# Adaptive Fuzzy Cooperative Control for Nonlinear Multiagent Systems with Unknown Control Coefficient and Actuator Fault

Authors: Xin Deng; Xiaoping Liu; Yang Cui; Cungen Liu
Journal: Complexity
 (2021)

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## Abstract

In this paper, an adaptive fuzzy containment condtrol is considered for nonlinear multiagent systems, in which it contains the unknown control coefficient and actuator fault. The uncertain nonlinear function has been approximated by fuzzy logic system (FLS). The unknown control coefficient and the remaining control rate of actuator fault can be solved by introducing a Nussbaum function. In order to avoid the repeated differentiations of the virtual controllers, first-order filters are added to the traditional backstepping control method. By designing the maximum norm of ideal adaptive parameters, only one adaptive parameter needs to be adjusted online for each agent itself. An adaptive fuzzy containment controller is constructed through the backstepping control technique and compensating signals. It is demonstrated that all the signals in nonlinear multiagent systems are bounded by designing adaptive fuzzy containment controller, and all followers can converge to the convex hull built by the leaders. The simulation studies can further confirm the effectiveness of the proposed control method in this paper.


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## Body

## 1. Introduction

With the development of science and technology, more and more complex systems can be described as multiagent systems (MASs), for instance, unmanned aerial vehicle formation, sensor networks, and disaster emergency response (see [1–3]). Therefore, the cooperative control problem of MASs has been extensively concerned by scholars and gradually become one of the hot issues in the control field. According to different control objectives, the cooperative control of MASs is classified as flocking control [4], consensus control [5, 6], formation control [7], synchronous control [8], containment control [9, 10], and so on. Because of its wide applications, containment control is a fundamental and important research subject of MASs control, such as [11, 12]. For containment control, its purpose is to design a controller so that all follower agents can converge to the convex hull built by the leaders [9–18]. For example, the path-guided containment maneuvering approach was proposed for networked two-wheeled mobile robots with multiple virtual leaders moving along multiple parameterized paths in [11]. An output feedback distributed containment control method was investigated for marine vessels guided by multiple parameterized paths with unmeasured velocity in [12].In literature [13–15], all containment control methods were investigated for linear MASs. But, for the containment problem of nonlinear MASs, the control methods in [13–15] were not feasible. In fact, nonlinear systems can better describe a practical engineering problem than linear systems, such as manipulator systems [19], inverted pendulum systems [20], and rigid robotic systems [21]. All systems models in [16–18] and practical application systems models in [19–26] are first- or second-order systems. For example, under the common assumption that each agent can only obtain the relative information of its neighbours intermittently, the containment control method was proposed for a class of second-order MASs with inherent nonlinear dynamics in [17]. However, many actual industrial systems models are often high-order systems. For instance, a jerk system and single-link flexible manipulators [23] were modeled as third-order and fourth-order nonlinear systems, respectively. So, it is of theoretical and practical value to study the containment control of high-order systems. Some containment control approaches were studied for high-order nonlinear MASs in [24–26]. Compared with low-order MASs, the containment control method for high-order nonlinear MASs is more difficult due to the complexity of high-order systems. In [25], for nonlinear nonaffine pure-feedback MASs, a distributed containment control was proposed based on directed graph topology. For strict-feedback nonlinear systems, the backstepping control technique is an effective approach to solve the cooperative control problem, such as [21, 27–29]. In [21], the finite-time containment control problem was considered for nonlinear MASs with unmeasurable states and input delay. Adaptive neural networks (NNs) backstepping control was studied for uncertain strict-feedback nonlinear systems with input delay in [29].It is well known that adaptive intelligent control (FLSs and neural networks (NNs)) is an effective way to solve the control problem of nonlinear systems with unknown nonlinear functions, due to the fact that FLSs and NNs can approximate an unknown nonlinear function with arbitrary precision in a compact set, which are called universal approximators [21, 27–34]. In the case of full-state constraints, an adaptive fuzzy controller was designed for nonlinear strict-feedback systems in [28]. Some containment control approaches based on FLSs or NNs as an approximator were investigated for nonlinear MASs in [35–38]. In [36, 37], the containment control approaches were investigated for uncertain nonlinear systems with actuator faults. Under the condition of full-state constraints, a fuzzy adaptive containment control method was investigated for nonlinear systems in [38]. But, in [35–38], the number of adaptive laws was too many, which results in the need to adjust too many parameters online, and the control design is too complicated. When the system order is high, the complexity of the control design increases exponentially. In addition, actuator faults are inevitable in industrial systems, especially for MASs. But, so far, there have been no results on adaptive fuzzy fault-tolerant containment control method for uncertain nonlinear MASs with unknown control coefficient and external disturbance.This article focuses on the adaptive fuzzy containment control problem for uncertain nonlinear MASs with unknown control coefficient and actuator fault. By using backstepping control technique, compensating signals, first-order filters, and Nussbaum function, some issues in the above control methods can be solved. The main contributions are summarized as follows:(1)
It is the first time to construct adaptive fuzzy containment controllers for uncertain nonlinear MASs with unknown control coefficient and actuator faults. Compared with the existing nonlinear MASs in [27, 39], assuming that there were no actuator faults and the control coefficient was 1, the nonlinear MASs considered in this paper have more practical significance. The unknown control coefficient and the remaining control rate of actuator faults in nonlinear MASs will be resolved by introducing a Nussbaum function.(2)
For the adaptive fuzzy containment control method in the literature [21], the number of designed adaptive laws was equal to or more than the order of the systems. But, in this paper, instead of estimating the optimal parameter vectors themselves but by estimating the maximum value of the norm of the optimal parameter vectors, the number of adaptive laws is greatly reduced.


