Event-triggered containment pulse control method based on large-scale nonlinear system

Abstract

The invention relates to the technical field of nonlinear system control, in particular to an event trigger containment pulse control method based on a large-scale nonlinear system. Comprising the following steps: constructing a nonlinear system model with time delay; designing an event triggering mechanism based on a Lyapunov function, and dynamically determining a pulse control moment; a containment control strategy is adopted, and pulse control is only applied to part of key nodes in the network at each pulse moment; through the Lyapunov stability theory, the consistent stability and the global asymptotic stability of the system are verified. According to the event trigger containment pulse control method based on the large-scale nonlinear system provided by the invention, a dual resource saving mechanism is constructed through collaborative design of an event trigger mechanism and containment control; the event triggering mechanism ensures that the control action is triggered only when the system state deviates from the expected track to a certain degree, unnecessary and frequent control updating in periodic control is avoided, and the communication bandwidth is greatly saved.

Description

Technical Field

[0001] This invention relates to the field of nonlinear system control technology, and in particular to an event-triggered restraint pulse control method for large-scale nonlinear systems. Background Technology

[0002] With the widespread application of large-scale nonlinear systems such as neural networks and multi-agent systems in fields such as information processing and secure communication, the problem of stable control has become increasingly prominent. Traditional control methods, such as continuous control or periodic pulse control, suffer from problems such as high control frequency, high resource consumption, and slow response.

[0003] In existing technologies, pulse control can effectively reduce control costs, but it mostly adopts time-triggered mechanisms, which are difficult to adapt to dynamic changes in the system; restraint control can achieve network stability by controlling some nodes, but it lacks a dynamic triggering mechanism; event-triggered control can execute control on demand, but it is difficult to guarantee stability and Zeno-free behavior in systems with large time delays.

[0004] Therefore, there is an urgent need for an efficient control method that can coordinate event triggering, pulse control, and restraint strategies to significantly reduce the consumption of communication and computing resources while ensuring system stability.

[0005] To address this issue, an event-triggered restraint pulse control method based on large-scale nonlinear systems is designed to provide an alternative technical solution. Summary of the Invention

[0006] Therefore, it is necessary to provide an event-triggered restraint pulse control method based on a large-scale nonlinear system to address the aforementioned technical problems.

[0007] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0008] An event-triggered restraint pulse control method based on a large-scale nonlinear system includes the following steps:

[0009] Construct a nonlinear system model with time delay, the system model being expressed as:

[0010] ;

[0011] ;

[0012] in, For system status, Due to time lag, The pulse moment;

[0013] Design an event triggering mechanism based on Lyapunov functions to dynamically determine the pulse control timing. The triggering condition is:

[0014] ;

[0015] in, It is a Lyapunov function. For trigger parameters;

[0016] A restraint control strategy is adopted, in which pulse control is applied only to some key nodes in the network at each pulse moment;

[0017] Using Lyapunov stability theory, the uniform stability and global asymptotic stability of the system are verified, and it is proven that the event-triggered mechanism is free from Zeno behavior.

[0018] As a preferred embodiment of the event-triggered restraint pulse control method based on a large-scale nonlinear system provided by the present invention, a forced pulse sequence is introduced into the event triggering mechanism. To ensure the minimum pulse interval and prevent Zeno's phenomenon.

[0019] As a preferred embodiment of the event-triggered restraint pulse control method based on a large-scale nonlinear system provided by the present invention, in the restraint control strategy, the selection of the controlled node is based on dynamic error assessment, and the control law is defined as:

[0020] ;

[0021] in, For the set of controlled nodes, This is the pulse gain.

[0022] As a preferred embodiment of the event-triggered restraint pulse control method based on a large-scale nonlinear system provided by the present invention, the Lyapunov function V(t) satisfies the following condition:

[0023] ;

[0024] ;

[0025] ;

[0026] in, for Class function, , It is a positive real number.

[0027] An event-triggered restraint pulse control method based on large-scale nonlinear systems is applied to the synchronization control of chaotic neural networks to achieve image encryption and decryption.

[0028] A computer-readable storage medium having a computer program stored thereon, which, when executed, implements an event-triggered restraint pulse control method based on a large-scale nonlinear system.

[0029] An electronic device includes a processor and a memory, the memory storing a computer program, and the processor executing the program to implement an event-triggered restraint pulse control method based on a large-scale nonlinear system.

