Trigonometric spline and spectral bounds for the solution of linear time-periodic systems. (English) Zbl 1385.65045
Authors’ abstract: Linear time-periodic systems arise whenever a nonlinear system is linearized about a periodic trajectory. Examples include anisotropic rotor-bearing systems and parametrically excited systems. The structure of the solution to linear time-periodic systems is known due to Floquet’s theorem. We use this information to derive a new norm which yields two-sided bounds on the solution and in this norm vibrations of the solution are suppressed. The obtained results are a generalization for linear time-invariant systems. Since Floquet’s theorem is non-constructive, the applicability of the aforementioned results suffers in general from an unknown Floquet normal form. Hence, we discuss trigonometric splines and spectral methods that are both equipped with rigorous bounds on the solution. The methodology differs systematically for the two methods. While in the first method the solution is approximated by trigonometric splines and the upper bound depends on the approximation quality, in the second method
the linear time-periodic system is approximated and its solution is represented as an infinite series. Depending on the smoothness of the time-periodic system, we formulate two upper bounds which incorporate the approximation error of the linear time-periodic system and the truncation error of the series representation. Rigorous bounds on the solution are necessary whenever reliable results are needed, and hence they can support the analysis and, e.g., stability or robustness of the solution may be proven or falsified. The theoretical results are illustrated and compared to trigonometric spline bounds and spectral bounds by means of three examples that include an anisotropic rotor-bearing system and a parametrically excited Cantilever beam.
Reviewer: Maria Gousidou-Koutita (Thessaloniki)
MSC:
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65D07 | Numerical computation using splines |
| 93C05 | Linear systems in control theory |
| 65K10 | Numerical optimization and variational techniques |
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 34A30 | Linear ordinary differential equations and systems |
| 65L70 | Error bounds for numerical methods for ordinary differential equations |
Keywords:
time-periodic system; initial value problem; weighted norm; trigonometric approximation; Chebyshev projection; Floquet’s theorem; spectral methods; error estimates; rotor-bearing systems; trigonometric splines; truncation error; stability; Cantilever beamReferences:
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