A state estimation method for denial-of-service attack on lipschitz nonlinear systems

Abstract

The application discloses a state estimation method for a denial-of-service attack Lipschitz nonlinear system, aiming at the DoS attack possibly occurring in network transmission and the gain perturbation in the implementation of the estimator. First, an event-triggered communication mechanism is constructed, and sampling data is sent only when the trigger error energy exceeds a set threshold. Then, Bernoulli distribution is used to model whether single data transmission is blocked by the attack, and a non-fragile state estimator with additive perturbation is designed. By defining a state estimation error system, a sufficient matrix inequality is derived based on Lyapunov stability theory to ensure random stability and H performance, which is then converted into a linear matrix inequality to solve the event-triggered weight matrix and the estimator gain parameter at one time, thereby realizing the joint design of the trigger condition and the estimator gain. Simulation shows that even in the case of simultaneous DoS attack and gain perturbation, the method can accurately and quickly estimate the system state, and significantly improve the non-fragility of the system and the network resource utilization.

Description

TECHNICAL FIELD

[0001] The application relates to a state estimation method for a Lipschitz nonlinear system under denial-of-service attack, and belongs to the cross field of control theory and network security. BACKGROUND

[0002] Lipschitz nonlinear systems exist widely in the fields of aircraft attitude control, robot trajectory tracking, chemical reaction process analysis, power system stability control and biological system dynamic behavior analysis. System state information is crucial for system operation, and when the system state cannot be directly measured, a state estimator needs to be designed to estimate the system state. Due to the influence of environmental changes, non-ideal characteristics of components and other factors, the state estimator has gain perturbation when implemented.

[0003] With the expansion of the system scale, system components exchange information through a network, which has become a development trend. The introduction of a network into the system can improve the scalability and collaboration of the system, but also faces network security threats. If the system is subjected to a network attack, the system performance will be severely affected. Denial-of-service attack is a common network attack, and attackers send a large amount of requests or data to the target network, causing the network bandwidth to be exhausted, and the system components cannot exchange information. In order to reduce the network communication burden, the defender can use an event-triggered communication mechanism. When the data to be exchanged meets the trigger condition, it is transmitted through the network. However, the existing state estimation of Lipschitz nonlinear systems does not consider denial-of-service attack, event-triggered communication mechanism and gain perturbation of the state estimator when implemented, and the non-vulnerability is poor. SUMMARY

[0004] The application is to solve the problems existing in the prior art and provides a state estimation method for a Lipschitz nonlinear system under denial-of-service attack.

[0005] The technical scheme adopted by the application is as follows:

[0006] A state estimation method for a Lipschitz nonlinear system under denial-of-service attack, comprising the following steps:

[0007] S1: The sensor samples the measurement output of the Lipschitz nonlinear system, and the event trigger transmits the sampled data meeting the trigger condition to the non-vulnerable state estimator through the network;

[0008] S2: The Bernoulli distribution is used to describe whether the sampled data meeting the trigger condition is subjected to denial-of-service attack at the transmission moment, and the non-vulnerable state estimator receives the data according to the attack situation;

[0009] S3: Define the state estimation error as the difference between the state of the Lipschitz nonlinear system and the output state of the non-fragile state estimator, establish the dynamic model of the state estimation error system, and derive the sufficient condition for the state estimation error system to be stochastically stable and have H ∞ performance based on Lyapunov stability theory.

[0010] S4: Convert the matrix inequality into a linear matrix inequality, solve to obtain the symmetric positive definite matrix of the event trigger and the gain parameter of the non-fragile state estimator, thereby completing the joint design of the event trigger condition and the gain of the non-fragile state estimator.

