Abstract
An implementable robust containment control algorithm is proposed for a group of underactuated ships in the presence of hydrodynamic parameter uncertainties and external disturbances. The control objective is to drive all the followers into the convex hull spanned by the virtual leaders, whose state information is available only to a subset of the followers. For this purpose, the ship model is primarily transformed to a strict-feedback form. In the kinematic design, a virtual containment controller, requiring the state information from its neighbors, is presented based on the results obtained from graph theory. In the dynamic design, a robust containment controller is developed through utilizing upper-to-up sliding mode control. In addition, in order to simplify the implementations of the control law, the command filtered backstepping (CFBP) method is introduced to prevent the analytic differentiations of the virtual law from each design step of the backstepping (BP) method. Subsequently, it is well proven that all the tracking errors could converge to and remain small neighborhoods of the equilibrium point. Finally, several simulation experiments are conducted to demonstrate the performance of the proposed control algorithm.
1. Introduction
Over the past few years, a huge and rapidly growing body of research has focused on the cooperative control of the multiple vehicle systems within a motion-control community because of their broad applications, for example, wheeled mobile robot, manipulator, spacecraft, and ship [1–4]. Although an individual vehicle can be used to complete a specific task, some benefits including greater robustness, lower cost, and better performance can be achieved through a group of vehicles working cooperatively. In the marine domain, many applications using cooperative control include various tasks such as environmental monitoring as well as rescue and reconnaissance operations [4–7]. To perform these tasks, an increasing number of studies have focused on the cooperative control of multiple ships.
These studies have included behavior-based method [8], virtual structure method [9–14], and leader-follower (L-F) method [4, 5, 15–19]. On the basis of null space methods, a formation control strategy has been developed to achieve multiple tasks in [8]. In [15], an output synchronization control strategy has been designed for the L-F control of ships. The authors in [9, 10] introduce passivity theory as a practical design tool for solving the formation problem, which includes group coordination and path following. A robust cooperative control method has also been applied to form a desired geometric pattern with the aid of a neural network (NN), a backstepping (BP) method, and graph theory [11]. In [8–11, 15], the vehicles are assumed to be fully actuated.
In actuality, most ships are underactuated, meaning that they are not actuated in the sway axis [20]. Since the dynamics of each underactuated ship do not satisfy the necessary conditions of Brockett for stabilization [21], the cooperative control law for each ship cannot be smooth functions with theirs variables [16]. In [12], a decentralized formation control method has been presented to address such an issue by using the nonlinear cascaded system theory and line-of-sight guidance to ensure a straight-line path following for the formations of ships. On the basis of Lyapunov theory, the BP method, and the graph theory, a path-following coordination controller (PFCC) is proposed in [13] and its control strategy also considers time delayed communication among ships. The work of [14] is extended further in [13]. In [14], a dynamic surface control (DSC) technique is proposed to estimate the virtual control at each step of the BP method. An additional virtual control is adopted to solve the difficult problems when designing the formation controller for ships in [22]. The L-F method for ship has been also reported in [4, 5, 16–19]. In [16, 17], continuous time-varying cooperative control laws are designed to perform a geometric pattern by using suitable transformations. It is noted that, in [16, 17], the yaw velocity is assumed to be nonzero, which is referred to as the “persistent excitation” (PE) condition. Under this condition, a straight line cannot be tracked for underactuated ships. The authors in [18] address the design of the nonlinear model predictive formation controller, where the relative distances and orientations between the follower and the leader can be stabilized. To cope with the uncertainties of the model, two different robust L-F formation control laws have been designed that combine NN with the DSC technique in [4, 19]. More recently, a robust formation control algorithm is proposed to force ships to maintain the desired orientations and positions relative to one leading vessel, considering the limited magnitude of the control signal in [5]. In [4, 5, 19], the radial basis function NN and adaptive control method can approximate the nonlinear uncertainty systems, which increases the complexity of the online computing. In addition, in all the works on L-F formation control, a common trait is that only one leader exists in the group.
