# Robust Adaptive Sliding Mode Consensus of Multiagent Systems with Perturbed Communications and Actuators

Authors: Xiao-Zheng Jin; Zhong-Hu Yuan
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper deals with the asymptotic consensus problem for a class
of multiagent systems with time-varying additive actuator faults
and perturbed communications. TheL2 performance of systems is
also considered in the consensus controller designs. The upper and
lower bounds of faults and perturbations in actuators and
communications and controller gains are assumed to be unknown but
can be estimated by designing some indirect adaptive laws. Based on
the information from the adaptive estimation mechanism, the
distributed robust adaptive sliding mode controllers are constructed
to automatically compensate for the effects of faults and
perturbations and to achieve any given level of L2 gain
attenuation from external disturbance to consensus errors. Through
Lyapunov functions and adaptive schemes, the asymptotic consensus of
resulting adaptive multiagent system can be achieved with a
specified performance criterion in the presence of perturbed
communications and actuators. The effectiveness of the proposed
design is illustrated via a decoupled longitudinal model of F-18
aircraft.


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## Body

## 1. Introduction

The consensus behavior of multiagent systems has received significant attention over the last few years. It involves in lots of practical control systems and many physical phenomenon, such as synchronization of coupled oscillators [1], rendezvous in space [2], aircraft formation control [3], and flocking theory [4]. Recently, lots of researchers have focused on the development of methodologies to solve consensus problem with some usual issues existed in communications such as time-delays [5, 6], perturbations [7–15], and faulty communication links [16–18].As we know, state consensus may not be guaranteed under the influence of network perturbations. In [7], the bounded tracking results were obtained for multiagent consensus with an active leader in the presence of bounded perturbations. A stochastic model for distributed average consensus with zero mean noise was presented in [8]. Recently, the authors of [9] utilized an adaptive method to deal with a class of known upper bound disturbances. In addition, from the Laplacian eigenvalue standpoint, paper [10] focused on maximizing the second smallest eigenvalue of a state-dependent graph Laplacian to improve the robustness of the dynamic systems. The recent paper [11] gave a lazy consensus protocol against unknown but bounded disturbances and achieved bounded consensus results. Note that the aforementioned systems cannot guarantee asymptotic consensus when the disturbance always exists in the systems. Therefore, the capability of disturbance rejection for the previous systems seems weak. Recently, an asymptotic consensus result has been obtained in [12] under a matching condition in the presence of time-delays and disturbances in communications. But the design is complicated and the performances of systems are not considered in the paper.It is well known that actuators play an important role in the consensus of multiagent systems. Many researchers were devoted to the study of fault-tolerant control against actuator/sensor faults to ensure the stability and performance optimization of systems (e.g., see [19–21]). Paper [17] considered consensus problem with missing data in actuators and Markovian communication failure, and the communication failure process was reduced to a Bernoulli process. The recent paper [22] studied fault diagnosis for a class of discrete time-delayed complex interconnected networks with linear coupling in the case of actuator faults. The authors of [23] analyzed the performances of a team of unmanned vehicles with some actuator fault types, such that loss of effectiveness and lock-in-place. In [18], the bias actuators have been considered in synchronization of master-slave systems using an indirect adaptive method. Here, the similar additive actuator faults which can also be considered as perturbed actuators are treated by using an adaptive sidling mode method.It should be noted that the system performances, such asH∞, H2, L2 performances, are rarely considered in the consensus designs. In [13], the H∞ consensus control problem was considered in directed networks of delayed and nondelayed agents using a matrix inequality method. The H∞ consensus filtering problem was dealt with in [14] by a difference linear matrix inequalities over a finite-horizon for sensor networks with multiple missing measurements. The authors of  [15] utilized a model transformation approach and matrix theory to solve the H∞ consensus for second-order multiagent systems with multiple asymmetric time-varying delays. Different from those papers using matrix inequality methods, this paper mainly considers the studies of L2 performance of multiagent systems by an adaptive method.In this paper, we consider the consensus problem of multiagent systems in the presence of perturbed communications and actuators with a specified performance criterion. Here, the bounds of additive faults and the size of perturbations in communications are not necessary to be known. Based on the Lyapunov stability theory, a novel adaptive sliding mode control strategy is developed to achieve asymptotic consensus of the multiagent systems. On the basis of this proposed method, an integral sliding manifold is developed for average consensus and varying consensus. Some adaptive schemes are proposed to estimate the bounds of faults and perturbations and controller gains. Then, adaptive sliding mode controllers are constructed relying on the updated gains. By using the designed controller, the faulty and perturbed factors effects can be completely compensated and the asymptotic consensus can be achieved in the finite time with any given level ofL2 gain attenuation. Besides, it should be noted that the new proposed adaptive design method is not necessary for the estimations to give the exact information.The asymptotic consensus problem formulation is described in Section2. In Section 3, the adaptive sidling mode state feedback controllers are developed. Section 4 gives an example and simulation. Finally, conclusions are given in Section 5.


