IEEE Circuits and Systems Magazine - Q2 2019 - 13

This system is state controllable because rank (Q) =
4 = N as long as the parameters b 1, a 21, a 32 and a 43
are all nonzero. That is, its controllability is independent of the specific values of b 1, a 21, a 32 and a 43 .
Therefore, this system is also strongly structural ly controllable.
The linear system shown in Fig. 3(b) is described by
Ro
V R
Sx 1 (t)W S 0
Sxo 2 (t)W Sa 21
So
W =S
Sx 3 (t)W Sa 31
Sxo 4 (t)W S 0
T
X T

0
0
0
a 42

0
0
0
a 43

R
V
0VW Sx 1 (t)W RSb 1VW
0W Sx 2 (t)W S 0 W
S
W+
u 1 (t) .
0W Sx 3 (t)W S 0 W
WW
SS WW
0X Sx 4 (t)W T 0 X
T
X

xo i (t) =

Its controllability matrix is given by
R1
S
S0
Q =[B, AB, A 2 B, A 3 B] = b 1 S
0
SS
T0

0
0
0
a 21
0
a 31
0 a 42 a 21 + a 43 a 31

/

a ij (x j (t) - x i (t)), i =1, 2, f, N,

(6)

j ! Ni

0VW
0W
.
0W
WW
0X

This system is state uncontrollable because rank (Q) =
3 1 N. Further note that this system is always state and
structurally uncontrollable no matter how the parameters b 1, a 21, a 31, a 42 and a 43 are chosen.
The linear system shown in Fig. 3(c) is described by
xo 1 (t)
0 0 0 x 1 (t)
b1
o
x
(
t
)
=
a
0
a
x
(
t
)
+
23H > 2
H > 0 Hu 1 (t).
> 2 H > 21
0
xo 3 (t)
a 31 a 32 0 x 3 (t)
Its controllability matrix is given by
1 0
0
Q =[B, AB, A 2 B] = b 1 >0 a 21 a 23 a 31H .
0 a 31 a 32 a 21
This system will be state controllable if one choose the
detailed parameters such that rank (Q) = 3 = N. In the
special case of a 23 a 231 = a 32 a 221, one has rank (Q) = 2 1 N,
so the system becomes state uncontrollable. Therefore,
this system is structurally controllable. Clearly, it is not
strongly structurally controllable. Figure 3(c) illustrates
the relationship between state controllability and structural controllability.
■
3. Analysis of Network Controllability
In this section, various approaches to evaluating the
controllability of a complex networked dynamical system are presented. Fundamental concepts and selected theoretical results on both state controllability and
structural controllability for different types of complex networked systems are summarized. In particular, how the network topology, node-system dynamics,
external control inputs and inner dynamical interactions, altogether or respectively affect the controllability of a general network of linear dynamical systems
are discussed.
sEcOnd QuartEr 2019

3.1 Kalman Rank Criterion
In a networked dynamical system, agents as nodes are
endowed with state variables and dynamics, and are
interconnected via a communication network in a certain
topology. The controllability issue in such a networked
multi-agent system under the so-called leader-follower
framework was first considered in [11], where the problem is formulated as the classical state controllability of
a single-input linear system.
Specifically, consider a networked multi-agent system described by

where x i (t) ! R is the state vector of the ith agent at time t,
a ij is the (i, j )th element of the adjacency matrix of an undirected graph G that denotes the information flows among
the agents, a ij =1 if agents i and j ( j ! i ) are neighbors
(i.e., connected to each other), and a ij = 0 otherwise (herein, self-loops and multi-connected edges are excluded), and
N i is the neighbor set of agent i, which is assumed fixed,
therefore the interconnection graph G is time invariant.
System (6) can be written in the Laplacian dynamics form
xo (t) =-Lx (t),

(7)

where x (t) =[x 1 (t), x 2 (t), f, x N (t)] T ! R N denotes the aggregated state vector of the agents and L = (L ij) ! R N # N
is the Laplacian matrix defined by
-a ij,
j ! i,
L ij = * / a ij, j = i.

(8)

j ! Ni

The agents in the network can be divided into two
groups: leaders and followers, where external control
inputs are injected only to the leaders. Denote the set of
controlled agents as the leader set, Vl, and the remaining agents as the follower set, Vf , where the subscripts
l and f denote the leaders and followers, respectively. It is clear that Vl ' Vf =V (the whole network), and
Vl ( Vf = Q. Define the follower graph G f to be the subgraph induced by Vf and the leader graph G l the subgraph induced by Vl . Obviously, G l and G f are disjoint.
Without loss of generality, one can reorganize the indices of the agents in such a way that the first N f (1 1 N f 1 N )
agents are followers; that is, one can label the followers
from 1 to N f and the leaders from N f +1 to N. The associated Laplacian matrix L is thereby partitioned as
L f L fl
E,
L =; T
L fl L l

(9)

where L f and L l are N f # N f and (N - N f ) # (N - N f ) matrices, respectively. However, these two sub-matrices
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