Stabilization of USVs Under Mismatched Condition Based on Fixed-Time Observer

This study investigates the problem of asymptotic stabilization of underactuated surface vehicles (USVs) in the presence of simultaneous matched and mismatched disturbances. Within the framework of back-stepping control and by introducing a periodic term in the control laws of virtual yaw angle and surge velocity, a control design method is developed to stabilize the USVs into the bounded region. By the induction method, a novel fixed-time disturbance observer based on higher-order sliding mode differentiators are introduced. Then a control law based on the proposed observers is proposed, by which the errors between the virtual and actual yaw angle and surge velocity can converge to zero, and the state of the USVs converge to a bounded region. Numerical simulations are carried out to verify the proposed control schemes.

SECTION I. Introduction

Stabilization of underactuated surface vehicles (USVs) is widely applied in maritime rescue, self-positioning, autonomous docking and other ones [1]. As shown in [2], the dynamics of underactuated vehicles contains no gravitational field component and, hence, does not satisfy Brockett’s necessary condition [3]. In this case, the vehicles cannot be asymptotically stabilized to the desired equilibrium point with time-invariant continuous feedback laws. For these reasons, stabilization of USVs has attracted great attentions in recent years [4]. Pettersen and Egeland gave a stabilization method in [5] that can exponentially stabilize the USVs to a vicinity of the equilibrium. References [1], [6], [7] gave control methods with different concerns that can guarantee global asymptotic stability of USVs. Additionally, finite-time stabilization methods have been used in USVs such as [15]. These works are commonly based on the assumption that no disturbance exists.

USV is susceptible to disturbances and unknown parameters. Without special treatment, it reduces the control accuracy of the USV and affects the stability of the closed-loop system. Therefore, the problem of interference suppression and compensation is the USV stabilization control important question. To this end, [8] adopts adaptive sliding mode design, which can use an adaptive function to estimate unknown disturbance. Reference [9] proposed a new model predictive control algorithm. Literature [10] designed an adaptive fuzzy stable controller; the adaptive fuzzy system is combined with the auxiliary dynamic function to approximate the unknown disturbance. These controllers rely on intelligent control methods such as model predictive control, neural network control, adaptive fuzzy control, etc., with high computational cost and unknown solvability parameters. Thus, we are interested in designing a non-intelligent method to stabilize USVs with disturbances.

Traditional robust control methods usually are not accurate enough to model disturbances; hence it is difficult to solve complex disturbance models. In such a case, disturbance observer-based control (DOBC) can play an important role; see [11], [12], and the references therein. By adjusting the feedback controller and the disturbance observer, the closed-loop stability performance and the interference compensation performance can be improved respectively, thereby effectively improving the anti-interference performance of the closed-loop system and reducing the conservativeness. Compared with asymptotic control approaches, the finite-time DOBC method has not only faster convergence speed but also better disturbance rejection properties and robustness [16], [17]. Further, a fixed-time DOBC strategy in [21] can guarantee states tracking preset trajectories within a designated time independent of the initial conditions. Hence, finding out a method combining fixed-DOBC and nonlinear control laws is needed for USVs stabilization.

The existing works on USV stabilization rarely consider mismatched disturbance rejection, which is still a challenging issue worth investigating. Most DOBC methods are only available for systems with matched disturbances [13], i.e., the disturbances and the control inputs enter the system via the same channel. However, mismatched disturbances are frequently encountered in practical USVs. By designing appropriate disturbance compensation gains, some DOBC methods, after extension, can cope with mismatched disturbances for nonlinear systems [13], [14]. However, due to the coupling between velocities and yaw angle in USVs, mismatched disturbances in the USV system cannot be eliminated using most DOBC methods. Thus, to design control laws for USVs with non-matching disturbances, is also a challenging problem that needs to be studied.

This article aims at giving a new method with a faster convergence rate and a more robust disturbance rejection performance for the stabilization of USVs. We want to find out a fixed-time DOBC method for USVs in the presence of simultaneous mismatched and matched disturbances. A higher-order sliding mode differentiator is an efficient DOBC method to estimate disturbances, which shows many attractive properties, including its insensitivity to external and internal disturbances, ultimate accuracy, and finite-time convergent performance [14]. How to adopt the higher-order sliding mode differentiator in the fixed-time observers requires extensive research. Most USV stabilization methods rely on the environment with no perturbance and coordinate transformations with some singular features. In the presence of disturbances, the closed-loop system has no robustness and may diverge to infinity. Even periodic time-varying methods avoid this defect; control laws may not work when the introduced auxiliary terms are near zero points so that the anti-interference ability of periodic time-varying control law is also intermittent. We first give a virtual yaw angle and surge velocity within the back-stepping method with a periodic term. It is illustrated that the bound of the USV system depends on the parameters of virtual states by the scaling method. Second, a fixed-time observer is introduced so that after the settling time, unknown disturbances can be seen as known. At last, by combining the observer and back-stepping method, a control law is proposed. The errors between actual and virtual yaw angles and surge velocities can converge in a given fixed time, ensuring the global boundedness of the original USV system.

Compared with the previous works, our main contributions are summarized as follows: (i) Mismatched disturbances are addressed. Based on the back-stepping method and a periodic term, the performance bound of the USV system depends solely on the controller parameters. (ii) Based on a novel higher-order sliding mode differentiator, the proposed observer can estimate disturbances in a fixed time instead of finite time. Unlike fixed time observer in [21], the proposed one can estimate high order derivatives of disturbances.

The rest of the work is structured as below. Section 2 gives the preliminaries and presents the problems to be solved. Section 3 and Section 4, respectively, shows the fixed-time disturbance observers and stabilization control algorithm for USVs. In Section 5, simulations validate the effectiveness of the proposed protocols. In Section 6, we conclude the results.

SECTION II. Problem Statements and Preliminaries

A. Modeling and Objective

Consider a USV as depicted in Figure 1 with an inertial frame $\{O_{G}\}$ and a body-fixed frame $\{O_{E}\}$ . Denote $x$ and $y$ as positions of the USV in an inertial frame, and $\psi$ as the yaw angle relative to the geographic north. The kinematics and of the USV with mismatched and matched disturbances are given as follows,

$$\begin{align*}&\begin{cases} \dot {x} = u\mathrm {\cos }\psi - v\mathrm {\sin }\psi + \xi _{x},\\ \dot {y} = u\mathrm {\sin }\psi + v\mathrm {\cos }\psi + \xi _{y},\\ \dot {\psi }=r, \end{cases}\tag{1a}\\&\begin{cases} \dot {u}=-\dfrac {d_{11}}{m_{11}}u + \dfrac {m_{22}}{m_{11}}vr + \dfrac {\tau _{1}}{m_{11}} + \xi _{1},\\ \dot {v}=-\dfrac {d_{22}}{m_{22}}v-\dfrac {m_{11}}{m_{22}}ur+\xi _{v},\\ \dot {r}=-\dfrac {d_{33}}{m_{33}}r - \dfrac {m_{22} - m_{11}}{m_{33}}uv + \dfrac {\tau _{2}}{m_{33}} + \xi _{2}, \end{cases}\tag{1b}\end{align*}$$ Here $u$ and $v$ are velocities of the vehicle in the surge and sway, $r$ is the yaw velocity, parameters $m_{11},~m_{22}$ and $m_{33}$ are known constants representing inherent and added mass coefficients, $d_{11},d_{22}$ and $d_{33}$ are known constants representing hydrodynamic damping coefficients, $\tau _{1}$ and $\tau _{2}$ are the force and torque inputs, $\xi _{x}$ and $\xi _{y}$ are unknown mismatched disturbances, $\xi _{v}$ , $\xi _{1}$ and $\xi _{2}$ are unknown matched disturbances. Table 1 shows all the variables involved in the USV model.

TABLE 1 The Stabilization Control Algorithm of USVs

FIGURE 1.

The underactuated surface vehicle.

Assumption 1:

Disturbances $\xi _{x}$ , $\xi _{y}$ , $\xi _{v}$ , $\xi _{1}$ and $\xi _{2}$ satisfy that $|\dot {\xi }_{x}| < \sigma _{x}$ , $|\dot {\xi }_{y}| < \sigma _{y,1}$ , $|\dot {\xi }_{v}| < \sigma _{v}$ , $|\dot {\xi }_{1}| < \sigma _{1}$ , $|\dot {\xi }_{2}| < \sigma _{2}$ , $|\ddot {\xi }_{y}| < \sigma _{y,2}$ , $|\xi _{x}| < \xi _{xmax}$ and $|\xi _{y}| < \xi _{ymax}$ . Here $\sigma _{x}$ , $\sigma _{y,1}$ , $\sigma _{y,2}$ , $\sigma _{v}$ , $\sigma _{1}$ , $\sigma _{2}$ , $\xi _{xmax}$ and $\xi _{ymax}$ are known positive constants [21].

Ocean disturbances are due to low-frequency waves, currents and wind, whose derivatives can be seen bounded. Here, we need not compensate for high-frequency disturbances, since they cause back-and-forth rocking motions of USVs [22].

Assumption 2:

The sway velocity of the USV is bounded by $|v| \leq v_{\max }$ .

The assumption of passive-boundedness for the sway velocity is a mild one widely adopted in the literature. For hostile enough sea conditions that may threaten the mission of vehicles, we are unable to attenuate the disturbances with actuators and even the most advanced control technologies may fail. So, tough disturbances are beyond our interest here [23].

