[Studies in Computational Intelligence 481] Artur Babiarz, Robert Bieda, Karol Jędrasiak, Aleksander Nawrat (auth.), Aleksander Nawrat, Zygmunt Kuś (eds.) - Vision Based Systemsfor UAV Applications (2013, Sprin

322 A. Czornik, A. Nawrat, and M. Niezabitowski
[
1
|0| +
2 ∣ 0 + 2
∣ ∣ ∣ ∣∣∣ n 3 + 10∣ + 2 ∣∣∣ n 3 + 10 + 0 1
+
∣(n 3 + 10) 2 + 1 ∣∣∣∣
(n 3 + 10) 2 +
∣ 1
∣(n 2 + 20) 2 + 1 ∣∣∣∣ ∣ (n 2 + 20) 2 +
2 ∣∣∣
∣n 2 + 20 + 0 +
∣ 0 + 2
] n 2 + 20∣ + |0| =
[
∣ ∣ ∣
1
4
1
∣∣∣∣ 2 ∣n 3 + 10∣ + 2 1 ∣∣∣∣ 1 ∣∣∣∣ ]
(n 3 + 10) 2 + 2
∣(n 2 + 20) 2 + 4
1
∣n 2 + 20∣
=
2
n 3 + 10 + 1
(n 3 + 10) 2 + 1
(n 2 + 20) 2 + 2
n 2 + 20 = 2n3 + 21
(n 3 + 10) 2 + 2n2 + 41
(n 2 + 20) 2
tends to zero it implies simultaneously that for any solution x(n,x 0 ) of system (1)
the estimates (7) and (8)
‖x 0 ‖
n
∏
p=0
√
(1 − b(p)) ≤‖x(n + 1,x0 )‖≤‖x 0 ‖
n
∏
p=0
√
(b(p)+1)
are valid for all n. The numerical values of the solution and bounds are presented in
Figure (4)
Fig. 4. Our upper and lower estimates, and values of norms of matrix products
By contrast with previous 3 examples here we can both estimate a growth rates
and, what is more important, decide about stability of system (1).