﻿ 基于量子行为鸽群优化的无人机紧密编队控制
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Tight formation flight control of UAVs based on pigeon inspired algorithm optimization by quantum behavior
XU Bo, ZHANG Dalong
College of automation, Harbin Engineering University, Harbin 150001, China
Abstract: UAV tight formation refers to the formation whose lateral distance between the UAVs is within one to two times of the wingspan. It has attracted considerable attention because of its effective improvement in the aerodynamic performance of UAVs in formation. In this paper, the aerodynamic coupling effect of the UAVs in tight formation is studied, and the state-space equation in three-dimensional space is established to describe the relative motion of two UAVs. The optimal formation configuration of two UAVs in close formation is deduced, where the combination of the artificial potential field method and formation control is used as the indirect control loop in the control system, and the basic pigeon inspired opbimization algorithm is calculated. The optimization defect of the algorithm is improved by quantum behavior rules. The improved opbimization pigeon inspired algorithm and the UAV control variables are combined as the direct control loop in the control system. Finally, the effectiveness of the control system is verified by simulation comparisons.
Keywords: UAV    tight formation    aerodynamic coupling    artificial potential field    improved pigeon inspired opbimization

1 编队飞行系统建模

1.1 无人机自动驾驶仪模型

 ${\dot V = - \frac{1}{{{\tau _V}}}V + \frac{1}{{{\tau _V}}}{V_{\rm{c}}}}$ （1）
 ${\dot \psi = - \frac{1}{{{\tau _\psi }}}\psi + \frac{1}{{{\tau _\psi }}}{\psi _{\rm{c}}}}$ （2）
 ${\ddot h = - \left( {\frac{1}{{{\tau _{{h_{\rm{a}}}}}}} + \frac{1}{{{\tau _{{h_{\rm{b}}}}}}}} \right)\dot h - \frac{1}{{{\tau _{{h_{\rm{a}}}}}}} \cdot \frac{1}{{{\tau _{{h_{\rm{b}}}}}}}h + \frac{1}{{{\tau _{{h_{\rm{a}}}}}}} \cdot \frac{1}{{{\tau _{{h_{\rm{b}}}}}}}{h_{\rm{c}}}}$ （3）

 参数 数值 平均气动压q/(N·m-1) 2 272.2 机翼面积S /m2 27.9 翼展b /m 9.1 展弦比AR 3 升力曲线斜率a 5.3 垂尾面积Svt /m2 5.1 垂尾高度 hz /m 36.6 垂尾升力曲线斜率avt /(°)-1 0.09 气动效率因子η 0.95 速度时间常数τv /s 5 航向时间常数τψ /s 0.75 高度时间常数τha /s 0.307 5 高度时间常数τhb/s 3.85 总重量W /N 111 139 总质量m /kg 11 340.7 空气流速(无人机航速) V /(m·s-1) 251.5
1.2 无人机编队运动学模型

 图 1 无人机旋转参考系 Fig. 1 Rotating reference frame of wing aircraft
 ${\frac{{{\rm{d}}x}}{{{\rm{d}}t}} = {V_{\rm{L}}}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\psi _{\rm{E}}} + {{\dot \psi }_{\rm{W}}}y - {V_{\rm{W}}}}$ （4）
 ${\frac{{{\rm{d}}y}}{{{\rm{d}}t}} = {V_{\rm{L}}}{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\psi _{\rm{E}}} + {{\dot \psi }_{\rm{W}}}x}$ （5）

