Event-triggered fixed-time distributed output feedback sliding mode cooperative control for multiple surface vessels with input saturation

Abstract

Aiming at the problems of cooperative control for multiple surface vessels (MSVs), such as unknown environmental disturbances, unavailable velocities, model uncertainties, actuator saturation, and limited action times of the actuator, an event-triggered fixed-time distributed output feedback sliding mode cooperative control method with input saturation is proposed. The scheme ensures the practical fixed-time stability of the cooperative control system. First, a fixed-time extended state observer (FxESO) is designed to provide the estimations of velocities and lumped disturbances including unknown environmental disturbances and model uncertainties. Second, to deal with the actuator saturation, a fixed-time auxiliary dynamic system is designed. Third, a fixed-time non-singular terminal sliding mode manifold (FxNTSMM) is introduced to effectively eliminate singularity and improve chattering of the system. Finally, an event-triggered distributed controller based on FxNTSMM and FxESO is proposed. An event-triggered condition can avoid unnecessary actions of the actuator, and Zeno phenomenon is avoided by theoretical proof. In addition, the upper bound of the convergence time is independent of the initial state. Simulation results are given to demonstrate the effectiveness of the proposed control scheme.

Keywords

Multiple surface vessels fixed-time cooperative control extended state observer sliding mode control event-triggered controller input saturation

Introduction

With the increasing types and number of marine engineering operations, surface vessels need to complete more complex or larger marine operations (Xia et al., 2021a; Zhang et al., 2019). Some of these complex ocean tasks, such as rescue operations, cannot be performed effectively with a single surface vessel. Therefore, if the cooperative operation of multiple surface vessels (MSVs) is adopted, the rescue mission can be carried out quickly and effectively, and a series of losses can be greatly reduced (Xia et al., 2019b).

Since some urgent missions need to be completed quickly and accurately, the convergence rate is considered as an important performance index to judge the dynamic behaviors of MSVs. As is known to all, most cooperative control systems are asymptotically stable, which means that cooperative control can achieve stability only when time approaches infinity (Huang and Jia, 2017). To improve the convergence performance, the finite-time control method comes into being. The method has the advantages of faster convergence, higher precision, and better robustness to parameter uncertainties and disturbances, and it is preferable to asymptotic stability control method. In the work by Wang et al. (2016, 2017), the finite-time theory was applied to the surface vehicle to solve the trajectory tracking problem. Nevertheless, the upper bound of the convergence time by the above finite-time control strategy depends on the initial state of the system, that is, the upper bound of the convergence time cannot be obtained in advance when the initial state is not available. Therefore, the fixed-time control scheme was proposed for the first time in Polyakov (2012) to overcome the shortcoming of the finite-time control scheme, that is, the convergence time was bounded no matter what the initial state was. In recent years, the fixed-time control scheme has been applied to the fully actuated surface vessel tracking control system to achieve faster convergence and higher precision (Zhang et al., 2019, 2020b).

There are uncertainties and disturbances in MSVs’ cooperative control system, which affect the cooperative control performance of MSVs. Therefore, in addition to convergence rate, robustness is also an important indicator to judge the dynamic performance of MSVs. It is well known that in nonlinear control, sliding mode control method has desirable robustness against uncertainties and disturbances (Cui et al., 2016; Mishra et al., 2020; Van et al., 2019; Yan and Yu, 2018). However, chattering problem exists in traditional sliding mode control, which seriously affects the performance of the control system. To solve the problem, a finite-time terminal sliding mode control method was proposed in Shao et al. (2017). This method not only ensured the fast convergence of sliding mode manifold but also ensured the great performance of the control system. However, if the above methods are extended to the second-order system, the singularity problem is encountered in the control design. Therefore, a non-singular terminal sliding mode control (Zhu and Fei, 2017) was proposed to solve the singularity problem. To achieve faster convergence rate, a non-singular fast terminal sliding mode control was designed in Van et al. (2019). Also, in the work by Zhang et al. (2020a), a novel non-singular terminal sliding mode control was proposed for the tracking control system of the surface vessel, which not only could avoid the singularity problem but also had faster convergence performance than fast terminal sliding mode.

Among some methods, the robustness of the sliding mode control system was used to deal with unknown environmental disturbances and model uncertainties, which meant that this approach sacrificed its nominal control performances. Therefore, in recent years, to improve the performance and adaptive ability of the sliding mode control system, many researchers used disturbance observers to deal with disturbances, such as the linear disturbance observer, the sliding mode disturbance observer (Xu et al., 2014), the high-gain disturbance observer (Yang et al., 2014), and nonlinear disturbance observer (Liu, 2017). Furthermore, the finite-time disturbance observer in Fu et al. (2019b) and the fixed-time disturbance observer in Yao (2022) were proposed to improve the convergence rate of estimation errors. However, the above proposed observers only estimate the environmental disturbances without considering the model uncertainties and unmeasured velocities. To advance the estimation of lumped disturbances and system states to compensate the controller, the extended state observer (ESO) was proposed in Xiong et al. (2015). Also, the finite-time extended state observer (FTESO) in Fu et al. (2019a) was proposed to improve convergence performance of estimation errors and stronger disturbance rejection performance. Furthermore, the fixed-time extended state observer (FxESO) in Wang et al. (2022) was proposed to achieve faster convergence, and the convergence time is independent of the initial state. This scheme not only obtained the disturbance attenuation performance but also maintained the nominal performance.

During the cooperative control of MSVs, the actuators’ high-frequency action may lead not only to mechanical abrasion but also to a short service life (Xia et al., 2020). In the work by Deng et al. (2019), an event-triggered strategy was applied into robust fuzzy control method to promise the high fidelity of the path following control system. In the work by Su et al. (2021a), a static event-triggered mechanism was applied to the integral sliding mode control system, which limited the action frequency of actuators, but this mechanism does not balance system control performance and effectively make use of resources. Therefore, to tackle with the disadvantage, the dynamic event-triggering mechanism was proposed in Su et al. (2021b). However, the event-triggered strategy is not currently considered in the non-singular terminal sliding mode control scheme. In the work by Song et al. (2021), Zeno behavior was avoided because the action frequency of the actuator is bounded.

Since the fixed-time cooperative control strategy of MSVs requires fast instantaneous response, the actuator may be saturated due to large control torque (Wang and Su, 2021). Actuator saturation may lead to instability of the control system; therefore, it is necessary to consider actuator saturation (Khatibi and Haeri, 2019). In the work by Su et al. (2021a), the system uncertainty caused by actuator saturation was treated as compound disturbances, which was estimated by the observer. This approach imposed computational burden on the observer, and the additional disturbances caused by actuator saturation could greatly affect the dynamics of the system. In the work by Cao et al. (2021), a fixed-time nonlinear anti-windup compensator was designed to compensate for the saturation effect of the joint torque actuator in real time. However, in the above scheme, if there is a slight disturbance which makes the sliding mode surface being not zero and the actuator being not saturated, the saturation compensator will still compensate. For the above problems, in the work by Sai et al. (2021), a novel fixed-time actuator saturation compensator was proposed. In the case of actuator saturation, the controller could adaptively compensate the control input.

As discussed above, few existing studies simultaneously consider the system uncertainties, unavailable velocities, and external time-varying disturbances in the context of fixed-time sliding mode control, system singularities, event-triggered strategy, and actuator saturation. Consequently, motivated by the aforementioned analyses and inspired by Xia et al. (2021c), Zhang et al. (2020a), Wu et al. (2021), and Liang et al. (2020), paper designs a novel fixed-time distributed event-triggered output feedback sliding mode cooperative control scheme for MSVs with input saturation. First, an FxESO is designed to deal with unmeasurable velocities, external time-varying disturbances, and model uncertainties. Also, the estimated errors can converge in fixed time. Second, a fixed-time auxiliary dynamic system is designed to deal with actuator saturation. Third, the introduced fixed-time non-singular terminal sliding mode manifold (FxNTSMM) not only eliminates the singularity but also achieves fast convergence. Finally, a fixed-time event-triggered controller is designed to avoid the unnecessary actions of the actuator and to make the system stable within a fixed time. The main contribution of this paper is that combining with FxNTSMM and directed topology with cooperative control strategy, distributed event-triggered fixed-time output feedback controller with anti-saturation ability are applied for MSVs cooperative control research. In comparison with Liang et al. (2020), the convergence rate of the proposed sliding mode manifold is faster, and the tracking errors of the proposed controller can converge in fixed time. Compared with the state feedback control scheme in Zhang et al. (2020a), the proposed scheme is an output feedback control scheme, which does not require known velocity information. In comparison with Zhang et al. (2020b), the proposed scheme takes the input saturation and the event-triggered control strategy into account to avoid actuator saturation and reduce the action frequency of the actuator.

The rest of the article is organized as follows. Section “Preliminaries and problem formulation” describes the preliminaries and problem formulation. Section “Design of FxESO” describes observer design. Section “Controller design” gives controller design and stability analysis. Simulation results are presented in section “Simulation results.” Section “Conclusion” concludes this paper.

Preliminaries and problem formulation

Notations

$R^{n \times n}$ are the $n \times n$ dimensional Euclidean Space. $| \cdot |$ denotes the absolute value. $‖ \cdot ‖$ is the Euclidean norm. Denoting $x = [x_{1}, x_{2}, . . ., x_{n}]^{T} \in R^{n}$ and $si g^{α} (x) = [si g^{α} (x_{1}), si g^{α} (x_{2}), . . ., si g^{α} (x_{n})]^{T}$ , where $si g^{α} (x_{i}) = {| x_{i} |}^{α} sign (x_{i}), (i = 1, 2, . . ., n)$ , $x_{i} \in R$ , $α \in (0, 1)$ . $sign (\cdot)$ is expressed as

sign (x) = {\begin{matrix} - 1, & if & x = 0 \\ 0, & if & x = 0 \\ 1, & if & x > 0 \end{matrix}

(1)

Graph theory

The graph theory will be used to describe the communication topology of $n$ MSVs. An ordered dyadic array $G = (V_{G}, E)$ is used to describe a set of nodes. The dyadic array $G$ is called a graph. $V_{G} = {1, 2, . . ., n}$ represents the set of nodes. $E = {(i, j) \in V_{G} \times V_{G}} . (i, j) \in E$ represents that the information of vessel $i$ is available to vessel $j$ , and the vessel $i$ is a neighbor of vessel $j$ . $N_{i} = {j \in V_{G}, (i, j) \in E}$ describes the neighbors of vessel $i$ . Define an adjacency matrix $A = [a_{ij}] \in R^{n \times n}$ , where $a_{ij} = 1$ , if $(i, j) \in E$ ; otherwise $a_{ij} = 0$ . If $a_{ij} = a_{ji}$ , the graph is undirected; otherwise is directed. The Laplacian matrix $L = [ℓ_{ij}] \in R^{n \times n}$ is a matrix description of the graph, which is defined as $L = D - A$ , where $D = diag {d_{i}} \in R^{n \times n}$ with $d_{i} = \sum_{j \in N_{i}} a_{ij}$ . A diagonal matrix $A_{0} = diag {a_{i 0}}$ is defined to describe a leader adjacency matrix. If the $i th$ vessel can obtain the reference signal of the virtual leader vessel, define $a_{i 0} = 1$ , otherwise define $a_{i 0} = 0$ . Finally, the information exchange matrix is defined as $H = L + A_{0}$ .

