Switching system security control method based on two-stage multi-attacker stackelberg game

Abstract

The application discloses a switching system security control method based on a two-stage multi-attacker Stackelberg game, which comprises the following steps: constructing a discrete switching system model and a controller, and obtaining a controller mode switching signal, system state information under SC deception attack and control input information under CA deception attack; constructing a two-stage multi-attacker Stackelberg game model according to a Stackelberg game algorithm, and obtaining index functions of SC deception attackers, the controller and CA deception attackers according to the game model, the switching system, the switching signal and the system state information; combining the index functions, and adopting a backstepping method to construct a two-stage multi-attacker Stackelberg equilibrium strategy, which comprises an optimal attack strategy of the CA deception attackers, an optimal control strategy of the controller and an optimal tampering strategy of the SC deception attackers; the equilibrium solution of the two-stage game can be used to guide the behaviors of two attackers and one defender, and the effect of the active security control of the switching system based on the game theory under deception attack is realized.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of switching system security control, and particularly relates to a switching system security control method based on two-level multi-attacker Stackelberg game. BACKGROUND

[0002] Modern society has a higher demand for informatization, and the stability and security of the system become the top priority. Due to the long-distance transmission of information, networked control systems have emerged. Networked control systems are a kind of control systems in which sensors, controllers and actuators communicate through a network, and have the advantages of light weight, low power consumption, high reliability, strong flexibility, easy installation and maintenance, etc. In the past few decades, it has attracted widespread attention from the scientific community. Networked switching systems are composed of a limited number of subsystems and switching laws that determine how these subsystems operate. Due to its special structure, networked switching systems have important applications in describing actual industrial systems with multi-channel, multi-modal, multi-sensor and unstable dynamic system models, such as aircraft systems, marine vehicle systems and autonomous robot systems. Because the network structure is open, it is easy to be damaged by network attacks, which destroys the information transmitted in the network and further affects the system performance or even instability. Among network attacks, deception attacks are the most difficult to detect and have the greatest destructive effect on the system, which makes the security control problem of networked switching systems under deception attacks a research hotspot. However, when the switching signals in the switching system are also tampered with by deception attacks, it will produce more complex asynchronous switching behavior compared to general control systems. Obviously, passive control schemes in the past cannot completely solve the active security control problem of switching systems under deception attacks. Because game theory has unique advantages in simulating the confrontation between malicious attackers and defenders, it has become an effective tool for analyzing the security problems of networked control systems. However, existing game theory can only handle the behavior of one attacker and one defender, and cannot effectively solve the problem when multiple attackers appear. SUMMARY

[0003] The present application provides a switching system security control method based on two-level multi-attacker Stackelberg game to overcome the technical problem that the existing Stackelberg game model cannot handle the game behavior of a controller containing two attacks at the same time when the switching system is attacked by two attacks.

[0004] To achieve the above purpose, the technical scheme of the present application is:

[0005] A switching system security control method based on two-level multi-attacker Stackelberg game, comprising:

[0006] S1: constructing a discrete switching system model and a controller, and obtaining a controller mode switching signal, system state information under SC deception attack, and control input information under CA deception attack;

[0007] S2: constructing a two-level multi-attacker Stackelberg game model according to a Stackelberg game algorithm, the attackers including an SC deception attacker and a CA deception attacker, and obtaining an index function of the SC deception attacker, the controller and the CA deception attacker according to the two-level multi-attacker Stackelberg game model, the switching system model, the controller mode switching signal, the system state information and the control input information, the index function being used to determine optimal strategies of the SC deception attacker, the controller and the CA deception attacker;

[0008] S3: combining the index functions of the CA deception attacker, the controller and the SC deception attacker, and constructing a two-level multi-attacker Stackelberg equilibrium strategy by using a backstepping method, the two-level multi-attacker Stackelberg equilibrium strategy including an optimal attack strategy of the CA deception attacker, an optimal control strategy of the controller and an optimal tampering strategy of the SC deception attacker, the two-level multi-attacker Stackelberg equilibrium strategy being used to control behaviors of the SC deception attacker, the CA deception attacker and the controller;

[0009] S4: controlling behaviors of the SC deception attacker, the controller and the CA deception attacker according to the two-level multi-attacker Stackelberg equilibrium strategy, and realizing active security control of the switching system under deception attack.

[0010] Further, S1 constructs a discrete switching system model and a controller, and obtains a controller mode switching signal under SC deception attack, system state information and control input information under CA deception attack, and includes the following steps.

[0011] S11: constructing a discrete switching system model, as shown in formula (1),

[0012] x(k+1)=A σ(k) x(k)+B σ(k) u(k) (1)

[0013] wherein, x(k) is system state, u(k) is system input, A σ(k) is state matrix, B σ(k) is input matrix, σ(k)∈S, S∈{0, 1,..., s}, represents a switching signal, s represents the number of subsystems, K(0, k m )={k1, k2,..., k m} represents a set of switching time points, wherein km is the mth switching time;

[0014] S12, constructing a controller, as shown in formula (2),

[0015]

[0016] wherein c(k) represents the switching signal transmitted to the controller, is the input information of the controller, i.e., the system state information, H c(k) represents the control gain to be designed;

[0017] S13, obtaining the controller modal switching signal and the system state information under the SC spoofing attack, the controller modal switching signal and the system state information are as shown in formula (3) and (4),

[0018]

[0019] wherein, represents the switching signal tampered by the spoofing attack, a σ(k)c(k) (k) represents the tampering information of the system state by the SC spoofing attack, and respectively represent that σ(k) is subjected to the SC spoofing attack and is not subjected to the SC spoofing attack, and the mathematical expectation of satisfies represents the tampering rate of σ(k) by the SC spoofing attack;

[0020] S14, obtaining the control input information of the controller under the CA spoofing attack, as shown in formula (5),

[0021]

[0022] wherein φ c(k) (k) = 1 and φ c(k) (k) = 0 respectively represent that the controller is subjected to the information transmission failure and the information correct transmission caused by the CA spoofing attack, wherein the expectation of φ c(k) (k) satisfies E{φ c(k) (k)} = ρ c(k) , E[φ c(k) (k) = 1] = ρ c(k) , E[φ c(k) (k) = 0] = 1-ρ c(k) , wherein ρ c(k) is the blocking rate of the control input information by the CA spoofing attack.