2 无人机气动耦合效应分析

2.1 无人机涡流数学模型

 图 2 无人机涡流模型 Fig. 2 Vortex model of UAV
 ${b^\prime } = \frac{\pi }{4}b$ （6）

 $W = \frac{{{\bar \varPhi} \varGamma }}{{2\pi {r_{\rm{c}}}}}$ （7）

 图 3 双机编队示意图 Fig. 3 View of twin aircraft formation
 $\begin{array}{l} {V_{{\rm{U}}{{\rm{W}}_{{\rm{avg}}}}}} = \frac{{{\Gamma _{\rm{L}}}}}{{4\pi b}}\left[ {{\rm{ln}}\frac{{{y^{\prime 2}} + {z^{\prime 2}} + {\mu ^2}}}{{{{\left( {{y^\prime } - \frac{\pi }{4}} \right)}^2} + {z^{\prime 2}} + {\mu ^2}}} - } \right.\\ \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{ln}}\frac{{{{\left( {{y^\prime } + \frac{\pi }{4}} \right)}^2} + {z^{\prime 2}} + {\mu ^2}}}{{{y^{\prime 2}} + {z^{\prime 2}} + {\mu ^2}}}} \right]( - \hat z) \end{array}$ （8）
 $\begin{array}{l} {V_{{\rm{S}}{{\rm{W}}_{{\rm{avg}}}}}} = \frac{{{\Gamma _{\rm{L}}}}}{{4\pi {h_z}}}\left[ {{\rm{ln}}\frac{{{{\left( {{y^\prime } - \frac{\pi }{8}} \right)}^2} + {z^{\prime 2}} + {\mu ^2}}}{{{{\left( {{y^\prime } - \frac{\pi }{8}} \right)}^2} + {{\left( {{z^\prime } + \frac{{{h_z}}}{b}} \right)}^2} + {\mu ^2}}} - } \right.\\ {\kern 1pt} \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{ln}}{\kern 1pt} \frac{{{{\left( {{y^\prime } + \frac{\pi }{8}} \right)}^2} + {z^{\prime 2}} + {\mu ^2}}}{{{{\left( {{y^\prime } + \frac{\pi }{8}} \right)}^2} + {{\left( {{z^\prime } + \frac{{{h_z}}}{b}} \right)}^2} + {\mu ^2}}}} \right]\hat y \end{array}$ （9）

 图 4 无人机受力变化侧视图 Fig. 4 Side view of drone force changes
 $\begin{array}{l} \Delta {C_{{L_{\rm{W}}}}} = \frac{{{a_W}}}{{\pi {A_{\rm{R}}}}}{C_{{L_{\rm{L}}}}}\frac{2}{{{\pi ^2}}}\left[ {{\rm{ln}}\frac{{{y^{\prime 2}} + {z^{\prime 2}} + {\mu ^2}}}{{{{\left( {{y^\prime } - \frac{\pi }{4}} \right)}^2} + {z^{\prime 2}} + {\mu ^2}}} - } \right.\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left. {{\rm{ln}}\frac{{{{\left( {{y^\prime } + \frac{\pi }{4}} \right)}^2} + {z^{\prime 2}} + {\mu ^2}}}{{{y^{\prime 2}} + {z^{\prime 2}} + {\mu ^2}}}} \right] \end{array}$

 $\begin{array}{l} \Delta {C_{{D_{\rm{W}}}}} = \frac{1}{{\pi {A_{\rm{R}}}}}{C_{{L_{\rm{L}}}}}{C_{{L_{\rm{W}}}}}\frac{2}{{{\pi ^2}}}\left[ {{\rm{ln}}\frac{{{y^{\prime 2}} + {z^{\prime 2}} + {\mu ^2}}}{{{{\left( {{y^\prime } - \frac{\pi }{4}} \right)}^2} + {z^{\prime 2}} + {\mu ^2}}} - } \right.\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left. {{\rm{ln}}\frac{{{{\left( {{y^\prime } + \frac{\pi }{4}} \right)}^2} + {z^2} + {\mu ^2}}}{{{y^{\prime 2}} + {z^{\prime 2}} + {\mu ^2}}}} \right] \end{array}$

 $\begin{array}{l} \Delta {C_Y} = \eta \frac{{{S_{{\rm{vt}}}}}}{S} \cdot \frac{{{a_{{\rm{vt}}}}}}{V} \cdot \frac{{{\varGamma _{\rm{L}}}}}{{4\pi {h_z}}} \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left[ {{\rm{ln}}\frac{{{{\left( {{y^\prime } - \frac{\pi }{8}} \right)}^2} + {z^{\prime 2}} + {\mu ^2}}}{{{{\left( {{y^\prime } - \frac{\pi }{8}} \right)}^2} + {{\left( {{z^\prime } + \frac{{{h_z}}}{b}} \right)}^2} + {\mu ^2}}} - } \right.\\ \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{ln}}\frac{{{{\left( {{y^\prime } + \frac{\pi }{8}} \right)}^2} + {z^{\prime 2}} + {\mu ^2}}}{{{{\left( {{y^\prime } + \frac{\pi }{8}} \right)}^2} + {{\left( {{z^\prime } + \frac{{{h_z}}}{b}} \right)}^2} + {\mu ^2}}}} \right] \end{array}$

