CN116520692A - Intermittent control-based index synchronous control method of complex-valued neural network - Google Patents

Intermittent control-based index synchronous control method of complex-valued neural network Download PDF

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CN116520692A
CN116520692A CN202310374032.7A CN202310374032A CN116520692A CN 116520692 A CN116520692 A CN 116520692A CN 202310374032 A CN202310374032 A CN 202310374032A CN 116520692 A CN116520692 A CN 116520692A
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李小凡
黄鑫
李慧媛
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Wuxi Xiangyuan Information Technology Co ltd
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Yancheng Institute of Technology
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Abstract

The invention belongs to the technical field of new generation information, and particularly relates to an index synchronous control method of a complex-valued neural network based on intermittent control. The method comprises the following steps: step S1: establishing a driving system and a response system based on a complex-valued neural network with mixed time-varying time lags; step S2: constructing real and imaginary subsystems of the driving system and the response system; step S3: setting a synchronization error, and establishing a real part subsystem and an imaginary part subsystem of a synchronization error system; step S4: an aperiodic intermittent synchronous controller is designed and acts on the response system so that the response system is exponentially synchronous with the driving system. The invention introduces mixed time-varying time lag into the complex-valued neural network, adopts the non-periodic intermittent synchronous controller, and provides a novel control method for realizing exponential synchronization of the complex-valued neural network with the mixed time-varying time lag.

Description

Intermittent control-based index synchronous control method of complex-valued neural network
Technical Field
The invention relates to the technical field of new generation information, in particular to an index synchronous control method of a complex-valued neural network based on intermittent control.
Background
In 1982, the american scientist john Hopfield proposed Hopfield neural networks, which have become research hotspots since the rise of artificial intelligence. The synchronous dynamic characteristics of the artificial neural network are widely applied to the technical fields of new generation information such as pattern recognition, automatic control, prediction estimation, information security, associative memory, model prediction, secure communication and the like.
Compared with a real-value neural network, the complex-value neural network has complex neuron states, connection weights, activation functions and external inputs, and can solve more complex problems, such as: XOR problems, associative memory, timing processing, etc. Therefore, it is important to study the synchronization problem of complex-valued neural networks.
Non-periodic intermittent control is more compatible with real life applications than periodic intermittent control, such as: wind power generation, solar power generation, stock fluctuation, and the like have uncertainty, so that it is of great importance to study non-periodic intermittent control.
Disclosure of Invention
In view of the above, the present invention aims to provide an exponential synchronization control method for a complex-valued neural network based on intermittent control, which can realize the exponential synchronization control of the complex-valued neural network with mixed time-varying time lags.
The invention is realized by adopting the following scheme: an exponential synchronization control method of a complex-valued neural network based on intermittent control comprises the following steps:
step S1: establishing a driving system and a response system based on a complex-valued neural network with mixed time-varying time lags; the specific contents of the step S1 are as follows:
the driving system and the response system for establishing the complex-valued neural network with the mixed time-varying time lag are respectively as follows:
wherein, the time t is more than or equal to 0; drive system state x (t) = (x 1 (t)，x 2 (t)，…，x n (t)) T ，z(t)＝(z 1 (t)，z 2 (t)，…，z n (t)) T ，x i The real and imaginary parts of (t) are respectivelyAnd->z i The real and imaginary parts of (t) are +.>Andresponse system state y (t) = (y) 1 (t)，y 2 (t)，…，y n (t)) T ，v(t)＝(v 1 (t)，v 2 (t)，…，v n (t)) T ，y i The real and imaginary parts of (t) are +.>And->v i The real and imaginary parts of (t) are +.>And->n represents the number of neurons in the neural network; A. b is a real number connection weight matrix, a=diag { a } 1 ，a 2 ，…，a n }，B＝diag{b 1 ，b 2 ，…，b n }，a i And b i Is a constant; C. d, W is a complex value connection weight matrix, c= (C) ij ) n×n ，D＝(d ij ) n×n ，W＝(w ij ) n×n Complex number c ij 、d ij 、w ij The real parts of (2) are>Complex number c ij 、d ij 、w ij Imaginary parts of (2) are->Activation function f (x (t)) without time lag= (f) 1 (x 1 (t))，f 2 (x 2 (t))，…，f n (x n (t))) T ，f(y(t))＝(f 1 (y 1 (t))，f 2 (y 2 (t))，…，f n (y n (t))) T ，f j (x j The real and imaginary parts of (t)) are +.>And->f j (y j The real and imaginary parts of (t)) are +.>And->Activation function f (x (t- σ (t))) = (f) containing time lags 1 (x 1 (t-σ(t)))，f 2 (x 2 (t-σ(t)))，…，f n (x n (t-σ(t)))) T ，f(y(t-σ(t)))＝(f 1 (y 1 (t-σ(t)))，f 2 (y 2 (t-σ(t)))，…，f n (y n (t-σ(t)))) T ，f j (x j The real and imaginary parts of (t- σ (t))) are respectivelyAnd->f j (y j The real and imaginary parts of (t- σ (t))) are +.>And->Each of the aboveThe activation function satisfies the lipschitz condition, namely:
wherein the method comprises the steps ofIs a lipschitz constant; sigma (t) and tau (t) are time-varying discrete time lag and time-varying distributed time lag, respectively, and satisfy 0<σ(t)<σ，0<τ(t)<τ, σ and τ are positive constants and let +.>η is an integral variable; complex-valued neural network external input γ (t) = (γ) 1 (t)，γ 2 (t)，…，γ n (t)) T And gamma is i The real and imaginary parts of (t) are +.>And->Ξ＝diag{ξ 1 ，ξ 2 ，…，ξ n }，ξ i Is constant and the matrix xi needs to satisfy the inequality +.> Wherein->And->Respectively are provided withRepresentation matrix->And->P-norm of> p=1, 2 or ≡e 2n Is a 2 n-order identity matrix, E 4n Is a 4 n-order identity matrix-> U (t) is an aperiodic intermittent synchronous controller, U (t) = (U) 1 (t)，U 2 (t)，…，U n (t)) T ，U i The real and imaginary parts of (t) are +.>And->In the above, i=1, 2, …, n; j=1, 2, …, n;
step S2: constructing real and imaginary subsystems of the driving system and the response system; the specific contents of the step S2 are as follows:
performing variable separation on the driving system and the response system established in the step S1, and constructing real part subsystems and imaginary part subsystems of the driving system and the response system respectively as follows:
wherein,,
step S3: setting a synchronization error, and establishing a real part subsystem and an imaginary part subsystem of a synchronization error system;
step S4: an aperiodic intermittent synchronous controller is designed and acts on the response system so that the response system is exponentially synchronous with the driving system.
Further, the step S3 specifically includes the following steps:
step S31: setting the synchronous error of the driving system and the response system as follows:
wherein e 1 The real and imaginary parts of (t) are respectivelyAnd->e 2 The real and imaginary parts of (t) are +.>Andnamely: />
Step S32: according to the driving system and the response system and the synchronization error set in step S31, real part and imaginary part subsystems of the synchronization error system are established, and the real part and imaginary part subsystems are respectively:
step S33: according to the real part and the imaginary part subsystems of the synchronous error system established in the step S32, constructing a synchronous error system in a vector form as follows:
wherein:
further, the step S4 specifically includes the following steps:
step S41: according to step S3, establishing real part and imaginary part subsystems of a synchronous error system, designing an aperiodic intermittent synchronous controller, and realizing real part U R (t) and imaginary part U I (t) are respectively designed as follows:
where r is the number of control cycles, i.e. r=0, 1,2, …; t is t r For the start time of the controller in the (r) th period, s r Indicating the controller stop time in the (r) th period, t r Sum s r The requirements are as follows: delta and->Is constant and satisfies->K 1 、K 2 、Ω 1 、Ω 2 Gain matrix K for non-periodic intermittent synchronous controller 1 ＝diag{k 11 ，k 12 ，…，k 1n }，K 2 ＝diag{k 21 ，k 22 ，…，k 2n }，Ω 1 ＝diag{η 11 ，η 12 ，…，η 1n }，Ω 2 ＝diag{η 21 ，η 22 ，...，η 2n -a }; the parameters of the aperiodic intermittent synchronous controller satisfy the following inequality:
wherein phi is a constant For matrix->Matrix measure of->μ p (H) For matrix measure of matrix H, +.> Respectively as a matrixP-norms of (2); ρ is the exponential convergence rate; ζ is equation-> Is the only positive root of (2);
step S42: and applying a designed non-periodic intermittent synchronous controller to the response system so that the response system is exponentially synchronous with the driving system.