In practical applications, for a group of vehicles multiple leaders might exist. In this case, the control objective is to drive the followers of the group into the convex hull spanned by the leaders, which is called the containment control problem [23]. Thus, containment control can be regarded as a special L-F formation control, and the study of containment control is motivated by its possible applications. For example, a group of ships are guided by another group, and some ships are regarded as leaders, which are all equipped with sensors to detect the hazardous obstacles in the marine environment [24]. As such, the leaders can form a safe area in which the followers can converge [25]. Recent works on the containment control of multiple vehicle systems focus on single integrator systems [25], double integrator systems [26], strict-feedback form systems [23], general linear dynamics [27], and Lagrange dynamics [28]. However, these proposed methods cannot be applied directly to an underactuated mechanical system, and the uncertainties from the model might affect the control performance. Accordingly, how to achieve the containment control objective in the presence of uncertain dynamics and external disturbances needs to be investigated further.
For nonlinear systems, various methods have been adopted, and, among these, the BP method can be regarded as a major design tool. However, this method suffers from the “explosion of complexity” problem, which is caused by the repeated derivatives of the virtual control signals [29]. To overcome this drawback, the command filtered backstepping (CFBP) method, which introduces, at each design step of the BP method, a command filter to prevent the derivative of nonlinear functions, has also been applied to the strict-feedback form system in [30, 31]. The CFBP method is regarded as an improved version of standard BP method and is widely used in different fields [32–34]. However, these works cannot prove the boundedness of the tracking errors, which implies that the stability of the closed-loop system cannot be analyzed quantifiably. In addition, most current CFBP adopted second-order command filter (SOCF), and the first-order command filter (FOCF) with simpler structure is rarely mentioned.
In conclusion, the containment control of underactuated ships involved the following main difficulties: () the underactuated surface ship lacks control input at side, so it belongs to a class of underactuated system; namely, the general nonlinear control theory could not be directly applied; () in complex sea cases, ship is often influenced by the external disturbances and parameter uncertainties (EDPU), which makes it complicated to accurately control the motion of ship; () the current research results on containment control are applied only in full-actuated system such as single integrator systems, double integrator systems, and Lagrange systems. However, it is not mentioned how to achieve containment control by combining the knowledge of graph theory and the underactuated system, which is one of the research difficulties. In this paper, a robust containment control strategy, inspired by the previous works, which is performed by using the standard BP method, the FOCF technique, the sliding mode control (SMC) method, the Lyapunov stability theory, and the results from graph theory, is developed for a group of ships in the presence of the EDPU. Compared with the existing results, the main features of this paper can be summarized as follows: () for the first time, this paper considers the containment problem of underactuated ships; () by incorporating the FOCF technique, the BP commands are simplified; () different from the previous conclusions about SOCF [30, 32–34], the stability of all the closed-loop systems is first analyzed quantifiably; () by introducing a polar coordinate, the model is transformed to a strict- feedback form system, and the PE condition from [16, 17] is avoided; () compared with the results in [4, 5, 19], the development, using upper-to-up SMC to design a robust containment controller, is more practical because the online computation of the uncertainties is avoided.
2. Control Problem Formulation
2.1. Notations
To prepare for the subsequent control design, some notations are standard as below: denotes a set of Euclidean matrices, and denotes -dimensional Euclidean space; denotes a diagonal matrix with entries ; the maximum and the minimum eigenvalues of a square matrix are described as and , respectively; let be the absolute value of a scalar; let be the Euclidean norm of a vector; let be the Kronecker product of matrices and ; denotes that is defined as .
2.2. Concepts in Graph Theory
Suppose that a group of underactuated ships interact with each other through a communication network to perform a containment control task. It is natural to model the communication topology among these ships by using graph theory. Without loss of generality, we assume that the individual ship is a node, and the interaction for ships can be described by a directed graph , where denotes a set of nodes and denotes a set of edges with element , which implies that node can receive information from node . Here, we say that node is a neighbor of node , and the notation is described as the set of all neighbors of node. Let the adjacency matrix be defined such as if and otherwise; Note that we assume for all the nodes. The Laplacian matrix of the directed graph is defined such thatA directed path from to in the graph is a set of edges: , where all the nodes in this path are different [35].
