﻿ 基于模糊自适应滑模方法的AUV轨迹跟踪控制
 舰船科学技术  2017, Vol. 39 Issue (12): 53-58 PDF

1. 三峡大学 水电机械设备设计与维护湖北省重点实验室，湖北 宜昌 443002;
2. 中国船舶重工集团公司第七一〇研究所，湖北 宜昌 443003

Trajectory-tracking control of autonomous underwater vehicles based on fuzzy adaptive sliding mode method
SUN Qiao-mei1, CHEN Jin-guo2, YU Wan2
1. Hubei Key Laboratory of Hydroelectric Machinery Design and Maintenance, China Three Gorges University, Yichang 443002, China;
2. The 710 Research Institute of CSIC, Yichang 443003, China
Abstract: In this paper, a control system is developed based on fuzzy adaptive sliding mode method for Autonomous Underwater Vehicles (AUV) precisely trajectory-tracking. With AUV dynamic model established, variable structure sliding mode strategy and fuzzy logic methods are adopted to design the path tracking controller. Environmental disturbances are compensated by the adaptive fuzzy algorithm. The stability of the control system is discussed using Lyapunov stability theory. Numerical simulations are conducted on Matlab/Simulink and show that the proposed approach can achieve smooth continuous outputs in the condition of environment disturbances. The ability of restraining interference is enhanced. The performance of the proposed control law is evaluated by simulation results.
Key words: trajectory tracking     sliding mode strategy     fuzzy control     adaptive control
0 引　言

AUV轨迹跟踪控制的目标是设计有效的控制律，使其从初始状态跟踪参考轨迹，并保证跟踪位置误差的全局一致渐进稳定[12]。目前的研究成果采用的控制方法主要有传统PID控制方法、滑模控制方法、反演控制方法、神经网络法。由于传统PID参数需要适应模型参数的变化，而AUV动力学模型参数存在不确定性，因此很难满足需要。

1 AUV数学模型

 $\left\{ \begin{array}{l}{\dot{\eta }} = {\mathit{\boldsymbol{J}}}\left( {\mathit{\boldsymbol{\eta }}} \right){\mathit{\boldsymbol{\nu }}}\text{，}\\{\mathit{\boldsymbol{M\dot \nu }}} + {\mathit{\boldsymbol{C}}}\left( {\mathit{\boldsymbol{\nu }}} \right){\mathit{\boldsymbol{\nu }}} + {\mathit{\boldsymbol{D}}}\left( {\mathit{\boldsymbol{\nu }}} \right){\mathit{\boldsymbol{\nu }}} + {\mathit{\boldsymbol{g}}}\left( {\mathit{\boldsymbol{\eta }}} \right) = {\mathit{\boldsymbol{\tau }}} + {\mathit{\boldsymbol{d}}}\text{。}\end{array} \right.$ (1)

 ${{\mathit{\boldsymbol{M}}}_\eta }\left( {\mathit{\boldsymbol{\eta }}} \right){\ddot{\eta }} + {{\mathit{\boldsymbol{C}}}_\eta }\left( {{\mathit{\boldsymbol{\nu }}},{\mathit{\boldsymbol{\eta }}}} \right){\dot{\eta }} + {{\mathit{\boldsymbol{D}}}_\eta }\left( {{\mathit{\boldsymbol{\nu }}},{\mathit{\boldsymbol{\eta }}}} \right){\dot{\eta }} + {{\mathit{\boldsymbol{g}}}_\eta }\left( {\mathit{\boldsymbol{\eta }}} \right) = {{\mathit{\boldsymbol{\tau }}}_\eta }\text{。}$ (2)

${{\mathit{\boldsymbol{M}}}_\eta }\left( {\mathit{\boldsymbol{\eta }}} \right) = {{\mathit{\boldsymbol{J}}}^{ - {\rm T}}}\left( {\mathit{\boldsymbol{\eta }}} \right){\mathit{\boldsymbol{M}}}{{\mathit{\boldsymbol{J}}}^{ - 1}}\left( {\mathit{\boldsymbol{\eta }}} \right)$

