Robust adaptive neural network integrated fault-tolerant control for underactuated surface vessels with finite-time convergence and event-triggered inputs

1.   Introduction

With the continuous development of the marine economy, intelligent unmanned surface vessels have received special attention. To further improve the degree of automation of USVs, many studies focus on improving the tracking performance of the vessel control system ^[1,2,3]. Trajectory tracking control is a very typical application scenario, which is often used to verify the effectiveness of control schemes ^[4,5,6]. And in actual engineering, many engineering tasks are unavoidable to avoid making the vessels track the predetermined trajectory, such as oil exploration, submarine cable laying, etc. The tracking performance of the vessel is usually limited by the following factors:

1) Internal uncertain dynamics and external unmeasurable unknown environmental disturbances lead to the failure to obtain accurate model information in the control scheme design.

2) For USVs performing missions in extreme environments, frequent actuator actions, as well as the physical limitations of propulsion and steering devices, lead to actuator failures, which seriously affect the tracking performance of the system.

The USVs trajectory tracking system has typical non-holonomic constraint characteristics. Under the constraints physical, constraints of the actuator, the design of the control scheme needs to consider more factors. (Pettersen et al. ^[7] and Do ^[8]) respectively used differential homeomorphism and additional control methods to solve the underactuation problem. The method in ^[7] converted the mathematical model of USVs into two chained subsystems through changes and stabilized the system errors indirectly by designing control laws. In ^[8], by constructing a virtual transverse drive vector, the virtual control variables of the original propulsion system were decomposed through a coordinate transformation to achieve the purpose of applying part of the longitudinal drive to the transverse direction. However, both were control schemes designed based on accurate models. In other words, prior knowledge of the model must be obtained in advance. (Zhang et al. ^[9]) carried out the state transition for the system and considered the uncertain terms in the model. However, this method imposes a norm limit on transverse and longitudinal errors in the design process, and the implicit assumption is that the heading error must be less than 0.5 PI. In ^[10], new kinematic and dynamic equations were obtained by using the definition of hands, and the control scheme was designed by combining the vector method. However, both the underactuated transformation and the NNs algorithm used by the system may bring huge computational load to the system.

In practice, the vessel's control system cannot execute arbitrary control instructions. This is caused by the saturation limit of the actuator. The steady state of the control system is challenged by the non-smooth property of the saturation function. In numerous studies, there are two main types of saturation. One method is to use the strong robustness of the control system to force the system error to stabilize. The other is to adopt the active compensation method. For example, auxiliary system design, smooth function approximation, and other methods. In ^[11], the smooth property of the Gaussian error function was used to replace the non-linearity of the saturation function. However, the control scheme is designed on the basis of a fully actuated system. In addition, (Zheng ^[12]) used a hyperbolic tangent function to replace the nonlinearity caused by the non-smooth saturation function, but this method increases the complexity of the control scheme design.

In the actual ocean voyage, due to the limitation of modeling technology and the unmeasurable ocean environment, the tracking system cannot obtain the model dynamic parameters and the prior information of unknown external disturbances in advance. Therefore, the vessel will inevitably be affected by internal and external uncertainties. In view of the dynamic uncertainty of the USVs, it is a very popular way to use intelligent control algorithms such as fuzzy or NNs to reconstruct the vessel. (Zhou et al.^[13] and Kong et al. ^[14]) designed the control scheme combined with the adaptive theory without obtaining the prior information of the model. (Huang et al. ^[15]) combined with a disturbance observer to compensate for the unmeasurable disturbances and further improved the tracking accuracy on the basis of ^[13] and ^[14]. Although the control effect of an intelligent algorithm is very considerable, we cannot ignore the huge load faced by the system. In other words, these burdens are unbearable in real control systems. In order to solve this problem, (Zhang et al. ^[16]) converted the uncertainty of the system into a single parameter form by combining the minimum learning parameter, which effectively reduced the computational pressure of the system. However, the norm calculation method tended to be conservative obviously, that is to say, we must inevitably sacrifice part of the tracking accuracy. To solve this problem, a nonlinear function was introduced in ^[17] to filter the error through nonlinear feedback.