B. Objective

The USV stabilization problem considered is stated as: under Assumptions 1 and 2, design a robust controller for the USV (1), so that system states $x$ , $y$ , $\psi$ , $u$ and $r$ are globally bounded.

Remark 1:

In this article, we consider disturbances containing both matched and mismatched disturbances on the velocities and positions. This problem is very challenging but rarely considered in previous works. Note that uncertainties are included in the disturbances.

SECTION III. Fixed-Time Disturbance Observers

Before designing disturbance observers, the following definitions and lemma are introduced. Consider the following system

$$\begin{equation*} \dot {z}=f(z,t)+\xi, \tag{2}\end{equation*}$$ where $z$ is the real state, $f(z,t)$ is the known function about $z$ and $t$ , $\xi$ is the lumped disturbance satisfying the following condition $|\xi ^{(i)}| < L_{i}$ , where $L_{i}$ are known constants, $i=1,2,\ldots n+1$ .

Definition 1[26]:

The system (2) is said to be finite-time stable, if it is globally asymptotically stable and any solution $z\left ({t,z_{0} }\right)$ reaches the equilibria at some finite-time moment, i.e., $z\left ({t,z_{0} }\right) = 0$ , $\forall t \geq T\left ({z_{0} }\right)$ , where $T$ :$R^{n} \rightarrow R^{+} \cup \left \{{ 0 }\right \}$ is the settling-time function.

Definition 2[26]:

The system (2) is said to be fixed-time stable if it is finite-time stable and the settling time function $T\left ({z_{0} }\right)$ is bounded, i.e., $\exists ~T_{\max } > 0:T\left ({z_{0} }\right) < T_{\max }, \forall z_{0} \in R^{n}$ .

Define the symbol $\lceil \cdot \rfloor ^{\lambda }$ as $| \cdot |^{\lambda }\mathbf {Sign}(\cdot)$ .

Lemma 1[21]:

For the system (2), let $\epsilon _{\xi }$ be the estimation state of $\xi$ . Disturbance $\xi$ can be estimated within a fixed time $T_{\xi }$ by the following observer

$$\begin{equation*} \dot {\epsilon }_{\xi }=\kappa _{3}\mathbf {sign}(z-\hat {z}),\tag{3}\end{equation*}$$ where $\hat {z}$ is an auxiliary estimation state which is given by $$\begin{equation*} \dot {\hat {z}}= \epsilon _{\xi } + \kappa _{1}\lceil \tilde {z}\rfloor ^{\frac {1}{2}} + \kappa _{3}\lceil \tilde {z}\rfloor ^{p}.\tag{4}\end{equation*}$$ Here $\tilde {z}=z-\hat {z}$ , $p>1$ , $k_{1}>\sqrt {2\kappa _{3}}$ , $\kappa _{2}>0$ , $\kappa _{3}>4L_{1}$ are parameters to be chosen, the settling time $T_{\xi }$ satisfies that $$\begin{align*}&\hspace {-0.5pc}T_{\xi }\leq \left \lbrack{ \frac {1}{\kappa _{2}(p - 1)(2^{1/4}\kappa _{1}/\kappa _{2})^{\frac {p- 1}{p + 1/2}}} + \frac {2(2^{3/4}\kappa _{1}/\kappa _{2})^{1/2}}{\kappa _{1}} }\right \rbrack \\&\qquad\qquad\qquad\quad \displaystyle {\times \,\left \lbrack{ 1 + \frac {\kappa _{3} + \sigma }{(\kappa _{3} - \sigma)(1 - \sqrt {2\kappa _{3}}/\kappa _{1})} }\right \rbrack.} \tag{5}\end{align*}$$

Compared with finite-time/asymptotic method, the observer in [21] can estimate the disturbance in a fixed time. The existence of mismatched disturbance make it necessary to estimate the higher-order derivatives of the disturbance, which is quite difficult. Here, we give our results on fixed-time observers based on high-order sliding mode differentiator.

Theorem 1:

For the system (2), let $\epsilon _{\xi,0}$ , $\epsilon _{\xi,1},\ldots \epsilon _{\xi,i}, \ldots \epsilon _{\xi,n}$ be the estimation states of $\xi$ , $\dot {\xi }, \ldots, \xi ^{(i)}, \ldots, \xi ^{(n)}$ . Then, the following observer can estimate disturbance $\xi$ within a fixed time,

$$\begin{align*} \dot {\epsilon }_{\xi,0}=&\kappa _{3,0}\mathbf {sign}(z-\hat {z}), \\ \dot {\epsilon _{\xi,i}}=&\kappa _{3,i}\mathbf {sign}(\epsilon _{\xi,i-1}-\hat {\xi }_{i-1}), \\ \dot {\epsilon _{\xi,n}}=&\kappa _{3,n}\mathbf {sign}(\epsilon _{\xi,n-1}-\hat {\xi }_{n-1}),\tag{6}\end{align*}$$ where $\hat {z}$ , $\hat {\xi }_{0}$ , $\hat {\xi }_{1},\ldots,\hat {\xi }_{i}$ ,…$\hat {\xi }_{n-1}$ are given by $$\begin{align*} \dot {\hat {z}}=&f(z,t)+\epsilon _{\xi,0}+\kappa _{0,1}\lceil z-\hat {z}\rfloor ^{\frac {1}{2}} + \kappa _{0,2}\lceil z-\hat {z}\rfloor ^{p_{\xi,0}},\\ \dot {\hat {\xi }}_{0}=&\epsilon _{\xi,1}+\kappa _{1,1}\lceil \epsilon _{\xi,0}-\hat {\xi }_{0}\rfloor ^{\frac {1}{2}} + \kappa _{1,2}\lceil \epsilon _{\xi,0}-\hat {\xi }_{0}\rfloor ^{p_{\xi,1}},\\ \dot {\hat {\xi }}_{i-1}=&\epsilon _{\xi,i}+\kappa _{i,1}\lceil \epsilon _{\xi,i-1}-\hat {\xi }_{i-1}\rfloor ^{\frac {1}{2}} + \kappa _{i,2}\lceil \epsilon _{\xi,i-1}-\hat {\xi }_{i-1}\rfloor ^{p_{\xi,i}},\\ \dot {\hat {\xi }}_{n-1}=&\kappa _{n,1}\lceil \epsilon _{\xi,n-1}-\hat {\xi }_{n-1}\rfloor ^{\frac {1}{2}} + \kappa _{n,2}\lceil \epsilon _{\xi,n-1}-\hat {\xi }_{n-1}\rfloor ^{p_{\xi,n}},\end{align*}$$ with constants $p_{\xi,j}>1$ , $\kappa _{j,1}>\sqrt {2\kappa _{j,3}}$ , $~\kappa _{j,2} > 0$ , $\kappa _{\xi,j}>4L_{j+1}$ , $j=0,1,2,\ldots n$ .

Proof:

According to Lemma 1, $\epsilon _{\xi,0}$ can estimate the state $\xi$ within a fixed time

$$\begin{align*}&\hspace {-1pc}T_{\xi,0} \\=&\left [{\!\! \frac {1}{\kappa _{2}(p_{\xi,0}\!-\!1)(2^{1/4}\kappa _{0,1}/\kappa _{0,2})^{\frac {p_{\xi,0}\!- \! 1}{p_{\xi,0} + 1/2}}}\!+\! \frac {2(2^{3/4}\kappa _{0,1}/\kappa _{0,2})^{1/2}}{\kappa _{1}} \!}\right] \\&{\times \,\left \lbrack{ 1 + \frac {\kappa _{3,0} + L_{1}}{(\kappa _{3,0} - L_{1})(1 - \sqrt {2\kappa _{0,3}}/\kappa _{0,1})} }\right \rbrack.}\end{align*}$$ Next, based on the estimation of $\xi$ , we show how to observe the $n$ th derivative of the disturbance by inductive method.

Case 1. According to the equation (6), we can have

$$\begin{equation*} \dot {\epsilon _{\xi,1}}=\kappa _{3,1}\mathbf {sign}(\dot {\xi }-\hat {\xi }_{0}),\tag{7}\end{equation*}$$ with $\dot {\hat {\xi }}_{0}=\epsilon _{\xi,1}+\kappa _{1,1}\lceil \xi -\hat {\xi }_{0}\rfloor ^{\frac {1}{2}} + \kappa _{1,2}\lceil \xi -\hat {\xi }_{0}\rfloor ^{p_{\xi,1}}$ . This indicates that $\epsilon _{\xi,1}$ can estimate the state $\dot {\xi }$ within the fixed time $$\begin{align*}&\hspace {-1pc}T_{\xi,1} \\=&\left [{\!\! \frac {1}{\kappa _{2}(p_{\xi,1}\!-\!1)(2^{1/4}\kappa _{1,1}/\kappa _{2,1})^{\frac {p_{\xi,1}- 1}{p_{\xi,1} + 1/2}}}\!+\! \frac {2(2^{3/4}\kappa _{1,1}/\kappa _{2,1})^{1/2}}{\kappa _{1,1}} \!\!}\right]\\&{\times \,\left \lbrack{ 1 + \frac {\kappa _{1,3} + L_{2}}{(\kappa _{3,1} - L_{2})(1 - \sqrt {2\kappa _{3,1}}/\kappa _{1,1})} }\right \rbrack.}\end{align*}$$

