Switching system security control method based on two-stage multi-defender starkbog game

Abstract

The application discloses a kind of based on two-level multi-defender Stackelberg game switching system safety control method, comprising: constructing discrete switching system model and controller, obtains the controller modal switching signal and system state information under deception attack;According to the construction of Stackelberg game algorithm two-level multi-defender Stackelberg game model, according to two-level multi-defender Stackelberg game model, switching system model, controller modal switching signal and system state information obtain the index function of switch, deception attacker and controller;Combined with index function, using backtracking method, construct two-level multi-defender Stackelberg equilibrium strategy, including the optimal control strategy of controller, the optimal attack strategy of deception attacker and the optimal switching strategy of switch;It can use the equilibrium solution of two-level game to guide the behavior of an attacker and two defenders, realizes the effect of active safety control of switching system based on game theory under deception attack.

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-defender Stackelberg game. BACKGROUND

[0002] With the increasing complexity of control objects in various actual industrial systems, switching systems capable of describing multi-modal complex models have attracted more and more attention. When the information between the switching system and the controller needs to be transmitted through the network, the networked switching system emerges as the times require. The networked switching system is composed of a limited number of subsystems and switching laws that determine how these subsystems operate. Open networks are vulnerable to network attacks, and among network attacks, deception attacks are the most difficult to detect and the most destructive to the system, which makes the security control problem of networked switching systems under deception attacks a research hotspot. When the stability of a subsystem running alone is destroyed by external disturbances, the switching system can maintain the stability of the entire system through the switching between subsystems. Therefore, the stability of the switching system is inevitably related to the design of the switching strategy of the subsystem switcher. The transmission of information between the subsystem and the controller in the networked switching system will be affected by network attacks. Deception attacks can tamper with switching signals, input signals, output signals and the like transmitted in the network to achieve the purpose of destroying the stability of the switching system. Thus, many scholars have researched the security control strategy of the switching system under network attacks. However, for the switching system, since the switching signal transmitted in the network will also be destroyed by the network attack, the active security control strategy of the switching system based on game theory under network attacks is often more difficult to design than non-switching systems. The passive control scheme in the past cannot completely solve the active security control problem of the switching system under deception attacks tampering with the switching signal. Since game theory has unique advantages in simulating the confrontation between malicious attackers and defenders, it has become an effective tool for analyzing the security problem of networked control systems. However, the existing game theory can only deal with the behavior of one attacker and one defender, and cannot obtain an effective solution when multiple defenders appear. SUMMARY

[0003] The present application provides a switching system security control method based on two-level multi-defender Stackelberg game, to overcome the technical problem that the existing Stackelberg game model cannot be used to simultaneously process the game behavior of the switcher and the controller under deception attacks when the switching system is attacked.

[0004] In order to achieve the above-mentioned purpose, the technical scheme of the present application is as follows:

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

[0006] S1: Construct a discrete switching system model and controller, and obtain controller mode switching signals and system state information under deception attacks;

[0007] S2: Construct a two-level multi-defender Stackelberg game model based on the Stackelberg game algorithm, where the attacker is a deception attacker, and the defenders are a switcher and a controller. Obtain the index functions of the switcher, the deception attacker, and the controller based on the two-level multi-defender Stackelberg game model, the switching system model, the controller mode switching signal, and the system state information. These index functions are used to determine the optimal strategies for the switcher, the deception attacker, and the controller.

[0008] S3: Combining the index functions of the controller, the deceptive attacker, and the switch, a two-level multi-defender Stackelberg equilibrium strategy is constructed using a backward reasoning method. The two-level multi-defender Stackelberg equilibrium strategy includes the optimal control strategy of the controller, the optimal attack strategy of the deceptive attacker, and the optimal switching strategy of the switch. The two-level multi-defender Stackelberg equilibrium strategy is used to control the behavior of the switch, the deceptive attacker, and the controller.

[0009] S4: Control the behavior of the switcher, the deception attacker, and the controller according to the two-level multi-defender Stackelberg equilibrium strategy to achieve proactive security control of the switching system under deception attack.

[0010] Furthermore, S1 constructs a discrete switching system model and controller, and acquires controller mode switching signals and system state information under spoofing attacks, including:

[0011] S11. Construct 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] in, For system status, For system input, A σ(k) For the state matrix, B σ(k) Let σ(k)∈S, S∈{0,1,...,s}, represent the switching signal, s represent the number of subsystems, and K(0,k) represent the input matrix. m )={k1,k2,...,k m} represents the set of switching moments, where k m This is the m-th switching time.

[0014] S12. Construct the 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, that is, the system state information transmitted to the controller, H c(k) represents the control gain to be designed;

[0017] S13, obtaining the controller mode switching signal and the system state information transmitted to the controller under the deception attack, as shown in formulas (3) and (4),

[0018]

[0019] wherein represents the mode switching signal tampered by the deception attack, aσ(k)c(k) represents the tampering information of the system state by the deception attack, and respectively represent σ(k) under the deception attack and not under the deception attack, and the mathematical expectation of satisfies represents the tampering rate of the deception attack on the mode switching signal; a σ(k)c(k) (k) represents the deception information of the system state by the deception attack, a σ(k)c(k) (k) = 0 represents that the system state is not attacked.