[0023] Further, S2 constructs a two-level multi-attacker Stackelberg game model according to a Stackelberg game algorithm, the attackers include an SC deception attacker and a CA deception attacker, and obtains an index function of the SC deception attacker, the controller and the CA deception attacker according to the two-level multi-attacker Stackelberg game model, the switching system model, the controller modal switching signal, the system state information and the control input information, the index function being used to determine an optimal strategy of the SC deception attacker, the controller and the CA deception attacker, comprising:

[0024] S21, constructing a two-level multi-attacker Stackelberg game model according to a Stackelberg game algorithm;

[0025] S22, obtaining an information set of the SC deception attacker, the controller and the CA deception attacker, as shown in formulas (6), (7) and (8),

[0026]

[0027] wherein, I SC represents the information set of the SC deception attacker, I u I C represents the information set of the controller, I CA represents the information set of the CA deception attacker, u SC represents a tampering strategy of the SC deception attacker, a control strategy of the controller and an attack strategy of the CA deception attacker, J SC represents an index function of the SC deception attacker, an index function of the controller and an index function of the CA deception attacker;

[0028] S23, obtaining a control gain and an action of the CA deception attacker according to the information set of the SC deception attacker, and constructing a switching system equation under the SC deception attack, as shown in formula (9),

[0029] x1(k+1)=A σ(k) x(k)+(1-φ c(k) (k))B σ(k) H c(k) (x(k)+a σ(k)c(k) ) (9)

[0030] wherein, σ(k) represents a switching signal in the information set of the SC deception attacker, c(k) represents a switching signal transmitted to the controller in the information set, the information set being the information set of the SC deception attacker, H c(k) represents the control gain, φ c(k) (k) satisfies E{φ c(k) (k)}=ρ c(k); p c(k) φ c(k) (k) represents the action of the CA spoofing attacker;

[0031] According to the information set of the SC spoofing attacker and the switching system equation under the SC spoofing attack, the index function of the SC spoofing attacker is constructed as shown in formulas (10) and (11),

[0032]

[0033] wherein P σ(k) and S σ(k) are the weight matrices of the SC spoofing attacker and P σ(k) > 0, S σ(k) > 0, and γ d is a constant related to the signal-to-noise ratio of the communication network channel and γ d > 0;

[0034] S24, according to the information set of the controller, the action of the CA spoofing attacker is obtained and the switching system equation of the controller is constructed as shown in formula (12),

[0035]

[0036] represents the system state information in the information set, represents the controller in the information set, represents the control input information under the CA spoofing attack; c(k) represents the switching signal transmitted to the controller in the information set; the information set here is the information set of the controller;

[0037] According to the information set of the controller and the switching system equation of the controller, the index function of the controller is constructed as shown in formula (13),

[0038]

[0039] wherein Q c(k) and U c(k) are the weight matrices of the controller and Q c(k) > 0, U c(k) > 0;

[0040] S25, according to the information set of the CA spoofing attacker, the switching system equation under the CA spoofing attack is constructed as shown in formula (14),

[0041]

[0042] Wherein, the switching system equation of the controller is consistent with the switching system equation under the CA spoofing attack, and c(k) represents the switching signal transmitted to the controller in the information set; the information set here is the information set of the CA spoofing attacker;

[0043] According to the information set of the CA spoofing attacker and the switching system equation under the CA spoofing attack, an index function of the CA spoofing attacker is constructed, as shown in formula (15),

[0044]

[0045] Wherein, G c(k) and R c(k) are weight matrices of the CA spoofing attacker, and G c(k) > 0 and R c(k) > 0.

[0046] Further, in the two-level multi-attacker Stackelberg game model, the controller and the CA spoofing attacker are modeled as a leader and a follower in the first-level Stackelberg game, and the SC spoofing attacker is modeled as a leader in the second-level game, and the controller and the CA spoofing attacker are jointly modeled as two followers in the second-level game.

[0047] Further, S3 combines the index functions of the CA spoofing attacker, the controller and the SC spoofing attacker, and adopts a back-stepping method to construct a two-level multi-attacker Stackelberg equilibrium strategy, which includes an optimal attack strategy of the CA spoofing attacker, an optimal control strategy of the controller and an optimal tampering strategy of the SC spoofing attacker; the two-level multi-attacker Stackelberg equilibrium strategy is used to control the behaviors of the SC spoofing attacker, the CA spoofing attacker and the controller, including:

[0048] S31, the CA spoofing attacker minimizes the index function of the CA spoofing attacker according to the switching system equation of the CA spoofing attacker, as shown in formula (16),

[0049]

[0050] Suppose that the current controller mode is c(k), and formula (16) is solved to obtain the optimal attack strategy for minimizing the index function of the CA spoofing attacker, as shown in formula (17),

[0051]

[0052] S32, the controller as a leader, obtains the optimal attack strategy of the CA spoofing attacker, and minimizes the index function of the controller according to the optimal attack strategy of the CA spoofing attacker and the switching system equation of the controller, as shown in formula (18),

[0053]

[0054] According to the optimal attack strategy of the CA spoofing attacker and the index function of the controller, formula (18) is solved to obtain the optimal strategy, as shown in formula (19),

[0055]

[0056] Satisfy the constraint condition

[0057] S33, the SC spoofing attacker as a leader common to the controller and the CA spoofing attacker, obtains the optimal attack strategy of the CA spoofing attacker and the optimal tampering strategy of the controller, and minimizes the index function of the SC spoofing attacker, as shown in formulas (20) and (21),

[0058]

[0059] s.t.x1(k+1)=A σ(k) x(k)+(1-φ c(k) (k))B σ(k) H c(k) (x(k)+a σ(k)c(k) ), (20)

