Fixed time synchronization method of discontinuous complex value memory neural network with time delay

Abstract

The invention discloses a fixed time synchronization method of a discontinuous complex-valued memory neural network with time delay, relates to the technical field of synchronization control, and solves the problems that an analysis result is easy to dynamically disconnect from an actual system, and the time delay is poor due to random hopping of a Markov chain description network topology or parameters. Meanwhile, an analysis result may fail due to sliding mode motion or a boundary layer effect caused by neglecting discontinuity, and even a wrong stability conclusion is caused. A continuous time Markov chain is utilized to describe a topological structure of a bidirectional associative memory neural network with a time-delay discontinuous complex value, and node comprehensive importance is calculated in real time based on node dynamic attitude centrality and weight contribution of a Markov chain hopping network. The preset proportion node with the highest comprehensive importance is screened as the dynamic important node, the controller is only applied to the nodes, and the traction control strategy set for the controller can be dynamically adjusted and controlled according to the network state on the basis of the node importance evaluation result.

Description

Technical Field

[0001] This invention belongs to the field of synchronization control, specifically a fixed-time synchronization method with a time-delay discontinuous complex-valued memory neural network. Background Technology

[0002] Complex-valued neural networks process signals through the complex domain, with their state variables, weight matrices, and activation functions all in complex form. This structure offers unique advantages in complex signal processing, optoelectronics, and grayscale image processing, enabling the simultaneous capture of both amplitude and phase information of signals. For example, in radar imaging, complex-valued networks can efficiently handle the phase modulation characteristics of echo signals. In practical systems, signal transmission or inter-neuron communication often involves time delays, including constant, time-varying, or distributed time delays. Time delays can complicate the dynamic behavior of systems, such as causing oscillations or instability. For instance, in neurocryptography, time delays can disrupt the synchronization of encrypted signals, leading to decryption failures. Therefore, a fixed-time synchronization method is needed for time-delayed discontinuous complex-valued memory neural networks. In real-time signal processing, fixed-time synchronization ensures that nodes synchronize within constraints, avoiding delays caused by differences in initial states.

[0003] Existing fixed-time synchronization methods for memory neural networks typically assume a fixed network structure or slowly changing parameters, failing to utilize Markov chains to describe network topology or random parameter jumps. This leads to analysis results that are easily decoupled from the dynamics of the actual system. Complex-valued memory neural networks often contain discontinuous activation functions; without employing Filippov's differential inclusion theory, traditional analysis methods based on the assumption of continuous systems cannot be directly applied. Analysis results may become invalid due to neglecting sliding mode motion or boundary layer effects caused by discontinuities, or even lead to erroneous stability conclusions. Summary of the Invention

[0004] This invention aims to at least solve one of the technical problems existing in the prior art. To this end, this invention proposes a fixed-time synchronization method for a time-delayed discontinuous complex-valued memory neural network to solve the technical problem that the lack of Markov chain description of network topology or random jumps in parameters leads to the analysis results being easily decoupled from the actual system dynamics. At the same time, the analysis results may fail due to neglecting the sliding mode motion or boundary layer effect caused by discontinuity, or even lead to erroneous stability conclusions.

[0005] To address the aforementioned problems, a first aspect of the present invention provides a fixed-time synchronization method for a time-delayed discontinuous complex-valued memory neural network, comprising the following steps: Based on external disturbances and mixed time delays, and by introducing Markov jump parameters, a time-delayed discontinuous complex-valued bidirectional associative memory neural network model is established. The complex-valued variables and disturbances of the complex-valued neural network are separated into real and imaginary parts. A time-varying and time-delay influence factor is introduced to transform it into an equivalent real-valued subsystem. The error system is then obtained based on the real-valued subsystem. The topology of the time-delay discontinuous complex-valued bidirectional associative memory neural network is described by a continuous-time Markov chain, and the overall importance of nodes is evaluated in real time based on the dynamic degree centrality and weight contribution of the nodes in the Markov chain jump network. Select the node with the highest overall importance as the dynamic important node, apply a controller to the dynamic important node, and set a traction control strategy for the controller based on importance assessment. Combining Markov jump, hybrid time delay, and traction control characteristics, a Lyapunov function containing time delay integral terms and state-dependent weights is constructed. Based on Filippov differential inclusion theory and Jensen integral inequality, it is used to detect whether the derivative of the Lyapunov function is less than or equal to the controller's attenuation term suppression and time delay coupling term, and to ensure fixed-time convergence by adjusting the control parameters.