Case $j$ . By the proposed observers, $\epsilon _{\xi,j-1}=\xi ^{(j-1)}$ and Lemma 1, state $\xi ^{(j)}$ can be estimated by $\epsilon _{\xi,j}$ within the fixed time

$$\begin{align*}&\hspace {-1pc}T_{\xi,j} \\=&\left [{\!\! \frac {1}{\kappa _{2}(p_{\xi,j}\!-\!1)(2^{1/4}\kappa _{1,j}/\kappa _{2,j})^{\frac {p_{\xi,j}- 1}{p_{\xi,j}\! +\! 1/2}}}\!+ \!\frac {2(2^{3/4}\kappa _{1,j}/\kappa _{2,j})^{1/2}}{\kappa _{1,j}} \!\!}\right]\\&{\times \,\left \lbrack{ 1 + \frac {\kappa _{3,j} + L_{j+1}}{(\kappa _{3,j} - L_{j+1})(1 - \sqrt {2\kappa _{3,j}}/\kappa _{1,j})} }\right \rbrack.}\end{align*}$$

Case $n$ . After the settling time of the estimation of $\xi ^{(n-1)}$ , state $\xi ^{(n)}$ can also be estimated by $\epsilon _{\xi,n}$ within the fixed time

$$\begin{align*} T_{\xi,n}=&\Bigg \lbrack \frac {1}{\kappa _{2}(p_{\xi,n}-1)(2^{1/4}\kappa _{1,n}/\kappa _{2,n})^{\frac {p_{\xi,n}- 1}{p_{\xi,n} + 1/2}}}\\&+\, \frac {2(2^{3/4}\kappa _{1,n}/\kappa _{2,n})^{1/2}}{\kappa _{1,n}} \Bigg \rbrack \\&\times \,\left \lbrack{ 1+\frac {\kappa _{3,n} + L_{n+1}}{(\kappa _{3,n}-L_{n+1})(1-\sqrt {2\kappa _{3,n}}/\kappa _{1,n})}}\right \rbrack.\end{align*}$$

In summary, the final estimation time $T_{sum}$ of states $\xi$ , $\dot {\xi },\ldots,\xi ^{(i)}$ ,…$\xi ^{(n)}$ can be calculated as

$$\begin{align*} T_{sum}\leq&\sum \limits _{j=0}^{n}\Bigg \{\Bigg \lbrack \frac {1}{\kappa _{2}(p_{\xi,j}-1)(2^{1/4}\kappa _{1,j}/\kappa _{2,j})^{\frac {p_{\xi,j}- 1}{p_{\xi,j} + 1/2}}} \\&+\, \frac {2(2^{3/4}\kappa _{1,j}/\kappa _{2,j})^{1/2}}{\kappa _{1,j}} \Bigg \rbrack \\&\times \,\left \lbrack{ 1 + \frac {\kappa _{3,j} + L_{j+1}}{(\kappa _{3,j} - L_{j+1})(1 - \sqrt {2\kappa _{3,j}}/\kappa _{1,j})} }\right \rbrack \Bigg \}. \\\tag{8}\end{align*}$$ The proof is completed.

Remark 2:

In most existing observer methods, the estimation error can be eliminated asymptotically or within a finite time [24]. The proposed observer can estimate disturbances within a fixed time. Compared with the fixed-time observer in [21], the new method can estimate the $nth$ order derivative of the disturbance, which is realized by using a high-order sliding mode differentiator. Specifically, we first estimate $\xi$ by the observer introduced in [21]. Then, on this basis, in Case 1, we can estimate the time derivative of disturbances utilizing the acquired disturbance estimation states. Case $j$ illustrates that we can estimate the $jth$ order derivative in a fixed time. By induction method and using high-order sliding mode differentiator, the $n$ th order derivative can be estimated in a fixed time.

The above result can be extended for fixed-time disturbance observer design for USVs. Let $\epsilon _{x}$ , $\epsilon _{y}$ , $\epsilon _{v}$ , $\epsilon _{1}$ , $\epsilon _{2}$ and $\epsilon _{y,1}$ be the estimation states of $\xi _{x}$ , $\xi _{y}$ , $\xi _{v}$ , $\xi _{1}$ , $\xi _{2}$ and $\dot {\xi }_{y}$ . Then based on Theorem 1, we give the following observers:

$$\begin{align*} {\dot {\epsilon }}_{x}=&\kappa _{{\tilde {x}}_{3}}\mathbf {sign}(\tilde {x}),\quad {\dot {\epsilon }}_{y} = \kappa _{{\tilde {y}}_{3}}\mathbf {sign}(\tilde {y}),~{\dot {\epsilon }}_{v} = \kappa _{{\tilde {v}}_{3}}\mathbf {sign}(\tilde {v}), \\ \tag{9a}\\ \dot {\epsilon }_{1}=&\kappa _{{\tilde {u}}_{3}}\mathbf {sign}(\tilde {u}),\quad {\dot {\epsilon }}_{2} = \kappa _{{\tilde {r}}_{3}}\mathbf {sign}(\tilde {r}),~\dot {\epsilon }_{y,1} = \kappa _{\tilde {\xi }_{y,3}}\mathbf {sign}(\tilde {y}), \\\tag{9b}\end{align*}$$ where $\tilde {x}=x-\hat {x}$ , $\tilde {y}=y-\hat {y}$ , $\tilde {v}=v-\hat {v}$ , $\tilde {u}=u-\hat {u}$ , $\tilde {r}=r-\hat {r}$ , with $\dot {\hat {x}} = u\,\,cos\psi + \epsilon _{x} + \kappa _{{\tilde {x}}_{1}}\lceil \tilde {x}\rfloor ^{\frac {1}{2}} + \kappa _{{\tilde {x}}_{2}}\lceil \tilde {x}\rfloor ^{p_{x}}$ , $\dot {\hat {y}} = u\,\,sin\psi + \epsilon _{y} + \kappa _{{\tilde {y}}_{1}}\lceil \tilde {y}\rfloor ^{\frac {1}{2}} + \kappa _{y_{2}}\lceil \tilde {y}\rfloor ^{p_{y}}$ , $\dot {\hat {v}} = - \frac {d_{22}}{m_{22}}v + \epsilon _{v} + \kappa _{{\tilde {v}}_{1}}\lceil \tilde {v}\rfloor ^{\frac {1}{2}} + \kappa _{{\tilde {v}}_{2}}\lceil \tilde {v}\rfloor ^{p_{v}}$ , $\dot {\hat {u}}=- \frac {d_{11}}{m_{11}}u + \frac {m_{22}}{m_{11}}vr + \epsilon _{1} + \kappa _{{\tilde {u}}_{1}}\lceil \tilde {u}\rfloor ^{\frac {1}{2}}+ \kappa _{{\tilde {u}}_{2}}\lceil \tilde {u}\rfloor ^{p_{1}}+\frac {\tau _{1}}{m_{11}}$ , $\dot {\hat {r}} =-\frac {d_{33}}{m_{33}}r-\frac {m_{22}-m_{11}}{m_{33}}uv+\epsilon _{2} + \kappa _{{\tilde {r}}_{1}}\lceil {\tilde {r}}_{i}\rfloor ^{\frac {1}{2}}+\kappa _{{\tilde {r}}_{2}}\lceil \tilde {r}\rfloor ^{p_{2}}+\frac {\tau _{2}}{m_{33}}$ , $\dot {\hat {\xi }}_{y}=\kappa _{\tilde {\xi }_{y,1}}\lceil \epsilon _{y}-\hat {\xi }_{y}\rfloor ^{\frac {1}{2}}+\kappa _{\tilde {\xi }_{y,2}}\lceil \epsilon _{y}-\hat {\xi }_{y}\rfloor ^{p_{\tilde {\xi }_{y}}}+\epsilon _{y,1}$ . Constants $p_{x}>1$ , $p_{y}>1$ , $p_{v}>1$ , $p_{1}>1$ , $p_{2}> 1$ , $p_{\xi _{y}}>1$ , $\kappa _{{\tilde {x}}_{1}}>\sqrt {2~\kappa _{{\tilde {x}}_{3}}}$ , $~\kappa _{{\tilde {x}}_{2}} > 0$ , $\kappa _{{\tilde {x}}_{3}}>4\sigma _{x}$ , $\kappa _{{\tilde {y}}_{1}}>\sqrt {2\kappa _{{\tilde {y}}_{3}}}$ , $\kappa _{{\tilde {y}}_{2}}>0$ , $\kappa _{{\tilde {y}}_{3}}>4\sigma _{y,1}$ , $\kappa _{{\tilde {v}}_{1}}>\sqrt {2\kappa _{{\tilde {v}}_{3}}}$ , $\kappa _{{\tilde {v}}_{2}} > 0$ , $\kappa _{{\tilde {v}}_{3}}>4\sigma _{v,1}$ , $~\kappa _{{\tilde {u}}_{1}} > \sqrt {2\kappa _{{\tilde {u}}_{3}}}$ , $\kappa _{{\tilde {u}}_{2}} > 0$ , $\kappa _{{\tilde {u}}_{3}} > {4\sigma }_{1}$ , $\kappa _{{\tilde {r}}_{1}}> \sqrt {2\kappa _{{\tilde {r}}_{3}}}$ , $\kappa _{{\tilde {r}}_{2}} >0$ , $\kappa _{{\tilde {r}}_{3}} > {4\sigma }_{2}$ , $\kappa _{\tilde {\xi }_{y,1}}>\sqrt {2\kappa _{\tilde {\xi }_{y,3}}}$ , $\kappa _{\tilde {\xi }_{y,2}}>0$ , $\kappa _{\tilde {\xi }_{y,3}} > {4\sigma }_{y,2}$ .