For a USVs control system in normal operation, the stability of the system highly depends on the normal operation of the actuator. However, when the USVs are operating, harsh sea conditions are inevitable and the equipment is often subjected to seawater erosion. These adverse conditions may become potential failure factors for the actuator. Therefore, developing a fault-tolerant control scheme will undoubtedly improve the reliability of the vessel control system. In dynamic systems, adaptive control is a common method compensating for the fault factors acting on the actuator. (Cai et al. ^[18]) and (Tang et al. ^[19]) designed fault-tolerant control schemes for strictly nonlinear systems and multi-input multi-output systems, respectively. (Wang er al.^[20] and Zheng et al.^[21]) had applied the submethod in a marine surface vessel control system. However, reference ^[21] only considered bias faults.

References ^{[18,19,20,21]} are all based on time-triggered control strategies, which will lead to broadband congestion, aggravation of actuator wear, and other problems. On the contrary, an event-triggered policy transfers control commands to the actuator only if the trigger condition is violated. Avoid the huge communication burden brought about by the vessel control system in periodic control mode. At the same time, the actuator does not have to repeatedly receive control commands, so as to effectively reduce the failure rate of the equipment. Control schemes based on event-triggered control design had been widely used in trajectory-tracking control ^[22,23,24]. (Xing et al. ^[25]) designed a novel fault-tolerant control scheme based on an event trigger mechanism. Subsequently, (Zhang et al. ^[26] and Zhu et al.^[27]) had done similar research on course-keeping control and trajectory-tracking control, respectively. All of them effectively compensate for the uncertain effect of fault factors in the system. However, these control schemes only achieved convergence when the system time tends to infinity. FTC can effectively improve the transient response steady-state performance of the system, which cannot be ignored. Many authors have proved this conclusion ^{[28,29,30,31,32]}. If the idea of FTC is incorporated into the design of a fault-tolerant control scheme, the tracking accuracy of the system will be further guaranteed. Inspired by the above references, this paper designs a novel fault-tolerant control scheme. The specific contributions of this paper are as follows:

1) In this study, we consider both internal and external uncertain dynamics. Reconstructing the dynamic uncertainty with RBFNNs combined with MLPs reduces the computational load of the system. It is worth noting that the proposed control scheme does not require real-time updates of the neural network weights.

2) Without prior knowledge of the model, the effects of input saturation limits and actuator faults are considered. The relative threshold event trigger strategy is adopted to reduce the frequent update of control commands and reduce actuator wear. The unknown factors including bias fault factors and partial failure fault factors, as well as other unknown factors of the system are dynamically transformed into a single-parameter linearized form, which is compensated by an adaptive parameter updated online.

2.   Problem formulation and preliminaries

In general, the 3-DOF mathematical model of USV tracking control can be expressed in the following form ^[22,33]:

where $x$, $y$, $\psi$ represent the position and heading angle of the USV in the geodetic coordinate system, respectively. $u$, $v$, $r$ represent the surge velocity, sway velocity and yaw velocity, respectively. ${f_u}\left({u, v, r} \right)$, ${f_v}\left({u, v, r} \right)$, ${f_r}\left({u, v, r} \right)$ are the non-linear dynamics, respectively. ${m_i}$, $\left({i = u, v, r} \right)$ are the inertial mass. ${d_u}$, ${d_v}$, ${d_r}$ represent the unmeasurable unknown disturbances, respectively. $\tau_{_u}^f$, $\tau_{_r}^f$ are the surge control force and the yaw control force due to actuator failure, respectively. In this work, we focus on the Loss-of-effectiveness (LOE) and bias fault, the specific form is as follows:

if ${\varrho _i} = 1$, ${\sigma _i} = 0$, it means that the unmanned ship system is fault-free. If $0 < {\varrho _i} < 1$, ${\sigma _i} = 0$, it means that the actuator is suffering from a LOE fault. If ${\varrho _i} = 1$, $0 < {\sigma _i} < 1$, it means that the actuator is suffering from a bias fault.

In engineering practice, ship actuators inevitably suffer from physical limitations, which make the control inputs $\tau_{_u}^f$, $\tau_{_r}^f$ affected by input saturation, which are described as follows ^[27]:

In view of the design of control law, the following assumptions are made and relevant definitions and theorems are introduced:

Assumption 1. ${f_i}\left({u, v, r} \right)$, $\left({i = u, v, r} \right)$ are unknown. The external disturbance ${d_i}$, $\left({i = u, v, r} \right)$ unknown and bounded. So there are unknown positive constants ${\sigma _u}$, ${\sigma _v}$, ${\sigma _r}$ such that ${d_u}$, ${d_v}$, ${d_r}$ satisfy $\left| {{d_u}} \right| \le {\sigma _u}$, $\left| {{d_u}} \right| \le {\sigma _v}$, $\left| {{d_u}} \right| \le {\sigma _r}$.

Definition 1. ^[34,35] Nonlinear control systems are described as follows system (2.6)

where $x \in {R^n}$ is the state variable of the system, ${\Omega _0}$ is a spherical domain containing the origin, and $f\left(x \right)$ is a continuous function. For any initial condition ${x_0}$, if there is a constant $\Im > 0$ and a regulating time function $0 < T\left({{x_o}} \right) < \infty$ such that $\left\| {x\left(t \right)} \right\| \le \Im$, $t \ge T\left({{x_0}} \right)$, then the system (2.6) can be said to be semi-globally practical finite-time stable.

Lemma 1. ^[36] For the nonlinear system (2.6), assuming that there is a positive definite Lyapunov function $V\left(x \right)$: ${\Omega _0} \to R$ and any scalar $a > 0$, $b > 0$ and $0 < \kappa < 1$ such that the inequality $\dot V(x) + aV(x) + b{V^\kappa }(x) \le 0$ holds, the system (2.6) is stable in finite time, and its adjustment time satisfies:

where $V\left({{x_0}} \right)$ is the initial value of $V\left({{x}} \right)$.

Lemma 2. ^[37,38] For any given continuous smooth function defined on the compact set $\Omega \subset {R^n}$

where $\varepsilon$ is the approximation error, and for all $x \in \Omega$, here is a vector ${\varepsilon ^*} > 0$, and $\left| \varepsilon \right| \le {\varepsilon ^*}$ is satisfied. $s\left(x \right)$ is the NN basis function, which is represented by a Gaussian function, as follows:

where ${{\bf{c}}_i}$ is the center vector, and ${\omega _i}$ is the width of the Gaussian function. ${W^*}$ is the weight vector under ideal conditions. Usually the ideal neural network weight vector is an unknown vector and needs to be estimated. It can be understood that $\left| \varepsilon \right|$ can minimize ${\omega}$ on $x \in \Omega \subset {R^n}$, that is

Assumption 2. In the compact set ${\Omega _x} \subset {R^n}$, the weight ${W^*}$ of the RBFNNs used to approximate the unknown vector is bounded, that is, $\left\| {{W^*}} \right\| \le {W_M}$, where ${W_M}$ is a positive constant.

Lemma 3. ^[39] For any constant $\varsigma > 0$ and any scalar $\rho \in R$, inequality (2.11) holds

Lemma 4. ^[40] For any $a > 0$ and $x \in R$, the following relation is satisfied.

3.   Control design and stability analysis

First, define the tracking error as follows

where ${x^*}$, ${y^*}$, are the reference position of the USV.

Taking the derivation of Eq (3.1), one can get

where ${g_u}\left(\psi \right) = \left[ \begin{array}{l} \cos \left(\psi \right)\\ \sin \left(\psi \right) \end{array} \right]$, ${g_v}\left(\psi \right) = \left[ \begin{array}{l} - \sin \left(\psi \right)\\ \cos \left(\psi \right) \end{array} \right]$.

The virtual control law is designed to stabilize the error ${z_e}$ as follows:

where $k_{11}$, $k_{12}$ are positive definite parameters.

It is not difficult to obtain the relationship in Eq (3.4) by further calculating Eq (3.3)

where ${u^*}$ is the reference surge velocity, ${\psi ^*}$ is the reference heading angle.

The heading angle error is defined as follows:

Taking the derivation of Eq (3.5), we can get

To stabilize the error ${\psi _e}$ a virtual control variables of the following form is designed:

where $k_{31}$ and $k_{32}$ are positive definite parameters.

Filter ${u^*}$ and ${r^*}$ by using dynamic stability control (DSC) technology

where the dynamic surface error ${e_{f, u}} = {\beta _u} - {u_d}$, ${e_{f, r}} = {\beta _r} - {r_d}$.

According to Eqs (3.2)–(3.8), it can be obtained

where ${\Delta _e} = u{g_u}\left(\psi \right) - {u^*}{g_u}\left({{\psi ^*}} \right)$.

The velocity error variable is defined as follows

Taking the derivation of Eq (3.10), one can get

Since ${f_u}\left({u, v, r} \right)$ and ${f_r}\left({u, v, r} \right)$ are unknown, they cannot be used in the design of the controller. Therefore, the RBFNNs are used to approximate the unknown nonlinear functions.

where ${W_u}$ and ${W_r}$ are the neural network weights. ${\sigma_u}\left(\eta \right)$ and ${\sigma_r}\left(\eta \right)$ are the NN functions. ${\varepsilon _u}$ and ${\varepsilon _r}$ are approximation errors. Substituting Eq (3.12) into Eq (3.11), one can get

Next, by combining the idea of the relative threshold event triggering mechanism, the following dynamic event triggering mechanism is designed ^[25]

where ${e_u}\left(t \right)$, ${e_r}\left(t \right)$ are the measurement errors, and ${\eta _{11}}$, ${\eta _{12}}$, ${\eta _{21}}$, ${\eta _{22}}$ re the positive definite design parameters. $t_k$ is the update time of the controller. During the time interval $t \in \left[ {{t_k}, {t_{k + 1}}} \right]$, the control input ${\omega _u}\left({{t_k}} \right)$ and ${\omega _r}\left({{t_k}} \right)$ remains unchanged. When the moment switches to ${t_{k + 1}}$, a new control command will be sent to the actuator.

Under this event triggering mechanism, $\left| {{\omega _u}\left(t \right) - {\tau _u}\left(t \right)} \right| \le {\eta _{12}}\left| {{\tau _u}\left(t \right)} \right| + {\eta _{11}}$ and $\left| {{\omega _r}\left(t \right) - {\tau _r}\left(t \right)} \right| \le {\eta _{22}}\left| {{\tau _r}\left(t \right)} \right| + {\eta _{21}}$ are satisfied within any time. From this, when $\left| {{\lambda _{u, 1}}\left(t \right)} \right| \le 1$, $\left| {{\lambda _{u, 2}}\left(t \right)} \right| \le 1$, $\left| {{\lambda _{r, 1}}\left(t \right)} \right| \le 1$, $\left| {{\lambda _{r, 2}}\left(t \right)} \right| \le 1$, it is not difficult to draw the following relationship

Further, the following relationship can be obtained

According to Eqs (3.14)–(3.17), the velocity error becomes

where $\frac{{{\varrho _u}}}{{1 + {\lambda _{u, 1}}\left(t \right){\eta _{12}}}}$ and $\frac{{{\varrho _r}}}{{1 + {\lambda _{r, 1}}\left(t \right){\eta _{22}}}}$ are nonlinear bounded functions with positive definite upper bound less than $1$.

