Networked Predictive Trajectory Tracking Control for Underactuated USV with Time-Varying Delays

Abstract

This study explores the control framework for the trajectory tracking problem concerning unmanned surface vessels (USVs) in the presence of time-varying communication delays. To address the aforementioned problem, a novel networked predictive sliding mode control architecture is proposed by integrating a discrete sliding mode control technique and predictive control scheme. By leveraging a first-order forward Euler discretization approach, a discrete-time model of USVs was initially formulated. Then, a virtual velocity controller was developed to convert the position tracking into expected velocity tracking, which was achieved by utilizing a sliding mode control. Subsequently, a networked predictive control technique was performed to compensate for the time-varying delays. Finally, theoretical analysis and extensive comparative simulation tests demonstrated that the proposed control scheme guaranteed complete compensation for time-varying delays while ensuring the stability of the closed-loop system.

1. Introduction

In recent years, unmanned surface vehicles (USVs), such as marine unmanned equipment, have received widespread attention and have had applications in military and civilian fields, such as marine resource exploration [1], maritime attack and defense [2], water quality monitoring [3], maritime rescue [4], and maritime mapping [5]. On the one hand, with the rapid development of wireless sensor network technology [6], microelectronics technology [7], Internet of Things technology [8], and advanced control theory [9], the functions of USVs are greatly expanding, and the level of automation is increasing day by day. On the other hand, the adoption of the aforementioned technologies in the USV industry, in particular, the introduction of communication networks, introduces new challenges to the design of USV control systems. Since the communication network serves as a bridge to integrate various control devices into a single whole, control signals and feedback signals are transmitted via the network. The control system of USVs demonstrates the attributes typical of a cyber–physical system (CPS). It is well known that time delays widely exist in CPSs, which not only presents challenges for control system design but also detrimentally impacts the system performance, potentially leading to instability [10]. Therefore, for a special CPS such as a USV, the trajectory tracking control issue associated with time-varying communication delays is worth attention.

Trajectory tracking control plays an important role in engineering applications of USVs, which ensures that USVs move along the desired trajectory in order to fulfill given tasks. Numerous control technologies have been introduced into USV trajectory tracking control, and fruitful results have been reported. Sliding mode variable structure control (i.e., sliding mode control, SMC) [11,12,13,14,15] is a frequently used strategy because it is insensitive to unknown disturbances and parameter disturbances and is very effective for USVs with nonlinear dynamic characteristics. However, while SMC can effectively handle trajectory tracking to a certain extent, its inherent switching characteristic can cause system chattering, which further leads to excessive wear and shortens the life of the actuator. To overcome the negative effects of the SMC, some improved schemes based on SMC have been proposed. For instance, super-twisting SMC was proposed by Liu to solve the trajectory tracking control of USVs [16]; with the help of super-twisting technology, the chattering effect caused by SMC was reduced. More studies on super-twisting SMC for USV tracking control can be found in [17,18,19]. Another well-liked alternative is the adaptive control scheme [20,21], which was introduced into SMC and has the ability to effectively suppress chattering. In [20], the authors designed an adaptive sliding mode controller to achieve surge velocity and heading tracking and thus finally achieve accurate trajectory tracking of a USV by combining the approach with a LOS guidance algorithm. From the above discussion, it is shown that the adaptive control approach introduced into the sliding mode controller simultaneously mitigates chattering and is robust to bounded uncertainties/disturbances. Additionally, the trajectory tracking control of USVs can also be realized through other control strategies such as model-free control [22], robust control [23], fuzzy control [24], reinforcement learning control [25], model predictive control [26], and backstepping control [27]. It should be noted that the controller of a USV is usually digital; thus, a discrete-time controller is more attractive. Studies related to the application of discrete-time control designs include [28,29,30]. Although the aforementioned control methods can effectively address the trajectory tracking control problem for USVs, time-varying delays have yet to be taken into account, and studies are rare in this field.

In order to overcome the negative effects of time-varying delays, two major types of control strategies have been proposed—namely, passive compensation strategies and active compensation strategies. The central idea of passive compensation is to apply Lyapunov stability theory to construct the Lyapunov function of the control system [31,32]. As this approach emphasizes the stability of the control system, it inevitably requires a compromise between the control performance and stability [33]. From the above perspective, the control design of the system is naturally conservative, which means that control performance will be sacrificed to ensure that the system maintains stability. Regarding the active compensation strategy, the main study object is predictive control technology. Predictive control technology is based on a model prediction scheme, which is an optimal control strategy determined by using current and past information of the controlled object to predict system performance in the future [34,35]. In other words, model predictive control technology possesses the inherent benefit of conquering time delays. Thus, the active compensation strategy based on predictive control technology is particularly suitable for time-delay systems. In addition, with the aim of fully compensating for the network delays of CPSs, network predictive control has been presented in the literature [36,37]. According to the aforementioned investigation, after introducing networked predictive control technology to a delayed system, its performance was comparable to that of a system without delays. It is noteworthy that the research object of the active compensation strategy mentioned above is fully actuated systems, which cannot directly cope with the control issue of underactuated USVs with time-varying network delays. Hence, with regard to trajectory tracking for underactuated USV under time-varying communication delays, the control architecture based on network prediction needs to be reconstructed.

At present, networked predictive control technology is widely employed in trajectory tracking control under communication constraints due to its ability to actively compensate for system delays. Aiming at position tracking control of unmanned vehicles under a network environment, in [38], Zhang designed a tracking controller based on networked predictive control technology to achieve accurate trajectory tracking under time-varying communication delays. In this study, only fixed delays were considered, and time-varying delays were not considered. Chen et al. introduced networked predictive control into collaborative path tracking of unmanned vehicles [39], and the time-varying delay was accurately compensated with the help of the designed two data buffers and networked predictive control strategy. However, the control objects involved in these studies were all fully actuated. To the best of our understanding, for typical nonlinear underactuated systems such as USVs, there is no research on using the networked predictive control strategy to tackle the trajectory tracking control issue for USVs encountering time-varying delays. Motivated by the proposal in [37] to design a networked predictive control architecture to actively compensate for the delays, this work designed a predictive control scheme building on discrete-time SMC to suppress the detrimental impact of time-varying delays on the USV system.

Taking inspiration from the discussion above, we aimed to develop a solution based on networked predictive control to deal with the trajectory tracking control of the USV suffering from time-varying network delays. Considering that motion controllers today are almost exclusively implemented using digital control methods, the control design involved in this study was constructed within the discrete domain. With the application of dual-loop technology, the controller design was divided into an inner-loop controller design and an outer-loop controller design. The outer-loop controller is composed of a discrete-time surge virtual velocity control law and a discrete-time sway virtual velocity control law. In line with that, the inner-loop controller consists of a thrust control law in the surge direction and a steering torque in the the heading direction, both of which are discrete-time sliding mode controllers. Furthermore, a networked predictive control strategy was implemented for the abovementioned dual-loop controller to achieve accurate compensation for time-varying network delays, thereby ultimately realizing accurate tracking for the USV’s desired trajectory.

The main contributions of this paper are summarized as follows:

• An improved discrete-time virtual speed control law was proposed. Compared with the one proposed in the literature [30], the improved discrete-time virtual velocity controller is simpler and requires less calculation. The purpose of introducing the virtual velocity controller is to convert the trajectory tracking problem into speed tracking.
• In the light of the USV’s discrete-time dynamic mathematical model and discrete-time sliding mode control theory, the surge thrust control law and steering torque control law were constructed to realize asymptotic tracking for the virtual velocities.
• Networked predictive control was introduced to nonlinear underactuated USV for the first time. Benefiting from networked predictive control, the time-varying delays existing in the communication network were completely compensated.

The rest of this paper is structured as follows. The problem construction including USV modeling, the influence of the network environment on USV control, and the control objectives are stated in Section 2. Section 3 gives the detailed design procedure for the networked predictive sliding mode control strategy and the stability proof of the developed controller. The numerical simulation and result analysis for USV trajectory tracking are described in Section 4. Finally, Section 5 outlines the conclusions of this study.

