The Development of Neural Network Based System Identification ...

114 CHAPTER 4 NEURAL NETWORK BASED SYSTEM IDENTIFICATION
Initialize
Parameters Vector,
Forgetting factor, 0
and matrix
P 0 Q 0 I
ˆ 0
Update Training
Data at time,
Calculate Jacobian Matrix
t |
Calculate step (1) – (5) in Potter’s Square
Root Algorithm
Calculate prediction error e(t)
and compute,
tˆ t
Update
t
ˆ t
Qt
End Condition
END
Figure 4.13
The recursive Gauss-Newton (rGN) algorithm with Potter’s square root factorisation.
nearly singular if the model set contains too many parameters or if the input signal is
not general enough [Ljung and Soderstrom, 1983]. This problem can be overcome by
introducing a lower and upper bounds on the eigenvalues of P (t). Several variations
of recursive Gauss-Newton algorithms such as Constant Trace (CT) and Exponential
Forgetting and Resetting Algorithm (EFRA) have been proposed in various examples to
overcome the unstable numerical P (t) recursion [Norgaard, 2000, Salgado et al., 1988].
By using the CT method, Step 6 in Table 4.1 is introduced to bound the eigenvalues of
P (t). The ρ max and ρ min are the maximum and minimum eigenvalues respectively, and
the values are selected so that ρ max /ρ min ≃ 10 5 . The initial Q(0) should be selected as