CN1885216A - Chaos generation method based on direct delay feedback - Google Patents

Abstract

The invention relates to a method for generating infinite-dimension chaotic signal, which is based on selected non-chaotic non-linear system and the real condition to control the injection point, while the control method uses simple direct delay feedback, wherein the control parameter can be selected by the chaotic branch pattern obtained by MATLAB and Poincare section, and real demand; then by analyzing the time wave shape, the power spectrum and the Liapunouph index, the generation of chaotic signal can be fixed. Compared with present technique, the invention is more simple and flexible, with better effect.

Description

Chaos production method based on direct delay feedback

Technical field

The present invention relates to a kind of chaotic signal producing method, particularly a kind of chaos production method based on direct delay feedback.

Background technology

People have had the history in 40 years to the research of chaos, to the research emphasis of chaos problem also from initial discovery with explain that chaos phenomenon develops into control in recent years and utilizes chaos phenomenon ^[1]For the control problem of chaos, from nineteen ninety, famous OGY method ^[2]After being suggested, people have proposed a lot of control methods and have obtained widespread use ^[3-10]When the existence of chaos was harmful to system, these methods can be eliminated chaos.Opposite with the control problem of chaos, the generation of chaos is meant existence when chaos to system when useful, has to produce artificially or strengthen chaos.For example: having of chaos is beneficial to the performance that improves neural network ^[11]Utilize the code efficiency in chaos raising signal and the image transmission ^[12,13]Chaos advection (Chaotic Advection) can obtain better effect in liquid mixing process and the occasion that relates to heat interchange ^[14]Chaotic vibration can improve street roller and muller efficient ^[15,16]Deng.The potential application of these of chaos phenomenon makes chaotic signal generation problem become new research focus.

With respect to Study of chaos control, the research of chaos production method is started late.1998, Chen etc. proposed to produce at discrete system the feedback of chaos, made discrete system produce chaos phenomenon under the Li-Yorke meaning ^[17]1999, Wang etc. provided the theoretical proof of this method ^[18]2000, proposition such as Wang and Chen has the delay FEEDBACK CONTROL of small magnitude sine function form---postpone FEEDBACK CONTROL indirectly, and be applied to make these systems produce chaos phenomenon in linear minimum phase system, CHUA circuit, LORENZ system and the induction motor ^[19-23]Calendar year 2001, people such as Tang and Chen propose to adopt absolute value feedback generation chaos again, and this mechanism is applied to linear system, DUFFERING oscillator and brushless direct current motor ^[24-26]Recent years, people propose the chaos production method based on feedback linearization again ^[27], based on the chaos production method of fuzzy neural network ^[28], adopt the chaos production method of piecewise linearity control function ^[29-32]The engineering that these chaos revertive control methods are chaos is used and is benefited our pursuits.

Below be the list of references that the applicant provides:

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[20]Xiao?Fan?Wang?and?Guanrong?Chen，Chaotifying?a?Stable?LTI?Systemby?Tiny?Feedback?Control，IEEE?Trans.on?Circuits?and?Systems?1，Vol.47(3)，2000，410-415。

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[23] Zhu Hailei etc. utilizes to postpone feedback and carries out the chaos revertive control of asynchronous motor, Proceedings of the CSEE, Vol.24 (12), 2004, pp.156-159.

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[25]Guo-Qun?Zhong，Kim?F?Man，Guanrong?Chen，Generating?Chaos?Viaa?Dynamical?Controller，International?Journal?of?Bifurcation?and?Chaos，Vol.11(3)，2001，865-869。

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[27] close new equality, continuous time, Acta Physica Sinica, Vol.51 (10), 2002, pp.2216-2219 were studied in the chaos revertive control of stable linear system.