## 2. Problem Description and Preliminaries

### 2.1. Problem Description

Consider nonlinear MASs withN followers and M leaders. The dynamic equations for the τ-th follower are described as follows:(1)x˙τ,i=Fτ,ix¯τ,i+xτ,i+1+ϕτ,it,1≤i≤n−1,x˙τ,n=Fτ,nx¯τ,n+gτ,nx¯τ,nuτt+ϕτ,nt,yτ=xτ,1,where τ=1,2,…,N. x¯τ,i=xτ,1,xτ,2,…,xτ,iT∈Ri, and x¯τ,n=xτ,1,xτ,2,…,xτ,nT∈Rn are the state vectors. yτ∈R is the output of the τ-th follower. Fτ,ix¯τ,i is an unknown smooth function. ϕτ,it is a bounded unknown external disturbance. gτ,nx¯τ,n represents an unknown smooth nonlinear function. uτt denotes the input of the considered system. The leader’s signal yrj is a sufficiently smooth function; yrj and y˙rj are all bounded, where j=N+1,…,N+M.Remark 1.
In the literature [7, 24], it is assumed that the MASs will not show actuator faults at any time. However, due to machine aging or working environment, actuator faults are inevitable in actual industrial production. Therefore, it is more practical to design the cooperative control of nonlinear MASs with actuator faults in this paper.
To model the information exchange between agents, the graph theory is introduced. The exchange of information between the followers and the leaders is portrayed byG=v,ϱ,A¯. v=n1,…nN,nN+1,…,nN+M is the set of agents, where the followers are labeled as τ=1,…,N, while the leaders are labeled as τ=N+1,…,N+M. Thus, in the set v, the first N agents are the followers, and the last M agents are the leaders. ϱ=nτ,nj∈v×v represents the edge set, and the adjacency matrix is defined as A¯=aτj∈RN+M×N+M. nτ,nj∈ϱ means follower j is able to gain the information of its neighbor agent τ. If nτ,nj∉ϱ, aτj=0; if not, aτj=1. L=Lτj∈RN+M×N+M=D−A¯ stands for Laplacian matrix, and Lτj=−aτj if τ≠j; D=diagd1,…,dN,dN+1,…,dN+M denotes degree matrix of agent τ, and dτ=∑j=1,j≠τN+Maτj. Let vf=n1,…,nN be the follower agents’ set, and let vL=nN+1,…,nN+M denote the leader agents’ set. Every follower agent owns at least a neighbor. On the contrary, leaders do not own neighbors. For this condition, the Laplacian L of the graph is partitioned as(2)L=L1L20M×N0M×M.
It is supposed that there exists at least a leader who possesses a directed path to every follower agent. Then, each eigenvalue of matrixL1 has a positive real part, the sum of each line of −L1−1L2 is equal to 1, and all the entries of −L1−1L2 are nonnegative. Letting yl=yN+1,…,yN+MT, the convex hull built by the leaders can be defined as ydt=−L1−1L2yl. Then, the containment error can be denoted as y−ydt, where y=y1,…,yNT.
However, actuators may become faulty in practical engineering. Bias faults and gain faults are two kinds of actuator faults that commonly occur in the practice, expressed as [40](3)uτft=ρτx¯τ,nuτt+ζτt,where the remaining control rate ρτ of gain faults satisfies 0<ρτx¯τ,n≤1 and ζτt represents a bounded signal of bias faults. Here, ρτx¯τ,n at the failure time instant tf is assumed to be unknown.
Considering (3), the follower dynamic equation (1) can be rewritten as(4)x˙τ,i=Fτ,ix¯τ,i+xτ,i+1+ϕτ,it,1≤i≤n−1,x˙τ,n=Fτ,nx¯τ,n+Γτuτt+Φτ,n,yτ=xτ,1,where Γτ=gτ,nx¯τ,nρτx¯τ,n and Φτ,n=gτ,nx¯τ,nζτt+ϕτ,nt.Assumption 1 (see [40, 41]).
There exist positive constantsg¯τ and g¯τ, and τ=1,…,N, such that g¯τ≤gτ,nx¯τ,n≤g¯τ. Without losing generality, it is assumed that gτ,nx¯τ,n is positive in the following adaptive fault-tolerant containment controller design.Assumption 2.
All the signalsζτt and ϕτ,it are bounded for the actual industrial control, so it is assumed that ζτt≤ζτ∗ and ϕτ,it≤ϕτ,i∗ with ζτ∗ and ϕτ,i∗ being positive constants. Because gτ,nx¯τ,n is bounded, we shall assume that Φτ,n≤Φτ,n∗ with Φτ,n∗ being a positive constant.
Nussbaum Type Gain: a continuous functionNs is a Nussbaum-type function if(5)lims⟶+∞sup1s∫0sNςdς=+∞,lims⟶+∞inf1s∫0sNςdς=−∞.
For example,ς2cosς, ς2sinς, and eς2cosπ/2ς are Nussbaum-type functions. For clarity, the even Nussbaum-type function Nς=eς2cosπ/2ς is used throughout this paper.Lemma 1 (see [42]).
For any positive semidefinite smooth functionVt and any smooth function ςt with t∈0,tf, if, for an even smooth Nussbaum-type function Nς, the inequality,(6)0≤Vt≤c0+e−c1t∫0tgxτNςς˙ec1τdτ+e−c1t∫0tς˙ec1τdτ,∀t∈0,tf,is true with positive constants c1 and c0 and a function gxt taking values in the unknown closed intervals I:=l−,l+ with 0∉I, then Vt, ςt and ∫0tgxτNςς˙dτ are bounded on 0,tf.


