Stabilization Of Unstable Periodic Orbits Research Articles

STABILIZING THE PERIODIC ORBITS IN A CHAOTIC MAPPING DESCRIBING THE DISCRETE HEALTH SYSTEMS VIA PREDICTION-BASED CONTROL

In the paper the problem of location and stabilization of unstable periodic orbits (UPOs) in discrete systems is investigated via the prediction-based control (PBC). It involves using the state of the free system one period ahead as reference for the control signal. Two types of control gains are tested, the first requiring the knowledge of the UPO to be stabilized and the second depending only on the actual state of the trajectory. The effectiveness of PBC is demonstrated on a chaotic mapping describing the malignant tumor growth. When the results obtained with the two control laws are compared with each other, it is found that the second variant is qualitatively superior, both in terms of convergence and the number of stabilized UPOs, especially for long-period orbits.

Feedback control modulation for controlling chaotic maps

Based on existing feedback control methods such as OGY and Pyragas, alternative new schemes are proposed for stabilization of unstable periodic orbits of chaotic and hyperchaotic dynamical systems by suitable modulation of a control parameter. Their performances are improved with respect to: (i) robustness, (ii) rate of convergences, (iii) reduction of waiting time, (iv) reduction of noise sensitivity. These features are analytically investigated, the achievements are rigorously proved and supported by numerical simulations. The proposed methods result successful for stabilizing unstable periodic orbits in some classical discrete maps like 1-D logistic and standard 2-D Hénon, but also in the hyperchaotic generalized n-D Hénon-like maps.

Average Predictive Control for Nonlinear Discrete Dynamical Systems

We explore the problem of stabilization of unstable periodic orbits in discrete nonlinear dynamical systems. This work proposes the generalization of predictive control method for resolving the stabilization problem. Our method embodies the development of control method proposed by B.T. Polyak. The control we propose uses a linear (convex) combination of iterated functions. With the proposed method auxiliary, the problem of robust cycle stabilization for various cases of its multipliers localization is solved. An algorithm for finding a given length cycle when its multipliers are known is described as a particular case of our method application. Also, we present numerical simulation results for some well-known mappings and the possibility of further generalization of this method.

Stabilization of chaotic systems via act-and-wait delayed feedback control using a high-precision direct integration method

A technique for the numerical analysis of the stability problem in chaotic systems via act-and-wait delayed feedback control is developed and justified. Recently, act-and-wait modification of a delayed feedback control method is proposed to stabilize unstable periodic orbits in dynamical systems. To overcome the difficulties in obtaining conditions under which the state of closed-loop system converges toward a periodic solution via the act-and-wait scheme, this paper presents a high-precision direct integration method for calculating the monodromy matrix corresponding to the closed-loop system based on the theory of Peano–Baker series. The capabilities of the proposed schemes in stabilization of unstable periodic orbits of chaotic systems are illustrated by numerical examples.

Stabilization of chaotic systems via fuzzy time-delayed controller approac

In this paper, we investigate the stabilization of unstable periodic orbits of continuous time chaotic systems usingfuzzy time-delayed controllers. For this aim, we present a control method that can achieve stabilization of an unstableperiodic orbit (UPO) without any knowledge of the system model. Our proposal is attained progressively. First, wecombine the input-to-state linearizing controller with the fuzzy method to obtain one that achieves UPO stabilization.Then, we use reduced order sliding observer to estimate the necessary state for the controller construction. Finally, theefficiency of the proposed methods is demonstrated using numerical simulations applied to Chua’s system.

State-dependent act-and-wait time-delayed feedback control algorithm

Time-delayed feedback control (TDFC) method is one of the most popular experimental tools used for stabilization of unstable periodic orbits in nonlinear dynamical systems. The theory of the method is difficult since systems controlled by time-delayed feedback evolve in an infinite-dimensional function space. Here we modify the TDFC by appending the feedback loop with a state-dependent switch. The switch closes and opens the feedback loop in the dependence of the phase of the output signal, which is measured by a specially designed phase detector. This modification represents a further development of the act-and-wait concept applied to time-delayed feedback controllers. The concept allows to reduce the phase space dimension of time-delayed feedback systems and to significantly simplify the theory. In previous works, the time-delayed feedback loop was tuned on and off periodically, independently of the state of the system. We discuss the advantages of our algorithm with respect to previously considered modifications. The efficacy of the proposed algorithm is numerically demonstrated with autonomous Rössler and Lorenz systems as well as a nonautonomous Duffing oscillator.

