Distributed Adaptive Coordinated Control of Multiple Euler–Lagrange Systems considering Output Constraints and Time Delays

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. *en, 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. *e 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.


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][2][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. e 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]. e 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.
e 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. is method can not only simplify the design difficulty of the control algorithm but also reduce energy consumption and cost. erefore, 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. erefore, 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][16][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 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. e 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. en, 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. e 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) e 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.

e Basic Knowledge.
e basic mathematical symbols and definitions in this paper are shown in Table 1.
e communication interactions among a virtual leader and n followers can be presented by the directed graph ζ � (υ, ε, A). In the directed graph ζ � (υ, ε, A), u 1, 2, . . . , n + 1 { } is the set of nodes, ε ⊆ υ × υ denotes the set of edges, and A � a ij ∈ R (n+1)×(n+1) represents the nonnegative adjacency matrix. In node set υ, υ i denotes the i th 2 Complexity follower. e edge υ, υ i ∈ ε represents that the j th agent can get the information from the i th agent. We call υ i as the parent node and its neighbor υ j is called as the child node. e path of a directed graph is the sequence of nodes υ il , . . . , υ in , which satisfies (υ ik , υ ik+1 ) ∈ ε. e 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.

Dynamics of Euler-Lagrange
Systems. e multiple EL systems contain n followers and a virtual leader. en, we utilize the EL equation to describe the dynamic model of the i th follower as where q i ∈ R p , i � 1, . . . , n denotes the generalized coordinate, τ i ∈ R p denotes the control input torque, M i (q i ) ∈ R p×p represents the inertia matrix which is symmetric and positive definite, C i (q i _ q i ) ∈ R p×p represent the centripetal and Coriolis torques, g i (q i ) ∈ R p denotes the gravitational force, and ω i ∈ R p is the external disturbance. We assume M i (q i ), C i (q i , _ q i ), and g i (q i ) are all unknown.
Assumption 2. e disturbance ω i is bounded, which one satisfies ‖ω i ‖ ≤ c, where c is a bounded positive constant. e following properties of EL systems (1) are useful. We assume that q n+1 can be represented as follows [35]: where v ∈ R p denotes the auxiliary state variable, S ∈ R p×p and F ∈ R n×p are constant real matrices, and q n+1 represents the state of the leader.

Time-Varying BLF Design.
e useful lemmas used in this study are shown as follows.
Lemma 1 (see [36]). If there is a continuous Lyapunov where k and c are positive constants, then the variable w(t) is bounded.
Lemma 2 (see [37]). For ∀B ∈ R p×p , if matrix B is symmetric and positive definite, the following inequality can be obtained: Lemma 4 (see [39]). Consider x ∈ R and |x| < |k a |, where k a 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. erefore, the output states of the followers should be limited. If the time-varying bounds are k c(t) � [k c 1 (t), . . . , k c n (t)] T and kc(t) � [kc 1 (t), . . . , kc n (t)] T , then the output q i (t) should remain in the region: Table 1: Mathematical symbols in this study.

Mathematical symbol
Definition R e set of real numbers R n e set of n-dimensional column vectors with all elements being real numbers R n×n e set of n × n real matrices Diagonal matrices whose diagonal elements are x 1 , . . . , x n ⊗ Kronecker product λ min (·) e minimum eigenvalue of a real symmetric matrix λ max (·) e maximum eigenvalue of a real symmetric matrix 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 q i satisfies the time-varying output constraints.
First, an auxiliary variable is defined as en, we define the following error variables: where r i denotes a virtual control. Inspired by He et al. [40], we set the time-varying bounds of Z 1i 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 as where we define h(i) as e output tracking error variables are transformed as follows: Substituting (11) into (9), we obtain From (12), we can know that V 1i (t) is positive definite and continuously differentiable when e virtual control r i is selected as where K 1i is defined as β is a positive constant, and it can ensure _ r i is bounded when _ ka i and _ kb i are zero, where the gain matrix K 1 � diag[k 11 , k 12 , . . . , k 1i , . . . , k 1n ] is symmetric positive definite. Substituting (14) and (15) into (13) yields where X i is defined as follows: and

