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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 6, JUNE 2009
Synchronized Output Regulation of
Linear Networked Systems
Ji Xiang, Wei Wei, and Yanjun Li
Abstract—This note addresses the problem of synchronized output regulation of linear networked system where all the nodes have their outputs
track signals produced by the same exosystem and the state of exosystem is
accessible only to leader nodes, while follower nodes regulate their outputs
via a distributed synchronous protocol. This problem can be decoupled into
two: one is the output regulation problem on the synchronous manifold and
the other is the stability problem of synchronous manifold. The proposed
synchronous protocol is independent of the network topology. The stability
of synchronous manifold is analyzed through the permissible eigenvalue
region and the requirements of information graph. Finally, a numerical example illustrates the efficacy of the presented results.
Index Terms—Information graph, networked systems, synchronized
output regulation.
I. INTRODUCTION
In recent years, much interest has been shown in synchronous
behaviors among coupled dynamical systems, such as synchronization
of coupled oscillators [1]–[3] and information consensus between
autonomous agents [4]–[6]. These systems are often called networked
systems in the sense that they are formed through interactions (the
edges in the network), of many smaller subsystems (the nodes in the
network), and are also referred to as large-scale systems or multi-agent
systems. Investigations into the fundamental theory of the emergence
of such coordinated behaviors have been widely reported. Particularly,
the method of master stability function is presented for the local
synchronizability of coupled nonlinear oscillators [1] with the tool
of Lyapunov exponent. The sufficient and necessary condition of
information consensus of linear integral multi-agent systems has
been independently developed in [4] and [5], i.e., there is a spanning
tree embedded in the underlying information graph. This condition
is also proved in the discrete-time domain, even in the dynamical
topology [7].
This note works on the linear networked systems with the focus
on “how to synchronize them onto a desired signal”, instead of on
“whether they are synchronized”. This can be thought as an extension
of output regulation on the networked systems, where all the nodes in
the network have their outputs track (or reject) the reference (or disturbance) signals produced by the same exosystem, for example, the multiple marine crafts navigating in a certain formation have to reject the
Manuscript received August 20, 2008; revised August 26, 2008 and January 12, 2009. First published May 27, 2009; current version published June
10, 2009. This work was supported by the Natural Science Fund for Distinguished Scholars of Zhejiang Province (R105341), the Natural Science Fund of
Zhejiang Province (Y106046), the Teacher Foundation of Zhejiang University
City College (581639), and the National Natural Science Foundation of China
(60704030). Recommended by Associate Editor Karl H. Johansson.
J. Xiang and W. Wei are with the Department of System Science and Engineering, College of Electrical Engineering, Zhejiang University, Hangzhou
310027, China (e-mail: [email protected]; [email protected]).
Y. Li is with the School of Information and Electrical Engineering, Zhejiang
University City College, Hangzhou 310027, China (e-mail: [email protected].
cn).
Color versions of one or more of the figures in this technical note are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TAC.2009.2015546
approximately same disturbances from wave, drift, currents and mean
wind forces of the slowly varying sea environment [8]. This problem
might firstly have been studied by Xiaoping [9], where the subsystem
and exosystem of every node both have the same interactions with all
the other nodes in the network which is a complete graph. The problem
similar to ours is the decentralized output feedback control of interconnected systems, where the primary focus of most researchers is, however, to cope with the negative effect of the interconnections [10]–[12].
The most relevant to our work are the recent works of Ren [13] and
Ren et al. [14], where the time-varying reference and model reference
tracking problem of multi-agent systems were studied respectively. In
the former the information of state derivative of neighboring nodes is
used and in the latter the reference model is the same as the node subsystem. Both of them can be thought of as the special cases of output
regulation if the time-varying reference signal can be produced by a
linear system.
In this note, the considered scenario is that the tracked signal or rejected disturbance is the same for all the nodes but only partial nodes
have the state information of exosystem, which we call leader nodes.
