The Bridge - Issue 2, 2018 - 20

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)



Table of Contents for the Digital Edition of The Bridge - Issue 2, 2018

Contents
The Bridge - Issue 2, 2018 - Cover1
The Bridge - Issue 2, 2018 - Cover2
The Bridge - Issue 2, 2018 - Contents
The Bridge - Issue 2, 2018 - 4
The Bridge - Issue 2, 2018 - 5
The Bridge - Issue 2, 2018 - 6
The Bridge - Issue 2, 2018 - 7
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The Bridge - Issue 2, 2018 - 44
https://www.nxtbook.com/nxtbooks/ieee/bridge_issue2_2022
https://www.nxtbook.com/nxtbooks/ieee/bridge_issue1_2022
https://www.nxtbook.com/nxtbooks/ieee/bridge_issue3_2021
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https://www.nxtbook.com/nxtbooks/ieee/bridge_2020_issue3
https://www.nxtbook.com/nxtbooks/ieee/bridge_2020_issue2
https://www.nxtbook.com/nxtbooks/ieee/bridge_2020_issue1
https://www.nxtbook.com/nxtbooks/ieee/bridge_2019_issue3
https://www.nxtbook.com/nxtbooks/ieee/bridge_2019_issue2
https://www.nxtbook.com/nxtbooks/ieee/bridge_2019_issue1
https://www.nxtbook.com/nxtbooks/ieee/bridge_2018_issue3
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https://www.nxtbook.com/nxtbooks/ieee/bridge_2018_issue1
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