欠驱动USV是一种新兴的水面无人舰艇，可实现海洋环境观测、资源勘查、水面运载等海洋作业；但是由于大多USV为复杂的欠驱动系统，各自由度之间存在较大的非线性耦合，给USV的运动控制带来了较大困难；另外，USV易受到恶劣海洋环境的影响，难以获得动力学参数，从而造成了欠驱动USV难以实现轨迹跟踪的难题.针对欠驱动USV跟踪问题，沈智鹏等^[1]通过设计神经网络观测器来获得船舶速度，仿真结果验证了该观测器的有效性，但是并未考虑未知缓慢时变海流干扰问题；杜佳璐等^[2]通过采用自适应律来估计外部干扰，提高了船舶定位精度；张天平等^[3]采用神经网络逼近系统中不确定函数，经过仿真验证了该控制算法的有效性.廖煜雷等^[4]采用滑模控制补偿外界干扰，经仿真验证，该控制系统具有较好的鲁棒性，但是该控制方法并未考虑难以获得准确水动力系数问题；王金强等^[5]通过设计自适应律补偿难以测定的水动力系数，经仿真验证了该控制系统的有效性，但是该方法只是针对位置跟踪问题；Teek等^[6]采用高增益观测器获得难以测定的船舶速度，经过仿真验证了该控制器的有效性；Shojaei^[7]将神经网络自适应方法运用到船舶编队方面，经过仿真验证了该方法的鲁棒性；Wang等^[8]运用RBF神经网络自适应方法逼近不确定函数，通过仿真验证了该控制器的稳定性与鲁棒性，该方法仍然未考虑未知海流干扰影响；Pan等^[9]通过设计神经网络动态模型的方法获取虚拟变量的导数，运用级联控制方法验证了该控制系统的稳定性；Wang等^[10]采用自适应动态面方法解决了水面船的协同跟踪问题；Liu等^[11-12]采用基于预估器的神经网络逼近不确定函数，取得了较好的效果，但是并未考虑海流干扰问题.

基于上述分析，对欠驱动USV受未知时变海流影响的研究较少，本文针对未知缓慢时变海流影响，提出了一种自适应海流观测器，并且采用神经网络逼近不确定函数，避免了因水动力学系数、速度等难以测定带来的困扰，通过采用动态面方法获取虚拟控制变量的导数，减少了直接求导的计算量，通过运用李雅普诺夫函数证明了该控制系统的稳定性，仿真验证了该控制器的有效性与鲁棒性.

1 问题描述 1.1 神经网络简述

RBF神经网络能在一个紧凑集和任意精度下逼近任何非线性函数.本文运用RBF神经网络逼近未知函数，RBF神经网络算法为:

$ {{h_j} = \exp \left( {\frac{{{{\left\| {\mathit{\boldsymbol{x}} - {c_j}} \right\|}^2}}}{{2b_j^2}}} \right),} $

$ {f = {W^{*{\rm{T}}}}\mathit{\boldsymbol{h}}(\mathit{\boldsymbol{x}}) + {\varepsilon _0}.} $

式中:x为神经网络输入；i为神经网络的输入个数；j为神经网络隐含层第j个节点；h=[h_j]^T为高斯函数的输出；W^*为神经网络的理想权值；μ为神经网络的逼近误差；且ε₀≤ε_N.

RBF神经网络的输出为

$ \hat f = {W^{\rm{T}}}\mathit{\boldsymbol{h}}(\mathit{\boldsymbol{x}}). $

式中: $\hat f$为f的估计值，W神经网络的实际权值.

1.2 欠驱动USV运动坐标系

对于欠驱动USV，设定固定坐标系为{O_E, X_E, Y_E}，随体坐标系为{O_B, X_B, Y_B}；对于USV而言，本文只研究水平面内运动控制问题，USV位姿及速度可表示为{x, y, ψ}和{u, v, r}，USV轨迹跟踪如图 1所示，期望运动轨迹为η_c(t)，运动轨迹误差为{x_e, y_e}.

1.3 欠驱动USV运动坐标系

对于欠驱动USV, 选取简化的USV运动数学方程和动力学方程为：

$ \begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot \eta }} = \mathit{\boldsymbol{J}}(\psi )\mathit{\boldsymbol{v}} + {\mathit{\boldsymbol{v}}_c},}\\ {\mathit{\boldsymbol{M\dot v}} + \mathit{\boldsymbol{C}}(v)\mathit{\boldsymbol{v}} + \mathit{\boldsymbol{Dv}} = \mathit{\boldsymbol{\tau }} + \mathit{\boldsymbol{d}}.} \end{array} $

(1)

式中：η=[x  y  ψ]^T∈R³为欠驱动USV惯性坐标系下的横坐标x、纵坐标y和偏航角ψ组成的向量；v=[u  v  r]^T∈R³为欠驱动USV体坐标系下的横向速度u、纵向速度v和偏航角速度r组成的向量；v_c=[v_cx  v_cy  0]^T∈R³为欠驱动USV惯性坐标下的横向海流速度v_cx和纵向海流速度v_cy组成的向量；τ=[τ_u  0  τ_r]^T∈R³为欠驱动USV体坐标系下推力τ_u和偏转力矩τ_r组成的向量；d=[d₁  d₂  d₃]^T∈ R³为欠驱动USV体坐标系下受到的横向干扰力d₁、纵向干扰力d₂和偏航干扰力矩d₃组成的向量；J(ψ)∈R^3×3为坐标系转换矩阵，且满足η_c=[x_c  y_c  ψ_c]^T和‖J(Ψ)‖=1;M∈ R^3×3为欠驱动USV惯性质量和水动力附加惯性矩阵；C(v)∈ R^3×3为科氏向心矩阵；D∈ R^3×3为非线性水动力阻尼矩阵.