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[35]Tetsushi?Ueta，Guanrong?Chen，Bifurcation?analysis?of?chen’sequation，International?Journal?of?Bifurcation?and?Chaos，Vol.10(8)，2000，pp.1917-1931。

[36]Jinhu?Lü，Tianshou?Zhou，Guanrong?Chen，Suochun?Zhang，Localbifurcation?of?the?Chen?system，International?Journal?of?Bifurcation?and?Chaos，Vol.12(10)，2002，pp.2257-2270。

[37]Thomas?S?Parker，Leon?O?Chua，Practical?numerical?algorithms?forchaotic?systems，Springer-Verlag，New?York，1993。

[38] Ren Haipeng, Liu Ding, Li Jie, the delay FEEDBACK CONTROL [J] of chaotic motion in the permasyn morot, Proceedings of the CSEE, 2003,23 (6), pp.175-178.

[39] Li Jie, Ren Haipeng, the partly decoupled of chaos phenomenon control in the permasyn morot, control theory and application, 2005, Vol.22 (4), pp.637-640.

In the existing chaos production method, adopt the chaotic signal that postpones the FEEDBACK CONTROL generation indirectly to compare with the chaotic signal that other method produces, the signal chaos degree of generation is stronger, has infinite dimension in theory.Be applied to chaotic secret communication and will have better secret effect, be applied to muller and street roller and have better grinding and consolidation effect.But this delay feedback has the following disadvantages: at first, when adopting this method to produce chaos, need be with inhibit signal through sine function conversion, control forms complexity; Secondly, this method must be carried out feedback linearization when nonlinear system is controlled, find suitable output function to make system can satisfy the feedback linearization condition, so just make the control function form of computation process and reality more complicated, use to engineering and bring considerable restraint.In addition, from the angle of control, those skilled in the art wishes to adopt a kind of unified method, when the needs chaos, produces chaos, when not needing chaos, eliminates chaos, gives the deviser dirigibility with maximum ^[19,31], this indirect delay FEEDBACK CONTROL also can't realize chaos controlling at present.

Summary of the invention

The objective of the invention is to, a kind of chaos production method based on direct delay feedback is provided.

In order to realize above-mentioned task, the present invention takes following technical solution:

A kind of chaos production method based on direct delay feedback is characterized in that, this method adopts in Continuous Nonlinear Systems and directly postpones the feedback generation chaotic signal, specifically comprises the following steps:

The first step: select controlled device

Controlled device is selected the three rank Continuous Nonlinear Systems of above (comprising three rank), and its system moves by dx/dt=f (x) mode, and wherein x is a state vector, and f () is the Nonlinear Vector function, and system is in non-chaos state when not adding input;

Second step: the injection mode of selecting external control at the characteristics of practical object

Injection mode is dx/dt=f (x)+u, the position that external control applies is selected according to the concrete condition of real system, can not select the unacceptable control of real system, the method of control adopts and directly postpones feedback system u=K (x (t)-x (t-τ)), wherein, u is the external control amount, and K is that enlargement factor, τ are time delay;

The 3rd step: utilize the controlled device mathematical model to carry out Computer Simulation

Select suitable Poincare section, obtain under specific external control injection mode, directly postpone feedback enlargement factor and the chaos bifurcation graphs of time delay.Under the SIMULINK of MATLAB environment, set up mathematics model, utilize M running paper SIMULINK program, by analyzing and the selection Poincare section is gathered in examination, adopt trace-point method to obtain about the enlargement factor K of direct delay control and the chaos bifurcation graphs of delay time T.Parameter that the value of controlled parameter generating fork and the system that makes enter chaos state from bifurcation graphs and corresponding controlled system are in the roughly amplitude of chaos state;

The 4th step: application requirements and chaos bifurcation graphs according to controlled system are determined directly to postpone feedback enlargement factor and time delay, and chaotic motion amplitude is based on the actual application requirements determined the roughly value that controlled variable is selected in bifurcation graphs, for the chaos degree that requires, estimate by the Li Ya spectrum promise husband index of calculating parameter correspondence;

The 5th step: utilize the time delay that chooses and directly postpone the feedback enlargement factor in non-chaos system, to produce chaotic signal;

The 6th step: chaos characteristic amounts such as the power spectrum of check chaotic signal, Li Ya spectrum promise husband index, confirm system works and chaos state.

