Global trajectory tracking control of underactuated surface vessels with non-diagonal inertial and damping matrices

Abstract

This paper investigates the tracking control problem of underactuated surface vessels with non-diagonal inertia and damping matrices. Based on the inherent cascaded structure of vessel dynamics and the cascaded system theory, the tacking control problem of vessels is converted to the stabilization control problem of two cascaded subsystems. Two trajectory tracking control laws are designed respectively for known and unknown model parameters. The first feedback control law is derived with the aid of Lyapunov method, realizing the global ${\mathscr {K}}$-exponential trajectory tracking of vessels with accurate model parameters. As all the state variables in the first law are independent of model parameters, the sliding mode technique is applied to extend the first control scheme to the case of unknown model parameters, resulting into an another tracking control law, which ensures the global ${\mathscr {K}}$-exponential stability of the closed-loop error system despite unknown model parameters. Effectiveness of the proposed controllers is demonstrated by numerical simulations.

Appendix

Lemma 4

Suppose that Assumption 1 holds, then the reference velocity variables $(u_r,v_r,r_r)$ are bounded, and $\lim \nolimits _{t\rightarrow \infty }{r_r} \ne 0$.

Proof

Verify the boundedness of $(u_r,v_r,r_r)$. Define the Lyapunov function candidate:

$$\begin{aligned} \begin{aligned} V_3= \mathbf{v}_r^\mathrm{T} \mathbf{M} \mathbf{v}_r. \end{aligned} \end{aligned}$$

(32)

Differentiating $V_3$ yields:

$$\begin{aligned} {\dot{V}}_3= & {} -\,\mathbf{v}_r^\mathrm{T} \left[ \mathbf{C}(\mathbf{v}_r)\mathbf{v}_r+\mathbf{D}{} \mathbf{v}_r-{\bar{\tau }}_r\right] \nonumber \\&-\left[ \mathbf{C}(\mathbf{v}_r)\mathbf{v}_r+\mathbf{D}{} \mathbf{v}_r-{\bar{\tau }}_r\right] ^\mathrm{T} \mathbf{v}_r\nonumber \\= & {} -\,\mathbf{v}_r^\mathrm{T} \left[ \mathbf{C}(\mathbf{v}_r)+\mathbf{C}^\mathrm{T}(\mathbf{v}_r)\right] \mathbf{v}_r\nonumber \\&-\mathbf{v}_r^\mathrm{T} \left( \mathbf{D}+\mathbf{D}^\mathrm{T}\right) \mathbf{v}_r +2\mathbf{v}_r^\mathrm{T}{\bar{\tau }}_r\nonumber \\= & {} -\,\mathbf{v}_r^\mathrm{T} \left( \mathbf{D}+\mathbf{D}^\mathrm{T}\right) \mathbf{v}_r+2\mathbf{v}_r^\mathrm{T}{\bar{\tau }}_r, \end{aligned}$$

(33)

where $\mathbf{v}_r^\mathrm{T} \left[ \mathbf{C}(\mathbf{v}_r)+\mathbf{C}^\mathrm{T}(\mathbf{v}_r)\right] \mathbf{v}_r=0$ due to $\mathbf{C}=-\mathbf{C}^\mathrm{T}$. Since $\mathbf{D}+\mathbf{D}^\mathrm{T} >0$ and $-\,\mathbf{v}_r^\mathrm{T} \cdot$ $\left( \mathbf{D}+\mathbf{D}^\mathrm{T}\right) \mathbf{v}_r<0$, it easily follows that the system $\mathbf{M}{\dot{\mathbf{v}}}_r+\mathbf{C}(\mathbf{v}_r)\mathbf{v}_r+\mathbf{D}{} \mathbf{v}_r={\bar{\tau }}_r$ is input-to-state stable with ${\bar{\tau }}_r$ as input. As a result, $\mathbf{v}_r=[u_r,v_r,r_r]^\mathrm{T}$ is bounded for any bounded ${\bar{\tau }}_r=[\tau _{u_r},0,\tau _{r_r}]^\mathrm{T}$.

Verify $\lim \nolimits _{t\rightarrow \infty }{r_r} \ne 0$ using contradiction. Suppose $\lim \nolimits _{t\rightarrow \infty }{r_r} = 0$ holds. Since $r_r\in {\mathscr {L}}_\infty$ and $\lim \nolimits _{t\rightarrow \infty } r_r=0$, one gets that $\lim \nolimits _{t\rightarrow \infty } \int _0^t {\dot{r}}_r(t)dt$ exists and remains finite. As $(u_r,v_r,r_r)$ and $(\tau _{u_r},\tau _{r_r})$ are bounded, one concludes that $({\dot{u}}_r,{\dot{v}}_r,{\dot{r}}_r)$ are bounded. Then, $\ddot{r}_r$ is bounded too due to the bounded $({\dot{u}}_r,{\dot{v}}_r,{\dot{r}}_r,\dot{\tau }_{r_r})$, and ${\dot{r}}_r$ is uniformly continuous. Thus, according to the Barbalat’s lemma, ${\dot{r}}_r$ is convergent to zero.

Under $\lim \nolimits _{t\rightarrow \infty }{r_r} = 0$, $({\bar{v}}_r,r_r)$-dynamics will become

$$\begin{aligned} \left\{ \begin{aligned} \dot{{\bar{v}}}_r&= -\frac{d_{22}}{m_{22}} {\bar{v}}_r,\\ {\dot{r}}_r&=\frac{1}{\varDelta }\left[ m_{22}(m_{11}-m_{22}) u_r {\bar{v}}_r\right. \\&\quad \left. +a_v {\bar{v}}_r +m_{22}\tau _{r_r} \right] \rightarrow 0. \end{aligned}\right. \end{aligned}$$

(34)

As $m_{22}/d_{22}>0$, ${\bar{v}}_r \rightarrow 0$, and ${\dot{r}}_r\rightarrow m_{22}\tau _{r_r}/\varDelta \rightarrow 0$, which is contradictive with $\lim \limits _{t\rightarrow \infty }\tau _{r_r} \ne 0$. Thus, $r_r$ must satisfy $\lim \limits _{t\rightarrow \infty }{r_r} \ne 0$.$\square$