# Global Adaptive Consensus Control for Multiagent Systems with Predefined Accuracy

Authors: Chunsheng Zhang; Jian Wu
Journal: Complexity
 (2021)

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## Abstract

This paper addresses a consensus problem for uncertain nonlinear multiagent systems with predefined precision under disturbance. By employing the neural networks method and backstepping technique, adaptive controllers for each agent are created. In contrast to the exiting global control methods for multiagent systems, global precision consensus control scheme is first put forward. Moreover, by using threenth-order continuous differentiable functions, adaptive tuning laws and virtual controllers and the real controller are designed. It is proved that the presented method can ensure that all signals are globally bounded and systems can be consistent with a given accuracy under disturbance. Finally, a practical simulation verifies the correctness for the devised control protocol.


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## Body

## 1. Introduction

Over the last few decades, multiagent systems (MASs) have become a hot research direction because of widespread application in various areas such as mobile robots, cooperative surveillance, spacecraft formation flying, and sensor networks [1–3]. The consensus is a key issue in the study of MASs; it requires that all agents’ states tend to a same value. If all agents converge to one of them, that is the leader-following problem. As a significance problem for MASs, the leader-following problem has received wide attention [4–7]. With the development of research, the study for leader-follower has appeared in many aspects of MASs. For instance, containment control for leader-following MASs [8–11], the consensus about leader-following for fractional-order MASs [12–15], event-based trigger control for leader-following MASs [16–20], and leader-following consensus for uncertain nonlinear MASs [21–23].Nonlinearity exists in all kinds of systems [24–29]. For uncertain nonlinear MASs, estimation-based strategies are often used to cope with unknown items in dynamic models [30–40]. These mean that both fuzzy logic systems (FLSs) and neural networks (NNs) can be utilized to estimate uncertain nonlinearities in MASs through the backstepping technique. To name a few, the radial basis function (RBF) NNs ξ∗TΦZ were applied to approximate nonlinear function fZ in [32]; the RBF NNs Wk∗TSkZ¯k were employed to estimate the unknown items fkx¯k in [33], i=1,…,n; in [34, 35], FLSs were used to approximate nonlinear uncertain items in dynamic models; the work in [36] showed that the approximator WTφx is devised to replace the unknown function yx. In aforementioned reports, adaptive control technique and backstepping technique were common approaches to design the controller and adaptive laws.Due to the influence of external factors such as noise, temperature, and electromagnetic pulse, disturbance is a factor that must be considered when studying MASs. Many practical MASs are often affected by disturbance, even the consensus of plants was compromised. Compared with single agent with disturbance [41–44], it is an interesting and challenging job for nonlinear MASs in the presence of disturbance, since more objects and transport information are considered. There are two control schemes which are often used to deal with disturbance. One is to make the disturbance to be bounded, see, e.g., [32, 45]. The other strategy is to dispose interference by utilizing disturbance observer as stated in [35, 36, 46]. The bounded disturbance is straightforward to solve, and observer-based control strategy for the disturbance needs to be considered when the interference is unbounded.In current reports of MASs, only a little literature reported that the consensus tracking errors converged to a range of a known precision such as [47]. However, the alteration for the scope of tracking errors relied on the adjustment of parameters. That will result in a waste of time and efficiency, as well as an increase in cost. In practical engineering applications, many MASs tend to have a predefined accuracy range for consensus tracking errors. Consequently, motivated by the above discussions, up to now, there are no reports about global consensus strategy for nonlinear MASs under a priori precision. In this note, we are committed to global adaptive neural consensus control protocol for nonlinear uncertain MASs with disturbance and predefined accuracy. The major contributions are summarized as follows.(i)
In contrast to the exiting global consensus conclusion for MASs [48, 49], the work in this paper is first to develop the scheme of the global consensus for uncertain MASs with a given precision. The merit of the proposed strategy can ensure system signals are globally bounded and the consensus errors can converge to a predefined accuracy.(ii)
Technically, in order to achieve control objective, three auxiliary functions are employed to devise appropriate controller by using the backstepping technique in this note. These three functions are particularly important that must be continuous and have derivatives up to thenth order. Benefitted from them, the desired adaptive NN controller can be devised.The remainder of this paper is organized as follows. In Section2, some preliminaries are given. Section 3 shows the design process and performance investigation. The consistency analysis is given in Section 4. The simulation and conclusion are developed in Sections 5 and 6.Notation. In this paper, R is the symbol of a real number set. Rn and RN×N represent n-dimensional and N×N-dimensional Euclidean space, respectively. sup⋅ means the minimum upper bound. argmin⋅ denotes minimized ⋅. ⋅ is the symbol for an absolute value. Cn stands for the set of functions which are continuous derivatives of order n. •˜ is the error and •˜=•−•^, where •^ is the estimate of •.


## 2. Preliminaries

### 2.1. Graph Theory

In this note, an undirected graphGg=P,F,W is utilized to express information transfer among agents in which the element of P=1,2,…,N is corresponding to each agent, F⊆P×P represents the set of edges, and W=sij∈RN×N (i = 1, …, N, j = 1, …, n) indicates the adjacency matrix. In the Gg, if the edge Pi,Pj∈F, then Pj,Pi∈F. The set of neighbors is denoted by Ei=Pj|Pj,Pi∈F,i≠j. If sij>1 in W, it indicates that the information for agent j can receive agent i; otherwise, sij=0. Obviously, sij=sji>1 is true in Gg. Every couple of different agents has an undirected sequence for edges because the graph Gg is connected [50]. The in-degree is expressed by di=∑j∈Eisij. ℒ=lij∈RN×N represents the Laplacian matrix for Gg, where lii=∑sij and lij=−sij, i≠j. Define the leader adjutancy matrix as U=diagbi∈RN×N which indicates an augmented matrix of the leader, and bi>1 denotes that agent i can receive the leader’s message; otherwise, bi=0.


