Cohen-Grossberg neural network construction method with quantitative control input

Abstract

The invention discloses a Cohen-Grossberg neural network construction method with quantitative control input, and the method comprises the following steps: 1, constructing a neural network model which comprises a driving system and a response system corresponding to the driving system; step 2, constructing a quantitative feedback controller of the response system; step 3, constructing definitions and ledges required for synchronous analysis of finite time and fixed time of the driving system and the response system; and step 4, carrying out finite time synchronization and fixed time synchronization analysis on the Cohen-Grossberg neural network with quantitative control input. According to the Cohen-Grossberg neural network construction method with quantitative control input, the theories of differential equation stability, finite time stability, quantitative control and the like are combined, and index synchronization, finite time synchronization and fixed time synchronization of the Cohen-Grossberg neural network under the same quantitative control strategy are researched.

Description

Technical Field

[0001] This invention belongs to the field of electronic information technology, and specifically relates to a method for constructing a Cohen-Grossberg neural network with quantized control input. Background Technology

[0002] Artificial Neural Networks (NNs) are complex systems comprised of a large number of interconnected, simple processing units similar to neurons. They are abstract simulations of the nervous system rather than realistic descriptions. Research on neural networks began in the early 1940s and has spanned over half a century. Since the 1980s, neural networks have been a hot topic in artificial intelligence research. NNs have wide-ranging practical applications, offering significant advantages in signal processing, still image processing, and pattern recognition. The Cohen-Grossberg Neural Network (CGNNs) model, first proposed by Cohen and Grossberg in 1983, incorporates many common biological and neuroscience models. Many well-known neural networks, such as Hopfield neural networks, cellular neural networks (CNNs), and bidirectional associative memory neural networks, can be considered special cases of CGNNs. CGNNs are widely used in many scientific and engineering fields and have strong practical value.

[0003] Synchronization, as one of the main dynamic behaviors of neural networks, has gradually become a research hotspot and focus of attention for scholars both domestically and internationally in recent years. Synchronization generally refers to the process by which two or more coupled systems triggered by different initial values ​​eventually reach the same motion state over time. To enable systems to achieve synchronization quickly, scholars have introduced finite-time control methods into the study of system synchronization. Finite-time control systems, with their good robustness and high anti-interference ability, can reduce costs and improve work efficiency in practical applications, and are now widely used in finance, aviation, communications, and other fields. In finite-time synchronization analysis, the key problem is how to effectively estimate the synchronization resting time and find its upper bound. It is worth noting that the estimation of the resting time in finite-time synchronization depends on the initial value of the system. In practical engineering applications, the initial value of the system is difficult to obtain in advance or even impossible to obtain, which brings many inconveniences to the estimation of the synchronization time. To this end, Polyakov proposed the concept of finite-time stability independent of the initial value—fixed-time stability—providing a theoretical basis for later scholars to study fixed-time synchronization.

[0004] Subsequently, with the advent of the information age, many networked control systems are implemented remotely through network channels with limited bandwidth; these are known as remote control systems. In such systems, communication links are typically shared by multiple different applications. Therefore, when designing a controller, the system's control performance and signal transmission constraints should be considered together. Signal quantization—a technique to reduce the burden of signal transmission—is the process of mapping an infinite set of continuous signals to a finite set of discrete signals. Therefore, during signal transmission, only the already discretized values ​​need to be encoded and transmitted, which greatly reduces the burden of signal transmission and improves transmission efficiency. Common quantizers include static quantizers and dynamic quantizers. Among the two, a quantizer that satisfies the bounded region condition is a static quantizer, and logarithmic quantizers belong to this category. Due to their simple design method and easily determined model, they have good practical value.

[0005] As research on synchronization deepens, different types of synchronization have been proposed, such as complete synchronization, projective synchronization, cluster synchronization, generalized synchronization, lag synchronization, and phase synchronization. In recent years, the synchronization control of CGNNs has attracted much attention due to its advantages in pattern recognition, associative memory, parallel computing, and optimization computation. CGNNs can achieve various results of exponential synchronization, lag synchronization, complete synchronization, and anti-synchronization under different control methods (such as adaptive control, feedback control, and discontinuous control). However, most existing results on neural network synchronization, including those mentioned above, are actually asymptotic results, meaning that synchronization is only achieved when time approaches infinity. In practical applications, systems usually need to achieve synchronization as quickly as possible within a finite time, i.e., finite-time synchronization.

[0006] Finite-time synchronization has important applications in many practical physical and engineering systems, such as secure communication, rapid observation and estimation, fast adaptation algorithms, and fast consensus protocols. Therefore, finite-time synchronization, as a time-optimal control method, has attracted increasing attention from scholars. For example, Jiang, Wang, and Mei et al. explored finite-time synchronization of memristor recurrent neural networks based on the Lyapunov method and the differential inclusion principle. Velmurugan, Rakkiyappan, and Cao studied how coupled neural networks can achieve finite-time synchronization under discontinuous feedback control strategies. Shen and Cao derived sufficient conditions for achieving finite-time synchronization of coupled neural networks through discontinuous controllers.

