#### Abstract

In this paper, we mainly investigate the coordinated tracking control issues of multiple Euler–Lagrange systems considering constant communication delays and output constraints. Firstly, we devise a distributed observer to ensure that every agent can get the information of the virtual leader. In order to handle uncertain problems, the neural network technique is adopted to estimate the unknown dynamics. Then, we utilize an asymmetric barrier Lyapunov function in the control design to guarantee the output errors satisfy the time-varying output constraints. Two distributed adaptive coordinated control schemes are proposed to guarantee that the followers can track the leader accurately. The first scheme makes the tracking errors between followers and leader be uniformly ultimately bounded, and the second scheme further improves the tracking accuracy. Finally, we utilize a group of manipulator networks simulation experiments to verify the validity of the proposed distributed control laws.

#### 1. Introduction

With the rapid development of industrial technology, the industrial tasks are gradually becoming complicated and large-scale. When solving some complex industrial tasks, multiagent systems (MASs) gradually become the first choice due to its high reliability and economy, such as multiple robotic manipulator systems, spacecraft formation flying, and unmanned underwater vehicles [1–3]. More and more experts and scholars focus on the distributed coordinated control of MASs [4].

So far, there mainly have been two consensus methods for the MASs coordinated control. The first control strategy is about the leaderless control method, which requires all state variables to gradually converge to a constant. For example, the authors used Lyapunov finite-time theory to propose a consensus control method to ensure the states converge to a constant based on undirected graphs in [5]. The authors in [6] proposed a leaderless consensus control strategy and analysed the stability problem for MAS. However, for the leaderless case, the convergence of the state variables of all agents is related to the initial state variables of each agent, which leads to the great restriction of the system movement. The second strategy is the leader-following control for MASs. In this case, followers utilize their own or neighbors’ information to track leader so that one only needs to design the movement of leader accurately. This method can not only simplify the design difficulty of the control algorithm but also reduce energy consumption and cost. Therefore, this approach is more suitable for MASs. In [7], a trajectory tracking algorithm was proposed by using the small gain feedback technique, so that the followers can effectively track the dynamic leader. In [8], the authors used the linear event-trigger feedback technique to make the state variables of followers and leaders tend to be consistent in finite time so as to achieve the tracking control. For the coordinated tracking control case, the leader’s motion is the most crucial part in MASs, which means that the whole system will stop working if leader fails. Actually, virtual leaders play the same role as the real leaders in MASs. At the same time, virtual leaders can change the number of leaders flexibly [9]. In [10], the authors discussed the current situation and illustrated the extensive application prospect of the leadership relationship within virtual working environment. Furthermore, the input signals were regarded as the virtual leaders and a distributed control algorithm was proposed by using Lyapunov theory; this control algorithm realized the coordinated movement of the followers and achieved the goal of tracking the target trajectory [11, 12].

