Page 189 - Neural Network Modeling and Identification of Dynamical Systems

180 5. SEMIEMPIRICAL NEURAL NETWORK MODELS OF CONTROLLED DYNAMICAL SYSTEMS

2
ˆ
∂ e(˜y, ˆ x,w) ∂ ˆ g(ˆ x,w) T ∂ ˆ g(ˆ x,w) ∂f(ˆ x(t,w),u(t),w) ∂ ˆ x(t,w) T
≈ , − λ(t,w)
∂w 2 ∂w ∂w ∂ ˆ x ∂w
2
∂ e(˜y, ˆ x,w) ∂ ˆ g(ˆ x,w) T ∂ ˆ g(ˆ x,w) ∂λ(t,w) T d ˆ x(t,w)
ˆ
≈ , (5.19) − − f(ˆ x(t,w),u(t),w) dt.
∂w∂ ˆ x ∂w ∂ ˆ x ∂w dt
2
∂ e(˜y, ˆ x,w) ∂ ˆ g(ˆ x,w) T ∂ ˆ g(ˆ x,w) (5.21)
≈ .
∂ ˆ x 2 ∂ ˆ x ∂ ˆ x
Integrating by parts, we obtain
Now, we describe a continuous time coun-
terpart of a backward-in-time BPTT algorithm.
¯ t
This method is based on the application of ad- d ∂ ˆ x(t,w) T
joint sensitivity analysis [22–24]. First, we treat dt ∂w λ(t,w)dt
the set of ODEs that describe the semiempiri- 0
cal model dynamics (5.2)asasetofconstraints. ∂ ˆ x(¯ t,w) T ∂ ˆ x(0,w) T
Next, we include them into the original indi- = ∂w λ(¯ t,w) − ∂w λ(0,w)
vidual trajectory error function along with the
¯ t
T
Lagrange multipliers λ to obtain the following ∂ ˆ x(t,w) dλ(t,w)
Lagrange function: − ∂w dt dt. (5.22)
0
¯ t

L(w) = e(˜y(t), ˆ x(t,w),w) If we substitute (5.22)into(5.21) and exclude

the term d ˆ x(t,w) − f(ˆ x(t,w),u(t),w) ,whichis
ˆ
0 dt

d ˆ x(t,w) identically zero, we get
T
ˆ
− λ(t,w) − f(ˆ x(t,w),u(t),w) dt.
dt
(5.20) ¯ t
∂L(w) ∂e(˜y(t), ˆ x(t,w),w)
=
Note that since the state variable predictions ˆ x ∂w ∂w
0
are in fact computed by solving the initial value
T
problem for the ODE system (5.2), these newly ∂f(ˆ x(t,w),u(t),w)
ˆ
imposed constraints are satisfied on the whole + ∂w λ(t,w)
time segment, and the second term of (5.20)is ∂ ˆ x(t,w) T dλ(t,w)
identically zero. Hence, the gradient and Hes- + ∂w
sian of the individual trajectory error function dt
T
are identical to the gradient and Hessian of the ∂f(ˆ x(t,w),u(t),w)
ˆ
Lagrange function. The Lagrange function gra- + ∂ ˆ x λ(t,w)
dient equals ∂e(˜y(t), ˆ x(t,w),w)
+ dt
∂ ˆ x
¯ t

∂L(w) ∂e(˜y(t), ˆ x(t,w),w) ∂ ˆ x(0,w) T ∂ ˆ x(¯ t,w) T
=
∂w ∂w + ∂w λ(0,w) − ∂w λ(¯ t,w).
0
(5.23)
T
∂ ˆ x(t,w) ∂e(˜y(t), ˆ x(t,w),w)
+
∂w ∂ ˆ x Again, since the artificial constraints are satis-

ˆ
d ∂ ˆ x(t,w) ∂f(ˆ x(t,w),u(t),w) fied on the whole time segment, the Lagrange
− −
dt ∂w ∂w multipliers are completely arbitrary. In order to