 $\begin{array}{l} {\sigma _{{\rm{UW}}}}({y^\prime }, {z^\prime }) = \frac{2}{{{\pi ^2}}}\left[ {{\rm{ln}}\frac{{{y^{\prime 2}} + {z^{\prime 2}} + {\mu ^2}}}{{{{\left( {{y^\prime } - \frac{\pi }{4}} \right)}^2} + {z^{\prime 2}} + {\mu ^2}}} - } \right.\\ \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{ln}}\frac{{{{\left( {{y^\prime } - \frac{\pi }{4}} \right)}^2} + {z^{\prime 2}} + {\mu ^2}}}{{{y^{\prime 2}} + {z^{\prime 2}} + {\mu ^2}}}} \right] \end{array}$
 $\begin{array}{l} {\sigma _{{\rm{SW}}}}({y^\prime }, {z^\prime }) = \frac{2}{\pi }\left[ {{\rm{ln}}\frac{{{{\left( {{y^\prime } - \frac{\pi }{8}} \right)}^2} + {z^2} + {\mu ^2}}}{{{{\left( {{y^\prime } - \frac{\pi }{8}} \right)}^2} + {{\left( {{z^\prime } + \frac{{{h_z}}}{b}} \right)}^2} + {\mu ^2}}} - } \right.\\ \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{ln}}\frac{{{{\left( {{y^\prime } + \frac{\pi }{8}} \right)}^2} + {z^{\prime 2}} + {\mu ^2}}}{{{{\left( {{y^\prime } + \frac{\pi }{8}} \right)}^2} + {{\left( {{z^\prime } + \frac{{{h_z}}}{b}} \right)}^2} + {\mu ^2}}}} \right] \end{array}$

 图 5 上洗和侧洗无量纲系数变化示意图 Fig. 5 Schematic diagram of up and side washing dimensionless coefficient changes

2.2 加入气动耦合效应无人机紧密编队数学模型

 $\frac{{\rm{d}}}{{{\rm{d}}t}}\left[ {\begin{array}{*{20}{c}} x\\ {{V_{\rm{W}}}}\\ y\\ {{\psi _{\rm{W}}}}\\ z\\ \zeta \end{array}} \right] = \mathit{\boldsymbol{A}}\left[ {\begin{array}{*{20}{c}} x\\ {{V_{\rm{W}}}}\\ y\\ {{\psi _{\rm{W}}}}\\ z\\ \zeta \end{array}} \right] + \mathit{\boldsymbol{B}}\left[ {\begin{array}{*{20}{c}} {{V_{{{\rm{W}}_{\rm{c}}}}}}\\ {{\psi _{{{\rm{W}}_{\rm{c}}}}}}\\ {{h_{{{\rm{W}}_{\rm{c}}}}}} \end{array}} \right] + \mathit{\boldsymbol{C}}\left[ {\begin{array}{*{20}{c}} {{V_{\rm{L}}}}\\ {{\psi _{\rm{L}}}}\\ {{h_{{{\rm{L}}_{\rm{c}}}}}} \end{array}} \right]$ （10）