[0060]

[0061] s.t.x1(k+1)=A σ(k) x(k)+(1-φ c(k) (k))B σ(k) H c(k) (x(k)+a σ(k)c(k) ) (21)

[0062] The SC spoofing attacker solves the minimum value of the index function of formulas (20) and (21) according to the optimal attack strategy of the CA spoofing attacker, the optimal tampering strategy of the controller and the index function of the SC spoofing attack, to obtain the optimal tampering strategy, as shown in formulas (22) and (23),

[0063]

[0064] Satisfy the constraint condition

[0065] The application designs a two-stage multi-attacker Stackelberg game model, obtains an index function, determines the optimal strategy of SC deception attackers and CA deception attackers, constructs a two-stage multi-attacker Stackelberg equilibrium strategy according to the index function, uses the equilibrium solution of the two-stage game to guide the behaviors of two attackers and one defender, and realizes the effect of active security control of the switching system based on game theory under deception attack. BRIEF DESCRIPTION OF DRAWINGS

[0066] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings needed to be used in the embodiments or prior art description. Obviously, the drawings in the following description are some embodiments of the present application, and other drawings can be obtained by those skilled in the art without creative labor under the premise of the drawings.

[0067] Figure 1 The method flow chart of the switching system security control method based on two-stage multi-attacker Stackelberg game of the present application;

[0068] Figure 2 The control system block diagram under deception attack;

[0069] Figure 3 The subsystem and controller modal diagram of one embodiment of the present application;

[0070] Figure 4 The Stackelberg equilibrium action broken line diagram of the CA deception attacker of one embodiment of the present application;

[0071] Figure 5 The Stackelberg equilibrium action broken line diagram of the SC deception attacker tampering with the switching signal of one embodiment of the present application;

[0072] Figure 6 The Stackelberg equilibrium action broken line diagram of the SC deception attacker tampering with the system state of one embodiment of the present application;

[0073] Figure 7 The system state trajectory diagram of one embodiment of the present application;

[0074] Figure 8 The Stackelberg equilibrium action broken line diagram of the controller of one embodiment of the present application. DETAILED DESCRIPTION

[0075] In order to make the purposes, technical solutions and advantages of the embodiments of the present application clearer, the technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are some but not all of the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by a person of ordinary skill in the art without creative work fall within the protection scope of the present application.

[0076] The embodiment provides a switching system security control method based on a two-level multi-attacker Stackelberg game, as shown in the formula (1), comprising the steps of: Figure 1

[0077] S1: Constructing a discrete switching system model and a controller, and obtaining a controller mode switching signal, system state information under SC spoofing attack and control input information under CA spoofing attack;

[0078] S2: Constructing a two-level multi-attacker Stackelberg game model according to a Stackelberg game algorithm, wherein the attackers include an SC spoofing attacker and a CA spoofing attacker, and obtaining an index function of the SC spoofing attacker, the controller and the CA spoofing attacker according to the two-level multi-attacker Stackelberg game model, the switching system model, the controller mode switching signal, the system state information and the control input information, wherein the index function is used to determine the optimal strategy of the SC spoofing attacker, the controller and the CA spoofing attacker;

[0079] S3: Combining the index functions of the CA spoofing attacker, the controller and the SC spoofing attacker, and constructing a two-level multi-attacker Stackelberg equilibrium strategy by using a backstepping method, wherein the two-level multi-attacker Stackelberg equilibrium strategy includes an optimal attack strategy of the CA spoofing attacker, an optimal control strategy of the controller and an optimal tampering strategy of the SC spoofing attacker, and the two-level multi-attacker Stackelberg equilibrium strategy is used to control the behaviors of the SC spoofing attacker, the CA spoofing attacker and the controller;

[0080] S4: Controlling the behaviors of the SC spoofing attacker, the controller and the CA spoofing attacker according to the two-level multi-attacker Stackelberg equilibrium strategy, so as to realize active security control of the switching system under spoofing attack.

[0081] ​Specifically, first, a discrete switching system model and a controller are constructed, and a controller mode switching signal, system state information under SC spoofing attack, and control input information under CA spoofing attack are obtained, which are helpful for the design of a subsequent active security control scheme; second, a two-level multi-attacker Stackelberg game model is constructed according to a Stackelberg game algorithm, the attackers include an SC spoofing attacker and a CA spoofing attacker, and an index function of the SC spoofing attacker, the controller and the CA spoofing attacker is obtained according to the two-level multi-attacker Stackelberg game model, the switching system model, the controller mode switching signal, the system state information and the control input information, the index function is used to determine the optimal strategy of the SC spoofing attacker, the controller and the CA spoofing attacker, and through the index function, how each party makes a decision according to its own strategy and the strategy of the opponent can be accurately described. The leader makes a decision first, and the follower optimizes its own strategy according to the decision of the leader; in combination with the index function of the CA spoofing attacker, the controller and the SC spoofing attacker, a two-level multi-attacker Stackelberg equilibrium strategy is constructed by using a backstepping method, the two-level multi-attacker Stackelberg equilibrium strategy includes an optimal attack strategy of the CA spoofing attacker, an optimal control strategy of the controller and an optimal tampering strategy of the SC spoofing attacker; the two-level multi-attacker Stackelberg equilibrium strategy is used to control the behaviors of the SC spoofing attacker, the CA spoofing attacker and the controller, and two-level multi-attacker Stackelberg game equilibrium analysis is helpful to find the optimal strategies of the attacker and the defender, so that the system can remain relatively stable when facing attacks; finally, the behaviors of the SC spoofing attacker, the controller and the CA spoofing attacker are controlled according to the two-level multi-attacker Stackelberg equilibrium strategy, and active security control of the switching system under spoofing attack is realized.

