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    Journals >Chinese Journal of Ship Research >Volume 20 >Issue 1 >Page 309 > Article
    • Chinese Journal of Ship Research
    • Vol. 20, Issue 1, 309 (2025)
    Adaptive neural control for marine autonomous surface ships in cross-water scenarios
    Xiang YE1, Chao CHEN1, Jian Xiong JIA2, and Hang CHEN2
    Author Affiliations
  • 1School of Naval Architecture and Maritime, Zhejiang Ocean University, Zhoushan 316022, China
  • 2Zhejiang Branch of China Classification Society, Ningbo 430060, China
    • show less
    DOI: 10.19693/j.issn.1673-3185.03609 Cite this Article
    Xiang YE, Chao CHEN, Jian Xiong JIA, Hang CHEN. Adaptive neural control for marine autonomous surface ships in cross-water scenarios[J]. Chinese Journal of Ship Research, 2025, 20(1): 309 Copy Citation Text show less

    Abstract

    Objective

    An adaptive neural control (ANC) scheme with specified performance is proposed for the tracking control of marine autonomous surface ships (MASS) subject to uncertain model parameters and unknown external environmental disturbances in cross-water scenarios.

    Methods

    Under the back-stepping design framework, a neural network is utilized to approximate the uncertain model parameters and unknown external environmental disturbances. A novel specified performance function is constructed and combined with the barrier Lyapunov function (BLF) to transform the cross-water design, while the dynamic surface control technique is employed to reduce the system's computational complexity. Stability analysis is then performed by means of Lyapunov theory to demonstrate that all signals within the control system are bounded.

    Results

    The simulation results show that the designed control scheme is not only capable of solving the cross-water tracking control of MASS, but that the tracking error can satisfy the convergence to a given bounded range within a predefined time offline.

    Conclusion

    The results of this study can solve the cross-water tracking control problems of MASS and provide valuable references for the tracking control of ships in restricted waters, giving them practical engineering significance.

    Keywords

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    $ \boldsymbolη˙=\boldsymbolR(ψ)\boldsymbolυ\boldsymbolM\boldsymbolυ˙+\boldsymbolC(\boldsymbolυ)\boldsymbolυ+\boldsymbolD(\boldsymbolυ)\boldsymbolυ=\boldsymbolτ+\boldsymbolτω  $(1)

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    $ 0 \leqslant \left| h \right| - h\tanh (h/\omega ) \leqslant 0.278\; 5\omega $(2)

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    $ ab \leqslant ({d^p}/p){\left| a \right|}^p + (1/d^p q){\left| b \right|}^q $(3)

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    $ {{\boldsymbol{F}}}({{\boldsymbol{X}}}) = {{\boldsymbol{W}}}^{\text{T}} {\xi }({{\boldsymbol{X}}}) + {\varepsilon } $(4)

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    $ {\xi _i}({\boldsymbol{X}}) = \exp \left[ {\frac{{ - {\left\| {({\boldsymbol{X}}- {\boldsymbol{L}}_i )} \right\|}^2 }}{{{w_i }^2 }}} \right],i = 1,2, \ldots ,n $(5)

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    $ \lg \left( {\frac{{k_{\text{x}}^2}}{{k_{\text{x}}^2 - S_0^2}}} \right) \leqslant \frac{{S_0^2}}{{k_{\text{x}}^2 - S_0^2}} $(6)

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    $ {\boldsymbol{e}} = {\boldsymbol{\eta }} - {\boldsymbol{\eta }}_{\text{d}} $(7)

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    $ - \rho _i (t) < e_i (t) < \rho _i (t),i = 1,2,3 $(8)

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    $ \rho (t) = (1 - \varphi (t))/(\iota t + \ell ) + \rho _{\text{T}} $(9)

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    $ \varphi (t) = \left\{ sin⁡(πt/2T),0⩽t<T1,t⩾T \right. $(10)

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    $ S_{ 1,i} = \varphi (t)e_i $(11)

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    $ - \rho _i (t) < - \rho _i (t)\varphi (t) < S_{ 1,i}(t) < \rho _i (t)\varphi (t) < \rho _i (t) $(12)

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    $ - \rho _i (0) < S_{ 1,i}(0) < \rho _i (0) $(13)

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    $ - \rho _i (t) < S_{ 1,i}(t) < \rho _i (t) $(14)