[0030] It is clear without a doubt that the technical solution described above in this application can solve the technical problem that this application aims to address.

[0031] Meanwhile, through the above technical solutions, the present invention has at least the following beneficial effects:

[0032] This invention provides an event-triggered restraint pulse control method for large-scale nonlinear systems. Through the collaborative design of the event-triggered mechanism (ETM) and restraint control, a dual resource-saving mechanism is constructed. The event-triggered mechanism ensures that control actions (pulses) are only triggered when the system state deviates from the desired trajectory to a certain extent, avoiding unnecessary and frequent control updates in periodic control and greatly saving communication bandwidth. At the same time, the restraint control strategy applies control inputs only to a few key nodes in the network, rather than all nodes, significantly reducing the space complexity and computational overhead required for each control action. This combination of "on-demand control" and "precise control" makes this invention particularly suitable for resource-constrained embedded systems, networked control systems, and large sensor networks.

[0033] This invention combines the Lyapunov-Razumikhin method for stability analysis, and the derived sufficient conditions can effectively compensate for the negative impact of time delay on system stability. This enables the proposed control strategy to maintain system stability and convergence performance even when faced with inherent delays caused by signal transmission and processing, thereby enhancing the applicability and robustness of the control system in real-world complex environments. Attached Figure Description

[0034] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the following description of the embodiments will be briefly introduced. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0035] Figure 1 This is a schematic diagram of the trajectory of the leader system of the present invention;

[0036] Figure 2 This is a schematic diagram of the network topology of the intelligent agent system of the present invention;

[0037] Figure 3 This is a schematic diagram of variable synchronization in the first state of the present invention;

[0038] Figure 4 This is a schematic diagram illustrating that the first state variable of the present invention has not been synchronized.

[0039] Figure 5 This is a schematic diagram of the event-driven pulse sequence of the present invention;

[0040] Figure 6 This is a schematic diagram of each event in the present invention at a fixed scale;

[0041] Figure 7 This is a schematic diagram of the original plaintext image of the present invention;

[0042] Figure 8 This is a schematic diagram of the encrypted text of the present invention after being encrypted by the leader;

[0043] Figure 9 This is a schematic diagram comparing the original image and the encrypted image of the present invention;

[0044] Figure 10 This is a schematic diagram of the decrypted image of the present invention. Detailed Implementation

[0045] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0046] To enable those skilled in the art to better understand the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings.

[0047] It should be noted that, unless otherwise specified, the embodiments and features and technical solutions in the present invention can be combined with each other.

[0048] It should be noted that similar labels and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.

[0049] An event-triggered restraint pulse control method based on large-scale nonlinear systems.

[0050] 1. Information cognition

[0051] Consider the following broad class of nonlinear systems affected by time delays:

[0052] (1)

[0053] Here, the state of the system is given by a vector. The left derivative is denoted as . Nonlinear functions and (from Mapped to The construction of ) satisfies and This condition ensures that the origin is an equilibrium point, thus guaranteeing that system (1) has a trivial solution. The pulse time series is composed of Given any element This represents a specific pulse moment. Furthermore, the parameters... ,and Represents a constant coupled time delay, where Defined as the maximum time delay value, i.e. .

[0054] To lay the foundation for the main theoretical development, we will first introduce some basic definitions and assumptions.

[0055] Definition 1: Let It is a local Lipschitz function. The upper left Dini derivative along the system trajectory is defined as:

[0056] ;

[0057] Definition 2: If for any given bound A corresponding radius can be found. This makes any point from the origin... Within the neighborhood The starting track All are guaranteed to remain in all future time periods Within the neighborhood, system (1) possesses the uniform stability (US) property:

[0058] ;

[0059] "Consistency" here means The choice is only The function, and the start time Irrelevant.

[0060] Definition 3: If the equilibrium point of system (1) at the origin is both Lyapunov stable and globally attractive, then it is said to have global asymptotic stability (GAS). Its definition is as follows:

[0061] (i) For each A corresponding one can be determined. Such that if the system starts from the origin within the radius ( Initially, its solution In all The inside is restricted to Within the radius, i.e. .

[0062] (ii) Each trajectory of the system, regardless of its initial position Regardless, everything converges to the origin over time. This is represented as... 0.

[0063] Definition 4: A sequence The average pulse interval is considered to be less than The premise is that a positive number can be found. and a positive integer This makes it possible for all The number of pulses satisfies the following lower bound:

[0064] ;

[0065] here, Represents a sequence In the time window The number of pulse events within.

[0066] Definition 5: In The pulse restraint rate at time is defined as

[0067] ;

[0068] The restraint control law is time-varying and depends on the pulse timing. The lower bound will be determined.

[0069] Assumption 1: For nonlinear functions For each i∈ Both have constants , so that for any The following inequalities hold:

[0070] (2)

[0071] Assumption 2: For nonlinear functions There exists a nonnegative constant. , such that for any ;

[0072] (3)

[0073] Symbol Explanation: We establish the following standard symbols. Let and These are the standard sets of nonnegative real numbers, natural numbers, and positive integers, respectively. Space It refers to having the Euclidean norm | The space of real numbers, and It refers to The space of matrices. Regarding the properties of matrices, Represents a symmetric matrix It is positive definite (negative definite). Represents an identity matrix of appropriate dimension. Complete a symmetric matrix by transposing it. In the function context, Indicates from the interval to the set The set of functions, which are continuous except that they are left-continuous at a finite number of points. The norm of a function on the delay interval is defined as... If the function If it is continuous, strictly increasing, and starts from the origin, then it is called... Class function; if it is also unbounded, then it is... Function class. Collection. It is a function that is positive definite and locally Lipschitz, contained in its state variables. Used to represent the maximum value of two scalars.

[0074] II. Main Results

[0075] The main focus is on establishing sufficient conditions to ensure the Lyapunov stability of system (1) through event-triggered control methods.

[0076] Lemma 1: If there exists a function And there exists a function And some normal numbers and A system is uniformly stable (US) if the following conditions are met:

[0077] (H1):

[0078] ;

[0079] ;

[0080] (H2):

[0081] ,when For all ;

[0082] (H3):

[0083] ;

[0084] (H4):

[0085] ;

[0086] (H5):

[0087] and ;

[0088] Among them, the pulse time sequence Generated by the following Lyapunov-based event triggering mechanism:

[0089] (4)

[0090] in, It is a Lyapunov function. For trigger parameters;

[0091] Proof: The key to proving the Lyapunov stability of system (1) lies in the fact that for any given Lyapunov... Find a suitable However, since the number of pulses is not predetermined under the event-triggered scheme, a comprehensive analysis is needed to address every possible scenario (zero, finite, or infinite pulses). We propose a specific... Candidate value, defined as The remainder of this proof is dedicated to proving that this choice is valid, ensuring that any choice made from... The starting trajectory for all They are all restricted to Within, and regardless of the number of events triggered.

[0092] Case I: The event never occurs. According to ETC(4), it can be observed that:

[0093] For all ;

[0094] Case II: Considering the system undergoes a finite number of ( (Next) pulse events, these events occur at time... , making ,in .

[0095] for This can be derived from the ETC strategy (4). ,and ,for .

[0096] From condition (H3), we can obtain:

[0097]

[0098]

[0099] for From the ETC strategy (4), we can obtain:

[0100] ,and , ;

[0101] exist Then, from (4) and (H3), we can obtain:

[0102] ;

[0103] ;

[0104] For general Assuming the result is correct Established. Then, from (1) and (H3), we can obtain:

[0105] ;

[0106] exist At that time, based on condition (H3), we have:

[0107] ;

[0108] ;

[0109] For general Assuming the result is correct It is established. So, for... have:

[0110] ;

[0111] exist At that time, based on condition (H3), we have:

[0112] ;

[0113] Therefore, the conclusion is:

[0114] ;

[0115] According to the principle of mathematical induction, this statement applies to all... Valid. For any This can be verified for 1.,2.,.,.N;

[0116] ;

[0117] Furthermore, no other events will be triggered in this situation, as agreed. The conclusion is:

[0118] ;

[0119] Overall, it can be verified from (H5)

[0120] (5)

[0121] And we can conclude that:

[0122] ;

[0123] Case III: Consider an event-triggered strategy to generate an infinite pulse sequence The main objective is to prove that Zeno behavior does not exist in system (1). Specifically, it will be proven that the time interval between any two consecutive pulses... There is a guaranteed positive lower bound. The proof begins with... The analysis of the intervals is detailed below:

[0124] ;

[0125] Therefore, we define:

[0126] but And there are:

[0127] , ;

[0128] This means that for all :

[0129] ;

[0130] for ,have .