Stabilizing periodic orbits of Chua’s system using adaptive fuzzy sliding mode controller

PurposeThe purpose of this paper is to investigate the stabilization of unstable periodic orbits of Chua’s system using adaptive fuzzy sliding mode controllers with moving surface.Design/methodology/approachFor this aim, the sliding mode controller and fuzzy systems are combined to achieve the stabilization. Then, the authors propose a moving sliding surface to improve robustness against uncertainties during the reaching phase, parameter variations and extraneous disturbances.FindingsAfterward, the authors design a sliding observer to estimate the unmeasurable states which are used in the previously designed controller.Originality/valueNumerical results are provided to show the effectiveness and robustness of the proposed method.

Delayed feedback control method for computing the cyclic steady states of evolution problems

This work is focused on fast techniques for computing the cyclic steady states of evolution problems in non-linear mechanics with space–time periodicity conditions. In industrial applications, in order to avoid the inversion of very large matrices, such a cyclic solution is usually computed as an asymptotic limit of the associated initial value problem with arbitrary initial data. However, when the relaxation time is high, convergence to the limit cycle can be very slow. In such cases nonetheless, one is not interested in the transient solution, but only in a fast access to the limit cycle. Thus, in this work we modify the problem, introducing the time-delayed feedback control, which is widely used for stabilization of unstable periodic orbits. In our framework it is applied to an initially stable system in order to accelerate its convergence to the limit cycle. Moreover, the control term, based on the space–time periodicity error, includes both shifts in time and in space. Our main result is the optimal form of the control term for a very general class of linear evolution problems, providing the fastest convergence to the cyclic solution, which has been further extended and studied in the non-linear case. Efficiency of the method increases with the problem’s relaxation time.The method has been tested using academic applications and compared to the non-controlled asymptotic convergence as well as to the Newton–Krylov shooting algorithm. Finally, the method has been implemented into the Michelin industrial code, applied to a full 3D rolling tyre model.

Stabilizing unstable periodic orbits with delayed feedback control in act-and-wait fashion

A delayed feedback control framework for stabilizing unstable periodic orbits of linear periodic time-varying systems is proposed. In this framework, act-and-wait approach is utilized for switching a delayed feedback controller on and off alternately at every integer multiples of the period of the system. By analyzing the monodromy matrix of the closed-loop system, we obtain conditions under which the closed-loop system’s state converges towards a periodic solution under our proposed control law. We discuss the application of our results in stabilization of unstable periodic orbits of nonlinear systems and present numerical examples to illustrate the efficacy of our approach.

Act-and-wait time-delayed feedback control of autonomous systems

Recently an act-and-wait modification of time-delayed feedback control has been proposed for the stabilization of unstable periodic orbits in nonautonomous dynamical systems (Pyragas and Pyragas, 2016 [30]). The modification implies a periodic switching of the feedback gain and makes the closed-loop system finite-dimensional. Here we extend this modification to autonomous systems. In order to keep constant the phase difference between the controlled orbit and the act-and-wait switching function an additional small-amplitude periodic perturbation is introduced. The algorithm can stabilize periodic orbits with an odd number of real unstable Floquet exponents using a simple single-input single-output constraint control.

Stabilizing unstable periodic orbits of dynamical systems using delayed feedback control with periodic gain

Two algorithms of unstable periodic orbits stabilization based on delayed feedback control and periodic gain are discussed. The efficiency of these algorithms for stabilization of unstable periodic orbits of the Lorenz and Rossler systems and visualization of stabilized periodic orbits are demonstrated.

Solar Radiation Pressure Hamiltonian Feedback Control for Unstable Libration-Point Orbits

This work investigates a Hamiltonian structure-preserving control that uses the acceleration of solar radiation pressure for the stabilization of unstable periodic orbits in the circular restricted three-body problem. This control aims to stabilize the libration-point orbits in the sense of Lyapunov by achieving simple stability. It also preserves the Hamiltonian nature of the controlled system. The Hamiltonian structure-preserving control is then extended to a general case in which complex and conjugate eigenvalues occur at high-amplitude orbits. High-amplitude orbits are currently of interest to the European Space Agency for future libration-point orbit missions because they require a lower insertion compared to low-amplitude orbits. Based on the design of the feedback control, the purpose of this work is to verify when the use of solar radiation pressure is feasible and to determine the structural requirements and the spacecraft’s pointing accuracy.

Nonlinear optimal stabilization of unstable periodic orbits

The framework for the optimal stabilization of unstable periodic orbits in nonlinear systems is proposed. First, the stable manifold method for approximately solving Hamilton-Jacobi equations in the optimal control problem is extended to nonlinear periodic systems. Then, this method is applied for optimal stabilization problem of error dynamics around an unstable periodic orbit. The effectiveness of the proposed method is evaluated via numerical simulations for the Rossler chaotic system.

Controlling a Shape Memory Alloy Two-Bar Truss Using Delayed Feedback Method

This work discusses the use of chaos control in smart structures. An archetypal model of a shape memory alloy (SMA) two-bar truss is treated. This system exhibits both constitutive and geometrical nonlinearities presenting a complex nonlinear dynamics response including either the snap-through or the chaotic behaviors. A constitutive model that presents a close agreement with experimental data is employed to describe the themomechanical SMA behavior. A variation of the continuous time-delayed feedback method is employed as a control strategy. This variable structure controller is applied to the stabilization of unstable periodic orbits of the SMA structure avoiding the snap-through behavior.

Continuous pole placement method for time-delayed feedback controlled systems

Continuous pole placement method is adapted to time-periodic states of systems with time delay. The method is applied for finding an optimal control matrix in the problem of stabilization of unstable periodic orbits of dynamical systems via time-delayed feedback control algorithm. The optimal control matrix ensures the fastest approach of a perturbed system to the stabilized orbit. An application of the pole placement method to systems with time delay meets a fundamental problem, since the number of the Floquet exponents is infinity, while the number of control parameters is finite. Nevertheless, we show that several leading Floquet exponents can be efficiently controlled. The method is numerically demonstrated for the Lorenz system, which until recently has been considered as a system inaccessible for the standard time-delayed feedback control due to the odd-number limitation. The proposed optimization method is also adapted for an extended time-delayed feedback control algorithm and numerically demonstrated for the Rossler system.

State space reconstruction applied to a multiparameter chaos control method

The idea of the chaos control is the stabilization of unstable periodic orbits (UPOs) embedded in chaotic attractors. The OGY method achieves system stabilization by using small perturbations promoted in the neighborhood of the desired orbit when the trajectory crosses a Poincare section. A generalization of this method considers multiple actuations of parameters and sections, known as semi-continuous multiparameter method. This paper investigates the state space reconstruction applied to this general method, allowing chaotic behavior control of systems with non-observable states using multiple control parameters from time series analysis, avoiding the use of governing equations. As an application of the proposed multiparameter general formulation it is presented an uncoupled approach where the control parameters do not influence the system dynamics when they are not active. This method is applied to control chaos in a nonlinear pendulum using delay coordinates to perform state space reconstruction. Results show that the proposed procedure can be applied together with delay coordinates providing UPO stabilization.

A pseudo-stable structure in a completely invertible bouncer system

It is shown that a pseudo-stable structure of non-asymptotic convergence may exist in a completely invertible bouncing ball model. Visualization of the pattern of H-ranks helps to identify this structure. It appears that this structure is similar to the stable manifold of non-invertible nonlinear maps which govern the non-asymptotic convergence to unstable periodic orbits. But this convergence to the unstable repeller of the bouncing ball problem is only temporary since non-asymptotic convergence cannot exist in completely invertible maps. This nonlinear effect is exploited for temporary stabilization of unstable periodic orbits in completely reversible nonlinear maps.