In this paper, an augmented graph $\bar{G} = ({\bar{V}}_{\bar{G}}, \bar{E})$ with $n$ + 1 nodes is introduced to describe the communication network relationship between the reference signal and several vessels, in which the reference signal $η_{d} \in R^{3}$ is treated as a virtual leader vessel (indexed by $n$ + 1), where ${\bar{N}}_{i} = {1, 2, . . ., n}$ and $\bar{E} = {(i, j) \in {\bar{V}}_{\bar{G}} \times {\bar{V}}_{\bar{G}}}$ . Then, the following assumption is made.

Assumption 1: The communication topology among MSVs is directed, and there is at least one directed path from the virtual leader vessel to each following vessel.

Remark 1: According to Xia et al. (2019a), Assumption 1 is to ensure the connectivity of the directed communication topology, which is the necessary condition for cooperative control of MSVs.

Definitions and lemmas

Consider the following system (Wang et al., 2016)

\overset{\cdot}{x} = f (x (t)), f (0) = 0, x (0) = 0

(2)

where $x = [x_{1}, x_{2}, . . ., x_{n}]^{T}$ . $f (\cdot)$ is a continuous nonlinear function.

Definition 1: For system (2), if it can satisfy the following requirements, then the origin $x = 0$ is called as a fixed-time equilibrium (Basin et al., 2017).

System (2) is globally finite-time stable;

There is a settling time function $T (x)$ ensuring $T (x) \leq T_{max}$ , where $T_{max}$ is a positive constant.

Lemma 1: Considering a Lyapunov function $V (x)$ , if it is defined on a neighborhood $D$ of the origin and satisfies $\overset{\cdot}{V} (x) \leq - (α V (x)^{p} + β V (x)^{g})^{k}$ , then the origin of system (2) is fixed-time stable (Zhang et al., 2018). It implies $V (x)$ is able to converge to $V (x) = 0$ from any initial values within the region $D$ in fixed time. Moreover, the settling time $T \leq \frac{1}{α^{k} (1 - pk)} + \frac{1}{β^{k} (gk - 1)}$ holds, where $α, β, p, g, k$ are positive constants, and $pk < 1, gk > 1$ .

Lemma 2: Considering a Lyapunov function $V (x)$ , if it satisfies $\overset{\cdot}{V} (x) \leq - (α V (x)^{p} + β V (x)^{g})^{k} + ϒ$ , where $α, β, p, g, k$ are positive constants, $pk < 1, gk > 1$ and $0 < ϒ < \infty$ , then the system is practical fixed-time stable (Huang and Jia, 2018). Also, the residual set is given by ${lim_{t \to T} x | V (x) \leq min {α^{- \frac{1}{p}} {(\frac{ϒ}{1 - θ^{k}})}^{\frac{1}{kp}}, β^{- \frac{1}{g}} {(\frac{ϒ}{1 - θ^{k}})}^{\frac{1}{kg}}}}$ , where $θ$ is a constant and satisfies $0 < θ \leq 1$ . The time $T$ is bounded by $T \leq \frac{1}{α^{k} θ^{k} (1 - pk)} + \frac{1}{β^{k} θ^{k} (gk - 1)}$ .

Lemma 3: If ${x_{1}, x_{2}, . . ., x_{n}} \in R$ , $p > 0$ , then $(| x_{1} | + | x_{2} | + \dots +$ $| x_{n} |)^{p} \leq \max (n^{p - 1}, 1) ({| x_{1} |}^{p} + {| x_{2} |}^{p} + \dots + {| x_{n} |}^{p})$ (Zhang et al., 2020a). If $p = m / n \leq 1$ , and $m, n$ are odd integers, then $| x_{1}^{p} - x_{2}^{p} | \leq 2^{1 - p} {| x_{1} - x_{2} |}^{p}$ .

Lemma 4: If there exits a continuous function $V : D \to R$ such that $V > 0$ , then there also exist real numbers $d_{F}$ and $θ_{F}$ satisfying $d_{F} > 0$ and $θ_{F} \in (0, 1)$ such that $\overset{\cdot}{V} (x) + d_{F} (V (t))^{θ_{F}} \leq 0, x \in U \ {0}$ , where $U \subset D$ of the origin is an open neighborhood (Wang et al., 2016). Then, the origin of system in equation (2) is finite-time stable, and the continuous settling time function $T_{F}$ satisfies $T_{F} \leq \frac{1}{d_{F} (1 - θ_{F})} (V (x))^{(1 - θ_{F})}$ .

System modeling and problem formulation

Consider a network of n MSVs, labeled as 1 to $n$ . Let $η_{i} = {[x_{i}, y_{i}, ψ_{i}]}^{T} \in R^{3}$ denote the position and heading angle vector of the $i th$ surface vessel expressed in earth-fixed frame $X_{E} O_{E} Y_{E}$ as shown in Figure 1. Let $υ_{i} = {[u_{i}, v_{i}, r_{i}]}^{T} \in R^{3}$ denote the velocity and yaw rate vector of the $i th$ surface vessel expressed in the body-fixed frame $X_{B} O_{B} Y_{B}$ . Then, the three degrees of freedom (3-DOF) horizontal motion mathematical model for the $i th$ marine surface vessel are given as Xia et al. (2019b)

{\overset{\cdot}{η}}_{i} = R_{i} (ψ_{i}) υ_{i}

(3)

M_{i} {\overset{\cdot}{υ}}_{i} = - D_{i} (υ_{i}) υ_{i} - C_{i} (υ_{i}) υ_{i} + τ_{i} + τ_{id}

(4)

where $τ_{id} = {[τ_{id 1}, τ_{id 2}, τ_{id 3}]}^{T} \in R^{3}$ represents the vector composed of the disturbing forces and torques caused by wind, current, and waves acting on the vessel, $τ_{i} = {[τ_{i 1}, τ_{i 2}, τ_{i 3}]}^{T} \in R^{3}$ represents the control force and the torque vector generated by the $i th$ vessel with constraint as

τ_{i} = {\begin{matrix} τ_{i max}, τ_{i max} < τ_{ic} \\ τ_{ic}, τ_{i min} \leq τ_{ic} \leq τ_{i max} \\ τ_{i min}, τ_{ic} < τ_{i min} \end{matrix}

(5)

where $τ_{i max} \in R^{3}$ and $τ_{i min} \in R^{3}$ are the maximum and minimum control forces and moment produced by the thruster of the $i th$ vessel. The mismatch function between input without saturation and with saturation is described as $Δ τ_{i} = τ_{i} - τ_{ic}$ , where $τ_{ic} = {[τ_{ic 1}, τ_{ic 2}, τ_{ic 3}]}^{T} \in R^{3}$ . $τ_{ic}$ is calculated by the proposed controller. $M_{i} \in R^{3 \times 3}$ is the inertia matrix of the $i th$ surface vessel, and $D_{i} (υ_{i}) \in R^{3 \times 3}$ is the damping matrix of the $i th$ surface vessel. $C_{i} (υ_{i}) \in R^{3 \times 3}$ is the Coriolis–centripetal force matrix caused by hydrodynamic forces. The $M_{i}$ , $C_{i} (υ_{i})$ , and $D_{i} (υ_{i})$ are given as

M_{i} = [\begin{matrix} m_{11} & 0 & 0 \\ 0 & m_{22} & m_{23} \\ 0 & m_{32} & m_{33} \end{matrix}]

(6)

C_{i} (υ_{i}) = [\begin{matrix} 0 & 0 & - m_{22} υ - m_{23} r \\ 0 & 0 & m_{11} u \\ m_{22} υ + m_{23} r & - m_{11} u & 0 \end{matrix}]

(7)

D_{i} (υ_{i}) = [\begin{matrix} d_{11} & 0 & 0 \\ 0 & d_{22} & d_{23} \\ 0 & d_{32} & d_{33} \end{matrix}]

(8)

For detailed definitions of the inertia matrix, the Coriolis–centripetal force matrix, and the damping matrix, please refer to Skjetne et al. (2005). $R_{i} (ψ_{i}) \in R^{3 \times 3}$ represents the rotation matrix between two coordinate systems, and its specific form is as follows (Xia et al., 2021b)

R_{i} (ψ_{i}) = [\begin{matrix} \cos ψ_{i} & - \sin ψ_{i} & 0 \\ \sin ψ_{i} & \cos ψ_{i} & 0 \\ 0 & 0 & 1 \end{matrix}]

(9)

with the following properties (Wang et al., 2016)

{\overset{\cdot}{R}}_{i} (ψ_{i}) = R_{i} (ψ_{i}) S_{i} (r_{i})

(10)

R_{i}^{T} (ψ_{i}) S_{i} (r_{i}) R_{i} (ψ_{i}) = R_{i} (ψ_{i}) S_{i} (r_{i}) R_{i}^{T} (ψ_{i}) = S_{i} (r_{i})

(11)

S_{i} (r_{i}) = [\begin{matrix} 0 & - r_{i} & 0 \\ r_{i} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]

(12)

R_{i}^{T} (ψ_{i}) R_{i} (ψ_{i}) = I_{3}

(13)

‖ R_{i} (ψ_{i}) ‖ = 1

(14)

For the convenience of writing, $ψ_{i}$ will be omitted when representing the rotation matrix and its transpose, that is $R_{i} = R_{i} (ψ_{i})$ and $S_{i} = S_{i} (r_{i})$ .

Assumption 2: The desired reference signal $η_{d}$ is smooth enough that the first and second derivatives of $η_{d}$ exist and are bounded.

