IEEE Computational Intelligence Magazine - August 2019 - 54

TABLE I Continuous-time models of ZND methodology corresponding to different
time-varying problems solving.
#

TIME-VARYING PROBLEM

CONTINUOUS-TIME SOLUTION MODEL

1

Time-Varying tth Root Finding
x t (t ) - a (t ) = 0

tx t -1(t ) xo (t ) = ao (t ) - pz(x t(t ) - a(t ))

2

Time-Varying Minimization
min nf ( x (t ), t ) ! R

22 f ( x (t ), t ) .
2 f ( x (t ), t )
22 f ( x (t ), t )
mx (t ) =-pU c
2x
2x 2t
2x 2x T

3

Time-Varying Nonlinear System
Solving f ( x (t ), t ) = 0 ! R n

2f ( x (t ), t ) .
2f ( x (t ), t )
x (t ) =-pU( f ( x (t ), t )) 2t
2x T

4

Time-Varying Matrix Inversion
A(t ) X (t ) - I = 0 ! R n # n

Xo (t ) =- X (t ) Ao (t ) X (t ) - pX (t ) ( A (t ) X (t ) - I )

x (t) ! R

min f (x(t ), t ) ! R, 6t ! [t 0, t final], (5)

x (t ) ! R n

where f ($ , $) : R n # [t 0, t final] " R is a
time-varying mapping function. For simplicity, f ( x(t ), t ) is guaranteed to be continuously differentiable and strictly
convex. When f ( x(t ), t ) achieves its minimum, the partial derivative of it with
respect to x(t ) is zero.Therefore, the error
function can be defined as [28], [29]:
e(t ) =

2f ( x(t ), t )
,
2x

(6)

which is expected to converge to zero
rapidly. Error function (6) can be substituted into the ZND design formula (3),
and the following continuous-time
model is obtained [30]:
2 2 f ( x(t ), t ) .
2f ( x(t ), t )
m
x(t ) =-pU c
2x
2x 2x T
2 2 f ( x(t ), t )
,
(7)
2x 2t
which is also shown in Table  I. If the
matrix (2 2 f ( x(t ), t ))/(2x 2x T ) is nonsingular, Eq. (7) can be rewritten as [30]:
2 2 f (x(t ), t ) -1
2f (x(t ), t )
m c pU c
m
T
2x
2x 2x
2 2 f (x(t ), t )
m.
+
(8)
2 x 2t

x(t ) =-c
.

Continuous-time model (8) is an efficient design for time-varying minimization problem solving. If the activation
function is a simple linear function,
Eq.  (8) can be reduced to a simpler
form given by [19]-[21], [28], [29]:

54

2 2 f (x(t ), t ) -1 2f (x(t ), t )
m cp
2x
2x 2x T
2
2 f (x(t ), t )
m.
+
(9)
2 x 2t

x(t ) =-c
.

Eq. (9) is also called continuous-time
ZLSF model [19]-[21], as proposed by
Zhao, Lu, Swamy and Feng. Evidently,
this model is a simplified form of the
continuous-time minimization-type
ZND model (8). Notably, the design
process of the ZLSF model is different
from that of the ZND model, in that
the former uses a non-extendable
attempt design of splicing type (translated from its original Chinese writing to
English in the Appendix), and then provides proof, whereas the latter uses a systematic design formula (3), which can
be applied to solve different time-varying problems.
III. New Discretization Formulas
and Various Discrete-Time Models

In real-time applications, continuoustime models are difficult to directly
implement on digital computers or circuits. Continuous-time models are supposed to be discretized for the purposes
of potential digital hardware implementation (e.g., digital circuits) and numerical algorithm development. Specifically,
the first-order time derivative should be
.
approximated, e.g., xo (t ), x(t ) and Xo (t )
in Table I.

{o (t k ) =

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2019

A. ZeaD Formulas

The numerical differentiation formulas,
which can be adopted to discretize continuous-time ZND models, should comply with some strict conditions. First, the
formula should be 1-step-ahead. The reason is that computation has to obtain the
predicted value using only the present and
previous data at each computational time
interval. Specifically, the predicted value at
time instant t k +1 should be obtained in
time interval [t k, t k +1 ) by using some of
the information at present and previous
time instants (e.g., t k, t k -1, f), where
t k +1 denotes the time instant at
t = (k + 1) g, in which g represents the
sampling gap. Moreover, the obtained discrete-time model should have 0-stability,
which is defined below [16], [17].
Lemma. The 0-stability of an
N-step model/method R Nr = 0 a r x k +r =
gR Nr = 0 b r v k +r is determined by the roots
of the characteristic polynomial PN (m) =
R Nr = 0 a r m r . If any root that is denoted by
m of the polynomial PN (m) satisfies
; m ; # 1 and the root whose modulus
equals one is simple, then the corresponding N-step model/method has
0-stability (i.e., is 0-stable).
For example, the Lagrange-type
1-step-ahead formulas in [16] are all
1-step-ahead, but they are unsuitable
to obtain stable models. Take the following Lagrange-type 1-step-ahead
for mula at the bot tom of the page
into consideration.
According to the Lemma at the bottom of the page, the characteristic polynomial P4 of the corresponding model is
P4 = 3m 4 + 10m 3 - 18m 2 + 6m - 1.
P4 has four roots in complex plane, i.e.,
m 1 = -4.7028, m 2 = 1, m 3 = 0.1847 +
0.1917j, and m 4 = 0.1847 - 0.1917j.
Therefore, the corresponding model
is unstable because m 1 is outside the
unit circle.
Almost all the traditional numerical
differentiation formulas are unsuitable
for time discretization due to these strict

3{(t k +1 ) + 10{(t k ) - 18{(t k -1) + 6{(t k -2) - {(t k -3)
+O( g 4 ).
12g
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