### 2.2. Fuzzy Logic Systems

On the basis of [28, 30], the knowledge base for FLS comprises a collection of fuzzy if-then rules as follows:(7)Rl:ifx1isF1landx2isF2land…andxnisFnl,thenyisGl,l=1,2,…,N,where x=x1,x2,…,xnT and y are the input and output of the FLS, respectively. The fuzzy sets Fτl and Gl are associated with the fuzzy membership functions μFτlxτ and μGτlxτ, respectively. N is the rules number.Through the singleton fuzzification, the center average defuzzification, and product inference engine, the FLS can be represented as(8)yx=∑l=1Ny¯l∏τ=1nμFτlxτ∑l=1N∏τ=1nμFτlxτ,where y¯l=maxy∈RμGτly.The fuzzy basis functions are designed as(9)φl=∏τ=1nμFτlxτ∑l=1N∏τ=1nμFτlxτ,and θT=y¯1,…,y¯N=θ1,…,θN and φx=φ1x,…,φNxT. Then, FLS (7) can be further expressed as yx=θTφx.


## 2.1. Problem Description

Consider nonlinear MASs withN followers and M leaders. The dynamic equations for the τ-th follower are described as follows:(1)x˙τ,i=Fτ,ix¯τ,i+xτ,i+1+ϕτ,it,1≤i≤n−1,x˙τ,n=Fτ,nx¯τ,n+gτ,nx¯τ,nuτt+ϕτ,nt,yτ=xτ,1,where τ=1,2,…,N. x¯τ,i=xτ,1,xτ,2,…,xτ,iT∈Ri, and x¯τ,n=xτ,1,xτ,2,…,xτ,nT∈Rn are the state vectors. yτ∈R is the output of the τ-th follower. Fτ,ix¯τ,i is an unknown smooth function. ϕτ,it is a bounded unknown external disturbance. gτ,nx¯τ,n represents an unknown smooth nonlinear function. uτt denotes the input of the considered system. The leader’s signal yrj is a sufficiently smooth function; yrj and y˙rj are all bounded, where j=N+1,…,N+M.Remark 1.
In the literature [7, 24], it is assumed that the MASs will not show actuator faults at any time. However, due to machine aging or working environment, actuator faults are inevitable in actual industrial production. Therefore, it is more practical to design the cooperative control of nonlinear MASs with actuator faults in this paper.
To model the information exchange between agents, the graph theory is introduced. The exchange of information between the followers and the leaders is portrayed byG=v,ϱ,A¯. v=n1,…nN,nN+1,…,nN+M is the set of agents, where the followers are labeled as τ=1,…,N, while the leaders are labeled as τ=N+1,…,N+M. Thus, in the set v, the first N agents are the followers, and the last M agents are the leaders. ϱ=nτ,nj∈v×v represents the edge set, and the adjacency matrix is defined as A¯=aτj∈RN+M×N+M. nτ,nj∈ϱ means follower j is able to gain the information of its neighbor agent τ. If nτ,nj∉ϱ, aτj=0; if not, aτj=1. L=Lτj∈RN+M×N+M=D−A¯ stands for Laplacian matrix, and Lτj=−aτj if τ≠j; D=diagd1,…,dN,dN+1,…,dN+M denotes degree matrix of agent τ, and dτ=∑j=1,j≠τN+Maτj. Let vf=n1,…,nN be the follower agents’ set, and let vL=nN+1,…,nN+M denote the leader agents’ set. Every follower agent owns at least a neighbor. On the contrary, leaders do not own neighbors. For this condition, the Laplacian L of the graph is partitioned as(2)L=L1L20M×N0M×M.