${{\mathit{\boldsymbol{C}}}_\eta }\left( {{\mathit{\boldsymbol{\nu }}},{\mathit{\boldsymbol{\eta }}}} \right) = {{\mathit{\boldsymbol{J}}}^{ - {\rm T}}}\left( {\mathit{\boldsymbol{\eta }}} \right)\left[ {{\mathit{\boldsymbol{C}}}\left( {\mathit{\boldsymbol{\nu }}} \right) - {\mathit{\boldsymbol{M}}}{{\mathit{\boldsymbol{J}}}^{ - 1}}\left( {\mathit{\boldsymbol{\eta }}} \right){\dot{J}}\left( {\mathit{\boldsymbol{\eta }}} \right)} \right]{{\mathit{\boldsymbol{J}}}^{ - 1}}\left( {\mathit{\boldsymbol{\eta }}} \right)$

${{\mathit{\boldsymbol{g}}}_\eta }\left( {\mathit{\boldsymbol{\eta }}} \right) = {{\mathit{\boldsymbol{J}}}^{ - {\rm T}}}\left( {\mathit{\boldsymbol{\eta }}} \right){\mathit{\boldsymbol{g}}}\left( {\mathit{\boldsymbol{\eta }}} \right)$

${{\mathit{\boldsymbol{D}}}_\eta }\left( {{\mathit{\boldsymbol{\nu }}},{\mathit{\boldsymbol{\eta }}}} \right) = {{\mathit{\boldsymbol{J}}}^{ - {\rm T}}}\left( {\mathit{\boldsymbol{\eta }}} \right){\mathit{\boldsymbol{D}}}\left( {\mathit{\boldsymbol{\nu }}} \right){{\mathit{\boldsymbol{J}}}^{ - 1}}\left( {\mathit{\boldsymbol{\eta }}} \right)$

${{\mathit{\boldsymbol{\tau }}}_\eta } = {{\mathit{\boldsymbol{J}}}^{ - {\rm T}}}\left( {\mathit{\boldsymbol{\eta }}} \right){\mathit{\boldsymbol{\tau }}}$

2 轨迹跟踪控制器设计

 ${\mathit{\boldsymbol{\sigma }}} = \left[ {{{\mathit{\boldsymbol{s}}}_1}\;\;{{\mathit{\boldsymbol{s}}}_2}} \right]\left[ \begin{array}{l}{{\mathit{\boldsymbol{e}}}_1}\\{{\mathit{\boldsymbol{e}}}_2}\end{array} \right] = \left[ {{{\mathit{\boldsymbol{s}}}_1}\;\;{{\mathit{\boldsymbol{s}}}_2}} \right]\left[ \begin{array}{l}{\mathit{\boldsymbol{\nu }}} - {{\mathit{\boldsymbol{\nu }}}_d}\\{\mathit{\boldsymbol{\eta }}} - {{\mathit{\boldsymbol{\eta }}}_d}\end{array} \right]\text{，}$ (3)

 $V = \frac{1}{2}{{\mathit{\boldsymbol{\sigma }}}^{\rm T}}{\mathit{\boldsymbol{\sigma }}}\text{。}$ (4)

 $\dot V = {{\dot{\sigma }}^{\rm T}}{\mathit{\boldsymbol{\sigma }}} < 0\text{。}$ (5)

 ${\dot \sigma _i} = - {k_i}{\mathop{\rm sgn}} \left( {{\sigma _i}} \right)\;\;\text{，}i = 1,2,3\text{。}$ (6)

 ${\dot \sigma _i} = - {k_i}\tanh \left( {{\sigma _i}/{\phi _i}} \right)\;\;\text{，}i = 1,2,3\text{，}$ (7)

