Active safety control method and device for switching system based on stackelberg game

Abstract

The application provides a switching system active security control method and device based on Stackelberg game, the method comprises the following steps: establishing a switching system model; if the switching system model is subjected to a network attack, analyzing the network attack to obtain an analysis result; based on the analysis result, establishing a two-level single-index Stackelberg game model and a two-level double-index Stackelberg game model; solving the two-level single-index Stackelberg game model and the two-level double-index Stackelberg game model to obtain an equilibrium solution; based on the equilibrium solution, obtaining the optimal strategy of the network attack and the controller; based on the optimal strategy, analyzing the sufficient condition for the exponential stability of the switching system, so as to realize active security control.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of aircraft network security control, and particularly relates to a switching system active security control method and device based on Stackelberg game. BACKGROUND

[0002] Modern society has a higher demand for informatization, and the stability and security of system operation become the top priority. Due to long-distance information transmission, 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 and strong flexibility, which have attracted widespread attention from scientists in the past decade. As a special networked control system, networked switching systems are composed of a limited number of subsystems and switching laws that determine how these subsystems switch. Due to its special structure, networked switching systems have important applications in describing actual industrial models with multiple channels, multiple modes and multiple sensors, such as aircraft systems, marine vehicle systems and intelligent robot systems. Because the network structure is open, it is easy to be attacked by network attacks, which can destroy the information transmitted in the network and further affect the system performance or even instability. This makes the security control problem of switching systems under network attacks a research hotspot.

[0003] The two main attack methods in the network are DoS attacks and deception attacks. When a network switching system is attacked by both of them, the passive security control scheme in the past must limit the duration and frequency of the attacks suffered by the system, and cannot completely solve the security control problem of switching systems under network attacks. SUMMARY

[0004] The purpose of the application is to analyze the security problem of networked control systems, and the application provides a switching system active security control method and device based on game theory.

[0005] In a first aspect, the application provides a switching system active security control method based on Stackelberg game, which comprises:

[0006] Step 1, establishing a switching system model and a controller model;

[0007] Step 2, if the switching system model is attacked by a network attack, analyzing the network attack to obtain an analysis result; wherein the network attack includes DoS attack and deception attack;

[0008] Step 3, based on the analysis result, establishing a two-level single-index Stackelberg game model and a two-level double-index Stackelberg game model;

[0009] Step 4, solving the two-stage single-index Stackelberg game model and the two-stage double-index Stackelberg game model to obtain the equilibrium solution;

[0010] Step 5, based on the equilibrium solution, obtaining the optimal strategy of the network attack and the controller;

[0011] Step 6, based on the optimal strategy, analyzing the sufficient condition for the exponential stability of the switching system.

[0012] Preferably, the step 1 comprises:

[0013] Establishing a switching system model to describe the switching signal, the system state, and the system input;

[0014] Establishing a controller model to describe the controller output.

[0015] Preferably, the step 2 comprises:

[0016] Analyzing the changes of the system state after the SC DoS attack, the changes of the switching signal after the deception attack, and the changes of the controller output information after the CA DoS attack.

[0017] Preferably, the step 3 comprises:

[0018] Establishing a two-stage single-index Stackelberg game model, which includes the participant description of the two-stage single-index Stackelberg game and the index function design of the two-stage single-index Stackelberg game, and giving the algorithm definition for solving the equilibrium solution of the two-stage single-index Stackelberg game;

[0019] Establishing a two-stage double-index Stackelberg game model, which includes the participant description of the two-stage double-index Stackelberg game and the index function design of the two-stage double-index Stackelberg game, and giving the algorithm definition for solving the equilibrium solution of the two-stage single-index Stackelberg game.

[0020] Preferably, the step 4 comprises:

[0021] According to the game equilibrium solution solving algorithm given in the previous step, solving the equilibrium solution of the SC DoS attack, the controller, and the CA DoS attack in the two-stage single-index Stackelberg game;

[0022] Solving the equilibrium solution of the deception attack, the controller, and the CA DoS attack in the two-stage double-index Stackelberg game.

[0023] Preferably, the step 5 comprises:

[0024] Based on the obtained equilibrium solution, the optimal attack strategy of the SC DoS attacker, the optimal attack strategy of the deception attacker, the optimal control strategy of the controller and the optimal attack strategy of the CA DoS attacker are obtained.