[0131] definition Make ,and:

[0132]

[0133] And we have:

[0134]

[0135] This means that for

[0136] ;

[0137] Applying a recursive method to the aforementioned analysis, we verify that for any This property holds true. Specifically, let's first examine... Establish this property, and then prove it through recursive steps. Established:

[0138] ;

[0139] ;

[0140] This means that for ;

[0141] ;

[0142] Suppose this statement is for a fixed It is valid. This proves its validity. This also holds true. And it is defined as follows:

[0143] ;

[0144] Depend on The continuity can be obtained

[0145] ;

[0146] Using the recursive structure assumed in the previous steps, we have

[0147] ;

[0148] And for all ;

[0149] ;

[0150] Now, for any ,estimate exist The value on. Because The following segment boundaries exist:

[0151] ;

[0152] Combine each From the recursion bound, we obtain:

[0153] ;

[0154] ;

[0155] Then, the uniformity limit becomes:

[0156] ;

[0157] By definition We have guaranteed the condition for consistent stability, namely For all This is true. Therefore, even with an infinite number of triggering events, this proves the uniform stability (US) of the pulse system (1).

[0158] By mathematical induction, this conclusion applies to all Both are true. Based on condition (H2), we have...

[0159] (6)

[0160] By setting and ,get .

[0161] and ;

[0162] We define Then (6) can be rewritten as

[0163] (7)

[0164] From both sides of (7) arrive By integration, we obtain:

[0165] ;

[0166] Notice that the left side is

[0167] ;

[0168] Therefore we get

[0169] ;

[0170] ;

[0171] On the other hand, according to the definition of pulse mapping,

[0172] ,so ;

[0173] turn out,

[0174] ;

[0175] Rewrite the inequality:

[0176] ;

[0177] right Summing, we get:

[0178] ;

[0179] Because when hour ,so

[0180] ;

[0181] Therefore, due to the pulse time sequence If the divergence extends to infinity, it can be concluded that the system does not exhibit the Zeno phenomenon.

[0182] Furthermore, if we consider the case of a continuous sequence of triggered events, and adopt a reasoning approach similar to that in case II, we can arrive at the same conclusion as in (5):

[0183] ;

[0184] ;

[0185] and

[0186] ;

[0187] Whether the number of pulses is zero, finite, or infinite, we have proven that for any ,choose All can be guaranteed Contains For all This result directly proves the uniform stability (US) of system (1).

[0188] To analyze the global asymptotic stability (GAS) of the impulsive system, an event-triggered mechanism (ETM) was employed. To explicitly guarantee the asymptotic convergence of all system trajectories, the standard ETM framework integrates a forced impulsive time series. Enhancements have been made. This modification resulted in the following revised ETM:

[0189] ;

[0190] Among them, candidate trigger time Given by the following formula

[0191] (8)

[0192] Forced pulse time series Assume the conditions are met.

[0193] ;

[0194] This condition guarantees that there exists a strictly positive minimum duration between any two consecutive pulses. For later use, the set of all pulse sequences that satisfy this property is denoted as . .

[0195] Lemma 2: If it exists Class function A locally Lipshitz-continuous Lyapunov function and positive numbers If conditions (H1)-(H5) and the following inequality (9) are satisfied, then the system operating under the event-triggered mechanism (ETM) is globally asymptotically stable (GAS). Event triggering time Determined by the designed ETM, and the sequence satisfy:

[0196] (9)

[0197] Proof: According to the definition of Event Triggered Mechanism (ETM), the system will generate an infinite sequence of impulse events, which we represent as follows: As a prerequisite for Lyapunov stability analysis, it must first be determined that the system does not exhibit the Zeno phenomenon. To this end, our proof considers the following three different cases:

[0198] Case (a): If the pulse sequence degenerates into a forced sequence Therefore, due to the existence of control conditions, Zeno's behavior is excluded:

[0199] ;

[0200] Case (b): Next, let's assume that at least some of the impulses are triggered by event conditions, which means... It is not empty. We will use proof by contradiction to rule out Zeno's behavior. Suppose that, conversely, there exists a finite Zeno time. This means that in the interval An infinite number of pulses occurred within it. We define... Based on this, we can determine a pulse time subsequence. Included in the interval Inside, for a certain integer , making when hour .