 $\mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{A}}_{11}}}&{{\mathit{\boldsymbol{A}}_{12}}}&{{\mathit{\boldsymbol{A}}_{13}}}\\ {{\mathit{\boldsymbol{A}}_{21}}}&{{\mathit{\boldsymbol{A}}_{22}}}&{{\mathit{\boldsymbol{A}}_{23}}} \end{array}} \right]$
 ${\mathit{\boldsymbol{A}}_{11}} = \left[ {\begin{array}{*{20}{c}} 0&{ - 1}\\ 0&{ - \frac{1}{{{\tau _{{V_{\rm{W}}}}}}}} \end{array}} \right]$
 ${\mathit{\boldsymbol{A}}_{12}} = \left[ {\begin{array}{*{20}{c}} {\frac{{\bar qS}}{{mV}}\Delta {C_{{Y_{{{\rm{W}}_y}}}}}, \bar y}&{ - \frac{{\bar y}}{{{\tau _{{V_{\rm{W}}}}}}}G}\\ {\frac{{\bar qS}}{m}\Delta {C_{{D_{{{\rm{W}}_y}}}}}}&0 \end{array}} \right]$
 ${\mathit{\boldsymbol{A}}_{13}} = \left[ {\begin{array}{*{20}{c}} {\frac{{\bar qS}}{{mV}}\Delta {C_{{Y_{{{\rm{W}}_z}}}}}, \bar y}&0\\ 0&0 \end{array}} \right]$
 ${\mathit{\boldsymbol{A}}_{21}} = \left[ {\begin{array}{*{20}{l}} 0&0\\ 0&0\\ 0&0\\ 0&0 \end{array}} \right]$
 ${\mathit{\boldsymbol{A}}_{22}} = \left[ {\begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{\bar qS}}{{mV}}\Delta {C_{{Y_{{{\rm{W}}_y}}}}}\bar x}&{\left( {\frac{{\bar x}}{{{\tau _{{\psi _{\rm{W}}}}}}} - {{\bar V}_{\rm{L}}}1} \right)G}\\ {\frac{{\bar qS}}{{mV}}\Delta {C_{{Y_{{{\rm{W}}_y}}}}}\frac{1}{G}}&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - \frac{1}{{{\tau _{{\psi _V}}}}}}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0}&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{\bar qS}}{m}\Delta {C_{{L_{{{\rm{W}}_y}}}}}}&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0} \end{array}} \right]$
 ${\mathit{\boldsymbol{A}}_{23}} = \left[ {\begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} - \frac{{\bar qS}}{{mV}}\Delta {C_{{Y_{{{\rm{W}}_z}}}}}\bar x}&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0}\\ {\frac{{\bar qS}}{{mV}}\Delta {C_{{Y_{{{\rm{W}}_z}}}}}\frac{1}{G}}&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0}&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - \frac{1}{{{\tau _{{h_{\rm{a}}}}}{\tau _{{h_{\rm{b}}}}}}}}&{ - \left( {\frac{1}{{{\tau _{{h_{\rm{a}}}}}}} + \frac{1}{{{\tau _{{h_{\rm{b}}}}}}}} \right)} \end{array}} \right]$
 $\mathit{\boldsymbol{C}} = \left[ {\begin{array}{*{20}{c}} 1&0&0\\ 0&0&0\\ 0&{{{\bar V}_{\rm{L}}}G}&0\\ 0&0&0\\ 0&0&0\\ 0&0&{ - \frac{1}{{{\tau _{{h_{\rm{a}}}}}{\tau _{{h_{\rm{b}}}}}}}} \end{array}} \right]$
 $\mathit{\boldsymbol{B}} = \left[ {\begin{array}{*{20}{c}} 0&{\frac{{\bar y}}{{{\tau _{{\psi _{\rm{W}}}}}}}G}&0\\ {\frac{1}{{{\tau _{{V_{\rm{W}}}}}}}}&0&0\\ 0&{ - \frac{{\bar x}}{{{\tau _{{\psi _{\rm{W}}}}}}}G}&0\\ 0&{\frac{1}{{{\tau _{{\psi _{\rm{W}}}}}}}}&0\\ 0&0&0\\ 0&0&{\frac{1}{{{\tau _{{h_{\rm{a}}}}}{\tau _{{{\rm{h}}_{\rm{b}}}}}}}} \end{array}} \right]$

3 紧密编队控制器设计 3.1 人工势场控制器设计

 $\left\{ {\begin{array}{*{20}{l}} {{{\dot x}^i} = {v^i}}\\ {{m_i}{{\dot v}^i} = {u^i} - {k_i}{v^i}} \end{array}} \right.i = 1, 2, \cdots , N$ （11）

 $\begin{array}{*{20}{l}} {{u^i} = - {K_v}\sum\limits_{j \in {{\bf{N}}_i}} {({v^i} - {v^j})} - {K_p}\sum\limits_{j \in {N_i}} {{\nabla _{{x^i}}}} {V^{ij}} - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {m_i}({v^i} - {v_{{\rm{ end }}}}) + {k_i}{v^i}} \end{array}$ （12）

