﻿ 考虑输入饱和的固定翼无人机自适应增益滑模控制
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1. 天津大学 电气自动化与信息工程学院, 天津 300072;
2. 中国电子科技集团公司电子科学研究院, 北京 100041

Adaptive-gain sliding mode control for fixed-wing UAVs with input saturation
ZHANG Chaofan1, DONG Qi2
1. School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China;
2. China Academy of Electronics and Information Technology, Beijing 100041, China
Abstract: Under the effects of complex flight environments, a control strategy for fixed-wing UAVs based on the adaptive sliding mode algorithm is proposed considering external disturbances and the input saturation. Firstly, a fixed-wing UAV model is introduced and further divided into an attitude subsystem and a velocity subsystem. Then, according to different characteristics and control requirements of the two subsystems, a novel adaptive multivariable twisting algorithm and a novel adaptive-gain fast super twisting sliding mode algorithm are proposed to design an attitude controller and a velocity controller for a fixed-wing UAV respectively. In this control strategy, the observer is not needed to estimate the disturbance. The stability of the closed loop systems is proved based on Lyapunov stability analysis. Finally, simulations are conducted to demonstrate the effectiveness and favorable control performance of the proposed strategy.
Keywords: fixed-wing UAVs    attitude control    velocity control    adaptive-gain    twisting sliding mode algorithms    fast super-twisting sliding mode algorithms

1 固定翼无人机模型

 图 1 固定翼无人机模型坐标系示意图 Fig. 1 Schematic diagram of referential frames configuration of fixed wing UAV model

 ${\mathit{\boldsymbol{I\dot \omega }} = - \mathit{\boldsymbol{\omega }} \times \mathit{\boldsymbol{I\omega }} + \mathit{\boldsymbol{M}}}$ （1）
 ${\mathit{\boldsymbol{ \boldsymbol{\dot \varTheta} }} = {\mathit{\boldsymbol{R}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}} \cdot \mathit{\boldsymbol{\omega }}}$ （2）
 ${\mathit{\boldsymbol{\dot v}} = (\mathit{\boldsymbol{F}} + \mathit{\boldsymbol{T}})/m + {\mathit{\boldsymbol{R}}_\mathit{\boldsymbol{I}}}\mathit{\boldsymbol{g}} - \mathit{\boldsymbol{\omega }} \times \mathit{\boldsymbol{v}}}$ （3）
 ${{{\mathit{\boldsymbol{\dot p}}}^n} = {\mathit{\boldsymbol{R}}_\mathit{\boldsymbol{I}}}\mathit{\boldsymbol{v}}}$ （4）

 ${\mathit{\boldsymbol{R}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}} = \left[ {\begin{array}{*{20}{c}} 0&{{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \phi }&{ - {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \phi }\\ 0&{\frac{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \phi }}{{{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta }}}&{\frac{{{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \phi }}{{{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta }}}\\ 1&{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \phi {\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta }&{{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \phi {\kern 1pt} {\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta } \end{array}} \right],\mathit{\boldsymbol{I}} = \left[ {\begin{array}{*{20}{c}} {{I_{xx}}}&0&{{I_{xz}}}\\ 0&{{I_{yy}}}&0\\ {{I_{zx}}}&0&{{I_{zz}}} \end{array}} \right]$
 ${\mathit{\boldsymbol{R}}_\mathit{\boldsymbol{I}}} = \left[ {\begin{array}{*{20}{c}} {{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi }&{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \phi - {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \phi }&{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \phi + {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta }\\ {{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi }&{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \phi + {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \phi }&{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \phi - {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \phi }\\ { - {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta }&{{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \phi }&{{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \phi } \end{array}} \right]$
2 基于自适应滑模算法的控制器设计

