OGY Control Research Articles

Fractional order prey–predator model incorporating immigration on prey: Complexity analysis and its control

In this paper, the Caputo fractional derivative is assumed to be the prey–predator model. In order to create Caputo fractional differential equations for the prey–predator model, a discretization process is first used. The fixed points of the model are categorized topologically. We identify requirements for the fixed points of the suggested prey–predator model’s local asymptotic stability. We demonstrate analytically that, under specific parametric conditions, a fractional order prey–predator model supports both a Neimark–Sacker (NS) bifurcation and a Flip bifurcation. We present evidence for NS and Flip bifurcations using central manifold and bifurcation theory. The parameter values and the initial conditions have been found to have a profound impact on the dynamical behavior of the fractional order prey–predator model. As the bifurcation parameter is increased, the system displays chaotic behavior. Numerical simulations are shown to demonstrate chaotic behaviors like bifurcations, phase portraits, invariant closed cycles, and attractive chaotic sets in addition to validating analytical conclusions. The suggested prey–predator dynamical system’s chaotic behavior will be controlled by the OGY and hybrid control methodology, which will also visualize the chaotic state for various biological parameters.

Controlling chaos and Neimark-Sacker bifurcation in a discrete-time predator-prey system

This article is about a discrete-time predator-prey model obtained by the forward Euler method. The stability of the fixed point of the model and the existence conditions of the Neimark-Sacker bifurcation are investigated. In addition, the direction of the Neimark-Sacker bifurcation is given. Moreover, OGY control method is to implement to control chaos caused by the Neimark-Sacker bifurcation. Finally, Neimark-Sacker bifurcation, chaos control strategy, and asymptotic stability of the only positive fixed point are verified with the help of numerical simulations. The existence of chaotic behavior in the model is confirmed by computing of the maximum Lyapunov exponents.

Fuzzy Modelling and Chaos Control in the Photogravitational Magnetic Binary Problem With Potential From a Belt

Control of chaotic systems has led to many fruitful results, such as the famous OGY and feedback control. Of course, controlling chaos is not limited to the approaches above and is not specifically reviewed one by one here. The representation of chaotic systems using fuzzy models has a unified approach. The fuzzy logic controllers had been proposed for a long time and were successfully applied to control the chaos in many systems. The main purpose of this study is to presents the new fuzzy model for the photogravitational magnetic-binary problem (PMBP) where the bigger primary is a source of radiation and the smaller primary is an oblate body; and they are encompassed by a homogeneous circular cluster of material points centred at the mass centre of the system (belt). It was shown that using a new fuzzy controller, it is possible to control of chaotic behaviour of the photogravitational magnetic-binary problem (PMBP). The simulation results have demonstrated that the proposed method can satisfy the control object and enhanced stability.

Controlling Chaos of the Ricker Population Model by OGY Method

The chaotic dynamics of the Ricker mapping are studied. Controlling chaos of the Ricker population model is searched by OGY control method. The dynamic behavior in the Ricker mapping is very complex in different values of a. When the value of a is changed from 0.455 to 90, the mapping goes through doubling bifurcation to Neimark-Sacker bifurcation. Several strange attractors coexist at last. Through the numerical simulation and analysis of the bifurcation and phase diagrams of the mapping, it is consistent with the theoretical analysis. Studies have shown that ecological balance can be achieved by appropriately adjusting the birth rate a, survival rate b.

Combination of Improved OGY and Guiding Orbit Method for Chaos Control

Chaotic systems have gathered much attention. When the OGY method is applied to control a chaotic system, chaos can be contained and target signals can be traced with satisfactory accuracy. However, the traditional control method have a low convergence speed, which may hamper the performance of the whole system. To solve this problem, the cuckoo search algorithm is used to guide the orbits of chaotic systems. Moreover, the OGY method is improved so that a chaotic system can be stabilized for different target points. Finally, the effectiveness of the proposed method is verified through several typical chaotic systems. The simulation results indicate that the modified method has a faster convergence speed and yields better performance than the traditional OGY control method.

Controlling Chaos and Neimark–Sacker Bifurcation in a Host–Parasitoid Model

AbstractIn this paper, a new density‐dependent host–parasitoid model is proposed. The modification is based on density‐dependent factor by introducing Hassell growth function in host population. Moreover, the permanence of solutions, existence and uniqueness of positive equilibrium, local asymptotic stability and global behavior of the positive equilibrium point are also investigated. It is demonstrated that system endures Neimark–Sacker bifurcation for wide range of bifurcation parameter. In order to control chaos due to emergence of Neimark–Sacker bifurcation, two feedback control strategies, that is, OGY and hybrid control methods are implemented. Finally, all mathematical analysis, particularly, Neimark–Sacker bifurcation, chaos control strategies, and global asymptotic stability of unique positive point are verified with the help of numerical simulations.