Definition 1. For a group of ships, ship is said to be a leader if , and ship is said to be a follower if , .
Definition 2. The real set is said to be convex if, , there exists a point that satisfies for any . The convex hull for a set of points in is the minimal convex set including all the points in and let be the convex hull of . In particular, is defined as follows [36]:
2.3. Underactuated Ship Modeling
Suppose that there are identical followers, labeled as ship 1 to . According to [37], for the containment control task we can neglect the motions in heave, pitch, and roll; hence, for a ship, the 3-DOF mathematical model can be formulated as (see Figure 1)wherewhere (, ) and denote the position and heading angle of the ship in the inertial reference frame ; , , and denote surge, sway, and yaw velocities expressed in the vessel-fixed reference frame , respectively; , , and describe ocean environment disturbances; and are used to describe actual control inputs; , , , , , , , and denote hydrodynamic damping parameters; , , and denote ship inertia including added mass.
Because no control signal is introduced in the sway direction, this model is an underactuated system. Inspired by the work of [38], we define the following polar coordinates to solve this problem:where is often recognized as side-angle and and are recognized as the speed and the course angle of ship . Then, substituting (5) into (3) yields
Then, suppose that the group contains leaders in the group, labeled as leaders to . To describe the communication among the followers and leaders in the group, let denote the Laplacian matrix. As each leader has no neighbors, can be described by the following:where and .
Assumption 3. The environment disturbances and , , and satisfywhere [38].
Assumption 4. For every follower of the group, there exists at least one leader that has a directed path to it [36].
Assumption 5. The trajectories of the virtual leaders satisfywhere , , and are all positive constants.
Remark 6. () Assumption 3 means that the disturbances induced by winds, waves, and currents have finite energy; () Assumption 4 indicates that the proposed control law relies just on a directed communication topology, which reduces the complexity of the communication among ships; () the boundedness of the first derivatives and second derivatives of the leaders will be introduced to analyze the stability of the closed-loop system.
Remark 7. Because the surge control in (7) is available only for the nonzero value of , the condition can guarantee that the side-angle satisfies ; that is, . In the next part, we will give a certain initial condition for ship such that is positive at all times.
Remark 8. Using the polar coordinate transformation (5), the underactuated models (3) are transformed to a strict-feedback form of (6) and (7). Then, one can apply the standard BP method to design the control input signals and , which achieves the containment control objective.
2.4. Command Filter
To overcome the problem of “explosion of complexity” which occurs in a strict-feedback form [39], a FOCF is introduced to eliminate the analytic differentiation of the virtual control signal in the standard BP design procedure [30]. The standard state space form of FOCF is presented aswhere the FOCF parameter is a constant and satisfies ; and are estimates of the input and its derivative, respectively; the initial conditions of states and are and ; the following lemma proves the input-to-state stability property of FOCF which will be used in Section 3 to analyze the stability of the closed-loop system.
Lemma 9. Consider the FOCF described by (11). Assume that the positive constants and exist, such that , for all . Then, the following properties of the FOCF can be given:(1) is bounded and satisfies(2) is bounded and satisfies
Proof. Define the tracking error . By using (11), one can obtainConsider a Lyapunov function candidate (LFC)Then, the time derivative of (15) isUsing inequality (16), one obtainsThus, for all ,Because , it follows that is bounded and satisfiesFrom (11) and (18), one obtainsThat is, and are bounded for all .
Remark 10. For a given input , the larger value of the FOCF parameter will decrease the estimated error , which leads to better final estimate accuracy.
Remark 11. To address the problem of excessive estimation error in the initial time, we assume that the FOCF initial values are in this paper. Combined with (11) and (18), the following inequalities hold for all :
2.5. Control Formulation
This article aims to design a containment control scheme such that all the followers in the group move into the convex hull formed by the leaders [28, 40]; that is,where is the position of the follower ; ; denotes the distance from a point to a set ; that is,
Lemma 12. Under Assumption 4, all the eigenvalues of have positive real parts. Furthermore, each of is nonnegative, and all the row sums of this matrix are equal to [41].
