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where
Y
is the output of the neuron,
X
represents the internal states of the neuron,
and
W
denotes weights assigned to each neural connection given as
Xxxx
= ( )
123
and
Wwww
T
1 2 3
, respectively. In the SNPID controller configuration, the adap-
tive weights
w
i
are analogous to the conventional PID gains,
K
p
,
K
i
, and
K
d
. From
Equation (5.8), the output of the SNPID controller is further expressed as
= (
,
)
3
∑
0
()
=
uk
Kwkx k
i
() ()
(5.10)
i
i
=
where
K
is the gain value of the neuron. Note that the input to the neuron at time
k
is
given by the following equations.
()
=
()
−−
xk ek
ek
(
1
)
(5.11)
1
2
()
=
()
xk ek
(5.12)
()
=
()
−
(
)
+−
xk ek
2
ek
−
1
ek
(
2
)
(5.13)
3
The errors at time
k
, (
k
- 1), (
k
- 2), etc., are represented by
e
(
k
),
e
(
k -
1), (
k
- 2), etc.
where:
()
=
()
−
()
ek
rk
yk
(5.14)
Hence the single neuron PID control law may be expressed as:
3
∑
()
=−
(
)
+
uk
uk
1
Kwkx k
i
′
() ()
(5.15)
i
i
=
0
whereby the weights are determined by the Hebb learning algorithm described
below from Equations (5.16) to (5.19).
()
()
wk
i
wk
′
()
=
(5.16)
i
∑
3
wk
i
i
=
1
(
)
()
=
(
)
+
()
(
)
()
−−
wk
wk
−
1
ρ
ekuk
−
1
2
ek
ek
(
1
)
(5.17)
1
1
1
()
=
(
)
+
()
(
)
(
()
−−
)
wk wk
−
1
ρ
ekuk
−
1
2
ek
ek
(
1
)
(5.18)
2
2
2
()
=
(
)
+
()
(
)
()
−−
(
)
wk
wk
−
1
ρ
ekuk
−
1
(
2
ek
ek
1
)
(5.19)
3
3
3
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