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 智能系统学报  2018, Vol. 13 Issue (2): 254-260  DOI: 10.11992/tis.201611011 0

### 引用本文

YANG Di, GUO Chen, ZHU Yuhua, et al. Neural network adaptive path tracking control for underactuated ships[J]. CAAI Transactions on Intelligent Systems, 2018, 13(2): 254-260. DOI: 10.11992/tis.201611011.

### 文章历史

1. 沈阳工业大学 化工过程自动化学院，辽宁 辽阳 111003;
2. 大连海事大学 船舶自动化仿真器研究所，辽宁 大连 116026

Neural network adaptive path tracking control for underactuated ships
YANG Di1, GUO Chen2, ZHU Yuhua1, FU Si1
1. College of Chemical Process Atomation, Shenyang University of Technology, Liaoyang 111003, China;
2. Institute of Ship Automation and Simulator, Dalian Maritime University, Dalian 116026, China
Abstract: Considering path-following problems of underactuated ships with parameter uncertainties, the nerve network technology was combined with the backstepping method for proposing a stable nerve-network adaptive control method. Firstly, based on kinematics error equations and linear transformation, auxiliary surge velocity and heading angle were determined; then the nerve network approximation technology was utilized to compensate for any uncertainties in the model, an adaptive control law was designed, so as to make actual surge velocity and heading angle converge to the auxiliary values respectively. By using the Lyapunov function, it was proved that the ultimately uniform boundedness of the error signals in the closed-loop path following system of ship. Numerical simulation results show that, the designed law can force underactuated ship to follow curve and straight path, it has strong robustness.
Key words: underactuated ship    parameter uncertainties    backstepping    adaptive control    neural networks    path following    Lyapunov function    ultimately uniform boundedness

1 船舶运动数学模型

 $\left\{ \begin{array}{l}\dot x = u\cos \psi - v\sin \psi \\\dot y = u\sin \psi + v\cos \psi \\\dot \psi = r\end{array} \right.$ (1-1)

 $\left\{ \begin{array}{l}\dot u = \displaystyle\frac{{{m_2}}}{{{m_1}}}vr - \frac{{{X_u}}}{{{m_1}}}u - \frac{{{X_{|u|u}}}}{{{m_1}}}|u|u + \frac{{{\tau _u}}}{{{m_1}}} + \frac{{{d_{wu}}}}{{{m_1}}}\\[7pt]\dot v = - \displaystyle\frac{{{m_1}}}{{{m_2}}}ur - \frac{{{Y_v}}}{{{m_2}}}v - \frac{{{Y_{|v|v}}}}{{{m_2}}}|v|v + \frac{{{d_{wv}}}}{{{m_2}}}\\[7pt]\dot r = \displaystyle\frac{{{m_1} - {m_2}}}{{{m_3}}}uv - \frac{{{N_r}}}{{{m_3}}}r - \frac{{{N_{|r|r}}}}{{{m_3}}}|r|r + \frac{{{\tau _r}}}{{{m_3}}} + \frac{{{d_{wr}}}}{{{m_3}}}\end{array} \right.$ (2)

 ${m_{i,{{Min}}}} < {m_i} < {m_{i,{{Max}}}},i = 1,2,3$ (3)

 ${{|}}{d_{wu}}{{|}} < {d_{u{{Max}}}},\;\;\;{{|}}{d_{wv}}{{|}} < {d_{v{{Max}}}},\;\;\;{{|}}{d_{wr}}{{|}} < {d_{r{{Max}}}}$ (4)

 $\left\{ \begin{array}{l}{{\dot x}_d} = {u_d}{{cos}}\,{\psi _d} - {v_d}\sin \,{\psi _d}\\{{\dot y}_d} = {u_d}\sin \,{\psi _d} + v\cos\, {\psi _d}\\{{\dot \psi }_d} = {r_d}\\{{\dot v}_d} = - \frac{{{m_1}}}{{{m_2}}}{u_d}{r_d} - \displaystyle\frac{{{Y_v}}}{{{m_2}}}{v_d} - \frac{{{Y_{|v|v}}}}{{{m_2}}}|{v_d}|{v_d}\end{array} \right.$ (5)

