#### 期刊菜单

Research on Ships Trajectory Tracking Algorithm Based on Adaptive Control
DOI: 10.12677/DSC.2023.122009, PDF , HTML, XML, 下载: 130  浏览: 357

Abstract: This paper presents a fixed-time sliding mode control algorithm based on adaptive law for ships with disturbance. In the design of the controller, a dual layer nested adaptive scheme is used to reduce the chattering phenomenon, and the value of the control gain changes dynamically according to the external disturbance. Applying the fixed-time control theory to the research field of ships trajectory tracking can make the trajectory error independent of the initial state and reach the sliding surface in fixed time. In addition, the proposed control method can ensure the boundedness of the system trajectory. Finally, a simulation experiment is designed to verify the effectiveness and availability of the proposed method.

1. 引言

2. 船舶系统模型

2.1. 动力学模型

$\begin{array}{l}\stackrel{˙}{q}=J\left(\theta \right)\upsilon \\ M\stackrel{˙}{\upsilon }=-C\left(\upsilon \right)\upsilon -D\left(\upsilon \right)\upsilon +\tau +{d}_{1}\end{array}$ (1)

$J\left(\theta \right)=\left[\begin{array}{ccc}\mathrm{cos}\left(\theta \right)& -\mathrm{sin}\left(\theta \right)& 0\\ \mathrm{sin}\left(\theta \right)& \mathrm{cos}\left(\theta \right)& 0\\ 0& 0& 1\end{array}\right]$ $M=\left[\begin{array}{ccc}m-{X}_{\stackrel{˙}{u}}& 0& 0\\ 0& m-{Y}_{\stackrel{˙}{v}}& m{x}_{g}-{Y}_{\stackrel{˙}{r}}\\ 0& m{x}_{g}-{N}_{\stackrel{˙}{v}}& {I}_{z}-{N}_{\stackrel{˙}{r}}\end{array}\right]$

$C\left(\upsilon \right)=\left[\begin{array}{ccc}0& 0& {c}_{13}\left(\upsilon \right)\\ 0& 0& {c}_{23}\left(\upsilon \right)\\ -{c}_{13}\left(\upsilon \right)& -{c}_{23}\left(\upsilon \right)& 0\end{array}\right]$ $D\left(\upsilon \right)=\left[\begin{array}{ccc}{d}_{11}\left(\upsilon \right)& 0& 0\\ 0& {d}_{22}\left(\upsilon \right)& {d}_{23}\left(\upsilon \right)\\ 0& {d}_{32}\left(\upsilon \right)& {d}_{33}\left(\upsilon \right)\end{array}\right]$

${d}_{22}\left(\upsilon \right)=-{Y}_{v}-{Y}_{|v|v}|v|-{Y}_{|r|v}|r|$${d}_{23}\left(\upsilon \right)=-{Y}_{r}-{Y}_{|v|r}|v|-{Y}_{|r|r}|r|$${d}_{32}\left(\upsilon \right)=-{N}_{v}-{N}_{|v|v}|v|-{N}_{|r|v}|r|$${d}_{33}\left(\upsilon \right)=-{N}_{r}-{N}_{|v|r}|v|-{N}_{|r|r}|r|$ ，m表示船舶的质量， ${x}_{g}$ 表示船舶的重心到坐标系原点的距离。旋转矩阵 $J\left(\theta \right)$ 满足 ${J}^{\text{T}}\left(\theta \right)J\left(\theta \right)=I$$\stackrel{˙}{J}=JQ\left(r\right)$ 以及 $J\left(\theta \right)Q\left(r\right){J}^{\text{T}}\left(\theta \right)=Q\left(r\right)$ ，其中矩阵 $Q\left(r\right)$ 具有以下形式：

$Q\left(r\right)=\left[\begin{array}{ccc}0& -r& 0\\ r& 0& 0\\ 0& 0& 0\end{array}\right]$

$\begin{array}{c}{\stackrel{˙}{q}}_{2}=JQ\upsilon +J{M}^{-1}\left[-\left({C}_{0}\left(\upsilon \right)+\Delta C\left(\upsilon \right)\right)\upsilon -\left({D}_{0}\left(\upsilon \right)+\Delta D\left(\upsilon \right)\right)\upsilon +\tau +{d}_{1}\right]\\ =Q{q}_{2}-J{M}^{-1}\left({C}_{0}\left(\upsilon \right)+{D}_{0}\left(\upsilon \right)\right)\upsilon +J{M}^{-1}\tau +d\\ =f\left(q\right)+g\left(q\right)\tau +d\end{array}$ (2)