[0082] Figure 2 The system block diagram of the switching system of the embodiment, the multi-attackers refer to the multiple attack behaviors of the SC spoofing attacker and the CA spoofing attacker;

[0083] In specific embodiments, the scheme of constructing a discrete switching system model and a controller, and obtaining a controller mode switching signal, system state information under SC spoofing attack, and control input information under CA spoofing attack is as follows:

[0084] S11, a discrete switching system model is constructed, as shown in formula (24),

[0085] x(k+1)=A σ(k) x(k)+B σ(k) u(k) (24)

[0086] wherein, is the system state, For system input, A σ(k) Let B be the state matrix, Bσ(k) be the input matrix, σ(k)∈S, S∈{0,1,...,s}, represent the switching signal, s represent the number of subsystems, and K(0,k) be the input matrix. m )={k1,k2,...,k m} represents the set of switching moments, where k m This is the m-th switching time.

[0087] S12. Construct the controller as shown in formula (25).

[0088]

[0089] Where c(k) represents the switching signal transmitted to the controller. The input information to the controller, i.e., the value of the system state when it reaches the controller, is the system state information, H. c(k) Indicates the control gain to be designed;

[0090] S13. Obtain the controller mode switching signal and system status information under the SC spoofing attack, as shown in formulas (26) and (27).

[0091]

[0092] in, This indicates the switching signal after being tampered with by a deception attack, a σ(k)c(k) (k) represents the information about the system state tampering caused by the SC spoofing attack. and Let represent σ(k) being subject to SC deception attack and not subject to SC deception attack, respectively. The mathematical expectation satisfies

[0093] It is the rate at which σ(k) is altered by an SC deception attack;

[0094] S14. In a CA network, when a CA spoofing attacker tampers with the controller, causing information failure, the control input u(k) will be cleared to zero, the system will operate in open loop, and the control input information of the controller under the CA spoofing attack will be obtained, as shown in formula (28).

[0095]

[0096] Where, φ c(k) (k)=1 and φ c(k) (k) = 0 indicates that the controller... Information transmission failure and correct information transmission caused by CA spoofing attacks; φ c(k) The expectation of (k) satisfies E{φc(k) (k)} = p c(k) ; E[φ c(k) (k) = 1] = p c(k) , E[φ c(k) (k) = 0] = 1 - p c(k) , where p c(k) is the blocking probability of the controller output information by the CA deception attack.

[0097] The application constructs a switching model and obtains switching signals and control input information under SC deception attacks and CA deception attacks, which helps the design of subsequent active security control schemes.

[0098] In specific embodiments, a two-level multi-attacker Stackelberg game model is constructed according to a Stackelberg game algorithm, the attackers including SC deception attackers and CA deception attackers, and an index function of the SC deception attackers, the controller and the CA deception attackers is obtained according to the two-level multi-attacker Stackelberg game model, the switching system model, the controller modal switching signal, the system state information and the control input information, the index function being used to determine a scheme of optimal strategies of the SC deception attackers, the controller and the CA deception attackers is:

[0099] S21, a two-level multi-attacker Stackelberg game model is constructed according to a Stackelberg game algorithm:

[0100] In the two-level multi-attacker Stackelberg game model, the controller and the CA deception attacker are modeled as a leader and a follower in a first-level Stackelberg game, the SC deception attacker is modeled as a leader in a second-level game, and the controller and the CA deception attacker are jointly modeled as two followers in the second-level game.

[0101] S22, information sets of the SC deception attackers, the controller and the CA deception attackers are obtained, as shown in formulas (29), (30) and (31),

[0102]

[0103] wherein, I u denotes an information set of the SC deception attackers, denotes an information set of the controller, denotes a tampering strategy of the SC deception attackers, a control strategy of the controller and an attack strategy of the CA deception attackers, The indicator function of the SC deception attacker, the indicator function of the controller and the indicator function of the CA deception attacker; due to the leading position in the game, the SC deception attacker can know the indicator function J u 、 and the strategy S u 、 The controller is in the middle position of the two-level multi-attacker Stackelberg game, and can know the indicator function J and the strategy S

[0104] S23, according to the information set of the SC deception attacker, obtain the control gain and the action of the CA deception attacker and build the switching system equation under the SC deception attack, as shown in formula (32),

[0105] x1(k+1)=A σ(k) x(k)+(1-φ c(k) (k))B σ(k) H c(k) (x(k)+a σ(k)c(k) ) (32)

[0106] Wherein, σ(k) represents the switching signal in the information set of the SC deception attacker, c(k) represents the switching signal transmitted to the controller in the information set, the information set here is the information set of the SC deception attacker, H c(k) represents the control gain, the expectation of φ c(k) (k) satisfies E{φ c(k) (k)}=ρ c(k) ; ρ c(k) φ c(k) (k) represents the action of the CA deception attacker, the SC deception attacker can obtain the indicator function J

[0107] According to the information set of the SC deception attacker and the switching system equation under the SC deception attack, the indicator function of the SC deception attacker is built, as shown in formula (33) and (34),

[0108]

[0109] Wherein, P σ(k) and S σ(k) are the weight matrix of the SC deception attacker and P σ(k) > 0, S σ(k) > 0, γ d is a constant related to the signal-to-noise ratio of the communication network channel and γ d > 0;

[0110] S24, according to the information set of the controller, obtaining the action of the CA spoof attacker and constructing the switching system equation of the controller, as shown in formula (35),

[0111]

[0112] representing the system state information in the information set, representing the controller in the information set, representing the control input information under the CA spoof attack; c(k) represents the switching signal transmitted to the controller in the information set; the information set here is the information set of the controller; the controller can obtain the index function and strategy of the CA spoof attacker, and the action of the CA spoof attacker can be derived through the index function and strategy of the CA spoof attacker;

[0113] According to the information set of the controller and the switching system equation of the controller obtained, the index function of the controller is constructed, as shown in formula (36),

[0114]

[0115] Wherein, Q c(k) , U c(k) is the weight matrix of the controller and Q c(k) > 0, U c(k) > 0;

[0116] S25, according to the information set of the CA spoof attacker, constructing the switching system equation under the CA spoof attack, as shown in formula (37),

[0117]

[0118] Wherein, the switching system equation of the controller is consistent with the switching system equation under the CA spoof attack, and c(k) represents the switching signal transmitted to the controller in the information set; the information set here is the information set of the CA spoof attacker;

[0119] According to the information set of the CA spoof attacker and the switching system equation under the CA spoof attack, the index function of the CA spoof attacker is constructed, as shown in formula (38),

[0120]

[0121] Wherein, G c(k) > 0 and R c(k) > 0 are the weight matrix of the CA spoof attacker.