The remained nodes, the so-called follower nodes, should depend on
the information communication with its neighboring nodes to force
their output “synchronize” on the desired signal. In this sense it is called
synchronized output regulation. Many real systems can be modeled as
such, for example: a group of soldiers or tanks parading on the ceremony ground where the route is unavailable to those in the middle of
alignment and they have to decide their actions by its neighboring ones;
a line of multiple mobile robots among which only the leader robots
have the information of desired trajectory while the other robots follow
their leader by state error information obtained from the vision sensor.
Our approach is to construct a distributed synchronous protocol to estimate the exosystem state for the follower nodes. Under this framework, the synchronized output regulation problem can be decoupled
into two: the output regulation problem on a synchronous manifold and
the asymptotical stability problem of the synchronous manifold.
Our main contribution is to give a synchronous protocol dependent
on solutions of the output regulation problem on the synchronous manifold but independent of information graphs, and then to present conditions the information graph should satisfy for the asymptotical stability
of synchronous manifold.
II. PROBLEM FORMULATION
A. Information Graph
A networked system can be formulated in an information graph G to
depict the information flow between nodes.
A digraph G is a pair of (N ; E ), where N = f1; 2; 1 1 1 ; N g is a node
set and E N 2N is an edge set of ordered pair of nodes, called edges.
The edge (i; j ) 2 E if the node j can receive the information of node
i. For the edge (i; j ), the node i is called the parent node while j is
called child node. In contrast to digraph, the edge (i; j ) in undirected
graph is unordered and denotes that the node i and node j can obtain
information from each other. All the parent nodes of node i compose
the in-neighboring set Nin (i) while all the child nodes compose the
out-neighboring set Nout (i). For an undirected graph, Nin = Nout
for all nodes.
A directed path of digraph is a sequence of edges with form
(i1 ; i2 ); (i2 ; i3 ); 1 1 1. A tree Gt = (Nt ; Et ) is a graph where every node
has exactly one parent node except for one node, so-called root node,
which has no parent node but has a directed path to every other node.
If there is a directed path from node i to node j , then node i can reach
node j . The graph Gs = (Ns ; Es ) is a subgraph of G if Ns N and
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Es E \ (Ns 2 Ns ). The tree Gt is a spanning tree of graph G iff Gt is
a subgraph of G with jNt j = jN j, where jS j denotes the size of set S .
The adjacency matrix A = [aij ] 2 Rn2n of digraph G with node
set N = f1; 2; 1 1 1 ; ng is defined as aij > 0 if (j; i) 2 E while
aij = 0 if (j; i) 62 E . The Laplacian matrix L = 3[lij ] 2 Rn2n
6 j , and lii = j 6=i aij . For a bidirecis defined as lij = 0aij if i =
tional graph, both the adjacency matrix and the Laplacian matrix are
symmetric. Given a matrix S 2 Rn2n , the digraph of S , denoted by
G (S ), is a graph whose Laplacian matrix L satisfies L = S .
1337
T
where X = (x1T ; xT2 ; 1 1 1 ; xTN ) 2 RNn is the concatenated state
vector. Our control law is designed as
ui = Kxi + F w;
ui = Kxi + F w^i = Kxi + F w + F we
B. Model
x_ i = Axi + Bui + P w
w_ = Sw
; i 2 IN
ei = Cxi + Qw
A2
A3
S w^i +
j =1
aij H (xj 0 xi ); i 2
BF
(P + BF )w)
I(N 0l) S We 0 0 I(N 0l) L H X
We
+ (1lN
(II.5)
where IN is the N 2 N identity matrix, and 1ls is the column vector of
dimension s with all the elements being 1. The exosystem is still
w_ = Sw
(II.6)
T
In our study, only parts of nodes are able to measure the state of
exosystem, w . We call such nodes leader node while the other nodes
follower node. The information available for every node is the state of
itself, xi , and the states of its parent nodes, xj (j 2 Nin (i)), while for
each leader node, it has additional state information w besides information every node has. The regulation problem is to design a suitable
distributed control law ui so that the networked dynamical system is
stable with ei ! 0 for all i 2 IN . For a leader node, this is a traditional output regulation problem; while for a follower node, this can
not be directly solved and one possible way to solve it is to rely on the
received information from its parent nodes.