[0006] Optionally, in one example of the above aspects, a time-delayed discontinuous complex-valued bidirectional associative memory neural network model is established based on external disturbances and mixed time delays, and by introducing Markov jump parameters, including the following steps: Based on external disturbances and mixed time delays (including discrete and distributed time delays), and introducing the Markov jump parameter γ(t) to describe the network topology parameters at different times during the Markov process, a time-delayed discontinuous complex-valued bidirectional associative memory neural network model is established. Optionally, in one example of the above aspects, the complex-valued variables and disturbances of the complex-valued neural network are separated into real and imaginary parts, and a time-varying delay influence factor is introduced to transform it into an equivalent real-valued subsystem, including the following steps: Since the state variables, connection weights, activation functions, and external inputs of a complex-valued neural network are all complex numbers, the complex-valued neural network is separated into real and imaginary parts to construct an equivalent real-valued subsystem. Define the synchronization error and the state variables after introducing time delay into the complex-valued neural network model.

[0007] Optionally, in one example of the above aspects, obtaining the error system from the real-valued subsystem includes the following steps: By introducing the Filippov selection set elements of the nonlinear function of the associated state corresponding to the Markov jump parameters of each associated state, as well as the real part time-delayed associated error term and the imaginary part time-delayed associated error term, the error system of the real-valued subsystem is obtained by differentiating the real part of the error.

[0008] Optionally, in one example of the above aspects, a comprehensive evaluation of node importance is performed in real time based on the dynamic degree centrality and weight contribution of nodes in the Markov chain transition network, including the following steps: Based on the value H={1,2,…,Nr} of the Markov jump parameter γ(t) in the Markov chain jump network, the network is divided into Nr topological states, and each state corresponds to a fixed connection weight matrix. For each topological state i∈H in the topology of a time-delayed discontinuous complex-valued bidirectional associative memory neural network, a comprehensive importance index Ih(i) for node h is defined. The state-dependent comprehensive importance index of nodes is calculated by combining the degree centrality and weight contribution of the nodes.

[0009] Optionally, in one example of the above aspects, selecting the node with the highest overall importance according to a preset ratio as the dynamically important node includes the following steps: For each topological state i, the comprehensive importance index Ih(i) of node h is sorted in descending order, and the top q% of nodes are selected as the candidate controlled node set for that state. Filter the intersection of candidate controlled node sets for all Nr topological states, and define the intersection as the node set with high influence; For nodes not defined as high-influence nodes in the candidate controlled node set, a high-influence node set is temporarily added only when the comprehensive importance index Ih(i) of the corresponding node exceeds the threshold. This enables real-time dynamic adjustment of the high-influence node set, and nodes in the high-influence node set are determined as dynamically important nodes.

[0010] Optionally, in one example of the above aspects, a controller is applied to dynamically important nodes, and a traction control strategy based on importance assessment is set for the controller, including: Real traction control strategy based on importance assessment and imaginary traction control strategy based on importance assessment.

[0011] Optionally, in one example of the above aspects, based on Filippov differential inclusion theory and Jensen integral inequalities, it is determined whether the derivative of the Lyapunov function is less than or equal to the controller's attenuation term suppression and time-delay coupling term, including: By using Filippov differential inclusion theory and Jensen's inequality, an inequality is constructed to check whether the derivative of the Lyapunov function is less than or equal to the controller's attenuation term and time delay coupling term. If the inequality holds, then time converges, and fixed-time synchronization is achieved; otherwise, time does not converge.

[0012] Optionally, in one example of the above aspects, ensuring fixed-time convergence by adjusting control parameters includes the following steps: If the inequality between the controller's attenuation suppression and time-delay coupling terms does not hold, and the time does not converge, then by constructing a discrete time-delay compensation term and a distributed time-delay compensation term in the Lyapunov function containing time-delay integral terms and state-dependent weights, the system can satisfy the condition that the derivative of the Lyapunov function is less than or equal to the controller's attenuation suppression and time-delay coupling terms.

[0013] Compared with the prior art, the beneficial effects of the present invention are: This invention utilizes continuous-time Markov chains to describe the topology of a time-delayed, discontinuous, complex-valued, bidirectional associative memory neural network, accurately capturing the random jump characteristics of connections between network nodes. This modeling approach is particularly suitable for scenarios in real-world systems where structural and parameter changes occur due to environmental noise and limitations of physical components.

[0014] Based on the dynamic degree centrality and weight contribution of nodes in the Markov chain transition network, the overall importance of nodes is calculated in real time. This mechanism enables the system to dynamically identify key nodes, avoiding the waste of control resources or the omission of core nodes due to static evaluation.

[0015] This invention achieves precise allocation of control resources by selecting nodes with the highest overall importance as dynamically important nodes and applying controllers only to these nodes. Compared to global control strategies, this method significantly reduces the number of controllers and computational burden while maintaining synchronization performance. The traction control strategy set for the controllers is based on the node importance assessment results and can dynamically adjust the control strength according to the network state. This strategy can maintain system stability even when node importance changes, avoiding overcontrol or undercontrol problems that may occur with fixed control parameters.

