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Accepted Manuscript
Solvability conditions and design for H & H2 almost state
synchronization of homogeneous multi-agent systems
Anton A. Stoorvogel, Ali Saberi, Meirong Zhang, Zhenwei Liu
PII:
DOI:
Reference:
S0947-3580(17)30260-1
https://doi.org/10.1016/j.ejcon.2018.08.001
EJCON 281
To appear in:
European Journal of Control
Received date:
Revised date:
Accepted date:
17 July 2017
25 July 2018
6 August 2018
Please cite this article as: Anton A. Stoorvogel, Ali Saberi, Meirong Zhang, Zhenwei Liu, Solvability
conditions and design for H & H2 almost state synchronization of homogeneous multi-agent systems,
European Journal of Control (2018), doi: https://doi.org/10.1016/j.ejcon.2018.08.001
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ACCEPTED MANUSCRIPT
Solvability conditions and design for H? & H2 almost state
synchronization of homogeneous multi-agent systems
a School
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Anton A. Stoorvogelb,?, Ali Saberia , Meirong Zhangc , Zhenwei Liua
of Electrical Engineering and Computer Science, Washington State University, Pullman, WA, USA
of Electrical Engineering, Mathematics and Computer Science, University of Twente, P.O. Box
217, Enschede, The Netherlands
c School of Engineering and Applied Science, Gonzaga University, Spokane, WA, USA
b Department
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Abstract
This paper studies the H? and H2 almost state synchronization problem for homogeneous multi-agent systems with general linear agents affected by external disturbances
and with a directed communication topology. Agents are connected via diffusive fullstate coupling or diffusive partial-state coupling. A necessary and sufficient condition
is developed for the solvability of the H? and H2 almost state synchronization problem.
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Moreover, a family of protocols based on either an algebraic Riccati equation (ARE)
method or a directed eigenstructure assignment method are developed such that the
impact of disturbances on the network disagreement dynamics, expressed in terms of
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the H? and H2 norm of the corresponding closed-loop transfer function, is reduced
to any arbitrarily small value. The protocol for full-state coupling is static, while for
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partial-state coupling it is dynamic.
Keywords: Multi-agent systems, H? and H2 almost state synchronization,
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Distributed control
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1. Introduction
Over the past decade, the synchronization problem of multi-agent system (MAS)
has received substantial attention because of its potential applications in cooperative
control of autonomous vehicles, distributed sensor network, swarming and flocking and
? Corresponding
author
Email addresses: A.A.Stoorvogel@utwente.nl (Anton A. Stoorvogel), saberi@eecs.wsu.edu
(Ali Saberi), zhangm@gonzaga.edu (Meirong Zhang), zhenwei.liu@wsu.edu (Zhenwei Liu)
Preprint submitted to Elsevier
August 14, 2018
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others. The objective of synchronization is to secure an asymptotic agreement on a
common state or output trajectory through decentralized protocols (see [1, 2, 3, 4] and
references therein).
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State synchronization inherently requires homogeneous MAS (i.e. agents have identical dynamics). Most works have focused on state synchronization based on diffu-
sive full-state coupling, where the agent dynamics progress from single- and double-
integrator dynamics (e.g. [5], [6], [7], [8], [9]) to more general dynamics (e.g. [10],
[11], [12], [13]). State synchronization based on diffusive partial-state coupling has
also been considered (e.g. [14], [15], [10], [16], [17], [18], [19]).
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Most research has focused on the idealized case where the agents are not affected
by external disturbances. In the literature where external disturbances are considered,
?-suboptimal H? design is developed for MAS to achieve H? norm from an external
disturbance to the synchronization error among agents less than an, a priori given, bound
?. In particular, [15], [20] considered the H? norm from an external disturbance to the
output error among agents. [21] considered the H? norm from an external disturbance
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to the state error among agents. These papers do not present an explicit methodology
for designing protocols. The papers [22] and [23] try to obtain an H? norm from a
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disturbance to the average of the states in a network of single or double integrators.
By contrast, [24] introduced the notion of H? almost synchronization for homoge-
neous MAS, where the goal is to reduce the H? norm from an external disturbance to
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the synchronization error, to any arbitrary desired level. But it requires an additional
layer of communication among distributed controllers, which is completely dispensed
in this paper. This work is extended later in [25], [26], and [27]. The paper [27], where
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heterogeneous MAS are considered, provides a solution for the case of right-invertible
agents with the addional objective beyond output synchronization that the agents track
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a regulated signal given to some or all of the agents. Although homogeneous MAS,
which are considered in this paper, are a subset of heterogeneous MAS, the results of
[27] cannot be directly applied to the case of full-state coupling since the agents are
not right-invertible. Secondly, the results for synchronization without regulation cannot
be obtained from results obtained for regulated synchronization. Thirdly, we consider
state synchronization instead of output synchronization in both full- and partial-state
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coupling. Finally, by restricting to homogeneous networks more explicit designs can
be obtained under weaker conditions.
In this paper, we will study H? almost state synchronization for a MAS with full-
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state coupling or partial-state coupling. We will also study H2 almost state synchronization, since it is closely related to the problems of H? almost state synchronization.
In H? we look at the worst case disturbance with the only constraints being the power,
while in H2 we only consider white noise disturbances which is a more restrictive
class. In both cases, disturbances or noises are restricted in the process, not in the
measurement. Our contribution in this paper is three-fold.
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? We obtain necessary and sufficient conditions for H? and H2 almost state synchronization for a MAS in the presence of external disturbances,
? We develop a protocol design for H? and H2 almost state synchronization based
on an algebraic Riccati equation (ARE) method,
? We develop a protocol design for H? and H2 almost state synchronization based
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on an asymptotic time-scale eigenstructure assignment (ATEA) method for the
full-state coupling case, and on the direct eigenstructure assignment method for
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the partial-state coupling case.
It is worth noting that our solvability conditions and protocol designs are developed for
a MAS associated with a set of network graphs. Specifically, only rough information
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of a network graph is utilized.
1.1. Notations and definitions
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Given a matrix A ? CmОn , A0 denotes its conjugate transpose, k Ak is the induced
2-norm. A square matrix A is said to be Hurwitz stable if all its eigenvalues are in
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the open left half complex plane. A ? B depicts the Kronecker product between A and
B. In denotes the n-dimensional identity matrix and 0n denotes n О n zero matrix;
sometimes we drop the subscript if the dimension is clear from the context. Given a
complex number ?, Re(?) is the real part of ? and Im(?) is the imaginary part of ?.
A weighted directed graph G is defined by a triple (V, E, A) where V = {1, . . . , N }
is a node set, E is a set of pairs of nodes indicating connections among nodes, and
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A = [ai j ] ? R N ОN is the weighting matrix, and ai j > 0 iff (i, j) ? E which denotes
an edge from node j to node i. In our case, we have aii = 0. A path from node i 1 to
i k is a sequence of nodes {i 1, . . . , i k } such that (i j+1, i j ) ? E for j = 1, . . . , k ? 1. A
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directed tree is a subgraph (subset of nodes and edges) in which every node has exactly
one parent node except for one node, called the root, which has no parent node. In this
case, the root has a directed path to every other node in the tree. A directed spanning
tree is a directed tree containing all the nodes of the graph. For a weighted graph G, a
matrix L = [` i j ] with
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PN
?
?
aik , i = j,
? k=1
`i j = ?
?
? ?ai j , i , j,
?
is called the Laplacian matrix associated with the graph G. The Laplacian L has all its
eigenvalues in the closed right half plane and at least one eigenvalue at zero associated
with right eigenvector 1. A specific class of graphs needed in this paper is presented
below:
N denote the set of directed graphs
Definition 1. For any given ? ? ? > 0, let G?,
?
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with N nodes that contain a directed spanning tree and for which the corresponding
Laplacian matrix L satisfies kLk < ? while its nonzero eigenvalues have a real part
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larger than or equal to ?.
2. Problem formulation
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Consider a MAS composed of N identical linear time-invariant agents of the form,
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x? i = Ax i + Bui + E?i ,
yi = C x i ,
(i = 1, . . . , N )
(1)
where x i ? Rn , ui ? Rm , yi ? R p are respectively the state, input, and output vectors
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of agent i, and ?i ? R? is the external disturbances.
The communication network provides each agent with a linear combination of its
own outputs relative to that of other neighboring agents. In particular, each agent
i ? {1, . . . , N } has access to the quantity,
?i =
N
X
j=1
ai j (yi ? y j ),
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(2)
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where ai j ? 0 and aii = 0 indicate the communication among agents. This communication topology of the network can be described by a weighted and directed graph G
with nodes corresponding to the agents in the network and the weight of edges given
rewritten as
?i =
N
X
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by the coefficient ai j . In terms of the coefficients of the Laplacian matrix L, ? i can be
(3)
`i j y j .
j=1
We refer to this case as partial-state coupling. Note that if C has full column rank then,
without loss of generality, we can assume that C = I, and the quantity ? i becomes
N
X
j=1
ai j (x i ? x j ) =
N
X
`i j x j .
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?i =
We refer to this case as full-state coupling.
(4)
j=1
If the graph G describing the communication topology of the network contains a
directed spanning tree, then it follows from [9, Lemma 3.3] that the Laplacian matrix L
has a simple eigenvalue at the origin, with the corresponding right eigenvector 1 and all
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the other eigenvalues are in the open right-half complex plane. Let ? 1, . . . , ? N denote
the eigenvalues of L such that ? 1 = 0 and Re(? i ) > 0, i = 2, . . . , N.
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Let N be any agent and define x? i = x N ? x i and
*. x? 1 +/
.
x? = ... .. ///
.
/
x?
N
?1
,
-
and
*. ?1 +/
.
? = ... .. /// .
. /
,? N -
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Obviously, synchronization is achieved if x? = 0. That is
lim (x i (t) ? x N (t)) = 0,
t??
?i, ? {1, . . . , N ? 1}.
(5)
We denote by T? x? , the transfer function from ? to x?
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Remark 1. Agent N is not necessarily a root agent. Obviously, (5) is equivalent to the
condition that
lim (x i (t) ? x j (t)) = 0,
t??
?i, j ? {1, . . . , N }.
We formulate below four almost state synchronization problems for a network with
either H2 or H? almost synchronization.
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Problem 1. Consider a MAS described by (1) and (4). Let G be a given set of graphs
such that G ? G N . The H? almost state synchronization problem via full-state
coupling (in short H? -ASSFS) with a set of network graphs G is to find, if possible, a
ui = F (?)? i ,
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linear static protocol parameterized in terms of a parameter ?, of the form,
(6)
such that, for any given real number r > 0, there exists an ? ? such that for any ? ? (0, ? ? ]
and for any graph G ? G, (5) is satisfied for all initial conditions in the absence of
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disturbances and the closed loop transfer matrix T? x? satisfies
kT? x? k? < r.
(7)
Problem 2. Consider a MAS described by (1) and (3). Let G be a given set of graphs
such that G ? G N . The H? almost state synchronization problem via partial-state
coupling (in short H? -ASSPS) with a set of network graphs G is to find, if possible, a
linear time-invariant dynamic protocol parameterized in terms of a parameter ?, of the
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form,
?? i = Ac (?) ? i + Bc (?)? i ,
(8)
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ui = Cc (?) ? i + Dc (?)? i ,
where ? i ? Rn c , such that, for any given real number r > 0, there exists an ? ? such that
for any ? ? (0, ? ? ] and for any graph G ? G, (5) is satisfied for all initial conditions in
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the absence of disturbances and the closed loop transfer matrix T? x? satisfies (7).
Problem 3. Consider a MAS described by (1) and (4). Let G be a given set of graphs
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such that G ? G N . The H2 almost state synchronization problem via full-state
coupling (in short H2 -ASSFS) with a set of network graphs G is to find, if possible,
a linear static protocol parameterized in terms of a parameter ?, of the form (6) such
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that, for any given real number r > 0, there exists an ? ? such that for any ? ? (0, ? ? ]
and for any graph G ? G, (5) is satisfied for all initial conditions in the absence of
disturbances and the closed loop transfer matrix T? x? satisfies
kT? x? k2 < r.
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(9)
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Problem 4. Consider a MAS described by (1) and (3). Let G be a given set of graphs
such that G ? G N . The H2 almost state synchronization problem via partial-state
coupling (in short H2 -ASSPS) with a set of network graphs G is to find, if possible,
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a linear time-invariant dynamic protocol parameterized in terms of a parameter ?, of
the form (8) such that, for any given real number r > 0, there exists an ? ? such that for
any ? ? (0, ? ? ] and for any graph G ? G, (5) is satisfied for all initial conditions in the
absence of disturbances and the closed loop transfer matrix T? x? satisfies (9).
Note that the problems of H? almost state synchronization and H2 almost state
synchronization are closely related. Roughly speaking, H2 almost synchronization is
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easier to achieve than H? almost synchronization. This is related to the fact that in H?
we look at the worst case disturbance with the only constraints being the power:
Z T
1
lim sup
?i0 (t)?i (t)dt < ?.
T ?? 2T ?T
while in H2 we only consider white noise disturbances which is a more restrictive class.
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3. MAS with full-state coupling
In this section, we establish a connection between the almost state synchronization
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among agents in the network and a robust H? or H2 almost disturbance decoupling
problem via state feedback with internal stability (in short H? or H2 -ADDPSS) (see
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[28]). Then, we use this connection to derive the necessary and sufficient condition and
design appropriate protocols.
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3.1. Necessary and sufficient condition for H? -ASSFS
The MAS system described by (1) and (4) after implementing the linear static
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protocol (6) is described by
x? i = Ax i + BF (?)? i + E?i ,
for i = 1, . . . , N. Let
*. x 1 +/
.
x = ... .. /// ,
. /
,x N -
*. ?1 +/
.
? = ... .. /// .
. /
,? N 7
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Then, the overall dynamics of the N agents can be written as
x? = (I N ? A + L ? BF (?))x + (I N ? E)?.
(10)
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We define the robust H? -ADDPSS with bounded input as follows. Given ? ? C,
there should exist M > 0 such that for any given real number r > 0, we can find a
parameterized controller
u = F (?)x
for the following subsystem,
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x? = Ax + ? Bu + B?,
(11)
(12)
such that for any ? ? ? the following hold:
1. The interconnection of the systems (12) and (11) is internally stable;
2. The resulting closed-loop transfer function T? x from ? to x has an H? norm less
than r.
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3. The resulting closed-loop transfer function T?u from ? to u has an H? norm less
than M.
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In the above, ? denotes all possible locations for the nonzero eigenvalues of the Laplacian matrix L when the graph varies over the set G. It is also important to note that M
is independent of the choice for r.
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In the following lemma we give a necessary condition for the H? -ASSFS. Moreover,
for sufficiency, we connect the H? -ASSFS problem to the robust H? -ADDPSS with
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bounded input problem which we will address later.
Lemma 1. Let G be a set of graphs such that the associated Laplacian matrices are
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uniformly bounded and let ? consist of all possible nonzero eigenvalues of Laplacian
matrices associated with graphs in G.
(Necessity) The H? -ASSFS for the MAS described by (1) and (4) given G is solvable
by a parameterized protocol ui = F (?)? i only if
im E ? im B.
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(13)
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(Sufficiency) The H? -ASSFS for the MAS described by (1) and (4) given G is solved
by a parameterized protocol ui = F (?)? i if the robust H? -ADDPSS with bounded input
for the system (12) with ? ? ? is solved by the parameterized controller u = F (?)x.
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Proof: Note that L has eigenvalue 0 with associated right eigenvector 1. Let
(14)
L = T SL T ?1,
with T unitary and SL the upper-triangular Schur form associated to the Laplacian
matrix L such that SL (1, 1) = 0. Let
*. ? 1 +/
.
? In )x = ... .. /// ,
. /
,? N -
*. ??1 +/
.