### 2.2. RBF NNs

In this subsection, RBF NNs are utilized to estimate nonlinear functiong′X which is unknown as follows:(1)g′X=WTSX,where X∈ΩX⊂Rq is the input vector and ΩX represents a compact set, W=W1,…,WqT∈Rq indicates the weight vector and q is the count of nodes for RBF NN, and SX=S1X,…,SqT is the vector of the basis function in which SlX is the Gaussian function, and the following equality holds:(2)SlX=exp−X−υlTX−υlμ2,l=1,…,q,where υl=υl1,…,υlqT is the center for the receptive field and μ is the width for SlX.Lemma 1 ([51]).
The RBF NN (1) can estimate any function gX with a given positive η and a big enough big q in ΩX⊂Rq; meanwhile, the following equality is true:(3)gX=W∗TSX+σX,∀X∈ΩX⊂Rq,where W∗ is an ideal constant weight vector and is set as(4)W∗≔argminW∈RqsupX∈ΩXgX−WTSX,and σX is an estimate error satisfying σX≤η.


### 2.3. Three Necessary Switching Functions

To facilitate the design of adaptive NN controller, as stated in [52], three important Cn-class functions are defined as follows:(5)Nnϵz=z−ϵnn!,z>ϵ,0,z≤ϵ,ℋnϵz=sgnz,z>ϵ,1−2cosnπ2sinnπ4ϵz+ϵ,z≤ϵ,where ϵ>0 is a constant and n>0 is an integer.(6)cz=1,z≤e1,cosn+1π2sinn+1π2z2−e12e22−e12,otherwise,0,z≥e2,where e1>0 and e2>0 are constants satisfying e2>e1. To simplify, e1 and e2 are defined as the minimum value and the maximum value of z, respectively. We define(7)CZ=∏q=1rczq,with Z=z1,…,zrT∈Rr. In what follows, it is an important basis for creating global controller.Lemma 2 ([52]).
The features of functionsNnϵz and ℋnϵz are expressed as follows:(i)
Nnϵz∈Cn and ℋnϵz∈Cn. What’s more, the following equality can be obtained:(8)dqNn+1ϵzdzq=Nn−q+1ϵzℋn−q+1ϵzq,q=1,2,…,n.(ii)
Nn+1ϵz is a nonnegative function, and only if z≤ϵ, Nn+1ϵz=0. Furthermore, for q=1,2,…,n, Nqϵzℋqϵz2=Nqϵz.Proof.
By applying the fundamental natures of derivatives, Lemma2 can be proved easily. Here, due to space limitation, the procedure of the proof is omitted.Remark 1.
Due to the property ofCn-class functions, Nn+1ϵz, ℋnϵz, and C(Z) are very appropriate to employ the backstepping technique. What’s more, both functions Nn+1ϵz and ℋn+1ϵz can be adopted to devise the desired controller which guarantees agents’ output converges to a predefined accuracy ϵ.


### 2.4. System Stability

Consider the following nonlinear system [51]:(9)χ˙=fχ,t,χt0=χ0,where χt∈Rn denotes the state for (9), f:Rn×t0,∞⟶Rn is a continuous function, t0 is the initial time, and χ0∈Rn is the initial state.Definition 1 ([51]).
If there existsΩ1⊂Rn (Ω1 is a compact set) satisfying all χt0=χ0∈Ω1, the solution for system (9) is said to be uniformly ultimately bounded (UUB). If there exists a positive κ and Tκ,χ0, for all t≥Tκ,χ0+t0, the inequality χt≤κ holds. Especially, if Ω1=Rn, we say the solution of (9) is globally uniformly ultimately bounded (GUUB).


## 2.1. Graph Theory

In this note, an undirected graphGg=P,F,W is utilized to express information transfer among agents in which the element of P=1,2,…,N is corresponding to each agent, F⊆P×P represents the set of edges, and W=sij∈RN×N (i = 1, …, N, j = 1, …, n) indicates the adjacency matrix. In the Gg, if the edge Pi,Pj∈F, then Pj,Pi∈F. The set of neighbors is denoted by Ei=Pj|Pj,Pi∈F,i≠j. If sij>1 in W, it indicates that the information for agent j can receive agent i; otherwise, sij=0. Obviously, sij=sji>1 is true in Gg. Every couple of different agents has an undirected sequence for edges because the graph Gg is connected [50]. The in-degree is expressed by di=∑j∈Eisij. ℒ=lij∈RN×N represents the Laplacian matrix for Gg, where lii=∑sij and lij=−sij, i≠j. Define the leader adjutancy matrix as U=diagbi∈RN×N which indicates an augmented matrix of the leader, and bi>1 denotes that agent i can receive the leader’s message; otherwise, bi=0.


## 2.2. RBF NNs

In this subsection, RBF NNs are utilized to estimate nonlinear functiong′X which is unknown as follows:(1)g′X=WTSX,where X∈ΩX⊂Rq is the input vector and ΩX represents a compact set, W=W1,…,WqT∈Rq indicates the weight vector and q is the count of nodes for RBF NN, and SX=S1X,…,SqT is the vector of the basis function in which SlX is the Gaussian function, and the following equality holds:(2)SlX=exp−X−υlTX−υlμ2,l=1,…,q,where υl=υl1,…,υlqT is the center for the receptive field and μ is the width for SlX.Lemma 1 ([51]).
The RBF NN (1) can estimate any function gX with a given positive η and a big enough big q in ΩX⊂Rq; meanwhile, the following equality is true:(3)gX=W∗TSX+σX,∀X∈ΩX⊂Rq,where W∗ is an ideal constant weight vector and is set as(4)W∗≔argminW∈RqsupX∈ΩXgX−WTSX,and σX is an estimate error satisfying σX≤η.