[0007] However, the initial values ​​of some real-world networks are difficult to obtain. Fixed-time synchronization not only guarantees synchronization within a finite time but also ensures that the synchronization time is independent of the initial values. Therefore, in certain specific situations, fixed-time synchronization is more practical than finite-time synchronization. Polyakov first studied fixed-time stability based on nonlinear feedback control. Wang, Zeng, and Hu et al. obtained sufficient conditions for achieving fixed-time stability and synchronization using adaptive control and state feedback control. Cao and Li used Lyapunov functions and inequality analysis techniques to study how delayed memristor recurrent neural networks can achieve fixed-time synchronization. Hua, Li, and Guan designed a new discrete finite-time controller and a fixed-time controller and derived some criteria for achieving finite-time stability and fixed-time stability. Wan, Cao, and Wen et al. discussed fixed-time synchronization of parameter uncertainties and time-varying delay CGNNs using Filippov discontinuity theory and discontinuity control criteria. Li, Jiang, and Hu studied multiple synchronization analyses of time-varying delay discontinuous CGNNs, including finite-time synchronization and fixed-time synchronization.

[0008] Based on the above research, it is clear that current research on finite-time synchronization and fixed-time synchronization of Cohen-Grossberg neural networks is relatively limited and requires further investigation. Secondly, as a special type of finite-time synchronization, fixed-time synchronization is rarely discussed in conjunction with fixed-time synchronization under the same control strategy. Furthermore, existing finite-time control methods rarely consider the impact of signal quantization. Summary of the Invention

[0009] To address the aforementioned technical problems, this invention provides a method for constructing a Cohen-Grossberg neural network with quantized control input. Combining theories of differential equation stability, finite-time stability, and quantized control, it studies exponential synchronization, finite-time synchronization, and fixed-time synchronization of the Cohen-Grossberg neural network under the same quantized control strategy.

[0010] The objective of this invention is achieved through the following technical solution: a method for constructing a Cohen-Grossberg neural network with quantized control input, comprising the following steps:

[0011] Step 1: Construct a neural network model, including a driving system and a corresponding response system;

[0012] Step 2: Construct a quantized feedback controller for the response system;

[0013] Step 3: Construct the definitions and lemmas required for finite-time and fixed-time synchronization analysis of the driving and response systems;

[0014] Step 4: Perform finite-time synchronization and fixed-time synchronization analysis on the Cohen-Grossberg neural network with quantized control input.

[0015] Preferably, in step 1, the network model of the driving system is constructed as follows:

[0016]

[0017] Where i, j∈Z={1,2,...,n}, n≥2 represents the number of neurons, x i (t) represents the state of the i-th neuron at time t, d i (·) represents the amplification function of the i-th neuron, a i (·) represents the behavioral function of the i-th neuron, b ij f represents the synaptic connection strength between the i-th neuron and the j-th neuron; j (·) is defined as the activation function corresponding to the neuron, I i It is an external input from the neural network.

[0018] Preferably, in step 1, the response system network model corresponding to the driving system network model is constructed as follows:

[0019]

[0020] Where y i (t) represents the state of the i-th neuron at time t, U i (t) is the quantization feedback controller.

[0021] Preferably, in step 2, the method for constructing the quantization feedback controller includes the following steps:

[0022] U i (t)=-α i q(ei(t))-β i sign(q(e i (t)))-γ i sign(q(e i (t)))|q(e i (t))| ω (3)

[0023] Where e i (t)=y i (t)-x i (t) represents the synchronization error, α i ,β i γ i Both ω and i are positive constants, i∈Z, and q(·) denotes the logarithmic quantizer.

[0024] Preferably, in step 2, the functional expression for the quantizer q(·) is:

[0025]

[0026] In the above formula, v is the input to the quantizer, and q(v) is the output of the quantizer. It is the quantization parameter; the quantizer q(·) can be further expressed as:

[0027] q(v)=(1+Δ)v

[0028] Where Δ∈[δ,δ] is the quantization error.

[0029] Preferably, in step 3, the definitions required for finite-time and fixed-time synchronization analysis include:

[0030] Definition 1: If there exists a constant M > 0, making The drive system and the response system then achieve exponential synchronization;

[0031] Definition 2: If there exists a constant T(e(0)) ≥ 0, such that...

[0032] lim t→T(e(0)) e(t) = 0 and e(t) = 0,

[0033] Where e(0)=y(0)-x(0)∈R n Then system (1) and system (2) are said to achieve finite-time synchronization. Furthermore, if for any initial value e(0)∈R... n Both have a constant T. max such that T(e(s))≤T max Then the drive system and the response system achieve fixed-time synchronization.