Remark 2: According to Xia et al. (2021c), in the cooperative control scheme, to make the vessel move smoothly, it is usually necessary to make the reference signal sufficiently smooth, and its first and second derivatives exist and are bounded. Therefore, the Assumption 2 is reasonable.

Assumption 3: Both time-varying marine environmental disturbance $τ_{id}$ and its first-derivative ${\overset{\cdot}{τ}}_{id}$ are bounded, that is, there are unknown normal numbers ${\bar{τ}}_{id}$ and $ϖ_{i}$ , so that $‖ τ_{id} ‖ \leq {\bar{τ}}_{id}$ and $‖ {\overset{\cdot}{τ}}_{id} ‖ \leq ϖ_{i}$ .

Remark 3: According to Xia et al. (2021b), the time-varying marine environmental disturbance is always considered as slowly varying and have finite energy. Therefore, the disturbances acting on surface vessel can be viewed as unknown finite change rates and bounded signals. Therefore, the Assumption 3 is reasonable.

Assumption 4: The parameter matrix $M_{i}$ of the $i th$ vessel mathematical model is known.

Remark 4: According to Zhang et al. (2020a), the parameter matrix $M_{i}$ of the $i th$ vessel is easy to measure. Therefore, the Assumption 4 is reasonable.

The control objective of this paper is to design a fixed-time distributed output feedback sliding mode cooperative control scheme for each MSV with model uncertainties, external disturbances, and unavailable velocities such that the reference signal $η_{d}$ can be tracked accurately in fixed time.

Design of FxESO

In this section, to simplify the following design, a new auxiliary velocity vector is presented as follows

w_{i} = R_{i} υ_{i}

(15)

where $w_{i} = {[w_{i 1}, w_{i 2}, w_{i 3}]}^{T} \in R^{3}$ .

According to equations (10)–(14), the $i th$ surface vessel mathematical models (3)–(4) are rewritten as Zhang et al. (2020a)

{\overset{\cdot}{η}}_{i} = w_{i}

(16)

{\overset{\cdot}{w}}_{i} = R_{i} M_{i}^{- 1} τ_{i} + Γ_{i}

(17)

{\overset{\cdot}{η}}_{d} = w_{d}

(18)

where $Γ_{i} = R_{i} S_{i} υ_{i} - R_{i} M_{i}^{- 1} C_{i} (υ_{i}) υ_{i} - R_{i} M_{i}^{- 1} D_{i} (υ_{i}) υ_{i} + R_{i} M_{i}^{- 1} τ_{i d}$ , $η_{d} = {[x_{d}, y_{d}, ψ_{d}]}^{T}$ is a desired position vector and $w_{d} = {[w_{d 1}, w_{d 2}, w_{d 3}]}^{T}$ is a desired auxiliary velocity vector.

Remark 5: According to Wang et al. (2016), $Γ_{i}$ is continuously differentiable and bounded and it includes model uncertainties and unknown environmental disturbances. Hence, there is a constant $Γ_{id}$ , $‖ \overset{\cdot}{Γ} (t) ‖ \leq H_{n} < Γ_{id}$ with $0 < Γ_{id} < \infty$ .

In this part, by designing an FxESO, it can re-establish the vessel’s velocity and estimate the compound disturbance $Γ_{i}$ . ${\hat{η}}_{i}$ is used to represent the estimate of position and heading vector $η_{i}$ . Thus, the FxESO is designed as follows (Basin et al., 2017)

{\begin{matrix} {\overset{\cdot}{\hat{η}}}_{i} = {\hat{w}}_{i} + μ_{1} si g^{α_{1}} ({\tilde{η}}_{i}) + ε_{1} si g^{β_{1}} ({\tilde{η}}_{i}) \\ {\overset{\cdot}{\hat{w}}}_{i} = {\hat{Γ}}_{i} + μ_{2} si g^{α_{2}} ({\tilde{η}}_{i}) + ε_{2} si g^{β_{2}} ({\tilde{η}}_{i}) + R_{i} M_{i}^{- 1} τ_{i} \\ {\overset{\cdot}{\hat{Γ}}}_{i} = μ_{3} si g^{α_{3}} ({\tilde{η}}_{i}) + ε_{3} si g^{β_{3}} ({\tilde{η}}_{i}) + Γ_{id} sign ({\tilde{η}}_{i}) \end{matrix}

(19)

where ${\tilde{η}}_{i} = η_{i} - {\hat{η}}_{i}$ , $α_{m} \in (0, 1)$ , $β_{m} \in (1, \infty)$ , $m = 1, 2, 3$ . $α_{m} = m \bar{α} - (m - 1)$ , $β_{m} = m \bar{β} - (m - 1)$ , $\bar{α} \in (1 - l_{1}, 1)$ , $\bar{β} \in (1, 1 + l_{2})$ with small enough constants $l_{1} > 0$ , $l_{2} > 0$ . The observer gains are designed to guarantee the following matrix Hurwitz

P_{1} = [\begin{matrix} - μ_{1} & 1 & 0 \\ - μ_{2} & 0 & 1 \\ - μ_{3} & 0 & 0 \end{matrix}], P_{2} = [\begin{matrix} - ε_{1} & 1 & 0 \\ - ε_{2} & 0 & 1 \\ - ε_{3} & 0 & 0 \end{matrix}]

Then, $w_{i}$ and $Γ_{i}$ can be observed by ${\hat{w}}_{i}$ and ${\hat{Γ}}_{i}$ , respectively.

Theorem 1: Under the Assumption 3 and Remark 3, it can be estimated about the velocity $w_{i}$ and lumped disturbance $Γ_{i}$ and the estimation errors can converge to a neighborhood of the origin within fixed time by FxESO (19).

Proof: The estimation errors are as follows

{\begin{matrix} {\tilde{e}}_{i 1} = η_{i} - {\hat{η}}_{i} \\ {\tilde{e}}_{i 2} = w_{i} - {\hat{w}}_{i} \\ {\tilde{e}}_{i 3} = Γ_{i} - {\hat{Γ}}_{i} \end{matrix}

(20)

The derivative of equation (20) can be written as

{\begin{matrix} {\overset{\cdot}{\tilde{e}}}_{i 1} = {\tilde{e}}_{i 2} - μ_{1} si g^{α_{1}} ({\tilde{e}}_{i 1}) - ε_{1} si g^{β_{1}} ({\tilde{e}}_{i 1}) \\ {\overset{\cdot}{\tilde{e}}}_{i 2} = {\tilde{e}}_{i 3} - μ_{2} si g^{α_{2}} ({\tilde{e}}_{i 1}) - ε_{2} si g^{β_{2}} ({\tilde{e}}_{i 1}) \\ {\overset{\cdot}{\tilde{e}}}_{i 3} = {\overset{\cdot}{Γ}}_{i} - μ_{3} si g^{α_{3}} ({\tilde{e}}_{i 1}) - ε_{3} si g^{β_{3}} ({\tilde{e}}_{i 1}) - Γ_{id} sign ({\tilde{e}}_{i 1}) \end{matrix}

(21)

According to Zhang et al. (2018), to prove that the estimation errors converging to zero in a fixed time, it can take the following steps:

The following error system that can converge to zero in fixed time will be proved

{\begin{matrix} {\overset{\cdot}{\tilde{e}}}_{i 1} = {\tilde{e}}_{i 2} - μ_{1} si g^{α_{1}} ({\tilde{e}}_{i 1}) - ε_{1} si g^{β_{1}} ({\tilde{e}}_{i 1}) \\ {\overset{\cdot}{\tilde{e}}}_{i 2} = {\tilde{e}}_{i 3} - μ_{2} si g^{α_{2}} ({\tilde{e}}_{i 1}) - ε_{2} si g^{β_{2}} ({\tilde{e}}_{i 1}) \\ {\overset{\cdot}{\tilde{e}}}_{i 3} = - μ_{3} si g^{α_{3}} ({\tilde{e}}_{i 1}) - ε_{3} si g^{β_{3}} ({\tilde{e}}_{i 1}) \end{matrix}

(22)

Proof: consider error system (22) separately as follows

{\begin{matrix} {\overset{\cdot}{\tilde{e}}}_{i 1} = {\tilde{e}}_{i 2} - μ_{1} si g^{α_{1}} ({\tilde{e}}_{i 1}) \\ {\overset{\cdot}{\tilde{e}}}_{i 2} = {\tilde{e}}_{i 3} - μ_{2} si g^{α_{2}} ({\tilde{e}}_{i 1}) \\ {\overset{\cdot}{\tilde{e}}}_{i 3} = μ_{3} si g^{α_{3}} ({\tilde{e}}_{i 1}) \end{matrix}

(23)

Define $φ_{i} = [{\tilde{e}}_{i 1}, {\tilde{e}}_{i 2}^{1 / α_{1}}, {\tilde{e}}_{i 3}^{1 / α_{2}}]^{T}$ and ${\tilde{e}}_{i} = [{\tilde{e}}_{i 1}, {\tilde{e}}_{i 2}, {\tilde{e}}_{i 3}]^{T}$ , $Ω_{1}$ is a symmetric positive-definite matrix such that

P_{1}^{T} Ω_{1} + Ω_{1} P_{1} = - Q_{1}

(24)

where $P_{1}$ is defined in equation (19) and $Q_{1}$ is a symmetric positive-definite matrix.

