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

Authors: Jie Lan; Tongyu Xu
Journal: Complexity
 (2021)

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## Abstract

This 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. The 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.


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## Body

## 1. 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. The 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 [1–7]. 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. The research directions include consensus control [9, 10], tracking control [11], aggregation control [12], formation control [13], and synchronous control [1, 14].The 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 [16–18]. 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 [19–21]. The neural networks (NNs) [19, 20] were related to the event trigger. The fuzzy logic systems (FLSs) [21–23] solved the finite time of stochastic nonlinear systems in [21]. Distributed nonlinear MASs among the literature [24–27] had been proved distinguished approximation ability to solve the unknown dynamics. The consensus control was discussed in [9], and consensus tracking control protocols were studied in [24–26]. The multileader consensus control method was researched in [27]. However, the above methods have not involved the problem of state time delays.The 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. Therefore, time delays would be an important consideration. Some achievements had been made for time delays in uncertain nonlinear systems in [28–31]. 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. The Barrier Lyapunov functions have been preferred as the primary candidate functions for designing controllers with constraints. The 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 [34–41], and nonlinear MASs in [27, 42]. Integral barrier Lyapunov functions have been employed in [39–41]. The 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. The 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. The authors of [45, 46] proposed the control strategies to solve both the state constraints and state time-varying delays. Therefore, 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. The 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. The 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. The 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.The proposed strategy is aimed at the study of the time-varying state of the agents, which has great realistic significance and further enriches the exploration of the uncertain nonlinear MASs.The organization of the article is as follows: the presentation and problem of the systems are described in Section2; 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.


## 2. Systems Presentation and Problem Description

### 2.1. Systems Description and Problem Presentation

Consider a class of uncertain nonlinear distributed consensus systems with time-varying delay and theith dynamic is described as(1)x˙it=fixit+ρixit−τit+uit,where xit=xi1,xi2,…,ximT∈Rm represents the state vector of ith agent, i=1,2,…,n, and n is the number of agents; fixi and ρixit−τit: Rm⟶Rm are unknown nonlinear continuous vector functions; τit represents the uncertain time-varying delay; and uit∈Rm is the control input vector.Assumption 1.
∀t≥0, the ith unknown smooth nonlinear time delayed function ρixit−τit, satisfies(2)ρixit−τit≤qixit−τit,where qixit−τit is the known positive smooth function.Remark 1.
The unknown time-varying delays functionρxit−τit contains the time-varying delayed state. The 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.
The unknown time-varyingτit should satisfy the following: (1) the τ˙it is bounded by a known constant τdmax and τ˙it≤τdmax≤1 and (2) the continuous τit should be uniformly bounded by a known constant τmax and τit≤τmax in [33, 45, 46].
The desired reference leader signal is described by the following dynamic function:(3)x˙lt=glt,where xlt∈Rm is the state of reference leader and glt∈Rm is a bounded smooth vector function.
The time-varying state constraint is considered, and the constraint boundary ofith agent meets requirements as follows:(4)−kcjt<xijt<kcjt,where xij represents the jth state of the ith agent; j=1,2,…,m; kcjt is a user-defined time-varying bound function in jth state; and kct∈Rm. The initial value should satisfy −kcj0<xij0<kcj0.
The requirement of (4) can be satisfied by choosing a nonlinear coordinate transformation as follows:(5)sijt=xijtkcjt+xijtkcjt−xijt,where sijt is a constructed state transition that converts the original xijt bounded within −kcjt,kcjt into a nonconstrained state, sit=si1,si2,…,simT∈Rm.
Then, (5) satisfies(6)limxit⟶−kctsit=−∞,limxit⟶kctsit=+∞.
Take the derivative ofsijt as follows: (7)s˙ijt=∂sijt∂xijtx˙ijt+∂sijt∂kcjtk˙cjt,where ∂sijt/∂xijt=kcj2t+xij2t/kcj2t−xij2t2 and ∂sijt/∂kcjt=−2xijtkcjt/kcj2t−xij2t2.
The state of the leader in (3) can be transformed by the same form of (5); it can be expressed as(8)αljt=xljtkcjt+xljtkcjt−xljt,where αljt is a state transition to transform the constrained xljt within −kcjt,kcjt into an unconstrained state, αit=αi1,αi2,…,αimT∈Rm. Then, (8) satisfies(9)limxlt⟶−kctαlt=−∞,limxlt⟶kctαlt=+∞.
Take the derivative of (8) as follows:(10)α˙ljt=∂αljt∂xljtx˙ljt+∂αljt∂kcjtk˙cjt,where ∂αljt/∂xljt=kcj2t+xlj2t/kcj2t−xlj2t2 and ∂αljt/∂kcjt=−2xljtkcjt/kcj2t−xlj2t2.Remark 2.
The statesxi and xl are converted into si and αl, respectively; ∂sijt/∂xijt>0 and ∂αljt/∂xljt>0 are invertible functions; and sijt and αljt are monotonically increasing. Just proving sijt and αljt are bounded, by criteria of monotone bounded, the original states xi and xl would converge and satisfy the constraint region.Assumption 3.
The time-varying bound vector functionskct and k˙ct satisfy the following: ∀t≥0; there exist positive constant vectors Kc∈Rm and Kc1∈Rm that satisfy kcjt≤Kcj and its time derivative k˙cjt satisfies k˙cjt≤Kcj1.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. The control process is time-varying, so it is significant to study the time-varying full state constraints.Remark 4.
The constraints based on BLFs (or IBLFs) require additional complex stability analysis to ensure the feasibility conditions of virtual controllers. The 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.
The 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. This 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.
This paper only shows the feasibility and effectiveness of constructing state time-varying constraints to distributed nonlinear and uncertain time-varying delayed MASs. The asymmetric forms would be further discussed in other papers.


