# Multiagent Consensus Control under Network-Induced Constraints

Authors: Won Il Kim; Rong Xiong; Qiuguo Zhu; Jun Wu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Mean consensus problem is studied using a class of discrete time multiagent systems in which information exchange is subjected to some network-induced constraints. These constraints include package dropout, time delay, and package disorder. Using Markov jump system method, the necessary and sufficient condition of mean square consensus is obtained and a design procedure is presented such that multiagent systems reach mean square consensus.


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## Body

## 1. Introduction

Cooperative control of networked multiagent systems by information exchange has received extensive attention presently, because of their extensive applications in flocking, swarming, distributed sensor fusion, attitude alignment, and so forth (see [1, 2] for surveys). An important problem for cooperative control is to design an appropriate control law such that a multiagent system reaches consensus in the presence of insecure information exchange. Distributed cooperative control of networked multiagent systems has been investigated in various perspectives [3–7]. In [3], the leaderless consensus problem is studied. The problem of consensus with leader node was researched in [4–7]. For networked multiagent systems of linear dynamics, consensus using state feedback or output feedback was analysed in [8, 9].Unmodelled time delay during the design phase is an important factor that may affect the performance of dynamical systems. It can even, in some situation, cause instability of a system. In these years, consensus in networked multiagent systems with time delay was discussed using linear matrix inequality (LMI) method [10–12]. In [10], the average-consensus problem for continuous-time multiagents with switching topology and time delay was studied. The work of [11] investigated the average consensus problem in undirected networks with fixed and switching topologies under time-varying communication delays. The consensus problem was solved in [12] on directed graphs of the multiagent system with model uncertainty and time delay.In the information exchange of network, there are not only time-delay but also other network-induced constraints. The other network-induced constraints, which include package dropout and package disorder, also affect the consensus of networked multiagent systems. However, not many works have studied multiagent systems with these network-induced constraints. Based on Markov jump system method [13–15], this paper considers mean square consensus of multiagent systems of first-order integrator under network-induced constraints such as package dropout, time delay, and package disorder. By system transformation, the necessary and sufficient condition of mean square consensus problem is provided and a corresponding design algorithm is given.The remainder of the paper is organized as follows. Section2 contains the formulation of the problem and terminology. The main results are presented in Section 3. Section 4 provides the numerical simulation and Section 5 draws conclusions.


