Swarm Control for Multiple Unmanned Surface Vehicles with Unknown Time-Varying Environmental Disturbance and Input Saturation

Xia, Guoqing; Sun, Xianxin

doi:https://doi.org/10.1155/2022/8962970

Journal of Control Science and Engineering

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 8962970 | https://doi.org/10.1155/2022/8962970

Swarm Control for Multiple Unmanned Surface Vehicles with Unknown Time-Varying Environmental Disturbance and Input Saturation

Guoqing Xia¹and Xianxin Sun¹

Academic Editor: Petko Petkov

Received02 Mar 2022

Revised05 Jun 2022

Accepted14 Jun 2022

Published11 Jul 2022

Abstract

This paper addresses the swarm tracking problem of multiple unmanned surface vehicles subjected to unknown time-varying environmental disturbance and input saturation. The main control objective of this paper is that USVs cluster to follow the virtual leader with the desired position and heading and are required to maintain a specified position separation relative to both neighbor vehicles. In order to achieve the design goal, we mainly focus on three aspects. Firstly, to estimate the external disturbance accurately and improve the convergence speed, a finite-time disturbance observer is designed. Secondly, an auxiliary dynamic system is introduced to solve the input saturation problem. Thirdly, an output feedback controller based on a finite-time disturbance observer and an auxiliary dynamic system is designed to achieve swarm control of multiple unmanned surface vehicles. The stability of the system is proved by the Lyapunov directly method. Finally, the simulation results show that the proposed control strategy is effective.

1. Introduction

In recent years, swarm control for multiple unmanned surface vehicles (USVs) has attracted increasing interest in many fields, such as search and track mission [1], rescue operations [2], and dynamic guarding [3, 4]. These have brought new challenges to USV cluster control, especially maintaining a desired position separation relative to what is often required when USVs swarm to perform the corresponding missions [5].

These challenges can be divided into two aspects: unknown time-varying environmental disturbance and input saturation. The environmental disturbance is caused by the wind, waves, and ocean currents in surge, sway, and yaw, respectively. Due to the environmental disturbance being unknown and time-varying, it cannot be measured directly and accurately. Therefore, some control strategies based on estimating disturbance or compensating for disturbance are proposed in [6–11]. Through these proposed control strategies, the observation error only accurately estimates rather than converges to a small neighborhood of an equilibrium state as soon as possible. Furthermore, to improve the convergence speed, the finite-time control technology is adopted in [12–16], which ensures the consistency of all states in the closed-loop system in a finite time. Therefore, the combination of disturbance observer and finite-time control technology is an executable program.

Except for environmental disturbance, another challenge to USVs’ cluster control is input saturation. Since the USV actuator cannot be infinite, there is a deviation between the expected input signal and the actual output of the actuator. Therefore, the amplitude of the control signal is usually limited to a certain range. In the process of designing the actuator, it is necessary to consider the physical constraints of the actuator, that is, input saturation. The existence of input saturation may lead to system oscillation and even system instability. In recent years, the input saturation problem of various systems has received the most extensive attention. In [17], an adaptive mechanism is devised to figure out the input saturation problem. In [18], a novel finite-time control approach is presented to overcome the effects of state constraints on system performance. The group consensus algorithms with input saturation are given in [19]. From the vessels’ work situation, it is very necessary to take input saturation into account in cluster control through the actual working conditions of the multiple USVs.

Based on the above research background, for the first challenge, a finite-time disturbance observer is proposed to resolve the first challenge, which can not only measure the disturbance accurately, but also improve the converge speed. Secondly, by using an auxiliary dynamic system to solve the second problem of input saturation. Furthermore, to solve the above two challenges more perfectly, an output feedback controller is proposed, which is mainly composed of a finite time-disturbance observer, an auxiliary dynamic system, and other control technologies. Meanwhile, it is proved that all signals in the closed-loop system are bounded by the Lyapunov method.

The main contributions of this paper are summarized as follows: firstly, different from existing disturbance observer approaches in [20], the unknown time-varying environmental disturbance can be measured accurately by the finite-time disturbance observer. Secondly, unlike the existing work ignoring the actuator constraints, an auxiliary dynamic system is introduced to resolve the input saturation problem. Thirdly, an output feedback controller is designed, which is mainly composed of a finite-time disturbance observer, an auxiliary dynamic system, and other control technologies.

A summary of recent works is outlined in Table 1, corresponding to different features and classification.

This paper is organized as follows: Section 2 describes some necessary preliminaries and mathematical modeling of USVs. Section 3 depicts the finite-time disturbance observer design. Section 4 describes the output feedback controller design and stability analysis. Simulation results and comparison results are discussed in Section 5 and Section 6 concludes this paper.