2. Problem Formulation

2.1. Modeling the Dynamics of a USV

The kinematic and dynamic equations of a USV are characterized by two reference frames as shown in Figure 1. The earth-fixed reference frame $O_E{X}_E{Y}_E$ is used to construct the kinematic equations of the USV, and the dynamic equations of the USV are presented in the body-fixed reference frame $O_B{X}_B{Y}_B$. By taking the advantage of the above two reference frameworks, the mathematical model of USV is expressed by

$$\begin{array}{c}\stackrel{·}{\mathit{\eta }}=\mathit{J}\left(\psi \right)\mathit{\nu }\end{array}$$

(1)

$$\begin{array}{c}\mathit{M}\stackrel{·}{\mathit{\nu }}+\mathit{C}\left(\mathit{\nu }\right)\mathit{\nu }+\mathit{D}\mathit{\nu }=\mathit{\tau }+\mathit{d},\end{array}$$

(2)

where $[x,y]$ and $\psi$ represent the position and heading of the USV in the earth-fixed coordinate system, i.e., $\mathit{\eta }={[x,y,\psi ]}^T$. The velocity vector is denoted as $\mathit{\nu }={[u,v,r]}^T$, which is composed of the linear speed vector ${[u,v]}^T$ and heading angular rate r. $\mathit{\tau }={[{\tau }_u,0,{\tau }_r]}^T$ is the control force and torque generated by the propulsion system of the USV. $\mathit{d}={[d_u,d_v,d_r]}^T$ is the external disturbance vector of the USV caused by wind, waves, and currents. The transformation matrix between the two coordinate frames mentioned above is defined as $\mathit{J}\left(\psi \right)$, which is expressed as follows:

$$\mathit{J}\left(\psi \right)=\left[\begin{array}{ccc}cos\left(\psi \right)& -sin\left(\psi \right)& 0\\ sin\left(\psi \right)& cos\left(\psi \right)& 0\\ 0& 0& 1\end{array}\right],$$

(3)

where $\mathit{M}$ denotes the inertia mass matrix. $\mathit{C}\left(\mathit{\nu }\right)$ is denoted as the Coriolis force centripetal matrix of the USV. The hydrodynamic coefficient matrix is expressed by $\mathit{D}\left(\mathit{\nu }\right)$. Specifically, $\mathit{M}$, $\mathit{C}\left(\mathit{\nu }\right)$ and $\mathit{D}\left(\mathit{\nu }\right)$ are defined as

$$\mathit{M}=\left[\begin{array}{ccc}{m}_{11}& 0& 0\\ 0& m_{22}& 0\\ 0& 0& m_{33}\end{array}\right],\mathit{C}\left(\mathit{\nu }\right)=\left[\begin{array}{ccc}0& 0& -m_{22}v\\ 0& 0& m_{11}u\\ m_{22}v& -m_{11}u& 0\end{array}\right],\mathit{D}\left(\mathit{\nu }\right)=\left[\begin{array}{ccc}{d}_{11}& 0& 0\\ 0& d_{22}& 0\\ 0& 0& d_{33}\end{array}\right].$$

(4)

To facilitate the implementation of the digital controller, the forward Euler method was used to discretize the (1) and (2) continuous-time system equations by making use of the sampling interval $T_c$. The discrete-time system equations are given in the following.

$$\begin{array}{c}\mathit{\eta }(k+1)=\mathit{\eta }\left(k\right)+T_c\mathit{J}\left(\psi \left(k\right)\right)\mathit{\nu }\left(k\right)\end{array}$$

(5)

$$\begin{array}{c}\mathit{\nu }(k+1)=\mathit{\nu }\left(k\right)-T_c{\mathit{M}}^{-1}\mathit{C}\left(\mathit{\nu }\left(k\right)\right)\mathit{\nu }\left(k\right)-T_c{\mathit{M}}^{-1}\mathit{D}\mathit{\nu }\left(k\right)+T_c{\mathit{M}}^{-1}\left(\mathit{\tau }\left(k\right)+\mathit{d}\left(k\right)\right)\end{array}$$

(6)

Property 1.

$\mathit{J}\left(\psi \right(k\left)\right)$  is an orthogonal matrix and satisfies $\mathit{J}\left(\psi \left(k\right)\right)={\mathit{J}}^{-1}\left(\psi \left(k\right)\right),\left|\mathit{J}\left(\psi \right(k\left)\right)\right|=1$. Each element on the diagonal of $\mathit{M}$ satisfies $m_{ii}>0$, i.e., matrix $\mathit{M}$ is positive definite.

2.2. Networked Control Architecture for the USV

A closed-loop control system that connects sensors, controllers, and actuators via digital networks is called a networked control system. Figure 2 depicts the networked control structure for the USV. From this figure, it is obvious that the controller is deployed on a remote shore base or mother ship (control station). Communication networks exist between the sensor and the control station, as well as the actuator and the control station. Note that the USV control system will unavoidably encounter time-varying delays due to the existence of the networks. The variable ${\mathit{y}}_c\left(k\right)$ in Figure 2 is the speed and position information of the USV at time k measured by the sensor (i.e., GPS (global positioning system) and IMU (inertial measurement units)). In order to better characterize the above time-varying delay, the communication link including the sensor, communication network and control station form the feedback channel, and its time-varying delays are denoted as ${\iota }_b$. In the same way, the forward channel is the communication link that transmits the signal from the control station to the actuator, and the time-varying delays of this channel are expressed as ${\rho }_a$.

2.3. Control Objective

Within this study, aiming at the USV trajectory tracking control problem under communication constraints, we developed a networked predictive sliding mode control technology to overcome the negative influence of the time-varying delays. Under the action of the designed control technology, the time-varying delays in the USV control system were completely compensated, and the position tracking error of the USV converged to a region near zero. To accomplish this objective, a cascaded dual-loop control architecture combined with networked predictive control technique was developed to design the trajectory tracking controller of the USV. First, the virtual velocity control law of the outer-loop was designed in the light of the position error generated by the USV’s desired position and actual position. Then, the position tracking control was converted into velocity tracking control by means of the outer-loop controller, and a discrete-sliding mode control scheme was introduced into the inner loop to realize virtual velocity tracking. Finally, networked predictive control technology was implemented in the abovementioned dual-loop controller to compensate for the time-varying delays in the communication link, thereby enabling the USV to achieve trajectory tracking under communication constraints.

Assumption 1.

The expected trajectory for the USV, referred to as curve ${\mathit{\eta }}_D\left(k\right)={[x_D\left(k\right),y_D\left(k\right)]}^T$, is sufficiently smooth, while the first derivative of this curve, referred to as curve ${\stackrel{·}{\mathit{\eta }}}_D\left(k\right)={[{\stackrel{·}{x}}_D\left(k\right),{\stackrel{·}{y}}_D\left(k\right)]}^T$, exists and remains bounded.

3. Networked Predictive Sliding Mode Controller Design

This section describes the design of a networked predictive control strategy for the USV in a network environment. The focus of this paper is the design of control strategies for the USV control system subjected to time-varying delays, so the control strategy design does not consider the external disturbance of the USV. This also means that we assume that the external environmental disturbance are known and can be fully compensated by the controller. Considering that the external disturbances can be quantitatively characterized by experiments and measurements, the above assumption is reasonable and feasible. As a result, the system disturbances ($d_u,d_v,d_r$) were not considered in the subsequent control strategy design. The detailed control scheme consists of three parts, i.e., the virtual velocity control law for the outer loop, the inner-loop sliding mode control strategy, and the networked predictive control strategy. The design of the controller and its stability analysis are discussed in detail below.

3.1. USV Predictive Control Framework in a Network Environment

Figure 3 illustrates the general architecture of the control scheme for the USV wtih time-varying delays. As shown in Figure 3, the networked predictive controller is comprised of three principal components, namely, the prediction controller, the communication compensator, and data buffers located on both sides of them. In accordance with the actual physical point of view, with the aim of avoiding the open loop of the networked closed-loop control system, assuming that only consecutive data losses with a finite number can be tolerated is acceptable. In addition, the following assumptions are established.