A Twenty-Year Review of Time-Delay Feedback Control and Recent Developments

The delayed feedback control (DFC) method has been invented in 1992 (this year is a 20th anniversary). Following the original paper by Pyragas [Phys. Lett. A, 170, 421, 1992], more than 1500 papers devoted or related to the DFC have been published. Many different modifications of the algorithm have been proposed, and significant achievements are attained in the theory of the DFC. Although this theory is non-trivial, currently the mechanism of the DFC action is rather well understood, and the main limitations of the algorithm are established. The DFC has been successfully implemented in a number of experimental systems of different physical nature. The aim of this talk is to present a brief review of important modifications of the DFC algorithm, significant theoretical results and experimental implementations attained during the past twenty years. The recent results concerning adaptive modifications of the DFC and analytical achievements based on phase reduction of time-delay systems will be discussed as well. 1. Brief review of experimental and theoretical results The DFC algorithm [1] is a simple, robust, and efficient method to stabilize unstable periodic orbits (UPOs) in chaotic systems. Nowadays, it becomes one of the most popular methods in the chaos control research [2]. The method allows a noninvasive stabilization of UPOs of dynamical systems in the sense that the control force vanishes when the target state is reached. The DFC algorithm is reference-free and makes use of a control signal K∆s(t) obtained from the difference ∆s(t) = s(t) − s(t − τ) between the current state s(t) of the system and the state of the system s(t−τ) delayed by one-period τ of the target orbit. The UPO may become stable under the appropriate choice of feedback strength K. Note that only the stability properties of the orbit are changed, while the orbit itself and its period remain unaltered. The controlled system can be treated as a black box, since the method does not require any exact knowledge of either the form of the periodic orbit or the system’s equations. The method is particularly appealing for experimentalists, since one does not need to know anything about the target orbit beyond its period τ. The DFC algorithm is notably superior to other control methods in fast dynamical systems, since it does not require any realtime computer processing. Successful implementation of the DFC algorithm has been attained in diverse experimental systems, including electronic chaotic oscillators, mechanical pendulums, lasers, gas discharge systems, a current-driven ion acoustic instability, a chaotic Taylor-Couette flow, chemical systems, high-power ferromagnetic resonance, helicopter rotor blades, and a cardiac system. (cf. [3] for review up to 2006). An important practical application of the DFC algorithm has been recently demonstrated by Yamasue et al. [4]. The authors have successfully implemented the DFC method in an atomic force microscope and managed to stabilize cantilever oscillations. As a result, they remove artifacts on a surface image. Another interesting application of the DFC for the analysis of bifurcations of periodic states in experimental systems has been recently considered in Ref. [5]. A reach variety of modifications of the DFC has been suggested in order to improve its performance (cf. [3]). Here we mention only the most important modification known as an extended DFC (EDFC), which has been introduced in Ref. [6]. The authors improved an original DFC scheme by using an information from many previous states of the system. The EDFC scheme achieves stabilization of UPOs with a greater degree of instability [7, 8]. The theory of DFC is difficult because the delayed feedback induces an infinite number of degrees of freedom. Even linear analysis of such systems is complicated due to the infinite number of Floquet exponents characterizing the stability of controlled orbits. Nevertheless, some analytical approaches have been developed in vicinity to various bifurcations of periodic orbits, such as the period doubling bifurcation [9, 10], the subcritical Hopf bifurcation [11, 12, 13] and the Nejmark-Sacker (discrete Hopf) bifurcation [14]. In 1997 Nakajima [15, 9] proved the so-called odd number limitation, which states that any UPOs with an odd number of real Floquet multipliers greater than unity can never be stabilized by any DFC technique. This limitation has been commonly accepted and intensively discussed in the literature. However, in 2007, Fiedler at al [16] have shown by a simple example that this limitation does not hold in general for autonomous systems (note that for non-autonomous systems it remains valid in general). Recently, a modified (corrected) proof of the limitation for autonomous systems has been presented by Hooton and 2012 International Symposium on Nonlinear Theory and its Applications NOLTA2012, Palma, Majorca, Spain, October 22-26, 2012

Inverted Pendulum Stabilization Via a Pyragas-Type Controller: Revisiting the Triple Zero Singularity

The use of Pyragas-Type controller proved interest in the stabilization of unstable periodic orbits. The stabilization problem of a balancing inverted pendulum on an horizontally moving cart by the use of such a controller is considered. The main objective of the paper is to propose delayed control law containing only proportional gains able to stabilize the inverted pendulum by avoiding the existence of a triple zero eigenvalue at the origin. We analyze the center dynamics described by a three dimensional system of ordinary differential equations (ODEs) with a codimension-three triple zero bifurcation. Furthermore, the stability analysis of the corresponding linear time invariant system with two delays describing the behavior around the equilibrium is also proposed. This analysis is done in order to characterize the possible local bifurcations. Finally, the proposed control scheme is numerically illustrated and discussed. Time-Delay, Stability, Delayed Feedback, Control, Center Manifold Theorem, Normal Forms, Local Bifurcation, Inverted Pendulum.

Adaptive fuzzy sliding mode control of a chaotic pendulum with noisy signals

AbstractChaotic response is related to a dense set of unstable periodic orbits (UPOs) and the system often visits the neighborhood of each one of them. Moreover, chaos has sensitive dependence to initial conditions, which implies that the system evolution may be altered by small perturbations. Chaos control is based on the richness of chaotic behavior and may be understood as the use of tiny perturbations for the stabilization of an UPO enclosed in a chaotic attractor. It makes this kind of behavior to be desirable in a variety of applications, since one of these UPOs can provide better performance than others in a particular situation. In this work, an adaptive fuzzy sliding mode controller is combined with the close return method for the stabilization of UPOs in a chaotic pendulum. The adaptive fuzzy inference system is embedded in a smooth sliding mode controller to cope with both structured and unstructured uncertainties. Since noise contamination is unavoidable in experimental data acquisition, this work also investigates the effect of noisy signals on the used control law, verifying their influence on the system stabilization and on the required control action. Numerical results are presented in order to illustrate the ability of the proposed control scheme to track UPOs even in the presence of modeling inaccuracies and noisy input signals.