Design and experiment of controlled bistable vortex induced vibration energy harvesting systems operating in chaotic regions

Abstract Vortex induced vibration based energy harvesting systems have gained interests in these recent years due to its potential as a low water current energy source. However, the effectiveness of the system is limited only at a certain water current due to the resonance principle that governs the concept. In order to extend the working range, a bistable spring to support the structure is introduced on the system. The improvement on the performance is essentially dependent on the bistable gap as one of the main parameters of the nonlinear spring. A sufficiently large bistable gap will result in a significant performance improvement. Unfortunately, a large bistable gap might also increase a chance of chaotic responses, which in turn will result in diminutive harvested power. To mitigate the problem, an appropriate control structure is required to stabilize the chaotic vibrations of a VIV energy converter with the bistable supporting structure. Based on the nature of the double-well potential energy in a bistable spring, the ideal control structure will attempt to drive the responses to inter-well periodic vibrations in order to maximize the harvested power. In this paper, the OGY control algorithm is designed and implemented to the system. The control strategy is selected since it requires only a small perturbation in a structural parameter to execute the control effort, thus, minimum power is needed to drive the control input. Facilitated by a wake oscillator model, the bistable VIV system is modelled as a 4-dimensional autonomous continuous-time dynamical system. To implement the controller strategy, the system is discretized at a period estimated from the subspace hyperplane intersecting to the chaotic trajectory, whereas the fixed points that correspond to the desired periodic orbits are estimated by the recurrence method. Simultaneously, the Jacobian and sensitivity matrices are estimated by the least square regression method. Based on the defined fixed point and the linearized model, the control gain matrix is calculated using the pole placement technique. The results show that the OGY controller is capable of stabilizing the chaotic responses by driving them to the desired inter-well period-one periodic vibrations and it is also shown that the harvested power is successfully improved. For validation purpose, a real-time experiment was carried out on a computer-based forced-feedback testing platform to validate the applicability of the controller in real-time applications. The experimental results confirm the feasibility of the controller to stabilize the responses.

Enhanced Energy Harvesting from Nonlinear Oscillators via Chaos Control

Abstract Modern electronic devices require less energy on-board and could be powered by energy harvested from the environment. Broadband vibration energy harvesting is one of the widely explored options for powering such devices. One such harvester that exploits this technique is the nonlinear inverted elastic pendulum. This system has a double-well potential and can undergo large amplitude inter-well oscillations. Piezoelectric patches, pasted near the base of the beam, convert these mechanical oscillations into electrical energy. However, the large amplitude responses can deteriorate over time into low energy chaotic oscillations, thereby reducing the harvested power. Under such conditions, forcing the system to follow a high energy orbit would improve the energy harvested from the system. However, it is not wise to invest a large control force to stabilize the high energy orbits. This manuscript aims to exploit the chaotic nature of the system to stabilize the unstable periodic orbits. The chaotic behavior of the system allows even small perturbations to alter its response dramatically. Taking advantage of this property, a low power controller, based on the method of Ott, Grebogi, and Yorke (OGY), is used to stabilize the unstable high energy periodic orbits. The control strategy is implemented on the linearized system, about the operating point corresponding to the chosen orbit. LQR is used to determine the optimum control force. The devised strategy is numerically simulated, which favors the implementation of OGY control to improve their energy harvesting capabilities.

Displayed phenomena in the semi-passive torso-driven biped model under OGY-based control method: Birth of a torus bifurcation

We consider the displayed phenomena in the semi-passive dynamic walking of a torso-driven biped robot under the OGY control method as it descends inclined surfaces. The desired torso angle is the accessible control parameter. Analysis of the controlled steady semi-passive gaits are carried out mainly by means of bifurcations diagrams and Poincaré sections as the slope parameter varies. Based on Floquet multipliers, we show that a torus bifurcation was born leading hence to the generation of a quasi-periodic bipedal walking. Analysis of the controlled semi-passive dynamic walking is investigated also through a constrained controlled Poincaré map. We demonstrate that a such controlled map and the controlled impulsive hybrid nonlinear model of the biped display almost the same phenomena. We show also that the controlled constrained Poincaré map exhibits a discontinuous period-doubling bifurcation.

Computation of the Lyapunov exponents in the compass-gait model under OGY control via a hybrid Poincaré map

This paper aims at providing a numerical calculation method of the spectrum of Lyapunov exponents in a four-dimensional impulsive hybrid nonlinear dynamics of a passive compass-gait model under the OGY control approach by means of a controlled hybrid Poincaré map. We present a four-dimensional simplified analytical expression of such hybrid map obtained by linearizing the uncontrolled impulsive hybrid nonlinear dynamics around a desired one-periodic passive hybrid limit cycle. In order to compute the spectrum of Lyapunov exponents, a dimension reduction of the controlled hybrid Poincaré map is realized. The numerical calculation of the spectrum of Lyapunov exponents using the reduced-dimension controlled hybrid Poincaré map is given in detail. In order to show the effectiveness of the developed method, the spectrum of Lyapunov exponents is calculated as the slope (bifurcation) parameter varies and hence used to predict the walking dynamics behavior of the compass-gait model under the OGY control.