## 2. Preliminaries and Problem Statement

In this paper, we consider a multiagent systemG composed of N interconnected linear time-invariant continuous time agents Gi, i=1,2,…,N, which can be illustrated as an undirected graph. Each edge (vj,vi) corresponds to an available information link from agent i to agent j. Then, the N nodes constitute a network as the following state-space equation:
(1)x˙i(t)=Axi(t)+∑j=1,j≠iN‍aijA12(xj(t)-xi(t)+dij(t))+B2(ui(t)+fi(t))+B1wi(t),
where xi(t)∈Rn is the state of node i, and ui(t)∈Rq is the control input; aij∈R represents the topological structure of the network, which is an element of Laplacian matrix [24] satisfying aii=-∑j=1,j≠iN‍aij, and A12∈Rn×n is the inner coupling matrix describing the interconnections among components; dij(t)∈Rn is the network perturbation in communication between the i agent and the j agent, which is satisfied by d_ij≤dij≤d-ij, where d_ij and d-ij denote the unknown amplitude size of dij; fi(t)∈Rq denotes the additive actuator faults. Here, we assume that fi(t) can be described by a nonlinear function and bounded by unknown lower and upper bounds f_i and f-i; wi(t)∈Rm is a continuous vector function denoting external disturbance and possible variations with respect to the nominal parameter values for the system; A, B1, and B2 are real constant matrices with appropriate dimensions.Then, by the Kronecker product, the multiagent system (1) can be rewritten as
(2)x˙(t)=(I⊗A+L⊗A12)x(t)+(I⊗A12)d(t)+(I⊗B2)u(t)+(I⊗B2)f(t)+(I⊗B1)w(t),
where x=(x1T,…,xNT)T, u=(u1T,…,uNT)T, f=(f1T,…,fNT)T, w=(w1T,…,wNT)T, d=(d1T,…,dNT)T, di=∑j=1,j≠iNaijdij, i=1,2,…,N, and L is called the graph Laplacian induced by the information flow G and is defined by
(3)lij={-∑k=1,k≠iN‍aik,j≠i,aij,j=i.Similar to [12], to ensure the achievement of average consensus objective, the following assumption in consensus design is also assumed to be valid.Assumption 1.
For multiagent system (2) and any appropriate dimension matrix K1, there exist matrix functions H, Z of appropriate dimensions such that