[0025] Preferably, the step 6 comprises:

[0026] In combination with the optimal strategies of the SC DoS attacker, the deception attacker, the controller and the CA DoS attacker in the game equilibrium, the exponential stability of the switching system under the average residence time condition is proved.

[0027] In a second aspect, the application further provides a switching system active security control device based on Stackelberg game, the device comprising:

[0028] A first modeling unit is configured to establish a switching system model and a controller model.

[0029] A first analysis unit is configured to analyze the network attack to obtain an analysis result if the switching system model is subjected to the network attack, wherein the network attack comprises a DoS attack and a deception attack.

[0030] A second modeling unit is configured to establish a two-level single-index Stackelberg game model and a two-level double-index Stackelberg game model based on the analysis result.

[0031] A first solving unit is configured to solve the two-level single-index Stackelberg game model and the two-level double-index Stackelberg game model to obtain an equilibrium solution.

[0032] A second solving unit is configured to obtain the optimal strategies of the network attack and the controller based on the equilibrium solution.

[0033] A second analysis unit is configured to analyze the sufficient condition for the exponential stability of the switching system based on the optimal strategies.

[0034] Preferably, the first modeling unit is further configured to establish a switching system model to describe a switching signal, a system state and a system input.

[0035] The first modeling unit is further configured to establish a controller model to describe a controller output.

[0036] Preferably, the first analysis unit is further configured to analyze the change of the system state after the SC DoS attack, the change of the switching signal after the deception attack and the change of the controller output information after the CA DoS attack.

[0037] Preferably, the second modeling unit is further configured to establish a two-stage single-index Stackelberg game model, wherein a participant description of the two-stage single-index Stackelberg game and an index function design of the two-stage single-index Stackelberg game are included, and an algorithm definition for solving an equilibrium solution of the two-stage single-index Stackelberg game is given.

[0038] The second modeling unit is further configured to establish a two-stage double-index Stackelberg game model, wherein a participant description of the two-stage double-index Stackelberg game and an index function design of the two-stage double-index Stackelberg game are included, and an algorithm definition for solving an equilibrium solution of the two-stage single-double-index Stackelberg game is given.

[0039] Preferably, the first solving unit is further configured to solve the equilibrium solution of the SC DoS attack, the controller and the CA DoS attack in the two-stage single-index Stackelberg game and solve the equilibrium solution of the fraud attack, the controller and the CA DoS attack in the two-stage double-index Stackelberg game according to the game equilibrium solution solving algorithm given in the previous step.

[0040] Preferably, the second solving unit is further configured to obtain the optimal attack strategy of the SC DoS attacker, the optimal attack strategy of the fraud attacker, the optimal control strategy of the controller and the optimal attack strategy of the CA DoS attacker based on the obtained equilibrium solution.

[0041] Preferably, the second analysis unit is further configured to prove the exponential stability of the switching system under the average residence time condition in combination with the optimal strategies of the SC DoS attacker, the fraud attacker, the controller and the CA DoS attacker under the game equilibrium.

[0042] The beneficial technical effects of the present application are as follows:

[0043] 1) For the case that the network switching system is subjected to DoS attacks and fraud attacks, compared with the previous passive compensation scheme, the present application constructs a two-stage single-index Stackelberg game model and a two-stage double-index Stackelberg game model, and realizes active security control of the switching system under network attacks.

[0044] 2) The equilibrium solving algorithm of the two-stage single-index Stackelberg game and the two-stage double-index Stackelberg game is invented, and the optimal strategies of each participant in the control system under the game equilibrium can be obtained. BRIEF DESCRIPTION OF DRAWINGS

[0045] Figure 1 is a control system block diagram under network attacks provided by an embodiment of the present application;

[0046] Figure 2 is a two-stage Stackelberg game model schematic diagram provided by an embodiment of the present application;

[0047] Figure 3 is a switching RLC circuit schematic diagram provided by the embodiment of the application;

[0048] Figure 4 is a subsystem and controller modal schematic diagram provided by the embodiment of the application;

[0049] Figure 5 is a system state trajectory schematic diagram provided by the embodiment of the application;

[0050] Figure 6 is a Stackelberg equilibrium action schematic diagram of the controller provided by the embodiment of the application;

[0051] Figure 7 is a Stackelberg game equilibrium action schematic diagram of the deception attack provided by the embodiment of the application;