Assumption 2. (1) The delay existing in the feedback channel is ${\iota }_b$, and its upper bound is ${\iota }_U$ (i.e., ${\iota }_b\le {\iota }_U$). (2) The feedback channel has a delay of ${\rho }_a$, and its upper bound exists and is ${\rho }_U$ (i.e., ${\rho }_a\le {\rho }_U$). (3) $s_U$ is the upper bound on the successive data dropouts for both the forward and feedback channels of the system. (4) Each data point is timestamped as it is sent via the network.

Moreover, ensuring clock synchronization among all nodes poses a significant challenge in networked control systems. Currently, a variety of approaches can be used to achieve clock synchronization for digital devices. Since the control problem of the USV under time-varying delays is the focus of this research, it is reasonable to presume that the clocks of each component in a networked control system have been synchronized.

Remark 1.

The clock synchronization of networked communication systems belongs to the research scope of the communication field and has yielded fruitful results. Commonly used time synchronization technologies in networks include network time protocol, precision time protocol, clock synchronization message protocol, and IEEE 1588 protocol. Given that IEEE 1588 possesses good openness and scalability, it is widely used for time synchronization in the industrial control field. It can be easily transplanted or modified based on the design standards and implementation plans it provides. Therefore, in view of the state-of-the-art clock synchronization technologies, it is reasonable and feasible for this article to assume that the clocks of various components in the control system have already been synchronized.

In order to cope with the adverse effects caused by time-varying delays and data dropout, the following data sending mechanism was employed.

• The velocity and position information of the USV is measured by the sensors and transmitted in the following data packets.

  $${\left[{\mathit{y}}_c{\left(k\right)}^T,{\mathit{y}}_c{(k-1)}^T,\cdots {\mathit{y}}_c{(k-s_U)}^T\right]}^T,$$

  (7)

  where ${\mathit{y}}_c{(k-n)}^T$ is a composite column vector consisting of position and velocity vectors. That is to say, ${\mathit{y}}_c{(k-n)}^T=\left[\mathit{\eta }{(k-n)}^T,\nu {(k-n)}^T\right]$, and $0\le n\le S_U$.
• In the same way, to prevent data dropout in the forward channel, the control sequence ${\widehat{\tau }}_{FS}\left(k\right)$ of future moments is obtained by performing multi-step predictive control calculations at time k.

  $${\widehat{\mathit{\tau }}}_{FS}\left(k\right)={\left[\widehat{\mathit{\tau }}{\left(k|k-{\iota }_b\right)}^T,\widehat{\mathit{\tau }}{\left(k+1|k-{\iota }_b\right)}^T,\cdots ,\widehat{\mathit{\tau }}{\left(k+{{\rm Y}}_a|k-{\iota }_b\right)}^T\right]}^T$$

  (8)
  $\widehat{\mathit{\tau }}{\left(k|k-{\iota }_b\right)}^T,\widehat{\mathit{\tau }}{\left(k+1|k-{\iota }_b\right)}^T,\cdots ,\widehat{\mathit{\tau }}{\left(k+{{\rm Y}}_a|k-{\iota }_b\right)}^T$ are the predictions of the control input sequence at the future instants, which are calculated by using the predictive control algorithm through the received motion information of the USV delayed by ${\iota }_b$ steps, where ${{\rm Y}}_a={\rho }_U+s_U$.

In terms of the feedback signals and control prediction, which are sequences accompanied by timestamps, two data buffers need to be set up to reorder the received data on the basis of timestamps. One is data buffer A, located on the predictive controller side, and the other is buffer B, located at the communication compensator side. Hence, under Assumption 2, the relevant data for the output of the predictive controller and the input of the actuator are always present and available.

Buffer B stores the predictive control sequences required by the USV actuators. In accordance with the timestamp technology and round-trip time measurement technology, communication delays can be identified. Then, at the time k, the communication compensator selects the most suitable component from the control prediction sequence (8) to serve as the control input for the actuator, based on the current timestamp. It is assumed that the delays for the forward and feedback channels, as measured at the current moment using timestamp technology, are ${\rho }_a$ and ${\iota }_b$, respectively. As a result, one can obtain

$$\mathit{\tau }\left(k\right)=\widehat{\mathit{\tau }}\left(k\right|k-{\rho }_a-{\iota }_b)=\widehat{\mathit{\tau }}(k+{\rho }_a|k-{\iota }_b).$$

(9)

3.2. Design of the Networked Predictive Controller for the USV

The networked predictive controller for the USV comprised two components: the predictive controller and the communication compensator. The specific design procedure of the above parts is detailed in the following.

3.2.1. Prediction Generator

A. Design of the virtual velocity control law

According to the explanation in Figure 3, the motion state information for the USV, including the speed and position, is received by the controller after a delay of ${\iota }_b$ steps at time k. Therefore, the available feedback information on the controller side is $\mathit{\eta }(k-{\iota }_b)$ and $\mathit{\nu }(k-{\iota }_b)$. On the basis of such feedback information, the control input $\mathit{\tau }(k-{\iota }_b)$ is derived by using a discrete sliding mode scheme. The specific design procedure is given later. Using $\mathit{\eta }(k-{\iota }_b)$, $\mathit{\nu }(k-{\iota }_b)$ and $\mathit{\tau }(k-{\iota }_b)$, we performed the following multi-step output predictor based on the nominal model, which yielded the following:

$$\left\{\begin{array}{c}\widehat{\mathit{\eta }}\left(k-{\iota }_b+1|k-{\iota }_b\right)=\mathit{\eta }\left(k-{\iota }_b\right)+T_c\mathit{J}\left(\psi \left(k-{\iota }_b\right)\right)\mathit{\nu }\left(k-{\iota }_b\right)\hfill \\ \widehat{\mathit{\nu }}\left(k-{\iota }_b+1|k-{\iota }_b\right)=\mathit{\nu }\left(k-{\iota }_b\right)+T_c\mathit{G}\left(\mathit{\nu }\left(k-{\iota }_b\right)\right)+T_c{\mathit{M}}^{-1}\mathit{\tau }\left(k-{\iota }_b\right)\hfill \\ \widehat{\mathit{\eta }}\left(k-{\iota }_b+2|k-{\iota }_b\right)=\widehat{\mathit{\eta }}\left(k-{\iota }_b+1|k-{\iota }_b\right)\hfill \\ +T_c\mathit{J}\left(\widehat{\psi }\left(k-{\iota }_b+1|k-{\iota }_b\right)\right)\widehat{\mathit{\nu }}\left(k-{\iota }_b+1|k-{\iota }_b\right)\hfill \\ \widehat{\mathit{\nu }}\left(k-{\iota }_b+2|k-{\iota }_b\right)=\widehat{\mathit{\nu }}\left(k-{\iota }_b+1|k-{\iota }_b\right)+T_c\mathit{G}\left(\widehat{\mathit{\nu }}\left(k-{\iota }_b+1|k-{\iota }_b\right)\right)\hfill \\ +T_c{\mathit{M}}^{-1}\widehat{\mathit{\tau }}\left(k-{\iota }_b+1|k-{\iota }_b\right)\hfill \\ ⋮\hfill \\ \widehat{\mathit{\eta }}\left(k+{{\rm Y}}_a|k-{\iota }_b\right)=\widehat{\mathit{\eta }}\left(k+{{\rm Y}}_a-1|k-{\iota }_b\right)\hfill \\ +T_c\mathit{J}\left(\widehat{\psi }\left(k+{{\rm Y}}_a-1|k-{\iota }_b\right)\right)\widehat{\mathit{\nu }}\left(k+{{\rm Y}}_a-1|k-{\iota }_b\right)\hfill \\ \widehat{\mathit{\nu }}\left(k+{{\rm Y}}_a|k-{\iota }_b\right)=\widehat{\mathit{\nu }}\left(k+{{\rm Y}}_a-1|k-{\iota }_b\right)+T_c\mathit{G}\left(\widehat{\mathit{\nu }}\left(k+{{\rm Y}}_a-1|k-{\iota }_b\right)\right)\hfill \\ +T_c{\mathit{M}}^{-1}\widehat{\mathit{\tau }}\left(k+{{\rm Y}}_a-1|k-{\iota }_b\right)\hfill \end{array},\right.$$