[0034] Preferably, in step 3, the lemmas required for finite-time and fixed-time synchronization analysis include:

[0035] Lemma 1: Assume the function V(x) It is regular, and It is absolutely continuous on any compact interval [0, ∞). Let v(t) = V(x(t)), if there exists a continuous function And when σ(0,∞) γ(σ)>0, then For any t > 0, v(t) is differentiable at t, and γ(·) satisfies:

[0036]

[0037] When t > t1, we have v(t) = 0. If for any σ ∈ (0, ∞), there exists γ(σ) = Qσ. μ If the following holds true, where μ∈(0,1) and Q>0, then the synchronization time can be estimated as:

[0038]

[0039] Lemma 2: If x1,...,x n If k is a real number and k>1, then the following inequalities hold:

[0040]

[0041] Lemma 3: If there exists a regular, positive definite, radially unbounded function Make:

[0042]

[0043] Wherein, any solution x(t) satisfies the inequality:

[0044]

[0045] Among them, a, b, δ, κ>0, θ≥0, and δκ>1, θκ<1; x∈R n f:R n →R n At this point, the origin of equation (6), i.e., x(0) = x0, is stable at a fixed time, and its resting time T(x0) is estimated as follows:

[0046]

[0047] in:

[0048]

[0049] Preferably, in step 3, when constructing the definitions and lemmas required for the finite-time and fixed-time synchronization analysis of the driving system and the response system, the following assumptions are introduced for all i,j∈Z:

[0050] Assumption 1: d i (x) is continuous and has positive constants. d i , Make

[0051]

[0052] Assumption 2: There exists a positive constant a i , so that:

[0053]

[0054] Assumption 3: The activation function is Lipschitz continuous, meaning there exists a positive constant L such that:

[0055] |f(s1)-f(s2)|≤L|s1-s2|, s1,s2∈R.

[0056] Preferably, in step 4, the method for analyzing finite-time synchronization and fixed-time synchronization of the Cohen-Grossberg neural network with quantized control input is as follows:

[0057] According to Theorem 1: If Assumptions 1-3 hold, and:

[0058] min i {d i (a i +α i (1-δ))}-ξ≥0

[0059] This holds for any i, j ∈ Z, where Under a feedback controller, the following statement holds:

[0060] If 0 < ω < 1, the CGNNs driving system and response system achieve finite-time synchronization, and its resting time is estimated as follows:

[0061]

[0062] in

[0063] If ω = 1, the CGNNs driving system and response system achieve exponential synchronization;

[0064] If ω > 1, the CGNNs driving system and response system achieve fixed-time synchronization, and their resting time can be estimated as follows:

[0065]

[0066] in ρ2=min 1≤i≤n {β i};

[0067] Preferably, the Cohen-Grossberg neural network construction method with quantized control input further includes:

[0068] Step 5: When the signal is not quantized before being transmitted to the controller, perform finite-time synchronization and fixed-time synchronization analysis on the Cohen-Grossberg neural network.

[0069] Compared with the prior art, the present invention has the following advantages:

[0070] This invention provides a method for constructing a Cohen-Grossberg neural network with quantized control input. This method combines theories such as differential equation stability, finite-time stability, and quantized control to study exponential synchronization, finite-time synchronization, and fixed-time synchronization of the Cohen-Grossberg neural network under the same quantized control strategy. Attached Figure Description

[0071] Figure 1 This is a flowchart of a method for constructing a Cohen-Grossberg neural network with quantized control input according to an embodiment of the present invention;

[0072] Figure 2 This is a schematic diagram of the quantizer in an embodiment of the present invention;

[0073] Figure 3 This is a schematic diagram of the framework of inference 1 in step 5 of the present invention;

[0074] Figure 4 This is a schematic diagram of the framework for step 5, inference 2, in an embodiment of the present invention;

[0075] Figure 5 This is a time evolution diagram of the system constructed by equation (16) in an embodiment of the present invention;

[0076] Figure 6 This is a schematic diagram of the time evolution of x1(t) and y1(t) when ω=0.001 in an embodiment of the present invention;

[0077] Figure 7 This is a schematic diagram of the time evolution of x2(t) and y2(t) when ω=0.001 in an embodiment of the present invention;

[0078] Figure 8 This is a schematic diagram of the time evolution of x3(t) and y3(t) when ω=0.001 in an embodiment of the present invention;

[0079] Figure 9 This is a schematic diagram of the asymptotic behavior of the synchronization error when ω = 0.001 in an embodiment of the present invention;

[0080] Figure 10 This is a schematic diagram of the time evolution of x1(t) and y1(t) when ω=1 in an embodiment of the present invention;

[0081] Figure 11 This is a schematic diagram of the time evolution of x2(t) and y2(t) when ω=1 in an embodiment of the present invention;

[0082] Figure 12 This is a schematic diagram of the time evolution of x3(t) and y3(t) when ω=1 in an embodiment of the present invention;

[0083] Figure 13 This is a schematic diagram of the asymptotic behavior of the synchronization error when ω = 1 in an embodiment of the present invention;

[0084] Figure 14 This is a schematic diagram of the time evolution of x1(t) and y1(t) when ω=2.5 in an embodiment of the present invention;

[0085] Figure 15 This is a schematic diagram of the time evolution of x2(t) and y2(t) when ω=2.5 in an embodiment of the present invention;

[0086] Figure 16 This is a schematic diagram of the time evolution of x3(t) and y3(t) when ω=2.5 in an embodiment of the present invention;

[0087] Figure 17 This is a schematic diagram of the asymptotic behavior of the synchronization error when ω = 2.5 in an embodiment of the present invention. Detailed Implementation

[0088] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and not intended to limit it. Furthermore, it should be noted that, for ease of description, the accompanying drawings show only the parts relevant to the present invention, and not all of the structures.