[0016] This invention constructs a Lyapunov function that satisfies specific conditions, enabling explicit estimation of the upper bound of a fixed convergence time T. This explicit estimation provides a time reference for real-time control in practical systems, enhancing the method's practicality. It overcomes the limitation of separating real and imaginary parts in traditional complex-valued neural network analysis, proposing a stability criterion for the non-separation case and expanding the research boundaries of time-delay complex-valued neural networks. Furthermore, by combining Markov jumps with fixed-time control, it provides a new approach to the stability analysis of stochastic hybrid systems. Attached Figure Description

[0017] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0018] Figure 1 This is a schematic diagram of the method flow of the present invention. Detailed Implementation

[0019] The technical solution of the present invention will be clearly and completely described below with reference to the embodiments. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0020] Please see Figure 1 The first aspect of this invention provides a fixed-time synchronization method with a time-delayed discontinuous complex-valued memory neural network, comprising the following steps: Based on external disturbances and mixed time delays, and by introducing Markov jump parameters, a time-delayed discontinuous complex-valued bidirectional associative memory neural network model is established. The complex-valued variables and disturbances of the complex-valued neural network are separated into real and imaginary parts. A time-varying and time-delay influence factor is introduced to transform it into an equivalent real-valued subsystem, and the error system is obtained based on the real-valued subsystem. Since the state variables, connection weights, activation functions and external inputs of the complex-valued neural network are all complex numbers, the complex-valued neural network is separated into real and imaginary parts to construct an equivalent real-valued subsystem. The topology of the time-delay discontinuous complex-valued bidirectional associative memory neural network is described by a continuous-time Markov chain, and the overall importance of nodes is evaluated in real time based on the dynamic degree centrality and weight contribution of the nodes in the Markov chain jump network. Select the node with the highest overall importance as the dynamic important node, apply a controller to the dynamic important node, and set a traction control strategy for the controller based on importance assessment. Combining Markov jump, hybrid time delay, and traction control characteristics, a Lyapunov function containing time delay integral terms and state-dependent weights is constructed. Based on Filippov differential inclusion theory and Jensen integral inequality, it is used to detect whether the derivative of the Lyapunov function is less than or equal to the controller's attenuation term suppression and time delay coupling term, and to ensure fixed-time convergence by adjusting the control parameters.

[0021] Specifically, in this embodiment, by separating the real and imaginary parts of the complex-valued neural network, the original problem is transformed into an equivalent real-valued subsystem. This operation avoids the complexity of directly processing complex-valued variables, allowing traditional real-valued neural network analysis tools to be directly applied, significantly reducing the difficulty of theoretical derivation and numerical calculation.

[0022] By introducing a time-varying delay influence factor, the delay effect is explicitly embedded in the dynamic equations of the real-valued subsystem. This approach not only preserves the full impact of time delay on system behavior but also simplifies the analysis process of time-delay-related stability through a real-valued framework.

[0023] Using continuous-time Markov chains to describe the topology of discontinuous complex-valued bidirectional associative memory neural networks with time delays can accurately capture the random jump characteristics of the connection relationships between network nodes. This modeling approach is particularly suitable for scenarios in real-world systems where structural and parameter changes occur due to environmental noise and limitations of physical components.

[0024] Based on the dynamic degree centrality and weight contribution of nodes in the Markov chain transition network, the overall importance of nodes is calculated in real time. This mechanism enables the system to dynamically identify key nodes, avoiding the waste of control resources or the omission of core nodes due to static evaluation.

[0025] By selecting nodes with the highest overall importance as dynamically important nodes and applying controllers only to these nodes, precise allocation of control resources is achieved. Compared to global control strategies, this method significantly reduces the number of controllers and computational burden while maintaining synchronization performance. The traction control strategy set for the controllers is based on node importance assessment results and can dynamically adjust the control strength according to the network state. This strategy can maintain system stability even when node importance changes, avoiding over-control or under-control problems that may occur with fixed control parameters. By designing a nonlinear delay controller and synchronization criteria, the system is ensured to achieve synchronization within a fixed time, and the settling time is independent of the initial conditions. This characteristic provides reliable protection for applications requiring strict time constraints. By explicitly modeling time-varying delays and discrete discontinuities, the system can resist disturbances caused by time delay fluctuations and sudden changes in node states. Combining dynamic node selection and traction control further enhances the robustness of the system in complex environments.

[0026] By explicitly embedding time-delay integral terms into the Lyapunov function, the cumulative effect of mixed time delays on system dynamics can be accurately captured. This approach avoids the simplification or approximation of time-delay terms in traditional methods, thus reducing the conservatism of the analysis results. The weight matrix is ​​designed as a function of the state variables, enabling the Lyapunov function to adapt to changes in system state. Combined with Markov jump characteristics, the weight matrix can further depend on jump modes, thereby dynamically reflecting the impact of network topology or parameter mutations on system stability.

[0027] Based on Filippov's differential inclusion theory, the derivative of the Lyapunov function can be decomposed into a controller decay term and a time-delay coupling term. By adjusting the control parameters, the derivative can be ensured to satisfy the time stability condition.

[0028] By using Jensen's inequality to scale the time-delay integral term, the derivative analysis can be further simplified while maintaining an accurate estimate of the time-delay effect. This approach provides a theoretical basis for the design of parameters that converge at fixed time.

