CN105610574A - Combination synchronization method for functions of three complex chaos systems - Google Patents

Abstract

The invention discloses a combination synchronization method for functions of three complex chaos systems; the method comprises the following steps: S1: selecting the three complex chaos systems, and taking any two complex chaos systems as driving systems and the other one chaos system as a response system; S2: carrying out combination on corresponding variables of the two driving systems in the step S1; S3: carrying out corresponding subtraction on a complex of the two driving system and the combination of the corresponding variables of the response system obtained in the step S2 to obtain an error system; S4: according to the driving systems and the response system, designing a control law of the combination; and S5: loading the control law in the step S4 to the response system, thereby finishing synchronization of the combination system of the two driving systems and function combination of the response system. According to the combination synchronization method disclosed by the invention, the safety of secret communication can be improved and a proportion function matrix can be freely selected so that a code breaker is more difficult to decode; and thus, the secret communication is safer than the traditional secret communication scheme.

Description

A kind of combined synchronization method of three ignorant system functions of compound

Technical field

The invention belongs to signal processes and secret communication field, particularly a kind of combined synchronization method of three ignorant system functions of compound.

Background technology

Since Pei Kaola (Pecora) and Karol (Carroll) provide the definition of Complete Synchronization, up to the present the mixed pure synchronous research of large portion is all only confined to only have the model of a drive system and a responding system, is referred to as man-to-man system. Can two (or multiple) drive systems and (or multiple) responding system be realized is synchronously a significant problem. Such as only having in the mixed pure secure communication of the unified responding system of drivetrain, the signal that transmit is all to be sent by a drive system, a responding system reception. The anti-interference of such mode signal and anti-decoding are not high.

Therefore, need now a kind of combined synchronization method of three ignorant system functions of compound badly, can be in the model that contains two drive systems and a responding system, the information that will send can be divided into several different parts, and be written into respectively different drive systems, or the time sending is divided into the different stages, the information that will send in the different stages is sent by different drive systems respectively, and combination of function synchronously adopts complex variable that variable is doubled, key space increases, the more difficult decoding of code breaker, the security of raising secret communication. Proportion function matrix is free to select, and makes code breaker more be difficult to decode, thereby makes secret communication safer than traditional communication schemes.

Summary of the invention

The present invention proposes a kind of combined synchronization method of three ignorant system functions of compound, has solved anti-interference and the not high problem of anti-decoding of signal in prior art.

Technical scheme of the present invention is achieved in that the combined synchronization method of three ignorant system functions of compound, comprises the steps:

S1: choose three multiple chaos systems, so that wherein any two multiple chaos systems are as drive system, another chaos system is as responding system;

S2: by two drive systems in step S1 dependent variable is combined;

S3: the corresponding variable combination of the compound and responding system of two drive systems that obtain in step S2 is carried out to correspondence poor: obtain error system;

S4: according to drive system and responding system, the control law of composite design;

S5: the control law in step S4 is carried on responding system, completes the combined system of two drive systems and the combination of function of responding system is synchronizeed.

As one preferred embodiment, in step S1, first the multiple chaos system as drive system is:

Wherein, x=(x₁，x₂，…，x_n)^TThe multiple state variable of first multiple chaos system, x=x^r+jxⁱ, definition x₁＝u₁+ju₂，x₂＝u₃+ju₄，x_n＝u_2n-1+ju_2n, and x^r＝(u₁，u₃，…，u_2n-1)，xⁱ＝(u₂，u₄，…，u_2n)^T. F (x) is the complex matrix of n × n, and its element is the continuous function that contains complex variable, f=(f₁，f₂，…，f_2n)^TNonlinear vectorial complex function, A=(a₁，a₂，…，a_n)^TBe the real vector of n × 1, r and i represent respectively real part and the imaginary part of multiple state variable.

As one preferred embodiment, in step S1, second multiple chaos system as drive system is:

Wherein y=(y₁，y₂，…，y_n)^TThe multiple state variable of second multiple chaos system, y=y^r+jyⁱ, definition y₁＝u′₁+ju′₂，y₂＝u′₃+ju′₄，y_n＝u′_2n-1+ju′_2n, and y^r＝(u′₁，u′₃，…，u′_2n-1)，yⁱ＝(u′₂，u′₄，…，u′_2n）^T, G (y) is the complex matrix of n × n, its element is the continuous function that contains complex variable, g=(g₁，g₂，…，g_n)^TNonlinear vectorial complex function, B=(b₁，b₂，…，b_n)^TIt is the real vector of n × 1.