(10)

where $\widehat{\mathit{\eta }}\left(k-{\iota }_b+j|k-{\iota }_b\right)$ and $\widehat{\mathit{\nu }}\left(k-{\iota }_b+j|k-{\iota }_b\right)$, respectively, denote the predicted values for the velocity $\nu \left(k\right)$ and position $\mathit{\eta }\left(k\right)$ at time $\left(k-{\iota }_b+j\right)$ using the velocity and position of the USV up to time $\left(k-{\iota }_b\right)$, and $0\le j\le {{\rm Y}}_a+{\iota }_b$. $-T_c{\mathit{M}}^{-1}\left[\mathit{C}\left(\widehat{\mathit{\nu }}\left(⊛\right)\right)\widehat{\mathit{\nu }}\left(⊛\right)+\mathit{D}\widehat{\mathit{\nu }}\left(⊛\right)\right]$ is represented as $\mathit{G}\left(\widehat{\mathit{\nu }}\left(⊛\right)\right)$. The variable labeled with ’$^$’ indicates that it is a predicted value derived from historical information.

On the controller side, by implementing the aforementioned prediction process, the velocity and position information for the USV at time k can be obtained. Next, the virtual velocity control law was devised. The error arising from the predicted position of the USV to the desired position is defined as follows:

$$\left\{\begin{array}{c}{\epsilon }_x\left(k\right)=\widehat{x}\left(k\right|k-{\iota }_b)-x_D\left(k\right|k-{\iota }_b)\hfill \\ {\epsilon }_y\left(k\right)=\widehat{y}\left(k\right|k-{\iota }_b)-y_D\left(k\right|k-{\iota }_b)\hfill \end{array}.\right.$$

(11)

In accordance with the expansion of (5) and (11), this leads to

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{\epsilon }_x(k+1)\\ {\epsilon }_y(k+1)\end{array}\right]& =\left[\begin{array}{cc}cos\widehat{\psi }\left(k|k-{\iota }_b\right)& sin\widehat{\psi }\left(k|k-{\iota }_b\right)\\ -sin\widehat{\psi }\left(k|k-{\iota }_b\right)& cos\widehat{\psi }\left(k|k-{\iota }_b\right)\end{array}\right]\left[\begin{array}{c}\widehat{u}\left(k|k-{\iota }_b\right)\\ \widehat{v}\left(k|k-{\iota }_b\right)\end{array}\right]\hfill \\ \hfill & -\left[\begin{array}{c}{x}_D\left(k+1\right)-x_D\left(k\right)-{\epsilon }_x\left(k\right)\\ y_D\left(k+1\right)-y_D\left(k\right)-{\epsilon }_y\left(k\right)\end{array}\right].\hfill \end{array}$$

(12)

In equality (12), $\widehat{u}\left(k|k-{\iota }_b\right)$ and $\widehat{v}\left(k|k-{\iota }_b\right)$ are defined as the virtual velocity control variables that are to be designed. For the purpose of realizing the tracking error of the USV’s position converging to a region near zero, the control law for the virtual velocity variables was constructed as

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{\widehat{\mu }}_u\left(k|k-{\iota }_b\right)\\ {\widehat{\mu }}_v\left(k|k-{\iota }_b\right)\end{array}\right]& =\left[\begin{array}{cc}cos\widehat{\psi }\left(k|k-{\iota }_b\right)& sin\widehat{\psi }\left(k|k-{\iota }_b\right)\\ -sin\widehat{\psi }\left(k|k-{\iota }_b\right)& cos\widehat{\psi }\left(k|k-{\iota }_b\right)\end{array}\right]\hfill \\ \hfill & \times \left[\begin{array}{c}{\displaystyle \frac{x_D\left(k+1\right)-x_D\left(k\right)}{T_c}}-{\displaystyle \frac{{\Gamma }_u{\epsilon }_x\left(k\right)}{\sqrt{{\epsilon }_x^2\left(k\right)+{\epsilon }_y^2\left(k\right)+\Lambda }}}\\ {\displaystyle \frac{y_D\left(k+1\right)-y_D\left(k\right)}{T_c}}-{\displaystyle \frac{{\Gamma }_v{\epsilon }_x\left(k\right)}{\sqrt{{\epsilon }_y^2\left(k\right)+{\epsilon }_y^2\left(k\right)+\Lambda }}}\end{array}\right],\hfill \end{array}$$

(13)

where the variables ${\widehat{\mu }}_u\left(k|k-{\iota }_b\right)$ and ${\widehat{\mu }}_v\left(k|k-{\iota }_b\right)$ denote the control laws for the virtual velocity variables, which can also be referred to as the desired velocities for the USV including the surge and sway directions, respectively. Variables ${\Gamma }_u$, ${\Gamma }_v$, and $\Lambda$ are positive control parameters that need to be designed. The purpose of introducing the term $\sqrt{{\epsilon }_y^2\left(k\right)+{\epsilon }_y^2\left(k\right)+\Lambda }$ in Equation (13) is to prevent the virtual velocity control law from becoming overly large. If it becomes excessively large, the virtual velocity may surpass the USV’s normal sailing speed, resulting in a failure to track the trajectory tracking. Since the controller stores the desired trajectory information, both $x_D(k+1)$ and $y_D(k+1)$ in equality (13) can be obtained on the controller side.

Theorem 1.

For the virtual velocity controller (13) designed for virtual velocity variables $\widehat{u}\left(k|k-{\iota }_b\right)$ and $\widehat{v}\left(k|k-{\iota }_b\right)$, if the predicted velocity of the USV aligns with the virtual velocity, then the trajectory tracking error in (11) asymptotically converges to zero.

Proof. 

Substituting (13) into (12) results in

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{\epsilon }_x(k+1)-{\epsilon }_x\left(k\right)\\ {\epsilon }_y(k+1)-{\epsilon }_y\left(k\right)\end{array}\right]& =T_c\left[\begin{array}{cc}cos\widehat{\psi }\left(k|k-{\iota }_b\right)& sin\widehat{\psi }\left(k|k-{\iota }_b\right)\\ -sin\widehat{\psi }\left(k|k-{\iota }_b\right)& cos\widehat{\psi }\left(k|k-{\iota }_b\right)\end{array}\right]\left[\begin{array}{c}\widehat{u}\left(k\right|k-{\iota }_b)-{\widehat{\mu }}_u\left(k\right|k-{\iota }_b)\\ \widehat{v}\left(k\right|k-{\iota }_b)-{\widehat{\mu }}_v\left(k\right|k-{\iota }_b)\end{array}\right]\hfill \\ \hfill & \times \left[\begin{array}{c}-{\displaystyle \frac{{\Gamma }_u{T}_c{\epsilon }_x\left(k\right)}{\sqrt{{\epsilon }_x^2\left(k\right)+{\epsilon }_y^2\left(k\right)+\Lambda }}}\\ -{\displaystyle \frac{{\Gamma }_v{T}_c{\epsilon }_x\left(k\right)}{\sqrt{{\epsilon }_y^2\left(k\right)+{\epsilon }_y^2\left(k\right)+\Lambda }}}\end{array}\right].\hfill \end{array}$$

(14)

From the above equality, it is easy to conclude that if $\left[\widehat{u}\left(k\right|k-{\iota }_b)-{\widehat{\mu }}_u\left(k\right|k-{\iota }_b)\right]\to 0$ and $\left[\widehat{v}\left(k\right|k-{\iota }_b)-{\widehat{\mu }}_v\left(k\right|k-{\iota }_b)\right]\to 0$ are satisfied, then (14) can be simplified as

$$\left[\begin{array}{c}{\epsilon }_x(k+1)\\ {\epsilon }_y(k+1)\end{array}\right]=\left[\begin{array}{c}\left(1-{\displaystyle \frac{{\Gamma }_u{T}_c}{N}}\right){\epsilon }_x\left(k\right)\\ \left(1-{\displaystyle \frac{{\Gamma }_u{T}_c}{N}}\right){\epsilon }_y\left(k\right)\end{array},\right]$$

(15)

where $N=\sqrt{{\epsilon }_y^2\left(k\right)+{\epsilon }_y^2\left(k\right)+\Lambda }$.

