Projection quasi-synchronization method of complex valued recurrent neural network based on matrix measurement

Abstract

The invention relates to a projection quasi-synchronization method of a complex valued recurrent neural network (CVRNN) based on matrix measurement, which is based on a Lyapunov stability criterion and adopts a nonlinear Lipschitz activation function and a matrix measurement method. A controller is reasonably designed, a matrix measurement method is properly described, and sufficient conditions of projection synchronization criteria of the non-isomorphic CVRNN model are provided. For the CVRNN with mismatching parameters and proportion delay, an important theoretical result is obtained. And finally, verifying the effectiveness and durability of a theoretical result through numerical simulation, and displaying the results under different specific conditions by using a graph. The method provided by the invention can be applied to the actual fields of signal precision processing, medical image signal accuracy analysis and the like in wireless communication, and has effectiveness and potential in theoretical and actual application.

Description

Technical Field

[0001] The present invention relates to complex-valued recurrent neural networks (CVRNNs), and in particular to a method for realizing projection quasi-synchronization between CVRNNs with proportional delay and parameter mismatch, and belongs to the field of controller technology. Background Art

[0002] In the past few decades, complex-valued neural networks (CVNNs) have attracted great interest from engineers, scientists, and researchers due to their wide applications in many fields such as optoelectronics, filtering, quantum devices, computer vision, remote sensing, speech synthesis, etc. CVNNs are better than real-valued neural networks (RVNNs) in solving various real-world problems. For example, the well-known XOR problem can be handled by CVNNs, but RVNNs cannot. Complex-valued neurons have more powerful processing power and higher information processing capabilities. The research of Amin and Murase proved that a single-layer CVNN can show better performance than a multi-layer RVNN. Compared with the RVNN model, some problems can be solved in the CVRNN model, but cannot be solved in the RVNN model. Usually, in the RVNN model, the activation function is bounded, differentiable, and non-constant. However, this does not apply to CVNNs because the Liouville theorem states that every bounded and complete function must be a constant in the complex domain. This theory shows that if an analytical function in the complex domain is used, it is a constant on the entire complex plane and is therefore not suitable for the CVRNN model. Therefore, the main challenge facing the CVRNN model is to design the activation function.

[0003] In the literature, proportional delay is defined as an unbounded time-varying delay, which is less conservative and monotonically increasing than bounded discrete delay. This delay plays an important role in many fields such as control, physical and biological networks. Proportional delay is widely present in the real world, such as in routing decisions and web service quality. Researchers have begun to focus on the dynamic characteristics of neural networks with proportional delay terms. Most studies on neural network synchronization involve asymptotic proportional delay terms, and many researchers have proposed synchronization schemes that assume that the drive-response system is the same in CVNNs with or without time delays. However, in practical applications, there is often a mismatch between the drive system and the response system, which makes the synchronization error usually not asymptotically tend to zero. Most synchronization results are mainly based on the Lyapunov functional method, while the matrix measure method, as an alternative method, offers a wider range of potential advantages. The matrix measure considers both the positive and negative components of the connection weight matrix, which correspond to the excitation and inhibition of neurons, respectively. Although most synchronization results of CVNNs assume that the drive and response systems are the same, in practical applications, the frequent mismatch between the drive system and the response system will destroy the synchronization state, limiting the effective application of the synchronization process.

[0004] So far, various results have been achieved on quasi-projection and projection synchronization of traditional RVNN and fractional-order neural networks, mainly based on the Lyapunov function method. Another method, the matrix measurement method, has obvious advantages because it considers both the positive and negative components of the connection weight matrix. There are also some research results on quasi-synchronization and projection synchronization. In recent years, CVNN has attracted widespread attention due to its superior data representation ability to RVNN, and related research has continued to increase. However, research based on quasi-projection and quasi-projection synchronization is still relatively rare. In particular, in the projection synchronization of complex-valued neural networks with mismatched parameters, relevant research results are even more scarce. This is because the simultaneous existence of projection coefficients and parameter mismatches makes the error system more complicated, making theoretical derivation more difficult. Summary of the invention