Method of the present invention adopts when producing infinite dimension chaos and directly postpones the feedback generation chaotic signal, and this method is compared with indirect delay feedback chaos signal generating method, is easy to realization more simply more.This method can produce chaos when needed, can eliminate chaos when not required, so that provide greater flexibility for the deviser.The controller parameter bifurcation diagram that numerical simulation obtains has provided the selection foundation of controller parameter.The chaotic motion that this method produces has infinite dimension in theory, is applied to secret communication and will has better secret effect.This also is that existing other chaos production methods are not available.

Description of drawings

Fig. 1 directly postpones the feedback structure block diagram;

Fig. 2 stablizes CHEN system phasor, and wherein (a) is original state (0.1,1,0.1), (b) is original state (0.1 ,-10 ,-0.1);

Fig. 3 is τ ₁During=1.2s, controlled variable k ₁₁The chaos bifurcation graphs, wherein figure (b), (c), (d) is the partial enlarged drawing of figure (a);

Fig. 4 is τ ₁=1.2s, k ₁₁=-23 o'clock, the three-dimensional phase path figure of system (5), wherein (a) is time series, (b) is power spectrum;

Fig. 5 is k ₁₁=-50, τ ₁During=1.2s, postpone the chaotic attractor that FEEDBACK CONTROL produces;

Fig. 6 is the chaos time sequence of corresponding diagram 5 chaotic attractors and the power spectrum of this sequence;

Fig. 7 works as k ₁₁=-10 o'clock, delay time T ₁The chaos bifurcation graphs;

Fig. 8 works as τ ₃=0.3 o'clock, parameter k ₃₃The chaos bifurcation graphs;

Fig. 9 works as k ₃₃=2, parameter τ ₃The chaos bifurcation graphs;

Figure 10 is k ₃₃=3.8, τ ₃=0.3 o'clock, the chaotic attractor of system;

Figure 11 is Figure 10 time corresponding sequence and its power spectrum;

Figure 12 does not have the permasyn morot of control to move to the phase-plane diagram of equilibrium point from original state (3,0.01,3);

Figure 13 directly postpones the chaotic signal phase-plane diagram that the FEEDBACK CONTROL permasyn morot produces;

Figure 14 is the time domain waveform and the power spectrum chart of the motor angular velocity of Figure 13 correspondence, and wherein (a) is time domain waveform and (b) is power spectrum chart.

The present invention is described in further detail below in conjunction with specific embodiment that accompanying drawing and inventor provide.

Embodiment

Method of the present invention adopts and directly postpones the feedback generation chaotic signal, at first, select suitable nonlinear system, and system is as follows:

dx/dt＝f(x) (1)

X ∈ R wherein ⁿBe system state, f (x)) be the Nonlinear Vector function of x, system is in non-chaos state.

Secondly,, select the injection mode of external control signal, can apply control to one or more in the vector equation that can apply control according to the actual conditions of system.Following form is adopted in concrete control:

u＝K(x(t)-x(t-τ)) (2)

Wherein K is the square formation of n * n.

Adopt directly to postpone feedback structure as shown in Figure 1, by nonlinear function f (x), controlled quentity controlled variable u, time delay, n * n square formation K and comparer are formed.

The 3rd, according to the control injection mode that chooses, utilize the mathematical model of system and MATLAB to carry out emulation, obtain the systematic parameter bifurcation graphs.

The 4th, according to the motion conditions and the actual demand of controlled system described in the parameter bifurcation graphs, select controlled variable.

The 5th, control according to the controlled variable of choosing, obtain chaotic signal.

The 6th, the chaos characteristic of check chaotic signal after confirming to have produced chaos, can be applied to real system with the chaotic signal that produces.

CHEN circuit and permasyn morot system with non-chaos is example below, further specifies the technique effect that this control method produces chaos.

