Stabilization Of Unstable Periodic Orbits Research Articles

Experimental time-delayed feedback control with variable and distributed delays

We report on several improvements of the classical time-delayed feedback control method for stabilization of unstable periodic orbits or steady states. In an electronic circuit experiment, we were able to realize time-varying and distributed delays in the control force leading to successful control for large parameter sets, including large time delays. The presented techniques make advanced use of the natural torsion of the orbits, which is also necessary for the original control method to work.

Further stability results for a generalization of delayed feedback control

In this paper, we consider the stabilization of unstable periodic orbits for one-dimensional and discrete time chaotic systems. Various control schemes for this problem are available and we consider a recent generalization of delayed control scheme. We prove that if a certain condition, which depends only on the period number, is satisfied then the stabilization is always possible. We will also present some simulation results.

Adaptive modification of the delayed feedback control algorithm with a continuously varying time delay

We propose a simple adaptive delayed feedback control algorithm for stabilization of unstable periodic orbits with unknown periods. The state dependent time delay is varied continuously towards the period of controlled orbit according to a gradient-descent method realized through three simple ordinary differential equations. We demonstrate the efficiency of the algorithm with the Rössler and Mackey–Glass chaotic systems. The stability of the controlled orbits is proven by computation of the Lyapunov exponents of linearized equations.

Chaos control applied to heart rhythm dynamics

The dynamics of cardiovascular rhythms have been widely studied due to the key aspects of the heart in the physiology of living beings. Cardiac rhythms can be either periodic or chaotic, being respectively related to normal and pathological physiological functioning. In this regard, chaos control methods may be useful to promote the stabilization of unstable periodic orbits using small perturbations. In this article, the extended time-delayed feedback control method is applied to a natural cardiac pacemaker described by a mathematical model. The model consists of a modified Van der Pol equation that reproduces the behavior of this pacemaker. Results show the ability of the chaos control strategy to control the system response performing either the stabilization of unstable periodic orbits or the suppression of chaotic response, avoiding behaviors associated with critical cardiac pathologies.

New feedback laws for stabilization of unstable periodic orbits

In this note a gain tuning scheme for prediction-based chaos control of discrete-time systems is proposed, extending previous work by T. Ushio and S. Yamamoto. The derived control laws are proved to be stabilizing. Three different time-invariant or time-varying laws are proposed, leading to different convergence rates and sizes for the basins of attraction. The results are illustrated by numerical simulations. A parallel between finding unstable periodic orbits and chaos control is done.

Time-delayed feedback control of chaos in a GaAs/AlGaAs heterostructure

A theoretical model has been developed to study nonlinear behaviors in a GaAs/AlGaAs heterostructure. We show that the system can exhibit chaotic oscillations under transverse magnetic fields and an electric field. The time-delayed feedback method was applied to stabilize the unstable periodic orbits (UPOs) embedded in the chaotic attractor. A bifurcation specified by a successive decrease in the number of UPOs with delayed time and feedback strength was revealed, indicating the stabilization of UPOs under feedback control. Noticeably, the introduction of a feedback perturbation will sometimes induce chaotic states that do not exist in the unperturbed system.

Observer-Based Delayed Feedback Control Stabilizing Unstable Periodic Orbits of Continuous-Time Chaotic Systems

The most important topic in chaos control is the stabilization of unstable periodic orbits (UPOs). The original delayed feedback control (DFC) method has an inherent weak point, called the odd number limitation. In this paper, we introduce a design method involving the use of nonlinear time-delayed observers, which can overcome the odd number limitation, in continuous-time chaotic systems. The proposed method is a natural extension of the discrete-time observer-based DFC method , which is insusceptible to the odd number limitation.

