Controllability of leader-following multi-agent systems
M. I. GARC´
IA-PLANAS
Universitat Polit`
ecnica de Catalunya
Departament de Matem`
atica Aplicada
Mineria 1, 08038 Barcelona
SPAIN
Abstract: In this work, the controllability of a class of multi-agent linear systems that are interconnected via
communication channels is studied. Condition for controllability have been presented and described in terms of
the topology of the followers agents, in the case where the followers agents have the same linear dynamics.
Key–Words: Multiagent linear systems, graph, leaser-following, controllability.
Received: December 21, 2022. Revised: August 27, 2023. Accepted: October 4, 2023. Published: October 27, 2023.
1 Introduction
A multi-agent system is a system made up of sev-
eral agents that interact with each other where the dy-
namics of each agent and the leader is a linear sys-
tem. Multi-agent systems can be used to solve prob-
lems that are difficult or impossible to solve in a sin-
gle agent. Recently, the study of multi-agent sys-
tems has attracted the attention of many researchers
because this class of systems appears in various ar-
eas of knowledge, such as the cooperative control of
unmanned aerial vehicles, the consensus problem of
communication networks, the training control of mo-
bile robots, neural networks modeling the brain struc-
ture, Etc., [1], [2], [3], [4],[5], [6], [7].
An exciting topic is the study of a group of agents
with a leader, where the leader is a special agent
whose movement is affected by that of all the oth-
ers, but it does influence the rest of the agents, which
is why we speak of a leading agent who is followed
by all the others, [8]. In this sense, [9] examines
the stability of leader-follower multi-agent systems
with general linear dynamics and switching network
topologies.
It is known that the human brain can be inter-
preted mathematically as a multi-agent linear dynamic
system that moves through various cognitive regions,
promoting more or less complicated behaviors. The
dynamics of the cerebral neuronal system play a con-
siderable role in cognitive function and are, therefore,
of interest in the attempt to understand the processes
and evolution of possible disorders. The controllabil-
ity of a system refers to the possibility of manipulat-
ing its components to drive the system along a desired
trajectory: a set of states culminating in a target state
chosen for its functional utility (leader). The study of
system controllability could be a mechanism of cog-
nitive control in critical locations within the anatom-
ical system acting as drivers that move the system
(brain) towards specific modes of action (cognitive
functions), [3], [10].
In this paper, the leader-following multi-agent
systems are considered. In [2], the author analyzes
the case of multi-agent systems without a leader
where the significant difference in the study is that
in this case the graph considered is undirected and
connected, assuming, therefore, the symmetry of the
Laplacian matrix property that, in general, is not ful-
filled in the case of leader-following multi-agent sys-
tems.
2 Preliminaries
Let us consider a group of kagents. The following lin-
ear dynamical systems give the dynamic of each agent
˙x0(t) = A0x0(t) + C0v0(t)
˙x1(t) = A1x1(t) + B1u1(t) + C1v1(t)
.
.
.
˙xk(t) = Akxk(t) + Bkuk(t) + C1vk(t)
(1)
Ai∈Mn(IC),Bi∈Mn×m(IC),Ci∈Mn×p(IC),
xi(t)∈ICn,ui(t) = fi(x1(t), . . . , xk(t)) ∈ICm,
vi(t)∈ICp,1≤i≤k.
Sometimes, the considered internal controls ui
are given by means a communication topology de-
fined by a graph with
i) Vertex set: V={0,1, . . . , k}
ii) Edge set: E={i, j)|i, j ∈V} ⊂ V×V
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.34
M. I. Garcia-Planas
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iii) Neighbor of i:Ni={j∈V|(i, j)∈E}. (In
the case where j=ithe edge is called self-loop).
defining the communication topology among agents.
The leader is represented by vertex 0, and the
leader sends information to the agents located in the
leader’s neighbors. Then there are edges (0, j)but not
(j, 0) for j= 0
Associate to the graph, we have the Laplacian ma-
trix defined in the following manner.