#### 2.1.1. Control Objective

Devise an adaptive fuzzy decentralized tracking controllerui 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) The full states would not violate the time-varying constraint boundaries in (5) and (8). (iii) All the signals in the closed-loop systems are SUUB.The following assumptions and lemmas should be satisfied in order to establish a cooperative control objective.


### 2.2. Algebraic Graph Theory and Notations

An undirected connected topology expresses asG=V,E,A, where V=v1,v2,…,vn denotes a set of nodes and vi is ith node. E⊆V×V denotes the set of edges, ℘ij=vi,vj∈E refers to the edge from node i to node j, and ℘ij=℘ji. A=aij∈Rn×n is a weighted adjacency matrix that a weight corresponds to the edge, and when ℘ij∈Rn×n, aij=1. Otherwise, aij=0. In particular, define aii=0.The graph Laplacian matrixL is given in the following definition:(11)L=D−A,where L=lij⊂Rn×n; D=diagd1,d2,…,dn∈Rn×n defines an in-degree matrix; di=∑j∈Niaij represents the in-degree of node i; Ni=j:j,i∈E; L is an irreducible matrix; and 1,1,…,1T∈Rn is a right eigenvector with the eigenvalue zero.Lemma 1.
The graph Laplacian matrixL is a semidefinite symmetric matrix; all eigenvalues are nonnegative real; and its n real eigenvalues can be listed in ascending order as follows:(12)0=λ1<λ2≤λ3≤…≤λn≤C,where C=2max1≤i≤ndi is algebraic connectivity and it is used to analyze the rate of consensus convergence.Lemma 2.
∀t>0, the continuous function Vt satisfies Vt>0, and V0 is bounded. If it can prove the inequality V˙t≤−ℓVt+ℏ where ℓ>0, ℏ>0, then(13)V˙t≤V0e−ℓt+ℏℓ1−e−ℓt.Assumption 4.
In the graph topologyG, for the leader, there is at least one directed path from the root node to follower nodes.