 $\begin{array}{*{20}{l}} {m\dot V = {T_x}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha {\rm{cos}}{\kern 1pt} {\kern 1pt} \beta - D - mg[ - {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha {\rm{cos}}{\kern 1pt} {\kern 1pt} \beta {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{sin}}{\kern 1pt} {\kern 1pt} \beta {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \phi {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta + {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha {\rm{cos}}{\kern 1pt} {\kern 1pt} \beta {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \phi {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta ]} \end{array}$ （5）

 图 2 固定翼无人机控制结构图 Fig. 2 Structure of control scheme of fixed wing UAV
2.1 姿态控制器设计

 ${\mathit{\boldsymbol{e}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}} = \mathit{\boldsymbol{ \boldsymbol{\varTheta} }} - {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_{\rm{d}}}$ （6）

 ${\mathit{\boldsymbol{\dot e}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}} = \mathit{\boldsymbol{ \boldsymbol{\dot \varTheta} }} - {\mathit{\boldsymbol{ \boldsymbol{\dot \varTheta} }}_{\rm{d}}} = {\mathit{\boldsymbol{R}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}} \cdot \mathit{\boldsymbol{\omega }} - {\mathit{\boldsymbol{ \boldsymbol{\dot \varTheta} }}_{\rm{d}}}$ （7）

$\dot{\boldsymbol{e}}_{\boldsymbol{\varTheta}}=\boldsymbol{z}_{\boldsymbol{\varTheta}}$，则

 ${\mathit{\boldsymbol{\dot z}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}} = {\mathit{\boldsymbol{\dot R}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}} \cdot \mathit{\boldsymbol{\omega }} + {\mathit{\boldsymbol{R}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}{\mathit{\boldsymbol{I}}^{ - 1}}( - \mathit{\boldsymbol{\omega }} \times \mathit{\boldsymbol{I\omega }} + \mathit{\boldsymbol{M}} + \Delta {\mathit{\boldsymbol{d}}_\omega }) - {\mathit{\boldsymbol{ \boldsymbol{\ddot \varTheta} }}_{\rm{d}}}$ （8）

 ${\mathit{\boldsymbol{\dot z}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}} = {\mathit{\boldsymbol{\dot R}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}} \cdot \mathit{\boldsymbol{\omega }} + {\mathit{\boldsymbol{R}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}{\mathit{\boldsymbol{I}}^{ - 1}}( - \mathit{\boldsymbol{\omega }} \times \mathit{\boldsymbol{I\omega }} + {\bf{sa}}{{\bf{t}}_M} + \Delta {\mathit{\boldsymbol{d}}_\omega }) - {\mathit{\boldsymbol{ \boldsymbol{\ddot \varTheta} }}_{\rm{d}}}$ （9）

 $\begin{array}{*{20}{l}} {{f_i}(i = 1,2,3) = }\\ {\;\:\left\{ {\begin{array}{*{20}{c}} { {\rm{sign}} ({M_i}){M_{{\rm{max}}}} - {M_i}}\\ 0 \end{array}\;\:\begin{array}{*{20}{l}} {|{M_i}| \ge {M_{{\rm{max}}}}}\\ {|{M_i}| < {M_{{\rm{max}}}}} \end{array}} \right.} \end{array}$ （10）

 $\left\{ {\begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\dot e}}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}} = {\mathit{\boldsymbol{z}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}}\\ {{{\mathit{\boldsymbol{\dot z}}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}} = {\mathit{\boldsymbol{G}}_0} + {\mathit{\boldsymbol{R}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}{\mathit{\boldsymbol{I}}^{ - 1}}\mathit{\boldsymbol{M}} + \mathit{\boldsymbol{\rho }}} \end{array}} \right.$ （11）