Generalized chaos control and synchronization by nonlinear high-order approach

This paper investigates the generalized control and synchronization of chaotic dynamical systems. First, we show that it is possible to stabilize the unstable periodic orbits (UPOs) when we use a high-order derivation of the OGY control that is known as one of useful methods for controlling chaotic systems. Then we examine synchronization of identical chaotic systems coupled in a master/slave manner. A rigorous criterion based on the transverse stability is presented which, if satisfied, guarantees that synchronization is asymptotically stable. The Rössler attractor and Chen system are used as examples to demonstrate the effectiveness of the developed approach and the improvement over some existing results.

Controlling Discrete Time T-S Fuzzy Chaotic Systems via Adaptive Adjustment

Abstract In order to overcome typical drawbacks of the OGY control, i.e. the long waiting time for control to be applied and the accessible turning system parameter in advance, this paper presents a new chaos control method based on Takagi- Sugeno (T-S) fuzzy model and adaptive adjustment. This method represents a chaotic system by linear models in different state space regions based on T-S fuzzy model and then stabilize the linear models in different state space regions by the adaptive adjustment mechanism. An example for the Henon map is given to demonstrate the effectiveness of the proposed method.

Chaotic period-doubling and OGY control for the forced Duffing equation

In this paper we consider the Duffing equation forced with a pulse function, whose moments of discontinuity depend on the initial data. Existence of the chaos through period-doubling cascade is proved, and the OGY control method is used to stabilize the periodic solutions. Appropriate simulations of the chaos and stabilized periodic solutions are presented.

Extracting Chaos Control Parameters from Time Series Analysis

We present a simple method to analyze time series, and estimate the parameters needed to control chaos in dynamical systems. Application of the method to a system described by the logistic map is also shown. Analyzing only two 100-point time series, we achieved results within 2% of the analytical ones. With these estimates, we show that OGY control method successfully stabilized a period-1 unstable periodic orbit embedded in the chaotic attractor.

Sliding mode control and synchronization of chaotic systems with parametric uncertainties

In this paper, we propose a result on controlling and synchronizing chaos. Our approach consists in using the sliding mode control theory. We first compare the OGY control method to the sliding mode control method. We next present a new sliding mode controller, we show that for the class of chaotic systems that can be stabilized using a smooth feedback controller, a sliding manifold can be easily constructed using a Lyapunov function. Moreover, we prove that if the states are confined to the sliding surface, then the originally chaotic trajectory will slide towards the equilibrium. We also prove that the proposed controller is robust to mismatched parametric uncertainties. To diminish the unwanted chattering phenomenon resulting from high sliding gains, an adaptive sliding controller is finally designed to present a robust model independent controller that achieves stabilization of the equilibrium points as well as synchronization of two systems. All these results will be confirmed through numerical simulations on a modified Chua’s system.

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) .

Controlling chaos for nonautonomous systems by detecting unstable periodic orbits

Designing the Ott–Grebogi–Yorke controller for controlling a nonautonomous chaotic system with forcing input requires detecting the unstable periodic orbits (UPOs) embedded in the chaotic motion. A simple detection approach based on the shooting method is presented in this study. This improvement of the close returns (CR) method can find multiple unstable periodic orbits of the same period and unstable periodic orbits of long periods by a short time series. Thus the UPOs are quickly determined. For real applications, system models are unknown. To obtain mathematical model, this work applies a nonlinear identification method, based on the harmonic balance principle, to a chaotic unknown system with a noisy environment. A Duffing equation is presented as a numerical example in this study. To simulate experimental conditions, random data are added to a sampled time series. Furthermore, the OGY controller is designed based on the data evaluated by the shooting method. The simulation results demonstrate that the proposed method is both accurate and feasible.

Optimal multi-dimensional OGY controller

A technique which generalizes the standard OGY control scheme and may be applied to higher dimensional systems to stabilize both saddle nodes and fully unstable fixed points is presented. The proposed method is designed for nonlinear flows rather than maps and also provides a powerful technique to assess the controllability of nonlinear systems with OGY type controllers. An optimal OGY control scheme designed for larger dimensional systems having improved convergence properties compared to the standard OGY controller is also presented. Numerical examples for the well known Van der Pol–Duffing oscillator are presented to illustrate the proposed control schemes.

Stabilization of Unstable Limit Cycles in Systems with Limited Controllability: Expanding the Basin of Convergence of OGY-Type Controllers

The underlying geometric structure of the standard OGY control scheme is analyzed. Some of the main mechanisms that under certain conditions lead to failure of the control algorithm are revealed. The limited controllability available in a system is investigated and it is shown that it may lead to serious problems that will significantly enlarge the state space region of failure of the standard OGY controller. A minimal distance algorithm is analyzed and shown to be, for some problems, more advantageous than the standard OGY technique. Nevertheless, for a broad category of problems, the minimal distance scheme is also shown to fail. As a solution for these problems, two new techniques are proposed: the penalized minimal distance and the multi-step OGY-type scheme. The standard OGY and minimal distance algorithms are particular cases of the new techniques proposed. Finally, we give a necessary condition that estimates the region of controllability under the multi-step OGY-type control. We demonstrate a significantly improved basin of convergence for the new multi-step OGY-type algorithm.