[0011] Further, in S1, the state space expression of the Lipschitz nonlinear system is as follows:

[0012] ,

[0013] wherein, is the system state vector of the Lipschitz nonlinear system, representing the state of the system at discrete time k, with a dimension of n;

[0014] is the system measurement output of the Lipschitz nonlinear system, with a dimension of p;

[0015] is an external disturbance vector, belonging to space, with a dimension of m;

[0016] is the state to be estimated, with a dimension of q;

[0017] , , , , , and are known constant matrices of corresponding dimensions,

[0018] nonlinear function and satisfy:

[0019] , ,

[0020] ,

[0021] ,

[0022] wherein, n f and n g are functions and dimension, , , x1(k) and x2(k) are two arbitrary state vectors of a Lipschitz nonlinear system at discrete time k; and are known constant matrices with corresponding dimensions.

[0023] Further, in S1, the triggering condition is as follows:

[0024] ,

[0025] where δ(k) is the triggering error vector, , is the last time when data is transmitted over the network, is the last transmitted sample measurement output over the network, is the symmetric positive definite matrix to be solved, is the given event-triggered threshold; T denotes transpose.

[0026] Further, in S2, the Bernoulli distribution is used to describe the denial-of-service attack, as follows:

[0027] The random variable subject to Bernoulli distribution is used to describe the denial-of-service attack, denotes the occurrence of the denial-of-service attack, denotes no denial-of-service attack, The mathematical expectation of

[0028] ,

[0029] where is a known constant.

[0030] Further, in S2, the non-vulnerable state estimator receives data according to the attack situation, as follows:

[0031] For a given Lipschitz nonlinear system, the following non-vulnerable estimator is designed to perform state estimation:

[0032] ,

[0033] where , , respectively correspond to the state, input and output of the estimator, , , , , and For the parameters of the estimator to be designed, , , , and Given a real constant matrix of corresponding dimension, To meet The time-varying matrix, I, is the identity matrix, and the estimator input is:

[0034] .

[0035] Furthermore, in S3, a dynamic model of the state estimation error system is established, as follows:

[0036] set up , , Then the estimation error system is:

[0037] ,

[0038] The coefficient matrices are respectively , , , , , , , , , , , , , , , , , These correspond to the coefficient matrix respectively. , , , , , , , , The nominal value, , , , These correspond to the structure matrix respectively. , , , and The block constant matrix.

[0039] Furthermore, in S3, it is obtained that the state estimation error system is stable and has H ∞ The specific performance conditions are as follows:

[0040] For a given scalar , , Nonfragile state estimator parameters , and The state estimation error system is stochastically stable and has performance If a matrix exists and Makes the following matrix inequalities hold:

[0041] ,

[0042] The matrix elements are respectively , , , , , , , , , , , , , , , , * denotes a symmetric element in the matrix.

[0043] Furthermore, in S4, triggers and nonfragile state estimators for Lipschitz nonlinear systems subjected to denial-of-service attacks are designed, as follows:

[0044] For a given scalar , and The state estimation error system is stochastically stable and has performance If a scalar exists ,matrix , , , , and Makes the following linear matrix inequality hold:

[0045] ,

[0046] ,

[0047] The matrix elements are respectively , , , , , , , , , , , , , , , , , , * denotes a symmetric element in the matrix, so the symmetric positive definite matrix in the trigger condition can be obtained as follows: The parameters of the nonfragile state estimator are: , , .

[0048] The present invention has the following beneficial effects:

[0049] This invention establishes a mathematical model of a Lipschitz nonlinear system subjected to a denial-of-service attack, which can more accurately reflect the actual dynamic characteristics of the system. The event-triggered communication mechanism can guarantee the system's performance while saving network resources and reducing network communication burden. Based on the state estimation error system, a non-vulnerable state estimator is designed to reduce the impact of network attacks and gain perturbations during estimator implementation on state estimation, thereby improving the system's non-vulnerability. Attached Figure Description

[0050] Figure 1 This is a flowchart of the present invention.

[0051] Figure 2 This is the state estimation structure diagram of a Lipschitz nonlinear system subjected to a denial-of-service attack.

[0052] Figure 3 This is a diagram illustrating the trigger interval.

[0053] Figure 4 This is a diagram illustrating the moment a denial-of-service attack occurs.