[0089] like Figure 1 As shown, the technical solution of the present invention provides a method for constructing a Cohen-Grossberg neural network with quantized control input, comprising the following steps:

[0090] Step 1: Construct a neural network model, including a driving system and a corresponding response system;

[0091] Step 2: Construct a quantized feedback controller for the response system;

[0092] Step 3: Construct the definitions and lemmas required for finite-time and fixed-time synchronization analysis of the driving and response systems;

[0093] Step 4: Perform finite-time synchronization and fixed-time synchronization analysis on the Cohen-Grossberg neural network with quantized control input.

[0094] In one embodiment of the present invention, in step 1, the network model of the driving system is constructed as follows:

[0095]

[0096] Where i, j∈Z={1,2,...,n}, n≥2 represents the number of neurons, x i (t) represents the state of the i-th neuron at time t, di (·) represents the amplification function of the i-th neuron, a i (·) represents the behavioral function of the i-th neuron, b ij f represents the synaptic connection strength between the i-th neuron and the j-th neuron; j (·) is defined as the activation function corresponding to the neuron, I i It is an external input from the neural network.

[0097] In one embodiment of the present invention, in step 1, the response system network model corresponding to the driving system network model is constructed as follows:

[0098]

[0099] Where y i (t) represents the state of the i-th neuron at time t, U i (t) is the quantization feedback controller.

[0100] In one embodiment of the present invention, in step 2, in order to ensure synchronization between the drive system and the response system, the response system is designed with the following quantization feedback controller:

[0101] U i (t)=-α i q(e i (t))-β i sign(q(e i (t)))-γ i sign(q(e i (t)))|q(e i (t))| ω (3)

[0102] Where e i (t)=y i (t)-x i (t) represents the synchronization error, α i ,β i γ i Both ω and i are positive constants, i∈Z, and q(·) denotes the logarithmic quantizer.

[0103] As a further optimization of the above embodiment, in step 2, the quantizer divides the defined interval into different intervals, where each interval corresponds to a quantized value, so it can be regarded as a mapping of piecewise constants; the logarithmic quantizer used in this embodiment is a type of non-uniform quantizer, which has advantages over uniform quantizers in reducing the amount of transmitted data. Based on this, the logarithmic quantizer... Structure such as Figure 2 As shown, its quantization level (quantized values ​​of a finite set of values) set Λ can be described as:

[0104] Λ={±λ j :λ j =ρ j λ0,j=0,±1,±2,…}∪{0}

[0105] Where λ0>0 is the scaling parameter, and 0<ρ<1 is the quantization density. The larger the quantization density, the more accurate the quantization result; conversely, the lower the quantization density, the coarser the quantization. The quantizer q(·) maps each interval to a quantization series, and its function expression is:

[0106]

[0107] In the above formula, v is the input to the quantizer, and q(v) is the output of the quantizer. It is the quantization parameter; the quantizer q(·) can be further expressed as:

[0108] q(v)=(1+Δ)v

[0109] Where Δ∈[δ,δ] is the quantization error.

[0110] From the quantizer (4), the controller (3) can be written in the following form:

[0111] U i (t)=-α i (1+Δ)e i (t)-β j sign((1+Δ)e i (t))-γ j sign((1+Δ)e i (t))(1+Δ) ω |e i (t)| ω

[0112] From the definition of Δ, we can obtain (1+Δ)∈(0,2), which means that

[0113] sign((1+Δ)e i (t))=sign(e i (t))

[0114] Therefore, the quantization feedback controller can ultimately be written as:

[0115] U i (t)=-α i (1+Δ)e i (t)-β i sign(e i (t))-γ i sign(e i(t))(1+Δ) ω |e i (t)| ω (5)

[0116] In one embodiment of the present invention, step 3, the preliminary knowledge, and the definitions required for finite-time and fixed-time synchronization analysis include:

[0117] Definition 1: If there exists a constant M >>, making The drive system and the response system then achieve exponential synchronization;

[0118] Definition 2: If there exists a constant T(e(0)) ≥ 0, such that...