Without loss of generality, it is assumed the first l nodes are leader
nodes with 1 l N . Let Il be the index set of leader nodes and Il
the index set of follower nodes. Our method is, for the follower nodes,
using the state errors with their parent nodes to estimate the exosystem
state w
N
0
I(N 0l)
T
Np
and the collective tracking error E = (e1T ; e2T ; 1 1 1 ; eN
) 2R
is
(S ) 2 C+ ;
pair (A; B ) is stabilizable;
A P
is detectable.
pair [C Q],
0 S
w^_ i =
W_ e =
(II.1)
where xi 2 Rn is the state vector, ui 2 Rm is the input vector,
and w 2 Rq models both the reference signal to be tracked and the
disturbance to be rejected, and ei 2 Rp is the tracking error. IN =
f1; 2; 1 1 1 ; N g is the index set and is also the node set of underlying
information graph G . A, B , C , Q, P , and S are the constant matrices
of the appropriate dimensions satisfying the following three standard
assumptions of linear regulator theory [15].
A1
(II.4)
Substituting (II.4) into (II.1) and then combining with the estimation
error system (II.3) yield the following collective dynamics of closedloop system:
X_ = (IN (A + BK )) X +
Assume that the linear networked system consists of N nodes with
the same dynamics. The reference signal the node subsystem should
track or the disturbance it should reject is the same for every node. The
dynamics of each node subsystem can be described as
i 2 Il
; i 2 Il .
Il
(II.2)
where aij is the element of Adjacency matrix A and H 2 R(q2n) is
the distributed synchronous protocol gain (SPG) to be determined later.
T
Define the estimation error vector by We = (weT ) 2 R(N 0l)q with
we = w^i+l 0 w for i = 1; 1 1 1 ; N 0 l. And the dynamics of estimation
error is
E = (IN C )X + (1lN Q)w:
(II.7)
C. Synchronous Manifold
The following coordinate transformation is introduced:
Xs
Xt
Es
Et
1
0
1
0
=
01lN 01 IN 01
=
01lN 01 IN 01
In X
(II.8)
Ip E
(II.9)
where Xs = x1 corresponds to the dynamics on the synchronous manifold and Xt 2 R(N 01)n denotes the transversal error with respect
to the synchronous manifold. Application of (II.8) to the closed-loop
system (II.5) yields
X_ s = (A + BK )Xs + (P + BF )w
X_ t = (IN 01 (A + BK )) Xt
(II.10a)
0
BF We
(II.10b)
I(N 0l)
(II.10c)
W_ e = I(N 0l) S We 0 (L2t H )Xt
where L2t 2 R(N 0l)2(N 01) is the block matrix of Laplacian matrix
L1
with L1 2 Rl2N . Correspondingly,
L such that L =
L2s L2t
+
the collective tracking error (II.7) becomes,
Es = CXs + Qw;
Et = (IN 01 C )Xt :
(II.11a)
(II.11b)
The considered output regulation problem can now be decoupled
into two problems:
P1): the output regulation problem on the synchronous manifold,
Es = CXs + Qw ! 0
(II.12)
with
X_ s = (A + BK )Xs + (P + BF )w
w_ = Sw:
(II.13)
P2): the stability problem of synchronous manifold
W_ e = I(N 0l) S We 0
0 I(N 0l)
L H X
(II.3)
Xt ! 0
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(II.14)
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X
W
0
I 01 A BK X
BF W
I( 0 ) S W 0 L2 H X :
_
t = ( n
_
e =
(
N
+
))
(
e
l
t +
I
)
t
e
(II.15)
t
E
H
H
w
W
X
;
S A P B
(II.16)
C Q
A BK is Hurwitz;
2) find a matrix K such that A
3) obtain the matrix gain F
0K .
