1
CHAPTER 15
Dynamically Driven Recurrent Networks
Problem 15.1
Referring to the simple recurrent neural network of Fig. 15.3, let the vector u(n) denote the input
signal, the vector x(n) denotes the signal produced at the output of the hidden layer, and the vector
y(n) denotes the output signal of the whole network. Then, treating x(n) as the state of the
network, we may describe the state-space model of the network as follows:
where and are vector-valued functions of their respective arguments.
Problem 15.2
Referring to the recurrent MLP of Fig. 15.4, we note the following:
(1)
(2)
(3)
where , , and are vector-valued functions of their respective arguments.
Substituting (1) into (2), we write
(4)
Define the state of the system at time n as
(5)
Then, from (4) and (5) we immediately see that
(6)
where f is a new vector-valued function. Define the output of the system as
x n 1+()fxn()u n(),()=
y n() gxn()()=
f
.
() g
.
()
x
I
n 1+()f
1
x
I
n()u n(),()=
x
II
n 1+()f
2
x
II
n()x
I
n 1+(),()=
x
0
n 1+()f
3
x
0
n()x
II
n 1+()()=
f
1
.
() f
2
.
() f
3
.
()
x
II
f
2
x
II
f
1
x
I
u n(),(),()=
x
II
n 1+()
x
II
n()
x
0
n 1–()
=
x n 1+()fxn()u n(),()=
