Switching system security control method under DoS attack considering subsystem importance

Abstract

A security control method for switching systems under DoS attacks considering subsystem importance is proposed. Considering the switching characteristics of the physical subsystem, a dynamic switching weight coefficient construction method is proposed to characterize the difference between the subsystem characteristics before and after switching. The dynamic switching weight coefficient is embedded in the attacker's cost function to adjust the energy allocation strategy of the attacker to hinder the transmission of the switching signal, maximizing the potential damage of the attack. Considering multiple denial-of-service attackers, a Stackelberg game model is established to describe the interaction between multiple denial-of-service attackers and defenders. A non-dominated sorting genetic algorithm with knee point selection mechanism is proposed to obtain the Pareto-Stackelberg equilibrium strategy. Under the equilibrium condition, the worst-case channel packet loss rate is obtained based on digital communication theory. In the physical layer, the optimal controller and average dwell time switching law are designed to ensure the exponential mean square stability of the networked switching system.

Description

Technical Field

[0001] This invention relates to the fields of networked handover systems and security control, and particularly to a security control method for handover systems under DoS attacks that take into account the importance of subsystems. Background Technology

[0002] Switching systems consist of a class of subsystems, typically described by differential or difference equations and coordinated by switching laws. With the rapid development of communication technology, shared networks play a crucial role in data transmission, driving the widespread application of networked switching systems, such as automotive transmission systems, robot control systems, aircraft automatic navigation control systems, and advanced traffic management systems. For switching systems, traditional point-to-point control is costly, inflexible, and difficult to maintain, while networked control technology can transmit data via wireless networks, enabling resource sharing, remote operation, and control. Therefore, networked switching systems, combining network control technology and the characteristics of switching systems, have attracted widespread attention. However, the openness of networks exposes networked switching systems to potential malicious attacks, such as denial-of-service attacks, spoofing attacks, and replay attacks. Among these network attacks, denial-of-service attacks are particularly noteworthy because they can cause significant system fluctuations and instability without prior knowledge of the system. To address these issues, it is urgent to explore security control strategies that can guarantee the security of networked switching systems under denial-of-service attacks.

[0003] As denial-of-service attacks become increasingly complex and severe, coordinated attacks by multiple attackers are becoming more common. To effectively capture the interactions between multiple attackers or the competition between attackers and defenders, game theory methods have been widely adopted in cross-layer frameworks. However, existing cross-layer security control research rarely considers the impact of physical layer system characteristics on the network layer. This weakens the connection between the two layers, making the established game model less accurate and the obtained equilibrium solution more conservative. Furthermore, in general, the attacker acts first and possesses all the information of the defender; the worst-case attack strategy can be obtained based on the established game equilibrium. This invention considers the switching characteristics of the physical layer and proposes a novel cross-layer security control strategy for networked switching systems. A dynamic switching weight representing the difference in subsystem characteristics before and after the switch is introduced into the attacker's cost function to adjust energy allocation. When the difference is large, more energy is allocated to block the transmission of the switching signal to maximize the attacker's destructive power. Subsequently, a non-dominated sorting genetic algorithm with a knee selection mechanism is proposed to obtain a Pareto-Staklberg equilibrium with a worst-case attack strategy. In addition, controllers and switching rules were designed to ensure the mean square exponential stability of the networked switching system under worst-case attack conditions. Summary of the Invention

[0004] This invention employs a novel hierarchical dynamic game to describe the interaction between the defender and multiple attackers, and incorporates dynamic switching weights into the attackers' cost function. Furthermore, a security control strategy is adopted at the physical layer to ensure the mean square exponential stability of the switching system. To achieve the above objectives, this invention adopts the following technical solution: A method for switching system security control under a DoS attack that considers the importance of subsystems includes: Step 1: Construct a closed-loop mathematical model of the discrete-time networked switching system and characterize random packet loss behavior using Bernoulli distribution; Step 2: Construct a Stakolberg cooperative game model in the network layer that considers the switching characteristics of the physical layer, including: Attacker and Defender Action Sets: Energy Allocation Strategies of Multi-Denial Attackers and Defenders on Dual Channels; Cost functions for attackers and defenders: Multi-denial-of-service attackers and defenders construct cost functions with the goal of optimizing channel signal-to-interference-plus-noise ratio and energy consumption; Dynamic adjustment strategy: Based on the switching characteristics of subsystems in the physical layer, dynamically adjust the attacker's energy allocation strategy on dual channels to construct the worst-case denial-of-service attack scenario; Step 3: Based on the Stakolberg cooperative game model established in Step 2, a solution algorithm based on the knee selection mechanism is proposed at the network layer to obtain the Pareto-Stakolberg equilibrium strategy, which guides the energy allocation of multi-denial-of-service attackers and defenders on the dual channels. Step 4: Consider the scenario where multiple denial-of-service attackers at the network layer cause dynamic packet loss in the physical layer, and solve for the optimal controller gain under the condition of modal synchronization of the subsystem and controller; comprehensively consider the open-loop and modal asynchronous operation of the system caused by dynamic packet loss, and design the average dwell time condition to ensure the exponential mean square stability of the networked switching system.

