舰船科学技术  2017, Vol. 39 Issue (10): 66-69 PDF

Dynamic surface second order sliding mode control based on approximation for course-keeping control of ship
ZHANG Kai, LI Tie-shan, ZHAO Rong
Dalian Maritime University, Navigational College, Dalian 116026, China
Abstract: In this paper, a novel dynamic surface second order sliding model control method is proposed for course-keeping control of ship in the presence of uncertain errors. The controller is constructed by " dynamic surface control” technique to solve the problems of " explosion of complexity” in the traditional Lyapunov stability theory. A novel second order sliding model control method is proposed in this paper, which is not only capable of strengthening robustness of the system, but also attenuating inherent chattering of classical sliding mode control method effectively. And then the radial basis function neural network approximation technique is used for approximating modeling errors, meanwhile the " minimum learning parameter” technique is used to reduce the computational burden of the algorithm. The controller guarantees that all the close-loop signals are uniform ultimate bounded (UUB) and that the tracking er-rors converge to a small neighborhood of the desired trajectory. Finally, simulation results are given to illustrate the effectiveness of the proposed algorithm.
Key words: ship course control     dynamic surface control (DSC)     second order sliding mode control     radial basis function neural network (RBENN)     minimum learning parameter (MLP)
0 引　言

1 问题的描述

 $\ddot \varphi + \frac{1}{T}H\left( {\dot \varphi } \right) = \frac{K}{T}\delta + \Delta ,$ (1)

 $H\left( {\dot \varphi } \right) = {e_1}\dot \phi + {e_2}{\dot \phi ^3},$ (2)

 ${\dot x_1} = {x_2},$ (3)
 ${\dot x_2} = {f_2}\left( {{{\bar x}_2}} \right) + \frac{K}{T}u + \Delta {\text{。}}$ (4)

2 控制器设计

 ${z_{_1}} = {x_{_1}} - {x_d},$ (5)
 ${\dot z_{_1}} = {x_{_2}} - {\dot x_d}{\text{。}}$ (6)

 ${V_1} = \frac{1}{2}z_1^2,$ (7)
 ${\dot V_1} = {z_1}{\dot z_1}{\text{。}}$ (8)

 $\tau {\dot a_1} + {a_1} = a_1^0,\;\;\;{a_1}\left( 0 \right) = a_1^0\left( 0 \right){\text{。}}$ (9)

 ${h_1} = {a_1} - a_1^0{\text{。}}$ (10)

 ${z_2} = {x_2} - {a_1},$ (11)

 ${\dot z_2} = {\dot x_2} - {\dot a_1}{\text{。}}$ (12)

 ${\dot z_1} = {z_2} + {a_1} - {\dot x_d}{\text{。}}$ (13)

 ${V_2} = {V_1} + \frac{1}{2}z_2^2 + \frac{1}{2}h_1^2{\text{。}}$ (14)

 $\begin{split}\!\!\!\!\! {{\dot V}_2} & = {z_1}({z_2} + {h_1} + a_1^0 - {{\dot x}_d}) + {z_2}({f_2}({{\bar x}_2}) + \frac{K}{T}u + \Delta - {{\dot a}_1}) + \!\!\!\!\! \\& \quad{h_1}(\frac{{ - {h_1}}}{\tau } + {c_1}{{\dot z}_1} - {{\ddot x}_d}) = \\& \quad {z_1}({z_2} + {h_1} + a_1^0 - {{\dot x}_d}) + {z_2}({f_2}({{\bar x}_2}) + \frac{K}{T}u + \Delta - {{\dot a}_1}) + \!\!\!\!\! \\& \quad{h_1}(\frac{{ - {h_1}}}{\tau } + {B_2}){\text{。}}\end{split}$ (15)

 $u = {u_{st}} + {u_{eq}}{\text{。}}$ (16)

 ${u_{st}} = \int { - p{\mathop{\rm sgn}} ({z_2})} - k{\left| {{z_2}} \right|^{0.5}}{\mathop{\rm sgn}} ({z_2}),$ (17)
 $u_{eq}^* = \frac{T}{K}( - {c_2}{z_2} - {f_2}({\bar x_2}) - \Delta + {\dot a_1}){\text{。}}$ (18)