## 2.3. Three Necessary Switching Functions

To facilitate the design of adaptive NN controller, as stated in [52], three important Cn-class functions are defined as follows:(5)Nnϵz=z−ϵnn!,z>ϵ,0,z≤ϵ,ℋnϵz=sgnz,z>ϵ,1−2cosnπ2sinnπ4ϵz+ϵ,z≤ϵ,where ϵ>0 is a constant and n>0 is an integer.(6)cz=1,z≤e1,cosn+1π2sinn+1π2z2−e12e22−e12,otherwise,0,z≥e2,where e1>0 and e2>0 are constants satisfying e2>e1. To simplify, e1 and e2 are defined as the minimum value and the maximum value of z, respectively. We define(7)CZ=∏q=1rczq,with Z=z1,…,zrT∈Rr. In what follows, it is an important basis for creating global controller.Lemma 2 ([52]).
The features of functionsNnϵz and ℋnϵz are expressed as follows:(i)
Nnϵz∈Cn and ℋnϵz∈Cn. What’s more, the following equality can be obtained:(8)dqNn+1ϵzdzq=Nn−q+1ϵzℋn−q+1ϵzq,q=1,2,…,n.(ii)
Nn+1ϵz is a nonnegative function, and only if z≤ϵ, Nn+1ϵz=0. Furthermore, for q=1,2,…,n, Nqϵzℋqϵz2=Nqϵz.Proof.
By applying the fundamental natures of derivatives, Lemma2 can be proved easily. Here, due to space limitation, the procedure of the proof is omitted.Remark 1.
Due to the property ofCn-class functions, Nn+1ϵz, ℋnϵz, and C(Z) are very appropriate to employ the backstepping technique. What’s more, both functions Nn+1ϵz and ℋn+1ϵz can be adopted to devise the desired controller which guarantees agents’ output converges to a predefined accuracy ϵ.


## 2.4. System Stability

Consider the following nonlinear system [51]:(9)χ˙=fχ,t,χt0=χ0,where χt∈Rn denotes the state for (9), f:Rn×t0,∞⟶Rn is a continuous function, t0 is the initial time, and χ0∈Rn is the initial state.Definition 1 ([51]).
If there existsΩ1⊂Rn (Ω1 is a compact set) satisfying all χt0=χ0∈Ω1, the solution for system (9) is said to be uniformly ultimately bounded (UUB). If there exists a positive κ and Tκ,χ0, for all t≥Tκ,χ0+t0, the inequality χt≤κ holds. Especially, if Ω1=Rn, we say the solution of (9) is globally uniformly ultimately bounded (GUUB).


## 3. Problem Formulation

### 3.1. System Description

Consider nonlinear uncertain MASs with disturbance consisting ofN agents. The dynamic model for the ith agent is described as follows:(10)ξ˙i,m=ξi,m+1+fi,mξ¯i,m+di,mt,ξ˙i,n=ui+fi,nξ¯i,n+di,nt,yi=ξi,1,where ξ¯i,m=ξi,1,…,ξi,nT∈Rm; yi and ui are the state vector, system output and input with m=1,…,n and i=1,…,N, respectively. Both nonlinear functions fi,m⋅ and fi,n⋅ are smooth and unknown. di,mt and di,nt are bounded disturbances. There is a leader in MASs, and its output trajectory is yrt. For convenience, yrt is abbreviated as yr later.Unlike the semiglobal control objective in [47], the objective of this article is formulated as follows:(i)
A suitable consensus controller is devised that can make sure agents’ output trajectories synchronise toyr of a leader.(ii)
All signals of plants (10) are globally bounded and the consensus errors converge to a prior accuracy ϵ>0 range.


### 3.2. Assumptions and Lemmas

To realize this goal, the following information is necessary.Assumption 1 ([45]).
The undirected graphGg is considered to be fixed and connected. The trajectory of the leader yr has continuous derivatives up to the nth order and all of them are bounded.Assumption 2 ([32]).
The unknown disturbancedi,mt is bounded with i=1,…,N and m=1,…,n, and there exists a positive constant ϱi,m satisfying di,mt≤ϱi,m.Assumption 3 ([52]).
For the functionfi,mξ¯i,mt, assume that there exists a function Ψi,mξ¯i,mt which is smooth and known such that fi,mξ¯i,mt≤ρi,mΨi,mξ¯i,mt, where ρi,m is the unknown positive constant, i=1,…,N, and m=1,…,n.Remark 2.
As stated in [52, 53], Assumption 3 plays an important role in global consistency analysis. In what follows, because of the upper limit of the function fi,mξ¯i,mt, the robust controller part αi,m2 can be created. It is responsible for coarse tuning throughout the control process.Lemma 3 (Barbalat’s lemma [54]).
The functionKt is differentiable and satisfies limt⟶∞K˙t=0 if its first derivative is uniformly continuous; meanwhile, limt⟶∞ht is finite.Lemma 4 (Young’s inequality [55]).
For∀p,q∈R2, there exists the following inequality:(11)pq≤xssps+qttxt,where x>0, s>1, t>1, and s−1t−1=1.Lemma 5 ([50]).
If there exists a connected and undirected graphGg, and W∈Rp×p is its adjacency matrix, then the symmetric positive definite matrix is expressed as ℒ+diagϕ1,…,ϕp in which there is at least one ϕk>0k=1,…,p.


## 3.1. System Description

Consider nonlinear uncertain MASs with disturbance consisting ofN agents. The dynamic model for the ith agent is described as follows:(10)ξ˙i,m=ξi,m+1+fi,mξ¯i,m+di,mt,ξ˙i,n=ui+fi,nξ¯i,n+di,nt,yi=ξi,1,where ξ¯i,m=ξi,1,…,ξi,nT∈Rm; yi and ui are the state vector, system output and input with m=1,…,n and i=1,…,N, respectively. Both nonlinear functions fi,m⋅ and fi,n⋅ are smooth and unknown. di,mt and di,nt are bounded disturbances. There is a leader in MASs, and its output trajectory is yrt. For convenience, yrt is abbreviated as yr later.Unlike the semiglobal control objective in [47], the objective of this article is formulated as follows:(i)
A suitable consensus controller is devised that can make sure agents’ output trajectories synchronise toyr of a leader.(ii)
All signals of plants (10) are globally bounded and the consensus errors converge to a prior accuracy ϵ>0 range.