[0005] Furthermore, Step 1 constructs an attack and defense mathematical model of the network handover system under a denial-of-service attack, and uses the Bernoulli distribution to characterize the random behavior of packet loss, including: Step 1.1 Construct the following networked handover system, as shown in formula (1). (1) in, It is the system status. It is a control input. and Represents the system state matrix and control input matrix, where and These represent the dimensions of the system state vector and the control input vector, respectively. Indicates a switching signal, where It is the number of subsystems; when When, it indicates the first The subsystem in the first It is always in active mode; in addition... This indicates the function for switching dependencies, and its value range is... To maintain symbol consistency, the initial activity mode is defined as follows: ; Step 1.2 During data transmission in a networked handover system, sensors, acting as defenders, sample the system's state and handover signals, determining their transmission energy on the channel. Multiple denial-of-service attackers similarly allocate energy on the channel, disrupting data transmission. In this process, sensors and multiple attackers jointly determine the channel signal-to-noise ratio. Therefore, the first... The signal-to-interference-plus-noise ratio of the channel is (2) in, and They are in and Under the conditions, the defender and the first The energy allocated by an attacker on the channel; in addition... and These are the channel gains for the defender and the attacker, respectively. It is the Gaussian white noise energy in the environment; based on orthogonal amplitude modulation and digital communication theory, the relationship between the channel signal-to-interference-plus-noise ratio and the data transmission rate is established as follows: (3) in The given parameters are Gaussian. The function is ,in For Gauss The independent variable of the function, i.e., the lower limit of integration, For integration variables; Step 1.3 Under the above conditions, packet transmission indication functions are defined for system status and switching signals, respectively. and ; in the time, and These represent system status loss and successful reception, respectively. and These represent the switching signals at the 1st, 2nd, and 3rd respectively. Time loss and successful reception, among which Indicates the first The discrete moments of each mode switch; since the system state and the switching signal are transmitted separately, therefore, and They are independent random variables and follow the following Bernoulli distribution; (4) in, and These represent the packet delivery rate of the system status and the switching signal, respectively.

[0006] The discrete-time switching linear system mathematical model constructed in Step 1 of this invention provides a foundation for the subsequent construction of game theory models.

