Simple robust technique using time delay estimation for the control and synchronization of Lorenz systems

This work presents two simple and robust techniques based on time delay estimation for the respective control and synchronization of chaos systems. First, one of these techniques is applied to the control of a chaotic Lorenz system with both matched and mis-matched uncertainties. The nonlinearities in the Lorenz system is cancelled by time delay estimation and desired error dynamics is inserted. Second, the other technique is applied to the synchronization of the Lü system and the Lorenz system with uncertainties. The synchronization input consists of three elements that have transparent and clear meanings. Since time delay estimation enables a very effective and efficient cancellation of disturbances and nonlinearities, the techniques turn out to be simple and robust. Numerical simulation results show fast, accurate and robust performance of the proposed techniques , thereby demonstrating their effectiveness for the control and synchronization of Lorenz systems. A chaotic system is sensitive to initial conditions, highly nonlinear, irregular, and complex. Chaotic behaviors have been studied extensively since the first classical chaotic attractor was introduced by Lorenz [1]. Since the pioneering research of controlling chaos [2], some of the research moved from the pure analysis of chaos to the control and synchronization of chaos. In most engineering systems, chaotic behavior is undesirable, and the goal of chaos control is to suppress or remove chaotic behavior, and to provide the system with stable and predictable behaviors. On the other hand, in the applications of secure communications, biological systems, chemical reactions, and information processing, prescribed chaotic behaviors are wanted, and the goal of chaos synchronization is to make the chaotic states of the system to track the desired chaotic trajectory. As an example of chaotic systems to be controlled, the Lorenz system is popular because the Lorenz system is simple among many chaotic systems, yet captures many features of chaotic dynamics [3,4]. Various methods have been introduced to control or synchronize the Lorenz system. For example, bang–bang control [3], sliding mode control [4], feedback linearization [5], adaptive control [6–8], backstepping control [9,10], neural networks [11,12], and others in [13]. Recently, fusions of aforementioned control methods have been carried out to achieve more sophisticated control performance. For example, fuzzy logic and adaptive control is merged in [14,15]; adaptive and backstep-ping technique are merged in [16]; advantages of the adaptive control, neural network and sliding mode control are combined in [17]; fuzzy adaptive sliding mode is used in [18]; …