[0005] The core purpose of the present invention is to solve the problem of quasi-synchronization of projection of complex-valued recurrent neural networks (CVRNNs) under the conditions of proportional delay and parameter mismatch. By adjusting the controller parameters, the method can efficiently control the dynamic response of CVRNNs and ensure that the network operates within the predetermined performance indicators. The method not only improves the synchronization accuracy, but also enhances the stability of the system. The present invention also has a broad application space in practical fields. For example, in the field of wireless communications, signals are often transmitted in a complex electromagnetic environment and are easily affected by interference and noise. The projection quasi-synchronization method of the complex-valued recurrent neural network (CVRNN) based on matrix measurement in the present invention can be used to identify and eliminate interference caused by multipath effects, improve signal demodulation accuracy and communication reliability. In addition, in the field of medical image signal analysis, the CVRNN projection quasi-synchronization method based on matrix measurement in the present invention can be applied to the processing of magnetic resonance imaging (MRI) or computed tomography (CT) image signals to improve the resolution and contrast of the image. By synchronously processing image data from different angles, the method helps to improve the accuracy of image reconstruction, thereby better identifying and locating the lesion area. Therefore, the present invention is of great significance in the practical application of improving signal accuracy and reliability. In the case of mismatched parameters and proportional delays, the proposed CVRNN system is able to achieve projection synchronization, that is, signal coordination or synchronization under complex environmental interference.

[0006] To achieve the above object, the technical solution specifically adopted by the present invention is: a projection quasi-synchronous method of a complex-valued recurrent neural network based on matrix measurement, which mainly includes the following steps:

[0007] Step 1: Considering the impact of time delay, establish a complex-valued recurrent neural network (CVRNNs) model, introduce state update rules, and consider the impact of time delay: Establish the following CVRNN model

[0008]

[0009] In the formula, γ(t)=(γ 1 (t), γ 2 (t),...,γ n (t)) T ∈C n represents the state of n neurons at time t, R = diag (r 1 , r 2 ,...,r n )∈R n×n , and r n > 0 indicates the self-inhibition of the nth neuron. P∈C n×n and Q∈C n×n They represent the connection weight matrix respectively. The external input vector is represented by S(t)=(s 1 (t), s 2 (t),...,s n (t)) T It means that g(γ(t))=(g 1 (γ 1 (t)), g 2 (γ 2 (t)),...,g n (γ n (t))) T :C n →C n represents the activation function without time delay, and h(γ(qt))=(h 1 (γ 1 (qt)), h 2 (γ 2 (qt)),...,h n (γ n (qt))) T :C n →C n represents an activation function with proportional delay, the proportional delay term is q∈(0,1), and qt=t-τ(t), where τ(t)=(1-q)t is a continuous time-varying function that satisfies τ(t)→∞ when t→+∞.

[0010] The initial condition of the system is given as γ(s) = φ(s). 0 -τ,t 0 ], where φ(s)=(φ 1 (s), φ 2 (s),...,φ n (s))∈C n , and the real part of φ(s) φ R (s) and the imaginary part φI (s) in the interval [t 0 -τ,t 0 ] is continuous. Assume that γ(t)=α(t)+iβ(t), where α(t), β(t)∈R n , then the real and imaginary parts of the drive system are:

[0011]

[0012] Its initial condition is

[0013]

[0014] In the formula and Respectively represent the norm of the real and imaginary parts of φ(s). Corresponding to the driving system, the response system is constructed as:

[0015]

[0016] Where D(t)=(D 1 (t), D 2 (t),...,D n (t)) T ∈C n represents coupling control, represents the state of n neurons at time t, R′=diag(r′ 1 , r′ 2 ,...,r′ n )∈R n×n , and r′ n > 0 indicates the self-inhibition of the nth neuron. P′ and Q′∈C n×n represents the weight matrix with and without delay. S′(t)=(s′ 1 (t), s′ 2 (t),...,s′ n (t)) T represents the external input vector. The initial condition of the response system is:

[0017]

[0018] In the formula and is in the interval [t 0 -τ,t 0 ] is a continuous function on . The real and imaginary parts of the above response system are given as follows:

[0019]

[0020] And the initial conditions are:

[0021]

[0022] Step 2: Based on the system model and control objectives in step 1, design the mathematical expression of the controller and error function: Consider the controller as

[0023] D R (t) = Ωθ R (t)

[0024] D I (t) = Ωθ I (t),

[0025] Where D R (t) and D I (t) is the control input vector, and

[0026]

[0027] The control gain matrix is ​​defined as Ω∈R n×n .

[0028] Assumption 1. Let w = u + iv, where v ∈ R. l (w) and g 1 (w) is described as

[0029]

[0030] where l = 1, 2, ..., n, and Satisfies the Lipschitz condition:

[0031]

[0032] where r i ,s i , m l and n 1 is the Lipschitz constant, v and η∈Rn.

