Multiagent Consensus Control under Network-Induced Constraints

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.


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][4][5][6][7].In [3], the leaderless consensus problem is studied.The problem of consensus with leader node was researched in [4][5][6][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][11][12].In [10], the averageconsensus 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 timevarying 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 networkinduced constraints.Based on Markov jump system method [13][14][15], this paper considers mean square consensus of multiagent systems of first-order integrator under networkinduced 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.Section 2 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.

Problem Formulation and Preliminaries
In this paper, Z + is used to denote the set of all nonnegative integers.The × identity matrix is denoted by   .The th row of   is denoted by (, ).If a matrix  is positive (negative) definite, it is denoted by  > 0(< 0).The notation # within a matrix represents the symmetric term of the matrix.The expected value is represented by Here a discrete-time system is considered that it consists of 2 agents.Each agent is a first-order integrator, which, where  1 () ∈  and  2 () ∈  are the state at time step , 1 () ∈  and  2 () ∈  are the control at time step , and  ∈  is constant.The two agents exchange their state through two communication channels: channel no. 1 and channel no. 2. At each , agent 1 transmits  1 () to agent 2 through channel no. 1. Agent 2 utilizes () as the information obtained from channel no. 1 at . Due to random package dropout, time delay, and package disorder in communication, the receiving scenarios in the side of agent 2 at  are various.Agent 2 may receive one package  1 ( − ) from channel no. 1 at .The package  1 ( − ) is sent by agent 1 at  −  no later than .After received,  1 ( − ) is examined to see whether it is of disorder (i.e., whether agent 2 has received any packages sent later than  − ).
From the above mechanism, it is seen that () =  1 ( −   ) with some random   ∈  + constrained by On   , we adopt an assumption which is made by some researchers in networked control [16]; that is,   is assumed to be a Markov chain taking values in a finite set {0, 1, . . ., } with transition probabilities: where  is a given nonnegative integer.The transition probability matrix is also known.The expression (4) of Φ displays that, for the reason of constraint (2),  , = 0 when  >  + 1.Thus, we have described the communication in channel no.
The goal of agents 1 and 2 is a prescribed state  * ∈ .
Step It should be pointed out that the above method is easy to be extended to  agents when  > 2. Among  agents, there are ( − 1) communication channels.We utilize ( − 1) independent Markov chains to describe communication in these channels and can arrive at a similar result as Theorem 3.

Numerical Example
In the numerical example, we give  = 3,  = 3,  = 0.8, and transition probability of   and   is given as It can be seen that  1 and  2 converge at  * .

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.

Figure 1 :
Figure 1: State response of two agents.
the design steps in Section 3, we get ℎ = −0.6215.
1 using Markov chain   .The same method is applied to describe the communication in channel no. 2. Agent 1 obtains  2 ( −   ) from channel no. 2 at , where   is a Markov chain taking values in a known set {0, 1, . . ., } with a known transition probability matrix