﻿ 基于对角递归神经网络的AUV非线性<i>H</i><sub>∞</sub>控制
 舰船科学技术  2022, Vol. 44 Issue (24): 100-106    DOI: 10.3404/j.issn.1672-7649.2022.24.021 PDF

Nonlinear H control of AUV based on diagonal recursive neural network
TAO Jian-long, WANG Jun-xiong
State Key Laboratory of Ocean Engineering, Shanghai Jiaotong University, Shanghai 200240, China
Abstract: To improve the motion control accuracy and robustness of an autonomous underwater vehicle (AUV) during operation, a motion control strategy combining HJI (Hamilton-Jacobi-Isaacs) theory and recursive neural network was proposed. The dynamic model was formulated by considering various disturbance factors, and the diagonal recursive neural network was introduced to realize the effective compensation of multiple uncertainties and control input constraints in the system. The robust control law is designed based on HJI inequality, and the stability of the designed controller was proved by the second method of Lyapunov. The simulation results show the feasibility and effectiveness of the method proposed in this paper. Compared with the comparison method, the position tracking error average is reduced by more than 50%. It has higher control precision and anti-interference ability, and realizes the stable tracking control of nonlinear trajectory.
Key words: AUV     trajectory tracking     hamilton-jacobi-isaacs ( HJI ) theory     recurrent neural network
0 引　言

1 AUV数学模型建立 1.1 运动学模型

 图 1 作业型AUV坐标系 Fig. 1 Operational AUV coordinate system

 $\dot \eta = J(\eta )v = \left[ {\begin{array}{*{20}{c}} {{J_1}(\eta )}&{{{\text{0}}_{3 \times 3}}} \\ {{{\text{0}}_{3 \times 3}}}&{{J_2}(\eta ){\text{ }}} \end{array}} \right]v。$ (1)

 ${J_1}(\eta )= \left[ {\begin{array}{*{20}{c}} {{\rm{c}}\psi {\rm{c}}\theta }&{{\rm{c}}\psi {\rm{s}}\theta {\rm{s}}\phi - {\rm{s}}\psi {\rm{c}}\phi }&{{\rm{c}}\psi {\rm{s}}\theta {\rm{c}}\phi + {\rm{s}}\psi {\rm{s}}\phi } \\ {{\rm{s}}\psi {\rm{c}}\theta }&{{\rm{s}}\psi {\rm{s}}\theta {\rm{s}}\phi + {\rm{c}}\psi {\rm{c}}\phi }&{{\rm{s}}\psi {\rm{s}}\theta {\rm{c}}\phi - {\rm{c}}\psi {\rm{s}}\phi } \\ { - {\rm{s}}\theta }&{{\rm{c}}\theta {\rm{s}}\phi }&{{\rm{c}}\theta {\rm{c}}\phi } \end{array}} \right]，$ (2)
 ${J_2}(\eta )=\left[ {\begin{array}{*{20}{c}} 1&{{\rm{s}}\phi {\rm{t}}\theta }&{{\rm{c}}\phi {\rm{t}}\theta } \\ 0&{{\rm{c}}\phi }&{ - {\rm{s}}\phi } \\ 0&{{\rm{s}}\phi /{\rm{c}}\theta }&{{\rm{c}}\phi /{\rm{c}}\theta } \end{array}} \right]。$ (3)

1.2 动力学模型

 图 2 动力学模型结构图 Fig. 2 The structure diagram of dynamic model

 $\begin{gathered} {M_0}\dot v + {C_0}(v)v + {D_0}(v)v + {g_0}(\eta ) + {\tau _s} = \tau ，\\ {\tau _s} = \Delta M\dot v + (\Delta C(v) + \Delta D(v))v + \Delta g(\eta ) + {\tau _d}。\\ \end{gathered}$ (4)