(4)A12+B2K1=B2H,A=B2Z,
respectively.
Define(5)α=1N∑i=1N‍xi(0),
where xi(0), i=1,2,…,N, are the known initial values of system states. Then, for the sake of solving the consensus problems of multiagent systems in the presence of external faults, the local consensus protocol is considered as follows:
(6)ui(t)=-K1[∑j=1,j≠iN‍aij(xi(t)-xj(t)+dij)+ci(xi(t)-α){∑j=1,j≠iN‍}]+vi(t),
where i,j=1,2,…,N, aij is defined as in (1), ci is a positive constant, K1 is the control gain, and vi(t) is an adaptive control function which will be designed in later.
Then, by the Kronecker product, (6) can be rewritten as
(7)u(t)=-K1(L+C)⊗Ix(t)+K1(C1N)⊗α+(I⊗K1)d+v,
where u=(u1T,…,uNT)T, v=(v1T,…,vNT)T, and C is an N×N diagonal matrix whose ith diagonal element is ci, while 1N represents [1,1,…,1]NT.
Then, substituting (7) into (2), the closed-loop system model is stated as
(8)x˙(t)=(I⊗A+L⊗A12)x(t)-(I⊗B2)K1(L+C)⊗Ix(t)+(I⊗B2)K1(C1N)⊗α+(I⊗(A12+B2K1))d(t)+(I⊗B2)f(t)+(I⊗B2)v(t)+(I⊗B1)w(t).
Now, let(9)x-(t)=x(t)-1Nα.
Due to the characteristic of Laplacian matrix L, see (3), we have
(10)(I⊗B2)K1(L+C)⊗Ix-(t)=(I⊗B2)K1(L+C)⊗Ix(t)-(I⊗B2)K1(C1N)⊗α(t).
Then, following (10) and substituting (9) into (8), the closed-loop system can be rewritten as
(11)x-˙(t)=(I⊗A+L⊗A12-(I⊗B2)K1(L+C)⊗I)x-(t)+(I⊗B2)v(t)+(I⊗(A12+B2K1))d(t)+(I⊗B2)f(t)+(I⊗A)α+(I⊗B1)w(t).
According to Assumption1, we have
(12)x-˙(t)=(I⊗A+L⊗A12-(I⊗B2)K1(L+C)⊗I)x-(t)+(I⊗B2)v(t)+(I⊗B2)e(t)+(I⊗B1)w(t),
where  e=(e1T,…,eNT)T, ei(t)=fi(t)+Hdi(t)+Zα. Since fi(t), di(t), and α are bounded signals, we know that ei is also a bounded signal and denote e_i and e-i as unknown larger constants than lower and upper bounds of ei, respectively.
Then, the objective of this paper is to make sure that system (12) is asymptotically stable; namely,
(13)limt→∞x-(t)=0,
that is,
(14)limt→∞x(t)=1Nα,
with any given L2 performance index under the influence of perturbed communications and actuators.