? I)? = ... .. ///
. /
,?? N -
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? := (T
?1
?? = (T
?1
where ? i ? Cn and ??i ? Cq . In the new coordinates, the dynamics of ? can be written
as
which is rewritten as
N
X
s1 j BF (?)? j + E ??1,
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?? 1 = A? 1 +
(15)
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??(t) = (I N ? A + SL ? BF (?))? + (T ?1 ? E)?,
j=2
?? i = ( A + ? i BF (?))? i +
N
X
s i j BF (?)? j + E ??i ,
(16)
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j=i+1
?? N = ( A + ? N BF (?))? N + E ?? N ,
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for i ? {2, . . . , N ? 1} where SL = [s i j ]. The first column of T is an eigenvector of
?
L associated to eigenvalue 0 with length 1, i.e. it is equal to ▒1/ N. Using this we
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obtain:
*. x N ? x 1 +/ *.*.?1 0 и и и 0 1+/
+/
..
.. /
..
..
// ....
//
x N ? x 2 / .. 0 ?1
.
.
. //
// (T ? I )?
/
.
.
/
x? = ...
=
?
I
n
n
..
.. /
// .... ..
//
..
..
..
/
.
.
.
.
0
.
// ....
//
//
.
x
?
x
0
и
и
и
0
?1
1
N ?1 , N
,,
= 0 V ? In ?,
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for some suitably chosen matrix V . Therefore we have
(17)
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T
*. ? 2 +/
.
x? = (V ? In ) ... .. /// ,
. /
,? N -
Note that since T is unitary, also the matrix T ?1 is unitary and the matrix V is uniformly
bounded. Therefore the H? norm of the transfer matrix from ? to x? can be made
arbitrarily small if and only if the H? norm of the transfer matrix from ?? to ? can be
made arbitrarily small.
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In order for the H? norm from ?? to ? to be arbitrarily small we need the H? norm
from ?? N to ? N to be arbitrarily small. From classical results (see [29, 30]) on H?
almost disturbance decoupling we find that this is only possible if (13) is satisfied.
Conversely, suppose u = F (?)x solves the robust H? -ADDPSS with bounded input
for (12) and assume (13) is satisfied. We show next that ui = F (?)? i solves the
H? -ASSFS for the MAS described by (1) and (4). Let X be such that E = BX.
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The fact that u = F (?)x solves the robust H? -ADDPSS with bounded input for (12)
implies that for small ? we have that A + ? BF (?) is asymptotically stable for all ? ? ?.
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In particular, A + ? i BF (?) is asymptotically stable for i = 2, . . . , N which guarantees
that ? i ? 0 for i = 2, . . . , N for zero disturbances and all initial conditions. Therefore
we have state synchronization.
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Next, we are going to show that for any r? > 0, we can choose ? sufficiently small
such that the transfer matrix from ?? to ? i is less than r? for i = 2, . . . , N. This guarantees
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that we can achieve (7) for any r > 0. We have that
T?? x (s) = (sI ? A ? ? BF (?)) ?1 B,
?
T?u
(s) = F (?)(sI ? A ? ? BF (?)) ?1 B.
For a given M and parameter ?, the following is satisfied
kT?? x k? < r? ? ,
?
kT?u
k? < M
for all ? ? ? where r? ? is a parameter depending on ? with the property that lim??0 r? ? = 0.
Denote ?i = F (?)? i .
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When i = N, it is easy to find that,
иии
0
and hence
?N
T??? N = T?u
0
X ,
kT??? N k? < r?,
kT??? N k? < M?N
provided
k X k r? ? < r?,
k X k M < M?N .
иии
0
X
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T??? N = T?? xN 0
(18)
Recall that we can make r? ? arbitrarily small by reducing ? without affecting the bound
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M. Assume
kT??? j k? < r?,
holds for j = i + 1, . . . , N. We have:
kT??? j k? < M? j
M
??
??
N
X
s i j T??? j (s) ???
T??? i (s) = T?? ix (s) ???ei ? X +
??
??
j=i+1
??
??
N
X
?
?i
T??? i (s) = T?u (s) ??ei ? X +
s i j T??? j (s) ???
??
??
j=i+1
where ei is a row vector of dimension N with elements equal to zero except for the ith
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component which is equal to 1. Since
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N
N
X
X
e ? X +
< k X k +
s
T
|s i j | M? j
i j ??? j i
j=i+1
j=i+1
?
CE
we find:
kT??? i k? < r?,
kT??? i k? < M?i
(19)
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provided:
*. k X k +
,
N
X
|s i j | M? j +/ r? ? < r?, *. k X k +
j=i+1
,
N
X
|s i j | M? j +/ M? < M?i .
j=i+1
-
(20)
Note that s i j depends on the graph in G but since the Laplacian matrices associated to
graphs in G are uniformly bounded we find that also the s i j are uniformly bounded. In
this way for any arbitrary r?, we can recursively obtain the bounds in (19) for i = 2, . . . , N
provided we choose ? sufficiently small such that the corresponding r? ? satisfies (18)
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and (20) for i = 2, . . . , N ? 1. Hence, we can choose ? sufficiently small such that
the transfer matrix from ?? to ? i is less than r? for i = 2, . . . , N. As noted before this
guarantees that we can achieve (7) for any r > 0.
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N for some given ?, ? > 0, we
For the case where the set of graphs G equals G?,
?
develop necessary and sufficient conditions for the solvability of the H? -ASSFS for
MAS as follows:
Theorem 1. Consider a MAS described by (1) and (4) with an associated graph from
N for some ?, ? > 0.
the set G = G?,
?
Then, the H? -ASSFS is solvable if and only if (13) is satisfied and ( A, B) is
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stabilizable.
Proof: From Lemma 1, we note that (13) is actually a necessary condition for H? -
ASSFS. Clearly, also ( A, B) stabilizable is a necessary condition. Sufficiency is a direct
result of Theorems 3 or Theorem 5 for H? -ASSFS.
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3.2. Necessary and sufficient conditions for H2 -ASSFS
We define the robust H2 -ADDPSS with bounded input as follows. Given ? ? C,
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there should exist M > 0 such that for any given real number r > 0, we can find a
parameterized controller (11) for the system, (12) such that the following holds for any
? ? ?:
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1. The interconnection of the systems (11) and (12) is internally stable;
2. The resulting closed-loop transfer function T? x from ? to x has an H2 norm less
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than r.
3. The resulting closed-loop transfer function T?u from ? to u has an H? norm less
AC
than M.
In the above, ? denotes all possible locations for the nonzero eigenvalues of the Laplacian matrix L when the graph varies over the set G. It is also important to note that M
is independent of the choice for r. Note that we need to consider two aspects in our
controller H2 disturbance rejection and robust stabilization (because of a set of network
N ). The latter translates in the H norm constraint from ? to u.
graphs G?,
?
?
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Lemma 2. Let G be a set of graphs such that the associated Laplacian matrices are
uniformly bounded and let ? consist of all possible nonzero eigenvalues of Laplacian
matrices associated with graphs in G.
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(Necessity) The H2 -ASSFS for the MAS described by (1) and (4) given G is solvable
by a parameterized protocol ui = F (?)? i only if (13) is satisfied.
(Sufficiency)The H2 -ASSFS for the MAS described by (1) and (4) given G is solvable
by a parameterized protocol ui = F (?)? i if the robust H2 -ADDPSS with bounded input
for the system (12) with ? ? ? is solved by the parameterized controller u = F (?)x.
Proof: The proof is similar to the proof of Lemma 1. This time we need the H2 norm
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from ?? N to ? N to be arbitrarily small and also H2 almost disturbance decoupling then
immediately yields that we need that (13) is satisfied.
The rest of the proof follows the same lines except that we require the H2 norm from
?? to ? j arbitrarily small while we keep the H? norm from ?? to ? j bounded. Recall that
for any two stable, strictly proper transfer matrices T1 and T2 we have:
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kT1T2 k2 ? kT1 k2 kT2 k?
which we need in the modifications of the proof of Lemma 1.
ED
N (with given ?, ? > 0), we develop
For the case with a set of graph G = G?,
?
necessary and sufficient conditions for the solvability of the H2 -ASSFS for MAS as
follows:
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Theorem 2. Consider a MAS described by (1) and (4) with an associated graph from
N for some ?, ? > 0.
the set G = G?,
?
CE
Then, the H2 -ASSFS is solvable if and only if (13) is satisfied and ( A, B) is stabilizable.
Proof: We have already noted before that (13) is actually a necessary condition for
AC
H2 -ASSFS. Clearly, also ( A, B) being stabilizable is a necessary condition. Sufficiency
for H2 -ASSFS, is a direct result of either Theorem 4 or Theorem 6.
3.3. Protocol design for H? -ASSFS and H2 -ASSFS
We present below two protocol design methods for both H? -ASSFS and H2 -ASSFS
problems. One relies on an algebraic Riccati equation (ARE), and the other is based on
13
ACCEPTED MANUSCRIPT
an asymptotic time-scale eigenstructure assignment (ATEA) method.
3.3.1. ARE-based method
( A, B) is stabilizable. We consider the protocol,
ui = ?F? i ,
where ? =
1
?
CR
IP
T
Using an algebraic Riccati equation, we can design a suitable protocol provided
(21)
and F = ?B 0 P with P being the unique solution of the continuous-time
algebraic Riccati equation
AN
US
A0 P + P A ? 2 ?PBB 0 P + I = 0,
(22)
where ? is a lower bound for the real part of the non-zero eigenavlues of all Laplacian
N .
matrices associated with a graph in G = G?,
?
The main result regarding H? -ASSFS is stated as follows.
M
Theorem 3. Consider a MAS described by (1) and (4) such that (13) is satisfied. Let
any real numbers ?, ? > 0 and a positive integer N be given, and hence a set of network
N be defined.
graphs G?,
?
ED
N is
If ( A, B) is stabilizable then the H? -ASSFS stated in Problem 1 with G = G?,
?
solvable. In particular, for any given real number r > 0, there exists an ? ? , such that
for any ? ? (0, ? ? ), the protocol (21) achieves state synchronization and the resulting
PT
system from ? to x i ? x j has an H? norm less than r for any i, j ? 1, . . . , N and for
CE
N .
any graph G ? G?,
?
Proof: Using Lemma 1, we know that we only need to verify that u = ?F x solves
the robust H? -ADDPSS with bounded input for the system (12) with ? ? ?. Given
AC
N , we know that ? ? ? implies Re ? ? ?. Clearly, the Laplacian matrices are
G ? G?,
?
uniformly bounded since kLk ? ?.
Consider the interconnection of (12) and u = ?F x. We define
V (x) = x 0 Px
14
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and we obtain:
V? = x 0 ( A ? ?? BB 0 P) 0 Px + ? 0 B 0 Px + x 0 P( A ? ?? BB 0 P)x + x 0 PB?
CR
IP
T
= x 0 PBB 0 Px ? x 0 x ? 2? ?x 0 PBB 0 Px + 2x 0 PB?
? (1 ? ?? )x 0 PBB 0 Px ? x 0 x + ?? ? 0?
? ? ?2 ?u 0u ? x 0 x + ?? ? 0?
which implies that the system is asymptotically stable and the H? norm of the transfer
function from ? to x is less that ?/ ? while the H? norm of the transfer function from
AN
US
? to u is less that 2/ ? 2 . Therefore, u = ?F x solves the robust H? -ADDPSS with
bounded input for the system (12) as required.
For H2 -ASSFS we have the following classical result:
Lemma 3. Consider an asymptotically stable system:
M
p? = A1 p + B1 ?
The H2 norm from ? to p is less than ? if there exists a matrix Q such that:
ED
A1 Q + Q A10 + B1 B10 ? 0,
Q < ?I
The main result regarding H2 -ASSFS is stated as follows.
PT
Theorem 4. Consider a MAS described by (1) and (4) such that (13) is satisfied. Let
any real numbers ?, ? > 0 and a positive integer N be given, and hence a set of network
CE
N be defined.
graphs G?,
?
N is
If ( A, B) is stabilizable then the H2 -ASSFS stated in Problem 3 with G = G?,
?
solvable. In particular, for any given real number r > 0, there exists an ? ? , such that
AC
for any ? ? (0, ? ? ), the protocol (21) achieves state synchronization and the resulting
system from ? to x i ? x j has an H? norm less than r for any i, j ? 1, . . . , N and for
N .
any graph G ? G?,
?
Proof: Using Lemma 2, we know that we only need to verify that u = ?F x solves the
robust H2 -ADDPSS with bounded input for the system (12) with ? ? ?. We use the
15
ACCEPTED MANUSCRIPT
same feedback as in the proof of Theorem 3. In the proof of Theorem 3 it is already
shown that the closed loop system is asymptotically stable and the H? norm of the
transfer function from ? to u is bounded. The only remaining part of the proof is to
CR
IP
T
show that the H2 norm from ? to x can be made arbitrarily small. Using the algebraic
Riccati equation it is easy to see that we have:
( A ? ?? BB 0 P) 0 P + P( A ? ?? BB 0 P) + ? ?PBB 0 P ? 0
for large ?. But then we have:
AN
US
Q? ( A ? ?? BB 0 P) 0 + ( A ? ?? BB 0 P)Q? + BB 0 ? 0
for Q? = ? ? ?1 P?1 . Then Lemma 3 immediately yields that we can make the H2 norm
from ? to x arbitrarily small by choosing a sufficiently small ?.
3.3.2. ATEA-based method
The ATEA-based design is basically a method of time-scale structure assignment
M
in linear multivariable systems by high-gain feedback [31]. In the current case, we do
not need the full structure presented in the above method. It is sufficient to note that
ED
there exists non-singular transformation matrix Tx ? RnОn (See [32, Theorem 1]) such
PT
that
x? 1
x? = *. +/ = Tx x,
, x? 2 -
(23)
CE
and the dynamics of x? is represented as
x?? 1 = A?11 x? 1 + A?12 x? 2,
x?? 2 = A?21 x? 1 + A?22 x? 2 + ? B?u + B??,
(24)
AC
with B? invertible, and that ( A, B) is stabilizable implies that ( A?11, A?12 ) is stabilizable.
Choose F1 such that A?11 + A?12 F1 is asymptotically stable. In that case a suitable
protocol for (1) is
(25)
ui = F? ? i ,
where F? is designed as
F? =
1 ?1 B?
F1
?
16
?I Tx
(26)
ACCEPTED MANUSCRIPT
The main result regarding H? -ASSFS is stated as follows. The result is basically
the same as Theorem 3 except for a different design protocol.
Theorem 5. Consider a MAS described by (1) and (4) such that (13) is satisfied. Let
CR
IP
T
any real numbers ?, ? > 0 and a positive integer N be given, and hence a set of network
N be defined.
graphs G?,
?
N is
If ( A, B) is stabilizable then the H? -ASSFS stated in Problem 1 with G = G?,
?
solvable. In particular, for any given real number r > 0, there exists an ? ? , such that
for any ? ? (0, ? ? ), the protocol (25) achieves state synchronization and the resulting
system from ? to x i ? x j has an H? norm less than r for any i, j ? 1, . . . , N and for
AN
US
N .
any graph G ? G?,
?
Proof: Similarly to the proof of Theorem 3, we only need to establish that u = F? x
solves the robust H? -ADDPSS with bounded input for the system (12) with ? ? ?.
N , we know that ? ? ? implies Re ? ? ?.
Given G ? G?,
?
After a basis transformation, the interconnection of (12) and u = F? x is equal to
M
the interconnection of (24) and (25). We obtain:
x?? 1 = A?11 x? 1 + A?12 x? 2,
Define
ED
? x?? 2 = (? A?21 + ?F1 ) x? 1 + (? A?22 ? ?I) x? 2 + ? B??.
PT
x? 1 = x? 1,
(27)
x? 2 = x? 2 ? F1 x? 1 .
CE
Then we can write this system (27) in the form:
x?? 1 = A?11 x? 1 + A?12 x? 2,
? x?? 2 = ? A?21 x? 1 + (? A?22 ? ?I) x? 2 + ? B??,
(28)
AC
where
A?11 = A?11 + A?12 F1,
A?12 = A?12,
A?21 = A?21 ? F1 A?11 + A?22 ? F1 A?12,
A?22 = A?22 ? F1 A?12 .
