IEEE Systems, Man and Cybernetics Magazine - October 2023 - 55

XX XX2222 ff !
12 f !
2
23
m
vv ,, ,, ,, ).vv vvvv R(/ )( )Tnn 1
+
1 nn n
2
2
1
!
#
and
2
12
X Rnn
2
!
XX is a positive semidefinite matrix. In denotes the
-
12
n-dimensional identity matrix.
Preliminaries
Graph Theory
Consider an undirected graph GI EAN
I {, ,, }N12N
= f and EI INN
=
= (, ,) in which
and the edge set, respectively, and A []a Rij
denotes the weighted adjacency matrix with a
(, ),Eij ! and a 0ij
!
=
with da ,Nij ij
3 # denote the vertex set
NN
#
= otherwise. We assume that a 0ii
ij 02 if
= .
The set of neighbors of agent i and the degree matrix are
defined as {:(, )}NI Eji jiN
= R ! i
then expressed as LD .A=- Denote Gg
the pinning matrix, where g
the information of the leader marked as 0, and g 0i
ij ,! there is a path from i to j.
= !! and D ()ddiag i
respectively. The Laplacian matrix is
= diag i () as
= otherwise.
The undirected graph G is called connected if, for
each pair (i, j) with
Assumption 1
The undirected graph G is connected and at least one
agent can receive the leader's information.
System Dynamics
Consider a class of MASs composed of N followers and one
leader. The ith agent is described by the following dynamics:
in which () ,(),uti
m
ii ii N
() () ()
i RRn
!!
(),
!
i
xt Ax tBut Ct i I
xt !! and ()t Rp
o =+ ~+
AB nm##RR and C Rnp
#
nn ,, !
(1)
~ ! are the
system state, control input, and external input, respectively,
and
are the unknown
constant matrices. Assume that the pair (A, B) is stabilizable.
The leader system is modeled by
xt Ax t00o ()
=
where () ! denotes the state of the leader.
xt Rn
Define the local consensus tracking error for agent i as
()
f =- +!
iij
xt xt gx tx t
j N
ta (()( )) (()( )).
|
i
From (1) and (2), the local error dynamics can be
described by
ffo =+ ++ ~
|
-+
!
j Ni
jj
~
ii ii ii
ij
() () ()(()( ))
(()( )).
tA td gBut Ct
aBut Ct
(3)
Nondistributed Nash Equilibrium
Herein we review the Nash equilibrium of a class of multiagent
differential graphical games and discuss the
ij ii
Formulation of Distributed Minmax Strategy
The following local performance index for agent i is considered
as:
Jr (( ), (),( ), (),( ))
ii tu tu tt tdt
= f~ ~
+3
#
cc ~~
=+ +ii
ii ii
i
-!!
f~
~
ff
i
T
ii
ij
au tR ut atRt() ().
~
ii i
ii
22
ii
j
T
ij j
i
j
|| (6)
jNN
ij
j
T
c~
ij
T
2
j
-ii
rt
ut ut tt
tQ td gu tR ut
ii ii i
-(5)
in
which
(( ), (),( ), (),( ))
() () ()(()()()( ))
() ()
T tR t
~
i ~i
October 2023 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE 55
()
(2)
Formulation and Stability Analysis
of the Distributed Minmax Strategy
In networked dynamical systems, the expected control input
for every agent is distributed, constructed only by using limited
local information. This makes the Nash solution unattractive,
prompting us to develop an alternative, i.e., a distributed
minmax strategy. Such a strategy is obtained by solving a
new multiagent differential graphical game.
d f)
r
# , XX represents that == () {( ):ut jt tj
r
12
$
ij ii ji
-!!
~~
T
+!!
au
tR ut atRt
jNN
ii ii ii
i
f~ ~
ff
ii +- () i ~ ()
|| () ij
-T
i
ii
j
T
ij
j
c~ ~
=
2
i
c~
ij
2
ii
j
T
~
~
ii
j
c
i 02
R 0ij
r )
is a scalar, and QR RR00 00ii i
22 2 ,, ,
~
~ $ are constant matrices.
The control goal of each agent is to determine
(( ), (),( ))
minmax
Vt u
u x ~x
x$t
i()
x$t
In light of [11], the Nash equilibrium is calculated as
{( ), ():}I with
ut ti ! N
t 00
))~
$$
i
i
t
i 02 if the agent i can receive ))
-1
ut dg RB Vt
td gR CV t
i
~ =+ -1
2ci
))~
2 ii i
1
i
where (( )) (( ),ff_Vt Vti
i
))
rr
i
i
i
i ii ii ii
ff
i
-+ ()
j N
|
!
ij jj
~
rt ut ut tt i0
r
ii ii ii
(( ), (),( ), (),( )) ,.
())
f~ ~
++
))
)) ))
-!
IN
+ ~
() () (( ))
() ()()
=2
1
ii
i
+
Td r fi
i
Td r (( ))
i fi
u () , ~x)
)
-i x x$t
Vt At dg Bu tC t
aBut Ct
+=
)) )
(( ))(( )( )()( )
()
-i () )t
x$
^h
and
Vti (( )) satisfies the following coupled HJB equations:
d r T
(4)
From (4) one can observe that the solutions for the
agents are coupled with each other. As discussed in [15]
and [16], coupled HJB equations (4) may, generally, not
have a set of solutions {( ()):
group of distributed optimal policies.
r
Vt i I}
) f
!
i iN that provides a
i ii
-+
xx
3
t
i()
ru ud(( ), (),( ), (),( )) .
ri ii ii i
#
t
-i
=
$$t
ij
$
and
T tR t
i
j()
rt ut ut tt
tQ tu tR ut
() ()
(( ), (),( ), (),( ))
() () () ()
ij
vv vv Rn
Tn
,, ,, ). For a vec t or
= (, ,, ), denote
23ff !
2mm R(/ )( )Tmm 1
12
+
vecv () (, ,,vv vv1
2
=
$
12
f
denotes the
two-norm for a matrix. For two symmetric matrices
X Rnn
difficulties in obtaining Nash solutions. Consider the
local performance function for agent i as follows:
r
where ut NN}
Jr(( ), (),( ), (),( ))
}, () {( ):
ii tu tu tt tdt
r
= f~ ~
+3
#
ii ii i
-fx
~x
fx xx ~x ~x x
IEEE Systems, Man and Cybernetics Magazine - October 2023

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