[0020] Further, S2 constructs a two-level multi-defender Stackelberg game model according to a Stackelberg game algorithm, the attacker is a deception attacker, and the defenders are the switcher and the controller, and obtains an index function of the switcher, the deception attacker and the controller according to the two-level multi-defender Stackelberg game model, the switching system model, the controller mode switching signal and the system state information; the index function is used to determine the optimal strategy of the switcher, the deception attacker and the controller, comprising:

[0021] S21, constructing a two-level multi-defender Stackelberg game model according to a Stackelberg game algorithm; the attacker is a deception attacker, and the defenders are the switcher and the controller;

[0022] S22, obtaining an information set of the switcher, the deception attacker and the controller, as shown in formulas (5), (6) and (7),

[0023]

[0024] wherein I sw represents the information set of the switcher, I d represents the information set of the deception attacker, and I u represents the information set of the controller, respectively represent the switching strategy of the switcher, the attack strategy of the deception attack and the control strategy of the controller, σ(k) represents the modal switching signal designed by the switcher; a represents the tampering rate of the deception attack on the modal switching signal, a σ(k)c(k) a represents the deception value of the deception attack tampering with the system state; the control strategy of the controller is where H c(k) is the control gain of the controller; respectively represent the index function of the switcher, the deception attack and the controller;

[0025] S23, according to the information set of the switcher, the deception value of the deception attack tampering with the system state and the control gain of the controller are obtained and the switching system equation of the switcher is constructed, as shown in formula (8),

[0026] x1(k+1)=A σ(k) x(k)+B σ(k) H c(k) (x(k)+a σ(k)c(k) ) (8)

[0027] According to the information set of the switcher and the switching system equation of the switcher, the index function is constructed as shown in formula (9),

[0028]

[0029] where, R σ(k) represents the loss when the system is not normally operated, R σ(k) is the weight matrix of the switcher and R σ(k) > 0, σ(k) represents the controller modal switching signal in the information set, and the information set here is the information set of the switcher;

[0030] S24, according to the information set of the deception attack, the control gain is obtained and the switching system equation of the deception attack is constructed, as shown in formula (10),

[0031] x1(k+1)=A σ(k) (k)+B σ(k) H c(k) (x(k)+a σ(k)c(k) ) (10)

[0032] The switching system equation of the deception attack is consistent with the switching system equation of the switcher;

[0033] According to the information set of the deception attack and the switching system equation of the deception attack, the index function of the deception attack is constructed as shown in formula (11) and (12),

[0034]

[0035] where σ(k) represents the controller mode switching signal in the information set, c(k) represents the switching signal transmitted to the controller in the information set, formula (11) represents the index function of the spoofing attack tampering with the system state, and formula (12) represents the index function of the spoofing attack tampering with the system switching signal; represents the energy consumed by the spoofing attack tampering with the system state, and P σ(k) is a symmetric weight matrix of the energy consumed by the attack system state, and P σ(k) > 0; the single-step system state performance represents the benefit obtained after the spoofing attack destroys the system state, and S σ(k) is a symmetric weight matrix of the attack benefit, and S σ(k) > 0, represents the tampering rate of the spoofing attack on the mode switching signal in the information set, and the information set here is the information set of the spoofing attacker; is the quadratic term of the tampering rate of the attacker on the switching signal, and represents that the attacker will consume a certain amount of energy as a cost when tampering with the switching signal, represents the benefit obtained by the attacker when tampering with the switching signal to make the system mode asynchronous;

[0036] S25, constructing a switching system equation of the controller according to the information set of the controller, as shown in formula (13),

[0037]

[0038] where u(k) represents the system input;

[0039] constructing an index function of the controller according to the information set of the controller and the switching system equation of the controller, as shown in formula (14),

[0040]

[0041] where c(k) represents the switching signal transmitted to the controller in the information set, and the information set here is the information set of the controller, represents the energy consumed by the controller output information, and Qc(k) is a symmetric weight matrix of the controller energy consumption and Qc(k) > 0; represents the single-step system state performance, U c(k) is a weight matrix and U c(k) > 0.

[0042] Further, in the two-level multi-defender Stackelberg game model, the deception attacker and the controller are modeled as a leader and a follower in the first-level Stackelberg game, the switcher is modeled as a leader in the second-level game, and the deception attacker and the controller are jointly modeled as two followers in the second-level game.

[0043] Further, S3 combines the index functions of the controller, the deception attacker and the switcher, adopts a backstepping method, and constructs a two-level multi-defender Stackelberg equilibrium strategy, which includes an optimal control strategy of the controller, an optimal attack strategy of the deception attacker and an optimal switching strategy of the switcher; the two-level multi-defender Stackelberg equilibrium strategy is used to control the behaviors of the switcher, the deception attacker and the controller, including:

[0044] S31, minimizing the index function of the controller according to the switching system equation of the controller, the input information of the controller and the switching signal of the controller, as shown in formula (15),

[0045]

[0046] Supposing that the current controller mode is c(k), the optimal control strategy of the controller index function minimization is obtained by solving formula (15), as shown in formula (16),

[0047]

[0048] wherein H*c(k) is an optimal control gain, satisfying the constraint condition

