Anticontrol of Chaos in Continuous time Systems via Time delay Feedback
Spectrum optimization-based chaotification using time-delay feedback control
Highlights
► A time-delay feedback controller is designed for chaotification. ► A spectrum optimization method is proposed to determine chaotification parameters. ► Numerical examples verify the spectrum optimization- based chaotification method. ► Engineering application in line spectrum reconfiguration is demonstrated.
Introduction
Chaotification [1], [2] (called also chaotization and anti-control of chaos) has drawn the growing attention of many researchers over the last decade, and its engineering applications involve information encryption [3], broadband communication [4], liquid mixing [5]. Recently, an important application for improving the concealment capability of underwater vehicles involves the technique of chaotification that has been employed to blur and disfigure line spectrum emitted from machinery vibration as reported by Yu et al. [6] and Wen et al. [7]. In conclusion, there are two categories of methodology for chaotification, namely synchronization and feedback control.
Synchronization is a collaborative behavior between coupled systems. It includes complete synchronization (CS) [8], [9] between two identical systems, and generalized synchronization (GS) [10] between different systems. In engineering practice, CS almost cannot be carried out, since it is difficult to guarantee that the response system is exactly identical to the drive system. GS without requirement of identical dynamics in master–slave systems is hence usually used to drive a mechanical system chaotic. Based on unidirectional coupling mode of GS, Yu et al. [6] proposed a control scheme to generate or maintain chaos in the nonlinear vibration isolation system (VIS). A chaotic time series generated from a Duffing system was taken as the driving signal, and some parameters of the slave system were defined as a function of chaotic driving signals, and then chaotification in the nonlinear VIS were realized at a particular setting of parameters demonstrated in their numerical example. However, the persistence of chaotification is not guaranteed since this method [6] is sensitive to parameters settings. In a similar way, Wen et al. [7] employed a modified projective synchronization for chaotification where the Duffing system as the master system to drive a nonlinear VIS (response system) chaotic through a control. However, it requires a large control and is seemingly impractical for applications [11].
Another widely used method for chaotification is feedback control. For a stable linear time-invariant and discrete-time system, Wang and Chen [12] designed a nonlinear feedback controller with tiny amplitude, such as a modulo function of system states. Chaotification is realized by controlling the largest Lyapunov exponent positive meanwhile keeping system states uniformly bounded. Based on this concept, Konishi [13], [14] proposed a control method to chaotify a damped linear harmonic oscillator with or without excitation. The key step of this method was to discretize continuous-time systems into discrete-time systems. The discretization was carried out by constructing the map between x((n + 1)T) and x(nT) via the integration with respect to the time from nT to (n + 1)T, where T is half of the natural period of the damped oscillator. And then the controller was designed according to Wang and Chen's method [12]. On the contrary, time-delay feedback anti-control of chaos can handle continuous-time systems directly. Wang et al. [1] designed a simple time-delay feedback controller with small amplitude to drive a system chaotic, when the system had an exponentially stable equilibrium point and was controllable. Xu and Chung [15] also pointed out that the time-delay feedback control can be designed as a switch for the choice of system behaviors, namely chaotic or non-chaotic motions. In our previous work [11], the stability of a two degree of freedoms (DOFs) vibration isolation floating raft system with a time-delay feedback control was studied systematically, which gave a guideline of the design of the linear time-delay feedback controller for chaotification. However, to find a set of suitable parameters of the time-delay feedback controller for chaotification, it still relies on bifurcation diagram analysis. From the standpoint of computation efficiency, plotting of a fine-scale bifurcation diagram is very expensive.
In this paper, we attempt to achieve chaotification based on a spectrum optimization method, different from the methods of bifurcation analysis and calculation of the largest Lyapunov exponents. A time-delay feedback controller is introduced due to its useful feature of enhancing system complexity. In the sense of mathematics, a time-delay dynamic system possesses infinite dimensions, makes it much easier to generate chaos even in a first-order linear system [1]. A spectrum performance index will be especially designed in a way of characterizing spectrum spikes and frequency band of the steady state response of a 2-DOFs mechanical system. The genetic algorithm is used for determination of optimal parameters of the time-delay and feedback gain to minimize the index, to which the smaller index corresponds to the result of suppressing spectrum spikes and broadening frequency band, namely chaotification. This approach allows us to easily achieve the chaotification and make a harmonically excited mechanical system be chaotic. Several numerical simulations about the system driven by excitation with different frequencies will be carried out to verify both the feasibility of the spectrum optimization-based chaotification method and the effectiveness of the anti-control of chaos.