## 3.2. Assumptions and Lemmas

To realize this goal, the following information is necessary.Assumption 1 ([45]).
The undirected graphGg is considered to be fixed and connected. The trajectory of the leader yr has continuous derivatives up to the nth order and all of them are bounded.Assumption 2 ([32]).
The unknown disturbancedi,mt is bounded with i=1,…,N and m=1,…,n, and there exists a positive constant ϱi,m satisfying di,mt≤ϱi,m.Assumption 3 ([52]).
For the functionfi,mξ¯i,mt, assume that there exists a function Ψi,mξ¯i,mt which is smooth and known such that fi,mξ¯i,mt≤ρi,mΨi,mξ¯i,mt, where ρi,m is the unknown positive constant, i=1,…,N, and m=1,…,n.Remark 2.
As stated in [52, 53], Assumption 3 plays an important role in global consistency analysis. In what follows, because of the upper limit of the function fi,mξ¯i,mt, the robust controller part αi,m2 can be created. It is responsible for coarse tuning throughout the control process.Lemma 3 (Barbalat’s lemma [54]).
The functionKt is differentiable and satisfies limt⟶∞K˙t=0 if its first derivative is uniformly continuous; meanwhile, limt⟶∞ht is finite.Lemma 4 (Young’s inequality [55]).
For∀p,q∈R2, there exists the following inequality:(11)pq≤xssps+qttxt,where x>0, s>1, t>1, and s−1t−1=1.Lemma 5 ([50]).
If there exists a connected and undirected graphGg, and W∈Rp×p is its adjacency matrix, then the symmetric positive definite matrix is expressed as ℒ+diagϕ1,…,ϕp in which there is at least one ϕk>0k=1,…,p.