## 3. Main Results

In this section, we develop the adaptive laws to estimate unknown controller gains for designing robust adaptive sliding mode controllers to eliminate the effects of communication perturbations and actuator faults and, simultaneously, to achieve any givenL2 performance criterion of the closed-loop system (12).The composite sliding surface for the closed-loop system (12) is chosen as
(15)s(x-(t))=0,
with s(x-(t))≡:(s1(x-1(t)),s2(x-2(t)),…,sN(x-N(t)))T and
(16)si(x-i(t))=x-i(t)-x-i(t0)-∫t0t‍[Ax-i(τ)+∑j=1,j≠iN‍aijA12(x-j(τ)-x-i(τ))-B2K1(∑j=1,j≠iN‍aij(x-j(τ)-x-i(τ))+cix-i(τ)+cix-i(τ){∑j=1,j≠iN‍}+cix-i(τ){∑j=1,j≠iN‍})]dτ,
where K1 is the controller gain proposed in (6) which is obtained by solving the following linear matrix inequality:
(17)(I⊗A+L⊗A12-B2K1(L+C)⊗I)T(I⊗P)+(I⊗P)(I⊗A+L⊗A12-B2K1(L+C)⊗I<0,
where P is a positive definite matrix. Note that the matrix K1 is designed such that the nominal fault-free system (12) is stable and some prescribed specifications would also be satisfied via this nominal state feedback control. Here, the term of x-i(t0) achieves the nice property that σi(x-i(t0))=0 such that the reaching phase is eliminated.Now, consider the controller model (6) with controller gain K1 solved by (17). We design control function vi(t) as follows:
(18)vi=ϕ^i∥siTPB2∥sgn(siTPB2)T+K2[(I-ρi)e-^i(t)+ρie^_i(t)],
where ϕi is an existed but unknown large enough positive constant satisfying
(19)2ϕi∥siTPB2∥2≥∥siTPB1∥2+γi∥si∥2,
where γi is any given L2 performance index, and ϕ^i is the estimation of ϕi updated by the following adaptive laws:
(20)dϕ^i(t)dt=ri∥siTPB2∥2,
where ri>0 is the weight of adaptive law ϕ^i(t). The sign function sgn(siTPB2)T=[sgn(bi1),…,sgn(biq)]T, where bil, l=1,2,…,q, is the l element of the vector siTPB2 and sgn(bil) defined by
(21)sgn(bil)={-1,ifbil>0,1,ifbil<0,0,ifbil=0,
and P is a positive symmetric matrix designed in (17); ρi is the switching factor defined between constants 0 and 1 defined by
(22)ρi=diag[ρi1,ρi2,…,ρiq],ρil={0,bil≥0,1,bil<0.e-^il(t) and e^_il(t) are the estimations of e-il and e_il, respectively, updated by the following adaptive laws:
(23)de-^il(t)dt=tilbil,de^_il(t)dt=tilbil,
where til>0 are the adaptive law gains to be designed according to practical application.Let(24)ϕ~i(t)=ϕ^i(t)-ϕi,e-~i(t)=e-^i(t)-e-i,e~_i(t)=e^_i(t)-e_i,
where e-i=[e-i1,e-i2,…,e-in]T, e_i=[e_i1,e_i2,…,e_in]T, i=1,2,…,N.Because toϕi, e-i, e_i are constants, the error system can be written as the following equations:
(25)ϕ~˙i(t)=ϕ^˙i(t),e-~˙i(t)=e-^˙i(t),e~_˙i(t)=e^_˙i(t).Thus, for the multiagent system described by (12), we propose the adaptive robust local control scheme (7) with the control gain function K2 given by
(26)K2=-I.Hence, the following theorem can be obtained, which shows the uniform ultimate boundedness of the closed-loop system (12) and the error system (25).Theorem 2.
