Attack detection method and device for singular cubic nonlinear spring vibration isolation system and medium

Abstract

The invention belongs to the technical field of attack detection, and relates to a singular cubic nonlinear spring vibration isolation system attack detection method and device and a medium, and the method comprises the steps: constructing a state-space equation of a singular cubic nonlinear spring vibration isolation system; acquiring a first output variable, a second output variable and a difference value between the first output variable and the second output variable when the input signal of the system is not subjected to the network attack and is subjected to the network attack so as to construct a network attack equation; performing coordinate transformation on the system state variables to obtain system state variable difference upper and lower limits caused by network attack upper and lower limits; and performing coordinate conversion on the state-space equation by using a non-singular matrix, constructing an interval observer based on the state-space equation after coordinate conversion so as to output system state variable upper and lower limit estimation values, and judging whether the system is subjected to network attack based on the system state variable upper and lower limit estimation values and the system state variable difference upper and lower limits. According to the scheme, whether the singular cubic nonlinear spring vibration isolation system is attacked or not can be accurately detected.

Description

Technical Field

[0001] The present invention relates to the field of attack detection technology, and in particular to a method and device for detecting an attack of a singular cubic nonlinear spring vibration isolation system, and a computer-readable storage medium. Background Art

[0002] The spring vibration isolation system with singular cubic nonlinear damping characteristics is a nonlinear mechanical vibration system. Theory shows that the singular cubic nonlinear damping has little effect on the high and low frequency output spectra of the system. Compared with the traditional linear damping spring vibration isolation system, the singular cubic nonlinear damping can effectively improve the force transmission rate in the resonance frequency zone without affecting the transmission rate in the isolation frequency zone. Therefore, the spring vibration isolation system with singular cubic nonlinear damping characteristics can reduce the vibration effect occurring during the operation of the equipment by virtue of its excellent vibration isolation performance, thereby effectively protecting the equipment. It has a wider application and importance in the actual production process.

[0003] When the singular cubic nonlinear damping characteristic spring vibration isolation system is connected to the industrial network, it becomes a nonlinear physical information system in the industrial network. In practical applications, there are always network attacks on physical information systems. These network attacks will cause some data streams to be tampered, generated or injected with false data streams, thus affecting the safe and stable operation of the entire industrial network. Therefore, the state estimation problem of nonlinear physical information systems under network attacks is of great significance.

[0004] With the continuous development of control theory, interval observers are generally considered to be an effective tool for realizing system state estimation. Most of the existing technologies model the system to be observed, build a corresponding interval observer based on the model of the system to be observed, and use the interval observer to stably estimate the system state based on the input and output data of the system to obtain the estimated interval of the system state variable. Whether the system is attacked by a network is detected based on whether the estimated interval of the system state variable is within the normal range.

[0005] However, the prior art has not studied the state estimation and attack detection problems of the spring vibration isolation system with singular cubic nonlinear damping characteristics. In addition, in the attack detection method provided in the prior art, when the observed system is attacked by a network attack and its input data is tampered with, the interval observer will obtain an erroneous system state variable estimation interval based on the erroneous input data. This system state variable estimation interval cannot reflect the current true state of the system. Therefore, the detection result obtained based on the system state variable estimation interval is also inaccurate. Summary of the invention

[0006] To this end, the technical problem to be solved by the present invention is to overcome the problem that there is no research on state estimation and attack detection of singular cubic nonlinear spring vibration isolation systems in the prior art, and the existing system attack detection method ignores the problem of inaccurate attack detection results caused by network attacks on system input data.

[0007] In order to solve the above technical problems, the present invention provides a singular cubic nonlinear spring vibration isolation system attack detection method, comprising: Construct the state space equations of the singular cubic nonlinear spring isolation system; Based on the state space equation, the first output variable and the second output variable of the system when the input signal is not subjected to the network attack and when the input signal is subjected to the network attack are obtained; based on the difference between the first output variable and the second output variable, an equation of the network attack on the system state variable is constructed; Taking the constraint that the difference of system state variables caused by the upper limit of network attack is greater than the difference of system state variables caused by the lower limit of network attack, coordinate transformation is performed on system state variables to obtain the upper and lower limits of network attack, as well as the upper and lower limits of system state variable difference caused by the upper and lower limits of network attack; The state space equation is transformed by using a non-singular matrix, and the interval observer is constructed based on the Lyapunov function. The interval observer is used to output the estimated upper and lower limits of the system state variables in real time to determine whether the difference between the estimated upper and lower limits of the system state variables and the state variables when the system is not attacked by the network exceeds the upper and lower limits of the system state variables, thereby determining whether the system is attacked by the network.

[0008] Preferably, constructing the state space equation of the singular cubic nonlinear spring vibration isolation system includes: The force analysis of the singular cubic nonlinear spring vibration isolation system is carried out to obtain the system's dynamic function, and the initial state space equation of the system is constructed based on the system's dynamic function; The sensor fault equation and interference factor used to collect the system output variables are used as the augmented state of the system state variables to update the initial state space equation; The updated initial state-space equation is transformed nonlinearly using the Pershitsky model to obtain the state-space equation.

[0009] Preferably, the dynamics function of the system is expressed as: , in, Indicates the system weight; A function that represents the displacement of the object in the system over time; express The first derivative of ; express The second derivative of (t); represents the linear damping coefficient; represents the cubic damping coefficient; represents the spring constant; A function that represents the time-varying force on the objects in the system; Indicates time; The initial state space equation of the system is expressed as: , , in, is the system initial state variable, ; for The first derivative of ; is the initial linear matrix; For the The initial nonlinear matrix of nonlinear terms; For the A nonlinear function of a nonlinear term; For the The initial nonlinear function matrix of nonlinear terms; is the initial input matrix; is the output variable; is the initial linear mapping matrix; is the initial nonlinear mapping matrix of the first nonlinear term; For the The initial nonlinear mapping matrix of nonlinear terms; is the number of nonlinear terms; is the nonlinear function of the first nonlinear term; For the A nonlinear function of a nonlinear term; is the initial nonlinear function matrix of the first nonlinear term; For the The initial nonlinear function matrix of nonlinear terms; The state space equation is expressed as: ， , in, is the rate of change of system state; is the system state variable, , is the sensor fault equation, ; for The first derivative of ; is a linear matrix; For the A nonlinear matrix with nonlinear terms; For the The nonlinear function matrix of nonlinear terms; is the input matrix; To express The impact matrix on the system state change rate; is the interference factor; is the linear mapping matrix; is the nonlinear mapping matrix of the first nonlinear term; For the The nonlinear mapping matrix of nonlinear terms; is the nonlinear function matrix of the first nonlinear term; For the The nonlinear function matrix of nonlinear terms.