## 4. Consistency Analysis

In what follows, we will give a detailed procedure for the desired controller and adaptive tuning laws. Firstly,n error variables are necessary and defined as follows:(12)zi,1=∑j∈Eisijyi−yj+biyi−yr,(13)zi,m=ξi,m−αi,m−1,where i=1,2,…,N, m=2,…,n, αi,m−1 indicates the virtual variable, and bi>0 is the pinning gain.Step 1.
Combining the system model (10) and formula (11), we can get the following information:(14)z˙i,1=di+biξi,2+fi,1ξi,1+di,1t−∑j∈Eisijξj,2+fj,1ξj,1+dj,1t−biy˙r.
Choose the Lyapunov function as(15)Vi,1=n!Ni,n+1ϵzi,1.
Also, the derivative ofVi,1 along (14) can be obtained(16)dVi,1dt=dNi,n+1ϵzi,1dzi,1n!z˙i,1=Ni,nϵzi,1ℋi,nϵzi,1−n!di+biξi,2+fi,1ξi,1+di,1t−∑j∈Eisijξj,2+fj,1ξj,1+dj,1t−biy˙r.
Based on Lemma2 and Assumption 2, we have(17)di+biNi,nϵzi,1ℋi,nϵzi,1n!di,1t≤di+biNi,nϵzi,1ℋi,nϵzi,12n!ϱi,1,(18)−Ni,nϵzi,1ℋi,nϵzi,1n!∑j∈Eisijdj,1t≤Ni,nϵzi,1ℋi,nϵzi,12n!∑j∈Eisijϱj,1.
Combining with (17) and (18), one has(19)dVi,1dt≤Ni,nϵzi,1ℋi,nϵzi,1n!di+biξi,2+fi,1ξi,1+ℋi,nεzi,1ϱi,1−∑j∈Eisijξj,2+fj,1ξj,1−ℋi,nεzi,1ϱj,1−biy˙r≤Ni,nϵzi,1ℋi,nϵzi,1n!di+bizi,2+αi,1+gi,1Xi,1−biy˙r,where the function gi,1Xi,1=di+bifi,1ξi,1+ℋi,nϵzi,1ϱi,1−∑j∈Eisijξj,2+fj,1ξj,1−ℋi,nϵzi,1ϱj,1 is unknown with Xi,1=ξi,1,ξ¯j,2T∈ΩX1⊂R3 and ΩX1 is a compact set. Employing Lemma 1, there are a given scalar ηi,1>0 and a NN estimator Wi,1TSi,1Xi,1 conforming to(20)gi,1Xi,1=Wi,1TSi,1Xi,1+σi,1Xi,1,σi,1Xi,1≤ηi,1,where σi,1Xi,1 is an estimation error. In addition, based on Assumption 3, an unknown constant ζi,1>0 and a known smooth function φi,1Xi,1>0 can be chosen such that(21)gi,1Xi,1≤ζi,1φi,1Xi,1.
Motivated by the above information, the following inequality can be obtained:(22)dVi,1dt≤Ni,nϵzi,1ℋi,nϵzi,1n!di+bizi,2+αi,1+CXi,1Wi,1TSi,1Xi,1+σi,1Xi,1+1−CXi,1gi,1Xi,1−biy˙r.
Design the virtual controller as(23)αi,1=−1di+biki,1Ni,nϵzi,1ℋi,nϵzi,1n!+CXi,1αi,11+1−CXi,1αi,12+biy˙r−ϵ+1ℋi,nϵzi,1−14Ni,nϵzi,1ℋi,nϵzi,1n!,where(24)αi,11=−W^i,1TSi,1Xi,1−η^i,1ℋi,nϵzi,1,αi,12=−ζ^i,1φi,1Xi,1ℋi,nϵzi,1.
ki,1>0 is an adjustable parameter.
For further study, we employ the Lyapunov function such as(25)V¯i,1=Vi,1+12W˜i,1Tϒi,1−1W˜i,1+12βi,1η˜i,12+12γi,1ζ˜i,12,where ϒi,1>0, βi,1>0, and γi,1>0 are three adaptive parameters.Remark 3.
It is worthy to remind that the virtual controller (23) is composed of two parts. One is the traditional adaptive controller αi,11 governing the estimation active area. The other is the robust controller αi,12 which is in charge of the outer part of approximation area. It is nontrivial to add the robust part αi,12 in (23). Let us see, when C⋅=1, the controller (23) evolves into a typical adaptive control strategy which ensures that all signals for dynamic system are semiglobally bounded. However, when neural inputs exceed the estimation domain, the robust controller αi,12 works. Then, the actual controller of the structure is designed later that can guarantee all signals are GUUB.
Taking the derivative of (25), one has(26)dV¯i,1dt≤Ni,nϵzi,1ℋi,nϵzi,1n!di+bizi,2+αi,1+CXi,1Wi,1TSi,1Xi,1+σi,1Xi,1+1−CXi,1gi,1Xi,1−biy˙r−W˜i,1Tϒi,1−1W^⋅i,1−1βi,1η^i,1η^⋅i,1−1γi,1ζ^i,1ζ^⋅i,1≤Ni,nϵzi,1ℋi,nϵzi,1n!di+bizi,2−ki,1Ni,nϵzi,1ℋi,nϵzi,1n!+W^i,1TCXi,1Si,1Xi,1+η^i,1CXi,1ℋi,nϵzi,1+ζ^i,11−CXi,1φi,1Xi,1ℋi,nϵzi,1−ε+1ℋi,nϵzi,1−14di+biNi,nϵzi,1ℋi,nϵzi,1n!−W˜i,1Tϒi,1−1W^⋅i,1−1βi,1η^i,1η^⋅i,1−1γi,1ζ^i,1ζ^⋅i,1.
Design adaptive laws are as(27)W^⋅i,1=Υi,1Ni,nϵzi,1ℋi,nϵzi,1n!CXi,1Si,1Xi,1,η^⋅i,1=βi,1Ni,nϵzi,1n!CXi,1,ζ^⋅i,1=γi,1Ni,nϵzi,1n!1−CXi,1φi,1Xi,1.
Substituting (27) into (26) gives(28)dV¯i,1dt≤−ki,1Ni,nϵzi,1n!2−14di+biNi,nϵzi,1n!2+di+biNi,nϵzi,1n!zi,2−ϵ+1.Step m.
(m = 2, 3, …, n−1). According to the definition in (13), one has(29)z˙i,m=ξ˙i,m−α˙i,m−1=ξi,m+1+fi,mξ¯i,m+di,mt−∑q=1m−1∂αi,m−1∂ξi,qξi,q+1+fi,qξ¯i,q+di,qt−∑q=1m−1∑j∈Ei∂αi,m−1∂ξj,qξj,q+1+fj,qξ¯j,q+dj,qt−λi,m−1,where λi,m−1=∑q=1m−1∂αi,m−1/∂W^i,qW^⋅i,q+∑q=1m−1∂αi,m−1/∂η^i,qη^⋅i,q+∑q=1m−1∂αi,m−1/∂ζ^i,qζ^⋅i,q+∑q=0m−1∂αi,m−1/∂yrqyrq+1.
Select a nonnegative function as follows:(30)Vi,m=V¯i,m−1+n−m+1!Ni,n−m+2ϵzi,m.
Then, we gain(31)dVi,mdt=dV¯i,m−1dt+Ni,n−m+1ϵzi,mℋi,n−m+1ϵzi,mn−m+1!zi,m+1+αi,m+fi,mξ¯i,m+di,mt−∑q=1m−1∂αi,m−1∂ξi,qξi,q+1+fi,qξ¯i,q+di,qt−∑q=1m−1∑j∈Ei∂αi,m−1∂ξj,qξj,q+1+fj,qξ¯j,q+dj,qt−λi,m−1.
