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
Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q2 2019
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IEEE Circuits and Systems Magazine - Q2 2019 - Cover1
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