Notations: the following notations will be used throughout this paper. denotes all elements that belong to and not to . represents the absolute value of a scalar. denotes the Euclidean norm. represents the dimensional Euclidean Space. denotes a block-diagonal matrix with being the ith diagonal element. and represent the transpose and inverse of a matrix, respectively. denotes the Kronecker product of matrix. and represent minimum and maximum of eigenvalues a matrix, respectively. , represents the Symbolic function, i.e., , ; , ; , . represents the dimensional identity matrix. is used to denote the subscript of USVs.

2. Preliminaries and Mathematical Modeling of USVs

2.1. Algebraic Graph Theory

Graph theory is used to describe the communication topology of follower USVs and a virtual leader vehicle (denoted by 0). A directed graph consists of a vertex set and the set of edges . A directed edge is not only the incoming edge of node but also the outgoing edge of node . If , node is an adjacent node of node . The set of all adjacent nodes of node is represents by , see [21].

Consider a directed graph composed of nodes, the adjacency matrix is used to represent the link relationship between nodes, where , if ; , otherwise. If , the graph is undirected; otherwise is directed. The Laplacian matrix associates with the graph is defined as where with . Laplacian matrix always has a right eigenvector of associated with eigen value .

In particular, a diagonal matrix is defined as a leader adjacency matrix, where , if and only if the th USV receives information from the virtual leader vehicle; , otherwise. Finally, the information exchange matrix is defined as .

Assump 1. The graph is directed, and there is at least one spanning tree from the root node to the leader node, i.e., the is a positive definite matrix.

2.2. Finite Time Stability

Lemma 1 (see [13]). Consider the system of differential equationswhere is continuous on an open neighborhood of the origin. A continuously differentiable function is said to be a solution of formula (1) on the interval if satisfies formula (1) for all .
Suppose there exists a continuous positive definite function , real numbers , and an open neighborhood of the origin such that

Then the origin is a finite-time-stable equilibrium of (1).

Moreover, if , is proper, and takes negative values on , then the origin is a globally finite-time-stable equilibrium of (1). And is called the convergence-time function, it satisfies

2.3. USVs Modeling

Consider a group of USVs swarm consisting a virtual leader vehicle (subscript is 0) and follower USVs (subscript is ), the 3 degrees of freedom (DOFs) kinematics and dynamics equations of the th USV can be expressed in vector form as follows:[23]where is the rotation matrix, it is given as follows:with properties: and . is the position and yaw angle in the earth-fixed frame (see Figure 1). is the velocity vector in the body-fixed frame . The system inertia matrix is positive definite and constant, where and . The damping matrix is also symmetric and positive define. is the control input, which is produced by the propellers. is unknown time-varying environmental disturbance, which caused by the wind, waves, and ocean currents in surge, sway, and yaw, respectively.

In this paper, considering the input saturation constraint, the control forces and moment produced via the propellers are limited. The input saturation constraint can be described as follows:where and are the maximum and minimum control forces and moment of th vehicle, respectively. is calculated by the output feedback controller.

The main goal of this paper is to design a control law so that the USVs can track the desired reference point while maintaining a fixed formation, i.e.,where is the desired reference point. represents the expected relative deviation position and heading between the th USV and the desired reference point. In order to maintain fixed formation for USVs cluster, thus is a constant vector. is a positive constant.

Assump 2. The reference signal is smooth and differentiable everywhere. Its first derivative and second derivative exist and are bounded.

Assump 3. The time-varying environmental disturbances is unknown but bounded, and its first derivative exists and is bounded, i.e.,where and are positive constants.

Assump 4. The system inertia matrix and damping matrix are known.

2.4. Environmental Disturbances Modeling.

Unmodeled external forces and moment due to wind, ocean currents, and second waves are lumped together into an earth-fixed slowly-varying bias term [24, 25]. A widely used bias model for USVs is the first-order Markov process [26–28]. In this paper, the environmental disturbances are modeled as follows:where represents the first-order Markov process, it is given as follows:where is the diagonal matrix of positive bias time constants, is the diagonal matrix scaling the amplitude of the , is a vector of zero-mean Gaussian while noise.

3. Finite-Time Disturbance Observer Design

Owing the environmental disturbance is unknown and time-varying, the accurate value is difficult to receive directly. To solve this problem, a finite-time disturbance observer (FTDO) is designed as follows:where is the estimate of the unknown time-varying environmental disturbance , is the estimate of .

A new variable is defined as follows:

The update law of FTDO is obtained as follows:where and are positive define diagonal matrices. and are positive constants and satisfy and , respectively.

