Bounds on the Solution of Linear Time‐Periodic Systems

Abstract

Linear time-periodic systems have been an active area of research in the last decades. They arise in various applications such as anisotropic rotor-bearing systems and nonlinear systems linearized about a periodic trajectory. Rigorous bounds support the transient analysis of these systems. Optimal constants are determined by the differential calculus for norms of matrix functions. Bounds based on trigonometric spline approximations of the solution are introduced and convergence results for the approximations are stated. Bounds are illustrated by means of an anisotropic rotor-bearing system. Linear time-periodic systemsWe are considering a set of linear ordinary differential equations (ODEs) with time-periodic coefficients with periodicity T and a given initial condition as a linear time-periodic systeṁwhere x ∈ R n and A : R → R n×n . Under well known assumptions a unique solution to (1) is guaranteed. The solution can then be represented by the following theorem due to Floquet [1], but in a different form.Theorem 1.1 A fundamental matrix Φ(t) of (1) can be represented aswhere R, Z(t) ∈ C n×n and Z(t) = Z(t + T ) is nonsingular ∀t ∈ R.The solution x(t) is a linear combination of n linearly independent solutions w.r.t. the initial condition x 0 : x(t) = Φ(t)x 0 = Z(t)e Rt x 0 with Φ(0) = I. Determining Φ(t) in the interval [0, T ] is sufficient due to the semigroup property of the solution. For the sake of simplicity we consider equidistant nodes t i = ih for i = −1, 0, 1, . . . , r with h = T r and we use quadratic trigonometric splines T i (t) [2] to approximate the solution. . .whereT denotes a vector of the i-th coefficients α (j)i of the spline approximation s j (t) for j = 1, . . . , n. Quadratic trigonometric splines with compact support are chosen in order to simplify the computation of the coefficientswhere θ = 1 sin h sin h 2 . Solving ODEs by spline approximations has been investigated by F. R. Loscalzo and T. D. Talbot [3] and many others, e.g. A. Nikolis [4]. The focus on trigonometric splines is due to the periodicity of (1) and furthermore, we will equip the computation with rigorous upper and lower bounds on the solution. Demanding that the approximation s fulfills (1) at the nodes t i , yields a series of r + 1 linear systems A

Spectral Bounds on the Solution of Linear Time‐Periodic Systems

“…We investigate our proposed rigorous bounds on two examples: a simple IVPẋ(t) = | sin(2πt)| 3 x(t), x(0) = 1 (Figure 1) and a Jeffcott rotor on an anisotropic shaft, supported by anisotropic bearings (Figure 2, see [6]) which can be modeled as a linear time-periodic system (1). For the Jeffcott rotor the same parameter values as in [7] are chosen. This is an asymptotically stable system since the maximal Lyapunov exponent is ν = −0.002.…”

“…This is an asymptotically stable system since the maximal Lyapunov exponent is ν = −0.002. We illustrate and compare our new rigorous upper bounds based on Chebyshev projection for m := m 1 = m 2 to trigonometric spline approximation bounds [7]. The trigonometric spline bound depends quadratically on the node distance [7] which in our case is equidistant h = T r , where r is the number of nodes.…”

Trigonometric spline and spectral bounds for the solution of linear time-periodic systems

“…They used trigonometric B-splines of second and third order to solve a nonlinear ODE. We use a modified approach in order to apply it to a linear system of ODEs and further equip the computation with rigorous bounds [4]. The unknown quantities are the coefficients of the trigonometric splines.…”