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Explicit (Forward) vs. Implicit (Backward) Discretization Schemes:
Definitions
Consider a first-order differential equation,
d
x
(
t
)
d
t
=
f
[
x
(
t
)]
.
An explicit scheme discretizes it to
x
[(
k
+1)
T
]=
ϕ
[
x
(
kT
)
,T
]
,
•
where
T
is the discretization (or integration) step, which is usually the
sampling period of experimental data,
•
where
k
is a positive integer,
•
and where function
ϕ
depends on the discretization technique (examples
are shown below).
An implicit scheme discretizes the same differential equation to
x
[(
k
+1)
T
]=
ψ
[
x
[(
k
+1)
T
]
,x
(
kT
)
,T
]
.
The main difference is the fact that the quantity
x
[(
k
+1)
T
] appears in the
left-hand side only if an explicit scheme is used, whereas it appears on both
sides if an implicit scheme is used. Therefore, if a one-step-ahead predictor for
x
is to be designed, the computation of
x
[(
k
+1)
T
]from
x
[
kT
] is trivial if an
explicit scheme is used, whereas it requires solving a nonlinear equation if an
implicit scheme is used.
More generally, consider a set of state-space equations written in vector
form,
d
x
d
t
=
f
[
x
(
t
)
,
u
(
t
)]
.
If an explicit discretization scheme is used, the discretized equations can be
written under the general form,
K
[
x
(
kT
)]
x
[(
k
+1)
T
]+
Ψ
[
x
(
kT
)
,
u
(
kT
)
,T
]=0
,
where
K
is a matrix and
Ψ
is a vector function, whereas, if an implicit dis-
cretization scheme is used, the discretized equation can be written under the
general form,
K
[
x
[(
k
+1)
T
]]
x
[(
k
+1)
T
]+
Ψ
[
x
[(
k
+1)
T
]
,
x
(
kT
)
,
u
[(
k
+1)
T
]
,T
]=0
.
Again, the computation of
x
[(
k
+1)
T
]from
x
[
kT
] is trivial if an explicit
scheme is used (provided matrix
K
is invertible):
K
−
1
[
x
(
kT
)]
Ψ
[
x
(
kT
)
,
u
(
kT
)
,T
]
,
x
[(
k
+1)
T
]=
−
whereas it requires solving a system of nonlinear equations if an implicit
scheme is used.
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