[0033] In addition, the error function expressions of the driving system and the response system designed in step S2 are:

[0034]

[0035]

[0036] If θ R (t)→0 and θ I (t)→0When t→∞, the drive system and the response system will be synchronized. The error system of the controller is expressed as:

[0037]

[0038] In the formula

[0039] and

[0040]

[0041] Step 3: Construct Lyapunov function to analyze the system projection synchronization, use matrix measurement method to derive the upper bound of system error dynamics, and construct Lyapunov function as

[0042] V 1 (θ(t))=||θ R (t)|| w +||θ I (t)| | w.

[0043] Where V 1 (θ(t)) is the upper right edge error system, and its Dini derivative is

[0044]

[0045] Assumption 1. Let w = u + iv, where v ∈ R. l (w) and g l (w) is described as

[0046]

[0047] where l = 1, 2, ..., n, and Satisfies the Lipschitz condition:

[0048]

[0049] where r i ,s i , m l and n l is the Lipschitz constant, v and η∈R n .

[0050] According to hypothesis 1, we can get

[0051]

[0052] Likewise,

[0053]

[0054] Definition 1. Matrix N = (N lm )n×n The matrix metric is defined as

[0055]

[0056] Where N is a real n×n matrix and I is the identity matrix. The matrix norm is defined as

[0057]

[0058] The matrix metric is defined as

[0059]

[0060] Using Definition 1, D + V 1 (θ(t)) becomes:

[0061]

[0062] set up

[0063] k 1 =-{μw ( -R′+Ω)+(r k +s k )(||P′ R || w +||P′ I || w )},

[0064] k 2 =(m k +n k )(||Q′ R || w +||Q′ I || w ),

[0065] ζ=ζ 1 +ζ 2 .

[0066] Then according to hypothesis 1, we know that 0 <k 2 <k 1 , that is, the condition for the existence of the controller gain matrix Ω is obtained. Therefore,

[0067]

[0068] The solution δ is calculated as

[0069] δ=k 1 -k 2 e δτ

[0070] =-{μ w(-R′+Ω)+(r k +s k )(||P′ R || w +||P′ I || w )}-(m k +n k )(||Q′ R || w +||Q′O|| w ) δτ .

[0071] Therefore, the upper bound of the error θ(t) is

[0072] Therefore, the above derivation shows that the difference between the system states will decrease over time, thus achieving synchronization. It can be concluded that the proposed drive system and response system will achieve quasi-projective synchronization in the presence of the controller.

[0073] An electronic device comprises a memory, a processor and a computer program stored in the memory and executable on the processor. When the processor executes the program, the projection quasi-synchronization method of a complex-valued recurrent neural network based on matrix measurement is implemented.

[0074] A computer-readable storage medium stores computer instructions, which, when executed by a processor, implement the projection quasi-synchronization method of a complex-valued recurrent neural network based on matrix measurement.

[0075] Compared with the prior art, the present invention has the following advantages:

[0076] 1. The present invention is directed to complex-valued recurrent neural networks (CVRNNs), which can more effectively process complex-valued data containing real and imaginary parts than conventional real-valued neural networks (RVNNs). In the field of signal processing, such as wireless communications and radar systems, this capability enables the present invention to excel in identifying and separating complex-valued signals, especially in environments with severe multipath effects and noise interference, improving the accuracy and robustness of signal processing.

[0077] 2. In the face of the time-varying proportional delay and parameter mismatch problems commonly encountered in practical applications, the present invention provides an innovative projection quasi-synchronization method. In fields such as financial risk assessment and bioinformatics, the dynamic characteristics of the network and the uncertainty of parameters pose challenges to the synchronization performance of the system. The present invention ensures stable synchronization of the network under these complex conditions through controller design, thereby providing reliable data support for decision-making.

[0078] 3. The CVRNN model and synchronization strategy of the present invention are not limited to a specific network structure or application scenario, but have good versatility and scalability. In practical applications, this means that the system can be quickly adjusted and optimized according to different task requirements without large-scale hardware upgrades or redesigns. This flexibility gives the present invention a significant advantage in a rapidly changing technological environment. BRIEF DESCRIPTION OF THE DRAWINGS

[0079] Figure 1 is the state trajectory diagram of the system in three-dimensional space, where (a) α 1 (t), β 1 (t), (b)α 2 (t), β 2 (t); Figure 2 is the state waveform of the system at time t in Case 1, where (a) (b)

[0080] Figure 3 is the state waveform of the system at time t in Case 1, where (a) (b)