 $\Delta = {\bar S_2}\left( {{{\bar x}_2}} \right){A_2}{\bar x_2} = {\bar S_2}\left( {{{\bar x}_2}} \right){\omega _2}{b_2} + {\bar S_2}\left( {{{\bar x}_2}} \right){A_2}{\left[ {{x_d}\,,\,{a_1}} \right]^{\rm T}},$ (19)

 $\begin{split}& {z_2}{{\bar S}_2}\left( {{{\bar x}_2}} \right){A_2}{\left[ {{x_d}\,,\,{a_1}} \right]^{\rm T}} \leqslant \lambda _2^{\rm{T}}\left\| {{{\bar S}_2}\left( {{{\bar x}_2}} \right)} \right\|\left| {{z_2}} \right| \leqslant \\& \qquad \tilde \lambda _2^{\rm{T}}\left\| {{{\bar S}_2}\left( {{{\bar x}_2}} \right)} \right\|\left| {{z_2}} \right| + \hat \lambda _2^{\rm{T}}\left\| {{{\bar S}_2}\left( {{{\bar x}_2}} \right)} \right\|\left| {{z_2}} \right|,\end{split}$ (20)
 $\begin{split}& {z_2}{{\bar S}_2}\left( {{{\bar x}_2}} \right){\omega _2}{b_2} \leqslant \frac{{z_2^2{{\bar S}_2}\left( {{{\bar x}_2}} \right)\bar S_2^{\rm T}\left( {{{\bar x}_2}} \right)\tilde \lambda _2^{\rm{T}}}}{{4r_2^2}} + \\& \qquad\frac{{z_2^2{{\bar S}_2}\left( {{{\bar x}_2}} \right)\bar S_2^{\rm T}\left( {{{\bar x}_2}} \right)\hat \lambda _2^{\rm T}}}{{4r_2^2}} + r_2^2\omega _2^{\rm T}{\omega _2},\end{split}$ (21)

 $\begin{split}u & = \int { - p{\mathop{\rm sgn}} ({z_2})} - k{\left| {{z_2}} \right|^{0.5}}{\mathop{\rm sgn}} ({z_2}) + \\& \quad \frac{\rm T}{K}\left( { - {k_2}{z_2} - {f_2}\left( {{{\bar x}_2}} \right) + {{\dot a}_2}} \right) - \\& \quad \frac{\rm T}{K}\left( {\frac{{{z_2}{{\bar S}_2}\left( {{{\bar x}_2}} \right)\bar S_2^{\rm T}\left( {{{\bar x}_2}} \right)\hat \lambda _2^{\rm T}}}{{4r_2^2}} - \hat \lambda _2^{\rm T}\left\| {{{\bar S}_2}\left( {{{\bar x}_2}} \right)} \right\|{\mathop{\rm sgn}} \left( {{z_2}} \right)} \right){\text{。}}\!\!\!\!\end{split}$ (22)
3 稳定性分析

 ${V_3} = {V_2} + \frac{1}{2}\tilde \lambda _2^{\rm T}\varGamma _2^{ - 1}{\tilde \lambda _2},$ (23)

 ${\dot V_3} = {\dot V_2} - \tilde \lambda _2^{\rm{T}}\Gamma _2^{ - 1\dot \wedge }{\lambda _2}$ (24)

 ${c_1} \geqslant 1 \! + \! {a_0} \! + \! r_2^2,{r_2} > 0,{c_2} \geqslant \frac{1}{2} \! + \! {a_0} \! + \! r_2^2,\frac{1}{\tau } \geqslant \frac{1}{2} \! + \! \frac{{M_2^2}}{2} \! + \! {a_0}{\text{。}}$

 $\begin{array}{l} {{\dot V}_3} \le - {a_0}z_1^2 + \frac{1}{2} - r_2^2z_1^2 - {a_0}z_2^2 - ({a_0} + \frac{{M_2^2}}{2} - \frac{{B_2^2}}{2})h_1^2 + \\ \frac{{z_2^2{{\bar S}_2}\left( {{{\bar x}_2}} \right)\bar S_2^{\rm{T}}\left( {{{\bar x}_2}} \right)\tilde \lambda _2^{\rm{T}}}}{{4r_2^2}} + r_2^2\omega _2^{\rm{T}}{\omega _2} + \tilde \lambda _2^{\rm{T}}\left\| {{{\bar S}_2}\left( {{{\bar x}_2}} \right)} \right\|\left| {{z_2}} \right| - \\ \tilde \lambda _2^{\rm{T}}{\rm{ }}\Gamma _2^{ - 1\dot \wedge }{\lambda _2} - \frac{K}{{\rm{T}}}\left( {z_2^{\rm{T}}(\int { - p{\rm{sgn}}({z_2})} ) - k{z_2}{{\left| {{z_2}} \right|}^{0.5}}{\rm{sgn}}({z_2})} \right) \end{array}$ (25)