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    $ {{\dot {\boldsymbol{S}}}_1} = \dot \varphi {\boldsymbol{e}} + \varphi ({\boldsymbol{R}}(\psi ){\boldsymbol{\upsilon }} - {{\dot {\boldsymbol{\eta}} }}_{\text{d}} ) $(15)

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    $ {{\boldsymbol{\alpha}} } = - {\boldsymbol{R}}^{\text{T}} (\psi )({{\boldsymbol{K}}_1} + {\hat {\boldsymbol{\varTheta}} {\boldsymbol{\varPhi}} }) {\boldsymbol{e}} $(16)

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    $ {\dot {\hat {\boldsymbol{\varTheta}}} _i} = {c_2}S_{ 1,i}^2{\varPhi _i}/{(\rho }_i^2 - S_{ 1,i}^2) - {c_1}{\hat \varTheta _i} $(17)

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    $ {T_{\text{d}}}{{\dot {\boldsymbol{s}}}_{\text{d}}} + {{\boldsymbol{s}}_{\text{d}}} = {{\boldsymbol{\alpha}} },\;\;{{\boldsymbol{s}}_{\text{d}}}(0) = {{\boldsymbol{\alpha}} }(0) $(18)

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    $ {V_1} = \frac{1}{2}\sum\limits_{i = 1}^3 {\left( {\lg [\rho _i ^2/(\rho _i ^2 - S_{ 1,i}^2)] + (1/{c_2}){{\tilde \varTheta }_i}^2} \right)} $(19)

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    $ V˙1=∑i=13S1,iρi2−S1,i2φAi+∑i=13S1,iρi2−S1,i2(φZi−φ\boldsymbolη˙d,i+φ˙ei−ρ˙iρiS1,i)−1c2∑i=13Θ~iΘ^˙i $(20)

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    $ {{\boldsymbol{A}}} = {\boldsymbol{R}}(\psi ){\alpha } , \;\;{A_i} = - ({K_{1,i}} + {\hat \varTheta _i}{\varPhi _i})e_i , i = 1,2,3 $()

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    $ \boldsymbolZ=\boldsymbolR(ψ)\boldsymbolS2,Z1=cos⁡ψS2,1−sin⁡ψS2,2,Z2=sin⁡ψS2,1+cos⁡ψS2,2,Z3=S2,3$()

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    $ \varXi = \sum\limits_{i = 1}^3 {\left( {\frac{{S_{ 1,i}}}{{\rho _i ^2 - S_{ 1,i}^2}}\left(\varphi {Z_i} - \varphi {\dot \eta }_{{\text{d}},i} + \dot \varphi e_i - \frac{{{{\dot \rho }_i}}}{{\rho _i }}S_{ 1,i}\right)} \right)} $()

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    $ \sum\limits_{i = 1}^3 {\frac{{S_{ 1,i}\dot \varphi e_i }}{{\rho _i ^2 - S_{ 1,i}^2}}} \leqslant \sum\limits_{i = 1}^3 {\frac{{\beta S_{ 1,i}^2\varphi _{\text{m}} ^2e_i ^2}}{{{(\rho _i ^2 - S_{ 1,i}^2)}^2 }}} + \frac{3}{{4\beta }} $(21)

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    $ \sum\limits_{i = 1}^3 {\frac{{S_{ 1,i}( - \varphi ){\dot \eta }_{{\text{d}},i} }}{{\rho _i ^2 - S_{ 1,i}^2}}} \leqslant \sum\limits_{i = 1}^3 {\frac{{\beta S_{ 1,i}^2\varphi ^2 \eta _{\text{m}} ^2}}{{{(\rho _i ^2 - S_{ 1,i}^2)}^2 }}} + \frac{3}{{4\beta }} $(22)

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    $ \sum\limits_{i = 1}^3 {\frac{{S_{ 1,i}\varphi {Z_i}}}{{\rho _i ^2 - S_{ 1,i}^2}}} \leqslant \sum\limits_{i = 1}^3 {\left( {\frac{{\beta S_{ 1,i}^2\varphi ^2 }}{{{(\rho _i ^2 - S_{ 1,i}^2)}^2 }} + \frac{{S_{ 2,i}^2}}{{4\beta }}} \right)} $(23)