Further, the convergence rate of the exponential synchronization is
The invention provides an index synchronous control method of a complex-valued neural network based on intermittent control, which has the beneficial effects that compared with the prior art, the method has the following steps:
1. the invention uses the matrix measure theory to construct the Lyapunov function, so that the analysis difficulty is reduced, and the conservation of the obtained result is lower.
2. Compared with other periodic intermittent synchronous controllers, the non-periodic intermittent synchronous controller adopted by the invention is more suitable for actual production and application.
3. The invention considers the complex-valued neural network with wider application, and considers the influence of time-varying discrete time lag and time-varying finite distribution time lag, so that the obtained result is more in line with the actual engineering application.
Drawings
FIG. 1 is a flow chart of an exponential synchronization control method of a complex-valued neural network based on intermittent control of the present invention;
FIG. 2 shows a driving system state x without the synchronous controller in embodiment 2 of the present invention 1 The real part of (t)And responsive to system state y 1 (t) real part->Trajectory comparison of (2);
FIG. 3 shows a driving system state x without the synchronous controller in embodiment 2 of the present invention 1 Imaginary part of (t)And responsive to system state y 1 Imaginary part of (t)>Trajectory comparison of (2);
FIG. 4 shows a driving system state x without the synchronous controller in embodiment 2 of the present invention 2 The real part of (t)And responsive to system state y 2 (t) real part->Trajectory comparison of (2);
FIG. 5 shows a driving system state x without the synchronous controller in embodiment 2 of the present invention 2 Imaginary part of (t)And responsive to system state y 2 Imaginary part of (t)>Trajectory comparison of (2);
FIG. 6 shows a synchronization error without the synchronization controller in embodiment 2 of the present invention Is a change trace diagram of (1);
FIG. 7 shows a specific embodiment of the present inventionIn example 2, the synchronization error is not caused by the synchronization controller Is a change trace diagram of (1);
FIG. 8 shows a synchronization error under the action of an aperiodic intermittent synchronous controller according to embodiment 2 of the present invention Is a change trace diagram of (1);
FIG. 9 shows a synchronization error under the action of an aperiodic intermittent synchronous controller according to embodiment 2 of the present invention Is a change trace diagram of (a).
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments.
All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without any inventive effort, are intended to be within the scope of the invention.