[0049] S32, the deception attacker as the leader of the controller in the first-level Stackelberg game can obtain the optimal control strategy of the controller, and minimizes the index function of the deception attacker according to the optimal control strategy of the controller and the switching system equation of the deception attacker, as shown in formula (17) and (18),

[0050]

[0051] s.t.x1(k+1)=A σ(k) x(k)+B σ(k) H c(k) (x(k)+a σ(k)c(k) ) (17)

[0052]

[0053] s.t.x1(k+1)=A σ(k) x(k)+B σ(k) Hc(k) (x(k)+a σ(k)c(k) ) (18)

[0054] According to the optimal control gain of the controller and the index function of the deception attacker tampering with the system state, formula (17) is solved to obtain the optimal attack strategy of the deception attack on the system state, as shown in formula (19),

[0055]

[0056] The constraint condition is satisfied

[0057] According to the optimal control gain of the controller and the index function of the deception attacker tampering with the switching signal, formula (18) is solved to obtain the optimal attack strategy of the deception attacker on the switching signal, as shown in formula (20),

[0058]

[0059] Wherein, The optimal modal switching signal of the controller is represented by x(k+1) and x(k) respectively.

[0060] S33, the switcher is a leader shared by the deception attacker and the controller, obtains the optimal attack strategy of the deception attacker and the optimal control strategy of the controller, and minimizes the index function of the switcher, as shown in formula (21),

[0061]

[0062] s.t.x1(k+1)=A σ(k) x(k)+B σ(k) H c(k) (x(k)+a σ(k)c(k) ) (21)

[0063] Formula (20) and (21) satisfy the constraint condition

[0064] According to the index function of the switcher, formula (21) is solved to obtain the optimal switching strategy, as shown in formula (22),

[0065]

[0066] The present application designs a two-level multi-defender Stackelberg game model, obtains the index function, determines the optimal strategies of the switcher, the deception attacker and the controller, constructs a two-level multi-attacker Stackelberg equilibrium strategy according to the index function, and uses the equilibrium solution of the two-level game to guide the behaviors of one attacker and two defenders, so that the effect of the active security control of the switching system based on game theory under the deception attack is realized. BRIEF DESCRIPTION OF DRAWINGS

[0067] 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 described below are only some embodiments of the present application, and all other drawings obtained by those of ordinary skill in the art without creative labor based on these drawings also belong to the protection scope of the present application.

[0068] Figure 1 The method flowchart of the switching system security control method based on two-level multi-defender Stackelberg game of the present application is shown in the figure.

[0069] Figure 2 The control system block diagram under the deception attack is shown in the figure.

[0070] Figure 3 The Stackelberg equilibrium action broken line graph of the switcher in one embodiment of the present application is shown in the figure.

[0071] Figure 4 The Stackelberg equilibrium action broken line graph of the switcher in one embodiment of the present application is shown in the figure.

[0072] Figure 5 The Stackelberg equilibrium action broken line graph of the switcher in one embodiment of the present application is shown in the figure.

[0073] Figure 6 The Stackelberg equilibrium action broken line graph of the switcher in one embodiment of the present application is shown in the figure.

[0074] Figure 7 The system state trajectory graph in one embodiment of the present application is shown in the figure.

[0075] Figure 8 The system state trajectory graph in one embodiment of the present application is shown in the figure. DETAILED DESCRIPTION

[0076] In order to make the purpose, technical solutions and advantages of the embodiments of the present application more clear, the following will combine the drawings in the embodiments of the present application to clearly and completely describe the technical solutions in the embodiments of the present application. Obviously, the described embodiments are only some embodiments of the present application, not all embodiments. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative labor also belong to the protection scope of the present application.

[0077] The present embodiment provides a switching system security control method based on two-level multi-defender Stackelberg game, as shown in the figure, which includes: Figure 1

[0078] ​S1: Construct a discrete switching system model and a controller, and obtain a controller mode switching signal and system state information under a deception attack;

[0079] S2: Construct a two-level multi-defender Stackelberg game model according to a Stackelberg game algorithm, an attacker is a deception attacker, and defenders are a switcher and a controller, and obtain an index function of the switcher, the deception attacker and the controller according to the two-level multi-defender Stackelberg game model, the switching system model, the controller mode switching signal and the system state information; the index function is used to determine optimal strategies of the switcher, the deception attacker and the controller;

[0080] S3: Combined with the index functions of the controller, the deception attacker and the switcher, a two-level multi-defender Stackelberg equilibrium strategy is constructed by using a backstepping method, the two-level multi-defender Stackelberg equilibrium strategy includes an optimal control strategy of the controller, an optimal attack strategy of the deception attacker and an optimal switching strategy of the switcher; the two-level multi-defender Stackelberg equilibrium strategy is used to control behaviors of the switcher, the deception attacker and the controller;

[0081] S4: Behaviors of the switcher, the deception attacker and the controller are controlled according to the two-level multi-defender Stackelberg equilibrium strategy, and active security control of a switching system under a deception attack is realized.