### 2.3. Distributed Controller

The proposed distributed control algorithm can make arbitraryith 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.The synchronization erroreit∈Rm consists of the ith local consensus error and the tracking error among the leader and followers, and that can be defined as follows:(14)eit=∑j=1Niaijsit−sjt+bisit−αlt,where aij represents the element in adjacency matrix A. bi is the communication weight when only ith agent exchanges information with leader bi>0; otherwise bi=0, and B=diagb1,b2,…,bn is the communication weight matrix and satisfies b1+b2+⋯+bn>0:Let(15)L˜=L+B,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 anm−multiplicity eigenvalue of L˜⊗Im; ⊗ and Im, respectively, stand for the Kronecker product and identity matrix of dimension m×m. Let φ11,φ12,…,φ1m,…,φ2m,…,φnm be the eigenvectors and λ˜2,…,λ˜n eigenvalues of L˜⊗Im such there can be a set of orthogonal bases of Rnm. The definition is given by graph theory:(16)L˜⊗Im=PΛPT,where PTP=PPT=Inm, Inm denoting identity matrix of dimension nm×nm and P=φ11,φ12,…φnm∈Rnm×nm.Define the tracking error vector as the following form:(17)ςit=sit−αlt,where ςit=ςi1t,ςi2t,…,ςimtT∈Rm represents the tracking error between the ith follower agent and the leader agent. Then, (14) can be rewritten as(18)eit=∑j=1Niaijςit−ςjt+biςit,where eit=ei1t,ei2t,…,eimtT.The unknown functions and variables in multiagent systems can be approximated by FLSs as follows:(19)FijXij=ωijTΨijXij+εij,where ωij∈Rlj is the optimal fuzzy weight vector, lj denotes the number of fuzzy rules, ΨijXij is the fuzzy basis vector function, εij shows estimation error satisfied εij≤ε¯ij, and ε¯ij is an arbitrary indeterminate positive constant.


## 2.1. Systems Description and Problem Presentation

Consider a class of uncertain nonlinear distributed consensus systems with time-varying delay and theith dynamic is described as(1)x˙it=fixit+ρixit−τit+uit,where xit=xi1,xi2,…,ximT∈Rm represents the state vector of ith agent, i=1,2,…,n, and n is the number of agents; fixi and ρixit−τit: Rm⟶Rm are unknown nonlinear continuous vector functions; τit represents the uncertain time-varying delay; and uit∈Rm is the control input vector.Assumption 1.
∀t≥0, the ith unknown smooth nonlinear time delayed function ρixit−τit, satisfies(2)ρixit−τit≤qixit−τit,where qixit−τit is the known positive smooth function.Remark 1.
The unknown time-varying delays functionρxit−τit contains the time-varying delayed state. The 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.
The unknown time-varyingτit should satisfy the following: (1) the τ˙it is bounded by a known constant τdmax and τ˙it≤τdmax≤1 and (2) the continuous τit should be uniformly bounded by a known constant τmax and τit≤τmax in [33, 45, 46].
The desired reference leader signal is described by the following dynamic function:(3)x˙lt=glt,where xlt∈Rm is the state of reference leader and glt∈Rm is a bounded smooth vector function.
The time-varying state constraint is considered, and the constraint boundary ofith agent meets requirements as follows:(4)−kcjt<xijt<kcjt,where xij represents the jth state of the ith agent; j=1,2,…,m; kcjt is a user-defined time-varying bound function in jth state; and kct∈Rm. The initial value should satisfy −kcj0<xij0<kcj0.
The requirement of (4) can be satisfied by choosing a nonlinear coordinate transformation as follows:(5)sijt=xijtkcjt+xijtkcjt−xijt,where sijt is a constructed state transition that converts the original xijt bounded within −kcjt,kcjt into a nonconstrained state, sit=si1,si2,…,simT∈Rm.
Then, (5) satisfies(6)limxit⟶−kctsit=−∞,limxit⟶kctsit=+∞.
Take the derivative ofsijt as follows: (7)s˙ijt=∂sijt∂xijtx˙ijt+∂sijt∂kcjtk˙cjt,where ∂sijt/∂xijt=kcj2t+xij2t/kcj2t−xij2t2 and ∂sijt/∂kcjt=−2xijtkcjt/kcj2t−xij2t2.
The state of the leader in (3) can be transformed by the same form of (5); it can be expressed as(8)αljt=xljtkcjt+xljtkcjt−xljt,where αljt is a state transition to transform the constrained xljt within −kcjt,kcjt into an unconstrained state, αit=αi1,αi2,…,αimT∈Rm. Then, (8) satisfies(9)limxlt⟶−kctαlt=−∞,limxlt⟶kctαlt=+∞.
Take the derivative of (8) as follows:(10)α˙ljt=∂αljt∂xljtx˙ljt+∂αljt∂kcjtk˙cjt,where ∂αljt/∂xljt=kcj2t+xlj2t/kcj2t−xlj2t2 and ∂αljt/∂kcjt=−2xljtkcjt/kcj2t−xlj2t2.Remark 2.
The statesxi and xl are converted into si and αl, respectively; ∂sijt/∂xijt>0 and ∂αljt/∂xljt>0 are invertible functions; and sijt and αljt are monotonically increasing. Just proving sijt and αljt are bounded, by criteria of monotone bounded, the original states xi and xl would converge and satisfy the constraint region.Assumption 3.
The time-varying bound vector functionskct and k˙ct satisfy the following: ∀t≥0; there exist positive constant vectors Kc∈Rm and Kc1∈Rm that satisfy kcjt≤Kcj and its time derivative k˙cjt satisfies k˙cjt≤Kcj1.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. The control process is time-varying, so it is significant to study the time-varying full state constraints.Remark 4.
The constraints based on BLFs (or IBLFs) require additional complex stability analysis to ensure the feasibility conditions of virtual controllers. The 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.
The 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. This 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.
This paper only shows the feasibility and effectiveness of constructing state time-varying constraints to distributed nonlinear and uncertain time-varying delayed MASs. The asymmetric forms would be further discussed in other papers.