 ${{\mathit{\boldsymbol{G}}_0} = {{\mathit{\boldsymbol{\dot R}}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}} \cdot \mathit{\boldsymbol{\omega }} + {\mathit{\boldsymbol{R}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}{\mathit{\boldsymbol{I}}^{ - 1}}( - \mathit{\boldsymbol{\omega }} \times \mathit{\boldsymbol{I\omega }} + \mathit{\boldsymbol{f}}) - {{\mathit{\boldsymbol{ \boldsymbol{\ddot \varTheta} }}}_{\rm{d}}}}$
 ${{\eta _1} = {\rm{min}}\left( {{\eta _0},\frac{{{\alpha _1}}}{{\sqrt {2{\beta _1}} }}} \right)}$

||ρ||≤DD>0且未知。

 $\mathit{\boldsymbol{M}} = \mathit{\boldsymbol{R}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}^{ - 1}\mathit{\boldsymbol{I}}\left[ { - {\mathit{\boldsymbol{G}}_0} - {k_1}\left( {\frac{{{\mathit{\boldsymbol{e}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}}}{{\left\| {{\mathit{\boldsymbol{e}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}} \right\|}} + 0.5 \times \frac{{{\mathit{\boldsymbol{z}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}}}{{\left\| {{\mathit{\boldsymbol{z}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}} \right\|}}} \right)} \right]$ （12）

 ${\dot k_1} = \left\{ {\begin{array}{*{20}{c}} {{\alpha _1}\sqrt {\frac{{{\beta _1}}}{2}} }&{\left\| {{\mathit{\boldsymbol{e}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}} \right\| \ge {\varepsilon _1}}\\ 0&{\left\| {{\mathit{\boldsymbol{e}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}} \right\| < {\varepsilon _1}} \end{array}} \right.$ （13）

 $\left\{ {\begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\dot e}}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}} = {\mathit{\boldsymbol{z}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}}\\ {{{\mathit{\boldsymbol{\dot z}}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}} = - {k_1}\left( {\frac{{{\mathit{\boldsymbol{e}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}}}{{\left\| {{\mathit{\boldsymbol{e}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}} \right\|}} + 0.5 \times \frac{{{\mathit{\boldsymbol{z}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}}}{{\left\| {{\mathit{\boldsymbol{z}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}} \right\|}}} \right) + \mathit{\boldsymbol{\rho }}} \end{array}} \right.$ （14）

 ${V_1} = \underbrace {\left\| {{\mathit{\boldsymbol{e}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}} \right\|{\mathit{\boldsymbol{z}}^{\rm{T}}}\mathit{\boldsymbol{Az}} + \frac{1}{4}{{(\mathit{\boldsymbol{z}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}^{\rm{T}}{\mathit{\boldsymbol{z}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}})}^2}}_{{V_{10}}} + \underbrace {\frac{1}{{{\beta _1}}}{{({k_1} - k_1^*)}^4}}_{{V_{{\rm{1ad}}}}}$ （15）

V10进行稳定性分析。由不等式λmin{A}||z||2zTAzλmax{A}||z||2(λmin{A}为矩阵的A最小特征值，λmax{A}为矩阵A的最大特征值)可得

 $\begin{array}{*{20}{l}} {{V_{10}} \le \frac{3}{2}{\lambda _{{\rm{max}}}}(\mathit{\boldsymbol{A}}){{\left\| {{\mathit{\boldsymbol{e}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}} \right\|}^2} + \frac{1}{2}({\lambda _{{\rm{max}}}}(\mathit{\boldsymbol{A}}) + \frac{1}{2}){{\left\| {{\mathit{\boldsymbol{z}}_\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}} \right\|}^4}}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \le {\mathit{\boldsymbol{\chi }}^{\rm{T}}}{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{\chi }} \le {\lambda _{{\rm{max}}}}({\mathit{\boldsymbol{P}}_1})({{\left\| {{\mathit{\boldsymbol{e}}_\theta }} \right\|}^2} + {{\left\| {{\mathit{\boldsymbol{z}}_\theta }} \right\|}^4})} \end{array}$ （16）

 ${\dot V_{10}} \le - \frac{\vartheta }{{{2^{2/3}}\lambda _{{\rm{max}}}^{3/4}\{ {\mathit{\boldsymbol{P}}_1}\} }}V_{10}^{3/4} = - {\eta _0}V_{10}^{3/4}$ （17）