If $\bar{α}$ is selected as 1, the corresponding error system becomes ${\overset{\cdot}{\tilde{e}}}_{i} (t) = P_{1} {\tilde{e}}_{i} (t)$ . Choose $V_{io 1} ({\tilde{e}}_{i} (t)) = {\tilde{e}}_{i}^{T} P_{1} {\tilde{e}}_{i}$ as a Lyapunov function, then ${\overset{\cdot}{V}}_{io 1} ({\tilde{e}}_{i} (t)) = {\overset{\cdot}{\tilde{e}}}_{i}^{T} Ω_{1} {\tilde{e}}_{i} + {\tilde{e}}_{i}^{T} Ω_{1} {\overset{\cdot}{\tilde{e}}}_{i} = {\tilde{e}}_{i}^{T} (P_{1}^{T} Ω_{1} + Ω_{1} P_{1}) {\tilde{e}}_{i} = - {\tilde{e}}_{i}^{T} Q_{1} {\tilde{e}}_{i} \leq 0$ can be easily obtained. From Basin et al. (2017), we can obtain that if there exists a small constant $ς_{1}$ such that for $\bar{α} \in (1 - ς_{1}, 1)$ , then there exists a Lyapunov function $V_{io 1} (φ_{i}) = φ_{i}^{T} Ω_{1} φ_{i} > 0$ and satisfies ${\overset{\cdot}{V}}_{io 1} (φ_{i}) = φ_{i}^{T} (P_{1}^{T} Ω_{1} + Ω_{1} P_{1}) φ_{i} = - φ_{i}^{T} Q_{1} φ_{i} \leq 0$ . Moreover, ${\overset{\cdot}{V}}_{io 1} (φ_{i})$ satisfies

{\overset{\cdot}{V}}_{io 1} (φ_{i}) \leq - \frac{λ_{min} (Q_{1})}{λ_{max} (Ω_{1})} V_{io 1}^{\bar{α}} (φ_{i})

(25)

where $λ_{max} (Ω_{1})$ is positive and represents the maximum eigenvalue of $Ω_{1}$ , $λ_{min} (Q_{1})$ is positive and represents the minimum eigenvalue of $Q_{1}$ . Then according to Lemma 4, the settling time is

T_{1 a} \leq \frac{λ_{max} (Ω_{1})}{λ_{min} (Q_{1}) (1 - \bar{α})} V_{io 1}^{\bar{α}} (φ_{i})

(26)

Consider the second part of equation (23), that is

{\begin{matrix} {\overset{\cdot}{\tilde{e}}}_{i 1} = {\tilde{e}}_{i 2} - ε_{1} si g^{β_{1}} ({\tilde{e}}_{i 1}) \\ {\overset{\cdot}{\tilde{e}}}_{i 2} = {\tilde{e}}_{i 3} - ε_{2} si g^{β_{2}} ({\tilde{e}}_{i 1}) \\ {\overset{\cdot}{\tilde{e}}}_{i 3} = - ε_{3} si g^{β_{3}} ({\tilde{e}}_{i 1}) \end{matrix}

(27)

If $\bar{β}$ is selected as 1, the corresponding error system becomes ${\overset{\cdot}{\tilde{e}}}_{i} (t) = P_{2} {\tilde{e}}_{i} (t)$ . Choose $V_{io 2} ({\tilde{e}}_{i} (t)) = {\tilde{e}}_{i}^{T} Ω_{2} {\tilde{e}}_{i}$ as a Lyapunov function. $Ω_{2}$ is a symmetric positive-definite matrix such that $P_{2}^{T} Ω_{2} + Ω_{2} P_{2} = - Q_{2}$ , where $Q_{2}$ is a symmetric positive-definite matrix. Thus, ${\overset{\cdot}{V}}_{io 2} ({\tilde{e}}_{i} (t))$ satisfies $4 {\overset{\cdot}{V}}_{io 2} ({\tilde{e}}_{i} (t))$ $= {\overset{\cdot}{\tilde{e}}}_{i}^{T} Ω_{2} {\tilde{e}}_{i} + {\tilde{e}}_{i}^{T} Ω_{2} {\overset{\cdot}{\tilde{e}}}_{i} = {\tilde{e}}_{i}^{T} (P_{2}^{T} Ω_{2} + Ω_{2} P_{2}) {\tilde{e}}_{i} = - {\tilde{e}}_{i}^{T} Q_{2} {\tilde{e}}_{i} \leq 0$ .

Define $ψ_{i} = [{\tilde{e}}_{i 1}, {\tilde{e}}_{i 2}^{1 / β_{1}}, {\tilde{e}}_{i 3}^{1 / β_{2}}]^{T}$ . If there exists a small constant $ς_{2}$ such that for $\bar{β} \in (1, 1 + ς_{2})$ . Choose the Lyapunov function as $V_{io 2} (ψ_{i}) = ψ_{i}^{T} Ω_{2} ψ_{i}$ . For equation (27), the right side is a homogeneous vector field of degree $ϖ_{β} = β_{0} - 1 > 0$ with respect to dilations $β_{1} = \bar{β}, β_{2} = 2 \bar{β} - 1, β_{3} = 3 \bar{β} - 2$ . If $β_{0}$ is sufficiently close to 1, the full-time derivative of the Lyapunov function $V_{io 2} (ψ_{i})$ is homogeneous in ${\tilde{e}}_{i 1}, {\tilde{e}}_{i 2}, {\tilde{e}}_{i 3}$ of degree $1 + ϖ_{β}$ with respect to the same weights $β_{1} = \bar{β}, β_{2} = 2 \bar{β} - 1, β_{3} = 3 \bar{β} - 2$ . Combining the Theorem in Basin et al. (2017) and Lemma 4, the following inequality can be obtained

{\overset{\cdot}{V}}_{io 2} (ψ_{i}) \leq - \frac{λ_{min} (Q_{2})}{λ_{max} (Ω_{2})} V_{io 2}^{\bar{β}} (ψ_{i})

(28)

where $λ_{max} (Ω_{2})$ is positive and represents the maximum eigenvalue of $Ω_{2}$ , $λ_{min} (Q_{2})$ is positive and represents the minimum eigenvalue of $Q_{2}$ .

Thus, for error system (22) according to the Theorem 2 in Basin et al. (2017), the observer errors will converge to zero by a bounded time as

T_{1 b} \leq \frac{λ_{max}^{1 - \bar{α}} (Ω_{1})}{γ_{1} (1 - \bar{α})} + \frac{1}{γ_{2} (\bar{β} - 1) ℏ^{\bar{β} - 1}}

(29)

where $γ_{1} = λ_{min} (Q_{1}) / λ_{max} (Ω_{1})$ , $γ_{2} = λ_{min} (Q_{2}) / λ_{max} (Ω_{2})$ , the positive constant $ℏ \leq λ_{min} (Ω_{2})$ .

2. Once ${\tilde{e}}_{i 1} = 0$ occurs after the time $T_{1 b}$ , ${\tilde{e}}_{i 1}$ will stay in ${\tilde{e}}_{i 1} = 0$ in future time. Then, we can obtain ${\tilde{e}}_{i 1} = {\overset{\cdot}{\tilde{e}}}_{i 1} = 0$ . That is to say, there exists a bound time $T_{1 b}$ , such that ${\tilde{e}}_{i 1} = {\tilde{e}}_{i 2} = {\tilde{e}}_{i 3} = 0$ for $t \geq T_{1 b}$ , which implies the following identity holds (Zhang et al., 2018)

\begin{matrix} {\overset{\cdot}{\tilde{e}}}_{i 3} = {\overset{\cdot}{Γ}}_{i} - Γ_{id} sign ({\tilde{e}}_{i 1}) = 0, & t \geq T_{1 b} \end{matrix}

(30)

It should be noted that it is impossible to achieve the identity ${\tilde{e}}_{i 3} \equiv 0$ due to the impacts of imperfections, such as sampling noise, sampling step, small delays, and disturbance. Huang and Jia (2017) present a small convergence region ${\tilde{e}}_{i 3} |_{τ}$ , allowing ${\tilde{e}}_{i 3}$ to reach the region ${\tilde{e}}_{i 3} |_{τ}$ in a smaller time less than

T_{1 τ} = \frac{{\tilde{e}}_{i 3} |_{τ}}{Γ_{id} - H_{n}}

(31)

where $H_{n}$ is defined in Remark 5. Therefore, the upper bound of convergence time for the FxESO is

T_{1} \leq T_{1 b} + T_{1 τ}

(32)

Theorem 1 is proved.

Controller design

An event-triggered fixed-time non-singular terminal sliding mode controller (FxNTSMC) with actuator saturation combined with the FxESO is designed. The control law design process is divided into the following steps. The control system structure block diagram of the $i th$ vessel is shown in Figure 2.

Step 1: In this step, a fixed-time auxiliary dynamic system is designed to deal with the saturation constraint of the MSVs’ control input. The designed auxiliary dynamic system can limit control input saturation and improve system stability. Define the fixed-time auxiliary dynamic system of $i th$ surface vessel as follows

{\overset{\cdot}{ξ}}_{i} = - Asi g^{r_{01}} (ξ_{i}) - Bsi g^{r_{02}} (ξ_{i}) - {\bar{Δ}}_{i} G_{i} sign (ξ_{i}) + R_{i} M_{i}^{- 1} Δ τ_{i}

(33)

where $Δ τ_{i}$ represents the difference between before and after the saturation constraint of control input, which is defined as $Δ τ_{i} = τ_{i} - τ_{ic}$ , $τ_{ic}$ being calculated by the proposed controller. Suppose $Δ τ_{i}$ is bounded and there exists a normal number ${\bar{Δ}}_{i}$ such that $‖ Δ τ_{i} ‖ \leq {\bar{Δ}}_{i}$ . $ξ_{i} \in R^{3}$ is the auxiliary state generated by the auxiliary dynamic system. $A, B \in R^{3 \times 3}$ are designed positive-definite diagonal gain matrices. $r_{01}$ and $r_{02}$ are positive constants, $r_{01} \in (0, 1)$ , $r_{02} > 1$ . $G_{i} = diag {‖ g_{i} ‖} \in R^{3 \times 3}$ , with $g_{i}$ being the $i th$ row of the control gain matrix $R_{i} M_{i}^{- 1}$ .

Theorem 2: The auxiliary state converges to zero within a fixed-time $T_{sa}$ .

Proof: Select the following Lyapunov function

V_{i 1} = \frac{1}{2} ξ_{i}^{T} ξ_{i}

(34)

The derivative of equation (34) yields

\begin{matrix} {\overset{\cdot}{V}}_{i 1} = ξ_{i}^{T} {\overset{\cdot}{ξ}}_{i} \\ = ξ_{i}^{T} (- Asi g^{r_{o 1}} (ξ_{i}) - Bsi g^{r_{o 2}} (ξ_{i}) - {\bar{Δ}}_{i} G_{i} sign (ξ_{i}) + a_{id} R_{i} M_{i}^{- 1} Δ τ_{i}) \\ \leq - 2^{\frac{r_{o 1} + 1}{2}} λ_{min} (A) {(\frac{1}{2} ξ_{i}^{T} ξ_{i})}^{\frac{r_{o 1} + 1}{2}} - 2^{\frac{r_{o 2} + 1}{2}} λ_{min} (B) {(\frac{1}{2} ξ_{i}^{T} ξ_{i})}^{\frac{r_{o 2} + 1}{2}} \\ - {\bar{Δ}}_{i} \sum_{m = 1}^{3} g_{im} | ξ_{im} | + a_{id} ξ_{i}^{T} R_{i} M_{i}^{- 1} Δ τ_{i} \\ \leq - 2^{\frac{r_{o 1} + 1}{2}} λ_{min} (A) {V_{i 1}}^{\frac{r_{o 1} + 1}{2}} - 2^{\frac{r_{o 2} + 1}{2}} λ_{min} (B) {V_{i 1}}^{\frac{r_{o 2} + 1}{2}} \end{matrix}

(35)

where $λ_{min} (A)$ represents the minimum eigenvalue of $A$ and $λ_{min} (B)$ represents the minimum eigenvalue of $B$ .

