Attack Detection Method, Device and Medium for Singular Cubic Nonlinear Spring Vibration Isolation System

Abstract

The present invention belongs to the technical field of attack detection, and relates to an attack detection method, device and medium for a singular cubic nonlinear spring vibration isolation system, including constructing a state space equation of the singular cubic nonlinear spring vibration isolation system; obtaining a first output variable, a second output variable and the difference between the two of the system when the input signal is not under a cyber attack and under a cyber attack, so as to construct a cyber attack equation; performing a coordinate transformation on the system state variables to obtain the upper and lower limits of the difference of the system state variables caused by the upper and lower limits of the cyber attack; using a non-singular matrix to perform a coordinate transformation on the state space equation, and constructing an interval observer based on the state space equation after the coordinate transformation, so as to output the estimated values of the upper and lower limits of the system state variables, and determining whether the system is under a cyber attack based on the estimated values of the upper and lower limits of the system state variables and the upper and lower limits of the difference of the system state variables. This solution can accurately detect whether the singular cubic nonlinear spring vibration isolation system is under attack.

Description

Technical Field

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

[0002] A singular cubic nonlinear damping characteristic spring vibration isolation system is a nonlinear mechanical vibration system. The 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 ratio in the resonance frequency region without affecting the transmission ratio in the vibration isolation frequency region. Therefore, the singular cubic nonlinear damping characteristic spring vibration isolation system can reduce the vibration during the operation of the equipment with its excellent vibration isolation performance, thereby effectively protecting the equipment and having more extensive applications 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 the physical information system. These network attacks will cause some data streams to be tampered with, false data streams to be generated or injected, thus affecting the safe and stable operation of the entire industrial network. Therefore, the state estimation problem of the nonlinear physical information system under network attacks is of great significance.

[0004] With the continuous development of control theory, the interval observer is generally considered to be an effective tool for realizing system state estimation. In the prior art, most of them model the system to be observed, construct 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, obtain the estimation interval of the system state variables, and detect whether the system is under network attack based on whether the estimation interval of the system state variables obtained is within the normal interval range.

[0005] However, the prior art has not studied the state estimation and attack detection problems of the singular cubic nonlinear damping characteristic spring vibration isolation system. In addition, in the attack detection methods provided in the prior art, when the input data of the system to be observed is tampered with due to network attacks, the interval observer will obtain an incorrect estimation interval of the system state variables based on the incorrect input data. This estimation interval of the system state variables cannot reflect the current true state of the system. Therefore, the detection result obtained based on this estimation interval of the system state variables 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 lack of research on state estimation and attack detection of a singular cubic nonlinear spring vibration isolation system in the prior art, and the problem that the existing system attack detection methods ignore the inaccurate attack detection results caused by cyber attacks on system input data.

[0007] To solve the above technical problem, the present invention provides an attack detection method for a singular cubic nonlinear spring vibration isolation system, including:

[0008] Construct the state space equation of the singular cubic nonlinear spring vibration isolation system;

[0009] 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 under cyber attack and under cyber attack; construct an equation of the cyber attack with respect to the system state variables based on the difference between the first output variable and the second output variable;

[0010] Taking the difference in system state variables caused by the upper limit of the cyber attack being greater than the difference in system state variables caused by the lower limit of the cyber attack as a constraint, perform a coordinate transformation on the system state variables, so as to obtain the upper and lower limits of the cyber attack, and the upper and lower limits of the difference in system state variables caused by the upper and lower limits of the cyber attack;

[0011] Use a non-singular matrix to perform a coordinate transformation on the state space equation, and construct an interval observer based on the Lyapunov function for the coordinate-transformed state space equation;

[0012] Use the interval observer to output the upper and lower limit estimates of the system state variables in real time, and determine whether the difference between the upper and lower limit estimates of the system state variables and the state variables of the system when not under cyber attack exceeds the upper and lower limits of the difference in system state variables, so as to determine whether the system is under cyber attack.

[0013] Preferably, constructing the state space equation of the singular cubic nonlinear spring vibration isolation system includes:

[0014] Perform a force analysis on the singular cubic nonlinear spring vibration isolation system to obtain the dynamic function of the system, and construct the initial state space equation of the system based on the dynamic function of the system;

[0015] Take the sensor fault equation for collecting the system output variables and the interference factor as the augmented state of the system state variables, and update the initial state space equation;

[0016] Use the Pershitsky model to perform a nonlinear transformation on the updated initial state space equation to obtain the state space equation.