It is worthy to note that the existing chaotification methods are only effective on the basis of clearly understanding the system characteristics and under the known conditions. On the contrary, in this paper, the performance index directly characterizes the dynamic behavior based on the Fourier spectrum of steady state responses, and optimization determines favorable control parameters to minimize the performance index until chaos appears. This enables us to flexibly handle the cases where the system's operational conditions are unknown, or variable, which are typically useful for real applications in system chaotification.
This paper is organized as follow. Section 2 gives the controller design and illustrates the spectrum optimization method. Numerical simulations are carried out in Section 3 to verify the proposed methodology and illustrate the anti-control of chaos in the 2-DOFs mechanical system driven by excitation with different frequencies. Potential engineering applications will be briefly discussed in Section 4. Finally, Section 5 gives some conclusions of the present work.
Section snippets
Chaotification method
In this section, the time-delay feedback controller will be designed by using Wang and Chen's method [1], and then the spectrum optimization method for chaotification parameters of the controller will be demonstrated. A typical vibration isolation system can be considered as a 2-DOFs mass-spring system [6]. The time-delay feedback control scheme for chaotify the VIS is illustrated in Fig. 1. m 1 and m 2 denotes the isolated equipment and the floating raft, respectively. Both of them are supported
Numerical examples
The 2-DOFs linear system is driven by a harmonic excitation at three typical frequencies, namely low frequency (LF), intermediate frequency (IF), and high frequency (HF). Parameters of the 2-DOFs linear system and the harmonic excitation are listed in Table 1. Substituting those parameters into the expression of the matrix Θ gives and the rank of the matrix Θ is 4, and hence this system without
Discussion on potential engineering applications
Line spectra emitted from harmonic vibration of machinery systems degrade the concealment of underwater vehicles as this feature signals a strong identity [18]. Lou et al. [18], [21] introduced the nonlinear vibration isolation system (VIS) to covert the single frequency input to broad-band chaotic output. However, chaotic oscillations are sensitive to system parameters and excitation conditions, and hence the manufacturing error or variation of excitation conditions will lead to the
Conclusions
In this work, a spectrum optimization-based chaotification method in conjunction with time-delay feedback has been proposed to make a 2-DOFs linear mechanical system driven by harmonic excitation chaotic. The nonlinear time-delay feedback controller was employed. Then, a performance index was designed to characterize the Fourier spectrum. Minimization of the performance index generates optimal controlling parameters, i.e. time delay and the parameter σ, which lead to desirable chaotic state.
Acknowledgement
The authors would like to appreciate Professor Shijian Zhu for his offering of modeling parameters of the 2-DOFs linear system. This research work was supported by National Natural Science Foundation of China (11102062, 11072075), China Postdoctoral Science Foundation (20100480938), Specialized Research Fund for the Doctoral Program of Higher Education (20110161120040), Fundamental Research Funds for the Central Universities, and the research Grant (71075004) from the State Key Lab of Advanced
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Currently, not only is better protection from vibration required from the vibration isolation means, they also have to comply with additional constraints, such as vibration isolation in very low frequency, installment of highly vibration sensitive equipment in light and low-stiffness structures, exposure to high temperatures, or very space-restrictive packaging, which have become more and more difficult to accommodate. On the study of chaos-based vibration isolation via time delay control [2,3], we found the linear stiffness of the vibration isolation system plays a vital role in determining the minimum control energy, which draws our attention to the QZS isolators [4–6]. The mechanism of a QZS isolation system is to make the negative stiffness offset the positive stiffness at the equilibrium state, yielding a localized zero stiffness at the equilibrium state.
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The most terrific feature we found through this study is the availability of using tiny control for chaotification. This advantage not only reduces the control input by at least ten times in comparison with the previous methods [17–19] but also significantly improve the quality of chaotification in terms of reducing the intensity of line spectra. This method also extends the accessibility of chaotification to a wider parametric domain.
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