According to Lemma 1, the auxiliary system state will converge to zero within a fixed-time $T_{sa}$

T_{sa} \leq \frac{1}{2^{\frac{r_{o 1} + 1}{2}} λ_{min} (A) (1 - \frac{r_{o 1} + 1}{2})} + \frac{1}{2^{\frac{r_{o 2} + 1}{2}} λ_{min} (B) (\frac{r_{o 2} + 1}{2} - 1)}

(36)

This completes the proof of Theorem 2.

Step 2: According to the communication relationship among MSVs, graph theory, the position information of adjacent vessels, and auxiliary state in equation (33), define the tracking errors of $i th$ surface vessel as follows

z_{i 1} = \sum_{j \in N_{i}} a_{i j} (η_{i} - η_{j} - ϑ_{i j}) + a_{i 0} (η_{i} - η_{d} - ϑ_{i})

(37)

{\hat{z}}_{i 2} = a_{id} {\hat{w}}_{i} - \sum_{j \in N_{i}} a_{ij} {\hat{w}}_{j} - a_{i 0} w_{d} - ξ_{i}

(38)

where the definitions of $N_{i}$ , $a_{ij}$ , and $a_{i 0}$ refer to the graph theory part, $ϑ_{i}$ , a constant vector, represents the expected relative position between the $i th$ surface vessel and the reference point, $ϑ_{ij}$ , a constant vector as well, represents the expected position relative to the neighboring $i th$ vessel and $a_{id} = a_{i 0} + d_{i}$ . ${\hat{z}}_{i 2}$ is an estimate of $z_{i 2}$ $z_{i 2} = a_{id} w_{i} - \sum_{j \in N_{i}} a_{ij} w_{j} - a_{i 0} w_{d} - ξ_{i}$ .

The derivative of equation (38) is given by

{\overset{\cdot}{\hat{z}}}_{i 2} = R_{i} M_{i}^{- 1} τ_{i} + Γ_{i} - \sum_{j \in N_{i}} a_{ij} {\overset{\cdot}{\hat{w}}}_{j} - a_{i 0} {\overset{\cdot}{w}}_{d} - {\overset{\cdot}{ξ}}_{i}

(39)

Remark 6: Because of the topology in this paper, each vessel receives communication information from only one of other vessels, for simplicity, $a_{id} = 1$ . Moreover, based on the FxESO in equation (19) and the auxiliary system in equation (33), ${\hat{z}}_{i 2} = z_{i 2}$ and $ξ_{i} = 0$ for $t \geq T_{1} + T_{sa}$ , thus, equation (38) will be transferred as

{\hat{z}}_{i 2} = z_{i 2} = {\overset{\cdot}{z}}_{i 1} = w_{i} - \sum_{j \in N_{i}} a_{ij} w_{j} - a_{i 0} w_{d}

(40)

Step 3: Design FxNTSMM.

In this paper, the FxNTSMM (Huang and Jia, 2018) of $i th$ surface vessel is designed as follows

S_{i} = β (z_{i 1}) z_{i 1} + si g^{r_{2}} ({\hat{z}}_{i 2})

(41)

where ${\hat{z}}_{i 2} = {\overset{\cdot}{z}}_{i 1} = w_{i} - \sum_{j \in N_{i}} a_{ij} w_{j} - a_{i 0} w_{d}$ , $β (z_{i 1}) = diag ([β_{1} (z_{i 11}), β_{2} (z_{i 12}), β_{3} (z_{i 13})])$ , $β_{m} (z_{i 1 m}), (m = 1, 2, 3)$ as follows

β_{m} (z_{i 1 m}) = {\begin{matrix} {(a_{s} {| z_{i 1 m} |}^{r_{1} - \frac{1}{r_{2}}} + b_{s})}^{r_{2}}, & | z_{i 1 m} | \geq Θ \\ (a_{s 1} {| z_{i 1 m} |}^{r_{1} - \frac{1}{r_{2}}} + a_{s 2} {| z_{i 1 m} |}^{2 - \frac{1}{r_{2}}} + b_{s}), & | z_{i 1 m} | < Θ \end{matrix}

(42)

where $a_{s} > 0$ , $b_{s} > 0$ , $a_{s 1} = a_{s} (2 - r_{1}) Θ^{(r_{1} - 1)}$ , $a_{s 2} = a_{s} (r_{1} - 1) Θ^{(r_{1} - 2)}$ , $r_{1} > 1$ , $1 < r_{2} < 2$ , and $0 < Θ \leq 1$ .

Theorem 3: With FxESO (19) and auxiliary system (33), if $S_{i} = 0$ , then $z_{i 1}$ and ${\hat{z}}_{i 2}$ converge to zero within a fixed convergence time $T_{2}$ .

Proof: The proof does not differ much from that of Lemma 3 in Huang and Jia (2018) and thus is omitted here for space. If $S_{i} = 0$ , then $z_{i 1}$ and ${\hat{z}}_{i 2}$ converge to zero within a fixed convergence time $T_{2}$

T_{2} \leq T_{1} + T_{sa} + \frac{Λ^{- r_{ζ 1}}}{a_{s} (r_{1} - 1)} + \frac{r_{2}}{r_{2} - 1} \frac{1}{a_{s 1}} \ln (\frac{a_{s 1} Λ}{b_{s}} + 1)

(43)

Remark 7: It follows from equation (41) that the designed FxNTSMM is actually consisted of two fixed-time terminal sliding mode manifolds: $S_{1} = \overset{\cdot}{x} + a_{s} si g^{r_{1}} (x) + b_{s} si g^{1 / r_{2}} (x), | x_{m} | \geq Θ$ and $S_{2} = \overset{\cdot}{x} + a_{s 1} x + a_{s 2} sign (x) x^{2} + b_{s} si g^{1 / r_{2}} (x), | x_{m} | < Θ$ . When $| x_{m} | < Θ$ , the system state switches smoothly from the sliding modes $S_{1}$ to $S_{2}$ . From the sliding mode $S_{2}$ , it can be seen that the nonlinear term $si g^{1 / r_{2}} (x), (r_{2} > 1)$ is regarded as the dominating term rather than a quadratic function $a_{s 1} x + a_{s 2} \underset{(}{sign} x) x^{2}$ replaced $si g^{1 / r_{2}} (x), (r_{2} > 1)$ in Zou et al. (2011) to avoid the singularity problem. In addition, the nonlinear term $si g^{r_{1}} (x), (r_{1} > 1)$ is substituted by a quadratic function $a_{s 1} x + a_{s 2} sign (x) x^{2}$ , which dominates over $s i g^{r_{1}} (x), (r_{1} > 1)$ nearby the origin. Thus, the designed FxNTSMM has the advantages of faster convergence performance than fixed-time terminal sliding mode manifold and non-singularity property, simultaneously.

Step 4: Design the event-triggered fixed-time non-singular sliding mode controller based on the event-triggered strategy.

In this part, according to the FxESO in equation (19), the auxiliary system in equation (33), and FxNTSMM in equation (41), a non-singular fixed-time sliding mode control law is designed as follows

\begin{matrix} ω_{τ i} (t) = - M_{i} R_{i}^{- 1} \\ [\begin{matrix} \frac{1}{r_{2}} (\tilde{β} (z_{i 1}) + β (z_{i 1})) si g^{2 - r_{2}} ({\hat{z}}_{i 2}) + \frac{1}{r_{2}} diag {ρ ({| {\hat{z}}_{i 2} |}^{r_{2} - 1})} \\ diag {{| {\hat{z}}_{i 2} |}^{1 - r_{2}}} (α si g^{r_{3}} (S_{i}) + β si g^{r_{4}} (S_{i})) + {\hat{Γ}}_{i} - \sum_{j \in N_{i}} a_{ij} {\overset{\cdot}{\hat{w}}}_{j} \\ - a_{i 0} {\overset{\cdot}{w}}_{d} + Asi g^{r_{o 1}} (ξ_{i}) + Bsi g^{r_{o 2}} (ξ_{i}) + {\bar{Δ}}_{i} G_{i} sign (ξ_{i}) \end{matrix}] \end{matrix}

(44)

where $α$ , $β$ , $r_{3}$ , and $r_{4}$ are positive constants, $r_{3} \in (0, 1)$ and $r_{4} \in (1, + \infty)$

ρ ({| {\hat{z}}_{i 2 m} |}^{r_{2} - 1}) = {\begin{matrix} \sin (\frac{π}{2} \frac{{| {\hat{z}}_{i 2 m} |}^{r_{2} - 1}}{ι}), {| {\hat{z}}_{i 2 m} |}^{r_{2} - 1} < ι \\ 1, {| {\hat{z}}_{i 2 m} |}^{r_{2} - 1} \geq ι \end{matrix}

(45)

where $ι > 0$ is a design parameter. $ρ ({| {\hat{z}}_{i 2 m} |}^{r_{2} - 1})$ is a nonnegative function and satisfies $ρ ({| {\hat{z}}_{i 2 m} |}^{r_{2} - 1}) / {| {\hat{z}}_{i 2 m} |}^{r_{2} - 1} \to (π / 2 ι)$ when ${| {\hat{z}}_{i 2 m} |}^{r_{2} - 1} \to 0$

{\tilde{β}}_{m} (z_{i 1 m}) = {\begin{matrix} r_{2} a_{s} (r_{1} - \frac{1}{r_{2}}) (a_{s} {| z_{i 1 m} |}^{r_{1} - \frac{1}{r_{2}}} + b_{s})^{r_{2} - 1} {| z_{i 1 m} |}^{r_{1} - \frac{1}{r_{2}}}, | z_{i 1 m} | \geq Θ \\ r_{2} (a_{s 1} {| z_{i 1 m} |}^{1 - \frac{1}{r_{2}}} + a_{s 2} {| z_{i 1 m} |}^{2 - \frac{1}{r_{2}}} + b_{s})^{r_{2} - 1} \\ {a_{s 1} (1 - \frac{1}{r_{2}}) {| z_{i 1 m} |}^{1 - \frac{1}{r_{2}}} + a_{s 2} (2 - \frac{1}{r_{2}}) {| z_{i 1 m} |}^{2 - \frac{1}{r_{2}}}}, | z_{i 1 m} | < Θ \end{matrix}

(46)

The triggering event is chosen as

τ_{icm} (t) = ω_{τ im} (t_{km}), \forall t \in [t_{km}, t_{(k + 1) m}), (m = 1, 2, 3)

(47)

with

t_{km} = inf {t \in R : | e_{im} (t) | \geq σ_{m}}, t_{1 m} = 0, (m = 1, 2, 3)

(48)

where $e_{im} (t) = ω_{τ im} (t) - τ_{icm} (t)$ , $σ_{m}$ is a positive constant.