Example 1:
as shown in fig. 1, the present embodiment provides an exponential synchronization control method of a complex-valued neural network based on intermittent control, including the following steps:
step S1: establishing a driving system and a response system based on a complex-valued neural network with mixed time-varying time lags; the specific contents of the step S1 are as follows:
the driving system and the response system for establishing the complex-valued neural network with the mixed time-varying time lag are respectively as follows:
wherein, the time t is more than or equal to 0; drive system state x (t) = (x 1 (t)，x 2 (t)，...，x n (t)) T ，z(t)＝(z 1 (t)，z 2 (t)，…，z n (t)) T ，x i The real and imaginary parts of (t) are respectivelyAnd->z i The real and imaginary parts of (t) are +.>And->Response system state y (t) = (y) 1 (t)，y 2 (t)，…，y n (t)) T ，v(t)＝(v 1 (t)，v 2 (t)，…，v n (t)) T ，y i The real and imaginary parts of (t) are +.>And->v i The real and imaginary parts of (t) are +.>And->n represents the number of neurons in the neural network; A. b is a real number connection weight matrix, a=diag { a } 1 ，a 2 ，…，a n }，B＝diag{b 1 ，b 2 ，...，b n }，a i And b i Is a constant; C. d, W is a complex value connection weight matrix, c= (C) ij ) n×n ，D＝(d ij ) n×n ，W＝(w ij ) n×n Complex number c ij 、d ij 、w ij The real parts of (2) are>Complex number c ij 、d ij 、w ij Imaginary parts of (2) are->Activation function f (x (t)) without time lag= (f) 1 (x 1 (t))，f 2 (x 2 (t))，…，f n (x n (t))) T ，f(y(t))＝(f 1 (y 1 (t))，f 2 (y 2 (t))，...，f n (y n (t))) T ，f j (x j The real and imaginary parts of (t)) are +.>And->f j (y j The real and imaginary parts of (t)) are +.>And->Activation function f (x (t- σ (t))) = (f) containing time lags 1 (x 1 (t-σ(t)))，f 2 (x 2 (t-σ(t)))，…，f n (x n (t-σ(t)))) T ，f(y(t-σ(t)))＝(f 1 (y 1 (t-σ(t)))，f 2 (y 2 (t-σ(t)))，…，f n (y n (t-σ(t)))) T ，f j (x j The real and imaginary parts of (t- σ (t))) are respectivelyAnd->f j (y j The real and imaginary parts of (t- σ (t))) are +.>And->Each activation function satisfies the lipschitz condition, namely:
wherein the method comprises the steps ofIs a lipschitz constant; sigma (t) and tau (t) are time-varying discrete time lag and time-varying distributed time lag, respectively, and satisfy 0<σ(t)<σ，0<τ(t)<τ, σ and τ are positive constants and let +.>η is an integral variable; complex-valued neural network external input γ (t) = (γ) 1 (t)，γ 2 (t)，…，γ n (t)) T And gamma is i The real and imaginary parts of (t) are respectivelyAnd->Ξ＝diag{ξ 1 ，ξ 2 ，…，ξ n }，ξ i Is constant and the matrix xi needs to satisfy the inequality Wherein->And->Respectively represent matrix->And->Is used for the p-norm of (c), p=1, 2 or ≡e 2n Is a 2 n-order identity matrix, E 4n Is a 4 n-order identity matrix, u (t) is an aperiodic intermittent synchronous controller, U (t) = (U) 1 (t)，U 2 (t)，...，U n (t)) T ，U i The real and imaginary parts of (t) are +.>And->In the above, i=1, 2, …, n; j=1, 2, …, n;
step S2: constructing real and imaginary subsystems of the driving system and the response system; the specific contents of the step S2 are as follows:
performing variable separation on the driving system and the response system established in the step S1, and constructing real part subsystems and imaginary part subsystems of the driving system and the response system respectively as follows:
wherein,,
step S3: setting a synchronization error, and establishing a real part subsystem and an imaginary part subsystem of a synchronization error system;
step S4: an aperiodic intermittent synchronous controller is designed and acts on the response system so that the response system is exponentially synchronous with the driving system.