1. in non-chaos CHEN circuit, produce chaotic signal

1999, CHEN etc. found a new chaos system with LORENZ system topological non-equivalence, are called as the CHEN system ^[34]List of references ^[35,36]Parameter bifurcated and chaos phenomenon in the CHEN system have further been studied.The equation of CHEN system is as follows

$\left\{\begin{array}{c}\stackrel{\·}{x}=a(y-x)\\ \stackrel{\·}{y}=(c-a)x=\mathrm{xz}+\mathrm{cy}\\ \stackrel{\·}{z}=\mathrm{xy}-\mathrm{bz}\end{array}\right.---\left(3\right)$

By list of references ^[35,36]Analyze as can be known, work as a=35, b=3, system is in non-chaos state during c=18.Under this group parameter, three equilibrium points of system are respectively

$\left\{\begin{array}{c}O=\left(\mathrm{0,0,0}\right)\\ C^{-}=(-\sqrt{3},-\sqrt{3},1)\\ C^{+}=(\sqrt{3},\sqrt{3},1)\end{array}\right.---\left(4\right)$

The characteristic root that calculates each Jacobian of equilibrium point place matrix as can be known, O is a unstable equilibrium point, and two other is a stable equilibrium point.From the arbitrary initial state, system will be stabilized to corresponding equilibrium point place according to the domain of convergence difference at starting condition place.When system initial state was respectively (0.1,1,0.1), (0.1 ,-10 ,-0.1), system state finally was stabilized to one of two stable equilibrium points, as shown in Figure 2.

Select parameter to be in the CHEN system of non-chaotic region as controlled system.

This system can adopt circuit to realize, applies external control therefore can for any one differential equation in this system, that is to say that the decanting point of control can be determined arbitrarily.

For simplicity, will directly postpone FEEDBACK CONTROL only is applied on first equation of system (3), that is:

$K=\left[\begin{array}{ccc}{k}_{11}\≠0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

At this moment, directly postponing feedback control system is expressed as follows:

$\left\{\begin{array}{c}\stackrel{\·}{x}=a(y-x)+k_{11}(x\left(t\right)-x(t-{\mathrm{\τ}}_1))\\ \stackrel{\·}{y}=(c-a)x-\mathrm{xz}+\mathrm{cy}\\ \stackrel{\·}{z}=\mathrm{xy}-\mathrm{bz}\end{array}\right.---\left(5\right)$

The method of utilizing MATLAB and inventor to carry can obtain working as τ ₁=1.2 controlled variable k ₁₁Bifurcation graphs (as shown in Figure 3), the figure among Fig. 3 (b), figure (c) and (d) for scheming the partial enlarged drawing of (a).In the time of can obtaining the parameter controlled variable clearly and be different value by Fig. 3, the variation of the motion state of system.As seen from Figure 3, along with k ₁₁Increase, system carries out the transition to the chaotic motion state gradually from limit cycle motion, the border of chaotic motion enlarges gradually simultaneously, there are some period windows in motion change process more complicated in leading to the approach of chaos.By Fig. 3 (d) as can be known, work as k ₁₁=-23 exist high period windows, and figure among Fig. 4 (a) and figure (b) have provided three-dimensional phase path figure and time series and the power spectrum thereof of this moment respectively.Fig. 4 has illustrated that this moment, system was in the periodic motion state.

According to the chaos bifurcation graphs, produce chaos state if wish system, it is just passable only need to get the control parameter value that is in the chaotic region.For example getting controlled variable is k ₁₁=-50, τ ₁=1.2s, system will show chaotic motion, and the chaotic attractor of system is as shown in Figure 5.The chaos time sequence of output this moment and the power spectrum of this sequence are as shown in Figure 6.The pairing maximum Lyapunov exponent of chaotic attractor is 0.1938, by Fig. 5,6 and its corresponding Lyapunov index can verify to adopt and directly postpone feedback, can make stable system produce chaotic motion.

Work as k ₁₁=-10 o'clock, delay time T ₁The chaos bifurcation graphs as shown in Figure 7, can select controlled variable to make controlled system produce chaos according to Fig. 3 and Fig. 7.