Combined with the design idea of the robust adaptive method of depth information, using the RBFNNs and the MLPs, we can further obtain

where $\left\{ \begin{array}{l} {\wp _u} = \max \left\{ {\left\| {{W_u}^T} \right\|, \left| {{\varepsilon _u} + {d_u} - {m_u}{{\dot \beta }_u}} \right|} \right\}\\ {\wp _r} = \max \left\{ {\left\| {{W_r}^T} \right\|, \left| {{\varepsilon _r} + {d_r} - {m_r}{{\dot \beta }_r}} \right|} \right\} \end{array} \right.$, $\left\{ \begin{array}{l} {\zeta _u}\left(Z \right) = \left\| {{\sigma _u}\left(\eta \right)} \right\| + 1\\ {\zeta _r}\left(Z \right) = \left\| {{\sigma _r}\left(\eta \right)} \right\| + 1 \end{array} \right.$. With the transformation, the adaptive learning parameters are significantly reduced.

To stabilize the influence of the fault factor on the system, the following variables is designed ^[25,26]

where ${\delta _u} = \frac{{{\varrho _u}}}{{{m_u}\left({1 + {\lambda _{u, 1}}\left(t \right){\eta _{12}}} \right)}}$, ${\varpi _u} = \frac{{1 + {\lambda _{u, 1}}\left(t \right){\eta _{12}}}}{{{\varrho _u}}}\left({\frac{{{\varrho _u}{\lambda _{u, 2}}\left(t \right){\eta _{11}}}}{{1 + {\lambda _{u, 1}}\left(t \right){\eta _{12}}}} - {\sigma _u} + {\wp _u}{\zeta _u}\left(Z \right)} \right)$, ${\delta _r} = \frac{{{\varrho _r}}}{{{m_r}\left({1 + {\lambda _{r, 1}}\left(t \right){\eta _{22}}} \right)}}$, ${\varpi _r} = \frac{{1 + {\lambda _{r, 1}}\left(t \right){\eta _{22}}}}{{{\varrho _r}}}\left({\frac{{{\varrho _r}{\lambda _{r, 2}}\left(t \right){\eta _{21}}}}{{1 + {\lambda _{r, 1}}\left(t \right){\eta _{22}}}} - {\sigma _r} + {\wp _r}{\zeta _r}\left(Z \right)} \right)$.

Through analysis, the dynamic error equation of the system becomes

In view of this, the following control law is designed

The adaptive law as follows

Introduce the following Lyapunov function

where ${\tilde {\bar {\varpi}} _{i}} = {\bar \varpi _i} - {\hat {\bar {\varpi}} _{i}}$, $i = u, r$.

Then, derivation of Eq (3.24) can be obtained

According to Eqs (3.1)–(3.9), $\left| {{\beta _u} - {u^*}} \right| \le {\gamma _u}$, $\left| {{\beta _r} - {r^*}} \right| \le {\gamma _r}$ and Young's inequality, we can get

In view of Eqs (3.10)–(3.23), and from Lemma 4, we can get

Substitute Eqs (3.26)–(3.28) into Eq (3.24), and according to $0 < {\delta _i} < 1$, $i = u, r$, we can get

where ${\kappa _{11}} = \min \left\{ {\left({{k_{11}} - 0.1} \right), {k_{31}}} \right\}$, ${\kappa _{12}} = \min \left\{ {{k_{12}}, {k_{32}}} \right\}$, ${\kappa _{21}} = \min \left\{ {{\delta _u}{k_{21}}, {\delta _r}{k_{41}}} \right\}$, ${\kappa _{22}} = \min \left\{ {{\delta _u}{k_{22}}, {\delta _r}{k_{42}}} \right\}$.