[0007] In a specific embodiment, the Stakolberg cooperative game model constructed at the network layer includes information sets, action sets, cost functions, and dynamic adjustment strategies for both the defender and the attacker, thereby dynamically adjusting the attacker's total energy based on the switching behavior characteristics in the physical layer. When considering... and In this case, the specific implementation plan is as follows: Step 2.1 Identify the participants in the Stakolberg cooperative game as multiple denial-of-service attackers and defenders; Step 2.2 The action sets of both sides are the energy allocation on two channels, specifically, the defender and the first... The action set of an attacker is defined as: (5) in, This represents a set of denial-of-service attackers. Number of attackers; This represents the joint energy allocation strategy of all attackers under the current mode switching scenario. This represents the set of all combined actions of the attackers; and These represent the total available transmission energy of the defender and the total available attack energy of all attackers, respectively. Step 2.3 Determine the cost functions for both attackers and defenders. Both attackers and defenders will consider signal-to-interference-plus-noise ratio (SIR) and energy consumption in their cost functions. Since the range of SIR and energy consumption is uncertain, normalization is required. Considering the impact of physical layer switching characteristics, a dynamic switching weight coefficient is added to the attacker's cost function to dynamically adjust the energy allocation across the dual channels. Therefore, the first... The cost function for each attacker and defender is: (6) in, and ,and It dynamically switches the weighting coefficients. Indicates the defender in the The energy cost coefficient corresponding to allocating a unit of transmission energy on a channel. Indicates the attacker in the The energy cost coefficient corresponding to the allocation of a unit of attack energy on each channel, where , Represents the system state channel. Indicates switching the signal channel; Step 2.4 The introduction of dynamic handover weight coefficients aims to quantify the degree to which subsystem characteristics change with handover behavior, and its definition is as follows; For networked handover systems, the first The representative eigenvalue of a subsystem is defined as: ,in It is the first Subsystem One eigenvalue; These are adjustment coefficients used to handle cases where zero appears in the denominator of eigenvalue operations; the subscript set of the stable subsystem is defined as... The subscript set of an unstable subsystem is defined as and ,in, To stabilize the total number of subsystems, Let be the total number of unstable subsystems, and satisfy . ; This indicates the index of the stable subsystem within the original system. Indicates the index of the unstable subsystem within the original system; and They represent the first The stable subsystem and the first Representative eigenvalues ​​of an unstable subsystem; when and When, it indicates that the system is at the twentieth ... Time by the first The subsystem switched to the first Each subsystem dynamically switches weight coefficients. for: (1) For ,and ,have (7) (2) For ,and ,have (8) (3) For ,and ,have (9) (4) For ,and ,have (10) Based on the game theory model established in the aforementioned network layer, an algorithm is designed in the network layer to obtain the Pareto-Stakkelberg equilibrium strategy. Under this equilibrium condition, it guides the energy allocation of multiple attackers and defenders in the dual channels, specifically including: Step 3.1, Input switching signal The dynamic switching weight coefficients obtained from Step 2 and network layer parameters , , , , , , ,in ; Step 3.2, according to and Based on the switching situation, determine the corresponding dynamic switching weight coefficient. Then, substitute it into the attacker cost function to obtain the multi-attack cost function under the current mode switching scenario; Step 3.3: Given a multi-attack joint energy allocation strategy Under the given conditions, calculate the defender's optimal response mapping. The optimal response of the defender is obtained by solving the following constrained optimization problem: ; Step 3.4: Map the defender's optimal response. Substituting the cost function of multiple attackers, we obtain the result only regarding the joint strategy of multiple attackers. The multi-objective optimization problem is solved, and the Pareto front of the multi-attacker energy allocation problem is obtained by using a non-dominated sorting genetic optimization algorithm. Step 3.5: Normalize the candidate solutions in the Pareto front and determine the compromise solution based on the knee selection mechanism; Step 3.6, Based on the attacker's load balancing strategy Calculate the defender equilibrium strategy ; Step 3.7, Output: Pareto-Stakkelberg Equilibrium .

[0008] Based on the equilibrium strategy obtained above, the security control problem of networked handover systems under worst-case denial-of-service attacks can be solved at the physical layer. The specific solution is as follows: Step 4.1 Design controllers for each mode in the physical layer to ensure optimal performance of each subsystem during modal synchronization; and design switching signals based on average dwell time conditions to ensure system stability of the networked switching system under asynchronous conditions; assuming subsystems All are controllable, and the state feedback control input is... ,in Given the controller gain, the specific form of the closed-loop networked switching system is as follows: (7) Define the current mode when the subsystem and controller modes are consistent. The cost function is as follows (8) in and These are the weight matrices for the system state and the control input, respectively. This represents the mathematical expectation of a random packet loss process. Indicates the system at the 1st Always keep the first Transmission indicator variables for system status data packets during modal runtime; Step 4.2 According to the network layer game equilibrium strategy, the packet loss rate of the networked handover system during data transmission is closely related to its handover dynamics; when no handover occurs in the networked handover system, only the system state channel experiences packet loss, and its packet loss rate is expressed as follows: When the networked switching system switches from mode Switch to mode At that time, the packet loss rates of the system state channel and the switching signal channel were respectively and Based on this, considering the dynamics of the closed-loop networked switching system described by equation (7), and combining the subsystem cost function defined by equation (8), for the ideal case where the subsystem and controller modes remain consistent, the dynamic programming method is used to solve the optimal control problem of each subsystem, where, This represents the balanced data packet delivery rate of the state channel when no mode switching occurs in the system; This indicates that the system remains at the first Each subsystem operates, that is At that time, the balanced delivery rate of state data packets on the state channel; The system is represented by the first The subsystem switched to the first When there are multiple subsystems, the balanced delivery rate of system state data packets on the state channel; The system is represented by the first The subsystem switched to the first Subsystem, namely , At that time, the equalization transmission rate of the switching signal data packets on the switching signal channel; Step 4.3 For the obtained channel packet loss rate, if for all There exists a positive definite matrix. Satisfying matrix inequalities (9) and The solution to the following algebraic Ricardi equation (10) Then the optimal controller for modal synchronization exists, and the gain matrix is... (11) Step 4.4 Based on Step 4.3, design the corresponding optimal controller for each subsystem to ensure optimal performance under modal synchronization. At the same time, determine the worst-case channel packet loss rate based on the Pareto-Stakkelberg equalization of the network layer. In view of the problem that the loss of system state and handover signal may cause the networked handover system to operate in an open-loop state, which may lead to modal asynchrony, design a handover signal with average dwell time to construct sufficient conditions to ensure the exponential mean square stability of the networked handover system. Consider a networked switching system, assuming its network layer is in Pareto-Stakkelberg equilibrium, and using the controller designed in Step 4.3. If a constant exists... , , , Defined as (12) in, and Let represent the minimum and maximum eigenvalues ​​of a symmetric matrix, respectively. And switch signals The following average length of stay conditions must be met (13) in, , , , Then the networked switching system is globally exponentially stable.

