Adaptive Fuzzy Consensus Tracking Control for Nonlinear Multiagent Systems with Time-Varying Delays and Constraints

)is paper proposes an adaptive fuzzy distributed consensus tracking control scheme for a class of uncertain nonlinear dynamic multiagent systems (MASs) with state time-varying delays and state time-varying constraints. )e existing controllers with Lyapunov–Krasovskii functions (LKFs) were not suitable to address time-varying delays and time-varying constraints in nonlinear MASs simultaneously. State constraints further increase the difficulty of controller design and stability analysis, especially for nonstrict feedback systems. Fuzzy logic systems (FLSs) tackle the approximation of unknown dynamics functions and parameters. Especially when the distributed consensus tracking error is infinitely close to the origin, although there is no singular value, it would lead to the rapid growth of control rate or uncontrollability. Constructing appropriate piecewise functions can effectively avoid the above occurrence and accelerate convergence. Based on Lyapunov stability theory and algebraic graph theory, the constructed tracking control can ensure states within defined time-varying constraint bounds and eliminate the influence of time delays. All signals in closed-loop systems can be guaranteed semiglobally uniformly ultimately bounded (SUUB). Finally, the validity of the theoretical method is verified by the simulation.


Introduction
Over the past few decades, inspired by flocking behavior in nature, related research on multiagent systems (MASs) had developed rapidly due to potential military and civilian applications. e literature involved many aspects such as satellites, flights, distributed computing, robotics, power systems, surveillance and reconnaissance systems, multimissile coordinated attacks, and intelligent transportation systems in [1][2][3][4][5][6][7]. In [8], based on the urgent needs of intelligent agriculture, the bioinspired coordination protocol was employed to achieve refinement in agricultural management. Consensus control is the most fundamental and most valuable core research of multiagent. e research directions include consensus control [9,10], tracking control [11], aggregation control [12], formation control [13], and synchronous control [1,14]. e consensus framework is proposed by Olfatil-Saber and Murray in [14,15]. Previous literature mainly focused on linear MASs and acquired large numbers of breakthrough achievements. Communication security, information samples, edge maps, and other related methods have also made some achievements in [16][17][18]. In practical mechanical systems, the problems of inherent nonlinearity and uncertainty would be further enhanced on account of the complex cooperative tasks, the limitation or lack of system dynamic cognition, and so forth. Several remarkable consensus control approaches of linear MASs would not directly be employed to nonlinear MASs. It is indispensable to apply adaptive technology and intelligent control methods to solve uncertain nonlinear distributed MASs.
Adaptive technology had become an effective method to address various nonlinear systems in the literature [19][20][21]. e neural networks (NNs) [19,20] were related to the event trigger. e fuzzy logic systems (FLSs) [21][22][23] solved the finite time of stochastic nonlinear systems in [21]. Distributed nonlinear MASs among the literature [24][25][26][27] had been proved distinguished approximation ability to solve the unknown dynamics. e consensus control was discussed in [9], and consensus tracking control protocols were studied in [24][25][26]. e multileader consensus control method was researched in [27]. However, the above methods have not involved the problem of state time delays. e differences in controller performance are often accompanied by state time delays which would reduce the speed of convergence, affect system performance, and lead to instability or ineffectiveness. Meanwhile, once time delays exceed the threshold, they would affect the whole systems. erefore, time delays would be an important consideration. Some achievements had been made for time delays in uncertain nonlinear systems in [28][29][30][31]. Particularly, [30,31] solved the time delayed switched systems. In [32,33], LKFs were employed to eliminate time delays in cooperative tracking control with visual leader and leader-following, respectively. Nevertheless, it is obvious that the above omitted the constraints. In particular, it is unavoidable to consider the physical limits and safety performance of each agent, the mutual internal collision avoidance, and other operation restrictions. If ignoring constraints, system performance would be damaged or reduced and even would cause fatal accidents. From a practical point of view, the theoretical study of constraints becomes crucially important in MASs.
In the design of constrained controllers, reliable methods needed to be applied. e Barrier Lyapunov functions have been preferred as the primary candidate functions for designing controllers with constraints. e study in [34] presented the method to indirectly address the constraint by constructing barrier functions. Many BLFs, integral BLF (IBLF), and ABLF based adaptive approximate controllers were designed for different categories of the nonlinear systems in [34][35][36][37][38][39][40][41], and nonlinear MASs in [27,42]. Integral barrier Lyapunov functions have been employed in [39][40][41]. e study in [27] proposed an adaptive fuzzy containment control based on the distributed observer and distributed sliding mode estimators with state constraints via multiple BLFs. Note that the above constraint strategies are indirect methods, which are applicable to the case that the feasibility of virtual controllers needs to be satisfied and the constraint boundaries are not infinitesimal. e nonlinear coordinate transformation function was used to construct the constraint completely dependent on states in [43] and [44], and the complicated feasibility analysis was reduced. e authors of [45,46] proposed the control strategies to solve both the state constraints and state time-varying delays. erefore, direct constraints on output or states are worth further investigation of the constrained cooperative control.
Motivated by the above analysis, this paper schedules a novel adaptive fuzzy distributed cooperative tracking control approach for a class of uncertain nonlinear MASs with state time-varying delays and constraints. e main contributions of the paper are as follows: (1) Contrasting with the existing literature on time delays in [32,33], time delays in the case of time variation are further investigated. e most important one is to overcome the failure of the decomposition theorem in nonstrict feedback. Meanwhile, state time-varying constraints are introduced into distributed time delayed MASs since the state time-varying delays and constraints further increase the difficulty of controller design, which is an enormous challenge. (2) Distinguished from BLFs, the constraints construct by a nonlinear coordinate transformation in [43] that directly constrain on states, which are applicable to avoid complex feasibility analysis of virtual controllers. e constructed appropriate piecewise functions can effectively avoid the rapid growth and uncontrollability of the control rate and accelerate the convergence when the synchronization error is infinitely close to the origin. e proposed strategy is aimed at the study of the timevarying state of the agents, which has great realistic significance and further enriches the exploration of the uncertain nonlinear MASs. e organization of the article is as follows: the presentation and problem of the systems are described in Section 2; the adaptive fuzzy distributed cooperative control design is provided in Section 3; simulation is given in Section 4; and the conclusion is given in Section 5.