[0010] Preferably, the network attack equation about the system state variable is expressed as: , in, Equations that represent cyber attacks with respect to system state variables; represents the second output variable; represents the first output variable; An input signal indicating a cyber attack; An input signal indicating no cyber attack; The upper limit of network attack is expressed as: , in, Indicates the upper limit of network attack; represents the system state variables after coordinate transformation, , represents the coordinate transformation matrix, express The inverse matrix of represents the positive part of the matrix; represents the upper limit of the system state variables after coordinate transformation; represents the lower limit of the system state variable after coordinate transformation; represents the negative part of the matrix; The lower limit of network attack is expressed as: , in, Indicates the lower limit of network attack; The upper limit of the difference of system state variables is expressed as: , in, Indicates the upper limit of the difference of system state variables; represents the first linear real matrix; Indicates the upper limit of the system's initial state variables; Represents the lower limit of the system's initial state variables; The lower limit of the difference of system state variables is expressed as: , in, Indicates the lower limit of the system state variable difference.

[0011] Preferably, constructing an interval observer based on the Lyapunov function for the state space equation after coordinate transformation includes: Construct a radially unbounded and positive definite Lyapunov function, and obtain the first-order Lyapunov function by differentiating the Lyapunov function; Based on Lyapunov stability theory, the positive definite matrix inequality of the first-order Lyapunov function is constructed, and the positive definite matrix is ​​obtained by solving the positive definite matrix inequality. Based on the positive definite matrix and the observation error of the interval observer, the observation gain matrix inequality is constructed, and the observation gain matrix of the interval observer is obtained by solving the observation gain matrix inequality. Based on the observation gain matrix, the upper limit observation equation of the system state variables and the lower limit observation equation of the system state variables are constructed to obtain the interval observer.

[0012] Preferably, the state space equation after coordinate transformation is expressed as: ， , in, Represents the system state variables after coordinate transformation; express The first derivative of ; Represents the linear matrix after coordinate transformation; Indicates the coordinate transformation after The nonlinear matrix in the nonlinear terms; Indicates the coordinate transformation after The nonlinear function matrix in the nonlinear terms; Represents the input matrix after coordinate transformation; represents the first linear real matrix after coordinate transformation; represents the second real matrix; express right The impact matrix; Represents the linear mapping matrix after coordinate transformation; Indicates the coordinate transformation after The nonlinear function matrix in the nonlinear terms; Indicates the coordinate transformation after The nonlinear function matrix in the nonlinear terms; The interval observer is expressed as: , in, Represents the upper limit observation equation of the system state variables; represents the observation gain matrix of the interval observer for the linear terms of the system; Represents the upper limit of the state variable of the interval observer; represents the number of nonlinear functions handled using the positive system method; Indicates the coordinate transformation after The nonlinear matrix in the nonlinear terms; represents the observation gain matrix of the interval observer for the tth nonlinear term of the system; Indicates The nonlinear mapping matrix in the nonlinear terms; represents the nonlinear function in the tth nonlinear term; Indicates the coordinate transformation after The nonlinear function matrix in the nonlinear terms; Indicates The nonlinear function matrix in the nonlinear terms; Indicates the coordinate transformation after The nonlinear matrix in the nonlinear terms; It means that the interval observer is The observation gain matrix of nonlinear terms; Indicates The nonlinear mapping matrix in the nonlinear terms; Indicates Nonlinear functions in nonlinear terms; Indicates the coordinate transformation after The nonlinear function matrix in the nonlinear terms; express The positive part of express The negative part of Represents the lower limit of the state variable of the interval observer; Indicated by , , The observation gain matrix of the interval observer composed of; Indicated by and The system output variables composed of; It represents the upper limit of the input signal of the interval observer affected by the interference factor; represents the upper limit of the interval observer input signal affected by the network attack; , in, Represents the lower limit observation equation of the system state variables; It represents the lower limit of the input signal of the interval observer affected by the interference factor; Represents the lower limit of the interval observer input signal affected by the network attack.

[0013] Preferably, after determining that the system is attacked by a network, the method further includes: Construct the state variable dynamic differential equation of interval observer based on the upper and lower limits of the state variable of interval observer; Based on the state variable dynamic differential equation of interval observer, the state space equation of interval observer is obtained, and the state space equation of interval observer is used as the target state space equation of singular cubic nonlinear spring vibration isolation system.

[0014] Preferably, the state variable dynamic differential equation of the interval observer is expressed as:

[0015] in, The state variable dynamic differential equations representing the interval observer; represents the input signal of the interval observer, , represents the upper limit of the input signal of the interval observer; represents the lower limit of the input signal of the interval observer; The linear matrix of state variables representing the interval observer and the linear terms of the system; Represents the upper limit of the state variable of the interval observer; Represents the lower limit of the state variable of the interval observer; represents the nonlinear matrix of state variables related to the interval observer and the t-th linear term of the system; Represents a variable vector in a nonlinear function; represents the interval observer and the system The nonlinear matrix of state variables related to linear terms; , , The state space equation of the interval observer is expressed as: , in, Represent the state space equation of the interval observer; The linear matrix in the state space equation representing the interval observer; ; The state space equation representing the interval observer is The nonlinear matrix in the nonlinear terms; The state space equation representing the interval observer is Nonlinear functions in nonlinear terms; The state space equation representing the interval observer is The nonlinear function matrix in the nonlinear terms.