 ${M_\eta }(\eta )\ddot \eta + {C_\eta }(v,\eta )\dot \eta + {D_\eta }(v,\eta )\dot \eta + {G_\eta }(\eta ) + {\tau _{\eta s}} = {\tau _T} 。$ (5)

 \begin{aligned}[b] & {M_\eta }(\eta ) = {J^{ - {\rm{T}}}}{M_0}{J^{ - 1}} ，\\ & {C_\eta }(v,\eta ) = {J^{ - {\rm{T}}}}({C_0}(v) - {M_0}{J^{ - 1}}\dot J){J^{ - 1}}，\\ & {D_\eta }(v,\eta ) = {J^{ - {\rm{T}}}}({D_0}(v)){J^{ - 1}} ，\\ &{G_\eta }(\eta ) = {J^{ - {\rm{T}}}}{g_0}(\eta )，\\ & {\tau _{\eta s}} = {J^{ - {\rm{T}}}}{\tau _s} ，\\ & {\tau _T} = {J^{ - {\rm{T}}}}\tau 。\end{aligned} (6)
2 HJI理论及神经网络设计 2.1 HJI理论

 $\left\{ \begin{gathered} \dot x = \alpha (x) + \beta (x)\delta (t)，\\ \Lambda = h(x) 。\\ \end{gathered} \right.$ (7)

 ${\left\| {\delta (t)} \right\|_2} = {\left\{ {\int_0^\infty {{\delta ^{\rm{T}}}(t)\delta (t){\rm{d}}t} } \right\}^{{\text{1/2}}}}。$ (8)

 $J=\mathop {{\text{sup}}}\limits_{\left\| \delta \right\| \ne 0} \frac{{{{\left\| \Lambda \right\|}_{\text{2}}}}}{{{{\left\| {\delta (t)} \right\|}_{\text{2}}}}}。$ (9)

 $\frac{{\partial V}}{{\partial x}}\alpha (x) + \frac{1}{{2{\gamma ^2}}}\frac{{\partial V}}{{\partial x}}\beta (x){\beta ^{{\rm{T}}}}(x){\left( {\frac{{\partial V}}{{\partial x}}} \right)^{\rm{T}}}{\text{ + }}\frac{1}{2}{h^{\rm{T}}}(x)h(x) \leqslant 0 ，$ (10)

 \begin{aligned}[b] \dot V =& \frac{{\partial V}}{{\partial x}}\dot x = \frac{{\partial V}}{{\partial x}}\alpha (x) + \frac{{\partial V}}{{\partial x}}\beta (x)\delta (t)\leqslant \\ & \frac{1}{2}\left\{ {{\gamma ^2}\left\| \delta \right\|_{}^2 - \left\| \Lambda \right\|_{}^2} \right\}{\text{ (}}\forall \delta {\text{)}}。\end{aligned} (11)

Jγ，说明作业型AUV控制系统既满足渐近稳定，也满足有界输入—有界输出稳定。

2.2 对角递归神经网络设计

 图 3 对角递归神经网络结构示意图 Fig. 3 The structure diagram of diagonal recurrent neural network
 ${S_m}(k) = V_m^{}{h_m}(k - 1) + \sum\limits_{i = 1}^2 {v_{mi}^{}{{\bar x}_i}(k)}，$ (12)

 ${h_m}(k) = g({S_m}(k)) ，$ (13)

 ${y_n} = \sum\limits_{m = 1}^7 {w_{nm}^{} \cdot {h_m}} ,{\text{ }}n = 1,2, \cdots ,6 ，$ (14)

 $\begin{gathered} V = \left[ {\begin{array}{*{20}{c}} {{v_{11}}}&{{v_{11}}}& \cdots &{{v_{1m}}} \\ {{v_{21}}}&{{v_{22}}}& \cdots &{{v_{2m}}} \end{array}} \right]，\\ {\text{ }}W = \left[ {\begin{array}{*{20}{c}} {{w_{11}}}&{{w_{12}}}& \cdots &{{w_{1n}}} \\ {{w_{21}}}&{{w_{22}}}& \cdots &{{w_{2n}}} \\ \vdots & \vdots & \ddots & \vdots \\ {{w_{m1}}}&{{w_{m2}}}& \cdots &{{w_{mn}}} \end{array}} \right], m = 7,n = 6 ，\\ \end{gathered}$ (15)
 $h({V^T}x) = \left[ {\begin{array}{*{20}{c}} {{h_1}} \\ {{h_2}} \\ \vdots \\ {{h_7}} \end{array}} \right],{\tau _{\eta s}} = \left[ {\begin{array}{*{20}{c}} {{\text{ }}{y_1}} \\ {{\text{ }}{y_2}} \\ \vdots \\ {{\text{ }}{y_6}} \end{array}} \right] 。$ (16)

