit{\boldsymbol{\sigma }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }}} \right)}^{(1 + {a_i})/2}}} \end{array}\\ {c_i}{\alpha _i}{({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i})^{\rm{T}}}{\mathit{\boldsymbol{y}}_i} \end{array} \end{array} \end{array}$ （51）

 $\begin{array}{l} {\alpha _i}{({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i})^{\rm{T}}}{\mathit{\boldsymbol{y}}_i} = - {({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i})^{\rm{T}}}({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i} - {\mathit{\boldsymbol{\varepsilon }}_i}) \le \\ - \frac{{2{\delta _1} - 1}}{{2{\delta _1}}}{({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i})^{\rm{T}}}({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i}) + \frac{{{\delta _1}}}{2}\mathit{\boldsymbol{\varepsilon }}_i^{\rm{T}}{\mathit{\boldsymbol{\varepsilon }}_i} \end{array}$ （52）

 $\begin{array}{l} {{\dot V}_3} \le - {2^{(1 + {a_i})/2}}{k_i}{\left( {\frac{1}{2}{\mathit{\boldsymbol{\sigma }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }}} \right)^{(1 + {a_i})/2}} - {c_i}\frac{{2{\delta _1} - 1}}{{2{\delta _1}}} \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {{{({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i})}^{\rm{T}}}({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i}) + \frac{{{\delta _1}{c_i}}}{2}\mathit{\boldsymbol{\varepsilon }}_i^{\rm{T}}{\mathit{\boldsymbol{\varepsilon }}_i} = }\\ \begin{array}{l} - {2^{(1 + {a_i})/2}}{k_i}{\left( {\frac{1}{2}{\mathit{\boldsymbol{\sigma }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }}} \right)^{(1 + {a_i})/2}} - {c_i}\frac{{2{\delta _1} - 1}}{{{\delta _1}}} \cdot \\ \begin{array}{*{20}{l}} {{{\left[ {\frac{1}{2}{{({\mathit{\boldsymbol{\varepsilon }}_i} - {\mathit{\boldsymbol{y}}_i})}^{\rm{T}}}({\mathit{\boldsymbol{\varepsilon }}_i} - {\mathit{\boldsymbol{y}}_i})} \right]}^{(1 + {a_i})/2}} + {c_i}\frac{{2{\delta _1} - 1}}{{{\delta _1}}} \cdot }\\ \begin{array}{l} \left\{ {{{\left[ {\frac{1}{2}{{({\mathit{\boldsymbol{\varepsilon }}_i} - {\mathit{\boldsymbol{y}}_i})}^{\rm{T}}}({\mathit{\boldsymbol{\varepsilon }}_i} - {\mathit{\boldsymbol{y}}_i})} \right]}^{(1 + {a_i})/2}} - } \right.\\ \left. {\left[ {\frac{1}{2}{{({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i})}^{\rm{T}}}({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i})} \right]} \right\} + \frac{{{\delta _1}{c_i}}}{2}\mathit{\boldsymbol{\varepsilon }}_i^{\rm{T}}{\mathit{\boldsymbol{\varepsilon }}_i} \end{array} \end{array} \end{array} \end{array} \end{array}$ （53）

 $\begin{array}{l} {\left[ {\frac{1}{2}{{({\mathit{\boldsymbol{\varepsilon }}_i} - {\mathit{\boldsymbol{y}}_i})}^{\rm{T}}}({\mathit{\boldsymbol{\varepsilon }}_i} - {\mathit{\boldsymbol{y}}_i})} \right]^{(1 + {a_i})/2}} - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left[ {\frac{1}{2}{{({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i})}^{\rm{T}}}({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i})} \right] \le 0 \end{array}$ （54）

 ${\left[ {\frac{1}{2}{{({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i})}^{\rm{T}}}({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i})} \right] \le {\delta _2}}$ （55）

 $\begin{array}{l} {\left[ {\frac{1}{2}{{({\mathit{\boldsymbol{\varepsilon }}_i} - {\mathit{\boldsymbol{y}}_i})}^{\rm{T}}}({\mathit{\boldsymbol{\varepsilon }}_i} - {\mathit{\boldsymbol{y}}_i})} \right]^{(1 + {a_i})/2}} - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left[ {\frac{1}{2}{{({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i})}^{\rm{T}}}({\mathit{\boldsymbol{\varepsilon }}_i} - {\alpha _i}{\mathit{\boldsymbol{y}}_i})} \right] \le {\delta _2} \end{array}$ （56）

 $\begin{array}{*{20}{l}} {{{\dot V}_3} \le - {\delta _3}{{\left( {\frac{1}{2}\mathit{\boldsymbol{\sigma }}_i^{\rm{T}}{\mathit{\boldsymbol{\sigma }}_i}} \right)}^{(1 + {a_i})/2}} - {\delta _3} \cdot }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\left[ {\frac{1}{2}{{({\mathit{\boldsymbol{\varepsilon }}_i} - {\mathit{\boldsymbol{y}}_i})}^{\rm{T}}}({\mathit{\boldsymbol{\varepsilon }}_i} - {\mathit{\boldsymbol{y}}_i})} \right]}^{(1 + {a_i})/2}} + {c_i}\frac{{2{\delta _1} - 1}}{{{\delta _1}}}{\delta _2} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{{\delta _1}{c_i}}}{2}\mathit{\boldsymbol{\varepsilon }}_i^{\rm{T}}{\mathit{\boldsymbol{\varepsilon }}_i} \le - {\delta _3}V_3^{(1 + {a_i})/2} + {\delta _4}} \end{array}$ （57）