In the absence of the external disturbances, the above system (28) is asymptotically
stable for small enough ?.
17
ACCEPTED MANUSCRIPT
Since A?11 = A?11 + A?12 F1 is Hurwitz stable, there exists P > 0 such that the Lyapunov
0 P = ?I holds. For the dynamics x? , we define a Lyapunov function
equation P A?11 + A?11
1
V1 = x? 10 P x? 1 . Then the derivative of V1 can be bounded
CR
IP
T
0
V?1 ? ?k x? 1 k 2 + x? 20 A?12
P x? 1 + x? 10 P A?12 x? 2
? ?k x? 1 k 2 + 2 Re( x? 10 P A?12 x? 2 )
? ?k x? 1 k 2 + r 1 k x? 1 kk x? 2 k,
where 2kP A?12 k ? r 1 . Now define a Lyapunov function V2 = ? x? 20 x? 2 for the dynamics
AN
US
x? 2 . The derivative of V2 can then also be bounded.
V?2 ? ?2 Re(?)k x? 2 k 2 + 2? Re( x? 20 A?21 x? 1 ) + 2? x? 20 A?22 x? 2 + 2? Re( x? 20 B??)
? ?2 Re(?)k x? 2 k 2 + ?r 2 k x? 1 kk x? 2 k + ?r 3 k x? 2 k 2 + ?r 4 k?kk x? 2 k
? ? ?k x? 2 k 2 + ?r 2 k x? 1 k k x? 2 k + ?r 4 k?k k x? 2 k
for a small enough ?, where we choose r 2, r 3, r 4 such that
2k A?22 k ? r 3,
M
2k A?21 k ? r 2,
and
2k B?k ? r 4 .
ED
Let V = V1 + ?V2 for some ? > 0. Then, we have
V? ? ?k x? 1 k 2 + r 1 k x? 1 kk x? 2 k ? ? ?k x? 2 k 2 + ??r 2 k x? 1 kk x? 2 k + ??r 4 k?kk x? 2 k.
CE
PT
We have that
r 1 k x? 1 k k x? 2 k ? r 12 k x? 2 k 2 + 41 k x? 1 k 2,
??r 2 k x? 1 k k x? 2 k ? ? 2 ? 2 r 22 k x? 1 k 2 + 14 k x? 2 k 2,
??r 4 k?k k x? 2 k ? ? 2 ? 2 r 42 k?k 2 + 14 k x? 2 k 2 .
AC
Now we choose ? such that ? ? = 1 + r 12 and r 5 = ?r 4 . Then, we obtain
V? ? ? 12 k x? 1 k 2 ? 12 k x? 2 k 2 + ? 2 r 52 k?k 2
? ? 21 k x?k 2 + ? 2 r 52 k?k 2,
for a small enough ?. From the above, we have that kT? x? k? < 2?r 5 , which immediately
leads to kT? x k? < r for any real number r > 0 as long as we choose ? small enough.
18
ACCEPTED MANUSCRIPT
and hence:
T?u (s) = ? ?1 0
B??1 T? x? (s)
kT?u k? ? k B??1 kr 5 .
CR
IP
T
On the other hand:
Therefore, u = F? x solves the robust H? -ADDPSS with bounded input for the system
(12) as required.
The main result regarding H2 -ASSFS is stated as follows.
Theorem 6. Consider a MAS described by (1) and (4) such that (13) is satisfied. Let
AN
US
any real numbers ?, ? > 0 and a positive integer N be given, and hence a set of network
N be defined.
graphs G?,
?
N is
If ( A, B) is stabilizable then the H2 -ASSFS stated in Problem 1 with G = G?,
?
solvable. In particular, for any given real number r > 0, there exists an ? ? , such that
for any ? ? (0, ? ? ), the protocol (25) achieves state synchronization and the resulting
N .
graph G ? G?,
?
M
system from ? to x i ? x j has an H2 norm less than r for any i, j ? 1, . . . , N and for any
ED
Proof: Using Lemma 2, we know that we only need to verify that the feedback solves
the robust H2 -ADDPSS with bounded input for the system (12) with ? ? ?. We use
the same feedback as in the proof of Theorem 5. In the proof of Theorem 5 it is already
PT
shown that the closed loop system is asymptotically stable and the H? norm of the
transfer function from ? to u is bounded. The only remaining part of the proof is
CE
to show that the H2 norm from ? to x can be made arbitrarily small. This clearly is
equivalent to showing that the system (28) has an arbitrary small H2 norm from ? to
AC
x? 1 and x? 2 for sufficiently small ?. Choose Q such that
0
Q A?11
+ A?11 Q = ?I
19
ACCEPTED MANUSCRIPT
In that case we have:
?
0 + * ?Q
? /+.
?I - , 0
0 + 0
*0
? / Acl + .
?I ,0
?
? *.?
0
, ?( A?12 + A?21 Q)
for sufficiently small ? where:
A?11
Acl = *.
, A?21
A?12
A?22 ?
+/
?
? I-
?
0 )
?( A?12 + Q A?21
+/
? ??? I
-
AN
US
and we used that
0 +
/
B? B? 0?
CR
IP
T
?
?Q
*
Acl .
, 0
? + ? 0 ? 2 ?.
We then obtain for sufficiently small ? that:
?
0 + * ?Q
? /+.
?I - , 0
M
?
?Q
*
Acl .
, 0
0 + 0
*0
? / Acl + .
?I ,0
0 +
/ ?0
B? B? 0-
Then Lemma 3 immediately yields that we can make the H2 norm from ? to x arbitrarily
ED
small by choosing a sufficiently small ?.
4. MAS with partial-state coupling
PT
In this section, similar to the approach of the previous section, we show first
that the almost state synchronization among agents in the network with partial-state
CE
coupling can be solved by equivalently solving a robust H? or H2 almost disturbance
decoupling problem via measurement feedback with internal stability (in short H? or
H2 -ADDPMS). Then, we design a controller for such a robust H? or H2 -ADDPMS
AC
with bounded input.
20
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4.1. Necessary and sufficient condition for H? -ASSPS
The MAS system described by (1) and (3) after implementing the linear dynamical
protocol (8) is described by
for i = 1, . . . , N, where
(29)
AN
US
xi
x? i = *. +/ .
, ?i -
CR
IP
T
?
?
A BCc (?) +
BDc (?) +
E
?
?
?
?
/ x? i + *.
/? i + *. +/ ?i ,
x?? i = *.
?
?
?
?
Ac (?) ?
?
, 0
, Bc (?) ,0
?
?
? y =
C 0 x? i ,
i
?
?
?
?
?
N
?
X
?
?
?
?
?i =
`i j y j,
?
?
?
j=1
?
Define
M
*. x? 1 +/
.
x? = ... .. /// ,
. /
, x? N -
and
*. ?1 +/
.
? = ... .. /// ,
. /
,? N -
ED
A BCc (?) +
BDc (?) +
E
/ , B? = *.
/ , E? = *. +/ ,
A? = *.
, 0 Ac (?) , Bc (?) ,0Then, the overall dynamics of the N agents can be written as
0 .
(30)
PT
x?? = (I N ? A? + L ? B?C?) x? + (I N ? E?)?.
C? = C
We define a robust H? -ADDPMS with bounded input as follows. Given ? ? C,
CE
there should exist M > 0 such that for any given real number r > 0, we can find a
AC
parameterized controller
?? = Ac (?) ? + Bc (?)y,
u = Cc (?) ? + Dc (?)y,
(31)
where ? ? Rn c , for the following system,
x? = Ax + ? Bu + B?,
y = Cx
such that the following holds for any ? ? ?:
21
(32)
ACCEPTED MANUSCRIPT
1. The closed-loop system of (31) and (32) is internally stable
2. The resulting closed-loop transfer function T? x from ? to x has an H? norm less
than r.
CR
IP
T
3. The resulting closed-loop transfer function T?u from ? to u has an H? norm less
than M.
In the above, ? denotes all possible locations for the nonzero eigenvalues of the Laplacian matrix L when the graph varies over the set G. It is also important to note that M
is independent of the choice for r.
Lemma 4. Consider the system:
AN
US
In order to obtain our main result, we will need the following lemma:
x? = Ax + Bu + E?,
y = Cx
z=x
M
with ( A, B) stabilizable and (C, A) detectable. The H? -ADDPMS for the above system
is defined as the problem to find for any r > 0 a controller of the form (31) such that
the closed loop system is internally stable while the H? norm from ? to z is less than
ED
r. The H? -ADDPMS is solvable if and only if:
1. im E ? im B,
PT
2. ( A, E, C, 0) is left-invertible,
3. ( A, E, C, 0) is minimum-phase.
CE
Proof: From [29] we immediately obtain that the H? -ADDPMS is solvable if and only
AC
if:
1. im E ? im B
2. ( A, E, C, 0) is at most weakly non-minimum-phase and left-invertible.
3. For any ? > 0 and every invariant zero s0 of ( A, E, C, 0), there exists a matrix K
such that sI ? A ? BKC is invertible and
k(s0 I ? A ? BKC) ?1 E k? < ?
22
(33)
ACCEPTED MANUSCRIPT
Choose a suitable basis such that:
A11
A = *.
, A21
A12 +
/,
A22 -
B1
B = *. +/ ,
, B2 -
E1
E = *. +/ ,
, E2 -
C= I
0
*. sI ? A11
..
.. ?A21
I
,
?A12
sI ? A22
0
E1 +
//
E2 //
/
0-
CR
IP
T
Assume s0 is an imaginary axis zero of ( A, E, C, 0). In that case the rank of the matrix:
AN
US
drops for s = s0 . This implies the existence of p , 0 and q , 0 such that
*. ?A12 +/ p = *. E1 +/ q.
, s0 I ? A22 , E2 -
The final condition for H? almost disturbance decoupling requires for any ? > 0 the
(s0 I ? A ? BKC) ?1 Eq
s0 I ? A11 ? B1 K
= *.
, ?A21 ? B2 K
PT
0
= *. +/ ,
, p-
?1
?A12 +
/
s0 I ? A22 -
ED
s0 I ? A11 ? B1 K
= *.
, ?A21 ? B2 K
M
existence of a K such that (33) is satisfied. However:
?1
?A12 +
/
s0 I ? A22 -
*. ?A12 +/ p
, s0 I ? A22 -
*. s0 I ? A11 ? B1 K
, ?A21 ? B2 K
?A12 + *0+
/. /
s0 I ? A22 - , p-
kpk > ?kqk.
AC
CE
which yields a contradiction if ? is such that
Therefore we cannot have any invariant zeros in the imaginary axis. In other words,
the system ( A, E, C, 0) needs to be minimum-phase instead of weakly minimum-phase.
Conversely, if ( A, E, C, 0) is minimum-phase it is easy to verify that for any ? > 0 there
exists K such that (33) is satisfied.
23
ACCEPTED MANUSCRIPT
Theorem 7. Consider the MAS described by (1) and (3) with ( A, B) stabilizable and
(C, A) detectable.
N be defined. Then,
(Part I) Let ?, ? > 0 be given such that a set of graphs G?,
?
protocol (8) for any ? > ? > 0 if and only if
im E ? im B
CR
IP
T
N is solvable by a parameterized
the H? -ASSPS for the MAS with any graph G ? G?,
?
(34)
while ( A, E, C, 0) is minimum phase and left-invertible.
(Part II) Let G be a set of graphs such that the associated Laplacian matrices are
AN
US
uniformly bounded and let ? consist of all possible nonzero eigenvalues of Laplacian
matrices associated with graphs in G. Then, the H? -ASSPS for the MAS with any
graph G ? G is solved by a parameterized protocol (8) if the robust H? -ADDPMS with
bounded input for the system (32) with ? ? ? is solved by the parameterized controller
(31).
*. ? 1 +/
.
? In ) x? = ... .. /// ,
. /
,? N -
?? = (T
?1
ED
? := (T
?1
M
Proof: By using L = T SL T ?1 , we define
*. ??1 +/
.
? I)? = ... .. ///
. /
,?? N -
where ? i ? Cn+n c and ??i ? Cq . In the new coordinates, the dynamics of ? can be
PT
written as
??(t) = (I N ? A? + SL ? B?C?? + (T ?1 ? E)?,
(35)
AC
CE
which is rewritten as
?? 1 = A?? 1 +
N
X
s1 j B?C?? j + E? ??1,
j=2
?? i = ( A? + ? i B?C?)? i +
N
X
s i j B?C?? j + E? ??i ,
j=i+1
?? N = ( A? + ? N B?C?)? N + E? ?? N ,
with i ? {2, . . . , N ? 1} where
0
E? = *. +/ ,
,E -
SL = [s i j ].
24
(36)
ACCEPTED MANUSCRIPT
As in the case of full-state coupling, we can show that:
(37)
CR
IP
T
*. ? 2 +/
.
x? = (V ? In ) ... .. /// ,
. /
,? N -
for some suitably chosen matrix V which is uniformly bounded. Therefore the H?
norm of the transfer matrix from ? to x? can be made arbitrarily small if and only if the
H? norm of the transfer matrix from ?? to ? can be made arbitrarily small.
In order for the H? norm from ?? to ? to be arbitrarily small we need the H? norm
AN
US
from ?? N to ? N to be arbitrarily small. In other words, the robust H? -ADDPMS with
bounded input has to be solvable for the system
x? = Ax + ? Bu + E?,
y = Cx
From the results of Lemma 4, we find that this is only possible if (34) is satisfied and
( A, E, C, 0) is left-invertible and minimum phase.
M
On the other hand, suppose (31) solves the robust H? -ADDPMS with bounded input
of (32) and assume (34) is satisfied. We need to show that (8) solves the H? -ASSFS for
ED
the MAS described by (1) and (3). This follows directly from arguments very similar
to the approach used in the proof of Lemma 1.
PT
4.2. Necessary and sufficient condition for H2 -ASSPS
The MAS system described by (1) and (3) after implementing the linear dynamical
CE
protocol (8) is described by (29) for i = 1, . . . , N, and, as before, the overall dynamics
AC
of the N agents can be written as
x?? = (I N ? A? + L ? B?C?) x? + (I N ? E?)?.
(38)
We define a robust H2 -ADDPMS with bounded input as follows. Given ? ? C,
there should exist M > 0 such that for any given real number r > 0, we can find a
parameterized controller
?? = Ac (?) ? + Bc (?)y,
u = Cc (?) ? + Dc (?)y,
25
(39)
ACCEPTED MANUSCRIPT
where ? ? Rn c , for the following system,
x? = Ax + ? Bu + B?,
(40)
such that the following holds for any ? ? ?:
CR
IP
T
y = Cx
1. The closed-loop system of (39) and (40) is internally stable
2. The resulting closed-loop transfer function T? x from ? to x has an H2 norm less
than r.
than M.
AN
US
3. The resulting closed-loop transfer function T?u from ? to u has an H? norm less
In the above, ? denotes all possible locations for the nonzero eigenvalues of the Laplacian matrix L when the graph varies over the set G. It is also important to note that M
is independent of the choice for r.
The following lemma, provides a necessary condition for the H2 -ADDPMS:
M
Lemma 5. Consider the system:
x? = Ax + Bu + E?,
ED
y = Cx
z=x
PT
with ( A, B) stabilizable and (C, A) detectable. The H2 -ADDPMS for the above system
is defined as the problem to find for any r > 0 a controller of the form (39) such that
the closed loop system is internally stable while the H? norm from ? to z is less than
CE
r. The H2 -ADDPMS is solvable only if:
AC
1. im E ? im B
2. ( A, E, C, 0) is at most weakly non-minimum-phase and left-invertible.
Proof: This follows directly from [29].
Theorem 8. Consider the MAS described by (1) and (3) with ( A, B) stabilizable and
(C, A) detectable.
26
ACCEPTED MANUSCRIPT
N be defined. Then,
(Part I) Let ?, ? > 0 be given such that a set of graphs G?,
?
N is solvable by a parameterized
the H2 -ASSPS for the MAS with any graph G ? G?,
?
protocol (8) for any ? > ? > 0 only if
(41)
CR
IP
T
im E ? im B
while ( A, E, C, 0) is at most weakly non-minimum phase and left-invertible .