[0081] Figure 4 The waveform diagram of the system in Case 1, where (a) there is no controller and (b) there is a controller;

[0082] Figure 5 is the state waveform of the system at time t in Case 1, where (a) (b)

[0083] Figure 6 is the state waveform of the system at time t in Case 2, where (a) (b)

[0084] Figure 7 The waveform diagram of the system in Case 2, where (a) there is no controller and (b) there is a controller;

[0085] Figure 8 The waveform diagram of the system in Case 3, where (a) is without controller and (b) is with controller. DETAILED DESCRIPTION

[0086] The following describes the implementation of the present invention in conjunction with the accompanying drawings. The present invention is verified by Matlab simulation. An example including three cases is given to demonstrate the effectiveness and feasibility of the proposed projection quasi-synchronous scheme.

[0087] Example 1. Consider a complex-valued recurrent neural network (CVRNN) under proportional delay and parameter mismatch, which has the form:

[0088]

[0089] where the parameters are as follows:

[0090]

[0091] q = 0.1, τ(t) = 0.9exp(t).

[0092] The activation function is taken as:

[0093]

[0094] Figure 1 The three-dimensional trajectory plot of the system (1) at time t under parameter mismatch and proportional delay is shown. The real and imaginary parts of the trajectory exhibit chaotic behavior. The corresponding response system is:

[0095]

[0096] where the parameters are as follows:

[0097]

[0098] where represents the controller matrix. Under these parameters, Assumption 1 holds. Then the other parameters of the system are r k = 0.5, s k = 0.25, m k = 0.707 and n k = 1.414. The initial conditions are set as w 1 (s) = -0.15 + 0.45i, w 2 (s) = -0.25 + 0.15i,

[0099] Case 1. When 0 < r < 1 ≠ 1, assume r = 0.2. One can find k 1 = 110.8352k 2 = 3.2334, that is, k 1 > k 2 , ζ = 7.6532. Thus, all the conditions of Theorem 1 are satisfied. Therefore, the system (1) will achieve quasi-projective synchronization with the system (2) with an estimated error level of 0.2481.

[0100] Figure 2 and Figure 3The state evaluation curve of the system over time t under the controller with mismatched parameters is shown. It can be seen that the state variables of the driving system and the response system cannot achieve the same behavior over time, so there is always a difference. In other words, the system cannot achieve complete synchronization. Figure 4 (a) depicts the evolution of the synchronization error without a controller, indicating that quasi-projective synchronization is not achieved. Figure 4 (b) shows the quasi-projective synchronization under the action of the controller, with an error limit of 0.2481.

[0101] Case 2. When r = 1, it can be verified that k l =110.8352, k 2 =3.2334, that is, k 1 >k 2 , ζ=38.2662. Therefore, all the conditions of Theorem 1 are satisfied here. Therefore, the driving system (1) and the response system (2) will achieve quasi-synchronization, and the estimated error level is 1.2409.

[0102] Figure 5 and Figure 6 The state evaluation curves of systems (1) and (2) over time t under the controller with mismatched parameters are shown in the case of r = 1. It can be seen that the state variables of the driving system and the response system cannot achieve the same behavior over time, and there are always some minor differences, that is, the system cannot achieve complete synchronization. Figure 7 (a) shows the evaluation curve of synchronization error without a controller, indicating that quasi-synchronization is not achieved. Figure 7 (b) shows the quasi-synchronization under the controller considering the proportional delay and the error system with mismatched parameters, and the error bound is 1.2409.

[0103] Case 3. If r>1, or more precisely r=3.5, we can calculate k1=110.8352, k2=3.2334, i.e. k1>k2, and ζ=133.9316. This shows that all the conditions of Theorem 1 are satisfied. Therefore, the drive system (1) and the response system (2) will achieve quasi-synchronization, with an estimated error bound of 4.1925. Figure 8 (a) shows the evaluation curve of synchronization error without a controller, indicating that quasi-synchronization is not achieved. Figure 8 (b) shows the quasi-synchronization under the controller with the error system considering proportional delay and mismatched parameters, and the error bound is 4.1925.

[0104] It should be noted that the above embodiments are not intended to limit the protection scope of the present invention, and equivalent changes or substitutions made on the basis of the above technical solutions all fall within the protection scope of the claims of the present invention.

