P. Wielanda · F. Allgöwera
An Internal Model Principle for Consensus in
Heterogeneous Linear Multi-Agent Systems
Stuttgart, December 2009
a
Institute for Systems Theory and Automatic Control
University of Stuttgart
Pfaffenwaldring 9
70569 Stuttgart
Germany
{wieland,allgower}@ist.uni-stuttgart.de
www.ist.uni-stuttgart.de
Abstract The problem of reaching consensus in a heterogeneous multi-agent system is considered.
The agents are modeled as linear time-invariant systems with potentially different state dimension
and different dynamics. The interconnection topology between the agents is modeled as a directed
and weighted graph. We propose an internal model principle for consensus translating in necessary
conditions for existence of solutions to the output and state consensus problem.
Preprint Series
Stuttgart Research Centre for Simulation Technology (SRC SimTech)
SimTech – Cluster of Excellence
Pfaffenwaldring 7a
70569 Stuttgart
[email protected]
www.simtech.uni-stuttgart.de
Issue No. 2010-54
An Internal Model Principle for Consensus
in Heterogeneous Linear Multi-Agent
Systems
Peter Wieland ∗ Frank Allgöwer ∗
∗
Institute for Systems Theory and Automatic Control, University of
Stuttgart, Germany (e-mail: {wieland,allgower}@ist.uni-stuttgart.de)
Abstract: The problem of reaching consensus in a heterogeneous multi-agent system is
considered. The agents are modeled as linear time-invariant systems with potentially different
state dimension and different dynamics. The interconnection topology between the agents is
modeled as a directed and weighted graph. We propose an internal model principle for consensus
translating in necessary conditions for existence of solutions to the output and state consensus
problem.
Keywords: Multi-Agent System; Consensus; Internal Model Principle
1. INTRODUCTION
Recently, the consensus problem among multi-agent systems (MAS) has received a lot of attention in the literature. The interest in this problem is mainly motivated by
the huge variety of applications in various areas, e.g. unmanned aerial vehicles, mobile robots, satellites, formation
control, and sensor networks, to name only a few. For a
nice overview of recent results on the topic, see Ren et al.
(2007); Olfati-Saber et al. (2007); Tanner et al. (2007);
Wieland et al. (2008) and the references therein. Given
a group of N dynamically decoupled agents with outputs
zi ∈ Rq , i = 1, . . . , N , the problem under consideration
is to find local control laws which take a weighted sum of
P
differences δi = N
j=1 aij (zi − zj ), i = 1, . . . , N as inputs
and achieve consensus, i.e. limt→∞ kzi − zj k = 0, i, j =
1, . . . , N , where aij , i, j = 1, . . . , N are positive scalars
and δi ∈ Rq , i = 1, . . . , N .
Numerous results have been proposed on consensus for
MASs. Originally, mainly single and double integrator
dynamics have been considered (Jadbabaie et al., 2003;
Tanner et al., 2003). Results for integrator chains of length
greater than two have been dealt with in (Ren et al., 2006).
Solutions for general LTI systems have been proposed in
Fax and Murray (2004) and Wieland et al. (2008). It is
common to all the aforementioned results that they deal
with identical agent dynamics, the so-called homogeneous
MAS consensus problem. However, in reality, no two systems are perfectly the same. Even systems which are structurally the same may vary in parameter values, e.g. due
to different friction and damping coefficients in dynamical
systems, different or maybe changing masses, change of
material properties caused by abrasion, and the like. There
exist some cooperative control approaches, which deal with
different agent dynamics by using agent level controls
which transform the agents in some canonical form which
is the same for all agents (Qu et al., 2008). However, this
is not always possible due to required system properties
or information constraints. Therefore, it is an interesting
question to ask what is the minimum requirement on the
agents such that consensus may be possible.
For the case of one leader with one or more independent
followers, the problem corresponds to the well known
tracking problem. A famous result related to this problem
is the Internal Model Principle of Control Theory (Francis
and Wonham, 1976). The problem can be seen as a special
case of the output regulation problem, which has been
solved e.g. in Knobloch et al. (1993) and Byrnes et al.
(1997). However, the existing theory is limited to the very
specific case of one exosystem being tracked by one plant
by means of a feedback controller.
In MAS consensus, the situation is more complex. A
MAS is generally a group of dynamical systems whose
interconnections are modeled using graphs (cf. de Gennaro
and Jadbabaie (2006); Ren et al. (2006); Ren and Atkins
(2005); Olfati-Saber (2006)). In the most general setup,
arbitrary directed and weighted graphs may be used to
model the interconnection topology. As a consequence,
there is no exosystem to be tracked by individual agents
but the agents are mutually influencing each other. To the
authors’ best knowledge, there exists no internal model
principle in the literature for such a situation.
In view of the current status of consensus research, this
paper presents an internal model principle for consensus
in heterogeneous MAS. This internal model principle leads
to necessary conditions for solvability of the consensus
problem. It applies to state consensus as well as output
consensus. The conditions will be imposed on agent and
controller dynamics. No assumptions are made on the
graph representing the interconnection topology.
The remainder of this paper is organized as follows:
We start in Section 2 by giving some basic definitions
and setting up the problem under consideration. Section
3 presents the main result, namely the internal model
principle for consensus for the case of output consensus
as well as state consensus. Finally, Section 4 concludes the
paper.
2. PROBLEM SETUP
2.1 Notation
with some matrix Ci of appropriate dimension. The interconnection topology given by G thus determines the
information available to controllers (2). Note that the
special case of mi = 0 corresponds to a static feedback
controller for the ith agent.
2.3 The consensus problem
−
Throughout the paper, we will write C for the open
+
left-half complex plane and C for the closed right-half
complex plane. Given a matrix A ∈ Rn×n , we will denote
the spectrum or set of eigenvalues of A as σ(A). The
Laplacian matrix of some graph will usually be denoted
L with eigenvalues λi , i = 1, . . . , N which we assume to
be ordered by increasing real part, i.e. Re(λ1 ) ≤ · · · ≤
Re(λN ). We will write 1 for the all ones vector and I to
denote the identity matrix of appropriate dimension.
The problem under consideration in this paper is the
following:
Consensus. Given agents (1) described by matrices
Ai , Bi , Ei for i = 1, . . . , N and some interconnection topology described by a graph G, find, if possible, controllers (2)
described by matrices Fi , Gi , Hi , Ki for i = 1, . . . , N such
that the closed loop of (1) and (2) satisfies
lim kzi (t) − zj (t)k = 0, i, j = 1, . . . , N.
(4)
t→∞
2.2 Definition of the MAS
independent of initial conditions xi,0 and ξi,0 for i =
1, . . . , N .
We consider a MAS consisting of N linear agents
ẋi = Ai xi + Bi ui
i = 1, . . . , N
(1)
zi = Ei xi
with states xi ∈ Rni , inputs ui ∈ Rpi , outputs zi ∈ Rq ,
and initial conditions xi (0) = xi,0 ∈ Rni . All agents
potentially have different dynamics, input dimension, and
state dimension. The output dimension is the same for all
agents which is needed to formulate the output consensus
problem. The agents are dynamically decoupled but couplings will be introduced through interactions between the
agents.
If (4) is satisfied, a function η : R → Rq such that
lim zi (t) = η(t), i = 1, . . . , N.
Namely, we suppose that the interaction topology between
the agents is given by some weighted and directed graph
G = {V, E, W }. Each vertex in the vertex set V represents
one agent, each directed arc in the arc set E represents a
communication link between two agents, and the function
W assigns positive weights to the arcs in E. The arcs are
directed according to the directions of information flows,
i.e. information from the tail of an arc is available to its
head. The graph can be represented using the adjacency
matrix
W (j, i) if (j, i) ∈ E,
A = [aij ] ∈ RN ×N , aij =
0
if (j, i) 6∈ E
or the Laplacian matrix
L = [lij ] ∈ RN ×N ,
lij =
X

