Page 202 - Neural Network Modeling and Identification of Dynamical Systems

5.5 OPTIMAL DESIGN OF EXPERIMENTS FOR SEMIEMPIRICAL ANN-BASED MODELS 193
controlled dynamical systems in off-line setting for the set of predicted reference trajectories, i.e.,
according to the optimality criterion that does

not depend on the form of the network and also
allows to account for the constraints on the con- h(ξ) =− p(z)lnp(z)dz, (5.57)
trol and state variables. We assume that the total U×X
number of reference maneuvers P and their du-
rations ¯ t (p) are given. Thus, we need to find the where p is the probability density function of ξ.
optimal set of reference maneuvers of the fol- This criterion was proposed and analyzed in [45,
lowing form: 46].
Since the probability density function p is un-

P known, we cannot compute differential entropy
¯ x (p) (0), ¯ u (p) , (5.56)
p=1 using (5.57). Instead, we estimate it from a sam-
n z
ple Z = {z i } i=1 using the Kozachenko–Leonenko
where ¯ x (p) (0) ∈ X and ¯ u (p) :[0, ¯ t (p) ]→ U are the method [47], i.e.,
initial state and the control signal of the pth ref-
erence trajectory, respectively. n z
n u + n x
The corresponding set of true reference tra- h(ξ) = lnρ i + ln(n z − 1)
ˆ
P n z
jectories has the form x (p) , where x (p) : i=1
p=1 & '
n u +n x
(p)
[0, ¯ t (p) ]→ X satisfy dx (t) = f(x (p) (t),u (p) (t)) π 2 (5.58)
dt + ln n u +n x + γ,
and x (p) (0) = ¯ x (p) (0). However, the true function 2 + 1

f is unknown. Moreover, we do not have an ex- ρ i = min ρ z i ,z j ,
perimental data set to build an empirical or a j=1,...,n z
j =i
semiempirical model yet. Therefore, we utilize a
ˆ
theoretical model of the system f to obtain a set where ρ i is the distance between the ith sample
(p) P
of predicted reference trajectories ˆ x .As point and its nearest neighbor according to some
p=1
already mentioned, this theoretical model might metric ρ, is the gamma function, and γ is the
be a crude approximation to an unknown sys- Euler–Mascheroni constant. The sample Z is ob-
tem. Fortunately, the accuracy requirements for tained by numerical solution of a set of initial
the problem of reference maneuvers design are value problems for the predicted reference tra-
not too strict: the associated reference trajecto- jectories ˆ x (p) , using the theoretical model f and
ˆ
ries are required only to reach some area of in- the set of reference maneuvers (5.56).
terest within the state space. Then, the problem of optimal design of refer-
Now, we need to define the optimality crite- ence maneuvers may be viewed as an optimal
rion for a set of predicted reference trajectories. control problem, i.e.,
$ %
We treat each point ¯ u (p) (t), ˆ x (p) (t) as a sample
point of (n u + n x )-dimensional random vector ξ ˆ
and assume that we want it to have a uniform $ minimize − h
% P
¯ x (p) (0),¯ u (p)
distribution on U × X. Since the random vec- p=1
subject to ¯ u (p) (t) ∈ U,t ∈[0, ¯ t (p) ],
tor uniformly distributed on compact set U × X
achieves the maximum value of differential en- p = 1,...,P, (5.59)
tropy among all continuous distributions sup- (p) (p)
ˆ x (t) ∈ X,t ∈[0, ¯ t ],
ported in U × X, it seems reasonable to utilize
differential entropy as the optimality criterion p = 1,...,P.