[0017] Preferably, the dynamic function of the system is expressed as:

[0018] ,

[0019] Among them, represents the system weight; represents the function of the displacement of the block in the system changing with time; represents the first derivative of; represents (t) the second derivative of; represents the linear damping coefficient; represents the cubic damping coefficient; represents the spring coefficient; represents the function of the force on the block in the system changing with time; represents time;

[0020] The initial state - space equation of the system is expressed as:

[0021] ,

[0022] ,

[0023] Among them, is the initial state variable of the system, ; is the first derivative of; is the initial linear matrix; is the initial non - linear matrix of the th non - linear term; is the non - linear function of the th non - linear term; is the initial non - linear function matrix of the th non - linear term; is the initial non - linear mapping matrix of the th non - linear term; is the non - linear function of the th non - linear term; is the initial non - linear function matrix of the

[0024] The state - space equation is expressed as:

[0025] ，

[0026] ,

[0027] Among them, is the system state change rate; is the system state variable, , is the sensor fault equation, ; is 's first derivative; is the linear matrix; is the th non - linear matrix of the non - linear term; is the th non - linear function matrix of the non - linear term; is the input matrix; represents 's influence matrix on the system state change rate; is the interference factor; is the linear mapping matrix; is the non - linear mapping matrix of the 1st non - linear term; is the th non - linear mapping matrix of the non - linear term; is the non - linear function matrix of the 1st non - linear term; is the th non - linear function matrix of the non - linear term.

[0028] Preferably, the equation of the cyber - attack with respect to the system state variable is expressed as:

[0029] ,

[0030] Among them, represents the equation of the cyber - attack with respect to the system state variable; represents the second output variable; represents the first output variable; represents the input signal under cyber - attack; represents the input signal without cyber - attack;

[0031] The upper limit of the cyber - attack is expressed as:

[0032] ,

[0033] Among them, represents the upper limit of the cyber - attack; represents the system state variable after coordinate transformation, , represents the coordinate transformation matrix, represents 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 variables after coordinate transformation; represents the negative part of the matrix;

[0034] The lower limit of the cyber - attack is expressed as:

[0035] ,

[0036] wherein, represents the lower limit of the cyber - attack;

[0037] The upper limit of the difference of the system state variables is expressed as:

[0038] ,

[0039] wherein, represents the upper limit of the difference of the system state variables; represents the first linear real matrix; represents the upper limit of the initial system state variables; represents the lower limit of the initial system state variables;

[0040] The lower limit of the difference of the system state variables is expressed as:

[0041] ,

[0042] wherein, represents the lower limit of the difference of the system state variables.

[0043] Preferably, constructing an interval observer based on the Lyapunov function for the state - space equation after coordinate transformation includes:

[0044] Constructing a radially unbounded and positive - definite Lyapunov function, and taking the derivative of the Lyapunov function to obtain the first - order Lyapunov function;

[0045] Based on the Lyapunov stability theory, constructing a positive - definite matrix inequality for the first - order Lyapunov function, and solving the positive - definite matrix inequality to obtain a positive - definite matrix;

[0046] Based on the positive - definite matrix and the observation error of the interval observer, constructing an observation gain matrix inequality, and solving the observation gain matrix inequality to obtain the observation gain matrix of the interval observer;

[0047] Based on the observation gain matrix, constructing an upper - limit observation equation and a lower - limit observation equation for the system state variables, thereby obtaining an interval observer.

[0048] Preferably, the state - space equation after coordinate transformation is expressed as:

[0049] ，

[0050] ,

[0051] where, represents the system state variable after coordinate transformation; represents the first - order derivative of; represents the linear matrix after coordinate transformation; represents the non - linear matrix in the th non - linear term after coordinate transformation; represents the non - linear function matrix in the th non - linear term after coordinate transformation; represents the input matrix after coordinate transformation; represents the first linear real matrix after coordinate transformation; represents the second real matrix; represents the influence matrix on; ; represents the linear mapping matrix after coordinate transformation; represents the non - linear function matrix in the th non - linear term after coordinate transformation; represents the non - linear function matrix in the th non - linear term after coordinate transformation;

[0052] The interval observer is expressed as:

[0053] ,

[0054] where, represents the upper - bound observation equation of the system state variable; represents the observation gain matrix of the interval observer for the linear term of the system; represents the upper - bound value of the state variable of the interval observer; represents the number of non - linear functions processed by the positive - system method; represents the non - linear matrix in the th non - linear term after coordinate transformation; represents the observation gain matrix of the interval observer for the th non - linear term of the system; represents the non - linear mapping matrix in the th non - linear term; represents the non - linear function in the th non - linear term; represents the The non - linear function matrix in the denotes the non - linear function matrix in the th non - linear term; denotes the non - linear matrix in the th non - linear term after coordinate transformation; denotes the observation gain matrix of the interval observer for the th non - linear term of the system; denotes the non - linear mapping matrix in the th non - linear term; denotes the non - linear function in the th non - linear term; denotes the non - linear function matrix in the th non - linear term after coordinate transformation; denotes the positive part of; denotes the negative part of; denotes the lower limit value of the state variable of the interval observer; denotes the , , composed of the observation gain matrix of the interval observer; denotes the system output variable composed of and ; denotes the upper limit of the input signal of the interval observer affected by the interference factor; denotes the upper limit of the input signal of the interval observer affected by the network attack;

