A security analysis and control method for cyber-physical systems under periodic denial of service attacks

Abstract

A method for security analysis and control of an information-physical system under a periodic denial-of-service attack belongs to the field of information-physical systems. The present invention solves the problem of poor security analysis and control performance of existing methods. The present invention provides conditions for ensuring exponential stability of a closed-loop system. Independent of any assumptions and the residence time method used in the switching system, a critical value of the amount of denial-of-service attacks within an attack cycle can be derived through direct analysis, and the information-physical system can maintain stability when the critical value is higher than the critical value. Compared with existing methods, the method of the present invention has lower conservatism and improves the security analysis and control performance of information-physical systems. The method of the present invention can be applied to the security analysis and control of information-physical systems.

Description

Technical Field

[0001] The present invention belongs to the field of information-physical systems, and specifically relates to a security analysis and control method for information-physical systems under periodic denial-of-service attacks. Background Art

[0002] Cyber-physical systems integrate computing resources, communication networks, and physical systems, and are a new generation of intelligent control systems. The system has been applied in various fields, such as smart grids, aerospace, power systems, etc. Cyber-physical systems involve interactions between the network layer and the physical layer, which makes the system very vulnerable to attacks by malicious adversaries. By performing malicious attacks, the performance of the cyber-physical system may be degraded or even destroyed, or the adversary may illegally benefit from the malicious attack. There are many examples of attacks in recent years, such as the Stuxnet computer worm attack. Therefore, it is of great significance to propose a security control algorithm for cyber-physical systems against malicious attacks.

[0003] There are various research results on cyber-physical systems under malicious attacks. As an effective but easy-to-implement attack, denial of service attacks have received considerable attention. At present, experts and scholars have conducted in-depth research on the security state estimation problem and security control problem under denial of service attacks modeled using random models. However, it is worth noting that the denial of service attacks modeled using random models rely on an assumption that the malicious attacker has no energy constraints, which is difficult to achieve in reality because continuous malicious attacks consume a lot of attack energy and increase the probability of the attack being detected. Therefore, another model to describe denial of service attacks is proposed, namely, periodic denial of service attacks. Periodic denial of service attacks can reduce the probability of being detected, and by designing attack parameters that meet certain conditions, denial of service attacks are almost undetectable. Therefore, from the attacker's perspective, periodic denial of service attacks are more practical. At present, the filtering and control problems of cyber-physical systems under periodic denial of service attacks have received some attention, but the existing research results model the physical controlled objects of cyber-physical systems as switching systems, and use the switching control method to design control strategies, while the use of the residence time method in the switching system to analyze the stability of the control system will produce a certain degree of conservatism. Therefore, the safety analysis and control performance of existing methods is still poor, and it is necessary to propose a new safety analysis and control method. Summary of the invention

[0004] The purpose of the present invention is to solve the problem of poor security analysis and control performance of existing methods, and to propose a security analysis and control method for information-physical systems under periodic denial of service attacks.

[0005] The technical solution adopted by the present invention to solve the above technical problems is:

[0006] A security analysis and control method for a cyber-physical system under a periodic denial of service attack, the method specifically comprising the following steps:

[0007] Step 1: construct a physical controlled object model of the cyber-physical system without external disturbance;

[0008] The physical controlled object model is a linear discrete time system of formula (1):

[0009] x k+1 =Ax k +Bu k (1)

[0010] Among them, x k+1 is the system state at time k+1, x k is the system state at time k, u k is the control signal received by the actuator at time k, is a real number, m and n are dimensions, A and B are system matrices;

[0011] Step 2: Model the periodic denial of service attack, and construct a closed-loop system under the periodic denial of service attack based on the modeling results and the physical controlled object model;

[0012] Step 3: Construct the exponential stability condition of the closed-loop system under periodic denial of service attack, and give the critical condition of the periodic denial of service attack cycle;

[0013] If the critical condition of the periodic denial of service attack period is met, then execute step 4, otherwise end;

[0014] Step 4: Based on the exponential stability condition of the closed-loop system under periodic denial of service attacks, design a controller for the cyber-physical system without external disturbances under periodic denial of service attacks.

[0015] A security analysis and control method for a cyber-physical system under a periodic denial of service attack, the method specifically comprising the following steps:

[0016] Step 1: construct a physical controlled object model of the cyber-physical system without external disturbance;

[0017] The physical controlled object model is a linear discrete time system of formula (1):

[0018] x k+1 =Ax k +Bu k (1)

[0019] Among them, x k+1is the system state at time k+1, x k is the system state at time k, u k is the control signal received by the actuator at time k, is a real number, m and n are dimensions, A and B are constant matrices;

[0020] Step 2: Model the periodic denial of service attack, and construct a closed-loop system under the periodic denial of service attack based on the modeling results and the physical controlled object model;

[0021] The specific process of step 2 is as follows:

[0022] Design the controller u of equation (2) for the system of equation (1) ko :

[0023] u ko =-K′x k (2)

[0024] Among them, K′ represents the controller gain matrix to be designed, and the denial of service attack is composed of the periodic model Described as:

[0025]

[0026] Where T represents the attack period, n represents the nth attack period, is an integer, τ is the duration without denial of service attack in each attack cycle, and τ<T;

[0027] Under periodic denial of service attacks, the control signal u k for:

[0028]

[0029] The closed-loop system is:

[0030]

[0031] Step 3: Design a controller for the closed-loop system based on reinforcement learning;

[0032] Step 4: Conduct a safety analysis on the controller designed in step 3.