[0016] The present invention also provides a singular cubic nonlinear spring vibration isolation system attack detection device, comprising: State-space equation building module, used to build the state-space equations of singular cubic nonlinear spring isolation systems; A network attack equation building module, used for obtaining the first output variable and the second output variable of the system when the input signal is not subjected to the network attack and is subjected to the network attack based on the state space equation; and building the network attack equation about the system state variable based on the difference between the first output variable and the second output variable; The system state variable difference interval acquisition module is used to perform coordinate transformation on the system state variables, taking the system state variable difference caused by the network attack upper limit as a constraint greater than the system state variable difference caused by the network attack lower limit, so as to obtain the network attack upper and lower limits, as well as the system state variable difference upper and lower limits caused by the network attack upper and lower limits; An interval observer construction module is used to transform the state space equation using a non-singular matrix, and construct an interval observer based on the Lyapunov function for the state space equation after the coordinate transformation; The network attack detection module is used to use the interval observer to output the upper and lower limit estimates of the system state variables in real time, and to determine whether the difference between the upper and lower limit estimates of the system state variables and the state variables when the system is not attacked by the network exceeds the upper and lower limits of the system state variable difference, thereby determining whether the system is attacked by the network.

[0017] The present invention also provides a computer-readable storage medium, on which a computer program is stored. When the computer program is executed by a processor, the steps of the above-mentioned singular cubic nonlinear spring vibration isolation system attack detection method are implemented.

[0018] The singular cubic nonlinear spring vibration isolation system attack detection method provided by the present application has the following beneficial effects: 1. After constructing the state-space equation of the singular cubic nonlinear spring vibration isolation system, the present application first obtains the output variables of the system when the input signal is subjected to a network attack and when it is not subjected to a network attack based on the state-space equation, and constructs the relationship between the network attack and the system state variables based on the difference in the output variables of different input signals; since different degrees of network attacks have different effects on the system state variables, the present application uses the difference in the system state variables caused by the upper limit of the network attack as a constraint to be greater than the difference in the system state variables caused by the lower limit of the network attack, and performs coordinate transformation on the system state variables in the state-space equation, thereby obtaining the network attack degree interval and the system state variable difference interval after the input signal is tampered with by different degrees of network attacks; at the same time, since the singular cubic nonlinear spring vibration isolation system has a nonlinear term introduced by the cubic nonlinear damping term, this nonlinear term causes the input and output of the system to not be a simple linear relationship, and directly The constructed interval observer is difficult to accurately match the actual dynamic characteristics of the system. To this end, the present application introduces a non-singular matrix to perform coordinate transformation on the system state variables, converting the system from one state space description to another equivalent state space description, making the complex nonlinear relationship in the system simpler, thereby constructing an interval observer that effectively reflects the system state; finally, the interval observer is used to output the system state variable estimation interval in real time, and the system state variable estimation interval is compared with the system state variable when the input signal of the system is not attacked. Whether the system is attacked is determined based on whether the difference between the two exceeds the system state variable difference interval. Since the system state variable difference interval reflects the range of influence of the network attack on the input signal on the system state variable, even if the observation value output by the interval observer fluctuates when the input signal is tampered with, whether the system is attacked can be accurately detected by whether the degree of fluctuation is within the normal fluctuation range.

[0019] 2. In the prior art, the Lipschitz model and the one-side Lipschitz model are usually used to perform nonlinear transformation on the state space equation of the nonlinear system, so as to process the nonlinear terms of the system and construct the state space equation of the nonlinear system. However, when the scope of the nonlinear function in the singular cubic nonlinear spring vibration isolation system is expanded, the constants in the Lipschitz model and the one-side Lipschitz model will increase sharply, resulting in the linear matrix inequality sufficient condition for ensuring the stability of the system being unsolvable when constructing the interval observer based on the state space equation of the transformed system, so that the interval observer parameters that meet the system stability requirements cannot be found, resulting in the inability to construct an effective interval observer; in this regard, the present application considers for the first time using the Persidskii model to process the nonlinear terms in the singular cubic nonlinear spring vibration isolation system and construct the state space equation of the system to reduce the conservatism of the linear matrix inequality sufficient condition, so that an interval observer that meets the system stability requirements can be constructed. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] In order to make the content of the present invention more clearly understood, the present invention is further described in detail below according to specific embodiments of the present invention in conjunction with the accompanying drawings, wherein: Figure 1 A flow chart of the attack detection method for the singular cubic nonlinear spring vibration isolation system provided in this application; Figure 2 A schematic diagram of the structure of the singular cubic nonlinear spring vibration isolation system provided in this application; Figure 3 This is a schematic diagram of the structure of the attack detection device for the singular cubic nonlinear spring vibration isolation system provided in this application. DETAILED DESCRIPTION

[0021] The present invention is further described below in conjunction with the accompanying drawings and specific embodiments so that those skilled in the art can better understand the present invention and implement it, but the embodiments are not intended to limit the present invention.

[0022] See also Figure 1 , Figure 1 The figure shows a flow chart of a singular cubic nonlinear spring vibration isolation system attack detection method provided by the present application, which specifically includes: S10: Construct the state-space equations for the singular cubic nonlinear spring isolation system.

[0023] S20: Based on the state space equation, obtain the first output variable and the second output variable of the system when the input signal is not subjected to the network attack and when the input signal is subjected to the network attack; and construct an equation of the network attack on the system state variable based on the difference between the first output variable and the second output variable.

[0024] S30: Taking the system state variable difference caused by the network attack upper limit as a constraint that is greater than the system state variable difference caused by the network attack lower limit, coordinate transformation is performed on the system state variables to obtain the network attack upper and lower limits, and the system state variable difference upper and lower limits caused by the network attack upper and lower limits.

[0025] S40: Use a non-singular matrix to perform coordinate transformation on the state space equation, and construct an interval observer based on the Lyapunov function for the state space equation after coordinate transformation.

[0026] S50: Using the interval observer to output the estimated upper and lower limits of the system state variables in real time, determine whether the difference between the estimated upper and lower limits of the system state variables and the state variables when the system is not attacked by the network exceeds the upper and lower limits of the system state variable difference, thereby determining whether the system is attacked by the network.