### 2.1.1. Control Objective

Devise an adaptive fuzzy decentralized tracking controllerui 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) The full states would not violate the time-varying constraint boundaries in (5) and (8). (iii) All the signals in the closed-loop systems are SUUB.The following assumptions and lemmas should be satisfied in order to establish a cooperative control objective.


## 2.1.1. Control Objective

Devise an adaptive fuzzy decentralized tracking controllerui 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) The full states would not violate the time-varying constraint boundaries in (5) and (8). (iii) All the signals in the closed-loop systems are SUUB.The following assumptions and lemmas should be satisfied in order to establish a cooperative control objective.


## 2.2. Algebraic Graph Theory and Notations

An undirected connected topology expresses asG=V,E,A, where V=v1,v2,…,vn denotes a set of nodes and vi is ith node. E⊆V×V denotes the set of edges, ℘ij=vi,vj∈E refers to the edge from node i to node j, and ℘ij=℘ji. A=aij∈Rn×n is a weighted adjacency matrix that a weight corresponds to the edge, and when ℘ij∈Rn×n, aij=1. Otherwise, aij=0. In particular, define aii=0.The graph Laplacian matrixL is given in the following definition:(11)L=D−A,where L=lij⊂Rn×n; D=diagd1,d2,…,dn∈Rn×n defines an in-degree matrix; di=∑j∈Niaij represents the in-degree of node i; Ni=j:j,i∈E; L is an irreducible matrix; and 1,1,…,1T∈Rn is a right eigenvector with the eigenvalue zero.Lemma 1.
The graph Laplacian matrixL is a semidefinite symmetric matrix; all eigenvalues are nonnegative real; and its n real eigenvalues can be listed in ascending order as follows:(12)0=λ1<λ2≤λ3≤…≤λn≤C,where C=2max1≤i≤ndi is algebraic connectivity and it is used to analyze the rate of consensus convergence.Lemma 2.
∀t>0, the continuous function Vt satisfies Vt>0, and V0 is bounded. If it can prove the inequality V˙t≤−ℓVt+ℏ where ℓ>0, ℏ>0, then(13)V˙t≤V0e−ℓt+ℏℓ1−e−ℓt.Assumption 4.
In the graph topologyG, for the leader, there is at least one directed path from the root node to follower nodes.