According to Lemmas2 and 4 and Assumption 2, the items in (31) are developed into(32)Ni,n−m+1ϵzi,mℋi,n−m+1ϵzi,mn−m+1!di,mt≤Ni,n−m+1ϵzi,mℋi,n−m+1ϵzi,m2n−m+1!ϱi,m,−Ni,n−m+1ϵzi,mℋi,n−m+1ϵzi,mn−m+1!∑q=1m−1∂αi,m−1∂ξi,qdi,qt≤Ni,n−m+1ϵzi,mℋi,n−m+1ϵzi,m2n−m+1!∑q=1m−1∂αi,m−1∂ξi,qϱi,q≤Ni,n−m+1ϵzi,mℋi,n−m+1ϵzi,m2n−m+1!∑q=1m−1ϑi,m22∂αi,m−1∂ξi,q2+Ni,n−m+1ϵzi,2ℋi,n−m+1ϵzi,22n−m+1!∑q=1m−112ϑi,m2ϱi,q2−Ni,n−m+1ϵzi,mℋi,n−m+1ϵzi,mn−m+1!∑q=1m−1∑j∈Ei∂αi,1∂ξj,1dj,1t≤Ni,n−1ϵzi,2ℋi,n−1ϵzi,22n−m+1!∑q=1m−1∑j∈Ei∂αi,m−1∂ξj,qϱj,q≤Ni,n−1ϵzi,2ℋi,n−1ϵzi,22n−m+1!∑q=1m−1∑j∈Eiσi,m22∂αi,m−1∂ξj,q2+Ni,n−m+1ϵzi,m∑q=1m−1∑j∈Ei12σi,m2ϱj,q2.
Let(33)gi,mXi,m=fi,mξ¯i,m−∑q=1m−1∂αi,m−1∂ξi,qξi,q+1+fi,qξi,q−∑q=1m−1∑j∈Ei∂αi,m−1∂ξj,qξj,q+1+fj,qξj,q+ℋi,n−m+1ϵzi,mϱi,m+ℋi,n−m+1ϵzi,m∑q=1m−1ϑi,m22∂αi,m−1∂ξi,q2+ℋi,n−m+1ϵzi,m∑q=1m−112ϑi,m2ϱi,q2+ℋi,n−m+1ϵzi,m∑q=1m−1∑j∈Eiϑi,m22∂αi,m−1∂ξj,q2+ℋi,n−m+1ϵzi,m∑q=1m−1∑j∈Ei12ϑi,m2ϱj,q2,with Xi,m=ξ¯i,m,ξ¯j,mT∈ΩXm⊂R2m and a compact set ΩXm.
Employing the previous steps, we can easily get the following knowledge:(34)dVi,mdt=dV¯i,m−1dt+Ni,n−m+1εzi,mℋi,n−m+1ϵzi,mn−m+1!zi,m+1+αi,m+gi,mXi,m−λi,m−1,where gi,mXi,m is an unknown function. We can use an estimator and estimation error to replace it such as gi,mXi,m=Wi,mTSi,mXi,m+δi,mXi,m,δi,mXi,m≤εi,m in which εi,m>0 is a constant. As previously mentioned, there are a known smooth function φi,mZi,m and an uncertain constant ζi,m>0 which can be found satisfying(35)gi,mXi,m≤ζi,mφi,mXi,m.
Substituting (35) into (34), we can conclude(36)dVi,mdt≤dV¯i,m−1dt+Ni,n−m+1ϵzi,mℋi,n−m+1ϵzi,m×n−m+1!zi,m+1+αi,m+CXi,mWi,mTSi,mXi,m+δi,mXi,m+1−CXi,mgi,mXi,m−λi,m−1.
Create the virtual controller as, form=2,(37)αi,2=−ki,2+14Ni,n−1ϵzi,2ℋi,n−1ϵzi,2n−1!−di+biNi,n−1ϵzi,2ℋi,n−1ϵzi,2n−1!+CXi,2αi,21+1−CXi,2αi,22−ϵ+1ℋi,n−1ϵzi,2+λi,1,and for m=3,…,n−1,(38)αi,m=−ki,m+54Ni,n−m+1ϵzi,mℋi,n−m+1ϵzi,mn−m+1!+CXi,mαi,m1+1−CXi,mαi,m2−ϵ+1ℋi,n−m+1ϵzi,m+λi,m−1,where(39)αi,m1=−W^i,mTSi,mXi,m−η^i,mℋi,n−m+1ϵzi,m,αi,m2=−ζ^i,mφi,mXi,mℋi,n−m+1ϵzi,m,m=2,3,…,n−1.
ki,2 and ki,m are positive constants which need to be designed.Remark 4.
Becausezi,1 and zi,m have different definition forms, a special formula bi+di is developed. In order to continue employing the backstepping technique, the virtual controllers αi,2 and αi,m must have different forms.
Further choose the Lyapunov function as(40)V¯i,m=Vi,m+12W˜i,mTϒi,m−1W˜i,m+12βi,mη˜2+12γi,mζ˜i,m2,where constants ϒi,m>0, βi,m>0, and γi,m>0 need to be devised.
Taking the derivative ofV¯i,m, we have(41)dV¯i,mdt≤dVi,mdt+Ni,n−m+1ϵzi,mℋi,n−m+1ϵzi,mn−m+1!zi,m+1−ℋi,n−m+1ϵzi,mn−m+1!W^i,mTCXi,m×Si,mXi,m−η^i,mCXi,mℋi,n−m+1ϵzi,m−ζ^i,m1−CXi,mφi,mXi,mℋi,n−m+1ϵzi,m−ϵ+1ℋi,n−m+1ϵzi,m+Wi,mTCXi,mSi,mXi,m+ηi,mCXi,mℋi,n−m+1ϵzi,m+ζi,m1−CXi,mφi,mXi,mℋi,n−m+1ϵzi,m−W˜i,mTϒi,m−1W^⋅i,m−1βi,mη^i,mη^⋅i,m−1γi,mζ^i,mζ^⋅i,m.
Adaptive laws are chosen as(42)W^⋅i,m=Γi,mNi,n−m+1ϵzi,mℋi,n−m+1ϵzi,mn−m+1!CXi,mSi,mXi,m,ε^⋅i,m=βi,mNi,n−m+1ϵzi,mn−m+1!CXi,m,ξ^⋅i,m=γi,mNi,n−m+1ϵzi,mn−m+1!1−CXi,mφi,mXi,m.
Taking advantage of the above laws, we have(43)dV¯i,mdt≤−∑q=1mki,qNi,n−q+1ϵzi,qn−q+1!2+ℏi,m−14Ni,n−m+1ϵzi,mn−m+1!2+Ni,n−m+1ϵzi,mn−m+1!zi,m+1−ϵ+1,where for m=2, ℏi,2=−1/4di+biNi,nϵzi,1n!2+di+biNi,nϵzi,1n!zi,2−ϵ+1−di+biNi,n−1ϵzi,2n−1!2, and for 3≤m≤n−1, ℏi,m=−1/4Ni,n−m+2ϵzi,m−1n−m+2!2+Ni,n−m+2ϵzi,m−1n−m+2!zi,m−ϵ+1−Ni,n−m+1ϵzi,mn−m+1!2.
Obviously, whenzi,2≤ϵ+1 and zi,m≤ϵ+1, ℏi,2≤0 and ℏi,m≤0 hold. When zi,2,zi,m>ϵ+1, according to Young’s inequality, we know, for m=2,(44)di+biNi,nϵzi,1n!zi,2−ϵ+1≤di+bi14Ni,nϵzi,1n!2+di+bizi,2−ϵ+12.
Therefore, one has(45)ℏi,2≤di+bizi,2−ϵ+12−di+biNi,n−1ϵzi,2n−1!2=di+bizi,2−ϵ+12−zi,2−εn−1n−1!n−1!2≤di+bizi,2−ϵ+12−zi,2−ϵ2n−1=di+bizi,2−ϵ21−zi,2−ϵ2n−1≤0,and for m=3,…,n−1,(46)Ni,n−m+2ϵzi,m−1n−m+2!zi,m−ϵ+1≤14Ni,n−m+2ϵzi,m−1n−m+2!2+zi,m−ϵ+12.
Then, we know(47)ℏi,m≤zi,m−ϵ+12−Ni,n−m+1ϵzi,mn−m+1!2=zi,m−ϵ+12−zi,m−ϵn−m+1n−m+1!n−m+1!2≤zi,m−ϵ+12−zi,m−ϵ2n−m+1=zi,m−ϵ21−zi,m−ϵ2n−m+1≤0.Remark 5.
Because of the different forms betweenzi,1 and zi,m, ℏi,2 and ℏi,m have different expressions. The inequalities (45) and (47) are necessary.
Based on the results of (45) and (47), we get the following conclusion:(48)dV¯i,mdt≤−∑q=1mki,qNi,n−q+1ϵzi,qn−q+1!2−14Ni,n−m+1ϵzi,mn−m+1!2+Ni,n−m+1ϵzi,mn−m+1!zi,m+1−ϵ+1.Step n.