Remark 13. Let be the positions of the dynamic leaders. From Definition 2 and Assumption 4, Lemma 12 implies that every point of is in the convex hull spanned by the leaders .
With this previous notation, the control problem under study can be formulated as below.
Control Objective. Consider the model given by (6) and (7) under Assumptions 3–5, and the objective of this work is to design a robust cooperative controller for each ship on the basis of its local states and the information from neighbors and a portion of the leaders such thatwhere is a positive constant, denotes the actual position of follower , and denotes the desired position of follower which satisfieswhere .
Remark 14. Because of the movement of the ship affected by the ocean environment and the filtering error induced by FOCF, the tracking error cannot converge to zero but can be uniformly ultimately bounded by , which will be discussed in detail in the next part.
3. Control Design
3.1. Kinematic Loop Design
Step 1. First, to prepare for the design, the following error variables are introduced:where and denote the virtual control laws of and , respectively. Then, it is convenient to expand (6) intowhereIt is obvious that is the virtual control law for (27) which will be specified later. To solve the control problem (24), based on its local states and the information from neighbors and a portion of the leaders is designed as below:where is a constant. To make the presentation clearer, the following notations are used:Then, the following lemma is presented.
Lemma 15. Consider the model described by (27) canceling the terms , with the virtual control law (29). If the proposed control system satisfies Assumptions 4 and 5, the following properties can be given: (1) the control objective (24) is achieved; that is, , as , where is defined as (25); (2) the positive constants and exist, such that and for all , respectively; (3) the norm is strictly positive for all if the control parameter satisfieswhere .
Proof. This proof is divided into three parts.
() By substituting (29) into (27), canceling the term , and multiplying both sides of (27) by , one obtainsFrom (32), it follows thatHence, defining , (33) can be rewritten in a matrix form as below:where and . From (34), it can be observed that exponentially converges to zero as . Because exists [42], converges to as ; that is, the objective (24) is reached.
() Multiplying both sides of (34) with results in the following matrix form:From (30) and (35), it follows thatBy use of the triangle inequality, one obtains the upper bounds of the virtual control and its derivativewhere and are positive constants. Thus, and are bounded for all .
() Then, with the aid of the triangle inequality, the norm of can be expressed as follows:where is a constant. Clearly, if condition (31) holds, it would imply the strict positiveness of the norm of for all . This calculation completes this proof.
Remark 16. To facilitate the analysis of the properties of the proposed containment control law (29), the tracking error variable is omitted, which will be considered in the following BP design procedure.
At this point, one should also notice that the design of this control system will utilize the virtual control of and in the following design steps. To facilitate the computation of the virtual control, let in (29) be expressed aswhereThen, combing with (28), (29), and (39), the expressions of and can be given by
Lemma 17. For and described by (41), if in (36) satisfies condition (31), then the positive constants , , , and exist, such that
Proof. From (28), is bounded byDifferentiating both sides of the first equation of (41) givesBy use of Lemma 9, one can obtainAccording to the expression of in (41), is bounded byAs inequality (45), satisfies the following:This calculation completes this proof.
Step 2. In this work, the BP method is applied to the strict-feedback nonlinear system of (6) and (7). However, it uses the derivatives of the virtual control such as and , which increases the complexity of calculations and needs the acceleration information from its neighbors. To avoid the problems described previously, FOCF is introduced to estimate each virtual control law and its derivative [30, 43].