Define the estimation error of disturbances . The time derivative of (13) is given as follows:

Define a new variable vector , where . Thus, the formula (15) is rewritten as follows:

If a new closed-loop system is constructed by the formulas (16) and (17), then and can be regarded as the internal states of this closed-loop system. Under these circumstances, if and in finite time, then and in finite time.

Therefore, the following theorem holds.

Theorem 1. The disturbance observer composed by the formulas (12) and (14), the unknown time-varying environmental disturbance can be estimated in finite time, and the disturbance estimation error converges to a neighborhood of the equilibrium point in finite time.

Proof. The Lyapunov function is chosen as follows:where the vector and positive define matrix are designed as follows, respectively.Note that Lyapunov function is continuous and differentiable everywhere except and positive definite. Then, we getwhereThe time derivative of Lyapunov function is given as follows:whereFrom formula (21), the following inequality holds:Substituting the formula (24) into (22), the inequation (22) is written as follows:where .
To ensure the coefficient , the following inequality is holdsFrom formulas (20) and (26), the term is bounded, i.e.,whereAccording to the Lemma 1, the new closed-loop system composed of formulas (16) and (17) converges to a neighborhood of the equilibrium point in finite time.
This completes the proof.

4. Output Feedback Controller Design

In this chapter, inspired by references [29, 30], the output feedback controller is designed. The structure diagram of multiple unmanned surface vehicles swarm control is shown in Figure 2.

4.1. Auxiliary Dynamic System Design

In this subsection, the auxiliary dynamic system (ADS) is introduced to settle the input saturation problem [22]. For ADS, as shown in Figure 2, the input of the system is the difference value between the control forces and moments with saturation and those without saturation constraint . The output of ADS is the compensation of position and heading tracking error and the compensation of velocity tracking error for th USV. The purpose of designing ADS is to compensate for the specified variables in the closed-loop system.

Among them, and are position and heading compensation and velocity compensation for and , respectively. The ADS is designed as follows:where and are the output states of the ADS. and are diagonal positive matrices. . , , is a positive constant.

4.2. Output Feedback Controller Design

In this subsection, the output feedback controller of multiple USVs is designed by using dynamic surface control technology [20]. The design process of output feedback controller is divided into the following steps.

Step 1. The first tracking error of the th USV in earth-fixed frame is defined as follows:where , and are defined in Section 2.2. , and are explained in formula (8), and have similar definitions.
The time derivative of , and using the formula (4) and (29), we obtain as follows:Choosing as a virtual input in formula (32), by using dynamic surface control technology, the kinematic control law is designed as follows:where is the positive definite diagonal matrix.
From the kinematic control law obtained, it can be found that it is complex to calculate the time derivative of the . Therefore, a first-order low-poss filter is introduced to settle the matter.
Let pass through a first-order low-poss filterwhere is a time constant, is the output vector of the first-order low-poss filter.

Step 2. According to the velocity of the th USV, the output of the first-order low-pass filter, and the state of ADS, the second tracking error of the th USV is defined as follows:By differentiating and using the formula (5) and (30), we get the formula as follows:Furthermore, the term in formula (36) is replaced by the estimated value of unknown time-varying environmental disturbance produced via the FTDO. Then, by using dynamic surface control method, the dynamic control law is designed as follows:where is the positive definite diagonal matrix.
By substituting formula (33) and (37) into (31) and (35), the error subsystem of the th USV is obtained as follows:where , is explained in formula (15).
According to the definition of vector , by differentiating , and using the formula (34), we get the equation as follows:By integrating the two sides of formula (40), we have the following formula:Further calculation, the following inequality holds:Owing to the control input of adjacent USVs are bounded, i.e., . In fact, since all systems are energy consuming system, the output of all USVs is bounded. Therefore, the following inequality holds:Therefore, is bounded and .

4.3. Stability Analysis

Theorem 2. A close-loop system is considered, which is composed of USVs cluster with input saturation, FTDO (12) and (14) with unknown time-varying environmental disturbance, ADS (29) and (30), output feedback controller (33) and (37). The swarm control scheme guarantees that all error signals in this close-loop system are bounded, and the tracking error converges to a neighborhood of the equilibrium point in finite time.