Systems Description and Problem Presentation.
Consider a class of uncertain nonlinear distributed consensus systems with time-varying delay and the i th dynamic is described as . , x im ] T ∈ R m represents the state vector of i th agent, i � 1, 2, . . . , n, and n is the number of agents; f i (x i ) and ρ i (x i (t − τ i (t))): R m ⟶ R m are unknown nonlinear continuous vector functions; τ i (t) represents the uncertain time-varying delay; and u i (t) ∈ R m is the control input vector.
where q i (x i (t − τ i (t))) is the known positive smooth function.

Remark 1.
e unknown time-varying delays function ρ(x i (t − τ i (t))) contains the time-varying delayed state. e separation technique is used to decompose the unknown with all the time-varying delays into positive continuous functions in order to compensate for the time delays by LKFs in [45].

Assumption 2.
e unknown time-varying τ i (t) should satisfy the following: (1) the _ τ i (t) is bounded by a known constant τ dmax and _ τ i (t) ≤ τ dmax ≤ 1 and (2) the continuous 2 Complexity τ i (t) should be uniformly bounded by a known constant τ max and τ i (t) ≤ τ max in [33,45,46]. e desired reference leader signal is described by the following dynamic function: where x l (t) ∈ R m is the state of reference leader and g l (t) ∈ R m is a bounded smooth vector function. e time-varying state constraint is considered, and the constraint boundary of i th agent meets requirements as follows: where x ij represents the j th state of the i th agent; j � 1, 2, . . . , m; k c j (t) is a user-defined time-varying bound function in j th state; and k c (t) ∈ R m . e initial value should satisfy −k c j (0) < x ij (0) < k c j (0). e requirement of (4) can be satisfied by choosing a nonlinear coordinate transformation as follows: where s ij (t) is a constructed state transition that converts the original Take the derivative of s ij (t) as follows: where 2 . e state of the leader in (3) can be transformed by the same form of (5); it can be expressed as where α lj (t) is a state transition to transform the con- Take the derivative of (8) as follows: where

Remark 2.
e states x i and x l are converted into s i and α l , respectively; zs ij (t)/zx ij (t) > 0 and zα lj (t)/zx lj (t) > 0 are invertible functions; and s ij (t) and α lj (t) are monotonically increasing. Just proving s ij (t) and α lj (t) are bounded, by criteria of monotone bounded, the original states x i and x l would converge and satisfy the constraint region.