Consider the closed-loop multiagent system described by (12) satisfying Assumption 1. By using the control scheme u(t) described in (7) with adaptive laws (20) and (23) and control gain functions (26), one can guarantee that all closed-loop system signals are bounded and limt→∞xi(t)=α with any given L2 performance index γi for any initial value x(t0), if there exists a symmetric matrix P>0 in (17).Proof.
For the adaptive robust closed-loop system described by (12), we first define a Lyapunov functional candidate as
(27)V(t)=sT(I⊗P)s+∑i=1N‍ϕ~i2ri+∑i=1N∑l=1n(1-ρil)e-~il2til+∑i=1N∑l=1nρile~_il2til.
From (12) and (16), the derivative of s(x-(t)) with respect to time can be calculated as follows:
(28)s˙(x-(t))=x-˙(t)-x-˙(t0)-(I⊗A+L⊗A12-K1(L+C)⊗I)x-(t)=(I⊗B2)(sgn(sTϕ^s⊗PB2)T)+(I⊗B2)(I⊗K2)[(I⊗(I-ρ))e-^(t)+(I⊗ρ)e^_(t)]+(I⊗B2)e(t)+(I⊗B1)w(t),
where e-^=(e-^1T,e-^2T,…,e-^NT)T, e^_=(e^_1T,e^_2T,…,e^_NT)T, ρ=diag[ρ1,ρ2,…,ρN], ϕ^s=diag[ϕ^1∥s1TPB2∥,ϕ^2∥s2TPB2∥,…,ϕ^N∥sNTPB2∥].
Then, following Assumption1, the time derivative of V(t) for t>0 can be described as
(29)V˙(t)=+2sT(I⊗P)(I⊗B2)(sgn(sTϕ^s⊗PB2)T)+2sT(I⊗P)(I⊗B2)e+2sT(I⊗P)(I⊗B1)w+2sT(I⊗P)(I⊗B2)(I⊗K2)(e-^+ρe^_-ρe-^)+∑i=1N2ϕ~iϕ~˙iri+∑i=1N∑l=1n2(1-ρil)e-~ile-~˙iltil+∑i=1N∑l=1q2ρile~_ile~˙_iltil=2∑i=1Nϕ^i∥siTPB2∥siTPB2sgn(siTPB2)T+2∑i=1NsiTPB1wi+2sT(I⊗P)∑i=1N∑l=1qBieil+2sT(I⊗P)∑i=1N∑l=1qBik2l(e-^il+ρile^_il-ρile-^il)+∑i=1N2ϕ~iϕ~˙iri+∑i=1N∑l=1q2(1-ρil)e-~ile-~˙iltil+∑i=1N∑l=1q2ρile~_ile~˙_iltil,
where Bi is the ith column of I⊗B2, i=1,2,…,N.
Note that(30)sT(I⊗P)∑i=1N∑l=1qBieil≤sT(I⊗P)∑i=1N∑l=1qBi(e-il+ρile_il-ρile-il),
where ρil are denoted in (22). By the adaptive laws chosen in (23) and controller gain function chosen in (26), then (29) can be rewritten as
(31)V˙(t)≤2∑i=1Nϕ^i∥siTPB2∥siTPB2sgn(siTPB2)T+2∑i=1NsiTPB1wi+2sT(I⊗P)×∑i=1N∑l=1qBi(e-il+ρile_il-ρile-il)+2sT(I⊗P)×∑i=1N∑l=1qBik2l(e-^il+ρile^_il-ρile-^il)+∑i=1N2ϕ~iϕ~˙iri+∑i=1N∑l=1q2(1-ρil)e-~ile-~˙iltil+∑i=1N∑l=1q2ρile~_ile~˙_iltil≤2∑i=1Nϕ^i∥siTPB2∥siTPB2sgn(siTPB2)T+2∑i=1N∥siTPB1∥∥wi∥+∑i=1N2ϕ~iϕ~˙iri.
From the fact thatsiTPB2sgn(siTPB2)T≤-∥siTPB2∥ and the adaptive law proposed in (20) with inequality (19), then we have
(32)V˙(t)≤-2∑i=1Nϕ^i∥siTPB2∥2+2∑i=1N∥siTPB1∥∥wi∥+∑i=1N2ϕ~iϕ~˙iri≤-∑i=1N(2ϕ^i∥siTPB2∥2-∥siTPB1∥2-γi∥si∥2)+∑i=1N(∥wi∥2-γi∥si∥2)+∑i=1N2ϕ~iϕ~˙iri=∑i=1N(∥wi∥2-γi∥si∥2).
Obviously, we haveV˙(t)<0 if ∥si∥>∥wi∥/γi. It means that the sliding manifold signal is uniformly ultimately bounded by limt→∞∥si∥∈{si∈Rn∣∥si∥≥∥wi∥/γi}.
When inequality (32) is integrated over the interval [t0,t], we obtain
(33)V(t)-V(t0)≤∑i=1N(-∫t0tγi∥si(τ)∥2d(τ)+∫t0t∥wi(τ)∥2d(τ)).
Then, we have(34)∑i=1N∫t0tγi∥si(τ)∥2d(τ)≤V(t0)+∑i=1N∫t0t∥wi(τ)∥2d(τ),
where V(t0)=s(t0)T(I⊗P)s(t0)+∑i=1N(ϕ~i2(t0)/ri)+∑i=1N∑l=1n((1-ρil)e-~il2(t0)/til)+∑i=1N∑l=1n(ρile~_il2(t0)/til).
If the initial conditions are chosen to be zero, then theL2 gain becomes clear such that
(35)γi∫t0t∥si(τ)∥2d(τ)≤∫t0t∥wi(τ)∥2d(τ)+ϕ~i2(t0)ri+∑l=1n(1-ρil)e-~il2(t0)til+∑l=1nρile~_il2(t0)til.