[0029] By constructing Lyapunov functions that satisfy specific conditions, the upper bound of a fixed convergence time T can be explicitly estimated. This explicit estimation provides a time reference for real-time control in practical systems, enhancing the practicality of the method. It overcomes the limitation of separating real and imaginary parts in traditional complex-valued neural network analysis, proposing a stability criterion for the non-separation case and expanding the research boundaries of time-delay complex-valued neural networks. Furthermore, by combining Markov jumps with fixed-time control, it provides a new approach to the stability analysis of stochastic hybrid systems.

[0030] In one embodiment of the present invention, a time-delayed discontinuous complex-valued bidirectional associative memory neural network model is established based on external disturbances and mixed time delays, and by introducing Markov jump parameters, including the following steps: Based on external disturbances and mixed time delays (including discrete and distributed time delays), and introducing the Markov jump parameter γ(t) to describe the network topology parameters at different times during the Markov process, a time-delayed discontinuous complex-valued bidirectional associative memory neural network model is established. Where xk(t) represents the k-th state variable in the driving system, yj(t) represents the j-th associated state variable in the driving system; ak is the damping coefficient of the k-th state variable, dj is the damping coefficient of the j-th associated state variable; bkj is the instantaneous coupling coefficient of the j-th associated state yj(t) to the k-th state variable xk(t); ckj is the time-delay coupling coefficient of the j-th associated state yj(t) to the k-th state variable xk(t); wjk is the instantaneous coupling coefficient of the k-th state variable xk(t) to the j-th associated state yj(t); hjk is the time-delay coupling coefficient of the k-th state variable xk(t) to the j-th associated state yj(t); n is the total number of associated state variables, m is the total number of state variables, j∈(1,2,…,n), k∈(1,2,…,m); dkj is the integral coupling coefficient of the j-th associated state to the main state in the integral term, and ljk is the integral coupling coefficient of the main state to the associated state in the integral term; the larger the absolute value of the coefficient, the stronger the coupling effect; the positive and negative signs indicate the direction of action, positive for promotion and negative for inhibition.

[0031] fj(yj(t)) and fj(yj(t-τ)) are nonlinear transformation functions of the j-th associated state, such as saturation functions, sigmoid functions, power-law functions, etc., which map the value of yj(t) to the nonlinear effect on the master state xk(t), thus characterizing the nonlinear characteristics of the system.

[0032] gk(xk(t)) and gk(xk(t-τ)) are nonlinear transformation functions of the k-th state variable; the value of xk(t) is mapped to the nonlinear effect on the associated state yj(t), reflecting the nonlinear coupling mechanism between states; τ and σ are time delay parameters; Ik is a constant external input acting on the k-th state variable, and Jj is a constant external input acting on the j-th associated state; Dk(t) represents the time-varying perturbation / noise acting on the k-th state variable, and Ej(t) represents the time-varying perturbation / noise acting on the j-th associated state. τ>0, where τ is the discrete time delay, and σ>0, where σ is the upper bound of the distributed time delay.

[0033] In one embodiment of the present invention, the complex-valued variables and disturbances of the complex-valued neural network are separated into real and imaginary parts, and a time-varying delay influence factor is introduced to transform it into an equivalent real-valued subsystem, including the following steps: Since the state variables, connection weights, activation functions, and external inputs of a complex-valued neural network are all complex numbers, separating the complex-valued neural network into real and imaginary parts and constructing an equivalent real-valued subsystem includes: Define synchronization error: ; After incorporating time delay into the complex-valued neural network model, the state variables are: ; Where xk(t) represents the k-th state variable in the driving system. This represents the error corresponding to the k-th state variable in the driving system. For systematic error, Let be the real part of the error. Let zj(t) be the imaginary part of the error, zj(t) be the synchronization error of the j-th associated state, and yj(t) be the j-th associated state variable in the driving system. This represents the error corresponding to the j-th associated state variable in the driving system. Let be the real part of the synchronization error of the j-th associated state. Let be the imaginary part of the synchronization error of the j-th associated state.

[0034] In one embodiment of the present invention, obtaining the error system based on the real-valued subsystem includes the following steps: By introducing the Filippov selection set elements of the nonlinear function of the associated state corresponding to the Markov jump parameters of each associated state, as well as the real-part time-delayed associated error term and the imaginary-part time-delayed associated error term, the error system of the real-valued subsystem is obtained by differentiating the real part of the error: Where ak is the damping coefficient of the k-th state variable, bkj is the instantaneous coupling coefficient of the j-th associated state yj(t) to the k-th state variable xk(t); ckj is the time-delay coupling coefficient of the j-th associated state yj(t) to the k-th state variable xk(t); dkj is the integral coupling coefficient of the j-th associated state to the main state in the integral term; the superscripts R / I of bkj, ckj and dkj distinguish the real part / imaginary part coupling strength, with R indicating the real part coupling strength and I indicating the imaginary part coupling strength. The sign of the coefficient indicates the direction of action, with positive indicating promoting error growth and negative indicating inhibiting it. Uk R (t) represents the control input with the real part of the error, Δk R (t) represents the disturbance cancellation term; and The Filippov selection set elements are the associated state nonlinear functions corresponding to the Markov jump parameters of the j-th associated state. For the Filippov selection set elements of the imaginary part nonlinear function corresponding to the Markov jump parameter of the j-th associated state; τ is a time delay parameter, representing the delay in the impact of historical states on the current error. The real part of the time-delay related error term, The time delay associated error term is the imaginary part. This is an integral coupling term, where σ is the integration window length and s is a dummy variable in the time integration. The integral term reflects the cumulative effect of the error over the time interval [t-σ,t]. , , The (s) added after the parameter indicates the element value of the corresponding parameter at time s in the time integral.