[0055] ,

[0056] wherein, denotes the lower - limit observation equation of the system state variable; denotes the lower limit of the input signal of the interval observer affected by the interference factor; denotes the lower limit of the input signal of the interval observer affected by the network attack.

[0057] Preferably, after determining that the system is under a network attack, it further includes:

[0058] Constructing the dynamic differential equation of the state variable of the interval observer based on the upper and lower limits of the state variable of the interval observer;

[0059] Based on the dynamic differential equation of the state variable of the interval observer, obtaining the state - space equation of the interval observer, and taking the state - space equation of the interval observer as the target state - space equation of the singular cubic non - linear spring vibration isolation system.

[0060] Preferably, the dynamic differential equation of the state variable of the interval observer is expressed as:

[0061]

[0062] Among them, represents the dynamic differential equation of the state variables of 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; represents the state variable linear matrix of the interval observer related to the linear term of the system; represents the upper limit value of the state variables of the interval observer; represents the lower limit value of the state variables of the interval observer; represents the state variable nonlinear matrix of the interval observer related to the t-th linear term of the system; represents the variable vector in the nonlinear function; represents the interval observer and the state variable nonlinear matrix related to the t-th linear term of the system;

[0063] ,

[0064] ,

[0065] The state space equation of the interval observer is expressed as:

[0066] ,

[0067] Among them, represents the state space equation of the interval observer; represents the linear matrix in the state space equation of the interval observer; ; represents the nonlinear matrix in the th nonlinear term in the state space equation of the interval observer; represents the nonlinear function in the th nonlinear term in the state space equation of the interval observer; represents the nonlinear function matrix in the th nonlinear term in the state space equation of the interval observer.

[0068] The present invention also provides an attack detection device for a singular cubic nonlinear spring vibration isolation system, including:

[0069] A state space equation construction module for constructing the state space equation of the singular cubic nonlinear spring vibration isolation system;

[0070] A network attack equation construction module, configured to obtain a first output variable and a second output variable of the system when the input signal is not under a network attack and when it is under a network attack based on the state space equation; construct an equation of the network attack with respect to the system state variables based on the difference between the first output variable and the second output variable;

[0071] A system state variable difference interval acquisition module, configured to perform a coordinate transformation on the system state variables with the constraint that the difference in the system state variables caused by the upper limit of the network attack is greater than the difference in the system state variables caused by the lower limit of the network attack, so as to obtain the upper and lower limits of the network attack and the upper and lower limits of the difference in the system state variables caused by the upper and lower limits of the network attack;

[0072] An interval observer construction module, configured to perform a coordinate transformation on the state space equation using a non-singular matrix, and construct an interval observer for the coordinate-transformed state space equation based on the Lyapunov function;

[0073] A network attack detection module, configured 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 of the system when it is not under a network attack exceeds the upper and lower limits of the difference in the system state variables, so as to determine whether the system is under a network attack.

[0074] The present invention also provides a computer-readable storage medium, on which a computer program is stored, and when the computer program is executed by a processor, the steps of the above-mentioned attack detection method for a singular cubic nonlinear spring vibration isolation system are implemented.

[0075] The attack detection method for a singular cubic nonlinear spring vibration isolation system provided by the present application has the following beneficial effects:

[0076] 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 under cyber - attack and not under cyber - attack based on this state - space equation, and constructs the relationship between cyber - attack and system state variables based on the difference in output variables of different input signals. Since cyber - attacks of different degrees have different effects on system state variables, therefore, the present application takes the difference in system state variables caused by the upper limit of cyber - attack being greater than the difference in system state variables caused by the lower limit of cyber - attack as a constraint, and performs coordinate transformation on the system state variables in the state - space equation, so as to obtain the cyber - attack degree interval and the interval of the difference in system state variables after the input signal is attacked and tampered by cyber - attacks of different degrees. At the same time, because the singular cubic nonlinear spring vibration isolation system has a nonlinear term introduced by the cubic nonlinear damping term, this nonlinear term makes the relationship between the input and output of the system not a simple linear relationship, and it is difficult for the directly constructed interval observer to accurately match the actual dynamic characteristics of the system. In response to this, 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, and thus constructing an interval observer that can effectively reflect the system state. Finally, using this interval observer to output the estimated interval of system state variables in real - time, comparing the estimated interval of system state variables with the system state variables when the input signal of the system is not attacked, and determining whether the system is attacked according to whether the difference between the two exceeds the interval of the difference in system state variables. Since the interval of the difference in system state variables reflects the influence range of cyber - attacks on the input signal on system state variables, even if the observed values output by the interval observer fluctuate when the input signal is tampered, it can accurately detect whether the system is attacked by whether the degree of fluctuation is within the normal fluctuation range.