 ${\tau _{\eta s}} = {W^{\rm{T}}}h({V^{\rm{T}}}\bar x) + \varepsilon 。$ (17)

 ${\hat \tau _{\eta s}} = {\hat W^T}h({\hat V^{\rm{T}}}\bar x)。$ (18)

3 运动控制设计及稳定性分析 3.1 运动控制设计

 $\Lambda = \dot e + {\boldsymbol{c}}e ，$ (19)

 ${M_\eta }\dot \Lambda = {\tau _{\rm{T}}} - {W^{\rm{T}}}h({V^T}\bar x) - \varepsilon - {C_x}\Lambda + \omega - {M_\eta }{\ddot \eta _d} 。$ (20)

 ${C_x} = {C_\eta }(v,\eta ) + {D_\eta }(v,\eta )，$ (21)
 $\omega = {C_x}ce - {C_x}{\dot \eta _d} - {G_\eta }(\eta ) + {M_\eta }c\dot e 。$ (22)

 $h({V^{\rm{T}}}\bar x) = h({\hat V^{\rm{T}}}\bar x) + h'({\hat V^{\rm{T}}}\bar x){\tilde V^{\rm{T}}}\bar x + O{({\tilde V^{\rm{T}}}\bar x)^2} ，$ (23)

 \begin{aligned}[b] & {W^{\rm{T}}}h({V^{\rm{T}}}\bar x) + \varepsilon = {{\hat W}^{\rm{T}}}h({{\hat V}^{\rm{T}}}\bar x) + {{\tilde W}^{\rm{T}}}h({{\hat V}^{\rm{T}}}\bar x) + \\ & {{\hat W}^{\rm{T}}}h'({{\hat V}^{\rm{T}}}\bar x){{\tilde V}^{\rm{T}}}\bar x + \varsigma ，\end{aligned} (24)
 $\varsigma = \tilde W^{\rm{T}} h'({\hat V^{\rm{T}}}\bar x){\tilde V^{\rm{T}}}\bar x + {W^{\rm{T}}}O{({\tilde V^{\rm{T}}}\bar x)^2} + \varepsilon 。$ (25)

 $\left\{ \begin{gathered} \dot x = \left[ \begin{gathered} {x_2} - c{x_1} \\ M_\eta ^{ - 1}({\tau _T} - {{\hat W}^{\rm{T}}}h'({{\hat V}^{\rm{T}}}\bar x){{\tilde V}^{\rm{T}}}\bar x + \\ {{\tilde W}^{\rm{T}}}h({{\hat V}^{\rm{T}}}\bar x) + {{\hat W}^{\rm{T}}}h({{\hat V}^{\rm{T}}}\bar x) + \\ \omega - {C_x}{x_2} - {M_\eta }{{\ddot \eta }_d}) \\ \end{gathered} \right] + \left[ \begin{gathered} 0 \\ - M_\eta ^{ - 1} \\ \end{gathered} \right] \varsigma ，\\ z = {x_2} 。\\ \end{gathered} \right.$ (26)

 图 4 融合HJI理论和递归神经网络控制结构框图 Fig. 4 The control structure block diagram of combining HJI theory and recursive neural network
 ${\tau _T} = - \omega + \hat W^{\rm{T}} h({\hat V^{\rm{T}}}\bar x) + {C_x}{x_2} + {M_\eta }\left({\ddot \eta _d} - \frac{1}{{2{\gamma ^2}}}\Lambda - \frac{1}{2}\Lambda \right) ，$ (27)
 $\dot {\hat W} = - {\Gamma _1}M_\eta ^{ - 1}h({\hat V^{\rm{T}}}\bar x){\Lambda ^{\rm{T}}} ，$ (28)
 $\dot {\hat V} = - {\Gamma _2}\bar x{\Lambda ^{\rm{T}}}M_\eta ^{ - 1}{\hat W^{\rm{T}}}h'({\hat V^{{\rm{T}}}}\bar x)。$ (29)
3.2 稳定性分析