[0027] Specifically, if the difference between the upper limit estimate of the system state variables and the state variables when the system is not attacked by the network, and the difference between the lower limit estimate of the system state variables and the state variables when the system is not attacked by the network do not exceed the upper and lower limits of the system state variable difference, then it is determined that the system has not been attacked by the network; if the difference between the upper limit estimate of the system state variables and the state variables when the system is not attacked by the network exceeds the upper and lower limits of the system state variable difference, and / or the difference between the lower limit estimate of the system state variables and the state variables when the system is not attacked by the network exceeds the upper and lower limits of the system state variable difference, then it is determined that the system has been attacked by the network.

[0028] Furthermore, the step of constructing the state space equation of the singular cubic nonlinear spring vibration isolation system in step S10 includes: S100: Perform force analysis on the singular cubic nonlinear spring vibration isolation system to obtain the system's dynamic function, and construct the system's initial state space equation based on the system's dynamic function.

[0029] Specifically, Figure 2 The figure shows the structure schematic diagram of the singular cubic nonlinear spring vibration isolation system provided by the present application. The meanings of the parameters in the figure are shown in Table 1: Table 1 symbol meaning Parameter Description Displacement of the block relative to the wall m <![CDATA[ k c ]]> Spring constant N / m Nonlinear Damping N System weight kg The force exerted on an object N Through force analysis, we can get: , in, Indicates the system weight; An equation that represents the displacement of the objects in the system as a function of time; express The first derivative of ; express The second derivative of represents the linear damping coefficient; represents the cubic damping coefficient; represents the spring constant; An equation that represents how the forces on the objects in the system vary with time.

[0030] The nonlinear damping term in this system is expressed as , ; for The first derivative of .

[0031] Will As the state variable of the system, the dynamic function of the system can be obtained as: , in, Indicates the system weight; A function that represents the displacement of the object in the system over time; express The first derivative of ; express The second derivative of (t); represents the linear damping coefficient; represents the cubic damping coefficient; represents the spring constant; A function that represents the time-varying force on the objects in the system; Indicates time; Then the initial state space equation of the system is expressed as: , , in, is the system initial state variable, ; for The first derivative of ; is the initial linear matrix; For the The initial nonlinear matrix of nonlinear terms; For the A nonlinear function of a nonlinear term; For the The initial nonlinear function matrix of nonlinear terms; is the initial input matrix; is the output variable; is the initial linear mapping matrix; is the initial nonlinear mapping matrix of the first nonlinear term; For the The initial nonlinear mapping matrix of nonlinear terms; is the number of nonlinear terms; is the nonlinear function of the first nonlinear term; For the A nonlinear function of a nonlinear term; is the initial nonlinear function matrix of the first nonlinear term; For the The initial nonlinear function matrix of nonlinear terms.

[0032] S101: Using the sensor fault equation and the interference factor used to collect the system output variable as the augmented state of the system state variable, the initial state space equation is updated.

[0033] Specifically, since there may be sensor failures and interference factors in the actual measurement process of system parameters, the present application uses the sensor failure equation as the augmented state of the system's initial state variables, and uses the interference factor as the augmented state of the first-order derivative of the system's initial state variables, thereby updating the initial state space equation, so that the updated state space equation can more accurately fit the actual operating state of the system.

[0034] S102: Using the Perschitzky model, a nonlinear transformation is performed on the updated initial state space equation to obtain the state space equation.

[0035] Specifically, the prior art generally uses the Lipschitz model and the one-side Lipschitz model to perform nonlinear transformation on the state space equation of the nonlinear system, thereby processing the nonlinear terms of the system. However, since the singular cubic nonlinear spring vibration isolation system has nonlinear terms introduced by the cubic nonlinear damping terms, it is difficult to express it using traditional mathematical models. When the scope of the nonlinear function in the system is expanded, the constants in the Lipschitz model and the one-side Lipschitz model will increase sharply, resulting in the linear matrix inequality sufficient condition used to ensure the stability of the system being unsolvable when constructing an interval observer based on the state space equation of the transformed system, and thus it is impossible to find the interval observer parameters (positive definite matrix, observation gain matrix) that meet the system stability requirements, resulting in the inability to construct an effective interval observer.

[0036] Based on the above reasons, this application considers using the Persidskii model to deal with nonlinear terms in the system. The first study on the Persidskii model proposed a Lyapunov function represented by a linear combination of nonlinear integrals. Subsequently, the absolute value of the Persidskii model was integrated into the Lyapunov function. In addition, the Persidskii model is also widely used in neural networks, biological populations, and sliding mode control. This application uses it for the first time to deal with nonlinear terms in a singular cubic nonlinear spring isolation system to reduce the conservatism of the sufficient conditions of the linear matrix inequality, so that an interval observer that meets the system stability requirements can be constructed.

[0037] Specifically, the state space equation is expressed as: ， , in, is the rate of change of system state; is the system state variable, , is the sensor fault equation, ; for The first derivative of ; is a linear matrix; For the A nonlinear matrix with nonlinear terms; For the The nonlinear function matrix of nonlinear terms; is the input matrix; To express The impact matrix on the system state change rate; is the interference factor; is the linear mapping matrix; is the nonlinear mapping matrix of the first nonlinear term; For the The nonlinear mapping matrix of nonlinear terms; is the nonlinear function matrix of the first nonlinear term; For the The nonlinear function matrix of nonlinear terms.

[0038] Furthermore, when the singular cubic nonlinear spring vibration isolation system is connected to the industrial network, there is a hidden network attack in the information transmission process. The network attack will be injected during the input signal transmission process, resulting in output variable deviation. If it is established, it means that the network attack is hidden, and the output variable of the system is: , Indicates a network attack. Represents the output variable when the system is attacked by the network, Indicates the input signal when the system is attacked by the network. represents the output variable when the system is not attacked, An input signal that indicates that the system is not under a network attack.

[0039] Generally speaking, a network attacker will inject deviations into the input signal, causing deviations in the subsequent system output variables and the control decisions made based on the output variables, thereby affecting the system performance. Therefore, this application first studies the problem of the deviation interval of the system state variables caused by network attacks, and in order to relax conservatism to ensure the construction of an effective interval observer, this application first performs a coordinate transformation on the system state variables.

