﻿ 涵道风扇式无人机的优先级控制分配
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1. 华南理工大学 自主系统与网络控制教育部重点实验室, 广州 510640;
2. 华南理工大学 广东省无人机系统工程技术研究中心, 广州 510640

Prioritized control allocation for ducted fan UAVs
MENG Chaoheng1,2, PEI Hailong1,2, CHENG Zihuan1,2
1. Key Laboratory of Autonomous Systems and Networked Control, South China University of Technology, Guangzhou 510640, China;
2. Unmanned Aerial Vehicle Systems Engineering Technology Research Center of Guangdong, South China University of Technology, Guangzhou 510640, China
Abstract: Ducted fan UAVs, aircraft with redundant control surfaces, usually solve the control allocation problem via the pseudo-inverse method. However, this method cannot return admissible control for all the moments in the Attainable Moment Set (AMS), therefore sacrificing partial control ability of redundant control surfaces. To solve this problem, this paper proposes a prioritized control allocation method, which first decomposes the desired moments into a sequence of prioritized partitions, followed by solution of the constrained optimization problem to obtain admissible control. Compared with the pseudo-inverse method, the proposed method can return admissible control for a larger range of desired moments. In addition, when the desired moments are unattainable, it can prevent the system from output coupling caused by actuator saturation. The proposed method is applied to the control allocation of ducted fan UAVs. Simulation and experiments verify the effectiveness of the method.
Keywords: ducted fan UAVs    overactuated systems    control allocation    prioritized allocation    linear programming

1 控制分配问题 1.1 过驱动系统的控制分配问题

 $\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot x}} = \mathit{\boldsymbol{f}}(\mathit{\boldsymbol{x}}) + \mathit{\boldsymbol{g}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{\delta }})}\\ {\mathit{\boldsymbol{y}} = \mathit{\boldsymbol{h}}(\mathit{\boldsymbol{x}})} \end{array}} \right.$ （1）

 $\mathit{\boldsymbol{g}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{\delta }}) = \mathit{\boldsymbol{GM}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{\delta }})$ （2）

 $\mathit{\boldsymbol{\tau }} = \mathit{\boldsymbol{M}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{\delta }})$ （3）

 $\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot x}} = \mathit{\boldsymbol{f}}(\mathit{\boldsymbol{x}}) + \mathit{\boldsymbol{G\tau }}}\\ {\mathit{\boldsymbol{y}} = \mathit{\boldsymbol{h}}(\mathit{\boldsymbol{x}})} \end{array}} \right.$ （4）

 ${\mathit{\boldsymbol{\tau }} = \mathit{\boldsymbol{B}}(\mathit{\boldsymbol{\delta }} - {\mathit{\boldsymbol{\delta }}_0}) + {\mathit{\boldsymbol{\tau }}_0}}$ （5）
 ${{\mathit{\boldsymbol{\tau }}_0} \buildrel \Delta \over = \mathit{\boldsymbol{M}}(\mathit{\boldsymbol{x}},{\mathit{\boldsymbol{\delta }}_0})}$ （6）
 $\mathit{\boldsymbol{B}} \buildrel \Delta \over = \frac{{\partial \mathit{\boldsymbol{\tau }}}}{{\partial \mathit{\boldsymbol{\delta }}}} = \left[ {\begin{array}{*{20}{c}} {\frac{{\partial {\tau _1}}}{{\partial {\delta _1}}}}&{\frac{{\partial {\tau _1}}}{{\partial {\delta _2}}}}& \cdots &{\frac{{\partial {\tau _1}}}{{\partial {\delta _p}}}}\\ {\frac{{\partial {\tau _2}}}{{\partial {\delta _1}}}}&{\frac{{\partial {\tau _2}}}{{\partial {\delta _2}}}}& \cdots &{\frac{{\partial {\tau _2}}}{{\partial {\delta _p}}}}\\ \vdots & \vdots & \ddots & \vdots \\ {\frac{{\partial {\tau _m}}}{{\partial {\delta _1}}}}&{\frac{{\partial {\tau _m}}}{{\partial {\delta _2}}}}& \cdots &{\frac{{\partial {\tau _m}}}{{\partial {\delta _p}}}} \end{array}} \right]$ （7）