J(ψ)、M和D分别为：

$ \mathit{\boldsymbol{J}}(\psi ) = \left[ {\begin{array}{*{20}{c}} {\cos \psi }&{ - \sin \psi }&0\\ {\sin \psi }&{\cos \psi }&0\\ 0&0&1 \end{array}} \right], $

$ \mathit{\boldsymbol{M}} = \left[ {\begin{array}{*{20}{c}} {m - {X_{\dot u}}}&0&0\\ 0&{m - {Y_{\dot v}}}&0\\ 0&0&{m - {N_{\dot r}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{m_{11}}}&0&0\\ 0&{{m_{22}}}&0\\ 0&0&{{m_{33}}} \end{array}} \right], $

$ \mathit{\boldsymbol{D}} = \left[ {\begin{array}{*{20}{c}} {{X_u} + {X_{u|u}}|u|}&0&0\\ 0&{{Y_v} + {Y_{v|v|}}|v|}&0\\ 0&0&{{N_r} + {N_{r|r}}|r|} \end{array}} \right]. $

式中:m₁₁, m₂₂, m₃₃分别为欠驱动USV的惯性矩阵在船体坐标系上的3个分量；X_u、Y_v、N_r分别为欠驱动USV阻尼矩阵在体坐标系上的3个分量；X_u|u、Y_v|v、N_r|r分别为欠驱动USV的非线性阻尼项.

2 控制器设计 2.1 建立轨迹跟踪误差方程

定义欠驱动USV在惯性坐标系下的期望位置和偏航角为η_c=[x_c y_c ψ_c]^T，则欠驱动USV在惯性坐标下的轨迹跟踪误差为:

$ {x_e} = {x_c} - x,{y_e} = {y_c} - y,{\psi _e} = {\psi _c} - \psi . $

(2)

根据式(2)，可得体坐标下轨迹跟踪误差为

$ \left\{ {\begin{array}{*{20}{l}} {{e_x} = {x_e}\cos \psi + {y_e}\sin \psi ,}\\ {{e_y} = - {x_e}\sin \psi + {y_e}\cos \psi ,}\\ {{e_\psi } = {\psi _e},} \end{array}} \right. $

(3)

对式(3)求导，可得:

$ \left\{ {\begin{array}{*{20}{l}} {{{\dot e}_x} = - u + {v_m}\cos {\psi _e} + r{e_y} - {v_{cx}}\cos \psi - {v_{cy}}\sin \psi ,}\\ {{{\dot e}_y} = - v + {v_m}\sin {\psi _e} - r{e_x} + {v_{cx}}\sin \psi - {v_{cy}}\cos \psi ,}\\ {{{\dot e}_\psi } = {{\dot \psi }_e},} \end{array}} \right. $

式中，期望合速度为${v_m} = \sqrt {\dot x_c^2 + \dot y_c^2} $.

2.2 稳定e_x和e_y

定义一个李雅普诺夫函数为

$ {V_1} = \frac{1}{2}(e_x^2 + e_y^2), $

(4)

对式(4)求导，可得:

$ \begin{array}{l} {{\dot V}_1} = {e_x}( - u + {v_m}\cos {\psi _e} + r{e_y} - {v_{cx}}\cos \psi - {v_{cy}}\sin \psi ) + \\ {\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} {e_y}( - v + {v_m}\sin {\psi _e} + r{e_x} + {v_{cx}}\sin \psi - {v_{cy}}\cos \psi ). \end{array} $

(5)

定义一个新的变量: κ=v_msin ψ_e.

为保证${\dot V_1}$为负值，以u和κ为虚拟控制输入，选择其期望值为

$ \left\{ {\begin{array}{*{20}{l}} {{u_c} = {\alpha _1}{e_x} + {v_m}\cos {\psi _e} - {{\hat v}_{cx}}\cos \psi - {{\hat v}_{cy}}\sin \psi ,}\\ {{\kappa _c} = - {\alpha _2}{e_y} + v - {{\hat v}_{cx}}\sin \psi + {{\hat v}_{cy}}\cos \psi ,} \end{array}} \right. $

(6)

式中，α₁＞0，α₂＞0，且均为正实数.

将式(6)代入式(5)，可得:

$ {\dot V_1} = - {\alpha _1}e_x^2 - {\alpha _2}e_y^2 + \xi , $

(7)

式中，ξ=v_cx^e(e_ysin ψ－e_xcos ψ)－v_cy^e(e_xsin ψ+e_ycos ψ).

由于对虚拟控制变量直接求导较为复杂，为减小直接求导的复杂性，将u_c、κ_c和后文的r_c通过一阶滤波器，即

$ \left\{ {\begin{array}{*{20}{l}} {{\gamma _u}{\kern 1pt} {\kern 1pt} {{\dot u}_{cf}}(t) + {u_{cf}}(t) = {u_c}(t),}\\ {{\gamma _\kappa }{\kern 1pt} {\kern 1pt} {{\dot \kappa }_{cf}}(t) + {\kappa _{cf}}(t) = {\kappa _c}(t),}\\ {{\gamma _r}{\kern 1pt} {\kern 1pt} {{\dot r}_{cf}}(t) + {r_{cf}}(t) = {r_c}(t),} \end{array}} \right. $

其中初始值为:

$ {u_{cf}}(0) = {u_c}(0),{\kappa _{cf}}(0) = {\kappa _c}(0),{r_{cf}}(0) = {r_c}(0). $

式中：u_cf、v_cf、r_cf为虚拟控制变量通过一阶滤波器后的滤波值；γ_u＞0, γ_κ＞0, γ_r＞0且均为正实数.