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    $ \varXi \leqslant \sum\limits_{i = 1}^3 {\left( {\frac{{S_{ 1,i}^2{\varTheta _i}{\varPhi _i}}}{{\rho _i ^2 - S_{ 1,i}^2}} + \frac{{S_{ 2,i}^2}}{{4\beta }} + \frac{3}{{2\beta }}} \right)} $(24)

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    $ {\tilde \varTheta _i}{\hat \varTheta _i} \leqslant - {\tilde \varTheta _i}^2/2 + \varTheta _i^2/2 $(25)

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    $ {{\dot V}_1} \leqslant - \sum\limits_{i = 1}^3 {\frac{{{K_{1,i}}S_{ 1,i}^2}}{{\rho _i^2 - S_{ 1,i}^2}}} + \frac{{{{\left\| {{{\boldsymbol{S }}_2}} \right\|}^2}}}{{4\beta }} - {c_1}\sum\limits_{i = 1}^3 {\frac{{{{\tilde \varTheta }_i}^2}}{{2{c_2}}}} + \frac{{{c_1}{{\left\| {\varTheta } \right\|}^2}}}{{2{c_2}}} + \frac{3}{{2\beta }} $(26)

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    $ {{\boldsymbol{S }}_2} = {\boldsymbol{\upsilon }} - {{\boldsymbol{s}}_{\text{d}}} $(27)

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    $ {{\boldsymbol{M}}}{{\dot {\boldsymbol{S}}}_2} = - {\boldsymbol{C}}({\boldsymbol{\upsilon }}){\boldsymbol{\upsilon }} - {\boldsymbol{D}}({\boldsymbol{\upsilon }}){\boldsymbol{\upsilon }} - {M}{{\dot {\boldsymbol{s}}}_{\text{d}}} + {\boldsymbol{\tau }} + {{\boldsymbol{\tau }}_{\text{ω }}} $(28)

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    $ {\boldsymbol{F}}({\boldsymbol{X}}) = {\boldsymbol{W}}_{\text{a}}^{\text{T}}{\xi }({\boldsymbol{X}}) + {{{\boldsymbol{\varepsilon}} }_{\text{a}}} $(29)

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    $ ξ(\boldsymbolX)=[ξ1(\boldsymbolX);ξ2(\boldsymbolX),ξ3(\boldsymbolX)]T\boldsymbolWa=diag(\boldsymbolWa,1T,\boldsymbolWa,2T,\boldsymbolWa,3T) $()

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    $ \boldsymbolS2T[\boldsymbolda−Tanh(\boldsymbolS2\boldsymbolσ)\boldsymbolθ]⩽\boldsymbolθT[⌈\boldsymbolS2⌉−Tanh(\boldsymbolS2\boldsymbolσ)\boldsymbolS2]⩽0.2785\boldsymbolσT\boldsymbolθ $(30)

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    $ Tanh(\boldsymbolS2/\boldsymbolσ)=diag(tanh⁡(S2,1/σ1),tanh⁡(S2,2/σ2),tanh⁡(S2,3/σ3))⌈\boldsymbolS2⌉=[|S2,1|,|S2,2|,|S2,3|]T。 $()

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    $ {\boldsymbol{\tau }}= - {{\boldsymbol{K}}_2}{{\boldsymbol{S }}_2} - {\hat {\boldsymbol{W}}}_{\text{a}}^{\text{T}}{\xi }({\boldsymbol{X}}) - {\rm{Tanh}}({{\boldsymbol{S }}_2}/{\boldsymbol{\sigma }}){\hat {\boldsymbol{\theta}} } $(31)

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    $ {{\dot {\hat {\boldsymbol{W}}}}_{\text{a}}} = {{\boldsymbol{\varGamma }}_{\text{a}}}[{\mathrm{tr}}({\xi }({\boldsymbol{X}}){\boldsymbol{S }}_2^{\text{T}}) - \chi {{\hat {\boldsymbol{W}}}_{\text{a}}}] $(32)

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    $ {\dot {\hat {\boldsymbol{\theta}}} } = {\boldsymbol{\varLambda }}[{\rm{Tanh}}({{\boldsymbol{S }}_2}/{\boldsymbol{\sigma }}){{\boldsymbol{S }}_2} - \kappa {\hat {\boldsymbol{\theta}} }] $(33)