 $\mathit{\boldsymbol{\tau }} - {\mathit{\boldsymbol{\tau }}_{\rm{L}}} = \mathit{\boldsymbol{B}}(\mathit{\boldsymbol{\delta }} - {\mathit{\boldsymbol{\delta }}_{\rm{L}}}) \Rightarrow \Delta \mathit{\boldsymbol{\tau }} = \mathit{\boldsymbol{B}}\Delta \mathit{\boldsymbol{\delta }}$ （8）

 $\mathit{\boldsymbol{\tau }} = \mathit{\boldsymbol{B\delta }}$ （9）

 图 1 基于控制分配的控制系统 Fig. 1 Control system based on control allocation
1.2 涵道风扇式无人机的控制分配问题

 图 2 惯性系和机体坐标系 Fig. 2 Inertial and body-fixed frames
 $\mathit{\boldsymbol{\dot \omega }} = {\mathit{\boldsymbol{I}}^{ - 1}}(\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} - \mathit{\boldsymbol{\omega }} \times \mathit{\boldsymbol{I\omega }})$ （10）

 ${\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{{\rm{cs}}}} = \left[ {\begin{array}{*{20}{l}} {{\varGamma _x}}\\ {{\varGamma _y}}\\ {{\varGamma _z}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - {l_1}}&0&{{l_1}}&0\\ 0&{ - {l_1}}&0&{{l_1}}\\ {{l_2}}&{{l_2}}&{{l_2}}&{{l_2}} \end{array}} \right]{k_\delta }{V_{\rm{d}}}\mathit{\boldsymbol{\delta }}$ （11）
 图 3 力臂示意图 Fig. 3 Diagram of arm

 $\mathit{\boldsymbol{B}} = {k_\delta }{V_{\rm{d}}}\left[ {\begin{array}{*{20}{c}} { - {l_1}}&0&{{l_1}}&0\\ 0&{ - {l_1}}&0&{{l_1}}\\ {{l_2}}&{{l_2}}&{{l_2}}&{{l_2}} \end{array}} \right]$ （12）

1) 假设已经获得系统的控制律，并且在控制律下式(10)系统渐进稳定。本文的重点是在已知控制律的基础上研究控制分配问题。

2) 假设执行器动态响应时间远远小于被控对象。该无人机所用的力矩舵机带宽远大于被控对象，因此假设是合理的。

3) 假设各个状态下的控制效率矩阵B已经通过仿真或试验数据获得。控制向量和力矩曲线M(x, δ)以离散数据的形式保存在计算机中，系统运行时将通过查表，利用插值法获取B矩阵的元素。

2 控制分配方法

2.1节先介绍常用的2种控制分配方法：伪逆法和直接分配法，将其应用到涵道风扇式飞行器的控制分配问题中, 并对比2种分配方法的可达集。2.2节针对2种常用方法的不足，提出一种优先级分配方法。

2.1 常用控制分配方法

 $\mathrm{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mathit{\boldsymbol{{ }\!\!\delta\!\!\text{ } }}}}= \left[ {\begin{array}{*{20}{c}} {{\delta _{{\rm{1,min}}}}}\\ {{\delta _{{\rm{2,min}}}}}\\ \vdots \\ {{\delta _{p,{\rm{min}}}}} \end{array}} \right],\mathit{\boldsymbol{\bar \delta }} = \left[ {\begin{array}{*{20}{c}} {{\delta _{{\rm{1,max}}}}}\\ {{\delta _{{\rm{2,max}}}}}\\ \vdots \\ {{\delta _{p,{\rm{max}}}}} \end{array}} \right],\mathit{\boldsymbol{\delta }} = \left[ {\begin{array}{*{20}{c}} {{\delta _1}}\\ {{\delta _2}}\\ \vdots \\ {{\delta _p}} \end{array}} \right]$ （13）

