Jay Prakash Singh

Second order adaptive time varying sliding mode control for synchronization of hidden chaotic orbits in a new uncertain 4-D conservative chaotic system

The objectives of the paper are (i) to develop a new 4-D conservative chaotic system with hidden chaotic orbits, (ii) to design a second order adaptive time varying sliding mode control for the synchronization between two identical proposed chaotic systems in the presence of matched disturbances and (iii) to compare the performances of the proposed controller with two available controllers which have been published recently. The chaotic nature of the proposed system is validated using theoretical and numerical tools like divergence property, Lyapunov exponents, Lyapunov spectrum, bifurcation diagram, phase portrait, Poincaré map and a frequency spectrum. The new conservative chaotic system exhibits the coexistence of hidden chaotic orbits with no equilibrium point. The new system is synchronized with itself using the proposed second-order adaptive time varying sliding mode control technique in the presence of matched disturbances and by considering different initial conditions. During synchronization, the parameters of both the systems, gains of the first order and second order sliding surfaces and the gains of the switching laws are considered as unknown and estimated adaptively. Only two control inputs are used to synchronize all the four states of the system. The effectiveness of the proposed controller is compared with two available controllers for the synchronization of chaotic systems and it is found that the proposed controller performs much better than the two available controllers.

NAC-based Synchronisation and Anti-synchronisation Between Hyperchaotic and Chaotic Systems, Its Analogue Circuit Design and Application

ABSTRACT The objectives of this paper are synchronisation and anti-synchronisation between a hyperchaotic and a chaotic systems using nonlinear active control (NAC) scheme, its analogue circuit design and application to secure communication. Hyperchaotic Lorenz–Stenflo system is used as a master and the chaotic (a hyperchaotic system in its chaotic state) Chen system is used as a slave in this paper. Chaotic slave system is controlled to make it synchronised and anti-synchronised with the hyperchaotic master system by designing a modified NAC law via suitable forcing. Proposed scheme is applied in communication to ensure more security of the message signal through hyperchaotic masking. The MATLAB environment is used to simulate synchronisation, anti-synchronisation of the proposed hyperchaotic-chaotic pair using the modified NAC law and its application to secure communication application. The simulation results reflect the successful achievement of the objectives. Further, analogue circuit designs of the hyperchaotic Lorenz–Stenflo, the chaotic Chen systems as well as their synchronisation and anti-synchronisation of proposed pair using modified NAC are presented in this paper and realised in NI Multisim. NI Multisim circuit realisation results validate the simulation results and the fulfilment of the desired objectives of the paper.

Hybrid Synchronization of Lu and Bhalekar-Gejji Chaotic Systems Using Nonlinear Active Control

Abstract Hybrid synchronization based on the active control theory between two different chaotic systems, i.e. Lu and Bhalekar-Gejji is reported in this paper. Lu chaotic system is taken as master and Bhalekar-Gejji chaotic system as slave. Stabilization of error dynamics is achieved by satisfying the Lyapunov stability conditions. Control is designed by using the relevant variables of master and slave systems. Simulation is presented for verification of proposed scheme.

Synchronization between hyperchaotic systems using PI Based Sliding Mode Control

In this paper, design of SMC scheme based controller is proposed for synchronization of two different hyperchaotic systems. Synchronization is achieved using the Lyapunov stability theory. A proportional integral (PI) switching surface is used to ensure the stability of the closed-loop error dynamics in sliding motion. This SMC scheme is effective and guarantees the occurrence of sliding motion and achieves synchronization of master hyperchaotic Xu and slave hyperchaotic Lorenz system. Combinations of two different hyperchaotic systems are used for the synchronization reflecting the novelty of paper. Finally, numerical simulations are performed to demonstrate the effectiveness of the proposed control strategy.

Nonlinear Active Control Based Hybrid Synchronization between Hyperchaotic and Chaotic Systems

Abstract In this paper hybrid synchronization between hyperchaotic and chaotic systems is presented. The hyperchaotic Lorenz-Stenflo system is used as master and the chaotic Chen system is used as slave. The global hybrid synchronization between master and slave systems is achieved by using nonlinear active control. Nonlinear controller is designed by using the relevant variables of master and slave systems. The stability of the proposed hybrid synchronization scheme is shown by using Lyapunov stability theory. MATLAB simulation is carried out to find the effectiveness of the proposed scheme. Results obtained validate the proposed theory.

Hybrid synchronization of Lu and Bao hyperchaotic systems using sliding mode control

This paper investigates hybrid synchronization between two non-identical hyperchaotic systems using sliding mode control (SMC) strategy. Hyperchaotic Lu system is considered as master and hyperchaotic Bao system as slave system. The stability results are established using PI (Proportional Integral) switching surface and Lyapunov stability theory for hybrid synchronization scheme. SMC is used to reduce the influence of disturbance and improve system robustness which helps to achieve the convergence of error dynamics easily. Finally, numerical simulations are performed to demonstrate the effectiveness of proposed hybrid synchronization strategy of two non-identical hyperchaotic systems.