[0119] lim t→T(e(0)) e(t) = 0 and e(t) ≡ 0,

[0120] Where e(0)=y(0)-x(0)∈R n Then system (1) and system (2) are said to achieve finite-time synchronization. Furthermore, if for any initial value e(0)∈R... n Both have a constant T. max such that T(e(s))≤T max Then the drive system and the response system achieve fixed-time synchronization.

[0121] In one embodiment of the present invention, step 3, the lemma required for finite-time and fixed-time synchronization analysis includes:

[0122] Lemma 1: Hypothesis Function It is regular, and It is absolutely continuous on any compact interval [0,∞). Let v(t) = V(x(t)), if there exists a continuous function... And when σ∈(0,∞) γ(σ)>0, then For any t>0, v(t) is differentiable at t, and γ(·) satisfies:

[0123]

[0124] When t>t1, we have v(t)=0, if for any σ∈(0,∞), there exists γ(σ)=Qσ. μ If the following holds true, where μ∈(0,1) and Q>0, then the synchronization time can be estimated as:

[0125]

[0126] Lemma 2: If x1, ..., x n If k is a real number and k > 1, then the following inequalities hold:

[0127]

[0128] Lemma 3: If there exists a regular, positive definite, radially unbounded function Make:

[0129]

[0130] Wherein, any solution x(t) satisfies the inequality:

[0131]

[0132] Among them, a, b, δ, κ>0, θ≥0 and δκ>1, θκ<1; x∈R n f:R n →R n At this point, the origin of equation (6), i.e., x(0) = x0, is stable at a fixed time, and its resting time T(x0) is estimated as follows:

[0133]

[0134] in:

[0135]

[0136] In one embodiment of the present invention, in step 3, when constructing the definitions and lemmas required for finite-time and fixed-time synchronization analysis of the driving system and the response system, the following assumptions are introduced for all i,j∈Z:

[0137] Assumption 1: d i (x) is continuous and has positive constants. d i , Make

[0138]

[0139] Assumption 2: There exists a positive constant a i , so that:

[0140]

[0141] Assumption 3: The activation function is Lipschitz continuous, meaning there exists a positive constant L such that:

[0142] |f(s1)-f(s2)|≤L|s1-s2|, s1,s2∈R.

[0143] In one embodiment of the present invention, step 4 involves performing finite-time synchronization and fixed-time synchronization analysis on a Cohen-Grossberg neural network with quantized control input as follows:

[0144] According to Theorem 1: If Assumptions 1-3 hold, and:

[0145] min i {d i (a i +α i (1-δ))}-ξ≥0

[0146] This holds for any i, j ∈ Z, where Under a feedback controller, the following statement holds:

[0147] If 0 < ω < 1, the CGNNs driving system and response system achieve finite-time synchronization and their resting time is estimated as follows:

[0148]

[0149] in

[0150] If ω = 1, the CGNNs driving system and response system achieve exponential synchronization;

[0151] If ω > 1, the CGNNs driving system and response system achieve fixed-time synchronization, and their resting time can be estimated as follows:

[0152]

[0153] in ρ2=min 1≤i≤n {β i};

[0154] Proof of Theorem 1:

[0155] Construct the following Lyapunov function:

[0156]

[0157] Based on systems (1) and (2), the derivative of V(t) can be calculated to obtain...

[0158]

[0159] On the one hand, the inequality can be derived from assumption 1:

[0160]

[0161] On the other hand, based on system (1), (2) and controller (5), we can obtain

[0162]

[0163] Based on assumptions 2-3, the following inequality can be obtained:

[0164]

[0165] Substituting (10) into (9), we get

[0166]

[0167] Combining with (8) below, we have

[0168]

[0169] Based on the conditions of (8), Lemma 2, and Theorem 1, we can obtain

[0170] in From equations (7), (11), and (13), we can obtain

[0171]

[0172] Combining (8) and (14), and lemmas 1-3, we can obtain the following conclusion:

[0173] If 0 < ω < 1:

[0174]

[0175] in:

[0176]

[0177] The ultimate goal is to prove the existence of a constant T. * >0, such that:

[0178]

[0179] It is not difficult to see that for t > 0, V(t) is differentiable at t, and

[0180]

[0181] Therefore, according to Lemma 1, its synchronization time estimate can be derived as follows:

[0182]

[0183] That is, there exists a constant T. * =t1>0, such that

[0184]

[0185] According to Definition 2, the CGNN driving system and the response system achieve synchronization within a finite time t1:

[0186] If ω=1:

[0187]

[0188] Where ρ * =min{(1-δ)γ i d i}

[0189] When t≥0, integrating both sides of the inequality on [0, t], we have: Where ρ * V(0) are all constants, and ρ * >0,V(0)>0, according to definition 1, at this time CGNNs(1) and (2) achieve exponential synchronization;

[0190] If ω>1:

[0191]

[0192] in ρ2=min 1≤i≤n {β i}

[0193] It is not difficult to obtain that ρ1, ρ2, κ=1>0, and Therefore, according to Lemma 3, we can obtain:

[0194]

[0195] For any initial value e(0)∈R n Both have a constant T. max such that T(e(s))≤T max Therefore, according to Definition 2, the CGNNs driving system and response system achieve fixed-time synchronization. This completes the proof of Theorem 1.