 $\begin{array}{l}{\bf{\dot \sigma }} = \left[ {{{\bf{s}}_1}\;\;{{\bf{s}}_2}} \right]\left[ \begin{array}{l}{{\bf{M}}^{ - 1}}{\bf{\tau }} - {{\bf{M}}^{ - 1}}{\bf{C}}\left( {\bf{\nu }} \right){\bf{\nu }} - {{\bf{M}}^{ - 1}}{\bf{D}}\left( {\bf{\nu }} \right){\bf{\nu }} - \\{{\bf{M}}^{ - 1}}{\bf{g}}\left( {\bf{\eta }} \right) + {{\bf{M}}^{ - 1}}{\bf{d}} - {{{\bf{\dot \nu }}}_d}\\{\bf{J}}\left( {\bf{\eta }} \right){\bf{\nu }} - {{{\bf{\dot \eta }}}_d}\end{array} \right]\;\\ \;\;\;\;\;= - {\bf{F}}\left( {{\bf{\sigma }},\phi } \right)\end{array}$ (8)
 $\begin{array}{l}{{\mathit{\boldsymbol{s}}}_1}\left[ {{M^{ - 1}}\left( {{\mathit{\boldsymbol{\tau }}} - {\mathit{\boldsymbol{C}}}\left( {\mathit{\boldsymbol{\nu }}} \right){\mathit{\boldsymbol{\nu }}} - {\mathit{\boldsymbol{D}}}\left( {\mathit{\boldsymbol{\nu }}} \right){\mathit{\boldsymbol{\nu }}} - {\mathit{\boldsymbol{g}}}\left( {\mathit{\boldsymbol{\eta }}} \right) + {\mathit{\boldsymbol{d}}}} \right) - {{{\mathit{\boldsymbol{\dot \nu }}}}_d}} \right] + \\\;\;\;\;\;\;\;\;\;\;\;\;\;{{\mathit{\boldsymbol{s}}}_2}\left[ {{\mathit{\boldsymbol{J}}}\left( {\mathit{\boldsymbol{\eta }}} \right){\mathit{\boldsymbol{\nu }}} - {{{\mathit{\boldsymbol{\dot \eta }}}}_d}} \right] = - {\mathit{\boldsymbol{F}}}\left( {{\mathit{\boldsymbol{\sigma }}},\phi } \right)\text{。}\end{array}$ (9)

 ${{\mathit{\boldsymbol{\tau }}}_{smc}} = {{\mathit{\boldsymbol{\tau }}}_1} + {{\mathit{\boldsymbol{\tau }}}_2} + {{\mathit{\boldsymbol{\tau }}}_3}\text{，}$ (10)
 ${{\bf{\tau }}_1}\! =\! {\left( {{{\mathit{\boldsymbol{s}}}_1}{{{\mathit{\boldsymbol{\hat M}}}}^{ - 1}}} \right)^{ - 1}}\left[ \begin{array}{l}\!\!{{\mathit{\boldsymbol{s}}}_1}{{{\mathit{\boldsymbol{\hat M}}}}^{ - 1}}{\mathit{\boldsymbol{\hat {C}}}}\left( {\bf{\nu }} \right){\bf{\nu }} + {{\mathit{\boldsymbol{s}}}_1}{{{\mathit{\boldsymbol{\hat {M}}}}}^{ - 1}}{\mathit{\boldsymbol{\hat {D}}}}\left( {\bf{\nu }} \right){\bf{\nu }} + \\\!\!{{\mathit{\boldsymbol{s}}}_1}{{\mathit{\boldsymbol{M}}}^{ - 1}}{\mathit{\boldsymbol{\hat g}}}\left( {\bf{\eta }} \right) - {{\mathit{\boldsymbol{s}}}_1}{{{\mathit{\boldsymbol{\hat {M}}}}}^{ - 1}}{\mathit{\boldsymbol{\hat {d}}}} + {{\mathit{\boldsymbol{s}}}_1}{{{\bf{\dot \nu }}}_d}\end{array} \right]\text{，}$ (11)
 ${{\mathit{\boldsymbol{\tau }}}_2} = {\left( {{{\mathit{\boldsymbol{s}}}_1}{{{\hat{M}}}^{ - 1}}} \right)^{ - 1}}\left[ {{{\mathit{\boldsymbol{s}}}_2}{{{\dot{\eta }}}_d} - {{\mathit{\boldsymbol{s}}}_2}{\mathit{\boldsymbol{J}}}\left( {\mathit{\boldsymbol{\eta }}} \right){\mathit{\boldsymbol{\nu }}}} \right]\text{，}$ (12)
 ${{\mathit{\boldsymbol{\tau }}}_3} = - {\left( {{{\mathit{\boldsymbol{s}}}_1}{{{\hat{M}}}^{ - 1}}} \right)^{ - 1}}{\mathit{\boldsymbol{F}}}\left( {{\mathit{\boldsymbol{\sigma }}},\phi } \right)\text{。}$ (13)
3 模糊系统设计及稳定性分析

 ${\rm {if}}\;{\sigma _i}\;{\rm {is}}\;{A_{ir}}\;{\rm {is}}\;{\hat d_{\rm {then}}} = {\hat B_{ir}}\text{。}$ (14)