Controlling chaos in a nonlinear pendulum using an extended time-delayed feedback control method

Chaos control is employed for the stabilization of unstable periodic orbits (UPOs) embedded in chaotic attractors. The extended time-delayed feedback control uses a continuous feedback loop incorporating information from previous states of the system in order to stabilize unstable orbits. This article deals with the chaos control of a nonlinear pendulum employing the extended time-delayed feedback control method. The control law leads to delay-differential equations (DDEs) that contain derivatives that depend on the solution of previous time instants. A fourth-order Runge–Kutta method with linear interpolation on the delayed variables is employed for numerical simulations of the DDEs and its initial function is estimated by a Taylor series expansion. During the learning stage, the UPOs are identified by the close-return method and control parameters are chosen for each desired UPO by defining situations where the largest Lyapunov exponent becomes negative. Analyses of a nonlinear pendulum are carried out by considering signals that are generated by numerical integration of the mathematical model using experimentally identified parameters. Results show the capability of the control procedure to stabilize UPOs of the dynamical system, highlighting some difficulties to achieve the stabilization of the desired orbit.

Chaos control using an adaptive fuzzy sliding mode controller with application to a nonlinear pendulum

Chaos control may be understood as the use of tiny perturbations for the stabilization of unstable periodic orbits embedded in a chaotic attractor. The idea that chaotic behavior may be controlled by small perturbations of physical parameters allows this kind of behavior to be desirable in different applications. In this work, chaos control is performed employing a variable structure controller. The approach is based on the sliding mode control strategy and enhanced by an adaptive fuzzy algorithm to cope with modeling inaccuracies. The convergence properties of the closed-loop system are analytically proven using Lyapunov’s direct method and Barbalat’s lemma. As an application of the control procedure, a nonlinear pendulum dynamics is investigated. Numerical results are presented in order to demonstrate the control system performance. A comparison between the stabilization of general orbits and unstable periodic orbits embedded in chaotic attractor is carried out showing that the chaos control can confer flexibility to the system by changing the response with low power consumption.

Stabilizability and Stability Robustness of State Derivative Feedback Controllers

We study the stabilizability of a linear controllable system using state derivative feedback control. As a special feature the stabilized system may be fragile, in the sense that arbitrarily small modeling and implementation errors may destroy the asymptotic stability. First, we discuss the pole placement problem and illustrate the fragility of stability with examples of a different nature. We also define a notion of stability, called p-stability, which explicitly takes into account the effect of small modeling and implementation errors. Next, we investigate the effect on the fragility of including a low-pass filter in the control loop. Finally, we completely characterize the stabilizability and p-stabilizability of linear controllable systems using state derivative feedback. In the stabilizability characterization the odd number limitation, well known in the context of the stabilization of unstable periodic orbits using Pyragas-type time-delayed feedback, plays a crucial role.

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A design problem of DFC (delayed feedback control) satisfying H∞ performance specification is considered for sampled-data systems. It is shown that stabilization of UPOs (unstable periodic orbits) can be cast into an H∞ control synthesis with an unstable weight. The nonstandard synthesis problem is then reduced to a standard one under mild condition, which also concludes that the DFC structure is necessary for the stabilization of UPOs. A synthesis problem is formulated by putting extra sampled-data H∞ specifications to the problem related to the stabilization of UPOs, and the properties including the order of the resultant controller are discussed. The validity of the proposed method is confirmed by numerical examples.

Automatic control and tracking of periodic orbits in chaotic systems

Based on an automatic feedback adjustment of an additional parameter of a dynamical system, we propose a strategy for controlling periodic orbits of desired periods in chaotic dynamics and tracking them toward the set of unstable periodic orbits embedded within the original chaotic attractor. The method does not require information on the system to be controlled, nor on any reference states for the targets, and it overcomes some of the difficulties encountered by other techniques. Assessments of the method's effectiveness and robustness are given by means of the application of the technique to the stabilization of unstable periodic orbits in both discrete- and continuous-time systems.

Stabilization of unstable periodic orbit in chaotic Duffing-Holmes oscillator by second order resonant negative feedback

Stabilization of unstable periodic orbit in chaotic Duffing-Holmes oscillator by second order resonant negative feedback

Chaos and its control in an impulsive differential system

Abstract In this paper, the existence of chaos and its control in an autonomous impulsive differential system are discussed both theoretically and numerically. The existence of a snap-back repeller, as well as the chaos in the sense of Li–Yorke, is proved based on the qualitative analysis using the Poincare map and the Lambert W -function. Moreover, the existence of the period-3 periodic window embedded in the chaotic region is also demonstrated. An algorithm of chaos control to stabilize the unstable periodic solutions is proposed. Detailed numerical results of chaotic attractors and stabilization of unstable periodic orbits by the impulsive effects, which are illustrated by an example, are in good agreement with the theoretical analysis.