[0040] Specifically, the equation of network attack on system state variables is expressed as: , in, Equations that represent cyber attacks with respect to system state variables; represents the second output variable; represents the first output variable; An input signal indicating a cyber attack; An input signal indicating no cyber attack; The upper limit of network attack is expressed as: , in, Indicates the upper limit of network attack; represents the system state variables after coordinate transformation, , represents the coordinate transformation matrix, express The inverse matrix of represents the positive part of the matrix; represents the upper limit of the system state variables after coordinate transformation; represents the lower limit of the system state variable after coordinate transformation; represents the negative part of the matrix; The lower limit of network attack is expressed as: , in, Indicates the lower limit of network attack; The upper limit of the difference of system state variables is expressed as: , in, Indicates the upper limit of the difference of system state variables; represents the first linear real matrix; Indicates the upper limit of the system's initial state variables; Represents the lower limit of the system's initial state variables; The lower limit of the difference of system state variables is expressed as: , in, Indicates the lower limit of the system state variable difference.

[0041] In order to further relax the conservatism, this application introduces a non-singular matrix to perform coordinate transformation on the system state variables again, converting the system from one state space description to another equivalent state space description. In the new state space, the dynamic characteristics of the system have not changed, but the complex nonlinear relationships or strong coupling relationships in the system have become simpler or decoupled, so that the sufficient conditions for the linear matrix inequality used to ensure the stability of the system when constructing the interval observer can be solved.

[0042] Specifically, the state space equation after coordinate transformation is expressed as: ， , in, Represents the system state variables after coordinate transformation; express The first derivative of ; Represents the linear matrix after coordinate transformation; Indicates the coordinate transformation after The nonlinear matrix in the nonlinear terms; Indicates the coordinate transformation after The nonlinear function matrix in the nonlinear terms; Represents the input matrix after coordinate transformation; represents the first linear real matrix after coordinate transformation; represents the second real matrix; express right The impact matrix; Represents the linear mapping matrix after coordinate transformation; Indicates the coordinate transformation after The nonlinear function matrix in the nonlinear terms; Indicates the coordinate transformation after The nonlinear function matrix in the nonlinear terms.

[0043] In step S40, constructing an interval observer based on the Lyapunov function for the state space equation after coordinate transformation specifically includes: S400: Construct a radially unbounded and positive definite Lyapunov function, and differentiate the Lyapunov function to obtain a first-order Lyapunov function.

[0044] S401: Based on Lyapunov stability theory, construct the positive definite matrix inequality of the first-order Lyapunov function, and solve the positive definite matrix inequality to obtain the positive definite matrix.

[0045] S402: constructing an observation gain matrix inequality based on the positive definite matrix and the observation error of the interval observer, and solving the observation gain matrix inequality to obtain the observation gain matrix of the interval observer.

[0046] S403: constructing an upper limit observation equation of the system state variables and a lower limit observation equation of the system state variables based on the observation gain matrix, thereby obtaining an interval observer.

[0047] Specifically, the interval observer constructed in the embodiment of the present application is: , in, Represents the upper limit observation equation of the system state variables; represents the observation gain matrix of the interval observer for the linear terms of the system; Represents the upper limit of the state variable of the interval observer; represents the number of nonlinear functions handled using the positive system method; Indicates the coordinate transformation after The nonlinear matrix in the nonlinear terms; represents the observation gain matrix of the interval observer for the tth nonlinear term of the system; Indicates The nonlinear mapping matrix in the nonlinear terms; represents the nonlinear function in the tth nonlinear term; Indicates the coordinate transformation after The nonlinear function matrix in the nonlinear terms; Indicates The nonlinear function matrix in the nonlinear terms; Indicates the coordinate transformation after The nonlinear matrix in the nonlinear terms; It means that the interval observer is The observation gain matrix of nonlinear terms; Indicates The nonlinear mapping matrix in the nonlinear terms; Indicates Nonlinear functions in nonlinear terms; Indicates the coordinate transformation after The nonlinear function matrix in the nonlinear terms; express The positive part of express The negative part of Represents the lower limit of the state variable of the interval observer; Indicated by , , The observation gain matrix of the interval observer composed of; Indicated by and The system output variables composed of; It represents the upper limit of the input signal of the interval observer affected by the interference factor; represents the upper limit of the interval observer input signal affected by the network attack; , in, Represents the lower limit observation equation of the system state variables; It represents the lower limit of the input signal of the interval observer affected by the interference factor; Represents the lower limit of the interval observer input signal affected by the network attack.

[0048] The interval observer can be used to observe the state variables of the singular cubic nonlinear spring vibration isolation system in real time, thereby realizing attack detection.

[0049] Furthermore, when it is found that the singular cubic nonlinear spring vibration isolation system is under cyber attack, the singular cubic nonlinear spring vibration isolation system can be reconstructed to ensure that the correct output variables can be obtained based on the reconstructed system when the system is under cyber attack, thereby ensuring the effectiveness and accuracy of the subsequent control strategy. Specifically, for the above interval observer, define ,for have and ;definition , , , , , .

[0050] Furthermore, after determining that the system has been attacked by a network, the following steps are also included: S60: constructing a state variable dynamic differential equation of the interval observer based on the upper and lower limits of the state variable of the interval observer. Specifically, the state variable dynamic differential equation of the interval observer is expressed as: , in, The state variable dynamic differential equations representing the interval observer; represents the input signal of the interval observer, , represents the upper limit of the input signal of the interval observer; represents the lower limit of the input signal of the interval observer; The linear matrix of state variables representing the interval observer and the linear terms of the system; Represents the upper limit of the state variable of the interval observer; Represents the lower limit of the state variable of the interval observer; represents the nonlinear matrix of state variables related to the interval observer and the t-th linear term of the system; Represents a variable vector in a nonlinear function; represents the interval observer and the system The nonlinear matrix of state variables related to linear terms; , .