 ${\hat d_i}({\sigma _i}) = \left( {\sum\limits_{r = 1}^R {{w_{ir}} \cdot {{\hat B}_{ir}}} } \right)/\sum\limits_{r = 1}^R {{w_{ir}}} \text{。}$ (15)

 ${\hat d_i}({\sigma _i}) = {{\hat{B}}^{\rm T}}_i{{\mathit{\boldsymbol{\varepsilon }}}_i}\left( {{\sigma _i}} \right)\text{，}$ (16)

${{\hat{B}}^{\rm T}}_i = \left[ {{{\hat B}_{i1}},{{\hat B}_{i2}}, \cdots ,{{\hat B}_{iR}}} \right]$ ${{\mathit{\boldsymbol{\varepsilon }}}_i}\left( {{\sigma _i}} \right) = \left[ {{\varepsilon _{i1}},{\varepsilon _{i2}}, \cdots ,{\varepsilon _{iR}}} \right]$ ${\varepsilon _{ir}}\left( {{\sigma _i}} \right) = {w_{ir}}/\displaystyle\sum\limits_{r = 1}^R {{w_{ir}}}$

 ${\dot {\mathop {\mathit{\boldsymbol{B}}} \limits^ \frown }^{}}_i = - {\theta _i}{\sigma _i}{{\bf{\varepsilon }}_i}\left( {{\sigma _i}} \right)\text{，}$ (17)

 ${{\hat{B}}^*}_i = \arg \;\min \left[ {\sup \left| {{{\hat B}_i}({\sigma _i}) - {d_i}} \right|} \right]\text{。}$ (18)

${\mathit{\boldsymbol{\Delta }}} = {\hat{d}} - {{\hat{D}}^*}$ ，Lyapunov函数设为

 $V = \frac{1}{2}\left( {{{\mathit{\boldsymbol{\sigma }}}^{\rm T}}{\mathit{\boldsymbol{\sigma }}} + \frac{1}{\gamma }{{\mathit{\boldsymbol{\Delta }}}^{\rm T}}{\mathit{\boldsymbol{\Delta }}}} \right)\text{，}$ (19)

 $\begin{split}&\dot V = {{{\bf{\dot \sigma }}}^{\rm T}}{\bf{\sigma }} + \displaystyle\frac{1}{\gamma }{{{\bf{\dot \Delta }}}^{\rm T}}{\bf{\Delta }}=\\& \left\{ \begin{array}{l}{{\bf{s}}_1}\left[ {{{\bf{M}}^{ - 1}}\left( {{\bf{\tau }} \!\!-\!\! {\bf{C}}\left( {\bf{\nu }} \right){\bf{\nu }} \!\!-\!\! {\bf{D}}\left( {\bf{\nu }} \right){\bf{\nu }}\!\! -\!\! {\bf{g}}\left( {\bf{\eta }} \right) + {\bf{d}}} \right) - {{{\bf{\dot \nu }}}_d}} \right] \!\!+\!\! \;\\{{\bf{s}}_2}\left[ {{\bf{J}}\left( {\bf{\eta }} \right){\bf{\nu }} - {{{\bf{\dot \eta }}}_d}} \right]\end{array} \right\}{\bf{\sigma }} +\\& \displaystyle\frac{1}{\gamma }{{{\bf{\dot \Delta }}}^{\rm T}}{\bf{\Delta }}\text{。}\end{split}$ (20)