According to Young's inequality, we can get

Substitute Eq (3.30) into Eq (3.29), and further obtain

where ${\rho _1} = \min \left\{ {2{\kappa _{11}}, 2{\kappa _{21}}, \frac{{{\imath _1}{c_1}}}{2}, \frac{{{\imath _2}{c_2}}}{2}} \right\}$, ${\rho _2} = {2^{\frac{1}{2}}}\min \left\{ {{\kappa _{12}}, {\kappa _{22}}, \frac{{{\imath _1}{c_1}}}{4}, \frac{{{\imath _2}{c_2}}}{4}} \right\}$, $\Theta = \frac{{{\imath _3}}}{2}{\bar \varpi _u}^2 + \frac{{{\imath _4}}}{2}{\bar \varpi _r}^2 + \frac{{{\gamma _u} + {\gamma _r}}}{2} + 0.2785\left({{\varepsilon _u}{{\bar \varpi }_u} + {\varepsilon _r}{{\bar \varpi }_r}} \right) + \frac{5}{2}{\Delta _e}^2 + \frac{{{\imath _1} + {\imath _2}}}{{16}}$.

According to Eq (3.31), one can get

where $0 < \ell < 1$.

According to Eq (3.32), if $V > \frac{\Theta }{{\ell {\rho _1}}}$, one can get

According to Lemma 1, it can be known that the system will stabilize to the region c in finite time, and the stabilization time is

where $V\left(0 \right)$ is the initial value of $V$.

According to the measurement error ${e_u} = {\omega _u}\left(t \right) - {\tau _u}\left(t \right)$, ${e_r} = {\omega _r}\left(t \right) - {\tau _r}\left(t \right)$, one can get

Since all the variables that make up ${\omega _u}\left(t \right)$, ${\omega _r}\left(t \right)$ are globally bounded, ${\omega _u}\left(t \right)$, ${\omega _r}\left(t \right)$ are continuous. Therefore, there must be positive definite constants ${\zeta _u}$, ${\zeta _r}$, satisfying the conditions $\left| {\dot \omega _u}\left(t \right) \right| \le {\zeta _u}$, $\left| {\dot \omega _r}\left(t \right) \right| \le {\zeta _r}$.When $t = {t_k}$, there are ${e_u}\left({{t_k}} \right) = 0$, ${e_r}\left({{t_k}} \right) = 0$, $\mathop {\lim }\limits_{t \to {t_{k + 1}}} {e_u}\left(t \right) = {\eta _{12}}\left| {{\tau _u}\left(t \right)} \right| + {\eta _{11}}$ and $\mathop {\lim }\limits_{t \to {t_{k + 1}}} {e_r}\left(t \right) = {\eta _{22}}\left| {{\tau _r}\left(t \right)} \right| + {\eta _{21}}$. Therefore, there must be time intervals ${t_u}^*$ and ${t_r}^*$ satisfying ${t_u}^* \ge \frac{{{\eta _{12}}\left| {{\tau _u}\left(t \right)} \right| - {\eta _{11}}}}{{{\zeta _u}}}$, ${t_r}^* \ge \frac{{{\eta _{22}}\left| {{\tau _r}\left(t \right)} \right| - {\eta _{21}}}}{{{\zeta _r}}}$, which can avoid Zeno behavior.

4.   Simulation

In this section, we select Cybership 2 model for simulation, and its parameters are detailed in ^[41]. In order to verify the effectiveness of the designed ETC trajectory tracking control scheme, the time-varying disturbance of Eq (4.1) is selected to simulate the external uncertain disturbance in the actual voyage, which is detailed as follows

In addition, the specific forms of LOE fault and bias fault are as follows

In order to quantitatively analyze the tracking performance of the control scheme designed in this paper, the integrated absolute error (IAE) and mean integrated absolute control (MIAC) of Eq (4.4) are used to evaluate the steady-state performance and energy consumption performance.

In this paper, two sets of simulation experiments are carried out under the circular reference trajectory and the trapezoidal reference trajectory.

4.1.   Simulation experiment 1

The circular reference trajectory is as follows

The controller parameters under the circular trajectory are shown in Table 1.

