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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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