 $\begin{array}{l} {{\dot V}_3} \le - {a_0}z_1^2 + \frac{1}{2} - r_2^2z_1^2 - {a_0}z_2^2 - ({a_0} + \frac{{M_2^2}}{2} - \frac{{B_2^2}}{2})h_1^2 + \\ \frac{{z_2^2{{\bar S}_2}\left( {{{\bar x}_2}} \right)\bar S_2^{\rm{T}}\left( {{{\bar x}_2}} \right)\tilde \lambda _2^{\rm{T}}}}{{4r_2^2}} + r_2^2\omega _2^{\rm{T}}{\omega _2} + \\ \tilde \lambda _2^{\rm{T}}\left\| {{{\bar S}_2}\left( {{{\bar x}_2}} \right)} \right\|\left| {{z_2}} \right| - \tilde \lambda _2^{\rm{T}}\Gamma _2^{ - 1\dot \wedge }{\lambda _2} \end{array}$ (26)

 ${^{\dot \wedge }}{\lambda _2} = {\Gamma _2}\left( {\frac{{z_2^2{{\bar S}_2}\left( {{{\bar x}_2}} \right)\bar S_2^T\left( {{{\bar x}_2}} \right)}}{{4r_2^2}} + \left\| {{{\bar S}_2}\left( {{{\bar x}_2}} \right)} \right\|\left| {{z_2}} \right| - {\sigma _2}\left( {{{\hat \lambda }_2} - \lambda _2^0} \right)} \right),$ (27)

 $\begin{split}{\sigma _2}\tilde \lambda _2^{\rm T}& \left( {{{\hat \lambda }_2} - \lambda _2^0} \right) \leqslant - \frac{{{\sigma _2}\tilde \lambda _2^{\rm T}{{\tilde \lambda }_2}}}{2} + \frac{{{\sigma _2}{{\left( {{\lambda _2} - \lambda _2^0} \right)}^2}}}{2}\\& \leqslant - {a_0}\tilde \lambda _2^{\rm T}\Gamma _2^{ - 1}{{\tilde \lambda }_2} + \frac{{{\sigma _2}{{\left( {{\lambda _2} - \lambda _2^0} \right)}^2}}}{2}{\text{。}}\end{split}$ (28)

 $\begin{split}& {{\dot V}_3} \leqslant - {a_0}z_1^2 - {a_0}z_2^2 - {a_0}h_1^2 - r_2^2z_1^2 - \\& \quad r_2^2z_2^2 + r_2^2\omega _2^T{\omega _2} - {a_0}\tilde \lambda _2^T\Gamma _2^{ - 1}{{\tilde \lambda }_2} + D{\text{。}}\end{split}$ (29)

 $r_2^2\omega _2^T{\omega _2}\leqslant r_2^2{\left\| {A_2^m} \right\|^2}{\left\| {{{\bar z}_2}} \right\|^2} \leqslant r_2^2{\left\| {{{\bar z}_2}} \right\|^2}\leqslant r_2^2z_1^2 + r_2^2z_2^2,$ (30)

 ${\dot V_3} \leqslant - 2{a_0}{V_3} + D \leqslant - 2{a_0}{V_3} + D{\text{。}}$ (31)

 ${V_3} \leqslant \frac{D}{{2{a_0}}} + \left( {{V_3}\left( 0 \right) - \frac{D}{{2{a_0}}}} \right){e^{ - 2{a_0}t}}{\text{。}}$ (32)

4 计算机仿真

“育龙”轮的参数为：船长126.0 m，船宽20.8 m，满载吃水8.0 m，方形系数0.681，船速7.7 m/s。通过计算得船舶运动非线性数学模型参数K=0.478，T=216，e1=1，e2=30。

 图 1 船舶航向历时曲线 Fig. 1 Time response of ship course

 图 2 舵角历时曲线 Fig. 2 Time response of rudder angle

 图 3 自适应律历时曲线 Fig. 3 Time response of ${\hat \lambda _2}$
5 结　语

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