[0054] Figure 5 This is a schematic diagram of the state estimation error trajectory. Detailed Implementation

[0055] The invention will now be further described with reference to the accompanying drawings.

[0056] like Figure 1 and Figure 2The diagrams shown are a flowchart and a structural diagram of a state estimation method for a Lipschitz nonlinear system subjected to a denial-of-service attack. The present invention provides a state estimation method for a Lipschitz nonlinear system subjected to a denial-of-service attack, comprising the following steps:

[0057] S1: The sensor samples the measurement output of the Lipschitz nonlinear system, and the event trigger transmits the sampled data that meets the triggering conditions to the nonfragile state estimator via the network.

[0058] S2: The Bernoulli distribution is used to describe the denial-of-service attacks that occur during data transmission, and the non-fragile state estimator receives data according to the attack situation;

[0059] S3: Establish a dynamic model of the state estimation error system. Based on Lyapunov stability theory, the state estimation error system is stable and possesses H0. ∞ Performance conditions;

[0060] S4: Design triggers and nonfragile state estimators for Lipschitz nonlinear systems subjected to denial-of-service attacks using the linear matrix inequality method.

[0061] S1 specifically includes the following:

[0062] The state-space expression for a Lipschitz nonlinear system is as follows:

[0063] ,

[0064] in, Let n be the system state vector of the Lipschitz nonlinear system, representing the state of the system at discrete time k, with dimension n.

[0065] The system measurement output of the Lipschitz nonlinear system has a dimension of p;

[0066] The external disturbance vector belongs to Space, with dimensions m;

[0067] Let q be the state to be estimated;

[0068] , , , , , and Given a real constant matrix of corresponding dimension, the nonlinear function and satisfy:

[0069] , ,

[0070] ,

[0071] ,

[0072] Where, n f and n g For function and Dimensions , Let be two arbitrary state vectors of a Lipschitz nonlinear system at discrete time k; and Let be a real constant matrix of known dimension.

[0073] The event trigger transmits sampled data that meets the triggering conditions to the non-fragile state estimator over the network. The triggering conditions are as follows:

[0074] ,

[0075] Where δ(k) is the trigger error vector, , This refers to the last time data was transmitted over the network. This is the output of the sampling measurement transmitted over the network last time. Let be the symmetric positive definite matrix to be found. The given event trigger threshold; T represents the transpose.

[0076] S2 specifically includes the following:

[0077] Use random variables that follow a Bernoulli distribution Describe a denial-of-service attack. This indicates a denial-of-service attack has occurred, and the sampled data cannot be transmitted over the network. This indicates that there is no denial-of-service attack and the sampled data can be transmitted over the network. The mathematical expectation is:

[0078] ,

[0079] in, It is a known constant.

[0080] For a given Lipschitz nonlinear system, design the following nonfragile estimator for state estimation:

[0081] ,

[0082] in, , , These correspond to the state, input, and output of the estimator, respectively. , , , , and For the parameters of the estimator to be designed, , , , and Given a real constant matrix of corresponding dimension, To meet The time-varying matrix, I, is the identity matrix, and the estimator input is:

[0083] .

[0084] S3 specifically includes the following:

[0085] Establish a dynamic model for the state estimation error system:

[0086] set up , , Then the estimation error system is:

[0087] ,

[0088] The coefficient matrices are respectively , , , , , , , , , , , , , , , , , These correspond to the coefficient matrix respectively. , , , , , , , , The nominal value, , , , These correspond to the structure matrix respectively. , , , and The block constant matrix.

[0089] According to Lyapunov's stability theory, the state estimation error system is stable and possesses H0. ∞ The specific performance conditions are as follows:

[0090] Choose a Lyapunov functional: , It is a symmetric positive definite matrix.

[0091] Lyapunov functional difference ,

[0092] in, Representing mathematical expectation, vector ,

[0093] matrix The elements in the matrix are respectively

[0094] ,

[0095] , ,

[0096] , ,

[0097] , * denotes a symmetric element in the matrix.