[0196] In one embodiment of the present invention, the method for constructing a Cohen-Grossberg neural network with quantized control input further includes:

[0197] Step 5: When the signal is not quantized before being transmitted to the controller, perform finite-time synchronization and fixed-time synchronization analysis on the Cohen-Grossberg neural network. Specifically:

[0198] The signal is not quantized before it is transmitted to the controller, that is:

[0199] U i (t)=-α i e i (t)-β i sign(e i (t))-γ i sign(e i (t))|e i (t)| ω (15)

[0200] Theorem 1 can then degenerate into the following corollary:

[0201] like Figure 3 As shown, Corollary 1: If Assumptions 1-3 are true, and

[0202] min i { d i (a i +α i )}-ξ≥0

[0203] For any i, j∈Z, this holds true, where Under controller (15), the following statements hold true:

[0204] If 0 < ω < 1, the CGNNs driving system and response system achieve finite-time synchronization, and its resting time is estimated as follows:

[0205]

[0206] in

[0207] If ω = 1, the CGNNs driving system and response system achieve exponential synchronization;

[0208] If ω > 1, the CGNNs driving system and response system achieve fixed-time synchronization, and its resting time can be estimated as follows:

[0209]

[0210] in ρ2=min 1≤i≤n {β i}

[0211] When the amplification function d i If (·) = 1, the Cohen-Grossberg neural network will degenerate into a cellular neural network (CNN). Therefore, Theorem 1 can be reduced to the following corollary:

[0212] like Figure 4 As shown, Corollary 2: If Assumptions 1-3 are true, and:

[0213] min i {a i +α i (1-δ)}-ξ≥0

[0214] This holds for any i, j ∈ Z, where Under controller (3), the following statements are all true:

[0215] If 0 < ω < 1, CNNs achieve finite-time synchronization, and their resting time is estimated as follows:

[0216]

[0217] in

[0218] If ω = 1, CNNs achieve exponential synchronization;

[0219] If ω > 1, the resting time of CNNs achieving fixed-time synchronization can be estimated as follows:

[0220]

[0221] Where ρ1=min 1≤i≤n {γ i (1-δ) ω}, ρ2=min 1≤i≤n {β i}

[0222] The beneficial effects of the Cohen-Grossberg neural network construction method with quantized control input provided in the embodiments of the present invention are discussed below:

[0223] Consider the following three-dimensional CGNNs:

[0224]

[0225] Among them: i=1, 2, 3, d1(x)=5-2sinx, d2(x)=2+cos x, d3(x)=3-sin x, a1(x)=a2(x)=a3(x)=2x, f j (x)=tanh(x), j=1,2,3, I1=0.5, I2=1, I3=1.9, B=[6.5 3.5 5.7; 3.92.5 4.5; 3.7 3.4 0.16].

[0226] Figure 5 The time evolution of the three-dimensional CGNNs system under the initial conditions x1(0)=1.3, x2(0)=0.6, x3(0)=-1.5 in Equation (16) was simulated.

[0227] Using CGNNs(16) as the driving system, the corresponding response system is:

[0228]

[0229] Where i = 1, 2, 3, y1(0) = -2.3, y2(0) = -1.6, y3(0) = 1.5. Other parameters are the same as system (16);

[0230] choose:

[0231] L=1, ρ=0.55, δ≈0.2903, β1=β2=β3=1, γ1=γ2=γ3=2

[0232] a1 = a2 = a3 = 2, d 1 = 3, d 2 = 1, d 3 = 2,

[0233] According to the conditions in Theorem 1, α1 = 13.2997, α2 = 43.9394, and α3 = 20.9596; therefore, according to Theorem 1, when 0 < ω < 1, systems (16) and (17) can achieve finite-time synchronization; and Figure 6-8 as well as Figure 9 The time evolution of the system and the asymptotic behavior of the synchronization error were simulated when the parameter ω = 0.001, and the resting time was estimated to be t1 = 0.7572. Figure 10-12 and Figure 13 The time evolution and asymptotic behavior of the system and the synchronization error were simulated when the parameter ω = 1. It is easy to verify that when ω = 1, the systems (16) and (17) achieve exponential synchronization; when ω > 1, according to Theorem 1, the systems (16) and (17) achieve fixed-time synchronization. Figure 14-16 and Figure 17 The time evolution of the system and the synchronization behavior of the error were simulated when ω = 2.5, and its resting time was estimated to be T. max =1.7739.

[0234] Within the framework of control theory, when the quantized feedback controller designed in Theorem 1 degenerates into the feedback controller described in Corollary 1, this process essentially corresponds to a transformation in signal processing, namely, the change from a discretized quantized signal form to a continuous signal mode. During this process, the response mechanism of the control system changes significantly; the control command no longer follows the update rule of discrete time intervals, but instead achieves continuous and uninterrupted updates.