It is supposed that there exists at least a leader who possesses a directed path to every follower agent. Then, each eigenvalue of matrixL1 has a positive real part, the sum of each line of −L1−1L2 is equal to 1, and all the entries of −L1−1L2 are nonnegative. Letting yl=yN+1,…,yN+MT, the convex hull built by the leaders can be defined as ydt=−L1−1L2yl. Then, the containment error can be denoted as y−ydt, where y=y1,…,yNT.
However, actuators may become faulty in practical engineering. Bias faults and gain faults are two kinds of actuator faults that commonly occur in the practice, expressed as [40](3)uτft=ρτx¯τ,nuτt+ζτt,where the remaining control rate ρτ of gain faults satisfies 0<ρτx¯τ,n≤1 and ζτt represents a bounded signal of bias faults. Here, ρτx¯τ,n at the failure time instant tf is assumed to be unknown.
Considering (3), the follower dynamic equation (1) can be rewritten as(4)x˙τ,i=Fτ,ix¯τ,i+xτ,i+1+ϕτ,it,1≤i≤n−1,x˙τ,n=Fτ,nx¯τ,n+Γτuτt+Φτ,n,yτ=xτ,1,where Γτ=gτ,nx¯τ,nρτx¯τ,n and Φτ,n=gτ,nx¯τ,nζτt+ϕτ,nt.Assumption 1 (see [40, 41]).
There exist positive constantsg¯τ and g¯τ, and τ=1,…,N, such that g¯τ≤gτ,nx¯τ,n≤g¯τ. Without losing generality, it is assumed that gτ,nx¯τ,n is positive in the following adaptive fault-tolerant containment controller design.Assumption 2.
All the signalsζτt and ϕτ,it are bounded for the actual industrial control, so it is assumed that ζτt≤ζτ∗ and ϕτ,it≤ϕτ,i∗ with ζτ∗ and ϕτ,i∗ being positive constants. Because gτ,nx¯τ,n is bounded, we shall assume that Φτ,n≤Φτ,n∗ with Φτ,n∗ being a positive constant.
Nussbaum Type Gain: a continuous functionNs is a Nussbaum-type function if(5)lims⟶+∞sup1s∫0sNςdς=+∞,lims⟶+∞inf1s∫0sNςdς=−∞.
For example,ς2cosς, ς2sinς, and eς2cosπ/2ς are Nussbaum-type functions. For clarity, the even Nussbaum-type function Nς=eς2cosπ/2ς is used throughout this paper.Lemma 1 (see [42]).
For any positive semidefinite smooth functionVt and any smooth function ςt with t∈0,tf, if, for an even smooth Nussbaum-type function Nς, the inequality,(6)0≤Vt≤c0+e−c1t∫0tgxτNςς˙ec1τdτ+e−c1t∫0tς˙ec1τdτ,∀t∈0,tf,is true with positive constants c1 and c0 and a function gxt taking values in the unknown closed intervals I:=l−,l+ with 0∉I, then Vt, ςt and ∫0tgxτNςς˙dτ are bounded on 0,tf.


## 2.2. Fuzzy Logic Systems

On the basis of [28, 30], the knowledge base for FLS comprises a collection of fuzzy if-then rules as follows:(7)Rl:ifx1isF1landx2isF2land…andxnisFnl,thenyisGl,l=1,2,…,N,where x=x1,x2,…,xnT and y are the input and output of the FLS, respectively. The fuzzy sets Fτl and Gl are associated with the fuzzy membership functions μFτlxτ and μGτlxτ, respectively. N is the rules number.Through the singleton fuzzification, the center average defuzzification, and product inference engine, the FLS can be represented as(8)yx=∑l=1Ny¯l∏τ=1nμFτlxτ∑l=1N∏τ=1nμFτlxτ,where y¯l=maxy∈RμGτly.The fuzzy basis functions are designed as(9)φl=∏τ=1nμFτlxτ∑l=1N∏τ=1nμFτlxτ,and θT=y¯1,…,y¯N=θ1,…,θN and φx=φ1x,…,φNxT. Then, FLS (7) can be further expressed as yx=θTφx.