Corollary 1:

For the USV system in (1) with Assumption 1, the observers in (9a-9b) can estimate disturbances $\xi _{x}$ , $\xi _{y}$ , $\xi _{v}$ , $\xi _{1}$ , $\xi _{2}$ and $\dot {\xi }_{y}$ within a fixed time.

Proof:

Let $e_{x} = \xi _{x} - \epsilon _{x}$ . The dynamics of $[\tilde {x},e_{x}]$ can be represented as

$$\begin{align*} \dot {\tilde {x}}=-\kappa _{{\tilde {x}}_{1}}\lceil \tilde {x}\rfloor ^{\frac {1}{2}}-\kappa _{{\tilde {x}}_{2}}\lceil \tilde {x}\rfloor ^{p_{x}}+e_{x},~{\dot {e}}_{x}=-\kappa _{{\tilde {x}}_{3}}\mathbf {sign}(\tilde {x})+{\dot {\xi }}_{x}. \\\tag{10}\end{align*}$$ By the proof of Theorem 1, if the observer gains $\kappa _{{\tilde {x}}_{1}}$ , $\kappa _{{\tilde {x}}_{2}}$ and $\kappa _{{\tilde {x}}_{3}}$ satisfy that $$\begin{equation*} p_{x}>1,~\kappa _{{\tilde {x}}_{1}}>\sqrt {2\kappa _{{\tilde {x}}_{3}}},~\kappa _{{\tilde {x}}_{2}}>0,~\kappa _{{\tilde {x}}_{3}}>4\sigma _{x},\tag{11}\end{equation*}$$ then $\tilde {x}$ and ${\tilde {\xi }}_{x}$ can converge to the origin within the following fixed-time: $$\begin{align*} T_{\tilde {x}} \!\leq \! \left \lbrack{ \frac {1}{\kappa _{{\tilde {x}}_{2}}(p_{x} - 1)(2^{1/4}\kappa _{{\tilde {x}}_{1}}/\kappa _{{\tilde {x}}_{2}})^{\frac {p_{x} - 1}{p_{x} + 1/2}}} + \frac {2(2^{3/4}\kappa _{{\tilde {x}}_{1}}/\kappa _{{\tilde {x}}_{2}})^{1/2}}{\kappa _{{\tilde {x}}_{1}}} }\right \rbrack \end{align*}$$

Similarly, we can prove that $\tilde {y}$ , ${\tilde {\xi }}_{y}$ , $\tilde {u}$ , ${\tilde {\xi }}_{u}$ , $\tilde {v}$ , ${\tilde {\xi }}_{v}$ , $\tilde {r}$ and ${\tilde {\xi }}_{r}$ can converge to zero in fixed time. Therefore, the observer errors can converge to zeros in a fixed time, or namely, we can have that $\epsilon _{y}=\xi _{y}$ , $\forall t>T_{\tilde {y}}$ . This means that $\xi _{y}-\hat {\xi }_{y}=\epsilon _{y}-\hat {\xi }_{y}$ , and hence $\epsilon _{y,1}-\dot {\xi }_{y}$ can converge to zero in the fixed time:

$$\begin{align*} T_{d\tilde {y}}\leq&\Bigg \lbrack \frac {1}{\kappa _{d{\tilde {y}}_{2}}(p_{y} - 1)(2^{1/4}\kappa _{{d\tilde {y}}_{1}}/\kappa _{{d\tilde {y}}_{2}})^{\frac {p_{\xi _{y}} - 1}{p_{\xi _{y}} + 1/2}}} \\&+\,\frac {2(2^{3/4}\kappa _{{d\tilde {y}}_{1}}/\kappa _{{d\tilde {y}}_{2}})^{1/2}}{\kappa _{{d\tilde {y}}_{1}}} \Bigg \rbrack \\&\times \,\left \lbrack{ 1 + \frac {d\kappa _{{d\tilde {y}}_{3}} + \sigma _{y,2}}{(\kappa _{{d\tilde {y}}_{3}} - \sigma _{y,2})(1 - \sqrt {2\kappa _{{d\tilde {y}}_{3}}}/\kappa _{{d\tilde {y}}_{1}})} }\right \rbrack.\end{align*}$$

The proof is completed.

Remark 3:

Note that the estimation of $\dot {\xi }_{y}$ is required (details will be given in Subsection IV-B). Therefore, the fixed-time observer in [21] is not applicable. Corollary 1 tells us that fixed-time estimation of $\dot {\xi }_{y}$ can be obtained based on the estimation of $\xi _{y}$ .

SECTION IV. Stabilization of USVs

In this section, a time-varying control law is proposed for (1) in two steps on the basis of back-stepping techniques and the proposed fixed-time observer.

A. Stabilization of Subsystem $[x,y]$

In the first step, we consider the stabilization problem of subsystem $[x,y]$ . Since it is difficult to control the sway velocity $v$ , the surge velocity needs to provide persistent excitations for both states $x$ and $y$ . To this end, define $e_{x}$ as $e_{x} = x - {\lambda y}\sin \left ({{\alpha t} }\right)$ , whose convergence means that $x$ is driven to the trajectory of ${\lambda y}\sin \left ({{\alpha t} }\right)$ by the virtual surge velocity. Hence, $x$ can converge to zero by the convergence $e_{x}$ and $y$ . According to system (1), the dynamics of subsystem $[e_{x},y]$ can be stated as:

$$\begin{align*} \dot {e_{x}}=&\delta _{u}u - \delta _{v}v - \lambda \alpha y\cos \left ({{\alpha t} }\right) + \xi _{x} - \lambda \sin \left ({{\alpha t} }\right)\xi _{y}, \qquad \tag{12a}\\ \dot {y}=&u\mathrm {\sin }\psi + v\mathrm {\cos }\psi + \xi _{y},\tag{12b}\end{align*}$$ where $\delta _{u} = \cos \psi - \lambda \sin \psi \sin \left ({\alpha t + \beta }\right)$ , $\delta _{v} = \sin \psi + \lambda \cos {\psi \sin \left ({\alpha t + \beta }\right)}$ .

Let $\psi =\psi _{d}$ and $u=u_{d}$ be the virtual inputs, with

$$\begin{align*} \psi _{d}=&-\arctan {\left \lbrack{ \frac {\kappa _{\psi }}{\lambda }\frac {y^{2}\cos \left ({{\alpha t} }\right)}{\beta + y^{2}} }\right \rbrack,~} \\ u_{d}=&\frac {~\lambda \alpha y\cos \left ({{\alpha t} }\right)-\kappa _{u}\left \lbrack{ x - \lambda y\sin \left ({{\alpha t} }\right) }\right \rbrack }{\cos \left ({\psi _{d} }\right)\left [{ 1 - \lambda \tan \left ({\psi _{d} }\right)\sin \left ({{\alpha t} }\right) }\right]},\tag{13}\end{align*}$$ where $\kappa _{u}$ , $\alpha$ , $\beta$ , $\lambda$ , and $\kappa _{\psi } < 2$ are positive constants.

Theorem 2:

For subsystem (1a) with Assumptions 1 and 2 and virtual inputs $\psi _{d}$ and $u_{d}$ , the following properties hold:

1. Position states $x$ and $y$ satisfy that

   $$\begin{align*} \sup \limits _{t \rightarrow \infty }\left |{x}\right | \leq \frac {\Upsilon _{1}}{\kappa _{u}} + \lambda S_{M} + \frac {\lambda \pi }{2\alpha }\Upsilon _{3},~\sup \limits _{t \rightarrow \infty }|y| \leq S_{M} + \frac {\pi }{2\alpha }\Upsilon _{3}. \\\tag{14}\end{align*}$$

2. The yaw angle and surge velocity $\psi$ and $u$ satisfy that

   $$\begin{align*} \sup \limits _{t \rightarrow \infty }|\psi |\leq&\arctan \left ({\frac {\kappa _{\psi }}{\lambda }}\right), \\ \sup \limits _{t \rightarrow \infty }|u|\leq&\frac {\lambda \alpha \left ({S_{M}+\frac {\pi \Upsilon _{3}}{2\alpha }}\right)+\Upsilon _{1}}{\left ({1-\frac {\kappa _{\psi }}{2}}\right)\cos \left ({\frac {\kappa _{\psi }}{\lambda }}\right)}.\tag{15}\end{align*}$$ Here $\Upsilon _{1}= \left ({\lambda + 1 }\right)v_{\max }+\xi _{xmax}+ \lambda \xi _{ymax}, \Upsilon _{3} = \frac {\Upsilon _{1}}{1 - \kappa _{\psi }} + v_{\max } + \xi _{ymax}$ , $S_{M}$ is the maximum real solution to the following equation about $S$ , $$\begin{equation*} \left ({\frac {\pi }{\alpha }-2\rho }\right)\frac {\kappa _{\psi }\alpha \sin ^{2}(\alpha \rho)}{1 + \kappa _{\psi }}\frac {S^{3}}{\beta + S^{2}} = \frac {\pi }{\alpha }\Upsilon _{3},\tag{16}\end{equation*}$$ $0 < \rho < \frac {\pi }{2\alpha }$ is the positive constant to be chosen.