Assumption 3.
e time-varying bound vector functions k c (t) and _ k c (t) satisfy the following: ∀t ≥ 0; there exist positive constant vectors K c ∈ R m and K 1 c ∈ R m that satisfy Remark 3. In numbers of practical cooperative control of multiagent systems, inevitably, constraints need to be addressed due to physical limitations or performance. If the constraints were ignored, the systems may be degraded or damaged by interagent influences, and even fatal accidents would occur. State constraints are more effective in solving states of transient or steady real systems than only with output constraints. e control process is timevarying, so it is significant to study the time-varying full state constraints.

Remark 4.
e constraints based on BLFs (or IBLFs) require additional complex stability analysis to ensure the feasibility conditions of virtual controllers. e state-dependent transition method in [43,44] which can circumvent the above problems, flexibly solve initial conditions, and reduce the difficulty of stability analysis.

Remark 5.
e nonlinear coordinate transformation function is constructed and purely dependent on the constraint states. Introducing a new coordinate transformation for the backstepping design of MASs, it directly responses to state constraints and completely avoids the harsh feasibility conditions. is will get rid of the tedious feasibility verification. Let the designer have more freedom and be humanized to choose the designed parameters in implementations. Apply a wider range of initial conditions. is paper only shows the feasibility and effectiveness of constructing state time-varying constraints to distributed nonlinear and uncertain time-varying delayed MASs. e asymmetric forms would be further discussed in other papers.

Control Objective.
Devise an adaptive fuzzy decentralized tracking controller u i that, for the dynamic of agents described in (1) and (3), can guarantee the following: (i) All followers converge to the desired small neighborhood around the origin by tracking the leader. (ii) e full states would not violate the time-varying constraint boundaries in (5) and (8). (iii) All the signals in the closedloop systems are SUUB.
e following assumptions and lemmas should be satisfied in order to establish a cooperative control objective.

Algebraic Graph eory and
Notations. An undirected connected topology expresses as ∈ R n×n is a weighted adjacency matrix that a weight corresponds to the edge, and when ℘ ij ∈ R n×n , a ij � 1. Otherwise, a ij � 0. In particular, define a ii � 0. e graph Laplacian matrix L is given in the following definition: L is an irreducible matrix; and [1, 1, . . . , 1] T ∈ R n is a right eigenvector with the eigenvalue zero.

Lemma 1.
e graph Laplacian matrix L is a semidefinite symmetric matrix; all eigenvalues are nonnegative real; and its n real eigenvalues can be listed in ascending order as follows: where C � 2(max 1≤i≤n d i ) is algebraic connectivity and it is used to analyze the rate of consensus convergence.
Assumption 4. In the graph topology G, for the leader, there is at least one directed path from the root node to follower nodes.

Distributed Controller.
e proposed distributed control algorithm can make arbitrary i th agent feedback to itself and communicate with neighbor agents via adjacent matrix. Since fixed topology graphs are the basis for different types of switching connection topology and are widely applied such as [27,28], this paper is based on the fixed topological form.
e synchronization error e i (t) ∈ R m consists of the i th local consensus error and the tracking error among the leader and followers, and that can be defined as follows: where a ij represents the element in adjacency matrix A. b i is the communication weight when only i th agent exchanges information with leader b i > 0; otherwise b i � 0, and B � diag b 1 , b 2 , . . . , b n is the communication weight matrix and satisfies b 1 + b 2 + · · · + b n > 0: Let where L is a symmetrical positive definite matrix and its real eigenvalues λ 1 , λ 2 , . . . , λ n satisfy Lemma 1. Based on the matrix theory and graph theory, it can be readily concluded that zero is an m−multiplicity eigenvalue of L ⊗ I m ; ⊗ and I m , respectively, stand for the Kronecker product and identity matrix of dimension m × m. Let φ 11 , φ 12 , . . . , φ 1m , . . . , φ 2m , . . . , φ nm be the eigenvectors and λ 2 , . . . , λ n eigenvalues of L ⊗ I m such there can be a set of orthogonal bases of R nm . e definition is given by graph theory: where P T P � PP T � I nm , I nm denoting identity matrix of dimension nm × nm and P � [φ 11 , φ 12 , . . . φ nm ] ∈ R nm×nm . Define the tracking error vector as the following form: where ς i (t) � [ς i1 (t), ς i2 (t), . . . , ς im (t)] T ∈ R m represents the tracking error between the i th follower agent and the leader agent. en, (14) can be rewritten as where e i (t) � [e i1 (t), e i2 (t), . . . , e im (t)] T . e unknown functions and variables in multiagent systems can be approximated by FLSs as follows: where ω ij ∈ R l j is the optimal fuzzy weight vector, l j denotes the number of fuzzy rules, Ψ ij (X ij ) is the fuzzy basis vector function, ε ij shows estimation error satisfied ‖ε ij ‖ ≤ ε ij , and ε ij is an arbitrary indeterminate positive constant.