 $\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} = \{ \mathit{\boldsymbol{\delta }}|\mathrm{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mathit{\boldsymbol{{ }\!\!\delta\!\!\text{ } }}}} \le \mathit{\boldsymbol{\delta }} \le \mathit{\boldsymbol{\bar \delta }}\}$ （14）

 $- T{\mathit{\boldsymbol{u}}_{\rm{m}}} \le \mathit{\boldsymbol{\delta }} - {\mathit{\boldsymbol{\delta }}_{\rm{L}}} \le T{\mathit{\boldsymbol{u}}_{\rm{m}}}$ （15）

 ${{{\mathit{\boldsymbol{\bar \delta }}}^\prime } = {\rm{min}}\{ \mathit{\boldsymbol{\bar \delta }} - {\mathit{\boldsymbol{\delta }}_{\rm{L}}},\quad T{\mathit{\boldsymbol{u}}_{\rm{m}}}\} }$ （16）
 ${{\mathrm{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mathit{\boldsymbol{{ }\!\!\delta\!\!\text{ } }}}}^\prime } = {\rm{max}}\{ \mathrm{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mathit{\boldsymbol{{ }\!\!\delta\!\!\text{ } }}}} - {\mathit{\boldsymbol{\delta }}_{\rm{L}}}, - T{\mathit{\boldsymbol{u}}_{\rm{m}}}\} }$ （17）

 $\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} = \{ \mathit{\boldsymbol{\delta }}|{\mathrm{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mathit{\boldsymbol{{ }\!\!\delta\!\!\text{ } }}}}^\prime } \le \mathit{\boldsymbol{\delta }} \le {\mathit{\boldsymbol{\bar \delta }}^\prime }\}$ （18）

1) 伪逆法

2) 直接分配法

 $\begin{array}{l} \mathop {{\rm{max}}}\limits_{\mathit{\boldsymbol{\delta }},\alpha } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha \\ {\rm{s}}{\rm{.}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{t}}{\rm{.}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\{ {\begin{array}{*{20}{l}} {\mathrm{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mathit{\boldsymbol{{ }\!\!\delta\!\!\text{ } }}}} \le \mathit{\boldsymbol{\delta }} \le \mathit{\boldsymbol{\bar \delta }}}\\ {\alpha {\mathit{\boldsymbol{\tau }}_{\rm{c}}} = \mathit{\boldsymbol{B\delta }}}\\ {0 \le \alpha \le 1} \end{array}} \right. \end{array}$ （19）

 $\mathit{\boldsymbol{B}} = \left[ {\begin{array}{*{20}{c}} { - 0.539{\kern 1pt} {\kern 1pt} {\kern 1pt} 3}&0&{0.539{\kern 1pt} {\kern 1pt} {\kern 1pt} 3}&0\\ 0&{ - 0.539{\kern 1pt} {\kern 1pt} {\kern 1pt} 3}&0&{0.539{\kern 1pt} {\kern 1pt} {\kern 1pt} 3}\\ {0.209{\kern 1pt} {\kern 1pt} {\kern 1pt} 9}&{0.209{\kern 1pt} {\kern 1pt} {\kern 1pt} 9}&{0.209{\kern 1pt} {\kern 1pt} {\kern 1pt} 9}&{0.209{\kern 1pt} {\kern 1pt} {\kern 1pt} 9} \end{array}} \right]$ （20）

 $\mathrm{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mathit{\boldsymbol{{ }\!\!\delta\!\!\text{ } }}}} = \left[ {\begin{array}{*{20}{l}} { - {{20}^\circ }}\\ { - {{20}^\circ }}\\ { - {{20}^\circ }}\\ { - {{20}^\circ }} \end{array}} \right],\mathit{\boldsymbol{\bar \delta }} = \left[ {\begin{array}{*{20}{c}} {{{20}^\circ }}\\ {{{20}^\circ }}\\ {{{20}^\circ }}\\ {{{20}^\circ }} \end{array}} \right]$ （21）

 图 4 伪逆法和直接分配法的可达集对比 Fig. 4 Comparison of subsets of attainable moments using pseudo-inverse method and direct allocation method
2.2 优先级控制分配方法