As one preferred embodiment, in step S1, the 3rd the multiple chaos system as responding system is:

Wherein, z=(z₁，z₂，…，z_n)^TThe plural state variable of the 3rd multiple chaos system, z=z^r+jzⁱ, definition z₁＝u″₁+j₁u″₂，z₂＝u″₃+j₁u″₄，，z_n＝u″_2n-1+j₁u″_2n，z^r＝(u″₁，u″₃，…，u″_2n-1)，zⁱ＝(u″₂，u″₄，…，u″_2n), H (z) is the complex matrix of n × n, its element is the continuous function that contains complex variable, h=(h₁，h₂，…，h_n)^TNonlinear vector quantity complex function, C=(c₁，c₂，…，c_n)^TIt is the real vector of n × 1.

As one preferred embodiment, multiple to first multiple chaos system and second chaos system being carried out to combination of function is D₁(t)x+D₂(t) y; Wherein, D₁(t)，D₂(t)∈R^n×nTwo matrix functions, D₁(t)＝diag(d₁(t)，d₂(t)，…，d_n(t))

As one preferred embodiment, the error system obtaining is:

e(t)＝e^r(t)+jeⁱ(t)＝z-D₁(t)x-D₂(t)y，

Wherein e^r(t)＝(e₁(t)，e₃(t)，…，e_2n-1(t))^T，eⁱ(t)＝(e₂(t)，e₄(t)，…，e_2n(t))^T；

Error system further arranges:

As one preferred embodiment, combination control law is:

Wherein, constant k > 0.

As one preferred embodiment, after step S5, be further provided with step S6: the combined system of two drive systems is synchronizeed and verified with the combination of function of responding system.

As one preferred embodiment, described checking comprises error system (4) differentiate, obtains:

Then separate real part and imaginary part in equation (6), obtain:

Select following liapunov function to be:

Differentiate obtains:

Control law (5) formula is updated in (9) formula, obtains:

Because V is positive definite,Be negative semidefinite, according to Lyapunov theorem of stability, combination of function realizes synchronous.

Adopt after technique scheme, the invention has the beneficial effects as follows: the combination of function synchronous method that the invention discloses three multiple chaos systems. For a multiple drive system and a complex response system synchronization model in synchronous, study by Self Adaptive Control and realized two multiple combinations of drive system and synchronizeing of complex response system. Based on Lyapunov Theory of Stability, design adaptive law, makes multiple chaos system realize combination of function synchronous. Wherein, two chaos systems are as drive system, and other chaos system, as responding system, then, by corresponding poor to the combination of two drive systems and responding system, obtains error system; According to drive system and responding system, carry out design control law again; Finally control law is carried on responding system, according to Lyapunov Theory of Stability, the combined system of two drive systems is synchronizeed with responding system self adaptation. Multiple chaos system adopts complex variable that variable is doubled, and key space increases, the more difficult decoding of code breaker, the security of raising secret communication; Due to the increase of drive system, make load signal can more freely cut apart the state variable of load driver system, code breaker is difficult to decode more, improves to a certain extent the security of secret communication; In combination, proportionality constant is changed to proportion function, makes hybrid system more complicated, and dynamic behavior is abundanter. Comprehensive above three kinds of advantages, make our synchronization scenario have more great development potentiality, are expected to break through the bottleneck of information security.

Brief description of the drawings

In order to be illustrated more clearly in the embodiment of the present invention or technical scheme of the prior art, to the accompanying drawing of required use in embodiment or description of the Prior Art be briefly described below, apparently, accompanying drawing in the following describes is only some embodiments of the present invention, for those of ordinary skill in the art, do not paying under the prerequisite of creative work, can also obtain according to these accompanying drawings other accompanying drawing.

Curve time response that first state variable function combination real part of the multiple drive system (17) of Fig. 1 and (18) is synchronizeed with first state variable real part of complex response system (19);

Curve time response that first state variable function combination imaginary part of the multiple drive system (17) of Fig. 2 and (18) is synchronizeed with first state variable imaginary part of complex response system (19);

Curve time response that the multiple drive system (17) of Fig. 3 and (19) second state variable real parts of (18) second state variable function combination real parts and complex response system are synchronizeed;

Curve time response that the multiple drive system (17) of Fig. 4 and (19) second state variable imaginary parts of (18) second state variable function combination imaginary parts and complex response system are synchronizeed;

The multiple drive system (17) of Fig. 5 and (18) the 3rd state variable functions combine curve time response of synchronizeing with the 3rd state variable of complex response system (19).