[0033] A security analysis and control method for a cyber-physical system under a periodic denial of service attack, the method specifically comprising the following steps:

[0034] Step 1: construct a physical controlled object model of the cyber-physical system with external disturbances;

[0035] The physical controlled object model is a linear discrete time system of formula (29):

[0036]

[0037] Among them, x k+1 is the system state at time k+1, x k is the system state at time k, u k is the control signal received by the actuator at time k, A and B are system matrices, C is the output matrix, E is the disturbance matrix, ω k represents the external disturbance at time k, y k is the control output at time k, is a real number, m, n, p and q are dimensions;

[0038] Step 2: Model the periodic denial of service attack, and construct a closed-loop system under the periodic denial of service attack based on the modeling results and the physical controlled object model;

[0039] The specific process of step 2 is as follows:

[0040] Design the controller u of equation (30) for the system of equation (29) ko :

[0041] u ko =-Kx k (30)

[0042] Where K represents the controller gain matrix to be designed;

[0043] Denial of service attacks are described by the periodic model:

[0044]

[0045] Where T represents the attack period, n represents the nth attack period, is an integer, τ is the duration without denial of service attack in each attack cycle, and τ<T;

[0046] Under periodic denial of service attacks, the control signal u k for:

[0047]

[0048] According to equation (31) and equation (29), the closed-loop system is:

[0049]

[0050] Step 3: Construct the exponential stability condition of the closed-loop system under periodic denial of service attack, and give the critical condition of the periodic denial of service attack cycle;

[0051] If the critical condition of the periodic denial of service attack period is met, then execute step 4, otherwise end;

[0052] Step 4: Based on the exponential stability condition of the closed-loop system under periodic denial of service attacks, design a controller for the cyber-physical system with external disturbances under periodic denial of service attacks.

[0053] The beneficial effects of the present invention are:

[0054] The present invention studies the security analysis and security control of cyber-physical systems under periodic denial of service attacks, and provides conditions to ensure exponential stability of closed-loop systems. Independent of any assumptions and the residence time method used in the switching system, the critical value of the amount of denial of service attacks within an attack cycle can be derived through direct analysis, and the cyber-physical system can maintain stability when it is above this critical value. Compared with existing methods, the method of the present invention has lower conservatism and improves the security analysis and control performance of cyber-physical systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 This is a block diagram of the cyber-physical system under periodic denial of service attacks;

[0056] Figure 2 Schematic diagram of a two-wheeled mobile robot;

[0057] Figure 3 The actual trajectory and reference trajectory of the mobile robot when satisfying Theorem 1;

[0058] Figure 4 The actual heading angle and target heading angle of the mobile robot when Theorem 1 is satisfied;

[0059] Figure 5 The actual trajectory and expected trajectory of the mobile robot when satisfying Theorem 3;

[0060] Figure 6 The actual heading angle and target heading angle of the mobile robot when Theorem 3 is satisfied. DETAILED DESCRIPTION

[0061] Specific implementation method 1: Combination Figure 1 This embodiment describes a method for security analysis and control of a cyber-physical system under periodic denial of service attacks, the method specifically comprising the following steps:

[0062] Step 1: construct a physical controlled object model of the cyber-physical system without external disturbance;

[0063] The physical controlled object model is a linear discrete time system of formula (1):

[0064] x k+1 =Ax k +Bu k (1)

[0065] Among them, x k+1 is the system state at time k+1, x k is the system state at time k, u k is the control signal received by the actuator at time k, is a real number, m and n are dimensions, A and B are system matrices; the matrix pair (A, B) is controllable, the state x k It is measurable.

[0066] Step 2: Model the periodic denial of service attack, and construct a closed-loop system under the periodic denial of service attack based on the modeling results and the physical controlled object model;

[0067] Step 3: Construct the exponential stability condition of the closed-loop system under periodic denial of service attack, and give the critical condition of the periodic denial of service attack cycle;

[0068] If the critical condition of the periodic denial of service attack period is met, then execute step 4, otherwise end;

[0069] Step 4: Based on the exponential stability condition of the closed-loop system under periodic denial of service attacks, design a controller for the cyber-physical system without external disturbances under periodic denial of service attacks.

[0070] When the closed-loop system does not meet the critical conditions of the periodic denial of service attack cycle under a periodic denial of service attack, it is necessary to improve the ability of the information layer of the cyber-physical system to transmit signals (to overcome the powerful energy of the periodic denial of service attack) so that the closed-loop system meets the conditions. Then, the controller can be designed using the method in step four.