Proof:

1. According to system (12) and equation (13), the dynamics of subsystem $\left \lbrack{ e_{x},y }\right \rbrack$ can be described as

   $$\begin{align*} \dot {e_{x}}=&- \kappa _{u}e_{x} - \delta _{v}v + \xi _{x} - \lambda \sin \left ({\alpha t }\right)\xi _{y},\tag{17}\\ \dot {y}=&- \frac {\kappa _{\psi }\alpha \cos ^{2}\left ({{\alpha t} }\right)}{\lambda \Delta }\frac {y^{3}}{\beta + y^{2}} + \frac {\kappa _{u}}{\Delta }\tanh ^{2}{\left ({y }\right)\cos \left ({{\alpha t} }\right)} \\&\times \,e_{x}+v\cos \psi + \xi _{y},\tag{18}\end{align*}$$ where $\Delta = 1/\lambda - \tan \left ({\psi _{d} }\right)\sin \left ({{\alpha t} }\right)$ . Equation (17) implies that $e_{x}$ can be obtained as $$\begin{align*}&\hspace {-0.5pc}e_{x} = e^{- \kappa _{u}t}e_{x}\left ({0 }\right) + e^{- \kappa _{u}t}\int _{0}^{t}e^{\kappa _{u}\sigma }[- \delta _{v}\left ({\sigma }\right)v\left ({\sigma }\right) + \xi _{x}\left ({\sigma }\right) \\&\qquad\qquad\qquad\qquad\qquad\quad \displaystyle {-\,\lambda \sin \left ({\alpha \sigma }\right)\xi _{y}\left ({s }\right)]{d\sigma }.} \tag{19}\end{align*}$$ Since $\left |{ v }\right | \leq v_{\max }$ , $\xi _{x} \leq \xi _{xmax}$ and $\xi _{y} \leq \xi _{ymax}$ , we can have that $$\begin{equation*} \sup \limits _{t \rightarrow \infty }\left |{ e_{x} }\right | \leq \frac {\left \lbrack{ \left ({\lambda + 1 }\right)v_{\max } + \xi _{xmax} + \lambda \xi _{ymax} }\right \rbrack }{\kappa _{u}}. \tag{20}\end{equation*}$$

   Let $\Upsilon _{1} = \left ({\lambda + 1 }\right)v_{\max } + \xi _{xmax} + \lambda \xi _{ymax}$ and $S = \left |{ y }\right |$ . According to equation (20), one has

   $$\begin{align*}&\hspace {-0.5pc}\lim \limits _{t\rightarrow \infty }\dot {S} \leq - \frac {\kappa _{\psi }\alpha \cos ^{2}\left ({{\alpha t}}\right)}{1 + \kappa _{\psi }}\frac {S^{3}}{\beta + S^{2}} + \frac {\kappa _{u}}{1 - \kappa _{\psi }}\Upsilon _{1} \\&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \displaystyle {+\,v_{\max } + \xi _{ymax}.} \tag{21}\end{align*}$$ Since the term $\cos ^{2}\left ({{\alpha t}}\right)$ on the right hand side of equation (21) is periodic, the effect of term $\frac {\kappa _{\psi }\alpha \cos ^{2}\left ({{\alpha t} }\right)}{1 + 1/\lambda }\frac {S^{3}}{\beta + S^{2}}$ is time-varying depending on the value of $\cos ^{2}\left ({{\alpha t}}\right)$ . Hence to facilitate calculations, introduce a constant 0$< \rho < \pi /2\alpha$ as the piecewise point. Then for any integer $k$ , by the method of enlarging and reducing, we can divide the control process into two pieces $W$ and $D$ at the point $\rho$ , where $$\begin{align*} W=&\left \lbrack{ \frac {k}{\alpha }\pi + \frac {\pi }{2\alpha } + \rho,\frac {k}{\alpha }\pi + \frac {3\pi }{2\alpha } - \rho }\right \rbrack,\tag{22a}\\ D=&\left ({\frac {k}{\alpha }\pi + \frac {\pi }{2\alpha } - \rho,\frac {k}{\alpha }\pi + \frac {\pi }{2\alpha } + \rho }\right),\tag{22b}\end{align*}$$ where $k = 1,2,\ldots,n$ . With the above statements and inequality (21), for any integer $k$ , we can have $$\begin{align*} \dot {S} \leq \begin{cases} - \dfrac {\kappa _{\psi }\alpha \Upsilon _{2}}{1 + 1/\lambda }\dfrac {S^{3}}{\beta + S^{2}} + \Upsilon _{3},& t \in W,\\ \Upsilon _{3}, &t \in D, \end{cases}\tag{23}\end{align*}$$ where $\Upsilon _{3} = \frac {\Upsilon _{1}}{1 - \kappa _{\psi }} + v_{\max } + \xi _{ymax}$ . Define $\Upsilon _{2}=\sin ^{2}(\alpha \rho)$ . Then, in time interval $W$ , since $\cos ^{2}\left ({{\alpha t} }\right)$ is larger than $\Upsilon _{2}$ , by the method of enlarging and reducing, we can obtain that inequality $\left ({\ref {a6} }\right)$ holds. In time interval $D$ , since $\cos ^{2}\left ({{\alpha t} }\right)$ is smaller than $\Upsilon _{2}$ , the worst case where no beneficial effect exists on the stability of $S$ is considered. This implies that the term $\cos ^{2}\left ({{\alpha t} }\right)$ can be seemed as zero and hence the inequality $\left ({~\ref {a6} }\right)$ is obtained. To help the reader grasp the essence of our method, the working principle is shown in the Figure 2. In the region $D$ , the virtual control laws can approximate to zero, and in the region $W$ , virtual laws $u_{d}$ and $\psi _{d}$ take effect.

   Since that $S_{M}$ is the maximum solution to the equation $\left ({\frac {\pi }{\alpha } - 2\rho }\right)\left \lbrack{ \frac {\kappa _{\psi }\alpha \Upsilon _{2}}{1 + \kappa _{\psi }}\frac {S^{3}}{\beta + S^{2}} }\right \rbrack = \frac {\pi }{\alpha }{\Upsilon }_{3}$ , when $S > S_{M}$ , we can have

   $$\begin{equation*} \left ({\frac {\pi }{\alpha } - 2\rho }\right)\left \lbrack{ \frac {\kappa _{\psi }\alpha \Upsilon _{2}}{1 + \kappa _{\psi }}\frac {S^{3}}{\beta + S^{2}} - \Upsilon _{3} }\right \rbrack > {2\rho \Upsilon }_{3}. \tag{24}\end{equation*}$$ The equations (23–​24) means that $$\begin{align*}&\hspace {-1.2pc}\int _{\frac {k}{\alpha }\pi + \frac {\pi }{2\alpha } -\rho }^{\frac {k}{\alpha }\pi + \frac {3\pi }{2\alpha } - \rho }{\dot {S}\left ({\sigma }\right){d\sigma }} \\=&\int _{\frac {k}{\alpha }\pi + \frac {\pi }{2\alpha } + \rho }^{\frac {k}{\alpha }\pi + \frac {3\pi }{2\alpha } -\rho }{\dot {S}\left ({\sigma }\right){d\sigma }} + \int _{\frac {k}{\alpha }\pi +\frac {\pi }{2\alpha }-\rho }^{k\pi + \frac {\pi }{2\alpha }+\rho }{\dot {S}\left ({\sigma }\right){d\sigma }} \\\leq&{2\rho \Upsilon }_{3}-\left ({\frac {\pi }{\alpha } - 2\rho }\right)\left \lbrack{ \frac {\kappa _{\psi }\alpha \Upsilon _{2}}{1 + \kappa _{\psi }}\frac {S^{3}}{\beta + S^{2}} - \Upsilon _{3} }\right \rbrack < 0.\tag{25}\end{align*}$$ Thus if $S\left ({\frac {k + 1}{\alpha }\pi + \frac {3\pi }{2\alpha } - \rho }\right) > S_{M}$ , the value of $S\left ({\frac {k + 1}{\alpha }\pi + \frac {3\pi }{2\alpha } - \rho }\right)$ is smaller than $S\left ({\frac {k}{\alpha }\pi + \frac {3\pi }{2\alpha } - \rho }\right)$ . In conclusion, we can obtain that $$\begin{equation*} \lim _{k \rightarrow \infty }S\left ({\frac {k}{\alpha }\pi + \frac {3\pi }{2\alpha } - \rho }\right) < S_{M}. \tag{26}\end{equation*}$$

   Review the inequality (23), since $\dot {S} \leq \Upsilon _{3}$ in the region $D$ , according to the inequality (26), we conclude that for any $t\in D$ ,

   $$\begin{equation*} \lim _{k \rightarrow \infty }S\left ({t }\right) < S_{M} + 2\rho \Upsilon _{3}. \tag{27}\end{equation*}$$ In the region $W$ , according to the inequality (27), for any $S\left ({t }\right) > S_{M} + 2\rho \Upsilon _{3}$ , it is established that $\dot {S} < 0$ . Conclusively, whenever $t\in W$ or $t\in D$ , we can have $$\begin{equation*} \lim _{t \rightarrow \infty }\left |{ y\left ({t }\right) }\right |=\lim _{t \rightarrow \infty }S \leq \frac {\pi }{2\alpha }\Upsilon _{3} + S_{M} + {2\Upsilon }_{3}\rho S_{M}. \tag{28}\end{equation*}$$ where $\Upsilon _{3} = \frac {\Upsilon _{1}}{1 - \kappa _{\psi }} + v_{\max } + \xi _{ymax}$ , $\Upsilon _{1} = \left ({\lambda + 1 }\right)v_{\max } + \xi _{xmax} + \lambda \xi _{ymax}$ .