[0052] Figure 8 is a Stackelberg game equilibrium action schematic diagram of the SC DoS attack provided by the embodiment of the application;

[0053] Figure 9 is a Stackelberg game equilibrium action schematic diagram of the CA DoS attack provided by the embodiment of the application. DETAILED DESCRIPTION

[0054] It should be noted that, due to the unique advantages of game theory in simulating the confrontation between malicious attackers and defenders, it has become an effective tool for analyzing the security problems of networked control systems. In the game theory-based active security control scheme of networked control systems, the confrontation process between attackers and defenders such as sensors and controllers is described by the model of the game, and the equilibrium solution of the game is used to guide the action strategy of the attack and defense sides, so as to realize the active security control of networked control systems under network attacks.

[0055] Please refer to Figures 1-9 , the application designs a switching system active security control method based on game theory. When the communication network of the switching system is attacked by network attacks, two-level single-index Stackelberg game and two-level double-index Stackelberg game models are proposed, and the solving algorithms of the two game models are designed. By calculating the equilibrium solution of the two games, the optimal strategy of the network attack and the controller is obtained, considering the system open loop and asynchronous switching caused by DoS attack and deception attack, and the sufficient condition for the exponential stability of the control system is given. Finally, the effectiveness of the method is verified by the switching RLC circuit system simulation model.

[0056] 4.1 System model

[0057] Consider the discrete switching system model as

[0058] x(k+1)=Aσ(k) x(k)+B σ(k) u(k), (1)

[0059] in, For system status, For system input. A σ(k) For the state matrix, B σ(k) Let K(k) be the input matrix. σ(k) represents the switching signal, and σ(k)∈S, S∈{0,1,...,s}, where s represents the number of subsystems. Let K(0,k) be the input matrix. m )={k1,k2,…,k m} represents the set of switching moments, where k m This represents the m-th switching moment.

[0060] The controller output is represented as

[0061]

[0062] Where c(k) represents the switching signal transmitted to the controller. H is the input information to the controller, that is, the value of the system state when it reaches the controller. c(k) The control gain to be designed. System state transmitted to the controller under spoofing and SC DoS attacks. The switching signal c(k) can be expressed as follows:

[0063]

[0064] Under a CA DoS attack, the control input transmitted to the actuator can be represented as:

[0065]

[0066] Wherein, the packet delivery function of system state x(k) and the transmission accuracy of switching signal σ(k) are respectively defined as and And satisfy and Controller output information The packet delivery function is defined as φ c(k) (k)∈{0,1}, and E{φ c(k) (k)}=ρ c(k) That is, when x(k) was successfully transmitted without being attacked; when When σ(k) is not tampered with, it is transmitted correctly; when φ c(k) When (k) = 1, Transmission proceeded smoothly without obstruction.

[0067] Definition 1 Let N i(k0, k) denotes the number of times the i-th subsystem switches within the interval (k0, k). For the switching signal σ(k) = i, there exists N i0 , τ iA such that

[0068]

[0069] where N i0 is the tremor bound, τ iA is the average residence time of the i-th subsystem, and T i (k, k0) is the residence time length of subsystem i from k0 to k.

[0070] 4.2 Stackelberg game

[0071] In this section, two-stage single-index Stackelberg game models are established to describe the game decision-making process of CA DoS attacker, controller, SC DoS attacker, and two-stage double-index Stackelberg game models are established to describe the game decision-making process of CA DoS attacker, controller, and fraud attacker. And the SC DoS attacker and the fraud attacker will also interact information to achieve the maximum damage to the system, which is reflected in the design of the two index functions. The two game models are designed as follows.

[0072] 4.2.1 Two-stage single-index Stackelberg game

[0073] (A) Participants of two-stage single-index Stackelberg game

[0074] (1) SC DoS attacker Acting on the SC network, the SC DoS attacker obtains the information transmitted in the system earlier than the controller and the CA DoS attacker, which makes the action of the SC DoS attacker in a leading position, so it is modeled as the leader of the controller and the CA DoS attacker in the second stage of Stackelberg game.

[0075] (2) Controller (P u ): Among the three participants, it is in the middle of information transmission. The controller makes action before the CA DoS attacker, and is modeled as the leader of the CA DoS attacker in the first stage of Stackelberg game. And the decision of the controller is after the SC DoS attacker, and is modeled as the follower of the SC DoS attacker in the second stage of Stackelberg game.