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    $ {V_2} = \frac{1}{2}{{\boldsymbol{S }}_2}^{\text{T}}{{\boldsymbol{M}}}{{\boldsymbol{S }}_2} + \frac{1}{2}{{\boldsymbol{Y}}^{\text{T}}}{\boldsymbol{Y}} + \frac{1}{2}{{\tilde {\boldsymbol{W}}}_{\text{a}}}^{\text{T}}{{\boldsymbol{\varGamma }}_{\text{a}}}^{ - 1}{{\tilde {\boldsymbol{W}}}}_{\text{a}} + \frac{1}{2}{{\tilde {\boldsymbol{\theta}} }^{\text{T}}}{{\boldsymbol{\varLambda }}^{ - 1}}{\tilde {\boldsymbol{\theta}} } $(34)

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    $ V˙2=\boldsymbolS2T\boldsymbolτ+\boldsymbolS2T\boldsymbolWaTξ(\boldsymbolX)−\boldsymbolS2T\boldsymbolW~aTξ(\boldsymbolX)+χ\boldsymbolW~aT\boldsymbolW^a−\boldsymbolθ~TTanh(\boldsymbolS2/\boldsymbolσ)\boldsymbolS2+κ\boldsymbolθ~T\boldsymbolθ^++\boldsymbolS2T\boldsymbolda+\boldsymbolYT\boldsymbolY˙ $(35)

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    $ {{\boldsymbol{Y}}^{\text{T}}}{\dot {\boldsymbol{Y}}} \leqslant - ({{\boldsymbol{Y}}^{\text{T}}}{\boldsymbol{Y}}/{T_{\text{d}}}) + ({\left\| {\boldsymbol{Y}} \right\|^2}{\left\| {{\boldsymbol{H}}( \cdot )} \right\|^2}/2) + 1/2 $(36)

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    $ \boldsymbolH(⋅)=r\boldsymbolEα−\boldsymbolRT(ψ)[(\boldsymbolK1++\boldsymbolΦ\boldsymbolΘ^)⋅(\boldsymbolR(ψ)\boldsymbolυ−\boldsymbolη˙d)+\boldsymbolΦ˙\boldsymbolΘ^\boldsymbole+\boldsymbolΦ\boldsymbolΘ^˙\boldsymbole] $()

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    $ {{\boldsymbol{S }}}_{\text{2}}^{\text{T}}{{\boldsymbol{d}}_{\text{a}}} \leqslant {{\boldsymbol{\theta }}^{\text{T}}}{\rm{Tanh}}({{\boldsymbol{S }}_2}/{\boldsymbol{\sigma }}){{\boldsymbol{S }}_2} + 0.278\;5{\boldsymbol{\sigma }}^{\text{T}} {\boldsymbol{\theta }} $(37)

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    $ \chi {{\tilde {\boldsymbol{W}}}_{\text{a}}}^{\text{T}}{{\hat {\boldsymbol{W}}}_{\text{a}}} \leqslant - (3\chi /4){{\tilde {\boldsymbol{W}}}_{\text{a}}}^{\text{T}}{{\tilde {\boldsymbol{W}}}_{\text{a}}} + \chi {\boldsymbol{W}}_{\text{o}}^2 $(38)

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    $ \kappa {{\tilde {\boldsymbol{\theta}} }^{\text{T}}}{\hat{\boldsymbol{ \theta}} } \leqslant - (3\kappa /4){{\tilde {\boldsymbol{\theta}} }^{\text{T}}}{\tilde {\boldsymbol{\theta}} } + \kappa {\left\| {\boldsymbol{\theta }} \right\|}^2 $(39)

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    $ V˙2⩽−\boldsymbolK2\boldsymbolS2T\boldsymbolS2−3χ4\boldsymbolW~aT\boldsymbolW~a−3κ4\boldsymbolθ~T\boldsymbolθ~−1Td\boldsymbolYT\boldsymbolY+12+12‖\boldsymbolY‖2‖\boldsymbolH(⋅)‖2+χ\boldsymbolWo2+κ‖\boldsymbolθ‖2+0.2785\boldsymbolσT\boldsymbolθ $(40)