(Part II) Let G be a set of graphs such that the associated Laplacian matrices are
uniformly bounded and let ? consist of all possible nonzero eigenvalues of Laplacian
matrices associated with graphs in G. Then, the H2 -ASSPS for the MAS with any
AN
US
graph G ? G is solved by a parameterized protocol (8) if the robust H2 -ADDPMS with
bounded input for the system (40) with ? ? ? is solved by the parameterized controller
(39).
Proof: Similar, to the proof of Theorem 7, the dynamics can be written in the form (36).
Using (37), we note the H2 norm of the transfer matrix from ? to x? can be made
made arbitrarily small.
M
arbitrarily small if and only if the H2 norm of the transfer matrix from ?? to ? can be
In order for the H2 norm from ?? to ? to be arbitrarily small we need the H2 norm
ED
from ?? N to ? N to be arbitrarily small. In other words, the robust H2 -ADDPMS with
bounded input has to be solvable for the system
PT
x? = Ax + ? Bu + E?,
y = Cx
From the results of Lemma 5, we find that this is only possible if (41) is satisfied,
CE
( A, E, C, 0) is left-invertible and at most weakly non-minimum phase.
On the other hand, suppose (39) solves the robust H2 -ADDPMS with bounded input
of (40) and assume (41) is satisfied. We need to show that (8) solves the H2 -ASSFS for
AC
the MAS described by (1) and (3). This follows directly from arguments very similar
to the approach used in the proof of Lemma 1.
4.3. Protocol design for H? -ASSPS
We present below two protocol design methods based on robust stabilization for the
case E = B and therefore the case where ( A, B, C, 0) is minimum-phase. One relies
27
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on an algebraic Riccati equation (ARE) method, and the other is based on the direct
eigenstructure assignment method.
4.3.1. ARE-based method
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Using an algebraic Riccati equation, we can design a suitable protocol. As in the
full-state coupling case, we choose F = ?B 0 P with P = P 0 > 0 being the unique
solution of the continuous-time algebraic Riccati equation
A0 P + P A ? 2 ?PBB 0 P + I = 0,
(42)
where ? is a lower bound for the real part of the non-zero eigenvalues of all Laplacian
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N .
matrices associated with a graph in G?,
?
Since ( A, B, C, 0) is minimum-phase then for any ? there exists ? small enough such
that
AQ + Q A0 + BB 0 + ? ?4 Q2 ? ??2 QC 0CQ = 0
(43)
has a solution Q > 0. We then consider the following protocol:
M
?? i = ( A + K? C) ? i ? K? ? i ,
ui = F? ? i ,
where
F? = ? ?1 B 0 P,
(44)
ED
K? = ? ?12 QC 0
The main result in this section is stated as follows.
PT
Theorem 9. Consider a MAS described by (1) and (3) with ( A, B) stabilizable and
(C, A) detectable. Let any real numbers ?, ? > 0 and a positive integer N be given,
CE
N be defined.
and hence a set of network graphs G?,
?
N is solvable. In particular, for
The H? -ASSPS stated in Problem 2 with G = G?,
?
any given real number r > 0, there exists an ? ? , such that for any ? ? (0, ? ? ), the
AC
protocol (44) achieves state synchronization and the resulting system from ? to x i ? x j
N .
has an H? norm less than r for any i, j ? 1, . . . , N and for any graph G ? G?,
?
Proof: Using Theorem 7, we know that we only need to verify that
?? = ( A + K? C) ? ? K? y,
u = F? ?,
28
(45)
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solves the robust H? -ADDPMS with bounded input for the system (32) with ? ? ?.
N , we know that ? ? ? implies Re ? ? ?. Obviously A + BF and
Given G ? G?,
?
?
A + K? C are both asymptotically stable by construction and hence the intersection of
equal to:
T? x = I
where:
I
*
0 .
,T?3
?1
?T2 +
/
I -
*. T1 +/
,?T?4 -
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T1 (s) = (sI ? A ? ? BF? ) ?1 B
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(32) and (45) is asymptotically stable. The closed loop transfer function from ? to x is
(46a)
T2 (s) = ?(sI ? A ? ? BF? ) ?1 BF?
(46b)
T?3 (s) = ?(sI ? A ? K? C + ? BF? ) ?1 BF?
(46c)
T?4 (s) = (sI ? A ? K? C + ? BF? ) ?1 B
(46d)
As argued in the proof of Theorem 3, we have:
?
,
?
kT2 k? <
M
kT1 k? <
2|?|
2?
? 2.
2
?
?
PT
where
ED
On the other hand, (43) implies according to the bounded real lemma:
kT3 k? < ? 2
T3 (s) = (sI ? A ? K? C) ?1 B
CE
Note that:
T?3 = ?(I + ?T3 F? ) ?1T3 F?
AC
which yields, using (47), that
kT?3 k? < (1 ? ?M1 ) ?1 ?M1 < 2?M1
for small ? where M1 is such that:
|?|kB 0 Pk < ?kB 0 Pk = M1
29
(47)
ACCEPTED MANUSCRIPT
The above yields:
kT? x k? < ?M2
to:
I
*
F? .
,T?3
T?u = F?
?T2 +
/
I -
which yields using similar arguments as above that:
?1
*. T1 +/
,?T?4 -
kT?u k? < M3
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for some suitable constant M2 . The closed loop transfer function from ? to u is equal
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for some suitable constant M3 independent of ?. Therefore the H? norm of the transfer
matrix T? x becomes arbitrarily small for sufficiently small ? while the H? norm of the
transfer matrix T?u remains bounded.
4.3.2. Direct method
For ease of presentation, we only consider the case q = 1, i.e. the case where we
M
have a scalar measurement. We consider the state feedback gain F? given in (26), that
is
ED
F? =
where Tx is defined in (23).
1 ?1 B?
F1
?
?I Tx ,
Next, we consider the observer design. Note that the system ( A, B, C, 0) is minimum-
PT
phase and left-invertible. In that case there is a nonsingular matrix ?x such that, by
AC
CE
defining x? = ?x x, we obtain the system
x? a = Aa x a + L ad y,
x? d = Ad x d + Bd (u + ? + Eda x a + Edd x d ),
y = Cd x d .
where
xa
x? = ?x x = *. +/ ,
,xd -
30
(48)
ACCEPTED MANUSCRIPT
with x a ? Rn?? and x d ? R ? and where the matrices Ad ? R ?О? , Bd ? R ?О1 , and
Cd ? R1О? have the special form
1
..
.
иии
иии
иии
..
.
0
0
0+
.. ///
./
// ,
1//
/
0-
*.0+/
.. .. //
.
Bd = ... /// ,
..0//
. /
,1-
Cd = 1 0
0 .
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*.0
.. ..
.
Ad = ...
..0
.
,0
иии
(49)
Furthermore, the eigenvalues of Aa are the invariant zeros of ( A, B, C) and hence
Aa is asymptotically stable. The transformation ?x can be calculated using available
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software, either numerically [33] or symbolically [34].
Next, define a high-gain scaling matrix
S? := diag(1, ? 2, . . . , ? 2??2 ),
and define the output injection matrix
M
0
+/ .
K? = ?x *.
?2
?1
,? S? K -
(50)
(51)
protocol:
ED
where K is such that Ad + Bd K is asymptotically stable. We then consider the following
?? i = ( A + K? C) ? i ? K? ? i ,
(52)
PT
ui = F? ? i ,
The main result in this section is stated as follows.
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Theorem 10. Consider a MAS described by a SISO system (1) and (3). Let any real
numbers ?, ? > 0 and a positive integer N be given, and hence a set of network graphs
AC
N be defined.
G?,
?
N is
If ( A, B) is stabilizable then the H? -ASSPS stated in Problem 2 with G = G?,
?
solvable. In particular, for any given real number r > 0, there exists an ? ? , such that
for any ? ? (0, ? ? ), the protocol (52) achieves state synchronization and the resulting
system from ? to x i ? x j has an H? norm less than r for any i, j ? 1, . . . , N and for
N .
any graph G ? G?,
?
31
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Proof: We use a similar argument as in the proof of Theorem 10. We know that we
only need to verify that
?? = ( A + K? C) ? ? K? y,
(53)
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u = F? ?,
solves the robust H? -ADDPMS with bounded input for the system (32) with ? ? ?.
N , we know that ? ? ? implies Re ? ? ?. Obviously A + BF and
Given G ? G?,
?
?
A + K? C are both asymptotically stable by construction and hence the intersection of
(32) and (45) is asymptotically stable. As in the proof of Theorem 9, the closed loop
transfer function from ? to x is equal to:
T? x = I
I
*
0 .
,T?3
?1
?T2 +
/
I -
*. T1 +/
,?T?4 -
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(54)
where, as before, we use the definitions in (46) but with our modified F? and K? . As
argued in the proof of Theorem 5, we have:
M
kT1 k? < M1 ?,
kT2 k? < M2 .
for suitable constants M1, M2 > 0. Finally
ED
sI ? Aa
T3 (s) = ?x *.
,?Bd Eda
PT
where
?1
L ad Cd +
/
Z1 -
*. 0 +/
, Bd -
Z1 = sI ? Ad ? ? ?2 S??1 KCd ? Bd Edd
AC
CE
We obtain:
with
I
T3 (s) = ? ?x *.
,0
2n
0 + * sI ? Aa
/.
S??1 - ,? 2n Bd Eda
?1
L ad Cd +
/
Z2 -
*. 0 +/
, Bd -
Z2 = sI ? ? ?2 Ad ? ? ?2 KCd + ? 2n Bd Edd S??1,
using that
? ?2 Ad = S? Ad S??1,
S? Bd = ? 2n Bd
32
and
Cd S??1 = Cd .
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Note that Ed? is bounded for ? < 1. Next, we note that:
L ad Cd
+/
?2
sI ? ? ( Ad + KCd ) -
?1
*. 0 +/
, Bd -
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sI ? Aa
X? (s) = *.
, 0
?(sI ? Aa ) ?1 L ad + 2
/ (? sI ? Ad ? KCd ) ?1 Bd
= ? 2 *.
I
,
-
From the above we can easily conclude that there exists M such that k X? k? < M? 2 .
We have:
where
0 +
/ (I + ? 2 X? Ed? ) ?1 X?
S??1 -
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I
T3 = ? 2n ?x *.
,0
Ed? = ? 2n?2 Eda
? 2n?2 Edd S??1
which is clearly bounded for ? < 1. This clearly implies, using our bounds for X? and
Ed? , that there exists M3 > 0 such that:
for small ? since
M
kT3 k ? ? 2 M3
PT
ED
I
? 2n *.
,0
Our bound for T3 guarantees that
0 +
/ < ? 2 I.
S??1 -
kT?3 k? < ?M4,
kT?4 k? < ?M5,
CE
for suitable M4 and M5 . Moreover
kF? k < ? ?1 M0
AC
Given our bounds, we immediately obtain from (54) that there exists M6 such that
kT? x k? < M6 ?.
The closed loop transfer function from ? to u is equal to:
T?u = F?
I
*
F? .
,T?3
33
?T2 +
/
I -
?1
*. T1 +/
,?T?4 -
ACCEPTED MANUSCRIPT
which yields, using similar arguments as above, that:
kT?u k? < M7
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for some suitable constant M7 independent of ?. In other words, the transfer function
from ? to x is arbitrarily small for sufficiently small ? while the transfer function from
? to u is bounded which completes the proof.
4.4. Protocol design for H2 -ASSFS
We present below two protocol design methods based on robust stabilization for the
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US
case E = B. The necessary condition provided earlier shows that ( A, B, C, 0) need only
be at most weakly non-minimum-phase. The following designs are provided under the
stronger assumption that ( A, B, C, 0) is minimum-phase.
4.4.1. ARE-based method
We consider the protocol (44) already used in the case of H? -ASSFS. It is easy to
M
verify using a similar proof that this protocol also solves the robust H2 -ADDPMS with
bounded input and therefore solves H2 -ASSPS for the MAS. Using the same notation
ED
as before, this relies on the fact that we have N1 such that
kT1 k2 < ?N1
PT
which follows directly from the full-state coupling case. On the other hand we have N2
CE
such that
kT3 k2 < ? 2 N2
AC
since Q ? 0 for ? ? 0 and
( A ? KC)Q + Q( A ? KC) 0 + BB 0 ? 0.
It is then easily shown that
kT?4 k2 < ? 2 N3
for some N3 > 0. The rest of the proof is then as before in the case of H? -ASSPS.
34
Figure 1: The communication topology
4.4.2. Direct method
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We consider the protocol (52) already used in the case of H? -ASSFS. It is easy to
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verify using a similar proof that this protocol also solves the robust H2 -ADDPMS with
bounded input and therefore solves H2 -ASSPS for the MAS. Using the same notation
as before, this relies on the fact that we have N1 such that
kT1 k2 < ?N1
M
which follows directly from the full-state coupling case. On the other hand we have N2
ED
such that
kT?4 k2 < ?N2
using that X? has an H2 norm of order ?. The rest of the proof is then as before in the
PT
case of H? -ASSPS.
5. Example
CE
In this section, we illustrate our results on a homogeneous MAS of N = 6 agents.
We consider the H? almost state synchronization problem via partial-state coupling.
AC
The agent model is given by:
*.?2
A = ... 2
.
,5
0 0+
//
2 0// ,
/
4 2-
*.0+/
B = ...3/// ,
. /
,2-
35
*.0+/
C 0 = ...1/// ,
. /
,1-
*.0+/
E = ...3/// ,
. /
,2-
ACCEPTED MANUSCRIPT
with disturbances
?1 = sin(3t), ?2 = cos(t), ?3 = 0.5,
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?4 = sin(2t) + 1, ?5 = sin(t), ?6 = cos(2t).
The communication topology is shown in Figure 1 with the Laplacian matrix
?1
2
?1
0
?2
0
0
0
?2
0
4
0
?3
2
0
0+
//
0 0//
/
0 0//
//
?2 0//
//
2 0//
/
?1 1-
AN
US
*.1
..
..0
..0
L = ..
..0
..
..0
.
,0
0
0
0
0
We design a controller of the form (44) based on an ARE-based method. The feedback
gain F? = ? ?1 B 0 P with P given by the algebraic Riccati equation (22) and K? = ? ?12 QC 0
given by the algebraic Riccati equation (43). When choosing ? = 0.3 and ? = 0.01, we
M
get the controller
PT
ED
0
0
*.?2
+/
*. 0 +/
.
/
?? i = .. 2 ?299.214 ?301.214// ? i + ...301.214/// ? i ,
.
/
.
/
203.194, 5 ?199.194 ?201.194,
ui = ?34.068 ?30.5702 ?27.0943 ? i ;
AC
CE
while when choosing ? = 0.01 and ? = 0.0001, the controller is
0
0 +
*.?2
*. 0 +/
//
.
?? i = .. 2 ?29999 ?30001// ? i + ...30001/// ? i ,
.
/
.
/
5
?19999
?20001
20003
,
- ,
ui = ?1022 ?917.1 ?812.8 ? i .
The results are shown in Figure 2. It is clear that when ? goes smaller, the H? norm
from the disturbance to the relative error between the states of the different agents gets
smaller. The controller inputs for all agents are shown in Figure 3.
36
=0.3 and =0.01
10
5
0
-5
-10
0
5
10
15
20
25
30
=0.01 and =0.0001
10
5
0
-5
-10
0
5
10
15
35
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State errors among 6 agents
Time
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State errors among 6 agents
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20
25
30
40
35
40
30
35
40
30
35
40
Time
M
Figure 2: State errors among N = 6 agents
=0.3 and =0.01
ED
500
0
-500
-1000
PT
Control inputs of 6 agents
1000
-1500
0
10
15
20
25
Time
=0.01 and =0.0001
10 4
1
AC
CE
Control inputs of 6 agents
5
0.5
0
-0.5
-1
-1.5
-2
0
5
10
15
20
25
Time
Figure 3: The controller inputs of N = 6 agents
37
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6. Conclusion
In this paper, we have studied H? and H2 almost state synchronization for MAS with
identical linear agents affected by external disturbances. The communication network
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is directed and coupled through agents? states or outputs. We have first developed the
necessary and sufficient conditions on agents? dynamics for the solvability of H? and
H2 almost state synchronization problems. Then, we have designed protocols to achieve
H? and H2 almost state synchronization among agents based on two methods. One
is ARE-based method and the other is ATEA-baed method. The future work could
be to extend the results of this paper to nonlinear agents, that is, H? and H2 almost
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state synchronization for MAS with identical nonlinear agents affected by external
disturbances.