## 2.3. Distributed Controller

The proposed distributed control algorithm can make arbitraryith 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.The synchronization erroreit∈Rm consists of the ith local consensus error and the tracking error among the leader and followers, and that can be defined as follows:(14)eit=∑j=1Niaijsit−sjt+bisit−αlt,where aij represents the element in adjacency matrix A. bi is the communication weight when only ith agent exchanges information with leader bi>0; otherwise bi=0, and B=diagb1,b2,…,bn is the communication weight matrix and satisfies b1+b2+⋯+bn>0:Let(15)L˜=L+B,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 anm−multiplicity eigenvalue of L˜⊗Im; ⊗ and Im, respectively, stand for the Kronecker product and identity matrix of dimension m×m. Let φ11,φ12,…,φ1m,…,φ2m,…,φnm be the eigenvectors and λ˜2,…,λ˜n eigenvalues of L˜⊗Im such there can be a set of orthogonal bases of Rnm. The definition is given by graph theory:(16)L˜⊗Im=PΛPT,where PTP=PPT=Inm, Inm denoting identity matrix of dimension nm×nm and P=φ11,φ12,…φnm∈Rnm×nm.Define the tracking error vector as the following form:(17)ςit=sit−αlt,where ςit=ςi1t,ςi2t,…,ςimtT∈Rm represents the tracking error between the ith follower agent and the leader agent. Then, (14) can be rewritten as(18)eit=∑j=1Niaijςit−ςjt+biςit,where eit=ei1t,ei2t,…,eimtT.The unknown functions and variables in multiagent systems can be approximated by FLSs as follows:(19)FijXij=ωijTΨijXij+εij,where ωij∈Rlj is the optimal fuzzy weight vector, lj denotes the number of fuzzy rules, ΨijXij is the fuzzy basis vector function, εij shows estimation error satisfied εij≤ε¯ij, and ε¯ij is an arbitrary indeterminate positive constant.