aij
if i = j,
j6=i
−aij
if i 6= j,
both of which uniquely define G in the absence of selfloops. Details on algebraic properties of graphs can be
found e.g. in Godsil and Royle (2004).

Each agent has a local controller of the form
ξ̇i = Fi ξi + Gi δi
i = 1, . . . , N
(2)
ui = Hi ξi + Ki δi
with controller state ξi ∈ Rmi and initial condition ξi (0) =
ξi,0 ∈ Rmi . The controllers of different agents may have
different state dimensions mi . In (2), δi ∈ Rq is a relative
measurement defined as
N
N
X
X
lij zj
(3)
aij (zi − zj ) = Ci
δi = Ci
j=1
j=1
t→∞
is called a consensus trajectory for the closed loop (1), (2).
The controllers (2) are said to trivially solve the consensus
problem if η(t) ≡ 0 is a consensus trajectory for (1),
(2) independent of initial conditions xi,0 and ξi,0 for i =
1, . . . , N .
To simplify notation, we will use the following abbreviations in the sequel:
Ai Bi Hi
Bi Ki
A∗i :=
, Bi∗ :=
, Ei∗ := ( Ei 0 ) .
0 Fi
Gi
Before proceeding with the main results, we impose the
following explicit assumption on the agents and their
controllers:
Assumption 1. (Detectability). The pair
(A∗i , Ei∗ )
is detectable for i = 1, . . . , N .
In fact, Assumption 1 can be imposed without loss of
generality. If Assumption 1 was not satisfied for some i,
there would exist a matrix Ti such that


Ãi B̃i H̃i 0
Ai Bi Hi
Ti−1
Ti =  0 F̃i 0  ,
0 Fi
∗
∗ ∗
( Ei 0 ) Ti = Ẽi 0 0 ,
and the pair
Ãi B̃i H̃i
, Ẽi 0
0 F̃i
is detectable. The asterisks in the transformed matrices
stand for submatrices without any interest for the result.
Transforming Gi and Ki accordingly yields G̃i and K̃i .
Neither substituting Ãi , B̃i , and Ẽi for Ai , Bi , and Ei ,
nor substituting F̃i , G̃i , H̃i , and K̃i for Fi , Gi , Hi , and Ki
has any impact on (4). That is, we can solve the problem
for a reduced system which satisfies Assumption 1 and
thereby obtain a solution for the original problem.
We are now ready to state the main result of this paper,
namely a necessary condition for the solvability of the
consensus problem imposed on the agent and controller
dynamics stated as an internal model principle.
3. NECCESSARY CONDITIONS FOR SOLVABILITY
OF THE CONSENSUS PROBLEM
In this section, we will derive an internal model principle
for consensus in a heterogeneous MAS. Just as the well
known internal model principle for control theory (Francis and Wonham, 1976) requires an internal model of a
reference or disturbance to be present in the open loop
system, our result shows that each agent with its controller
needs an internal model of the consensus dynamics if
the consensus problem has a solution. Hence, the result
imposes necessary conditions on the agent and controller
dynamics, i.e. a minimum requirement which needs to be
satisfied by all agents with their controllers in order to
solve the consensus problem. We will derive the result for
the case of output consensus first and then treat the special
case of state consensus.
3.1 Output Consensus
The main result of this section is stated in the following
Theorem:
Theorem 2. Given agents (1) described by matrices Ai ,
Bi , Ei for i = 1, . . . N and some interconnection topology
described by a graph G. Suppose a solution (2) described
by matrices Fi , Gi , Hi , Ki for i = 1, . . . , N to the
consensus problem has been found and Assumption 1 is
satisfied. Then there exists full-rank matrices Πi , i =
1, . . . , N and matrices S and R, such that
A∗i Πi = Πi S
i = 1, . . . , N.
(5)
Ei∗ Πi = R
Proof. We begin by writing the closed loop system of all
agents together with their controllers as
ζ̇ = Acl ζ
z = Ecl ζ
where ζ is the stacked vector of agent and controller states
PN
T T
ζ = (xT1 ξ1T · · · xTN ξN
) ∈ Rν with ν = i=1 (ni + mi ) and
T T
z is the stacked vector of outputs z = (z1T · · · zN
) ∈ RN q .
The matrices Acl and Ecl are thus given as
 ∗
  ∗

A1
B1 C1

 