## 3. Adaptive Fuzzy Containment Control Design and Stability Analysis

In this section, an adaptive fuzzy fault-tolerant containment controller will be constructed by using the backstepping control technique. Due to the fact that FLS can approximate an unknown nonlinear function in a compact set, which is called universal approximator, nonlinear functionFτ,ix¯τ,i can be approximated by FLS. It is assumed that(10)F^τ,ix¯τ,i|θτ,i=θτ,iTφτ,ix¯τ,i,and the optimal parameter vectorθτ,i∗ is designed as(11)θτ,i∗=argminθτ,i∈Ωτ,isupx¯τ,i∈Uτ,iF^τ,ix¯τ,i|θτ,i−Fτ,ix¯τ,i,where Ωτ,i is a bounded region of θτ,i and Uτ,i is a bounded region of x¯τ,i. At the same time, the approximation error ετ,i is designed as(12)ετ,i=Fτ,ix¯τ,i−F^τ,ix¯τ,i|θτ,i∗,where ετ,i satisfies ετ,i≤ετ,i∗ with ετ,i∗ being a positive constant.According to [43], in order to reduce the number of adaptive laws, a new constant Wτ∗ is designed as follows:(13)Wτ∗=maxθτ,i∗2:i=1,2,…,n.Obviously,Wτ∗ is an unknown positive constant.In order to accomplish the control objective, we design the following coordinate transformations:(14)sτ,1=∑j=1Naτjyτ−yj+∑j=N+1N+Maτjyτ−yrjt,sτ,i=xτ,i−α¯τ,i,i=2,…,n,where sτ,i denotes error surface and α¯τ,i represents the output of the first-order filter.(15)ητ,iα¯˙τ,i+α¯τ,i=ατ,i,α¯τ,i0=ατ,i0,with a positive constantητ,2, which is introduced to avoid the repeated differentiation of the virtual control signal ατ,i.(i)
Step 1: from (14), s˙τ,1 is expressed as(16)s˙τ,1=dτθτ,1∗Tφτ,1+dτετ,1+dτsτ,2+dτα¯τ,2+dτϕτ,1−∑j=1Naτjθj,1∗Tφj,1+εj,1+xj,2+ϕj,1−∑j=N+1N+Maτjy˙rjt.The following compensating dynamics is introduced to compensate the negative impact of the filter errorα¯τ,2−ατ,2:(17)q˙τ,1=−cτ,1qτ,1+dτα¯τ,2−dτατ,2+dτqτ,2,qτ,10=0,with a positive constantcτ,1.It follows from (16) and (17) that s¯τ,1=sτ,1−qτ,1 satisfies the following differential equation:(18)s¯˙τ,1=dτθτ,1∗Tφτ,1+dτετ,1+dτϕτ,1+dτsτ,2+cτ,1qτ,1−dτqτ,2+dτατ,2−∑j=1Naτjθj,1∗Tφj,1+εj,1+xj,2+ϕj,1−∑j=N+1N+Maτjy˙rjt.The first virtual controlατ,2 is designed as(19)ατ,2=1dτ−cτ,1sτ,1+∑j=N+1N+Maτjy˙rj−12s¯τ,1Wτφτ,1Tφτ,1+∑j=1Naτjxj,2+∑j=1Naτjθj,1Tφj,1,wherecτ,1 is a positive design parameter.Then,s¯˙τ,1 can be further written as(20)s¯˙τ,1=dτθτ,1∗Tφτ,1+dτετ,1+dτs¯τ,2−cτ,1s¯τ,1+dτϕτ,1−12s¯τ,1Wτφτ,1Tφτ,1−∑j=1Naτjθ˜j,1Tφj,1−∑j=1Naτjεj,1−∑j=1Naτjϕj,1,wheres¯τ,2=sτ,2−qτ,2 and θ˜j,1=θj,1∗−θj,1.The Lyapunov function candidate is considered in the following:(21)Vτ,1=12s¯τ,12+12∑j=1Naτ,j1rj,1θ˜j,1Tθ˜j,1,whererj,1 is a positive design parameter.From equation (21), V˙τ,1 can be expressed as(22)V˙τ,1=s¯τ,1s¯˙τ,1−∑j=1Naτj1rj,1θ˜j,1Tθ.j,1=s¯τ,1dτθτ,1∗Tφτ,1+dτετ,1+dτs¯τ,2−cτ,1s¯τ,1+dτϕτ,1−12s¯τ,1Wτφτ,1Tφτ,1−∑j=1Naτjθ˜j,1Tφj,1−∑j=1Naτjεj,1−∑j=1Naτjϕj,1−∑j=1Naτj1rj,1θ˜j,1Tθ.j,1.It can be easily verified that(23)s¯τ,1dτθτ,1∗Tφτ,1≤12s¯τ,12Wτ∗φτ,1Tφτ,1+12dτ2,s¯τ,1dτετ,1+dτϕτ,1−∑j=1Naτ,jεj,1−∑j=1Naτjϕj,1≤s¯τ,12+dτ2ετ,1∗2+dτ2ϕτ,1∗2+∑j=1Naτjεj,1∗2+∑j=1Naτjϕj,1∗2.From (23), V˙τ,1 is expressed as(24)V˙τ,1≤−cτ,1−1s¯τ,12+dτs¯τ,1s¯τ,2+12s¯τ,12W˜τφτ,1Tφτ,1−∑j=1Naτj1rj,1θ˜j,1Tθ.j,1+rj,1s¯τ,1φj,1+12dτ2+dτ2ετ,1∗2+dτ2ϕτ,1∗2+∑j=1Naτjεj,1∗2+∑j=1Naτjϕj,1∗2,whereW˜τ=Wτ∗−Wτ.The adaptive law is designed as(25)θ.j,1=rj,1−s¯τ,1φj,1−δj,1θj,1.Then, one has(26)V˙τ,1≤−cτ,1−1s¯τ,12+dτs¯τ,1s¯τ,2+ωτ,1+12s¯τ,12W˜τφτ,1Tφτ,1+∑j=1Naτjδj,1θ˜j,1Tθj,1,whereωτ,1=1/2dτ2+dτ2ετ,1∗2+dτ2ϕτ,1∗2+∑j=1Naτjεj,1∗2+∑j=1Naτjϕj,1∗2.(ii)