 ${V^{ij}}(|{x^{ij}}|) = {\rm{ln}}|{x^{ij}}{|^2} + \frac{{d_{ij}^2}}{{|{x^{ij}}{|^2}}}$ （13）

3.2 改进鸽群控制器设计

 图 6 指南针算子模型 Fig. 6 Compass operator model
 $\mathit{\boldsymbol{V}}_i^{{N_{\rm{c}}}} = \mathit{\boldsymbol{V}}_i^{{N_{\rm{c}}} - 1}{{\rm{e}}^{ - R \times {N_{\rm{c}}}}} + {\rm{rand}} ({\mathit{\boldsymbol{X}}_{{\rm{gbest}}}} - \mathit{\boldsymbol{X}}_i^{{N_{\rm{c}}} - 1})$ （14）
 $\mathit{\boldsymbol{X}}_i^{{N_{\rm{c}}}} = \mathit{\boldsymbol{X}}_i^{{N_{\rm{c}}} - 1} + \mathit{\boldsymbol{V}}_i^{{N_{\rm{c}}}}$ （15）

 图 7 地标算子模型 Fig. 7 Landmark operator model
 $\mathit{\boldsymbol{X}}_{{\rm{ center }}}^{{N_{\rm{c}}} - 1} = \frac{{\sum\limits_{i = 1}^{{N^{{N_{\rm{c}}} - 1}}} {\mathit{\boldsymbol{X}}_i^{{N_{\rm{c}}} - 1}} F(\mathit{\boldsymbol{X}}_i^{{N_{\rm{c}}} - 1})}}{{{N^{{N_{\rm{c}}} - 1}}\sum\limits_{i = 1}^{{N^{{N_{\rm{c}}} - 1}}} {F(\mathit{\boldsymbol{X}}_i^{{N_{\rm{c}}} - 1})} }}$ （16）
 ${N^{{N_{\rm{c}}}}} = \frac{{{N^{{N_{\rm{c}}} - 1}}}}{2}$ （17）
 $\mathit{\boldsymbol{X}}_i^{{N_{\rm{c}}}} = \mathit{\boldsymbol{X}}_i^{{N_{\rm{c}}} - 1} + {\rm{rand }}(\mathit{\boldsymbol{X}}_{{\rm{center}}}^{{N_{\rm{c}}} - 1} - \mathit{\boldsymbol{X}}_i^{{N_{\rm{c}}} - 1})$ （18）

 $\alpha = \frac{{(1 - 0.5)({T_1} - t)}}{{{T_1}}} + 0.5$ （19）
 $\mathit{\boldsymbol{P}} = \frac{{ {\rm{ran}}{{\rm{d}}_1}\mathit{\boldsymbol{X}}_{{\rm{pbes}}{{\rm{t}}_i}}^{{N_{\rm{c}}} - 1} + {\rm{ran}}{{\rm{d}}_2}\mathit{\boldsymbol{X}}_{{\rm{gbest}}}^{{N_{\rm{c}}} - 1}}}{{ {\rm{ran}}{{\rm{d}}_1} + {\rm{ran}}{{\rm{d}}_2}}}$ （20）
 $\mathit{\boldsymbol{X}}_i^{{N_{\rm{c}}}} = \mathit{\boldsymbol{P}} \pm \alpha |\mathit{\boldsymbol{X}}_{{\rm{mbest}}}^{{N_{\rm{c}}} - 1} - \mathit{\boldsymbol{X}}_i^{{N_{\rm{c}}} - 1}|{\rm{ln}}\left( {\frac{1}{{{\rm{rand}}}}} \right)$ （21）