2.2.1 控制律的优先级分解

 $\left\{ {\begin{array}{*{20}{l}} {\ddot \varphi = {f_1}(\varphi ,\theta ,\psi ,\dot \varphi ,\dot \theta ,\dot \psi ) + {W_1} + I_x^{ - 1}{\tau _x}}\\ {\ddot \theta = {f_2}(\varphi ,\theta ,\psi ,\dot \varphi ,\dot \theta ,\dot \psi ) + {W_2} + I_y^{ - 1}{\tau _y}}\\ {\ddot \psi = {f_3}(\varphi ,\theta ,\psi ,\dot \varphi ,\dot \theta ,\dot \psi ) + {W_3} + I_z^{ - 1}{\tau _z}}\\ {\mathit{\boldsymbol{y}} = {{\left[ {\begin{array}{*{20}{l}} \varphi &\theta &\psi \end{array}} \right]}^{\rm{T}}}} \end{array}} \right.$ （22）

 $\left\{ {\begin{array}{*{20}{l}} {e = [\varphi (k) - {z_{{\rm{r1}}}}(k)]/{\varepsilon ^2}}\\ {{z_{{\rm{r1}}}}(k + 1) = {z_{{\rm{r1}}}}(k) + T({z_{{\rm{r2}}}}(k) + \varepsilon {g_1}(e))}\\ {{z_{{\rm{r2}}}}(k + 1) = {z_{{\rm{r2}}}}(k) + T({z_{{\rm{r3}}}}(k) + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g_2}(e) + I_x^{ - 1}{\tau _x}(k))}\\ {{z_{{\rm{r3}}}}(k + 1) = {z_{{\rm{r3}}}}(k) + T{g_3}(e)/\varepsilon } \end{array}} \right.$ （23）
 $\left\{ \begin{array}{l} {v_{{\rm{r1}}}}(k + 1) = {v_{{\rm{r1}}}}(k) + T{v_{{\rm{r2}}}}(k)\\ {v_{{\rm{r2}}}}(k + 1) = {v_{{\rm{r2}}}}(k) + T{\rm{fst}}({v_{{\rm{r1}}}}(k) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\varphi _{\rm{c}}},{v_{{\rm{r2}}}}(k),r,h) \end{array} \right.$ （24）
 $\left\{ {\begin{array}{*{20}{l}} {{e_1} = {v_{{\rm{r1}}}}(k) - {z_{{\rm{r1}}}}(k)}\\ {{e_2} = {v_{{\rm{r2}}}}(k) - {z_{{\rm{r2}}}}(k)}\\ {{\tau _{x{\rm{c}}}} = {k_1}{\rm{fal}}({e_1},{\sigma _1},\xi ) + {k_2}{\rm{fal}}({e_2},{\sigma _2},\xi ) - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {I_x}{z_{{\rm{r3}}}}(k)} \end{array}} \right.$ （25）

 $\left\{ {\begin{array}{*{20}{l}} {{g_i}(e) = {{\rm{ \mathsf{ β} }}_i}{{[e]}^{i{\zeta _i} - (i - 1)}}}&{i = 1,2,3}\\ {{{[e]}^\chi } \buildrel \Delta \over = {\rm{sign}} (e)|e{|^\chi }} \end{array}} \right.$ （26）
 $\left\{ \begin{array}{l} {\rm{fst}} ({v_{{\rm{r1}}}},{v_{{\rm{r2}}}},r,h) = - \left\{ {\begin{array}{*{20}{l}} {ra/d}&{|a| \le d}\\ {r {\rm{sign}} (a)}&{|a| > d} \end{array}} \right.\\ a \buildrel \Delta \over = \left\{ {\begin{array}{*{20}{l}} {{v_{{\rm{r2}}}} + {\rm{sign}} (n)({a_0} - d)/2}&{|n| > {d_0}}\\ {{v_{{\rm{r2}}}} + n/h}&{|n| \le {d_0}} \end{array}} \right.\\ \begin{array}{*{20}{l}} {d \buildrel \Delta \over = rh,{d_0} \buildrel \Delta \over = dh,n \buildrel \Delta \over = {v_{{\rm{r1}}}} + h{v_{{\rm{r2}}}}}\\ {{a_0} \buildrel \Delta \over = {{({d^2} + 8r|n|)}^{1/2}}} \end{array} \end{array} \right.$ （27）
 ${\rm{fal}}(e,\sigma ,\xi ) = \left\{ {\begin{array}{*{20}{l}} {|e{|^\sigma } {\rm{sign}} (e)}&{|e| > \xi }\\ {e/{\xi ^{1 - \sigma }}}&{|e| \le \xi ,} \end{array}\xi > 0} \right.$ （28）