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    $ V˙⩽−∑i=13K1,iS1,i2ρi2−S1,i2−c1∑i=13Θ~i22c2−[λmin(\boldsymbolK2)−14β]/λmax(\boldsymbolM)\boldsymbolS2T\boldsymbolM\boldsymbolS2−3χ4λmin(\boldsymbolΓa)\boldsymbolW~aT\boldsymbolW~a−3κ4λmin(\boldsymbolΛ)\boldsymbolθ~T\boldsymbolθ~−1Td\boldsymbolYT\boldsymbolY+c1‖\boldsymbolΘ‖22c2+32β+12+12‖\boldsymbolY‖2‖\boldsymbolH(⋅)‖2+χ\boldsymbolWo2+κ‖\boldsymbolθ‖2+0.2785\boldsymbolσT\boldsymbolθ⩽−ϑV+ϖ $(41)

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    $ ϑ=min{2K1,i,c1,2[λmin(\boldsymbolK2)−1/(4β)]/λmax(\boldsymbolM),1.5χλmin(\boldsymbolΓa),1.5κλmin(\boldsymbolΛ),2/Td}$()

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    $ ϖ=(c1‖\boldsymbolΘ‖2)/(2c2)+1/2+3/(2β)+χWo2+κ‖\boldsymbolθ‖2+0.2785\boldsymbolσT\boldsymbolθ+1.5‖\boldsymbolY‖2‖\boldsymbolH(⋅)‖2 $()

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    $ [{\lambda _{\min } ({\boldsymbol{K}}}_2 )]\beta > 0.25 $(42)

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    $ V \leqslant \varpi /\vartheta + (V(0) - \varpi /\vartheta ){\mathrm{e}}^{ - \vartheta t} $(43)

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    $ {{{\boldsymbol{\alpha}} }_{\text{o}}} = - {\boldsymbol{R}}^{\text{T}} (\psi )({{\boldsymbol{K}}_1} {\boldsymbol{e}} + {{\dot {\boldsymbol{\eta}} }_{\text{d}}}) $(44)

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    $ {{\boldsymbol{\tau }}_{\text{o}}} = - {{\boldsymbol{K}}_2}{{\boldsymbol{S }}_2} - {{\hat {\boldsymbol{W}}}_{\text{a}}}^{\text{T}}{\xi }({\boldsymbol{X}}) - {\rm{Tanh}}({{\boldsymbol{S }}_2}/{\boldsymbol{\sigma }}){\hat {\boldsymbol{\theta}} } $(45)

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    $ {{\dot {\boldsymbol{\eta}} }_{\text{d}}} = {\boldsymbol{R}}({\psi _{\text{d}}}){{\boldsymbol{\upsilon }}_{\text{d}}}{M}{{\dot \upsilon }_{\text{d}}} + {\boldsymbol{C}}({{\boldsymbol{\upsilon }}_{\text{d}}}){{\boldsymbol{\upsilon }}_{\text{d}}} + {\boldsymbol{D}}({{\boldsymbol{\upsilon }}_{\text{d}}}){{\boldsymbol{\upsilon }}_{\text{d}}} = {{\boldsymbol{\tau }}_{\text{d}}} + {{\boldsymbol{\tau }}_{\text{ω }}} $(46)

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    $ \boldsymbolη(0)=[−0.5,0.5,0.5]T,\boldsymbolυ(0)=[0.1,0.1,0.01]T\boldsymbolΘ^(0)=[3,\;3,\;3]T $()

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    $ {{\boldsymbol{K}}_1} ={\mathrm{ diag}}(4.5,4.5,4.5) ,\; {c_1} = 15 , \;{c_2} = 0.01 ,\; \beta = 0.008 $()

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    $ {{\boldsymbol{K}}_2} = {\mathrm{diag}}(40,40,40) ,\; {{\boldsymbol{\varGamma }}_{\text{a}}} = 10{{{\boldsymbol{I}}}_{30 \times 30}} ,\; \chi = 0.08 $()

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    $ {\boldsymbol{\varLambda }} = {\mathrm{diag}}(3,5,5) ,\; \kappa = 0.02 , \;{T_{\text{d}}} = 0.01 ,\; {T_i} = 10 $()

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    $ {\iota _i} = 0.2 , \;{\ell _i} = 3 , \;\rho _{{\text{T}},i} = 0.06（ i = 1,2,3 ） $()

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    Xiang YE, Chao CHEN, Jian Xiong JIA, Hang CHEN. Adaptive neural control for marine autonomous surface ships in cross-water scenarios[J]. Chinese Journal of Ship Research, 2025, 20(1): 309
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