..
..
Acl = 
+
 (L ⊗ I)Ecl
.
.
A∗N


Ecl = 
E1∗
..
∗
BN
CN

.
∗
EN

.
Denote V − and V + the invariant subspaces of Acl corresponding to eigenvalues in the open left-half complex
plane and the closed right-half complex plane respectively.
Assume V − is spanned by the columns of some matrix
Σ ∈ Rν×(ν−µ) and V + is spanned by the columns of
some matrix Π ∈ Rν×µ . Together, the matrices Π and
Σ transform the closed loop system in the following way:
S 0
−1
( Π Σ ) Acl ( Π Σ ) =
0 ∗
Ecl ( Π Σ ) = ( Q ∗ )
for some matrices S ∈ Rµ×µ and Q ∈ RN q×µ . Again, the
asterisks stand for some matrices which are not of interest
in the sequel. We obtain
Acl Π = ΠS
Ecl Π = Q.
The spectrum of S satisfies σ(S) ⊂ C
+
by construction.
Denote E the subspace where kzi −zj k = 0, i, j = 1, . . . , N .
This subspace can be written as
E = ker ((J ⊗ I)Ecl )
T
where J = I − 11N ∈ RN ×N is the Laplacian matrix
of a full graph on N vertices with the property that
ker J = span{1}. 1 By assumption, (4) is satisfied, i.e.
lim (J ⊗ I)Ecl ζ = 0.
t→∞
This can only be satisfied if the invariant subspace of
Acl corresponding to eigenvalues in the closed right-half
complex plane is contained in E, i.e. V + ⊆ E. Since V + is
spanned by the columns of Π this can be expressed as
(J ⊗ I)Ecl Π = 0.
Hence, we have that (J ⊗ I)Q = 0 which implies that
there exists some matrix R ∈ Rq×µ such that Q = 1 ⊗ R.
Consider Π to be partitioned as


Π1


Πi ∈ R(ni +mi )×µ , i = 1, . . . , N.
Π =  ...  ,
ΠN
It follows from Ecl Π = Q that
Ei∗ Πi = R
Since (J ⊗ I)Ecl Π = 0 implies (L ⊗ I)Ecl Π = 0 for any
Laplacian matrix L, it follows that
 

 ∗
Π1 S
A1 Π1
  .. 

..
Acl Π = 
 =  . .
.
A∗N ΠN
ΠN S
It remains to show that the matrices Πi , i = 1, . . . , N have
full rank. To that end, note that




R
Ei∗
∗ ∗

 RS 
 Ei Ai
 , i = 1, . . . , N.
 Πi = 

..
.