Hidden Chaotic Path Planning and Control of a Two-Link Flexible Robot Manipulator

Robotics is an emerging and interesting area in many fields of technical science. In general, a robot manipulator (rigid/flexible) is a more focused research direction in comparison with other areas of robotics. Specifically, flexible manipulators are more applicable in many fields when compared with its rigid counterparts because of many advantages like lightweight, more workspace, lower energy consumption, smaller in size, mobility, etc. These advantages give rise to many control challenges like underactuation, nonminimum phase, noncollocation, control spillover, uncertainties, nonlinearities, complex dynamical behaviours, etc. Path planning or trajectory tracking problem is considered as an interesting and challenging control problem for a flexible manipulator in comparison with the regulation problem. In recent decades, the theory of chaos is used in various technical fields. Aperiodic long time, highly sensitive to initial conditions, unpredictable behaviours, etc. are the fundamental properties of a chaotic signal arising out of a deterministic nonlinear system. Many continuous/discrete/fractional order autonomous and non-autonomous chaotic dynamical systems are available in the literature. In the recent past, more attention has been given to the design and applications of hidden chaotic dynamical systems. The path planning problem of a flexible manipulator requires a reference signal. Various reference signals are used in the literature. Recently, a chaotic signal is used as a reference signal for path planning. However, we have not found any paper wherein a reference signal using a hidden chaotic system is used for path planning. The use of a signal from a hidden chaotic attractor for path planning of a flexible manipulator can provide a new domain of research. Hidden chaotic path planning/trajectory tracking of a two-link flexible manipulator is the aim of this chapter. Use of hidden chaotic attractors as a path/trajectory reference creates extra challenges and complexity in controlling the flexible manipulator. Thus, controlling a flexible manipulator in such a scenario is a challenging task. The dynamics of a two-link flexible manipulator is first modelled using assumed modes method and divided into two parts using two-time scale separation principle (singular perturbation). One subsystem is called as the slow subsystem involving with the rigid parts and another subsystem is called as the fast subsystem which incorporates the flexible dynamics. Separate control techniques are applied to each subsystem. An adaptive sliding mode control technique is designed for the slow subsystem which tackles the uncertainties and helps in fast tracking of the desired hidden chaotic trajectory. A backstepping controller is designed for the fast subsystem system for quick suppression of tip deflections and vibration suppressions. The proposed control techniques are validated using a reference chaotic signal generated from a 3-D hidden attractors chaotic system in MATLAB simulation environment and results are demonstrated. The results reveal that the objective of the chapter is achieved successfully by the proposed control techniques.

5-D Hyperchaotic and Chaotic Systems with Non-hyperbolic Equilibria and Many Equilibria

In the present decade, chaotic systems are used and appeared in many fields like in information security, communication systems, economics, bioengineering, mathematics, etc. Thus, developing of chaotic dynamical systems is most interesting and desirable in comparison with dynamical systems with regular behaviour. The chaotic systems are categorised into two groups. These are (i) system with self-excited attractors and (ii) systems with hidden attractors. A self-excited attractor is generated depending on the location of its unstable equilibrium point and in such case, the basin of attraction touches the equilibria. But, in the case of hidden attractors, the basin of attraction does not touch the equilibria and also finding of such attractors is a difficult task. The systems with (i) no equilibrium point and (ii) stable equilibrium points belong to the category of hidden attractors. Recently chaotic systems with infinitely many equilibria/a line of equilibria are also considered under the cattegory of hidden attractors. Higher dimensional chaotic systems have more complexity and disorders compared with lower dimensional chaotic systems. Recently, more attention is given to the development of higher dimensional chaotic systems with hidden attractors. But, the development of higher dimensional chaotic systems having both hidden attractors and self-excited attractors is more demanding. This chapter reports three hyperchaotic and two chaotic, 5-D new systems having the nature of both the self-excited and hidden attractors. The systems have non-hyperbolic equilibria, hence, belong to the category of self-excited attractors. Also, the systems have many equilibria, and hence, may be considered under the category of a chaotic system with hidden attractors. A systematic procedure is used to develop the new systems from the well-known 3-D Lorenz chaotic system. All the five systems exhibit multistability with the change of initial conditions. Various theoretical and numerical tools like phase portrait, Lyapunov spectrum, bifurcation diagram, Poincare map, and frequency spectrum are used to confirm the chaotic nature of the new systems. The MATLAB simulation results of the new systems are validated by designing their circuits and realising the same.

Control of a many equilibria hyperchaotic system

This paper presents a control technique for a many equilibria hyperchaotic system. Controlling a many equilibria hyperchaotic system is a challenging task. An adaptive proportional integral SMC is proposed for controlling of a hyperchaotic system. In the proposed control technique, the sliding surfaces are designed with unknown constant gains. Although there are four states but only two control inputs are used for regulating the states of the system to zero. The parameters of the system are estimated. It is shown that the control of a many equilibria hyperchaotic system is achieved in the presence of unknown parameters of the system and sliding gains of the sliding surfaces.

An LMI Based Integral SMC for Tracking Control of a New 4-D Conservative Chaotic System

The objectives of the paper are (i) to develop a new 4-D conservative chaotic system and (ii) to design an LMI based integral SMC for tracking control of the proposed system. The proposed system has hidden chaotic orbits with no equilibrium point and transient chaotic behaviour. The proposed controller is designed for controlling the transient chaotic behaviour of the system. The gain of the sliding surface is determined using LMI toolbox in MATLAB. The performance of the proposed controller is evaluated in terms of low chattering and small tracking error.