 $\begin{split}\dot V &\leqslant - [{{\mathit{\boldsymbol{s}}}_1}{{\mathit{\boldsymbol{M}}}^{ - 1}}{\left( {{{\mathit{\boldsymbol{s}}}_1}{{{\mathit{\boldsymbol{\hat {M}}}}}^{ - 1}}} \right)^{ - 1}}{\mathit{\boldsymbol{K}}}{\mathop{\rm sgn}} ({\bf{\sigma }}) - {{\mathit{\boldsymbol{s}}}_1}{{\mathit{\boldsymbol{M}}}^{ - 1}}\times\\&\;\;\;\;\;\;\; \left( {{\mathit{\boldsymbol{\hat {C}}\nu }} - {\mathit{\boldsymbol{{C}}\nu }} + {\mathit{\boldsymbol{\hat {D}}\nu }} - {\mathit{\boldsymbol{{D}}\nu }} + {\mathit{\boldsymbol{\hat {g}}}} - {\mathit{\boldsymbol{g}}} - {{\bf{\Delta }}^{\rm T}}{\bf{\varepsilon }} + {\bf{\xi }}} \right)-\\&\;\;\;\; \left( {{{\mathit{\boldsymbol{s}}}_1}{{\mathit{\boldsymbol{M}}}^{ - 1}}{\mathit{\boldsymbol{\hat {M}}}}{{{\bf{\dot \nu }}}_d} - {{\mathit{\boldsymbol{s}}}_1}{{{\bf{\dot \nu }}}_d}} \right) - \left( {{{\mathit{\boldsymbol{s}}}_1}{{\mathit{\boldsymbol{M}}}^{ - 1}}{\mathit{\boldsymbol{\hat {M}}}}{S_1}^{ - 1} - I} \right)\times\\&\;\;\;\;\;\;\;\;\;{{\mathit{\boldsymbol{s}}}_2}\left( {{{{\bf{\dot \eta }}}_d} - {\mathit{\boldsymbol{{J}}\nu }}} \right)]{\bf{\sigma }} + \displaystyle\frac{1}{\gamma }{{\bf{\Delta }}^{\rm T}}\dot {\mathop {\mathit{\boldsymbol{B}}} \limits^ \frown }=\\& - [{{\mathit{\boldsymbol{s}}}_1}{{\mathit{\boldsymbol{M}}}^{ - 1}}{\left( {{{\mathit{\boldsymbol{s}}}_1}{{{\mathit{\boldsymbol{\hat {M}}}}}^{ - 1}}} \right)^{ - 1}}{\mathit{\boldsymbol{K}}}{\mathop{\rm sgn}} ({\bf{\sigma }}) - {{\mathit{\boldsymbol{s}}}_1}{{\mathit{\boldsymbol{M}}}^{ - 1}}\times\\&\;\;\;\;\;\;\;\left( {{\mathit{\boldsymbol{\hat {C}}\nu }} - {\mathit{\boldsymbol{{C}}\nu }} + {\mathit{\boldsymbol{\hat {D}}\nu }} - {\mathit{\boldsymbol{{D}}\nu }} + {\mathit{\boldsymbol{\hat {g}}}} - {\mathit{\boldsymbol{g}}} + {\bf{\xi }}} \right)-\\&\;\;\;\; \left( {{{\mathit{\boldsymbol{s}}}_1}{{\mathit{\boldsymbol{M}}}^{ - 1}}{\mathit{\boldsymbol{\hat {M}}}}{{{\bf{\dot \nu }}}_d} - {{\mathit{\boldsymbol{s}}}_1}{{{\bf{\dot \nu }}}_d}} \right) - \left( {{{\mathit{\boldsymbol{s}}}_1}{{\mathit{\boldsymbol{M}}}^{ - 1}}{\mathit{\boldsymbol{\hat {M}}}}{S_1}^{ - 1} - I} \right) \times\\&\;\;\;\;\;\;\;\;\;{{\mathit{\boldsymbol{s}}}_2}\left( {{{{\bf{\dot \eta }}}_d} - {\mathit{\boldsymbol{{J}}\nu }}} \right)]{\bf{\sigma }} + {{\bf{\Delta }}^{\rm T}}{\bf{\varepsilon \sigma }}\left( {{{\mathit{\boldsymbol{s}}}_1}{{\mathit{\boldsymbol{M}}}^{ - 1}} - \displaystyle\frac{1}{\gamma }{\bf{\theta }}} \right)\text{。}\end{split}$ (21)

$\gamma = {\left( {{{\mathit{\boldsymbol{s}}}_1}{{\mathit{\boldsymbol{M}}}^{ - 1}}} \right)^{ - 1}}{\mathit{\boldsymbol{\theta }}}$ ，则有