Claims

1. A projection quasi-synchronous method for complex-valued recurrent neural networks based on matrix measurement, characterized in that: The method comprises the following steps: S1: Considering the influence of time delay, a complex-valued recurrent neural network (CVRNNs) model is established, the driving system and the response system are constructed, and the state update rules are introduced; S2: Design the controller and the mathematical expressions of the error function and error system according to the constructed drive system and response system; S3: Construct Lyapunov function to analyze the system projection synchronization, introduce the conditions of control gain matrix, and use matrix measurement method to derive the upper bound of system error dynamics.

2. According to the projection quasi-synchronous method of complex-valued recurrent neural network based on matrix measurement in claim 1, in step S1, the established complex-valued recurrent neural network (CVRNNs) driving system model is In the formula, γ(t)=(γ1(t), γ2(t),…,γ n (t)) T ∈C n represents the state of n neurons at time t, R = diag (r1, r2, ..., r n )∈R n×n , and r n > 0 indicates the self-inhibition of the nth neuron, P∈C n×n and Q∈C n×n They represent the connection weight matrix respectively, and the external input vector is s(t)=(s1(t),s2(t),…,s n (t)) T means, g(γ(t))=(g1(γ1(t)),g2(γ2(t)),…,g n (γ n (t))) T :C n →C n represents the activation function without time delay, and h(γ(qt))=(h1(γ1(qt)),h2(γ2(qt)),…,h n (γ n (qt))) T :C n →C n represents an activation function with proportional delay, the proportional delay term is q∈(0,1), and qt=t-τ(t), where τ(t)=(1-q)t is a continuous time-varying function that satisfies τ(t)→∞ when t→+∞, The initial condition of the system is given as γ(s) = φ(s), on the interval [t0-τ, t0], where φ(s) = (φ1(s), φ2(s), …, φ n (s))∈C n , and the real part of φ(s) φ R (s) and the imaginary part φ I (s) is continuous on the interval [t0-τ,t0], assuming that γ(t)=α(t)+iβ(t), where α(t),β(t)∈R n , then the real and imaginary parts of the drive system are: Its initial condition is In the formula and Respectively represent the norms of the real and imaginary parts of φ(s), corresponding to the driving system, the response system is constructed as: Where D(t)=(D1(t),D2(t),…,D n (t)) T ∈C n represents coupling control, represents the state of n neurons at time t, R′=diag(r′1,r′2,…,r′ n )∈R n×n , and r′ n > 0 indicates the self-inhibition of the nth neuron, P′ and Q′∈C n×n represents the weight matrix with and without delay, S′(t)=(s′1(t),s′2(t),…,s′ n (t)) T represents the external input vector, and the initial condition of the response system is: In the formula and is a continuous function on the interval [t0-τ, t0], and the real and imaginary parts of the above response system are given as follows: And the initial conditions are:

3. According to the projection quasi-synchronization method of complex-valued recurrent neural network based on matrix measurement according to claim 1, the mathematical expression of the controller designed in step S2 is: D R (t)=Ωθ R (t) D I (t)=Ωθ I (t), Where D R (t) and D I (t) is the control input vector, and The control gain matrix is ​​defined as Ω∈R n×n ,In addition, the error function expression of the driving system and the response system designed in step S2 is: If θ R (t)→0 and θ I (t)→0When t→∞, the drive system and the response system will be synchronized, and the error system of the controller can be expressed as: In the formula and H(t)=r(R-R′)α(t)+r(P ′R -P R )g R (γ(t))+r(P I -P ′I )g I (γ(t))+r(Q ′R -Q R )h R (γ(qt)) +r(Q I -Q ′I )h I (γ(qt))+S ′R (t)-rS ′R (t), H′(t)=r(R-R′)β(t)+r(P ′R -P R )g I (γ(t))+r(P ′I -P I )g R (γ(t))+r(Q ′R -Q R )h I (γ(qt)) +r(Q ′I -Q I )h R (γ(qt))+S ′I (t)-rS ′I (t). Next, the Lyapunov function is established to analyze the projection synchronization of the system, and the upper bound of the system error dynamics is derived using the matrix metric method. The synchronization error is calculated and its variation with time is determined, the convergence of the error bound is proved, and the conditions for the control gain matrix are introduced. It is proved that the proposed CVRNN system can achieve projection synchronization in the case of mismatched parameters and proportional delay.

4. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the processor executes the program, the projection quasi-synchronous method of the complex-valued recurrent neural network based on matrix measurement as described in any one of claims 1 to 3 above is implemented.

5. A computer-readable storage medium having computer instructions stored thereon, characterized in that: When the computer instructions are executed by a processor, a projection quasi-synchronous method for a complex-valued recurrent neural network based on matrix measurement as described in any one of claims 1 to 3 is implemented.

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