$\left\{\begin{array}{l}{\stackrel{˙}{q}}_{1}={q}_{2}\\ {\stackrel{˙}{q}}_{2}=f\left(q\right)+g\left(q\right)\tau +d\end{array}$ (3)

2.2. 轨迹误差模型

$\left\{\begin{array}{l}{\stackrel{˙}{q}}_{d1}={q}_{d2}\\ {\stackrel{˙}{q}}_{d2}={f}_{d}\left({q}_{d},{\tau }_{d}\right)\end{array}$ (4)

$\left\{\begin{array}{l}{\stackrel{˙}{x}}_{1}={x}_{2}\\ {\stackrel{˙}{x}}_{2}=f\left(q\right)+g\left(q\right)\tau +d-{f}_{d}\left({q}_{d},{\tau }_{d}\right)\end{array}$ (5)

3. 基于自适应的固定时间滑模控制算法

$s=c{x}_{1}+{x}_{2}$

$\tau \left(x\right)=-g{\left(q\right)}^{-1}\left[f\left(q\right)-{f}_{d}+c{x}_{2}+{k}_{1}\mathrm{sgn}\left(s\right)+{k}_{2}{‖s‖}^{\lambda }\mathrm{sgn}\left(s\right)+{k}_{3}{‖s‖}^{\mu }\mathrm{sgn}\left(s\right)+{k}_{4}s\right]$ (6)

$T\le \frac{1}{\left({k}_{1}-\kappa \right)/\sqrt{2}}+\frac{1}{{k}_{2}\left(\lambda -1\right){2}^{\frac{\lambda -1}{2}}}$ (7)

$\begin{array}{c}\stackrel{˙}{s}=f\left(q\right)+g\left(q\right)\tau +d-{f}_{d}+c{x}_{2}\\ =-\left({k}_{1}\mathrm{sgn}\left(s\right)+{k}_{2}{‖s‖}^{\lambda }\mathrm{sgn}\left(s\right)+{k}_{3}{‖s‖}^{\mu }\mathrm{sgn}\left(s\right)+{k}_{4}s\right)+d\end{array}$ (8)

$\begin{array}{c}\stackrel{˙}{V}=-{s}^{\text{T}}\left({k}_{1}\mathrm{sgn}\left(s\right)+{k}_{2}{‖s‖}^{\lambda }\mathrm{sgn}\left(s\right)+{k}_{3}{‖s‖}^{\mu }\mathrm{sgn}\left(s\right)+{k}_{4}s\right)+{s}^{\text{T}}d\\ \le -\left({k}_{1}-\kappa \right)s-{k}_{2}{‖s‖}^{\lambda +1}-{k}_{3}{‖s‖}^{\mu +1}-{k}_{4}{s}^{\text{T}}s\end{array}$ (9)

$\stackrel{˙}{V}\le -{2}^{\frac{1}{2}}\left({k}_{1}-\kappa \right){V}^{\frac{1}{2}}-{2}^{\frac{\lambda +1}{2}}{k}_{2}{V}^{\frac{\lambda +1}{2}}$ (10)

$T\le \frac{1}{\left({k}_{1}-\kappa \right)/\sqrt{2}}+\frac{1}{{k}_{2}\left(\lambda -1\right){2}^{\frac{\lambda -1}{2}}}$ (11)

${\stackrel{˙}{x}}_{1}={x}_{2}=-c{x}_{1}$ (12)

$\tau \left(x\right)=-g{\left(q\right)}^{-1}\left[f\left(q\right)-{f}_{d}+c{x}_{2}+k\left(t\right)\mathrm{sgn}\left(s\right)+{k}_{2}{‖s‖}^{\lambda }\mathrm{sgn}\left(s\right)+{k}_{3}{‖s‖}^{\mu }\mathrm{sgn}\left(s\right)+{k}_{4}s\right]$ (13)

