On the stability of delayed feedback controllers

Abstract

We consider the stability of delayed feedback control (DFC) scheme for one-dimensional discrete time systems. We first construct a map whose fixed points correspond to the periodic orbits of the uncontrolled system. Then the stability of the DFC is analyzed as the stability of the corresponding equilibrium point of the constructed map. For each periodic orbit, we construct a characteristic polynomial whose Schur stability corresponds to the stability of DFC. By using Schur-Cohn criterion, we can find bounds on the gain of DFC to ensure stability.

Stability Results for Some Periodic Feedback Controllers

“…Note that this condition is not satis£ed in many chaotic orbits, and classical DFC cannot be used in their stabilization. Also note that even if this necessary condition is satis£ed, stabilization by classical DFC is not guaranteed, see (Morgül, 2003). On the other hand, the schemes presented in this paper always yield stabilization provided that λ = 1 is not an eigenvalue of J i , (i.e.…”

“…(Ushio, 1996) , (Morgül, 2003). A set of necessary and suf£cient conditions to guarantee exponential stabilization can be found in (Morgül, 2003).…”

“…It is known that classical DFC scheme has some inherent limitations, and it can be shown that it cannot stabilize Σ m if the number of real eigenvalues of J i (which is the multiplication of the Jacobians of f evaluated at the periodic points) greater than 1 is odd, see e.g. (Ushio, 1999), (Morgül, 2003). Note that this condition is not satis£ed in many chaotic orbits, and classical DFC cannot be used in their stabilization.…”

“…Despite its simplicity, a detailed stability analysis of DFC is very dif£cult, (Pyragas, 2001), (Ushio, 1996). For some recent stability results related to DFC, see (Ushio, 1996), (Nakajima, 1997), (Pyragas, 2001), (Morgül, 2003). For more details as well as various applications of DFC, see (Pyragas, 2001), (Fradkov and Evans, 2002) and the references therein.…”

Stability of delayed feedback controllers for discrete time systems

“…However, for the case n = 1, these polynomials are identical, since in this case both the Jacobians and the gain are scalars, which necessarily commute. We also note that for the case n = 1, the polynomial p m (λ) given by (28)-(31) is the same as given in [13].…”