L= (lij ) =
|Ni|if i=j
−1if j∈ Ni
0otherwise
(2)
Example:
Consider the graph in figure 1.
Figure 1: Graph with leader node
The Laplacian matrix is:
L=
2−1−1 0 0
0 1 0 0 −1
0 0 1 0 −1
0 0 0 1 −1
0−1−1−1 3
(3)
We observe that Lis a block upper triangular ma-
trix ℓ0L
0L1
where the first diagonal block is a 1×1matrix. When
the subgraph with vertices set V′={1, . . . , k}is
undirected and connected, the second diagonal block
is symmetric. This submatrix is the Laplacian matrix
corresponding to the subgraph V′.
We use the following control law for agent i:
ui=−KiX
j∈Ni
(xj−xi), i = 0,1, . . . , k,
where Ki∈Mm×n(IR) are feedback matrices to be
designed. (We are interested in the case where Ki=
Kfor i= 0, . . . , k.
Writing X= (x0, x1, . . . , xk)t,U=
(0, ui, . . . , uk)tand V= (v0, v1, . . . , vk)t,
A=diag (A0, A1, . . . , Ak)
∈Mn(IC) ×k
. . . ×Mn(IC)
B=diag (0, B1, . . . , Bk)
∈Mn×m(IC) ×k
. . . ×Mn×m(IC)
C=diag (C1, . . . , Ck)
∈Mn×p(IC) ×k
. . . ×Mn×p(IC)
and X= (x0, x1, . . . , xk)t,U= (L ⊗ In)X=
(0, u1, . . . , uk)tand V= (v0, v1, . . . , vk)t, and in
the case where the communication topology for in-
ternal control is considered, the control is written as
U= (L ⊗ In)X, and K=diag (K0, K1, . . . , Kk)the
system 1 can be described as
˙
X(t)=(A+BK(L ⊗ In))X(t) + CV(t).(4)
Remember that A= (aij )∈Mn×m(IC) and B=
(bij )∈Mp×q(IC) the Kronecker product is defined as
follows:
Definition 1 Let A= (ai
j)∈Mn×m(IC) and B∈
Mp×q(IC) be two matrices, the Kronecker product of
Aand B, write A⊗B, is the matrix
A⊗B=
a1
1B a1
2B . . . a1
mB
a2
1B a2
2B . . . a2
mB
.
.
..
.
..
.
.
an
1B an
2B . . . an
mB
∈Mnp×mq(IC)
(See, [11] for Kronecker product properties).
3 Controllability
The importance of the qualitative property of dynamic
systems in the control theory, known as controllabil-
ity, is well known.
The controllability concept involves taking the
system from any initial state to any final state in finite
time, regardless of the path or input. Let us consider
the multi-agent system 1.
In our particular setup, the objective of leader-
following multi-agent control is to make the state of
each following agent consistent with that of the leader.
For every agent 1≤i≤k, a external control vi(t)is
required to realize
lim
t→∞ ∥xi(t)−x0(t)∥= 0,1≤i≤k
It is important to emphasize that various defini-
tions of controllability are derived, depending to a
large extent on the class of dynamic systems and the
form of allowable controls, [12].
In our particular setup, the controllability charac-
ter can be described as
rank A−λIn×kB C =nk
If a fixed B-feedback Kand a fixed topology com-
munication is considered, the controllability character
is described as
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DOI: 10.37394/23203.2023.18.34
M. I. Garcia-Planas
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Volume 18, 2023
rank A+BK(L ⊗ In)C=nk
The controllability of the system can be analyzed
by computing the rank of the controllability matrix:
(C(A+BK)(L ⊗ In))C)
. . . (A+BK)(L ⊗ In))nk−1C)
The rank of this matrix is invariant under feed-
back, that is to say
Proposition 2 The matrix controllability of the sys-
tem 1 is invariant under external feedback
Proof:
rank A+BK)(L ⊗ In) + CF C=
rank A+BK)(L ⊗ In)CI
FI
⊓⊔
We are going to carry out the study for a particular
case in which all the systems have the same dynamics,
that is, Ai=A,Bi=B,Ci=Cand Ki=Kfor
all 1≤i≤k; and the graph defining the topology
relating to the systems is undirected and connected.