[0051] S70: Based on the state variable dynamic differential equation of the interval observer, the state space equation of the interval observer is obtained, and the state space equation of the interval observer is used as the target state space equation of the singular cubic nonlinear spring vibration isolation system. Specifically, the state space equation of the interval observer is expressed as: , in, Represent the state space equation of the interval observer; The linear matrix in the state space equation representing the interval observer; ; The state space equation representing the interval observer is The nonlinear matrix in the nonlinear terms; The state space equation representing the interval observer is Nonlinear functions in nonlinear terms; The state space equation representing the interval observer is The nonlinear function matrix in the nonlinear terms.

[0052] The attack detection method for the singular cubic nonlinear spring vibration isolation system provided in the above embodiment will be described below with reference to specific examples: It is worth noting that, in this embodiment, and denote the real (non-negative real) vector space and dimensional real matrix; It means starting from 1 and continuing to finite sequence of integers, similarly ; For a vector z, Indicates that the values ​​of all components are less than or equal to (or greater than or equal to) 0; and function Under the strictly increasing condition, the function belong Class function; if Class Functions Increasing to infinity, the function that is Class function; for matrix T, , ; is a continuously differentiable function The derivative of ; In addition, This means that T is a negative definite (positive definite) matrix. For simplicity, the matrix is assumed to be of full row rank, so the real matrix and , so that .

[0053] Based on the above definition, the singular cubic nonlinear spring vibration isolation system is transformed into: ， , in, for The first derivative of , then the system can be equivalent to: ， , Before designing the interval observer for the singular cubic nonlinear spring vibration isolation system, this embodiment ensures the monotonicity of the nonlinear function by making the following assumptions: Assumption 1. For any and , the following inequality holds: , According to hypothesis 1, there is , satisfying all and , then: , and exists , satisfying for all and , then: , Assumption 2. For any and , the following inequality holds: , Assumption 3. The external disturbance in the system is bounded, then: , in, and are all constant vectors.

[0054] Furthermore, in order to make the designed interval observer eventually stable and consistent, this embodiment proposes the following definition: Definition 1. For the state space equation of the system: , , If for all and exist and , so that: , , Then the Lyapunov function A system is input-state stable if and only if it admits an input-state stable Lyapunov function.

[0055] In order to prove the non-negativity of the designed interval observer, this embodiment also gives the following lemma: Lemma 1. Given a matrix M and a vector variable Satisfy the inequality ,but: , Lemma 2. Given a matrix equation ,in , , ,if , the solution to this equation is as follows: , in, for The pseudo-inverse matrix of , represents an arbitrary matrix.

[0056] Furthermore, the following lemma states the non-negativity condition for nonlinearity in the interval observer: Lemma 2. Considering the nonlinear part of the Persitzky system Assumption 1 is satisfied if for all , one of the following properties holds: matrix The i-th row and matrix The i-th column of is greater than or equal to 0; matrix The i-th row and matrix The i-th column of is less than or equal to 0; Then the nonlinear part is non-negative, so this embodiment uses the following symbols to simplify the non-negativity condition: .

[0057] Lemma 3. If Assumption 1 holds, and Established, then: , In addition, if for , , exist , then according to Lemma 1 we can get the inequality Established, therefore, we can get: , Theorem 1. Assume that Assumptions 1, 2, and 3 are satisfied. If have and , then for any have , Proof: After coordinate transformation, the system is equivalent to: ， , in, , , , , , , , ; is a nonlinear matrix; The estimated error of the interval observer is defined as and , therefore, the dynamic expression of the estimation error can be obtained as: , , in, , , , , According to Assumption 3 and Lemma 2, for any ,satisfy: , Furthermore, since for any have and , according to Lemma 3, we can get and ; Due to Lemma 1 and Lemma 4, for all have and ; These all show the non-negativity of the interval observer and .

[0058] Although The existence of is not necessary, but if it is not possible to construct a matrix that guarantees the non-negativity of the system after the coordinate transformation, a nonlinear interval inclusion method is required, which increases the design complexity of the interval observer.

[0059] Theorem 2. Assuming that Assumptions 1, 2, and 3 are satisfied, if there exists a positive definite matrix , a diagonal matrix ,for With Matrix ,for and With Matrix ,for With Matrix and positive definite matrices , so that: , , And make the positive definite matrix , represents a positive definite matrix OK, represents a positive definite matrix Columns; in, , , , , , , , , in, represents a diagonal matrix; express The set of non-negative diagonal matrices of order; express The set of non-negative diagonal matrices of order; express The set of non-negative diagonal matrices of order; express The matrix of ; Indicates The matrix ; express The diagonal matrix when ; Then it can be proved is uniformly ultimately bounded.

[0060] Proof: Definition ,for have and ; , , , , , ; Then the dynamic differential equation of the state variable of the interval observer is: , in, , , Represents a variable vector in a nonlinear function; , in, Represents a new input signal whose boundaries are No matter, definition , , then the state space equation of the reconstructed new system is expressed as: , The Lyapunov function used in this embodiment is: , Based on assumption 1 and Lyapunov function The condition that is radially unbounded and positive definite is true for the Lyapunov function Taking the derivative, we get: , Therefore, under the equation we can get: , in, The nonlinear function representing the first nonlinear term in the Lyapunov first-order derivative function; represents the first derivative function of Lyapunov A nonlinear function of a nonlinear term; represents the first derivative function of Lyapunov A nonlinear function of a nonlinear term; represents the first derivative function of Lyapunov A nonlinear function with a nonlinear term.

[0061] According to the above inequality and assumption 1, the right side of the inequality is similar to the form of the inequality in definition 1. have: .

[0062] Therefore, it can be obtained that the Lyapunov function V has the property of input state stability, which can ensure the input state stability of the system. At the same time, it also proves that , thus obtaining a non-negative and eventually stable and consistent interval observer.

[0063] Based on the singular cubic nonlinear spring vibration isolation system attack detection method provided in the above embodiment, the embodiment of the present application also provides a singular cubic nonlinear spring vibration isolation system attack detection device, such as Figure 3 As shown, the device specifically includes: The state space equation construction module 10 is used to construct the state space equation of the singular cubic nonlinear spring vibration isolation system.