 $\begin{split}\dot V \leqslant& - [{{\mathit{\boldsymbol{s}}}_1}{{\mathit{\boldsymbol{M}}}^{ - 1}}{\mathit{\boldsymbol{\hat {M}}}}{{\mathit{\boldsymbol{s}}}_1}^{ - 1}{\mathit{\boldsymbol{K}}}{\mathop{\rm sgn}} ({\bf{\sigma }}) - {{\mathit{\boldsymbol{s}}}_1}{{\mathit{\boldsymbol{M}}}^{ - 1}}\times\\&\left( {{\mathit{\boldsymbol{\hat {C}}\nu }} - {\mathit{\boldsymbol{{C}}\nu }} + {\mathit{\boldsymbol{\hat {D}}\nu }} - {\mathit{\boldsymbol{{D}}\nu }} + {\mathit{\boldsymbol{\hat {g}}}} - {\mathit{\boldsymbol{g}}} + {\bf{\xi }}} \right)-\\&\;\;\;\;\;\;\; \left( {{{\mathit{\boldsymbol{s}}}_1}{{\mathit{\boldsymbol{M}}}^{ - 1}}{\mathit{\boldsymbol{\hat {M}}}}{{{\bf{\dot \nu }}}_d} - {{\mathit{\boldsymbol{s}}}_1}{{{\bf{\dot \nu }}}_d}} \right) - \left( {{{\mathit{\boldsymbol{s}}}_1}{{\mathit{\boldsymbol{M}}}^{ - 1}}{\mathit{\boldsymbol{\hat {M}}}}{S_1}^{ - 1} - I} \right)\times\\&{{\mathit{\boldsymbol{s}}}_2}\left( {{{{\bf{\dot \eta }}}_d} - {\mathit{\boldsymbol{{J}}\nu }}} \right)]{\bf{\sigma }}\text{。}\end{split}$ (22)

 $\begin{split}K \geqslant &{{\mathit{\boldsymbol{s}}}_1}{{{\mathit{\boldsymbol{\hat {M}}}}}^{ - 1}}\left( {{\bf{\alpha }} + {\bf{\beta }} + {\bf{\kappa }} + {\bf{\xi }}} \right)\; + \left( {{{\mathit{\boldsymbol{s}}}_1} - {{\left( {{{\mathit{\boldsymbol{M}}}^{ - 1}}{\mathit{\boldsymbol{\hat {M}}}}} \right)}^{ - 1}}} \right){{{\bf{\dot \nu }}}_d} + \\&\left( {{{\mathit{\boldsymbol{s}}}_1}{{\mathit{\boldsymbol{M}}}^{ - 1}}{\mathit{\boldsymbol{\hat {M}}}}{{\mathit{\boldsymbol{s}}}_1}^{ - 1} - I} \right){{\mathit{\boldsymbol{s}}}_2}{{{\bf{\dot e}}}_2}\text{。}\end{split}$ (23)
 图 1 隶属度函数 Fig. 1 Degree of membership function

 图 2 外界干扰 Fig. 2 External disturbances
4 仿真验证

 $\begin{split}{\bf{d}}\left( t \right) =& [{\rm{0}}{\rm{.1sin(0}}{\rm{.1}}t - \displaystyle\frac{\pi }{4}{\rm{),0}}{\rm{.1sin(0}}{\rm{.1}}t - \displaystyle\frac{\pi }{6}{\rm{)}},\;\\&{\rm{0}}{\rm{.1sin(0}}{\rm{.1}}t - \displaystyle\frac{\pi }{3}{\rm{)}}]\end{split}$ (24)

 $\begin{split}\!\!\!\!\left( {x,y,\psi } \right) \!=\! ({\rm{3sin(0}}{\rm{.04}}\pi {{t)}},{\rm{3cos(0}}{\rm{.04}}\pi {{t)}},{\rm{0}}{\rm{.3sin(}}\pi {{t/60)}})\text{。}\!\!\!\!\end{split}$ (25)

 图 3 滑模控制下AUV水平面轨迹 Fig. 3 AUV horizontal path using sliding mode method

 图 4 滑模控制AUV的跟踪曲线 Fig. 4 AUV trajectory curve using sliding mode method

 图 5 滑模控制器输出 Fig. 5 Sliding mode control output

 图 6 滑模控制下航迹误差 Fig. 6 Track error using sliding mode method

 图 7 模糊自适应滑模控制下AUV水平面轨迹 Fig. 7 AUV horizontal path using fuzzy adaptive SMC

 图 8 模糊自适应滑模控制AUV的跟踪曲线 Fig. 8 AUV trajectory curve using fuzzy adaptive SMC

 图 9 模糊自适应滑模控制器输出 Fig. 9 Fuzzy adaptive SMC control output

 图 10 模糊自适应滑模控制下航迹误差 Fig. 10 Track error using fuzzy adaptive SMC

5 结　语

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