[0122] In this embodiment, the index functions of the SC cheating attacker, the controller and the CA cheating attacker are designed. Through the index functions, how each party makes decisions according to its own strategy and the strategy of the opponent can be accurately described. The leader makes a decision first, and the follower optimizes its strategy according to the decision of the leader.

[0123] In specific embodiments, in combination with the index functions of the CA cheating attacker, the controller and the SC cheating attacker, a back-stepping method is adopted to construct a two-level multi-attacker Stackelberg equilibrium strategy, which includes an optimal attack strategy of the CA cheating attacker, an optimal control strategy of the controller and an optimal tampering strategy of the SC cheating attacker. The scheme for controlling the behaviors of the SC cheating attacker, the CA cheating attacker and the controller using the two-level multi-attacker Stackelberg equilibrium strategy is:

[0124] S31, constructing a constraint condition of the two-level multi-attacker Stackelberg equilibrium strategy:

[0125] (1) for a given the follower CA cheating attacker of the first-level and second-level Stackelberg game there is a that can minimize the index function that is, the follower there is always a mapping satisfying the condition

[0126]

[0127] wherein, is the optimal response of the follower in the first-level Stackelberg game;

[0128] (2) for a given the follower P of the second-level Stackelberg game u the controller (the leader of the first level) has a that can minimize the index function that is, the follower of the second-level Stackelberg game always has a mapping satisfying the condition

[0129]

[0130] wherein, is called the optimal response of the follower in the second-level Stackelberg game;

[0131] (3) in the game of the second level, the leader SC cheating attacker also has a strategy that can minimize its index function such that for all satisfying conditions

[0132]

[0133] is the optimal leader strategy in the second stage Stackelberg game, is the optimal leader strategy in the first stage Stackelberg game and the optimal follower strategy in the second stage Stackelberg game, is the optimal follower strategy in the first stage and the second stage Stackelberg game;

[0134] S32, the CA spoofing attacker minimizes the index function of the CA spoofing attacker according to the switching system equation of the CA spoofing attacker, as shown in formula (39),

[0135]

[0136] Suppose the current controller mode is c(k), solve formula (39) to obtain the optimal attack strategy of the CA spoofing attacker minimizing the index function, as shown in formula (40),

[0137]

[0138] satisfying (1) in the constraint condition;

[0139] Proof: Substitute into the objective function and simplify to obtain formula (41),

[0140]

[0141] wherein let Z c(k) The minimization problem of the CA spoofing attacker index function in formula (39) is converted into the minimum value problem of Z c(k) under the independent variable p c(k) , let Z c(k) be the derivative of p c(k) , and formula (42) is obtained,

[0142]

[0143] Considering that the packet loss rate p c(k) should be between (0, 1) in actual situations, the optimal strategy of the CA spoofing attacker in the two-stage multi-attacker Stackelberg game is formula (40);

[0144] S33, the controller as a leader, obtains the optimal attack strategy of the CA fraud attacker, and minimizes the index function of the controller according to the optimal attack strategy of the CA fraud attacker and the switching system equation of the controller, as shown in formula (43),

[0145]

[0146] According to the optimal attack strategy of the CA fraud attacker and the index function of the controller, formula (43) is solved to obtain the optimal strategy, as shown in formula (44),

[0147]

[0148] Where the constraint condition and constraint condition (2) is satisfied;

[0149] Proof: Substitute into J u and simplify to obtain formula (45)

[0150]

[0151] According to the constraint condition, the independent variable The second derivative of the quadratic function is greater than zero, and the minimization problem of the controller index function in formula (43) can be converted into the optimal solution problem that the first derivative of is equal to 0 at the independent variable , and let The derivative of is equal to 0, and formula (46) is obtained,

[0152]

[0153] Solving the above formula can obtain the optimal strategy of the controller in the two-level multi-attacker Stackelberg game as formula (44);

[0154] S34, the SC fraud attacker as a leader of the controller and the CA fraud attacker, obtains the optimal attack strategy of the CA fraud attacker and the optimal tampering strategy of the controller, and minimizes the index function of the SC fraud attacker, as shown in formula (47) and (48),

[0155]

[0156] The SC fraud attacker solves the minimum values of the index functions of formula (47) and (48) according to the optimal attack strategy of the CA fraud attacker, the optimal tampering strategy of the controller and the index function of the SC fraud attacker, and obtains the optimal tampering strategy, as shown in formula (49) and (50),

[0157]

[0158] Among them, satisfying the constraints and constraints (3);

[0159] Proof: Find the optimal information that the SC attacker can manipulate to deceive the system state. Let x(k+1)=A σ(k) x(k)+(1-φ c(k) (k))B σ(k) H c(k) (x(k)+a σ(k)c(k) Substitute into Formula (51) is obtained from this.

[0160]

[0161] From the constraints, we know that a σ(k)c(k) The coefficient before the quadratic term is positive, and the index function in formula (47) is positive. The minimization problem can be transformed into finding The information a is tampered with by the independent variable. σ(k)c(k) The problem of finding the optimal solution where the derivative equals 0, let For a σ(k)c(k) Differentiating by 0 yields formula (52).

[0162]

[0163] Formula (52) shows that formula (49) is true;

[0164] Find the optimal tampering strategy for SC deception attackers to manipulate the handover signal. Suppose that the current σ(k)th subsystem is activated, and when the switching signal is tampered with, the controller switches to... in Let x1(k+1)=A σ(k) x(k)+(1-φ c(k) (k))B σ(k) H c(k) (x(k)+a σ(k)c(k) Substitute into After simplification, we obtain formula (53).