[0009] The beneficial effects of this invention are: This invention provides an effective solution to the security problems in networked switching systems by introducing game theory to analyze and solve them, significantly improving the system's resistance to attacks and stability. Especially in the face of malicious network attacks such as denial-of-service attacks, game theory provides optimal strategic decisions for multiple participants (including attackers and defenders), thereby enhancing the system's defensiveness and resilience. The application of this method enables networked switching systems to maintain high security and performance in complex attack environments, effectively reducing system failures or performance degradation caused by network attacks. Specifically, its practical applications can be summarized as follows: Improving the security of networked switching systems: Applicable to critical areas such as automotive transmission systems, robot control systems, aircraft automatic navigation systems, and advanced traffic management systems, maintaining system stability and normal operation when facing malicious attacks; Enhancing the ability to resist multiple attacks: Game theory can simulate complex adversarial environments, ensuring that the system can make the best response in situations where multiple attackers and defenders are engaged in a game, protecting the system from impact; Optimizing network resource management: By analyzing the strategies of attackers and defenders, game theory can provide theoretical support for the management and scheduling of network resources, preventing network resource exhaustion and system crashes caused by denial-of-service attacks and other similar behaviors. Attached Figure Description

[0010] Figure 1 This is a block diagram of a networked CSTR system.

[0011] Figure 2 A schematic diagram of the subsystems and controllers.

[0012] Figure 3 This is a system state trajectory diagram.

[0013] Figure 4 This is the controller trajectory diagram.

[0014] Figure 5 This is a diagram showing the specific loss conditions for system states and switching signals.

[0015] Figure 6 This is a diagram of the cross-layer information transmission structure of a networked switching system. Detailed Implementation

[0016] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be described in detail below. Obviously, the described embodiments are merely some embodiments of this invention, and not all embodiments. Based on the embodiments of this invention, all other implementation methods obtained by those skilled in the art without creative effort are within the scope of protection of this invention.