[0098] According to Schur's complement lemma Equivalent to:

[0099] matrix ,

[0100] The elements in the matrix are respectively

[0101] ,

[0102] ,

[0103] ,

[0104] ,

[0105] ,

[0106] , ,

[0107] , ,

[0108] , * denotes a symmetric element in the matrix.

[0109] From the above matrix inequalities, we can obtain:

[0110] , , ,

[0111] in, for The smallest eigenvalue.

[0112] make ,but:

[0113] ,

[0114] Where, vector In conclusion:

[0115] For a given scalar , , Nonfragile state estimator parameters , and The state estimation error system is stochastically stable and has performance If a matrix exists and Makes the following matrix inequalities hold:

[0116] ,

[0117] The matrix elements are respectively , , , , , , , , , , , , , , , , * denotes a symmetric element in the matrix.

[0118] S4 specifically includes the following:

[0119] Designing triggers and nonfragile state estimators for Lipschitz nonlinear systems subjected to denial-of-service attacks using the linear matrix inequality method:

[0120] For matrix inequalities Multiply by the diagonal matrix on the left and right respectively. We can obtain:

[0121]

[0122] The matrix elements are respectively , , , , , * denotes a symmetric element in the matrix.

[0123] The above matrix inequality is equivalent to:

[0124] ,

[0125] The matrix elements are respectively , , , , , , * denotes a symmetric element in the matrix.

[0126] Let matrix , , In conclusion:

[0127] For a given scalar , and The state estimation error system is stochastically stable and has performance If a scalar exists ,matrix , , , , and Makes the following linear matrix inequality hold:

[0128] ,

[0129] ,

[0130] The matrix elements are respectively , , , , , , , , , , , , , , , , , , * denotes a symmetric element in the matrix, so the symmetric positive definite matrix in the trigger condition can be obtained as follows: The parameters of the nonfragile state estimator are: , , .

[0131] The embodiments of the present invention verify the effectiveness of the proposed state estimation method through simulation examples, as follows:

[0132] The parameters of the Lipschitz nonlinear system are:

[0133] , , , , , , , , , .

[0134] Trigger trigger conditions , Figure 3 A schematic diagram of the trigger interval is provided, which shows that event-triggered data transmission saves network resources compared to periodic data transmission.

[0135] Denial-of-Service Attack Probability , Figure 4 A diagram illustrating the moment a denial-of-service attack occurs is provided.

[0136] In the non-fragile state estimator:

[0137] , , , , , performance .

[0138] Solving the linear matrix inequality yields the trigger condition for the trigger. In the nonfragile state estimator , , , Figure 5 A schematic diagram of the state estimation error trajectory is given, showing that even in the presence of denial-of-service attacks and gain perturbations, the designed state estimator can accurately estimate the state of the Lipschitz nonlinear system.

[0139] The above description is only a preferred embodiment of the present invention. It should be noted that those skilled in the art can make several improvements without departing from the principle of the present invention, and these improvements should also be considered within the scope of protection of the present invention.

Claims

1. A method for state estimation of a Lipschitz nonlinear system subjected to a denial-of-service attack, characterized in that: Includes the following steps: S1: The sensor samples the measurement output of the Lipschitz nonlinear system, and the event trigger transmits the sampled data that meets the triggering conditions to the nonfragile state estimator via the network. S2: The Bernoulli distribution is used to describe whether the sampled data that meets the triggering condition is subjected to a denial-of-service attack at the moment of transmission. The non-fragile state estimator receives data according to the attack situation. S3: Define the state estimation error as the difference between the state of the Lipschitz nonlinear system and the output state of the nonfragile state estimator. Establish a dynamic model of the state estimation error system and, based on Lyapunov stability theory, derive a model that guarantees the stochastic stability of the state estimation error system and possesses H0. ∞ Performance sufficiency matrix inequalities; S4: Transform the matrix inequality into a linear matrix inequality, solve for the symmetric positive definite matrix of the event trigger and the gain parameter of the non-fragile state estimator, thereby completing the joint design of the event trigger condition and the gain of the non-fragile state estimator.

