[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 Adam Czornik, Aleksander Nawrat, and Michał Niezabitowski Abstract. In this paper we propose the upper, and in some cases, lower bounds for norm of solution of linear discrete time-varying system by coefficients matrices. We show that our estimates have two merits. Firstly, we do not need to know eigenvalues or spectral norm of matrices because we can simply calculate it by matrix coefficients. Secondly, our bounds are also valid in case of singular matrices. Moreover, in the paper we present an upper estimate for stationary system. Keywords: time-varying discrete linear systems, growth bounds, stability, Lyapunov exponents. 1 Introduction Consider the linear discrete time-varying system x(n + 1)=A(n)x(n),n ≥ 0 (1) where A =(A(n)) n∈N , A(n)=[a ij (n)] i, j=1,...,s is a sequence of s-by-s real matrices. By ‖·‖ we denote the Euclidean norm in R s and the induced operator norm. The transition matrix is defined as A (m,k)=A(m − 1)...A(k) for m > k and A (m,m)=I, whereI is the identity matrix. For an initial condition x 0 the solution of (1) is denoted by x(n,x 0 ) so Adam Czornik · Aleksander Nawrat · Michał Niezabitowski Silesian University of Technology, Institute of Automatic Control, Akademicka 16 Street, 44-101 Gliwice e-mail: {Adam.Czornik,Aleksander.Nawrat, Michal.Niezabitowski}@polsl.pl A. Nawrat and Z. Kuś(Eds.):Vision Based Systems for UAV Applications, SCI 481, pp. 311–326. DOI: 10.1007/978-3-319-00369-6_20 c○ Springer International Publishing Switzerland 2013
312 A. Czornik, A. Nawrat, and M. Niezabitowski x(n,x 0 )=A (n,0)x 0 . The main result of this paper is to present an upper, and in some cases, a lower bound for norm of solution of system (1) by elements of matrices A(n). Estimation problem of the solutions of continuous linear systems in terms of the coefficients is discussed in literature (for example see [1], Chapter IV, §6). The author shows on examples that is impossible to judge objectively, which of the presented there estimates is in general better. This problem is also discussed in [3], [15] and [16]. Estimation problem for the solution of dynamical systems is closely related to the concept of Lyapunov exponents. Now we present the definitions. Let a =(a(n)) n∈N be a sequence of real numbers. The numbers (or the symbol ±∞)definedas 1 λ (a)=limsup n→∞ n ln|a(n)| is called the characteristic exponent of sequence (a(n)) n∈N . For x 0 ∈ R s ,x 0 ≠ 0 the Lyapunov exponent λ A (x 0 ) of (1) is defined as characteristic exponent of (‖x(n,x 0 )‖) n∈N , therefore 1 λ A (x 0 )=limsup n→∞ n ln‖x(n,x 0)‖. It is well known [4] that the set of all Lyapunov exponents of system (1) contains at most s elements, say −∞ < λ 1 (A) < λ 2 (A) < ... < λ r (A) < ∞, r ≤ s and the set {λ 1 (A),λ 2 (A),...,λ r (A)} is called the spectrum of (1). For exhaustive presentation of theory of Lyapunov exponents we recommend the following literature positions: [2], [4]. Problem of estimation of Lyapunov exponents is discussed in [5]-[9], [12]-[14] and [18]. In the paper we will also use the following conclusion from publication [10] and [11]. Theorem 1. Suppose, that for system (1) we have lim A(n)=A. (2) n→∞ Then, Lyapunov exponents of system (1) coincides with logarithms of moduli’s eigenvalues of matrix A. 2 MainResults Let introduce the following notations: { aij (p) for i ≠ j a ij (p)= a ii (p) − 1fori = j , (3)
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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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[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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