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41
structure assignment method are developed such that the
impact of disturbances on the network disagreement dynamics, expressed in terms of
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the H? and H2 norm of the corresponding closed-loop transfer function, is reduced
to any arbitrarily small value. The protocol for full-state coupling is static, while for
PT
partial-state coupling it is dynamic.
Keywords: Multi-agent systems, H? and H2 almost state synchronization,
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Distributed control
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1. Introduction
Over the past decade, the synchronization problem of multi-agent system (MAS)
has received substantial attention because of its potential applications in cooperative
control of autonomous vehicles, distributed sensor network, swarming and flocking and
? Corresponding
author
Email addresses: A.A.Stoorvogel@utwente.nl (Anton A. Stoorvogel), saberi@eecs.wsu.edu
(Ali Saberi), zhangm@gonzaga.edu (Meirong Zhang), zhenwei.liu@wsu.edu (Zhenwei Liu)
Preprint submitted to Elsevier
August 14, 2018
ACCEPTED MANUSCRIPT
others. The objective of synchronization is to secure an asymptotic agreement on a
common state or output trajectory through decentralized protocols (see [1, 2, 3, 4] and
references therein).
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State synchronization inherently requires homogeneous MAS (i.e. agents have identical dynamics). Most works have focused on state synchronization based on diffu-
sive full-state coupling, where the agent dynamics progress from single- and double-
integrator dynamics (e.g. [5], [6], [7], [8], [9]) to more general dynamics (e.g. [10],
[11], [12], [13]). State synchronization based on diffusive partial-state coupling has
also been considered (e.g. [14], [15], [10], [16], [17], [18], [19]).
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Most research has focused on the idealized case where the agents are not affected
by external disturbances. In the literature where external disturbances are considered,
?-suboptimal H? design is developed for MAS to achieve H? norm from an external
disturbance to the synchronization error among agents less than an, a priori given, bound
?. In particular, [15], [20] considered the H? norm from an external disturbance to the
output error among agents. [21] considered the H? norm from an external disturbance
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to the state error among agents. These papers do not present an explicit methodology
for designing protocols. The papers [22] and [23] try to obtain an H? norm from a
ED
disturbance to the average of the states in a network of single or double integrators.
By contrast, [24] introduced the notion of H? almost synchronization for homoge-
neous MAS, where the goal is to reduce the H? norm from an external disturbance to
PT
the synchronization error, to any arbitrary desired level. But it requires an additional
layer of communication among distributed controllers, which is completely dispensed
in this paper. This work is extended later in [25], [26], and [27]. The paper [27], where
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heterogeneous MAS are considered, provides a solution for the case of right-invertible
agents with the addional objective beyond output synchronization that the agents track
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a regulated signal given to some or all of the agents. Although homogeneous MAS,
which are considered in this paper, are a subset of heterogeneous MAS, the results of
[27] cannot be directly applied to the case of full-state coupling since the agents are
not right-invertible. Secondly, the results for synchronization without regulation cannot
be obtained from results obtained for regulated synchronization. Thirdly, we consider
state synchronization instead of output synchronization in both full- and partial-state
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coupling. Finally, by restricting to homogeneous networks more explicit designs can
be obtained under weaker conditions.
In this paper, we will study H? almost state synchronization for a MAS with full-
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state coupling or partial-state coupling. We will also study H2 almost state synchronization, since it is closely related to the problems of H? almost state synchronization.
In H? we look at the worst case disturbance with the only constraints being the power,
while in H2 we only consider white noise disturbances which is a more restrictive
class. In both cases, disturbances or noises are restricted in the process, not in the
measurement. Our contribution in this paper is three-fold.
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? We obtain necessary and sufficient conditions for H? and H2 almost state synchronization for a MAS in the presence of external disturbances,
? We develop a protocol design for H? and H2 almost state synchronization based
on an algebraic Riccati equation (ARE) method,
? We develop a protocol design for H? and H2 almost state synchronization based
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on an asymptotic time-scale eigenstructure assignment (ATEA) method for the
full-state coupling case, and on the direct eigenstructure assignment method for
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the partial-state coupling case.
It is worth noting that our solvability conditions and protocol designs are developed for
a MAS associated with a set of network graphs. Specifically, only rough information
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of a network graph is utilized.
1.1. Notations and definitions
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Given a matrix A ? CmОn , A0 denotes its conjugate transpose, k Ak is the induced
2-norm. A square matrix A is said to be Hurwitz stable if all its eigenvalues are in
AC
the open left half complex plane. A ? B depicts the Kronecker product between A and
B. In denotes the n-dimensional identity matrix and 0n denotes n О n zero matrix;
sometimes we drop the subscript if the dimension is clear from the context. Given a
complex number ?, Re(?) is the real part of ? and Im(?) is the imaginary part of ?.
A weighted directed graph G is defined by a triple (V, E, A) where V = {1, . . . , N }
is a node set, E is a set of pairs of nodes indicating connections among nodes, and
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A = [ai j ] ? R N ОN is the weighting matrix, and ai j > 0 iff (i, j) ? E which denotes
an edge from node j to node i. In our case, we have aii = 0. A path from node i 1 to
i k is a sequence of nodes {i 1, . . . , i k } such that (i j+1, i j ) ? E for j = 1, . . . , k ? 1. A
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directed tree is a subgraph (subset of nodes and edges) in which every node has exactly
one parent node except for one node, called the root, which has no parent node. In this
case, the root has a directed path to every other node in the tree. A directed spanning
tree is a directed tree containing all the nodes of the graph. For a weighted graph G, a
matrix L = [` i j ] with
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PN
?
?
aik , i = j,
? k=1
`i j = ?
?
? ?ai j , i , j,
?
is called the Laplacian matrix associated with the graph G. The Laplacian L has all its
eigenvalues in the closed right half plane and at least one eigenvalue at zero associated
with right eigenvector 1. A specific class of graphs needed in this paper is presented
below:
N denote the set of directed graphs
Definition 1. For any given ? ? ? > 0, let G?,
?
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with N nodes that contain a directed spanning tree and for which the corresponding
Laplacian matrix L satisfies kLk < ? while its nonzero eigenvalues have a real part
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larger than or equal to ?.
2. Problem formulation
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Consider a MAS composed of N identical linear time-invariant agents of the form,
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x? i = Ax i + Bui + E?i ,
yi = C x i ,
(i = 1, . . . , N )
(1)
where x i ? Rn , ui ? Rm , yi ? R p are respectively the state, input, and output vectors
AC
of agent i, and ?i ? R? is the external disturbances.
The communication network provides each agent with a linear combination of its
own outputs relative to that of other neighboring agents. In particular, each agent
i ? {1, . . . , N } has access to the quantity,
?i =
N
X
j=1
ai j (yi ? y j ),
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(2)
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where ai j ? 0 and aii = 0 indicate the communication among agents. This communication topology of the network can be described by a weighted and directed graph G
with nodes corresponding to the agents in the network and the weight of edges given
rewritten as
?i =
N
X
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by the coefficient ai j . In terms of the coefficients of the Laplacian matrix L, ? i can be
(3)
`i j y j .
j=1
We refer to this case as partial-state coupling. Note that if C has full column rank then,
without loss of generality, we can assume that C = I, and the quantity ? i becomes
N
X
j=1
ai j (x i ? x j ) =
N
X
`i j x j .
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?i =
We refer to this case as full-state coupling.
(4)
j=1
If the graph G describing the communication topology of the network contains a
directed spanning tree, then it follows from [9, Lemma 3.3] that the Laplacian matrix L
has a simple eigenvalue at the origin, with the corresponding right eigenvector 1 and all
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the other eigenvalues are in the open right-half complex plane. Let ? 1, . . . , ? N denote
the eigenvalues of L such that ? 1 = 0 and Re(? i ) > 0, i = 2, . . . , N.
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ED
Let N be any agent and define x? i = x N ? x i and
*. x? 1 +/
.
x? = ... .. ///
.
/
x?
N
?1
,
-
and
*. ?1 +/
.
? = ... .. /// .
. /
,? N -
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Obviously, synchronization is achieved if x? = 0. That is
lim (x i (t) ? x N (t)) = 0,
t??
?i, ? {1, . . . , N ? 1}.
(5)
We denote by T? x? , the transfer function from ? to x?
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Remark 1. Agent N is not necessarily a root agent. Obviously, (5) is equivalent to the
condition that
lim (x i (t) ? x j (t)) = 0,
t??
?i, j ? {1, . . . , N }.
We formulate below four almost state synchronization problems for a network with
either H2 or H? almost synchronization.
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Problem 1. Consider a MAS described by (1) and (4). Let G be a given set of graphs
such that G ? G N . The H? almost state synchronization problem via full-state
coupling (in short H? -ASSFS) with a set of network graphs G is to find, if possible, a
ui = F (?)? i ,
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linear static protocol parameterized in terms of a parameter ?, of the form,
(6)
such that, for any given real number r > 0, there exists an ? ? such that for any ? ? (0, ? ? ]
and for any graph G ? G, (5) is satisfied for all initial conditions in the absence of
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disturbances and the closed loop transfer matrix T? x? satisfies
kT? x? k? < r.
(7)
Problem 2. Consider a MAS described by (1) and (3). Let G be a given set of graphs
such that G ? G N . The H? almost state synchronization problem via partial-state
coupling (in short H? -ASSPS) with a set of network graphs G is to find, if possible, a
linear time-invariant dynamic protocol parameterized in terms of a parameter ?, of the
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form,
?? i = Ac (?) ? i + Bc (?)? i ,
(8)
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ui = Cc (?) ? i + Dc (?)? i ,
where ? i ? Rn c , such that, for any given real number r > 0, there exists an ? ? such that
for any ? ? (0, ? ? ] and for any graph G ? G, (5) is satisfied for all initial conditions in
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the absence of disturbances and the closed loop transfer matrix T? x? satisfies (7).
Problem 3. Consider a MAS described by (1) and (4). Let G be a given set of graphs
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such that G ? G N . The H2 almost state synchronization problem via full-state
coupling (in short H2 -ASSFS) with a set of network graphs G is to find, if possible,
a linear static protocol parameterized in terms of a parameter ?, of the form (6) such
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that, for any given real number r > 0, there exists an ? ? such that for any ? ? (0, ? ? ]
and for any graph G ? G, (5) is satisfied for all initial conditions in the absence of
disturbances and the closed loop transfer matrix T? x? satisfies
kT? x? k2 < r.
6
(9)
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Problem 4. Consider a MAS described by (1) and (3). Let G be a given set of graphs
such that G ? G N . The H2 almost state synchronization problem via partial-state
coupling (in short H2 -ASSPS) with a set of network graphs G is to find, if possible,
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a linear time-invariant dynamic protocol parameterized in terms of a parameter ?, of
the form (8) such that, for any given real number r > 0, there exists an ? ? such that for
any ? ? (0, ? ? ] and for any graph G ? G, (5) is satisfied for all initial conditions in the
absence of disturbances and the closed loop transfer matrix T? x? satisfies (9).
Note that the problems of H? almost state synchronization and H2 almost state
synchronization are closely related. Roughly speaking, H2 almost synchronization is
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easier to achieve than H? almost synchronization. This is related to the fact that in H?
we look at the worst case disturbance with the only constraints being the power:
Z T
1
lim sup
?i0 (t)?i (t)dt < ?.
T ?? 2T ?T
while in H2 we only consider white noise disturbances which is a more restrictive class.
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3. MAS with full-state coupling
In this section, we establish a connection between the almost state synchronization
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among agents in the network and a robust H? or H2 almost disturbance decoupling
problem via state feedback with internal stability (in short H? or H2 -ADDPSS) (see
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[28]). Then, we use this connection to derive the necessary and sufficient condition and
design appropriate protocols.
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3.1. Necessary and sufficient condition for H? -ASSFS
The MAS system described by (1) and (4) after implementing the linear static
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protocol (6) is described by
x? i = Ax i + BF (?)? i + E?i ,
for i = 1, . . . , N. Let
*. x 1 +/
.
x = ... .. /// ,
. /
,x N -
*. ?1 +/
.
? = ... .. /// .
. /
,? N 7
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Then, the overall dynamics of the N agents can be written as
x? = (I N ? A + L ? BF (?))x + (I N ? E)?.
(10)
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We define the robust H? -ADDPSS with bounded input as follows. Given ? ? C,
there should exist M > 0 such that for any given real number r > 0, we can find a
parameterized controller
u = F (?)x
for the following subsystem,
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x? = Ax + ? Bu + B?,
(11)
(12)
such that for any ? ? ? the following hold:
1. The interconnection of the systems (12) and (11) is internally stable;
2. The resulting closed-loop transfer function T? x from ? to x has an H? norm less
than r.
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3. The resulting closed-loop transfer function T?u from ? to u has an H? norm less
than M.
ED
In the above, ? denotes all possible locations for the nonzero eigenvalues of the Laplacian matrix L when the graph varies over the set G. It is also important to note that M
is independent of the choice for r.
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In the following lemma we give a necessary condition for the H? -ASSFS. Moreover,
for sufficiency, we connect the H? -ASSFS problem to the robust H? -ADDPSS with
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bounded input problem which we will address later.
Lemma 1. Let G be a set of graphs such that the associated Laplacian matrices are
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uniformly bounded and let ? consist of all possible nonzero eigenvalues of Laplacian
matrices associated with graphs in G.
(Necessity) The H? -ASSFS for the MAS described by (1) and (4) given G is solvable
by a parameterized protocol ui = F (?)? i only if
im E ? im B.
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(13)
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(Sufficiency) The H? -ASSFS for the MAS described by (1) and (4) given G is solved
by a parameterized protocol ui = F (?)? i if the robust H? -ADDPSS with bounded input
for the system (12) with ? ? ? is solved by the parameterized controller u = F (?)x.
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Proof: Note that L has eigenvalue 0 with associated right eigenvector 1. Let
(14)
L = T SL T ?1,
with T unitary and SL the upper-triangular Schur form associated to the Laplacian
matrix L such that SL (1, 1) = 0. Let
*. ? 1 +/
.
? In )x = ... .. /// ,
. /
,? N -
*. ??1 +/
.
? I)? = ... .. ///
. /
,?? N -
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? := (T
?1
?? = (T
?1
where ? i ? Cn and ??i ? Cq . In the new coordinates, the dynamics of ? can be written
as
which is rewritten as
N
X
s1 j BF (?)? j + E ??1,
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?? 1 = A? 1 +
(15)
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??(t) = (I N ? A + SL ? BF (?))? + (T ?1 ? E)?,
j=2
?? i = ( A + ? i BF (?))? i +
N
X
s i j BF (?)? j + E ??i ,
(16)
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j=i+1
?? N = ( A + ? N BF (?))? N + E ?? N ,
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for i ? {2, . . . , N ? 1} where SL = [s i j ]. The first column of T is an eigenvector of
?
L associated to eigenvalue 0 with length 1, i.e. it is equal to ▒1/ N. Using this we
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obtain:
*. x N ? x 1 +/ *.*.?1 0 и и и 0 1+/
+/
..
.. /
..
..
// ....
//
x N ? x 2 / .. 0 ?1
.
.
. //
// (T ? I )?
/
.
.
/
x? = ...
=
?
I
n
n
..
.. /
// .... ..
//
..
..
..
/
.
.
.
.
0
.
// ....