 $L = \frac{1}{2}{\Lambda ^{\rm{T}}}\Lambda + \frac{1}{2}tr({\tilde W^{\rm{T}}}\Gamma _1^{ - 1}\tilde W) + \frac{1}{2}tr({\tilde V^{\rm{T}}}\Gamma _2^{ - 1}\tilde V)，$ (30)

 $\dot L = {\Lambda ^{\rm{T}}}\dot \Lambda + tr({\tilde W^{\rm{T}}}\Gamma _1^{ - 1}\dot {\tilde W}) + tr({\tilde V^{\rm{T}}}\Gamma _2^{ - 1}\dot {\tilde V}) ，$ (31)

 \begin{aligned}[b] \dot L = & {\Lambda ^{\rm{T}}}\Bigg( - \frac{1}{{2{\gamma ^2}}}\Lambda - \frac{1}{2}\Lambda - \varsigma \Bigg) + \\ & tr({{\tilde W}^{\rm{T}}}(\Gamma _1^{ - 1}\dot {\tilde W} - M_\eta ^{ - 1}h({{\hat V}^{\rm{T}}}\bar x){\Lambda ^{\rm{T}}})) + \\ & tr({{\tilde V}^{\rm{T}}}(\Gamma _2^{ - 1}\dot {\tilde V} - \bar x{\Lambda ^{\rm{T}}}M_\eta ^{ - 1}{{\hat W}^{\rm{T}}}h'({{\hat V}^{\rm{T}}}\bar x)))= \\ & {\Lambda ^{\rm{T}}}\Bigg( - \frac{1}{{2{\gamma ^2}}}\Lambda - \frac{1}{2}\Lambda - \varsigma \Bigg) 。\end{aligned} (32)

 $H = \dot L - \frac{1}{2}{\gamma ^2}\left\| \varsigma \right\|_{}^2 + \frac{1}{2}\left\| \Lambda \right\|_{}^2，$ (33)

 $H = {\Lambda ^{\rm{T}}}\left( - \frac{1}{{2{\gamma ^2}}}\Lambda - \frac{1}{2}\Lambda - \varsigma \right) - \frac{1}{2}{\gamma ^2}\left\| \varsigma \right\|_{}^2 + \frac{1}{2}\left\| \Lambda \right\|_{}^2 ，$ (34)

 $\left\{ \begin{gathered} {\Lambda ^{\rm{T}}}\Bigg( - \frac{1}{{2{\gamma ^2}}}\Lambda - \varsigma \Bigg) - \frac{1}{2}{\gamma ^2}\left\| \varsigma \right\|_{}^2 = - \frac{1}{2}\left\| {\frac{1}{\gamma }\Lambda + \gamma \varsigma } \right\|_{}^2 \leqslant 0 ，\\ - \frac{1}{2}{\Lambda ^{\rm{T}}}\Lambda + \frac{1}{2}\left\| \Lambda \right\|_{}^2 = 0。\\ \end{gathered} \right.$ (35)

 $\dot L \leqslant \frac{1}{2}{\gamma ^2}\left\| \varsigma \right\|_{}^2 - \frac{1}{2}\left\| \Lambda \right\|_{}^2 。$ (36)

4 系统数值仿真结果

 $\left\{ \begin{gathered} {x_d}(t) = \sin (0.2t)\;{\rm{m}} ，\\ {y_d}(t) = \cos (0.2t)\;{\rm{m}}，\\ {z_d}(t) = 0.2t\;{\rm{m}} ，\\ {\phi _d}(t) = 0\;{\rm{rad}}，\\ {\theta _d}(t) = 0\;{\rm{rad}}，\\ {\psi _d}(t) = 0.2t\;{\rm{ rad}} 。\\ \end{gathered} \right.$ (37)

 $\left\{ \begin{gathered} {\tau _{\eta s1}} = - 5\sin (0.6t){\text{ N}} ，\\ {\tau _{\eta s2}} = - 10\sin (0.5t){\text{ N}}，\\ {\tau _{\eta s3}} = - 10\sin (0.4t) - 5\cos (0.2t){\text{ N}}，\\ {\tau _{\eta s4}} = - 4\cos (0.2t){\text{ N}} \cdot {\text{m}}，\\ {\tau _{\eta s5}} = - 2\cos (0.2t) - 3\cos (0.3t){\text{ N}} \cdot {\text{m}}，\\ {\tau _{\eta s6}} = - 6\cos (0.1t){\text{ N}} \cdot {\text{m}}。\\ \end{gathered} \right.$ (38)