   Together with the inequality (20), we can obtain that

   $$\begin{equation*} \lim \limits _{t \rightarrow \infty }\left |{ x }\right | \leq \frac {\Upsilon _{1}}{\kappa _{u}} + \lambda S_{M} + \frac {\lambda \pi }{2\alpha }\Upsilon _{3}. \tag{29}\end{equation*}$$ The equations (28) and (29) show the results in (i).

2. According to the equation (13), with the assumption that $\psi =\psi _{d}$ we can have

   $$\begin{equation*} \psi =-\arctan {\left \lbrack{ \frac {\kappa _{\psi }}{\lambda }\frac {y^{2}\cos \left ({{\alpha t} }\right)}{\beta + y^{2}} }\right \rbrack }.\tag{30}\end{equation*}$$ Since $|\frac {y^{2}\cos \left ({{\alpha t} }\right)}{\beta + y^{2}} | < 1$ , the property that $\sup \limits _{t \rightarrow \infty }|\psi | < \arctan \left ({\frac {\kappa _{\psi }}{\lambda }}\right)$ can be obtained. Additionally, if the surge velocity $u=u_{d}$ , it can be concluded that $$\begin{equation*} u = \frac {~\lambda \alpha y\cos \left ({{\alpha t} }\right)-\kappa _{u}e_{x}}{\cos \left ({\psi _{d} }\right)\left ({1 - \lambda \tan \left ({\psi _{d} }\right)\sin \left ({{\alpha t} }\right) }\right)},\tag{31}\end{equation*}$$ where $e_{x} = x - {\lambda y}\sin \left ({{\alpha t} }\right)$ .

   Additionally, according to the equation (20), $e_{x}$ satisfies that

   $$\begin{equation*} \sup \limits _{t \rightarrow \infty }\left |{ e_{x} }\right | \leq \frac {\left \lbrack{ \left ({\lambda + 1 }\right)v_{\max } + \xi _{xmax} + \lambda \xi _{ymax} }\right \rbrack }{\kappa _{u}}.\tag{32}\end{equation*}$$ This implies that $$\begin{equation*} \sup \limits _{t \rightarrow \infty }|u|\leq \frac {\lambda \alpha \left({S_{M}+\frac {\pi \Upsilon _{3}}{2\alpha }}\right)+\Upsilon _{1}}{\left({1-\frac {\kappa _{\psi }}{2}}\right)\cos \left({\frac {\kappa _{\psi }}{\lambda }}\right)}.\tag{33}\end{equation*}$$

FIGURE 2.

The phase plot of the closed-loop system.

Remark 4:

Since no static smooth control law can stabilize the USV system, similarly with others’ time-varying methods, we introduce the periodic function. In contrast to the previous works for the stabilization of USVs, to solve the problem of the mismatched disturbances, this article uses the piecewise programming by which the virtual states are divided into two pieces according to the value of the periodic function. By the method of enlarging and reducing, the functions in the two parts are proved to be equivalent to the positive constant and zero respectively. Note that for simplification, we perceive the sway velocity as the mismatched disturbances in this article.

B. Control Law Design

In the second step, based on the proposed virtual inputs, we need to design the actual control laws. Define $T_{B}$ as the settling time of observers (9), according to Corollary 1, after time $T_{B}$ , disturbances can be regarded as known variables. Define $u_{e}=u-u_{d}$ , $\psi _{e} = \psi - \psi _{d}$ and $r_{e} = r - \dot {\psi _{d}}$ . The dynamics of the subsystem $\lbrack u_{e}, \psi _{e},r_{e}\rbrack$ can be stated as:

$$\begin{align*} {\dot {u}}_{e}=&f_{1}(t)+g_{1}\tau _{1}, \tag{34a}\\ {\dot {\psi }}_{e}=&r_{e},,~{\dot {r}}_{e}=f_{2}\left ({t}\right)+g_{2}\tau _{2}, \tag{34b}\end{align*}$$ Here $f_{1}(t)$ , $f_{2}(t)$ , $g_{1}$ and $g_{2}$ can be presented as $f_{1}(t) =u_{s} - {\dot {u}}_{d},\,\,f_{2}\left ({t }\right) = r_{s}- {\ddot {\psi }}_{d}, g_{1} = \frac {1}{m_{11}},\,\,g_{2} = \frac {1}{m_{33}}$ , where $u_{s}=-\frac {d_{11}}{m_{11}}u+\frac {m_{22}}{m_{11}}vr+\epsilon _{1}, r_{s}=\frac {d_{33}}{m_{33}}r+\frac {m_{22}-m_{11}}{m_{33}}uv+\epsilon _{2},\,\,\epsilon _{1}$ and $\epsilon _{2}$ are presented in Corollary 1, states ${\dot {u}}_{d}$ , $\dot {\psi _{d}}$ and ${\ddot {\psi }}_{d}$ can be stated as: $$\begin{align*} \dot {\psi }_{d}=&\frac {\psi _{d,3}}{\psi _{d,0}},\quad \ddot {\psi }_{d} = \frac {\psi _{d,1}}{\psi _{d,0}}-\frac {\psi _{d,2}\psi _{d,3}}{\psi _{d,0}^{2}}, \\ \dot {u}_{d}=&\frac {u_{d,1}}{u_{d,0}} - \frac {~u_{d,2}u_{d,3}}{u_{d,0}^{2}}.\end{align*}$$ In the above equations, $\psi _{d,0}$ , $\psi _{d,1}$ , $\psi _{d,2}$ , $\psi _{d,3}$ , $u_{d,0}~u_{d,1}$ , $u_{d,2}$ and $u_{d,3}$ are presented as: $$\begin{align*} \psi _{d,0}=&\lambda ^{2}\left ({\beta + y^{2} }\right)^{2} + \kappa _{\psi }^{2}y^{4}\cos ^{2}\left ({{\alpha t} }\right),\\ u_{d,0}=&cos \left ({\psi _{d} }\right)\lbrack 1 - \lambda \tan \left ({\psi _{d} }\right)\sin \left ({{\alpha t} }\right)\rbrack,\\ \psi _{d,1}=&2\alpha \lambda \kappa _{\psi }y\left ({\beta + {2y}^{2} }\right)\sin \left ({{\alpha t} }\right)y_{s} \\&+\,\alpha ^{2}\lambda \kappa _{\psi }\left ({\beta + y^{2} }\right) y^{2}\cos \left ({{\alpha t} }\right) -2\beta \lambda \kappa _{\psi }y_{s}^{2}\cos \left ({{\alpha t} }\right)\\&+\,2\alpha \beta \lambda \kappa _{\psi }{yy_{s}\sin }{\left ({{\alpha t} }\right) - 2\beta \lambda \kappa _{\psi }{y\dot {y}}_{s}\cos (\alpha t)},\\ \psi _{d,2}=&4\lambda ^{2}\left ({\beta + y^{2} }\right)yy_{s} + 4\kappa _{\psi }^{2}y^{3}{y_{s}\cos ^{2}}\left ({{\alpha t} }\right)\\&-\,\alpha \kappa _{\psi }^{2}y^{4}\sin \left ({2\alpha t }\right),\\ \psi _{d,3}=&\alpha \lambda \kappa _{\psi }\left ({\beta + y^{2} }\right)y^{2}\sin \left ({{\alpha t} }\right) - 2\beta \lambda \kappa _{\psi }y\cos \left ({{\alpha t} }\right),\\ u_{d,1}=&- \kappa _{u}\left \lbrack{ x_{s} - \lambda y_{s}\sin \left ({{\alpha t} }\right) - \lambda {\alpha y}_{s}\cos }\right \rbrack + \lambda \alpha y_{s}\cos \left ({{\alpha t} }\right)\\&-\,\lambda \alpha ^{2}y\sin \left ({{\alpha t} }\right),\\ u_{d,2}=&\lambda \alpha y\cos \left ({{\alpha t} }\right) - \kappa _{u}\lbrack x - \lambda y\sin \left ({{\alpha t} }\right)\rbrack,\\ u_{d,3}=&- \sin \left ({\psi _{d} }\right){\dot {\psi }}_{d} - \lambda cos\left ({\psi _{d} }\right){\dot {\psi }}_{d~}\sin \left ({{\alpha t} }\right)\\&-\, \alpha \lambda \sin \left ({\psi _{d} }\right)\cos \left ({{\alpha t} }\right),\end{align*}$$ where $\alpha$ , $\beta$ , $\lambda$ and $\kappa _{\psi }$ are constants defined in Theorem 2, $x_{s}$ , $y_{s}$ , $v_{s}$ and $\dot {y}_{s}$ can be stated as $x_{s}=u\mathrm {\cos }\psi - v\mathrm {\sin }\psi + \epsilon _{x},\,\,y_{s} = u\sin \psi + v\cos \psi + \epsilon _{y}, {\dot {y}}_{s}= ur\cos \psi -u_{s}\sin \psi -v_{s}\cos \psi {-} vr\sin \psi +\epsilon _{y,1}, v_{s}=-\frac {d_{22}}{m_{22}}v-\frac {m_{11}}{m_{22}}ur+\epsilon _{v},\,\,\epsilon _{x}$ , $\epsilon _{y}$ , $\epsilon _{y,1}$ , $\epsilon _{v}$ , $\kappa _{{\tilde {y}}_{3}}$ and $\tilde {y}$ are defined in Corollary 1.

To stabilize the system (34) in the fixed time, in this article, the following useful lemmas are introduced from [25] and [26].