To facilitate the expressions of the variables involved in filtered BP method, a particular notation is introduced in this paper. For example, let go through FOCF depicted in Figure 2 to obtain and , which represent the estimated values of and , respectively. Similarly, we can also define the variables , , , and . For the sake of the design work, the following tracking errors are defined as
Remark 18. It is noted the boundedness of , , , and ensures the boundedness of , , and by Lemma 9. According to Lemma 17, , , and are bounded by , , , and , respectively, and then the constants , , , and exist and satisfy
With (48) applied, the kinematic model (6) can be rewritten aswhere , , , , , and are defined as (28), . Then, substituting (29) into (50) and multiplying both sides of (50) with giveClearly, if the last two terms converge to zero, exponentially converges to zero. To prepare for the following step, define the compensating tracking error vector aswhere , , and denote the compensating signals of , , and , respectively. In order to remove the effect of the nonlinear term in (51), a compensating signal is selected aswhere the initial conditions are and . From (51), (52), and (53), one obtainsnoting that the last nonlinear term of (54) will be compensated in the next step. Then, considering the first LFCTaking the time derivative of along the solution of (54) gives
Step 3. At this step, will be considered as a control input to stabilize the tracking error . With notation (48), the kinematic model for can be rewritten as below:where is the virtual control for . To stabilize the tracking error in (48) and to compensate the last term for (56), the virtual control law is designed aswhere is a constant. Then, from (34) and (35), it follows thatSelect the compensating signal for aswhere . By the definition of in (52) and (59) and (60), the compensating tracking error can be expressed asConsider the second LFC:whose time derivative along (56) and (61) isIt should be noted that the last two terms of (63) will be compensated in the dynamic design.
3.2. Dynamic Loop Design
This section aims to design the robust control inputs and which make the compensated tracking errors and asymptotically stable.
Step 1 (surge control design). The definition of in (52) is chosen as the sliding surface:Differentiating both sides of (64) along (7), one can obtain dynamics asTo ensure that is always equal to zero, the equivalent control input can be obtained by solving without the parameter uncertainties and external disturbances. Thus, can be chosen aswhere “” denotes the estimated ship parameters. It could be easily found that the equivalent control input cannot guarantee when the EDPU are considered; to eliminate the effect of the uncertainties of the control system and to ensure the convergence of to zero, an additional control input is introduced:where is a positive constant and is the control parameter, determined from the following bounds of the ship parameters in (7):Moreover, to compensate for the last term of (63), a compensated control input is set to beCombined with the previously mentioned control input, the actual yaw control input can be chosen asTo determine , consider the LFC for aswhose time derivative along (65) and (70) isThen, a reasonable choice for can be given asConsidering the third LFC and taking the time derivative of along (63) give
Step 2 (yaw control design). Similar to Step , a sliding surface for is introduced:Taking the time derivative of along (7) results inSimilar to the previous step, the equivalent control input can be designed aswhere “” denotes the estimated ship parameters. To compensate for the last term of inequality (74) and reject the uncertainties, the actual surge control input can be chosen aswhere is a constant; can be determined by the following bounds of the ship parameters in (7):and, to determine , we consider the LFC for aswhose time derivative along (76) and (78) isAccording to (80) and (82), computing the first-order derivative of results inTo stabilize at the origin, can be defined asSubstituting (84) into (83), one obtainsConsider the overall LFC ; take the time derivative of along (74) and (85) gives
3.3. Stability Analysis of Closed-Loop System
This section consists of four steps.
Step 1 (boundedness of and ). By using equalities (68) and (80), (86) can be derived aswhere .
Then, one can obtainFrom equality (88), one can conclude that , (where ) for all .
Step 2 (boundedness of ). The definition of in (52) and the definition of in (48) can be rewritten asThen, the bounds of , , and giveIt is obviously concluded that the strict positiveness of can be guaranteed if satisfies the conditionThus, ; that is, , for all . Because the determinant of in (78) is given by the term , condition (91) can guarantee that are well defined for all .
Remark 19. Notice that, with being bounded by 0.5π, the constants and exist, such that , all the time.
Step 3 (boundedness of ). Prior to analyzing the boundedness the boundedness of , should be discussed first. From the definition of in (58) and the Young equality, one obtainswhich implies the constant exists which satisfies . Combining with Lemma 9, it is seen that , where is a constant. Then, consider the LFC aswhereand , , and satisfyWith equality (93), one obtainsIt could easily be found from equality (96) that for all .