Let us construct the following Lyapunov function:

$$V_v\left(k\right)={\displaystyle \frac{1}{2}}{\epsilon }_x^2\left(k\right)+{\epsilon }_y^2\left(k\right).$$

(16)

In view of (15) and (16), the following relationship is derived:

$$\begin{array}{cc}\hfill \Delta V_v\left(k\right)& =\Delta V_v(k+1)-\Delta V_v\left(k\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\epsilon }_x^2\left(k\right)\left[{\displaystyle \frac{{\Gamma }_u^2{T}_c^2}{N^2}}-{\displaystyle \frac{2{\Gamma }_u{T}_c}{N}}\right]+{\displaystyle \frac{1}{2}}{\epsilon }_y^2\left(k\right)\left[{\displaystyle \frac{{\Gamma }_v^2{T}_c^2}{N^2}}-{\displaystyle \frac{2{\Gamma }_v{T}_c}{N}}\right].\hfill \end{array}$$

(17)

Owing to ${\Gamma }_u<2N/T_c$ and ${\Gamma }_u<2N/T_c$, the following inequality is obtained:

$$\Delta V_v<0.$$

(18)

Hence, we can infer from Equation (18) that if ${\Gamma }_u<2N/T_c$ and ${\Gamma }_u<2N/T_c$ hold, the tracking error of the USV relative to the desired trajectory will asymptotically converge to zero. □

B. Design of the predictive discrete-time SMC

Building on the state prediction and the surge virtual velocity control law, the velocity tracking error in the surge direction is defined as

$${\epsilon }_u\left(k\right|k-{\iota }_b)=\widehat{u}\left(k\right|k-{\iota }_b)-\widehat{\mu }\left(k\right|k-{\iota }_b).$$

(19)

According to (19), we construct the following integral form of the discrete sliding surface

$${\zeta }_1\left(k\right|k-{\iota }_b)={\epsilon }_u\left(k\right|k-{\iota }_b)+{\chi }_1{T}_c\sum _{i=0}^{{\iota }_b-1}{\epsilon }_u(k+i-{\iota }_b|k-{\iota }_b),$$

(20)

where ${\chi }_1>0$, $T_c$ is the sampling time.

In the light of discrete-time sliding mode theory, under ideal sliding mode, one obtains

$${\zeta }_1(k+1|k-{\iota }_b)={\zeta }_1\left(k\right|k-{\iota }_b).$$

(21)

Substituting (19) and (20) into (21), the following equality is derived:

$${\epsilon }_u(k+1|k-{\iota }_b)+({\chi }_1{T}_c-1){\epsilon }_u\left(k\right|k-{\iota }_b)=0.$$

(22)

Combining (19) and (22), and using the expansion of (10), the following equivalent control in the surge direction is obtained.

$$\begin{array}{cc}\hfill {\widehat{\tau }}_{ue}\left(k\right|k-{\iota }_b)& =-m_{22}\widehat{v}\left(k|k-{\iota }_b\right)\widehat{r}\left(k|k-{\iota }_b\right)-{\chi }_1{m}_{11}{\epsilon }_u\left(k|k-{\iota }_b\right)\hfill \\ \hfill & +d_{11}\widehat{u}\left(k|k-{\iota }_b\right)\widehat{r}\left(k|k-{\iota }_b\right)+{\displaystyle \frac{m_{11}}{T_c}}\left[{\widehat{\mu }}_u(k+1|k-{\iota }_b)-{\widehat{\mu }}_u\left(k\right|k-{\iota }_b)\right]\hfill \end{array}$$

(23)

With the aim of stabilizing the system’s state on the switching hyperplane, the switching control law ${\widehat{\tau }}_{us}\left(k|k-{\iota }_b\right)=-{\kappa }_{1}sgn\left(s_1\left(k|k-{\iota }_b\right)\right)$ is selected, where ${\kappa }_1$ is a constant. Therefore, at the controller side, the overall control law in the surge direction for the time k can be obtained by adding the equivalent control and the switching control law, and its specific expression is

$${\widehat{\tau }}_u\left(k|k-{\iota }_b\right)={\widehat{\tau }}_{ue}\left(k|k-{\iota }_b\right)+{\widehat{\tau }}_{us}\left(k|k-{\iota }_b\right).$$

(24)

Hence, the thrust control law in the surge direction has been accomplished. Next, the details of the steering torque control law in the sway direction are given.

In view of the predicted information regarding the sway velocity and the virtual velocity, the velocity tracking error in the sway direction is defined as follows:

$${\epsilon }_v\left(k|k-{\iota }_b\right)=\widehat{v}\left(k|k-{\iota }_b\right)-{\widehat{\mu }}_v\left(k|k-{\iota }_b\right).$$

(25)

Considering the heading is controlled by the steering torque, to facilitate the appearance of the control torque term ${\widehat{\tau }}_r\left(k\right)$ in the sliding mode surface, we established the following discrete form of the differential sliding mode surface.

$${\zeta }_2\left(k|k-{\iota }_b\right)={\displaystyle \frac{{\epsilon }_v\left(k+1|k-{\iota }_b\right)-{\epsilon }_v\left(k|k-{\iota }_b\right)}{T_c}}+{\chi }_2{\epsilon }_v\left(k|k-{\iota }_b\right),$$

(26)

where ${\chi }_2>0$. In light of ${\zeta }_2\left(k+1|k-{\iota }_b\right)={\zeta }_2\left(k|k-{\iota }_b\right)$, the following equality is obtained:

$${\epsilon }_v\left(k+2|k-{\iota }_b\right)-{\epsilon }_v\left(k+1|k-{\iota }_b\right)+\left({\chi }_2{T}_c-1\right)\left({\epsilon }_v\left(k+1|k-{\iota }_b\right)-{\epsilon }_v\left(k|k-{\iota }_b\right)\right)=0.$$

(27)

Using the expansion of (10) and substituting it into (27), one can conclude that

$$\begin{array}{cc}\hfill & \left(-m_{11}\widehat{u}\left(k+1|k-{\iota }_b\right)\widehat{r}\left(k+1|k-{\iota }_b\right)-d_{22}\widehat{v}\left(k+1|k-{\iota }_b\right)\right)+{\displaystyle \frac{m_{22}}{T_c}}{\widehat{\mu }}_v\left(k+1|k-{\iota }_b\right)\hfill \\ \hfill & -{\displaystyle \frac{m_{22}}{T_c}}{\widehat{\mu }}_v\left(k+2|k-{\iota }_b\right)+\left({\chi }_2{m}_{22}-{\displaystyle \frac{m_{22}}{T_c}}\right)\left({\widehat{\mu }}_v\left(k|k-{\iota }_b\right)-{\widehat{\mu }}_v\left(k+1|k-{\iota }_b\right)\right)\hfill \\ \hfill & +\left({\chi }_2{T}_c-1\right)\left(-m_{11}\widehat{u}\left(k|k-{\iota }_b\right)\widehat{r}\left(k|k-{\iota }_b\right)-d_{22}\widehat{v}\left(k|k-{\iota }_b\right)\right)=0,\hfill \end{array}$$

(28)

where $\widehat{\mu }\left(k+2|k-{\iota }_b\right)$ and $\widehat{\mu }\left(k+1|k-{\iota }_b\right)$ are the predictions of the virtual swaying velocities at the moments $(k+2)$ and $(k+1)$, respectively, which are jointly determined by the prediction of the position and the desired position at the moments $(k+2)$ and $(k+1)$.