[0201] Assume it exists For a certain According to the design of ETM, there can only be one such sequence. For all We, on the other hand, have .

[0202] Now, for any and Using a similar argument to that in (H2) and (4), we obtain:

[0203] ;

[0204] This means:

[0205] ;

[0206] Therefore, through recursion:

[0207] ;

[0208] Given the above circumstances, this led to when hour This is related to This contradicts the assumption that there are finite elements.

[0209] If in sequence There is no such thing in China. ,Right now A similar contradiction would also occur. Therefore, Zeno's phenomenon is excluded in case (b).

[0210] Case (c): If the pulse sequence Complete with event triggering sequence The case of overlap has already been covered in the analysis of Theorem 1, which has proven to exclude the Zeno phenomenon.

[0211] Based on the foregoing analysis of all possible scenarios, we can conclude that the system running under the proposed ETM does not exhibit Zeno behavior.

[0212] To establish a GAS, start from the triggering conditions. It can be observed that:

[0213] ;

[0214] Applying a recursive argument similar to that in Theorem 1, for any We got

[0215] (10)

[0216] Therefore, based on the above inequalities and conditions (H1) and (H5), the standard argument is sufficient to conclude that the system (1) operating under the proposed ETM is globally asymptotically stable (GAS).

[0217] Example 2

[0218] Reference Figures 1-10 An application is disclosed based on the above embodiment one.

[0219] I. Specific Applications

[0220] Consider the following nonlinear control system:

[0221] (11)

[0222] make The desired trajectory is described by the following formula:

[0223] ;

[0224] Define error signal The pulse restraint controller is as follows:

[0225] ;

[0226] Pulse time sequence It has two properties: it is strictly increasing, that is... And it is non-Zeno, that is The control strategy is defined as follows: at time... The set of nodes whose pulses are restrained is denoted as and constant pulse gain .item This represents the standard Dirac function used to simulate pulses.

[0227] Using the proposed pulse control (13), the error system is described by the following equation:

[0228] (14)

[0229] in and exist The time is left-continuous.

[0230] The initial condition for (14) is defined by the following equation:

[0231] ;

[0232] in, and . Indicates the interval Mapped to A continuous vector-valued function space.

[0233] Theorem 1: Assume the average pulse interval is less than Then, under assumption 1, if there exists a symmetric positive definite matrix... Positive definite diagonal matrix and positive numbers If the following inequality holds, then the error system (14) is uniformly stable:

[0234] (16)

[0235] (17)

[0236] (18)

[0237] in ;

[0238] Proof: We choose the following Lyapunov function as a candidate:

[0239] (19)

[0240] because Obviously satisfy:

[0241] ;

[0242] We calculate The trajectory along (14) is in The derivative over time.

[0243] ;

[0244] For any matrix Using standard inequalities For a positive definite diagonal matrix We have:

[0245]

[0246] ;

[0247] Based on assumption 1, we can define the nonlinear term as follows:

[0248] ;

[0249] Similarly,

[0250] ;

[0251] Combining these inequalities, we obtain:

[0252] ;

[0253] LMI conditions Therefore, the first part of the inequality is negative definite. For the second part, using condition (18), We have:

[0254] ;

[0255] Given that (16) is negative definite, the inequality This is true. It is noteworthy that this result is true without invoking the Lyapunov-Razumikhin condition, and is therefore stronger than a necessary condition. Therefore, condition (H2) is satisfied. At the pulse time... The value of the Lyapunov function becomes:

[0256] Using pinning rate From the definition, we can write:

[0257]

[0258] Substituting it back, we get:

[0259] ;

[0260] Applying the condition of theorem (18), we get Therefore, (H3) is satisfied. Since these conditions have been shown to ensure that all requirements (H1)-(H5) of Lemma 1 are satisfied, we conclude that the origin of the error system (14) is uniformly stable. Proof complete.

[0261] II. Numerical Examples

[0262] Example 1: Synchronization of multi-agent systems.

[0263] To verify the aforementioned theoretical conclusions, the leader system is described by the following neural network:

[0264] (20)

[0265] The system's dynamic equations can be compactly expressed as:

[0266] (twenty one)

[0267] in, and .