2 控制律的设计

2.1 运动学部分设计

 ${x_e} = x - {x_d}, \;{y_e} = y - {y_d}$ (6)

 $\begin{array}{c}{{\dot x}_{{e}}} + {k_1}\tanh ({k_{{3}}}{x_e}) = {D_x}\\{{\dot y}_{{e}}} + {k_2}\tanh ({k_{{4}}}{y_e}) = {D_y}\end{array}$ (7)

 $\begin{array}{c}{D_{{x}}} = u\cos \, \psi - v\sin \, \psi - {u_d}\cos \, {\psi _d} + {v_d}\sin \, {\psi _d} + \\\;\;\;\;\;\;\;\; {k_1}\tanh ({k_{{3}}}{x_e})\\{D_{{y}}} = u\sin \, \psi + v\cos \, \psi - {u_d}\sin \, {\psi _d} - {v_d}\cos \, {\psi _d}+ \\\;\;\;\;\;\;\;\; {k_{{2}}}\tanh ({k_{{4}}}{{{y}}_e})\end{array}$ (8)

 ${E_u} = u - {\alpha _u},\;{\psi _e} = \psi - {\psi _d},\;{E_\psi } = {\psi _e} - {\alpha _{{\psi _e}}}$ (9)

 $\begin{array}{c}\left[ {\begin{array}{*{20}{c}}{{D_x}}\\{{D_y}}\end{array}} \right] = J(\psi )\left[ {\begin{array}{*{20}{c}}{{E_u}}\\0\end{array}} \right] + J(\psi ) \times \\[9pt]\;\;\;\;\;\;\;\;\;\;\;\left[ {\begin{array}{*{20}{c}}{{\alpha _u} - \cos ({E_\psi } + {\alpha _{{\psi _e}}}){{\bar D}_x} - \sin ({E_\psi } + {\alpha _{{\psi _e}}}){{\bar D}_y}}\\[3pt]{v + \sin ({E_\psi } + {\alpha _{{\psi _e}}}){{\bar D}_x} - \cos ({E_\psi } + {\alpha _{{\psi _e}}}){{\bar D}_y}}\end{array}} \right]\end{array}$ (10)

 $\begin{array}{c}J(\psi ) = \left[ {\begin{array}{*{20}{c}}{\cos \, \psi }&{ - \sin \, \psi }\\{\sin \, \psi }&{\cos \, \psi }\end{array}} \right]\\{{\bar D}_x} = {u_d} - \cos ({\psi _d}){k_1}\tanh ({k_{{3}}}{x_e}) - \\\;\;\;\;\;\;\sin ({\psi _d}){k_2}\tanh ({k_{{4}}}{y_e})\\{{\bar D}_y} = {v_d} + \sin ({\psi _d}){k_1}\tanh ({k_{{3}}}{x_e}) - \\\;\;\;\;\;\;\cos ({\psi _d}){k_2}\tanh ({k_{{4}}}{y_e})\end{array}$ (11)

 $\begin{array}{c}{\alpha _u} = \cos \, {\alpha _{{\psi _e}}}{{\bar D}_x} + \sin \, {\alpha _{{\psi _e}}}{{\bar D}_y}\\{\alpha _{{\psi _e}}} = \theta - \varphi \end{array}$ (12)

 $\theta = \arctan ({\bar D_y}/{\bar D_x}),\phi = \arctan (v/u)$ (13)

 ${k_1} > 0,{k_2} > 0,{k_1} + {k_2} < {u_d}$ (14)

 ${\alpha _u} = \cos ({\alpha _{{\psi _e}}} - \theta )\sqrt {\bar D_x^2 + \bar D_y^2}$ (15)

 $v = u\beta ({\alpha _{{\psi _e}}})$ (16)

 $\begin{array}{c}\beta ({\alpha _{{\psi _e}}}) = ({{\bar D}_y}\cos \, {\alpha _{{\psi _e}}} - {{\bar D}_x}\sin \, {\alpha _{{\psi _e}}})/\\\;\;\;\;\;\;\;\;\;\;\;\;\;({{\bar D}_x}\cos \, {\alpha _{{\psi _e}}} + {{\bar D}_y}\sin \, {\alpha _{{\psi _e}}})\end{array}$ (17)

 $v = {\bar D_y}\cos \, {\alpha _{{\psi _e}}} - {\bar D_x}\sin \, {\alpha _{{\psi _e}}} + \beta ({\alpha _{{\psi _e}}}){E_u}$ (18)

 $\begin{array}{c}\left[ {\begin{array}{*{20}{c}}{{{\dot x}_e}}\\{{{\dot y}_e}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - {k_1}\tanh ({k_3}{x_e})}\\{ - {k_2}\tanh ({k_4}{y_e})}\end{array}} \right] + J(\psi )\left[ {\begin{array}{*{20}{c}}{{E_u}}\\{\beta ({\alpha _{{\psi _e}}}){E_u}}\end{array}} \right] + 2J(\psi )\; \times \\[9pt]\left[ {\begin{array}{*{20}{c}}{\sin ({E_\psi }/2 + {\alpha _{{\psi _e}}})}&{ - \cos ({E_\psi }/2 + {\alpha _{{\psi _e}}})}\\{\cos ({E_\psi }/2 + {\alpha _{{\psi _e}}})}&{\sin ({E_\psi }/2 + {\alpha _{{\psi _e}}})}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{{\bar D}_x}}\\{{{\bar D}_y}}\end{array}} \right] \times \\[9pt] \sin ({E_\psi }/2)\end{array}$ (19)

 ${S_\psi } = {\dot E_\psi } + {k_\psi }{E_\psi }$ (20)

 $\begin{array}{c}{{\dot E}_u} = \displaystyle\frac{1}{{{m_{1,M}}\Delta {m_1}}}{\tau _u} + {f_u}\\{{\dot S}_\psi } = \displaystyle\frac{1}{{{m_{3,M}}\Delta {m_3}}}{\tau _r} - {{\dot r}_d} + {k_\psi }(r - {r_d}) + {f_r}\end{array}$ (21)

 $\begin{array}{c}{f_u} = \displaystyle\frac{{{m_2}}}{{{m_1}}}vr - \frac{{{X_u}}}{{{m_1}}}u - \frac{{{X_{|u|u}}}}{{{m_1}}}|u|u + \frac{{{d_{wu}}}}{{{m_1}}} - {{\dot \alpha }_u}\\{f_r} = \displaystyle\frac{{{m_1} - {m_2}}}{{{m_3}}}uv - \frac{{{N_r}}}{{{m_3}}}r - \frac{{{N_{|r|r}}}}{{{m_3}}}|r|r + \frac{{{d_{wr}}}}{{{m_3}}} - {{\ddot \alpha }_{{\psi _e}}} - {k_\psi }{{\dot \alpha }_{{\psi _e}}}\\{m_{1,M}} = \sqrt {{m_{1,{{Min}}}}{m_{1,{{Max}}}}} \;\;\;{m_{3,M}} = \sqrt {{m_{3,{{Min}}}}{m_{3,{{Max}}}}} \end{array}$ (22)

 $\begin{array}{c}\displaystyle\frac{{{m_{1,{{Min}}}}}}{{{m_{1,M}}}} \leqslant \Delta {m_1} \leqslant \frac{{{m_{1,{{Max}}}}}}{{{m_{1,M}}}}\\\displaystyle\frac{{{m_{3,{{Min}}}}}}{{{m_{3,M}}}} \leqslant \Delta {m_3} \leqslant \frac{{{m_{3,{{Max}}}}}}{{{m_{3,M}}}}\end{array}$ (23)

2.2 动力学部分设计

 $\begin{array}{c}{f_u} = {{{W}}_u}^{{T}}{{\sigma}} ({\eta }) + {\varepsilon _u}\\{f_r} = {{{W}}_r}^{{T}}{{\sigma}} ({\eta }) + {\varepsilon _r}\end{array}$ (24)

