Robust synchronization of Sprott circuits using sliding mode control
Introduction
Synchronization can be defined as a phenomenon where two or more appropriately coupled systems undergo resembling evolution in time. Pecora and Carroll [6] showed that synchronization can be achieved in dynamical systems on chaotic regime. They constructed an unidirectional master/slave configuration where the slave is a suitably altered copy of the master system and the synchronization is achieved in terms of the time evolution of the difference between the states of the master and those of the slave system. Then, if the error dynamics are asymptotically stable, synchronization is guaranteed. From the seminal paper by Pecora and Carroll [6], a large number of alternative approaches to achieve synchronization have been studied (see for example [5], [8]). However, chaos synchronization is far from being straightforward because different aspects affect significantly the ability of the systems to synchronize. A particularly important aspect to consider is the presence of disturbances and uncertainty, which are unavoidable in any practical implementation. Therefore, robustness is a very desirable characteristic to demand to any synchronization technique.
Chaos synchronization has important applications; for example, many techniques of controlled synchronization have been applied to synchronize chaotic circuits to develop private communication systems. In this application, the objective is to encode or encrypt information through a chaotic signal that will be sent to a receiver, where a chaotic system is synchronized to re-create the information (see for example [2] and references therein).
Simplicity is always a desirable characteristic to consider in a practical implementation. As shown by Sprott [7], a practical chaos generator can be constructed with a very simple circuit where the nonlinear term is a piecewise linear function. At the same time, the piecewise linear nature of the system simplifies the analysis because in this case the nonlinear system can be reformulated as a variable structure system consisting of linear parts with a given switching logic. There are some proposals to synchronize this kind of chaotic systems, see for example [4], [1], where two Sprott circuits are synchronized using a feedback linearization and the Open-Plus-Closed-Loop (OPCL) control techniques, respectively. In these works, the synchronization is achieved provided no external disturbances or parametric perturbations are present. These conditions are not realistic in practice due to the tolerance in electronic components and devices, therefore, a robustness analysis to ensure stability and convergence to zero of the error dynamics must be performed.
A control technique that is robust to parametric variations and external disturbances is the sliding mode control; this technique is used in [9] to synchronize chaotic systems with uncertainties. Another method to cope with parametric uncertainties is adaptive control; for example, an adaptive PID controller has been proposed in [3] to synchronize mechanical systems. These techniques are not designed for variable structure systems; therefore the synchronization of chaotic variable structure systems with disturbances is still an open problem.
In this paper, we propose a robust control technique to synchronize two Sprott systems connected in a master/slave scheme, and obtain conditions to ensure zero steady-state synchronization error under the presence of parameter uncertainty and external disturbances. The systems may be different and can be affected by parameter variations or matched bounded perturbations. We use the sliding mode control technique to design the coupling signal; therefore, the closed loop system is robust to external perturbations and parametric variations. In theory, we can attain asymptotic identical synchronization in spite of the existence of this kind of disturbances. However, in practice, as a result of a discontinuous coupling signal, there will be a small chattering component in the synchronization errors. Nevertheless, in many applications this error may be negligible.
The paper will be outlined as follows. The problem statement is given in Section 2, where the type of systems considered is defined, as well as the synchronization criterion. In Section 3, the design of the coupling signal using the sliding mode technique is described. In Section 4, we present numerical and experimental results to illustrate the proposed synchronization technique. Finally, the conclusions are given in Section 5.
Section snippets
The synchronization problem
The Sprott circuits are defined by [7]where the dots above the variable x means time derivatives (first, second and third), Gi(x) can take one of the following forms:and sign (·) is the signum function. A state representation of system (1) can be obtained by defining x1 = x, and .
Now define a master system, denoted with the subscript m, given in the state form
Design of the coupling signal
We design the coupling signal v based on the sliding mode control technique. Consider a discontinuity surface in defined bywhere , i = 1, 2, 3 are constants. We define a coupling signal v given bywhere F(·) is, in general, a piecewise smooth function.
In the following subsections, we present a way to design the surface S (4) and the coupling signal v(·) (5) so that the objective (2) will be reached.
Synchronization of two Sprott circuits
In this section, we illustrate the performance of the proposed synchronization technique with numerical and experimental results. Consider the master and slave systems given bywhere v is a coupling signal and δm = δs = 0.1. Define the error variables e1 = x1,m − x1,s, e2 = x2,m − x2,s, and e3 = x3,m − x2,s, with dynamics given by
Conclusions
In this paper, we have proposed an algorithm to synchronize two piecewise linear chaotic systems called Sprott systems. The conditions to apply this algorithm is that the perturbations satisfy the matching conditions.
In theory, this algorithm guarantees a zero steady-state synchronization error; however, due to high frequency components in the coupling signal, in practice the synchronization errors display a very small chattering, giving an approximate synchronization. The magnitude of the
Acknowledgements
This research was supported in part by CONACyT and PROMEP, México.
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