Theorem 4: Consider the system consisting of the MSVs dynamic in equations (3) and (4), the FxESO in equation (19), the auxiliary system in equation (33), the FxNTSMM in equation (41), and control law (47), with unknown environmental disturbances under Assumptions 1–4. Then the sliding mode $S_{i}$ and the tracking errors $z_{i 1}$ and ${\hat{z}}_{i 2}$ finally converge into the small regions within a fixed time. The upper bound of the convergence time can be obtained as $T \leq T_{2} + T_{3}$ .

Proof: Select the following Lyapunov function

V_{i 2} = \frac{1}{2} S_{i}^{T} S_{i}

(49)

The derivative of equation (49) yields

\begin{matrix} {\overset{\cdot}{V}}_{i 2} = S_{i}^{T} {\overset{\cdot}{S}}_{i} \\ = S_{i}^{T} [\frac{d β (z_{i 1})}{dt} z_{i 1} + β (z_{i 1}) {\hat{z}}_{i 2} + r_{2} diag {{| {\hat{z}}_{i 2} |}^{r_{2} - 1}} {\overset{\cdot}{\hat{z}}}_{i 2}] \end{matrix}

(50)

Substituting equation (39) into equation (50) yields

{\overset{\cdot}{V}}_{i 2} = S_{i}^{T} [\begin{matrix} \frac{d β (z_{i 1})}{dt} z_{i 1} + β (z_{i 1}) {\hat{z}}_{i 2} + r_{2} diag {{| {\hat{z}}_{i 2} |}^{r_{2} - 1}} \\ (R_{i} M_{i}^{- 1} τ_{i} + {\hat{Γ}}_{i} - \sum_{j \in N_{i}} a_{ij} {\overset{\cdot}{\hat{w}}}_{j} - a_{i 0} {\overset{\cdot}{w}}_{d} - {\overset{\cdot}{ξ}}_{i}) \end{matrix}]

(51)

Substituting equation (33) into equation (51) yields

{\overset{\cdot}{V}}_{i 2} = S_{i}^{T} [\begin{matrix} \frac{d β (z_{i 1})}{dt} z_{i 1} + β (z_{i 1}) {\hat{z}}_{i 2} + r_{2} diag {{| {\hat{z}}_{i 2} |}^{r_{2} - 1}} \\ {\begin{matrix} R_{i} M_{i}^{- 1} τ_{ic} + {\hat{Γ}}_{i} - \sum_{j \in N_{i}} a_{ij} {\overset{\cdot}{\hat{w}}}_{j} - a_{i 0} {\overset{\cdot}{w}}_{d} + \\ Asi g^{r_{o 1}} (ξ_{i}) + Bsi g^{r_{o 2}} (ξ_{i}) + {\bar{Δ}}_{i} G_{i} sign (ξ_{i}) \end{matrix}} \end{matrix}]

(52)

According to equations (47) and (48), one has $| ω_{τ im} (t_{km}) - τ_{icm} (t) | \leq σ_{m}$ . There exist time-varying functions $μ_{m} (m = 1, 2, 3)$ satisfying $| μ_{m} | \leq 1$ . Then, $ω_{τ im} = τ_{icm} + μ_{m} σ_{m} (m = 1, 2, 3)$ . Equation (52) is converted into the following

{\overset{\cdot}{V}}_{i 2} = S_{i}^{T} [\begin{matrix} \frac{d β (z_{i 1})}{dt} z_{i 1} + β (z_{i 1}) {\hat{z}}_{i 2} + r_{2} diag {{| {\hat{z}}_{i 2} |}^{r_{2} - 1}} \\ {\begin{matrix} R_{i} M_{i}^{- 1} (ω_{τ i} - μ σ) + {\hat{Γ}}_{i} - \sum_{j \in N_{i}} a_{ij} {\overset{\cdot}{\hat{w}}}_{j} - a_{i 0} {\overset{\cdot}{w}}_{d} \\ + Asi g^{r_{o 1}} (ξ_{i}) + Bsi g^{r_{o 2}} (ξ_{i}) + {\bar{Δ}}_{i} G_{i} sign (ξ_{i}) \end{matrix}} \end{matrix}]

(53)

where $μ = diag [μ_{1}, μ_{2}, μ_{3}]$ , $σ = {[σ_{1}, σ_{2}, σ_{3}]}^{T}$ , and $σ_{m} (m = 1, 2, 3)$ is defined in equation (48).

Applying equation (44) yields

{\overset{\cdot}{V}}_{i 2} = S_{i}^{T} [\begin{matrix} \frac{d β (z_{i 1})}{dt} z_{i 1} + β (z_{i 1}) {\hat{z}}_{i 2} - r_{2} diag {{| {\hat{z}}_{i 2} |}^{r_{2} - 1}} \\ {\begin{matrix} \frac{1}{r_{2}} (\tilde{β} (z_{i 1}) + β (z_{i 1})) si g^{2 - r_{2}} ({\hat{z}}_{i 2}) + \frac{1}{r_{2}} diag {ρ ({| {\hat{z}}_{i 2} |}^{r_{2} - 1})} \\ diag {{| {\hat{z}}_{i 2} |}^{1 - r_{2}}} (α si g^{r_{3}} (S_{i}) + β si g^{r_{4}} (S_{i})) + R_{i} M_{i}^{- 1} μ σ \end{matrix}} \end{matrix}]

(54)

Subsequently, we have

{\dot{V}}_{i 2} = S_{i}^{T} [\begin{matrix} \frac{d β (z_{i 1})}{d t} z_{i 1} + β (z_{i 1}) {\hat{z}}_{i 2} - d i a g {{| {\hat{z}}_{i 2} |}^{r_{2} - 1}} \\ {(\tilde{β} (z_{i 1}) + β (z_{i 1})) s i g^{2 - r_{2}} ({\hat{z}}_{i 2})} \\ - {d i a g {ρ ({| {\hat{z}}_{i 2} |}^{r_{2} - 1})} (α s i g^{r_{3}} (S_{i}) + β s i g^{r_{4}} (S_{i})) \\ + r_{2} d i a g {{| {\hat{z}}_{i 2} |}^{r_{2} - 1}} R_{i} M_{i}^{- 1} μ σ} \end{matrix}]

(55)

According to $\frac{d β (z_{i 1})}{dt} z_{i 1} = \tilde{β} (z_{i 1}) z_{i 2}$ , equation (55) can be rewritten as follows

\begin{matrix} {\overset{\cdot}{V}}_{i 2} = S_{i}^{T} [- diag {ρ ({| {\hat{z}}_{i 2} |}^{r_{2} - 1})} (α si g^{r_{3}} (S_{i}) + β si g^{r_{4}} (S_{i})) \\ - r_{2} diag {{| {\hat{z}}_{i 2} |}^{r_{2} - 1}} (R_{i} M_{i}^{- 1} μ σ)] \end{matrix}

(56)

According to $- S_{i}^{T} r_{2} diag {{| {\hat{z}}_{i 2} |}^{r_{2} - 1}} R_{i} M_{i}^{- 1} μ σ \leq | S_{i}^{T} r_{2} diag {{| {\hat{z}}_{i 2} |}^{r_{2} - 1}} R_{i} M_{i}^{- 1} μ σ | = ϒ_{i}$ , where $0 < ϒ_{i} < \infty$ , yeilds

\begin{array}{l} {\dot{V}}_{i 2} \leq S_{i}^{T} [- d i a g {ρ ({| {\hat{z}}_{i 2} |}^{r_{2} - 1})} (α s i g^{r_{3}} (S_{i}) + β s i g^{r_{4}} (S_{i}))] + Υ_{i} \\ \leq - α \sum_{m = 1}^{3} ρ ({| {\hat{z}}_{i 2 m} |}^{r_{2} - 1}) {| S_{i m} |}^{1 + r_{3}} - β \sum_{m = 1}^{3} ρ ({| {\hat{z}}_{i 2 m} |}^{r_{2} - 1}) \\ {| S_{i m} |}^{1 + r_{4}} + Υ_{i} \end{array}

(57)

According to Lemma 2, we have

{\overset{\cdot}{V}}_{i 2} \leq - c_{1} V_{i 2}^{\frac{1 + r_{3}}{2}} - c_{2} V_{i 2}^{\frac{1 + r_{4}}{2}} + ϒ_{i}

(58)

where $c_{1} = \min {α ρ ({| {\hat{z}}_{i 2 m} |}^{r_{2} - 1}) 2^{\frac{1 + r_{3}}{2}}}$ and $c_{2} = \min {β ρ ({| {\hat{z}}_{i 2 m} |}^{r_{2} - 1}) 2^{\frac{1 + r_{4}}{2}} 3^{\frac{1 - r_{4}}{2}}}$ .

To show that the system states are practical fixed-time stability, the state space $z_{i 1}$ and ${\hat{z}}_{i 2}$ are divided into the following two areas

Π_{1} = {(z_{i 1 m}, {\hat{z}}_{i 2 m}) | {| {\hat{z}}_{i 2 m} |}^{r_{2} - 1} \geq ι}

(59)

Π_{2} = {(z_{i 1 m}, {\hat{z}}_{i 2 m}) | {| {\hat{z}}_{i 2 m} |}^{r_{2} - 1} < ι}

(60)

Then, the following analysis will be divided into two cases.

