[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

Estimation of Solution of Discrete Linear Time-Varying System 323
∞
It is well known (see [17]), that an infinite product ∏ (x(p)+1) with positive
p=0
x(p) converges if and only if the series ∑
∞ x(p) converges. Also similarly, if 0 <
p=0
x(p) < 1 holds, then ∏
∞ (1 − x(p)) approaches a non-zero limit if and only if the
p=0
series ∑
∞ x(p) converges. Therefore, from Theorem 2 we obtain the following result
p=0
about stability.
Corollary 1. If the series ∑
∞ b(p), where b(p) are given by (6), converges, then (1)
p=0
is stable.
It is clearly that in Example (4) the series ∑
∞ b(p) is convergent, and therefore
p=0
system (1) is stable.
Observe that from Theorem 1 we know that Lyapunov exponents of system (1)
with coefficients given by (9) are all equal to zero. But it doesn’t imply that the
system is stable.
Bound of norm of a matrix power by coefficients is a particular case of our estimate.
Our result is also valid for stationary systems and we will ilustrate it by an
example of 2-by-2 matrix.
Example 5. Consider system (1) with all A(n) equal to A,where
[ ] ef
A = .
gh
Namely, its elements are as follows
a 11 (n)=ea 12 (n)= f
a 21 (n)=ga 22 (n)=h .
Applying the above-mentioned elements of A to (5) and (3) we obtain (5) in the
following form
a (1)
11 (n) =a 11(n)ã 11 (n)=(a 11 (n)) 2 − 1 = e 2 − 1
a (1)
12 (n) =a 11(n)ã 12 (n)=(a 11 (n) − 1) a 12 (n)=(e − 1) f
a (1)
21 (n) =a 12(n)ã 11 (n)=a 12 (n)((a 11 (n)) + 1)=(e + 1) f
a (1)
22 (n) =a 12(n)ã 12 (n)=(a 12 (n)) 2 = f 2
a (2)
11 (n) =a 21(n)ã 21 (n)=(a 21 (n)) 2 = g 2
a (2)
12 (n) =a 21(n)ã 22 (n)=a 21 (n)(a 22 (n)+1) =g(h + 1)
a (2)
21 (n) =a 22(n)ã 21 (n)=(a 22 (n) − 1) a 21 (n)=g(h − 1)
a (2)
22 (n) =a 22(n)ã 22 (n)=(a 22 (n)) 2 − 1 = h 2 − 1