2.3 稳定e_u和z_u

定义欠驱动USV系统的速度误差变量:

$ {e_u} = u - {u_{cf}},{z_u} = {u_{cf}} - {u_c}. $

(8)

将式(8)代入方程(7)，可得:

$ {\dot V_1} = - {\alpha _1}e_x^2 - {\alpha _2}e_y^2 - {e_u}{e_x} + {e_\kappa }{e_y} + \xi . $

定义一个新的李雅普诺夫函数为

$ {V_2} = {V_1} + \frac{1}{2}(e_u^2 + z_u^2), $

(9)

对式(9)求导，可得:

$ \begin{array}{*{20}{l}} {{{\dot V}_2} = {{\dot V}_1} + {e_u}{{\dot e}_u} + {z_u}{{\dot z}_u} = }\\ {{\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} - {\alpha _1}e_x^2 - {\alpha _2}e_y^2 - {e_u}{e_x} + {e_\kappa }{e_y} + {e_u}{{\dot e}_u} + {z_u}{{\dot z}_u} + \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} {\kern 1pt} - {\alpha _1}e_x^2 - {\alpha _2}e_y^2 + {e_u}({{\dot e}_u} - {e_x}) + {e_\kappa }{e_y} + {z_u}{{\dot z}_u} + \xi .} \end{array} $

2.4 稳定e_ψ、e_κ和z_κ

定义欠驱动USV系统的速度误差变量:

$ {e_\kappa } = \kappa - {\kappa _{cf}},{z_\kappa } = {\kappa _{cf}} - {\kappa _c}. $

(10)

定义一个新的李雅普诺夫函数为

$ {V_3} = {V_2} + \frac{1}{2}(e_\psi ^2 + e_\kappa ^2 + z_\kappa ^2), $

(11)

对式(11)求导，可得:

$ \begin{array}{l} {{\dot V}_3} = {{\dot V}_2} + {e_\psi }{{\dot e}_\psi } + {e_\kappa }{{\dot e}_\kappa } + {z_\kappa }{{\dot z}_\kappa } = \\ {\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} \begin{array}{*{20}{l}} { - {\alpha _1}e_x^2 - {\alpha _2}e_y^2 + {e_u}({{\dot e}_u} - {e_x}) + {z_u}{{\dot z}_u} + {z_\kappa }{{\dot z}_\kappa } + }\\ {{e_\psi }{{\dot e}_\psi } + {e_\kappa }{{\dot e}_\kappa } + {e_\kappa }{e_y} + \xi = - {\alpha _1}e_x^2 - {\alpha _2}e_y^2 + } \end{array}\\ {\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} \begin{array}{*{20}{l}} {{e_u}({{\dot e}_u} - {e_x}) + {z_u}{{\dot z}_u} + {z_\kappa }{{\dot z}_\kappa } + \xi + }\\ {{e_\psi }\left[ {{e_y} + {v_m}\sin {\psi _e} + ({v_m}\cos {\psi _e} + \frac{{{e_\psi }}}{{{e_\kappa }}})({{\dot \psi }_d} - r) - {{\dot \kappa }_{cf}}} \right],} \end{array} \end{array} $

(12)

式中r为虚拟控制变量.

可设定r的期望值为

$ {r_c} = - {\alpha _3}{e_\kappa } + {\dot \psi _c} + \frac{{{e_y} + {{\dot v}_m}\sin {\psi _e} - {{\dot \kappa }_{cf}}}}{{{v_m}\cos {\psi _e} + {e_\psi }/{e_\kappa }}}, $

(13)

式中，α₃>0，且为正实数.

将式(13)代入式(12)，可得:

$ \begin{array}{*{20}{l}} {{{\dot V}_3} = - {\alpha _1}e_x^2 - {\alpha _2}e_y^2 - {\alpha _3}e_\kappa ^2 + {e_u}({{\dot e}_u} - {e_x}) + }\\ {{\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} {z_u}{{\dot z}_u} + {z_\kappa }{{\dot z}_\kappa } + \xi .} \end{array} $

(14)

2.5 稳定e_r和z_r

定义速度误差变量为:

$ {e_r} = r - {r_{cf}},{z_r} = {r_{cf}} - {r_c}, $

(15)

将式(15)代入式(14)，可得:

$ \begin{array}{*{20}{l}} {{{\dot V}_3} = - {\alpha _1}e_x^2 - {\alpha _2}e_y^2 - {\alpha _3}e_\kappa ^2 + {e_u}({{\dot e}_u} - {e_x}) - }\\ {{\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} {v_m}{e_\kappa }{e_r}\cos {\psi _e} + {z_u}{{\dot z}_u} + {z_\kappa }{{\dot z}_\kappa } + \xi .} \end{array} $

定义一个新的李雅普诺夫函数为

$ {V_4} = {V_3} + \frac{1}{2}(e_r^2 + z_r^2), $

(16)

对式(16)求导，可得:

$ \begin{array}{l} {{\dot V}_4} = {{\dot V}_3} + {e_r}{{\dot e}_r} + {z_r}{{\dot z}_r} = - {\alpha _1}e_x^2 - {\alpha _2}e_y^2 - {\alpha _3}e_\kappa ^2 + \\ {\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} {e_u}({{\dot e}_u} - {e_x}) + {e_r}({{\dot e}_r} - {v_m}{e_\kappa }{e_r}\cos {\psi _e}) + \\ {\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} {z_u}{{\dot z}_u} + {z_\kappa }{{\dot z}_\kappa } + {z_r}{{\dot z}_r} + \xi . \end{array} $

