Feature E[F i (x i )], E[G i (x i )], E[H i (x i )] r γ r γ r γ i i i i i i are networked MAS dynamics with ||G i (z i )||F ≤ G i,M and ||H i (z i (k))||F ≤ H i,M , where ||*|| F denotes the Frobenius norm, G i,M , H i,M are positive constants, and E { * } is the mean operator [20]. Figure 1. An undirected graph topology for MAS with five agents III. DISTRIBUTED STOCHASTIC OPTIMAL FLOCKING CONTROL A. Two-Player Zero-Sum Game Formulation Considering the original nonlinear continuous-time MAS dynamics given as x i(t)=f i,(x i)+g i(x i)u i(t)+h i(x i)d i(t), ∀ i=1,...,L (1) where f i(x i)∈ℜ m,g i(x i)∈ℜ m,h i(x i) ∈ℜ m, ∀i=1,...,L denote the heterogeneous nonlinear MAS dynamics, xi =[ piT viT ]T ∈ℜm,ui ∈ℜm,di ∈ℜm, ∀i=1,...,L are the MAS states, control inputs, and disturbance, where pi , vi represent ith agent's position and velocity, respectively, and is the number of agents. Next, based on the Assumption 1 and [20]-[21], the network-induced delays and packet dropout are incorporated into the MAS dynamics as x i(t)=f i,(x i)+γ i(t)g i(x i)u i(t-τ i)+ γ i(t)h i(x i)d i(t-τ i), (2) { with γ i(t)= t I mxm if control is r`eceived by the actuator at the time 0 mxm if control is lost, and is identity matrix, and τ i = τ i,sc + τ i,ca. According to [20], by discretizing equation (2) with network-induced delays and packet dropout, the networked MAS dynamics can be derived as E[z i (k+1)]=E[F i (z i (k))]+E[G i (z i (k))]u i (k) r γ r γ r γ i i i i i i +E[H i (z i (k))]d i (k), ∀i=1,2,...,L r γ i i with augment state z i(k)=[x (k) u (k-1) ... T u Ti (k-b) d Ti (k-1)] , T i THE BRIDGE T i (3) B. The Novel NN-based Identifier Design In diverse recent NDP literatures, e.g., [15]-[17], either partial or complete system dynamics of the networked MAS (i.e. F i( * ), G i( * ), H i( * )∀i=1,2,...,N are needed for attaining the optimal flocking control. However, due to uncertainties and modelling inaccuracy, the networked MAS dynamics are very difficult to be known beforehand. To circumvent this challenge, we propose a novel online NN-based identifier as follows. According to the universal function approximation property from neural network (NN), the networked MAS heterogeneous nonlinear system dynamics (4) can be represented as (4) where W F,i ∈ℜ 1Fx2m,W G,i ∈ℜ 1Gx2m,W H,i ∈ℜ 1Hx2m, ∀i=1,2,...,L represent the target weights, σ F,i (z i)∈ℜ 1F,σ G,i (z i)∈ℜ 1Gxm,σ H,i (z i)∈ℜ 1Hxm,∀i=1,...,L are the activation functions ε F,i ∈ℜ 2m, ε G,i ∈ℜ 2m, ε H,i ∈ℜ 2m,∀i=1,2,...,L denote the reconstruction errors, and lF,lG,lH, are the number of neurons. Based on relevant NN literature [14]-[15], the networked MAS system state, z i(k)∀i=1,...,L, can be approximated as (5) where WI,i (k)∈ℜ (1F+1G+1H)x2m,∀i=1,2,...,L is the estimated weights of the NN-based identifier at time kTs. Using (4) and (5), the networked MAS system state identification error can be derived as (6)

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