Computational Intelligence - November 2012 - 63
and X (t) ! R m # m denotes the neuralnetwork state matrix corresponding to
the time-varying theoretical inverse
A -1 (t) . Ao (t) and Xo (t) denote the
time-derivatives of A (t) and X (t) ,
v11
respectively. In addition, the scalar-valv12
1
..
R
ued design parameter c is a simplifica.
v1m
·a
tion of C in (2) (i.e., C = cI), which
11
should be set as large as the hardware
x1j
a· 12
1
-1
∫
R
R
..
would permit [24], or selected appro.
a· 1m
priately for experimental purposes. For
-d1j
a11
better readability and potential hard-c
a12
1
ware implementation, the network
R
z(·)
..
.
structure of NRNN model (4) is illusa1m
trated in Figure 1.
It is worth pointing out that, by
v21
exploiting NRNN design formula (2)
v22
1
to solve different time-varying probR
..
.
lems (e.g., time-varying linear system,
v2m
a· 21
time-varying linear matrix equation,
x2j
a· 22
-1
time-varying matrix square root, and
1
∫
R
R
..
time-varying convex quadratic pro·a.
2m
gram), different NRNN models [16]-
-d2j
a21
[18], [21]-[23] are established with
a22
-c
1
z(·)
R
their models formulation shown in
..
.
a2m
Table 1 (where superscript T denotes
..
..
..
..
.
.
.
.
the transpose of a matrix/vector). As
seen from the table, the NRNN modvm1
els are all depicted in implicit dynamvm2
ics and exploit the time-derivative
1
R
..
.
information of coefficients. In sumvmm
a· m1
mary, NRNN design formula (2) can
x
a· m2
be viewed as a comprehensive alternamj
-1
1
R
R
..
∫
.
tive for a rich repertoire of time-varya· mm
ing problems solving.
-dmj
am1
On the other hand, by employing
-c
am2
1
z(·)
R
the
gradient-based design method
..
.
amm
(with a conventional norm-based scalar-valued positive or lower-bounded
energy function constructed), we
Figure 1 Structure of the jth column of neurons in NRNN model (4) for time-varying matrix
obtain the following generalized
inversion. For better understanding on the neural-network structure, (4) is expressed in
m
m
m
o
o
o
^
GNN (or HNN) model
x
=
/
v
x
/
a
x
cz
/
a
x
kj - d ij h
the ijth neuron form (with i, j d " 1, 2, g, m , ): ij
k = 1 ik kj
k = 1 ik kj
k = 1 ik nonlinear
m
cz ^/k = 1 a ik x kj - d ij h , where x ij denotes the ijth neuron of (4) corresponding to the ijth
[16], [18], [20]:
entry of state matrix X(t), time-varying weights a ij and ao ij denote respectively the ijth entries
o ^ t h , d ij denotes the ijth entry of identity matrix I, and v ij = d ij - a ij .
of A ^ t h and A
Xo (t) = - cA T (t) U (A (t) X (t) - I) (5)
is established by defining the error
for time-varying matr ix inversion.
while the latter exploits a scalar-valued
function E (t) = A (t) X (t) - I ! R m # m:
Note that other GNN models solving
norm-based energy function [16]-[18],
time-varying problems (e.g., time[20]. Thus, the resultant NRNN models
linear system and time-varyare quite different from the conventional
A (t) Xo (t) = - Ao (t) X (t) - cU (A (t) X (t) -varying
I),
ing
convex
quadratic program) can
GNN (or HNN) models [16]-[18],
o
o
A (t) X (t) = - A (t) X (t) - cU (A (t) X (t) - I), (4)
also be obtained by using gradient[20]-[22]. For instance, when we apply
descent design method, but omitted
NRNN (2) to the time-varying matrix
where A (t) ! R m # m is smoothly timehere due to space limitation.
inversion, the following NRNN model
varying, I denotes the identity matrix,
november 2012 | Ieee ComputatIonal IntellIgenCe magazIne
63
Table of Contents for the Digital Edition of Computational Intelligence - November 2012
Computational Intelligence - November 2012 - Cover1
Computational Intelligence - November 2012 - Cover2
Computational Intelligence - November 2012 - 1
Computational Intelligence - November 2012 - 2
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Computational Intelligence - November 2012 - Cover3
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