If select other control decanting point, can obtain different control forms, obtain different parameter bifurcation graphs, adopt above-mentioned steps, also can produce chaotic signal.If select to postpone feedback matrix be $K=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& k_{33}\≠0\end{array}\right],$ Then controlled CHEN system equation is:

$\left\{\begin{array}{c}\stackrel{\·}{x}=a(y-x)\\ \stackrel{\·}{y}=(c-a)x-\mathrm{xz}+\mathrm{cy}\\ \stackrel{\·}{z}=\mathrm{xy}-\mathrm{bz}+k_{33}(z\left(t\right)-z(t-{\mathrm{\τ}}_3))\end{array}\right.---\left(6\right)$

Directly postpone feedback control system (6) about parameter k ₃₃And τ ₃The chaos bifurcation graphs shown in Fig. 8 and 9.When controlled variable is k ₃₃=3.8, τ ₃=0.3 o'clock, the phase path figure of system as shown in figure 10, time corresponding sequence and power spectrum are as shown in figure 11.At this moment, the Lyapunov index of system is 1.0214.The CHEN circuit can be applied to secret communication by the chaotic signal that postpones feedback generation, and the infinite dimension of its chaos will obtain better secret effect.

2. in the permasyn morot system, produce chaotic motion

Permasyn morot model through conversion ^[38,39]For

$\left\{\begin{array}{c}d{\tilde{i}}_d/\mathrm{dt}=-{\tilde{i}}_d+\widetilde{\mathrm{\ω}}{\tilde{i}}_q+{\tilde{u}}_d\\ d{\tilde{i}}_q/\mathrm{dt}=-{\tilde{i}}_q-\widetilde{\mathrm{\ω}}{\tilde{i}}_d+\mathrm{\γ}\widetilde{\mathrm{\ω}}+{\tilde{u}}_q\\ d\widetilde{\mathrm{\ω}}/\mathrm{dt}=\mathrm{\σ}({\tilde{i}}_q-\widetilde{\mathrm{\ω}})-{\tilde{T}}_L\end{array}\right.---\left(7\right)$

In the formula,

Be respectively direct-axis current, hand over shaft current and motor angular velocity through conversion, With Be respectively d through conversion, q shaft voltage and load torque, σ and γ are systematic parameter.

For the permasyn morot system, the decanting point of control can not be selected arbitrarily, because the real external control of permasyn morot is

In the 3rd equation of (7), can't inject external control.Therefore selecting the decanting point of control is first and second equations, directly postpones feedback controller as the formula (8)

${\tilde{u}}_d=K_d({\tilde{i}}_d\left(t\right)-{\tilde{i}}_d(t-{\mathrm{\τ}}_d))$

${\tilde{u}}_q=K_q({\tilde{i}}_q\left(t\right)-{\tilde{i}}_q(t-{\mathrm{\τ}}_q))---\left(8\right)$

Controlled permasyn morot system is suc as formula (9)

$\left\{\begin{array}{c}\frac{d{\tilde{i}}_d}{\mathrm{dt}}=-{\tilde{i}}_d+\widetilde{\mathrm{\ω}}{\tilde{i}}_q+K_d({\tilde{i}}_d\left(t\right)-{\tilde{i}}_d(t-{\mathrm{\τ}}_d))\\ \frac{d{\tilde{i}}_q}{\mathrm{dt}}=-{\tilde{i}}_q-\widetilde{\mathrm{\ω}}{\tilde{i}}_d+\mathrm{\γ}\widetilde{\mathrm{\ω}}+K_q({\tilde{i}}_q\left(t\right)-{\tilde{i}}_q(t-{\mathrm{\τ}}_q))\\ \frac{d\widetilde{\mathrm{\ω}}}{\mathrm{dt}}=\mathrm{\σ}({\tilde{i}}_q-\widetilde{\mathrm{\ω}})-{\tilde{T}}_L\end{array}\right.---\left(9\right)$