[0082] Specifically, first, a discrete switching system model and a controller are constructed, and a controller mode switching signal and system state information under a deception attack are obtained, which helps the design of a subsequent active security control scheme; second, a two-level multi-defender Stackelberg game model is constructed according to a Stackelberg game algorithm, an attacker is a deception attacker, and defenders are a switch and a controller, and an index function of the switch, the deception attacker and the controller is obtained according to the two-level multi-defender Stackelberg game model, the switching system model, the controller mode switching signal and the system state information; the index function is used to determine the optimal strategy of the switch, the deception attacker and the controller, 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 functions of the controller, the deception attacker and the switch, a two-level multi-defender Stackelberg equilibrium strategy is constructed by using a backstepping method, the two-level multi-defender Stackelberg equilibrium strategy includes an optimal control strategy of the controller, an optimal attack strategy of the deception attacker and an optimal switching strategy of the switch; the two-level multi-defender Stackelberg equilibrium strategy is used to control the behaviors of the switch, the deception attacker and the controller, and two-level multi-defender Stackelberg game equilibrium analysis helps 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 switch, the deception attacker and the controller are controlled according to the two-level multi-defender Stackelberg equilibrium strategy, and active security control of the switching system under a deception attack is realized.

[0083] The system block diagram of the switching system of the embodiment is shown in Figure 2 The switching strategy in the figure is the switching strategy of the switch, and the multi-defender refers to multiple different switching strategies and control behaviors of the switch and the controller;

[0084] In specific embodiments, the scheme of constructing a discrete switching system model and a controller, and obtaining a controller mode switching signal and system state information under a deception attack is as follows:

[0085] S11, a discrete switching system model is constructed, as shown in formula (23),

[0086] x(k+1)=A σ(k) x(k)+B σ(k) u(k)(23)

[0087] wherein, is the system state, For the system input, Aσ(k) is the state matrix, Bσ(k) is the input matrix, σ(k)∈S, S∈{0,1,...,s}, representing the switching signal, s represents the number of subsystems, and K(0,k) represents the input signal. m )={k1,k2,...,k m} represents the set of switching moments, where k m This is the m-th switching time.

[0088] S12. Construct the controller as shown in formula (24).

[0089]

[0090] Where c(k) represents the switching signal transmitted to the controller. H is the input information for the controller, that is, the system status information transmitted to the controller. c(k) Indicates the control gain to be designed;

[0091] S13. Obtain the controller mode switching signal and system status information under deception attack, as shown in formulas (25) and (26).

[0092]

[0093] in, Let aσ(k)c(k)(k) represent the mode switching signal after being tampered with by the deception attack, and let aσ(k)c(k)(k) represent the information about the tampering of the system state by the deception attack. and Let represent σ(k) being subjected to a deception attack and not being subjected to a deception attack, respectively, and The mathematical expectation satisfies This indicates the rate at which a deception attack alters the mode-switching signal; a σ(k)c(k) (k) represents the deceptive information about the system state obtained through a deception attack, a σ(k)c(k) (k) = 0 indicates that the system state is not subject to attack;

[0094] This invention constructs a switching model and obtains controller mode switching signals and system state information under deception attacks, which is helpful for the design of subsequent active security control schemes.

[0095] In a specific embodiment, a two-level multi-defender Stackelberg game model is constructed based on the Stackelberg game algorithm. The attacker is a deception attacker, and the defenders are a switcher and a controller. Based on the two-level multi-defender Stackelberg game model, the switching system model, the controller mode switching signal, and the system state information, index functions for the switcher, the deception attacker, and the controller are obtained. The scheme for determining the optimal strategy for the switcher, the deception attacker, and the controller using these index functions is as follows:

[0096] S21, constructing a two-level multi-defender Stackelberg game model according to a Stackelberg game algorithm: wherein an attacker is a deception attacker, a defender is a switcher and a controller, modeling the deception attacker and the controller as a leader and a follower in a first-level Stackelberg game, modeling the switcher as a leader in a second-level game, and modeling the deception attacker and the controller together as two followers in the second-level game;

[0097] S22, obtaining information sets of the switcher, the deception attacker and the controller, as shown in formulas (27), (28) and (29),

[0098]

[0099] wherein I sw represents the information set of the switcher, I d represents the information set of the deception attacker, and I u represents the information set of the controller, respectively represent a switching strategy of the switcher, an attack strategy of the deception attacker and a control strategy of the controller, σ(k) represents a modal switching signal designed by the switcher; represents a tampering rate of the deception attacker on the modal switching signal, and aσ(k)c(k) represents a deception value of the deception attacker tampering with the system state; the control strategy of the controller is wherein H c(k) is a control gain of the controller; respectively represent index functions of the switcher, the deception attacker and the controller; since the switcher is in a leading position in the game, the index functions of the deception attacker and the controller can be obtained and strategies The deception attacker is in an intermediate position in the two-level multi-defender Stackelberg game and can obtain the index function of the controller and strategy The controller cannot obtain the strategies of the two previous participants because it is at the back end of the network, and can only make an optimal strategy based on its own index function under

[0100] S23, constructing a switching system equation of the switcher according to the information set of the switcher, the deception value of the deception attacker tampering with the system state and the control gain of the controller, as shown in formula (30),

[0101] x1(k+1)=A σ(k) x(k)+B σ(k) H​c(k) (x(k)+a σ(k)c(k) ) (30)

[0102] According to the information set of the switcher and the switching system equation of the switcher, an index function is constructed as shown in formula (31),

[0103]