 图 8 改进地标算子模型 Fig. 8 Improved landmark operator model
 ${\beta = {\rm{round}} (1 + {\rm{rand}} )}$ （22）
 ${{\mathit{\boldsymbol{X}}_{{\rm{ne}}{{\rm{w}}_i}}} = \mathit{\boldsymbol{X}}_i^{{N_{\rm{c}}} - 1} + {\rm{rand}} (\mathit{\boldsymbol{X}}_{{\rm{gbest}}}^{{N_{\rm{c}}} - 1} - \beta \mathit{\boldsymbol{X}}_{{\rm{mbest}}}^{{N_{\rm{c}}} - 1})}$ （23）
 $\mathit{\boldsymbol{X}}_i^{{N_{\rm{c}}}} = \left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{X}}_{{\rm{ne}}{{\rm{w}}_i}}}}&{F({\mathit{\boldsymbol{X}}_{{\rm{ne}}{{\rm{w}}_i}}}) < F(\mathit{\boldsymbol{X}}_i^{{N_{\rm{c}}} - 1})}\\ {\mathit{\boldsymbol{X}}_i^{{N_{\rm{c}}} - 1}}&{F({\mathit{\boldsymbol{X}}_{{\rm{ne}}{{\rm{w}}_i}}}) > F(\mathit{\boldsymbol{X}}_i^{{N_{\rm{c}}} - 1})} \end{array}} \right.$ （24）
3.3 无人机紧密编队控制

 图 9 无人机紧密编队控制系统 Fig. 9 UAV tight formation control system

 $\mathit{\boldsymbol{X}} = {[{V_{{{\rm{W}}_{\rm{c}}}}}, {\psi _{{{\rm{W}}_{\rm{c}}}}}, {h_{{{\rm{W}}_{\rm{c}}}}}]^{\rm{T}}}$ （25）

 $J = {({\mathit{\boldsymbol{X}}_{{\rm{ Fnext}}}} - \mathit{\boldsymbol{X}}_{{\rm{ Fnext }}}^\prime )^{\rm{T}}}({\mathit{\boldsymbol{X}}_{{\rm{ Fnext }}}} - \mathit{\boldsymbol{X}}_{{\rm{ Fhext }}}^\prime )$ （26）

3.4 仿真实验及结果分析

1) 基本鸽群优化(PIO)算法[16]

2) 加入混沌、反向以及柯西扰动的自适应鸽群优化(IPIO)算法[22]

3) 加入收缩因子的自适应鸽群优化(CFPIO)算法[23]

4) 基于量子行为改进的鸽群优化(QPIO)算法。

 参数 数值 粒子维数D 10 种群个数Np 5D 收缩-扩张系数α 0.5~1(均匀递减) 指南针算子迭代次数T1 800 地标算子迭代次数T2 200

 函数 函数名 函数表达式 F1 Sphere $f(x) = \sum\limits_{i = 1}^n {{x_i}}$ F2 Schwefel $\begin{array}{l} f(X) = 418.982{\kern 1pt} {\kern 1pt} {\kern 1pt} 9n + \\ \mathop \sum \limits_{i = 1}^n [ - {x_i}{\rm{sin}}(\sqrt {|{x_i}|} )] \end{array}$ F3 Rastrigin $f(X) = 10n + \mathop \sum \limits_{i = 1}^n [x_i^2 - 10{\rm{cos}}(2\pi {x_i})]$ F4 Griewangk $f(X) = \frac{1}{{4{\kern 1pt} {\kern 1pt} {\kern 1pt} 000}}\sum\limits_{i = 1}^n {x_i^2} + \prod\limits_{i = 1}^n {{\rm{cos}}} (\frac{{{x_i}}}{{\sqrt i }}) + 1$ F5 Schwefel2.22 $f(X) = \sum\limits_{i = 1}^n | {x_i}| + \prod\limits_{i = 1}^n | {x_i}|$ F6 Schwefel2.21 $f(X) = \mathop {{\rm{max}}}\limits_i \{ |{x_i}|, 1 \le i \le n\}$