In this embodiment, the step S3 specifically includes the following steps:
step S31: setting the synchronous error of the driving system and the response system as follows:
wherein e 1 The real and imaginary parts of (t) are respectivelyAnd->e 2 The real and imaginary parts of (t) are +.>Andnamely:
step S32: according to the driving system and the response system and the synchronization error set in step S31, real part and imaginary part subsystems of the synchronization error system are established, and the real part and imaginary part subsystems are respectively:
step S33: according to the real part and the imaginary part subsystems of the synchronous error system established in the step S32, constructing a synchronous error system in a vector form as follows: 00000
Wherein:
in this embodiment, the step S4 specifically includes the following steps:
step S41: according to step S3, establishing real part and imaginary part subsystems of a synchronous error system, designing an aperiodic intermittent synchronous controller, and realizing real part U R (t) and imaginary part U I (t) are respectively designed as follows:
where r is the number of control cycles, i.e., r=0, 1,2,; t is t r For the start time of the controller in the (r) th period, s r Indicating the controller stop time in the (r) th period, t r Sum s r The requirements are as follows: delta and->Is constant and satisfies->K 1 、K 2 、Ω 1 、Ω 2 Gain matrix K for non-periodic intermittent synchronous controller 1 ＝diag{k 11 ，k 12 ，...，k 1n }，K 2 ＝diag{k 21 ，k 22 ，...，k 2n }，Ω 1 ＝diag{η 11 ，η 12 ，…，η 1n }，Ω 2 ＝diag{η 21 ，η 22 ，…，η 2n -a }; the parameters of the aperiodic intermittent synchronous controller satisfy the following inequality:
wherein phi is a constant For matrix->Matrix measure of->μ p (H) For matrix measure of matrix H, +.> Respectively is a matrix->P-norms of (2); ρ is the exponential convergence rate; ζ is equation-> Is the only positive root of (2);
step S42: and applying a designed non-periodic intermittent synchronous controller to the response system so that the response system is exponentially synchronous with the driving system.
In this embodiment, the convergence rate of the exponential synchronization is
It is worth to say that the invention combines matrix measure theory to construct Lyapunov function, so that the analysis difficulty is reduced, and the conservation of the obtained result is lower. Most of the prior inventions are based on periodic intermittent synchronous controllers, and the non-periodic intermittent synchronous controllers are more suitable for practical production and application than the periodic intermittent synchronous controllers, and the complex-valued neural network is considered, so that the complex-valued neural network can solve the problem of more complexity compared with the traditional real-valued neural network, and meanwhile, time-varying discrete time lag and time-varying distributed time lag are introduced into the complex-valued neural network, so that the complex-valued neural network is more suitable for practical application.
Example 2:
the embodiment mainly comprises two parts of contents:
one is to carry out theoretical demonstration on the effectiveness of an exponential synchronization control method of a complex-valued neural network based on intermittent control as presented in example 1.
Secondly, simulation verification is carried out on the synchronous performance of the complex-valued neural network driving system and the response system with the mixed time-varying time lags, which are constructed in the embodiment 1, by a numerical simulation method.
(neither theoretical demonstration nor simulation experiments are intended to limit the invention, in other embodiments, simulation experiments may not be performed, and other experimental schemes may be used to perform experiments to verify the performance of the neural network system.
1. Proof of theory
The quotation that will be adopted in the certification process is given below:
lemma 1: for any real number x,0y, p>0，The following inequality holds:
(|x|+|y|) p ≤c p (|x| p +|y| p )
and (4) lemma 2: for a non-negative continuous function y (t) over [ - τ, +), and the normal number r 1 ，r 2 ，r 3 ，r 4 And τ, φ (t) is less than or equal to τ, if r is satisfied 1 >r 3 +r 4 τ、ρ＝ζ-γψ>0. And satisfies the following:
then for any t is greater than or equal to 0, there is
Wherein γ=r 1 +r 2 Psi is a constant and 0<ψ<1，ζ>0 is an equationIs the only positive root of (c).
Next, according to the matrix measure theory and the lyapunov stability theory, constructing a lyapunov functional:
V(t)＝||e(t)|| p
wherein e (t) = (e (t), μ (t)) T ；||e(t)|| p Representing the p-norm of vector e (t);
then consider the time t.epsilon.t r ，s r ) In the range, the constructed lyapunov functional is solved for the dily derivative:
taylor expansion is performed on e (t+h), and the following can be obtained:
wherein:
and is also provided with
/>
Discarding an infinitesimal amount o (h) higher than h gives:
from the p-norm definition, it is possible to:
again because: i E (t) I p ≤||e(t)|| p ，||∈(t-σ(t))|| p ≤||e(t-σ(t))|| p The method comprises the steps of carrying out a first treatment on the surface of the And then based on the activation function meeting the lipschitz condition and the primer 1, then:
using the same analytical method, one can obtain:
||g 2 (∈(t))|| p ≤2l I ||e(t)|| p ，
||g 1 (∈(t-σ(t)))|| p ≤2l R ||e(t-σ(t))|| p ，
||g 2 (∈(t-σ(t)))|| p ≤2l I ||e(t-σ(t))|| p ，
then:
it is possible to further obtain:
similarly, it can be demonstrated that when s r ≤t<t r+1 When the method is used, the following steps are included:
according to the parameters of the non-periodic intermittent synchronous controller, the following steps are included: then by means of lemma 2, it is possible to obtain:
wherein ρ is the convergence rate of the exponential synchronization,ζ is equation-> Is the only positive root of (c).