[0077] 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 a 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 expands, the constants in the Lipschitz model and the one - side Lipschitz model will increase sharply. As a result, when constructing an interval observer based on the state - space equation of the transformed system, the sufficient condition of the linear matrix inequality for ensuring the stability of the system is unsolvable, and thus it is impossible to find the interval observer parameters that meet the system stability requirements, leading to the inability to construct an effective interval observer. In response to this, the present application first considers 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, so as to reduce the conservatism of the sufficient condition of the linear matrix inequality, and thus be able to construct an interval observer that meets the system stability requirements. BRIEF DESCRIPTION OF THE DRAWINGS

[0078] In order to make the content of the present invention easier to be clearly understood, the following further details the present invention according to specific embodiments of the present invention in conjunction with the accompanying drawings, where:

[0079] Figure 1 is a flowchart of the attack detection method for the singular cubic nonlinear spring vibration isolation system provided by the present application;

[0080] Figure 2 is a schematic structural diagram of the singular cubic nonlinear spring vibration isolation system provided by the present application;

[0081] Figure 3 is a schematic structural diagram of the attack detection device for the singular cubic nonlinear spring vibration isolation system provided by the present application. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0082] The following further illustrates the present invention in conjunction with the accompanying drawings and specific embodiments, so that those skilled in the art can better understand the present invention and be able to implement it, but the embodiments cited do not limit the present invention.

[0083] Please refer to Figure 1 , Figure 1 shown is a flowchart of the attack detection method for the singular cubic nonlinear spring vibration isolation system provided by the present application. The method specifically includes:

[0084] S10: Construct the state - space equation of the singular cubic nonlinear spring vibration isolation system.

[0085] S20: Obtain the first output variable and the second output variable of the system when the input signal is not under cyber - attack and when it is under cyber - attack based on the state - space equation; construct an equation of the cyber - attack with respect to the system state variables based on the difference between the first output variable and the second output variable.

[0086] S30: With the constraint that the difference in system state variables caused by the upper limit of the cyber - attack is greater than the difference in system state variables caused by the lower limit of the cyber - attack, perform a coordinate transformation on the system state variables, so as to obtain the upper and lower limits of the cyber - attack, and the upper and lower limits of the difference in system state variables caused by the upper and lower limits of the cyber - attack.

[0087] S40: Use a non - singular matrix to perform a coordinate transformation on the state - space equation, and construct an interval observer for the coordinate - transformed state - space equation based on the Lyapunov function.

[0088] S50: 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 of the system when it is not under cyber - attack exceeds the upper and lower limits of the difference in system state variables, so as to determine whether the system is under cyber - attack.

[0089] Specifically, if neither the difference between the estimated upper limit of the system state variables and the state variables of the system when it is not under cyber - attack nor the difference between the estimated lower limit of the system state variables and the state variables of the system when it is not under cyber - attack exceeds the upper and lower limits of the difference in system state variables, it is determined that the system is not under cyber - attack; if the difference between the estimated upper limit of the system state variables and the state variables of the system when it is not under cyber - attack exceeds the upper and lower limits of the difference in system state variables, and / or the difference between the estimated lower limit of the system state variables and the state variables of the system when it is not under cyber - attack exceeds the upper and lower limits of the difference in system state variables, it is determined that the system is under cyber - attack.

[0090] Furthermore, the steps for constructing the state - space equation of the singular cubic non - linear spring vibration isolation system in step S10 include:

[0091] S100: Conduct a force analysis on the singular cubic non - linear spring vibration isolation system to obtain the dynamic function of the system, and construct the initial state - space equation of the system based on the dynamic function of the system.

[0092] Specifically, as Figure 2 shown in the structural schematic diagram of the singular cubic non - linear spring vibration isolation system provided by this application, the meanings of the parameters in the figure are shown in Table 1:

[0093] Table 1

[0094] Symbol Meaning Parameter Description Displacement of the block relative to the wall m <![CDATA k c > Spring coefficient N / m Nonlinear damping N System weight kg Force applied to the object N

[0095] Through force analysis, it can be obtained that:

[0096] ,

[0097] Among them, represents the system weight; represents the equation of the displacement of the block in the system changing with time; represents the first derivative of; represents the second derivative of; represents the linear damping coefficient; represents the cubic damping coefficient; represents the spring coefficient; represents the equation of the force on the block in the system changing with time.

[0098] The non - linear damping term in this system is expressed as , ; is the first derivative of.

[0099] Taking as the state variable of this system, the dynamic function of the system can be obtained as:

[0100] ,

[0101] Among them, represents the system weight; represents the function of the displacement of the block in the system changing with time; represents the first derivative of; represents (t) the second derivative of; represents the linear damping coefficient; represents the cubic damping coefficient; represents the spring coefficient; represents the function of the force on the block in the system changing with time; represents time;

[0102] Then the initial state - space equation of the system is expressed as:

[0103] ,

[0104] ,

[0105] Among them, is the initial state variable of the system, ; is the first derivative of; is the initial linear matrix; is the The initial nonlinear matrix of the nonlinear terms; is the nonlinear function of the is the initial nonlinear function matrix of the 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; is the initial nonlinear mapping matrix of the is the number of nonlinear terms; is the nonlinear function of the first nonlinear term; is the nonlinear function of the is the initial nonlinear function matrix of the first nonlinear term; is the initial nonlinear function matrix of the

[0106] S101: Use the sensor fault equation and interference factor for collecting the system output variable as the augmented state of the system state variable to update the initial state space equation.

[0107] Specifically, since there may be sensor faults and interference factors during the actual measurement of system parameters, therefore, in this application, the sensor fault equation is used as the augmented state of the system initial state variable, and the interference factor is used as the augmented state of the first derivative of the system initial state variable, so as to update the initial state space equation, making the updated state space equation able to more accurately fit the actual operating state of the system.

[0108] S102: Use the Pershitzky model to perform a nonlinear transformation on the updated initial state space equation to obtain the state space equation.

[0109] 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.

[0110] 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.

[0111] Specifically, the state space equation is expressed as:

[0112] ，

[0113] ,

[0114] 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 an interference factor; is a linear mapping matrix; is the nonlinear mapping matrix of the first nonlinear term; is the nonlinear mapping matrix of the is the nonlinear function matrix of the first nonlinear term; is the nonlinear function matrix of the

[0115] Furthermore, when the singular cubic nonlinear spring vibration isolation system is connected to the industrial network, there are hidden network attacks during the information transmission process. For the network attacks, they will be injected during the input signal transmission process, resulting in output variable deviation. If holds, it indicates that the network attack is hidden. At this time, the output variable of the system is: , represents the network attack, represents the output variable when the system is under network attack, represents the input signal when the system is under network attack, represents the output variable when the system is not under attack, represents the input signal when the system is not under network attack.

[0116] Generally speaking, network attackers will inject deviations into the input signal, resulting in deviations in the output variables of the subsequent system and the control decisions 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 the conservatism to ensure the construction of an effective interval observer, this application first performs a coordinate transformation on the system state variables.

[0117] Specifically, the equation of the network attack with respect to the system state variables is expressed as:

[0118] ,

[0119] where, represents the equation of the network attack with respect to the system state variables; represents the second output variable; represents the first output variable; represents the input signal under network attack; represents the input signal without network attack;

[0120] The upper limit of the network attack is expressed as:

[0121] ,

[0122] where, Indicates the upper limit of network attack; Indicates the system state variables after coordinate transformation, , Indicates the coordinate transformation matrix, Indicates The inverse matrix of; Indicates the positive part of the matrix; Indicates the upper limit of the system state variables after coordinate transformation; Indicates the lower limit of the system state variables after coordinate transformation; Indicates the negative part of the matrix;

[0123] The lower limit of network attack is expressed as:

[0124] ,

[0125] where, Indicates the lower limit of network attack;

[0126] The upper limit of the difference in system state variables is expressed as:

[0127] ,

[0128] where, Indicates the upper limit of the difference in system state variables; Indicates the first linear real matrix; Indicates the upper limit of the initial system state variables; Indicates the lower limit of the initial system state variables;

[0129] The lower limit of the difference in system state variables is expressed as:

[0130] ,

[0131] where, Indicates the lower limit of the difference in system state variables.

[0132] To further relax the conservativeness, this application introduces a nonsingular matrix to perform a 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 remain unchanged, but the complex nonlinear relationships or strong coupling relationships in the system become simpler or decoupled, so that the sufficient condition of the linear matrix inequality used to ensure the stability of the system when constructing the interval observer is solvable.

[0133] Specifically, the state space equation after coordinate transformation is expressed as:

[0134] ，

[0135] ,

[0136] Among them, represents the system state variable after coordinate transformation; represents the first derivative of; represents the linear matrix after coordinate transformation; represents the th non - linear matrix in the non - linear term after coordinate transformation; represents the th non - linear function matrix in the non - linear term after coordinate transformation; represents the input matrix after coordinate transformation; represents the first linear real matrix after coordinate transformation; represents the second real matrix; represents the influence matrix on ; represents the linear mapping matrix after coordinate transformation; represents the th non - linear function matrix in the non - linear term after coordinate transformation; represents the th non - linear function matrix in the non - linear term after coordinate transformation.

[0137] Specifically, constructing an interval observer based on the state - space equation after coordinate transformation in step S40 specifically includes:

[0138] S400: Construct a radially unbounded and positive - definite Lyapunov function, and take the derivative of the Lyapunov function to obtain the first - order Lyapunov function.

[0139] S401: Based on the Lyapunov stability theory, construct a positive - definite matrix inequality for the first - order Lyapunov function, and solve the positive - definite matrix inequality to obtain a positive - definite matrix.

[0140] S402: Based on the positive - definite matrix and the observation error of the interval observer, construct an observation gain matrix inequality, and solve the observation gain matrix inequality to obtain the observation gain matrix of the interval observer.

[0141] S403: Based on the observation gain matrix, construct an upper - bound observation equation for the system state variable and a lower - bound observation equation for the system state variable, thereby obtaining the interval observer.

[0142] Specifically, the interval observer constructed in the embodiment of the present application is:

[0143] ,

[0144] Among them, represents the upper - bound observation equation of the system state variable; Denotes the observation gain matrix of the interval observer for the linear term of the system; Denotes the upper limit value of the state variable of the interval observer; Denotes the number of nonlinear functions processed by the positive system method; Denotes the nonlinear matrix in the Denotes the observation gain matrix of the interval observer for the t-th nonlinear term of the system; Denotes the nonlinear mapping matrix in the Denotes the nonlinear function in the t-th nonlinear term; Denotes the nonlinear function matrix in the Denotes the nonlinear function matrix in the Denotes the nonlinear matrix in the Denotes the observation gain matrix of the interval observer for the -th nonlinear term of the system; Denotes the nonlinear mapping matrix in the Denotes the nonlinear function in the Denotes the nonlinear function matrix in the Denotes positive part of; Denotes negative part of; Denotes the lower limit value of the state variable of the interval observer; Denotes the , , observation gain matrix of the interval observer composed of; Denotes the and system output variable composed of; Denotes the upper limit of the input signal of the interval observer affected by the disturbance factor; Denotes the upper limit of the input signal of the interval observer affected by the network attack;

[0145] ,

[0146] wherein, Denotes the lower limit observation equation of the system state variable; Denote the lower limit of the input signal of the interval observer affected by the interference factor; Denote the lower limit of the input signal of the interval observer affected by the cyber-attack.

[0147] Using this interval observer, the state variables of the singular cubic nonlinear spring vibration isolation system can be observed in real time, thus realizing attack detection.

[0148] 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 also be reconstructed, so as to ensure that the correct output variables can be obtained based on the reconstructed system when the system is under cyber-attack, and to ensure the effectiveness and accuracy of the subsequent control strategy. Specifically, for the above interval observer, define , for there is and ; define , , , , , .

[0149] Furthermore, after determining that the system is under cyber-attack, it also includes:

[0150] S60: Construct the dynamic differential equation of the state variables of the interval observer based on the upper and lower limits of the state variables of the interval observer. Specifically, the dynamic differential equation of the state variables of the interval observer is expressed as:

[0151] ,

[0152] where, denote the dynamic differential equation of the state variables of the interval observer; denote the input signal of the interval observer, , denote the upper limit of the input signal of the interval observer; denote the lower limit of the input signal of the interval observer; denote the state variable linear matrix of the interval observer related to the linear term of the system; denote the upper limit value of the state variables of the interval observer; denote the lower limit value of the state variables of the interval observer; denote the state variable nonlinear matrix of the interval observer related to the t-th linear term of the system; denote the variable vector in the nonlinear function; denote the state variable nonlinear matrix of the interval observer related to the -th linear term of the system;

[0153] ,

[0154] .