[0165]

[0166] By γ d >0, the index function in formula (48) The minimization problem is then transformed into finding Packet loss function for switching signal σ(k) The optimal value problem of derivation equals 0. The derivation equals 0, and formula (54) is obtained,

[0167] According to formula (54), formula (50) is the optimal tampering strategy of the SC deception attacker to sigma (k) ;

[0168]

[0169] In summary, combined with the constraint condition, the optimal attack strategy of the SC deception attacker is The optimal control strategy of the controller as the defender is The optimal attack strategy of the CA deception attacker is And Together constitute the equilibrium strategy under the two-level multi-attacker Stackelberg game;

[0170] The present application constructs the equilibrium strategy under the two-level multi-attacker Stackelberg game, which can guide the behaviors of the attacker and the controller. The controller can maximize the security and stability of the system under the guidance of the equilibrium strategy when the attacker adopts the optimal attack strategy. The equilibrium analysis of the two-level multi-attacker Stackelberg game helps to find the optimal strategies of the attacker and the defender, so that the system can maintain relative stability when facing attacks.

[0171] In specific embodiments, according to the two-level multi-attacker Stackelberg equilibrium strategy, the states of the SC deception attacker, the controller and the CA deception attacker are controlled, and a scheme for active security control of the switching system under deception attacks is realized.

[0172] The behaviors of the three participants all follow the guidance of the Stackelberg game equilibrium solution, assuming that the current ith subsystem is activated, the controller is in mode j, the switching signal under deception attack is tampered into p, and The switching system is represented as formula (55),

[0173] x(k+1)=A i x(k)+(1-φ j (k))B i H j (x(k)+a ij (k)) (55)

[0174] Theorem 1: for given κ i ∈(-∞, 0), μ i ∈(1, ∞), if there is a positive definite matrix ​satisfying the conditions of formulas (56), (57), (58), (59) and (60),

[0175]

[0176]

[0177] The switching system of formula (24) is exponentially mean-square stable under the conditions of satisfying three optimal strategies;

[0178] Proof: According to whether the SC spoofing attack tampers with σ(t) to cause system mode asynchrony, it can be discussed in the following two forms of synchronization and asynchronization.

[0179] (1) When the switching signal σ(k) is not attacked (case A), The system mode is synchronous, and the controller mode j=i, and the system state equation is shown in formula (61),

[0180]

[0181] (2) When the switching signal σ(k) transmitted in the SC network is attacked by a spoofing attack (case B), The system mode is asynchronous, the controller mode j=p, and the system state equation under the system mode asynchronization is shown in formula (62),

[0182]

[0183] Accordingly, the Lyapunov function is selected as shown in formula (63),

[0184]

[0185] Case A: When j=i, the spoofing attacker in the SC network does not tamper with the switching signal transmitted in the network, and the spoofing information of the system state by the spoofing attacker is Formula (63) under mode synchronization can be mainly divided into the following two cases:

[0186] A1: The spoofing attack in the CA network successfully blocks the controller output information The system is open-loop running;

[0187] A2: The spoofing attacker in the CA network fails to block The system is only affected by the behavior of the SC spoofing attacker;

[0188] When the system is in case A1, take φ i (k) = 1 into formula (60), and derive the difference Lyapunov function ΔV i (k) satisfies the condition

[0189]

[0190] i.e., formula (64),

[0191] V i (k+1)≤(1-κ i )V i (k) (64)

[0192] When the system is in case A2, φ i (k) = 0 is substituted into formula (60), and combined with formula (57), the difference Lyapunov function ΔV i (k) is derived as

[0193]

[0194] Simplifying, formula (65) is obtained,

[0195]

[0196] According to formulas (64) and (65), when the system mode is synchronized, the inequality of formula (66) is established

[0197]

[0198] Case B: When σ(k) is tampered with, The system mode is asynchronous, and j = p, which can be mainly divided into the following two cases:

[0199] B1: The CA spoofing attack successfully blocks the controller output information The system is open-loop running.

[0200] B2: The CA spoofing attacker does not attack The running state of the closed-loop system only depends on the behavior of the SC spoofing attack.

[0201] When the system is in case B1, φ p (k) = 1 is substituted into formula (62), combined with formula (56), formula (67) is obtained,

[0202] V i (k+1)≤(1-κ i )V i (k) (67)

[0203] When the system is in case B2, φ p (k) = 0 and are substituted into formula (62), combined with formula (58), the difference Lyapunov function ΔV i (k) is derived as

[0204]

[0205] i.e., formula (68),

[0206]

[0207] In summary, when the system is asynchronous, formula (69) is obtained

[0208]

[0209] where j = p. Since the tampering rate of the attacker on σ(k) is Combining formula (66) and (69), formula (70) is obtained,

[0210]

[0211] Considering the jump of formula (63) at the switching point, formula (59) is substituted into formula (70) to obtain formula (71),

[0212]

[0213]

[0214] Let denote the time when the qth switching to subsystem i occurs, denote the last time within its activation time, and formula (63) within the entire system running time can be derived as

[0215]

[0216] According to the above formula can be simplified as

[0217]

[0218] Similarly, according to φ j (k) ∈ {0, 1}, the above formula is further simplified as

[0219]

[0220] According to the optimal attack strategy of the CA cheating attacker, the optimal control strategy of the controller, and the optimal tampering strategy of the SC cheating attacker, it can be known that under the equilibrium strategy of the two-stage multi-attacker Stackelberg game, the optimal tampering rate of the SC cheating attacker on the switching signal σ(t) is The optimal blocking rate of the CA cheating attacker on the controller output information is Let denotes the residence time of the qth switch to subsystem i, the expectation of formula (62) satisfies

[0221]

[0222] where T i (k0, k) denotes the length of time that the ith subsystem is activated in the interval (k0, k). For the ith subsystem, it satisfies

[0223]

[0224] Definition Substituting and into the above formula respectively, formula (72) is obtained,

[0225]

[0226] Combined with formula (60) and (72), it can be concluded that the switching system formula (24) is mean-square exponentially stable under the condition that the switching signal σ(k) satisfies the mode-dependent average residence time switching condition.

