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African vultures optimization-based hybrid neural network–proportional-integral-derivative controller...
The second hidden layer, similar in structure
to the first, consists of four neurons. Each neu-
ron in this layer is fully connected to all outputs
of the neurons in the first hidden layer, in addi-
tion to a bias unit, through corresponding con-
nection weights. The output of each neuron is
computed by summing its total weighted inputs,
as described in Equations (44) and (45):

 Figure 4. The neural network of the self-tuning

 
P 2 proportional-integral-derivative controller
(k)
1  v11 v12 v13 v14 v15 
2   
P
2
 (k)   v21 v22 v23 v24 v25 
 =  
 P
 (k) 2   v31 v32 v33 v34 v35 
 
 3 
P 2  v41 v42 v43 v34
(k) v45 
4
v51 v52 v53 v54 v55
 1 
S (k)
1
1
 S (k) 
2
 1 
 S (k) 
3
 1 
S (k)
 
4
1
(44) Figure 5. Block diagram of the self-tuning neural
network–proportional-integral-derivative (PID)
control system
P 3 
(k)  
1 vv11 vv12 vv13 vv14 vv15
P
(k)  =  vv21 vv22 vv23 vv24 vv25 
 3
 2  P  2
S (k) 2 
1 (k) 1 P (k) 3 vv31 vv32 vv33 vv34 vv35
S (k)  (k) 
 2  P 2 3

 2  = (45)  2 
2  P 2  S (k)
 2  1
3  (k)
2
 S (k) 3 S (k)
2
S (k) P (k) 2 4  2 2 
4


 S (k) 
3
 2 
S (k)
 4 
1
(46)
P 2
where (k) is defined as the sum of input con-
i   P (k) 3 
nections for each neuron in the second hidden K p P 1
3
2
layer, while S (k) represents the i-th output of  K i  =  P (k) 2 3  (47)
i
the same layer’s neuron, and vij are weights of K d (k) 3
3
connections. where P (k) is defined as the sum of input con-
i
nections for each neuron in the output layer, while
The output layer, also known as the final
layer, consists of three neurons. Each neuron in (K p , K i , and K d ) are neurons’ outputs of the out-
this layer is connected to all outputs from the sec- put layer, and vvij are weights of connections.
ond hidden layer, as well as to a bias unit, through 3.3. Neural network–proportional-
weighted connections. The output of each neu- integral-derivative controller
ron is calculated as the weighted sum of its in-
puts. These three outputs correspond directly to The proposed hybrid NN–PID controller is de-
the PID controller parameters K p , K i , and K d as picted in Figure 6. A single neuron represent-
represented in Equations (46) and (47). These ing the error e ψi (k) between the desired and ac-
values are then applied in the same manner as in tual control variables constitutes the input layer.
a con-PID controller. The architecture of the self- Three neurons constitute the first hidden layer,
tuning PID controller and the complete feedback imitating the PID functions of a traditional PID
control system is illustrated in Figure 5. controller. The e ψi (k), multiplied by a weight
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