From the above, when t.fwdarw. +.infinity, the synchronization error converges to 0 at an exponential rate, i.e. the response system and the drive system achieve exponential synchronization.
2. Numerical simulation
In this embodiment, taking a driving system and a response system of a two-dimensional complex-valued neural network with mixed time-varying time lags as an example, the driving system and the response system are determined as follows:
wherein: x (t) = (x) 1 (t)，x 2 (t)) T ，z(t)＝(z 1 (t)，z 2 (t)) T ，f(x(t))＝(f 1 (x 1 (t))，f 2 (x 2 (t))) T ，f(x(t-σ(t)))＝(f 1 (x 1 (t-σ(t)))，f 2 (x 2 (t-σ(t)))) T ，γ(t)＝(γ 1 (t)，γ 2 (t)) T ；y(t)＝(y 1 (t)，y 2 (t)) T ，v(t)＝(v 1 (t)，v 2 (t)) T ，f(y(t))＝(f 1 (y 1 (t))，f 2 (y 2 (t))) T ，f(y(t-σ(t)))＝(f 1 (y 1 (t-σ(t)))，f 2 (y 2 (t-σ(t)))) T ，U(t)＝(U 1 (t)，U 2 (t)) T The method comprises the steps of carrying out a first treatment on the surface of the The matrix A, B and connection weight matrix C, D, W parameters are set as follows:
other specific parameters are set as follows: p=1;
/>
γ 1 (t)＝5sin(t)+i(5cos(t))，γ 2 (t) =5 sin (t) +i (5 cos (t)); σ (t) =τ (t) =1.01|sin (t) |; from the following components
The above parameters can be found:l R ＝l I ＝0.05。
furthermore, according to the inequalityThe condition sets the matrix xi to:
in accordance with the above-mentioned parameter settings,inequality of Non-periodic intermittent synchronous controller parameters can be selected: k (k) 11 ＝k 12 ＝k 21 ＝k 22 ＝1，η 11 ＝η 12 ＝η 21 ＝η 22 =5; according to ζ is equation-> And then solve for the unique positive root of: ζ=0.923; the control time sequence of the non-periodic intermittent control is as follows: [0,3.55]∪[3.59,7.4]∪[7.44,11.6]∪[11.65,15.8]∪[15.85,19.4]∪[19.44,23.4]∪[23.45,27.2]∪[27.24,31.1]∪[31.13,35]It can be known that: phi=0.0125, thereby calculating an exponential convergence rate ρ=0.908;
and the driving system, the response system and the aperiodic intermittent synchronous controller carry out numerical simulation experiments on the driving system, the response system and the aperiodic intermittent synchronous controller under the set parameters. The initial values of the drive system and the response system are set as follows: x is x 1 (0)＝1.5-1.5i，x 2 (0)＝-1.5+1.5i，z 1 (0)＝-1.5+1.5i，z 2 (0)＝-1.5+1.4i，y 1 (0)＝-0.5+0.5i，y 2 (0)＝0.5-0.4i，v 1 (0)＝1.5-1.5i，v 2 (0) =1.5-1.5 i; the specific simulation experiment results are as follows: FIG. 2 shows the driving system state x without the synchronous controller 1 The real part of (t)And responsive to system state y 1 (t) real part->Trajectory comparison of (2); FIG. 3 shows the state x of the driving system without the synchronous controller 1 Imaginary part of (t)>And responsive to system state y 1 Imaginary part of (t)>Trajectory comparison of (2); FIG. 4 shows the driving system state x without the synchronous controller 2 (t) real part->And responsive to system state y 2 (t) real part->Trajectory comparison of (2); FIG. 5 shows the driving system state x without the synchronous controller 2 Imaginary part of (t)>And responsive to system state y 2 Imaginary part of (t)>Trajectory comparison of (2); FIG. 6 shows synchronization error +.>Is a change trace diagram of (1); FIG. 7 shows synchronization error +.>Is a change trace diagram of (1); FIG. 8 shows synchronization error under the action of an aperiodic intermittent synchronous controller>Is a change trace diagram of (1); FIG. 9 shows synchronization error under the action of an aperiodic intermittent synchronous controller>Is a change trace diagram of (1); FIGS. 2-7 illustrate lack of synchronization controlUnder the action of the device, the driving system and the response system can not reach a synchronous state by themselves; while figures 8 and 9 show that the response system is synchronized to the drive system under the influence of the non-periodic intermittent synchronous controller, thereby verifying the correctness of the design of the non-periodic intermittent synchronous controller and the synchronization performance of the drive system and the response system.