[0227] In this embodiment, the effectiveness of the active safety control scheme in this chapter is verified by switching RLC circuit. The switching RLC circuit can be modeled as a continuous-time switching system, where x = [q c i L ] T , q c denotes the charge of the capacitor, i L is the magnetic flux in the inductor. Its corresponding system matrix can be represented as

[0228]

[0229] Select L = 1H, R = 1Ω, c1 = 1F, c2 = 0.5F, c3 = 0.67F, then the specific system matrix value is represented as

[0230]

[0231] The switching RLC system matrix of the continuous-time system is discretized accordingly, and the discrete time step is selected as T = 1s, and the discretized switching system is obtained, and its system matrix is

[0232] In the two-stage multi-attacker Stackelberg game, the weight matrices of the SC cheating attacker, the controller and the CA cheating attacker are set as

[0233] S1=S2=S3=I, U1=U2=U3=I, G1=G2=G3=I, Q1=Q2=Q3=1, R1=R2=R3=1,

[0234] Based on the obtained equilibrium solution of the two-level multi-attack Stackelberg game, the optimal control gain can be obtained. for

[0235] H1 = [0.3185 0.2275], H2 = [0.3770 0.2828], H3 = [0.0183 -0.1187]. The CA spoofing attacker outputs information to the controller. Optimal barrier ratio It can be represented as

[0236]

[0237] The parameters in Theorem 3.1 are chosen as κ1 = -0.1, κ2 = -0.2, and κ3 = -0.2. Given μ1 = 1.7, μ2 = 1.8, and μ3 = 1.4, the positive definite matrix and the modal correlation mean residence time can be calculated as follows;

[0238]

[0239] T 1A ≥5.0269, T 2A ≥2.5986, T 3A ≥3.0489.

[0240] The results of the simulation experiment are as follows Figures 3-8 As shown,

[0241] SC spoofing attack tamperes with the subsystem and controller modes under the switching signal σ(t), such as Figure 3 As shown, when σ(t) is tampered with, the system modes become asynchronous;

[0242] CA spoofing attackers manipulate controller output information Barrier rate like Figure 4 As shown in the figure, the blue solid line represents the optimal attack strategy for CA spoofing attacks on the controller output information, and the red dashed line represents the controller switching signal. It changes with the switching of controller modes. Based on the optimal strategy of the SC spoofing attacker, the optimal tampering rate of the SC spoofing attacker on the switching signal σ(k) is obtained. And attack information on system state x(k), as shown in Table 1. Figure 5 and Figure 6 As shown,

[0243] Table 1 SC spoofing attacker tampering with switching signal Stackelberg equilibrium action

[0244]

[0245] From Table 1, when the SC spoofing attacker steals the switching signal σ(k), it will choose to tamper σ(k) to other modalities different from the subsystem modality, and the probability of successful tampering to different modalities is also different.

[0246] The green solid line in the frame and σ(k) shows that when the subsystem is in modality 2, the SC spoofing attacker may tamper the switching signal to 1 or 3 at different times, and the tampering rate corresponding to tampering to different modalities is also different. The SC spoofing attacker uses the stolen x(k) information to imitate the action of x(k), thereby launching an attack, as shown in Figure 6 、 Figure 7 . Figure 6 The middle blue dashed line represents the optimal strategy of the SC spoofing attacker tampering with the system state x(k) The upper half of the matrix, the red solid line represents the optimal strategy of the SC spoofing attacker tampering with the system state x(k) The lower half of the matrix.

[0247] When x(k) tends to be stable, the SC spoofing attacker will also give up tampering with x(k), that is The controller reaches a stable state under the action guidance of the two-level switching Stackelberg game equilibrium, as shown in Figure 8 .

[0248] Finally, it should be noted that: the above embodiments are only used to illustrate the technical solutions of the present application, and not to limit them; although the present application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand: it can still modify the technical solutions recorded in the foregoing embodiments, or make equivalent replacement to part or all of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present application.

Claims

1. A switching system security control method based on a two-level multi-attacker Stackelberg game, characterized in that, The method comprises the following steps of: S1: constructing a discrete switching system model and a controller, and obtaining a controller mode switching signal under a SC deception attack, system state information, and control input information under a CA deception attack; the specific steps are as follows: S11, constructing a discrete switching system model, as shown in formula (1), （1） wherein is the system state, is the system input, is the state matrix, is the input matrix, , denotes the switching signal, denotes the number of subsystems, denotes the set of switching instants, wherein is the th switching instant; S12, constructing a controller, as shown in formula (2), （2） wherein, represents a switching signal transmitted to the controller, is input information of the controller, i.e., system state information, represents a control gain to be designed; S13, obtaining the controller mode switching signal and the system state information under the SC deception attack, and the controller mode switching signal and the system state information are as shown in formulas (3) and (4), （3） （4） wherein, denotes the switching signal tampered by the SC spoofing attack, denotes the tampering information of the system state by the SC spoofing attack, and denote respectively under the SC spoofing attack and not under the SC spoofing attack, and ; denotes the tampering rate of the SC spoofing attack on ; S14, obtaining the control input information of the controller under the CA deception attack, as shown in formula (5), （5） wherein, and respectively represent the controller information transmission failure and information correct transmission caused by CA spoofing attack, wherein the expectation satisfies , , wherein is the blocking rate of the controller output information by CA spoofing attack; S2: constructing a two-level multi-attacker Stackelberg game model according to a Stackelberg game algorithm, the attackers including a SC deception attacker and a CA deception attacker, and obtaining an index function of the SC deception attacker, the controller and the CA deception attacker according to the two-level multi-attacker Stackelberg game model, the switching system model, the controller mode switching signal, the system state information and the control input information, the index function being used for determining the optimal strategy of the SC deception attacker, the controller and the CA deception attacker; S3: combining the index functions of the CA deception attacker, the controller and the SC deception attacker, and adopting a backstepping method to construct a two-level multi-attacker Stackelberg equilibrium strategy, the two-level multi-attacker Stackelberg equilibrium strategy including an optimal attack strategy of the CA deception attacker, an optimal control strategy of the controller and an optimal tampering strategy of the SC deception attacker; the two-level multi-attacker Stackelberg equilibrium strategy being used for controlling the behaviors of the SC deception attacker, the CA deception attacker and the controller; S4: controlling the behaviors of the SC deception attacker, the controller and the CA deception attacker according to the two-level multi-attacker Stackelberg equilibrium strategy, so as to realize active security control of the switching system under deception attack.