[0076] (3) CA DoS attacker The action of the CA network, the action in the SC DoS attacker and the controller, while being modeled as the follower of the controller in the first-level Stackelberg game and the follower of the SC DoS attacker in the second-level Stackelberg game, can only be made after the action of the SC DoS attacker and the controller, and the optimal response based on its own index function.

[0077] (B) Two-level single-index Stackelberg game index function

[0078] Before designing the index function, the information set of the SC DoS attacker, the controller and the CA DoS attacker in the game needs to be understood. The basic information A σ(k) , B σ(k) , C σ(k) of the system is known to all three game participants, and other information needs to be judged according to the different positions of each participant in the system. The information set of the SC DoS attacker. Due to the leading position in the game, the SC DoS attacker can obtain the index functions J u , and strategies S u , The information set of the controller. The controller is in the middle position of the two-level multi-attacker Stackelberg game, and the index function J and strategy S of the CA DoS attacker are known. The decision information of the SC DoS attacker is unknown. As a follower of the SC DoS attacker and the controller, the information set of the CA DoS attacker is The CA DoS attacker can steal and c(k) through the SC network. Next, the index functions of the SC DoS attacker, the controller and the CA DoS attacker will be designed respectively.

[0079] i. Design of the index function of the CA DoS attacker:

[0080] From the information set , the CA DoS attacker cannot obtain the decision information of the SC DoS attacker and the controller, but can estimate the controller gain H c(k) from the stolen information and the received information. Under the known information set of the CA DoS attacker, the switching system (1) can be expressed as

[0081]

[0082] The index function is designed as

[0083]

[0084] where G c(k) > 0 and R c(k) > 0 are the weight matrices of the CA DoS attacker. x(k + 1) in (8) satisfies (7).

[0085] ii. Index function design of the controller:

[0086] According to the information set I u of the controller, the controller can obtain the information of p c(k) . Under the known information set I u of the controller, the switching system (1) can be described as

[0087]

[0088] The index function is designed as

[0089]

[0090] where Q c(k) > 0 and U c(k) > 0 are the weight matrices of the controller. x(k + 1) in (10) satisfies (9).

[0091] iii. Index function design of the SC DoS attacker:

[0092] According to the information set I of the SC DoS attacker, the attacker can obtain the information of H c(k) and p c(k) . Under the known information set I of the SC DoS attacker, the switching system (1) can be described as

[0093]

[0094] The index function is designed as

[0095]

[0096] Note: In the two-level single-index Stackelberg game, (12) of the DoS attacker not only contains the decision H c(k) of the controller, but also considers the decision of the CA DoS attacker. Moreover, there is information interaction between the SC DoS attacker and the deception attacker, i.e., the decision of the deception attacker is also considered in c(k).

[0097] Definition 2: For the designed strategy pair With a proper switching law design, if the following conditions are met, it can be called a two-level single-index Stackelberg equilibrium strategy.

[0098] (1) For a given S u The common followers of the first and second level Stackelberg games There exists a Minimize Followers There is always a mapping Make

[0099]

[0100] in, This represents the optimal response of the follower in a first-level Stackelberg game.

[0101] (2) For a given Follower P in Level 2 Stackelberg Game u (The first-level leader) exists as a Minimize That is, in the second-level Stackelberg game, the follower always has a mapping. Make

[0102]

[0103] in, This is known as the optimal response of the follower in a second-order Stackelberg game.

[0104] (3) In the second-order game, the leader There are also strategies It can minimize its index function This makes it possible for all ,have

[0105] (15)

[0106] This represents the optimal leadership strategy in a second-order Stackelberg game. It also serves as the optimal leadership strategy for the leader in the first-level Stackelberg game and the optimal following strategy for the follower in the second-level Stackelberg game. This represents the optimal following strategy in both Level 1 and Level 2 Stackelberg games.

[0107] 4.2.2 Two-level dual-index Stackelberg game

[0108] (A) Participants in a two-level, dual-indicator Stackelberg game

[0109] (1) Deceiver The deceiver acts on the SC network, and gets the information transmitted in the system earlier than the controller and the CA DoS attacker, which makes the deceiver's action in the leading position, so the deceiver is modeled as the leader of the second stage Stackelberg game of the controller and the CA DoS attacker.