//
//
.
x
?
x
0
и
и
и
0
?1
1
N ?1 , N
,,
= 0 V ? In ?,
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for some suitably chosen matrix V . Therefore we have
(17)
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T
*. ? 2 +/
.
x? = (V ? In ) ... .. /// ,
. /
,? N -
Note that since T is unitary, also the matrix T ?1 is unitary and the matrix V is uniformly
bounded. Therefore the H? norm of the transfer matrix from ? to x? can be made
arbitrarily small if and only if the H? norm of the transfer matrix from ?? to ? can be
made arbitrarily small.
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In order for the H? norm from ?? to ? to be arbitrarily small we need the H? norm
from ?? N to ? N to be arbitrarily small. From classical results (see [29, 30]) on H?
almost disturbance decoupling we find that this is only possible if (13) is satisfied.
Conversely, suppose u = F (?)x solves the robust H? -ADDPSS with bounded input
for (12) and assume (13) is satisfied. We show next that ui = F (?)? i solves the
H? -ASSFS for the MAS described by (1) and (4). Let X be such that E = BX.
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The fact that u = F (?)x solves the robust H? -ADDPSS with bounded input for (12)
implies that for small ? we have that A + ? BF (?) is asymptotically stable for all ? ? ?.
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In particular, A + ? i BF (?) is asymptotically stable for i = 2, . . . , N which guarantees
that ? i ? 0 for i = 2, . . . , N for zero disturbances and all initial conditions. Therefore
we have state synchronization.
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Next, we are going to show that for any r? > 0, we can choose ? sufficiently small
such that the transfer matrix from ?? to ? i is less than r? for i = 2, . . . , N. This guarantees
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that we can achieve (7) for any r > 0. We have that
T?? x (s) = (sI ? A ? ? BF (?)) ?1 B,
?
T?u
(s) = F (?)(sI ? A ? ? BF (?)) ?1 B.
For a given M and parameter ?, the following is satisfied
kT?? x k? < r? ? ,
?
kT?u
k? < M
for all ? ? ? where r? ? is a parameter depending on ? with the property that lim??0 r? ? = 0.
Denote ?i = F (?)? i .
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When i = N, it is easy to find that,
иии
0
and hence
?N
T??? N = T?u
0
X ,
kT??? N k? < r?,
kT??? N k? < M?N
provided
k X k r? ? < r?,
k X k M < M?N .
иии
0
X
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T??? N = T?? xN 0
(18)
Recall that we can make r? ? arbitrarily small by reducing ? without affecting the bound
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M. Assume
kT??? j k? < r?,
holds for j = i + 1, . . . , N. We have:
kT??? j k? < M? j
M
??
??
N
X
s i j T??? j (s) ???
T??? i (s) = T?? ix (s) ???ei ? X +
??
??
j=i+1
??
??
N
X
?
?i
T??? i (s) = T?u (s) ??ei ? X +
s i j T??? j (s) ???
??
??
j=i+1
where ei is a row vector of dimension N with elements equal to zero except for the ith
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component which is equal to 1. Since
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N
N
X
X
e ? X +
< k X k +
s
T
|s i j | M? j
i j ??? j i
j=i+1
j=i+1
?
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we find:
kT??? i k? < r?,
kT??? i k? < M?i
(19)
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provided:
*. k X k +
,
N
X
|s i j | M? j +/ r? ? < r?, *. k X k +
j=i+1
,
N
X
|s i j | M? j +/ M? < M?i .
j=i+1
-
(20)
Note that s i j depends on the graph in G but since the Laplacian matrices associated to
graphs in G are uniformly bounded we find that also the s i j are uniformly bounded. In
this way for any arbitrary r?, we can recursively obtain the bounds in (19) for i = 2, . . . , N
provided we choose ? sufficiently small such that the corresponding r? ? satisfies (18)
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and (20) for i = 2, . . . , N ? 1. Hence, we can choose ? sufficiently small such that
the transfer matrix from ?? to ? i is less than r? for i = 2, . . . , N. As noted before this
guarantees that we can achieve (7) for any r > 0.
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N for some given ?, ? > 0, we
For the case where the set of graphs G equals G?,
?
develop necessary and sufficient conditions for the solvability of the H? -ASSFS for
MAS as follows:
Theorem 1. Consider a MAS described by (1) and (4) with an associated graph from
N for some ?, ? > 0.
the set G = G?,
?
Then, the H? -ASSFS is solvable if and only if (13) is satisfied and ( A, B) is
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stabilizable.
Proof: From Lemma 1, we note that (13) is actually a necessary condition for H? -
ASSFS. Clearly, also ( A, B) stabilizable is a necessary condition. Sufficiency is a direct
result of Theorems 3 or Theorem 5 for H? -ASSFS.
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3.2. Necessary and sufficient conditions for H2 -ASSFS
We define the robust H2 -ADDPSS with bounded input as follows. Given ? ? C,
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there should exist M > 0 such that for any given real number r > 0, we can find a
parameterized controller (11) for the system, (12) such that the following holds for any
? ? ?:
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1. The interconnection of the systems (11) and (12) is internally stable;
2. The resulting closed-loop transfer function T? x from ? to x has an H2 norm less
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than r.
3. The resulting closed-loop transfer function T?u from ? to u has an H? norm less
AC
than M.
In the above, ? denotes all possible locations for the nonzero eigenvalues of the Laplacian matrix L when the graph varies over the set G. It is also important to note that M
is independent of the choice for r. Note that we need to consider two aspects in our
controller H2 disturbance rejection and robust stabilization (because of a set of network
N ). The latter translates in the H norm constraint from ? to u.
graphs G?,
?
?
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Lemma 2. Let G be a set of graphs such that the associated Laplacian matrices are
uniformly bounded and let ? consist of all possible nonzero eigenvalues of Laplacian
matrices associated with graphs in G.
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(Necessity) The H2 -ASSFS for the MAS described by (1) and (4) given G is solvable
by a parameterized protocol ui = F (?)? i only if (13) is satisfied.
(Sufficiency)The H2 -ASSFS for the MAS described by (1) and (4) given G is solvable
by a parameterized protocol ui = F (?)? i if the robust H2 -ADDPSS with bounded input
for the system (12) with ? ? ? is solved by the parameterized controller u = F (?)x.
Proof: The proof is similar to the proof of Lemma 1. This time we need the H2 norm
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from ?? N to ? N to be arbitrarily small and also H2 almost disturbance decoupling then
immediately yields that we need that (13) is satisfied.
The rest of the proof follows the same lines except that we require the H2 norm from
?? to ? j arbitrarily small while we keep the H? norm from ?? to ? j bounded. Recall that
for any two stable, strictly proper transfer matrices T1 and T2 we have:
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kT1T2 k2 ? kT1 k2 kT2 k?
which we need in the modifications of the proof of Lemma 1.
ED
N (with given ?, ? > 0), we develop
For the case with a set of graph G = G?,
?
necessary and sufficient conditions for the solvability of the H2 -ASSFS for MAS as
follows:
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Theorem 2. Consider a MAS described by (1) and (4) with an associated graph from
N for some ?, ? > 0.
the set G = G?,
?
CE
Then, the H2 -ASSFS is solvable if and only if (13) is satisfied and ( A, B) is stabilizable.
Proof: We have already noted before that (13) is actually a necessary condition for
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H2 -ASSFS. Clearly, also ( A, B) being stabilizable is a necessary condition. Sufficiency
for H2 -ASSFS, is a direct result of either Theorem 4 or Theorem 6.
3.3. Protocol design for H? -ASSFS and H2 -ASSFS
We present below two protocol design methods for both H? -ASSFS and H2 -ASSFS
problems. One relies on an algebraic Riccati equation (ARE), and the other is based on
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an asymptotic time-scale eigenstructure assignment (ATEA) method.
3.3.1. ARE-based method
( A, B) is stabilizable. We consider the protocol,
ui = ?F? i ,
where ? =
1
?
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Using an algebraic Riccati equation, we can design a suitable protocol provided
(21)
and F = ?B 0 P with P being the unique solution of the continuous-time
algebraic Riccati equation
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A0 P + P A ? 2 ?PBB 0 P + I = 0,
(22)
where ? is a lower bound for the real part of the non-zero eigenavlues of all Laplacian
N .
matrices associated with a graph in G = G?,
?
The main result regarding H? -ASSFS is stated as follows.
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Theorem 3. Consider a MAS described by (1) and (4) such that (13) is satisfied. Let
any real numbers ?, ? > 0 and a positive integer N be given, and hence a set of network
N be defined.
graphs G?,
?
ED
N is
If ( A, B) is stabilizable then the H? -ASSFS stated in Problem 1 with G = G?,
?
solvable. In particular, for any given real number r > 0, there exists an ? ? , such that
for any ? ? (0, ? ? ), the protocol (21) achieves state synchronization and the resulting
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system from ? to x i ? x j has an H? norm less than r for any i, j ? 1, . . . , N and for
CE
N .
any graph G ? G?,
?
Proof: Using Lemma 1, we know that we only need to verify that u = ?F x solves
the robust H? -ADDPSS with bounded input for the system (12) with ? ? ?. Given
AC
N , we know that ? ? ? implies Re ? ? ?. Clearly, the Laplacian matrices are
G ? G?,
?
uniformly bounded since kLk ? ?.
Consider the interconnection of (12) and u = ?F x. We define
V (x) = x 0 Px
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and we obtain:
V? = x 0 ( A ? ?? BB 0 P) 0 Px + ? 0 B 0 Px + x 0 P( A ? ?? BB 0 P)x + x 0 PB?
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= x 0 PBB 0 Px ? x 0 x ? 2? ?x 0 PBB 0 Px + 2x 0 PB?
? (1 ? ?? )x 0 PBB 0 Px ? x 0 x + ?? ? 0?
? ? ?2 ?u 0u ? x 0 x + ?? ? 0?
which implies that the system is asymptotically stable and the H? norm of the transfer
function from ? to x is less that ?/ ? while the H? norm of the transfer function from
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? to u is less that 2/ ? 2 . Therefore, u = ?F x solves the robust H? -ADDPSS with
bounded input for the system (12) as required.
For H2 -ASSFS we have the following classical result:
Lemma 3. Consider an asymptotically stable system:
M
p? = A1 p + B1 ?
The H2 norm from ? to p is less than ? if there exists a matrix Q such that:
ED
A1 Q + Q A10 + B1 B10 ? 0,
Q < ?I
The main result regarding H2 -ASSFS is stated as follows.
PT
Theorem 4. Consider a MAS described by (1) and (4) such that (13) is satisfied. Let
any real numbers ?, ? > 0 and a positive integer N be given, and hence a set of network
CE
N be defined.
graphs G?,
?
N is
If ( A, B) is stabilizable then the H2 -ASSFS stated in Problem 3 with G = G?,
?
solvable. In particular, for any given real number r > 0, there exists an ? ? , such that
AC
for any ? ? (0, ? ? ), the protocol (21) achieves state synchronization and the resulting
system from ? to x i ? x j has an H? norm less than r for any i, j ? 1, . . . , N and for
N .
any graph G ? G?,
?
Proof: Using Lemma 2, we know that we only need to verify that u = ?F x solves the
robust H2 -ADDPSS with bounded input for the system (12) with ? ? ?. We use the
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same feedback as in the proof of Theorem 3. In the proof of Theorem 3 it is already
shown that the closed loop system is asymptotically stable and the H? norm of the
transfer function from ? to u is bounded. The only remaining part of the proof is to
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show that the H2 norm from ? to x can be made arbitrarily small. Using the algebraic
Riccati equation it is easy to see that we have:
( A ? ?? BB 0 P) 0 P + P( A ? ?? BB 0 P) + ? ?PBB 0 P ? 0
for large ?. But then we have:
AN
US
Q? ( A ? ?? BB 0 P) 0 + ( A ? ?? BB 0 P)Q? + BB 0 ? 0
for Q? = ? ? ?1 P?1 . Then Lemma 3 immediately yields that we can make the H2 norm
from ? to x arbitrarily small by choosing a sufficiently small ?.
3.3.2. ATEA-based method
The ATEA-based design is basically a method of time-scale structure assignment
M
in linear multivariable systems by high-gain feedback [31]. In the current case, we do
not need the full structure presented in the above method. It is sufficient to note that
ED
there exists non-singular transformation matrix Tx ? RnОn (See [32, Theorem 1]) such
PT
that
x? 1
x? = *. +/ = Tx x,
, x? 2 -
(23)
CE
and the dynamics of x? is represented as
x?? 1 = A?11 x? 1 + A?12 x? 2,
x?? 2 = A?21 x? 1 + A?22 x? 2 + ? B?u + B??,
(24)
AC
with B? invertible, and that ( A, B) is stabilizable implies that ( A?11, A?12 ) is stabilizable.
Choose F1 such that A?11 + A?12 F1 is asymptotically stable. In that case a suitable
protocol for (1) is
(25)
ui = F? ? i ,
where F? is designed as
F? =
1 ?1 B?
F1
?
16
?I Tx
(26)
ACCEPTED MANUSCRIPT
The main result regarding H? -ASSFS is stated as follows. The result is basically
the same as Theorem 3 except for a different design protocol.
Theorem 5. Consider a MAS described by (1) and (4) such that (13) is satisfied. Let
CR
IP
T
any real numbers ?, ? > 0 and a positive integer N be given, and hence a set of network
N be defined.
graphs G?,
?
N is
If ( A, B) is stabilizable then the H? -ASSFS stated in Problem 1 with G = G?,
?
solvable. In particular, for any given real number r > 0, there exists an ? ? , such that
for any ? ? (0, ? ? ), the protocol (25) achieves state synchronization and the resulting
system from ? to x i ? x j has an H? norm less than r for any i, j ? 1, . . . , N and for
AN
US
N .
any graph G ? G?,
?
Proof: Similarly to the proof of Theorem 3, we only need to establish that u = F? x
solves the robust H? -ADDPSS with bounded input for the system (12) with ? ? ?.
N , we know that ? ? ? implies Re ? ? ?.
Given G ? G?,
?
After a basis transformation, the interconnection of (12) and u = F? x is equal to
M
the interconnection of (24) and (25). We obtain:
x?? 1 = A?11 x? 1 + A?12 x? 2,
Define
ED
? x?? 2 = (? A?21 + ?F1 ) x? 1 + (? A?22 ? ?I) x? 2 + ? B??.
PT
x? 1 = x? 1,
(27)
x? 2 = x? 2 ? F1 x? 1 .
CE
Then we can write this system (27) in the form:
x?? 1 = A?11 x? 1 + A?12 x? 2,
? x?? 2 = ? A?21 x? 1 + (? A?22 ? ?I) x? 2 + ? B??,
(28)
AC
where
A?11 = A?11 + A?12 F1,
A?12 = A?12,
A?21 = A?21 ? F1 A?11 + A?22 ? F1 A?12,
A?22 = A?22 ? F1 A?12 .
In the absence of the external disturbances, the above system (28) is asymptotically
stable for small enough ?.
17
ACCEPTED MANUSCRIPT
Since A?11 = A?11 + A?12 F1 is Hurwitz stable, there exists P > 0 such that the Lyapunov
0 P = ?I holds. For the dynamics x? , we define a Lyapunov function
equation P A?11 + A?11
1
V1 = x? 10 P x? 1 . Then the derivative of V1 can be bounded
CR
IP
T
0
V?1 ? ?k x? 1 k 2 + x? 20 A?12
P x? 1 + x? 10 P A?12 x? 2
? ?k x? 1 k 2 + 2 Re( x? 10 P A?12 x? 2 )
? ?k x? 1 k 2 + r 1 k x? 1 kk x? 2 k,
where 2kP A?12 k ? r 1 . Now define a Lyapunov function V2 = ? x? 20 x? 2 for the dynamics
AN
US
x? 2 . The derivative of V2 can then also be bounded.
V?2 ? ?2 Re(?)k x? 2 k 2 + 2? Re( x? 20 A?21 x? 1 ) + 2? x? 20 A?22 x? 2 + 2? Re( x? 20 B??)