[0071] Specific implementation method 2: This implementation method is different from the specific implementation method 1 in that the specific process of step 2 is as follows:

[0072] Design the controller u of equation (2) for the system of equation (1) ko :

[0073] u ko =-Kx k (2)

[0074] Where K represents the controller gain matrix to be designed. Figure 1In the example, the controller-actuator communication network is blocked by a malicious denial-of-service attack, which is caused by a periodic model Described as:

[0075]

[0076] Where T represents the attack period, n represents the nth attack period, is an integer, τ is the duration without denial of service attack in each attack cycle, and τ<T;

[0077] Under periodic denial of service attacks, Figure 1 The control signal u in k for:

[0078]

[0079] The closed-loop system is:

[0080]

[0081] The other steps and parameters are the same as those in the first embodiment.

[0082] Specific implementation method three: This implementation method is different from the specific implementation method two in that the index stability condition is:

[0083] Given scalar constants 0<μ1<1,μ2>0 and a gain matrix K, there exists a symmetric matrix P>0 satisfying inequalities (5) and (6):

[0084] (A-BK) T P(A-BK)-P+μ1P<0 (5)

[0085] A T PA-P-μ2P<0 (6)

[0086] The critical conditions of the periodic denial of service attack cycle are:

[0087]

[0088] Among them, ceil(·) represents rounding up, the superscript T represents transposition, and ln is the natural logarithm. is the critical value of τ.

[0089] When the system index is stable:

[0090]

[0091] Among them, λ M (P) is the maximum eigenvalue of the matrix P, λ m(P) is the minimum eigenvalue of the matrix P, ||·|| is the 2-norm, x0 is the state of the system at the initial moment,

[0092] The other steps and parameters are the same as those in the second specific implementation manner.

[0093] Specific implementation method 4: This implementation method is different from specific implementation method 3 in that the specific process of step 4 is as follows:

[0094] Solve the matrix F and the matrix that satisfy inequalities (26) and (27) And F>0:

[0095]

[0096]

[0097] The “*” in formula (26) represents The “*” in formula (27) represents (AF) T ;

[0098] Then the gain matrix K is:

[0099]

[0100] Among them, the superscript -1 is the inverse of the matrix;

[0101] Substituting the gain matrix K into equation (3), we obtain the controller of the cyber-physical system without external disturbance.

[0102] The other steps and parameters are the same as those in the third specific implementation manner.

[0103] The safety analysis performed using this implementation is less conservative than the existing methods.

[0104] Specific implementation method 5: A security analysis and control method for a cyber-physical system under a periodic denial of service attack described in this implementation method specifically includes the following steps:

[0105] Step 1: construct a physical controlled object model of the cyber-physical system without external disturbance;

[0106] The physical controlled object model is a linear discrete time system of formula (1):

[0107] x k+1 =Ax k +Bu k (1)

[0108] Among them, x k+1 is the system state at time k+1, xk is the system state at time k, u k is the control signal received by the actuator at time k, is a real number, m and n are dimensions, A and B are constant matrices; the matrix pair (A, B) is controllable, the state x k It is measurable.

[0109] Step 2: Model the periodic denial of service attack, and construct a closed-loop system under the periodic denial of service attack based on the modeling results and the physical controlled object model;

[0110] The specific process of step 2 is as follows:

[0111] Design the controller u of equation (2) for the system of equation (1) ko :

[0112] u ko =-K′x k (2)

[0113] Where K′ represents the controller gain matrix to be designed. Figure 1 In the example, the controller-actuator communication network is blocked by a malicious denial-of-service attack, which is caused by a periodic model Described as:

[0114]

[0115] Where T represents the attack period, n represents the nth attack period, is an integer, τ is the duration without denial of service attack in each attack cycle, and τ<T;

[0116] Under periodic denial of service attacks, Figure 1 The control signal u in k for:

[0117]

[0118] The closed-loop system is:

[0119]

[0120] Step 3: Design a controller for the closed-loop system based on reinforcement learning;

[0121] Step 4: Conduct a safety analysis on the controller designed in step 3.

[0122] Specific implementation method 6: This implementation method is different from the specific implementation method 5 in that the specific process of step 3 is: the optimal controller u koDesigned for:

[0123] u ko =-K′x k =-(R+B T PB) -1 B T PAx k (14)

[0124] Where R>0 is a symmetric performance matrix, and the matrix P satisfies the algebraic Riccati equation of formula (15):

[0125] -P+Q+A T PA-A T PB(R+B T PB) -1 B T PA=0 (15)

[0126] Among them, Q ≥ 0 is a symmetric performance matrix;

[0127] Design the Q-function of equation (16):

[0128]

[0129] Among them, Q(x k ,u ko ) is the Q-function;

[0130] Substituting the system of equation (1) into the Q-function of equation (16), we get:

[0131]

[0132] Definition xx =Q+A T PA, Υ xu =A T PB, Υ ux =B T PA, Υ uu =R+B T PB, then:

[0133]

[0134] Applying formula (18) have to:

[0135]

[0136] in, stands for gradient finding;

[0137] From formula (16), we can see that the Q-function satisfies the Bellman equation of formula (20):

[0138]

[0139] Among them, u k+1,o =-K′x k+1 ;

[0140] definition Then formula (18) is rewritten as:

[0141]

[0142] Substituting formula (21) into formula (20), we get:

[0143]

[0144] Then, the gain matrix K′ and matrix P of the controller based on reinforcement learning are calculated according to formula (22), and the gain matrix K′ is substituted into formula (14) to obtain the controller;

[0145] The calculation method of the matrix K′ and P is:

[0146] Step 1: Initialize the reinforcement learning based controller as

[0147] Step 2: Given the learning error ζ, initialize the number of iterations i = 0;

[0148] Step 3: According to the controller Calculate the controller for the i+1th iteration The specific process of step 3 is as follows:

[0149] Step 31: According to Calculate Υ i+1 :

[0150]

[0151] in, P i+1 is the matrix to be solved;

[0152] Step 32: According to i+1 calculate

[0153]

[0154] Step 4: Determine whether ||Υ is satisfied i+1 -Υ i ||<ζ, ||·|| is the 2-norm;

[0155] If it is satisfied, then according to the Υ solved in the last iteration i+1 Get the matrix P, according to the last iteration solution Get the matrix K′;

[0156] If not satisfied, set i=i+1 and return to step 3.

[0157] The other steps and parameters are the same as those in the fifth embodiment.

[0158] Specific implementation method 7: This implementation method is different from specific implementation method 6 in that the specific process of step 4 is as follows:

[0159] According to the gain matrix K′ and the matrix P, the scalar constants 0<μ1<1 and μ2>0 are solved to make inequalities (23) and (24) hold:

[0160] -QK′ T RK′+μ1P<0 (23)

[0161] Υ xx -QP-μ2P<0 (24)

[0162] And the solved μ1 and μ2 satisfy:

[0163]

[0164] ceil(·) represents rounding up, the superscript T represents transposition, and ln represents the natural logarithm. is the critical value of τ;

[0165] Therefore, the closed-loop system achieves exponential stability under the controller designed in step three, verifying the security of the cyber-physical system.

[0166] The other steps and parameters are the same as those in the sixth embodiment.

[0167] Specific implementation method eight: This implementation method is a security analysis and control method for a cyber-physical system under a periodic denial of service attack, and the method specifically includes the following steps:

[0168] Step 1: construct a physical controlled object model of the cyber-physical system with external disturbances;

[0169] The physical controlled object model is a linear discrete time system of formula (29):

[0170]

[0171] Among them, x k+1 is the system state at time k+1, x k is the system state at time k, Status x k It is measurable, k is the control signal received by the actuator at time k, A and B are system matrices, C is the output matrix, E is the disturbance matrix, and the matrix pair (A, B) is controllable, ω k represents the external disturbance at time k, y k is the control output at time k, is a real number, m, n, p and q are dimensions;

[0172] Step 2: Model the periodic denial of service attack, and construct a closed-loop system under the periodic denial of service attack based on the modeling results and the physical controlled object model;

[0173] The specific process of step 2 is as follows:

[0174] Design the controller u of equation (30) for the system of equation (29) ko :

[0175] u ko =-Kx k (30)

[0176] Where K represents the controller gain matrix to be designed;

[0177] Denial of service attacks are described by the periodic model:

[0178]

[0179] Where T represents the attack period, n represents the nth attack period, is an integer, τ is the duration without denial of service attack in each attack cycle, and τ<T;

[0180] Under periodic denial of service attacks, the control signal u k for:

[0181]

[0182] According to equation (31) and equation (29), the closed-loop system is:

[0183]

[0184] Step 3: Construct the exponential stability condition of the closed-loop system under periodic denial of service attack, and give the critical condition of the periodic denial of service attack cycle;

[0185] If the critical condition of the periodic denial of service attack period is met, then execute step 4, otherwise end;

[0186] Step 4: Based on the exponential stability condition of the closed-loop system under periodic denial of service attacks, design a controller for the cyber-physical system with external disturbances under periodic denial of service attacks.

[0187] When the closed-loop system does not meet the exponential stability condition under a periodic denial of service attack, it is necessary to improve the ability of the information layer of the cyber-physical system to transmit signals (to overcome the powerful energy of the periodic denial of service attack) so that the closed-loop system meets the exponential stability condition. Then, the controller can be designed using the method in step four.

[0188] Specific implementation method 9: This implementation method is different from specific implementation method 8 in that the index stability condition is:

[0189] Setting condition 1: given parameters When the external disturbance ω k = 0, the controller (31) is designed so that the closed-loop system (32) achieves exponential stability;

[0190] Condition 2: When the initial condition is 0, the control output y k satisfy

[0191]

[0192] If the above two conditions are met, the closed-loop system (32) is exponentially stable and has a guaranteed H ∞ Performance Level That is, according to conditions 1 and 2, we get the exponential stability condition:

[0193] Given a scalar constant 0<μ1<1,μ2>0,H ∞ Performance level parameters and gain matrix K, there exists a symmetric matrix P>0 satisfying inequalities (34) and (35):

[0194]

[0195]

[0196] Where, Λ1=(A-BK) T P(A-BK)-P+μ1P+C T C, Λ2=A T PA-P-μ2P+C T C, I are the identity matrices;

[0197] The critical conditions of the periodic denial of service attack cycle are:

[0198]

[0199] ceil(·) is rounded up, ln is the natural logarithm, is the critical value of τ.

[0200] The other steps and parameters are the same as those in the eighth embodiment.