..
.
Ei∗ (A∗i )µ−1
RS µ−1
Since the pair (S, R) is observable as a consequence of
Assumption 1, this implies ni + mi ≥ µ and rank(Πi ) = µ
for i = 1, . . . , N . 2
The key claim in Theorem 2 is given by (5). Equation (5)
actually states that the restriction of the systems defined
by {A∗i , Ei∗ } to the space spanned by the columns of Πi
corresponds to a system with dynamics matrix S and
output matrix R for i = 1, . . . , N . All agents thus contain
an internal model of the same system {S, R}. The meaning
of this system in the solution of the consensus problem is
highlighted in the next corollary.
Corollary 3. Assume the setup as in Theorem 2. Then for
given initial conditions of the agents and the controllers
1
The matrix J could be replaced by any other Laplacian matrix of
some quasi strongly connected graph on N vertices or any matrix L
such that ker L = span{1}.
xi,0 , ξi,0 , i = 1, . . . , N , there exists some unique vector
ω0 ∈ Rµ such that the solution η(t) to the exosystem
ω̇ = Sω, ω(0) = ω0
η = Rω
is a consensus trajectory.
Proof. It is always possible to find matrices Σi , i =
1, . . . , N such that
S 0
−1
( Πi Σi ) A∗i ( Πi Σi ) =
0 ∗
i = 1, . . . , N,
Ei∗ ( Πi Σi ) = ( R ∗ )
where the asteriks stand for some matrices without interest
in the sequel. As shown in the proof of Theorem 2,
(xTi , ξiT )T will converge to the subspace spanned by the
columns of Πi as t → ∞ for i = 1, . . . , N . Therefore,
condition (4) implies that there exists some ω0 ∈ Rµ such
that
lim zi = ReSt ω0 , i = 1, . . . , N.
t→∞
+
Since the pair (S, R) is observable and σ(S) ⊂ C , ω0 is
unique. 2
Ei Aki Πi,1 +
k−1
X
Ei Ali Bi Ki Fik−1−l Πi,2 = RS k ,
k ≥ 0.
l=0
Since ri is the minimum relative degree of the ith agent,
Ei Ali Bi = 0 for l = 0, . . . , ri −1. Consequently, for k ≤ ri −
1, condition (6) is recovered.
The second part of the corollary follows from the fact that
Πi spans an invariant subspace corresponding to eigenvalues in the closed right-half complex plane. Consequently,
if the eigenvalues of Fi have negative real part, necessarily
Πi,2 = 0. 2
The internal model of the exosystem {S, R} required
for consensus can be contained in the agent dynamics
or the controller dynamics. Corollary 4 states that the
relative degree of the agents imposes a limitation on the
possibility to include the internal model in the controller
dynamics. If in addition, the controllers are constrained
to be asymptotically stable, the internal model must be
contained in the agent dynamics alone.
Corollary 3 shows that {S, R} can be seen as an exosystem
producing consensus trajectories for the MAS (1) with
controllers (2). Consequently, Theorem 2 indeed represents
an internal model principle since it shows that a model of
{S, R} is contained in the dynamics of every agent together
with its controller {A∗i , Ei∗ }. This means that - if the
consensus problem admits a solution - all agents together
with their controllers necessarily contain the exosystem
generating the consensus trajectory as an internal model.
The assumption that the controllers are constrained to
be stable can be given a simple interpretation. It is quite
natural to require that the consensus problem is solved in a
way such that the consensus trajectories reflect properties
of the agents. One may impose that the class of possible
consensus trajectories, i.e. the exosystem generating the
consensus trajectories, depends on the agent dynamics
alone. This is exactly the case, if the controllers are asymptotically stable. If in contrast the controllers influence the
class of possible consensus trajectories, the meaning of the
consensus trajectories may need additional justification.
Nothing can be said though from Theorem 2 on whether
there actually exist controllers, that solve the consensus
problem. Hence, Theorem 2 gives a minimum requirement
for the possibilty to solve the consensus problem.
If in addition, one imposes that for specific initial conditions of the agents, the consensus trajectory is independent
of the controllers, this will in general only be possible for
static controllers, i.e. mi = 0, i = 1, . . . , N .
Condition (5) given in Theorem 2 is imposed on the
dynamics of the agent together with the controller. Since
the dynamics of the controllers are an object of design