Lemma 2:

Consider the following system:

$$\begin{equation*} \dot {z}=-\alpha z^{\frac {m}{n}}-\beta z^{\frac {p}{q}},\quad z(0)=z_{0}, \tag{35}\end{equation*}$$ where $\alpha, \beta >0$ , $m, n, p, q$ are positive odd integers satisfying $m>n$ and $p < q$ . Then the equilibrium point of system 35 is fixed-time stable and the settling time is upper bounded by: $T(z_{0})\leq \left [{1/\alpha ^{k}(1-pk)+1/\beta ^{k}(qk-1)}\right], \forall \,\,z_{0}\in \mathbb {R}^{n}$ .

Lemma 3:

Consider the system below,

$$\begin{equation*} \dot {z}_{1}=z_{2},~~\dot {z}_{2}=\tau, \tag{36}\end{equation*}$$ where $z_{1}$ , $z_{2}\in R$ are the system states, $\tau$ is the control input. Let $T_{max}>0$ , and the control input $\tau$ is formed as $$\begin{align*} \tau =-\dfrac {1}{2}(\alpha _{1}+3\beta _{1}z_{1}^{2})sign(\phi)-sig^{\frac {1}{2}}[\alpha _{2}\phi +\beta _{2}sig^{3}(\phi)].\! \\ \tag{37}\end{align*}$$ Here $\phi =z_{2}+sig^{\frac {1}{2}}(sig^{2}(z_{2})+\alpha _{1}z_{1}+\beta _{1}sig^{3}(z_{1}))$ and the parameters $\alpha _{1}/2=\alpha _{2}=\beta _{1}/2=\beta _{2}=(64/{T^{2}_{max}})$ . Then the origin of the closed-loop system (36) is globally fixed-time stable for all $z_{1}(0),z_{2}(0)\in R^{n}$ before the fixed-time $T\leq T_{max}$ .

Then based on the above two lemmas, we present the following control law

$$\begin{align*} \tau _{1}=&-\frac {1}{g_{1}}\Bigg \{\alpha sig^{p}(u_{e})+\beta sig^{q}(u_{e})+f_{1}(t)\Bigg \},\tag{38a}\\ \tau _{2}=&-\frac {1}{g_{2}}\Bigg \{\dfrac {1}{2}(\alpha _{1}+3\beta _{1}\psi _{e}^{2})sign(\varTheta)+f_{2}(t) \\&+\,sig^{\frac {1}{2}}\Big [\alpha _{2}\varTheta +\beta _{2}sig^{3}(\varTheta)\Big]\Bigg \}.\tag{38b}\end{align*}$$ where $\varTheta =r_{e}+sig^{\frac {1}{2}}\left [{sig^{2}(r_{e})+\alpha _{1}\psi _{e}+\beta _{1}sig^{3}(\psi _{e})}\right]$ , parameters $\frac {\alpha _{1}}{2}=\alpha _{2}=\frac {\beta _{1}}{2}=\beta _{2}=\frac {64}{T^{2}_{C_{2}}}$ , parameters $\kappa _{1}$ , $\kappa _{2}$ , $\gamma _{1}$ and $\gamma _{2}$ satisfy $\frac {1}{\kappa _{1}(1-\gamma _{1})}+\frac {1}{\kappa _{2}(\gamma _{2}-1)}=T_{C_{1}}$ where $T_{C_{1}}$ and $T_{C_{2}}$ are positive constants to be chosen.

C. Stability Analysis

Theorem 3:

Consider the USV system (1), under Assumptions 1 and 2, the control law (38) ensures properties (i) and (ii) in Theorem 2 hold. Here $T_{B}$ is the settling time of disturbance observers.

Proof:

According to Lemma 2, the state $u_{e}$ can be converged to zero in the fixed time $T_{C_{1}}$ for $t>T_{B}$ . By Lemma 3, the closed-loop system of $[\psi _{e},r_{2}]$ can be stabilized to zero in the fixed time $T_{C_{2}}$ for $t>T_{B}$ . Hence, $\forall t>T_{B}+\max \{T_{C_{1}},T_{C_{2}}\}$ , we can have $\psi =\psi _{d}$ and $u=u_{d}$ . This implies that the control law (38) ensures two properties in Theorem 2 hold. The proof is completed.

Remark 5:

The presented control law can solve matched and mismatched disturbances, which is the first of its kind. Moreover, the fixed-time higher-order observer is proposed to estimate the $n$ th order derivative of the disturbance by a improved sliding mode differentiator. Although, because the control law is formed by compounding, there is an error before the dynamic system ultimately tracks the virtual target. However, since the error system can converge to the origin within a fixed time, the cumulative error is bounded. Hence its influence on the USV system is also bounded. After the settling time, theoretically, this influence ceases to exist. During this period, the USV system will remain bounded because the error system is bounded. Following the error system converged, this effect gradually reduced.

According to Theorem 2, the bounds of states depend on values of parameters $\kappa _{\psi }$ , $\kappa _{u}$ , $\alpha$ , $\lambda$ and $S_{M}$ . The relationship between parameters $\kappa _{\psi }$ , $\kappa _{u}$ , $\alpha$ , $\lambda$ and boundaries of states can be directly perceived. Next, we present some analysis about $S_{M}$ , which is the max solution of the equation

$$\begin{equation*} \left ({\frac {\pi }{\alpha } - 2\rho }\right) \frac {\kappa _{\psi }\alpha \Upsilon _{2}}{1 + \kappa _{\psi }}\frac {S^{3}}{\beta + S^{2}} = \frac {\pi }{\alpha }\Upsilon _{3}. \tag{39}\end{equation*}$$ Since the right hand of the equation is positive, obviously, $S_{M}$ is also larger than zero. Thus if $S_{M}>1$ , according to the equation (39), we can have that $$\begin{equation*} {\frac {S_{M}^{3}}{\beta + S_{M}^{2}}\leq \frac {\pi (1+ \kappa _{\psi })\Upsilon _{3}}{\kappa _{\psi }}\cdot \frac {1}{\sin ^{2}\left ({{\alpha \rho } }\right)\alpha (\pi -2\rho \alpha)}.}\tag{40}\end{equation*}$$ In the above equation, the upper bound of $S_{M}$ depends on the maximum of term $\sin ^{2}\left ({{\alpha \rho } }\right)\alpha (\pi -2\rho \alpha)$ . Define constant $\mu$ as $\mu =2\frac {\alpha \rho }{\pi }$ , by which we can have $\rho =\frac {\mu \pi }{2\alpha }$ and $0 < \mu < 1$ . Then the term $\sin ^{2}\left ({{\alpha \rho } }\right)\alpha (\pi -2\rho \alpha)$ can be presented as $$\begin{equation*} \sin ^{2}\left ({{\alpha \rho } }\right)\alpha (\pi -2\rho \alpha)=\alpha \sin ^{2}\left ({\frac {\mu \pi }{2}}\right)(\pi -\mu \pi). \tag{41}\end{equation*}$$

Define $\Upsilon _{4}$ as $\Upsilon _{4}=\sin ^{2}\left ({\frac {\mu \pi }{2}}\right)(\pi -\mu \pi), \mu \in (0,1)$ . In the equation (41), the maximum of $\Upsilon _{4}$ is fixed and independent of $\alpha$ . Details are shown in Figure 3. Since the maximum of $\Upsilon _{4}$ is difficult to be achieved precisely and can not be changed by parameter designing, the further discussions about $\Upsilon _{4}$ are omitted. Hence, a lager $\alpha$ indicates the smaller bound of $S_{M}$ . This also means that we can control the bounds of $x$ and $y$ by designing the parameter $\alpha$ .

FIGURE 3.

The relation between $\Upsilon _{4}$ and $\mu$ .

If $S_{M}\leq 1$ , since $S_{M}^{4}\leq S_{M}^{3}$ , according to the equation (39), we can have

$$\begin{equation*} \frac {S_{M}^{4}}{\beta + S_{M}^{2}}\leq \frac {\pi (1 + \kappa _{\psi })\Upsilon _{3}}{\kappa _{\psi }}\cdot \frac {1}{\sin ^{2}\left ({{\alpha \rho } }\right)\alpha (\pi -2\rho \alpha)}.\tag{42}\end{equation*}$$ Similarly, the lager $\alpha$ makes smaller bounds $x$ and $y$ .

Remark 6:

A trade-off exists in selecting the parameter $\alpha$ . Indeed, the larger $\alpha$ implies the smaller upper bounds of $|x|$ and $|y|$ , however, it also means the larger $|u|$ . Physically, this situation can be also intuitively explained. The existence of mismatched disturbances prevents the USV from remaining almost entirely motionless at a certain point. The surge velocity is necessary to provide the excitation, in inverse proportion to the bounds of $x$ and $y$ .

Remark 7:

The proposed fixed-time based method not only holds the faster convergence speed but also has better disturbance rejection properties and better robustness against uncertainties [27]. However, since the existence of fractional power terms, control laws lose their smoothness. The fixed-time control law is only one option in the control designer’s toolbox and we do not claim its superiority concerning the asymptotic approaches.

D. Control Algorithm

To sum up, with the main results, we give the algorithm for implementation of the presented method in Table 2. The framework of proposed stabilization algorithm of USVs is described as Figure 4.

TABLE 2 The Stabilization Control Algorithm of USVs

FIGURE 4.

The framework of proposed stabilization algorithm of USVs.

