Optimal mean-square consensus for heterogeneous multi-agent system with probabilistic time delay

Funding information National Natural Science Foundation of China, Grant/Award Numbers: 61673080, 61773082; Science Funds of Chongqing CSTC, Grant/Award Number: cstc2019jcyj-msxmX0102; State Scholarship Foundation of China, Grant/Award Number: 201808500022; Science and Technology Research Program of Chongqing Municipal Education Commission, Grant/Award Number: KJZDK202000601; the Russian Ministry of Science and Education “Digital biodesign and personalised healthcare” Abstract This paper studies the mean-square consensus for heterogeneous multi-agent systems with probabilistic time delay. Each agent in the system has an objective function and only knows its own objective function. Control protocols for the system both over the fixed and the switched weighted-balanced topologies are designed. The consensus state of agents’ position can make the sum of objective functions minimum. By adopting probability statistics, stochastic process, matrix theory and some stability method, sufficient conditions for the consensus protocol are given. Several simulations are presented to illustrate the potential correctness of the results.


INTRODUCTION
Consensus of multi-agent systems (MAS) has attracted more and more attention because of its wide application [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Due to the complexity of environment, there are many occasions that require various kinds of agents cooperating with each other to complete tasks that cannot be done by a single agent. When different kinds of agents complete a task together, because of the influence of external environment, communication

PRELIMINARIES
In this paper, R denotes the real number set, R m×n the set of real matrix with m-row and n-column, I n the n-dimensional identity matrix, 1 = (1, … , 1) T a column vector with all elements being 1, 0 a zero matrix with a proper dimension, E[ ] the mathematical expectation of , and H > 0 indicates the matrix H is positive definite. C 2 function refers to the second-order continuous differentiable function. And other symbols are in the usual sense. Denote G = ( , Σ, A) a weighted graph, = {v 1 , … , v n } the node set with every node representing an agent, Σ ⊆ × the edge set of the graph G , Υ = {1, … , n} the node index set, and Υ 1 = {1, … , m}, Υ 2 = {m + 1, … , n}. In the digraph G , an edge (v i , v j ) ∈ Σ means that the j th agent can receive the information from the i th agent directly, and a directed path from v j to v i is a sequence of the edges (v j , v j 1 ), (v j 1 , v j 2 ), … , (v j k , v i ). If there is a directed path between any two distinct nodes, the graph is connected. The neighbour set of agent i is  i = { j |(v j , v i ) ∈ Σ}, i ∈ Υ, A = [a i j ] ∈ R n×n the adjacency matrix, a i j > 0 if (v i , v j ) ∈ Σ, otherwise a i j = 0, and a ii = 0 for all i ∈ Υ. D = diag{ ∑ n j =1 a 1 j , … , ∑ n j =1 a n j } is the degree matrix of the graph G and L = D − A the Laplacian matrix. A graph G is called weighted balanced if there are some w i > 0, w j > 0, such that the coupling weights of G satisfy w i a i j = w j a ji for all i, j ∈ Υ.
Consider the following HMAS with m (m < n) first-order agents and (n − m) second-order agents. Their dynamics arė with p i (t ) ∈ R, q i (t ) ∈ R, and u i (t ) ∈ R being the position, velocity, and the control input of agent i, respectively. Practically, systems are often subjected to time delay and the delay is usually random. That is there are two events as following Event (I) : There is time delay in the system. Event (II) : There is no time delay in the system. Define the following random variable 1, Event (I) occurs, 0, Event (II) occurs.
Let the mathematical expectation of (t ) be E[ (t ) = 1] =w ith 0 ≤̃≤ 1. Then E[ (t ) = 0] = 1 −̃. The objective of this work is to solve the mean-square consensus of systems (2.1) and (2.2) with the probabilistic time delay such that the consensus state makes the sum of objective functions g(p) = ∑ n i=1 g i (p i ) minimum, where function g i (p i ) is the objective function of agent i and is only known by agent i, i ∈ Υ.
Next, several necessary assumptions and lemmas are given for the further analysis. Assumption 2.1. Assume that the function g i : R n → R, i ∈ Υ, is strictly convex and twice differentiable. That is, for any x, y ∈ R, there is If and only if x = y, the equation holds. And the second derivative of function g i , i ∈ Υ, exists.