## 3. 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:(20)Vω=12ςtTQςt+12∑i=1ntrω˜iTΓi−1ω˜i,where ςt=ς1Tt,ς2Tt,…,ςnTtT∈Rnm is the consensus error vector that ςit has defined in (17), then ςt=L˜⊗Imet, Q=PΛ^−1PT, ω˜ij=ωij−ω^ij, ω^ij is the estimate of adaptive laws, and ωij and ω^ij will be given later.The inequality can be described as follows:(21)λminQ2≤12ςtTQςt≤λmaxQ2,where λminQ and λmaxQ are the smallest and largest eigenvalues of Q, respectively. From (16), it yields(22)12ςtTQςt=12eTtL˜⊗ImTPΛ^−1PTL˜⊗Imet=12eTtPΛPTPΛ^−1PTPΛPTet=12eTtL˜⊗Imet=12ςtTet=12∑i=1nςiTteit,where Λ=diag0Im,λ˜2Im,…,λ˜nIm.Redefine the Lyapunov candidate function as follows:(23)Vω=12∑i=1nςiTteit+12∑i=1ntrω˜iTΓi−1ω˜i.Take the derivative of (23). Then, V˙ω can yield(24)V˙ω=12ς˙tTQςt+12ςtTQς˙t+12∑i=1ntrω˜˙iTΓi−1ω˜i+12∑i=1ntrω˜iTΓi−1ω˜˙i=∑i=1neiTtς˙it−∑i=1ntrω˜iTΓi−1ω^˙i.V˙ω can be written as follows:(25)V˙ω=∑j=1mV˙ωj=∑j=1m∑i=1neijTtς˙ijt−ω˜ijTΓij−1ω^˙ij.The derivative ofςijt in the form of (17) can be described as(26)ς˙ijt=∂sijt∂xijtx˙ijt−∂αljt∂xljtx˙ljt+∂sijt∂kcjt−∂αljt∂kcjtk˙cjt.Combining systems (1), (3), and (26) into V˙ωj yields(27)V˙ωj≤∑i=1neijTt∂sijt∂xijtuijt+ρijxijt−τit+fijxijt+Kijt−∂αljt∂xljtgljt−∑i=1nω˜ijTΓij−1ω^˙ij,where Kijt is ith time-varying gain described as follows:(28)Kijt=sup∂sijt∂kcjtk˙cjt2+∂αljt∂kcjtk˙cjt2+βij.βij is a positive parameter to be given.Using Young’s inequality and Assumption1, in order to separate eij and uncertainties ρijxijt−τit, the inequality gets(29)eijTt∂sijt∂xijtρijxijt−τit≤12veijt2Mij2+12vqij2xijt−τit,where v=1−τdmax has been illustrated in Remark 4 and Mij=∂sijt/∂xijt.The unknown smooth functionFijXij is approximated by FLSs to tackle the unknown dynamics functions and parameters. FijXij is denoted as(30)FijXij=fijxijt+121veijt∂sijt∂xijt,where Xi1=xi1,xi2,kc1,ei1T, Xi2=xi1,xi2,kc2,ei2T, and eijt2Mij2 takes the norm of eijt∂sijt/∂xijt in (29).Equation (30) can be estimated by (19) as follows:(31)FijXij=ωijTΨijXij+εij.Combining (28), (29), and (30) by FLSs, then (27) can yield(32)V˙ωj≤∑i=1neijTt∂sijt∂xijtuijt+ωijTΨijXij+εij+Uijt+12∑i=1nvqij2xijt−τit−∑i=1nω˜ijTΓij−1ω^˙ij,where Uijt=Kijt−∂αljt/∂xljtgljt.By using Young’s inequality, we obtain the following:(33)∑i=1neijTt∂sijt∂xijtεij≤12∑i=1neijt2Mij2+12∑i=1nε¯ij2.Combining (33) into (32), we obtain(34)V˙ωj≤∑i=1neijTt∂sijt∂xijtuijt+ωijTΨijXij+Uijt+12∑i=1nvqij2xijt−τit−∑i=1nω˜ijTΓij−1ω^˙ij+12∑i=1neijt2Mij2+∑i=1n12ε¯ij2.The Lyapunov–Krasovskii function is constructed to compensate uncertain time-varying delays as follows:(35)Vρt=∑i=1mVρjt=12∑i=1m∑i=1n∫t−τittβijxijsds,where βijxijs≤qij2xijs has been defined under Assumption 1.Take the derivative ofVρjt. It obtains(36)V˙ρjt≤12∑i=1nqij2xijt−12v∑i=1nqij2xijt−τit.The Lyapunov function candidate is chosen by combining (23) and (35) as follows:(37)Vt=Vω+Vρt,∑j=1mVijt=∑j=1mVωj+∑i=1mVρjt,∑j=1mVijt=∑j=1m12∑i=1nςijTteijt+12∑i=1nω˜ijTΓij−1ω˜ij+12∑j=1m∑i=1n∫t−τittβijxijsds.According to (34) and (36), the derivative of Vijt can obtain(38)V˙ij≤∑i=1neijTt∂sijt∂xijtuijt+ωijTΨijXij+Uijt+12∑i=1nqij2xijt−∑i=1nω˜ijTΓij−1ω^˙ij+12∑i=1neijt2Mij2+12∑i=1nε¯ij2.The designed decentralized controller has been described as follows:(39)uijt=∂sijt∂xijt−1−kijteijt−ω^ijTΨijXij−12eijtqij2xijteijt2+χij−Uijt,eijt∈Ωδic,0,eijt∈Ωδi,where ∂sijt/∂xijt is a monotonically increasing function and the inverse must exist in Remark 1. kijt is the controller gain that will be defined later. Ωδi=eijt|eijt<δi is a compact set, Ωδic is its complement set, δi is an arbitrarily positive real number, and its definition will be given later. Once in Ωδi near the origin, there is no need to impose on control. The piecewise function χij satisfies the following:(40)χij=1,eijt∈Ωδic,0,eijt∈Ωδi.Remark 6.
Particularly, singularities may occur wheneit=0m in cooperative control strategy in 1/2eijt−1qij2xijt. The function limeijt⟶0eij−1t=∞ 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. Therefore, (40) is designed to prevent the occurrence and accelerate the convergence rate.