Detailed description of the invention

Below in conjunction with the accompanying drawing in the embodiment of the present invention, the technical scheme in the embodiment of the present invention is clearly and completely described, obviously, described embodiment is only the present invention's part embodiment, instead of whole embodiment. Based on the embodiment in the present invention, those of ordinary skill in the art, not making the every other embodiment obtaining under creative work prerequisite, belong to the scope of protection of the invention.

As shown in Figure 1, the combined synchronization method of these three ignorant system functions of compound, comprises the steps:

S1: choose three multiple chaos systems, so that wherein any two multiple chaos systems are as drive system, another chaos system is as responding system;

S2: by two drive systems in step S1 dependent variable is combined;

S3: the corresponding variable combination of the compound and responding system of two drive systems that obtain in step S2 is carried out to correspondence poor: obtain error system;

S4: according to drive system and responding system, the control law of composite design;

S5: the control law in step S4 is carried on responding system, completes the combined system of two drive systems and the combination of function of responding system is synchronizeed.

As one preferred embodiment, in step S1, first the multiple chaos system as drive system is:

Wherein, x=(x₁，x₂，…，x_n)^TThe multiple state variable of first multiple chaos system, x=x^r+jxⁱ, definition x₁＝u₁+ju₂，x₂＝u₃+ju₄，x_n＝u_2n-1+ju_2n, and x^r＝(u₁，u₃，…，u_2n-1)，xⁱ＝(u₂，u₄，…，u_2n)^T. F (x) is the complex matrix of n × n, and its element is the continuous function that contains complex variable, f=(f₁，f₂，…，f_2n)^TNonlinear vectorial complex function, A=(a₁，a₂，…，a_n)^TBe the real vector of n × 1, r and i represent respectively real part and the imaginary part of multiple state variable.

As one preferred embodiment, in step S1, second multiple chaos system as drive system is:

Wherein y=(y₁，y₂，…，y_n)^TThe multiple state variable of second multiple chaos system, y=y^r+jyⁱ, definition y₁＝u′₁+ju′₂，y₂＝u′₃+ju′₄，y_n＝u′_2n-1+ju′_2n, and y^r＝(u′₁，u′₃，…，u′_2n-1)，yⁱ＝(u′₂，u′₄，…，u′_2n)^T, G (y) is the complex matrix of n × n, its element is the continuous function that contains complex variable, g=(g₁，g₂，…，g_n)^TNonlinear vectorial complex function, B=(b₁，b₂，…，b_n)^TIt is the real vector of n × 1.

As one preferred embodiment, in step S1, the 3rd the multiple chaos system as responding system is:

Wherein, z=(z₁，z₂，…，z_n)^TThe plural state variable of the 3rd multiple chaos system, z=z^r+jzⁱ, definition z₁＝u″₁+j₁u″₂，z₂＝u″₃+j₁u″₄，，z_n＝u″_2n-1+j₁u″_2n，z^r＝(u″₁，u″₃，…，u″_2n-1)，zⁱ＝(u″₂，u″₄，…，u″_2n), H (z) is the complex matrix of n × n, its element is the continuous function that contains complex variable, h=(h₁，h₂，…，h_n)^TNonlinear vector quantity complex function, C=(c₁，c₂，…，c_n)^TIt is the real vector of n × 1.

As one preferred embodiment, multiple to first multiple chaos system and second chaos system being carried out to combination of function is D₁(t)x+D₂(t) y; Wherein, D₁(t)，D₂(t)∈R^n×nTwo matrix functions, D₁(t)＝diag(d₁(t)，d₂(t)，…，d_n(t))

As one preferred embodiment, the error system obtaining is:

e(t)＝e^r(t)+jeⁱ(t)＝z-D₁(t)_x-D₂(t)_y，

Wherein e^r(t)＝(e₁(t)，e₃(t)，…，e_2n-1(t))^T，eⁱ(t)＝(e₂(t)，e₄(t)，…，e_2n(t))^T；

Error system further arranges:

As one preferred embodiment, combination control law is:

Wherein, constant k > 0.

As one preferred embodiment, after step S5, be further provided with step S6: the combined system of two drive systems is synchronizeed and verified with the combination of function of responding system.