BF , to find
Our concerns can be reduced to: Given A and B
the qualifiable SPG H and Laplacian matrix L, if they exists, such that
X ! with the system (II.15) is stable. That is, the synchronous
5
=
0 =
5 +
5 +
+
0
;
K =
+
= 0
5
F =
K
0
t
TL2 2T 01
t
Remark 1: The requirement (II.14) in general is stronger than the
standard output regulation requirement t ! 0. Using (II.14) instead
of t ! 0, although this unavoidably results in some conservation,
will bring convenience and easiness in designing SPG . Moreover, it
can be seen in next section that a qualifiable satisfying (II.14) does
not require more assumptions than present ones.
Remark 2: The intention of the system (II.2) is to provide the estimation of exosystem state for the follower nodes, but e ! 0 is
not necessary for the asymptotical stability of synchronous manifold,
which only requires t ! 0.
The problem P1) can be straightly solved by the well-known linear
regulator theory [15]:
1) solve the matrix equations over the variables (5 0)
E
Proof: Let = ( ) t02 and = ( ) e where
is the transformation matrix by which 2t2 is converted to the Jordan
canonical form
X
with
manifold is asymptotically stable.
L
Xt01
Xt02
=
L?2t In
0 L2 I X
t
(III.1)
t
n
L
L
L L
I( 01) A X 01 0 L2? 2 B W
I( 0 ) A X 02 0 L2 2 B W
I( 0 ) S W I( 0 ) H X 02
where the matrices L2? 2 2 R( 01)2( 0 ) and L2 2 2 R(
X 01
X 02
W
_
_
t
=
l
t
=
N
l
e =
N
l
_
K
t
F
(
t
F)
N
l
t
K
t
e +
l
t
N
e
e
t
l
t
N
(III.2a)
(III.2b)
(III.2c)
0l)2(N 0l)
L2? ( 01)2( 0 ) :
L2
I 0
Lemma 1: If the matrix 0L2 2 is Hurwitz with all eigenvalues being
M2201B M1 with positive definite
real, then there exists a SPG H
2
and M2 2 R
satisfying
matrices M1 2 R
M1 A A M1 <
(III.3)
M2 S S M2 such that the system (II.15) is stable with X ! .
t
t
0 l
t
=
t
N
N
t
=
n
n
q
K +
+
T
q
T
F
0
0
t
w
w_
l
l
T
K
0
= ([0; ]
I )X 2 R and
= 0 such that x_ = A x + B W ,
1Or else, there is a projection in state space, x
= ( I )W 2 R with L
= Sw , which is bounded but does not converge to zero. This implies that
(II.14) is infeasible.
2If l
, X is null, and delete the first equation (III.2a).
=1
T
with
r N 0 l:
(III.4)
; ; ;N l
r
N l
N l
x A x 0 B w ; i ;111;N 0l
(III.5)
w H x Sw
where x 2 R and w 2 R are the state vector of the subsystem
corresponding to the eigenvalue . Consider the Lyapunov funcx M1 x w M2 w . Its time derivative along the
tion V
_i =
K i
_ i =
q
i
T
= i
= 1
i
n
i
F i
i
i +
i
T
i i
i +
i
system (III.5) is
V x M1 A
T
_
K +
= i
A M1 x w M2 S S M2 w T
K
T
i + i (
T
+
) i
0
(III.6)
which means that the solution of system (III.5) is bounded. Noting
that i 0 means that F i 0, by LaSalle’s theorem, it can
be obtained that i F i ! 0 as ! 1. Furthermore, we have
0
F)
e ! 0. Rewrite the system (III.2a) as
t2 ! 0 and ( x
X
=
t
Bw
t
x ;B w
I B W
I( 01) A X 01 0 L2? 2 I I B W
K
l
t
((
t
from which it follows that t01 ! 0 because
( F)
e ! 0.
C2: Some Jordan blocks i are non-diagonal.
