Information Technology Reference
In-Depth Information
K
[
x
[(
k
+1)
T
]]
x
[(
k
+1)
T
]+
Ψ
[
x
[(
k
+1)
T
]
,
x
(
kT
)
,
u
[(
k
+1)
T
]
,T
]=0
,
with
[1 +
Tx
1
[(
k
+1)
T
]+4
Tx
2
[(
k
+1)
T
]]
4
Tx
2
[(
k
+1)
T
]
K
[
x
[(
k
+1)]
T
]=
−
Tw
1
and
Ψ
[
x
[(
k
+1)
T
]
,
x
(
kT
)
,
u
[(
k
+1)
T
]
,T
]=
x
1
(
kT
)+
Tu
[(
k
+1)
T
]
x
2
(
kT
)
.
Explicit vs. Implicit Discretization Scheme: Impact on Stability
The above examples show that an explicit discretization scheme makes the
design of a semiphysical model simpler than an implicit scheme. The main
incentive for using implicit scheme is the stability issue: implicit schemes may
have better stability properties than implicit schemes. In order to illustrate
that, we discuss the simple first-order linear differential equation
d
u
(
t
)
d
t
=
−
αu
(
t
)
,α>
0
.
Euler's explicit method discretizes it to
u
[(
k
+1)
T
]
−
u
(
kT
)
=
−
αu
(
kT
)
,
T
or equivalently
αT
)
u
(
kT
)
.
Thus,
u
[(
k
+1)
T
] is computed from
u
(0) recursively, and the recursion con-
verges if and only if the magnitude of (1
u
[(
k
+1)
T
]=(1
−
αT
) is smaller than 1, or
T<
2
/α
.
The computation time necessary for integrating that equation numerically is
proportional to 1
/α
:if
α
is very large, numerical integration may become
impossible since the integration step
T
must be very small.
Now we consider the discretization of the same equation by Euler's implicit
method; one has
−
u
[(
k
+1)
T
]
−
u
(
kT
)
=
−
αu
(
kT
)
,
T
or equivalently,
u
[(
k
+1)
T
]=
1
(1 +
αT
)
u
(
kT
)
.
Because the denominator on the right-hand side is larger than 1, the compu-
tation of
u
[(
k
+1)
T
] converges irrespective of
α
.
However, the price to be paid is the fact that (in contrast to the previous
very simple example), the computation of the quantities of interest at time
(
k
+1)
T
requires the resolution of a nonlinear equation. This has important
consequences on the architecture of the corresponding neural model.
Search WWH ::
Custom Search