Floquet-based chaos control for continuous-time systems with stability analysis

In this Letter, a framework for controlling continuous-time chaotic systems is proposed. The framework is based on the Floquet theory of linear periodic differential equations and provides a practical method to stabilize unstable periodic orbits (UPOs) and a stability analysis of the closed loop systems. An example of controlling the circular restricted three-body problem known as halo orbits is illustrated. It is also reported that stabilization of UPOs can be effective by using the maximum principle to select a nominal orbit. It also turns out that the proposed framework enables us to give a theoretical account of the well-known occasional proportional feedback (OPF).

ITERATIONS OF PERTURBED TENT MAPS WITH APPLICATIONS TO CHAOS CONTROL

Iterations of 1D simple maps such as logistic, tent, cubic ones are very well studied. However perturbed versions of these maps (close in uniform norm but with strongly varying derivatives) can exhibit completely different behavior. We encounter such situation when dealing with chaos stabilization via small control. In this paper we present analytical investigation of this effect for one particular case –- piecewise linear perturbation of the tent map. Surprisingly, iterations of this map converge to the unique fixed point very fast for all initial points. The result is in sharp contrast with iterations of the original tent map but explains fast stabilization of unstable periodic orbits by predictive control, proposed in Polyak & Maslov (2005); Polyak (2005).

STATE SPACE RECONSTRUCTION USING EXTENDED STATE OBSERVERS TO CONTROL CHAOS IN A NONLINEAR PENDULUM

Chaos control may be understood as the use of tiny perturbations for the stabilization of unstable periodic orbits embedded in a chaotic attractor. Since chaos may occur in many natural processes, the idea that chaotic behavior may be controlled by small perturbations of some physical parameter allows this kind of behavior to be desirable in different applications. In general, it is not necessary to have a mathematical model to achieve the control goal since all control parameters may be resolved from time series analysis. Therefore, state space reconstruction is an important task related to chaos control. This contribution analyzes chaos control performed using a semi-continuous method based on OGY approach and proposes the use of extended state observers in order to perform state space reconstruction. The use of extended state observers allows a direct application of the control method. Comparing with the delay coordinates method, extended state observers avoids the calculation of parametric changes related to delayed Poincaré sections that influence the system dynamics. The proposed procedure is applied in the control of chaos in a nonlinear pendulum, showing that it may be used to control chaos in mechanical systems.

Integrity control of chaotic systems

In this Letter, the problem of the fault-tolerant control (FTC) of chaotic systems is investigated. A time-delayed integrity controller is proposed to ensure the stabilization of unstable periodic orbits (UPOs) in the case of sensor failures. If some parameters are chosen suitably, this controller can effectively stabilize UPOs for possible sensor failures. The famous Lorenz system is illustrated to show the effectiveness of this controller.

A GENERALIZATION OF THE OGY CONTROL TO CONTINUOUS-TIME SYSTEMS USING FLOQUET THEORY

In this paper, a framework to generalize the OGY control to continuous-time systems is proposed. The framework is based on Floquet theory of linear periodic differential equations and provides a practical method to stabilize unstable periodic orbits (UPOs) and a stability analysis of the closed loop systems. An example of controlling the circular restricted three-body system known as halo orbits is illustrated. It is also reported that stabilization of UPOs can be effective by using the maximum principle to select a nominal orbit.

Stability borders of feedback control of delayed measured systems

When stabilization of unstable periodic orbits or fixed points by the method given by Ott, Grebogi, and Yorke (OGY) must be based on a measurement delayed by tau orbit lengths, the performance of unmodified OGY method is expected to decline. For experimental considerations, it is desired to know the range of stability with minimal knowledge of the system. We find that unmodified OGY control fails beyond a maximal Lyapunov number of lambda(max) =1+ ( 1/tau ) . In this paper the area of stability is investigated both for OGY control of known fixed points and for difference control of unknown or inaccurately known fixed points. An estimated value of the control gain is given. Finally we outline what extensions must be considered if one wants to stabilize fixed points with Lyapunov numbers above lambda(max) .