The Bridge - Issue 2, 2018 - 21

Distributed Stochastic Optimal Flocking Control for Uncertain Networked Multi-Agent Systems

Next, putting equation (8) into the HJI equation, it
can be represented as
with
(9)
with
defined as the weight estimation error of the NNbased identifier. According to [20] and equation (6),
the main objective of the NN-based identifier design
is to force the identification errors and the NN weight
estimation errors close to zero. To accomplish this,
a novel update law is developed for the NN-based
identifier as

Δφ J,i (z i(k),z -i(k)) being defined as
Δφ J,i (z i(k),z -i(k))=φ J,i (z i(k+1),z -i(k+1))-
φ J,i (z i(k),z -i(k)),and Δε J,i(k)=ε J,i(k+1)-ε J,i(k).
Similar to [20], the ideal stochastic optimal cost
function (8) can be approximated by using the critic
NN as
(10)

(7)

where α I,i,∀i=1,2,...,L is the tuning parameter
for NN-based identifier, and ||*|| denotes the
2-norm [20]. NDP-based Flocking Cost Function
Approximation
According to [20], the ideal stochastic optimal cost
function of a networked MAS two-player zero-sum
game, J *i(x i)∀i =1,...,L can be represented by using
a novel critic NN as
(8)
where W J,i ∈ℜ 1J,∀i=1,...,L is the target weights of
the critic NN, with l J being the number of hiddenlayer neurons, φ J,i (z i(k),z -i(k))∈ℜ 1J,∀i=1,...,L
denotes the activation function of the critic NN
with z i(k) being the system state of ith agent,
z -i(k)= { z j(k) } j∈Ni being the system state from
the neighbors of the ith agent, and ε j,i∈ℜ,∀i=1
represents the reconstruction error of the critic NN.
According to relevant NN literatures [14]-[15],
[18], the reconstruction error can be forced close
to zero while increasing the number of
hidden-layer neurons.

where W J,i (k),∀i=1,...,L denotes the estimated
weights of the critic NN and φ J,i (z i(k),z -i(k)),
∀i=1,...,L is selected from an activation function set
whose elements in the set are linearly independent
[19]. However, substituting the estimated stochastic
optimal cost function (10) into the HJI equation
(9) does not hold. To evaluate the effects from
inaccurate stochastic optimal cost function
estimation, an HJI equation estimation error is
introduced and defined as
(11)

Furthermore, to incorporate the effects from the
terminal constraint, the relevant terminal constraint
estimation error
is defined as
(12)
where
are the estimated networked MAS
states, obtained by using the available system
information (i.e.
) at time kT s.
Similar to recent NDP literatures [14]-[15], [18],
the HJI equation estimation error and terminal

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21
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Table of Contents for the Digital Edition of The Bridge - Issue 2, 2018