It is worth noting that all the above research studies about MASs are based on linear systems. However, linear systems have great limitations in MASs due to the existence of nonlinear uncertainties. Therefore, it is necessary to study the coordinated control problem of nonlinear systems [13, 14]. In fact, Euler–Lagrange (EL) equation is widely used in the field of nonlinear systems, such as autonomous underwater robots and manipulating robots systems [15–17]. In [18], the authors focused on the distributed coordinated tracking issues and designed two kinds of novel control algorithms for multiple EL systems. In multiple EL systems, distributed observers are often used to ensure the normal operation of the systems when only partial agents could get the state information of the leader. In [19], the authors utilized a new dynamic velocity observer to ensure all agents could get the leader’s velocity variables. In [20], the leader’s information could be acquired by the followers through the velocity observer and guaranteed the goal of coordinated tracking control. At present, there usually exist communication delays among different agents due to the limitation of communication bandwidth and packet losses. Therefore, the communication delays cannot be ignored in the study of multiple EL systems’ coordinated control [21]. Considering the communication delays, a new aperiodic sampled-data cluster formation control algorithm was proposed for the cluster formation control problem for MASs in [22]. In [23], the authors investigated the leader-following consensus problem of multiple EL systems which considered communication delays under the switching network. In [24], in order to solve the problem of communication delays, the authors proposed a novel coordinated tracking control algorithm based on Lyapunov stability theory. In actual engineering projects, state constraint problems or output constraint problems are often considered to meet certain requirements. At present, the state constraint or output constraint methods for nonlinear systems include the prescribed performance (PP) methods and the barrier Lyapunov function (BLF) technique [25]. PP methods mainly utilize error transformation to achieve the performance constraint [26, 27], which may cause control law singularity problem. In contrast, the BLF technique mainly applies the performance constraint variables to construct constrained boundary functions, which greatly reduces the difficulty of designing the control algorithms. Thus BLF technique is widely used in the state constraint control problem for nonlinear systems. In [28], the authors utilized Integral BLFs to make all state errors converge to the neighbor of origin. The authors used an asymmetric BLF to solve the time-varying output error limitation problem in [29]. The authors in [30] utilized the asymmetric time-varying BLF to construct an adaptive controller and applied the Lyapunov stability theory to guarantee the output error within the excepted range. It is feasible to use the time-varying BLF method to deal with the output constraints of multiple EL systems.

Generally speaking, MASs are often affected by working environment, unknown dynamics, and unknown disturbances. The uncertainties will affect the work efficiency of MASs. To handle the uncertainties problem, the neural network (NN) technique is widely used for the MASs due to its good approximation ability [31, 32]. In [33], the authors designed a distributed adaptive NN controller to ensure the mobile robots can obtain the expected control effect. In the case that only partial agents could get the leader’s information, a control algorithm which used the NN technique to compensate the uncertainties and external disturbances was designed in [34]. It made the tracking errors among leader and followers tend to origin.

In this study, the coordinated control problems for nonlinear multiple EL systems considering communication delays and output constraints are investigated. Compared with the existing papers, it uses a distributed observer to solve the communication delay problem of multiple EL systems. Then, the BLF technique is used to guarantee the time-varying output errors of the systems within the prescribed constraint boundary. We also utilize adaptive NNs to deal with the uncertain dynamics and the unknown disturbances of the multiple EL systems. The main contributions of the paper are summarized as follows.(1)Considering communication delays among different followers, the input signal source is regarded as a virtual leader. In addition, a distributed observer is used to ensure that the virtual leader’s state information can be obtained by all followers.(2)Two distributed control schemes are designed to guarantee the tracking errors are UUB and asymptotically converge to origin.(3)The adaptive NN technique is utilized to compensate the uncertain dynamics of the multiple EL systems.(4)Based on the BLFs, an asymmetric BLF is designed to make the output errors satisfy the time-varying output constraint requirements.

In the following research, Section 2 introduces the dynamic model and some basic lemmas. In Section 3, two distributed adaptive control algorithms and stability analysis are formulated. Section 4 presents several simulation examples to prove the validity of the control algorithms. Section 5 summarizes the whole paper.

#### 2. Materials and Methods

##### 2.1. The Basic Knowledge

The basic mathematical symbols and definitions in this paper are shown in Table 1.

The communication interactions among a virtual leader and followers can be presented by the directed graph . In the directed graph is the set of nodes, denotes the set of edges, and represents the non-negative adjacency matrix. In node set , denotes the follower. The edge represents that the agent can get the information from the agent. We call as the parent node and its neighbor is called as the child node. The path of a directed graph is the sequence of nodes , which satisfies . The directed tree denotes a directed graph in which each node has only one parent node, but there is one root node which is different. When a directed tree contains all nodes of the graph, it can be described as a directed spanning tree. If the directed spanning tree is a part of the directed graph, it is said that the directed graph contains a directed spanning tree.

*Assumption 1. *A directed spanning tree is contained within .