[0110] (2) Controller (P u ): Among the three participants, the controller is in the middle of the information transmission. The controller makes the action before the CA DoS attacker, and is modeled as the leader of the first stage Stackelberg game of the CA DoS attacker. While the decision of the controller is after the deceiver, and is modeled as the follower of the second stage Stackelberg game of the deceiver.

[0111] (3) CA DoS attacker The CA DoS attacker acts on the CA network, and the action is after the deceiver and the controller. At the same time, the CA DoS attacker is modeled as the follower of the first stage Stackelberg game of the controller and the follower of the second stage Stackelberg game of the deceiver, and can only make the optimal response based on the own index function after the actions of the deceiver and the controller.

[0112] (B) Two-stage double-index Stackelberg game index function

[0113] Before designing the index function, the information set of the deceiver, the controller and the CA DoS attacker in the game needs to be understood. The basic information A σ(k) , B σ(k) , C σ(k) of the system of the three game participants is known, and for other information, it needs to be judged according to the different positions of each participant in the system.

[0114] The information set of the deceiver is Because of the leading position in the game, the deceiver can know the index functions J u , and strategies S u , The information set of the controller is and strategies The decision information of the SC DoS attacker is unknown. As the follower of the deceiver and the controller, the information set of the CA DoS attacker is The CA DoS attacker can steal the information transmitted through the SC network and c(k). Next, we design the index functions for the impostor attacker, the controller, and the CA DoS attacker, respectively.

[0115] (1) CA DoS attack index function, same as (8).

[0116]

[0117] (2) Controller index function, same as (10).

[0118]

[0119] (3) Impostor attack index function

[0120] According to the information set of the impostor attacker The attacker can obtain H c(k) , p c(k) Information. The switching system (1) under the information set of the impostor attacker can be described as

[0121]

[0122] The index function of the impostor attacker for the switching signal s(k) is designed as

[0123] First, find

[0124]

[0125] Second, find

[0126]

[0127] where x(k+1) in (17), (18) satisfies (16).

[0128] Definition 3: For the designed strategy pair where Under the appropriate switching law design, if the following conditions are met, it can be called a two-stage double index Stackelberg equilibrium strategy.

[0129] (1) For a given S u , the first and second stage Stackelberg games have a common follower There is a that can minimize That is, the follower There is always a mapping such that

[0130]

[0131] in, This represents the optimal response of the follower in a first-level Stackelberg game.

[0132] (2) For a given Follower P in Level 2 Stackelberg Game u (The first-level leader) exists as a Minimize That is, in the second-level Stackelberg game, the follower always has a mapping. Make

[0133]

[0134] in, This is known as the optimal response of the follower in a second-order Stackelberg game.

[0135] (3) In the second-order game, the leader There are also strategies It can minimize its index function This makes it possible for all ,have

[0136] (twenty one)

[0137] This represents the optimal leadership strategy in a second-order Stackelberg game. It also serves as the optimal leadership strategy for the leader in the first-level Stackelberg game and the optimal following strategy for the follower in the second-level Stackelberg game. This represents the optimal following strategy in both Level 1 and Level 2 Stackelberg games.

[0138] 4.3 Equilibrium Solution of Two-Level Stackelberg Game

[0139] According to Definition 2, Definition 3 calculates the equilibrium strategy for SC DoS attackers, controllers, CA DoS attackers, and spoofing attacks step by step.

[0140] Step 1: Based on definition 2(1), find the optimal strategy for SC DoS attacks.

[0141]

[0142] in, Definition 2 The minimization problem can be transformed into Z c(k) In the independent variable ρ c(k) The minimum value problem. Let Z c(k) For ρ c(k) Differentiating by 0, we get

[0143]

[0144] Considering the actual situation in the packet loss rate ρ c(k) Should be between (0, 1), get CA DoS attacker optimal strategy for

[0145]

[0146] Step2: According to the definition of 2 (2), the optimal strategy of the controller is calculated, and the controller knows the optimal decision information of the follower

[0147] If Holds, then

[0148]

[0149] To minimize the above formula, let J u The derivative of is equal to 0, and

[0150]

[0151] That is, the optimal strategy of the controller is

[0152]

[0153] Step3: According to the definition of 2 (3), the optimal strategy of SC DoS attack barrier x(k) is calculated, and if

[0154]

[0155] Holds, then

[0156]

[0157] To minimize the above formula, let The derivative of is equal to 0, and

[0158]

[0159] Considering the actual situation in the packet loss rate Should be between (0, 1), get

[0160]

[0161] Step4: According to the definition of 3 (3), the optimal attack strategy of the deception attacker to the switching signal σ(k) is calculated.