 函数 PIO IPIO CFPIO QPIO F1 Mean 0.486 9 0.128 5 0.041 9 0 STD 0.746 1 0.461 3 0.136 0 0 Min 0 0 0 0 Max 3.240 5 2.423 2 0.541 7 0 Tmean/s 1.063 5 0.929 7 0.996 4 1.732 8 F2 Mean 2.236 7 0.710 7 1.064 8 0.067 2 STD 4.788 7 1.582 8 2.688 9 0.141 7 Min/10-5 1.272 8 1.272 8 1.272 8 1.272 8 Max 24.922 0 7.430 6 14.153 3 0.647 5 Tmean/s 0.965 1 1.031 8 1.022 9 2.007 3 F3 Mean 0.195 6 0.014 5 0.004 3 0 STD 0.392 6 0.067 4 0.015 9 0 Min 0 0 0 0 Max 1.135 9 0.364 9 0.075 6 0 Tmean/s 0.938 0 1.016 1 1.081 8 1.814 1 F4 Mean 0.080 3 0.057 6 0.004 9 0.004 2 STD 0.121 8 0.122 5 0.005 9 0.004 2 Min 0.002 5 0.002 5 0.002 5 0.002 5 Max 0.557 6 0.586 5 0.032 2 0.024 0 Tmean/s 0.994 8 1.000 5 1.035 4 1.968 2 F5 Mean 1.811 5×10-42 7.820 0×10-27 8.537 5×10-53 3.364 6×10-34 STD 9.785 5×10-42 4.283 2×10-26 3.647 9×10-52 1.748 0×10-33 Min 0 0 1.294 5×10-98 1.647 3×10-64 Max 5.361 8×10-41 2.346 0×10-25 1.912 5×10-51 9.584 6×10-33 Tmean/s 1.007 8 1.042 2 1.135 4 1.926 0 F6 Mean 0.580 7 0.212 3 0.180 2 1.284 9×10-32 STD 0.755 6 0.415 8 0.569 6 7.037 6×10-32 Min 0 0 1.879 7×10-99 0 Max 2.938 2 1.584 5 2.925 5 3.854 7×10-31 Tmean/s 1.056 8 1.104 2 1.193 2 1.860 9
 图 10 各适应度函数收敛曲线 Fig. 10 Convergence curves of fitness functions

1) 在不同函数条件下本项目所提出的改进鸽群算法基本上在4个评价指标中均为最优，说明本项目所提出的算法寻优性能较其他算法有较大提升，且其寻优稳定性较好。

2) 在仿真所用平均时间这一项上，原始鸽群优化算法较其他算法表现更好，但基本相差不大，说明现存的改进鸽群优化算法基本上都是以牺牲一定的收敛时长来显著提升其收敛性能。这也说明了如何在保证收敛时长基本不变的条件下提升算法的寻优性能是此优化算法值得进一步研究的方向。

3) 由收敛曲线可以看出本项目所提出改进鸽群优化算法收敛速度较传统鸽群优化算法有较大提升，在几种改进算法的比较上也是速度最快的，基本上在迭代的前100次以内就可以满足一般的寻优要求。

 图 11 仿真结果 Fig. 11 Simulation results

 状态名称 控制量 长机 僚机 速度/(m·s－1) 251.5 0 0 航向/(°) -30 0 0 高度/m 13 716 0 0 横向间距/m 18.3 18.3 侧向间距/m 7.2 7.2 纵向间距/m 0 0

1) 仿真开始无人机编队可以很快到达指定高度，随着仿真时长增加，长机按照指定控制指令进行变化，而僚机根据所设计的控制器输出控制量进行相应编队。

2) 从图 11(a)可以看出在xy方向上编队收敛程度较好，而z方向上随着仿真时长的增加误差不断减小；从图 11(b)可以看出仿真时长结束后所形成的编队队形，10架无人机排成斜“一”字形编队。

3) 从图 11(c)可以看出机群速度拟合程度较好，在整个编队过程中僚机与长机的速度能始终保持一致；从图 11(d)可以看出对于航向方向上由于长机增加了扰动，僚机的航向存在一定的波动，但始终保持在误差允许范围内。

4 结论

1) 针对无人机紧密编队所产生的气动耦合效应进行了数学建模，分析了无人机紧密编队的理论最优队形。

2) 在此基础上设计了人工势场控制器以及改进鸽群控制器，经过仿真验证证实了该控制器对于无人机紧密编队控制的有效性，可以提高无人机紧密编队的可靠性，使无人机在不同控制指令下快速形成设定编队队形并保持队形不变。

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http://dx.doi.org/10.7527/S1000-6893.2020.23722

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#### 文章信息

XU Bo, ZHANG Dalong

Tight formation flight control of UAVs based on pigeon inspired algorithm optimization by quantum behavior

Acta Aeronautica et Astronautica Sinica, 2020, 41(8): 323722.
http://dx.doi.org/10.7527/S1000-6893.2020.23722