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 d i � 1/ n+1 j�1 a ij , η ir � R p represents the i th follower's estimate of v and T represents the constant time delays among different followers. We assume that η n+1 � v.
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 ‖Dη‖ > λ max (S) + ‖R e ‖ holds, we can draw a conclusion that η i − v is bounded, which can be expressed as where Remark 2. Since S and F are real matrices and the elements are independent of time and the state variables of the leader, we utilize matrices S and F to design the observer (18).
According to [42], we design following distributed adaptive control schemes of (1) as where α and β are two positive constants and K 2i is a gain matrix. W i and ϕ i 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 Z 1i is UUB with the distributed observer (18) and the distributed adaptive control laws (20) and (21). At the same time, q i (t) satisfies the time-varying output constraints, i.e., ∀t > 0, k c i (t) < q i (t) < kc i (t).
Proof. Differentiating (6) yields Substituting (22) into (1), we obtain where In this study, M i (q i ), C i (q i , _ q i ), g i (q i ) 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. erefore, the NN technique is used to solve the nonlinear e methods are shown as follows: where W i represents the ideal weighted matrix, ϕ i is the Gauss function, and Δ i denotes the approximation error. In this research, Δ i is assumed to satisfy ‖Δ i ‖ ≤ Δ Mi , where Δ Mi is a positive constant. e estimate of the nonlinear uncertainties d i for the i th follower can be written as where W i is the estimate of W i . e distributed adaptive control schemes are designed as (20) and (21). Consider a Lyapunov function as follows: where W i � W i − W i . Differentiating (27) and according to (20)- (26), we can obtain Since Δ i and ω i are bounded, there is a positive constant χ max satisfying ‖Δ i + ω i ‖ ≤ χ max . Furthermore, the inequality can be obtained as follows: where σ is a positive constant.
e matrices have the following properties: tr(BA) � tr(AB), Further we have Substituting (16), (29), and (33) into (28), we have us, (34) can be written as where We can know V 2i is UUB according to Lemma 1. Integrating (35) yields We can ensure κ 1 > 0 and υ 1 > 0 by properly selecting parameters, and we have It can be seen from (27) that Moreover, By substituting (10) and (11) into (40), we can obtain From (4), one has According to (19), (41), and (42), we can obtain From (46), we can conclude that the tracking errors among different followers and the virtual leader are bounded.

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 Z i (t) can converge to origin asymptotically. For the EL system (1), we propose following modified distributed adaptive control strategy based on controller 1 as where ι i is a positive constant, and it satisfies that where δ is a small positive constant.

Remark 3. By introducing the continuous function sat(Z 2i ),
it not only makes the tracking errors among different followers and the leader asymptotically converge to zero but 6 Complexity also avoids the chattering phenomenon caused by discontinuous sign function sgn(Z 2i /δ).
Substituting (45) and (46) into (1), we can obtain Select the same V 1i and V 2i as those in eorem 1. Taking the derivative of V 2i and from (16), (47), and (48), it can be obtained that Similar to (28)- (34), it holds that For a small positive constant δ, we can find that ‖Z 2i ‖ ≤ ‖Z 2i /δ‖. Substituting (47) into (51), we have Since k 1i is a positive constant and matrix K 2i is symmetric positive definite, it can be found that (53) Due to the fact that V 2i is bounded, we find Z μ , Z 2i ∈ L ∞ . According to (6) Integrating both sides of (54), it can be obtained that (55) erefore, we have Z 1i ∈ L 2 and Z 1i ∈ L 2 ∩ L ∞ . According to Barbalet lemma, we have lim t⟶∞ Z 1i (t) � 0. (56) From (56), it can be found that Following the similar steps as (42) and (43), we have e i th follower has a good tracking performance on the virtual leader, and the upper boundary of the tracking error is described by (58).
When ‖Z 2i /δ‖ < 1, it can be obtained that sat(Z 2i ) � Z 2i /δ. According to (49) and (50), we have . From the procedures in (53)-(58), we can find that the tracking error between the i th follower and the virtual leader is bounded.

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

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-offreedom robotic manipulators as an example for the simulation experiment. e communication topology is shown in Figure 1, where 5 represents the virtual leader and 1-4 represent the four followers. e structure of robotic manipulator is shown in Figure 2. e dynamic equation of the i th follower can be expressed as where Complexity 7 In this section, q 1i , q 2i represent the rotation angle of two joints of the manipulator.
In this section, we choose the communication delay as T � 0.5 s.

Simulation Performance for Controller 1.
For controller 1, we choose the parameters as k 1i � 10, K 2i � 20I 2 , c � 1, and v � 10. Figures 3-12 introduce the simulation results. Figures 3 and 4 illustrate the state variables q i1 and q i2 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 Z i1 and Z i2 are bounded. At the same time, the auxiliary variables Z i1 do not exceed 0.1 rad, and Z 2i 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.

Simulation Performance for Algorithm 2.
For the improved algorithm by using discontinuous sign function on Algorithm 1, let k 1i � 20, K 2i � 20I 2 , c � 1, ι i � 80, and δ � 0.5. Figures 13 and 14 show that the control input torque of all robotic manipulators has chattering problem.
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 Z 1i and Z 2i 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,          14 Complexity    and the tracking error always satisfies the time-varying constraints. In addition, the tracking errors Z 1i based on Algorithm 2 are smaller than those based on Algorithm 1, and Algorithm 2 avoids the unexpected chattering problem arising from sign function.

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