In this case, the system (III.5) becomes
X
I B W
A
K
F)
e)
(III.7)
is Hurwitz and
J
x A x 0 J B w ; i ;111;r
(III.8)
w H x Sw
where x 2 R
and w 2 R
are the state vector of the
subsystem corresponding to the Jordan block of J with s being
the size of J . Assume that the size of Jordan block J is s
and the form of J is
J :
K i
_ i =
(
i +
ns
i
F)
i
i
= 1
i
qs
i
i
i
i
satisfy
L2? 2
L2 2
i
J
_i =
where 2?t 2 R(l01)2(N 01) denotes the orthogonal complement matrix with full row rank of matrix 2t , i.e. 2t 2?tT = 0. Application
(III.1) to the system (II.15) yields2
r)
L
J _
Assume that 2t is of full row rank.1 Then introduce the following
coordinate transformation:
J1 ;J2 ; 1 1 1 ; J ;
= diag(
T IW
W
L
The following two cases are considered here, where all the eigen0.
values of 2t2 are denoted by i , = 1 2 1 1 1
C1: All the Jordan blocks i are diagonal, i.e., =
0 and
i =
i.
The subsystem consisting of (III.2b) and (III.2c) can be rewritten
as the following ( 0 ) decoupled subsystems
X 01
III. STABILITY OF SYNCHRONOUS MANIFOLD
T IX
i = 2
i
i
i =
i
1
0
i
x and w respectively as x
x 1; x 2
w ; w 2 yields
x 1 A x11 0 B w 1 0 B w 2
w 1 H x 1 Sw 1;
x2 A x20 B w2
w 2 H x 2 Sw 2:
Partition
i
col( i1
i )
i
_i
=
_ i
K
=
_i
i
=
_ i
i )
i = col( i
i
+
i
K i
=
i
F i
i
+
and
F i
i =
(III.9)
F i
i
w
(III.10)
x
By C1, it is known that the system (III.10) is stable with i2 ! 0
and F i2 ! 0. For the system (III.9), considering the
Lyapunov function = Ti1 1 i1 + i iT1 2 i1 , we have
Bw
V x M x w M w
(III.11)
V 0x 1 m3 x 1 x 1 M1 B w 2
A M1 <
where m3 is a positive scalar satisfying M1 A
0m3 I . The inequality (III.11) means that the system (III.9) is
bounded and the trajectory of x 1 will in finite time enter the
region
fx 1 kx 1 k kM1 k= m3 kB w 2 kg. Since
B w 2 ! , x 1 ! , on which B w 1 ! .
T
_
i
i
T
+ 2 i
F i
K +
n
i
=
F i
0
i
i
:
i
0
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(
2
F i
)
0
F i
T
K
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1339
This result can be easily extended to the situation of si > 2.
0 and (I BF )We
0.
Thus, it can be concluded that Xt02
0.
Subsequently, Xt01
Thus, the proof is completed.
By the linear regulator theory, the inequalities (III.3) are always feasible, therefore, the proposed SPG H under the requirement of (II.14)
0 does not need any more assumptions than those
instead of Et
needed in standard output regulation, as shown in the above Remark 1.
The following result gives the condition by which the estimation
error We will converge to zero. This is not guaranteed by Lemma 1.
Lemma 2: The stable system (III.5) is asymptotically stable if and
only if the matrix pair (BF ; S ) is detectable.
i
0 and BF w
0. At
Proof: (Sufficiency) By Lemma 1, x
i
the steady state, the second equation of (III.5) becomes
!
!
!
!
!
i ;
w
_ i = S w
!
i = 0:
with BF w
(III.12)
!
By the detectable property, it follows that w
i
0.
(Necessity) By the way of contradiction, assume that the matrix pair
(BF ; S ) is not detectable. Since (S ) C + , there is a nonzero eigenvector such that BF = 0 and S = 0. Thus, the closed-loop system
(III.5) has a 0 eigenvalue associated with the eigenvector [0; ]T . This
contradicts to the asymptotical stability of system (III.5).
+ , the detectability is equivalent to
Due to (S ) C
2
2
rank
BF
S
=q
0
0
2R
6 2C
A )x 0 0BB BB
= (I H )
x + (I S )w
w
_ i
2
K
i
i
i
F
i
2
F
i
2R
2R
2n
2q
where x
i
and w
i
are the state vector of the subsystem
corresponding to the eigenvalue pair i ji . What we need is that the
system (III.14) is asymptotically stable, but its characteristic matrix
6
(III.13)
which is very easy to test. Noting that BF = B (0 K 5) and that
K is in great degree of design freedom, this condition can be almost
always true. Since the asymptotical stability of subsystem (III.10) implies the asymptotical stability of the whole subsystem corresponding
to the non-diagonal Jordan block matrix, this sufficient and necessary
condition can work for the asymptotical stability of the system (II.15).