[0104] wherein, represents the loss when the system is not normally operated, R σ(k) is the weight matrix of the switcher and R σ(k) > 0, and σ(k) represents the controller modal switching signal in the information set, wherein the information set is the information set of the switcher;

[0105] S24, according to the information set of the deception attacker, the control gain is obtained, and the switching system equation of the deception attacker is constructed as shown in formula (32),

[0106] x1(k+1)=A σ(k) (k)+B σ(k) H c(k) (x(k)+a σ(k)c(k) ) (32)

[0107] The switching system equation of the deception attacker is consistent with the switching system equation of the switcher;

[0108] According to the information set of the deception attacker and the switching system equation of the deception attacker, an index function of the deception attacker is constructed as shown in formula (33) and (34),

[0109]

[0110] wherein, σ(k) represents the controller modal switching signal in the information set, c(k) represents the switching signal transmitted to the controller in the information set, formula (33) represents the index function of the deception attack tampering with the system state, and formula (34) represents the index function of the deception attack tampering with the system switching signal, represents the energy consumed by the deception attack tampering with the system state, P σ(k) is the symmetric weight matrix of the energy consumed by the attack system state and P σ(k) > 0; the single-step system state performance represents the benefit obtained after the deception attack destroys the system state, and S σ(k) is the symmetric weight matrix of the attack benefit and S σ(k) > 0, represents the tampering rate of the deception attack on the modal switching signal in the information set, wherein the information set is the information set of the deception attacker; is the quadratic term of the tampering rate of the switching signal by the attacker, indicating that the attacker will consume a certain amount of energy as a cost when tampering with the switching signal, represents the benefit obtained by the attacker when tampering with the switching signal to make the system modal asynchronous;

[0111] S25, constructing a switching system equation of the controller according to the information set of the controller, the input information of the controller and the switching signal of the controller, as shown in formula (35),

[0112]

[0113] wherein u(k) represents the system input;

[0114] constructing an index function of the controller according to the information set of the controller and the switching system equation of the controller, as shown in formula (36),

[0115]

[0116] wherein c(k) represents the switching signal transmitted to the controller in the information set, and the information set is the information set of the controller, represents the energy consumed by the output information of the controller, and Qc(k) is a symmetric weight matrix of the energy consumption of the controller and Qc(k)>0; represents the single-step system state performance, and U c(k) is a weight matrix and U c(k) >0.

[0117] In this embodiment, the switcher, the controller and the index function of the controller are designed, 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 strategy according to the decision of the leader. The switching behavior of the switcher aims to maintain the system in a state close to the normal working point by using the switching between subsystems.

[0118] In specific embodiments, in combination with the index functions of the controller, the deception attacker and the switcher, a two-level multi-defender Stackelberg equilibrium strategy is constructed by using the backstepping method, the two-level multi-defender Stackelberg equilibrium strategy including an optimal control strategy of the controller, an optimal attack strategy of the deception attacker and an optimal switching strategy of the switcher; and a scheme for controlling the behaviors of the switcher, the deception attacker and the controller is:

[0119] S31, minimizing the index function of the controller according to the switching system equation of the controller, the input information of the controller and the switching signal of the controller, as shown in formula (37),

[0120]

[0121] Solve equation (37) to get the optimal control strategy that minimizes the controller index function, as shown in equation (38),

[0122]

[0123] where H*c(k) is the optimal control gain, satisfying the constraint condition

[0124] Proof: Substitute the system switching equation of the controller into the index function of the controller and simplify to get equation (39)

[0125]

[0126] According to the constraint condition, the coefficient of the quadratic term of u(k) in equation (39) is greater than zero, so the minimization problem of the controller index function in equation (37) can be converted into the optimal solution problem of the derivative of the independent variable u(k) being equal to 0, that is, Taking the derivative of u(k) equal to 0, equation (40) is obtained

[0127]

[0128] Moving the term to get equation (38) as the optimal control strategy of the controller, and H*c(k) is the optimal control gain.

[0129] S32, as the leader of the controller in the first-stage Stackelberg game, can obtain the optimal control strategy of the controller, and minimize the index function of the deception attacker according to the optimal control strategy of the controller and the switching system equation of the deception attacker, as shown in equations (41) and (42),

[0130]

[0131] s.t.x1(k+1)=A σ(k) x(k)+B σ(k) H c(k) (x(k)+a σ(k)c(k) ) (41)

[0132]

[0133] s.t.x1(k+1)=A σ(k) x(k)+B σ(k) H c(k) (x(k)+a σ(k)c(k)(42)

[0134] Based on the optimal control gain of the controller and the index function of the deception attacker tampering with the system state, equation (41) is solved to obtain the optimal attack strategy of the deception attack on the system state, as shown in equation (43).

[0135]

[0136] Meet the conditions

[0137] Proof: Substitute the switching equation for deceiving the attacker into the index function. In the middle, and simplified, we get formula (44),

[0138]

[0139] According to the constraints It can be known that a σ(k)c(k) The coefficient before the quadratic term is greater than 0, and the index function in formula (41) The minimization problem can be transformed into For independent variable a 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 zero yields formula (45).

[0140]

[0141] Further simplification of the above formula yields formula (43), which is the optimal attack strategy for deception attack on the system state.