(17)

2.6 控制输入设计

由于欠驱动USV速度难以准确测定以及水动力学系数难以确定，此处定义不确定函数:

$ \left\{ {\begin{array}{*{20}{l}} {{\vartheta _1} = {m_{22}}vr - {X_u}u - {X_{u|u|}}u|u|,}\\ {{\vartheta _3} = ({m_{11}} - {m_{22}})uv - {N_r}r - {N_{r|r}}r|r|.} \end{array}} \right. $

针对不确定函数，采用神经网络进行逼近，定义神经网络预测误差为:

$ {\dot \delta _1} = {\vartheta _1} - {\hat \vartheta _1},{\dot \delta _3} = {\vartheta _3} - {\hat \vartheta _3}. $

(18)

选取控制输入为

$ \left\{ {\begin{array}{*{20}{l}} {{\tau _u} = - {m_{11}}({\beta _1}{e_u} + {e_x} + {{\dot u}_{cf}}) + {{\mathit{\boldsymbol{\hat W}}}_1}h({\mathit{\boldsymbol{x}}_u}) + }\\ {{\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} {\delta _1}{\sigma _1} + {\eta _1} {\rm{sgn}} ({e_u}),}\\ {{\tau _r} = - {m_{33}}({\beta _3}{e_r} + {e_\kappa }{v_m}\cos {\psi _e} + {{\dot r}_{cf}}) + {{\mathit{\boldsymbol{\hat W}}}_3}h({\mathit{\boldsymbol{x}}_r}) + }\\ {{\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} {\delta _3}{\sigma _3} + {\eta _3} {\rm{sgn}} ({e_r}).} \end{array}} \right. $

(19)

式中：${\mathit{\boldsymbol{\hat W}}_1}、{\mathit{\boldsymbol{\hat W}}_3}$分别为神经网络权值W₁和W₃的估计值；β₁>0，σ₁>0，η₁>0，β₃>0，σ₃>0，η₃>0均为正实数.

选取自适应律为

$ \left\{ {\begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\hat W}}}_1} = {\chi _1}[({e_u} + {\lambda _1}{\delta _1})h({\mathit{\boldsymbol{x}}_u}) - {\rho _1}{{\mathit{\boldsymbol{\hat W}}}_1}],}\\ {{{\mathit{\boldsymbol{\hat W}}}_3} = {\chi _3}[({e_r} + {\lambda _3}{\delta _3})h({\mathit{\boldsymbol{x}}_r}) - {\rho _3}{{\mathit{\boldsymbol{\hat W}}}_3}],} \end{array}} \right. $

(20)

式中，χ₁>0、χ₃>0均为正实数.

将式(1)代入式(19)，可得:

$ \left\{ {\begin{array}{*{20}{l}} {{m_{11}}{{\dot e}_u} = - {m_{11}}({\beta _1}{e_u} + {e_x} + {{\dot u}_{cf}}) + }\\ {{\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} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\mathit{\boldsymbol{\hat W}}}_1}h({\mathit{\boldsymbol{x}}_u}) - {\vartheta _1} + {d_1} - {\eta _1} {\rm{sgn}} ({e_u}),}\\ {{m_{33}}{{\dot e}_r} = - {m_{33}}({\beta _3}{e_r} + {e_\kappa }{v_m}\cos {\psi _e} + {{\dot r}_{cf}}) + }\\ {{\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} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\mathit{\boldsymbol{\hat W}}}_3}h({\mathit{\boldsymbol{x}}_r}) - {\vartheta _3} + {d_3} - {\eta _3} {\rm{sgn}} ({e_r}).} \end{array}} \right. $

(21)

将式(21)代入式(17), 可得:

$ \begin{array}{l} {{\dot V}_4} = - {\alpha _1}e_x^2 - {\alpha _2}e_y^2 - {\alpha _3}e_\kappa ^2 - {\beta _1}e_u^2 - {\beta _3}e_r^2 + \\ {\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} \begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\tilde W}}}_1}h\left( {{\mathit{\boldsymbol{x}}_u}} \right){e_u} - {m_{11}}\left( {{\mathit{\boldsymbol{\eta }}_1}\left| {{e_u}} \right| - {d_1}{e_u}} \right) + }\\ {{{\mathit{\boldsymbol{\tilde W}}}_3}h\left( {{\mathit{\boldsymbol{x}}_r}} \right){e_r} - {m_{33}}\left( {{\mathit{\boldsymbol{\eta }}_3}\left| {{e_u}} \right| - {d_3}{e_r}} \right) + }\\ {{z_u}{{\dot z}_u} + {z_\kappa }{{\dot z}_\kappa } + {z_r}{{\dot z}_r} + \xi ,} \end{array} \end{array} $

(22)

式中，${\mathit{\boldsymbol{\tilde W}}_1}、{\mathit{\boldsymbol{\tilde W}}_3}$分别为神经网络权值W₁和W₃的估计误差，且有${\mathit{\boldsymbol{\widetilde W}}_1} = {\mathit{\boldsymbol{\widehat W}}_1} - \mathit{\boldsymbol{W}}_1^*, {\mathit{\boldsymbol{\widetilde W}}_3} = {\mathit{\boldsymbol{\widehat W}}_3} - \mathit{\boldsymbol{W}}_3^*$.

假设1  对所有神经网络理想权值矩阵W^*和逼近误差ε有界，即存在正常数W_M和有界函数ε_M，使得‖W^*‖≤W_M和‖ε‖≤ε_M.