[0035] Uk R (t) is the control input for the real part of the error, used to actively adjust the error dynamics, such as forcing the error to converge through feedback control. "Uncontrolled node Uk" R "(t)=0" indicates that a node without control has no external input.

[0036] Δk R (t) represents the disturbance cancellation term, Δk R (t)=0 indicates that the external disturbances of the driving system and the response system are completely matched, achieving disturbance cancellation and ensuring that the error dynamics are not affected by external interference.

[0037] γj R Filippov selects elements of the associated state nonlinear function, i.e., γj R ∈[FjR] is used to describe the generalized derivative at discontinuity points in a nonlinear system (such as switching systems, nonlinear characteristics with friction / saturation), where FjR is the set-valued mapping of the real part of the nonlinear function.

[0038] γj I Filippov selection set elements for the imaginary part nonlinear function characterize the nonlinear coupling relationship of the imaginary part error.

[0039] In one embodiment of the present invention, a comprehensive evaluation of node importance is performed in real time based on the dynamic degree centrality and weight contribution of nodes in the Markov chain transition network, including the following steps: Based on the value H={1,2,…,Nr} of the Markov jump parameter γ(t) in the Markov chain jump network, the network is divided into Nr topological states, and each state corresponds to a fixed connection weight matrix. For each topological state i∈H in the topology of a time-delayed discontinuous complex-valued bidirectional associative memory neural network, a comprehensive importance index Ih(i) for node h is defined. The state-dependent comprehensive importance index of nodes is calculated by combining the degree centrality and weight contribution of the nodes. Where bkj(i) is the instantaneous coupling coefficient of the j-th associated state yj(t) of topological state i to the k-th state variable xk(t); ckj(i) is the time-delay coupling coefficient of the j-th associated state yj(t) of topological state i to the k-th state variable xk(t); dkj(i) is the integral coupling coefficient of the j-th associated state to the main state in the integral term of topological state i; and wjk(i) is the maximum feedback weight of other nodes to node h.

[0040] The first three terms are the sum of the direct connection weights of node h to other nodes, including instantaneous, discrete time delay, and distributed time delay weights. The last term is the maximum feedback weight of other nodes to h, reflecting the "hub" status of the node in the topology.

[0041] In one embodiment of the present invention, selecting the node with the highest overall importance according to a preset ratio as the dynamically important node includes the following steps: For each topological state i, the comprehensive importance index Ih(i) of node h is sorted in descending order, and the top q% of nodes are selected as the candidate controlled node set for that state; in this embodiment, the top 20% of nodes are selected as the candidate controlled nodes for that state.

[0042] Filter the intersection of candidate controlled node sets for all Nr topological states, and define the intersection as the node set with high influence; For nodes not defined as high-influence nodes in the candidate controlled node set, a high-influence node set is temporarily added only when the comprehensive importance index Ih(i) of the corresponding node exceeds the threshold. This enables real-time dynamic adjustment of the high-influence node set, and nodes in the high-influence node set are determined as dynamically important nodes.

[0043] In one embodiment of the present invention, a controller is applied to dynamically important nodes, and a traction control strategy based on importance assessment is set for the controller, including: Real traction control strategy based on importance assessment: Imaginary part traction control strategy based on importance assessment: in, The traction control output is the real part of the action on the k-th state variable. The imaginary part of the traction control output acts on the k-th state variable. Let the real part of the error acting on the k-th state variable be denoted as . This represents the imaginary part of the error acting on the k-th state variable; Let be the fundamental control term function with respect to the real part of γ(t). Let be the imaginary part of the fundamental control term function with respect to γ(t); and This is the nonlinear gain coefficient, used to adjust the strength of the error nonlinear term; α and β are the nonlinear exponents corresponding to the error of the k-th state variable. α corresponds to the nonlinear exponent of the "sublinear term", β corresponds to the nonlinear exponent of the "superlinear term", and θk is the time delay compensation coefficient. The sublinear term α<1 can still provide sufficient control when the error is small, avoiding the "dead zone" effect; The superlinear term β>1 accelerates error decay and improves dynamic response speed when the error is large by amplifying the control effect.

