IEEE Circuits and Systems Magazine - Q2 2019 - 12

x8

x3
x7

x3

x8

x6

x6

x7

Bud

x5

x2

x9

x3

x8

x2

x9

x5

x4

x7

x4

u1

u1

(a)

x5

x1

x1

u1

x6
Bud

x2

x9

x4

x1

Stem

(b)

(c)

Figure 2. Inaccessibility, dilation and cactus. (a) the red node x 6 is inaccessible from the input node u 1 (blue node). (b) the red
nodes in the set S = {x 2, x 4} lead to a dilation. their common-neighbor set T (S) = {x 1} contains one node (less than 2). (c) a cactus
contains neither inaccessible nodes nor dilations. there are one stem (purple) and two buds (yellow).

u1

u1
b1

a21

b1

x1

a21

a14

x1

a31

x4 x2

x2
a32

x3
a42

a43

a43
x4

x3
(a)

(b)
u1
b1
x1

a31

a21
x2

a32

x3

a23
(c)
Figure 3. control diagrams of a directed network. (a) a directed cycle can be completely controlled by controlling only
one node therein. the controllability is independent of any
specific (nonzero) values of b 1, a 21, a 32 and a 43, so the networked system is strongly structurally controllable. (b) this
directed network can never be completely controlled by only
controlling the node x 1 . (c) this directed network is completely state controllable for almost all nonzero parameter
realizations, which will be uncontrollable only for some parameters satisfying the constraint of a 23 a 231 = a 32 a 221, thus this
networked system is structurally controllable.

a practical (parametrized) control system. It highlights
the importance of the system structures. In this setting,
furthermore, when the system is state controllable for
all nonzero parameter realizations, it is called strongly
structurally controllable [10]. Conversely, if a system is
12

IEEE cIrcuIts and systEms magazInE

state controllable then it is certainly structurally controllable (considering the given set of matrix values as a
particular parametric realization).
The advantage of structural controllability is due
to the fact that one can determine a network's controllability even if the exact weight values of some or all
edges are unknown. As will be demonstrated below, this
framework considerably expands the practical applicability of the classical control theory and techniques to
real-world networked systems with incomplete modeling or uncertainties.
It should be stressed that the two concepts and notions of "state controllability" and "structural controllability" are obviously closely related, but not equivalent. It is even possible that a networked system is not
state controllable but is structurally controllable. Here,
some illustrative examples are given to show the differences between state controllability, structural controllability, and strong structural controllability for
networked systems.
Example 1
The linear system shown in Fig. 3(a) 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 S 0
Sxo 4 (t)W S 0
T
X T

0
0
a 32
0

0
0
0
a 43

R
V
a 14VW Sx 1 (t)W RSb 1VW
0 W Sx 2 (t)W S 0 W
S
W+
u 1 (t) .
0 W Sx 3 (t)W S 0 W
WW
SS WW
S
W
0 X x 4 (t) T 0 X
T
X

The controllability matrix of this networked system is
given by
R1
S
S0
Q =[B, AB, A 2 B, A 3 B] = b 1 S
0
SS
T0

V
0
0
0
W
0
0
a 21
W
W.
0 a 32 a 21
0
W
0
0
a 43 a 32 a 21WX
sEcOnd QuartEr 2019
IEEE Circuits and Systems Magazine - Q2 2019

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