decisions and can be easily influenced, it is interesting
to ask what are the conditions imposed on the agent
dynamics directly. This is partly answered in the following
Corollary:
Corollary 4. Assume the setup as in Theorem 2. Denote ri
the minimum relative degree from any input to any output
of the ith agent for i = 1, . . . , N . There exist matrices
Πi,1 , i = 1, . . . , N such that
Similar to the results known for the output regulation
problem (cf. Knobloch et al. (1993)), this subsection
showed that an internal model of an exosystem is needed in
each agent together with its controller. In contrast to the
output regulation problem, the exosystem is no real system
but can be seen as a virtual system which only exists as
part of the MAS and generates consensus trajectories. The
output of the exosystem is not directly available to the
agents and their controllers.
Ei Aki Πi,1 = RS k , k = 0, . . . , ri − 1, i = 1, . . . , N. (6)
If the controllers are constrained to be asymptotically
stable, i.e. σ(Fi ) ⊂ C− , i = 1, . . . , N , then Πi,1 has full
rank and
Ai Πi,1 = Πi,1 S,
i = 1, . . . , N.
(7)
holds.
Proof. Condition (5) can be split up in
Ai Πi,1 + Bi Hi Πi,2 = Πi,1 S,
Fi Πi,2 = Πi,2 S,
Ei Πi,1 = R.
One can construct additional redundant conditions multiplying S k , k ≥ 0 to the right of the last condition which
yields
The next subsection extends the results of this subsection
to the case of state consensus.
3.2 State Consensus
The problem of state consensus can be seen as a special
case of the output consensus problem treated so far where
ni = n, i = 1, . . . , N and Ei = I, i = 1, . . . , N . The
adaptation of Theorem 2 to the case of state consensus is
given in the next corollary:
Corollary 5. Assume the setup as in Theorem 2. Additionally, assume ni = n, i = 1, . . . , N and Ei = I, i = 1, . . . , N .
Then there exist matrices Πi , i = 1, . . . , N , S, and R such
that (RT ΠTi )T is a full rank matrix for i = 1, . . . , N and
Ai R + Bi Hi Πi = RS
i = 1, . . . , N.
(8)
Fi Πi = Πi S
Proof. Assume Π̃i , i = 1, . . . , N solves (5). With Ei = I,
Ei∗ Π̃i = R implies that
R
Π̃i =
Πi
for some Πi , i = 1, . . . , N . 2
The conditions for state consensus in Corollary 5 are
easily obtained from the conditions in Theorem 2. In fact,
Ei = I, i = 1, . . . , N simply imposes some additional
structure of the matrices Πi in (5). Yet, there is another
interesting relation between output and state consensus in
the case when all agents have the same state dimension.
This relation is stated in the next theorem.
Theorem 6. Assume the setup as in Theorem 2 (in particular Ei 6= I, i = 1, . . . , N ). Additionally, assume ni =
n, i = 1, . . . , N and all eigenvalues of Fi are contained
in the open left-half plane for i = 1, . . . , N . There exists transformations xi = Ti x̃i such that the transformed
agents reach state consensus.
Theorem 6 states that output consensus implies state consensus (except for state transformations) if all agents have
the same state dimension and all controllers are asymptotically stable. The key assumption in the Theorem is
Assumption 1, i.e. detectability of (A∗i , Ei∗ ), i = 1, . . . , N .
This ensures that the output trajectory zi (t) of an agent
determines the agent and controller dynamics restricted to
V + uniquely except for state transformations.
Theorem 6 could be stated equivalently without the requirement that all eigenvalues of Fi are contained in the
open left-half plane. In that case, the transformations
would be applied to agent and controller states at the
same time, possibly identifying controller states of one
agent with agent states of another agent. Generally speaking, state consensus might be of limited significance if
the states of different agents/controllers that are to be
compared have different physical interpretations. Theorem
6 suggests though, that decreasing the dimension of the
output zi on which agents want to agree does not decrease
problem complexity as long as Assumption 1 is satisfied.
Proof. By Theorem 2 and Corollary 4, there exist matrices Πi , i = 1, . . . , N such that
Ai Πi = Πi S,
4. CONCLUSIONS
Ei Aki Πi = RS k , k ≥ 0.
Since (S, R) is observable, there exists a matrix P ∈ Rµ×µq
with rank(P ) = µ such that