Lemma 3: Given the Hurwitz matrix L2t2 with all eigenvalues
being real and the SPG H = M201 BFT M1 with M1 , M2 from (III.3),
the system (II.15) is asymptotically stable if and only if the rank condition (III.13) holds.
Noting that the rank condition (III.13) is independent of the information graph, in order to judge whether the system (II.15) is asymptotically stable, it is enough to judge the asymptotical stability of system
(III.5) with any positive i > 0.
On the other hand, the scaled H by a positive factor is still a qualifiable synchronous protocol (M1 , M2 is also the solution of (III.3)).
Henceforth, it is assumed, without loss of generality, that the solution
of output regulation problem P1) is such that the the system (III.5)
is asymptotically stable under the synchronous protocol H for all
+
.
i ; The above results are based on the condition that L2t2 does not have
complex eigenvalues. We now turn our attention to the case that L2t2
.
has some complex eigenvalues, i = i ji
Certainly, we hope that the above designed matrix H independent
of the information graph remains qualifiable in this general case. Different from the subsystem (III.5), the considered subsystem in this case
will be
x
_ i = (I2
Fig. 1. (Colored online) Permissible region with three cases of " and eigenvalues locations of L
(red stars).
i
F
i
F
wi
(III.14)
AK
Ctest =
0
H
0
0
AK
0
H
0 B 0 B
B
0 B
i
i
F
F
S
0
i
F
i
F
0
(III.15)
S
is in general not Hurwitz. Noting that i = 0 is always a feasible
0
+
for all i ; , there at least exists a
solution of (Ctest )
Br (i ; ) of zero such that the permissible
neighboring domain i
0
can be expressed
range of eigenvalues of L2t2 to make (Ctest )
+
; Br (; ) . With and fixed,
by = = + j : the bound Br can be obtained by calculating the eigenvalues of Ctest .
Fig. 1 shows an illustration of , which is a transverse band dependent
on the scaled factor around the right half real axis.
In summary, the following result is established straightforwardly.
Theorem 4: The synchronized output regulation problem is solvable
for a networked system if the eigenvalues of L2t2 are all located in the
domain of .
f
2C
2R
j j
2C
2R j j
g
IV. REQUIREMENTS OF INFORMATION GRAPH
The eigenvalues of L2t2 depend on the underlying information
and
graph, however, it is difficult to find the relation between
the permissible eigenvalue region . In this section we, with some
special cases, discuss the information graph conditions such that the
eigenvalues of L2t2 locate in the domain of .
Lemma 5: All the eigenvalues of L2t2 locate on the right half open
plane if and only if for every follower node there is at least a leader
node able to reach it.
Proof: Concentrating all the leader nodes into one node with
the edges from follower nodes to leader nodes and the edges between
leader nodes to be dropped yields a new graph, new . It is clear that the
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G
G
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N l
v
nodes of Gnew consist of the 0 follower nodes and a new concentrated node, which is denoted by new . The Laplacian matrix of Gnew
has the form of
followernodes
v
Lnew =
0
0
L2t2
b
(IV.1)
where the elements of vector b 2 RN 0l corresponds to the number
of edges that the follower node received from the leader nodes. By the
well-known result [4], [5], the following two statements are equivalent:
S1) the new graph Gnew thus built contains a spanning tree rooted at
the node new ; S2) all the eigenvalues of new , except for one unique
0 eigenvalue, have positive real parts. The proof is completed.
For an undirected graph, 2t2 is a symmetric matrix, and therefore
has all eigenvalues real. Noting that the permissible range always
contains the positive real axis, we have the following.
Theorem 6: The synchronized output regulation problem is solvable for a networked system where the follower nodes are connected
bidirectionally if and only if there is, for every follower node, at least
a leader node having a directed path to reach it.