AUV初始位置和初始速度为[x, y, z, ϕ, θ, ψ]T=[0.5, 0.5, –0.5, 0, 0, 0]T，[u, v, w, p, q, r]T= [0, 0, 0, 0, 0, 0]T

 图 5 三维空间轨迹跟踪结果 Fig. 5 Trajectory tracking result in 3D space

 图 6 跟踪轨迹在X-Y平面投影 Fig. 6 Projection of the tracking result in X-Y profile

 图 7 跟踪轨迹X-Z平面投影 Fig. 7 Projection of the tracking result in X-Z profile

 图 8 跟踪轨迹Y-Z平面投影 Fig. 8 Projection of the tracking result in Y-Z profile

 图 9 位置跟踪误差对比曲线 Fig. 9 Comparison curve of position tracking error

AUV轨迹跟踪时线速度变化曲线如图10所示，满足时间约束要求。融合递归神经网络和HJI理论设计的控制器为迅速收敛至期望轨迹，从初始位置趋近于期望轨迹期间的速度大于对比方法。

 图 10 速度跟踪对比曲线 Fig. 10 Comparison curve of vehicle speed tracking

 图 11 姿态角跟踪对比曲线 Fig. 11 Comparison curve of attitude angle tracking

 图 12 推进器1-6的推力 Fig. 12 Thrust of thrusters 1-6
5 结　语

 [1] 曾晓光, 金伟晨, 赵羿羽, 等. 海洋开发装备技术发展现状与未来趋势研判[J]. 舰船科学技术, 2019, 41(9): 1-7. DOI:10.3404/j.issn.1672-7649.2019.09.001 [2] 李硕, 刘健, 徐会希, 等. 我国深海自主水下机器人的研究现状[J]. 中国科学:信息科学, 2018, 48(9): 1152-1164. DOI:10.1360/N112017-00264 [3] 李福正. 导管螺旋桨匹配性及动态性能研究[D]. 哈尔滨: 哈尔滨工业大学, 2017. [4] DING H, LIU K P, YOU W L. Study on a new nonlinear H∞ guidance law for autonomous underwater vehicle[C]//Advanced Research on Information Science, Automation and Material System/Advanced Materials Research. 2011: 219−220, 362−365. [5] ISMAIL Z H, DUNNIGAN M W. Nonlinear H∞ optimal control scheme for an underwater vehicle with regional function formulation[J]. Journal of Applied Mathematics, 2013, 2013: 732738. [6] MAHAPATRA S, SUBUDHI B. Nonlinear matrix inequality approach based heading control for an autonomous underwater vehicle with experimental realization[J]. IFAC Journal of Systems and Control, 2021, 16: 100138. DOI:10.1016/j.ifacsc.2021.100138 [7] MAHAPATRA S, SUBUDHI B, ROUT R, et al. Nonlinear H∞ control for an autonomous underwater vehicle in the vertical plane[J]. IFAC-PapersOnLine, 2016, 49(1): 391-395. DOI:10.1016/j.ifacol.2016.03.085 [8] DUAN K, FONG S, CHEN C L P. Reinforcement learning based model-free optimized trajectory tracking strategy design for an AUV[J]. Neurocomputing, 2022, 469: 289-297. DOI:10.1016/j.neucom.2021.10.056 [9] DONG B, AN T, ZHU X, et al. Zero-sum game-based neuro-optimal control of modular robot manipulators with uncertain disturbance using critic only policy iteration[J]. Neurocomputing, 2021, 450: 183-196. DOI:10.1016/j.neucom.2021.04.032 [10] WANG Y, SUN W, XIANG Y, et al. Neural network-based robust tracking control for robots[J]. Intelligent Automation and Soft Computing, 2009, 15(2): 211-222. DOI:10.1080/10798587.2009.10643026 [11] KUMAR R, SRIVASTAVA S, GUPTA J R P. Diagonal recurrent neural network based adaptive control of nonlinear dynamical systems using lyapunov stability criterion[J]. ISA Transactions, 2017, 67: 407-427. DOI:10.1016/j.isatra.2017.01.022