Being un undirected graph the matrix L1is symmet-
ric, then there exist an orthogonal matrix Psuch that
PL1Pt=D, and the connection ensures that 0 is a
simple eigenvalue of L1.
Proposition 3 Under these conditions, the system
can be described as
˙
X(t) =
(Ik⊗A)+(¯
Ik⊗BK)(L ⊗ In))X(t)+
(In⊗C)V(t)
(5)
where ¯
Ik=0
Ik−1.
In our particular setup, we have that there exists
Q=1 0
0P∈Gl(k, R)with Porthogonal such
that PLPt=D=diag (λ1, . . . , λk),
(λ1≥. . . ≥λk−1> λk= 0).
that is
1 0
0Pℓ0L
0L11 0
0Pt=ℓ0LP t
0D=T
For the matrix L1given in 3 the matrix Dis
D=
0.0000 0 0 0
0 1.0000 0 0
0 0 1.0000 0
0 0 0 4.0000
and Pis
P=
0.5000 −0.4082 0.7071 −0.2887
0.5000 −0.4082 −0.7071 −0.2887
0.5000 0.8165 −0.0000 −0.2887
0.5000 0 0 0.8660
Corollary 4 The system can be described in terms of
the matrices A,B,Cthe feedback Kand the eigen-
values of L.
Proof:
Following the properties of Kronecker product,
we have that.
(Q⊗In)(Ik⊗A)(t⊗In)=(Ik⊗A)
(Q⊗In)(¯
Ik⊗BK)(Qt⊗In) =
(¯
Ik⊗BK)
(P⊗In)(Ik⊗C)(Pt⊗Ik)=(Ik⊗C)
(Q⊗In)(L ⊗ In)(Qt⊗In)=(T ⊗ In)
and calling b
X= (Q⊗In)X, and b
V= (P⊗Ik)Vwe
have
˙
b
X=((Ik⊗A) + (Ik⊗BK)(D ⊗ In))b
X+ (Ik⊗C)b
V.
Calling LP t=ℓ1. . . ℓkthe about equation
is written in the following form
˙
b
X=
AA
...A
!+ 0
B
...BK!
ℓ0Inℓ1In... ℓkIn
λ1In
...
λkIn
b
X+
CC
...C!b
V
That is to say
˙
b
X=
AA+λ1BK
...
A+λkIn
b
X+ CC
...C!b
V
⊓⊔
Using this description, the analysis of controlla-
bility is easier.
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Proposition 5 The system (5) is controllable if and
only if the systems (A, C)and (A+λiBK, C)are
controllable, for each 1≤i≤k.
If each system (A, C) (A+λiBK, C)are con-
trollable for 1≤i≤kthere exist external feed-
backs Fiin such a way that each system A+CF0)
and A+λiBK +CFiarrives to a final state preset,
and that in this case is the same for each system.
When the multi-agent system is not controllable,
one can try to change the proportionality of the inter-
action between the agents, that is, change the matrix
K, looking for one that makes the final system con-
trollable.
4 Conclusion
This work examines the controllability of leader-
following multi-agent systems communicated by a
graph, playing an essential role in describing the in-
teraction topologies. A necessary and sufficient con-
dition for controllability has been presented and de-
scribed in terms of the eigenvalues of the subgraph
defined by the topology of the follower’s agents hav-
ing the same linear dynamics. Based on these results,
the author proposes considering the multi-agent linear
system containing perturbation terms as future work.
Furthermore, we want to take advantage of the theo-
retical results to investigate how the structural char-
acteristics of a brain network determine the temporal
characteristics of cognitive dynamics.
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Contribution of Individual Authors to the Creation
of a Scientific Article (Ghostwriting Policy)
The author contributed in the present research, at
all stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this
study.
Conflict of Interest
The author has no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0 (Attri-
bution 4.0 International, CC BY 4.0)
This article is published under the terms
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Volume 18, 2023