For the sake of subsequent derivations, let $\left(\widehat{u}\left(k+1|k-{\iota }_b\right)-\widehat{u}\left(k|k-{\iota }_b\right)\right)$ be denoted as ${\widehat{\delta }}_u\left(k|k-{\iota }_b\right)$. Thus, the item $\widehat{u}\left(k+1|k-{\iota }_b\right)\widehat{r}\left(k+1|k-{\iota }_b\right)$ in (28) can be represented as the following equality.

$$\begin{array}{cc}\hfill \widehat{u}\left(k+1|k-{\iota }_b\right)\widehat{r}\left(k+1|k-{\iota }_b\right)& =\left[{\widehat{\delta }}_u\left(k|k-{\iota }_b\right)+\widehat{u}\left(k|k-{\iota }_b\right)\right]\hfill \\ \hfill & \times {\displaystyle \frac{T_c}{m_{33}}}\left[\left(m_{11}-m_{22}\right)\widehat{u}\left(k|k-{\iota }_b\right)\widehat{r}\left(k|k-{\iota }_b\right)\right.\hfill \\ \hfill & \left.-d_{33}\widehat{r}\left(k|k-{\iota }_b\right)+{\widehat{\tau }}_r\left(k|k-{\iota }_b\right)+\widehat{r}\left(k|k-{\iota }_b\right)\right]\hfill \end{array}$$

(29)

For the purpose of simplicity, the last four terms on the left in (28) are denoted as $\widehat{\sigma }\left(k|k-{\iota }_b\right)$, i.e.,

$$\begin{array}{cc}\hfill \widehat{\sigma }\left(k|k-{\iota }_b\right)& ={\displaystyle \frac{m_{22}}{T_c}}{\widehat{\mu }}_v\left(k+1|k-{\iota }_b\right)-{\displaystyle \frac{m_{22}}{T_c}}{\widehat{\mu }}_v\left(k+2|k-{\iota }_b\right)\hfill \\ \hfill & +\left({\chi }_2{m}_{22}-{\displaystyle \frac{m_{22}}{T_c}}\right)\left({\widehat{\mu }}_v\left(k|k-{\iota }_b\right)-{\widehat{\mu }}_v\left(k+1|k-{\iota }_b\right)\right)\hfill \\ \hfill & +\left({\chi }_2{T}_c-1\right)\left(-m_{11}\widehat{u}\left(k|k-{\iota }_b\right)\widehat{r}\left(k|k-{\iota }_b\right)-d_{22}\widehat{v}\left(k|k-{\iota }_b\right)\right).\hfill \end{array}$$

(30)

Substituting (29) and (30) into (28), it can be learned that

$$\begin{array}{cc}\hfill & \left\{{\displaystyle \frac{m_{11}{T}_c}{m_{33}}}\left[{\widehat{\delta }}_u\left(k|k-{\iota }_b\right)+\widehat{u}\left(k|k-{\iota }_b\right)\right]\right\}{\widehat{\tau }}_r\left(k|k-{\iota }_b\right)\hfill \\ \hfill & =-m_{11}\left({\widehat{\delta }}_u\left(k|k-{\iota }_b\right)+\widehat{u}\left(k|k-{\iota }_b\right)\right){\displaystyle \frac{T_c}{m_{33}}}\hfill \\ \hfill & \times \left[\left(m_{11}-m_{22}\right)\widehat{u}\left(k|k-{\iota }_b\right)\widehat{v}\left(k|k-{\iota }_b\right)-d_{33}\widehat{r}\left(k|k-{\iota }_b\right)+\widehat{r}\left(k|k-{\iota }_b\right)\right]\hfill \\ \hfill & -d_{22}\widehat{v}\left(k+1|k-{\iota }_b\right)+{\displaystyle \frac{m_{22}}{T_c}}\widehat{\sigma }\left(k|k-{\iota }_b\right).\hfill \end{array}$$

(31)

Let the term inside the curly braces on the left side of the above equality be represented as ${\widehat{w}}_L\left(k|k-{\iota }_b\right)$. Accordingly, ${\widehat{w}}_R\left(k|k-{\iota }_b\right)$ represents the term on the right side of Equation (31). Then, by using (31), the following equivalent steering torque control law can be formulated as

$${\widehat{\tau }}_{re}\left(k|k-{\iota }_b\right)={\widehat{w}}_R\left(k|k-{\iota }_b\right)/{\widehat{w}}_L\left(k|k-{\iota }_b\right).$$

(32)

Similar to the thrust control law, ${\widehat{\tau }}_{rs}\left(k|k-{\iota }_b\right)=-{\kappa }_{2}sgn\left(s_2\left(k|k-{\iota }_b\right)\right)$ is the switching control law, which is used to stabilize the sliding mode surface in the surge direction, where ${\kappa }_2$ is the control parameter to be designed. Consequently, by adding ${\widehat{\tau }}_{re}\left(k|k-{\iota }_b\right)$ and ${\widehat{\tau }}_{rs}\left(k|k-{\iota }_b\right)$, the final overall Steering control torque is derived as follows:

$${\widehat{\tau }}_r\left(k|k-{\iota }_b\right)={\widehat{\tau }}_{re}\left(k|k-{\iota }_b\right)+{\widehat{\tau }}_{rs}\left(k|k-{\iota }_b\right).$$

(33)

Hence, at time k, the design work for the predictive sliding mode controllers including the surge and heading directions are completed. In view of (33), it is obvious that ${\widehat{\tau }}_u\left(k|k-{\iota }_b\right)$ and ${\widehat{\tau }}_v\left(k|k-{\iota }_b\right)$ completely compensate the delays existing in the feedback channel. In order to achieve the goal of addressing the delays existing in the forward channel at time k, $\left({{\rm Y}}_a-1\right)$ steps control prediction was implemented according to (10)–(13), (19)–(33), and thus the following predictive control sequence was obtained on the controller side.

$${\widehat{\mathit{\tau }}}_{SEQ}\left(k\right)={\left[\widehat{\mathit{\tau }}{\left(k|k-{\iota }_b\right)}^T,\widehat{\mathit{\tau }}{\left(k+1|k-{\iota }_b\right)}^T,\cdots ,\widehat{\mathit{\tau }}{\left(k+{{\rm Y}}_a|k-{\iota }_b\right)}^T\right]}^T,$$

(34)

where $\widehat{\mathit{\tau }}\left(k+a|k-{\iota }_b\right)={\left[{\widehat{\tau }}_u\left(k+a|k-{\iota }_b\right),{\widehat{\tau }}_r\left(k+a|k-{\iota }_b\right)\right]}^T$, and $0\le a\le {{\rm Y}}_a$.

3.2.2. Communication Compensator

To obtain proper control input, a communication compensator is provided in the actuator of the USV. The predictive control sequence (34) on the controller side is packed with timestamps and transmitted to buffer B on the communication compensator side. Then, the buffer is used to preserve the received control sequence and reorder them according to timestamp. Finally, the proper predictive control is selected in the light of the system delay and sent to the actuator of the USV. We assume that the feedback and forward channels’ delays at time k are ${\iota }_b$ and ${\rho }_a$, respectively. Thus, at time k, the control input, as follows, is selected by the communication compensator and sent to the actuator side for the USV.

$$\mathit{\tau }\left(k\right)=\widehat{\mathit{\tau }}\left(k|k-\rho -{\iota }_b\right)=\widehat{\mathit{\tau }}\left(k+{\rho }_a|k-{\iota }_b\right)$$

(35)

3.3. Stability Analysis of the Controller

The closed-loop stability of the networked predictive controller is analyzed in this section. It is important to note that the examination of the closed-loop system comprised two parts: one involved assessing the stability of the sliding mode controller, while the other involved conducting stability analysis of the closed-loop system after the implementation of predictive control.

Firstly, focusing on the sliding mode controller, its stability was comprehensively analyzed and evaluated.