[0268] The activation function is defined as

[0269] (twenty two)

[0270] And the external input vector is .therefore, . Figure 1 The leader was drawn from the initial state. The initial chaotic trajectory.

[0271] We studied a... A network composed of individual cellular neural subsystems; Figure 2 The topology shown defines the coupling matrix. According to Theorem 1, a feasible solution can be obtained by applying the MATLAB LMI toolbox:

[0272] ;

[0273] ;

[0274] In the simulation, we use the obtained matrix To construct the Lyapunov function, we set the impulse gain. and event triggering parameters The coupling strength is set to... ,and At each pulse moment, 4 out of 8 follower agents are pinned. The initial state of the follower agents is... Randomly selected from within the range.

[0275] Simulation results are presented in Figure 3 In the middle, and Figure 4 This shows the behavior of the agent under the first state variable without any control. It is clear that the agent's trajectory does not converge to the leader's trajectory, indicating that control is necessary for synchronization. Figure 3 The evolution of the first state variables of the leader and all eight follower agents over time is shown. It is clearly observed that, despite their initial random positions, all follower agents converge rapidly and synchronize with the leader system's trajectory. This result strongly demonstrates the effectiveness of the proposed event-triggered pinning control strategy.

[0276] Figure 5The sequence of control pulses generated by the designed Event Triggered Mechanism (ETM) is demonstrated. It is evident that the control action is aperiodic, occurring only when Lyapunov conditions determine that intervention is necessary to maintain synchronization, thus confirming the resource-saving characteristics of the proposed control scheme.

[0277] at last, Figure 6 The pinning rate at each pulse moment was plotted. The pinning rate is defined as the ratio of the sum of squared errors of the pinned nodes to the total sum of squared errors of all nodes. The consistently high values ​​indicate that the dynamic pinning strategy effectively targets the agent that contributes the most to the total error, thereby maximizing control efficiency.

[0278] Example 2: Application in image encryption

[0279] To demonstrate a practical application, we designed a secure image encryption scheme based on a leader-follower synchronization paradigm using a chaotic cellular neural network. The basic principle is that the leader system generates a chaotic sequence to encrypt the image, while the follower system independently regenerates the same sequence for decryption by achieving high-precision synchronization.

[0280] The dynamics of both the leader (encryption) and follower (decryption) systems are controlled by the same set of matrices:

[0281] ;

[0282] An event-triggered pulse control strategy with the same parameters as in Example 1. This ensures that followers can keep pace with the leader.

[0283] The leader's encryption process: The encryption process begins at the leader's end. (Selection of standard) Grayscale images (such as) Figure 7 (As shown) is the plaintext to be encrypted. The leader system generates a chaotic sequence. Then, this sequence is used to perform permutation and diffusion operations on the plaintext image. The result of this process is... Figure 8 The ciphertext shown is visually indistinguishable from random noise, completely obscuring the content of the original image.

[0284] Histogram analysis was performed to quantitatively assess the security of the leader's encryption. Figure 9 (a) shows the histogram of the original image, which has a highly non-uniform and predictable structure. The histogram of the encrypted text (e.g.) Figure 9 (b) becomes approximately flat and uniformly distributed. This transformation eliminates the statistical pattern of the plaintext, thus providing strong resistance to statistical analysis attacks.

[0285] The feasibility of the entire scheme depends on whether the followers can perfectly decrypt the ciphertext. This is achieved through synchronization. Although the follower system starts from different initial states, it receives common system parameters and synchronizes its dynamics with the leader through an event-triggered control mechanism. This synchronization allows the followers to locally regenerate a secret sequence with the leader. Identical chaotic sequences .

[0286] Use this self-generated sequence The follower receives the ciphertext ( Figure 8 Perform inverse permutation and inverse diffusion operations. The decryption result is as follows: Figure 10 As shown, the decrypted image is a lossless and perfect reconstruction of the original plaintext image. This successful decryption ultimately verifies the accuracy of the synchronization mechanism and the integrity of the cryptographic system.

[0287] In summary, simulation results demonstrate that the proposed encryption scheme is both secure and practical. The leader effectively encrypts the data, while the follower, through robust synchronization, can reliably decrypt the data without sharing the chaotic key beforehand.