In this step, an actual consensus controllerui is designed for agent i. Firstly, we take derivative of zi,n such that(49)z˙i,n=ξ˙i,n−α˙i,n−1=ui+fi,nξ¯i,n+di,nt−∑q=1n−1∂αi,n−1∂ξi,qξi,q+1+fi,qξ¯i,q+di,qt−∑q=1n−1∑j∈Ei∂αi,n−1∂ξj,qξj,q+1+fj,qξ¯j,q+dj,qt−λi,n−1,where λi,n−1=∑q=1n−1∂αi,n−1/∂W^i,qW^⋅i,q+∑q=1n−1∂αi,n−1/∂η^i,qη^⋅i,q+∑q=1n−1∂αi,n−1/∂ζ^i,qζ^⋅i,q+∑q=0n−1∂αi,n−1/∂yrqyrq+1.
Choose the Lyapunov function as(50)V¯i,n=Vi,n+12W˜i,nTϒi,n−1W˜i,n+12βi,nη˜2+12γi,nζ˜i,n2,Vi,n=V¯i,n−1+Ni,2ϵzi,n,where three positive adaptive parameters are defined as ϒi,n, βi,n, and γi,n.
Since the following process is similar to Stepm, we have only given some major steps. From the aforementioned approaches, the time derivative of Vi,n is(51)dVi,ndt≤dV¯i,n−1dt+Ni,1ϵzi,nℋi,1ϵzi,nui+gi,nZi,n−λi,n−1,where(52)gi,nXi,n=fi,nξ¯i,n−∑q=1n−1∂αi,n−1∂ξi,qξi,q+1+fi,qξi,q−∑q=1n−1∑j∈Ei∂αi,n−1∂ξj,qξj,q+1+fj,qξj,q+ℋi,1ϵzi,nϱi,n+ℋi,1ϵzi,n∑q=1n−1ϑi,n22∂αi,n−1∂ξi,q2+ℋi,1ϵzi,n∑q=1n−112ϑi,n2ϱi,q2+ℋi,1ϵzi,n∑q=1n−1∑j∈Eiϑi,n22∂αi,n−1∂ξj,q2+ℋi,1ϵzi,n∑q=1n−1∑j∈Ei12ϑi,n2ϱj,q2%,with Zi,n=x¯i,n,x¯j,nT∈ΩXn⊂R2n and a compact set ΩXn, and ϑi,n is a positive constant. Considering Lemma 1 and Assumption 3, we can find an NN approximator Wi,nTSi,nXi,n, an approximation error δi,nxi,n, a known smooth function φXi,n, and an unknown positive constant ζi,n satisfying(53)gi,nXi,n=Wi,nTSi,nXi,n+δi,nXi,n,δi,nXi,n≤εi,n,gi,nXi,n≤ζi,nφi,nXi,n.
Then, we create the practical controller and adaptive tuning laws as(54)ui=−ki,n+1Ni,1ϵzi,nℋi,1ϵzi,n+CXi,nαi,n1+1−CXi,nαi,n2−ϵ+1ℋi,nϵzi,1+λi,n−1,with(55)αi,n1=−W^i,nTSi,nXi,n−η^i,nℋi,1ϵzi,n,αi,n2=−ζ^i,nφi,nZi,nℋi,1ϵzi,n,(56)W^⋅i,n=Γi,nNi,1ϵzi,nℋi,1ϵzi,nCXi,nSi,nXi,n,η^⋅i,n=βi,nNi,1ϵzi,nCXi,n,ζ^⋅i,n=γi,nNi,1ϵzi,n1−CXi,nφi,nXi,n,where ki,n is a positive adjustable parameter.
From the above presentation, we can conclude the following result(57)dV¯i,ndt≤−∑q=1nki,qNi,n−q+1ϵzi,qn−q+1!2−14Ni,2ϵzi,n−12!2−Ni,1εzi,n2+Ni,2ϵzi,n−12!zi,n−ϵ+1≤−∑q=1nki,qNi,n−q+1ϵzi,qn−q+1!2.Remark 6.
From the aforementioned procedure, it can be concluded that the method in this paper is very different from those existing control schemes for nonlinear uncertain MASs which only guarantee all signals for plants which are semiglobally uniformly ultimately bounded (SGUUB). However, the functionC⋅ is added in the actual consensus controller (54) that can broaden the scope of consistency for system signals. For the controller (54), if the inputs for RBF NNs go beyond the compact set ΩXn, the function C⋅ is to cut off the controller αi,n1, then the rest controller αi,n2 starts to run. Consequently, the nonlinear uncertain MASs are GUUB which is the main advantage of this paper.
The conclusion of this paper is generalized as follows.Theorem 1.
With Assumptions1–3, consider the nonlinear uncertain MASs (10) under the virtual controllers (23), (37), and (38), the real controller (54), and adaptive laws (27), (42), and (56), the following descriptions hold.(i)
All signals for plant (10) are globally bounded;(ii)
The tracking errors remain a predefined accuracyϵ>0, that is, zi,1≤ϵ as t⟶∞ with i=1,2,…,N.Proof.
(i) Observing the inequality (57), we know that the function V¯i,n is nonincreasing. Consequently, it easily concluded that zi,m, W˜i,m, η˜i,m, and ζ˜i,m are bounded with m=1,2,…,n and i=1,2,…,N. Due to the boundedness of W˜i,m, η˜i,m, and ζ˜i,m, we conclude that W^i,m=Wi,m−W˜i,m, η^i,m=ηi,m−η˜i,m, and ζ^i,m=ζi,m−ζ˜i,m are bounded. According to (12) and Lemma 5, yi is bounded leading to the boundedness of ξi,1. Then, it can be seen that αi,1 is bounded from (23). It is concluded zi,2 is bounded from (13). Repeating the above process, we can conclude all signals are bounded.
(ii) From (57), we have(58)Θt=∫0tΠϖdϖ≤V¯i,n0ci,where Π=∑q=1nNi,n−q+1ϵzi,qn−q+1!2 and ci=minki,1,ki,2,…,ki,n. Therefore, Θt is bounded. According to the derivative of zi,m, we know that it is uniformly bounded. Clearly, zi,m is uniformly continuous. Based on the above description, Πϖ is also uniformly continuous. Employing Lemma 3, we have limt⟶∞=0. Using Lemma 2(ii), zi,m≤ϵ holds when t⟶∞ with m=1,2,…,n and i=1,2,…,N. The proof is finished.Remark 7.
Unlike the traditional proof of error convergence, the convergence proof of the predefined accuracy depends on the Barbalat’s lemma and the switching functions’ property. Meanwhile, according to Definition1, all signals of MASs (10) are GUUB.