Lemma 1

(Lei, T., et al. [40]). For the discrete SMC system to be stable, the sliding mode surface $\zeta \left(k\right)$ of the system must satisfy the following necessary and sufficient conditions:

$$\left|\zeta \left(k+1\right)|<|\zeta \left(k\right)\right|.$$

(36)

In light of (10, (20), and (24), the following equality is obtained.

$${\zeta }_1\left(k+1|k-{\iota }_b\right)=-{\kappa }_1{\displaystyle \frac{T_c}{m_{11}}}sgn\left({\zeta }_1\left(k|k-{\iota }_b\right)\right)+{\zeta }_1\left(k|k-{\iota }_b\right)$$

(37)

Considering (37), for the sliding surface ${\zeta }_1\left(k|k-{\iota }_b\right)$, it is noticed that the holding of the Lemma 1 can be ensured by the following inequality.

$$|{\zeta }_1\left(k+1|k-{\iota }_b\right){|}^2<{\left|{\zeta }_1\left(k|k-{\iota }_b\right)\right|}^2$$

(38)

Substituting (37) into (38) yields

$$0<{\kappa }_1<2{\displaystyle \frac{m_{11}}{T_c}}\left|{\zeta }_1\left(k|k-{\iota }_b\right)\right|$$

(39)

As a result, the surge thrust controller is stable if the above inequality holds. In what follows, the steering torque controller’s stability analysis is provided. For the purpose of that, a similar practice was implemented. By substitution of (26) and (33) into (10), one obtains

$${\zeta }_2\left(k|k-{\iota }_b\right)=-{\kappa }_2{w}_l\left(k|k-{\iota }_b\right)sgn\left({\zeta }_2\left(k|k-{\iota }_b\right)\right)+{\zeta }_2\left(k|k-{\iota }_b\right).$$

(40)

In order to ensure that the sliding mode surface ${\zeta }_2\left(k|k-{\iota }_b\right)$ holds with respect to Lemma 1, the following inequality must be satisfied.

$$|{\zeta }_2\left(k+1|k-{\iota }_b\right){|}^2<{\left|{\zeta }_2\left(k|k-{\iota }_b\right)\right|}^2$$

(41)

Combining A and B, one can conclude that

$$w_l^2\left(k|k-{\iota }_b\right){\kappa }_2^2<2{\kappa }_2{w}_l\left(k|k-{\iota }_b\right)\left|{\zeta }_2\left(k|k-{\iota }_b\right)\right|.$$

(42)

From (42), it can be learned that

$$\left\{\begin{array}{c}0<{\kappa }_2<{\displaystyle \frac{2\left|{\zeta }_2\left(k|k-{\iota }_b\right)\right|}{w_l\left(k|k-{\iota }_b\right)}},\mathrm{if}{w}_l\left(k|k-{\iota }_b\right)>0\\ {\displaystyle \frac{2\left|{\zeta }_2\left(k|k-{\iota }_b\right)\right|}{w_l\left(k|k-{\iota }_b\right)}}<{\kappa }_2<0,\mathrm{if}{w}_l\left(k|k-{\iota }_b\right)<0\end{array}\right..$$

(43)

Therefore, if the above condition (43) is satisfied, the discrete-time steering torque controller is stable. In other words, the foregoing analysis demonstrates that the two sliding mode control laws including the surge thrust and steering torque are stable, and the tracking errors for the surge and sway velocities are convergent and bounded. That is to say, ${\epsilon }_u\left(k|k-{\iota }_b\right)$ and ${\epsilon }_v\left(k|k-{\iota }_b\right)$ are uniformly bounded.

Next, the closed-loop stability of the aforementioned discrete-time sliding mode control after introducing the networked predictive control scheme was analyzed. To facilitate performing stability analysis, the closed-loop equation of the system without considering the time delay was rewritten as

$$\left\{\begin{array}{c}\mathit{\eta }\left(k+1\right)=\mathit{\eta }\left(k\right)+T_c\mathit{J}\left(\psi \left(k\right)\right)\mathit{v}\left(k\right)\hfill \\ \mathit{\nu }\left(k+1\right)=\mathit{g}\left(\mathit{\nu }\left(k\right)\right)+T_c{\mathit{M}}^{-1}\mathit{\tau }\left(k\right)\hfill \\ \mathit{\tau }\left(k\right)=\mathit{f}\left(\mathit{\eta }\left(k\right),\mathit{\nu }\left(k\right),{\mathit{\eta }}_D\left(k\right)\right)\hfill \end{array}\right.,$$

(44)

where the term $\left\{\mathit{\nu }\left(k\right)-T_c{\mathit{M}}^{-1}\left[\mathit{C}\left(\mathit{\nu }\left(k\right)\right)\mathit{\nu }\left(k\right)-\mathit{D}\mathit{\nu }\left(k\right)\right]\right\}$ in the kinetic equation is denoted as $\mathit{g}\left(\mathit{\nu }\left(k\right)\right)$ in (44). It should be noted that the reason the control input is denoted as $\mathit{f}\left(\mathit{\eta }\left(k\right),\mathit{\nu }\left(k\right),{\mathit{\eta }}_D\left(k\right)\right)$ is because it is a function of the position, velocity, and desired position. Similarly, the control input after implementing a predictive control strategy is also a function of the predicted position, predicted velocity, and desired position. Accordingly, the closed-loop system considering time delay can be expressed as

$$\left\{\begin{array}{c}\mathit{\eta }\left(k+1\right)=\mathit{\eta }\left(k\right)+T_c\mathit{J}\left(\psi \left(k\right)\right)\mathit{v}\left(k\right)\hfill \\ \mathit{\nu }\left(k+1\right)=\mathit{g}\left(\mathit{\nu }\left(k\right)\right)+T_c{\mathit{M}}^{-1}\mathit{\tau }\left(k\right)\hfill \\ \mathit{\tau }\left(k\right)=\mathit{f}\left(\widehat{\mathit{\eta }}\left(k|k-{\iota }_b\right),\widehat{\mathit{\nu }}\left(k|k-{\iota }_b\right),{\mathit{\eta }}_D\left(k\right)\right)\hfill \end{array}\right..$$

(45)

To investigate this issue, Liu [37] provided a detailed proof of the stability of nonlinear systems after implementing networked predictive control strategy. Its stability was demonstrated by converting the control input to the actuator side. In addition, the proof outlined above indicates that the stability of the closed-loop system adopting the networked prediction strategy is tantamount to that of the ideal network system without time delays.

Theorem 2.

The stability of the system (45) with networked predictive sliding mode control strategy is equivalent to that of system (44).

Proof. 

The specific proof can be found in [37]. □

Therefore, from Theorem 2, if system (44) is stable, then system (45) with networked predictive control is stable. Recalling (39) and (43), the stability of the discrete SMC has been demonstrated previously, so the networked predictive sliding mode control scheme considering the time delays for the USV is stable.

4. Simulation Results

In this section, we present comparative numerical simulations to illustrate the effectiveness of the developed control strategy. The parameters of the mathematical model for the USV used in the simulation are shown in

Table 1. Model parameters of USV.

Parameter	Value	Unit
m	23.8	[kg]
L	1.225	[m]
$m_{11}$	25.8	[kg]
$m_{22}$	33.8	[kg]
$m_{33}$	2.76	[kg]
$d_{11}$	12	[kg/s]
$d_{22}$	17	[kg/s]
$d_{33}$	0.5	[kg/s]

, which can be found in [40]. The corresponding control parameters were selected, as listed in

Table 2. Control parameters.

Variable	Value
${\Gamma }_u$	10.5
${\Gamma }_v$	10
$\Lambda$	6
${\chi }_1$	1.05
${\chi }_2$	1.15

. The control parameters ${\kappa }_1$ and ${\kappa }_2$ are related to the sliding mode surfaces ${\zeta }_1$ and ${\zeta }_2$, which were determined by (39) and (44). The sampling frequency of the control system was set to 100 Hz, which means the sampling period $T_c$ was 10 ms.