Finally, it should be noted that: the foregoing is merely a preferred example of the present invention, and the present invention is not limited thereto, but it is to be understood that modifications and equivalents of some of the technical features described in the foregoing embodiments may be made by those skilled in the art, although the present invention has been described in detail with reference to the foregoing embodiments. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (4)

1. An exponential synchronization control method of a complex-valued neural network based on intermittent control is characterized by comprising the following steps:
step S1: establishing a driving system and a response system based on a complex-valued neural network with mixed time-varying time lags; the specific contents of the step S1 are as follows:
the driving system and the response system for establishing the complex-valued neural network with the mixed time-varying time lag are respectively as follows:
wherein, the time t is more than or equal to 0; drive system state x (t) = (x 1 (t)，x 2 (t)，…，x n (t)) T ，z(t)＝(z 1 (t)，z 2 (t)，…，z n (t)) T ，x i The real and imaginary parts of (t) are respectivelyAnd->z i The real and imaginary parts of (t) are +.>Andresponse system state y (t) = (y) 1 (t)，y 2 (t)，…，y n (t)) T ，v(t)＝(v 1 (t)，v 2 (t)，…，v n (t)) T ，y i The real and imaginary parts of (t) are +.>And->v i The real and imaginary parts of (t) are +.>And->n represents the number of neurons in the neural network; A. b is a real number connection weight matrix, a=diag { a } 1 ，a 2 ，…，a n }，B＝diag{b 1 ，b 2 ，…，b n }，a i And b i Is a constant; C. d, W is a complex value connection weight matrix, c= (C) ij ) n×n ，D＝(d ij ) n×n ，W＝(w ij ) n×n Complex number c ij 、d ij 、w ij The real parts of (2) are>Complex number c ij 、d ij 、w ij Imaginary parts of (2) are-> Activation function f (x (t)) without time lag= (f) 1 (x 1 (t))，f 2 (x 2 (t))，…，f n (x n (t))) T ，f(y(t))＝(f 1 (y 1 (t))，f 2 (y 2 (t))，…，f n (y n (t))) T ，f j (x j The real and imaginary parts of (t)) are +.>And->f j (y j The real and imaginary parts of (t)) are +.>And->Activation function f (x (t- σ (t))) = (f) containing time lags 1 (x 1 (t-σ(t)))，f 2 (x 2 (t-σ(t)))，…，f n (x n (t-σ(t)))) T ，f(y(t-σ(t)))＝(f 1 (y 1 (t-σ(t)))，f 2 (y 2 (t-σ(t)))，…，f n (y n (t-σ(t)))) T ，f j (x j The real and imaginary parts of (t- σ (t))) are respectivelyAnd->f j (y j The real and imaginary parts of (t- σ (t))) are +.>And-> Each activation function satisfies the lipschitz condition, namely:
wherein the method comprises the steps ofIs a lipschitz constant; sigma (t) and tau (t) are time-varying discrete time lag and time-varying distributed time lag, respectively, and satisfy 0<σ(t)<σ，0<τ(t)<τ, σ and τ are positive constants and let +.>η is an integral variable; complex-valued neural network external input γ (t) = (γ) 1 (t)，γ 2 (t)，…，γ n (t)) T And gamma is i The real and imaginary parts of (t) are +.>And->Ξ＝diag{ξ 1 ，ξ 2 ，…，ξ n }，ξ i Is constant and the matrix xi needs to satisfy the inequality +.>Wherein->And->Respectively represent matrix->Andp-norm of>p=1, 2 or ≡e 2n Is a 2 n-order identity matrix, E 4n Is a 4 n-order identity matrix-> U (t) is an aperiodic intermittent synchronous controller, U (t) = (U) 1 (t)，U 2 (t)，…，U n (t)) T ，U i The real and imaginary parts of (t) are +.>And->In the above, i=1, 2, …, n; j=1, 2, …, n;