We define the element , if and only if and otherwise . The element when A matrix is defined as , where , .

##### 2.2. Dynamics of Euler–Lagrange Systems

The multiple EL systems contain followers and a virtual leader. Then, we utilize the EL equation to describe the dynamic model of the follower aswhere denotes the generalized coordinate, denotes the control input torque, represents the inertia matrix which is symmetric and positive definite, represent the centripetal and Coriolis torques, denotes the gravitational force, and is the external disturbance. We assume , , and are all unknown.

*Assumption 2. *The disturbance is bounded, which one satisfies , where is a bounded positive constant.

The following properties of EL systems (1) are useful.

*Property 1. *The matrix is skew-symmetric, i.e., , .

*Property 2. * is bounded, and it satisfies , where and are positive constants.

We assume that can be represented as follows [35]:where denotes the auxiliary state variable, and are constant real matrices, and represents the state of the leader.

*Assumption 3. *When all followers can obtain and , and are bounded.

*Remark 1. *If the coefficient matrices , and the auxiliary state variable are chosen appropriately, any desired trajectory can be formed by the leader.

#### 3. The Distributed Adaptive Trajectory Tracking Control Law Design

##### 3.1. Time-Varying BLF Design

The useful lemmas used in this study are shown as follows.

Lemma 1. *(see [36]). If there is a continuous Lyapunov function which satisfies and the derivative satisfies , where and are positive constants, then the variable is bounded.*

Lemma 2. *(see [37]). For , if matrix is symmetric and positive definite, the following inequality can be obtained: , .*

Lemma 3. *(see [38]). is a continuously differentiable function. For , if hold, is bounded where is a small positive constant, it means that is bounded .*

Lemma 4. *(see [39]). Consider and , where is a positive constant, such that the inequality is shown as .*

In this study, if the trajectory tracking errors are too large, it may cause undesired losses. Therefore, the output states of the followers should be limited. If the time-varying bounds are and , then the output should remain in the region:

When considering the influence of tracking errors on multiple EL systems, a time-varying BLF is used in this study to ensure that the output satisfies the time-varying output constraints.

First, an auxiliary variable is defined as

Then, we define the following error variables:where denotes a virtual control.

Inspired by He et al. [40], we set the time-varying bounds of as

Consider that asymmetric BLF is an improvement of symmetric BLF, which can better adapt to the requirements of time-varying output constraints and ensure the high trajectory tracking accuracy for multiple EL systems. An asymmetric BLF is chosen aswhere we define as

The output tracking error variables are transformed as follows:

Substituting (11) into (9), we obtain

From (12), we can know that is positive definite and continuously differentiable when . Differentiating yields

The virtual control is selected aswhere is defined as

is a positive constant, and it can ensure is bounded when and are zero, where the gain matrix is symmetric positive definite. Substituting (14) and (15) into (13) yieldswhere is defined as follows:and .

##### 3.2. Distributed Adaptive NN Tracking Controller 1 Design

When only part of the followers can get the state information, we use the following distributed observer [35]:where , represents the follower’s estimate of and represents the constant time delays among different followers. We assume that .

Lemma 5. *(see [41]). In this research, we utilize the distributed observer (18) to ensure the leader’s information can be acquired by all agents when Assumptions 1–3 hold. If holds, we can draw a conclusion that is bounded, which can be expressed aswhere , , , denotes a positive constant, and is the defined neutral operator.*

*Remark 2. *Since and are real matrices and the elements are independent of time and the state variables of the leader, we utilize matrices and to design the observer (18).

According to [42], we design following distributed adaptive control schemes of (1) aswhere and are two positive constants and is a gain matrix. and are two related variables to NN.