[0064] The network attack equation construction module 20 is used to obtain the first output variable and the second output variable of the system when the input signal is not subjected to the network attack and is subjected to the network attack based on the state space equation; and construct the equation of the network attack on the system state variable based on the difference between the first output variable and the second output variable.

[0065] The system state variable difference interval acquisition module 30 is used to perform coordinate transformation on the system state variables, with the system state variable difference caused by the network attack upper limit being greater than the system state variable difference caused by the network attack lower limit as a constraint, so as to obtain the network attack upper and lower limits, as well as the system state variable difference upper and lower limits caused by the network attack upper and lower limits.

[0066] The interval observer construction module 40 is used to perform coordinate transformation on the state space equation using a non-singular matrix, and construct an interval observer based on the Lyapunov function for the state space equation after the coordinate transformation.

[0067] The network attack detection module 50 is used to use the interval observer to output the estimated upper and lower limits of the system state variables in real time, and determine whether the difference between the estimated upper and lower limits of the system state variables and the state variables when the system is not attacked by the network exceeds the upper and lower limits of the system state variable difference, thereby determining whether the system is attacked by the network.

[0068] An embodiment of the present application also provides a computer-readable storage medium, on which a computer program is stored. When the computer program is executed by a processor, the steps of the above-mentioned singular cubic nonlinear spring vibration isolation system attack detection method are implemented.

[0069] Obviously, the above embodiments are merely examples for the purpose of clear explanation and are not intended to limit the implementation methods. For those skilled in the art, other different forms of changes or modifications can be made based on the above description. It is not necessary and impossible to list all the implementation methods here. The obvious changes or modifications derived therefrom are still within the scope of protection of the present invention.

Claims

1. A singular cubic nonlinear spring vibration isolation system attack detection method, characterized in that: include: Construct the state space equations of the singular cubic nonlinear spring isolation system; Obtaining a first output variable and a second output variable of the system when the input signal is not attacked by a network and when the input signal is attacked by a network based on a state space equation; Constructing an equation of network attack on system state variables based on the difference between the first output variable and the second output variable; Taking the constraint that the difference of system state variables caused by the upper limit of network attack is greater than the difference of system state variables caused by the lower limit of network attack, coordinate transformation is performed on system state variables to obtain the upper and lower limits of network attack, as well as the upper and lower limits of system state variable difference caused by the upper and lower limits of network attack; The state space equation is transformed by using a non-singular matrix, and the interval observer is constructed based on the Lyapunov function. The interval observer is used to output the estimated upper and lower limits of the system state variables in real time to determine whether the difference between the estimated upper and lower limits of the system state variables and the state variables when the system is not attacked by the network exceeds the upper and lower limits of the system state variables, thereby determining whether the system is attacked by the network.

2. The singular cubic nonlinear spring vibration isolation system attack detection method according to claim 1 is characterized in that: The state space equations for constructing the singular cubic nonlinear spring isolation system include: The force analysis of the singular cubic nonlinear spring vibration isolation system is carried out to obtain the system's dynamic function, and the initial state space equation of the system is constructed based on the system's dynamic function; The sensor fault equation and interference factor used to collect the system output variables are used as the augmented state of the system state variables to update the initial state space equation; The updated initial state-space equation is transformed nonlinearly using the Pershitsky model to obtain the state-space equation.

3. The singular cubic nonlinear spring vibration isolation system attack detection method according to claim 2 is characterized in that: The dynamic function of the system is expressed as: ， in, Indicates the system weight; A function that represents the displacement of the object in the system over time; express The first derivative of ; express The second derivative of (t); represents the linear damping coefficient; represents the cubic damping coefficient; represents the spring constant; A function that represents the time-varying force on the objects in the system; Indicates time; The initial state space equation of the system is expressed as: ， ， in, is the system initial state variable, ; for The first derivative of ; is the initial linear matrix; For the The initial nonlinear matrix of nonlinear terms; For the A nonlinear function of a nonlinear term; For the The initial nonlinear function matrix of nonlinear terms; is the initial input matrix; is the output variable; is the initial linear mapping matrix; is the initial nonlinear mapping matrix of the first nonlinear term; For the The initial nonlinear mapping matrix of nonlinear terms; is the number of nonlinear terms; is the nonlinear function of the first nonlinear term; For the A nonlinear function of a nonlinear term; is the initial nonlinear function matrix of the first nonlinear term; For the The initial nonlinear function matrix of nonlinear terms; The state space equation is expressed as: ， ， in, is the rate of change of system state; is the system state variable, , is the sensor fault equation, ; for The first derivative of ; is a linear matrix; For the A nonlinear matrix with nonlinear terms; For the The nonlinear function matrix of nonlinear terms; is the input matrix; To express The impact matrix on the system state change rate; is the interference factor; is the linear mapping matrix; is the nonlinear mapping matrix of the first nonlinear term; For the The nonlinear mapping matrix of nonlinear terms; is the nonlinear function matrix of the first nonlinear term; For the The nonlinear function matrix of nonlinear terms.

4. The singular cubic nonlinear spring vibration isolation system attack detection method according to claim 3 is characterized in that: The equation of network attack on system state variables is expressed as: ， in, Equations that represent cyber attacks with respect to system state variables; represents the second output variable; represents the first output variable; An input signal indicating a cyber attack; An input signal indicating no cyber attack; The upper limit of network attack is expressed as: ， in, Indicates the upper limit of network attack; represents the system state variables after coordinate transformation, , represents the coordinate transformation matrix, express The inverse matrix of represents the positive part of the matrix; represents the upper limit of the system state variables after coordinate transformation; Represents the lower limit of the system state variable after coordinate transformation; represents the negative part of the matrix; The lower limit of network attack is expressed as: ， in, Indicates the lower limit of network attack; The upper limit of the difference of system state variables is expressed as: ， in, Indicates the upper limit of the difference of system state variables; represents the first linear real matrix; Indicates the upper limit of the system's initial state variables; Represents the lower limit of the system's initial state variables; The lower limit of the difference of system state variables is expressed as: ， in, Indicates the lower limit of the system state variable difference.