## 2. Problem Formulation and Preliminaries

In this paper,Z+ is used to denote the set of all nonnegative integers. The n×n identity matrix is denoted by In. The ith row of In is denoted by e(i,n). If a matrix P is positive (negative) definite, it is denoted by P>0(<0). The notation # within a matrix represents the symmetric term of the matrix. The expected value is represented by E[•].Here a discrete-time system is considered that it consists of 2 agents. Each agent is a first-order integrator, which,(1)x1(k+1)=x1(k)+bu1(k),x2(k+1)=x2(k)+bu2(k)k∈Z+,
where x1(k)∈R and x2(k)∈R are the state at time step k,u1(k)∈R and u2(k)∈R are the control at time step k, and b∈R is constant. The two agents exchange their state through two communication channels: channel no. 1 and channel no. 2. At each k, agent 1 transmits x1(k) to agent 2 through channel no. 1. Agent 2 utilizes z(k) as the information obtained from channel no. 1 at k. Due to random package dropout, time delay, and package disorder in communication, the receiving scenarios in the side of agent 2 at k are various. Agent 2 may receive one package x1(k-t) from channel no. 1 at k. The package x1(k-t) is sent by agent 1 at k-t no later than k. After received, x1(k-t) is examined to see whether it is of disorder (i.e., whether agent 2 has received any packages sent later than k-t). If x1(k-t) is not of disorder, z(k)←x1(k-t). If x1(k-t) is of disorder, x1(k-t) is discarded and z(k)←z(k-1). Agent 2 may receive severe package x1(k-t1), x1(k-t2),…,x1(k-td) from channel no. 1 at k. Except the newest x1(k-t*) with t*=min(t1,t2,…,td), these packages are discarded. If x1(k-t*) is not of disorder, z(k)←x1(k-t*). If x1(k-t*)  is of disorder, it is also discarded and z(k)←z(k-1). Agent 2 may receive no package from channel no. 1 at k. In this case, z(k)←z(k-1).From the above mechanism, it is seen thatz(k)=x1(k-αk) with some random αk∈Z+ constrained by
(2)αk+1≤αk+1∀k∈Z+.
On αk, we adopt an assumption which is made by some researchers in networked control [16]; that is, αk is assumed to be a Markov chain taking values in a finite set {0,1,…,m} with transition probabilities:
(3)ϕs,l=Pr(αk+1=l∣αk=s)∀s,l∈{0,1,…,m},
where m is a given nonnegative integer. The transition probability matrix
(4)Φ=[ϕ0,0ϕ0,10⋯0ϕ1,0ϕ1,1ϕ1,2⋯0⋮⋮⋮⋮⋮ϕm-1,0ϕm-1,1ϕm-1,2⋯ϕm-1,mϕm,0ϕm,1ϕm,2⋯ϕm,m]∈R(m+1)×(m+1)
is also known. The expression (4) of Φ displays that, for the reason of constraint (2), ϕs,l=0 when l>s+1. Thus, we have described the communication in channel no. 1 using Markov chain αk. The same method is applied to describe the communication in channel no. 2. Agent 1 obtains x2(k-βk) from channel no. 2 at k, where βk is a Markov chain taking values in a known set {0,1,…,n} with a known transition probability matrix
(5)Ψ=[φ0,0φ0,10⋯0φ1,0φ1,1φ1,2⋯0⋮⋮⋮⋮⋮φn-1,0φn-1,1φn-1,2⋯φn-1,nφn,0φn,1φn,2⋯φn,n]∈R(n+1)×(n+1).
The goal of agents 1 and 2 is a prescribed state x*∈R. In this paper, agent 1 is aware of x* while agent 2 is not. Consequently, agent 1 employs control law
(6)u1(k)=-h((x2(k-βk)-x1(k))-(x1(k)-x*))
but agent 2 employs control law
(7)u2(k)=-h(x1(k-αk)-x2(k)),
where h∈R is the control parameter.The above multiagent system is said to be mean square consensus if∀x1(0)∈R, ∀x2(0)∈R, ∀α0∈{0,1,…,m}, ∀β0∈{0,1,…,m},
(8)limk→∞E[(x1(k)-x*)2]=0limk→∞E[(x2(k)-x*)2]=0.
Our objective is to design h such that the two agents reach mean square consensus.