Adaptive Fuzzy Distributed Cooperative Control Design
In this section, an adaptive fuzzy distributed cooperative control algorithm will be designed for (1) and (3). Define the following candidate function: where ς(t) � (ς T 1 (t), ς T 2 (t), . . . , ς T n (t)) T ∈ R nm is the consensus error vector that ς i (t) has defined in (17), then , ω ij is the estimate of adaptive laws, and ω ij and ω ij will be given later. e inequality can be described as follows: where λ min (Q) and λ max (Q) are the smallest and largest eigenvalues of Q, respectively. From (16), it yields 4 Complexity where Λ � diag 0I m , λ 2 I m , . . . , λ n I m .
Redefine the Lyapunov candidate function as follows: Take the derivative of (23). en, _ V ω can yield _ V ω can be written as follows: e derivative of ς ij (t) in the form of (17) can be described as Combining systems (1), (3), and (26) where K ij (t) is i th time-varying gain described as follows: β ij is a positive parameter to be given. Using Young's inequality and Assumption 1, in order to separate e ij and uncertainties ρ ij (x ij (t − τ i (t))), the inequality gets where v � 1 − τ dmax has been illustrated in Remark 4 and M ij � ‖zs ij (t)/zx ij (t)‖. e unknown smooth function F ij (X ij ) is approximated by FLSs to tackle the unknown dynamics functions and parameters. F ij (X ij ) is denoted as where ] T , and ‖e ij (t)‖ 2 M 2 ij takes the norm of e ij (t)zs ij (t)/zx ij (t) in (29). Equation (30) can be estimated by (19) as follows: Combining (28), (29), and (30) by FLSs, then (27) can yield where U ij (t) � K ij (t) − zα lj (t)/zx lj (t)g lj (t). By using Young's inequality, we obtain the following: Combining (33) into (32), we obtain e Lyapunov-Krasovskii function is constructed to compensate uncertain time-varying delays as follows: β ij x ij (s) ds, (35) where β ij (x ij (s)) ≤ q 2 ij (x ij (s)) has been defined under Assumption 1.
Take the derivative of V ρj (t). It obtains e Lyapunov function candidate is chosen by combining (23) and (35) as follows: β ij x ij (s) ds.

(37)
According to (34) and (36), the derivative of V ij (t) can obtain e designed decentralized controller has been described as follows:

Complexity
where zs ij (t)/zx ij (t) is a monotonically increasing function and the inverse must exist in Remark 1. k ij (t) is the controller gain that will be defined later. Ω δ i � e ij (t)|‖e ij (t)‖ < δ i is a compact set, Ω c δ i is its complement set, δ i is an arbitrarily positive real number, and its definition will be given later. Once in Ω δ i near the origin, there is no need to impose on control. e piecewise function χ ij satisfies the following: Remark 6. Particularly, singularities may occur when e i (t) � [0] m in cooperative control strategy in 1/2‖e ij (t)‖ −1 q 2 ij (x ij (t)). e function lim e ij (t) ⟶ 0 e −1 ij (t) � ∞ means the limit of synchronization error function infinitely approaches the sphere near the origin. Even though it is not equal to zero and there is no singular value, it is still uncontrollable. erefore, (40) is designed to prevent the occurrence and accelerate the convergence rate. Figure 1 is the flow diagram of the control approach for a class of uncertain nonlinear MASs with state time-varying constraints and delays, which can clearly guide the construction process of the control strategy.
From Remark 3, the adaptive law for ω ij is designed as follows: where Γ ij is a positive symmetric gain. κ ij is the positive constant to improve the approximation error robustness of FLSs.