Step 4 (implementation of the control objective). By noting that, with the boundedness of , , the signal in (30) is bounded by , define the vector asThen, the upper bound for the norm of can be obtained as below:Because is invertible, equality (98) means thatThus, is bounded by for all , where is defined in (25). Subsequently, the containment control objective (24) is solved.
Theorem 20. Consider a group of ships given by (6) and (7), and suppose Assumptions 3–5 hold. For each follower, consider the robust cooperative controllers (70) and (78) and FOCF (11); then, the tracking errors are globally uniformly ultimately bounded (GUUB), if the control parameters satisfy conditions (31), (91), and (95).
Remark 21. In [16, 17, 44], the yaw velocity is required to be nonzero, which means that a straight-line reference cannot be tracked for ships. However, the PE condition is very restrictive from a practical point of view. By using a polar coordinate transformation, the proposed control strategy for ships can track not only straight-line paths but curved-line paths as well.
Remark 22. It is important to note that the traditional DSC method estimates the value of the virtual control law and its derivative only through first-order filter and fails to adjust its estimate through feedback. By contrast, FOCF method introduces compensation tracking error signal on the basis of traditional DSC method, realizes the real-time correction of virtual control law, and greatly improves the accuracy of the estimation.
Remark 23. The authors in [45] also adopt CFBP method to design the trajectory coordinated control strategy and use SOCF to estimate the virtual control law in the BP design procedure online. However, SOCF belongs to a class of second-order systems, and it is more tedious to implement than FOCF. It should be mentioned that there are many deduction errors in [45]. In addition, the authors in [45] adopt the adaptive method to estimate the hydrodynamic parameters online in the dynamic design, but it is difficult to implement in practice. Therefore, in the paper, we refer to the design structure framework of literature [45], use first-order filter to estimate the virtual control laws, solve the containment control objective, and analyze the stability of the closed-loop system by using the input-output boundedness property of FOCF. Moreover, in the dynamic design, upper-to-up sliding mode method is used to design the actual control inputs of ship. Rather than online estimation of hydrodynamic parameters, the method only needs to use the boundedness of hydrodynamic parameters and external disturbances and has the advantages of simple structure and strong robustness.
3.4. Stability Analysis of Sway Motion
Since there is no available control input in the sway direction of the individual ship (3), it is necessary to analyze the boundedness of the sway velocity . Consider the following LFC:From (100), the time derivative of can be obtained as
Because , , , and are bounded as shown in Section 3.2, one can obtainwhere It could be easily found that if , . That is to say, the sway velocity is uniformly ultimately bounded and will eventually converge to the invariant set .
4. Simulation Results
To verify the effectiveness of the proposed robust cooperative scheme, a networked group that consists of the following five ships with three virtual leaders is considered. The communication topology among the ships is shown in Figure 3 with and . It can be noted from Figure 3 that the network topology has a directed spanning tree. Assume that all the following ships have the identical structure and that mathematical model parameters are presented in Table 1. To satisfy conditions (31), (91), and (96), the cooperative control parameters for each following ship are chosen as follows:and the FOCF parameter is selected as . Next, four experiments will be presented to verify the effectiveness of the proposed algorithms.
4.1. Straight-Line Tracking without EDPU
In this section, an example validates the proposed containment control strategy without considering the disturbances for the followers. The predefined trajectories of the virtual leaders are chosen as , , and . The initial follower states are ; ; ; ; . It is noted that when the trajectories of virtual leaders are straight lines, the desired trajectories are straight lines. The simulation results are shown in Figures 4–7. Also, the total simulation time is 200 s. Figure 3 shows the motion traces of the followers. As shown in Figure 4, the five followers converge to the convex hull formed by those of the moving leaders. Figure 5 demonstrates that the norms of the tracking error for each follower converge asymptotically to zero as , which reveals that the containment control objective (24) is solved. Figure 6 shows the actual control inputs of each follower. Figure 7 shows the time evolution of the FOCF estimated errors:It is obvious from Figure 7 that FOCF can accurately estimate virtual control laws.