[0235] From the perspective of achieving the goal, both the quantization feedback controller based on Theorem 1 and the feedback controller using Corollary 1 enable the driver-response Cohen-Grossberg neural network to achieve synchronization. However, the quantization feedback controller in Theorem 1 shows significant advantages in cost control and resource optimization: it cleverly introduces a quantization mechanism, which effectively reduces the amount of data in the signal transmission and processing process and reduces redundant information in the control signal while ensuring the system's synchronization performance, thereby significantly saving the cost resources required for system operation.

[0236] Within the framework of the model, it is not difficult to find the amplification function d in the driver-response Cohen-Grossberg neural network. i The · symbol plays a special role. When it is set to 1, the driver-response Cohen-Grossberg neural network degenerates into a cellular neural network. This degeneration reveals the intrinsic connection between different types of neural network models. Similarly, using a quantized feedback controller, the cellular neural network can still achieve the synchronization goal, demonstrating the applicability of the proposed quantized feedback controller in neural network synchronization control.

[0237] An application scenario of the Cohen-Grossberg neural network construction method with quantized control input provided in this embodiment of the invention is described as follows:

[0238] When autonomous vehicle swarms collaborate to perform tasks, ensuring the rapid and stable formation of the swarm formation is crucial. This solution provides a reliable method to address this issue. Its application process begins with system modeling, where the lead vehicle in the swarm is regarded as the driving system, whose dynamics are described by nonlinear differential equations; while the following vehicles are regarded as the response system, and the motion state of each vehicle corresponds to the state of neurons in the neural network.

[0239] To achieve cluster synchronization, a quantization feedback controller is deployed on each following vehicle. The input to this controller is the state error between the lead vehicle and the following vehicles. Unlike traditional controllers that directly use continuous error signals, in this scheme, the continuous state error signal is first converted into a finite, discrete quantization level by a quantizer, and then transmitted through a bandwidth-limited wireless channel. After receiving the quantized error signal, the controller generates control commands according to a preset control law, dynamically adjusting the following vehicles to rapidly approximate the lead vehicle's state.

[0240] The synchronization type of the entire control process is determined by the key parameter ω. During the initial swarm assembly phase, if 0 < ω < 1, the system can achieve finite-time synchronization. This means the swarm can quickly take shape within a precisely estimated time point, making it ideal for tasks requiring urgent deployment. When ω = 1, the system achieves exponential synchronization with a still fast convergence speed. Furthermore, when ω > 1, the system can achieve fixed-time synchronization, with its synchronization time upper limit independent of the initial swarm formation. This enhances the system's robustness, ensuring that even if the initial positions of the autonomous vehicles are scattered or cannot be accurately determined, the target formation can be completed within a certain timeframe. This is of great significance for deploying autonomous vehicle swarms in unknown or dynamic environments.

[0241] Advantages of the technical solution: First, by quantizing the state error signal, the amount of data transmitted in the unmanned vehicle (UAV) is reduced. This makes the solution suitable for environments with limited communication bandwidth and susceptible to interference, laying the foundation for reliable large-scale UAV collaboration. Second, the solution provides fast convergence performance. Whether finite-time or fixed-time synchronization, it ensures that the cluster formation can reach the target state within a certain time range, rather than infinitely approximating as in traditional asymptotic synchronization theory. This deterministic convergence speed is crucial for task planning, ensuring timeliness. Third, the fixed-time synchronization characteristic makes it insensitive to the initial state, enabling it to easily cope with the impact of uncertain initial positions or sudden disturbances in UAVs. Furthermore, by adjusting the controller parameters, it can degenerate into a non-quantized version processing continuous signals, or be applied to simplified vehicle dynamics models, demonstrating good scalability. Finally, the quantization mechanism reduces communication burden and computational overhead. This makes the control strategy valuable for engineering applications and has significant potential for widespread adoption in modern unmanned systems that pursue high efficiency and low cost.

[0242] The above are preferred embodiments of the present invention. It should be noted that, for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A method for constructing a Cohen-Grossberg neural network with quantized control input, characterized in that: The method includes the following steps: Step 1: Construct a neural network model, including a driving system and a corresponding response system; Step 2: Construct a quantized feedback controller for the response system; Step 3: Construct the definitions and lemmas required for finite-time and fixed-time synchronization analysis of the driving and response systems; Step 4: Perform finite-time synchronization and fixed-time synchronization analysis on the Cohen-Grossberg neural network with quantized control input.

2. The method for constructing a Cohen-Grossberg neural network with quantized control input as described in claim 1, characterized in that: In step 1, the network model of the driving system is constructed as follows: Where i, j∈Z={1,2,...,n}, n≥2 represents the number of neurons, x i (t) represents the state of the i-th neuron at time t, d i (·) represents the amplification function of the i-th neuron, a i (·) represents the behavioral function of the i-th neuron, b ij This represents the synaptic connection strength between the i-th neuron and the j-th neuron; f j (·) is defined as the activation function corresponding to the neuron, I i It is an external input from the neural network.