2. The two-level multi-attacker Stackelberg game based switching system security control method of claim 1, wherein S2: constructing a two-level multi-attacker Stackelberg game model according to a Stackelberg game algorithm, the attackers including a SC deception attacker and a CA deception attacker, and obtaining an index function of the SC deception attacker, the controller and the CA deception attacker according to the two-level multi-attacker Stackelberg game model, the switching system model, the controller mode switching signal, the system state information and the control input information, the index function being used for determining the optimal strategy of the SC deception attacker, the controller and the CA deception attacker, comprising: S21, constructing a two-level multi-attacker Stackelberg game model according to a Stackelberg game algorithm; S22, obtaining an information set of the SC deception attacker, the controller and the CA deception attacker, as shown in formulas (6), (7) and (8), （6） （7） （8） wherein, denotes a set of information of the SC cheating attacker, denotes a set of information of the controller, denotes a set of information of the CA cheating attacker, , denotes a tampering strategy of the SC cheating attacker, a control strategy of the controller and an attack strategy of the CA cheating attacker, , denotes an index function of the SC cheating attacker, an index function of the controller and an index function of the CA cheating attacker; S23, obtaining a control gain and an action of the CA deception attacker according to the information set of the SC deception attacker, and constructing a switching system equation under the SC deception attack, as shown in formula (9), （9） wherein, represents a switching signal in the information set of the SC spoofing attacker, represents a switching signal in the information set transmitted to the controller, where the information set is the information set of the SC spoofing attacker, represents a control gain, the expectation is satisfied ; represents the action of the CA spoofing attacker; According to the information set of the SC spoofing attacker and the switching system equation under the SC spoofing attack, an index function of the SC spoofing attacker is constructed, as shown in formulas (10) and (11), （10） （11） wherein, and is a weight matrix of SC spoofing attackers and , , is a constant related to the communication network channel signal-to-noise ratio and ; S24, according to the information set of the controller, an action of the CA spoofing attacker is obtained and a switching system equation of the controller is constructed, as shown in formula (12), （12） system state information in the information set, controller in the information set, control input information under CA spoofing attack; switching signal transmitted to the controller in the information set; the information set here is the information set of the controller; According to the information set of the controller and the switching system equation of the controller, an index function of the controller is constructed, as shown in formula (13), （13） wherein is a weight matrix of the controller and ; S25, according to the information set of the CA spoofing attacker, a switching system equation under the CA spoofing attack is constructed, as shown in formula (14), （14） Wherein, the switching system equation of the controller is consistent with the switching system equation under the CA spoofing attack, The switching signal indicates the information set transmitted to the controller; here, the information set is the information set of the CA spoofing attacker. According to the information set of the CA spoofing attacker and the switching system equation under the CA spoofing attack, an index function of the CA spoofing attacker is constructed, as shown in formula (15), （15） wherein, and is a weight matrix for CA spoofing attackers and and .

3. The two-level multi-attacker Stackelberg game based switching system security control method of claim 2, wherein In the two-level multi-attacker Stackelberg game model, the controller and the CA spoofing attacker are modeled as a leader and a follower in the first-level Stackelberg game, the SC spoofing attacker is modeled as a leader in the second-level game, and the controller and the CA spoofing attacker are jointly modeled as two followers in the second-level game.

4. The switching system security control method based on two-level multi-attacker Stackelberg game according to claim 3, characterized in that, S3, combining the index functions of the CA spoofing attacker, the controller and the SC spoofing attacker, a two-level multi-attacker Stackelberg equilibrium strategy is constructed by using a backstepping method, the two-level multi-attacker Stackelberg equilibrium strategy including an optimal attack strategy of the CA spoofing attacker, an optimal control strategy of the controller and an optimal tampering strategy of the SC spoofing attacker; The two-level multi-attacker Stackelberg equilibrium strategy is used to control the behaviors of the SC spoofing attacker, the CA spoofing attacker and the controller, including: S31, the CA spoofing attacker minimizes the index function of the CA spoofing attacker according to the switching system equation of the CA spoofing attacker, as shown in formula (16), （16） Let the current controller mode be Solving equation (16), the optimal attack strategy that minimizes the CA cheating attacker index function is obtained as shown in equation (17), （17） S32, the controller, as a leader, obtains the optimal attack strategy of the CA spoofing attacker, and minimizes the index function of the controller according to the optimal attack strategy of the CA spoofing attacker and the switching system equation of the controller, as shown in formula (18), （18） According to the optimal attack strategy of the CA spoofing attacker and the index function of the controller, formula (18) is solved to obtain an optimal strategy, as shown in formula (19), （19） Satisfying a constraint condition ; S33, the SC spoofing attacker, as a common leader of the controller and the CA spoofing attacker, obtains the optimal attack strategy of the CA spoofing attacker and the optimal tampering strategy of the controller, and minimizes the index function of the SC spoofing attacker, as shown in formulas (20) and (21), （20） （21） The SC spoofing attacker solves the minimum values of the index functions of formulas (20) and (21) according to the optimal attack strategy of the CA spoofing attacker, the optimal tampering strategy of the controller and the index function of the SC spoofing attack, to obtain an optimal tampering strategy, as shown in formulas (22) and (23), （22） （23） Satisfying a constraint condition .

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