[0017] Continuous stirred tank reactors are widely used in industrial production for physical transformations and chemical reactions. With the application of network communication technology, networked continuous stirred tank reactors inevitably face network security issues. To verify the effectiveness of our proposed method, we considered a networked continuous stirred tank reactor system subjected to a denial-of-service attack, such as... Figure 1 As shown. A constant-volume continuous stirred tank reactor is connected to two different source streams via a selection valve. Since different input streams cause changes in system parameters, the system can be effectively simulated as a switching system. To facilitate remote control and reduce costs, system information is transmitted via a dual-channel wireless communication network. Unfortunately, industrial networks of this nature are easily vulnerable to malicious attacks due to their inherent openness. Therefore, security measures must be considered to effectively mitigate such threats. Each continuous stirred tank reactor model can be accurately described by the following differential equation, as shown in Equation (14). (14) in, Indicates reactant concentration. Indicates the rate of change of reactant concentration. This indicates the temperature of the material inside the reactor. This represents the rate of change of reactor temperature over time. Indicates the first Volumetric flow rate of each input stream Indicates the reactor volume. Indicates the first The inlet concentration of reactants in each input stream Indicates the first The inlet temperature of each input stream. Indicates the temperature of the cooling / heating medium. Indicates the activation energy of the reaction. Represents the ideal gas constant. This represents a constant related to the reaction rate. This represents a constant related to the heat effect of the reaction. This represents a constant related to the heat exchange process. This indicates two operating modes of the system. and These represent the concentration and temperature in the reaction vessel, respectively. The temperature of the coolant is a control input to the system. The rated operating conditions corresponding to the unstable equilibrium point are: , and ,in , and These represent the steady-state reactant concentration, steady-state reactor temperature, and steady-state cooling medium temperature under rated operating conditions, respectively. The initial state is taken as... and Define the state. and and control input Sampling time is The system matrix of the discretized model is The network layer parameters are set as follows: , , , The energy constraints for the defender and the attacker are respectively and The cost of each channel is respectively , To achieve Pareto-Nash equilibrium, channel 2 remains idle without switching, allowing defenders to concentrate all their energy on channel 1, leading to a coordinated attack on channel 1. Switching from subsystem 1 to subsystem 2 is more volatile than switching from subsystem 2 to subsystem 1, resulting in a lower packet delivery rate for channel 2.

[0018] At the physical layer, the weight parameters of the objective function are taken as follows: , Based on the data packet delivery rate obtained above, the positive matrix... and controller gain The calculation formula is According to the optimal strategy, the parameters of the theorem are defined as follows: , , , The mean dwell time condition for ensuring the stability of the networked handover system, calculated based on Step 4.4, is as follows: s. Additionally, the initial value is taken as... The following simulation results were obtained. Figure 2 The switching signal that meets the above definition is displayed. and the corresponding controller modes When two trajectories coincide, the system operates synchronously; conversely, if the two trajectories are separate, the system operates asynchronously. The system state trajectory is as follows: Figure 3 As shown, although potential state loss and asynchronous control can lead to state divergence, it will eventually tend to zero through compensation in the synchronous case. Figure 4 Although the control input shown fluctuates somewhat, it eventually tends towards zero. Figure 5The upper subplot shows the system state loss at each time step. The lower subplot shows the loss of the switching signal. This indicates that there is no switching behavior between subsystems, and and This indicates the signal loss during switching and successful reception.

[0019] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing examples, or equivalent substitutions can be made to some or all of the technical features therein. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the various examples of the present invention.

Claims

1. A system security control method for switching under a DoS attack that considers the importance of subsystems, characterized in that, Includes the following steps: Step 1: Construct a closed-loop mathematical model of the discrete-time networked switching system and characterize random packet loss behavior using Bernoulli distribution; Step 2: Construct a Stakolberg cooperative game model in the network layer that considers the switching characteristics of the physical layer, including: Attacker and Defender Action Sets: Energy Allocation Strategies of Multi-Denial Attackers and Defenders on Dual Channels; Cost functions for attackers and defenders: Multi-denial-of-service attackers and defenders construct cost functions with the goal of optimizing channel signal-to-interference-plus-noise ratio and energy consumption; Dynamic adjustment strategy: Based on the switching characteristics of subsystems in the physical layer, dynamically adjust the attacker's energy allocation strategy on the dual channels to construct the worst-case denial-of-service attack scenario; Step 3: Based on the Stakolberg cooperative game model established in Step 2, a solution algorithm based on the knee selection mechanism is proposed at the network layer to obtain the Pareto-Stakolberg equilibrium strategy, which guides the energy allocation of multi-denial-of-service attackers and defenders on the dual channels. Step 4: Consider the scenario where multiple denial-of-service attackers at the network layer cause dynamic packet loss in the physical layer, and solve for the optimal controller gain under the condition of modal synchronization of the subsystem and controller; comprehensively consider the open-loop and modal asynchronous operation of the system caused by dynamic packet loss, and design the average dwell time condition to ensure the exponential mean square stability of the networked switching system.