[0201] Specific implementation method 10: This implementation method is different from specific implementation methods 8 or 9 in that the specific process of step 4 is as follows:

[0202] Solve the matrix F and the matrix that satisfy inequalities (37) and (38) And F>0:

[0203]

[0204]

[0205] Then the control gain K is designed as:

[0206]

[0207] Among them, the superscript -1 represents the inverse of the matrix;

[0208] Substituting the gain matrix K into equation (31), we obtain the controller of the cyber-physical system with external disturbances.

[0209] The other steps and parameters are the same as those in the eighth or ninth embodiment.

[0210] Example

[0211] The present invention is described in detail below. The method of the present invention is mainly divided into the following steps:

[0212] (1) Description of the physical controlled object of the cyber-physical system without external disturbances

[0213] The physical controlled object of the cyber-physical system without external disturbance is described by the following linear discrete-time system:

[0214] x k+1 =Ax k +Bu k (1)

[0215] in is the system status, is the control input received by the actuator, A, B are constant matrices with appropriate dimensions. The matrix pair (A, B) is controllable, and the state x k It is measurable.

[0216] Design the following controller for system (1):

[0217] u ko =-Kx k (2)

[0218] Where K represents the controller gain value to be designed. Figure 1 It can be seen that the controller-actuator communication network is blocked by malicious denial-of-service attacks. These attacks are described by the following periodic model:

[0219]

[0220] Where T represents the attack period, Indicates the number of periodic DoS attacks. represents the time interval without denial of service attack, and τ<T. Under periodic denial of service attack, Figure 1 The control signal u in k for:

[0221]

[0222] The following closed-loop system can be obtained:

[0223]

[0224] (2) Security analysis of cyber-physical systems under periodic denial of service attacks

[0225] Security analysis method 1:

[0226] Theorem 1. Given scalar constants 0<μ1<1,μ2>0 and a gain matrix K, if there exists a symmetric matrix P>0 satisfying the following inequality, then the closed-loop system (4) can achieve exponential stability.

[0227] (A-BK) T P(A-BK)-P+μ1P<0 (5)

[0228] A T PA-P-μ2P<0 (6)

[0229]

[0230] Further

[0231]

[0232] in,

[0233] Safety analysis method 2: Safety assessment of cyber-physical systems based on reinforcement learning controllers

[0234] Using the control algorithm based on reinforcement learning, conditions (5) and (6) in Theorem 1 can be expressed by data, that is, Theorem 1 can be used to analyze the safety of cyber-physical systems under learning-based controllers. First, the controller design algorithm based on reinforcement learning is introduced. Consider the following cost function:

[0235]

[0236] Where Q ≥ 0, R > 0 is a symmetric performance matrix. The following lemma shows that the cost function (9) is k It is secondary.

[0237] Lemma 1. If the controller is designed as:

[0238] u io =-K′x i (10)

[0239] Then for some matrices P>0, the cost function (9) can be written as:

[0240]

[0241] Combining (9) and (11), we can get the following Bellman equation:

[0242]

[0243] Define the following Hamiltonian:

[0244]

[0245] Next, Lemma 2 is given to design the optimal controller.

[0246] Lemma 2. Using Lemma 1, the optimal controller design is as follows:

[0247] u ko =-K′x k =-(R+B T PB) -1 B T PAx k (14)

[0248] where P satisfies the following algebraic Riccati equation:

[0249] -P+Q+A T PA-A T PB(R+B T PB) -1 B T PA=0 (15) According to (12), the following Q-function is designed:

[0250]

[0251] Substituting system (1) into the Q-function (16) we obtain:

[0252]

[0253] Definition xx =Q+A T PA, Υ xu =A T PB, Υ ux =B T PA, Υ uu =R+B T PB, then:

[0254]

[0255] Application to (18) We can get:

[0256]

[0257] From (16), we can see that the Q-function satisfies the following Bellman equation:

[0258]

[0259] Among them, u k+1,o =-K′x k+1 .definition Then (18) can be rewritten as follows:

[0260]

[0261] Substituting (21) into (20), we can obtain:

[0262]

[0263] Based on the above analysis, the algorithm in Table 1 is proposed to learn the optimal control strategy u ko .

[0264] Table 1

[0265]

[0266] The following corollary establishes the relationship between Theorem 1 and reinforcement learning-based control. Using this corollary, the safety of cyber-physical systems based on reinforcement learning controllers can be evaluated.

[0267] Corollary 1: Use the algorithm in Table 1 to calculate the gain matrix K′ and the matrix P. If there exists a scalar constant 0<μ1<1,μ2>0 so that the following inequality holds, then the closed-loop system (4) can achieve exponential stability.

[0268] -QK′ T RK′+μ1P<0 (23)

[0269] Υ xx -QP-μ2P<0 (24)

[0270]

[0271] The second safety analysis method is to first design a controller based on reinforcement learning, and then perform safety analysis based on the controller.

[0272] (3) Safety controller design

[0273] Next, the controller gain K in (3) will be designed to ensure that the closed-loop system (4) is exponentially stable.

[0274] Corresponding to the first safety analysis method, the design theorem 2 calculates the controller gain:

[0275] Theorem 2: Given a scalar constant 0<μ1<1,μ2>0, if holds and there exists a matrix F>0 and a matrix If the following inequality holds, the closed-loop system (4) can achieve exponential stability.

