Page 199 - Neural Network Modeling and Identification of Dynamical Systems

190 5. SEMIEMPIRICAL NEURAL NETWORK MODELS OF CONTROLLED DYNAMICAL SYSTEMS

even for the slightest increase of τ.Toovercome ∂H(a,τ,w) ∂ E(a,τ,w)
2 ¯
this issue, we propose the following homotopy =
∂τ ∂w∂τ
for the error function which controls the predic- P 2 (p)
∂ E (τ,w)
tion horizon value: =−(w − a) + ,
∂w∂τ
p=1
2 P 2 ¯
w − a (p) ∂H(a,τ,w) ∂ E(a,τ,w)
¯
E(a,τ,w) = (1 − τ) + E (τ,w), = 2
2 ∂w ∂w
p=1
P 2 (p)
∂ E (τ,w)
(5.45) = (1 − τ)I + .
∂w 2
τ ¯ t (p) p=1

E (p) (τ,w) = e (p) (˜y (p) (t), ˆ x (p) (t,w),w)dt. (5.48)
0
Again, the individual trajectory error function
(5.46)
Hessian expressions (5.13)and (5.26) can be
2
adapted to compute ∂ E(τ,w) by replacing the
2
∂w
Thus, for τ = 0 the error function has a unique upper limit of integration from ¯ t to τ ¯ t.Inorder
stationary point – the global minimum w = a. to derive the expression for ∂ E(τ,w) , we apply
2
The prediction horizon for each trajectory of the ∂w∂τ
the Leibniz integral rule, which gives us
training set grows linearly with the parameter τ,
so that for τ = 1 the individual trajectory error 2
∂ E(τ,w) ∂e(˜y(τ ¯ t), ˆ x(τ ¯ t,w),w)
function (5.46) is identical to (5.8). = ¯ t
∂w∂τ ∂w
The corresponding total error function gradi-
T
ent homotopy has the form ∂ ˆ x(τ ¯ t,w) ∂e(˜y(τ ¯ t), ˆ x(τ ¯ t,w),w)
+ (5.49)
∂w ∂ ˆ x
¯
∂E(a,τ,w)
H(a,τ,w) = in the case of the forward-in-time method and
∂w
P (p)
∂E (τ,w) 2
= (1 − τ)(w − a) + . ∂ E(τ,w) ∂e(˜y(τ ¯ t), ˆ x(τ ¯ t,w),w)
∂w = ¯ t
p=1 ∂w∂τ ∂w
(5.47) ˆ T
∂f(ˆ x(τ ¯ t,w),u(τ ¯ t),w)
+ λ(τ ¯ t,w) (5.50)
∂w
As described in the previous section, the indi-
vidual trajectory error function gradient can be
in the case of the backward-in-time method.
computed either by taking a forward-in-time or Now, we discuss the numerical methods that
a backward-in-time approach. In fact, the cor-
allow us to trace the solution curve γ ⊂[0,1]×
responding expressions (5.12)and(5.25) remain R . If we parametrize the curve γ with re-
n w
almost the same for ∂E(τ,w) and the only dif-
∂w spect to some parameter s ∈ R,sothat γ(s) =
ference is the upper limit of integration, which τ(s) !
needs to be changed from ¯ t to τ ¯ t. w(s) , and then differentiate the equation sys-
Derivatives of the total error function gradi- tem H a (τ(s),w(s)) = 0 with respect to the pa-
ent homotopy (5.47)withrespect to τ and w are rameter s, we obtain the following system of dif-
as follows: ferential equations: