Delay-dependent stability criteria for time-delay chaotic systems via time-delay feedback control
Introduction
Time-delay systems are frequently encountered in engineering, biology, economy, and other disciplines. In the wake of intensive research on the robust stability and control theory, the stability and control of time-delay systems received renewed interests.
In 1963, Lorenz found the first a chaotic attractor in a simple three-dimensional autonomous system. So far there are many researchers who studied the chaos theory. During the last decades dynamic chaos theory has been deeply studied and applied to many fields extensively, such as secure communications, optical system, biology and so forth.
Chaos control is new field in explorations of chaotic motions and it is crucial in applications of chaos. Until now, many different techniques and methods have been proposed to achieve chaos control, such as OGY method, impulsive control method, differential geometric method and linear state space feedback, etc. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19].
Since Mackey and Glass [20], [21], [22], [23], [24], [25], [26], [27] first found chaos in time-delay system, there has been increasing interest in time-delay chaotic systems. One of the most frequent objectives consists in the stabilization of chaotic behaviors to one of unstable fixed points or unstable periodic orbits embedded within a chaotic attractor. That is, to design a suitable controller that guarantees the closed-loop system dynamics converges to the fixed point or periodic orbit. The famous OGY-method [1] describes that tracking problem admits a local feedback solution which achieves the desired tracking with small bounded control under suitable assumptions. The location of the desired periodic orbit, the linearized dynamics about the periodic orbit, and the dependence of the location of the periodic orbit on small variations of the control parameters are very important ideas, to present a feasible linearized error system under some suitable assumptions based on which Lyapunov stability analysis approach is used to solve the stabilization problem.
On the other hand, Pyragas [2] proposed a delayed feedback control (DFC) method that does not require a reference signal corresponding to the desired unstable periodic orbit. As is well known and easily verified, a DFC system may be viewed as a rather special case of the familiar auto-regressive moving-average model or a canonical state-space control system [28]. The authors have discussed the stabilization problem by using the DFC method [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39]. But seldom researchers consider if the DFC method can be used to control time-delay chaotic system. In paper [40], the authors have extended the DFC approach to time-delay chaotic system, and the sufficient conditions for stabilization and tracking problems via DFC have obtained from the results based on SFC, but these results are independent time-delay and conservative.
In this paper, we consider delay-dependent stability of time-delay chaotic systems via time-delay feedback control (DFC). Suppose that there exists an unstable fixed point. The delay-dependent stability criteria via DFC are derived from the results based on standard feedback control (SFC), the delay-dependent controller design method can be obtained to stabilize the system to an unstable fixed point.
This paper is organized as follows. Section 2 describes the time-delay chaotic system to be controlled and the two controller frameworks. Stability analysis and controller design to force system states to the fixed point are discussed in Section 3, an example is also discussed in this section to illustrate the advantage of the obtained result. Concluding remarks are provided in Section 4.
Section snippets
Controller structures of time-delay chaotic system
Consider the following chaotic system with an additional feedback force:where x(·)∈Rn is the state vector, u(·)∈Rn is feedback control input vector, A,B∈Rn×n are constant system matrices representing the linear parts of the system, f1,f2∈Rn are the nonlinear parts of the system, and τ>0 is the constant time-delay.
Suppose that the chaotic system (1) has an unstable fixed point or an unstable periodic orbit , and is currently in a chaotic state.
Stabilization problem
In order to understand to what extent the two different types of feedback control techniques are useful in time-delay chaotic systems, some analytic conditions are derived in this section based on Lyapunov stabilization arguments.
Firstly, we discuss the SFC technique. For the error system (6), since zero is a fixed point of F1(e(t),t)+F2(e(t−τ),t), we have a Taylor expansionwhere β0=F1′(e(t),t), β1=F2′(e(t−τ),t), [H.O.T.]1 is higher
Conclusion
This paper has dealt with stability analysis and controller design problems of time-delay chaotic system based on SFC and DFC methods. The established results for stabilization are dependent of the size of time-delay of the plant or controller. An example is discussed to illustrate the advantage of the obtained result.
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