[0162] For any switching signal, if

[0163]

[0164] Then the optimal strategy of the deception attack is as follows. First, we solve the first-order index

[0165]

[0166] Minimize

[0167]

[0168] Substitute into , we get the second-order index function

[0169]

[0170]

[0171]

[0172] Let the above equation equal to 0, we get

[0173]

[0174] Considering that the correct transmission rate σ(k) is between 0 and 1 in the actual situation, we have

[0175]

[0176] 4.4 Stability analysis

[0177] Suppose that the current i-th subsystem is activated and the controller is in mode j. In the two-level game, when the subsystem is in mode i, the switching signal is tampered into p, and i, j, p ∈ S, i ≠ p, j ∈ {i, p}.

[0178] Theorem 1 For given κ i ∈(0,1), α i ∈(1,∞), β i ∈(1,∞), μ i ∈(1,∞), if there exists a positive definite matrix P i such that

[0179] (A i +B i H i ) T P i (A i +B i H i )≤κ i ​​​​​P i (23)

[0180]

[0181] (A i +B i H p ) T P i (A i +B i H p )≤β i P i (25)

[0182] P i ≤μ i P j (26)

[0183] and the average dwell time satisfies

[0184]

[0185] The multiple Lyapunov functions are chosen as

[0186] V i (k)=x T (k)P i x(k) (28)

[0187] Suppose the current i-th subsystem is activated and the controller is in mode j, the switched system can be represented as

[0188]

[0189] The switching system under SC DoS attack, spoofing attack and CA DoS attack can be discussed in the following three cases.

[0190] A1 (system normal operation): x(k), σ(k), are not attacked,

[0191] x(k+1)=A i x(k)+B i H i x(k)

[0192] ΔV i (k)=V i (k+1)-V i (k)

[0193] =x T (k+1)P i x(k+1)-x T (k)Pi x(k)

[0194] = x T (k) (A i + B i H i ) T P i (A i + B i H i )x(k) - x T (k) P i x(k)

[0195] ≤ (κ i - 1) V i (k)

[0196] get

[0197] V i (k + 1) ≤ κ i V i (k)

[0198] A2 (system open-loop operation): σ(k) is not attacked, (x(k) and one is attacked or both are attacked)

[0199] x(k + 1) = A i x(k)

[0200] ΔV i (k) = V i (k + 1) - V i (k)

[0201] = x T (k + 1) P i x(k + 1) - x T (k) P i x(k)

[0202] = x T (k) A i T P i A i x(k) - x T (k) P i x(k)

[0203] ≤ (α i - 1) V i (k)

[0204] V i (k + 1) ≤ α i V i (k)

[0205] A3 (system asynchronous running): x(k) and Neither x(k) nor σ(k) is attacked

[0206] x(k+1) = A i x(k) + B i H p x(k)

[0207] ΔV i (k) = V i (k+1) - V i (k)

[0208] = x T (k+1)P i x(k+1) - x T (k)P i x(k)

[0209] = x T (k)(A i +B i H p ) T P i (A i +B i H p )x(k) - x T (k)P i x(k)

[0210] ≤ (β i -1)V i (k)

[0211] V i (k+1) ≤ β i V i (k)

[0212] Next, according to the system synchronous or asynchronous running, the average dwell time condition of the switched system is solved.

[0213] (1) When the system runs synchronously, it can be concluded that

[0214]

[0215] Substituting the switching point condition, we get

[0216]

[0217] (2) When the system runs asynchronously

[0218]

[0219] Substituting the switching point condition, we get

[0220]

[0221] Consider the switching signal packet function The Lyapunov function variation of the switching system is combined with the system synchronous and asynchronous

[0222]

[0223] Let denote the time of the qth switching to subsystem i, denote the last time within its activation time, and the entire system running time can be derived as

[0224]

[0225] According to The above formula can be simplified as

[0226]

[0227] Similarly, according to φ j (k)∈{0,1}, continue to simplify the above formula

[0228]

[0229] It is known that under the two-stage Stackelberg game equilibrium Let denote the residence time of the qth switching to subsystem i, then in the entire system running time, the expectation of V i (k) satisfies

[0230]

[0231] Where, T i (k0,k) denotes the length of time that the ith subsystem is activated within the interval (k0,k). For the ith subsystem, there is

[0232]

[0233] Combined with in Definition 1, we can get

[0234]

[0235] From the above formula, according to the switching system exponential stability condition, the average residence time condition is derived as

[0236]

[0237] The stability of the switching system is proved.