3. The method for constructing a Cohen-Grossberg neural network with quantized control input as described in claim 2, characterized in that: In step 1, the response system network model corresponding to the driving system network model is constructed as follows: Where y i (t) represents the state of the i-th neuron at time t, U i (t) is the quantization feedback controller.

4. The method for constructing a Cohen-Grossberg neural network with quantized control input as described in claim 3, characterized in that: In step 2, the method for constructing the quantization feedback controller includes the following steps: U i (t)=-α i q(e i (t))-β i sign(q(e i (t)))-γ i sign(q(e i (t)))|q(e i (t))| ω (3) Where e i (t)=y i (t)-x i (t) represents the synchronization error, α i ,β i ,r i Both ω and i are positive constants, i∈Z, and q(·) denotes the logarithmic quantizer.

5. The method for constructing a Cohen-Grossberg neural network with quantized control input as described in claim 4, characterized in that: In step 2, the function expression for the quantizer q(·) is: In the above formula, v is the input to the quantizer, and q(v) is the output of the quantizer. It is the quantization parameter; the quantizer q(·) can be further expressed as: q(v)=(1+Δ)v Where Δ∈[δ,δ] is the quantization error.

6. The method for constructing a Cohen-Grossberg neural network with quantized control input as described in claim 5, characterized in that: In step 3, the definitions required for finite-time and fixed-time synchronization analysis include: Definition 1: If there exists a constant Make The drive system and the response system then achieve exponential synchronization; Definition 2: If there exists a constant T(e(0)) ≥ 0, such that... lim t→T(e(0)) e(t) = 0 and Where e(0)=y(0)-x(0)∈R n Then system (1) and system (2) are said to achieve finite-time synchronization. Furthermore, if for any initial value e(0)∈R... n Both have a constant T. max such that T(e(s))≤T max Then the drive system and the response system achieve fixed-time synchronization.

7. The method for constructing a Cohen-Grossberg neural network with quantized control input as described in claim 6, characterized in that: In step 3, the lemmas required for finite-time and fixed-time synchronization analysis include: Lemma 1: Assume the function V(x) It is regular, and x(t) It is absolutely continuous on any compact interval [0,∞). Let v(t) = V(x(t)), if there exists a continuous function γ: And when σ∈(0,∞) γ(σ)>0, then For any t>0, v(t) is differentiable at t, and γ(·) satisfies: When t > t1, we have v(t) = 0. If for any σ ∈ (0, ∞), there exists γ(σ) = Qσ. μ If the condition is met, where μ∈(0,1) and Q>0, then the synchronization time can be estimated as: Lemma 2: If x1,...,x n If k is a real number and k>1, then the following inequalities hold: Lemma 3: If there exists a regular, positive definite, radially unbounded function V(x): Make: Wherein, any solution x(t) satisfies the inequality: Among them, a, b, δ, κ>0, θ≥0, and δκ>1, θκ<1; x∈R n f:R n →R n At this point, the origin of equation (6), i.e., x(0) = x0, is stable at a fixed time, and its resting time T(x0) is estimated as follows: in:

8. The method for constructing a Cohen-Grossberg neural network with quantized control input as described in claim 7, characterized in that: In step 3, when constructing the definitions and lemmas required for the finite-time and fixed-time synchronization analysis of the driving system and the response system, the following assumptions are introduced for all i,j∈Z: Assumption 1: d i (x) is continuous and has positive constants. d i , Make Assumption 2: There exists a positive constant a i , so that: Assumption 3: The activation function is Lipschitz continuous, meaning there exists a positive constant L such that: |f(s1)-f(s2|)≤L|s1-s2|, s1,s2∈R.

9. The method for constructing a Cohen-Grossberg neural network with quantized control input as described in claim 8, characterized in that: In step 4, the following methods are used to analyze finite-time synchronization and fixed-time synchronization of the Cohen-Grossberg neural network with quantized control input: According to Theorem 1: If Assumptions 1-3 hold, and: minutes i {d} i (a i +a i (1-d))}-ξ≥0 This holds for any i, j ∈ Z, where Under a feedback controller, the following statement holds: If 0 < ω < 1, the CGNNs driving system and response system achieve finite-time synchronization, and its resting time is estimated as follows: in If ω = 1, the CGNNs driving system and response system achieve exponential synchronization; If ω>1, the CGNNs driving system and response system achieve fixed-time synchronization, and its resting time can be estimated as follows: in ρ2=min 1≤i≤n {β i }; 10. The method for constructing a Cohen-Grossberg neural network with quantized control input as described in claim 9, characterized in that: The method for constructing a Cohen-Grossberg neural network with quantized control input also includes: Step 5: When the signal is not quantized before being transmitted to the controller, perform finite-time synchronization and fixed-time synchronization analysis on the Cohen-Grossberg neural network.

Images