For a general networked system, the above result is not true. Noting
the determinant formula of block matrix
v
L
L
B = det(A) det(D 0 BA01 C )
det CA D
(IV.2)
where A; B; C; D are matrices of appropriate dimensions, it can be
concluded that
Br (; ) = 1=Br (; 1)
(IV.3)
According to this property, the following result can be obtained.
Theorem 7: If r ( 1) is a monotone unbounded function with
respect to 2 (0 1), then there exists a positive scalar such that
solves the synchronized output regulation problem for the
the SPG
general networked system where for every follower node there is at
least a leader node able to reach it.
Proof: Complex eigenvalues of 2t2 are denoted by i = i 6
i , = 1 1 1 1 c with c being the number of complex eigenvalue
pair. Let 1 = maxi=1;111;n ( i i ), 2 = maxi=1;111;n i , and
3 = mini=1;111;n
i.
Firstly consider the case that r ( 1) is monotone decreasing.
In this case, r ( 1)
! 1 with ! 0. Thus, there exists a 3
such that r ( 3 1) = 1 3 . Take = 2 3 . It can be seen that
) 1 for all 2 (0 2 ]. This means
r ( ) = ( r ( 1)
that all the eigenvalues of 2t2 are located in the domain of .
Along the same line, the case of monotone increasing r ( 1)
can also be proved with = 3 3 . By Theorem 4, the proof is
completed.
Remark 3: Theorem 7 shows the case where the condition presented
in Lemma 5 is also qualifiable for the synchronized output regulation problem over a general information graph. But unfortunately the
r ( 1) is in general not a monotone function, e.g., in Fig. 1 there
exists at least a threshold point after which the monotone direction is
changed.
H
B ; =
;
L
j i ; ; n
n
= B ; =
B ; =
B ;
=
B ; B ; = L
=
;
B ; =
B ; =
V. EXAMPLE
Consider a networked system (II.1) with 100 nodes and 400
non-weighted directed edges selected randomly. In the simulation
performed here four leaders are selected randomly such that the
condition is satisfied that each follower node has at least a leader node
to reach it via a directed path, which ensures that all the eigenvalues
Fig. 2. (Color online) Histories of outputs of node subsystems and exosystem.
L
of 2t2 locate at the open right half plane. The node subsystem has
the following parameters [16]
5 0 01 6
0
0
01 0 0
A = 1 0 1 0 ; B = 01 ; P =
1 01 0 1
0
C = [0 0 1 1]; Q = [00:5 0]; S = 001 01 :
02
3
0 1
01 0
0 1
Solve the output regulation problem on the synchronous manifold
(II.16) to obtain
K = [116:000 0 39:514
F = [035:576 78:091]:
0 16:000
156:000]
Solve linear matrix inequalities (III.3) to obtain
74:0535
15:1128 12:2946 0162:5490 :
Numerical calculations of Br (; ) with the step length 1e 0 4
for three cases = 1; 0:1; 0:01 are executed under the condition
Re((Ctest)) < 00:05.3 Fig. 1 presents the regions of of three
cases and the distribution of eigenvalues of L2t2 with four nodes
H=
66:2289
0145:3738
06:8850 05:6011
selected as leader nodes. The tracking performance of networked
system is illustrated in Fig. 2. It can be seen that although only four
leader nodes have the state information of exosystem, the whole
network system is able to achieve a successful tracking performance.
VI. CONCLUSION
In this note, the synchronized output regulation problem of linear
networked systems, that every node has its output track the signal produced by the same exosystem, has been considered under the scenario
where only leader nodes have the information of the state of the exosystem. A distributed synchronous protocol is given for the follower
nodes to regulate its output by the estimation of the state of exosystem.
We have shown that this problem can be decoupled into two: one is
the standard output regulation problem on the synchronous manifold
and the other is the asymptotical stability problem of the synchronous
3In order to avoid the numerical error, the threshold
here.
Authorized licensed use limited to: Zhejiang University. Downloaded on June 18, 2009 at 22:41 from IEEE Xplore. Restrictions apply.
