IEEE Circuits and Systems Magazine - Q2 2019 - 14

generally do not have the Laplacian matrix properties.
The matrix L f is the principle diagonal submatrix of the
original Laplacian matrix L related to the followers, while
L l is the one related to the leaders, while L fl denotes the
connections between the followers and the leaders.
Note that the motion of each leader in this networked
multi-agent system is free; that is, its state variables can
change without being influenced by followers. The behavior of each follower obeys the local nearest-neighbor
rule and is dominated by the leaders, directly or indirectly. In this case, the trajectories of the leaders can
be viewed as exogenous control inputs to the followers.
Therefore, the followers evolve through the quasi-Laplacian-based dynamics, described by
xo f (t) =-L f x f (t) - L fl x l (t),

(10)

where x f (t) = [x 1 (t), x 2 (t), f, x Nf (t)]T ! R Nf represents
the state vector of the followers and x l (t) =[x Nf +1 (t),
x Nf +2 (t), f, x N (t)] T ! R N -Nf is that of the leaders.
The controllability of a networked multi-agent system
can be defined as follows: the followers can be steered by
the leaders from any initial states to arbitrary final states
in finite time.
Remark 1
There are two issues to be clarified. First, the notion of
controllability for networked multi-agent systems bears
some new physical features, depending on the application
background. In essence, it is a kind of formation control
[23]. It deals with multi-agent systems under the leaderfollower framework, where the agents are divided into two
groups, leaders and followers. The leaders are considered
as exogenous control inputs to the followers, while the
followers can be guided to pre-desired locations so as to
form some anticipated configurations due to the control
actions implemented by the leaders, while the leaders do
not participate in any network configuration changes and
updates. Second, if one equates x f with x and x l with u,

1

2

3

4
(a)

1

2

3

4
(b)

Figure 4. two leader-follower networks with Vf = {1, 2, 3} (red
nodes) and Vl = {4} (blue node). (a) the network is leadersymmetric with respect to {4}. (b) the network is leaderasymmetric with respect to {4}.
14

IEEE cIrcuIts and systEms magazInE

and identifies matrices A and B with -L f and -L fl, respectively, the leader-follower network (10) can be reformulated as (5). Note that the system matrix A in the LTI
system (5) is merely the adjacency matrix of the network,
while in the leader-follower setting described by (10) it is
a quasi-Laplacian matrix. The subtle difference between
these two matrices will lead to very different and even
contrary controllability results.
Based on the Kalman rank criterion, some necessary
and sufficient algebraic conditions are derived in [11], in
terms of the eigenvalues and eigenvectors of a submatrix
of the Laplacian matrix corresponding to the followers.
Theorem 3 [11]
The multi-agent system (10) with a single leader is state
controllable if and only if the following conditions are
satisfied simultaneously:
i) all the eigenvalues of L f are distinct;
ii) the eigenvectors of L f are not orthogonal to L fl .
Example 2
As examples, Fig. 4 shows two leader-follower networks
with Vf = {1, 2, 3} and Vl = {4}.
In Fig. 4(a), one has
R1
S
S-1
L =S
0
SS
T0

-1
3
-1
-1

0
-1
1
0

0 VW
1 -1 0
0
-1W
,
L
=
,
L
=
1
3
1
1H .
f
fl
H
>
>
0W
WW
0 -1 1
0
1X

The eigenvalues of L f are m (L f ) = {0.2679, 1, 3.7321},
which are distinct. Denote by y (L f ) their corresponding eigenvectors. Then, y T (L f ) L fl = {0.4597, 0, 0.8881}. That is,
one of the eigenvectors is orthogonal to L fl . It follows from
Theorem 3 that this single-leader network is uncontrollable.
In Fig. 4(b), one has
R2
S
S-1
L =S
0
SS
T 1

-1
2
-1
0

0
-1
1
0

-1VW
2 -1 0
-1
0W
, L f = >-1 2 -1H, L fl = > 0 H .
W
0
W
0 -1 1
0
1 WX

The eigenvalues of L f are m (L f ) = {0.1981, 1.5550, 3.2470},
which are distinct. As for the eigenvectors of L f , one has
y T (L f ) L fl = {0.328, -0.737, 0.591}. That is, none of the eigenvectors is orthogonal to L fl . Therefore, according to Theorem 3, this single leader network is state controllable.
■
Note that, although the conditions in Theorem 3 are derived for a single-leader system, they can be easily extended
to multi-leader cases (see Proposition 1 in [13]). In addition, in [12] a sufficient condition is given for multi-leader
controllability based on the algebraic characteristics of
a submatrix of the incidence matrix of the network. The
controllability of such undirected networked systems
depends on both the network topology and the number
sEcOnd QuartEr 2019
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