Proof. Consider the above closed-loop system, the Lyapunov function is chosen as follows:The time derivative of , we obtain the following equation:From the formula (32), and using Young’s inequality, we get the following equation:Similarly, the following inequalities hold:Substituting the formulas (46) and (47) into (45), then (45) is rewritten as follows:From the FTDO, the term converges to zero in finite time. According to the formula (43), . For the th USV, is a known positive constant matrix. is defined in Section 4.1. Thus, we have the following equation:where is bounded, i.e., .
To ensure the stability of the closed-loop system, the parameters , , , and should meet the following conditions:where , , , and are positive constants.
Substituting the formulas (49) and (50) into (48), then (48) is rewritten as follows:where .
From the formula (51), either , or , or , or , then holds. This implies that:It can be concluded that all signals in the closed-loop system are bounded. Thus, the values of variables and are given as:Next, it will be proved that the USV cluster can track the reference signal with the desired relative deviation.
The tracking error of the th USV in the earth-fixed frame is defined as , it satisfies the following constraints:According to the formula (31), we obtain:where , , and . is defined in Section 2.2.
According to the Assumption 1 of algebraic graph theory in the Section 2.2, it can be concluded that all eigenvalues of matrix have positive real parts. Furthermore, using the formulas (53) and (54), we get the following equation:According to the formulas (51) and (57), by adjusting the parameters of controller and disturbance observer, the upper bound of tracking error can be reduced and finally converges to a very small neighborhood close to zero.
This completes the proof.

5. Simulation Results

In this section, a USV swarm consisting of one virtual leader vehicle (indexed by 0) and six follower USVs (indexed by ) is considered to demonstrate the effectiveness of the proposed control strategy. The directed communication graph is shown in Figure 3.

In simulations, the model of the surface ship Cybership II is used [31]. The time-varying environmental disturbances are modeled as first-order Markov processes [24]. The control forces and moment are limited as , and . The desired reference point is set as . Some parameters setting are shown in Table 2.

5.1. Performance of Proposed Control Strategy

In this subsection, the simulation results are given to verify the performance of proposed control strategy.

The parameters of environmental disturbance are selected as , , , . The parameters of observer are selected as , , and . The design parameters of controller are chosen as , and .

The simulation results are shown in Figures 4–8. Figure 4 shows the trajectories of six USVs under the constraints of time-varying environmental disturbance and input saturation. It can be seen from Figure 4, the position and heading of six USVs can track the reference signal in the form of swarm motion and maintain the desired relative position with each other, although there is a deviation in the initial position of each USV. Figure 5 depicts the position and heading tracking errors of six USVs. It can be seen from the figure that the tracking errors of six USVs are convergent. Figure 6 shows the control forces and moments of six USVs with an auxiliary dynamic system. Through the analysis of Figure 6, the control inputs of six USVs are limited within the range of input constraints, and finally the constraint of input amplitude saturation is realized. Figure 7 depicts the true value of the time-varying environmental disturbance acting on the 1st USV and the estimated values obtained by methods FTDO and MDA, respectively. Figure 8 shows the disturbance error of six USVs between the actual value and estimation value via the FTDO. The specific values of simulation test and comparison test are shown in Table 3.

5.2. Comparison Group

The first comparison simulation is carried out by using a modular design approach (MDA) proposed in [20].

Simulation results on the proposed predictor module for estimating the value and estimation error of the unknown ocean disturbances are shown in Figures 7 and 9. Figure 7 shows the different time-varying environmental disturbance estimations of the 1st USV about actual value, MDA, and FTDO, respectively. Figure 9 shows the disturbance error of six USVs by using MDA. From Figures 7–9, the control inputs of two methods are both limited, but the more accurate estimation of the environmental disturbance is obtained by FTDO.

The other simulation results of the proposed state feedback controller without the auxiliary dynamic system and input saturation are illustrated in Figures 10 and 11, Figure 10 shows the control input of six follower USVs without considering the auxiliary dynamic system. Figure 11 shows the control input of six follower USVs without actuator constraint. Through comparing Figures 6, 10, and 11, the auxiliary dynamic system can effectively finish off the input saturation, and it is very necessary to take the physical constraints of the system into account when designing the controller.

6. Conclusion

This paper investigates swarm control for multiple USVs in the presence of an unknown time-varying environmental disturbance and input saturation. Firstly, an observer is designed to estimate disturbance by using finite time control technology. Secondly, an auxiliary dynamic system is introduced to finish off the input saturation problem. Thirdly, an output feedback controller is designed, which is mainly composed of a finite-time disturbance observer, an auxiliary dynamic system, and other control technologies. At the same time, the stability of the system is proved by the Lyapunov method. Finally, the effectiveness of the proposed control strategy is verified by simulation results.

However, in the design process of the controller, only input amplitude saturation and time-varying environmental disturbance are considered: collision avoidance and obstacle avoidance are ignored. Safe, reliable, and collision-free track planning is an important research direction in the field of unmanned surface vehicles. This is the focus of our future research.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This study is funded by the 7th Generation Ultra Deep Water Drilling Unit Innovation Project.

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Copyright

Copyright © 2022 Guoqing Xia and Xianxin Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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