[0288] III. Conclusion

[0289] In this invention, we successfully solve the stabilization problem for a class of large-scale nonlinear systems with arbitrary time delays. We propose and rigorously analyze a novel event-triggered pulse pinning control strategy, demonstrating the powerful synergy among three efficient control paradigms. The core of this strategy is a dual resource-saving mechanism: a Lyapunov-based event-triggered mechanism (ETM) that determines when to apply control; and a pinning scheme that strategically selects the location of control application, thereby conserving both communication and computational resources. Through rigorous theoretical analysis based on the Lyapunov-Razumikhin method, we establish sufficient conditions to ensure uniform stability and global asymptotic stability of the controlled system. A key theoretical contribution is the formal proof that the proposed ETM is free of the Zeno phenomenon, guaranteeing its physical realizability and practical relevance. Comprehensive numerical simulations verify the practical effectiveness of the theoretical framework. Application in chaotic neural network synchronization clearly demonstrates that the controller can guide the network to the desired state with high precision. Furthermore, simulation results highlight the efficiency of the strategy; the aperiodic pulse sequence and dynamic pinning rate confirm that the control action is applied intelligently and with restraint. Finally, the synchronization scheme was successfully applied to secure image encryption and decryption tasks, highlighting the potential of this method in solving practical engineering problems. In summary, this work provides a robust, efficient, and theoretically sound control framework for complex nonlinear systems, with broad application prospects in secure communication, multi-agent systems, and other fields.

[0290] Example 3

[0291] Based on the above embodiments one and two, a computer-readable storage medium is disclosed, on which a computer program is stored, which, when executed, implements an event-triggered restraint pulse control method based on a large-scale nonlinear system.

[0292] Example 4

[0293] Based on the above embodiments one and two, an electronic device is disclosed, including a processor and a memory. The memory stores a computer program, and when the processor executes the program, it implements an event-triggered restraint pulse control method based on a large-scale nonlinear system.

[0294] The preferred embodiments of the present invention disclosed above are merely illustrative of the invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the invention to the specific implementations described. Clearly, many modifications and variations can be made based on the content of this specification. This specification selects and specifically describes these embodiments to better explain the principles and practical applications of the invention, thereby enabling those skilled in the art to better understand and utilize the invention. The invention is limited only by the claims and their full scope and equivalents.

Claims

1. An event-triggered restraint pulse control method based on a large-scale nonlinear system, characterized in that, Includes the following steps: Construct a nonlinear system model with time delay, the system model being expressed as: ； ； in, For system status, Due to time lag, The pulse moment; Design an event triggering mechanism based on Lyapunov functions to dynamically determine the pulse control timing. The triggering condition is: ； in, It is a Lyapunov function. For trigger parameters; A restraint control strategy is adopted, in which pulse control is applied only to some key nodes in the network at each pulse moment; Using Lyapunov stability theory, the uniform stability and global asymptotic stability of the system are verified, and it is proven that the event-triggered mechanism is free from Zeno behavior.

2. The event-triggered restraint pulse control method based on a large-scale nonlinear system according to claim 1, characterized in that, Introducing a forced pulse sequence into the event triggering mechanism To ensure the minimum pulse interval and prevent Zeno's phenomenon.

3. The event-triggered restraint pulse control method based on a large-scale nonlinear system according to claim 1, characterized in that, In a restraint control strategy, the selection of the controlled node is based on dynamic error assessment, and the control law is defined as: ； in, For the set of controlled nodes, This is the pulse gain.

4. The event-triggered restraint pulse control method based on a large-scale nonlinear system according to claim 1, characterized in that, The Lyapunov function V(t) satisfies the following conditions: ； ； ； in, for Class function, , It is a positive real number.

5. An event-triggered restraint pulse control method based on a large-scale nonlinear system is applied to the synchronization control of a chaotic neural network to achieve image encryption and decryption.

6. A computer-readable storage medium having a computer program stored thereon, said program, when executed, implementing an event-triggered restraint pulse control method based on a large-scale nonlinear system as described in any one of claims 1 to 5.

7. An electronic device comprising a processor and a memory, the memory storing a computer program, wherein the processor, when executing the program, implements an event-triggered restraint pulse control method based on a large-scale nonlinear system as described in any one of claims 1 to 5.

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