Hence, it is easy to see thatV˙(t)<0 for any x-≠0. Thus, the solutions of closed-loop system are uniformly bounded, and the error x-(t) converges asymptotically to zero. Moreover, from (35) and the results of [25], the L2 gain level of the disturbance attenuation can be guaranteed to be a given small value by adjusting γi.Remark 3.
The first part of control functionvi(t) in (18) is designed for formulating the L2 performance index γi. The second part of vi(t) is constructed for eliminating the effects of the matched perturbations of actuators and communications. It should be noted that the method largely simplifies the designs in [12].
The average consensus problem has been solved in Theorem2. Actually, the system can also track the time-varying object by the proposed method. Here, we define that α(t)∈Rn is the given command of states in time t for the agents. Without loss of generality, we assume that the command is a differentiable continuous signal which satisfies
(36)α˙(t)=Cg(t),
where g(t) is bounded by an unknown positive constant g0 such that ∥g(t)∥≤g0, and C is a real constant matrix with appropriate dimensions and satisfies the condition C=B2M.
Without consideringwi(t), then the closed-loop system (12) can be rewritten as
(37)x-˙(t)=(I⊗A+L⊗A12-(I⊗B2)K1(L+C)⊗I)x-(t)+(I⊗B2)v(t)+(I⊗B2)e(t),
where e=(e1T,…,eNT)T, ei(t)=fi(t)+Hdi(t)+Mg(t)+Zα(t). Since fi(t), di(t), g(t), and α(t) are bounded signals, we know that ei is still a bounded signal.
Then, the following corollary is proposed to obtain asymptotic consensus results of perturbed communications and actuators.Corollary 4.
Suppose that Assumption1 holds. Consider the closed-loop multiagent system described by (37). Then, by using the control scheme u(t) described by
(38)ui(t)=-K1[∑j=1,j≠iNaij(xi(t)-xj(t)+dij)+ci(xi(t)-α){∑j=1,j≠iN‍}]+K2[(I-ρi)e-^i(t)+ρie^_i(t)]
with adaptive laws (23) and control gain functions (26), one can guarantee that all closed-loop multiagent system signals are bounded and limt→∞xi(t)=α(t) for any initial value x(0), if there exists a symmetric matrix P>0 in (17).Proof.
Similar to the proof of Theorem2 and according to (30), we have sT(I⊗P)∑i=1N∑l=1qBihleil=sT(I⊗P)∑i=1N∑l=1qBihl(e-il+ρile_il-ρile-il) if and only if si=0. Thus, it is obvious that V˙=0 if and only if si=0, i=1,2,…,N, and a set E can be found as E={(s,e^_,e-^):V˙=0}={(s,e^_,e-^):si=0}, where s=(s1,s2,…,sN)T, e^_=(e^_1,e^_2,…,e^_N)T, and e-^=(e-^1,e-^2,…,e-^N)T. Starting with arbitrary initial values si(0), e^_i(0), and e-^i(0), the orbit converges asymptotically to si=0, e^_i=e_i, and e-^i=e-i, where e_i, e-i are constants. Based on LaSalle invariance principle, the system trajectories converge to the largest positively invariant subset M={(s,e^_,e-^):s=0,e^˙_=0,e-^˙=0}. Moreover, according to (12) and (28), we know that the error si(t) converges asymptotically to zero if there exist K1 and P making the inequality (17) hold true.


