Page 229 - Neural Network Modeling and Identification of Dynamical Systems

220 6. NEURAL NETWORK SEMIEMPIRICAL MODELING OF AIRCRAFT MOTION

Algorithm 1 Generation of the test maneuvers.
Require: set of the admissible values for the state variables X ⊂ R n x and the control variables U ⊂
R ;
n u
Require: maximum number of maneuvers P, admissible maneuver duration limits [K min ,K max ];
Require: maximum number of candidate trajectory segments Q, minimum admissible quality of
a candidate trajectory segment d min , admissible candidate trajectory segment duration limits
[S min ,S max ] and number of trials R;
Require: dynamical system model right hand side f: X × U → R ;
n x
Require: metric ρ for comparison of sets of vectors;

Ensure: set of the test maneuvers M that contains pairs x 0 ,{u k } K , where x 0 ∈ X is an initial state,
k=1
u k ∈ U is a sequence of controls;
Ensure: set of the function f argument values A for the selected trajectories, which contains vectors
a ∈ R n x +n u ;
1: M ← ∅;
2: A ← ∅;
3: p ← 1;
4: while p P and S max >S min do
5: r ← 1;
6: while r< R do
¯
7: A ← A;
p
8: x ∼ U(X);
0
p
9: K ← 0;
p
10: while K <K max do
p
11: S ← min{S max ,K max − K };
p

p,q K +S−1

12: Generate a set of candidate maneuver segments u , q = 1,...,Q within U,
k k=K p
for example, a sequence of steps with uniformly distributed amplitudes and frequencies;
13: Numerically solve the corresponding initial value problems using the dynamical sys-
p

p,q K +S
tem model to obtain candidate trajectory segments x k k=K p , q = 1,...,Q;
p

p,q p,q T K +S−1
14: A ˜ p,q ← (x ,u ) , q = 1,...,Q;
k k k=K p
15: Evaluate fitness of each candidate maneuver segment d p,q =
⎧ p,q
p
p
⎪ 0, if ∃k ∈[K + 1,K + S]: x / ∈ X,
⎪ k
⎨
0, if maximum eigenvalue of the cov A ˜ p,q is too small, q = 1,...,Q;
⎪
⎪
⎩ ¯ ˜ p,q
ρ(A,A ), otherwise,
∗
∗
16: Find the best candidate maneuver segment q ← argmaxd p,q with fitness d ←
q
maxd p,q ;
q
∗
17: if d >d min then
p p,q ∗ p p
18: u ← u , k = K ,...,K + S − 1;
k k
p p,q ∗ p p
19: x ← x , k = K + 1,...,K + S;
k k
¯
¯
20: A ← A ∪ A ˜ p,q ∗ ;
p
p
21: K ← K + S;