 $\begin{array}{c}||{{{W}}_u}|{|_F} \leqslant {W_{u,M}},\;\;\;\;\;||{{{W}}_r}|{|_F} \leqslant {W_{r,M}}\\{\varepsilon _u} \leqslant {\varepsilon _{u,M}},\;\;\;\;\;{\varepsilon _r} \leqslant {\varepsilon _{r,M}}\end{array}$ (25)

 $\begin{array}{c}{{\hat f}_u} = {{\hat{ {W}}}^{{T}}}_u{\sigma }({\eta })\\{{\hat f}_r} = {{\hat{ {W}}}^{{T}}}_r{\sigma }({\eta })\end{array}$ (26)

 $\begin{array}{c}{{\dot E}_u} = - {k_u}{E_u} + {k_u}{E_u} + { {\hat{{W}}}_u^{{T}}}{\sigma }({{\eta} }) + \displaystyle\frac{1}{{{m_{1,M}}\Delta {m_1}}}{\tau _u} + \tilde{{{W}}_u}^{{T}}{\sigma }({\eta }) + {\varepsilon _u}\\{{\dot S}_\psi } = - {k_r}{S_\psi } + {k_r}{S_\psi } + \hat{{{W}}_r}^{{T}}{\sigma }({\eta }) + \displaystyle\frac{1}{{{m_{3,M}}\Delta {m_3}}}{\tau _r} - {{\dot r}_d} + {k_\psi }(r - \\\;\;\;\;\;\;\;\;{r_d}) + \tilde{{{W}}_r}^{{T}}\sigma ({\eta }) + {\varepsilon _u}\end{array}$ (27)

 $\begin{array}{c}{\tau _u} = {m_{1,M}}( - {k_u}{E_u} - \hat{ {{{W}}_u}}^{{T}}{\sigma }({\eta }) - \left| {\hat{ {{{W}}_u}}^{{T}}{\sigma }({\eta }) + {k_u}{E_u}} \right| \times \\\;\;\;\;\;\;\;{\mathop{ sgn}} ({E_u})(1 + {\delta _{u,{{Max}}}}))\end{array}$ (28)
 $\begin{array}{c}{\tau _r} = {m_{3,M}}( - {k_r}{S_\psi } - \hat{ {{{W}}_r}}^{{T}}{\sigma }({\eta }) + {{\dot r}_d} - {k_\psi }(r - {r_d}) - \\\;\;\;\;\;\;\left| {\hat{ {W_r}}^{{T}}{\sigma }({\eta }) + {k_r}{S_\psi } - {{\dot r}_d} + {k_\psi }(r - {r_d})} \right|{\mathop{ sgn}} ({S_\psi }) \times \\\;\;\;\;\;\;(1 + {\delta _{r,{{Max}}}}))\end{array}$ (29)
 ${\dot {\hat{ {{W}}_u}}}= {{{{F}}_u}}({E_u}{\sigma }({\eta }) - {\gamma _u}{\hat{ {{{W}}_u}}})$ (30)
 ${\dot{ \hat{{{{W}}_r}}}} = {{{{F}}_r}}({S_\psi }{\sigma }({\eta }) - {\gamma _r}{\hat{ {W}}_r})$ (31)

 ${\delta _{u,{{Max}}}} = \frac{{{m_{1,{{Max}}}}}}{{{m_{1,M}}}}, \,\, {\delta _{r,{{Max}}}} = \frac{{{m_{3,{{Max}}}}}}{{{m_{3,M}}}}$ (32)

 ${L_1} = 0.5{E_u}^{\!\!\!\! 2} + 0.5{S_\psi }^{\!\!\!\!\! 2} + 0.5\tilde{ {{{W}}_u}}^{{T}}{{{F}}_u}^{ - 1}{\tilde{ {W}}_u} + 0.5\tilde{ {{{W}}_r}}^{{T}}{{{F}}_r}^{ - 1}{\tilde{ {W}}_r}$ (33)

 $\begin{array}{c}{{\dot L}_1} = {E_u}[ - {k_u}{E_u} + {\varepsilon _u} + (1 - \displaystyle\frac{1}{{\Delta {m_1}}})({{\hat f}_u} + {k_u}{E_u}) - \displaystyle\frac{1}{{\Delta {m_1}}}\times \\[5pt] |{{\hat f}_u} + {k_u}{E_u}|{\mathop{ sgn}} ({E_u})(1 + {\delta _{u,{{max}}}})] + {S_\psi }[ - {k_r}{S_\psi } + \\ [5pt]{\varepsilon _r} + (1 - \displaystyle\frac{1}{{\Delta {m_3}}})({{\hat f}_r} + {k_r}{S_\psi } - {{\dot r}_d} + {k_4}(r - {r_d})] - \\ [5pt]\displaystyle\frac{1}{{\Delta {m_3}}}|{{\hat f}_r} + {k_r}{S_r} - {{\dot r}_d} + {k_4}(r - {r_d})|{\mathop{ sgn}} ({S_\psi }) \times \\ [5pt](1 + {\delta _{r,{{Max}}}})) + {\gamma _u}\tilde{ {{{W}}_u}}^{{T}}{\hat{ {{{W}}_u}}} + {\gamma _r}\tilde{ {{{W}}_r}}^{{T}}{\hat{ {{{W}}_r}}} \leqslant \\ [5pt]- {k_u}E_u^2 + {\varepsilon _{u,M}}|{E_u}| + \displaystyle\frac{{1 + \Delta {m_1}}}{{\Delta {m_1}}}|{{\hat f}_u} + {k_u}{E_u}||{E_u}| - \\[5pt]\displaystyle\frac{{1 + {\delta _{u,{{Max}}}}}}{{\Delta {m_1}}}|{{\hat f}_u} + {k_u}{E_u}||{E_u}| - {k_r}S_\psi ^2 + {\varepsilon _{r,M}}|{S_\psi }| + \\[5pt]\displaystyle\frac{{1 + \Delta {m_3}}}{{\Delta {m_3}}}|{{\hat f}_r} + {k_r}{S_\psi } - {{\dot r}_d} + {k_4}(r - {r_d})||{S_\psi }| - \\[5pt]\displaystyle\frac{{1 + {\delta _{r,{{Max}}}}}}{{\Delta {m_3}}}|{{\hat f}_r} + {k_r}{S_r} - {{\dot r}_d} + {k_4}(r - {r_d})||{S_\psi }| + \\[7pt]{\gamma _u}(\tilde{{{{W}}_u}}^{{T}}{{{{W}}_u}} - ||{\tilde{{{{W}}_u}}}|{|^2})\; + {\gamma _r}(\tilde{{{{W}}_r}}^{{T}}{{W}_r} - ||{\tilde{{{{W}}_r}}}|{|^2})\end{array}$ (34)

 $\begin{array}{c}{{\dot L}_1} \leqslant - {k_u}E_u^2 - {k_r}S_\psi ^2 + {\varepsilon _{u,M}}|{E_u}| + {\varepsilon _{r,M}}|{S_\psi }|\; + \\[5pt]{\gamma _u}\tilde{{{{W}}_u}}^{{T}}{\hat{{{{W}}_u}}} + {\gamma _r}\tilde{{{{W}}_r}}^{{T}}{\hat{{{{W}}_r}}}\end{array}$ (35)