2. The state estimation method for a Lipschitz nonlinear system subjected to a denial-of-service attack as described in claim 1, characterized in that: In S1, the state-space expression of the Lipschitz nonlinear system is as follows: ， in, Let n be the system state vector of the Lipschitz nonlinear system, representing the state of the system at discrete time k, with dimension n. The system measurement output of the Lipschitz nonlinear system has a dimension of p; The external disturbance vector belongs to Space, with dimensions m; Let q be the state to be estimated; , , , , , and Given a real constant matrix of corresponding dimension, nonlinear functions and satisfy: ， ， ， ， Where, n f and n g For function and Dimensions , x1(k) and x2(k) are two arbitrary state vectors of the Lipschitz nonlinear system at discrete time k; and Let be a real constant matrix of known dimension.

3. The state estimation method for a Lipschitz nonlinear system subjected to a denial-of-service attack as described in claim 2, characterized in that: In S1, the triggering conditions are as follows: ， Where δ(k) is the trigger error vector, , This refers to the last time data was transmitted over the network. This is the output of the sampling measurement transmitted over the network last time. Let be the symmetric positive definite matrix to be found. The given event trigger threshold; T represents the transpose.

4. The state estimation method for a Lipschitz nonlinear system subjected to a denial-of-service attack as described in claim 3, characterized in that: In S2, the Bernoulli distribution is used to describe denial-of-service attacks, as follows: Use random variables that follow a Bernoulli distribution Describe a denial-of-service attack. This indicates a denial-of-service attack has occurred. This indicates that there was no denial-of-service attack. The mathematical expectation is: ， in, It is a known constant.

5. The state estimation method for a Lipschitz nonlinear system subjected to a denial-of-service attack as described in claim 4, characterized in that: In S2, the non-fragile state estimator receives data based on the attack situation, as follows: For a given Lipschitz nonlinear system, design the following nonfragile estimator for state estimation: ， in, , , These correspond to the state, input, and output of the estimator, respectively. , , , , and For the parameters of the estimator to be designed, , , , and Given a real constant matrix of corresponding dimension, To meet The time-varying matrix, I, is the identity matrix, and the estimator input is: 。 6. The state estimation method for a Lipschitz nonlinear system subjected to a denial-of-service attack as described in claim 5, characterized in that: In S3, the dynamic model of the state estimation error system is established as follows: set up , , Then the estimation error system is: ， The coefficient matrices are respectively , , , , , , , , , , , , , , , , , These correspond to the coefficient matrix respectively. , , , , , , , , The nominal value, , , , These correspond to the structure matrix respectively. , , , and The block constant matrix.

7. The state estimation method for a Lipschitz nonlinear system subjected to a denial-of-service attack as described in claim 6, characterized in that: In S3, the state estimation error system is found to be stable and has H. ∞ The specific performance conditions are as follows: For a given scalar , , Nonfragile state estimator parameters , and The state estimation error system is stochastically stable and has performance If a matrix exists and Makes the following matrix inequalities hold: ， The matrix elements are respectively , , , , , , , , , , , , , , , , * denotes a symmetric element in the matrix.

8. The state estimation method for a Lipschitz nonlinear system subjected to a denial-of-service attack as described in claim 7, characterized in that: In S4, we design triggers and nonfragile state estimators for Lipschitz nonlinear systems subjected to denial-of-service attacks, as follows: For a given scalar , and The state estimation error system is stochastically stable and has performance If a scalar exists ,matrix , , , , and Makes the following linear matrix inequality hold: ， ， The matrix elements are respectively , , , , , , , , , , , , , , , , , , * denotes a symmetric element in the matrix, so the symmetric positive definite matrix in the trigger condition can be obtained as follows: The parameters of the nonfragile state estimator are: , , .

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