The data in Figures 1–10 and Table 2 show that the USVs system shows satisfactory tracking performance under the constraints of input saturation, actuator failure, communication congestion and other conditions. Under the two control schemes, the errors of the system are all within a reasonable range. First, under the constraints of actuator saturation, Figure 6 shows that the control inputs to the system are all within a reasonable range. According to the performance index data of MIAC in Table 2, the energy consumption of the ETC scheme is slightly higher than that of the continuous FTC scheme. However, within the simulation time, the controller updates for the continuous control scheme both are 20,000 times, compared to 2636 and 574 times for the ETC scheme. On the contrary, the ECT control scheme in this paper effectively reduces the frequency of updating of the controller instructions. Figures 4 and 5 show the tracking error trend of the system with time, respectively, which are bounded under the two fault-tolerant control schemes. According to the performance index given by IAE in Table 2, the tracking accuracy of the continuous FT fault-tolerant control scheme is slightly better than that of the ETC control scheme. Figure 9 shows the time interval of event triggering, we can clearly see that the control instruction does not trigger infinite times in a very short period of time. Figure 7 and 8 show that both the NN weights and the estimates of the uncertain terms are bounded.

4.2.   Simulation experiment 2

The trapezoidal reference trajectory is as follows

The controller parameters under the trapezoidal reference trajectory are shown in Table 3.

The simulation results under the trapezoidal trajectory are shown in Figures 11–20 and Table 4. Under the trapezoidal trajectory, the two fault-tolerant control schemes also show excellent tracking performance. Figures 14 and 15 show that the tracking error and velocity error of the system are within a small range. According to the performance indicators given by IAE in Table 4, we find that the tracking performance of the ETC control scheme is slightly better than that of the FTC control scheme. Figure 20 shows that under the FTC control scheme, the update of the controller are also 20,000 times, while the FTC are 4795 and 612 times. Furthermore, according to the performance index data in Table 4, the energy consumption of the ETC scheme is slightly higher than that of the FTC scheme. Figures 17 and 18 show that the estimates of NNs weights and uncertainty terms are bounded. The above data show that all signals in the closed-loop tracking system are bounded, and the proposed ETC control scheme effectively solves the problems of actuator failure, input saturation, and communication resource limitation.

In addition, the ETC scheme in this paper is essentially an active fault-tolerant control method, while the even FTC scheme is actually a passive fault-tolerant control method. FTC can improve the steady state of the system, and ETC can reduce the operating frequency of the controller. In the actual system, the introduction of ETC will inevitably sacrifice part of the control performance. However, the method of compensating fault factors by event-triggered mechanism in this paper finally obtains the control effect almost indistinguishable from the continuous FTC scheme, and effectively reduces the frequency of controller update and the frequency of actuator wear.

In summary, through the verification and analysis of two sets of simulation experiments, the ETC control scheme designed in this paper has demonstrated several superior performances in solving actuator faults, dealing with external interference, reducing the frequency of control commands and ensuring high-precision tracking effects. Moreover, the controller designed in this paper has a concise structure and is easier to apply in engineering.

5.   Conclusions

In this paper, a novel fault-tolerant control scheme is designed for USVs. By introducing MLPs technology, the uncertain dynamics, unknown interference, and fault factors of the system are converted into the form of single parameter. Since the controller involves only the adjustment of a single parameter, the structure of the controller in this paper is simple and easy to apply in practice. Upon non-respect of the response of the event-triggered mechanism, the communication resources occupied by the control scheme presented in this paper will be lower in the actual system. The theoretical analysis shows that the ETC control scheme designed in this paper ensures that all the error signals of the USVs system converge to a small set around the origin in a finite time, and all the signals in the tracking system are bounded.

Acknowledgments

The authors would like to acknowledge the National Natural Science Foundation of China (NSFC51779136), Science and Technology Commission of Shanghai Municipality (NO.20dz1206002).

Conflict of interest

The authors declare there are no conflict of interest.