 ${\dot V_1} \le - {\eta _1}V_1^{3/4} + \tilde \omega$ （18）

 ${\tilde \omega = \left( { - \frac{1}{{4{\beta _1}}}{{\dot k}_1} + \frac{{{\alpha _1}}}{{4\sqrt {2{\beta _1}} }}} \right)|{k_1} - k_1^*{|^3}}$

$\dot{k}_{1}=\alpha_{1} \sqrt{\beta_{1} / 2}$时，$\tilde{\omega}$为零，则

 ${\dot V_1} \le - {\eta _1}V_1^{3/4}$ （19）

2.2 速度控制器设计

 ${\dot e_v} = \dot V - {\dot V_d} = \gamma u + {\varPhi _1} + {\varPhi _2}$ （20）

 $\left\{ {\begin{array}{*{20}{l}} {u = - \frac{1}{\gamma }{\varPhi _1} - \frac{1}{\gamma }({k_{v1}}{\xi _1}({\mathit{\boldsymbol{e}}_v}) + {z_1})}\\ {{{\dot z}_1} = - {k_{v2}}{\xi _2}({\mathit{\boldsymbol{e}}_v})} \end{array}} \right.$ （21）

ξ1(ev)和ξ2(ev)表示如下：

 $\left\{ \begin{array}{l} {\xi _1}({\mathit{\boldsymbol{e}}_v}) = \frac{{{\mathit{\boldsymbol{e}}_v}}}{{{{\left\| {{\mathit{\boldsymbol{e}}_v}} \right\|}^{1/2}}}} + {k_{v3}}{\mathit{\boldsymbol{e}}_v}\;\:{k_{v3}} > 0\\ {\xi _2}({\mathit{\boldsymbol{e}}_v}) = \frac{{{\rm{d}}{\xi _1}({\mathit{\boldsymbol{e}}_v})}}{{{\rm{d}}{\mathit{\boldsymbol{e}}_v}}}{\xi _1}({\mathit{\boldsymbol{e}}_v}) = \frac{1}{2} \cdot \frac{{{\mathit{\boldsymbol{e}}_v}}}{{\left\| {{\mathit{\boldsymbol{e}}_v}} \right\|}} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{3}{2}{k_{v3}}\frac{{{\mathit{\boldsymbol{e}}_v}}}{{{{\left\| {{\mathit{\boldsymbol{e}}_v}} \right\|}^{1/2}}}} + k_{v3}^2{\mathit{\boldsymbol{e}}_v} \end{array} \right.$ （22）

kv3>0为固定增益；kv1, kv2为大于0的自适应增益，

 $\left\{ {\begin{array}{*{20}{l}} {{{\dot k}_{v1}} = \left\{ {\begin{array}{*{20}{c}} {{{\bar k}_{v1}} \cdot \left\| {{\mathit{\boldsymbol{e}}_v}} \right\| \cdot {\rm{sign}} (\left\| {{\mathit{\boldsymbol{e}}_v}} \right\| - {\varepsilon _2})}&{{k_{v1}} > \mu }\\ \mu &{{k_{v1}} \le \mu } \end{array}} \right.}\\ {{k_{v2}} = {\eta _v}{k_{v1}}} \end{array}} \right.$ （23）

 $\left\{ {\begin{array}{*{20}{l}} {{{\dot e}_v} = - {k_{v1}}{\xi _1}({\mathit{\boldsymbol{e}}_v}) + {z_v}}\\ {{{\dot z}_v} = - {k_{v2}}{\xi _2}({\mathit{\boldsymbol{e}}_v}) + \Delta {{\dot d}_v}} \end{array}} \right.$ （24）

 ${V_2} = \underbrace {{\zeta ^{\rm{T}}}\mathit{\boldsymbol{P \boldsymbol{\varsigma} }}}_{{V_{20}}} + \underbrace {\frac{1}{{2{\gamma _1}}}{{({k_{v1}} - k_{v1}^*)}^2} + \frac{1}{{2{\gamma _2}}}{{({k_{v2}} - k_{v2}^*)}^2}}_{{V_{{\rm{2ad}}}}}$ （25）

