The Bridge - Issue 2, 2018 - 19

First, to ensure the execution of the three
flocking rules in an optimal manner, we propose an
innovative flocking cost function which includes three
parts, i.e., cohesion cost function, separation cost
function, and alignment cost function. Subsequently,
the distributed stochastic optimal flocking control
for networked MAS can be obtained by minimizing
the proposed flocking cost function. According to
classical optimal control theory [8], the minimized
flocking cost function can be obtained by solving the
Hamiltonian-Jacobi-Isaac (HJI) equation. However,
due to nonlinearity, the HJI equation is very
difficult to be solved mathematically. Inspired from
computational intelligence, a novel neuro dynamic
programming (NDP) technique is developed to
approximate the HJI equation solution [9]. The
authors in [10]-[11] developed an NDP-based
near optimal control for nonlinear continuous-time
systems and discrete-time systems respectively.
Recently, the authors in [12]-[17] extended the
policy or value iteration-based NDP scheme to attain
the optimal control strategies when considering
more general nonlinear systems. However, to
compute the optimal solutions, these iteration-based
NDP schemes require a significant large number
of iterations which is not applicable for real-time
nonlinear system applications.
In order to overcome this issue, the authors in
[18]-[19] introduced a time-based NDP approach
to solve the optimal control of both nonlinear
continuous-time systems and nonlinear discrete-time
systems, which makes use of the system historical
information instead of iteration-based information.
However, these schemes require the exact
knowledge of the system dynamics, which cannot be
known beforehand.
In order to fill this research gap, in this paper a
novel time-based distributed stochastic optimal
flocking control is developed for networked MAS.
The proposed methodology engages the

NDP technique with the innovative flocking cost
function which includes cohesion, separation, and
alignment aspects. The main contributions of this
paper include: 1) the networked MAS optimal
flocking design problem is effectively formulated by
developing the flocking cost function in an intelligent
way, 2) a novel time-based NDP scheme is
proposed to obtain the distributed stochastic optimal
flocking control of the networked MAS even in the
presence of network imperfections and disturbances,
and 3) the requirement of exact knowledge of the
MAS dynamics is relaxed.

II. BACKGROUND
Similar to [3]-[4], distributed agents in a MAS are
connected through a communication network whose
topology is expressed by using graph theory. The
network communication graph can be represented
as G={V,C}. Here, V={1,...,i,...,N} denotes a set of
vertices, where is the ith agent, and the relevant edge
set C is defined as C ⊆ {(i,j):i,j∈V,i ≠ j}, where
the edges indicate the potential communication
links and sensing capabilities among the distributed
agents in the MAS.
Moreover, an unweighted adjacency matrix is
introduced as A=[aij]ℜNxN, ∀i , j=1,2...,N, with the
element aij defined as aij=1 if (i,j)∈C, and aij= 0
if (i,j)∉C. Then, the communication graph can be
defined as connected when there is a data exchange
path connecting each pair of distinct vertices. The
MAS with a connected communication graph can
be defined as a connected MAS [1]. Similar to [1][2], distributed agents in the MAS are assumed to
have identical omni-directional communication and
sensing capabilities, which indicates that there is a
mutual communication within the connected MAS.
Mathematically, the adjacency matrix is symmetric,
i.e. A T=A , and then the communication graph is
undirected. An undirected communication graph
topology example is provided in Figure 1.

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