Delayed Feedback Control by Commutative Gain Matrices | Zendy

Rinko Miyazaki | Zendy; Toshiki Naito | Zendy; Jong Son Shin | Zendy

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Delayed Feedback Control by Commutative Gain Matrices

Author(s) -

Rinko Miyazaki,

Toshiki Naito,

Jong Son Shin

Publication year - 2011

Publication title -

siam journal on mathematical analysis

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.882

H-Index - 92

eISSN - 1095-7154

pISSN - 0036-1410

DOI - 10.1137/090779450

Subject(s) - mathematics , differential equation , nonlinear system , mathematical analysis , control theory (sociology) , periodic orbits , characteristic equation , commutative property , feedback control , matrix (chemical analysis) , riccati equation , matrix difference equation , control (management) , pure mathematics , physics , materials science , management , quantum mechanics , control engineering , engineering , economics , composite material

textversion:publisherThe delayed feedback control (DFC) is a control method for stabilizing unstable periodic orbits in nonlinear autonomous differential equations. We give an important relationship between the characteristic multipliers of the linear variational equation around an unstable periodic solution of the equation and those of its delayed feedback equation. The key of our proof is a result about the spectrum of a matrix which is a difference of commutative matrices. The relationship, moreover, allows us to design control gains of the DFC such that the unstable periodic solution is stabilized. In other words, the validity of the DFC is proved mathematically. As an application for the Rössler equation, we determine the best range of $k$ such that the unstable periodic orbit is stabilized by taking a feedback gain $K=kE$

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