2. The method for switching system security control under a DoS attack considering the importance of subsystems, as described in claim 1, is characterized in that... Step 1 includes: Step 1.1 Construct the following networked handover system, as shown in formula (1). (1) in, It is the system status. It is a control input. and Represents the system state matrix and control input matrix, where and These represent the dimensions of the system state vector and the control input vector, respectively. Indicates a switching signal, where It is the number of subsystems; when When, it indicates the first The subsystem in the first It is always in active mode; in addition... This indicates the function for switching dependencies, and its value range is... To maintain symbol consistency, the initial activity mode is defined as follows: ; Step 1.2 During data transmission in a networked handover system, sensors, acting as defenders, sample the system's state and handover signals, determining their transmission energy on the channel. Multiple denial-of-service attackers similarly allocate energy on the channel, disrupting data transmission. In this process, sensors and multiple attackers jointly determine the channel signal-to-noise ratio. Therefore, the first... The signal-to-interference-plus-noise ratio of the channel is (2) in, and They are in and Under the conditions, the defender and the first The energy allocated by an attacker on the channel; in addition... and These are the channel gains for the defender and the attacker, respectively. It is the Gaussian white noise energy in the environment; based on orthogonal amplitude modulation and digital communication theory, the relationship between the channel signal-to-interference-plus-noise ratio and the data transmission rate is established as follows: (3) in The given parameters are Gaussian. The function is ,in For Gauss The independent variable of the function, i.e., the lower limit of integration, For integration variables; Step 1.3 Under the above conditions, packet transmission indication functions are defined for system status and switching signals, respectively. and ; in the time, and These represent system status loss and successful reception, respectively. and These represent the switching signals at the [number]th [time]. Time loss and successful reception, among which Indicates the first The discrete moments of each mode switch; since the system state and the switching signal are transmitted separately, therefore, and They are independent random variables and follow the following Bernoulli distribution; (4) in, and These represent the packet delivery rate of the system status and the switching signal, respectively.

3. A method for switching system security control under a DoS attack considering the importance of subsystems, as described in claim 2, is characterized in that... Step 2 specifically refers to: When considering and The specific situations include: Step 2.1 Identify the participants in the Stakolberg cooperative game as multiple denial-of-service attackers and defenders; Step 2.2 The action sets of both sides are the energy allocation on two channels, specifically, the defender and the first... The action set of an attacker is defined as: (5) in, This represents a set of denial-of-service attackers. Number of attackers; This represents the joint energy allocation strategy of all attackers under the current mode switching scenario. This represents the set of all combined actions of the attackers; and These represent the total available transmission energy of the defender and the total available attack energy of all attackers, respectively. Step 2.3 Determine the cost functions for both attackers and defenders. Both attackers and defenders will consider signal-to-interference-plus-noise ratio (SIR) and energy consumption in their cost functions. Since the range of SIR and energy consumption is uncertain, normalization is required. Considering the impact of physical layer switching characteristics, a dynamic switching weight coefficient is added to the attacker's cost function to dynamically adjust the energy allocation across the dual channels. Therefore, the first... The cost function for each attacker and defender is: (6) in, and ,and It dynamically switches the weighting coefficients. Indicates the defender in the The energy cost coefficient corresponding to the allocation of a unit of transmission energy on a channel. Indicates the attacker in the The energy cost coefficient corresponding to the allocation of a unit of attack energy on each channel, where , Represents the system state channel. Indicates switching the signal channel; Step 2.4 The introduction of dynamic handover weight coefficients aims to quantify the degree to which subsystem characteristics change with handover behavior, and its definition is as follows; For networked handover systems, the first The representative eigenvalue of a subsystem is defined as: ,in It is the first Subsystem One eigenvalue; These are adjustment coefficients used to handle cases where zero appears in the denominator of eigenvalue operations; the subscript set of the stable subsystem is defined as... The subscript set of an unstable subsystem is defined as and ,in, To stabilize the total number of subsystems, Let be the total number of unstable subsystems, and satisfy . ; This indicates the index of the stable subsystem within the original system. Indicates the index of the unstable subsystem within the original system; and They represent the first The stable subsystem and the first Representative eigenvalues ​​of an unstable subsystem; when and When, it indicates that the system is at the twentieth ... Time by the first The subsystem switched to the first Each subsystem dynamically switches weight coefficients. for: (1) For ,and ,have (7) (2) For ,and ,have (8) (3) For ,and ,have (9) (4) For ,and ,have (10)。 4. A method for switching system security control under a DoS attack considering the importance of subsystems, as described in claim 3, is characterized in that... Step 3 specifically includes: Step 3.1, Input switching signal The dynamic switching weight coefficients obtained from Step 2 and network layer parameters , , , , , , ,in ; Step 3.2, according to and Based on the switching situation, determine the corresponding dynamic switching weight coefficient. Then, substitute it into the attacker cost function to obtain the multi-attack cost function under the current mode switching scenario; Step 3.3: Given a multi-attack joint energy allocation strategy Under the given conditions, calculate the defender's optimal response mapping. The optimal response of the defender is obtained by solving the following constrained optimization problem: ; Step 3.4: Map the defender's optimal response. Substituting the cost function of multiple attackers, we obtain the result only regarding the joint strategy of multiple attackers. The multi-objective optimization problem is solved, and the Pareto front of the multi-attacker energy allocation problem is obtained by using a non-dominated sorting genetic optimization algorithm. Step 3.5: Normalize the candidate solutions in the Pareto front and determine the compromise solution based on the knee selection mechanism; Step 3.6, Based on the attacker's load balancing strategy Calculate the defender equilibrium strategy ; Step 3.7, Output: Pareto-Stakkelberg Equilibrium .