When σ=5.46, γ=3, ${\tilde{T}}_L=0,{\tilde{u}}_d=0,{\tilde{u}}_q=0$ There is three equilibrium points (0,0,0) saddle point in system, and other two is focus, and it is stable therefore not having the system of control, when initial condition is $({\tilde{i}}_d\left(0\right),\tilde{i}{\left(0\right)}_q,\widetilde{\mathrm{\ω}}\left(0\right))=(\mathrm{20,0.01}-5),$ Do not have control system three-dimensional phase-plane diagram as shown in figure 13, as seen what not have the system that controls will be stable at different focus places according to the domain of convergence at original state place.

Equally also can be by obtaining the chaos bifurcation graphs with the similar method of previous example, for the selection of controlled variable provides guidance, not repeating, only provide an example and do explanation here.

Get systematic parameter σ=5.46, γ=3, ${\tilde{T}}_L=0,$ Do not have the system of control this moment is stable, adopts directly to postpone FEEDBACK CONTROL, gets K in formula (9) _d=1, K _q=-0.1, τ _d=0.8, τ _q=0.9, the three-dimensional phase-plane diagram of controlled system as shown in figure 13.The time domain waveform of the motor angular velocity of corresponding process conversion and its power spectrum are as shown in figure 14.Through calculating its maximum Lyapunov exponent is 0.014.This has proved and has occurred chaos phenomenon in the motor.

Certainly, during practical application, controlled variable will be selected according to the velocity fluctuation scope of actual control system needs.Can simply realize chaotic vibration by the control that the inventor proposed, satisfy the practical application needs.

By top example as can be known, adopt and directly postpone FEEDBACK CONTROL, suitably select controlled variable can make the CHEN system and the PMSM Drive System of non-chaos chaotic motion occur, phase path figure, time series, power spectrum and Lyapunov exponent specification the chaos state of controlled system.

Claims (1)

1. the chaos production method based on direct delay feedback is characterized in that, this method adopts in Continuous Nonlinear Systems and directly postpones the feedback generation chaotic signal, specifically comprises the following steps:

The first step: select controlled device

Controlled device is selected three rank and above Continuous Nonlinear Systems thereof, and its system moves by dx/dt=f (x) mode, and wherein x is a state vector, and f () is the Nonlinear Vector function, and system is in non-chaos state when not adding input;

Second step: the injection mode of selecting external control at the characteristics of practical object

Injection mode is dx/dt=f (x)+u, the position that external control applies is selected according to the concrete condition of real system, can not select the unacceptable control of real system, the method of control adopts and directly postpones feedback system u=K (x (t)-x (t-τ)), wherein, u is the external control amount, and K is an enlargement factor, and τ is time delay;

The 3rd step: utilize the controlled device mathematical model to carry out Computer Simulation

Select suitable Poincare section, obtain under specific external control injection mode, directly postpone feedback enlargement factor and the chaos bifurcation graphs of time delay; Under the SIMULINK of MATLAB environment, set up mathematics model, utilize M running paper SIMULINK program, by analyzing and the selection Poincare section is gathered in examination, adopt trace-point method to obtain about the enlargement factor K of direct delay control and the chaos bifurcation graphs of delay time T, parameter that the value of controlled parameter generating fork and the system that makes enter chaos state from bifurcation graphs and corresponding controlled system are in the roughly amplitude of chaos state;

The 4th step: application requirements and chaos bifurcation graphs according to controlled system are determined directly to postpone feedback enlargement factor and time delay, and chaotic motion amplitude is based on the actual application requirements determined the roughly value that controlled variable is selected in bifurcation graphs, for the chaos degree that requires, estimate by the Li Ya spectrum promise husband index of calculating parameter correspondence;

The 5th step: utilize the time delay that chooses and directly postpone the feedback enlargement factor in non-chaos system, to produce chaotic signal;

The 6th step: chaos characteristic amounts such as the power spectrum of check chaotic signal, Li Ya spectrum promise husband index, confirm system works and chaos state.

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