[0142] Based on the optimal control gain of the controller and the tampering switching signal index function of the deceitful attacker, equation (39) is solved to obtain the optimal attack strategy of the deceitful attacker against the switching signal, as shown in equation (46).

[0143]

[0144]

[0145] in, Represents the optimal mode switching signal of the controller; prove that: by Can After simplification, we obtain formula (47).

[0146]

[0147] According to γ d >0, the index function in formula (42) The minimization problem can be transformed into finding For independent variable The problem of finding the optimal solution where the derivative equals 0. Let right Differentiating to zero and simplifying, we get formula (48).

[0148]

[0149] Rearranging the terms of the formula, we get formula (49).

[0150]

[0151] In reality, since the energy of an attack is finite, the rate of tampering with σ(t) by a deception attack is limited. It should be between (0, 1), and by combining formula (49), formula (46) is valid;

[0152] S34. The switch, acting as a leader for both the deception attacker and the controller, obtains the optimal attack strategy of the deception attacker and the optimal control strategy of the controller, and minimizes the index function of the switch, as shown in formula (50).

[0153]

[0154] stx1(k+1)=A σ(k) x(k)+B σ(k) H c(k) (x(k)+a σ(k)c(k) (50)

[0155] Based on the optimal strategies of the attacker and the controller and the metric function of the switcher, equation (50) is solved to obtain the optimal switching strategy, as shown in equation (51).

[0156]

[0157] Proof: Substituting the switching equation of the switch into the index function of the switch and simplifying it, we obtain formula (52).

[0158]

[0159] Optimal control gain Optimal attack strategy for deception attacks and tampering with system state Substituting into formula (52), we obtain a quadratic term in x(k), which is formula (53).

[0160]

[0161] Similarly, the optimal attack strategy for deceiving and tampering with the switching signal is... Substituting into formula (53) In the middle, we can further simplify to obtain formula (54).

[0162]

[0163] In formula (50), the index function The minimization problem can be transformed into minimizing the independent variable σ(k). The minimum problem is to find the switching behavior σ of the switcher under the equilibrium of a two-level multi-defender Stackelberg game. * (k) satisfies formula (51), which contains not only the optimal control gain of the controller. The optimal attack strategy to deceive the attacker was also considered. and This achieves proactive defense against deception attacks, as shown in formula (51). Represented as formula (55),

[0164]

[0165] This invention constructs an equilibrium strategy under a two-stage multi-defender Stackelberg game, which can guide the behavior of both the attacker and the controller. Guided by the equilibrium strategy, the controller can maximize the system's security and stability even when the attacker adopts the optimal attack strategy. The equilibrium analysis of the two-stage multi-defender Stackelberg game helps to find the optimal strategies for both the attacker and the defender, enabling the system to maintain relative stability when facing attacks.

[0166] In a specific embodiment, the scheme for proactive security control of the switching system under deception attacks, which controls the behavior of the switcher, the deception attacker, and the controller according to the two-level multi-defender Stackelberg equilibrium strategy, is as follows:

[0167] The actions of the three participants all follow the guidance of the equilibrium solution of the Stackelberg game. Suppose that the i-th subsystem is currently activated and the controller is in mode j. If the switching signal is not tampered with by a deception attack, the controller mode j = i. If the switching signal is tampered with by a deception attack, the controller mode is j = p, where i ≠ p, i, j, p ∈ S.

[0168] Theorem 1: For a given κ i ∈(1, ∞), η ij ≤0 if a positive definite matrix exists Meet the conditions

[0169]

[0170]

[0171] The switching system is mean-square exponentially stable under the condition that the three optimal strategies are satisfied;

[0172] Proof: Depending on whether the switching signal is tampered with by a deception attack, the system equations can be discussed in the following two cases. If The switching signal was not tampered with by the attacker, and the subsystem and controller were modally synchronized, j = i. The switching system can be rewritten as formula (56).

[0173] x4(k+1)=A i x(k)+B i H i (x(k)+a ii (k)) (56)

[0174] like The switching signal is tampered with by a deceitful attacker, and the subsystem and controller are modally asynchronous. At this time, j = p, and the switching system can be rewritten as formula (57).

[0175] x5(k+1)=A i x(k)+B i H p (x(k)+a ip (k)) (57)

[0176] In summary, the multi-Lyapunov functional is selected as formula (58).

[0177]

[0178] Based on the switcher index function According to the optimal switching strategy, at each switching moment, the system state always switches downwards, i.e., formula (59).

[0179]

[0180] The changes in formula (58) during the subsystem activation time are analyzed. For the case of modal synchronization between the subsystem and the controller, combining formulas (57) and (59), the differential Lyapunov function ΔV... i (k) can be expressed as formula (60),

[0181]

[0182] Combining Theorem 1, Condition 1, and formulas (59) and (60), we can obtain formula (61).

[0183]

[0184] For the case of modal asynchrony between the subsystem and the controller, combining formulas (58) and (59), the difference Lyapunov function ΔV i (k) can be expressed as formula (62),

[0185]

[0186] Substituting condition 2 from Theorem 1 and formula (26) into formula (62), we can obtain formula (63).

[0187] V i (k+1)≤κ i V i (k) (63)

[0188] Consider the optimal tampering strategy for switching signals in a deception attack. Combining formulas (61) and (63), we can obtain formula (64).