## 5. Simulation

In Section5, we focus on a practical pendulum system including five agents to explain the validity of the given control strategy. The communication topology is displayed in Figure 1 and the signal agent’s dynamic model is developed in [56].(59)ξ˙i,1=ξi,2−0.2ξi,1+0.01sint,ξ˙i,2=0.8sinξi,1+ui+0.01cost,yi=xi,1,i=1,…,4,where ξi,1 and ξi,2 are the angular position and the rate of agent i, respectively, ui represents the system input, and yi denotes the system output. Meanwhile, we choose the trajectory yt=sin0.5t+0.5sint for the leader.Figure 1
Communication topology.Based on the aforementioned design procedure, the virtual controller, real controller, and adaptive design laws are obtained as follows:(60)αi,1=−1di+bi2ki,1Ni,2ϵzi,1ℋi,2ϵzi,1+CXi,1αi,11+1−CXi,1αi,12+biy˙t−ϵ+1ℋi,2ϵzi,1−12Ni,2ϵzi,1ℋi,2ϵzi,1,ui=−ki,2+1Ni,1ϵzi,2ℋi,1ϵzi,2+CXi,2αi,21+1−CXi,2αi,22−ϵ+1ℋi,2ϵzi,1+λi,1,with(61)αi,m1=−W^i,mTSi,mXi,m−η^i,mℋi,3−mϵzi,mαi,m2=−ζ^i,mφi,mXi,mℋi,3−mϵzi,m,W^⋅i,m=Γi,2Ni,3−mϵzi,mℋi,3−mϵzi,m×3−m!CZi,mSi,mXi,m,η^⋅i,m=βi,mNi,3−mϵzi,m×3−m!CXi,m,ζ^⋅i,m=γi,mNi,3−mϵzi,m3−m!×1−CXi,mφi,2Zi,m,m=1,2,where λi,1 is defined in (29), Xi,1=ξi,1,ξj,1,ξj,2T, and Xi,2=ξi,1,ξi,2,ξj,1,ξj,2T. Clearly, the model (59) is in the form of the plant (10) and Assumptions 1–3 hold, and the functions in Assumption 3 are provided as follows:(62)φi,1=1+ξi,12+ξj,12+ξj,22,φi,1=1+ξi,12+ξi,22+ξj,12+ξj,22.Some NNs are developed in the simulation. NNsW^i,1TSi,1Xi,1 contain 675 nodes with centers spaced evenly in 1,3×−3,7×−3,7×−60,0×−100,100, and the widths are 15; NNs W^i,2TSi,2Xi,2 contain 729 nodes with centers spaced evenly in intervals −1,1×−1,1×−1,1×−1,1×−1,1×−1,1, and the widths are 2. The rest of parameters in this simulation are designed as k1,1=k1,2=1500, kc,1=kc,2=800, Γi,1=Γi,2=diag3,3, βi,1=2, βi,2=3, γi,1=γi,2=0.0002, ei,11=1.205, and ei,12=1.31, ei,21=4, ei,22=4.5, where i=1,…,4, c=2,…,4, and ei,∗∗ is the design parameter in function CXi,r with r=1,2. The accuracy is set to ϵ=0.05 and ϵ=0.07. The initial conditions are chosen as ξi,10,ξi,20T=1,1T; the remaining initial conditions are set as 0.The simulation results are presented in Figures2–12. The consensus trajectories are displayed in Figure 2, and it shows that the followers can follow with the leader. The consensus errors for four agents are shown in Figures 3 and 4 which indicates that none of the errors exceed 0.05 and 0.07, respectively. The boundedness of ui, W^i,j, η^i,j, and ζ^i,j is developed in Figures 4–10, where i=1,…,4 and j=1,2. Figures 11 and 12 denote trajectories of switching signals CXi,2 and CXi,2. All results displayed in Figures 2–12 are derived from Theorem 1.Figure 2
Trajectories of the leader and followers.Figure 3
(a) Tracking errorszi,1 for 1≤i≤4 and ε=0.05. (b) Tracking errors zi,1 for 1≤i≤4 and ε=0.07.
(a)(b)Figure 4
Inputsui for agent i, 1≤i≤4.Figure 5
Trajectories ofW^i,1, 1≤i≤4.Figure 6
Trajectories ofW^i,2, 1≤i≤4.Figure 7
Trajectories ofη^i,1, 1≤i≤4.Figure 8
Trajectories ofη^i,2, 1≤i≤4.Figure 9
Trajectories ofζ^i,1, 1≤i≤4.Figure 10
Trajectories ofζ^i,2, 1≤i≤4.Figure 11
Trajectories of the switching signalsCXi,1, 1≤i≤4.Figure 12
Trajectories of the switching signalsCXi,2, 1≤i≤4.Remark 8.
Compared to the existing control schemes for the pendulum system, the systems (59) are applied to verify the correctness of the proposed scheme in this paper, namely, GUUB and the precision convergence. Also, the simulation results support Theorem 1. Moreover, in contract to the control scheme in [48, 49], we consider the global and precision convergence strategy depend on three switching functions, and the common backstepping method is employed in this note.


## 6. Conclusions

In this note, a novel consensus control scheme is proposed for nonlinear MASs. The main advantages of this strategy are that the global consensus of plants under consideration is guaranteed and the tracking errors converge to a predefined accuracy on the basis of the backstepping technique. Furthermore, three keyCn-class functions are brought in to ensure the precision and global consensus problem for MASs. Finally, the practical example is introduced to demonstrate the effectiveness of the proposed scheme. In the future, we will study the precision consensus for MASs under fixed and switching topologies and switched systems consensus for MASs [57, 58].



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