[0155] 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:

[0156] ,

[0157] where, represents the state space equation of the interval observer; represents the linear matrix in the state space equation of the interval observer; ; represents the nonlinear matrix in the th nonlinear term in the state space equation of the interval observer; represents the th nonlinear function in the th nonlinear term in the state space equation of the interval observer; represents the nonlinear function matrix in the

[0158] The attack detection method for the singular cubic nonlinear spring vibration isolation system provided in the above embodiment will be described below with specific examples:

[0159] It should be noted that, in this embodiment, and represent the real number (non-negative real number) vector space and the -dimensional real matrix respectively; represents the finite integer sequence starting from 1 up to , and similarly ; for a vector z, means that all component values are less than or equal to (or greater than or equal to) 0; under the condition that and the function are strictly increasing, then the function belongs to the class of functions; if the class of functions increases to infinity, the function is the class of functions; for the matrix T, , ; is the derivative of the continuously differentiable function ; in addition, It means that T is a negative definite (positive definite) matrix. For simplicity, the matrix is assumed to be row full rank, so there are real matrices and such that .

[0160] Based on the above definition, the singular cubic nonlinear spring vibration isolation system is transformed into:

[0161] ，

[0162] ,

[0163] where is the first derivative of ; let , then the system can be equivalent to:

[0164] ，

[0165] ,

[0166] Before designing an interval observer for the singular cubic nonlinear spring vibration isolation system, this embodiment ensures the monotonicity of the nonlinear function through the following assumptions:

[0167] Assumption 1. For any and , the following inequality holds:

[0168] ,

[0169] According to Assumption 1, there exists satisfying all and conditions, then:

[0170] ,

[0171] And there exists satisfying for all and conditions, then:

[0172] ,

[0173] Assumption 2. For any and , the following inequality holds:

[0174] ,

[0175] Hypothesis 3. The external disturbance in the system is bounded, then:

[0176] ,

[0177] where, and are both constant vectors.

[0178] Furthermore, in order to make the designed interval observer finally stable and consistent, this embodiment proposes the following definition: Definition 1. For the state-space equation of the system:

[0179] ,

[0180] ,

[0181] If for all and there exist and such that:

[0182] ,

[0183] ,

[0184] then the Lyapunov function has input-to-state stability. If and only if the system allows a Lyapunov function with input-to-state stability, then the system is input-to-state stable.

[0185] To prove the non-negativity of the designed interval observer, this embodiment also gives the following lemmas:

[0186] Lemma 1. Given a matrix M and a vector variable satisfying the inequality , then:

[0187] ,

[0188] Lemma 2. Given a matrix equation , where , , , if , the solution of this equation is as follows:

[0189] ,

[0190] where, is the pseudo-inverse matrix of , representing an arbitrary matrix.

[0191] Furthermore, the following lemma illustrates the non - negative processing conditions for non - linearity in the interval observer:

[0192] Lemma 2. Considering the non - linear part of the Persidsky system satisfies Assumption 1. If for all , one of the following properties holds:

[0193] The \(i\) - th row of the matrix and the \(i\) - th column of the matrix are both greater than or equal to 0;

[0194] The \(i\) - th row of the matrix and the \(i\) - th column of the matrix are both less than or equal to 0;

[0195] Then the non - linear part is non - negative. Therefore, this embodiment uses the following symbols to simplify the non - negativity condition: .

[0196] Lemma 3. If Assumption 1 holds and holds, then:

[0197] ,

[0198] Furthermore, if for , , there exists , then according to Lemma 1, the inequality holds. Therefore, it can be obtained that:

[0199] ,

[0200] Theorem 1. Assume that Assumption 1, Assumption 2, and Assumption 3 are all satisfied. If for there are and , then for any there is ,

[0201] Proof: After the coordinate transformation, the system is equivalent to:

[0202] ，

[0203] ,

[0204] where, , , , , , , , ; is a non - linear matrix;

[0205] Define the estimation error of the interval observer as and , so the dynamic representation of the estimation error can be obtained as:

[0206] ,

[0207] ,

[0208] where,

[0209] ,

[0210] ,

[0211] ,

[0212] ,

[0213] According to Assumption 3 and Lemma 2, for any , it satisfies:

[0214] ,

[0215] Furthermore, since for any there is and , according to Lemma 3, we can get and ; due to Lemma 1 and Lemma 4, for all there is and ; these all show the non - negativity of the interval observer and .

[0216] Although the existence of is not necessary, if a matrix that can guarantee the non - negativity of the system after coordinate transformation cannot be constructed, then a non - linear interval inclusion method needs to be adopted, and this method will increase the design complexity of the interval observer.

[0217] Theorem 2. Assume that Assumption 1, Assumption 2 and Assumption 3 are satisfied. If there exist a positive - definite matrix , a diagonal matrix , for there is a matrix , for and there is a matrix , for There is a matrix and a positive definite matrix such that:

[0218] ,

[0219] ,

[0220] and the positive definite matrix , represents the rows of the positive definite matrix , represents the columns of the positive definite matrix ;

[0221] wherein,

[0222] ,

[0223] ,

[0224] ,

[0225] ,

[0226] ,

[0227] ,

[0228] ,

[0229] ,

[0230] wherein, represents a diagonal matrix; represents the set of non - negative diagonal matrices of order ; represents the set of non - negative diagonal matrices of order ; represents the matrix when ; represents the matrix when ; represents

[0231] Then it can be proved that is uniformly ultimately bounded.