$\tau ={\tau }_{1}+{\tau }_{2}$ ，其中

$\begin{array}{l}{\tau }_{1}=-g{\left(q\right)}^{-1}\left[f\left(q\right)-{f}_{d}+c{x}_{2}+{k}_{2}{‖s‖}^{\lambda }\mathrm{sgn}\left(s\right)+{k}_{3}{‖s‖}^{\mu }\mathrm{sgn}\left(s\right)+{k}_{4}s\right]\\ {\tau }_{2}=-g{\left(q\right)}^{-1}k\left(t\right)\mathrm{sgn}\left(s\right)\end{array}$ (14)

${\stackrel{˙}{\stackrel{¯}{\tau }}}_{eq}\left(t\right)=\frac{1}{\mathcal{l}}\left(-k\left(t\right)\mathrm{sgn}\left(s\right)-{\stackrel{¯}{\tau }}_{eq}\left(t\right)\right)$ (15)

$\frac{1}{\theta }‖{\stackrel{¯}{\tau }}_{eq}‖+\frac{\sigma }{2}>‖{\tau }_{eq}\left(t\right)‖=‖d\left(t\right)‖$ (16)

$\omega \left(t\right)=k\left(t\right)-\frac{1}{\theta }‖{\stackrel{¯}{\tau }}_{eq}‖-\sigma$ (17)

$\stackrel{˙}{k}\left(t\right)=-\epsilon \left(t\right)\mathrm{sgn}\left(\omega \left( t \right)\right)$

$\epsilon \left(t\right)={\nu }_{0}+\nu \left(t\right)$ (18)

$\beta \left(t\right)=\frac{h{\kappa }_{0}}{\theta }-\nu \left(t\right)$ (19)

$\stackrel{˙}{\nu }\left(t\right)=\alpha |\omega \left(t\right)|+{\nu }_{0}\sqrt{\alpha }\mathrm{sgn}\left(\beta \left(t\right)\right)$ (20)

${\stackrel{˙}{V}}_{0}=\omega \left(t\right)\stackrel{˙}{\omega }\left(t\right)+\frac{1}{\alpha }\beta \left(t\right)\stackrel{˙}{\beta }\left(t\right)$ (21)

$\begin{array}{c}\omega \left(t\right)\stackrel{˙}{\omega }\left(t\right)=\omega \left(t\right)\left(-\left({\nu }_{0}-\beta \left(t\right)+\frac{h{\kappa }_{0}}{\theta }\right)\mathrm{sgn}\left(\omega \left(t\right)\right)-\frac{1}{\theta }‖{\stackrel{˙}{\stackrel{¯}{\tau }}}_{eq}‖\right)\\ \le -{\nu }_{0}|\omega \left(t\right)|+\left(\beta \left(t\right)-\frac{h{\kappa }_{0}}{\theta }\right)|\omega \left(t\right)|+\frac{1}{\theta }|\omega \left(t\right)|{\stackrel{˙}{\stackrel{¯}{\tau }}}_{eq}\\ \le -{\nu }_{0}|\omega \left(t\right)|+\left(\beta \left(t\right)-\frac{h{\kappa }_{0}}{\theta }\right)|\omega \left(t\right)|+\frac{h{\kappa }_{0}}{\theta }|\omega \left(t\right)|\\ =-{\nu }_{0}|\omega \left(t\right)|+\beta \left(t\right)|\omega \left(t\right)|\end{array}$ (22)

$\beta \left(t\right)\stackrel{˙}{\beta }\left(t\right)=-\beta \left(t\right)\left(\alpha |\omega \left(t\right)|+{v}_{0}\sqrt{\alpha }\mathrm{sgn}\left(\beta \left(t\right)\right)\right)$ (23)

$\begin{array}{c}{\stackrel{˙}{V}}_{0}\le -{v}_{0}|\omega \left(t\right)|+\beta \left(t\right)|\omega \left(t\right)|-\frac{1}{\alpha }\beta \left(t\right)\left(\alpha |\omega \left(t\right)|+{v}_{0}\sqrt{\alpha }\mathrm{sgn}\left(\beta \left(t\right)\right)\right)\\ =-{v}_{0}|\omega \left(t\right)|-{v}_{0}\frac{1}{\sqrt{\alpha }}|\beta \left(t\right)|\\ \le -{v}_{0}\sqrt{{|\omega \left(t\right)|}^{2}+{\left(\frac{1}{\sqrt{\alpha }}|\beta \left(t\right)|\right)}^{2}}\\ =-{v}_{0}\sqrt{2{V}_{0}}\end{array}$ (24)