Considering that the complex desired trajectory can always be composed of straight lines and arcs, the desired trajectory was set as a curve consisting of a straight line and a circle. The preset trajectory for the USV was defined as follows.

$$\left\{\begin{array}{c}{x}_D\left(k\right)=\left\{\begin{array}{c}10k{T}_c-2300,\mathrm{if}0\le k{T}_c\le 190\hfill \\ 400\sqrt{2}cos\left(-{\displaystyle \frac{1}{40}}\left(k{T}_c-190\right)+{\displaystyle \frac{3}{4}}\pi \right),\mathrm{else}190\le k{T}_c\hfill \end{array}\right.\\ \\ y_D\left(k\right)=\left\{\begin{array}{c}10k{T}_c-1500,\mathrm{if}0\le k{T}_c\le 190\hfill \\ 400\sqrt{2}sin\left(-{\displaystyle \frac{1}{40}}\left(k{T}_c-190\right)+{\displaystyle \frac{3}{4}}\pi \right),\mathrm{else}190\le k{T}_c\hfill \end{array}\right.\end{array}\right.$$

(46)

Next, simulation tests of the three cases were implemented to better illustrate the usefulness of the developed strategy. The three cases mentioned here are those without delays, with invariable delays, and with random delays.

Case 1: Without delays

For case 1, the delays in the forward and feedback channels of the USV control system were both zero, i.e., ${\rho }_a=0$, ${\iota }_b=0$. According to Section 3.2.1, the following discrete-time sliding mode controller was designed using state information without delay.

$$\left\{\begin{array}{c}{\tau }_u\left(k\right)={\tau }_{ue}\left(k\right)+{\tau }_{us}\left(k\right)\hfill \\ {\tau }_r\left(k\right)={\tau }_{re}\left(k\right)+{\tau }_{rs}\left(k\right)\hfill \end{array}\right.$$

(47)

Case 2: Invariable delays

For this case, it was assumed that the forward and feedback channel delays of the control system were time-invariant. Here, we assumed ${\rho }_a=3$ and ${\iota }_b=3$, respectively. Following (24), (33) and (34), the networked predictive controller is given by

$$\left\{\begin{array}{c}{\tau }_u\left(k\right)={\widehat{\tau }}_{ue}\left(k+3|k-3\right)+{\widehat{\tau }}_{us}\left(k+3|k-3\right)\hfill \\ {\tau }_r\left(k\right)={\widehat{\tau }}_{re}\left(k+3|k-3\right)+{\widehat{\tau }}_{rs}\left(k+3|k-3\right)\hfill \end{array}\right..$$

(48)

Case 3: Random delays

In case 3, the delays ${\iota }_b$ and ${\rho }_a$ were time-varying, and their sum was assumed to be a random integer between 1 and 8. Similarly, the networked predictive controller is

$$\left\{\begin{array}{c}{\tau }_u\left(k\right)={\widehat{\tau }}_{ue}\left(k+{\rho }_a|k-{\iota }_b\right)+{\widehat{\tau }}_{us}\left(k+{\rho }_a|k-{\iota }_b\right)\hfill \\ {\tau }_r\left(k\right)={\widehat{\tau }}_{re}\left(k+{\rho }_a|k-{\iota }_b\right)+{\widehat{\tau }}_{rs}\left(k+{\rho }_a|k-{\iota }_b\right)\hfill \end{array}\right..$$

(49)

In the above three cases, the initial positions for the USV were constructed as $x\left(0\right)=-2350$ m, $y\left(0\right)=-1500$ m, and $\psi \left(0\right)=0$ rad. Correspondingly, the USV’s initial velocities were specified as $u\left(0\right)=0$ m/s, $v\left(0\right)=0$ m/s, and $r\left(0\right)=0.015$$\mathrm{rad}·s^{-1}$, respectively.

For the sake of comparison and analysis, the corresponding result curves for the three cases mentioned above were drawn together. Additionally, to facilitate the demonstration of the superiority and advancement of the control technique developed in this study, a comparative analysis was conducted between the three aforementioned cases and the SMC-TDC strategy proposed in [41].

The simulation results are depicted in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11. From Figure 4, Figure 5 and Figure 6, it can be seen that the developed networked predictive control strategy can effectively achieve trajectory tracking, whether it is dealing with fixed delays or time-varying delays. Figure 5 and Figure 6 illustrate the trajectory evolution in the x and $y-$ axis directions, respectively. It is evident that the control performance of the system suffering from network delay is almost equivalent to that of the system without delay after being compensated by the network predictive strategy, except for a slight difference in the position tracking error. Among all the curves shown in the Figure 7 and Figure 8, the tracking error of the position using the SMC-TDC scheme was the largest.

The mean absolute error $\left(z,\mathrm{MAE}={\displaystyle \frac{1}{t_{\infty }-t_0}}{\int }_{t_0}^{t_{\infty }}\left|z\right|dt\right)$ and root mean square error $\left(z,\mathrm{RMSE}=\sqrt{{\displaystyle \frac{{\sum }_{k=1}^n{(z\left(k\right)-0)}^2}{n}}}\right)$ were used to quantify the error characteristics of the three cases mentioned above; the mean absolute errors and root mean square errors after the system entered the stable tracking state are presented in

Table 3. MAE and RMSE in x and y directions.

Cases	MAE (x)	MAE (y)	RMSE (x)	RMSE (y)
Without delays	0.1398	0.1221	0.1422	0.1466
Invariable delays	0.1402	0.1223	0.1448	0.1489
Random delays	0.1410	0.1230	0.1496	0.1501
SMC-TDC scheme	0.2489	0.2478	0.3039	0.3032

.

Figure 4. The position curve on the $xy-$ plane.

Figure 4. The position curve on the $xy-$ plane.

Figure 5. Evolution of the tracking curve in the x direction.

Figure 5. Evolution of the tracking curve in the x direction.

From Table 3, it is evident that the networked predictive sliding mode control technique can effectively handle the USV tracking control problem under both constant and random delays, with performance close to the system without delays. Compared to the SMC-TDC strategy, the developed approach shows a smaller mean absolute error and root mean square error. Through the analysis of the MAE and RMSE in Table 3, it can be concluded that the strategy proposed in this study can fully eliminate the negative effects of delay, while the SMC-TDC strategy passively tolerates delays while ensuring system stability.

Figure 7 and Figure 8, respectively, depict the position tracking errors in the x and y directions. Furthermore, the analysis from Figure 7 and Figure 8 indicates that compared to the SMC-TDC, the strategy adopted in this study exhibits smaller error fluctuations, suggesting that the system has better dynamic performance.

Figure 6. The position of y changes over time.

Figure 6. The position of y changes over time.

Figure 7. Changes in tracking error ${\epsilon }_x$.

Figure 7. Changes in tracking error ${\epsilon }_x$.

The velocity variation curves of the three cases mentioned above and the SMC-TDC strategy are plotted in Figure 9. The subfigures in Figure 9 clearly reveal that, apart from the SMC-TDC, the velocity trends are almost identical with almost no jitter.

Figure 8. Change in error ${\epsilon }_y$.

Figure 8. Change in error ${\epsilon }_y$.

Figure 9. The velocity changes over time.

Figure 9. The velocity changes over time.

Figure 10. The curve of changes in the control signal.

Figure 10. The curve of changes in the control signal.

Figure 10 illustrates the control signals for the surge thrust and steering torque. The subplot in the figure clearly shows that the control signals of the SMC-TDC strategy exhibit frequent oscillations, whereas the proposed approach is smoother and more practically implementable.

Figure 11. The random delay (Case 3).

Figure 11. The random delay (Case 3).

Therefore, the above analysis and discussion demonstrate the effectiveness and advancement of the control scheme developed in this study for the USV trajectory tracking control with time-varying communication delays.

5. Conclusions

This study contributes to the development of networked predictive control technology, which can address the trajectory tracking control of underactuated USVs with time-varying delays. To deal with the adverse effects of time-varying communication delays, on the basis of the discrete-time sliding mode, a novel network predictive control approach is suggested. Benefiting from the discrete-time virtual velocity control law, the USV’s trajectory tracking control is converted into virtual velocity tracking control, and with the implementation of discrete-time sliding mode control technology, virtual velocity tracking control is achieved. Moreover, a networked predictive control strategy is designed to actively compensate for the time-varying communication delay of the system. For the networked sliding mode predictive control strategy proposed in this work, the comparative simulation results indicate that it not only achieves the expected tracking performance of the closed-loop system but also guarantees its stability. Considering the inevitable presence of unknown external disturbances in an actual USV system during trajectory tracking, future research will combine artificial intelligence (AI) technology and networked predictive control for further study and real ship experiments.