Theorem 1. *For the EL system (1) which considers communication delays and output constrains, if Assumptions 1–3 hold, the tracking error is UUB with the distributed observer (18) and the distributed adaptive control laws (20) and (21). At the same time, satisfies the time-varying output constraints, i.e., .*

*Proof. *Differentiating (6) yieldsSubstituting (22) into (1), we obtainwhereIn this study, are assumed to be unknown, and there are nonlinear uncertainties in the EL system (1). Considering that NN has good approximation ability for unknown nonlinear function, it is often used to deal with the uncertainty problem in nonlinear system. Therefore, the NN technique is used to solve the nonlinear uncertainties . The methods are shown as follows:where represents the ideal weighted matrix, is the Gauss function, and denotes the approximation error. In this research, is assumed to satisfy , where is a positive constant. The estimate of the nonlinear uncertainties for the follower can be written aswhere is the estimate of .

The distributed adaptive control schemes are designed as (20) and (21). Consider a Lyapunov function as follows:where . Differentiating (27) and according to (20)–(26), we can obtainSince and are bounded, there is a positive constant satisfying . Furthermore, the inequality can be obtained as follows:where is a positive constant.

Because is a scalar in (28), we have .

The matrices have the following properties:Further we haveSubstituting (16), (29), and (33) into (28), we haveThus, (34) can be written aswhereWe can know is UUB according to Lemma 1. Integrating (35) yieldsWe can ensure and by properly selecting parameters, and we haveIt can be seen from (27) thatMoreover,By substituting (10) and (11) into (40), we can obtainFrom (4), one hasAccording to (19), (41), and (42), we can obtainFrom (46), we can conclude that the tracking errors among different followers and the virtual leader are bounded.

From (5), (7), (8), and (41), we can obtainEquation (44) shows that the output satisfies the time-varying output constraints.

##### 3.3. Improved Distributed Adaptive NN Tracking Controller 2 Design

According to [42], the authors proposed a new algorithm to decrease the tracking errors in (47) by using a discontinuous sign function, and it can be proved by similar steps to (49)–(58). However, this approach will bring about the additional chattering. Considering the chattering problem, we propose an improved continuous control scheme to improve the tracking accuracy. At the same time, auxiliary variable can converge to origin asymptotically.

For the EL system (1), we propose following modified distributed adaptive control strategy based on controller 1 aswhere is a positive constant, and it satisfies thatwhere is a small positive constant.

*Remark 3. *By introducing the continuous function , it not only makes the tracking errors among different followers and the leader asymptotically converge to zero but also avoids the chattering phenomenon caused by discontinuous sign function .

Theorem 2. *For the EL system (1) which considers communication delays and output constrains, when Assumptions 1–3 hold, each follower can gradually track the leader and the tracking error asymptotically converging to origin with the distributed continuous control laws (45) and (46). The time-varying output constraints are not breached, that is, .*

*Proof. *When , we know . Substituting (45) and (46) into (1), we can obtainSelect the same and as those in Theorem 1. Taking the derivative of and from (16), (47), and (48), it can be obtained thatSimilar to (28)–(34), it holds thatFor a small positive constant , we can find that . Substituting (47) into (51), we haveSince is a positive constant and matrix is symmetric positive definite, it can be found thatMoreover, we haveDue to the fact that is bounded, we find . According to (6), we have . When , . Integrating both sides of (54), it can be obtained thatTherefore, we have and . According to Barbalet lemma, we haveFrom (56), it can be found thatFollowing the similar steps as (42) and (43), we haveThe follower has a good tracking performance on the virtual leader, and the upper boundary of the tracking error is described by (58).

When , it can be obtained that . According to (49) and (50), we haveIf , we have . From the procedures in (53)–(58), we can find that the tracking error between the follower and the virtual leader is bounded.

According to (5), (7), (8), and (56), we can obtain the same result as (44). It shows that the output satisfies the time-varying output constraints.

*Remark 4. *The results of this paper can provide some reference for formation control of MASs. In the future, the formation control problems for MASs considering time-varying communication delays and full-state constraints will be studied.