[0189]

[0190] Substituting condition 3 from Theorem 1 into formula (64), we can see that formula (58) is monotonically decreasing during the system's runtime. Next, we analyze the change of formula (58) at the switching point. From formula (59), we can see that formula (58) is decreasing at each switching moment, i.e., formula (65).

[0191]

[0192] According to formulas (63) and (65), (58) is decreasing both within the segment and at the switching point, thus proving the stability of the closed-loop system.

[0193] In this embodiment, the effectiveness of the proposed active safety control scheme is verified using a switching RLC circuit model. The switching RLC circuit can be modeled as a continuously switching system, wherein...

[0194]

[0195] Choosing c1 = 2F, c1 = 1.5F, c1 = 0.67F, R = 1Ω, and L = 1H, the system matrix can be expressed as follows:

[0196]

[0197] Since the switching RLC circuit is a continuous-time system, the system matrix of the discrete-time switching system needs to be discretized accordingly. Choosing a time step of T = 1s, the discretized switching system is obtained, and its system matrix is:

[0198]

[0199]

[0200] The weight matrices for the switcher, spoofing attacker, and controller metric functions are selected as follows:

[0201]

[0202] The controller gain matrix, the deception attack tampering system state matrix, the deception attack tampering switching signal error rate, and the subsystem switching strategy matrix are obtained from the controller's optimal control strategy, the deception attack tampering system's optimal attack strategy, and the switching system's optimal switching strategy, as well as the subsystem switching strategy matrix, are as follows:

[0203]

[0204] The initial value of the system state x(k) is x(0) = [4, 4]. T .

[0205] Specific simulation results are as follows Figures 3-8 As shown,

[0206] Figure 3 This is a Stackelberg equilibrium action of the switcher switching strategy. The subsystem's σ(k) is continuously transmitted to the controller along a fixed step size k. Under the equilibrium action of a two-stage multi-defender Stackelberg game, the subsystem mode σ(k) and the controller mode c(k) are as follows: Figure 3 As shown, since σ(k) may be tampered with by deception attacks during transmission in the network, when the trajectories of the two coincide, the subsystem and the controller are modally synchronized; otherwise, the system is asynchronous.

[0207] Figure 4 To deceive attackers into tampering with the handover signal, the Stackelberg equalization action is described in Table 1.

[0208] Table 2. Stackelberg Equilibrium Actions to Deceive Attackers by Tampering with Switching Signals

[0209]

[0210]

[0211] As shown in Table 1, spoofing attacks can alter the subsystem to different modes, and the alteration rate varies depending on the subsystem mode and the attacker's planned alteration mode. The alteration rate of the switching signal by a spoofing attack under different subsystem modes is as follows: Figure 4 As shown, when σ(k) = 1, the attacker will choose to tamper with the switching signal to or And the probability of tampering with these two. They are different;

[0212] The attacker's optimal Stackelberg equilibrium action on the system state is as follows: Figure 5 As shown, the controller's optimal following action is as follows: Figure 6 As shown. The system's operational trajectory is as follows. Figure 7 and Figure 8 As shown, even if the switching signals and system states transmitted in the network are spoofed and altered, causing the system to diverge at certain times, the system state eventually reaches stability thanks to the coordinated design of the switching law and the controller.

[0213] 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 embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A switching system security control method based on a two-level multi-defender Stackelberg game, characterized in that, The method comprises the following steps: S1: constructing a discrete switching system model and a controller, and obtaining a controller mode switching signal and system state information under a 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 system state information transmitted to the controller, represents a control gain to be designed; S13, obtaining a controller mode switching signal and system state information transmitted to the controller under a deception attack, as shown in formulas (3) and (4), （3） （4） wherein ), , denotes the tampered modal switching signal by the deception attack, denotes the tampering information of the system state by the deception attack, and denote under the deception attack and not under the deception attack, and the mathematical expectation of , denotes the tampering rate of the modal switching signal by the deception attack; denotes the deception information of the system state by the deception attack, denotes that the system state is not attacked; S2: constructing a two-level multi-defender Stackelberg game model according to a Stackelberg game algorithm, wherein the attacker is a deception attacker, and the defenders are the switcher and the controller, and obtaining an index function of the switcher, the deception attacker and the controller according to the two-level multi-defender Stackelberg game model, the switching system model, the controller mode switching signal and the system state information; the index function is used to determine the optimal strategy of the switcher, the deception attacker and the controller; S3: combining the index functions of the controller, the deception attacker and the switcher, and using a backstepping method to construct a two-level multi-defender Stackelberg equilibrium strategy, wherein the two-level multi-defender Stackelberg equilibrium strategy comprises an optimal control strategy of the controller, an optimal attack strategy of the deception attacker and an optimal switching strategy of the switcher; the two-level multi-defender Stackelberg equilibrium strategy is used to control the behaviors of the switcher, the deception attacker and the controller; S4: controlling the behaviors of the switcher, the deception attacker and the controller according to the two-level multi-defender Stackelberg equilibrium strategy, so as to realize active security control of the switching system under a deception attack.