TD以及NLSEF的参数 rhσ1σ2ξk1k2的选取原则可参考文献[22]。ESO的输出zr1zr2zr3将在有限时间内收敛于φ${\dot \varphi }$f1+W1，参数ε$\zeta i$βi的选取原则以及其收敛性证明参考文献[23]。对滚转通道的控制律τxc做优先级分解τxc=τx1+τx2，其中

 ${{\tau _{x1}} \buildrel \Delta \over = - {I_x}{z_{{\rm{r3}}}}(k)}$ （29）
 ${{\tau _{x2}} \buildrel \Delta \over = {k_1}{\rm{fal}}({e_1},{\sigma _1},\xi ) + {k_2}{\rm{fal}}({e_2},{\sigma _2},\xi )}$ （30）

 ${\mathit{\boldsymbol{\tau }}_{\rm{c}}} = {\mathit{\boldsymbol{\tau }}_1} + {\mathit{\boldsymbol{\tau }}_2}$ （31）
 ${\mathit{\boldsymbol{\tau }}_{\rm{c}}} \buildrel \Delta \over = \left[ {\begin{array}{*{20}{l}} {{\tau _{x{\rm{c}}}}}\\ {{\tau _{y{\rm{c}}}}}\\ {{\tau _{z{\rm{c}}}}} \end{array}} \right],{\tau _1} \buildrel \Delta \over = \left[ {\begin{array}{*{20}{l}} {{\tau _{x1}}}\\ {{\tau _{y1}}}\\ {{\tau _{z1}}} \end{array}} \right],{\tau _2} \buildrel \Delta \over = \left[ {\begin{array}{*{20}{l}} {{\tau _{x2}}}\\ {{\tau _{y2}}}\\ {{\tau _{z2}}} \end{array}} \right]$ （32）

2.2.2 优先级控制分配

 $\begin{array}{l} \mathop {{\rm{max}}}\limits_{\mathit{\boldsymbol{\delta }},\alpha } \alpha \\ {\rm{s}}{\rm{.t}}{\rm{.}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\{ {\begin{array}{*{20}{l}} {\mathrm{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mathit{\boldsymbol{{ }\!\!\delta\!\!\text{ } }}}} \le \mathit{\boldsymbol{\delta }} \le \mathit{\boldsymbol{\bar \delta }}}\\ {\mathit{\boldsymbol{B\delta }} = {\mathit{\boldsymbol{\tau }}_1} + {\mathit{\boldsymbol{\tau }}_2} + \cdots + \alpha {\mathit{\boldsymbol{\tau }}_k}}\\ {0 \le \alpha \le 1} \end{array}} \right. \end{array}$ （33）

1) 若式(33)有解

2) 若式(33)无解

 $\begin{array}{l} \mathop {{\rm{max}}}\limits_{\mathit{\boldsymbol{\delta }},\alpha } \alpha \\ {\rm{s}}{\rm{.t}}{\rm{.}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\{ {\begin{array}{*{20}{l}} {\mathrm{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mathit{\boldsymbol{{ }\!\!\delta\!\!\text{ } }}}} \le \mathit{\boldsymbol{\delta }} \le \mathit{\boldsymbol{\bar \delta }}}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{B\delta }} = {\mathit{\boldsymbol{\tau }}_1} + {\mathit{\boldsymbol{\tau }}_2} + \cdots + \alpha {\mathit{\boldsymbol{\tau }}_{k - 1}}}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \le \alpha \le 1} \end{array}} \right. \end{array}$ （34）