Theorem 1. For nonlinear dynamics (1) and (3) with Assumptions 1-3 and the controller gains in
(1) e synchronization error converges to the desired small neighborhood near the origin (2) e state time-varying constraints for agents given by (5) and (8)

Complexity
From (21), given the integral inequality in (48), _ V(t) yields For e ij ∈ Ω c δ i , integrate the above _ V(t) over [0, t]; then, it can obtain From (21), (37), and (56), the inequality obtains Given δ i > ������������ 2η 2 /λ min (Q)η 1 , there exists T > 0, for i � 1, . . . , n, and all t > T: where δ i is a small positive constant which determines tracking performance. e inequality can be obtained bounded by the Lyapunov eory and selecting the appropriate parameters in the closed-loop MASs. e proof is completed.  (1) and (3), Assumptions 1-3, the actual controller is designed in (39) and the adaptive law (41) guarantees that all the signals in the closed-loop systems are SUUB. e parameters selection guidelines for controller design with state time-varying constraints and delays in MASs are demonstrated: (i) Select appropriately designed parameters such that ij , and ε ij . Increasing the adaptive gain c ij and k 0 ij would result in better consensus tracking performance. Other parameters are selected according to experience. erefore, the parameter design of the controller should be adjusted according to the system performance and the tradeoff between tracking performance and constraints between agents such as collision avoidance.

Simulation Example
e following example is given to illustrate the effectiveness of the proposed adaptive fuzzy tracking control scheme for distributed uncertainties nonlinear MASs containing one leader with three followers. Assume that i th agent with timevarying delays is as follows: (59) e dynamic of the leader can be described as where u i is the input of the cooperative tracking control, x l (t) is the state of the leader, the designed trajectories are g l1 � 0.5 sin(0.8t), and g l2 � 0.1 + 0.5 sin(t). e state variables x i1 (t) and x i2 (t) are bounded by time-varying state constraints as follows: −k c 1 (t) < x i1 (t) < k c 1 (t), and k c 1 (t) � 1 + 0.3 sin(2t); −k c 2 (t) < x i2 (t) < k c 2 (t), and k c 2 (t) � 1.8 + 0.4 sin(0.5t); C i1 and C i2 are defined in Table 1; and D i1 and D i2 are defined in Table 2, i � 1, 2, 3. e timevarying delays are τ 1 (t) � 1.25 − 0.5 sin(0.5t), τ 2 (t) � 1.3 −0.6 sin(0.5t)sin(t), τ 3 (t) � 1.2 − 0.5 cos(0.5t), τ max � 2. e information interaction between agents is shown in Figure 2.
Apparently, Assumption 2 can be satisfied by choosing

e adaptive FLSs
Complexity controller u i is given by (39), and the fuzzy rule numbers are l 1 � l 2 � 10. Choose the fuzzy membership functions as follows: e simulation results are shown in Figures 3-7, and i � 1, 2, 3. e trajectories of the desired signal g l1 and full state x i1 (t) with time-varying symmetrical barrier functions −k c 1 (t) and k c 1 (t) are shown in Figure 4. e full state x i2 (t) constraints are bounded by time-varying symmetric functions −k c 2 (t) and k c 2 (t) and track the desired signal g l2 in Figure 5.
From Figures 3-5, it is easy to indicate that the tracking trajectory always holds within the time-varying constraints −k c j (t) < x ij (t) < k c j (t), j � 1, 2, and converges uniformly to the tracking signal. Figure 6 shows the synchronization errors. e i1 and e i2 fluctuate infinitely and approach the origin. Figure 7 shows the norms of the adaptive laws that can show that the effect of control is better. From the simulation, it is obvious that the proposed adaptive fuzzy distributed cooperative tracking control protocol is effective.

Conclusion
An adaptive fuzzy cooperative tracking control has been presented for a class of uncertain nonlinear MASs with state time-varying delays and time-varying constraints. LKFs have been used to compensate for unknown time-varying delays and overcome the difficulty of employing the variable separation technique in the nonfeedback systems. To guarantee that time-varying delay states would not violate the time-varying bounds by constructing nonlinear coordinate transformations, ulteriorly, additional piecewise constructors can speed up convergence and avoid uncontrollability. e simulation results can be sufficiently demonstrated effective control. is paper further enriches the study of nonlinear state constrained systems. e simulation results provide theoretical support for further practical application.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.  15 20 12 Complexity