[0044] (t-τ) represents the time delay. This represents the value of the real part of the error at the time delay of (t-τ). This represents the value of the imaginary part error at the (t-τ) time delay. By introducing the absolute value of the historical error, the control delay caused by the time delay is compensated, and the instability induced by the time delay is suppressed.

[0045] s is a dummy variable in the time integral. This shows the real part error value of the corresponding parameter at time s in the time integral. This represents the imaginary part error value of the corresponding parameter at time s during the time integral, and ρk is the integral gain coefficient, which controls the strength of the integral term.

[0046] In one embodiment of the present invention, combining Markov transition, hybrid time delay, and traction control characteristics, a Lyapunov function containing a time delay integral term and state-dependent weights is constructed, including: Where V(t) is a Lyapunov function that depends on time t and the Markov jump parameter γ(t). Let the real part of the error acting on the k-th state variable be denoted as . Let the imaginary part of the error acting on the k-th state variable be denoted as . Let be the real part of the error acting on the j-th associated state yj(t). Vj represents the imaginary part of the error acting on the j-th associated state yj(t); Vτ(t) is the discrete time delay compensation term, and Vσ(t) is the distributed time delay compensation term. |·|p is in p-norm form. In this embodiment, p>2 is used to quantify the error magnitude. Compared with the quadratic form (p=2), it is more robust and can adapt to nonlinear or non-smooth systems.

[0047] In one embodiment of the present invention, based on Filippov differential inclusion theory and Jensen integral inequality, detecting whether the derivative of the Lyapunov function is less than or equal to the controller's attenuation suppression and time-delay coupling terms includes: By using Filippov differential inclusion theory and Jensen's inequality, we can test whether the derivative of the Lyapunov function is less than or equal to the controller's attenuation term and time-delay coupling term, i.e., whether the following inequality holds: in, , 2. 3 and α is the attenuation coefficient; p is the norm in the p-norm form; α and β are the nonlinear exponents corresponding to the error of the k-th state variable. The term represents the time-delay disturbance, τ is the time-delay constant, and supV(s) represents the maximum energy within the time-delay interval, reflecting the effect of time-delay coupling on stability.

[0048] If the inequality holds, then time converges, and fixed-time synchronization is achieved; otherwise, time does not converge.

[0049] Used to measure the "energy" or "error magnitude" of a system's deviation from its equilibrium point. Its derivative... The rate of change of system energy is a core indicator for judging stability.

[0050] Linear decay term ‎: Corresponding to the damping term in the driving system, it reflects the inherent linear dissipation characteristics of the system and inhibits the unlimited growth of state variables.

[0051] Nonlinear decay term ‎: Related to nonlinear coupling terms in the drive system. This is the equivalent attenuation coefficient after adjusting the nonlinear control parameters. It is used to enhance the suppression effect under nonlinear dynamics. For example, superlinear attenuation can quickly suppress large errors.

[0052] This corresponds to the time-delay coupling term in the drive system. τ is the time-delay constant. The intensity of time-delayed disturbances reflects the potential instability of the system.

[0053] In one embodiment of the present invention, ensuring convergence at a fixed time by adjusting control parameters includes the following steps: If the inequality between the controller's attenuation suppression and time-delay coupling terms does not hold, and the time does not converge, then by constructing a discrete time-delay compensation term and a distributed time-delay compensation term in the Lyapunov function containing time-delay integral terms and state-dependent weights, the system can satisfy the condition that the derivative of the Lyapunov function is less than or equal to the controller's attenuation suppression and time-delay coupling terms.

[0054] The above embodiments are only used to illustrate the technical methods of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical methods of the present invention without departing from the spirit and scope of the technical methods of the present invention.

Claims

1. A fixed-time synchronization method for a discrete-valued recurrent neural network with time-delay, characterized in that, The method comprises the following steps: According to the external disturbance and mixed time delay, and introducing Markov jump parameters, a model of discontinuous complex-valued bidirectional associative memory neural network with time delay is established; The complex-valued variables and disturbances of the complex-valued neural network are separated into real and imaginary parts, the influence factor of time-varying time delay is introduced, and the error system is obtained according to the equivalent real-valued subsystem; The topology structure of the discontinuous complex-valued bidirectional associative memory neural network with time delay is described by a continuous-time Markov chain, and the dynamic importance of the nodes of the Markov chain jump network is evaluated in real time based on the dynamic degree centrality and weight contribution of the nodes; The nodes with the highest comprehensive importance in a preset proportion are selected as dynamic important nodes, a controller is applied to the dynamic important nodes, and a traction control strategy based on the importance evaluation is set for the controller; Combined with the characteristics of Markov jump, mixed time delay and traction control, a Lyapunov function containing an integral term with time delay and a state-dependent weight is constructed; Based on the Filippov differential inclusion theory and Jensen integral inequality, it is detected whether the derivative of the Lyapunov function is less than or equal to the attenuation term of the controller to suppress and time delay coupling term, and by adjusting the control parameters, the fixed-time convergence is ensured.