## 4. Numerical Example

In this section, an example of robust consensus control system design is given to demonstrate the proposed method. A multiagent system is composed of four dynamical agents, which have the same system matrices as follows:(39)A=[-10.210.1],B2=[2-1.51-1],A12=[0.4000.4],B1=[1.51-2-1],
and the topological structure matrix L in (3) described by
(40)L=[-10.20.50.30.1-10.60.30.20.5-10.30.10.30.6-1].To verify the effectiveness of the proposed adaptive sliding mode method, the simulations are given with the following parameters and initial conditions:(41)ri=100,til=100,e-^il(0)=10,e^_il(0)=-10,ci=1,x1(0)=[2,-1]T,x2(0)=[1,-0.5]T,x3(0)=[0.5,-1.5]T,x4(0)=[0.5,-3]T,i=1,2,3,4,l=1,2.Then, following the system state initial values, we get the average valueα=[1,-1.5]T.Then, solving LMI (17), we can obtain
(42)P=[0.1260-0.0156-0.01560.0262],K1=[1.8286-2.74291.8286-3.6571].The following case is considered in the simulations; that is, the communication perturbationsd12(t)=[-0.5,0.2sin(2t)]T, d23(t)=[0.4sin(0.2t),-0.5]T, d34(t)=[0.5,0.5cos(2t)]T, and d41(t)=[-0.2,0.3sin(20t)]T and the additive actuator faults f1(t)=[0.5,0.5cos(0.5t)]T, f2(t)=[-0.5,0.4sin(20t)]T, f3(t)=[0.5,-0.4sin(2t)]T, and f4(t)=[0.5sin(t),-1.5]T enter into the systems at the beginning (t≥0). Moreover, the external disturbances w1(t)=[-5,2sin(2t)]T, w2(t)=[4sin(0.2t),-5]T, w3(t)=[5,5cos(2t)]T, and w4(t)=[-2,3sin(20t)]T enter into the systems in t∈[0.8,0.9].Figures1 and 2 are, respectively, the response curves of the system’s states and consensus errors with adaptive robust feedback control scheme in aforementioned case. Figure 3 describes the estimate of upper and lower bounds of perturbations e-^i and e^_i, i=1,2,3,4, respectively. Figure 4 is the response curves of the estimate of controller gain ϕi, i=1,2,3,4. It is easy to see that the estimations can convergence but not converge to theirs true values. In our adaptive robust fault-tolerant control design, there is no need for the estimate of  e-^i, e^_i, ϕ^i to converge to their true values.Figure 1
Response curves of the system statesxi(t), i=1,2,3,4.Figure 2
Response curves of the consensus errorsx-i(t), i=1,2,3,4.Response curves of the estimate of upper bound and lower bound ofei(t), i=1,2,3,4.
(a)
(b)Figure 4
Response curves of the estimate ofϕi, i=1,2,3,4.Furthermore, the proposed method can also deal with the time-varying consensus problem. Here, we consider the previous system with the same initial conditions and perturbations and faults, but without considering the effects of external disturbances. The consensus target is described byα˙(t)=0.5B2[sin(t)cos(t)]T with initial values [1,-1.5]T. Then, Figure 5 illustrates the response curves of the consensus errors with adaptive sidling mode control scheme (38). The results indicated that the proposed method can also solve the time-varying consensus problem with some matched condition.Figure 5
Response curves of the consensus errorsx-i(t) with varying target; i=1,2,3,4.


## 5. Conclusions

In this paper, we have shown an adaptive design method to solve the robust asymptotic consensus problem for a class of multiagent systems with perturbed communications and faulted actuators. For the sake of automatically compensating for the effects of faults and networked perturbations and specifyingL2 performance criterion, the consensus protocol is constructed with the adaptive schemes, which are based on the updated adaptation laws to estimate the unknown bounds of perturbations and controller gains online. On the basis of Lyapunov stability theory, it has been shown that the resulting adaptive closed-loop multiagent system can be guaranteed to be asymptotic average and varying consensus with any given L2 performance index even in the presence of imperfect communications and actuators. A numerical example has been given to illustrate the effectiveness of the proposed method.



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*Source: 283196-2013-04-18.xml*