[0232] Proof: Define , for There is and ; , , , , , ;

[0233] Then the dynamic differential equation of the state variables of the interval observer is as follows:

[0234] ,

[0235] where, , , represents the variable vector in the nonlinear function;

[0236] ,

[0237] where, represents the new input signal, whose boundary is independent of , and define , , then the state - space equation of the reconstructed new system is expressed as:

[0238] ,

[0239] The Lyapunov function adopted in this embodiment is:

[0240] ,

[0241] Based on Assumption 1 and the condition that the Lyapunov function is radially unbounded and positive definite, the derivative of the Lyapunov function is obtained as:

[0242] ,

[0243] Therefore, under the equation, it can be obtained that:

[0244] ,

[0245] where, represents the nonlinear function of the first nonlinear term in the first - order derivative of the Lyapunov function; represents the nonlinear function of the th nonlinear term in the first - order derivative of the Lyapunov function; represents the nonlinear function of the th nonlinear term in the first - order derivative of the Lyapunov function; represents the A non - linear function with a non - linear term.

[0246] According to the above inequality and Assumption 1, the right - hand side of the inequality is similar to the form of the inequality in Definition 1. For some we have: .

[0247] Therefore, it can be obtained that the Lyapunov function V has input - to - state stability, which can guarantee the property of the input - to - state stability of the system. At the same time, it is also proved that , from which a non - negative and ultimately stable and uniform interval observer is obtained.

[0248] Based on the attack detection method for the singular cubic non - linear spring vibration isolation system provided in the above - mentioned embodiment, the embodiment of the present application also provides an attack detection device for the singular cubic non - linear spring vibration isolation system. As Figure 3 shown, the device specifically includes:

[0249] A state - space equation construction module 10 for constructing the state - space equation of the singular cubic non - linear spring vibration isolation system.

[0250] A network - attack equation construction module 20 for obtaining the first output variable and the second output variable of the system when the input signal is not under network attack and when it is under network attack based on the state - space equation; constructing an equation of the network attack with respect to the system state variables based on the difference between the first output variable and the second output variable.

[0251] A system - state - variable difference - interval acquisition module 30 for performing a coordinate transformation on the system state variables with the constraint that the difference in system state variables caused by the upper limit of the network attack is greater than the difference in system state variables caused by the lower limit of the network attack, so as to obtain the upper and lower limits of the network attack and the upper and lower limits of the difference in system state variables caused by the upper and lower limits of the network attack.

[0252] An interval - observer construction module 40 for performing a coordinate transformation on the state - space equation using a non - singular matrix, and constructing an interval observer based on the Lyapunov function for the coordinate - transformed state - space equation.

[0253] A network - attack detection module 50 for using the interval observer to output the estimated upper and lower limits of the system state variables in real - time, and determining whether the difference between the estimated upper and lower limits of the system state variables and the state variables of the system when not under network attack exceeds the upper and lower limits of the difference in system state variables, so as to determine whether the system is under network attack.

[0254] The 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 attack detection method for the singular cubic non - linear spring vibration isolation system are implemented.

[0255] Obviously, the above embodiments are merely examples given for clear illustration and are not limitations on the implementation manners. For those of ordinary skill 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 enumerate all the implementation manners here. And the obvious changes or modifications derived therefrom are still within the protection scope of the present invention.

Claims

1. A singular cubic nonlinear spring vibration isolation system attack detection method, characterized in that: include: The state space equation of the singular cubic nonlinear spring isolation system is constructed, which specifically 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; Based on the state space equation, a first output variable and a second output variable of the system when the input signal is not subjected to a network attack and when the input signal is subjected to a 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.

2. The singular cubic nonlinear spring vibration isolation system attack detection method according to claim 1 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.

3. The singular cubic nonlinear spring vibration isolation system attack detection method according to claim 2 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.

4. The singular cubic nonlinear spring vibration isolation system attack detection method according to claim 3 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.

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 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.

6. The singular cubic nonlinear spring vibration isolation system attack detection method according to claim 5, 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.

7. The singular cubic nonlinear spring vibration isolation system attack detection method according to claim 6, 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.

8. A singular cubic nonlinear spring vibration isolation system attack detection device, characterized in that: include: The state-space equation building module is used to build the state-space equation of the singular cubic nonlinear spring isolation system, which specifically 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; 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.

9. 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 7 are implemented.

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