5. A method for switching system security control under a DoS attack considering the importance of subsystems, as described in claim 4, is characterized in that... Step 4 specifically includes: Step 4.1 In the physical layer, design controllers for each mode to ensure optimal performance of each subsystem during modal synchronization; and design switching signals based on average dwell time conditions to ensure system stability of the networked switching system under asynchronous conditions; assuming subsystems All are controllable, and the state feedback control input is... ,in Given the controller gain, the specific form of the closed-loop networked switching system is as follows: (7) Define the current mode when the subsystem and controller modes are consistent. The cost function is as follows (8) in and These are the weight matrices for the system state and the control input, respectively. This represents the mathematical expectation of a random packet loss process. Indicates the system at the 1st Always keep the first Transmission indicator variables for system status data packets during modal runtime; Step 4.2 According to the network layer game equilibrium strategy, the packet loss rate of the networked handover system during data transmission is closely related to its handover dynamics; when no handover occurs in the networked handover system, only the system state channel experiences packet loss, and its packet loss rate is expressed as follows: When the networked switching system switches from mode Switch to mode At that time, the packet loss rates of the system state channel and the switching signal channel were respectively and Based on this, considering the dynamics of the closed-loop networked switching system described by equation (7), and combining the subsystem cost function defined by equation (8), for the ideal case where the subsystem and controller modes remain consistent, the dynamic programming method is used to solve the optimal control problem of each subsystem, where, This represents the balanced data packet delivery rate of the state channel when no mode switching occurs in the system; This indicates that the system remains at the first Each subsystem operates, that is At that time, the balanced delivery rate of state data packets on the state channel; The system is represented by the first The subsystem switched to the first When there are multiple subsystems, the balanced delivery rate of system state data packets on the state channel; The system is represented by the first The subsystem switched to the first Subsystem, namely , At that time, the equalization transmission rate of the switching signal data packets on the switching signal channel; Step 4.3 For the obtained channel packet loss rate, if for all There exists a positive definite matrix. Satisfying matrix inequalities (9) and The solution to the following algebraic Ricardi equation (10) Then the optimal controller for modal synchronization exists, and the gain matrix is... (11) Step 4.4 Based on Step 4.3, design the corresponding optimal controller for each subsystem to ensure optimal performance under modal synchronization. At the same time, determine the worst-case channel packet loss rate based on the Pareto-Stakkelberg equalization of the network layer. In view of the problem that the loss of system state and handover signal may cause the networked handover system to operate in an open-loop state, which may lead to modal asynchrony, design a handover signal with average dwell time to construct sufficient conditions to ensure the exponential mean square stability of the networked handover system. Consider a networked switching system, assuming its network layer is in Pareto-Stakkelberg equilibrium, and using the controller designed in Step 4.

3. If a constant exists... , , , Defined as (12) in, and Let represent the minimum and maximum eigenvalues ​​of a symmetric matrix, respectively. And switch signals The following average length of stay conditions must be met (13) in, , , , If so, the networked switching system is globally exponentially stable.

Images