SECTION V. Numerical Simulations

Numerical simulations are given to illustrate the effectiveness of the presented method. Let the USV have the following initial conditions:

$$\begin{align*} \lbrack x(0),y(0),\psi (0),u(0),v(0),r(0)\rbrack ^{T} = \lbrack 3,3,0,0,0,0\rbrack ^{T}.\!\! \\\tag{43}\end{align*}$$ Parameters of inertial and damping matrices in the USV model are given as $m_{11} = 25.8$ , $m_{22} = 33.8$ , $m_{33} = 2.76$ , $d_{11} = 12$ , $d_{22} = 17$ , and $d_{33} = 0.5$ . The parameters are chosen as $\kappa _{u}=3$ , $\kappa _{\psi }=0.7$ , $\beta =0.01$ , $\lambda =1$ , $T_{C_{1}}=30\,\,s$ , $T_{C_{2}}=30\,\,s$ , $\kappa _{1}=1$ , $\kappa _{2}=1$ , $\gamma _{1}=0.9$ , and $\gamma _{2}=1.1$ . To enhance the practicality of simulations, we choose the step size as 0.1s.

A. Stability Analysis

The stability performance of the USV with the proposed controller is shown in Figures 5(a) and 5(b). It is clear that states $x$ , $y$ , $\psi$ , $u$ , $v$ and $r$ converge to zero. Here we choose the parameter $\alpha$ as 1. Additionally, the figure 5(c) shows that inputs of the USV system can also converge to zero.

FIGURE 5.

Trajectories of the position states, yaw angle, velocities and inputs with no disturbance.

To further illustrate the stability of the USV system, we consider the case in which the disturbances can be presented as $\xi _{x} = 0.01\,\,m/s$ , $\xi _{v} = 0.01\sin \left ({0.3t }\right){\,\,N/kg}$ , $\xi _{\psi } = 0\,\,rad/s$ , $\xi _{1} = \left \lbrack{ 5\mathrm {\cos }\left ({0.2t }\right) + 10\,\,\mathrm {\sin }\left ({0.1t }\right) }\right \rbrack {\,\,N/kg}$ , $\xi _{y} = 0.01\,\,m/s$ , $\xi _{2} = \left \lbrack{ 4\mathrm {\cos }\left ({0.25t }\right) + 5\mathrm {\sin }\left ({0.2t }\right) }\right \rbrack \,\,N \cdot m/kg$ .

The trajectories of states $x$ , $y$ , $\psi$ , $u$ , $v$ , $r$ and control inputs $\tau _{1}$ , $\tau _{2}$ are shown in Figures 6(a), 6(b) and 6(c) that the USV system can be bounded with the presented method. Besides, measurement noise usually exists in actual situations. To increase the credibility of the algorithm simulation verification, we further added the random measurement noise with an absolute value of 0.1. Meanwhile, we increased the simulation step size to 0.2s to ensure simulations more coincident with the actual situation. Results in Figures 7(a), 7(b) and 7(c) show that although the stability of the closed-loop system is affected by disturbance and measurement noise, the proposed control law still guarantees states of USVs bounded. Here we choose the parameter $\alpha$ as 1.5.

FIGURE 6.

Trajectories of the position states, yaw angle, velocities and inputs with disturbances.

FIGURE 7.

Trajectories of the position states, yaw angle, velocities and inputs with disturbances and measurement noises.

B. Estimation Error

Observer parameters are chosen as $p_{1} = 2$ , $p_{2} = 2$ , $p_{x} = 2$ , $p_{y} = 2$ , $p_{\xi _{y}} = 2$ , $p_{v} = 2$ , $\kappa _{\tilde {u_{3}}} = 8$ , $\kappa _{{\tilde {r}}_{3}} = 8$ , $\kappa _{\tilde {x_{3}}} = 8$ , $\kappa _{{\tilde {y}}_{3}} = 8$ , $\kappa _{{\tilde {v}}_{3}} = 8$ , $\kappa _{{\tilde {v}}_{3}} = 8$ , $\kappa _{{\tilde {u}}_{2}} = 4$ , $\kappa _{\tilde {r_{2}}} = 4$ , $\kappa _{{\tilde {x}}_{2}} = 4$ , $\kappa _{\tilde {y_{2}}} = 4$ , $\kappa _{{\tilde {v}}_{2}} = 4$ , $\kappa _{{\tilde {v}}_{2}} = 4$ , $\kappa _{{\tilde {u}}_{1}} = 5$ , $\kappa _{{\tilde {r}}_{1}} = 5$ , $\kappa _{{\tilde {x}}_{1}} = 5$ , $\kappa _{{\tilde {y}}_{1}} = 5$ , $\kappa _{{\tilde {v}}_{1}} = 5$ , $\kappa _{{\tilde {v}}_{1}} = 5$ . According to Corollary 1, the estimation settling time of disturbances is $T_{B}=\max \{T_{\tilde {u}},T_{\tilde {v}},T_{\tilde {r}},T_{\tilde {x}},T_{\tilde {y}}+T_{\tilde {\xi _{y}}}\}=9.5196\,\,s$ .

For convenience, we define a sum of estimation errors such as $\xi _{e}=\sqrt {e_{x}^{2}+e_{y}^{2}+e_{v}^{2}+e_{1}^{2}+e_{2}^{2}}$ , where $e_{x}$ , $e_{y}$ , $e_{v}$ , $e_{1}$ and $e_{2}$ are estimation errors of disturbances $\xi _{x}$ , $\xi _{y}$ , $\xi _{\psi }$ , $\xi _{v},~\xi _{1}$ and $\xi _{2}$ . In Figure 8(a), the error states $\xi _{e}$ converges to zero before $T_{B}=9.5s$ implying the convergence of estimation errors with step size 0.001s. Even in the environment of 0.2s simulation step size, the observer error $\xi _{e}$ can still be stabilized to a small bound near the origin in Figure 8(b).

FIGURE 8.

The Trajectories of observers error performance.

C. Comparisons

To evaluate the effectiveness of our method, we consider comparisons of the performance between the proposed control law and approaches in [6], [15] under different initial conditions $C.1$ and $C.2$ . Details about situations $C.1$ and $C.2$ are stated in Table 3. Here $\Omega _{1}(t) = 5\cos \left ({0.2t }\right) + 10sin(0.1t)$ , $\Omega _{2}\left ({t }\right) = 0.2\sin \left ({0.3t }\right)$ , $\Omega _{3}\left ({t }\right) = 4\cos \left ({0.25t }\right) + 5sin(0.2)t$ , $\Omega _{4}(t)=10\cos \left ({0.1t }\right) + 5\sin (0.3t)$ , $\Omega _{5}\left ({t }\right) = 0.5\sin \left ({0.2t }\right)$ , $\Omega _{6}\left ({t }\right) = 7\cos \left ({0.2t }\right) + 8\sin (0.4t)$ .

TABLE 3 Cases

Define the performance $S$ and the square-root of control inputs $J$ as $S=\ln \sqrt {x^{2}+y^{2}+\psi ^{2}}$ and $J=\ln \int _{0}^{t}\sqrt {\tau _{1}^{2}(\delta)+ \tau _{2}^{2}}(\delta)d\delta$ . We now look at the differences in the performance $S$ and inputs $J$ between the presented method and approaches in [6], [15] under the case C.1, shown in Figures 9(a) and 9(b). It is clearly seen that the maximum value of $S_{1}$ in our method is the minimum one in the three methods. Moreover, the $J_{1}$ by our method is similar to the $J_{2}$ by the method in [6] and much smaller than the $J_{3}$ by the method in [15]. It means that we can achieve a better performance with smaller control effort or energy cost. It is worth mentioning that the reason for using the logarithmic function instead of the square form is the performances of our method and [15] are greatly different.In this situation, taking the logarithm of conventional indexes can show the performance relationship in more detail, avoiding the problem of unclear image information caused by too large gap.

FIGURE 9.

Comparisons of errors and inputs under case C.1.

To be specific, consider the comparison of our method and [6] under case C.2. Because the performance gap between the two methods does not affect the clarity of figures to display information, the performance and energy consumption indicators in the form of conventional square sums are used here. In this case, we redefine the performance index function and energy consumption function as: $Q=\sqrt {x^{2}+y^{2}+\psi ^{2}+u^{2}+v^{2}+r^{2}}$ and $P=\int _{0}^{t}\sqrt {\tau _{1}^{2}(\delta)+ \tau _{2}^{2}}(\delta)d\delta$ . Figures 10(a) and 10(b) show that the consumption and performance $Q_{1}$ by our method are much smaller.

FIGURE 10.

Comparisons of the errors and inputs under case C.2.

SECTION VI. Conclusion and Future Work

In this article, based on back-stepping method, the virtual yaw angle and surge velocity are presented to stabilize the USV into the bounded region. The bound of the USV system depends on the parameters of virtual states. Based on induction method, the higher-order fixed-time observer is proposed to estimate disturbances in the desired time. So that after the settling time, unknown disturbances can be seen as measurable. By presented observers, the fixed-time control law is proposed to drive the yaw angle and surge velocity to designed virtual states. This control law is proved to ensure the USV system bounded.

As actuator saturation due to mechanical constraints may have significant impacts on the system transient behavior and even stability, stabilization of USVs subject to actuator saturation is still an open problem. Energy efficient optimal control of USVs is another critical issue worthy of investigation in the future.

ACKNOWLEDGMENT

The authors would like to thank the associate editor and reviewers for all their very valuable suggestions and comments which have helped to improve the presentation and quality of the paper.