2. The fixed-time synchronization method of a discrete-valued memory neural network with time delay according to claim 1, characterized in that, According to the external disturbance and mixed time delay, and introducing Markov jump parameters, a model of discontinuous complex-valued bidirectional associative memory neural network with time delay is established, comprising the following steps: According to the external disturbance and mixed time delay, and introducing Markov jump parameters, a model of discontinuous complex-valued bidirectional associative memory neural network with time delay is established, comprising the following steps: Wherein, xk(t) represents the kth state variable in the driving system, yj(t) represents the jth associated state variable in the driving system; ak is the damping coefficient of the kth state variable, dj is the damping coefficient of the jth associated state variable; bkj is the instantaneous coupling coefficient of the jth associated state yj(t) to the kth state variable xk(t); ckj is the time delay coupling coefficient of the jth associated state yj(t) to the kth state variable xk(t); wjk is the instantaneous coupling coefficient of the kth state variable xk(t) to the jth associated state yj(t); hk is the time delay coupling coefficient of the kth state variable xk(t) to the jth associated state yj(t); n is the total number of associated state variables, m is the total number of state variables, j∈(1,2,…,n), k∈(1,2,…,m); dkj is the integral coupling coefficient of the jth associated state to the main state in the integral term, ljk is the integral coupling coefficient of the main state to the associated state in the integral term; fj(yj(t)), fj(yj(t-τ)) are the nonlinear conversion functions of the jth associated state; gk(xk(t)), gk(xk(t-τ)) are the nonlinear conversion functions of the kth state variable; τ and σ are time delay parameters; Ik is a constant external input acting on the kth state variable, Jj is a constant external input acting on the jth associated state; Dk(t) is a time-varying disturbance / noise acting on the kth state variable, Ej(t) is a time-varying disturbance / noise acting on the jth associated state.

3. The fixed-time synchronization method of a discrete-valued memory neural network with time delay according to claim 1, characterized in that, The complex-valued variables and disturbances of the complex-valued neural network are separated into real and imaginary parts, and the influence factor of time-varying time delay is introduced, and it is converted into an equivalent real-valued subsystem, including the following steps: Since the state variables, connection weights, activation functions and external inputs of the complex-valued neural network are all complex numbers, the complex-valued neural network is separated into real and imaginary parts, and an equivalent real-valued subsystem is constructed, including: The synchronous error is defined as: ； After introducing the time delay into the complex-valued neural network model, the state variable is: ； where xk(t) represents the kth state variable in the drive system, represents the error corresponding to the kth state variable in the drive system, is the system error, is the real part of the error, is the imaginary part of the error, zj(t) is the synchronization error of the jth associated state, yj(t) represents the jth associated state variable in the drive system, represents the error corresponding to the jth associated state variable in the drive system, is the real part of the synchronization error of the jth associated state, is the imaginary part of the synchronization error of the jth associated state.

4. The fixed-time synchronization method of a discrete-valued memory neural network with time delay according to claim 1, characterized in that, According to the real-valued subsystem, the error system is obtained, including the following steps: The Filippov selection set elements of the associated state nonlinear function corresponding to each associated state Markov jump parameter, and the real part time delay associated error term and the imaginary part time delay associated error term are introduced, and the error of the real part is derived to obtain the error system of the real-valued subsystem: Where ak is the damping coefficient of the kth state variable, bkj is the instantaneous coupling coefficient of the jth associated state yj(t) to the kth state variable xk(t); ckj is the time delay coupling coefficient of the jth associated state yj(t) to the kth state variable xk(t), dkj is the integral coupling coefficient of the jth associated state to the main state in the integral term, the superscripts R / I of bkj, ckj and dkj distinguish the real / imaginary coupling strength, the superscript R is the real part coupling strength, the superscript I is the imaginary part coupling strength, and the coefficient sign indicates the action direction; Uk R (t) is a control input for the error real part, Ak R (t) is a disturbance cancellation term; and is an element of the Filippov selection set of the associated state nonlinear function corresponding to the jth associated state Markov jump parameter, is an element of the Filippov selection set of the imaginary part nonlinear function corresponding to the jth associated state Markov jump parameter. τ is a time delay parameter representing the delay of the influence of the historical state on the current error, is a real part time delay associated error term, is an imaginary part time delay associated error term; is an integral type coupling term, where σ is the integral window length, s is a dummy variable in the time integral, and the integral term reflects the cumulative effect of the error over the time interval [t-σ, t], , The (s) added later indicates the element value of the corresponding parameter at time s in the time integral.