 $\begin{array}{*{20}{l}} {{{\dot V}_2} = {{\mathit{\boldsymbol{ \boldsymbol{\dot \varsigma} }}}^{\rm{T}}}\mathit{\boldsymbol{P \boldsymbol{\varsigma} }} + {{\mathit{\boldsymbol{ \boldsymbol{\dot \varsigma} }}}^{\rm{T}}}\mathit{\boldsymbol{P \boldsymbol{\varsigma} }} + \frac{1}{{{\gamma _1}}}({k_{v1}} - k_{v1}^*){{\dot k}_{v1}} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{{{\gamma _2}}}({k_{v2}} - k_{v2}^*){{\dot k}_{v2}}} \end{array}$ （26）

$k_{2}=k_{v 2}-\frac{\Delta \dot{d}_{v}}{\xi_{2}}$，则式(26)可简化为

 ${\dot V_{20}} = - 2\xi _1^\prime {\mathit{\boldsymbol{ \boldsymbol{\varsigma} }}^{\rm{T}}}\mathit{\boldsymbol{Q\zeta }}$ （27）

 ${{q_1} = {k_{v1}}{\lambda ^2} + 4{k_{v1}}\varepsilon - \lambda {k_2} = {a_1}{k_{v1}} - \lambda {k_2}}$
 $\begin{array}{l} {q_2} = \frac{1}{2}( - {\lambda ^2} - 4\varepsilon - \lambda {k_{v1}} + {k_2}) = \\ \;\:{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{2} \cdot ( - {a_1} - \lambda {k_{v1}} + {k_2}) \end{array}$
 ${{q_3} = \lambda }$

$\dot{V}_{20}$为负定，则矩阵Q为正定，即矩阵的行列式需要大于零。为保证det(Q)>0，求根判别式λk2(kv1-λ)要大于零。由于$\left|\Delta \dot{d}_{v}\right| \leqslant \bar{\delta}$$\left\|\frac{1}{\xi_{2}}\right\| \leqslant 2$，则可得到：

 ${k_2} > 0 \Rightarrow {k_{v2}} > 2\bar \delta \;\:{k_{v1}} > \lambda$ （28）

k2的取值范围为

 ${{k}_{2}}\in [{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{k}}_{2}},{{\bar{k}}_{2}}]=[{{k}_{v2}}-2\bar{\delta },{{k}_{v2}}+2\bar{\delta }]$ （29）

 ${{k}_{v1}}=\lambda +\tau \ \;\tau >0$ （30）

 $\left\{ \begin{array}{*{35}{l}} p_{1}^{+}=\lambda {{k}_{v1}}+{{k}_{2}}+2\sqrt{\lambda {{k}_{2}}({{k}_{v1}}-\lambda )} \\ p_{1}^{-}=\lambda {{k}_{v1}}+{{k}_{2}}-2\sqrt{\lambda {{k}_{2}}({{k}_{v1}}-\lambda )} \\ \end{array} \right.$ （31）

p1∈(p1max-, p1min+)时，det[Q]>0。为保证根的存在性，需要满足p1max- < p1min+。设k2=κ2，得到

 ${{\bar{\delta }}^{2}}<\lambda \tau {{\kappa }^{2}}\Rightarrow {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{k}}_{2}}>\frac{{{{\bar{\delta }}}^{2}}}{\lambda \tau }\Rightarrow {{k}_{2}}>\frac{{{{\bar{\delta }}}^{2}}}{\lambda \tau }+2\bar{\delta }$ （32）

 ${\dot V_{20}} = - 2\xi _1^\prime {\left\| \mathit{\boldsymbol{ \boldsymbol{\varsigma} }} \right\|^2}{\lambda _{{\rm{min}}}}\{ \mathit{\boldsymbol{Q}}\} \le - {\gamma _1}V_{20}^{1/2} - {\gamma _2}{V_20}$ （33）

