Continuous control of chaos by self-controlling feedback

Pyragas, K.

doi:10.1016/0375-9601(92)90745-8

Cited by 3,105 publications

(1,472 citation statements)

References 15 publications

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“…Now we can state our main results as follows. Let an m period UCPO of (1) (2) If p m (λ) has at least one unstable root, i.e., magnitude strictly greater than unity, then Σ m cannot be stabilized by (1) and (4). Hence the proposed method to test stability is not conclusive only if some roots of p m (λ) are on the unit disc, i.e., have unit magnitude, while the rest of the roots are strictly inside the unit disc.…”

Section: Remarkmentioning

confidence: 99%

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On the stability of delayed feedback controllers

Morgül

2003

Physics Letters A

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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.

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Section: Remarkmentioning

confidence: 99%

“…Among these, the delayed feedback control (DFC) scheme first proposed in [4] and is also known as Pyragas scheme, has gained considerable attention due to its various attractive features. In this technique the required control input is basically the difference between the current and one period delayed states, multiplied by a gain.…”

Section: Introductionmentioning

confidence: 99%

On the stability of delayed feedback controllers

Morgül

2003

Physics Letters A

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“…Various methods of controlling unstable and chaotic systems have been developed in the past 20 years and applied to real systems in physics, chemistry, biology, and medicine [9]. Pyragas [24] was the first who introduced delay feedback control (DFC) to stabilize UPO embedded in a chaotic attractor. This method, known as time delay autosynchronization (TDAS), bases on constructing the control force from the difference of the current state to the state one period in the past, so that the control signal vanishes when the stabilization of the target orbit is attained.…”

Section: Introductionmentioning

confidence: 99%

“…DFC method proposed by [24] works very well in case of the UPO if time delay is precisely equal to the period of the UPO and the feedback gain is strong enough. Note that only the stability properties of the orbit are changed, while the orbit itself and its period remain unaltered.…”

Section: Introductionmentioning

confidence: 99%

Time delay Duffing’s systems: chaos and chatter control

2014

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The effect of a delay feedback control (DFC), realized by displacement in the Duffing oscillator, for parameters which generate strange chaotic Ueda attractor is investigated in this paper. First, the classical Duffing system without time delay is analysed to find stable and especially unstable periodic orbits which can be stabilized by means of displacement delay feedback. The periodic orbits are found with help of the continuation method using the AUTO97 software. Next, the DFC is introduced with a time delay and a feedback gain parameters. The proper time delay and feedback gain are found in order to destroy the chaotic attractor and to stabilize the periodic orbit. Finally, chatter generated by time delay component is suppressed with help of an external excitation.

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“…Therefore direct approaches have been developed by explicitely taking into account either a Poincaré surface of section [23] or the explicit periodic orbit length [26]. This field of controlling chaos, or stabilization of chaotic systems, by small perturbations, in system variables [15] or control parameters [23], has developed to a widely discussed topic with applications in a broad area from technical to biological systems.…”

Section: Introduction 110mentioning

confidence: 99%

Poincaré‐Based Control of Delayed Measured Systems: Limitations and Improved Control

Claussen

2007

Handbook of Chaos Control

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6Poincaré-based control of delayed measured systems: Limitations and Improved Control 1 Jens Christian Claussen IntroductionWhat is the effect of measurement delay on Ott, Grebogi, and Yorke (OGY) chaos control? Which possibilities exists for improved control? These questions are addressed within this chapter, and the OGY control case is considered as well as a related control scheme, difference control; both together form the two main Poincaré-based chaos control schemes, where the control amplitude is computed once during the orbit after crossing the Poincaré section.If the stabilization of unstable periodic orbits or fixed points by the method given by Ott, Grebogi, Yorke [23] and Hübler [15] can only be based on a measurement delayed by τ orbit lengths, resulting in a control loop latency, the performance of unmodified OGY control is expected to decay. For experimental considerations, it is desired to know the range of stability with minimal knowledge of the system. In section 6.3, the area of stability is investigated both for OGY control and for difference control, yielding a delay-dependent maximal Lyapunov number beyond which control fails. Sections 6.3.4 to 6.4.3 address the question how the control of delayed measured chaotic systems can be improved, i.e., what extensions must be considered if one wants to stabilize fixed points with a higher Lyapunov number. Fortunately, the limitation can be overcome most elegantly by linear control methods that employ memory terms, as linear predictive logging control (Sec. 6.4.1) and memory difference control (Sec. 6.4.3). In both cases, one is equipped with an explicit deadbeat control scheme that allows, within linear approximation, to perform control without principal limitations in delay time, dimension, and Ljapunov numbers. The delay problem -time-discrete caseFor fixed point stabilization in time-continuous control, the issue of delay has been investigated widely in control theory, dating back at least to the Smith predictor [31].

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Continuous control of chaos by self-controlling feedback

Cited by 3,105 publications

References 15 publications

On the stability of delayed feedback controllers

On the stability of delayed feedback controllers

Time delay Duffing’s systems: chaos and chatter control

Poincaré‐Based Control of Delayed Measured Systems: Limitations and Improved Control

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