[0238] As Figure 1As shown, the switching signal σ(k) and the system state information x(k) can be affected by the deception attack and the SC DoS attack when transmitted in the Sensor to Controller (SC) network, and the controller output information The transmission in the Controller to Actuator (CA) network can be affected by the CA DoS attack, and the attacker can steal the c(k) and information transmitted in the CA network.

[0239] The present application respectively constructs a two-level single-index Stackelberg game model (CA DoS attack + controller + SC DoS attack) and a two-level double-index Stackelberg game model (CA DoS attack + controller + deception attack).

[0240] In the first-level game, the controller is modeled as a leader and the CA DoS attack is modeled as a follower.

[0241] In the second-level game, the two-level single-index Stackelberg game models the SC DoS attack as a leader and the controller and the CA DoS attack as followers.

[0242] The two-level double-index Stackelberg game models the deception attack as a leader and the controller and the CA DoS attack as followers, and the deception attack has a double-index function.

[0243] Simulation verification

[0244] In this section, the switching RLC circuit will be used to verify the effectiveness of the active security control scheme in this chapter. The switching RLC circuit can be modeled as a continuous-time switched system, where x = [q c i L ] T , q c represents the charge of the capacitor, and i L is the magnetic flux in the inductor. The corresponding system matrix can be represented as

[0245] σ = 1, 2, 3.

[0246] Select L = 1 H, R = 1 Ω, c1 = 1 F, c2 = 0.5 F, c3 = 0.67 F, then the specific system matrix value is represented as

[0247]

[0248] Since the invention discussed is a discrete-time system model, the switching RLC system matrix of the continuous-time system needs to be discretized accordingly. The discrete time step is selected as T = 1 s, and the discretized switching system (1) is obtained, and the system matrix is

[0249]

[0250] In the two-stage Stackelberg game, the weight matrix of the deception attacker, the SC DoS attacker, the controller, and the CA DoS attacker is set as S1 = S2 = S3 = I, Q1 = Q2 = Q3 = 1, R1 = R2 = R3 = 1, According to the two Stackelberg game equilibrium solutions obtained in the lemma.

[0251] The optimal packet rate of the controller output information under the CA DoS attacker The optimal packet rate of the controller output information under the CA DoS attacker The optimal packet rate of the controller output information under the CA DoS attacker

[0252]

[0253] The optimal control gain is

[0254] H1 = [0.0522 -0.2162], H2 = [0.2476 -0.0953], H3 = [0.1688 -0.1558].

[0255] The optimal packet rate of the system state x(k) under the SC DoS attack can be expressed as

[0256]

[0257] Under the deception attack, the benefits of tampering with the switching signal to different modes (the first re-index game)

[0258]

[0259] The minimum value of each row is obtained by comparison , that is, the optimal mode of tampering by the deception attacker, that is,

[0260]

[0261] Therefore, the optimal correct packet rate of the switching signal under the deception attack is

[0262]

[0263] From Figure 4 it can be seen that under the deception attack, the subsystem and the controller exist mode asynchrony. Figure 5 andFigure 6 It is shown that the system converges to a stable state under the active security control method based on Stackelberg game. Figure 7 Figure 9 is the equilibrium action of the cheating attacker, the SC DoS attacker and the CA DoS attacker under Stackelberg game.​