 $\begin{array}{l} \mathop {{\rm{max}}}\limits_{\mathit{\boldsymbol{\delta }},{\alpha _s}} {\alpha _s}\\ {\rm{s}}{\rm{.t}}{\rm{.}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\{ {\begin{array}{*{20}{l}} {\mathrm{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mathit{\boldsymbol{{ }\!\!\delta\!\!\text{ } }}}} \le \mathit{\boldsymbol{\delta }} \le \mathit{\boldsymbol{\bar \delta }}}\\ \begin{array}{l} \mathit{\boldsymbol{B\delta }} = {\alpha _1}{\mathit{\boldsymbol{\tau }}_1} + {\alpha _2}{\mathit{\boldsymbol{\tau }}_2} + \cdots + {\alpha _k}{\mathit{\boldsymbol{\tau }}_k}\\ {\alpha _j} = \left\{ {\begin{array}{*{20}{l}} 0&{j < s}\\ 1&{j > s} \end{array},j = 1,2, \cdots ,k} \right. \end{array}\\ {{\kern 1pt} 0 \le {\alpha _s} \le 1} \end{array}} \right. \end{array}$ （35）

 图 5 优先级分配法流程图 Fig. 5 Flowchart of prioritized allocation method

 图 6 m=2时优先级分配法和直接分配法对比 Fig. 6 Comparison of prioritized allocation method and direct allocation method when m=2
3 仿真验证

 图 7 伪控制指令响应曲线 Fig. 7 Virtual control command response curves

4 飞行试验

 图 8 飞行试验 Fig. 8 Flight test

 参数 物理意义 数值 Ix/(kg·m2) X轴转动惯量 0.025483 Iy/(kg·m2) Y轴转动惯量 0.025483 Iz/(kg·m2) Z轴转动惯量 0.00562 m/kg 质量 1.53 l1/m 滚转/俯仰通道力臂 0.17 l2/m 偏航通道力臂 0.06
 图 9 飞控系统框图 Fig. 9 Flight control system

τ1设为最高优先级，τ2设为低优先级。操纵面物理约束如式(21)所示，为方便后续试验中添加9°的操纵面扰动，试验时在控制程序中统一设定操纵面最大偏转角为11°。如前文所述，忽略执行器动力学，舵面偏转角用其执行器的指令δ近似。姿态角参考输入为滚转、俯仰通道20°的阶跃信号，航向角保持起飞时锁定的角度，统一为正北方向(零度偏航角)。

 图 10 在无外扰的情况下，分别使用伪逆、直接、优先级分配的阶跃响应 Fig. 10 Step responses to pseudo-inverse, direct allocation and prioritized allocation methods without external disturbance
 图 11 无外扰时的1~4号舵面偏转角 Fig. 11 Deflection angles of No. 1 to No. 4 surface vanes without external disturbance

 图 12 添加扰动，分别使用伪逆、直接分配的阶跃响应 Fig. 12 Step responses to pseudo-inverse and direct allocation methods with disturbance
 图 13 添加扰动，使用优先级分配的阶跃响应 Fig. 13 Step responses to prioritized allocation with disturbance
 图 14 添加扰动时的1~4号舵面偏转角 Fig. 14 Deflection angles of No.1 to No.4 surface vanes with disturbance

5 结论

1) 优先级分配法、直接分配法的品质因数都为100%，而常规的伪逆法仅为71.99%。

2) 优先级分配法可以保证控制律中高优先级分量尽可能无误差分配，而仅在低优先级分量上产生分配误差，具体表现为使系统可一定程度上防止因操纵面约束引起的系统输出耦合，尽可能保持输出解耦，这是直接分配法和伪逆法所没有的特性。

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http://dx.doi.org/10.7527/S1000-6893.2020.24017

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#### 文章信息

MENG Chaoheng, PEI Hailong, CHENG Zihuan

Prioritized control allocation for ducted fan UAVs

Acta Aeronautica et Astronautica Sinica, 2020, 41(10): 324017.
http://dx.doi.org/10.7527/S1000-6893.2020.24017