Assumption 2.2.
For any x, y ∈ R n , function g i : R n → R, i ∈ Υ, is a C 2 function and its partial differential on x i , that is, Then there is a positive i > 0, such that   ([38]). If the vector function (t ) ∈ R N is differentiable and the matrix Θ ∈ R N ×N is positive definite, then there is where d (t ) satisfies Assumption 2.3.

Systems over fixed topology
For systems (2.1) and (2.2), design the following control protocol for the first-order agents and the following protocol for the second-order agents For simplicity, denote the Laplacian matrix as , with (3.4). Then at this equilibrium point there is and The sum of (3.5) and (3.6) from 1 to n is From (3.7) to (3.10) one can get, at the equilibrium point of sys- and for i ∈ Υ 2 , there iṡ (3.12) Denote Then systems (3.11) and (3.12) can be described aṡỹ where holds where Proof. Choose the candidate Lyapunov function andỹ t is the short form of functionỹ(t ), the same as other similar symbols in the following. The infinitesimal operator  of function  (ỹ t ) is defined as Then along (3.13) there is and T (s)̇ỹ(s)ds.
(3.18) According to Assumption 2.3 and Lemma 2.1 one can obtain Based on (3.18) and (3.19) there is .

Systems over switching topology
Due to the complexity of environment, in practice, the communication between agents usually varies. Hence this section considers systems (2.1) and (2.2) over the switching topologies G k , k ∈ ℕ, which are connected and weighted balanced, ℕ = {1, 2, … , ℵ} is the index set and ℵ a finite positive integer.
That is for the adjacency matrix A k = [a k i j ] n×n of graph G k , there exists w k i such that w k i a k i j = w k j a k ji , k ∈ ℕ. For systems (2.1) and (2.2) over the switching weight balanced network G k , k ∈ ℕ, which switches in sequence G 1 , G 2 , … , G ℵ , design the following control protocol for the first-order agents and the following protocol for the second-order agents Then similar the analysis on the fixed topology, there is the following result.

Theorem 3.2. Suppose that Assumptions 2.1-2.3 hold and the network is switched among the weighted balanced graph G k , k ∈ ℕ.
Then under the control protocols (3.24) and (3.25) with some , , > 0, systems (2.1) and (2.2) can reach the optimal mean-square consensus with the objective function g(p) = ∑ n i=1 g i (p), if for any k ∈ ℕ, there exist symmetric matrices  k > 0,  k > 0, such that for certain d 0 > 0, the following inequalities The proof of Theorem 3.2 is similar to the argument in Theorem 3.1. Hence it is omitted here.
Remark 3.2. According to the control protocols (3.1) and (3.2), and (3.24) and (3.25),̃≡ 1 implies the delay case, that is Event (I) occurs with probability one. And̃≡ 0 implies the delay free case, that is Event (II) occurs with probability one. Hence the delay system and the delay free system are the special cases of this work. Remark 3.3. For simplicity, only one dimensional system is considered in this work. It should be pointed out that the result is also valid for the multi-dimensional system, which can be obtained by using the Kronecker product of matrix.

EXAMPLES
Example.
Note that graphs G 1 , G 2 and G 3 are connected and weighted balanced, and p * = −2 is the optimal point of the function  reach the optimal point p * = −2 and all the velocities reach together, while Figures 11 and 12 illustrate that the mean value of agents' positions and velocities cannot achieve together when d 0 > 0.017. (iv) For systems (2.1) and (2.2) over G 1 , under protocols (3.1) and (3.2) with d (t ) = 100|cos t | and̃= 0, the mean value trajectories of agents' state error are given in Figures 9 and  10, which show that the mean value of all agents' states achieve together.
Remark 4.1. Figures 1 and 2 and Figures 5 and 6 illustrate that, the mean values of agents' states achieve together, and the consensus state makes the objective function g(p) =  Remark 4.3. Figures 9 and 10 illustrate that if̃≡ 0, that is, the probability of time delay is zero, then the time delay does not affect the convergence.

CONCLUSIONS
This paper studies the optimal mean-square consensus for HMAS both over fixed and switched weighted-balanced topologies. By adopting probability statistics, stochastic process, matrix theory and stability method, the control protocol is designed and sufficient conditions for the optimal consensus are obtained. The presented simulations verify the potential correctness of the main results. The related problem of systems with noises or stochastic network is the future work to be done.