Neural Identification of Controlled Dynamical Systems and Recurrent Networks - Neural Networks: Methodology and Applications - page 282

we cannot go back to the past as in the BPTT algorithm. For instance at step
n − 1, we perform the computation

x ( n )= g [ u ( n − 1) , x ( n − 1) , w ( n − 1)]

instead of the computation

x ( n )= g [ u ( n − 1) , x ( n − 1) , w ( n )]

using a different state trajectory that was computed in real-time with the
time-varying weight trajectory w ( k ), instead of computing again a new state
trajectory with the single constant configuration w ( n ).
The idea is to update an approximation of ∇ w Φ 1 [ w ( n )] that is noted

ˆ

∇ w Φ 1 by the following recursive equation:
ˆ ˆ

∇ w Φ 1 = ∇ g [ u ( n− 1) , x ( n− 1) , w ( n− 1)] · ∇ w Φ n− 1 .

That approximation is proven mathematically using stochastic approxi-
mation theory in the framework of controlled Markov chains subject to as-
sumptions that will not be detailed here (see [Benveniste et al. 1987]).
For computational issues, it should be emphasized that RTRL does not use
the adjoint network. Indeed, in contrast to backpropagation, the full gradient
has to be computed explicitly. The computation must be performed from
inputs to outputs, instead of being performed backwards.

4.6.4 Application of Recurrent Networks to Measured Controlled
Dynamical System Identification

The applications of recurrent neural networks to identification, using undi-
rected or hybrid algorithms are not very common. Generally, academic exam-
ples are presented in the literature. The stability of the algorithms is more
di cult to guarantee than for linear models [Ljung 1996].
In practice, for the identification of nonlinear models, directed learning
algorithms should be tested first. In [Haykin 1999], it is shown that the iden-
tification using a NARX model of the time series sin( n + sin( n 2 )) outperforms
the identification using a semidirected algorithm with the same number of pa-
rameters. However, numerous examples of applications advocate for the oppo-
site conclusions. Generally, noise is essentially output noise and one has to use
semidirected or undirected learning algorithms (for examples, see Chap. 2).
It should be emphasized that, in a lot of published examples, the success of
directed algorithms rely essentially on the regularity of the functions to be ap-
proximated and that the dumb predictor (as defined in Chap. 2) outperforms
any directed learning algorithms.
For feed-forward networks, to the key problems that have to be faced are

• the selection of inputs,