[0276]

[0277]

[0278] Further, the gain matrix K in (3) is as follows:

[0279]

[0280] The first safety analysis method requires first conducting a safety analysis and then solving the controller gain matrix.

[0281] (4) Safety analysis and control of cyber-physical systems under external disturbances

[0282] The dynamic system under external disturbance can be described as:

[0283]

[0284] Among them, C and E are known constant matrices, represents the external disturbance, To control the output.

[0285] Design the controller u of equation (30) for the system of equation (29) ko :

[0286] uko =-Kx k (30)

[0287] Where K represents the controller gain matrix to be designed;

[0288] Denial of service attacks are described by the periodic model:

[0289]

[0290] Where T represents the attack period, n represents the nth attack period, is an integer, τ is the duration without denial of service attack in each attack cycle, and τ<T;

[0291] Under periodic denial of service attacks, the control signal u k for:

[0292]

[0293] Combining equation (31) and equation (29), the closed-loop system is:

[0294]

[0295] The problem that needs to be solved now is as follows: given the parameters When the external disturbance ω k = 0, the controller (31) is designed to make the closed-loop system (32) exponentially stable; when the initial condition is 0, the control output y k satisfy

[0296]

[0297] If the above two conditions are met, then system (32) is exponentially stable and has a guaranteed H ∞ Performance Level Next, we will solve the above two problems. The following theorem gives the stability conditions.

[0298] Theorem 3: Given a scalar constant 0<μ1<1,μ2>0, and gain matrix K, if there exists a symmetric matrix P>0 that satisfies the following inequality, then the closed-loop system (32) can be exponentially stable and has a guaranteed H ∞ Performance Level

[0299]

[0300]

[0301]

[0302] Where, Λ1=(A-BK) T P(A-BK)-P+μ1P+C T C,Λ2=A T PA-P0-μ2P+C T C.

[0303] (4) Design of safety controller for cyber-physical systems under external disturbances

[0304] Theorem 4. Given scalar constants 0<μ1<1,μ2>0, if condition (36) holds and there exists a matrix F>0 and If the following inequality is satisfied, the closed-loop system (32) can be exponentially stable and has a guaranteed H ∞ Performance Level

[0305]

[0306]

[0307] Furthermore, the controller gain K in (31) can be designed as follows:

[0308]

[0309] The present invention studies the security analysis and security control of information-physical systems under periodic denial of service attacks. Sufficient conditions for ensuring the exponential stability of closed-loop systems are derived. Independent of any assumptions and the residence time method used in switching systems, the critical value of the amount of denial of service attacks within an attack cycle is derived by directly analyzing the Lyapunov function difference with and without denial of service attacks. Below this critical value, the information-physical system cannot maintain stability. Compared with existing research results, this method has lower conservatism. The research block diagram of the present invention is shown in FIG. Figure 1 As shown by Figure 1 It can be seen that the controller and the actuator complete information exchange through the network layer. In order to prevent the transmission of control signals, attackers can implement malicious periodic denial of service attacks.

[0310] Experimental Section

[0311] In order to verify and demonstrate the effectiveness of the method proposed in this invention, a two-wheeled mobile robot will be used to illustrate the effectiveness of the proposed control algorithm. Figure 2 The motion of the mobile robot is represented by the vector [x, y, θ] T It is represented by (x, y) where (x, y) represents the position coordinates of the mobile robot and θ represents the heading angle of the mobile robot. The units of x and y are respectively m, and the units of θ are respectively rad.

[0312] The kinematic model of the mobile robot is as follows:

[0313]

[0314] where v and Denote the linear velocity and angular velocity of the mobile robot respectively. The goal of the mobile robot is to track a circular trajectory, which is generated by the following virtual robot:

[0315]

[0316] where [x r ,y r ,θ r ] is the position and heading angle of the reference robot, v r and denote the linear velocity and angular velocity of the reference robot, respectively.

[0317] According to (40) and (41), the tracking error system is as follows:

[0318]

[0319] where e x =(yy r )sinθ+(xx r )cosθ,e y =(yy r )cosθ-(xx r )sinθ,e θ =θ-θ r .

[0320] Next, the linear discrete-time model of system (42) is given:

[0321]

[0322] Among them, T s is the sampling period.

[0323] According to formula (43), the physical controlled object model x of the cyber-physical system without external disturbance is obtained: k+1 =Ax k +Bu k and the physical controlled object model of the cyber-physical system with external disturbances Among them, the system state x k It consists of the position coordinates of the mobile robot at time k and the heading angle of the mobile robot at time k.

[0324] (1) Safety analysis and controller design of cyber-physical systems without external disturbances

[0325] The effectiveness of the controller designed by Theorem 1 and Theorem 2 will be verified below.

[0326] Let T s =0.1, μ1=0.01, μ2=0.18, T=200. Using Theorem 1, we can get the critical value That is, the number of allowed denial-of-service attacks cannot exceed 11. Let τ = 192, it is obvious that condition (7) in Theorem 1 holds. By solving the linear matrix inequality in Theorem 2, the expected gain matrix K is obtained as follows:

[0327]

[0328] Select the initial conditions [x, y, θ] = [-2, -1.3, -0.1], Figure 3 and Figure 4 The figure shows the movement of the mobile robot. It can be seen that when the duration of the denial of service attack is less than The designed controller can realize the tracking control of the mobile robot.