2.7 海流观测器设计

针对海洋环境中未知时变海流，本文设计了一种指数收敛自适应观测器，目标为海流估计值指数趋近于未知时变海流，即

$ {\mathit{\boldsymbol{\hat v}}_c} = \mathit{\boldsymbol{\dot \eta }} - \mathit{\boldsymbol{J}}(\psi )\mathit{\boldsymbol{v}}. $

(23)

对于设计的海流观测器，可以选择:

$ {\mathit{\boldsymbol{\dot {\hat v}}}_c} = - \mathit{\boldsymbol{\hat K}}{\mathit{\boldsymbol{v}}_c} + \mathit{\boldsymbol{K\dot \eta }} - \mathit{\boldsymbol{KJ}}(\psi )\mathit{\boldsymbol{v}}. $

(24)

式中: ${\mathit{\boldsymbol{\hat v}}_c} = {\left[ {{{\hat v}_{cx}}\;{{\hat v}_{cy}}\;0} \right]^{\rm{T}}} \in {{\bf{R}}^3}$为海流观测器对未知海流的估计向量；K=diag[k₁  k₂  0]∈ R ^3×3为待定的正定参数对角阵向量.

将式(24)代入式(23)，可得:

$ \mathit{\boldsymbol{\dot v}}_c^e = {\mathit{\boldsymbol{\dot v}}_c} - \mathit{\boldsymbol{\dot {\hat v}}} = \mathit{\boldsymbol{\hat K}}{\mathit{\boldsymbol{v}}_c} - \mathit{\boldsymbol{K\dot \eta }} + \mathit{\boldsymbol{KJ}}(\psi )\mathit{\boldsymbol{v}} = - \mathit{\boldsymbol{Kv}}_c^e, $

(25)

式中，v_c^e=[v_cx^e  v_cy^e  0]^T∈ R ³为海流观测器观测误差组成的向量.

根据式(25)可知，设计的未知海流观测器是指数稳定的；但是，在欠驱动USV实际航行过程中，惯性坐标系下的速度$\mathit{\boldsymbol{\dot \eta }}$是难以测定的变量；此时，引入辅助变量ζ为$\zeta = {\mathit{\boldsymbol{\hat v}}_c} - \mathit{\boldsymbol{K\eta }}$, 选取:$\dot \zeta = - \mathit{\boldsymbol{K\zeta }} + \mathit{\boldsymbol{K}}\left( { - \mathit{\boldsymbol{K}} - \mathit{\boldsymbol{J}}\left( \psi \right)\mathit{\boldsymbol{v}}} \right)$, 则有

$ \begin{array}{l} \mathit{\boldsymbol{\dot v}}_c^e = {{\mathit{\boldsymbol{\dot v}}}_c} - {{\mathit{\boldsymbol{\dot {\hat v}}}}_c} = - \mathit{\boldsymbol{\dot \zeta }} - \mathit{\boldsymbol{K\dot \eta }} = \\ {\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} \mathit{\boldsymbol{K\zeta }} - \mathit{\boldsymbol{K}}( - \mathit{\boldsymbol{K\eta }} - \mathit{\boldsymbol{J}}(\psi )\mathit{\boldsymbol{v}}) - \mathit{\boldsymbol{K\dot \eta }} = \\ {\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} \mathit{\boldsymbol{K}}({{\mathit{\boldsymbol{\hat v}}}_c} - {\mathit{\boldsymbol{v}}_c}) = - \mathit{\boldsymbol{Kv}}_c^e. \end{array} $

综上所述，设计的海流观测器是指数收敛的.

3 稳定性分析

为稳定欠驱动USV轨迹跟踪控制器与海流观测器所组成的闭环系统，将整个控制系统分解为两个子系统.

定义第1个子系统∑₁为

$ \left\| {{\mathit{\boldsymbol{x}}_1}} \right\| = \left[ {\begin{array}{*{20}{l}} {{e_x}}&{{e_y}}&{{e_\psi }}&{{e_u}}&{{e_\kappa }}&{{e_r}}&{{{\tilde W}_1}}&{{{\tilde W}_3}}&{{\delta _1}}&{{\delta _3}} \end{array}} \right]. $

根据文献[13]可得:

$ \left\{ {\begin{array}{*{20}{l}} {{{\dot z}_u} = {{\dot u}_{cf}} - {{\dot u}_c} = - {z_u}/{\gamma _u} + {\rm{d}}{u_c}/{\rm{d}}t,}\\ {|{\rm{d}}{u_c}/{\rm{d}}t| \le {\zeta _u},} \end{array}} \right. $

(26)

式中，ζ_u>0，且为正实数.

将式(10)代入式(26)，可得:

$ {z_u}{\dot z_u} \le - \frac{{z_u^2}}{{{\gamma _u}}} + {\zeta _u}|{\zeta _u}| \le - \frac{{z_u^2}}{{{\gamma _u}}} + z_u^2 + \frac{1}{4}\zeta _u^2. $

(27)

将不等式(27)代入式(22)，可得:

$ \begin{array}{*{20}{l}} {{{\dot V}_4} \le - {\alpha _1}e_x^2 - {\alpha _2}e_y^2 - {\alpha _3}e_\kappa ^2 - {\beta _1}e_u^2 - {\beta _3}e_r^2 + }\\ {{\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} {{\mathit{\boldsymbol{\tilde W}}}_1}h({x_u}){e_u} - {m_{11}}({\eta _1}|{e_u}| - {d_1}{e_u}) + }\\ {{\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} {{\mathit{\boldsymbol{\tilde W}}}_3}h({x_r}){e_r} - {m_{33}}({\eta _3}|{e_r}| - {d_3}{e_r}) - }\\ {{\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} \left( {1 - \frac{1}{{{\gamma _u}}}} \right)z_u^2 - \left( {1 - \frac{1}{{{\gamma _\kappa }}}} \right)z_\kappa ^2 - \left( {1 - \frac{1}{{{\gamma _r}}}} \right)z_r^2 + }\\ {{\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}{4}(\zeta _u^2 + \zeta _\kappa ^2 + \zeta _r^2).} \end{array} $

式中，ζ_κ>0，ζ_r>0，且均为正实数.