5. The singular cubic nonlinear spring vibration isolation system attack detection method according to claim 4 is characterized in that: The state space equation after coordinate transformation is used to construct an interval observer based on the Lyapunov function, including: Construct a radially unbounded and positive definite Lyapunov function, and obtain the first-order Lyapunov function by differentiating the Lyapunov function; Based on Lyapunov stability theory, the positive definite matrix inequality of the first-order Lyapunov function is constructed, and the positive definite matrix is ​​obtained by solving the positive definite matrix inequality. Based on the positive definite matrix and the observation error of the interval observer, the observation gain matrix inequality is constructed, and the observation gain matrix of the interval observer is obtained by solving the observation gain matrix inequality. Based on the observation gain matrix, the upper limit observation equation of the system state variables and the lower limit observation equation of the system state variables are constructed to obtain the interval observer.

6. The singular cubic nonlinear spring vibration isolation system attack detection method according to claim 5, characterized in that: The state space equation after coordinate transformation is expressed as: ， ， in, Represents the system state variables after coordinate transformation; express The first derivative of ; Represents the linear matrix after coordinate transformation; Indicates the coordinate transformation after The nonlinear matrix in the nonlinear terms; Indicates the coordinate transformation after The nonlinear function matrix in the nonlinear terms; Represents the input matrix after coordinate transformation; represents the first linear real matrix after coordinate transformation; represents the second real matrix; express right The impact matrix; Represents the linear mapping matrix after coordinate transformation; Indicates the coordinate transformation after The nonlinear function matrix in the nonlinear terms; Indicates the coordinate transformation after The nonlinear function matrix in the nonlinear terms; The interval observer is expressed as: ， in, Represents the upper limit observation equation of the system state variables; represents the observation gain matrix of the interval observer for the linear terms of the system; Represents the upper limit of the state variable of the interval observer; represents the number of nonlinear functions handled using the positive system method; Indicates the coordinate transformation after The nonlinear matrix in the nonlinear terms; represents the observation gain matrix of the interval observer for the tth nonlinear term of the system; Indicates The nonlinear mapping matrix in the nonlinear terms; represents the nonlinear function in the tth nonlinear term; Indicates the coordinate transformation after The nonlinear function matrix in the nonlinear terms; Indicates The nonlinear function matrix in the nonlinear terms; Indicates the coordinate transformation after The nonlinear matrix in the nonlinear terms; It means that the interval observer is The observation gain matrix of nonlinear terms; Indicates The nonlinear mapping matrix in the nonlinear terms; Indicates Nonlinear functions in nonlinear terms; Indicates the coordinate transformation after The nonlinear function matrix in the nonlinear terms; express The positive part of express The negative part of Represents the lower limit of the state variable of the interval observer; Indicated by , , The observation gain matrix of the interval observer composed of; Indicated by and The system output variables composed of; It represents the upper limit of the input signal of the interval observer affected by the interference factor; represents the upper limit of the interval observer input signal affected by the network attack; ， in, Represents the lower limit observation equation of the system state variables; It represents the lower limit of the input signal of the interval observer affected by the interference factor; Represents the lower limit of the interval observer input signal affected by the network attack.

7. The singular cubic nonlinear spring vibration isolation system attack detection method according to claim 6, characterized in that: After determining that the system has been attacked by a network attack, it also includes: Construct the state variable dynamic differential equation of interval observer based on the upper and lower limits of the state variable of interval observer; Based on the state variable dynamic differential equation of interval observer, the state space equation of interval observer is obtained, and the state space equation of interval observer is used as the target state space equation of singular cubic nonlinear spring vibration isolation system.

8. The singular cubic nonlinear spring vibration isolation system attack detection method according to claim 7, characterized in that: The state variable dynamic differential equation of the interval observer is expressed as: ， in, The state variable dynamic differential equations representing the interval observer; represents the input signal of the interval observer, , represents the upper limit of the input signal of the interval observer; represents the lower limit of the input signal of the interval observer; The linear matrix of state variables representing the interval observer and the linear terms of the system; Represents the upper limit of the state variable of the interval observer; Represents the lower limit of the state variable of the interval observer; represents the nonlinear matrix of state variables related to the interval observer and the t-th linear term of the system; Represents a variable vector in a nonlinear function; represents the interval observer and the system The nonlinear matrix of state variables related to linear terms; ， ， The state space equation of the interval observer is expressed as: ， in, Represent the state space equation of the interval observer; The linear matrix in the state space equation representing the interval observer; ; The state space equation representing the interval observer is The nonlinear matrix in the nonlinear terms; The state space equation representing the interval observer is Nonlinear functions in nonlinear terms; The state space equation representing the interval observer is The nonlinear function matrix in the nonlinear terms.

9. A singular cubic nonlinear spring vibration isolation system attack detection device, characterized in that: include: State-space equation building module, used to build the state-space equations of singular cubic nonlinear spring isolation systems; A network attack equation building module, used for obtaining a first output variable and a second output variable of the system when the input signal is not subjected to a network attack and is subjected to a network attack based on a state space equation; Constructing an equation of network attack on system state variables based on the difference between the first output variable and the second output variable; The system state variable difference interval acquisition module is used to perform coordinate transformation on the system state variables, taking the system state variable difference caused by the network attack upper limit as a constraint greater than the system state variable difference caused by the network attack lower limit, so as to obtain the network attack upper and lower limits, as well as the system state variable difference upper and lower limits caused by the network attack upper and lower limits; An interval observer construction module is used to transform the state space equation using a non-singular matrix, and construct an interval observer based on the Lyapunov function for the state space equation after the coordinate transformation; The network attack detection module is used to use the interval observer to output the upper and lower limit estimates of the system state variables in real time, and to determine whether the difference between the upper and lower limit estimates of the system state variables and the state variables when the system is not attacked by the network exceeds the upper and lower limits of the system state variable difference, thereby determining whether the system is attacked by the network.

10. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores a computer program, and when the computer program is executed by a processor, the steps of the singular cubic nonlinear spring vibration isolation system attack detection method according to any one of claims 1 to 8 are implemented.

Images