As one preferred embodiment, described checking comprises error system (4) differentiate, obtains:

Then separate real part and imaginary part in equation (6), obtain:

Select following liapunov function to be:

Differentiate obtains:

Control law (5) formula is updated in (9) formula, obtains:

Because V is positive definite,Be negative semidefinite, according to Lyapunov theorem of stability, combination of function realizes synchronous.

Particularly,

Therefore,

Wherein,Real part and the imaginary part of first drive system (1) initial value.WithReal part and the imaginary part of second drive system (2) initial value.WithReal part and the imaginary part of responding system (3) initial value.

Inference 1: suppose two Jacobian matrix D₁And D (t)₂(t) be two constant matrices D₁And D₂Time, control law is designed to:

Then, responding system (3) can be realized combining with the combined system of two drive systems (1) and (2) and synchronize.

Inference 2: supposition D₁(t)=0 o'clock, particularly, d₁(t)＝d₂(t)＝…＝d_n(t)=0, design of control law is

Then, responding system (3) can be realized Function Projective Synchronization with drive system (2).

Inference 3: supposition D₂(t)=0 o'clock, particularly,Design of control law is

Then, responding system (3) can be realized Function Projective Synchronization with drive system (1).

Inference 4: suppose D₁(t)＝D₂(t)=0 o'clock, particularly, Design of control law is

Then, the equalization point of responding system is progressive stable.

Illustrate with object lesson below: the validity of utilizing three these schemes that chaos systems are verified as an example again.

First drive system is:

Second drive system is:

Responding system is:

According to middle driving and responding system model, complex system (17), (18) and (19), under the condition that there is no controller, can be written as:

Wherein, x=(x₁，x₂，x₃)^T，y＝(y₁，y₂，y₃)^T，z＝(z₁，z₂，z₃)^TIt is respectively the state vector of chaos complex system (17), (18) and (19). A=B=C=(a, b, c)^TThree chaos systems (17), (18) and (19) parameter vector, v₁，v₂，v₃，v₄，v₅And v₆It is control law to be designed.

The control law of design is:

Further arrange and be:

Real part and imaginary part in (21) formula of separation, obtain:

In order to verify the feasibility of combination of function synchronization scenario, study three identical multiple chaos systems (17), the simulation result of the combination of function synchronization scenario of (18) and (19). In numerical simulation, three parameters are chosen as a=1, b=1, and c=-1, initial value and the constant k of three chaos systems are chosen as respectively (x₁(0)，x₂(0)，x₃(0)，)^T＝(0.1+0.1j，0.1+0.1j，0.2)^T，(y₁(0)，y₂(0)，y₃(0)，)^T＝(2+2j，3+2j，1)^T，(z₁(0)，z₂(0)，z₃(0)，)^T＝(-0.1+0.2j，-0.3+0.2j，0.1)^T, and k=-0.7. Two function ratio matrixes are

And D₂(t)＝diag(Re[0.4+0.1x₁]，Re[0.3+0.2x₂]，Re[0.2+0.3x₃]). The synchronous corresponding Numerical Simulation Results of combination of function of multiple drive system (17) and (18) and complex response system (19) provides in Fig. 1-5. In the time of t → 0, the time response e of synchronous error_i→ 0 (i=1,2,3,4,5), show that multiple drive system (17) and (18) and the combination of complex response system (19) realizable function are synchronous. Wherein Re () and Im () are used for representing real and imaginary part.

The invention discloses the combination of function synchronous method of three multiple chaos systems, for a multiple drive system and a complex response system synchronization model in synchronous, study by Self Adaptive Control and realized two multiple combinations of drive system and synchronizeing of complex response system. Based on Lyapunov Theory of Stability, design adaptive law, makes multiple chaos system realize combination of function synchronous. Wherein, two chaos systems are as drive system, and other chaos system, as responding system, then, by corresponding poor to the combination of two drive systems and responding system, obtains error system; According to drive system and responding system, carry out design control law again; Finally control law is carried on responding system, according to Lyapunov Theory of Stability, the combined system of two drive systems is synchronizeed with responding system self adaptation. Multiple chaos system adopts complex variable that variable is doubled, and key space increases, the more difficult decoding of code breaker, the security of raising secret communication; Due to the increase of drive system, make load signal can more freely cut apart the state variable of load driver system, code breaker is difficult to decode more, improves to a certain extent the security of secret communication; In combination, proportionality constant is changed to proportion function, makes hybrid system more complicated, and dynamic behavior is abundanter. Comprehensive above three kinds of advantages, make our synchronization scenario have more great development potentiality, are expected to break through the bottleneck of information security.