根据式(18)，可得:

$ {\dot \delta _i} = {\vartheta _i} - {\hat \vartheta _i} = \mathit{\boldsymbol{\tilde W}}_i^{\rm{T}}h(\mathit{\boldsymbol{x}}) + {\varepsilon _i} - {\sigma _i}{\delta _i},(i = 1,3) $

(28)

式中，ε₁>0，ε₃>0，均为待定的正实数.

根据式(28)，可得:

$ {\delta _i}{\dot \delta _i} = {\delta _i}\mathit{\boldsymbol{\tilde W}}_i^{\rm{T}}h(\mathit{\boldsymbol{x}}) + {\delta _i}{\varepsilon _i} - {\sigma _i}\delta _i^2,(i = 1,3). $

定义一个新的李雅普诺夫函数:

$ \begin{array}{*{20}{l}} {{V_5} = {V_4} + \frac{1}{{2{\chi _1}}}\mathit{\boldsymbol{\tilde W}}_1^{\rm{T}}{{\mathit{\boldsymbol{\tilde W}}}_1} + \frac{1}{{2{\chi _3}}}\mathit{\boldsymbol{\tilde W}}_3^{\rm{T}}{{\mathit{\boldsymbol{\tilde W}}}_3} + }\\ {{\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}{2}{\lambda _1}\delta _1^2 + \frac{1}{2}{\lambda _3}\delta _3^2,} \end{array} $

(29)

对式(29)求导，可得:

$ \begin{array}{*{20}{l}} {{{\dot V}_5} = {{\dot V}_4} - \frac{1}{{{\chi _1}}}\mathit{\boldsymbol{\tilde W}}_1^{\rm{T}}{{\mathit{\boldsymbol{\dot {\hat W}}}}_1} - \frac{1}{{{\chi _3}}}\mathit{\boldsymbol{\tilde W}}_3^{\rm{T}}{{\mathit{\boldsymbol{\dot {\hat W}}}}_3} + {\lambda _1}{\delta _1}{{\dot \delta }_1} + {\lambda _3}{\delta _3}{{\dot \delta }_3} = }\\ {{\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} {{\dot V}_4} - \mathit{\boldsymbol{\tilde W}}_1^{\rm{T}}h({\mathit{\boldsymbol{x}}_u})({e_u} + {\lambda _1}{\delta _1}) + {\rho _1}\mathit{\boldsymbol{\tilde W}}_1^{\rm{T}}{{\mathit{\boldsymbol{\hat W}}}_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} {\kern 1pt} \mathit{\boldsymbol{\tilde W}}_3^{\rm{T}}h({\mathit{\boldsymbol{x}}_r})({e_r} + {\lambda _3}{\delta _3}) + {\rho _3}\mathit{\boldsymbol{\tilde W}}_3^{\rm{T}}{{\mathit{\boldsymbol{\hat W}}}_3} + }\\ {{\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} {\lambda _1}{\delta _1}\mathit{\boldsymbol{\tilde W}}_1^{\rm{T}}h({\mathit{\boldsymbol{x}}_u}) + {\lambda _1}{\delta _1}{\varepsilon _1} - {\lambda _1}{\sigma _1}\delta _1^2 + }\\ {{\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} {\lambda _3}{\delta _3}\mathit{\boldsymbol{\tilde W}}_3^{\rm{T}}h({\mathit{\boldsymbol{x}}_r}) + {\lambda _3}{\delta _3}{\varepsilon _3} - {\lambda _3}{\sigma _3}\delta _3^2.} \end{array} $

(30)

将式(20)代入式(30)，可得:

$ \begin{array}{l} \begin{array}{*{20}{l}} {{{\dot V}_5} \le - {\alpha _1}e_x^2 - {\alpha _2}e_y^2 - {\alpha _3}e_\kappa ^2 - {\beta _1}e_u^2 - {\beta _3}e_r^2 - }\\ {{\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} {m_{11}}({\eta _1}|{e_u}| - {d_1}{e_u}) - {m_{33}}({\eta _3}|{e_r}| - {d_3}{e_r}) - }\\ {{\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} {\rho _1}\mathit{\boldsymbol{\tilde W}}_1^{\rm{T}}{{\mathit{\boldsymbol{\tilde W}}}_1} + {\rho _1}\mathit{\boldsymbol{\tilde W}}_1^{\rm{T}}\mathit{\boldsymbol{W}}_1^* + {\lambda _1}{\delta _1}{\varepsilon _1} - {\lambda _1}{\sigma _1}\delta _1^2 - } \end{array}\\ \begin{array}{*{20}{l}} {{\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} {\rho _3}\mathit{\boldsymbol{\tilde W}}_3^{\rm{T}}{{\mathit{\boldsymbol{\tilde W}}}_3} + {\rho _3}\mathit{\boldsymbol{\tilde W}}_3^{\rm{T}}\mathit{\boldsymbol{W}}_3^* + {\lambda _3}{\delta _3}{\varepsilon _3} - {\lambda _3}{\sigma _3}\delta _3^2 - }\\ {{\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} \left( {1 - \frac{1}{{{\gamma _u}}}} \right)z_u^2 - \left( {1 - \frac{1}{{{\gamma _\kappa }}}} \right)z_\kappa ^2 - \left( {1 - \frac{1}{{{\gamma _r}}}} \right)z_r^2 + }\\ {{\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}{4}(\zeta _u^2 + \zeta _\kappa ^2 + \zeta _r^2).} \end{array} \end{array} $