## 3. Main Result

Define(9)y1(k)=x1(k)-x*y2(k)=x2(k)-x*∀k∈Z+.
Then from (1), (6), (7), and (9), we have
(10)y1(k+1)=(1+2bh)y1(k)-bhy2(k-βk)y2(k+1)=-bhy1(k-αk)+(1+bh)y2(k).
Further, denote
(11)Y(k)=[y1(k)⋮y1(k-m)y2(k)⋮y2(k-n)]T∈Rm+n+2.
Obviously, mean square consensus (8) is equivalent to limk→∞E[YT(k)Y(k)]=0. Using (11), system (10) is transformed into
(12)Y(k+1)=Gh(αk,βk)Y(k),
where
(13)Gh(αk,βk)=[Gh11Gh12(βk)Gh21(αk)Gh22]∈R(m+n+2)×(m+n+2),Gh11=[1+2bh0⋯001010⋱⋮10]∈R(m+1)×(m+1),Gh12(βk)=bh[-e(βk,n+1)0]∈R(m+1)×(n+1),Gh21(βk)=bh[-e(αk,m+1)0]∈R(n+1)×(m+1),Gh22=[1+bh0⋯001010⋱⋮10]∈R(n+1)×(n+1).
On system (12), [16] presented the following.Lemma 1.
Suppose that Markov chainsαk and βk are independent. System (12) achieves limk→∞E[YT(k)Y(k)]=0 if and only if there exist positive definite matrices P(α,β)∈R(m+n+2)×(m+n+2), α∈{0,1,…,m}, β∈{0,1,…,n} such that
(14)P(α,β)-GhT(α,β)(∑i=0m∑j=0nϕα,iφβ,jP(i,j))Gh(α,β)>0.Actually, the condition in Lemma1 can be converted using Schur complement.Lemma 2 (see [17]).
LetS=[S11S12S21S22] be given partitioned matrix. Then S>0 if and only if  S22>0 and S11-S12S22-1S21>0.Through the above converting, the necessary and sufficient condition is obtained from mean square consensus of the multiagent system.Theorem 3.
Suppose that Markov chainsαk and βk are independent. The multiagent system in Section 2 achieves mean square consensus if and only if there exist h∈R and positive definite matrices P(α,β)∈R(m+n+2)×(m+n+2), Q(α,β)∈R(m+n+2)×(m+n+2), α∈{0,1,…,m}, β∈{0,1,…,n} such that
(15)[P(α,β)#ϕα,0φβ,0Gh(α,β)Q(0,0)⋮⋱ϕα,mφβ,nGh(α,β)Q(m,n)]>0P(α,β)-Q-1(α,β)=0.In order to deal with the condition in Theorem3 using Cone Complementarity Linearisation algorithm [18], for r∈Z+, we construct LMI
(16)[Pr(α,β)#ϕα,0φβ,0Gh(α,β)Q(0,0)⋮⋱ϕα,mφβ,nGh(α,β)Q(m,n)]>0[Pr(α,β)IIQr(α,β)]>0(17)Pr(α,β)∈R(m+n+2)×(m+n+2),Qr(α,β)∈R(m+n+2)×(m+n+2),α∈{0,1,…,m},β∈{0,1,…,n}
which is denoted by L(Pr(α,β),Qr(α,β),h)>0. The following is our design steps:Step 1.
Specify an enough small real numberε>0 and an enough large integer T. Set r=0. Find   feasible P0(α,β), Q0(α,β), and h0, ∀α∈{0,1,…,m} and ∀β∈{0,1,…,n} satisfy   L(P0(α,β),Q0(α,β),h0)>0. If there is none, exit.Step 2.
Solve the LMI problem(18)μr+1=mintrace∑α=0m∑β=0nPr+1(α,β)Qr(α,β)+Pr(α,β)Qr+1(α,β)s.t.L(Pr+1(α,β),Qr+1(α,β),hr+1)>0,
and obtain Pr+1(α,β), Qr+1(α,β) and hr+1.Step 3.
If|μr+1-2(m+1)(n+1)(m+n+2)|<ε, let h=hr+1 and terminate. Otherwise, r←r+1  and go to Step 4.Step 4.
Ifr>T, exist. Otherwise, go to Step 2It should be pointed out that the above method is easy to be extended toq agents when q>2. Among q agents, there are q(q-1) communication channels. We utilize q(q-1) independent Markov chains to describe communication in these channels and can arrive at a similar result as Theorem 3.


## 4. Numerical Example

In the numerical example, we givem=3, n=3, b=0.8, and transition probability of αk and βk is given as
(19)Φ=[0.60.4000.250.30.4500.20.20.10.50.10.550.10.25],Ψ=[0.70.3000.550.350.100.20.40.250.150.30.350.150.2].
Using the design steps in Section 3, we get h=-0.6215. Figure 1 shows the state response of two agents with
(20)x1(0)=18,x2(0)=3,x*=100,α0=0,β0=0.
It can be seen that x1 and x2 converge at x*.Figure 1
State response of two agents.


## 5. Conclusion

The consensus control problem of multiagent systems of first-order integrator is studied under network-induced constraints. A new model is presented to describe the network communication with package dropout, time delay, and package disorder. For the new model, the definition of mean square consensus is given multiagent systems. Further, the necessary and sufficient condition of mean square consensus is proposed in the form of matrix inequalities. Based on this condition and Cone Complementarity Linearisation algorithm, a consensus control law can be designed to make systems reach mean square consensus.



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