[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 319 As previously, we can only estimate an upper bound for growth rate. Example 3. Consider system (1) with (A(n)) n∈N given by [ sinn A(n)= Namely, its elements are as follows 5 + 1 1 1 n+20 n+10 cosn 5 + 1 a 11 (n)= sinn 5 + 1 a 12 (n)= 1 n+10 a 21 (n)= n+20 1 a 22 (n)= cosn 5 + 1 . Applying the above-mentioned elements of (A(n)) n∈N to (3) and (4) we obtain (5) in the following form ] . Because, a (1) 11 (n) =a 11(n)ã 11 (n)=(a 11 (n)) 2 − 1 = sin2 n + 10sinn 25 a (1) 12 (n) =a 11(n)ã 12 (n)=(a 11 (n) − 1) a 12 (n)= sinn a (1) 21 (n) =a 12(n)ã 11 (n)=a 12 (n)((a 11 (n)) + 1)= a (1) 22 (n) =a 12(n)ã 12 (n)=(a 12 (n)) 2 = a (2) 11 (n) =a 21(n)ã 21 (n)=(a 21 (n)) 2 = 1 (n 3 + 10) 2 1 (n 2 + 20) 2 a (2) 12 (n) =a 21(n)ã 22 (n)=a 21 (n)(a 22 (n)+1) = 5n + 50 10 + sinn 5n + 50 10 + cosn 5n + 100 a (2) 21 (n) =a 22(n)ã 21 (n)=(a 22 (n) − 1) a 21 (n)= cosn 5n + 100 a (2) 22 (n) =a 22(n)ã 22 (n)=(a 22 (n)) 2 − 1 = cos2 n + 10cosn 25 b(n)= 1 2 s ∑ i, j,k=1 ∣ ∣a (i) ∣ ∣∣ jk (n)+a(i) kj (n) = 1 [∣ ∣ ∣ ∣ ∣ ∣ ∣∣a (1) ∣∣ ∣∣a 11 2 (n)+a(1) 11 (n) (1) ∣∣ ∣∣a + 12 (n)+a(1) 21 (n) (1) ∣∣+ + 21 (n)+a(1) 12 (n) ∣ ∣ ∣ ∣ ∣ ∣ ∣a (1) ∣∣ ∣∣a 22 (n)+a(1) 22 (n) (2) ∣∣ ∣∣a + 11 (n)+a(2) 11 (n) (2) ∣∣+ + 12 (n)+a(2) 21 (n) ∣ ∣ ∣] ∣ ∣a (2) ∣∣ ∣∣a 21 (n)+a(2) 12 (n) (2) ∣∣ + 22 (n)+a(2) 22 (n) =
320 A. Czornik, A. Nawrat, and M. Niezabitowski [∣ 1 ∣∣∣ sin 2 ∣ n + 10sinn + sin2 n + 10sinn ∣∣∣ 2 25 25 ∣ + sinn 10 + sinn + 5n + 50 5n + 50 ∣ + 10 + sinn ∣ 5n + 50 + sinn ∣ ∣ ∣∣∣∣ 5n + 50∣ + 1 (n 3 + 10) 2 + 1 ∣∣∣∣ (n 3 + 10) 2 + ∣ 1 ∣(n 2 + 20) 2 + 1 ∣∣∣∣ (n 2 + 20) 2 + 10 + cosn ∣ 5n + 100 + cosn 5n + 100∣ + ∣ cosn 10 + cosn ∣∣∣ ∣ + 5n + 100 5n + 100 ∣ + cos 2 ] n + 10cosn + cos2 n + 10cosn 25 25 ∣ = [ 1 2 ∣ sin 2 n + 10sinn ∣ 4 + 2 25 5n + 50 |5 + sinn| + 2 (n + 10) 2 + 2 (n + 20) 2 + 4 2 |5 + cosn| + ∣ cos 2 n + 10cosn ∣ ] = 5n + 100 25 1 ∣ sin 2 n + 10sinn ∣ 2 + 25 5n + 50 |5 + sinn| + 1 (n + 10) 2 + 1 (n + 20) 2 + 2 1 |5 + cosn| + ∣ cos 2 n + 10cosn ∣ 5n + 100 25 vary between 0.2 and 1 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 (3) In this example we are able to estimate also the lower bound for growth rate. Example 4. Consider system (1) with (A(n)) n∈N given by A(n)= Namely, its elements are as follows [ 1 1 1 n 2 +20 n 3 +10 a 11 (n)=1 a 12 (n)= 1 a 21 (n)= 1 a n 2 +20 22 (n)=1 1 ] . (9) n 3 +10 Applying the above-mentioned elements of (A(n)) n∈N to (5) and (3) 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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22 A. Babiarz et al. particular ang
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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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48 P. Demski et al. Fig. 1. Automat
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50 P. Demski et al. E-Grip firing m
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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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102 G. Bieszczad et al. Fig. 7. Pri
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104 G. Bieszczad et al. Fig. 10. Re
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106 G. Bieszczad et al. Fig. 13. Ti
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108 G. Bieszczad et al. Fig. 16. Op
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110 G. Bieszczad et al. has to make
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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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A Distributed Control Group of Mobi
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178 D. Bereska et al. The aim of th
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180 D. Bereska et al. 3 Control Alg
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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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210 M. Mellado and K. Skrzypczyk a
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Prototyping the Autonomous Flight A
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Feature Extraction and HMM-Based Cl
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248 K. Daniec et al. In this articl
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250 K. Daniec et al. Fig. 2. Applic
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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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