(31)

根据文献[14], 可得以下方程:

$ \left\{ \begin{array}{l} {\delta _i}{\varepsilon _i} - {\sigma _i}\delta _i^2 = - {\sigma _i}{({\delta _i} - \frac{{{\varepsilon _i}}}{{2{\sigma _i}}})^2} + \frac{1}{{4{\sigma _i}}}\varepsilon _i^2,(i = 1,3)\\ \mathit{\boldsymbol{\tilde W}}_i^{\rm{T}}\mathit{\boldsymbol{W}}_i^* - \mathit{\boldsymbol{\tilde W}}_i^{\rm{T}}{{\mathit{\boldsymbol{\tilde W}}}_i} = - {\left\| {{{\mathit{\boldsymbol{\tilde W}}}_i} - \frac{{\mathit{\boldsymbol{W}}_i^*}}{2}} \right\|^2} + \\ {\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} {\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} {\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} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{4}{\left\| {\mathit{\boldsymbol{W}}_i^*} \right\|^2},(i = 1,3) \end{array} \right. $

(32)

将式(32)代入式(31)，可得:

$ \begin{array}{l} {{\dot V}_5} \le - {\alpha _1}e_x^2 - {\alpha _2}e_y^2 - {\alpha _3}e_\kappa ^2 - {\alpha _4}e_u^2 - {\alpha _5}e_r^2 - \\ {\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} \begin{array}{*{20}{l}} {{m_{11}}({\eta _1}|{e_u}| - {d_1}{e_u}) - {m_{33}}({\eta _3}|{e_r}| - {d_3}{e_r}) - }\\ {{\rho _1}{{\left[ {\left\| {{{\mathit{\boldsymbol{\tilde W}}}_1} - \frac{{\mathit{\boldsymbol{W}}_1^*}}{2}} \right\| - \frac{1}{4}{{\left\| {\mathit{\boldsymbol{W}}_1^*} \right\|}^2}} \right]}^2} - }\\ {{\lambda _1}{{\left[ {{\sigma _1}{{\left( {{\delta _1} - \frac{{{\varepsilon _1}}}{{2{\sigma _1}}}} \right)}^2} - \frac{1}{{4{\sigma _1}}}\varepsilon _1^2} \right]}^2} - } \end{array}\\ {\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} \begin{array}{*{20}{l}} {{\rho _3}{{\left[ {{{\left\| {{{\mathit{\boldsymbol{\tilde W}}}_3} - \frac{{\mathit{\boldsymbol{W}}_3^*}}{2}} \right\|}^2} - \frac{1}{4}{{\left\| {\mathit{\boldsymbol{W}}_3^*} \right\|}^2}} \right]}^2} - }\\ {{\lambda _3}{{\left[ {{\sigma _3}{{\left( {{\delta _3} - \frac{{{\varepsilon _3}}}{{2{\sigma _3}}}} \right)}^2} - \frac{1}{{4{\sigma _3}}}\varepsilon _3^2} \right]}^2} - }\\ {\left( {1 - \frac{1}{{{\gamma _u}}}} \right)z_u^2 - \left( {1 - \frac{1}{{{\gamma _\kappa }}}} \right)z_\kappa ^2 - \left( {1 - \frac{1}{{{\gamma _r}}}} \right)z_r^2 + } \end{array}\\ {\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} \begin{array}{*{20}{l}} {\frac{1}{4}(\zeta _u^2 + \zeta _\kappa ^2 + \zeta _r^2) \le - {\alpha _1}e_x^2 - {\alpha _2}e_y^2 - {\alpha _3}e_\kappa ^2 - }\\ {{\alpha _4}e_u^2 - {\alpha _5}e_r^2 - {m_{11}}({\eta _1}|{e_u}| - {d_1}{e_u}) - }\\ {{m_{33}}({\eta _3}|{e_r}| - {d_3}{e_r}) - } \end{array}\\ {\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} \begin{array}{*{20}{l}} {_{\min }{\sigma _{\min }}\left[ {{{\left( {{\delta _1} - \frac{{{\varepsilon _1}}}{{2{\sigma _1}}}} \right)}^2} + {{\left( {{\delta _3} - \frac{{{\varepsilon _3}}}{{2{\sigma _3}}}} \right)}^2}} \right] - }\\ {\lambda {\rho _{\min }}\left[ {{{\left\| {{{\mathit{\boldsymbol{\tilde W}}}_1} - \frac{{\mathit{\boldsymbol{W}}_1^*}}{2}} \right\|}^2} + {{\left\| {{{\mathit{\boldsymbol{\tilde W}}}_3} - \frac{{\mathit{\boldsymbol{W}}_3^*}}{2}} \right\|}^2}} \right] + \mathit{\boldsymbol{ \boldsymbol{\varTheta} }}.} \end{array} \end{array} $

(33)

式中: $\begin{array}{*{20}{l}} {{\lambda _{\min }} = \min \left\{ {{\lambda _1}, {\lambda _3}} \right\}, {\sigma _{\min }} = \min \left\{ {{\sigma _1}, {\sigma _3}} \right\}, {\rho _{\min }} = } {\min \left\{ {{\rho _1}, {\rho _3}} \right\}, \mathit{\Theta } = \frac{{{\lambda _{\max }}}}{{2{\sigma _{\min }}}}\varepsilon _M^2 + \frac{{{\rho _{\max }}}}{2}W_M^2, {\lambda _{\max }} = \max \left\{ {{\lambda _1}} \right.}, {\left. {{\lambda _3}} \right\}, {\rho _{\max }} = \max \left\{ {{\rho _1}, {\rho _3}} \right\}} \end{array}$.

