Fast Generation of Stability Charts for Time-Delay Systems Using Continuation of Characteristic Roots
| Autor: | Thomas Uchida, Chandrika P. Vyasarayani, Surya Samukham |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
0209 industrial biotechnology
Implicit function Applied Mathematics Mechanical Engineering Characteristic equation 02 engineering and technology General Medicine Delay differential equation 01 natural sciences Stability (probability) 020901 industrial engineering & automation Numerical continuation Control and Systems Engineering Ordinary differential equation 0103 physical sciences Applied mathematics Galerkin method 010301 acoustics Eigenvalues and eigenvectors Mathematics |
| Zdroj: | Journal of Computational and Nonlinear Dynamics. 15 |
| ISSN: | 1555-1423 1555-1415 |
| DOI: | 10.1115/1.4048362 |
| Popis: | Many dynamic processes involve time delays, thus their dynamics are governed by delay differential equations (DDEs). Studying the stability of dynamic systems is critical, but analyzing the stability of time-delay systems is challenging because DDEs are infinite-dimensional. We propose a new approach to quickly generate stability charts for DDEs using continuation of characteristic roots (CCR). In our CCR method, the roots of the characteristic equation of a DDE are written as implicit functions of the parameters of interest, and the continuation equations are derived in the form of ordinary differential equations (ODEs). Numerical continuation is then employed to determine the characteristic roots at all points in a parametric space; the stability of the original DDE can then be easily determined. A key advantage of the proposed method is that a system of linearly independent ODEs is solved rather than the typical strategy of solving a large eigenvalue problem at each grid point in the domain. Thus, the CCR method can significantly reduce the computational effort required to determine the stability of DDEs. As we demonstrate with several examples, the CCR method generates highly accurate stability charts, and does so up to 10 times faster than the Galerkin approximation method. |
| Databáze: | OpenAIRE |
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