4.2. Straight-Line Tracking with EDPU
In this section, the robust containment control strategy in the presence of the disturbances is also examined. The trajectories of the virtual leaders are chosen as Section 4.1. The EDPU for the followers in the inertial reference frame are assumed to bewhere . The initial follower states are chosen as Section 4.1. The model parameters, used for simulation experiment, are given in the following, assuming that the model parameters are 1.5% parameter perturbation:The simulation results are depicted in Figures 8–11, which exhibit almost the same control performance as those given in Section 4.1. Figure 8 shows the motion traces of the five followers, from which it is evident that the followers also converge to the convex hull formed by those of the moving leaders. The norms of the tracking errors are plotted in Figure 9, and it reveals that the tracking error for each follower are bounded to a small neighborhood of the origin as . It is clearly seen from Figures 5 and 9 that the proposed controller yields slightly worse performance than the previous section due to the EDPU. However, the tracking errors are bounded, which means that the tracking errors for each follower are GUUB as proved in Theorem 20. Figure 10 shows the actual control inputs applied to each follower under the EDPU, whereas Figure 11 shows the time evolution of the FOCF estimated errors. Therefore, from the simulation results in this section, it is obvious to conclude that the proposed containment control strategy is also practical when each follower is exposed to the EDPU. Also, it is clear that the proposed containment control strategy for each follower can track straight lines without assuming the PE condition.
4.3. Curve Tracking without EDPU
In this section, the predefined trajectories of the virtual leaders are curved trajectories. This experiment validates the control strategy without considering the EDPU. The initial follower states are ; ; ; ; . In the following simulation, the curved trajectories of virtual leaders are chosen aswhere ; ; ; , . The simulation results are plotted in Figures 12–15. Figure 12 illustrates the motion traces of the five followers. It is obviously seen that the control strategy can also track the curved trajectories defined by the virtual leaders. The norms of the tracking errors are plotted in Figure 13. From this figure, we can observe that the tracking errors converge to a small neighborhood of the origin with fast convergence rate. The responses of the control inputs are displayed in Figure 14. In Figure 15, FOCF estimated errors converge to nearly zero.
4.4. Curve Tracking with EDPU
In this section, the simulation experiments are presented, which concern the robustness properties of the containment control law to the EDPU. The initial states of the followers and the curved predefined trajectories of leaders are the same as the previous section. These simulation results are demonstrated in Figures 16–19. Figure 16 shows the actual trajectories of the followers. The norms of the tracking errors are plotted in Figure 17. As seen from Figures 16 and 17, the containment control laws can force the followers to converge into the triangular region in the presence of the EDPU, and the norms of the tracking errors are all less than 0.75 m. In Figure 18, the control forces and are shown. Figure 19 illustrates the responses of the FOCF estimated errors.
In summary, from Figures 4–19, we can see that the proposed containment control law in this paper provided acceptable results as proven in Theorem 20 and all the tracking errors of the closed-loop system converge to a small neighborhood of the origin as time approaches infinity.
5. Conclusions
() Because the sway force is unavailable, the underactuated model is transformed to a strict-feedback form by introducing a polar coordinate transformation. In the kinematic loop design, a virtual controller is designed on the basis of the results from graph theory so as to achieve the control objective.
() To avoid the problem of “explosion of complexity” from the standard BP method, FOCF is introduced to estimate the virtual control laws and their derivatives. In addition, a compensating signal is introduced to compensate the nonlinear terms in BP design.
() Under the consideration that each follower is exposed to the EDPU, the actual control inputs are presented by using the SMC method in the dynamic loop design, which can guarantee that all the signals of the closed-loop system are GUUB and all the tracking errors eventually converge to a small neighborhood of the equilibrium point.
() Future works will be required to account for communication constraint among ships, such as packets dropouts and constant or time-varying communication delays as well as switching communication topology.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported partially by the National Natural Science Foundation of China (no. 51379044), the Fundamental Research Funds for the Central Universities (no. HEUCFX41304), and the Natural Science Foundation of Heilongjiang Province (no. F200916).