? ?2 Re(?)k x? 2 k 2 + ?r 2 k x? 1 kk x? 2 k + ?r 3 k x? 2 k 2 + ?r 4 k?kk x? 2 k
? ? ?k x? 2 k 2 + ?r 2 k x? 1 k k x? 2 k + ?r 4 k?k k x? 2 k
for a small enough ?, where we choose r 2, r 3, r 4 such that
2k A?22 k ? r 3,
M
2k A?21 k ? r 2,
and
2k B?k ? r 4 .
ED
Let V = V1 + ?V2 for some ? > 0. Then, we have
V? ? ?k x? 1 k 2 + r 1 k x? 1 kk x? 2 k ? ? ?k x? 2 k 2 + ??r 2 k x? 1 kk x? 2 k + ??r 4 k?kk x? 2 k.
CE
PT
We have that
r 1 k x? 1 k k x? 2 k ? r 12 k x? 2 k 2 + 41 k x? 1 k 2,
??r 2 k x? 1 k k x? 2 k ? ? 2 ? 2 r 22 k x? 1 k 2 + 14 k x? 2 k 2,
??r 4 k?k k x? 2 k ? ? 2 ? 2 r 42 k?k 2 + 14 k x? 2 k 2 .
AC
Now we choose ? such that ? ? = 1 + r 12 and r 5 = ?r 4 . Then, we obtain
V? ? ? 12 k x? 1 k 2 ? 12 k x? 2 k 2 + ? 2 r 52 k?k 2
? ? 21 k x?k 2 + ? 2 r 52 k?k 2,
for a small enough ?. From the above, we have that kT? x? k? < 2?r 5 , which immediately
leads to kT? x k? < r for any real number r > 0 as long as we choose ? small enough.
18
ACCEPTED MANUSCRIPT
and hence:
T?u (s) = ? ?1 0
B??1 T? x? (s)
kT?u k? ? k B??1 kr 5 .
CR
IP
T
On the other hand:
Therefore, u = F? x solves the robust H? -ADDPSS with bounded input for the system
(12) as required.
The main result regarding H2 -ASSFS is stated as follows.
Theorem 6. Consider a MAS described by (1) and (4) such that (13) is satisfied. Let
AN
US
any real numbers ?, ? > 0 and a positive integer N be given, and hence a set of network
N be defined.
graphs G?,
?
N is
If ( A, B) is stabilizable then the H2 -ASSFS stated in Problem 1 with G = G?,
?
solvable. In particular, for any given real number r > 0, there exists an ? ? , such that
for any ? ? (0, ? ? ), the protocol (25) achieves state synchronization and the resulting
N .
graph G ? G?,
?
M
system from ? to x i ? x j has an H2 norm less than r for any i, j ? 1, . . . , N and for any
ED
Proof: Using Lemma 2, we know that we only need to verify that the feedback solves
the robust H2 -ADDPSS with bounded input for the system (12) with ? ? ?. We use
the same feedback as in the proof of Theorem 5. In the proof of Theorem 5 it is already
PT
shown that the closed loop system is asymptotically stable and the H? norm of the
transfer function from ? to u is bounded. The only remaining part of the proof is
CE
to show that the H2 norm from ? to x can be made arbitrarily small. This clearly is
equivalent to showing that the system (28) has an arbitrary small H2 norm from ? to
AC
x? 1 and x? 2 for sufficiently small ?. Choose Q such that
0
Q A?11
+ A?11 Q = ?I
19
ACCEPTED MANUSCRIPT
In that case we have:
?
0 + * ?Q
? /+.
?I - , 0
0 + 0
*0
? / Acl + .
?I ,0
?
? *.?
0
, ?( A?12 + A?21 Q)
for sufficiently small ? where:
A?11
Acl = *.
, A?21
A?12
A?22 ?
+/
?
? I-
?
0 )
?( A?12 + Q A?21
+/
? ??? I
-
AN
US
and we used that
0 +
/
B? B? 0?
CR
IP
T
?
?Q
*
Acl .
, 0
? + ? 0 ? 2 ?.
We then obtain for sufficiently small ? that:
?
0 + * ?Q
? /+.
?I - , 0
M
?
?Q
*
Acl .
, 0
0 + 0
*0
? / Acl + .
?I ,0
0 +
/ ?0
B? B? 0-
Then Lemma 3 immediately yields that we can make the H2 norm from ? to x arbitrarily
ED
small by choosing a sufficiently small ?.
4. MAS with partial-state coupling
PT
In this section, similar to the approach of the previous section, we show first
that the almost state synchronization among agents in the network with partial-state
CE
coupling can be solved by equivalently solving a robust H? or H2 almost disturbance
decoupling problem via measurement feedback with internal stability (in short H? or
H2 -ADDPMS). Then, we design a controller for such a robust H? or H2 -ADDPMS
AC
with bounded input.
20
ACCEPTED MANUSCRIPT
4.1. Necessary and sufficient condition for H? -ASSPS
The MAS system described by (1) and (3) after implementing the linear dynamical
protocol (8) is described by
for i = 1, . . . , N, where
(29)
AN
US
xi
x? i = *. +/ .
, ?i -
CR
IP
T
?
?
A BCc (?) +
BDc (?) +
E
?
?
?
?
/ x? i + *.
/? i + *. +/ ?i ,
x?? i = *.
?
?
?
?
Ac (?) ?
?
, 0
, Bc (?) ,0
?
?
? y =
C 0 x? i ,
i
?
?
?
?
?
N
?
X
?
?
?
?
?i =
`i j y j,
?
?
?
j=1
?
Define
M
*. x? 1 +/
.
x? = ... .. /// ,
. /
, x? N -
and
*. ?1 +/
.
? = ... .. /// ,
. /
,? N -
ED
A BCc (?) +
BDc (?) +
E
/ , B? = *.
/ , E? = *. +/ ,
A? = *.
, 0 Ac (?) , Bc (?) ,0Then, the overall dynamics of the N agents can be written as
0 .
(30)
PT
x?? = (I N ? A? + L ? B?C?) x? + (I N ? E?)?.
C? = C
We define a robust H? -ADDPMS with bounded input as follows. Given ? ? C,
CE
there should exist M > 0 such that for any given real number r > 0, we can find a
AC
parameterized controller
?? = Ac (?) ? + Bc (?)y,
u = Cc (?) ? + Dc (?)y,
(31)
where ? ? Rn c , for the following system,
x? = Ax + ? Bu + B?,
y = Cx
such that the following holds for any ? ? ?:
21
(32)
ACCEPTED MANUSCRIPT
1. The closed-loop system of (31) and (32) is internally stable
2. The resulting closed-loop transfer function T? x from ? to x has an H? norm less
than r.
CR
IP
T
3. The resulting closed-loop transfer function T?u from ? to u has an H? norm less
than M.
In the above, ? denotes all possible locations for the nonzero eigenvalues of the Laplacian matrix L when the graph varies over the set G. It is also important to note that M
is independent of the choice for r.
Lemma 4. Consider the system:
AN
US
In order to obtain our main result, we will need the following lemma:
x? = Ax + Bu + E?,
y = Cx
z=x
M
with ( A, B) stabilizable and (C, A) detectable. The H? -ADDPMS for the above system
is defined as the problem to find for any r > 0 a controller of the form (31) such that
the closed loop system is internally stable while the H? norm from ? to z is less than
ED
r. The H? -ADDPMS is solvable if and only if:
1. im E ? im B,
PT
2. ( A, E, C, 0) is left-invertible,
3. ( A, E, C, 0) is minimum-phase.
CE
Proof: From [29] we immediately obtain that the H? -ADDPMS is solvable if and only
AC
if:
1. im E ? im B
2. ( A, E, C, 0) is at most weakly non-minimum-phase and left-invertible.
3. For any ? > 0 and every invariant zero s0 of ( A, E, C, 0), there exists a matrix K
such that sI ? A ? BKC is invertible and
k(s0 I ? A ? BKC) ?1 E k? < ?
22
(33)
ACCEPTED MANUSCRIPT
Choose a suitable basis such that:
A11
A = *.
, A21
A12 +
/,
A22 -
B1
B = *. +/ ,
, B2 -
E1
E = *. +/ ,
, E2 -
C= I
0
*. sI ? A11
..
.. ?A21
I
,
?A12
sI ? A22
0
E1 +
//
E2 //
/
0-
CR
IP
T
Assume s0 is an imaginary axis zero of ( A, E, C, 0). In that case the rank of the matrix:
AN
US
drops for s = s0 . This implies the existence of p , 0 and q , 0 such that
*. ?A12 +/ p = *. E1 +/ q.
, s0 I ? A22 , E2 -
The final condition for H? almost disturbance decoupling requires for any ? > 0 the
(s0 I ? A ? BKC) ?1 Eq
s0 I ? A11 ? B1 K
= *.
, ?A21 ? B2 K
PT
0
= *. +/ ,
, p-
?1
?A12 +
/
s0 I ? A22 -
ED
s0 I ? A11 ? B1 K
= *.
, ?A21 ? B2 K
M
existence of a K such that (33) is satisfied. However:
?1
?A12 +
/
s0 I ? A22 -
*. ?A12 +/ p
, s0 I ? A22 -
*. s0 I ? A11 ? B1 K
, ?A21 ? B2 K
?A12 + *0+
/. /
s0 I ? A22 - , p-
kpk > ?kqk.
AC
CE
which yields a contradiction if ? is such that
Therefore we cannot have any invariant zeros in the imaginary axis. In other words,
the system ( A, E, C, 0) needs to be minimum-phase instead of weakly minimum-phase.
Conversely, if ( A, E, C, 0) is minimum-phase it is easy to verify that for any ? > 0 there
exists K such that (33) is satisfied.
23
ACCEPTED MANUSCRIPT
Theorem 7. Consider the MAS described by (1) and (3) with ( A, B) stabilizable and
(C, A) detectable.
N be defined. Then,
(Part I) Let ?, ? > 0 be given such that a set of graphs G?,
?
protocol (8) for any ? > ? > 0 if and only if
im E ? im B
CR
IP
T
N is solvable by a parameterized
the H? -ASSPS for the MAS with any graph G ? G?,
?
(34)
while ( A, E, C, 0) is minimum phase and left-invertible.
(Part II) Let G be a set of graphs such that the associated Laplacian matrices are
AN
US
uniformly bounded and let ? consist of all possible nonzero eigenvalues of Laplacian
matrices associated with graphs in G. Then, the H? -ASSPS for the MAS with any
graph G ? G is solved by a parameterized protocol (8) if the robust H? -ADDPMS with
bounded input for the system (32) with ? ? ? is solved by the parameterized controller
(31).
*. ? 1 +/
.
? In ) x? = ... .. /// ,
. /
,? N -
?? = (T
?1
ED
? := (T
?1
M
Proof: By using L = T SL T ?1 , we define
*. ??1 +/
.
? I)? = ... .. ///
. /
,?? N -
where ? i ? Cn+n c and ??i ? Cq . In the new coordinates, the dynamics of ? can be
PT
written as
??(t) = (I N ? A? + SL ? B?C?? + (T ?1 ? E)?,
(35)
AC
CE
which is rewritten as
?? 1 = A?? 1 +
N
X
s1 j B?C?? j + E? ??1,
j=2
?? i = ( A? + ? i B?C?)? i +
N
X
s i j B?C?? j + E? ??i ,
j=i+1
?? N = ( A? + ? N B?C?)? N + E? ?? N ,
with i ? {2, . . . , N ? 1} where
0
E? = *. +/ ,
,E -
SL = [s i j ].
24
(36)
ACCEPTED MANUSCRIPT
As in the case of full-state coupling, we can show that:
(37)
CR
IP
T
*. ? 2 +/
.
x? = (V ? In ) ... .. /// ,
. /
,? N -
for some suitably chosen matrix V which is uniformly bounded. Therefore the H?
norm of the transfer matrix from ? to x? can be made arbitrarily small if and only if the
H? norm of the transfer matrix from ?? to ? can be made arbitrarily small.
In order for the H? norm from ?? to ? to be arbitrarily small we need the H? norm
AN
US
from ?? N to ? N to be arbitrarily small. In other words, the robust H? -ADDPMS with
bounded input has to be solvable for the system
x? = Ax + ? Bu + E?,
y = Cx
From the results of Lemma 4, we find that this is only possible if (34) is satisfied and
( A, E, C, 0) is left-invertible and minimum phase.
M
On the other hand, suppose (31) solves the robust H? -ADDPMS with bounded input
of (32) and assume (34) is satisfied. We need to show that (8) solves the H? -ASSFS for
ED
the MAS described by (1) and (3). This follows directly from arguments very similar
to the approach used in the proof of Lemma 1.
PT
4.2. Necessary and sufficient condition for H2 -ASSPS
The MAS system described by (1) and (3) after implementing the linear dynamical
CE
protocol (8) is described by (29) for i = 1, . . . , N, and, as before, the overall dynamics
AC
of the N agents can be written as
x?? = (I N ? A? + L ? B?C?) x? + (I N ? E?)?.
(38)
We define a robust H2 -ADDPMS with bounded input as follows. Given ? ? C,
there should exist M > 0 such that for any given real number r > 0, we can find a
parameterized controller
?? = Ac (?) ? + Bc (?)y,
u = Cc (?) ? + Dc (?)y,
25
(39)
ACCEPTED MANUSCRIPT
where ? ? Rn c , for the following system,
x? = Ax + ? Bu + B?,
(40)
such that the following holds for any ? ? ?:
CR
IP
T
y = Cx
1. The closed-loop system of (39) and (40) is internally stable
2. The resulting closed-loop transfer function T? x from ? to x has an H2 norm less
than r.
than M.
AN
US
3. The resulting closed-loop transfer function T?u from ? to u has an H? norm less
In the above, ? denotes all possible locations for the nonzero eigenvalues of the Laplacian matrix L when the graph varies over the set G. It is also important to note that M
is independent of the choice for r.
The following lemma, provides a necessary condition for the H2 -ADDPMS:
M
Lemma 5. Consider the system:
x? = Ax + Bu + E?,
ED
y = Cx
z=x
PT
with ( A, B) stabilizable and (C, A) detectable. The H2 -ADDPMS for the above system
is defined as the problem to find for any r > 0 a controller of the form (39) such that
the closed loop system is internally stable while the H? norm from ? to z is less than
CE
r. The H2 -ADDPMS is solvable only if:
AC
1. im E ? im B
2. ( A, E, C, 0) is at most weakly non-minimum-phase and left-invertible.
Proof: This follows directly from [29].
Theorem 8. Consider the MAS described by (1) and (3) with ( A, B) stabilizable and
(C, A) detectable.
26
ACCEPTED MANUSCRIPT
N be defined. Then,
(Part I) Let ?, ? > 0 be given such that a set of graphs G?,
?
N is solvable by a parameterized
the H2 -ASSPS for the MAS with any graph G ? G?,
?
protocol (8) for any ? > ? > 0 only if
(41)
CR
IP
T
im E ? im B
while ( A, E, C, 0) is at most weakly non-minimum phase and left-invertible .
(Part II) Let G be a set of graphs such that the associated Laplacian matrices are
uniformly bounded and let ? consist of all possible nonzero eigenvalues of Laplacian
matrices associated with graphs in G. Then, the H2 -ASSPS for the MAS with any
AN
US
graph G ? G is solved by a parameterized protocol (8) if the robust H2 -ADDPMS with
bounded input for the system (40) with ? ? ? is solved by the parameterized controller
(39).
Proof: Similar, to the proof of Theorem 7, the dynamics can be written in the form (36).
Using (37), we note the H2 norm of the transfer matrix from ? to x? can be made
made arbitrarily small.
M
arbitrarily small if and only if the H2 norm of the transfer matrix from ?? to ? can be
In order for the H2 norm from ?? to ? to be arbitrarily small we need the H2 norm
ED
from ?? N to ? N to be arbitrarily small. In other words, the robust H2 -ADDPMS with
bounded input has to be solvable for the system
PT
x? = Ax + ? Bu + E?,
y = Cx
From the results of Lemma 5, we find that this is only possible if (41) is satisfied,
CE
( A, E, C, 0) is left-invertible and at most weakly non-minimum phase.
