The Development of Neural Network Based System Identification ...

142 CHAPTER 5 NN BASED SYSTEM IDENTIFICATION: RESULTS AND DISCUSSION
1
Pitch Rate, q (rad/s)
0.5
0
−0.5
data
Elman NNID
−1
0 100 200 300 400 500 600 700 800 900 1000
time (samples)
(a)
0.05
Prediction Error
0
−0.05
0 100 200 300 400 500 600 700 800 900 1000
time (samples)
(b)
Figure 5.17 The prediction from the modified Elman network for pitch dynamics. (a) The one-step
ahead prediction overlaid with the measured helicopter responses. (b) The error plot between one-step
ahead prediction and the measurement data. The red dashed line indicates estimation from the modified
Elman network while solid blue line with ‘x’ marker represents the output measurement.
various noise levels indicate that an exceptional prediction performance is achieved up
until random noise with s = 0.5 added to the optimal weights.
Using the 300 set of weights generated, the 95% confidence intervals can be constructed
for the optimal weights using the standard statistical inference method. The
average range of the upper and lower output performance (NMCIW) for each noise level
case is given in Table 5.9. As can be seen from the result, the weights set added with
s = 0.7 random noise provides a wide average confidence interval (22.4513%) compare
Table 5.8
The Summaries of Error Statistics for Modified Elman Network.
Error Statistics
System Responses RMSE RMSE (%) R 2
One-Step Ahead Prediction
p 0.0604 14.3606 0.9792
q 0.0184 5.7498 0.9962
5-Step Ahead Prediction
p 0.0967 23.0956 0.9467
q 0.0468 15.6510 0.9755