Modeling with Neural Networks: Principles and Model Design Methodology - Neural Networks: Methodology and Applications - page 184

Examples

Consider again the first-order differential equation dx/dt = f [ x ( t ) , u ( t )].
Euler's explicit scheme consists in considering that function f is constant,
equal to f [ x ( kT )], between kT and ( k +1) T , so that the integration of the
differential equation between kT and ( k +1) T gives

x [( k +1) T ]= x ( kT )+ Tf [ x ( kT )] .

By contrast, Euler's implicit scheme consists in considering that function f
is constant, equal to f [ x (( k +1) T ] between kT and ( k +1) T , so that the
integration of the differential equation between kT and ( k +1) T gives

x [( k +1) T ]= x ( kT )+ Tf [ x [( k +1) T ]] .

Similarly, Tustin's scheme consists in considering that function f varies lin-
early between kT and ( k +1) T , so that the integration of the differential
equation between kT and ( k +1) T gives

x [( k +1) T ]= x ( kT )+ T

2 [ f [ x [( k +1) T ]] + f [ x ( kT )]] .

Because the values of quantities at time ( k +1) T are present on both sides
of the equation, the computation of x [( k +1) T ] requires solving a nonlinear
equation.

Application

We consider again the knowledge-based model described by the equations

d x 1 ( t )
d t

( x 1 ( t )+2 x 2 ( t )) 2 + u ( t )

= −

d x 2 ( t )
d t =8 . 32 x 1 ( t )
y ( t )= x 2 ( t ) .

Euler's explicit method discretizes it to

( x 1 ( kT )+2 x 2 ( kT )) 2 + u ( kT )]
x 2 [( k +1) T ]= x 2 ( kT )+ T (8 . 32 x 1 ( kT )) .

x 1 [( k +1) T ]= x 1 ( kT )+ T [ −

Its discretization by Euler's implicit scheme discretizes it to

[1 + Tx 1 [( k +1) T ]+4 Tx 2 [( k +1) T ]] x 1 [( k +1) T ]+4 Tx 2 [( k +1) T ]
= x 1 ( kT )+ Tu [( k +1) T ] x 2 [( k +1) T ] − T (8 . 32 x 1 [( k +1) T ])

= x 2 ( kT ) .

These equations are of the form