Case 1: For the case of $z_{i 1}$ and ${\hat{z}}_{i 2}$ in the region $Π_{1}$ , it follows from equation (45) that $ρ ({| {\hat{z}}_{i 2 m} |}^{r_{2} - 1}) = 1$ . Based on Lemma 2, the system states are practical fixed-time stability. Moreover, the system states convergent region can be given as

W = {lim_{t \to T_{3}} S_{i} | V_{i 2} \leq min {c_{1}^{- \frac{2}{1 + r_{3}}} {(\frac{ϒ_{i}}{1 - θ})}^{\frac{2}{1 + r_{3}}}, c_{2}^{- \frac{2}{1 + r_{4}}} {(\frac{ϒ_{i}}{1 - θ})}^{\frac{2}{1 + r_{4}}}}}

(61)

The time $T_{3}$ is bounded by

T_{3} \leq \frac{1}{c_{1} θ (1 - \frac{1 + r_{3}}{2})} + \frac{1}{c_{2} θ (\frac{1 + r_{4}}{2} - 1)}

(62)

where $θ$ is a scalar and satisfies $0 < θ \leq 1$ .

Case 2: For the case of $z_{i 1}$ and ${\hat{z}}_{i 2}$ in the region $Π_{2}$ , we will show that $Π_{2}$ is not an attractor except the origin. Substituting equation (47) into equation (39), we have

{\overset{\cdot}{\hat{z}}}_{i 2} = - \frac{π}{2 ι r_{2}} (α si g^{r_{3}} (S_{i}) + β si g^{r_{4}} (S_{i})) - R_{i} M_{i}^{- 1} μ σ

(63)

Thus, with the sufficient small parameter $ι$ selected, we have ${\overset{\cdot}{\hat{z}}}_{i 2} S_{i} < 0$ , which implies that $Π_{2}$ is not an attractor and the system states will leave the region $Π_{2}$ in a very short time (Huang and Jia, 2018). Thus, based on the analyses in Cases 1 and 2, it is concluded that the system states will converge into the region $W$ within a fixed-time $T_{3}$ . Therefore, the total convergence time is bounded by $T \leq T_{2} + T_{3}$ . This completes the proof of Theorem 4.

Theorem 5: The Zeno behavior can be avoided under event-triggered mechanisms (47) and (48), and the implementation intervals $t_{(k + 1) m} - t_{km} (m = 1, 2, 3)$ are lower bounded by a positive constant $t^{*}$ .

Proof: According to equation (48), one has that

\frac{d}{dt} | e_{im} (t) | = sign (e_{im}) {\overset{\cdot}{e}}_{im} \leq | {\overset{\cdot}{ω}}_{τ im} |, \forall t \in [t_{km}, t_{(k + 1) m})

(64)

where ${\overset{\cdot}{ω}}_{τ im}$ is the derivative of $ω_{τ im}$ because all signals in the closed-loop system are bounded. Therefore, there is a positive constant $\nabla_{im}$ satisfying $| {\overset{\cdot}{ω}}_{τ im} | \leq \nabla_{im}$ . Since

e_{im} (t_{km}) = 0, lim_{t \to t_{(k + 1) m}} | e_{im} (t) | = ς_{im}

(65)

there exists the lower bound of implementation interval $t^{*}$ and $t^{*} \geq ς_{im} / \nabla_{im}$ , Hence, no Zeno phenomenon will occur by the proposed control law.

Simulation results

In this section, simulation results are shown to verify the proposed fixed-time event-triggered sliding mode control method, and the vessel parameters are chosen from Skjetne et al. (2005). The time-varying environmental disturbances are modeled as $τ_{id} = R_{i}^{T} b_{i}$ with the first-order Markov process ${\overset{\cdot}{b}}_{id} + Ψ_{id} b_{id} = d_{i}$ , where $Ψ_{id} \in R^{3 \times 3}$ is a positive constant and $d_{i} \in R^{3}$ is a Gaussian white noise. Set the initial value of environmental disturbances as $b_{id} (0) = {[0, 0, 0]}^{T}$ , $Ψ_{id} = diag {0.001, 0.001, 0.001}$ and $d_{i} = {[0.5, 0.5, 0.2]}^{T}$ . Figure 3 shows the directed communication topology. In the figure, the 0 represents the virtual leader, and 1, 2, 3, 4, and 5 represent five follower vessels. The reference trajectory $η_{d} = {[x_{d}, y_{d}, ψ_{d}]}^{T}$ . The desired reference trajectory signal is $x_{d} = 0.06 t, y_{d} = 12 \sin (0.005 t)$ . Set the initial position of five follower vessels as $η_{1} = {[- 1.4, 0.2, \frac{π}{8}]}^{T}$ , $η_{2} = {[0.1, 1, \frac{π}{8}]}^{T}$ , $η_{3} = {[0.2, 0.1, \frac{π}{8}]}^{T}$ , $η_{4} = {[1.3, 0.2, \frac{π}{8}]}^{T}$ , and $η_{5} = {[0.1, - 1.5, \frac{π}{8}]}^{T}$ , respectively, and choose the desired deviation of five follower vessels as $ϑ_{1} = {[1.2, 0, 0]}^{T}$ , $ϑ_{2} = {[0, 1.2, 0]}^{T}$ , $ϑ_{3} = {[0, 0, 0]}^{T}$ , $ϑ_{4} = {[- 1.2, 0, 0]}^{T}$ and $ϑ_{5} = {[0, - 1.2, 0]}^{T}$ , respectively. The initial velocity $υ_{i} (0)$ , velocity estimate ${\hat{υ}}_{i} (0)$ , and position estimate ${\hat{η}}_{i} (0)$ of MSVs are chosen as ${[0, 0, 0]}^{T}$ . The total time of the simulation runs is 600 seconds, and the sampling time is 0.01 second.

The parameters of FxESO are chosen as $ε_{1} = μ_{1} = 40$ , $ε_{2} = μ_{2} = 150$ , $ε_{3} = μ_{3} = 100$ , $α_{1} = 0.8$ , $α_{2} = 0.6$ , $α_{3} = 0.4$ , $β_{1} = 1.2$ , $β_{2} = 1.4$ , and $β_{3} = 1.6$ , respectively. The parameters of the fixed-time auxiliary dynamic system are set as $r_{01} = 5 / 4$ , $r_{02} = 4 / 5$ , $A = diag {5, 6, 5}$ and $B = diag {2, 2, 5}$ . The actuator input limitation is set as $| τ_{i} | \leq [2, 2, 1.5]^{T}$ . The parameters of FxNTSMM and controller are chosen as $r_{1} = 13 / 11$ , $r_{2} = 15 / 13$ , $r_{3} = 0.7$ , $r_{4} = 1.8$ , $Θ = 0.1$ , $ι = 0.1$ , $a_{s} = 2$ , $b_{s} = 13 / 11$ , $α = β = 1$ , $σ_{1} = 0.2$ , $σ_{2} = 0.2$ , and $σ_{3} = 0.15$ .

Remark 8: According to equation (21), the nonlinear term $μ_{m} si g^{α_{m}} ({\tilde{e}}_{im}), 0 < α_{m} < 1, (m = 1, 2, 3)$ is regarded as the dominating term when estimation errors are at near zero. Correspondingly, the nonlinear term $ε_{m} si g^{β_{m}} ({\tilde{e}}_{im}), β_{m} > 1, (m = 1, 2, 3)$ is regarded as the dominating term when estimation errors are at far zero. $μ_{1}, μ_{2}, μ_{3}$ are the observer gains, which affect the estimation performance of the estimation errors of positions, velocities, and lumped disturbances, respectively. According to equation (29), the observer gain $μ_{m}, (m = 1, 2, 3)$ is related to the convergence time of the observer. The smaller $μ_{m}, (m = 1, 2, 3)$ is, the longer the convergence time and the worse the estimated performance. On the contrary, the larger $μ_{m}, (m = 1, 2, 3)$ is, the shorter the convergence time, but if it is too large, the estimation error will oscillate. $ε_{1}, ε_{2}, ε_{3}$ are the same as $μ_{1}, μ_{2}, μ_{3}$ . According to equation (29), parameters $α_{m}, (m = 1, 2, 3)$ and $β_{m}, (m = 1, 2, 3)$ are also related to the convergence time. The larger $α_{m}, (m = 1, 2, 3)$ is, the longer the convergence time and the worse the estimated performance. On the contrary, the smaller $α_{m}, (m = 1, 2, 3)$ is, the shorter the convergence time, but if it is too small, the estimation error will oscillate. The larger $β_{m}, (m = 1, 2, 3)$ is, the shorter the convergence time, but if it is too large, the estimation error will oscillate. On the contrary, the smaller $β_{m}, (m = 1, 2, 3)$ is, the longer the convergence time and the worse the estimated performance.

Remark 9: For the parameters of the fixed-time auxiliary dynamic system, according to equation (33), the nonlinear term $Asi g^{r_{o 1}} (ξ_{i}), 0 < r_{o 1} < 1$ is regarded as the dominating term when the auxiliary dynamic variable is at near zero, correspondingly, the nonlinear term $Bsi g^{r_{o 2}} (ξ_{i}), r_{o 2} > 1$ is regarded as the dominating term when the auxiliary dynamic variable is at far zero. $A$ and $B$ are the auxiliary dynamic system gains, which affect the auxiliary dynamic system performance. The method of adjusting parameters is similar to Remark 8.

Remark 10: According to equation (44), the nonlinear term $α si g^{r_{3}} (S_{i}), 0 < r_{3} < 1$ is regarded as the dominating term when the sliding mode manifold is at near zero, correspondingly, the nonlinear term $β si g^{r_{4}} (S_{i})), r_{4} > 1$ is regarded as the dominating term when the sliding mode manifold is at far zero. $α$ and $β$ are the controller gains, which affect the convergence performance of the sliding mode manifold. The method of adjusting parameters is similar to Remark 8. The event-triggered parameters $σ_{1}, σ_{2}, σ_{3}$ should be set as large as possible without affecting the system convergence time, dynamic performance, and steady-state errors, so as to ensure the actuators’ action frequency as low as possible.

The estimated performance of FxESO scheme:

To demonstrate FxESO’s estimated performance, this paper compares FxESO with FTESO and ESO, respectively. We take advantage of controller (44) proposed in this paper. FTESO and ESO are designed as follows

{\begin{matrix} {\overset{\cdot}{\hat{η}}}_{i} = {\hat{w}}_{i} + μ_{1} si g^{α_{1}} ({\tilde{η}}_{i}) \\ {\overset{\cdot}{\hat{w}}}_{i} = {\hat{Γ}}_{i} + μ_{2} si g^{α_{2}} ({\tilde{η}}_{i}) + R_{i} M_{i}^{- 1} τ_{i} \\ {\overset{\cdot}{\hat{Γ}}}_{i} = μ_{3} si g^{α_{3}} ({\tilde{η}}_{i}) \end{matrix}

(66)

where $μ_{1} = 40$ , $μ_{2} = 150$ , $μ_{3} = 100$ , $α_{1} = 0.8$ , $α_{2} = 0.8$ , and $α_{3} = 0.4$

{\begin{matrix} {\overset{\cdot}{\hat{η}}}_{i} = {\hat{w}}_{i} + μ_{1} {\tilde{η}}_{i} \\ {\overset{\cdot}{\hat{w}}}_{i} = {\hat{Γ}}_{i} + μ_{2} {\tilde{η}}_{i} + R_{i} M_{i}^{- 1} τ_{i} \\ {\overset{\cdot}{\hat{Γ}}}_{i} = μ_{3} {\tilde{η}}_{i} \end{matrix}

(67)

where $μ_{1} = 40$ , $μ_{2} = 150$ , and $μ_{3} = 100$ .

