Multiagent Consensus Control under Network-Induced Constraints

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.

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. 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.

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 𝐼_𝑛. The 𝑖th row of 𝐼_𝑛 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, 𝑥₁(𝑘+1)=𝑥₁(𝑘)+𝑏𝑢₁(𝑘),𝑥₂(𝑘+1)=𝑥₂(𝑘)+𝑏𝑢₂(𝑘)𝑘∈𝑍⁺, (1) 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−𝑡_𝑑) from channel no. 1 at 𝑘. Except the newest 𝑥₁(𝑘−𝑡^∗) with 𝑡^∗=min(𝑡₁,𝑡₂,…,𝑡_𝑑), 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 that 𝑧(𝑘)=𝑥₁(𝑘−𝛼_𝑘) with some random 𝛼_𝑘∈𝑍⁺ constrained by 𝛼_𝑘+1≤𝛼_𝑘+1∀𝑘∈𝑍⁺. (2) On 𝛼_𝑘, we adopt an assumption which is made by some researchers in networked control [16]; that is, 𝛼_𝑘 is assumed to be a Markov chain taking values in a finite set {0,1,…,𝑚} with transition probabilities: 𝜙_𝑠,𝑙=Pr𝛼_𝑘+1=𝑙∣𝛼_𝑘=𝑠∀𝑠,𝑙∈{0,1,…,𝑚}, (3) where 𝑚 is a given nonnegative integer. The transition probability matrix 𝜙Φ=^0,0𝜙_0,1𝜙0⋯0_1,0𝜙_1,1𝜙_1,2𝜙⋯0⋮⋮⋮⋮⋮_𝑚−1,0𝜙_𝑚−1,1𝜙_𝑚−1,2⋯𝜙_{𝑚−1,𝑚}𝜙_𝑚,0𝜙_𝑚,1𝜙_𝑚,2⋯𝜙_𝑚,𝑚∈𝑅^{(𝑚+1)×(𝑚+1)} (4) is also known. The expression (4) of Φ displays that, for the reason of constraint (2), 𝜙_𝑠,𝑙=0 when 𝑙>𝑠+1. Thus, we have described the communication in channel no. 1 using Markov chain 𝛼_𝑘. The same method is applied to describe the communication in channel no. 2. Agent 1 obtains 𝑥₂(𝑘−𝛽_𝑘) from channel no. 2 at 𝑘, where 𝛽_𝑘 is a Markov chain taking values in a known set {0,1,…,𝑛} with a known transition probability matrix 𝜑Ψ=^0,0𝜑_0,1𝜑0⋯0_1,0𝜑_1,1𝜑_1,2𝜑⋯0⋮⋮⋮⋮⋮_𝑛−1,0𝜑_𝑛−1,1𝜑_𝑛−1,2⋯𝜑_{𝑛−1,𝑛}𝜑_𝑛,0𝜑_𝑛,1𝜑_𝑛,2⋯𝜑_𝑛,𝑛∈𝑅^{(𝑛+1)×(𝑛+1)}. (5) 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 𝑢₁(𝑘)=−ℎ𝑥₂𝑘−𝛽_𝑘−𝑥₁(𝑘)−𝑥₁(𝑘)−𝑥^∗ (6) but agent 2 employs control law 𝑢₂(𝑘)=−ℎ𝑥₁𝑘−𝛼_𝑘−𝑥₂(𝑘), (7) 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}, lim_𝑘→∞𝐸𝑥₁(𝑘)−𝑥^∗2=0lim_𝑘→∞𝐸𝑥₂(𝑘)−𝑥^∗2=0. (8) Our objective is to design ℎ such that the two agents reach mean square consensus.

3. Main Result

Define 𝑦₁(𝑘)=𝑥₁(𝑘)−𝑥^∗𝑦₂(𝑘)=𝑥₂(𝑘)−𝑥^∗∀𝑘∈𝑍⁺. (9) Then from (1), (6), (7), and (9), we have 𝑦₁(𝑘+1)=(1+2𝑏ℎ)𝑦₁(𝑘)−𝑏ℎ𝑦₂𝑘−𝛽_𝑘𝑦₂(𝑘+1)=−𝑏ℎ𝑦₁𝑘−𝛼_𝑘+(1+𝑏ℎ)𝑦₂(𝑘). (10) Further, denote 𝑌(𝑘)=𝑦₁(𝑘)⋮𝑦₁()𝑦𝑘−𝑚₂(𝑘)⋮𝑦₂(𝑘−𝑛)^𝑇∈𝑅^𝑚+𝑛+2. (11) Obviously, mean square consensus (8) is equivalent to lim_𝑘→∞𝐸[𝑌^T(𝑘)𝑌(𝑘)]=0. Using (11), system (10) is transformed into 𝑌(𝑘+1)=𝐺_ℎ𝛼_𝑘,𝛽_𝑘𝑌(𝑘), (12) where 𝐺_ℎ𝛼_𝑘,𝛽_𝑘=𝐺_ℎ11𝐺_ℎ12𝛽_𝑘𝐺_ℎ21𝛼_𝑘𝐺_ℎ22∈𝑅^{(𝑚+𝑛+2)×(𝑚+𝑛+2)},𝐺_ℎ11=1+2𝑏ℎ0⋯001010⋱⋮10∈𝑅^{(𝑚+1)×(𝑚+1)},𝐺_ℎ12𝛽_𝑘𝛽=𝑏ℎ−𝑒_𝑘0,𝑛+1∈𝑅^{(𝑚+1)×(𝑛+1)},𝐺_ℎ21𝛽_𝑘𝛼=𝑏ℎ−𝑒_𝑘0,𝑚+1∈𝑅^{(𝑛+1)×(𝑚+1)},𝐺_ℎ22=1+𝑏ℎ0⋯001010⋱⋮10∈𝑅^{(𝑛+1)×(𝑛+1)}. (13) On system (12), [16] presented the following.