 E

R
i
 RS 
 Ei Ai 
 , E 0 := P 

Ei0 Πi = R0 , R0 := P 
..
.
i




..
.
This paper aims at providing insight into minimal requirements to be satisfied by agents with local controllers in a
heterogeneous multi-agent system such that a solution to
the consensus problem exists. One answer to this question
was given in the form of an internal model principle for
consensus. Namely we showed that each agent must contain a model of an exosystem which produces all consensus
trajectories. The result was derived for output consensus
and state consensus.
RS µ−1
Ei Aµ−1
i
0
with rank(R ) = rank(Ei ) = µ for i = 1, . . . , N . The
quantity zi0 = Ei0 xi ∈ Rµ can be seen as an output of dimension identical to the state dimension of the exosystem
{S, R} obtained from the original output zi and its time
derivatives in the same way for i = 1, . . . , N . Therefore,
lim kEi xi − Ej xj k = 0 ⇒ lim kEi0 xi − Ej0 xj k = 0 (9)
0
t→∞
t→∞
Additionally, we showed that state consensus, potentially
after some state transformation, is implied by output
consensus when the state dimension of all agents is the
same. We thus showed that requiring only some outputs
to reach consensus does not significantly reduce complexity
of the consensus problem.
for i, j = 1, . . . , N .
There exist matrices Ei00 ∈ R(n−µ)×n such that Ei00 Πi = 0
and
0
Ei
= n.
rank
Ei00
It was shown in the proof of Theorem 2, that (xTi , ξiT )T
converges to the subspace spanned by the columns of Πi ,
i.e. Ei00 Πi = 0 implies
lim Ei00 xi = 0
(10)
t→∞
for i = 1, . . . , N . Consider the transformation matrices
Ti , i = 1, . . . , N defined as
0 −1
Ei
Ti =
.
Ei00
Carrying out the transformation yields
0
0
Ei
R
−1
Ti = I, R̃ = Π̃i =
Ãi = Ti Ai Ti , Ẽi =
0
Ei00
with Ãi R̃ = R̃S for i = 1, . . . , N . Using (9) and (10),
lim kTi−1 xi − Tj−1 xj k = 0, i, j = 1, . . . , N
t→∞
follows. Consensus trajectories for the transformed agents
are produced by the exosystem given by {S, R̃}. 2
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