Step 2: from (14), the derivative of sτ,2 is expressed as(27)s˙τ,2=θτ,2∗Tφτ,2+ετ,2+xτ,3+ϕτ,2−α¯˙τ,2.The impact of the errorα¯τ,3−ατ,3 can be reduced by the following compensating dynamics:(28)q˙τ,2=−cτ,2qτ,2+α¯τ,3−ατ,3+qτ,3,qτ,20=0,with a positive constantcτ,2.Defines¯τ,3=sτ,3−qτ,3. Then, the derivative of s¯τ,2 can be determined by(29)s¯˙τ,2=θτ,2∗φτ,2+ετ,2+ϕτ,2+s¯τ,3−α¯˙τ,2+ατ,3+cτ,2qτ,2.Choose the following virtual controlατ,3:(30)ατ,3=−cτ,2sτ,2+α¯˙τ,2−12s¯τ,2Wτφτ,2Tφτ,2−dτs¯τ,1.From (30), the derivative of s¯τ,2 can be rewritten as(31)s¯˙τ,2=θτ,2∗2φτ,2+ετ,2+ϕτ,2+s¯τ,3−cτ,2s¯τ,2−12s¯τ,2Wτφτ,2Tφτ,2−dτs¯τ,1.A simple manipulation shows that the Lyapunov function candidate,(32)Vτ,2=12s¯τ,22+Vτ,1,has the following derivative:(33)V˙τ,2≤s¯τ,2s¯˙τ,2+V˙τ,1≤s¯τ,2θτ,2∗2φτ,2+ετ,2+ϕτ,2+s¯τ,3−cτ,2s¯τ,2−12s¯τ,2Wτφτ,2Tφτ,2−cτ,1−1s¯τ,12+ωτ,1+12s¯τ,12W˜τφτ,1Tφτ,1+∑j=1Naτjδj,1θ˜j,1Tθj,1.It is true that(34)s¯τ,2θτ,2∗Tφτ,2≤12s¯τ,22Wτ∗φτ,2Tφτ,2+12,s¯τ,2ετ,2+ϕτ,2≤s¯τ,22+12ετ,2∗2+12ϕτ,2∗2.Then, one can have(35)V˙τ,2≤−∑σ=12cτ,σ−1s¯τ,σ2+s¯τ,2s¯τ,3+ωτ,2+∑j=1Naτjδj,1θ˜j,1Tθj,1+12∑σ=12s¯τ,σ2W˜τφτ,σTφτ,σ,whereωτ,2=ωτ,1+1/2+1/2ετ,2∗2+1/2ϕτ,2∗2.(iii)
Stepi (3≤i≤n−1): the derivative of sτ,i can be expressed as(36)s˙τ,i=θτ,i∗2φτ,i+ετ,i+xτ,i+1+ϕτ,i−α¯˙τ,i.Similar to Step 2, to compensate the impact ofα¯τ,i+1−ατ,i+1, the following compensating dynamics is introduced:(37)q˙τ,i=−cτ,iqτ,i+α¯τ,i+1−ατ,i+1+qτ,i+1,qτ,i0=0,withcτ,i being a positive design parameter.It can be verified that the derivative ofs¯τ,i=sτ,i−qτ,i is given by(38)s¯˙τ,i=θτ,i∗φτ,i+ετ,i+ϕτ,i+s¯τ,i+1−α¯˙τ,i+ατ,i+1+cτ,iqτ,i,wheres¯τ,n+1=sτ,n+1−qτ,n+1.Define the following virtual controlατ,i+1:(39)ατ,i+1=−cτ,isτ,i+α¯˙τ,i−12s¯τ,iWτφτ,iTφτ,i−s¯τ,i−1.Then, the derivative ofs¯τ,i can be rewritten as(40)s¯˙τ,i=θτ,i∗2φτ,i+ετ,i+ϕτ,i+s¯τ,i+1−cτ,is¯τ,i−12s¯τ,iWτφτ,iTφτ,i−s¯τ,i−1.A simple manipulation shows that the Lyapunov function candidate,(41)Vτ,i=12s¯τ,i2+Vτ,i−1,has the following derivative:(42)V˙τ,i≤s¯τ,is¯˙τ,i+V˙τ,i−1≤s¯τ,iθτ,i∗2φτ,i+ετ,i+ϕτ,i+s¯τ,i+1−cτ,is¯τ,i−12s¯τ,iWτφτ,iTφτ,i−∑σ=1i−1cτ,σ−2s¯τ,σ2+ωτ,i−1+∑j=1Naτ,jδj,1θ˜j,1Tθj,1+12∑σ=1i−1s¯τ,σ2W˜τφτ,σTφτ,σ.By using Young’s inequality, the following inequalities can be verified:(43)s¯τ,iθτ,i∗Tφτ,i≤12s¯τ,i2Wτ∗φτ,iTφτ,i+12,s¯τ,iετ,i+ϕτ,i≤s¯τ,i2+12ετ,i∗2+12ϕτ,i∗2.Substituting (43) into (42), one has(44)V˙τ,i≤−∑σ=1icτ,σ−1s¯τ,σ2+∑j=1Naτjδj,1θ˜j,1Tθj,1+12∑σ=1is¯τ,σ2W˜τφτ,σTφτ,σ+s¯τ,is¯τ,i+1+ωτ,i,whereωτ,i=ωτ,i−1+1/2+1/2ετ,i∗2+1/2ϕτ,i∗2.(iv)