To display the simulation results more clearly, we chose the first vessel to compare the performance of the observer. Figures 4 and 5 show the simulation results of the estimated performance of FxESO compared to FTESO and ESO, respectively. Figure 4 shows the estimation performance of lumped disturbances estimated by three observers, where “actual” represents the actual lumped disturbances in the legend. It is clear that the lumped disturbances can be quickly identified by FxESO, and the local zooms show that FxESO’s estimation performance is superior to FTESO and ESO. Figure 5 shows the estimation errors of lumped disturbances of the three observers, where $Γ_{e 1} = Γ_{1} - {\hat{Γ}}_{1}$ . It can be known that compared with FTESO and ESO, FxESO has the fastest convergence performance and the best estimated performance. Figure 6 shows the velocity estimation error of the first vessel. It can be seen that FxESO’s velocity estimation error converges to zero within 1 second, and its estimation performance and convergence speed are far superior to FTESO and ESO. Figure 7 shows the cooperative tracking of reference signals by MSVs under model parameter uncertainty, time-varying marine environmental disturbance. Figure 8 demonstrates tracking errors of MSVs tracking reference trajectory. As can be seen from Figures 7 and 8, although the initial positions of the five vessels were different, after quick adjustment, the MSVs were able to track the reference trajectory while they maintained the desired relative positions. Figure 9 shows the control input for MSVs.

The proposed controller performance comparisons with Liang et al. (2020):

To better demonstrate the superiority of the proposed FxNTSMM and controller, we assume that lumped disturbances and velocity are known, and the event-triggered mechanism of the controller is canceled. Equations (41), (42), and (44)–(46) are rewritten as

S_{i} = β (z_{i 1}) z_{i 1} + si g^{r_{2}} (z_{i 2})

(68)

β_{m} (z_{i 1 m}) = {\begin{matrix} {(a_{s} {| z_{i 1 m} |}^{r_{1} - \frac{1}{r_{2}}} + b_{s})}^{r_{2}}, & | z_{i 1 m} | \geq Θ \\ (a_{s 1} {| z_{i 1 m} |}^{r_{1} - \frac{1}{r_{2}}} + a_{s 2} {| z_{i 1 m} |}^{2 - \frac{1}{r_{2}}} + b_{s}), & | z_{i 1 m} | < Θ \end{matrix}

(69)

\begin{matrix} τ_{i} = - M_{i} R_{i}^{- 1} \\ [\begin{matrix} \frac{1}{r_{2}} (\tilde{β} (z_{i 1}) + β (z_{i 1})) si g^{2 - r_{2}} (z_{i 2}) + \frac{1}{r_{2}} diag {ρ ({| z_{i 2} |}^{r_{2} - 1})} \\ diag {{| z_{i 2} |}^{1 - r_{2}}} (α si g^{r_{3}} (S_{i}) + β si g^{r_{4}} (S_{i})) + Γ_{i} - \sum_{j \in N_{i}} a_{ij} {\overset{\cdot}{w}}_{j} \\ - a_{i 0} {\overset{\cdot}{w}}_{d} + Asi g^{r_{o 1}} (ξ_{i}) + Bsi g^{r_{o 2}} (ξ_{i}) + {\bar{Δ}}_{i} G_{i} sign (ξ_{i}) \end{matrix}] \end{matrix}

(70)

ρ ({| z_{i 2 m} |}^{r_{2} - 1}) = {\begin{matrix} \sin (\frac{π}{2} \frac{{| z_{i 2 m} |}^{r_{2} - 1}}{ι}), {| z_{i 2 m} |}^{r_{2} - 1} < ι \\ 1, {| z_{i 2 m} |}^{r_{2} - 1} \geq ι \end{matrix}

(71)

{\tilde{β}}_{m} (z_{i 1 m}) = {\begin{matrix} r_{2} a_{s} (r_{1} - \frac{1}{r_{2}}) (a_{s} {| z_{i 1 m} |}^{r_{1} - \frac{1}{r_{2}}} + b_{s})^{r_{2} - 1} {| z_{i 1 m} |}^{r_{1} - \frac{1}{r_{2}}}, | z_{i 1 m} | \geq Θ \\ r_{2} (a_{s 1} {| z_{i 1 m} |}^{1 - \frac{1}{r_{2}}} + a_{s 2} {| z_{i 1 m} |}^{2 - \frac{1}{r_{2}}} + b_{s})^{r_{2} - 1} \\ {a_{s 1} (1 - \frac{1}{r_{2}}) {| z_{i 1 m} |}^{1 - \frac{1}{r_{2}}} + a_{s 2} (2 - \frac{1}{r_{2}}) {| z_{i 1 m} |}^{2 - \frac{1}{r_{2}}}}, | z_{i 1 m} | < Θ \end{matrix}

(72)

Then, the finite-time non-singular terminal sliding mode manifold (FTNTSMM) and the controller in Liang et al. (2020) are constructed as follows

S_{i} = z_{i 2} + k β (z_{i 1})

(73)

β (z_{i 1}) = {\begin{matrix} si g^{r_{1}} (z_{i 1}) if {\bar{S}}_{i} = 0 or {\bar{S}}_{i} \neq 0, z_{i 1} \geq Θ \\ κ_{1} z_{i 1} + κ_{2} si g^{2} (z_{i 1}) if {\bar{S}}_{i} \neq 0, z_{i 1} \leq Θ \end{matrix}

(74)

{\bar{S}}_{i} = z_{i 2} + k si g^{r_{1}} (z_{i 1})

(75)

\begin{matrix} τ_{i} = - M_{i} R_{i}^{- 1} [k \overset{\cdot}{β} (z_{i 1}) + k_{1} S_{i} + k_{2} si g^{r_{1}} (S_{i}) + Γ_{i} \\ - \sum_{j \in N_{i}} a_{ij} {\overset{\cdot}{w}}_{j} - a_{i 0} {\overset{\cdot}{w}}_{d} + A ξ_{i} + B {ξ_{i}}^{r_{o}} + {\bar{Δ}}_{i} G_{i} sign (ξ_{i})] \end{matrix}

(76)

where the parameters of the FTNTSMM and the controller are the same as the Liang et al. (2020).

Figures 10 –13 show the simulation results of the performance of the proposed controller and the comparative controller. Figure 10 shows the initial simulation comparison of the FxNTSMM and the FTNTSMM of the first vessel. It is seen that the convergence rate of the FxNTSMM is obviously faster than that of the FTNTSMM. Figure 11 describes the initial tracking errors of two control schemes tracking reference trajectory of the first vessel. It is known that the two control schemes can track the reference trajectory well, and the proposed control scheme has faster convergence and less oscillation. Figures 12 and 13 show the FxNTSMC law and the finite-time non-singular terminal sliding mode control law, respectively.

The proposed controller performance comparisons with Liang et al. (2020):

To verify the effectiveness of the controller with actuator saturation based on event-triggered strategy, we compare it with non-event-triggered strategy in this paper. Figure 14 shows the event-triggered time interval for the control input of first vessel. The horizontal axis denotes the moment of triggering, and the vertical axis represents the duration of the trigger controller. The length of the trigger interval determines the frequency of controller updates. The lower the frequency is, the less the action times and communication times of the controller. Figures 15 and 16 show the tracking errors of the two strategies, respectively. It is known that introducing event-triggered strategy does not affect its tracking performance compared with non-event-triggered strategy. In addition, to more clearly verify that the event-triggered strategy can reduce the execution rate of actuator and save communication resources without affecting the convergence performance, the results of the action times of three components of event-triggered strategy and non-event-triggered strategy for the five vessels are given in Table 1. The results show that the event-triggered controller has the advantages of reducing the update number of controller and reducing the mechanical loss of actuator, and there is no Zeno behavior.

Table 1.

The comparisons of event-triggered strategy and non-event-triggered strategy.

Vessel number	Event-triggered strategy	Non-event-triggered strategy
Vessel 1	5207,4918,2906	60000,60000,60000
Vessel 2	4763,4612,3005	60000,60000,60000
Vessel 3	4886,4517,2891	60000,60000,60000
Vessel 4	4996,4983,2987	60000,60000,60000
Vessel 5	4772,5295,3112	60000,60000,60000

Figure 17 shows the control inputs of MSVs with saturation based on event-triggered. Figure 18 presents the auxiliary dynamic variables of auxiliary dynamic system for MSVs. Due to the system needs to ensure fixed-time convergence, the actuator is saturated at the initial moment because of the need for a large torque. When the actuator is saturated, the controller can guarantee good performance. Figure 19 describes the estimation errors of MSVs of lumped disturbances. It can be seen that the estimation performance of lumped disturbances of MSVs is not affected under input saturation being considered. In addition, Figure 20 depicts the tracking errors of MSVs based on event-triggered strategy without saturation. The comparison with Figure 15 shows that the convergence performance of tracking error of MSVs is not affected under input saturation being considered. Figure 21 presents the trajectories of MSVs under cooperative control based on event-triggered strategy with saturation.

Conclusion

In this paper, a novel fixed-time sliding mode control method is proposed to solve the cooperative control problem of MSVs. The FxESO is designed to improve the robustness of the system against model uncertainties, unknown velocities, and unknown external disturbances. A novel fixed-time auxiliary dynamic system is designed to deal with the actuator saturation. An event-triggered distributed controller with anti-saturation ability is proposed to make the cooperative control system stable within a fixed time, and the control performance is superior. Finally, simulation results of 3-DOF fully actuated MSVs are presented to verify the effectiveness of the proposed controller. For the future work, the fixed-time output feedback cooperative control subject to actuator faults, sampling step, small delays, and sampling noise for MSVs will be considered.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the 7th Generation Ultra Deep Water Drilling Unit Innovation Project.

ORCID iD

Zheda Ren

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