2. The two-level multi-defender Stackelberg game based switching system security control method of claim 1, wherein S2: constructing a two-level multi-defender Stackelberg game model according to a Stackelberg game algorithm, wherein the attacker is a deception attacker, and the defenders are the switcher and the controller, and obtaining an index function of the switcher, the deception attacker and the controller according to the two-level multi-defender Stackelberg game model, the switching system model, the controller mode switching signal and the system state information; the index function is used to determine the optimal strategy of the switcher, the deception attacker and the controller, comprising: S21: constructing a two-level multi-defender Stackelberg game model according to a Stackelberg game algorithm; the attacker is a deception attacker, and the defenders are the switcher and the controller; S22: obtaining an information set of the switcher, the deception attacker and the controller, as shown in formulas (5), (6) and (7), （5） （6） （7） wherein denotes a set of information of the switcher, denotes a set of information of the deception attacker, denotes a set of information of the controller, denote a switching strategy of the switcher, an attack strategy of the deception attacker and a control strategy of the controller, respectively, , denotes a modal switching signal designed by the switcher; , denotes a tampering rate of the deception attacker to the modal switching signal, denotes a deception value of the deception attacker tampering the system state; the control strategy of the controller is wherein is a control gain of the controller; denote the index functions of the switcher, the deception attacker and the controller, respectively; S23: obtaining a deception value of a deception attack tampering with system state and a control gain of the controller according to the information set of the switcher, and constructing a switching system equation of the switcher, as shown in formula (8), （8） constructing an index function according to the information set of the switcher and the switching system equation of the switcher, as shown in formula (9), （9） wherein represents a loss when the system is not operating properly, is a weight matrix of the switcher and , represents a controller modal switching signal in the information set, where the information set is the information set of the switcher; S24: obtaining a control gain according to the information set of the deception attacker, and constructing a switching system equation of the deception attacker, as shown in formula (10), （10） the switching system equation of the deception attacker is consistent with the switching system equation of the switcher; According to the information set of the deception attacker and the switching system equation of the deception attacker, an index function of the deception attacker is constructed, as shown in formulas (11) and (12), （11） （12） wherein denotes the information centralized controller modal switching signal, denotes the information centralized transmission to controller switching signal, formula (11) denotes the index function of the fraud attack tampering system state, formula (12) denotes the index function of the fraud attack tampering system switching signal; denotes the energy consumed by the fraud attack tampering system state, is a symmetric weight matrix of the energy consumed by the attack system state and ; single-step system state performance denotes the benefit obtained after the fraud attack destroys the system state, and is a symmetric weight matrix of the attack benefit and , denotes the tampering rate of the fraud attack on the modal switching signal in the information set, and the information set here is the information set of the fraud attacker; is the quadratic term of the tampering rate of the attacker on the switching signal, and indicates that the attacker will consume a certain amount of energy as a cost when tampering with the switching signal, denotes the benefit obtained by the attacker when tampering with the switching signal to make the system modal asynchronous; S25, according to the information set of the controller, a switching system equation of the controller is constructed, as shown in formula (13), （13） wherein represents system input; 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 (14), （14） wherein denotes the switching signal for the collective transmission of information to the controller, the collective information here being the collective information of the controller, denotes the energy consumed by the output information of the controller, is a symmetric weight matrix for the energy consumption of the controller and ; denotes the performance of the system state in a single step, is a weight matrix and .

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

4. The two-level multi-defender Stackelberg game based switching system security control method of claim 3, wherein, S3, combining the index functions of the controller, the deception attacker and the switcher, an optimal two-level multi-defender Stackelberg equilibrium strategy is constructed by using a backstepping method, the two-level multi-defender Stackelberg equilibrium strategy including an optimal control strategy of the controller, an optimal attack strategy of the deception attacker and an optimal switching strategy of the switcher; The two-level multi-defender Stackelberg equilibrium strategy is used to control the behaviors of the switcher, the deception attacker and the controller, including: S31, according to the switching system equation of the controller, the input information of the controller and the switching signal of the controller, the index function of the controller is minimized, as shown in formula (15), （15） Solving formula (15) obtains the optimal control strategy of the controller for minimizing the index function of the controller, as shown in formula (16), （16） wherein is the optimal control gain satisfying the constraint condition ; S32, the deception attacker, as a leader of the controller in the first-level Stackelberg game, can obtain the optimal control strategy of the controller, and according to the optimal control strategy of the controller and the switching system equation of the deception attacker, the index function of the deception attacker is minimized, as shown in formulas (17) and (18), （17） （18） According to the optimal control gain of the controller and the index function of the deception attacker tampering with the system state, formula (17) is solved to obtain the optimal attack strategy of the deception attacker on the system state, as shown in formula (19), （19） Satisfying a constraint condition ; According to the optimal control gain of the controller and the index function of the deception attacker tampering with the switching signal, formula (18) is solved to obtain the optimal attack strategy of the deception attacker on the switching signal, as shown in formula (20), （20） wherein, represents the optimal modal switching signal of the controller; S33, the switcher, as a leader of the deception attacker and the controller, obtains the optimal attack strategy of the deception attacker and the optimal control strategy of the controller, and minimizes the index function of the switcher, as shown in formula (21), （21） Equations (20) and (21) satisfy the constraint condition ; According to the index function of the switcher, formula (21) is solved to obtain the optimal switching strategy, as shown in formula (22), （22）。

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