#### 4. Results and Discussion

##### 4.1. Parameter Setting

In this section, we utilize some examples to verify the validity of the proposed schemes in practical aspect. In this study, we consider 4 2-degree-of-freedom robotic manipulators as an example for the simulation experiment. The communication topology is shown in Figure 1, where 5 represents the virtual leader and 1–4 represent the four followers. The structure of robotic manipulator is shown in Figure 2.

The dynamic equation of the follower can be expressed aswhere

In this section, represent the rotation angle of two joints of the manipulator. , , , , , and denotes gravitational acceleration, where and denote masses of links, is the length of links, represents the distance from center of mass to motors, and and are moments of inertia.

Table 2 shows the indicators of robotic manipulators.

We set the time-varying output constraints as follows:

The boundary value of tracking error is expressed as

Thus, we have

The initial angle of robotic manipulators is set as follows.

, , , , , , , , and .

For the follower , the activation equation of the neural network system can be written as

We select Gauss function as the activation function, and its form iswhere . Suppose all followers utilize the same activation equation. denotes the NN center and it is distributed over . is Gaussian function’s width, and we define , .

The desired trajectory of the virtual leader is designed as follows:where , , , , , , and .

The state variable of the virtual leader can be expressed aswhere

In practical physical systems, we often need to limit the amplitude of the control inputs . The following saturation equation is used to limit the amplitude:where denotes a positive constant, and we let .

In this section, we choose the communication delay as .

##### 4.2. Simulation Performance for Controller 1

For controller 1, we choose the parameters as , , , and . Figures 3–12 introduce the simulation results.

Figures 3 and 4 illustrate the state variables and among the followers and the virtual leader, from which we can see that each follower can effectively track the leader after about 5 s. Figures 5 and 6 show that the auxiliary variables and are bounded. At the same time, the auxiliary variables do not exceed , and do not exceed 0.05 after 3 s. From Figures 7 and 8, it can be obtained that the control inputs are continuous and change within 10 Nm. From Figures 9–12, we can find that the time-varying output constraints of the robotic manipulators are always satisfied.

##### 4.3. Simulation Performance for Algorithm 2

For the improved algorithm by using discontinuous sign function on Algorithm 1, let , , , , and . Figures 13 and 14 show that the control input torque of all robotic manipulators has chattering problem.

For Algorithm 2, we set , , , , and . Figures 15–24 show the performance of simulation results.

From Figures 15 and 16, it is obvious that the tracking performance is better, and the tracking errors are smaller compared with the cases in Figures 5 and 6. Comparing Figures 17 and 18 with Figures 7 and 8, we can find that the fluctuation range of and is smaller after 1 s. From Figures 19 and 20, we can obtain that the control input torques of the robotic manipulators are stable and continuous. From Figures 21–24, we can see that under the control Algorithm 2, the error variables of the followers always satisfy the time-varying output constraints.

Based on the above simulation results, we can discover that all followers have very good tracking effect on the leader, and the tracking error always satisfies the time-varying constraints. In addition, the tracking errors based on Algorithm 2 are smaller than those based on Algorithm 1, and Algorithm 2 avoids the unexpected chattering problem arising from sign function.

#### 5. Conclusions

In this study, we propose two practical control strategies to address distributed coordinated tracking problem for the multiple EL systems subjected to communication delays and time-varying constraints. The distributed observer is used to cope with the communication delays for multiple EL systems. We utilize the NN technique to compensate nonlinear uncertainties. At the same time, an asymmetric BLF is used to guarantee that the output errors are always within the output constraints. The adaptive control Algorithm 1 is designed to ensure that the tracking errors is designed to ensure that the tracking errors among the followers and the virtual leader can be bounded. Based on Algorithm 1, the improved Algorithm 2 can make the tracking errors smaller. The simulation results indicate that the proposed methods can effectively solve the problem of communication delays and make the tracking errors meet the prescribed output constraints.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This study was supported by the National Natural Science Foundation of China (grant nos. 61803119, U1713205, and 51779058).