00.05 instead of 0 is used
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 6, JUNE 2009
1341
Activity Invariant Sets and Exponentially Stable
Attractors of Linear Threshold Discrete-Time
Recurrent Neural Networks
manifold. The feedback gains and synchronous protocol gain depend
on the output regulation problem. The permissible region of the eigenvalue distribution to ensure the stability of synchronous manifold is a
transversal band along the right real axis. A numerical example illustrates the efficacy of the presented theoretical analysis.
A natural extension of this work will be the error feedback case
which, in the classic output regulation problem, is solvable if the full information case is solvable. The difficulty is how to rationally formulate
the error feedback case in the distributed sense. One possible formulation is that leader nodes have the error information e = Cx + Qw
while follower nodes the sum of weighted output error with respect to
aij (Cxi 0 Cxj ). However, the trivial
its neighboring nodes e =
extension of the framework developed in this note is infeasible for such
a formulation. Whether there exists a framework in which the similar
results as in the classic output regulation problem still hold is an interesting future work.
Abstract—This technical note proposes to study the activity invariant
sets and exponentially stable attractors of linear threshold discrete-time recurrent neural networks. The concept of activity invariant sets deeply describes the property of an invariant set by that the activity of some neurons
keeps invariant all the time. Conditions are obtained for locating activity invariant sets. Under some conditions, it shows that an activity invariant set
can have one equilibrium point which attracts exponentially all trajectories starting in the set. Since the attractors are located in activity invariant
sets, each attractor has binary pattern and also carries analog information.
Such results can provide new perspective to apply attractor networks for
applications such as group winner-take-all, associative memory, etc.
REFERENCES
Index Terms—Activity invariant sets, discrete-time recurrent neural networks, exponentially stable attractors, linear threshold.
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Lei Zhang, Zhang Yi, Stones Lei Zhang, and Pheng Ann Heng
I. INTRODUCTION
In recent years, linear threshold recurrent neural networks (LT
networks) have been studied by many authors [7], [9], and [16].
The linear threshold transfer function is an unbounded function
with binary pattern. It has been used to model many cortical neural
networks [1]–[4]. Networks endowed with this transfer function form
a class of hybrid analog and digital networks that can implement a
form of hybrid analog-digital computation. Since the linear threshold
transfer function is essentially nonlinear, complex dynamic properties
may exist in such networks [12] and [17]–[19]. LT networks have
been got many applications, such as associative memory [10], [11],
winner-take-all [5], group selection [6], [14], feature binding [13], etc.
The main contributions of this technical note consist of two parts.
We fist present the concept of activity invariant set for discrete-time LT
networks. Discrete-time recurrent neural networks can provide direct
algorithms and easily be implemented by digital hardware [15]. Moreover, invariant sets play important roles in dynamics study of recurrent
neural networks. An invariant set restricts trajectories starting from the
set stay in the set. The concept of activity invariant set more deeply
describes the dynamic properties of invariant sets: the activity of some
neurons keeps invariant during the time evolution. Thus, neurons can
be divided into two classes by active neurons and inactive neurons. We
will derive conditions for locating activity invariant sets.
Manuscript received March 23, 2008; revised October 13, 2008. First published May 27, 2009; current version published June 10, 2009. This work was
supported by the Chinese 863 High-Tech Program under Grant 2007AA01Z321.
Recommended by Associate Editor C.-Y. Su.
L. Zhang is with the Department of Computer Science and Engineering,
The Chinese University of Hong Kong, Shatin, N.T., Hong Kong (e-mail:
[email protected]).
Z. Yi is with the College of Computer Science, Sichuan University, Chengdu
610065, China (e-mail: [email protected].)
S. L. Zhang is with the School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China
(e-mail: [email protected]).
P. A. Heng is with the Department of Computer Science and Engineering, The
Chinese University of Hong Kong, Shatin, N.T., Hong Kong and the School
of Computer Science and Engineering, University of Electronic Science and
Technology of China, Chengdu 610054, China (e-mail: [email protected].
hk).
Color versions of one or more of the figures in this technical note are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TAC.2009.2015552
0018-9286/$25.00 © 2009 IEEE
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