Stepn: the derivative of sτ,n is given by(45)s˙τ,n=θτ,n∗Tφτ,n+ετ,n+Γτuτt+Φτ,n−α¯˙τ,n.Defines¯τ,n=sτ,n−qτ,n with qτ,n being determined by(46)q˙τ,n=−cτ,nqτ,n,qτ,n0=0,where cτ,n is a positive design parameter.Then, it follows from (45) and (46) that(47)s¯˙τ,n=θτ,n∗Tφτ,n+ετ,n+Γτuτt+Φτ,n−α¯˙τ,n+cτ,nqτ,n.The actual adaptive fuzzy controller is chosen as(48)uτ=Nςτvτ,with(49)ς˙τ=γτs¯τ,nvτ,and(50)vτ=cτ,nsτ,n−α¯˙τ,n+12s¯τ,nWτφτ,nTφτ,n+s¯τ,n−1,where γτ is a positive design parameter.Substituting (48) and (50) into (47), s¯˙τ,n is obtained as(51)s¯˙τ,n=θτ,n∗Tφτ,n+ετ,n+ΓτNςτvτ+vτ+Φτ,n−cτ,ns¯τ,n−12s¯τ,nWτφτ,nTφτ,n−s¯τ,n−1.Consider the Lyapunov function in the following:(52)Vτ,n=12s¯τ,n2+12bτW˜τ2+Vτ,n−1,where bτ is a positive design parameter.The time derivative ofVτ,n becomes(53)V˙τ,n≤s¯τ,ns¯˙τ,n−1bτW˜τW˙τ+V˙τ,n−1≤s¯τ,nθτ,n∗Tφτ,n+ετ,n+ΓτNςτvτ+vτ+Φτ,n−cτ,ns¯τ,n−12s¯τ,nWτφτ,nTφτ,n−1bτW˜τW˙τ−∑σ=1n−1cτ,σ−1s¯τ,σ2+∑j=1Naτjδj,1θ˜j,1Tθj,1+12∑σ=1n−1s¯τ,σ2W˜τφτ,σTφτ,σ+ωτ,n−1.We have(54)s¯τ,nθτ,n∗Tφτ,n≤12s¯τ,n2Wτ∗φτ,nTφτ,n+12,s¯τ,nετ,n+Φτ,n≤s¯τ,n2+12ετ,n∗2+12Φτ,n∗2.Then,V˙τ,n can be estimated by(55)V˙τ,n≤−∑σ=1ncτ,σ−1s¯τ,σ2+∑j=1Naτjδj,1θ˜j,1Tθj,1−1bτW˜τW˙τ−12∑σ=1nbτs¯τ,σ2φτ,σTφτ,σ+ΓτNςτvτs¯τ,n+vτs¯τ,n+ωτ,n−1+12+12ετ,n∗2+12Φτ,n∗2.With the adaptive law,(56)W˙τ=12∑σ=1nbτs¯τ,σ2φτ,σTφτ,σ−pτWτ.V˙τ,n can be further estimated by(57)V˙τ,n≤−∑σ=1ncτ,σ−1s¯τ,σ2+∑j=1Naτjδj,1θ˜j,1Tθj,1+pτbτW˜τWτ+ΓτNςτvτs¯τ,n+vτs¯τ,n+ωτ,n−1+12+12ετ,n∗2+12Φτ,n∗2.It is true that(58)pτbτW˜τWτ≤pτ22bτ2Wτ∗2−pτbτ−12W˜τ2,∑j=1Naτjδj,1θ˜j,1Tθj,1≤12∑j=1Naτjδj,12θj,1∗Tθj,1∗−∑j=1Naτjδj,1−12θ˜j,1Tθ˜j,1.Then,V˙τ,n is obtained as follows:(59)V˙τ,n≤−∑σ=1ncτ,σ−1s¯τ,σ2−pτbτ−12W˜τ2−∑j=1Naτjδj,1−12θ˜j,1Tθ˜j,1+ΓτNςτvτs¯τ,n+vτs¯τ,n+ωτ,where ωτ=1/2∑j=1Naτjδj,12θj,1∗Tθj,1∗+pτ2/2bτ2Wτ∗2+ωτ,n−1+1/2+1/2ετ,n∗2+1/2Φτ,n∗2.Remark 2.
In the literature [21], by using backstepping control technique, the adaptive containment controller was designed for nonlinear MASs without actuator faults, and the number of adaptive laws was greater than or equal to the order of the system. Adaptive fuzzy containment controllers are investigated for nonlinear MASs with actuator faults in this paper. Meanwhile, compared with literature [21], the number of adaptive laws that need to be designed is less, and the amount of calculation is greatly reduced.
According to the above adaptive fuzzy fault-tolerant containment control, the main result can be summarized as a theorem as follows.Theorem 1.
Consider the adaptive fuzzy backstepping containment controller, composed of the compensating signals (17), (28), (37), and (46), the virtual controllers (19), (30), and (39), the actual controller (48), and adaptive laws (25) and (56). If there exist the parameters cτ,σ, δj,1, pτ, and bτ, such that cτ,σ−1>0, pτ/bτ−1/2>0, and δj,1−1/2>0, then all the signals in the closed-loop system are bounded and all followers can converge to the convex hull built by the leaders.Proof.
Definecτ=mini=1,…,n2cτ,i−1,2pτ−bτ,2δj,1rj,1−rj,1. From (59), one has(60)V˙τ,n≤−cτVτ,n+ΓτγτNςτς˙τ+1γτς˙τ+ωτ,which has a solution of the form(61)Vτ,nt≤c0+e−ct∫0tΓτγτNςτς˙τecϱdϱ+e−ct∫0t1γτς˙τecϱdϱ,where c0=Vτ,n0+2ωτ/c. According to Lemma 1, all the signals for follower τ are bounded. Similarly, the whole nonlinear multiagent systems are stable.