The foregoing is only preferred embodiment of the present invention, in order to limit the present invention, within the spirit and principles in the present invention not all, any amendment of doing, be equal to replacement, improvement etc., within all should being included in protection scope of the present invention.

Claims (9)

1. a combined synchronization method for three ignorant system functions of compound, is characterized in that, comprises the steps:

S1: choose three multiple chaos systems, so that wherein any two multiple chaos systems are as drive system, anotherA chaos system is as responding system;

S2: by two drive systems in step S1 dependent variable is combined;

S3: the corresponding variable of the compound and responding system of two drive systems that obtain in step S2 is combined intoRow is corresponding poor, obtains error system;

S4: according to drive system and responding system, the control law of composite design;

S5: the control law in step S4 is carried on responding system, completes the combination system of two drive systemsSystem is synchronizeed with the combination of function of responding system.

2. the combined synchronization method of three ignorant system functions of compound according to claim 1, its feature existsIn, first the multiple chaos system as drive system in step S1 is:

$\stackrel{\·}{x}=F\left(x\right)A+f\left(x\right)$

Wherein, x=(x₁，x₂，…，x_n)^TThe multiple state variable of first multiple chaos system, x=x^r+jxⁱ, definitionx₁＝u₁+ju₂，x₂＝u₃+ju₄，x_n＝u_2n-1+ju_2n, and x^r＝(u₁，u₃，…，u_2n-1)，xⁱ＝(u₂，u₄，…，u_2n)^T，F(x)Be the complex matrix of n × n, its element is the continuous function that contains complex variable, f=(f₁，f₂，…，f_2n)^TNonlinearVector complex function, A=(a₁，a₂，…，a_n)^TBe the real vector of n × 1, r and i represent respectively the real part of multiple state variableAnd imaginary part.

3. the combined synchronization method of three ignorant system functions of compound according to claim 2, its feature existsIn, second multiple chaos system as drive system in step S1 is:

$\stackrel{\·}{y}=G\left(y\right)B+g\left(y\right)---\left(2\right),$

Wherein y=(y₁，y₂，…，y_n)^TThe multiple state variable of second multiple chaos system, y=y^r+jyⁱ, definitiony₁＝u′₁+ju′₂，y₂＝u′₃+ju′₄，y_n＝u′_2n-1+ju′_2n, and y^r＝(u′₁，u′₃，…，u′_2n-1)，yⁱ＝(u′₂，u′₄，…，u′_2n)^T，G (y) is the complex matrix of n × n, and its element is the continuous function that contains complex variable, g=(g₁，g₂，…，g_n)^TIt is non-lineThe vectorial complex function of property, B=(b₁，b₂，…，b_n)^TIt is the real vector of n × 1.

4. the combined synchronization method of three ignorant system functions of compound according to claim 3, its feature existsIn, the 3rd the multiple chaos system as responding system in step S1 is:

$\stackrel{\·}{z}=H\left(z\right)C+h\left(z\right)+v---\left(3\right),$

Wherein, z=(z₁z₂，…，z_n)^TThe plural state variable of the 3rd multiple chaos system, z=z^r+jzⁱ, definitionz₁＝u″₁+j₁u″₂，z₂＝u″₃+j₁u″₄，，z_n＝u″_2n-1+j₁u″_2n，z^r＝(u″₁，u″₃，…，u″_2n-1)，zⁱ＝(u″₂，u″₄，…，u″_2n)，H (z) is the complex matrix of n × n, and its element is the continuous function that contains complex variable, h=(h₁，h₂，…，h_n)^TNon-linearVector quantity complex function, C=(c₁，c₂，…，c_n)^TIt is the real vector of n × 1.