 $\begin{array}{c}{{\dot L}_1} \leqslant - \displaystyle\frac{1}{2}({k_u}E_u^2 - \displaystyle\frac{{{\varepsilon ^2}_{u,M}}}{{{k_u}}}) - \frac{1}{2}({k_r}S_\psi ^2 - \displaystyle\frac{{{\varepsilon ^2}_{r,M}}}{{{k_r}}}) - \\[7pt]\displaystyle\frac{{{\gamma _u}}}{2}(||{\tilde{{{{W}}_u}}}|{|^2} - ||{{{{W}}_u}}|{|^2}) - \frac{{{\gamma _r}}}{2}(||{\tilde{{{{W}}_r}}}|{|^2} - ||{{{{W}}_r}}|{|^2}) \leqslant \\[7pt]- \eta {L_1} + (\displaystyle\frac{\eta }{2} - \displaystyle\frac{{{k_u}}}{2})E_u^2 + (\displaystyle\frac{\eta }{2} - \frac{{{k_r}}}{2})S_\psi ^2 + [\frac{\eta }{2}{\lambda _{\max }}({{{F}}_u}^{ - {{1}}}) - \\[7pt]\displaystyle\frac{{{\gamma _u}}}{2}]||{\tilde{{{{W}}_u}}}|{|^2} + [\frac{\eta }{2}{\lambda _{\max }}({F}_r^{ - 1}) - \displaystyle\frac{{{\gamma _r}}}{2}]||{\tilde{{{{W}}_r}}}|{|^2} + \mu \leqslant \\[5pt]- \eta {L_1} + \mu \end{array}$ (36)

 $\begin{array}{c}\eta \leqslant \min \left\{ {k_u},{k_r},\displaystyle\frac{{{\gamma _u}}}{{{\lambda _{\max }}({F}_u^{ - 1})}},\frac{{{\gamma _r}}}{{{\lambda _{\max }}({F}_r^{ - 1})}}\right\} \\[7pt]\mu = \displaystyle\frac{{{\gamma _u}}}{2}||{{{{W}}_u}}|{|^2} + \frac{{{\gamma _r}}}{2}||{{W}_r}|{|^2} + \frac{{{\varepsilon ^2}_{u,M}}}{{2{k_u}}} + \frac{{{\varepsilon ^2}_{r,M}}}{{2{k_\psi }}}\end{array}$ (37)

$\rho = \mu /\eta$ ，由式(34)得

 ${L_1} \leqslant \rho + ({L_1}(0) - \rho ){{{e}}^{ - \eta t}}$ (38)

 ${L_2} = \ln \{ \cosh ({k_3}{x_e})\} + \ln \{ \cosh ({k_4}{y_e})\}$ (39)

 ${\dot L_2} \leqslant - {k_1}{k_{{3}}}{\tanh ^2}({k_3}{x_e}) - {k_2}{k_{{4}}}{\tanh ^2}({k_4}{y_e}) + {\mu _2}$ (40)

 ${\mu _{{2}}} = ({k_3} + {k_4})((1 + \beta ({\alpha _{{\psi _e}}}))|{E_u}| + 4(|{\bar D_x}| + |{\bar D_y}|)|\sin ({E_\psi }/2)|)$ (41)

3 仿真研究

 $\begin{array}{c}{m_1} = 25.8\,\,{{kg}},\,{m_2} = 33.8\,\,{{kg}},\,{m_3} = 2.76\,\,{{kg}} \cdot {{{m}}^2},\\[3pt]{X_u} = 12\,\,{{kg/s}},\,{Y_v} = 17\,\,{{kg/s}},\,{N_r} = 0.5\,\,{{kg}} \cdot {{{m}}^2}/{{s}},\\[3pt]{X_{|u|u}} = 2.5\,\,{{kg/s}},\,{Y_{|v|v}} = 4.5\,{{kg/s}},\,{N_{|r|r}} = 0.1\,\,{{kg}} \cdot {{{m}}^2}\end{array}$
3.1 直线路径跟踪

 $\begin{array}{c}W' = \displaystyle\frac{{H \times S}}{{100}}\\[3pt]H = {H_0} \times (1 + {H_r} \times {{{Rand}}_b}) \times A{H_0}\end{array}$

 Download: 图 3 制输入 ${\tau _u}\text{、}{\tau _r}$ 对比曲线 Fig. 3 Comparsion of control efforts ${\tau _u},{\tau _r}$
3.2 曲线路径跟踪

 Download: 图 6 制输入 ${\tau _u}\text{、}{\tau _r}$ 对比曲线 Fig. 6 Comparsions of control efforts ${\tau _u},{\tau _r}$