5. The fixed-time synchronization method of a discrete-valued complex-valued recurrent neural network with time delay according to claim 1, characterized in that, Real-time node dynamic degree centrality and weight contribution based on Markov chain jump network are used to evaluate the comprehensive importance of nodes, including the following steps: According to the Markov chain jump network, the value of the Markov jump parameter γ(t) is H={1,2,…,Nr}, and the network is divided into Nr topological states, each state corresponds to a fixed connection weight matrix; For each topological state i∈H in the topological structure of the time-delay discontinuous complex-valued bidirectional associative memory neural network, the comprehensive importance index Ih(i) of node h is defined, which combines the degree centrality and weight contribution of the node to calculate the state-dependent comprehensive importance index of the node: Where bkj(i) is the instantaneous coupling coefficient of the jth associated state yj(t) to the kth state variable xk(t) in the topological state i; ckj(i) is the time delay coupling coefficient of the jth associated state yj(t) to the kth state variable xk(t) in the topological state i, dkj(i) is the integral coupling coefficient of the jth associated state to the main state in the integral term in the topological state i, and wjk(i) is the maximum feedback weight of other nodes to node h.

6. The fixed-time synchronization method of a discrete-valued memory neural network with time delay according to claim 5, characterized in that, The nodes with the highest comprehensive importance in a preset proportion are selected as dynamic important nodes, including the following steps: For each topological state i, the comprehensive importance index Ih(i) of node h is sorted in descending order, and the top q% nodes are selected as the candidate controlled node set in this state; Screening the intersection of all candidate sets of nodes in Nr topological states, and defining the intersection as the set of nodes with high influence; For all candidate nodes not defined as high influence nodes, only when the corresponding node has a comprehensive importance index Ih(i) exceeding the threshold, temporarily join the set of nodes with high influence, realize the real-time dynamic adjustment of the set of nodes with high influence, and the nodes in the set of nodes with high influence are determined as dynamic important nodes.

7. The fixed-time synchronization method of a discrete-valued complex-valued recurrent neural network with time delay according to claim 1, characterized in that, Apply a controller to the dynamic important nodes, and set a traction control strategy based on importance evaluation for the controller, including: Real part traction control strategy based on importance evaluation: Virtual part traction control strategy based on importance evaluation: wherein is the real part of the traction control output acting on the kth state variable, is the imaginary part of the traction control output acting on the kth state variable, is the real part of the error acting on the kth state variable, is the imaginary part of the error acting on the kth state variable; is the real part of the base control term function for γ(t), is the imaginary part of the base control term function for γ(t); and is a non-linear gain coefficient for adjusting the strength of the error non-linear term; α and β are the nonlinear indices corresponding to the error of the kth state variable, α corresponds to the nonlinear index of "sublinear term", β corresponds to the nonlinear index of "superlinear term", and θk is the time delay compensation coefficient; (t - τ) is the time lag, represents the value of the real part error at the time lag (t - τ), represents the value of the imaginary part error at the time lag (t - τ). s is a dummy variable in the time integration, Rek(s) represents the real part error value of the corresponding parameter at time integration s, Imk(s) represents the imaginary part error value of the corresponding parameter at time integration s, and pk is the integration gain coefficient.

8. The fixed-time synchronization method of a discrete-valued complex-valued recurrent neural network with time delay according to claim 1, characterized in that, Combined with Markov jump, mixed time delay and traction control characteristics, a Lyapunov function containing time delay integral term and state dependent weight is constructed, including: where V(t) is a Lyapunov function depending on time t and Markov jump parameter γ(t), is the real part of the error acting on the kth state variable, is the imaginary part of the error acting on the kth state variable, is the real part of the error acting on the jth associated state yj(t), is the imaginary part of the error acting on the jth associated state yj(t); Vτ(t) is a discrete time-delay compensation term, Vσ(t) is a distributed time-delay compensation term; | · |p is a p-norm form.

9. The fixed-time synchronization method of a discrete-valued memory neural network with time delay according to claim 8, characterized in that, Based on the theory of Filippov differential inclusion and Jensen integral inequality, it is detected whether the derivative of Lyapunov function is less than or equal to the attenuation term of the controller to suppress and delay coupling term, including: Through the theory of Filippov differential inclusion and Jensen inequality, it is detected whether the derivative of Lyapunov function is less than or equal to the attenuation term of the controller to suppress and delay coupling term, that is, whether the following inequality is established: wherein, , 2, 3 and are decay coefficients; p is a norm in the p-norm form, and a and β are non-linear indices corresponding to the error of the kth state variable. is the time delay disturbance term, τ is the time delay constant, and supV(s) represents the maximum energy in the time delay interval; If the inequality is established, the time converges, and the fixed time synchronization is completed; otherwise, the time does not converge.

10. The fixed-time synchronization method of a discrete-valued memristive neural network with time delay according to claim 8, characterized in that, By adjusting the control parameters, fixed time convergence is ensured, including the following steps: When the inequality that the derivative of Lyapunov function is less than or equal to the attenuation term of the controller to suppress and delay coupling term is not established, the time does not converge, then through the discrete time delay compensation term and distributed time delay compensation term in the constructed Lyapunov function containing time delay integral term and state dependent weight, the system satisfies the derivative of Lyapunov function is less than or equal to the attenuation term of the controller to suppress and delay coupling term.

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