 ${T \le \frac{{2V_{3,t = 0}^{(1 - {a_j})/2}}}{{{\delta _3}{\theta _1}(1 - {a_i})}}}$
 ${{D_S} = \left\{ {\mathit{\boldsymbol{\sigma }}:\left\| \mathit{\boldsymbol{\sigma }} \right\| \le \sqrt 2 {{\left[ {\frac{{{\delta _4}}}{{(1 - {\theta _1}){\delta _3}}}} \right]}^{1/(1 + {a_i})}}} \right\}}$

 ${V_4} = x_{ji}^2$ （58）

V4求导可得：

 $\begin{array}{*{20}{l}} {{{\dot V}_4} = {x_{ji}}{{\dot x}_{ji}} = - {x_{ji}}({\varphi _i}{x_{ji}} + {\lambda _i}|{x_{ji}}{|^{{r_i}}} {\rm{sign}} ({x_{ji}}) - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {s_{ji}}) \le - {\varphi _i}x_{ji}^2 - {\lambda _i}{{(x_{ji}^2)}^{\frac{{1 + {r_i}}}{2}}} + |{x_{ji}}{s_{ji}}|} \end{array}$ （59）

 ${{{\dot V}_4} \le - \left( {{\varphi _i} - \left| {\frac{{{s_{ji}}}}{{{x_{ji}}}}} \right|} \right)x_{ji}^2 - {\lambda _i}{{(x_{ji}^2)}^{\frac{{1 + {r_i}}}{2}}}}$ （60）
 ${{{\dot V}_4} \le - {\varphi _i}x_{ji}^2 - \left( {{\lambda _i} - \left| {\frac{{{s_{ji}}}}{{x_{ji}^{{r_i}}}}} \right|} \right){{(x_{ji}^2)}^{\frac{{1 + {r_i}}}{2}}}}$ （61）

 $\begin{array}{*{20}{l}} {{{\dot x}_{ji}} = - {\varphi _i}{x_{ji}} - {\lambda _i}f({x_{ji}}) + {\sigma _j}|{{\dot x}_{ji}}| \le |{\varphi _i}{x_{ji}}| + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} |{\lambda _i}f({x_{ji}})| + |{\sigma _j}| \le {\varphi _i}{\eta _i} + {\lambda _i}{\eta _i}^{{r_i}} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \varOmega \le {\varphi _i}{\varOmega _1} + {\lambda _i}\varOmega _1^{{r_i}} + \varOmega } \end{array}$ （62）

$\dot{x}_{j i}$在有限时间收敛到区域|$\dot{x}_{j i}$|≤Ω2=φiΩ1+λiΩ1ri+Ω，也就是说，视线角速率$\dot{\theta}_{\mathrm{L}i} $$\dot{\varphi}_{{\rm L} i} 分别在有限时间内收敛到区域 \left|\dot{\theta}_{\mathrm{L} i}\right| \leqslant \varOmega_{2}$$ \left|\dot{\varphi}_{\mathrm{L} i}\right| \leqslant \varOmega_{2}$，那么，结论2)和3)得证。至此，定理2得证。

3 仿真分析

 图 2 3枚导弹之间的通信拓扑 Fig. 2 Communication topologies for three missiles

 参数 导弹1 导弹2 导弹3 R/m 12 000 11 000 10 000 $\dot{R}$ -600 -680 -850 θL/(°) -50 -20 -0 $\dot{\theta}_{\mathrm{L}}$/((°)·s-1) -4 3 1.5 ϕL/(°) 70 20 0 $\dot{\phi}_{\mathrm{L}}$/((°)·s-1) -5 4 3 θLf/(°) -30 -10 5 ϕLf/(°) 50 10 5

 图 3 剩余时间 Fig. 3 Time-to-go
 图 4 弹目视线距离 Fig. 4 Relative distance between missile and target
 图 5 视线角速率 Fig. 5 Line of sight rate
 图 6 视线角 Fig. 6 Line of sight angle
 图 7 视线方向过载指令 Fig. 7 Acceleration command along LOS
 图 8 过载指令 Fig. 8 Acceleration command

4 结论

1) 基于积分滑模法设计了视线方向上的制导律，该制导律保证了所有导弹同时攻击目标。

2) 基于快速非奇异滑模法设计了视线法向上的制导律，该制导律保证了所有导弹以期望的视线角攻击目标。

3) 对所设计的制导律进行了数学仿真实验，仿真结果表明了所设计制导律的有效性及优越性。

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http://dx.doi.org/10.7527/S1000-6893.2019.23759

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#### 文章信息

SI Yujie, XIONG Hua, SONG Xun, ZONG Rui

Three dimensional guidance law for cooperative operation based on adaptive terminal sliding mode

Acta Aeronautica et Astronautica Sinica, 2020, 41(S1): 723759.
http://dx.doi.org/10.7527/S1000-6893.2019.23759