Figure1 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 Remark3, the adaptive law for ω^ij is designed as follows:(41)ω^˙ij=ΓijeijtΨijXij−κijω^ij,where Γij is a positive symmetric gain. κij is the positive constant to improve the approximation error robustness of FLSs.
Wheneijt∈Ωδic, combining (39), (40), and (41) into (38) yields(42)V˙ijt≤∑i=1n−kijteijt2+∑i=1nκijω˜ijTω^ij+12∑i=1neijt2Mij2+12∑i=1nε¯ij2.
Using Young’s inequality, it follows that(43)ω˜ijTω^ij=ω˜ijTωij−ω˜ij≤12ωij2−12ω˜ijTω˜ij.
According to (43) into (42), then V˙ijt obtains(44)V˙ijt≤∑i=1n−kijteijt2+12∑i=1nMij2eijt2−12κijω˜ijTω˜ij+ηj2,where ηj2=1/2∑i=1nκijωij2+1/2∑i=1nε¯ij2.
The definition of controller gainkijt is described briefly as follows:(45)kijt=kij0+kij1t,where kij0≥1/2Mij2, and kij1t satisfies(46)kij1t=γij2λmaxQ+1eijt2+χij∫t−τmaxtβijxijsds.
Substitute (45) and (46) into (44); it yields(47)V˙ijt≤−12∑i=1nγijλmaxQeijt2−12∑i=1nκijΓij−1ω˜ijTΓij−1ω˜ij−12∑i=1nγij∫t−τmaxtβijxijsds+ηj2,where βijxijs>0 has been defined; therefore, the integral inequality satisfies(48)12∑i=1n∫t−τmaxtβijxijsds≥12∑i=1n∫t−τittβijxijsds.
By (48), V˙ijt can obtain(49)V˙ijt≤−12∑i=1nγijλmaxQeijt2−12∑i=1nκijΓij−1ω˜ijTΓij−1ω˜ij−12∑i=1nγij∫t−τittβijxijsds+ηj2.
From (21), V˙ijt yields(50)V˙ijt≤−ηj1Vijt+ηj2,where ηj1=min1≤i≤nγij,κij/Γij−1.
Then, compute this sum and integral ofV˙ijt obtained that(51)Vt≤V0e−η1t+η2η11−e−η1t,where η1=min1≤j≤mηj1, η2=∑j=1mηj2.Figure 1
The block diagram illustrating the flow of the proposed control approach.Theorem 1.
For nonlinear dynamics (1) and (3) with Assumptions 1–3 and the controller gains in (39), the adaptive law is given by (41). By selecting appropriate parameters, the adaptive fuzzy cooperative control strategy can guarantee the following:(1)
The synchronization error converges to the desired small neighborhood near the origin(2)
The state time-varying constraints for agents given by (5) and (8) would not be violated(3)
The time-varying delays can be compensated by LKFs. The signals in the closed-loop systems can be verified SUUBProof.
Wheneijt∈Ωδic, choosing the Lyapunov function in (37), the derivative of Vt is as follows:(52)V˙t≤∑j=1m∑i=1ne¯ijTt∂sijt∂xijtuijt+ρijxijt−τit+fijxijt+Uijt−∑i=1nω˜ijTΓij−1ω^˙ij+∑i=1nqi2xit−12v∑i=1nqi2xit−τit.
Applying the results in (28), (29), (30), (33), and (36) into (52), it obtains(53)V˙t≤∑j=1m∑i=1neijTt∂sijt∂xijtuijt+ωijTΨijXij+Uijt+∑i=1n12qij2xijt−∑i=1nω˜ijTΓij−1ω^˙ij+12∑i=1nMij2eijt2+∑i=1n12ε¯ij2.
Substitute (39), (41), controller gain (45), (46) into (53); then, it yields(54)V˙t≤∑j=1m−12∑i=1nγijλmaxQeijt2−12∑i=1nγij∫t−τmaxtβijxijsds−12∑i=1nκijΓij−1ω˜ijTΓij−1ω˜ij+ηj2.
From (21), given the integral inequality in (48), V˙t yields(55)V˙t≤∑j=1m∑i=1n−12γijλmaxQeijt2−γij2∫t−τittβijxijsds−12κijΓij−1ω˜ijTΓij−1ω˜ij+ηj2≤−η1Vt+η2.
Foreij∈Ωδic, integrate the above V˙t over 0,t; then, it can obtain(56)0<Vt≤η2η1+V0−η2η1e−η1t.
From (21), (37), and (56), the inequality obtains(57)∑j=1m∑i=1nλminQ2eijt2≤Vt≤η2η1+V0−η2η1e−η1t≤η2η1+V0e−η1t∑j=1m∑i=1neijt2≤2η2λminDη1+2λminDV0e−η1t.
Givenδi>2η2/λminQη1, there exists T>0, for i=1,…,n, and all t>T:(58)eijt<δi,where δi is a small positive constant which determines tracking performance. The inequality can be obtained bounded by the Lyapunov Theory and selecting the appropriate parameters in the closed-loop MASs.
The proof is completed.Remark 8.
For the uncertain nonlinear MASs (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. The parameters selection guidelines for controller design with state time-varying constraints and delays in MASs are demonstrated:(i)
Select appropriately designed parameters such thatkij0, ωijT, βij, δi, Γij, κij, γij, ε¯ij. Meanwhile, a positive matrix Q is needed to satisfy Q=PΛ^−1PT and PΛPT=L˜⊗Im.(ii)
Decrease the control gainΓij, ωij2, and ε¯ij. Increasing the adaptive gain γij and kij0 would result in better consensus tracking performance. Other parameters are selected according to experience. Therefore, 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.


