Time-delay systems with a controlled time-varying delay Stability analysis and applications
Résumé
We study the stability of a linear system with a point-wise, time-varying delay. We assume that the delay varies around a nominal value in a deterministic way and investigate the influence of this variation on stability. More precisely we are interested in characterizing situations where the time-varying delay system is stable, whereas the system with constant delay is unstable. Our approach consists of relating the stability properties of a system with a fast varying point-wise delay with these of a time-invariant system with a distributed delay. Then we can use frequency domain methods to analyze the problem and to derive stability criteria. The results are first illustrated with two theoretical example. Then we study a model of a variable speed rotating cutting tool. Based on the developed theory, we thereby provide both a theoretical explanation and a quantitative analysis tool for the beneficial effect of a variation of the machine speed on enhancing stability properties, which was reported in the literature.
Mots clés
- Mechanical engineering
- Cutting tools
- Eigenvalues and eigenfunctions
- Frequency domain analysis
- Frequency modulation
- Linear systems
- Lyapunov methods
- Mathematical models
- Theorem proving
- Natural frequencies
- Quenching
- Set theory
- System stability
- Time varying systems
- Time-delay systems
- Time invariant systems
- Rotating cutting tool
- Quantitative analysis tools