5. the combined synchronization method of three ignorant system functions of compound according to claim 4, its feature existsIn, it is D that multiple to first multiple chaos system and second chaos system is carried out to combination of function₁(t)x+D₂(t)y；Wherein, D₁(t)，D₂(t)∈R^n×nTwo matrix functions, D₁(t)＝diag(d₁(t)，d₂(t)，…，d_n(t)) $D_2\left(t\right)=\mathrm{diag}(d_1^{*}\left(t\right),d_2^{*}\left(t\right),\·\·\·,d_n^{*}\left(t\right)).$

6. the combined synchronization method of three ignorant system functions of compound according to claim 5, its feature existsIn, the error system obtaining is:

e(t)＝e^r(t)+jeⁱ(t)＝z-D₁(t)x-D₂(t)y，

Wherein e^r(t)＝(e₁(t)，e₃(t)，…，e_2n-1(t))^T，eⁱ(t)＝(e₂(t)，e₄(t)，…，e_2n(t))^T；

Error system further arranges:

$\begin{array}{c}e\left(t\right)=z^r+{\mathrm{jz}}^i-D_1\left(t\right)(x^r+{\mathrm{jx}}^i)-D_2\left(t\right)(y^r+{\mathrm{jy}}^i)\\ =z^r-D_1\left(t\right)x^r-D_2\left(t\right)y^r+j(z^i-D_1(t)x^i-D_2\left(t\right)y^i)\end{array}---\left(4\right).$

7. the combined synchronization method of three ignorant system functions of compound according to claim 6, its feature existsIn, combination control law is:

$\begin{array}{c}v=v^r+{\mathrm{jv}}^i\\ ={\stackrel{\·}{D}}_1\left(t\right)x+D_1\left(t\right)F\left(x\right)A+D_1\left(t\right)f\left(x\right)+{\stackrel{\·}{D}}_2\left(t\right)y\\ +D_2\left(t\right)G\left(y\right)B+D_2\left(t\right)g\left(y\right)-H\left(z\right)C-h\left(z\right)-ke\\ ={\stackrel{\·}{D}}_1\left(t\right)x^r+D_1\left(t\right)\[F^r(u_1,u_3,...,u_{2n-1})A+f^r(u_1,u_3,...,u_{2n-1})\]\\ +{\stackrel{\·}{D}}_2\left(t\right)y^r+D_2\left(t\right)\[G^r(u_1^{\′},u_3^{\′},...,u_{2n-1}^{\′})B+g^r(u_1^{\′},u_3^{\′},...,u_{2n-1}^{\′})\]\\ -H^r(u_1^{\′\′},u_3^{\′\′},...,u_{2n-1}^{\′\′})C-h^r(u_1^{\′\′},u_3^{\′\′},...,u_{2n-1}^{\′\′})\\ -{\mathrm{ke}}^r+j\{{\stackrel{\·}{D}}_1\left(t\right)x^i+D_1\left(t\right)\[F^i(u_2,u_4,...,u_{2n})A+f^i(u_2,u_4,...,u_{2n})\]\\ +{\stackrel{\·}{D}}_2\left(t\right)y^i+{\stackrel{\·}{D}}_2\left(t\right)\mathrm{\[}{G}^i(u_2^{\′},u_4^{\′},...,u_{2n}^{\′})B+g^i(u_2^{\′},u_4^{\′},...,u_{2n}^{\′})\mathrm{\]}\\ -H^i(u_2^{\′\′},u_4^{\′\′},...,u_{2n}^{\′\′})C-h^i(u_2^{\′\′},u_4^{\′\′},...,u_{2n}^{\′\′})-{\mathrm{ke}}^i\}\end{array}---\left(5\right),$

Wherein, constant k > 0.

8. the combined synchronization method of three ignorant system functions of compound according to claim 1, its feature existsIn, after step S5, be further provided with step S6: the combined system to two drive systems and responding systemCombination of function is synchronously verified.

9. the combined synchronization method of three ignorant system functions of compound according to claim 8, its feature existsIn, described checking comprises error system (4) differentiate, obtains:

$\begin{array}{c}\stackrel{\·}{e}={\stackrel{\·}{e}}^r+j{\stackrel{\·}{e}}^i\\ =H\left(z\right)C+h\left(z\right)-{\stackrel{\·}{D}}_1\left(t\right)x-D_1\left(t\right)F\left(x\right)A-D_1\left(t\right)f\left(x\right)\\ -{\stackrel{\·}{D}}_2\left(t\right)y-D_2\left(t\right)G\left(y\right)B-D_2\left(t\right)g\left(y\right)+ke\\ =H^r(u_1^{\′\′},u_3^{\′\′},...,u_{2n-1}^{\′\′})C+h^r(u_1^{\′\′},u_3^{\′\′},...,u_{2n-1}^{\′\′})\\ -{\stackrel{\·}{D}}_2\left(t\right)\[G^r(u_1,u_3,...,u_{2n-1})A+f^r(u_1,u_3,...,u_{2n-1})\]-{\stackrel{\·}{D}}_2\left(t\right)y^r\\ -D_2\left(t\right)H^r(u_1^{\′},u_3^{\′},...,u_{2n-1}^{\′})B-g^r(u_1^{\′},u_3^{\′},...,u_{2n-1}^{\′})\\ +v^r+j\{H^i(u_2^{\′\′},u_4^{\′\′},...,u_{2n}^{\′\′})C+h^i(u_2^{\′\′},u_4^{\′\′},...,u_{2n}^{\′\′})\\ -{\stackrel{\·}{D}}_1\left(t\right)x^i+D_1\left(t\right)\[F^i(u_2,u_4,...,u_{2n})A+f^i(u_2,u_4,...,u_{2n})\]\\ -{\stackrel{\·}{D}}_2\left(t\right)y^i+D_2\left(t\right)\mathrm{\[}{G}^i(u_2^{\′},u_4^{\′},...,u_{2n}^{\′})B+g^i(u_2^{\′},u_4^{\′},...,u_{2n}^{\′})\]+v^i\}\end{array}---\left(6\right),$

Then separate real part and imaginary part in equation (6), obtain:

$\left\{\begin{array}{c}{\stackrel{\·}{e}}^r=H^r(u_1^{\′\′},u_3^{\′\′},...,u_{2n-1}^{\′\′})C+h^r(u_1^{\′\′},u_3^{\′\′},...,u_{2n-1}^{\′\′})\\ -{\stackrel{\·}{D}}_1\left(t\right)x^r-D_1\left(t\right)\[F^r(u_1,u_3,...,u_{2n-1})A+f^r(u_1,u_3,...,u_{2n-1})\]\\ -{\stackrel{\·}{D}}_2\left(t\right)y^r-D_2\left(t\right)\[G^r(u_1^{\′},u_3^{\′},...,u_{2n-1}^{\′})B+g^r(u_1^{\′},u_3^{\′},...,u_{2n-1}^{\′})\]+v^r\\ {\stackrel{\·}{e}}^i=H^i(u_2^{\′\′},u_4^{\′\′},...,u_{2n}^{\′\′})C+h^i(u_2^{\′\′},u_4^{\′\′},...,u_{2n}^{\′\′})\\ -{\stackrel{\·}{D}}_1\left(t\right)x^i-D_1\left(t\right)\[F^i(u_2,u_4,...,u_{2n})A+f^i(u_2,u_4,...,u_{2n})\]\\ -{\stackrel{\·}{D}}_2\left(t\right)y^i-D_2\left(t\right)\mathrm{\[}{G}^i(u_2^{\′},u_4^{\′},...,u_{2n}^{\′})B+g^i(u_2^{\′},u_4^{\′},...,u_{2n}^{\′})\]+v^i\end{array}\right.---\left(7\right),$

Select following liapunov function to be:

Differentiate obtains:

$\begin{array}{c}\stackrel{\·}{V}={\left({\stackrel{\·}{e}}^r\right)}^T{e}^r+{\left({\stackrel{\·}{e}}^i\right)}^T{e}^i\\ =\{H^r(u_1^{\′\′},u_3^{\′\′},...,u_{2n-1}^{\′\′})C+h^r(u_1^{\′\′},u_3^{\′\′},...,u_{2n-1}^{\′\′})\\ -{\stackrel{\·}{D}}_1\left(t\right)x^r-D_1\left(t\right)\[F^r(u_1,u_3,...,u_{2n-1})A+f^r(u_1,u_3,...,u_{2n-1})\]-{\stackrel{\·}{D}}_2\left(t\right)y^r\\ -D_2\left(t\right)\[G^r(u_1^{\′},u_3^{\′},...,u_{2n-1}^{\′})B+g^r(u_1^{\′},u_3^{\′},...,u_{2n-1}^{\′})\]+v^r{\}}^T{e}^r\\ +\{H^i(u_2^{\′\′},u_4^{\′\′},...,u_{2n}^{\′\′})C+h^i(u_2^{\′\′},u_4^{\′\′},...,u_{2n}^{\′\′})\\ -{\stackrel{\·}{D}}_1\left(t\right)x^i-D_1\left(t\right)\[F^i(u_2,u_4,...,u_{2n})A+f^i(u_2,u_4,...,u_{2n})\]-{\stackrel{\·}{D}}_2\left(t\right)y^i\\ -D_2\left(t\right)\mathrm{\[}{G}^i(u_2^{\′},u_4^{\′},...,u_{2n}^{\′})B+g^i(u_2^{\′},u_4^{\′},...,u_{2n}^{\′})\]+v^i{\}}^T{e}^i\end{array}---\left(9\right),$

