Neural Identification of Controlled Dynamical Systems and Recurrent Networks - Neural Networks: Methodology and Applications - page 233

t = kT . The controlled dynamical system time evolution is described by the
following evolution equation

x ( k +1)= f [ x ( k ) , u ( k )] ,

where f is the mapping from E F into E that allows us to infer the state

at time ( k +1) T from the state at time kT . It is possible to make this general
set-up more specific for particular systems.
The more classical model is the linear model. In that case, the state space
and the control space are vector spaces, A is a linear mapping from E to E , B
is a linear mapping from F to E and the evolution equation has the following
form:

×

x ( k +1)= Ax ( k )+ Bu ( k ) .

Because mathematical models are just an approximation of the real evo-
lution of physical devices, the modeling error is generally represented by a
random additional term. This term is often called the state noise.
For instance, in the stationary linear model, the model error is modeled by
an additive noise that is generally White and Gaussian. Then the evolution
equation takes the form

x ( k +1)= Ax ( k )+ Bu ( k )+ v ( k +1) ,

where the v ( k ) are gaussian centered (null expectation) independent random
vectors with covariance matrix Γ .
In that case, the state trajectory is partially random, and the process is
called a stochastic process. In the following, we provide some examples of
controlled dynamical systems, as illustrations for more formal considerations.

4.1.2 An Example of Discrete Dynamical System

First consider an example of a dynamical system with discrete state space. A
labyrinth with 18 possible positions is shown on Fig. 4.1.
The state space is an 18-element set

{ 12, 13, 14, 15, 21, 22, 24, 32, 33, 34,

35, 41, 42, 44, 52, 53, 54, 55 . The set of controls may be chosen as the set of

four directions (N, W, S, E). The evolution is given by the natural mapping
that associates to an initial position and a course order either the resulting
position if the order is feasible, or the initial state if it is not:

}

f (12 , N) = 12 ,f (13 , N) = 13 , ..., f (21 , N) = 21 ,f (22 , N) = 12 , ...,
f (12 , W) = 12 ,f (13 , W) = 12 , ..., f (21 , W) = 21 ,f (22 , W) = 21 , ...,
f (12 , S) = 22 ,f (13 , S) = 13 , ..., f (21 , S) = 21 ,f (22 , S) = 32 , ...,
f (12 , E) = 13 ,f (13 , E) = 14 , ..., f (21 , E) = 22 ,f (22 , E) = 22 , ....

Other modeling rules may be chosen, corresponding to other state repre-
sentations of the same problem. For instance, one may prefer characterizing