Knot Polynomial As a Tool for Fault Detection

Resumen: The present work proposes a novel methodology for failure analysis using knot theory. In this particular case a LC circuit of order four is studied, where the current and voltage signals are analyzed by means of the above mentioned method, and where a topological invariant is obtained under normal conditions. This procedure is also performed when a fault occurs in the dielectric of the capacitors. As a result, it is shown how the changes in the dielectric are manifested and associated to a certain topological invariant. This provides justification for this case based on topological invariants. An advantage of the proposed approach is that no equations describing the system are required to be implemented.

¿Cómo citar?

Daniel Enrique Rivas-Cisneros, David. A. Diaz-Romero & Efrain Alcorta Garcia. Knot Polynomial As a Tool for Fault Detection. Memorias del Congreso Nacional de Control Automático, pp. 273-278, 2021.

Palabras clave

Fault detection, Knot theory, Topological invariants, Knot polynomial, Dynamical Systems

• Adams, C.C. (1994). The knot book an elementary introduction to the mathematical theory of knots. W. H. Freeman and Company, New York.
• Alcorta Garcia, E. and Frank, P.M. (1999). A novel design of structured observer-based residuals for fdi. In Proceedings of the American Control Conference, 1341– 1345. San Diego, Cal. USA.
• Chen, J. and Patton, R.J. (1999). Robust model based fault diagnosis for dynamic systems. Kluwer Academic Publishers Group.
• Chiang, L., Russell, E., and Braatz, R. (2001). Fault Detection and Diagnosis in Industrial Systems. Springer, Great Bretain.
• Cromwell, P.R. (2004). Knots and Links. Cambridge University Press. doi:10.1017/CBO9780511809767.
• Ding, S.X. (2013). Model-based fault diagnosis techniques 2nd edition. Springer, 2nd edition.
• Ding, S.X. (2014). Data-driven Design of Fault Diagnosis and Fault-tolerant Control Systems. Advances in Industrial Control. Springer, London.
• Frank, P.M. (1990). Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy – a survey. Automatica, 26, 459–474.
• Freyd, P., Yetter, D., Hoste, J., Lickorish, W.B.R., Millett, K., and Ocneanu, A. (1985). A new polynomial invariant of knots and links. Bulletin of the American Mathematical Society, (12), 239–246. doi: 10.1090/s0273-0979-1985-15361-3.
• Isermann, R. (2006). Fault-diagnosis systems: An introduction from fault detection to fault tolerance. Springer, 1 edition.
• Kauffman, L.H. (1991). Knots and physics, volume 1 of Series on Knots and Everything. World Scientific Publishing Co. Inc., River Edge, NJ. doi:10.1142/9789812796226. URL http://dx.doi.org/10.1142/9789812796226.
• Parris, R. (1977). A three-dimensional system with knotted trajectories. The American Mathematical Monthly, 84(6), 468–469.
• David A. Diaz-Romero,Efrain Alcorta Garcia and Daniel Enrique Rivas-Cisneros. Aislamiento de fallas en sistemas dinámicos: un enfoque topológico, (Memorias del Congreso Nacional de Control Automtico,2019).
• R.Ghrist, P.J. Holmes and M.C. Sullivan, Knots and Links in the Three-Dimensional Flows, (Springer, 2011 ).
• R. Ghrist, Chaos, Solitons & Fractals 9(1998) 583.