Control law (5) formula is updated in (9) formula, obtains:

$\begin{array}{c}\stackrel{\·}{V}={\left({\stackrel{\·}{e}}^r\right)}^T{e}^r+{\left({\stackrel{\·}{e}}^i\right)}^T{e}^i\\ =\{H^r(u_1^{\′\′},u_3^{\′\′},...,u_{2n-1}^{\′\′})C+h^r(u_1^{\′\′},u_3^{\′\′},...,u_{2n-1}^{\′\′})\\ -{\stackrel{\·}{D}}_1\left(t\right)x^r-D_1\left(t\right)\[F^r(u_1,u_3,...,u_{2n-1})A+f^r(u_1,u_3,...,u_{2n-1})\]-{\stackrel{\·}{D}}_2\left(t\right)y^r\\ -D_2\left(t\right)\[G^r(u_1^{\′},u_3^{\′},...,u_{2n-1}^{\′})B+g^r(u_1^{\′},u_3^{\′},...,u_{2n-1}^{\′})\]+{\stackrel{\·}{D}}_1\left(t\right)x^r\\ +{\stackrel{\·}{D}}_1\left(t\right)x^r-D_1\left(t\right)\[F^r(u_1,u_3,...,u_{2n-1})A+f^r(u_1,u_3,...,u_{2n-1})\]+{\stackrel{\·}{D}}_1\left(t\right)x^r\\ +D_1\left(t\right)\[F^r(u_1,u_3,...,u_{2n-1})A+f^r(u_1,u_3,...,u_{2n-1})\]+{\stackrel{\·}{D}}_2\left(t\right)y^r\\ +D_2\left(t\right)\[G^r(u_1^{\′},u_3^{\′},...,u_{2n-1}^{\′})B+g^r(u_1^{\′},u_3^{\′},...,u_{2n-1}^{\′})\]\\ -H^r(u_1^{\′\′},u_3^{\′\′},...,u_{2n-1}^{\′\′})C\\ -h^r(u_1^{\′\′},u_3^{\′\′},...,u_{2n-1}^{\′\′})-{\mathrm{ke}}^r{\}}^T{e}^r\\ +\{H^i(u_2^{\′\′},u_4^{\′\′},...,u_{2n}^{\′\′})C+h^i(u_2^{\′\′},u_4^{\′\′},...,u_{2n}^{\′\′})\\ -{\stackrel{\·}{D}}_1\left(t\right)x^i-D_1\left(t\right)\[F^i(u_2,u_4,...,u_{2n})A+f^i(u_2,u_4,...,u_{2n})\]-{\stackrel{\·}{D}}_2\left(t\right)y^i\\ -D_2\left(t\right)\mathrm{\[}{G}^i(u_2^{\′},u_4^{\′},...,u_{2n}^{\′})B+g^i(u_2^{\′},u_4^{\′},...,u_{2n}^{\′})\mathrm{\]}\\ +{\stackrel{\·}{D}}_1\left(t\right)x^i+D_1\left(t\right)\[F^i(u_2,u_4,...,u_{2n})A+f^i(u_2,u_4,...,u_{2n})\]+{\stackrel{\·}{D}}_2\left(t\right)y^i\\ +{\stackrel{\·}{D}}_2\left(t\right)\mathrm{\[}{G}^i(u_2^{\′},u_4^{\′},...,u_{2n}^{\′})B+g^i(u_2^{\′},u_4^{\′},...,u_{2n}^{\′})\mathrm{\]}\\ -H^i(u_2^{\′\′},u_4^{\′\′},...,u_{2n}^{\′\′})C-h^i(u_2^{\′\′},u_4^{\′\′},...,u_{2n}^{\′\′})-{\mathrm{ke}}^i{\}}^T{e}^i\\ =-k\[{\left(e^r\right)}^T{e}^r+{\left(e^i\right)}^T{e}^i\]\end{array}---\left(10\right),$

Because V is positive definite,Be negative semidefinite, according to Lyapunov theorem of stability, combination of function is sameStep realizes.