[0329] (2) Safety analysis and controller design of cyber-physical systems under external disturbances

[0330] The following verifies the effectiveness of the controller designed by Theorem 3 and Theorem 4:

[0331] Select the sampling period T s =0.1, parameters μ1=0.02, μ2=0.26, T=200, Matrix C = [-3, -1, 0]. Matrix E and external disturbance signal are as follows:

[0332]

[0333] Based on the above parameters, the critical value can be obtained using Theorem 3 That is, the maximum number of allowed denial-of-service attacks shall not exceed 16. Let τ = 193, and by solving Theorem 4, we can obtain the expected gain matrix K in (30) as follows:

[0334]

[0335] Select the initial condition [x, y, θ] = [-1.5, -0.5, -0.1], The tracking effect of the designed controller is as follows Figure 5 and Figure 6 As shown. Figure 5 and Figure 6It can be seen that within the allowed duration of the denial of service attack, the designed controller can achieve tracking control of the mobile robot.

[0336] The above calculation examples of the present invention are only used to explain the calculation model and calculation process of the present invention in detail, and are not intended to limit the implementation methods of the present invention. For ordinary technicians in the relevant field, other different forms of changes or modifications can be made based on the above description. It is impossible to list all the implementation methods here. All obvious changes or modifications derived from the technical solution of the present invention are still within the scope of protection of the present invention.

Claims

1. A security analysis and control method for a cyber-physical system under periodic denial of service attacks, characterized in that: The method specifically comprises the following steps: Step 1: construct a physical controlled object model of the cyber-physical system without external disturbance; The physical controlled object model is a linear discrete time system of formula (1): x k+1 =Ax k +Bu k (1) Among them, x k+1 is the system state at time k+1, x k is the system state at time k, u k is the control signal received by the actuator at time k, is a real number, m and n′ are dimensions, A and B are system matrices; Step 2: Model the periodic denial of service attack, and construct a closed-loop system under the periodic denial of service attack based on the modeling results and the physical controlled object model; The specific process of step 2 is as follows: Design the controller u of equation (2) for the system of equation (1) ko : you ko =-Kx k (2) Among them, K represents the controller gain matrix to be designed, and the denial of service attack is composed of the periodic model Described as: Where T represents the attack period, n represents the nth attack period, is an integer, τ is the duration without denial of service attack in each attack cycle, and τ<T; Under periodic denial of service attacks, the control signal u k for: The closed-loop system is: Step 3: Construct the exponential stability condition of the closed-loop system under periodic denial of service attack, and give the critical condition of the periodic denial of service attack cycle; If the critical condition of the periodic denial of service attack period is met, then execute step 4, otherwise end; Step 4: Based on the exponential stability condition of the closed-loop system under periodic denial of service attacks, design a controller for the cyber-physical system without external disturbances under periodic denial of service attacks.

2. The method for security analysis and control of a cyber-physical system under periodic denial of service attacks according to claim 1 is characterized in that: The exponential stability condition is: Given scalar constants 0<μ1<1,μ2>0 and controller gain matrix K, there exists a symmetric matrix P>0 satisfying inequalities (5) and (6): (A-BK) T P(A-BK)-P+μ1P<0 (5) A T PA-P-μ2P<0 (6) The critical conditions of the periodic denial of service attack cycle are: Among them, ceil(·) represents rounding up, the superscript T represents transposition, and ln is the natural logarithm. is the critical value of τ.

3. The method for security analysis and control of a cyber-physical system under periodic denial of service attacks according to claim 2 is characterized in that: The specific process of step 4 is as follows: Solve the matrix F and the matrix that satisfy inequalities (26) and (27) And F>0: The "*" in formula (26) represents The "*" in formula (27) represents (AF) T ; Then the controller gain matrix K is: Among them, the superscript -1 is the inverse of the matrix; Substituting the controller gain matrix K into equation (3), we obtain the controller of the cyber-physical system without external disturbances.

4. A security analysis and control method for a cyber-physical system under periodic denial of service attacks, characterized in that: The method specifically comprises the following steps: Step 1: construct a physical controlled object model of the cyber-physical system without external disturbance; The physical controlled object model is a linear discrete time system of formula (1): x k+1 =Ax k +Bu k (1) Among them, x k+1 is the system state at time k+1, x k is the system state at time k, u k is the control signal received by the actuator at time k, is a real number, m and n′ are dimensions, A and B are constant matrices; Step 2: Model the periodic denial of service attack, and construct a closed-loop system under the periodic denial of service attack based on the modeling results and the physical controlled object model; The specific process of step 2 is as follows: Design the controller u of equation (2) for the system of equation (1) ko : u ko =-K′x k (2) Among them, K′ represents the controller gain matrix to be designed, and the denial of service attack is composed of the periodic model Described as: Where T represents the attack period, n represents the nth attack period, is an integer, τ is the duration without denial of service attack in each attack cycle, and τ<T; Under periodic denial of service attacks, the control signal u k for: The closed-loop system is: Step 3: Design a controller for the closed-loop system based on reinforcement learning; Step 4: Conduct a safety analysis on the controller designed in step 3.