## 4. Simulation Example

The 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 thatith agent with time-varying delays is as follows:(59)x˙i1=xi2tsinDi1xi1t+ρi1xi1t−τit+uit,x˙i2=xi1tcosDi2xi2t+ρi2xi2t−τit+uit.The dynamic of the leader can be described as(60)x˙lt=0.5sin0.8t0.1+0.5sint,where ui is the input of the cooperative tracking control, xlt is the state of the leader, the designed trajectories are gl1=0.5sin0.8t, and gl2=0.1+0.5sint. The state variables xi1t and xi2t are bounded by time-varying state constraints as follows: −kc1t<xi1t<kc1t, and kc1t=1+0.3sin2t; −kc2t<xi2t<kc2t, and kc2t=1.8+0.4sin0.5t; Ci1 and Ci2 are defined in Table 1; and Di1 and Di2 are defined in Table 2, i=1,2,3. The time-varying delays are τ1t=1.25−0.5sin0.5t, τ2t=1.3−0.6sin0.5tsint, τ3t=1.2−0.5cos0.5t, τmax=2.Table 1
Values ofCi1 and Ci2, i=1,2,3.
i123Ci113−2Ci21.52.70.8Table 2
Values ofDi1 and Di2, i=1,2,3.
i123Di10.6−36Di20.70.5−5The information interaction between agents is shown in Figure2.Figure 2
The communication topology for one leader and three followers.The topology illustrates the information interaction between agents in Figure2 and B=diag1,1,0. The Laplacian matrix L is given as L=0.7−0.70.0−0.71.5−0.80.0−0.80.8.The initial of three agents arex10=0.05,0.108T, x20=−0.13,0.1T, and x30=0.038,−0.135T. The designed parameters are chosen as Γi1=Γi2=1, κ11=0.7, κ21=0.4, κ31=0.4, κ12=0.6, κ22=0.6, κ32=0.6, k01=k02=270, δi=10−7, and γi=0.9, i=1,2,3.Apparently, Assumption2 can be satisfied by choosing qixit=Ci1xi1t2+Ci2xi2t2. The adaptive FLSs controller ui is given by (39), and the fuzzy rule numbers are l1=l2=10. Choose the fuzzy membership functions as follows:(61)μFi1l1Xi1=exp−xi1−0.5l128−xi2−0.5l128−kc1−0.5l128−ei1−0.5l128,μFi2l2Xi2=exp−xi1−0.5l228−xi2−0.5l228−kc2−0.5l228−ei2−0.5l228,Ψi1l1Xi1=μFi1l1Xi1∑l1=110μFi1l1Xi1,Ψi2l2Xi2=μFi1l2Xi2∑l210μFi1l2Xi2.The simulation results are shown in Figures3–7, and i=1,2,3. The trajectories of the desired signal gl1 and full state xi1t with time-varying symmetrical barrier functions −kc1t and kc1t are shown in Figure 4. The full state xi2t constraints are bounded by time-varying symmetric functions −kc2t and kc2t and track the desired signal gl2 in Figure 5.Figure 3
The three agents follow the trajectory of the leader.Figure 4
The trajectories of statexi1 with the reference signal gl1 in the time-varying constraint boundaries −kc1t and kc1t.Figure 5
The trajectories of statexi2 with the reference signal gl2 in the time-varying constraint boundaries −kc2t and kc2t.Figure 6
The trajectories of errorsei1 and ei2 with time-varying delays and constraints.Figure 7
The boundedness of the adaptive laws.From Figures3–5, it is easy to indicate that the tracking trajectory always holds within the time-varying constraints −kcjt<xijt<kcjt, j=1,2, and converges uniformly to the tracking signal. Figure 6 shows the synchronization errors. ei1 and ei2 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.


## 5. 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. The simulation results can be sufficiently demonstrated effective control. This paper further enriches the study of nonlinear state constrained systems. The simulation results provide theoretical support for further practical application.



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