On the other hand, suppose (39) solves the robust H2 -ADDPMS with bounded input
of (40) and assume (41) is satisfied. We need to show that (8) solves the H2 -ASSFS for
AC
the MAS described by (1) and (3). This follows directly from arguments very similar
to the approach used in the proof of Lemma 1.
4.3. Protocol design for H? -ASSPS
We present below two protocol design methods based on robust stabilization for the
case E = B and therefore the case where ( A, B, C, 0) is minimum-phase. One relies
27
ACCEPTED MANUSCRIPT
on an algebraic Riccati equation (ARE) method, and the other is based on the direct
eigenstructure assignment method.
4.3.1. ARE-based method
CR
IP
T
Using an algebraic Riccati equation, we can design a suitable protocol. As in the
full-state coupling case, we choose F = ?B 0 P with P = P 0 > 0 being the unique
solution of the continuous-time algebraic Riccati equation
A0 P + P A ? 2 ?PBB 0 P + I = 0,
(42)
where ? is a lower bound for the real part of the non-zero eigenvalues of all Laplacian
AN
US
N .
matrices associated with a graph in G?,
?
Since ( A, B, C, 0) is minimum-phase then for any ? there exists ? small enough such
that
AQ + Q A0 + BB 0 + ? ?4 Q2 ? ??2 QC 0CQ = 0
(43)
has a solution Q > 0. We then consider the following protocol:
M
?? i = ( A + K? C) ? i ? K? ? i ,
ui = F? ? i ,
where
F? = ? ?1 B 0 P,
(44)
ED
K? = ? ?12 QC 0
The main result in this section is stated as follows.
PT
Theorem 9. Consider a MAS described by (1) and (3) with ( A, B) stabilizable and
(C, A) detectable. Let any real numbers ?, ? > 0 and a positive integer N be given,
CE
N be defined.
and hence a set of network graphs G?,
?
N is solvable. In particular, for
The H? -ASSPS stated in Problem 2 with G = G?,
?
any given real number r > 0, there exists an ? ? , such that for any ? ? (0, ? ? ), the
AC
protocol (44) achieves state synchronization and the resulting system from ? to x i ? x j
N .
has an H? norm less than r for any i, j ? 1, . . . , N and for any graph G ? G?,
?
Proof: Using Theorem 7, we know that we only need to verify that
?? = ( A + K? C) ? ? K? y,
u = F? ?,
28
(45)
ACCEPTED MANUSCRIPT
solves the robust H? -ADDPMS with bounded input for the system (32) with ? ? ?.
N , we know that ? ? ? implies Re ? ? ?. Obviously A + BF and
Given G ? G?,
?
?
A + K? C are both asymptotically stable by construction and hence the intersection of
equal to:
T? x = I
where:
I
*
0 .
,T?3
?1
?T2 +
/
I -
*. T1 +/
,?T?4 -
AN
US
T1 (s) = (sI ? A ? ? BF? ) ?1 B
CR
IP
T
(32) and (45) is asymptotically stable. The closed loop transfer function from ? to x is
(46a)
T2 (s) = ?(sI ? A ? ? BF? ) ?1 BF?
(46b)
T?3 (s) = ?(sI ? A ? K? C + ? BF? ) ?1 BF?
(46c)
T?4 (s) = (sI ? A ? K? C + ? BF? ) ?1 B
(46d)
As argued in the proof of Theorem 3, we have:
?
,
?
kT2 k? <
M
kT1 k? <
2|?|
2?
? 2.
2
?
?
PT
where
ED
On the other hand, (43) implies according to the bounded real lemma:
kT3 k? < ? 2
T3 (s) = (sI ? A ? K? C) ?1 B
CE
Note that:
T?3 = ?(I + ?T3 F? ) ?1T3 F?
AC
which yields, using (47), that
kT?3 k? < (1 ? ?M1 ) ?1 ?M1 < 2?M1
for small ? where M1 is such that:
|?|kB 0 Pk < ?kB 0 Pk = M1
29
(47)
ACCEPTED MANUSCRIPT
The above yields:
kT? x k? < ?M2
to:
I
*
F? .
,T?3
T?u = F?
?T2 +
/
I -
which yields using similar arguments as above that:
?1
*. T1 +/
,?T?4 -
kT?u k? < M3
CR
IP
T
for some suitable constant M2 . The closed loop transfer function from ? to u is equal
AN
US
for some suitable constant M3 independent of ?. Therefore the H? norm of the transfer
matrix T? x becomes arbitrarily small for sufficiently small ? while the H? norm of the
transfer matrix T?u remains bounded.
4.3.2. Direct method
For ease of presentation, we only consider the case q = 1, i.e. the case where we
M
have a scalar measurement. We consider the state feedback gain F? given in (26), that
is
ED
F? =
where Tx is defined in (23).
1 ?1 B?
F1
?
?I Tx ,
Next, we consider the observer design. Note that the system ( A, B, C, 0) is minimum-
PT
phase and left-invertible. In that case there is a nonsingular matrix ?x such that, by
AC
CE
defining x? = ?x x, we obtain the system
x? a = Aa x a + L ad y,
x? d = Ad x d + Bd (u + ? + Eda x a + Edd x d ),
y = Cd x d .
where
xa
x? = ?x x = *. +/ ,
,xd -
30
(48)
ACCEPTED MANUSCRIPT
with x a ? Rn?? and x d ? R ? and where the matrices Ad ? R ?О? , Bd ? R ?О1 , and
Cd ? R1О? have the special form
1
..
.
иии
иии
иии
..
.
0
0
0+
.. ///
./
// ,
1//
/
0-
*.0+/
.. .. //
.
Bd = ... /// ,
..0//
. /
,1-
Cd = 1 0
0 .
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IP
T
*.0
.. ..
.
Ad = ...
..0
.
,0
иии
(49)
Furthermore, the eigenvalues of Aa are the invariant zeros of ( A, B, C) and hence
Aa is asymptotically stable. The transformation ?x can be calculated using available
AN
US
software, either numerically [33] or symbolically [34].
Next, define a high-gain scaling matrix
S? := diag(1, ? 2, . . . , ? 2??2 ),
and define the output injection matrix
M
0
+/ .
K? = ?x *.
?2
?1
,? S? K -
(50)
(51)
protocol:
ED
where K is such that Ad + Bd K is asymptotically stable. We then consider the following
?? i = ( A + K? C) ? i ? K? ? i ,
(52)
PT
ui = F? ? i ,
The main result in this section is stated as follows.
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Theorem 10. Consider a MAS described by a SISO system (1) and (3). Let any real
numbers ?, ? > 0 and a positive integer N be given, and hence a set of network graphs
AC
N be defined.
G?,
?
N is
If ( A, B) is stabilizable then the H? -ASSPS stated in Problem 2 with G = G?,
?
solvable. In particular, for any given real number r > 0, there exists an ? ? , such that
for any ? ? (0, ? ? ), the protocol (52) achieves state synchronization and the resulting
system from ? to x i ? x j has an H? norm less than r for any i, j ? 1, . . . , N and for
N .
any graph G ? G?,
?
31
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Proof: We use a similar argument as in the proof of Theorem 10. We know that we
only need to verify that
?? = ( A + K? C) ? ? K? y,
(53)
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u = F? ?,
solves the robust H? -ADDPMS with bounded input for the system (32) with ? ? ?.
N , we know that ? ? ? implies Re ? ? ?. Obviously A + BF and
Given G ? G?,
?
?
A + K? C are both asymptotically stable by construction and hence the intersection of
(32) and (45) is asymptotically stable. As in the proof of Theorem 9, the closed loop
transfer function from ? to x is equal to:
T? x = I
I
*
0 .
,T?3
?1
?T2 +
/
I -
*. T1 +/
,?T?4 -
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(54)
where, as before, we use the definitions in (46) but with our modified F? and K? . As
argued in the proof of Theorem 5, we have:
M
kT1 k? < M1 ?,
kT2 k? < M2 .
for suitable constants M1, M2 > 0. Finally
ED
sI ? Aa
T3 (s) = ?x *.
,?Bd Eda
PT
where
?1
L ad Cd +
/
Z1 -
*. 0 +/
, Bd -
Z1 = sI ? Ad ? ? ?2 S??1 KCd ? Bd Edd
AC
CE
We obtain:
with
I
T3 (s) = ? ?x *.
,0
2n
0 + * sI ? Aa
/.
S??1 - ,? 2n Bd Eda
?1
L ad Cd +
/
Z2 -
*. 0 +/
, Bd -
Z2 = sI ? ? ?2 Ad ? ? ?2 KCd + ? 2n Bd Edd S??1,
using that
? ?2 Ad = S? Ad S??1,
S? Bd = ? 2n Bd
32
and
Cd S??1 = Cd .
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Note that Ed? is bounded for ? < 1. Next, we note that:
L ad Cd
+/
?2
sI ? ? ( Ad + KCd ) -
?1
*. 0 +/
, Bd -
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sI ? Aa
X? (s) = *.
, 0
?(sI ? Aa ) ?1 L ad + 2
/ (? sI ? Ad ? KCd ) ?1 Bd
= ? 2 *.
I
,
-
From the above we can easily conclude that there exists M such that k X? k? < M? 2 .
We have:
where
0 +
/ (I + ? 2 X? Ed? ) ?1 X?
S??1 -
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I
T3 = ? 2n ?x *.
,0
Ed? = ? 2n?2 Eda
? 2n?2 Edd S??1
which is clearly bounded for ? < 1. This clearly implies, using our bounds for X? and
Ed? , that there exists M3 > 0 such that:
for small ? since
M
kT3 k ? ? 2 M3
PT
ED
I
? 2n *.
,0
Our bound for T3 guarantees that
0 +
/ < ? 2 I.
S??1 -
kT?3 k? < ?M4,
kT?4 k? < ?M5,
CE
for suitable M4 and M5 . Moreover
kF? k < ? ?1 M0
AC
Given our bounds, we immediately obtain from (54) that there exists M6 such that
kT? x k? < M6 ?.
The closed loop transfer function from ? to u is equal to:
T?u = F?
I
*
F? .
,T?3
33
?T2 +
/
I -
?1
*. T1 +/
,?T?4 -
ACCEPTED MANUSCRIPT
which yields, using similar arguments as above, that:
kT?u k? < M7
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for some suitable constant M7 independent of ?. In other words, the transfer function
from ? to x is arbitrarily small for sufficiently small ? while the transfer function from
? to u is bounded which completes the proof.
4.4. Protocol design for H2 -ASSFS
We present below two protocol design methods based on robust stabilization for the
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US
case E = B. The necessary condition provided earlier shows that ( A, B, C, 0) need only
be at most weakly non-minimum-phase. The following designs are provided under the
stronger assumption that ( A, B, C, 0) is minimum-phase.
4.4.1. ARE-based method
We consider the protocol (44) already used in the case of H? -ASSFS. It is easy to
M
verify using a similar proof that this protocol also solves the robust H2 -ADDPMS with
bounded input and therefore solves H2 -ASSPS for the MAS. Using the same notation
ED
as before, this relies on the fact that we have N1 such that
kT1 k2 < ?N1
PT
which follows directly from the full-state coupling case. On the other hand we have N2
CE
such that
kT3 k2 < ? 2 N2
AC
since Q ? 0 for ? ? 0 and
( A ? KC)Q + Q( A ? KC) 0 + BB 0 ? 0.
It is then easily shown that
kT?4 k2 < ? 2 N3
for some N3 > 0. The rest of the proof is then as before in the case of H? -ASSPS.
34
Figure 1: The communication topology
4.4.2. Direct method
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We consider the protocol (52) already used in the case of H? -ASSFS. It is easy to
AN
US
verify using a similar proof that this protocol also solves the robust H2 -ADDPMS with
bounded input and therefore solves H2 -ASSPS for the MAS. Using the same notation
as before, this relies on the fact that we have N1 such that
kT1 k2 < ?N1
M
which follows directly from the full-state coupling case. On the other hand we have N2
ED
such that
kT?4 k2 < ?N2
using that X? has an H2 norm of order ?. The rest of the proof is then as before in the
PT
case of H? -ASSPS.
5. Example
CE
In this section, we illustrate our results on a homogeneous MAS of N = 6 agents.
We consider the H? almost state synchronization problem via partial-state coupling.
AC
The agent model is given by:
*.?2
A = ... 2
.
,5
0 0+
//
2 0// ,
/
4 2-
*.0+/
B = ...3/// ,
. /
,2-
35
*.0+/
C 0 = ...1/// ,
. /
,1-
*.0+/
E = ...3/// ,
. /
,2-
ACCEPTED MANUSCRIPT
with disturbances
?1 = sin(3t), ?2 = cos(t), ?3 = 0.5,
CR
IP
T
?4 = sin(2t) + 1, ?5 = sin(t), ?6 = cos(2t).
The communication topology is shown in Figure 1 with the Laplacian matrix
?1
2
?1
0
?2
0
0
0
?2
0
4
0
?3
2
0
0+
//
0 0//
/
0 0//
//
?2 0//
//
2 0//
/
?1 1-
AN
US
*.1
..
..0
..0
L = ..
..0
..
..0
.
,0
0
0
0
0
We design a controller of the form (44) based on an ARE-based method. The feedback
gain F? = ? ?1 B 0 P with P given by the algebraic Riccati equation (22) and K? = ? ?12 QC 0
given by the algebraic Riccati equation (43). When choosing ? = 0.3 and ? = 0.01, we
M
get the controller
PT
ED
0
0
*.?2
+/
*. 0 +/
.
/
?? i = .. 2 ?299.214 ?301.214// ? i + ...301.214/// ? i ,
.
/
.
/
203.194, 5 ?199.194 ?201.194,
ui = ?34.068 ?30.5702 ?27.0943 ? i ;
AC
CE
while when choosing ? = 0.01 and ? = 0.0001, the controller is
0
0 +
*.?2
*. 0 +/
//
.
?? i = .. 2 ?29999 ?30001// ? i + ...30001/// ? i ,
.
/
.
/
5
?19999
?20001
20003
,
- ,
ui = ?1022 ?917.1 ?812.8 ? i .
The results are shown in Figure 2. It is clear that when ? goes smaller, the H? norm
from the disturbance to the relative error between the states of the different agents gets
smaller. The controller inputs for all agents are shown in Figure 3.
36
=0.3 and =0.01
10
5
0
-5
-10
0
5
10
15
20
25
30
=0.01 and =0.0001
10
5
0
-5
-10
0
5
10
15
35
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State errors among 6 agents
Time
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State errors among 6 agents
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20
25
30
40
35
40
30
35
40
30
35
40
Time
M
Figure 2: State errors among N = 6 agents
=0.3 and =0.01
ED
500
0
-500
-1000
PT
Control inputs of 6 agents
1000
-1500
0
10
15
20
25
Time
=0.01 and =0.0001
10 4
1
AC
CE
Control inputs of 6 agents
5
0.5
0
-0.5
-1
-1.5
-2
0
5
10
15
20
25
Time
Figure 3: The controller inputs of N = 6 agents
37
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6. Conclusion
In this paper, we have studied H? and H2 almost state synchronization for MAS with
identical linear agents affected by external disturbances. The communication network
CR
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is directed and coupled through agents? states or outputs. We have first developed the
necessary and sufficient conditions on agents? dynamics for the solvability of H? and
H2 almost state synchronization problems. Then, we have designed protocols to achieve
H? and H2 almost state synchronization among agents based on two methods. One
is ARE-based method and the other is ATEA-baed method. The future work could
be to extend the results of this paper to nonlinear agents, that is, H? and H2 almost
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US
state synchronization for MAS with identical nonlinear agents affected by external
disturbances.
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[1] H. Bai, M. Arcak, J. Wen, Cooperative control design: a systematic, passivity-
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based approach, Communications and Control Engineering, Springer Verlag,
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[2] M. Mesbahi, M. Egerstedt, Graph theoretic methods in multiagent networks,
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[5] R. Olfati-Saber, J. Fax, R. Murray, Consensus and cooperation in networked
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