[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

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Estimation of Solution of Discrete Linear Time-Varying System 313 { aik (p) for i ≠ k ã ik (p)= a ii (p)+1fori = k , (4) a (i) jk (p)=a ij(p)ã ik (p), (5) b(p)= 1 ∣ ∣ ∣a (i) ∣∣. 2 jk (p)+a(i) kj (p) (6) s ∑ i, j,k=1 The next Theorem contains the main result of our paper. Theorem 2. For any solution x(n,x 0 ) of system (1) the estimate ‖x(n + 1,x 0 )‖ 2 ≤‖x 0 ‖ 2 ∏ n p=0 is valid for any n ∈ N. Moreover, if b(n) < 1 for all n, then ‖x 0 ‖ 2 is also valid for all n ∈ N. Proof. Take a solution of (1) ∏ n p=0 with nonzero initial condition. We have and (b(p)+1), (7) (1 − b(p)) ≤‖x(n + 1,x 0 )‖ 2 (8) x(p,x 0 )=[x 1 (p),...,x s (p)] T x i (p + 1)= s ∑ j=1 x i (p + 1) − x i (p)= x i (p + 1)+x i (p)= Multiplying the last two identities we obtain x 2 i (p + 1) − x 2 i (p)= a ij (p)x j (p), i = 1,...,s s ∑ j=1 s ∑ k=1 a ij (p)x j (p), ã ik (p)x k (p). s s ∑ a ij (p)ã ik (p)x j (p)x k (p)= ∑ a (i) jk (p)x j(p)x k (p). j,k=1 j,k=1 Adding up these equalities for i = 1,...,s we get ‖x(p + 1)‖ 2 −‖x(p)‖ 2 = s ∑ i, j,k=1 a (i) jk (p)x j(p)x k (p)
314 A. Czornik, A. Nawrat, and M. Niezabitowski and therefore ∣ ∣‖x(p + 1)‖ 2 −‖x(p)‖ 2∣ ∣ 1 ≤ 4 1 2 1 2 s ∑ i, j,k=1 ( s s ∑ x j(p)) 2 ∑ j=1 s ∑ i, j,k=1 ∣ ∣a (i) ∣ ∣∣ jk (p)+a(i) kj (p) ( x 2 j (p)+x 2 k (p)) ≤ (∣ ∣ ) ∣∣a (i) ∣∣x jk (p)+a(i) kj (p) 2 j (p) ≤ ∣ ∣a (i) i, j,k=1 ∣ ∣∣ jk (p)+a(i) kj (p) = ‖x(p)‖ 2 b(p). Now, for natural p = 0,...,n the following inequality −‖x(p)‖ 2 b(p) ≤‖x(p + 1)‖ 2 −‖x(p)‖ 2 ≤‖x(p)‖ 2 b(p) holds. Dividing the last inequality by ‖x(p)‖ 2 we obtain and therefore n ∏ p=0 −b(p) ≤ (1 − b(p)) ≤ ‖x(p + 1)‖2 ‖x(p)‖ 2 ‖x(n + 1)‖2 ‖x(0)‖ 2 − 1 ≤ b(p), ≤ n ∏ p=0 (1 + b(p)) Finally, multiplying the last inequality by ‖x(0)‖ 2 , our upper (7) and lower (8) estimates follow immediately what ends the proof. The next four examples illustrate our estimates. Example 1. Consider system (1) with (A(n)) n∈N given by Namely, its elements are as follows [ √ 2sinn 1 A(n)= √n 3 +10 ]. 2cosn 1 n 2 +20 a 11 (n)= √ 2sinn a 12 (n)= 1 n 3 +10 a 21 (n)= 1 a n 2 +20 22 (n)= √ 2cosn . Applying the above-mentioned elements of (A(n)) n∈N to (3) and (4) we obtain (5) in the following form
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Studies in Computational Intelligen
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Aleksander Nawrat · Zygmunt Kuś E
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“In God We Trust, All others we o
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VIII Preface valuable information i
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Contents Part I: Design of Object D
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Contents XIII Technology Developmen
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XVI List of Contributors Adam Czorn
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XVIII List of Contributors Henryk M
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XX List of Contributors Józef Wron
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2 Design of Object Detection, Recog
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4 A. Babiarz et al. 1 Introduction
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6 A. Babiarz et al. The description
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8 A. Babiarz et al. a) b) Fig. 5. T
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10 A. Babiarz et al. o o o “micvo
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12 A. Babiarz et al. a) b) Fig. 7.
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14 A. Babiarz et al. • The camera
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16 A. Babiarz et al. the camera ser
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18 A. Babiarz et al. a) b) Fig. 11.
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20 A. Babiarz et al. patrol point i
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24 A. Babiarz et al. The sample cod
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Recognition and Location of Objects
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52 P. Demski et al. 4 Automatic Tar
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54 P. Demski et al. implementing su
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Object Tracking for Rapid Camera Mo
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Object Tracking in a Picture during
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94 Construction of Image Acquisitio
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96 G. Bieszczad et al. effectivenes
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98 G. Bieszczad et al. such a way t
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100 G. Bieszczad et al. Table 1. No
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112 G. Bieszczad et al. MTF functio
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114 G. Bieszczad et al. [11] Krupi
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116 D. Bereska et al. Presented in
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118 D. Bereska et al. Fig. 2. Image
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120 D. Bereska et al. Fig. 4. Ortho
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Omnidirectional Video Acquisition D
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Part III Design of Vision Based Con
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Vision System for Group of Mobile R
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182 D. Bereska et al. Fig. 5. Schem
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184 D. Bereska et al. 4.1 Mechanica
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186 D. Bereska et al. 4.1.3 Open-Lo
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188 D. Bereska et al. Fig. 16. Comp
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Probabilistic Approach to Planning
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206 Practical Applications of Class
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208 M. Mellado and K. Skrzypczyk an
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Prototyping the Autonomous Flight A
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Feature Extraction and HMM-Based Cl
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252 K. Daniec et al. by the ability
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254 K. Daniec et al. The idea of us
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Selection of Individual Gait Featur
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[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

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