IEEE Computational Intelligence Magazine - August 2019 - 56

B. Discrete-Time Models

verted into a discrete-time form given
by [29], [31]:

Continuous-time models can be discretized by exploiting ZeaD formulas.
Table III shows the discrete-time ZND
models corresponding to those in
Table I by using general formula (11). In
Table  III, the subscript k represents the
index of the variables at time instant
t = kg, and h = gp ! R + is termed as the
stepsize. Notably, the precision of the discretization formula affects the convergent degree of the error function. To
illustrate this point, we take the timevarying minimization problem as an
example again. The Euler forward-difference formula (10), 3-step ZeaD formula,
and 5-step ZeaD formula are used for
comparison [32], [33], [38]. Applying
Eq. (10), the second formula, and the
seventh formula in Table II to discretize
ZND model (9) yields three different
discrete-time ZND models [20], [28],
[29]. Then, let us consider the following
time-varying function for minimization
to compare these three models:

min f ( x k +1, t k +1 ) ! R,

x k +1 ! R n

where x k +1 should be obtained in time
interval [t k, t k +1). The corresponding
experimental results are shown in Fig. 1.
Specifically, Fig. 1(a) shows the trajectories of x k +1 and f ( x k +1, t k +1 ), wherein
the generated f ( x k +1, t k +1 ) by 3-step
model is smaller than that by Euler-type
model, while that by 5-step model is the
smallest. The residual errors < e(t k +1) <, in
which < $ < denotes the Euclidean norm,
are shown in Fig.  1(b). As illustrated in
Fig.  1(b), the residual error of 5-step
model is the smallest, while the residual
error of Euler-type model is the largest.
The convergence of the Euler-type
model is not as good as that of the
3-step model or 5-step model in terms
of residual error. Therefore, the discretization formula applied has a great effect
on the performance of the resultant discrete-time ZND model. Specifically, the
discretization formula with higher precision may help the discrete-time ZND
model to achieve a better convergence
performance in terms of residual error.

f (x(t ), t ) = (x 1(t ) -exp(cos (t ))) 2
+(x 2(t ) - x 1(t ) tanh(t ))2. (12)
Evidently, the theoretical solution is
x )(t ) = [exp(cos(t )), exp(cos(t )) tanh (t )]T,
and the theoretical minimum value of
f (x(t ), t ) is zero at any time instant. The
minimization problem (5) can be con-

C. Analysis of Convergence

The abovementioned experiment substantiates the assumption that the selected

TABLE III Discrete-time ZND models corresponding to different time-varying
problems solving with N-step ZeaD formula.
#

TIME-VARYING PROBLEM

1

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

2

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

DISCRETE-TIME SOLUTION MODEL
t
N
gao k - hz(x k - a k)
x k +1 =-/ ` a i x k -N +i j +
+ O( g l +1)
t -1
a N +1
a N +1 t x k
i =1

2
x k +1 =- 1 e 2 f T
a N +1 2x2x

-1

(x k , t k )

o e hU e 2f
2x

(x k , t k )

2
o +g 2 f
2x2t

(x k , t k )

-/ ` a i x k -N +i j + O( g l +1)
N

i =1

3

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

x k +1 =- 1 e 2fT
a N +1 2x

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

o e hU(f (x k, t k)) + g 2f
2t

-/ ` a i x k -N +i j + O( g l +1)
a N +1

X k +1 =- 1 ( gX k Ao k X k + hX k( A k X k - I ))
a N +1

-/ ` a i X k -N +i j + O( g l +1)
N

i =1

56

-1

(x k , t k )

N

i =1

4

a N +1

a N +1

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2019

(x k , t k )

o

o

discretization formula deeply affects the
performance of the corresponding discrete-time ZND model. As discussed earlier, the discretization formula should be
1-step-ahead, and the model obtained
should be 0-stable. Furthermore, the precision of the discretization formula affects
the performance of the resultant discretetime ZND model. The discrete-time
ZND model could have a better convergence performance in terms of residual
error for a more accurate discretization
formula, especially when the sampling gap
g becomes smaller. Moreover, other factors can influence the convergence performance of a discrete-time ZND model
in terms of residual error, e.g., the values
of the sampling gap g and stepsize h.
The sampling gap g has an important effect on the convergence performance. It can be improved with a
smaller sampling gap, while a larger sampling gap could even lead to divergence.
However, the number of calculations in
unit time will increase with respect to a
decrease in sampling gap g. Therefore,
the value of sampling gap g should be
set properly in prevalent digital-equipment applications to balance the performance and the number of calculations.
The value of stepsize h should also
be set appropriately in a discrete-time
ZND model. The stepsize h is related to
the convergence or convergence speed.
If the value of stepsize is set appropriately, the ZND model can be convergent in
terms of residual error. However, the
convergence speed may be different for
distinct values of stepsize. A relatively
large value of parameter h could lead to
divergence of the ZND model in terms
of residual error. Fig. 2 shows the residual errors of the 5-step model with different values of h. In Fig. 2(a), the residual
errors converge at different speeds. The
error converges relatively faster for
h = 0.16 and slowly for h = 0.32. In
Fig. 2(b), the residual errors diverge for
relatively large values of h, i.e., h = 0.35,
h = 0.40, h = 0.45, and h = 0.50.
According to previous investigations,
a domain of stepsize h exists in a discrete-time ZND model, e.g., the stepsize
domain of the 3-step ZND model used
in the previous subsection is h ! (0, 1)
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