根据式(33)可知，通过选取不同的控制器参数，可保证${\dot V_5} \le 0$，即子系统∑₁是收敛的.

定义第2个子系统∑₂为

$ \left\| {{\mathit{\boldsymbol{x}}_2}} \right\| = \left[ {\begin{array}{*{20}{l}} {v_{cx}^e}&{v_{cy}^e} \end{array}} \right]. $

定义一个新的李雅普诺夫函数为

$ {V_6} = \frac{1}{2}({(v_{cx}^e)^2} + {(v_{cy}^e)^2}), $

(34)

对式(34)求导，可得:

$ {\dot V_6} = v_{cx}^e\dot v_{cx}^e + v_{cy}^e\dot v_{cy}^e = - {k_1}{(v_{cx}^e)^2} - {k_2}{(v_{cy}^e)^2}, $

则子系统∑₂为指数稳定的.

定义一个新的李雅普诺夫函数为

$ V = {V_5} + {V_6}, $

则$\dot V$可进一步表示为

$ \dot V \le - 2\mu V + \varPhi , $

(35)

式中，μ、Φ分别为：

$ \begin{array}{*{20}{l}} {\mu = \min \{ {\alpha _1},{\alpha _2},{\alpha _3},{\alpha _4},{\alpha _5},{\rho _1},{\lambda _1},{\rho _3},{\lambda _3},}\\ {{\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} (1 - \frac{1}{{{\gamma _u}}}),(1 - \frac{1}{{{\gamma _\kappa }}}),(1 - \frac{1}{{{\gamma _r}}}),{k_1},{k_2}\} ,} \end{array} $

$ \mathit{\boldsymbol{ \boldsymbol{\varPhi} }} = \frac{1}{4}(\zeta _u^2 + \zeta _\kappa ^2 + \zeta _r^2) + \frac{{{\lambda _{\max }}}}{{2{\sigma _{\min }}}}\varepsilon _M^2 + \frac{{{\rho _{\max }}}}{2}W_M^2. $

对式(35)求解可得:

$ V \le \frac{\varPhi }{{2\mu }} + \left[ {V(0) - \frac{\varPhi }{{2\mu }}} \right]{{\rm{e}}^{ - 2\mu t}}. $

显然，当t→∞时，有V→Φ/2μ，则闭环系统的所有信号是半全局一致最终有界的，即所设计的控制器是稳定的.

4 仿真分析

本文选取文献[15]欠驱动USV系统模型和外界干扰，验证文中设计的控制算法, 取欠驱动USV初始条件为：

$ \mathit{\boldsymbol{\eta }}(0) = {\left[ {\begin{array}{*{20}{l}} 0&{17}&0 \end{array}} \right]^{\rm{T}}},v(0) = {\left[ {\begin{array}{*{20}{l}} {0.4}&0&0 \end{array}} \right]^{\rm{T}}}. $

欠驱动USV应用过程中，τ_u、τ_r是有限的，在控制器设计中，设定控制输入为有限值，即

$ 0 \le |{\tau _u}| \le 90{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{N}},0 \le |{\tau _r}| \le 20{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{N}} \cdot {\rm{m}}. $

控制器参数为:α₁=0.5, α₂=0.4, α₃=0.3, α₄=0.2, α₅=1.0, γ_u=0.1, γ_κ=0.1, γ_r=0.1, η₁=0.2, η₃=0.2, χ₁=0.1, χ₃=0.1, λ₁=0.15，λ₃=0.10, ρ₁=0.3, ρ₃=0.2, 神经网络的初始权值取0.1, c_j=[-1.0  -0.5  0  0.5  1.0]和b_j =5.0.

缓慢变化海流选取为

$ \left\{ {\begin{array}{*{20}{l}} {{v_{cx}} = ( - 0.1 - 0.1\sin (0.01t)),}\\ {{v_{cy}} = ( - 0.1 - 0.1\cos (0.01t)).} \end{array}} \right. $

为验证该控制器控制性能，选取预期圆轨迹为

$ \left\{ {\begin{array}{*{20}{l}} {{x_c}(t) = 15\sin (0.05t),}\\ {{y_c}(t) = 15\cos (0.05t).} \end{array}} \right. $

为便于仿真结果分析，仿真时间为200 s，可清晰地看出运动轨迹，仿真结果如图 2~图 7所示.

根据图 2可知，在欠驱动USV航行初始阶段受海流影响较大，但在较短时间内有效地实现了轨迹跟踪；根据图 3、4可得出，位置跟踪误差、速度跟踪误差均收敛到零附近的一个区域内；根据图 5可知，设计的神经网络自适应模型可有效逼近欠驱动USV系统未知函数；根据图 6可知，设计的自适应海流观测器可有效观测未知缓慢时变海流.根据图 7可知，控制力及控制力矩较为稳定，且在设定范围内.

5 结论

1) 设计了一种自适应海流观测器，有效地估计了未知缓慢时变海流.

2) 运用李雅普诺夫稳定性理论验证了欠驱动USV闭环控制系统的稳定性.

3) 仿真结果表明，欠驱动USV轨迹跟踪误差、速度跟踪误差均收敛到零附近的一个区域内，验证了该控制系统的有效性.