## 4. Simulation Example

In order to illustrate the feasibility of the designed adaptive fuzzy backstepping fault-tolerant containment control, the nonlinear MASs are considered with two leaders and three followers. Theτ-th follower is considered as(62)x˙τ,1=Fτ,1xτ,1+xτ,2+ϕτ,1t,x˙τ,2=Fτ,2x¯τ,2+gτ,2x¯τ,2uτt+ϕτ,2t,yτ=xτ,1,τ=1,2,3,where F1,1=0.5e−x1,1, F1,2=0.3sinx1,1cosx1,2, F2,1=−0.2x2,1e−x2,1, F2,2=x2,12cosx2,2, F3,1=x3,1cosx3,1, F3,2=0.4x3,2sinx3,1, g1,2=esinx1,1x1,2, g2,2=ecosx2,2sinx2,14, g3,2=ecosx3,2x3,14, ϕ1,1=0.5cost, ϕ1,2=sintcost, ϕ2,1=0.2sint, ϕ2,2=sin0.5t, ϕ3,1=cost, and ϕ3,2=0.5costsint.The flow of information among leaders and followers is elaborated through Figure1, in which F1, F2, and F3 denote followers and L1 and L2 represent leaders. The actuator bias faults are chosen as ζ1=0.2sint, ζ2=0.6cos0.5t, and ζ3=0.1sin0.1t, and the actuator fault appears at tf=12s.Figure 1
Directed communication graph.The unknown nonlinear remaining control rate coefficientsρτ are chosen as ρ1=1/1+e−0.2, ρ2=1/1+e−2sinx2,12+cosx2,22, and ρ3=1/1+e−0.2sinx3,2. The leaders’ signals are chosen as yr1=sint+1 and yr2=sint−0.7.From the directed communication graph, the adjacency matrix is obtained as follows:(63)A¯=011000011000010000000000.The initial values are chosen asx10=0.2,0.1T, x20=0.5,0T, and x30=−0.2,0.3T. The other initial values are set as zero. The design parameters are tuned to be c1,1=27, c1,2=28, c2,1=22, c2,2=27, c3,1=20, c3,2=27, η1,2=η2,2=η3,2=0.1, and p1=p2=p3=20 by trial and error.Through the proposed controller, the simulation results are given in Figures2–4. Figure 2 denotes the curves of yrj and yττ=1,2,3,j=4,5. From Figure 2, we can clearly see that all the followers F1, F2, and F3 can converge to the convex hull built by the leaders L4 and L5. That is to say, all the followers are running among the leaders. Figure 3 depicts the curves of the controller uτ. According to Figure 3, although the actuator fault appears at tf=12, the containment control performance of the proposed control method can still be guaranteed. Figure 4 demonstrates the curves of Wτ. Figures 2–4 show that the stability of every follower’s system is guaranteed through the designed adaptive fuzzy backstepping containment controller. Besides, all followers are able to converge to the convex hull built by the leaders.Figure 2
Curves ofyrj and yτ.Figure 3
Curves ofuτ.Figure 4
Curves ofW1, W2, and W3.


## 5. Conclusion

An adaptive fuzzy containment control method has been studied for nonlinear MASs with unknown control coefficient and actuator faults. The unknown control coefficient and the remaining control rate of actuator faults have been solved by introducing a Nussbaum-type function. The fuzzy logic system has been used as an approximator to approximate an unknown nonlinear function. The adaptive fuzzy containment controller has been designed by using the backstepping control technique and compensating signals. Only few adaptive parameters have been designed for each follower agent. It has been demonstrated that the designed adaptive fuzzy containment controller can ensure that all the signals in system are bounded and make all followers converge to the convex hull built by the leaders. In the future, we will investigate the adaptive cooperative controller design for practical application of nonlinear MASs. [44]



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