Abstract
In the paper we study the problem of the influence of the parametric uncertainties on the Bohl exponents of discrete time-varying linear system. We obtain formulas for the computation of the exact boundaries of lower and upper mobility for the supremum and infimum of the Bohl exponents under arbitrary small perturbations of system coefficients matrices on the basis of the transition matrix.
Highlights
Consider a discrete time-varying linear system x(n + 1) = A(n)x(n), x(n) ∈ Rs, n ∈ N0 ≡ N ∪ {0}, (1)where A = (A(n))n∈N0 is a bounded sequence of invertible s-by-s real matrices such that A−1(n) is bounded
Denote a = max sup A(n), sup A−1(n) n∈N0 n∈N0 and let ΦA (·, ·) denote the transition matrix of the system (1) which is defined as ΦA(n, k) = A(n − 1)...A(k), if n > k, ΦA(n, n) = Is, and ΦA(n, m) = Φ−A1(m, n) if m > n, where Is is the identity matrix of order s
In the paper it has been studied the influence of parametric uncertainties on Bohl exponents of the discrete linear time-varying systems
Summary
Consider a discrete time-varying linear system x(n + 1) = A(n)x(n), x(n) ∈ Rs, n ∈ N0 ≡ N ∪ {0}, (1). The solution of the system (1) with initial n∈N0 condition x(0) = x0 we will denote by (x(n, x0))n∈N0 and by (x(n, k, xk))n∈N0 we will denote the solution satisfying equality x(k, k, xk) = xk. Denote a = max sup A(n) , sup A−1(n) n∈N0 n∈N0 and let ΦA (·, ·) denote the transition matrix of the system (1) which is defined as ΦA(n, k) = A(n − 1)...A(k), if n > k, ΦA(n, n) = Is, and ΦA(n, m) = Φ−A1(m, n) if m > n, where Is is the identity matrix of order s. Discrete linear time-varying systems, characteristic exponents, Bohl exponents, general exponents, Millionschikov’s method of rotations. Discrete linear time-varying systems, characteristic exponents, Bohl exponents, general exponents, Millionschikov’s method of rotations. ∗ Corresponding author: Michal Niezabitowski
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