$\stackrel{˙}{v}\left(t\right)=\left\{\begin{array}{l}\alpha |\omega \left(t\right)|,\text{\hspace{0.17em}}\text{\hspace{0.17em}}|\omega \left(t\right)|>{\omega }_{0}\\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}其他\end{array}$ (25)

$\frac{1}{4}{\sigma }^{2}>{\omega }_{0}^{2}+\frac{1}{\alpha }{\left(\frac{h{\kappa }_{0}}{\theta }\right)}^{2}$ (26)

$|\omega \left(t\right)|>{\omega }_{0}$ 时， $\stackrel{˙}{v}\left(t\right)=\alpha |\omega \left(t\right)|$ 。对 $\beta \left(t\right)$ 进行求导，则有：

$\beta \left(t\right)\stackrel{˙}{\beta }\left(t\right)=-\beta \left(t\right)\nu \left(t\right)=-\alpha \beta \left(t\right)|\omega \left(t\right)|$ (27)

${\stackrel{˙}{V}}_{0}\le -{v}_{0}|\omega \left(t\right)|+\beta \left(t\right)|\omega \left(t\right)|+\frac{1}{\alpha }\left(-\alpha \beta \left(t\right)|\omega \left(t\right)|\right)=-{v}_{0}|\omega \left(t\right)|$ (28)

$|\omega \left(t\right)|\le {\omega }_{0}$ 时， $\stackrel{˙}{v}\left(t\right)=0$ 成立，于是有：

${\stackrel{˙}{V}}_{0}\le -{v}_{0}|\omega \left(t\right)|+\beta \left(t\right)|\omega \left(t\right)|$ (29)

${\stackrel{¯}{V}}_{0}=\left\{\left(\omega ,\beta \right):{V}_{0}<\frac{1}{2}{\omega }_{0}^{2}+\frac{1}{2\alpha }{\left(\frac{h{\kappa }_{0}}{\theta }\right)}^{2}\right\}$ (30)

$\begin{array}{c}{\int }_{0}^{+\infty }{\stackrel{˙}{V}}_{0}\text{d}t={V}_{0}\left(+\infty \right)-{V}_{0}\left(0\right)\\ =-{V}_{0}\left(0\right)\\ \le {\int }_{0}^{+\infty }-{v}_{0}|\omega \left(t\right)|\text{d}t\end{array}$ (31)

$\begin{array}{c}|k\left(t\right)|<|\omega \left(t\right)|+‖{\stackrel{¯}{\tau }}_{eq}\left(t\right)‖/\theta +\sigma \\ <|\omega \left(t\right)|+\left(1+{r}_{1}\right)\kappa /\theta +\sigma \end{array}$ (32)

$|\omega \left(t\right)|=|k\left(t\right)-‖{\stackrel{¯}{\tau }}_{eq}\left(t\right)‖/\theta -\sigma |<\sigma /2$

$k\left(t\right)>‖{\stackrel{¯}{\tau }}_{eq}\left(t\right)‖/\theta +\sigma /2>‖d\left(t\right)‖$ (33)

$|v\left(t\right)|=|h{\kappa }_{0}/\theta -\beta \left(t\right)|$ (34)

$|v\left(t\right)|<{\kappa }_{0}/\theta +|\beta \left(t\right)|$ (35)

4. 数值仿真

${q}_{d1}={\left[\mathrm{sin}\left(t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cos}\left(2t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}2\mathrm{sin}\left(2t\right)\right]}^{\text{T}}$

${q}_{d2}={\left[\mathrm{cos}\left(t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\mathrm{sin}\left(2t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}4\mathrm{cos}\left(2t\right)\right]}^{\text{T}}$

$c=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Figure 1. Sliding mode variable under different initial states

Figure 2. Trajectory error under different initial states

Figure 3. Control input under adaptive gain

Figure 4. Control gain and norm of disturbance

Figure 5. Sliding mode

Figure 6. The actual and desired trajectory of autonomous surface vehicle

Figure 7. The trajectory error of autonomous surface vehicle

5. 结论

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