step S2: constructing real and imaginary subsystems of the driving system and the response system; the specific contents of the step S2 are as follows:
performing variable separation on the driving system and the response system established in the step S1, and constructing real part subsystems and imaginary part subsystems of the driving system and the response system respectively as follows:
wherein,,
step S3: setting a synchronization error, and establishing a real part subsystem and an imaginary part subsystem of a synchronization error system;
step S4: an aperiodic intermittent synchronous controller is designed and acts on the response system so that the response system is exponentially synchronous with the driving system.
2. The method for exponential synchronization control of a complex-valued neural network based on intermittent control according to claim 1, wherein step S3 specifically comprises the steps of:
step S31: setting the synchronous error of the driving system and the response system as follows:
wherein e 1 The real and imaginary parts of (t) are respectivelyAnd->e 2 The real and imaginary parts of (t) are respectivelyAnd->Namely: />
Step S32: according to the driving system and the response system and the synchronization error set in step S31, real part and imaginary part subsystems of the synchronization error system are established, and the real part and imaginary part subsystems are respectively:
step S33: according to the real part and the imaginary part subsystems of the synchronous error system established in the step S32, constructing a synchronous error system in a vector form as follows:
wherein:
3. the method for controlling the exponential synchronization of the complex-valued neural network based on intermittent control according to claim 2, wherein the step S4 specifically comprises the steps of:
step S41: according to step S3, establishing real part and imaginary part subsystems of a synchronous error system, designing an aperiodic intermittent synchronous controller, and realizing real part U R (t) and imaginary part U I (t) are respectively designed as follows:
where r is the number of control cycles, i.e. r=0, 1,2, …; t is t r For the start time of the controller in the (r) th period, s r Indicating the controller stop time in the (r) th period, t r Sum s r The requirements are as follows: delta sumIs constant and satisfies->K 1 、K 2 、Ω 1 、Ω 2 Gain matrix K for non-periodic intermittent synchronous controller 1 ＝diag{k 11 ，k 12 ，…，k 1n }，K 2 ＝diag{k 21 ，k 22 ，…，k 2n }，Ω 1 ＝diag{η 11 ，η 12 ，…，η 1n }，Ω 2 ＝diag{η 21 ，η 22 ，…，η 2n -a }; the parameters of the aperiodic intermittent synchronous controller satisfy the following inequality:
wherein phi is a constant For matrix->Matrix measure of-> μ p (H) For matrix measure of matrix H, +.> Respectively is a matrix->P-norms of (2); ρ is the exponential convergence rate; ζ is equation-> Is the only positive root of (2);
step S42: and applying a designed non-periodic intermittent synchronous controller to the response system so that the response system is exponentially synchronous with the driving system.
4. The method for controlling exponential synchronization of a complex-valued neural network based on intermittent control according to claim 3, wherein the convergence rate of the exponential synchronization is
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