Neural Identification of Controlled Dynamical Systems and Recurrent Networks - Neural Networks: Methodology and Applications - page 234

Fig. 4.1. Alabyrinth

the state of a robot by the couple of the actual position and of the direction.
In our example, the state space would have then 18 4 = 72 elements and it

will be completed by a control set of three elements (A for move ahead, L for
move towards left, R for move towards right).
Software products that are used to exploit large database or to perform
data mining on the Web are facing a lot of problems that are formalized as
navigation problems in a graph. The state space is the set of the nodes of the
graph.

×

4.1.3 Example: The Linear Oscillator

Let us consider now the harmonic oscillator that is governed by the second-
order linear differential equation,

d 2 x
d t 2

= − x.

First, notice that the differential equation does not provide a genuine state
representation since it is a second-order equation. The associated continuous
state representation is

x 1
x 2

= x 2
−

,

d
d t

x 1

where the state incorporates the mobile current position x 1 and its speed x 2 .
In order to derive a discrete time evolution, we have to solve the differential
equation on the sampling period T . In that trivial example, the solution is