 ${\dot V_{20}} \le - {\gamma _1}V_{20}^{1/2}$ （34）

 $\begin{array}{l} \begin{array}{*{20}{l}} {{{\dot V}_2} \le - {\beta _\eta }V_2^{1/2} + }\\ {\underbrace {( - \frac{1}{{{\gamma _1}}} \cdot k_{v1}^ - \cdot |{\mathit{\boldsymbol{e}}_v}| \cdot {\rm{sign}} (|{\mathit{\boldsymbol{e}}_v}| - \varepsilon ) + {\beta _1}) \cdot |{k_{v1}} - k_{v1}^*|}_{{\zeta _1}} + } \end{array}\\ \underbrace {( - \frac{1}{{{\gamma _2}}} \cdot \eta \cdot k_{v1}^ - \cdot |{\mathit{\boldsymbol{e}}_v}| \cdot {\rm{ sign}} (|{\mathit{\boldsymbol{e}}_v}| - \varepsilon ) + {\beta _2}) \cdot |{k_{v2}} - k_{v2}^*|}_{{\zeta _2}} \end{array}$ （35）

 ${\gamma _1} < \frac{{{{\bar k}_{v1}} \cdot \varepsilon }}{{{\beta _1}}},{\gamma _2} < \frac{{\eta \cdot {{\bar k}_{v1}} \cdot \varepsilon }}{{{\beta _2}}}$ （36）

$\dot{V}_{2} \leqslant-\beta_{\eta} V_{2}^{1 / 2}+\zeta_{1}+\zeta_{2} \leqslant-\beta_{\eta} V_{2}^{1 / 2}$，即|ev|可在有限时间内收敛到|ev|>ε2区间内；当|ev| < ε2时，ζ1ζ2小于零，$\dot{V}_{2}$的正负不确定，增益变化率会变为-kv1·|ev|和-ηvkv1·|ev|，即增益会逐渐减小直至|ev|>ε2后，再以kv1·|ev|和ηvkv1·|ev|的斜率增长。

3 仿真分析 3.1 仿真参数设定

 $\mathit{\boldsymbol{I}} = \left[ {\begin{array}{*{20}{c}} {0.552{\kern 1pt} {\kern 1pt} {\kern 1pt} 8}&0&{0.001{\kern 1pt} {\kern 1pt} {\kern 1pt} 5}\\ 0&{0.633{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 5}&0\\ {0.001{\kern 1pt} {\kern 1pt} {\kern 1pt} 5}&0&{1.078{\kern 1pt} {\kern 1pt} {\kern 1pt} 3} \end{array}} \right]$

 姿态控制器 速度控制器 参数 数值 参数 数值 α1 12 μ 0.02 β1 2 ε2/(m·s-1) 0.001 ε1/(°) 0.002 ηv 2 kv1 10 kv3 10

3.2 仿真结果

 图 3 姿态角误差曲线 Fig. 3 Errors curves of attitude angle
 图 4 姿态角跟踪曲线 Fig. 4 Tracking curves of attitude angles

 图 5 力矩变化曲线 Fig. 5 Changing curves of control moments

 图 6 速度误差曲线 Fig. 6 Changing curves of airspeed errors
 图 7 速度跟踪曲线 Fig. 7 Tracking curves of airspeed

 图 8 速度控制器增益变化曲线 Fig. 8 Changing curves of adaptive-gain in controller

 图 9 推力变化曲线 Fig. 9 Changing curves of thrust
4 结论

http://dx.doi.org/10.7527/S1000-6893.2019.23755

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#### 文章信息

ZHANG Chaofan, DONG Qi

Adaptive-gain sliding mode control for fixed-wing UAVs with input saturation

Acta Aeronautica et Astronautica Sinica, 2020, 41(S1): 723755.
http://dx.doi.org/10.7527/S1000-6893.2019.23755