Information Technology Reference
In-Depth Information
Fig. 1.5.
A two-input recurrent neural network. Digits in
square boxes
are the
delay assigned to each connection, an integer multiple of the time unit (or sampling
period)
T
. The network features a cycle from 3 to 3 through 4
it computes its output
y
3
(
kT
); the inputs of neuron 4 are
u
2
(
kT
)and
y
3
(
kT
),
and it computes its output
y
4
(
kT
); the inputs of neuron 5 are
y
3
(
kT
)
,u
1
(
kT
)
et
y
4
[(
k−
1)
T
], and it computes its output, which is the output of the network
g
(
kT
).
The Canonical Form of Recurrent Neural Networks
Because recurrent neural networks are governed by recurrent discrete-time
equations, it is natural to investigate the relations between such nonlinear
models and the conventional dynamic linear models, as used in linear modeling
and control.
The general mathematical description of a linear system is the state
equations,
x
(
k
)=
A
x
(
k
−
1) +
B
u
(
k
−
1)
g
(
k
)=
C
x
(
k
−
1) +
D
u
(
k
−
1)
,
where
x
(
k
) is the state vector at time
kT,
u
(
k
) is the input vector at time
kT,
g
(
k
) is the output vector at time
kT
and
A,B,C,D
are matrices. The state
variables are the minimal set of variables such that their values at time (
k
+1)
T
can be computed if (i) their initial values are known, and if (ii) the values of
the inputs are known at all time from 0 to
kT
. The number of state variables
is the
order
of the system.
Similarly the canonical form of a nonlinear system is defined as
x
(
k
)=
Φ
[
x
(
k
−
1)
,
u
(
k
−
1)]
g
(
k
)=
Ψ
[
x
(
k
−
1)
,
u
(
k
−
1)]
,
Search WWH ::
Custom Search