5. The method for security analysis and control of a cyber-physical system under periodic denial of service attacks according to claim 4 is characterized in that: The specific process of step three is: Optimal controller u ko Designed for: u ko =-K′x k =-(R+B T PB) -1 B T PAx k (14) Where R>0 is a symmetric performance matrix, and the matrix P satisfies the algebraic Riccati equation of formula (15): -P+Q+A T PA-A T PB(R+B T PB) -1 B T PA=0 (15)where Q≥0 is a symmetric performance matrix; Design the Q-function of equation (16): Among them, Q(x k ,u ko ) is the Q-function; Substituting the system of equation (1) into the Q-function of equation (16), we get: Define Υ xx = Q + A T PA, Υ xu = A T PB, Υ ux = B T PA, Υ uu = R + B T PB, then we have: Applying formula (18) have to: in, stands for gradient finding; The Q-function satisfies the Bellman equation of formula (20): Among them, u k+1,o =-K′x k+1 ; definition Then formula (18) is rewritten as: Substituting formula (21) into formula (20), we get: Then, the gain matrix K′ and matrix P of the controller based on reinforcement learning are calculated according to formula (22), and the controller gain matrix K′ is substituted into formula (14) to obtain the controller; The calculation method of the matrix K′ and P is: Step 1: Initialize the reinforcement learning based controller as Step 2: Given the learning error ζ, initialize the number of iterations i = 0; Step 3: According to the controller Calculate the controller for the i+1th iteration The specific process of step 3 is as follows: Step 31: According to Calculate Υ i+1 : in, P i+1 is the matrix to be solved; Step 32: According to i+1 calculate Step 4: Determine whether ||Υ is satisfied i+1 -Υ i ||<ζ, ||·|| is the 2-norm; If it is satisfied, then according to the Υ solved in the last iteration i+1 Get the matrix P, according to the last iteration solution Get the matrix K′; If not satisfied, set i=i+1 and return to step 3.

6. The method for security analysis and control of a cyber-physical system under periodic denial of service attacks according to claim 5, characterized in that: The specific process of step 4 is as follows: According to the controller gain matrix K′ and the matrix P, the scalar constants 0<μ1<1 and μ2>0 are solved to make inequalities (23) and (24) hold: -Q-K′ T RK′+μ1P<0 (23) Y xx -QP-μ2P<0 (24) And the solved μ1 and μ2 satisfy: ceil(·) represents rounding up, the superscript T represents transposition, and ln represents the natural logarithm. is the critical value of τ; Therefore, the closed-loop system achieves exponential stability under the controller designed in step three.

7. A security analysis and control method for a cyber-physical system under periodic denial of service attacks, characterized in that: The method specifically comprises the following steps: Step 1: construct a physical controlled object model of the cyber-physical system with external disturbances; The physical controlled object model is a linear discrete time system of formula (29): Among them, x k+1 is the system state at time k+1, x k is the system state at time k, u k is the control signal received by the actuator at time k, A and B are system matrices, C is the output matrix, E is the disturbance matrix, ω k represents the external disturbance at time k, y k is the control output at time k, is a real number, m, n′, p and q are dimensions; Step 2: Model the periodic denial of service attack, and construct a closed-loop system under the periodic denial of service attack based on the modeling results and the physical controlled object model; The specific process of step 2 is as follows: Design the controller u of equation (30) for the system of equation (29) ko : you ko =-Kx k (30) Where K represents the controller gain matrix to be designed; Denial of service attacks are described by the periodic model: Where T represents the attack period, n represents the nth attack period, is an integer, τ is the duration without denial of service attack in each attack cycle, and τ<T; Under periodic denial of service attacks, the control signal u k for: According to equation (31) and equation (29), the closed-loop system is: Step 3: Construct the exponential stability condition of the closed-loop system under periodic denial of service attack, and give the critical condition of the periodic denial of service attack cycle; If the critical condition of the periodic denial of service attack period is met, then execute step 4, otherwise end; Step 4: Based on the exponential stability condition of the closed-loop system under periodic denial of service attacks, design a controller for the cyber-physical system with external disturbances under periodic denial of service attacks.

8. The method for security analysis and control of a cyber-physical system under periodic denial of service attacks according to claim 7, characterized in that: The exponential stability condition is: Given a scalar constant 0<μ1<1,μ2>0,H ∞ Performance level parameters and the controller gain matrix K, there exists a symmetric matrix P>0 satisfying inequalities (34) and (35): Where, Λ1=(A-BK) T P(A-BK)-P+μ1P+C T C, Λ2=A T PA-P-μ2P+C T C, I are the identity matrices; The critical conditions of the periodic denial of service attack cycle are: ceil(·) is rounded up, ln is the natural logarithm, is the critical value of τ.

9. The method for security analysis and control of a cyber-physical system under periodic denial of service attacks according to claim 8, characterized in that: The specific process of step 4 is as follows: Solve the matrix F and the matrix that satisfy inequalities (37) and (38) And F>0: Then the controller gain matrix K is designed as: Among them, the superscript -1 represents the inverse of the matrix; Substituting the controller gain matrix K into equation (31), we obtain the controller of the cyber-physical system with external disturbances.

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