Lemma 1. Suppose that Markov chains 𝛼_𝑘 and 𝛽_𝑘 are independent. System (12) achieves lim_𝑘→∞𝐸[𝑌^T(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 𝑃𝛼,𝛽−𝐺^T_ℎ𝛼,𝛽^𝑚𝑛_𝑖=0𝑗=0𝜙_𝛼,𝑖𝜑_𝛽,𝑗𝑃𝑖,𝑗𝐺_ℎ𝛼,𝛽>0. (14)

Actually, the condition in Lemma 1 can be converted using Schur complement.

Lemma 2 (see [17]). Let 𝑆=^𝑆11𝑆12_𝑆21𝑆22 be given partitioned matrix. Then 𝑆>0 if and only if 𝑆₂₂>0 and 𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆₂₁>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 chains 𝛼_𝑘 and 𝛽_𝑘 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 𝑃𝛼,𝛽#𝜙_𝛼,0𝜑_𝛽,0𝐺_ℎ𝛼,𝛽𝑄(0,0)⋮⋱𝜙_𝛼,𝑚𝜑_𝛽,𝑛𝐺_ℎ𝑄()𝑃𝛼,𝛽𝑚,𝑛>0𝛼,𝛽−𝑄⁻¹𝛼,𝛽=0. (15)

In order to deal with the condition in Theorem 3 using Cone Complementarity Linearisation algorithm [18], for 𝑟∈𝑍⁺, we construct LMI 𝑃_𝑟𝛼,𝛽#𝜙_𝛼,0𝜑_𝛽,0𝐺_ℎ𝛼,𝛽𝑄(0,0)⋮⋱𝜙_𝛼,𝑚𝜑_𝛽,𝑛𝐺_ℎ𝑄()𝑃𝛼,𝛽𝑚,𝑛>0_𝑟𝛼,𝛽𝐼𝐼𝑄_𝑟𝑃𝛼,𝛽>0_𝑟𝛼,𝛽∈𝑅^{(𝑚+𝑛+2)×(𝑚+𝑛+2)},𝑄_𝑟𝛼,𝛽∈𝑅^{(𝑚+𝑛+2)×(𝑚+𝑛+2)},𝛼∈{0,1,…,𝑚},𝛽∈{0,1,…,𝑛} (16)(17) which is denoted by 𝐿(𝑃_𝑟(𝛼,𝛽),𝑄_𝑟(𝛼,𝛽),ℎ)>0. The following is our design steps:

Step 1. Specify an enough small real number 𝜀>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 𝜇_𝑟+1=mintrace^𝑚𝑛_𝛼=0𝛽=0𝑃_𝑟+1𝛼,𝛽𝑄_𝑟𝛼,𝛽+𝑃_𝑟𝛼,𝛽𝑄_𝑟+1𝑃𝛼,𝛽s.t.𝐿_𝑟+1𝛼,𝛽,𝑄_𝑟+1𝛼,𝛽,ℎ_𝑟+1>0, (18) and obtain 𝑃_𝑟+1(𝛼,𝛽), 𝑄_𝑟+1(𝛼,𝛽) and ℎ_𝑟+1.

Step 3. If |𝜇_𝑟+1−2(𝑚+1)(𝑛+1)(𝑚+𝑛+2)|<𝜀, let ℎ=ℎ_𝑟+1 and terminate. Otherwise, 𝑟←𝑟+1 and go to Step 4.

Step 4. If 𝑟>𝑇, exist. Otherwise, go to Step 2

It should be pointed out that the above method is easy to be extended to 𝑞 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 give 𝑚=3, 𝑛=3, 𝑏=0.8, and transition probability of 𝛼_𝑘 and 𝛽_𝑘 is given as ,.Φ=0.60.4000.250.30.4500.20.20.10.50.10.550.10.25Ψ=0.70.3000.550.350.100.20.40.250.150.30.350.150.2 (19) Using the design steps in Section 3, we get ℎ=−0.6215. Figure 1 shows the state response of two agents with 𝑥₁(0)=18,𝑥₂(0)𝑥=3,_∗=100,𝛼₀=0,𝛽₀=0. (20) 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.