Claims

1. A method for active safety control of a switching system based on Stackelberg game, characterized in that, The method comprises: Step 1, establishing a switching system model; Step 2, if the switching system model is subjected to a network attack, analyzing the network attack to obtain an analysis result; wherein the network attack comprises a DoS attack and a deception attack; Step 3, based on the analysis result, establishing a two-level single-index Stackelberg game model and a two-level double-index Stackelberg game model; Step 4, solving the two-level single-index Stackelberg game model and the two-level double-index Stackelberg game model to obtain an equilibrium solution; Step 5, based on the equilibrium solution, obtaining an optimal strategy of the network attack and the controller; Step 6, based on the optimal strategy, analyzing a sufficient condition for exponential stability of the switching system; The step 3 comprises: establishing a two-level single-index Stackelberg game model, which comprises a participant description of the two-level single-index Stackelberg game and a design of an index function of the two-level single-index Stackelberg game, and an algorithm definition for solving an equilibrium solution of the two-level single-index Stackelberg game is given; establishing a two-level double-index Stackelberg game model, which comprises a participant description of the two-level double-index Stackelberg game and a design of an index function of the two-level double-index Stackelberg game, and an algorithm definition for solving an equilibrium solution of the two-level single-index Stackelberg game is given; The step 4 comprises: solving the equilibrium solution of the SC DoS attack, the controller and the CA DoS attack in the two-level single-index Stackelberg game according to the game equilibrium solution solving algorithm given in the previous step; solving the equilibrium solution of the deception attack, the controller and the CA DoS attack in the two-level double-index Stackelberg game.

2. The method of claim 1, wherein, The step 1 comprises: establishing a switching system model, describing a switching signal, a system state and a system input.

3. The method of claim 2, wherein, The step 2 comprises: analyzing a change of the system state after the SC DoS attack, analyzing a change of the switching signal after the deception attack, and analyzing a change of the controller output information after the CA DoS attack.

4. The method of claim 3, wherein, The step 5 comprises: obtaining an optimal attack strategy of the SC DoS attacker, an optimal attack strategy of the deception attacker, an optimal control strategy of the controller and an optimal attack strategy of the CA DoS attacker based on the equilibrium solution.

5. The method of claim 4, wherein, The step 6 comprises: combining the optimal strategies of the SC DoS attacker, the deception attacker, the controller and the CA DoS attacker under the game equilibrium, and proving the exponential stability of the switching system under the average residence time condition.

6. A device for active safety control of a switching system based on Stackelberg game, characterized in that, The device comprises: a first modeling unit configured to establish a switching system model; a first analysis unit configured to, if the switching system model is subjected to a network attack, analyze the network attack to obtain an analysis result; wherein the network attack comprises a DoS attack and a deception attack; a second modeling unit configured to, based on the analysis result, establish a two-level single-index Stackelberg game model and a two-level double-index Stackelberg game model; a first solving unit configured to solve the two-level single-index Stackelberg game model and the two-level double-index Stackelberg game model to obtain an equilibrium solution; a second solving unit, configured to obtain optimal strategies of the network attack and the controller based on the equilibrium solution; a second analyzing unit, configured to analyze sufficient conditions for exponential stability of the switching system based on the optimal strategies; The second modeling unit is further configured to establish a two-stage single-index Stackelberg game model, which includes a participant description of the two-stage single-index Stackelberg game and a design of an index function of the two-stage single-index Stackelberg game, and algorithm definitions for solving equilibrium solutions of the two-stage single-index Stackelberg game are given. The second modeling unit is further configured to establish a two-stage double-index Stackelberg game model, which includes a participant description of the two-stage double-index Stackelberg game and a design of an index function of the two-stage double-index Stackelberg game, and algorithm definitions for solving equilibrium solutions of the two-stage single-double-index Stackelberg game are given. The first solving unit is further configured to solve equilibrium solutions of SC DoS attacks, the controller and CA DoS attacks in the two-stage single-index Stackelberg game and solve equilibrium solutions of deception attacks, the controller and CA DoS attacks in the two-stage double-index Stackelberg game according to the game equilibrium solution solving algorithm given in the previous step.

7. The apparatus of claim 6, wherein The first modeling unit is further configured to establish a switching system model, which describes switching signals, system states and system inputs.

8. The apparatus of claim 7, wherein The first analyzing unit is further configured to analyze changes in system states after SC DoS attacks, analyze changes in switching signals after deception attacks, and analyze changes in controller output information after CA DoS attacks.

9. The apparatus of claim 8, wherein The second solving unit is further configured to obtain optimal attack strategies of SC DoS attackers, optimal attack strategies of deception attackers, optimal control strategies of the controller and optimal attack strategies of CA DoS attackers based on the obtained equilibrium solutions.

10. The apparatus of claim 9, wherein The second analyzing unit is further configured to prove exponential stability of the switching system under the average residence time condition in combination with the optimal strategies of the SC DoS attackers, the deception attackers, the controller and the CA DoS attackers in the game equilibrium.

Images